View source for BECO-85-230, Responds to 851121 Meeting Questions Re IE Bulletin 80-11, Masonry Walls. Encl Documents Rationale Re Cable Spreading Room & Radwaste Walls

{{Adams
| number = ML20136F660
| issue date = 12/31/1985
| title = Responds to 851121 Meeting Questions Re IE Bulletin 80-11, Masonry Walls. Encl Documents Rationale Re Cable Spreading Room & Radwaste Walls
| author name = Harrington W
| author affiliation = BOSTON EDISON CO.
| addressee name = Zwalinski J, Zwolinski J
| addressee affiliation = NRC OFFICE OF NUCLEAR REACTOR REGULATION (NRR)
| docket = 05000293
| license number = 
| contact person = 
| document report number = BECO-85-230, IEB-80-11, NUDOCS 8601070459
| document type = CORRESPONDENCE-LETTERS, INCOMING CORRESPONDENCE, UTILITY TO NRC
| page count = 58
}}

=Text=
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     -%      A                                                                                                 l EOSTON EDISON COMPANY B00 50VLaTON STREET 50sTON, MASSACHUSETTS 02199                                  j l
WILLIAM D. HARRINGTON c' ween wees passenesse rK abaan December 31, 1985 I BECo 85-230 Mr. John A. Zwolinski, Director BHR Project Directorate #1 Division of Licensing Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, D. C.       20555 License DPR-35 Docket 50-293 f
IE Bulletin 80-11:    Request for Additional Information On November 21, 1985 the NRC and Bostor. Edison (BECo) met to discuss the methodology used in BECo's efforts to address IE Bulletin 80-11 (IEB 80-11),
                   " Masonry Walls," and to examine documentation used in assessing walls. In addition, BECo verbally proposed an alternative to modifying three walls in the Cable Spreading Room (CSR), and explained the future reclassification of eight walls in the radwaste corridor which will make modification of the walls unnecessary.
Attachment 1 of this submittal documents the rationale concerning the CSR and radwaste walls.
l Attachment 2 provides documentation in answer to your questions which generated the November 21, 1985 meeting.
!_                 Both attachments are provided to aid in your review of actions taken by BECo
!                  in addressing IEB 80-11, and to document material and conversations pertinent to our meeting. Should you wish further information, please contact us.
Very truly yours, PMK/ns Attachments:      (1) Discussion of Specific Halls (2) Responses to NRC Questions 8601070459 851231
                                                        ^
                                                                                   '//
i PDR     ADOCK 05000293                                   I/
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f 8 ATTACHMENT 1 To comply with the IEB 80-11, Boston Edison's Masonry Wall Project qualified a part of PNPS Walls by analyses; and the walls which could not be qualified by analyses were/will be qualified by physical modifications with the following exceptions:
: 1. Three walls in the cable spreading room (194.17, 194.21 and 194.22) will not qualify for the tornado depressurization loads without considerable modification. Such a modification would require extensive relocation of safety related cable and conduit to provide access for the structural modification. This would require a station outage, involve a lengthy construction schedule, raise concerns about
            .the extensive interruptions to safety systeen and constitute a significant expenditure of funds. Boston Edison claims a hardship exemption on completing such modifications, and advances the following technical basis to demonstrate that nuclear safety will be maintained:
o    These walls only affect safety related components in the cable spreading room (CSR). The alternate shutdown system wholly backs up the affected components assuming a fire in the CSR. Since a masonary wall collapse is no more damaging than a fire, safe shutdown can be achieved and maintained.
o   BECo has completed procedural changes which require the initiation of shutdown upon receipt of a Tornado alert from the area load dispatcher. The procedure advises the operators that shutdown via the alternate shutdown process may be required.
o   The critical load, tornado, is a unidirectional load which would cause a failure away from the CSR. The walls are qualified as is to withstand all other design basis loads (i.e. seismic). Thus, even in the event of wall failure, the safety related components in the CSR would be expected to remain essentially intact.
: 2. Eight walls in the radwaste corridor (191.29,191.37,193.11,193.12, 193.5,193.6,193.7, & 193.B) are presently classified as safety related solely because of two safety related cables which are in the zone of influence of these walls. During the next refueling outage (RF0 #7), these two cables will be re-routed out of the zone of
;              influence of these walls due to Appendix R modifications. Upon completion of the above mentioned re-routing, these eight walls will be reclassified as non-safety related. Boston Edison will not attemot to " qualify" these walls for safety related service in the intea m.
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ATTACHMENT 2 RESPONSliS 'IO NRC QJESTIONS B0SION EDISON OJMPANY la) Explain how the cracking noment for a particular element was determined.
The cracking moment (Mer) is expressed as a function of the allowable ten-sile stress, Ft. 'Ihe value used for Ft (modulus of rupture) is 67 psi and 134 poi for the vertical and horizonal directions, respectively for hollow ansonry. The allowable cracking nreant is Mcit,FeIs
                                                      /
Where    Ft= modulus of rupture Ig=' moment of inertia of uncracked section y= distance of neutral plane from tension face.
Mcrx and Mcry are constant values in their respective directions for each analysis depending on the wall properties (reinforcing and thickness).
These values are used to determine the varying stiffnesses for particular elements in each direction using Branson's equation.                                                    ,
lb) Indicate how the offacts of the directional variation of principal stresses were determined and how they affect the wall stiffness.
The principal stresses concide with the two orthogonal directions. Since the mortar joints initiate cracking either parallel or normal to the bed joints, these are the principal directions [1,2] and the assunption of the special orthtropy is appropriate. Also, the mortar joints are aligned with the finite element x and y directions and this allows directional stiff-nesses to be independently specified in the analysis.
One of the couputer programs (a preprocessor) use'. in the blockwall project has the ability to determine Which elements havs ioments that exceed the allowable cracking noment. The program takes these elements and recalcula-tes the new inertia properties using Branson's equation (o sets them to zero in the horizontal direction). These properties are input into the analysis program and the iterative process is continued.
A note to question one, two issues should be mentioned. The first concerns the conservatism in cygna's method of analysis and the other is the use of reinf6rced concrete methodology for nasonry block walls.
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Cygna's method of analysis assumes the modulus of rupture (F                                  t ) equal to 67 psi. This is considered a conservative value. In reference 2, figure 7 indicates that the value for the modulus of rupture should be 120 pai.
Since the initial cracking will occur along the bed joints and the Pilgrim walls are grouted, the higher value of Ft is justified. Using Ft equal to 120 psi, test walls were analyzed and compared to an analysis using the
 !                   original Ft (see table 1). The results show that since Her increases significantly, the walls either crack less or don't crack at all. This onuses the walls to be stiffer, the frequency to be higher, and the m pents to be lower.
Table 1 shows the results.
Ft        Frequency          Nximum Wall
* Type (psi) @ Convergence             Moment (H2)           (in-#/in) 1              a     67          4.73              2713 2              a    120          6.60              1034 3             b      67          4.74              3265 4             b     120          8.19               621 TABIE 1                                                                   ,
* Wall types a - square wall pinned top and bottom b - rectangular a ll pinned on 4 sides The other concern is the sensitivity of Branson's equation within the range of deflections of the Pilgrim walls. Jofreit and McNeice [3] did studies using reinforced concrete. They conpiled experimental data and conpared
:                    the results to Branson's equation. The load-deflection curves show that Branson's equation and the experimental data are consistent to a displace-ment of 0.04 inches.
1 The displacements of a group of Pilgrim blockwalls are coupiled in Table 2.
g                    These displacements are converted to represent equivalent displacements used by Jofreit and McNeice. The naximum displacement is 0.0077 inches.
3 This falls in the lower part of the load-deflection curves. Therefore, the
;-                   results of the analysis are not sensitive to the use of Branson's 4                    equations.
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2a) Since the Branson equation was developed for structural members, justify its applicability to an element of the finite element model.
The finite element method enables each region of the msonry wall to be modelled separately with respect to flexural rigidities. The Branson equation is a method for determining the flexural rigidities of each ele-ment as cracking occurs. The equation
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* l' ,ni_                  It is for calculating the effective moment of inertia at any particular cross-section as a function of the bending noment, section properties, and concrete strength [4] as applied for masonry analysis, this was applied on an element-by-element basis. This was appropriate because the element size encoupassed multiple nasonry units.
While Branson's (quations were originally developed for beams, Jofreit and McNeice [3] demonstrated taeir applicability to the finite element analysis of one-way and two-way sla'as.      In addition to Jofreit's and McNeice's work, the Branson equations are permitted for use under the 1971 ACI code for two way slabs [5].                                                                                                  .
'l 2b) Based on an element stiffness, explain how the inertial forces were obtained taking into account the stiffness changes (i.e., frequency shift).
Explain how the stiffness of each element was combined in the evaluation of the inertial forces.)
The extent of cracking in the wall, Which is a function of the inertial forces, is taken into account by varying the stiffness on an element by element basis.
The finite element nodels are initially analyzed using the uncracked section properties of the wall. As load is applied and the noments exceed the cracking noment (Mer), the stiffness properties of the elements must be I                    recalculated.
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In the vertical direction, the stiffness properties are recalculated using Branson's equation. This allows each element to have a different stiffness as a function of the actual and cracking noment (bbr/Ma).
In ,the horizontial direction, if the moment evcaads the cracking moment, based on the face shell area only, the section is assumed cracked rnd s.#
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       ,        .-                                                                                                                                            l 0         0 unable to transmit any load. In the finite element model, the element stiffness in the horizontal direction is set to zero. This conservatively neglects any contribution from grout oores or joint wire reinforcement, and Branson's equation is not used.
(An exception to this occurs Where horizontal bond beams exist. The                                                                 ,
analyses of these walls used an average steel area cver the wall height for                                                         j a two-way analysis. A comparative analysis of wall 64.4/65.8 showed this                                                            l conservative with respect to considering the wall unreinforced horizontally except for a narrow % = strip containing the bond beam steel).
The recalculated stiffness properties are input into the finite elenunt model and reanalyzed. Since the overall stiffness of the nodel is                                                                   '
decreased, the frequency also decreases causing the frequencey shift. The iterative analysis process continues with the stiffness properties changing until covergence is obtained.
: 3)         Provide the total number of walls qualified by the two-way cracked analysis and indicate how many of these walls have horizontal bond % ma.
There were approximately 200 masonry wallai, analyzed for Boston Edison. Of the 200 walls, roughly ten percent had bond beams.
: 4)         For walls without horizontal bond beams, explain how the stiffness was evaluated for the case in Which the noment in the horizontial direction                                                         .
exceeded the unreinforced allowable (i.e., it was not clear in the Licensee's response (1) how cracking was represented in the model).
As mentioned in 2b, in the horizontal direction, if the moment exceeds the cracking moment, based on the face shell area only, the section is assumed cracked and unable to transmit any load. In the finite element model, the element stiffness in the horizontal direction is set to zero. This conscr-vatively neglects any contribution from grout oores or joint wire reinfor-cement, and Branson's equation is not used.
: 5)         For walls with horizontal loni beams, explain how the stiffness along the horizontal ,direetton was determined.
for horizontal bond beams in the masonry walls, the initial horizontal stiffness is based on the uncracked properties of the wall (the same as -
l                         the walls without bond M ma). As the load is applied and the moments
,                         exceed the cracking noment, Branson's equation is used to calculate a new
;                        horizontal stiffness for these elements. The cracked moment of interia, used in Branson's equation, is based on the area of steel in the bond beam distributed over the wall height.
The walls with bond M ma exhibit two way action in the analysis.
Therefore, both the vertical and horizontal stiffness of each element change as required during each iteration of the anaylsis.
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REFERENCES
: 1. Drysdale and Hamid, " Tension Failure Criteria for Plain Concrete Masonry",
Journal of the Structural Divison, ASCE, February 1984
: 2. Drysdale, Hamid, and Heidebrecht," Tensile Strength of Concrete Masonry",
Journal of the Structural Division, ASCE, July 1979.
: 3. Jofreit and McNeice , " Finite Element Analysis of Reinforced Concrete Slabs", Journal of the Structural Division, ASCE, March 1971
: 4. ACI Committee 435, " Deflections of Reinforced Concrete Flexural Mamhars",
ACI Manual of concrete Practice, ACI 435.2R-66, 1980
: 5. ACI Committee 435, " Deflection of Two-Way Reinforced Concrete Floor Systems: State-of-the-Art Report", ACI Manual of Concrete Practice.
ACI 435.6R-74,1980 I
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          .*                       TENSION FAII.URE h POR PLAIN                                                      l of weakness and is an obvious axis, the bed joint and the direction nor-CONCRETE MASONRY                                                            mal in it (head ioint) are considered to t e the pnncipal directions for the t
lormulation of tensile strengths
 )                                    37maa-t G. Drysdale' and Aluned A. Hamsids                                    .            Strength is investigated herein using a macromechanics approach, whach' is more adaptable to ir.w.yasting the analytical features that are im-Assinacr: Feaure citerris for the tensile strength of ungrouted ar.d grouted                        Portant for design than is the micromechanics                                 (9). Thus thy unmnsorced concrew mu omy are dmioped. These cnseria account ice the                                material is taken to be globally hama**nenus.
' -                                                                                                                                                                                               the effects of the con-savngth vananon dJe to the " M nature of masonry as a composiw me.
                                                                                                                     ,       st ituent matenals are detected only as averaged properties of the Coma wmt. The wnsanoensthe
                                         *'""8* * # normal msgret m me oea joints an descreed
                                                                                           " "                                     .te material. As is shown later, the strength of the assemblage may
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4.-..: offaGure    are creen the two soepossible sension fadure p.ranesmodes to the that  were andcons.idered  in the i
the raaure                                         .d joint     sor diasonal cr cians. The pmhcud ultimate streryths usmg the proposed expree-                                                                                          ,
mons are compared with empertmentas muits for spheims te.t. of unsrouted                            Tessas Smenom Nosmana.1o Ben Jouers
                        *ad s'*"*d =*="7         d'**   C"d as**'a8 '' 'ho"a. *ad i' 6' coaciudad                   ,
that the proposed cieeria can be used to evaluate the influence of the stress                            Under tensile stresses normal to the bed sints, separation occurs at
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onentation and the cuect of earytns the dinerent soensth and geometric prop.
ernes of the =
I       the block-mortar interfaces, including tensie failure of the grout col-
                                          ,       : on ew wasue navnsth of concww black masonry umns in the cores of the blocks for grouted assemblages. 'Ihis mode of                            '
hemonucnon failure was observed from tensde-splitting tests of masonry disks (2) of the type shown in Fig.1.
Previous tests by the writers (2-5) and others (7,10,11) have indicated                        i           For ungrouted masonry, the capecity is governed by the tensde bond i
that the tension-controlled capacity of masonry elements subject to in-                            '       strength of the mortar acting along the block-mortar contact area (bed plane forces is a variable quantity. Presented herein are proposed failure                                 joint). For grouted masonry, the grout contributes to the tensde strength
{                                                                                                                           of the assemblage by its tensile capacity, which is a function of the core 1
criteria that were developed to predict the tensile strength of unrem-                                     area and the grout tensile strength. It is proposed that the tensile strenath forced concete block masonry (This work would also be applicable to                                        normal to the bed joint, f,,, , be described by mcorporatme the resistances the uncracked stage of reinforced masonry.) Observed and potential fail-or me component matenals m (mortar and grout) as follows:
ure patterrts were analyzed. This resulted in criteria that are dependent
!                on the orientatinn nf teminn from the bed iomt (normal, carallel, and
* 1                diagonal) and on the strength and geometric charactenstics of the com-                                                                                     g,,e I
               -ponent matenals (block, mortar. and grout).                                              -
so          20             3o 200 I
In a previous study (5), it was found that no existing failure criteria                        {'                                              .           .               .
ad ately acenunr rar me arusuuvyit saaturr un tuutsete Dlock masonry.
1        "" ace"''                                                       y
                                                                                                                                                  .       AMmercurED Oleus. GeseD grouted blockwork, this antsuuvyis ucaiavior was snown to oc rur--                                                                                                                                 ,1
                                                                                                                                                .E         a_E7E y,,e s esonran                                           i ther complicated because the influen.ce of grouting varied considerably                                                          -
2 iso      %. .g ,,,                                 y,                   !
with the stress orientation (2,4). It is suggested that the proposed ten-sion failure criteria will help form a sound basis for interpreting test data                                                   g                                         a                               j and will help establish the influence of the parameters.                                                                        y                                                                         y
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.                                                                                                                                                                                                       E w       h''.*a"V The continuous bed joints that divide masonry into equal horizontal
                                                                                                                                               !t: SN /               e, e                          -os                   '
layers have been shown to be the weakest plane for tension for un-
;               grouted masonry (2,4,12). Because the bed joint tends to be a major plane                                                                             evares         _U_
                                                                                                                                                        ,                          d      -   h
                   ' Prof., Dept. of Civ. Engrg. and Engrg. Mech., McMaster Univ., Hamilton, 1
ontano, Canada, taS 412.
                   ' Assoc. Prof., Civ. Engrg. Dept , Drexel Univ., Philadelphia, Pa.                                                          g M
y 7 t
Note.--Discn==nri open until July 1,1984. To extend the dosing date one month,                                              g y
a written request must be filed with the ASCE Manager of Technical and Profes-                                                  *                ,       ,       ,         ,
sional Publications. The manuscript for this paper was submitted for review and                                                        o       10 0    200      300      400        Soo       -
possible pub ication on November 3,1962. '!has                                                                                        on0UT SpuTTsfG TDisti sTROeGTH, egg.pei
!              Striscrural Engueversag, Vol.110, No. 2, 1984.               February,   paper0733-9445/
CASCE,ISSN      is part 6f the foerneal of                                                                                                 ,
j              84/0002-0228/$01.00. Paper No.18590.
PltL L-Aassmhlage Teness Shengen 14emiel to Best leines     y         weresse Grout abungen
!                                                              22e                                                        -
:                                                                                                                                                                          2ae I

T t
N_, far
* A sC*a., + g(A - A h . . n o . . . m . . . . . . . . . . . . . . . . . . . . . . . ,                                                                                                              i
        *                                           ~                                                                   (}}           I WM                               *
                    .Eq.1 expresses the average tensile strength cf the assemblage normal                                             !
to          23        So
.                                                                                                                                                              ase          ,.                      ~
to the head joint,f,,, times the gmss area of the cross section along the bed joint plane, A,, in terms of the tensile bond strength of the mortar                                                                    g              +p          i aame (hl aied by splitting tests along the head joint of block masonry cou-8
                                                                                                                                                                                       - O*"'          -
LS
                                                                                                                                                                                                                       ~
Pl ets), a , tmws the net area of the block cross section in the bed joint                                                                     200-i            Se,        ,uns.
P ane, A., plus the grout's tensde strength (calculated from tensde-split-                                                                                 4 ting tests of block-moided grout prisms (5)], a,,, times the area of the                                                                    M                     mut /
grouted core, (A, - A.). Ghis implies that both materials fail at the                                                                       " '80 E      '    ,g so,,,
88 same stram.) Substituting the termy,for the ratio of A./A, results in                                                                                                         e'       -
  .                                                                                                                                                                  ,_,,,. ss,,,                                            -
f      M . a + (1 9. ) a,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)                                                                             *h
[
b co-                                           I A comparison of the strengths predacted using Eq.1 and the previ-ously reported (2) experimental results of splitting tests of masonry disks
                                                                                                                                                                        /
: y. **        as                  s is presented m Fig.1. Each data point represents the average of three                                                                        $
(
bo- S tests. Since the face shells and webs of blocks are tapered to be thicker                                                                                                                                    !
at the top of the block, the failure plane for ungrouted masonry will pass                                                                                                                                    1 thmugh the minimum contact area between the mortar and the block.                                                                                                                                            !
This will be the plane along the top of the bed joint and the bottom of the block. For very low strength grout, the failure plane should remain 5 o          g' g'            '      '
* 6 m this location. However, as the tensile strength of the gmut becomes larger than the tensile bond strength between the mortar and the block,                                                      N 2MeneAs W W W W to med m versus enest Teneas                                   i the failure plane should shift to the minimum area of grout and maxi-                                                                                                                                         ;
mum area of mortar contact (the plane along the bottom of the bed joint                                                                                                                                       i and the top of the block). In Fig.1, Eq. 2 is plotted for the two values
: 3. The block's mechanical properties have no direct effect on the_ma-of alonethe the netenn areaand ratio, hn**~m , or o.no and 0.75. representmg tailure planes'
                                             ,4 the bed gognt, respect ye1Y.                                          '
sonry tensue strengm normai to me - gum m. It is the other physical              ;
properties inat atiect the capacity. as >urtace roughness and initial rate The comparison indicates good agreement, m the w7riters' opinion,                                                         of absorption (IRA) affect the tensile bond strength of the mortar, and rticularly when it is considered that masonry is nonisotropic and that                                                   its net-to-gross-area ratio, as, affects the relative contribution of the mortar rge involved.variability may be expected because of the many parameters                                                           and the grout.
His pmposed formulation provides a simple method for quantifying the several factors influencing the behavior of concrete masonry under                                                        Teessite STnessam Panansi. To Ben Joerrs tensde stresses normal to the bed joints. He followinig observations were made as a result of the study:                                                                                                   Under tensile stresses parallel to the bed joints, there are two possible modes of failure, Modes I and II, defined as follows:
Failure Mode I.-nis mode is splitting failure alone a plane nassing
: 1. The mortar contributes to the tensile strength of masomy normal                                                         throuch the head inints and the block % tace shells for crouted and un-to the bed joints by its tensile bond strength, which is a characteristic of the physical piu                                                                                                           'eronta .n - nm u shown m Fic. M (For the usual case of blocEs bond strength resu A of the mortar and the block. Lhigher tensile                                                              Tvith two voids,'the failure plane through the head joints must deviate ts in a hieher tensile capacity of the as*emNace. The                                                  slightly at alternate courses to miss the central web of the block.) The mortar tensile bond strength is less sagruncant for grouted masonry, 'where                                                    Mode I failure plane is similar for both ungrouted and grouted masonry, the grout tensile strength is much higher and is the dominant param-                                                           because it passes between the columns of grout (2) at the mterface of eter. This would be expected to be particularly apparent for a lower net-the grout and the block cross. webs [see Fig. 3(a)]. The effect of any ten-to-gross-area ratio, m. De analytical results shown in Fig. 2 illustrate                                                       sion bond between the block and the grout has been shown to be very this where two different tensile bond strengths are compred for differ-                                                         small (5). Therefore, for the region where the crack passes through the ent values cf g.
block face shells, the tensile bond between the grout and the web of the
: 2. Grout significantly affects the assemblage'tensde capacity normal                                                        block was neglected in the analytical formulation. His dedsion is slightly to the bed joints, especially for lower values of the net-to-gross-area ra-                                          '
conservative, but was based on observations (5) and on the argument tios. riigher grout tensile strengths result in hieher assemblaec capaci-                                            ,'
ties. as snown m rig. 2.                                                                                   -
that shnnkage of the grout would weaken this bond. However, this con-servative approach may be the reason for the predemn of lower strengths 230                                                                               -
231

              '                                                                                                                                    than those shown in Fig. 4 for grouted blockwork.
4                                                c,.,jE%""                                        The c-oacity is governed by the tensile streneth of the blocks and the
* rz mortar's tensde bond strength alone the head ioints. For the contnbu s
[      "80'
                                                              %                        ~.         d          .lt %
m'E4.'*                                                                             -non of the block, the jomt thickness is added to the block height, and
                      -                                -                                           GI            l.+                        l 7
J-       m g(  spursue               --
y'
                                                                                                *]* 4} IA*               ..
I      the analytical formulation is based on the nommal height of the idock.
This simplification sems justified because the mortar's thickness is quite de**                  *!'E'                  "
                                                                                         -      k.,i                                          ,
small compared to the block height (about 5%). For the muumum path k,h ,%j;'-   q          -=                  ,
of the failure plane associated with Mode I (see Fi6. 3(a)], the 'tensile strength of the assemblage parallel to the bed ioint, f.a . was des"c"n'52 stenom a-a                                           ,    'a)         -
by mcorporatmg the resistance of the mmyonent materials (block and e rau me noots                                                 as mortar) as follows:
Ft  L                                             genne                                           2 A , fg = m , A ,a     + n ,. A , a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) s                                               _
                                                                                           .     .y ,p-.          7 ,
in which a = the splitting tensile strength of the block; q, = the ratio g
stan S{",b]'E a
M E
of the mortared area of the head joint to the gross area; and q,. = the
                                            ~
k antas
                                                                                      '.        l[.![*j h _)
                                                                                                     'y7 net-to-gross-area ratio of the block for a vertical section crossing the face shells just beside the middle web. Eq. 3 reduces to
                                                    ]                               ,
                                                                                       . lA,        t ><      .ar r
o f                                    y*g                                      ' )) ' -      *
* fg = (% e , + q ,.a. ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
WaumE monen                                                                                          ,
When flat-ended units are mortared with full head joints, the value of
                                                                                                                                             '       %, as equal to urutv. However, when the cells created by the adjoining' FIG. 3 N IIndes og                                    a                                      Tenego agineses persigeg       -Trogged ends of the blocks are grouted, failure takes place by debonding to and Joines                                                                                                                  between the columns of grout and the blocks. To be consistent with the Previous assumption, this bonding could be assumed to be negligible haud                                                     4 and not included in the formulation. However, several differences led 1                                   200 to               2o                  so             4o                       ;
to the decision to recommend inclusion of this effect for the head joint y
I               gM,e 4.,                                         '
strength. He most important is that, for debonding of the column of
                                                                                                                  /
: i.                                 c.              o -unis=ourro grout to occur, there must be not only a tensile debonding between the C                                 j                * - enourro                              ,
                                                                                                             /
                                                                                                               /
grout and the cross-web of the block, but also a shear debonding be-z~              tver s-seonran                                     /                                           tween the grout and the face shells forming frogged ends of the blocks.
     '.:                              y 850           ,,, o m                                         /                   -
88                     Shear bond strength is much larger than tensile bond streneth W An-g               w ee e,a                                   f eg                                                other factor as mat tne cell between the frogged ends of the blocks is
                                     *g                               **                      /
R                                                                                    /                                                           much smaller than the cells in the block. Therefore, the effect of shrink-t a
w y goo      .                               j/o                                            )I            ;
l age of the grout would be smaller. If the total bonding betwa a the grout and the block is assumed to be equivalent to the tensile bond strength y                                         /                                                                     between the mortar and the block, y,,, can be assumed to be unity.
7-                           9                              /
o ,o     r A comparison of the strengths predicted using Eq. 4 and emerimental g                                                                                   -05                           results of sphtting tests of masonry disks (4) is presented m Fir 4. Each
     ..f                            t-                          /                                    O
                                                              #                                      I                                              ~ data point is the average of at least four tests. The influence of including
      ;.                            dSo                   j                                      I                                           '        the bond between the grout and the frogged ends of the blocks is the s                              w                  /                                             ,'                                               difference between the predicted strength for grouted versus ungrouted i                             j          ,/                                                g        ,
specimens. Here, % = 1.0 and 0.52 for the grouted and ungrouted cases,
      .                             ig                                                               g respectively.
0                       200                          400-                 600                                 Assurning full head joints (q,,, = 1.0), Eq. 4 was used to assess the effects of different block characteristics on tensile strength. The results y                                                   m m st.E s m e .* n.smi are shown graphically in Fig. 5. "Ihey reveal the following behavioral m
pig, ,
                                       . Tenene W Paranel se med Jointe wereue gensk Teness                                                           characteristics of masonry (ungrouted and grouted) under tensile stresses 3
Q                                                                                                                                 parallel to the bed joints, when Failure Mode I governs:
232                                                                                                                   233 5
* h. fg, was described by incorporating the.seelstances_d
            .,                                                                                     PS                                      29              So 12se                                                                                                                            the component matenals (mortar rnd arout as 'anaws
: i.                                                                                                                             (2N)fu = (2N)n a + (2at3.)a , + 2(1 w.Xat)a. . -
        ~
                                                                  *                                                                                                                                                                                                        ~ ~ - (5)
                                                                .                                                                 suas                          - ts jE2co-                                                                                                                      f'   in which 6 = the nominal height of the block, includmg the morhr M I    t = the block thackness; 2a = the block length (a is the overlap of the               t R                            <a
[
                                                                                          '            cd''-'                                                                               !
blocks for running bond); and a,,,, = the sheer band seienge of he a
* mortar (6). As mentioned prevassly, in this egaatson se h emng&
                                                               *j , . -ai 33
                                                                          -a,,,- se ssi
                                                                                                  -seese LO.
I of the grout is taken to be egaal to the spidtang tenoue sWnge d he E                                                                                                                                 grout, a*. True shear fadure cannot occur because me sheer emesse W                             ,,,,.t.                                                          ,       j                          result in pnnciPal tension stresses that control the capacity (2,6). N g soo-                                                                                                  =                          ing through by 26t, Eq. 5 reduces to as                            fg=%e.+n                  a Ma=*+           (I M *)'e * * * * * " " * " * " " " " . (6)   ,
                                                             @80 - /
* Assuming fully filled head joints (n = 1.0), Fig. 6 illustrates the re-sults from Eq. 6 for severai mrrerent nmeant lhese resuRs andscate                       '
[
the followmg as xcts of behavior of masonry under tenede eheses per-Oo                                                                                                                                                                                                                        j soo       200                                        soo     400                           ,'     allel to the bed .oints when Failure Mode II governs:
g
: 1. The shear bond and tensile bond strengths of the mortar have con-              .
PIG. 5eEflect of Blocit TeneRe SIsengen en TeneBe Shengst of Itseenry Persant                                                                                            siderat>ie effect on the tensile capacny.
to Bed Joines (Fagure stede l}                                                                                                                                    ,                                                                                            +
e ane tensue strength of the grodt has a pronounced sheer strength along the bed joint and, therefore, also hasr a ' -                ---
: 1. The block's tensile strength sigmficantl                                                                                                                                                                                                            l of the assemblage, especially for higher                                                                                        q,.y affects the tension capacity values.                                                                                                                            '
: 2. The mortar tensile bond strength has a relatively small effect on the                                                                                                                                      usuis/
capacity, particularly for high strength block.                                                                                                                                           y               os         to       ts      to
: 3. Since the hinck's tensile strenath is the dc.%. ant narameter, it seems                                                                                                             :,                            atma appmpnate to try to a Im** the m2= nary tensale strcnMh to the hInck"s                                                                       j                                              g Tensde strength.                                                                                                                                        ~                                   *-                     .gfies 4.
200                           ,0*
                       -    - -The grout strength has no infhsence on the tensde capacity of grouted
                                        =y under tensile stresses parallel to the bed joints when failure                                                                                                                          ,                   *p Mode I is the gc~. .g mode, because the fadure plane does not pass                                                                                                                                                           ,
througit the grout columns (2,4).                                                                                                                                                          } ,3o_-'         .as ons g
: 5. The net-to-gross-area ratio of the block for a vertical cross section                                                                                      ,                         ,,,                                   **     .
5 passing through the face shells just beside the                                                                                   ' nuddle web has a sig-                                   "                                                 4 nificant effect on the tensde strength.                                                                                                                                                                                                       I
                                                                                                                                                                                                                  $ soo/ <s,o' Failure Mode II-This mode is a stepped f=ihire =- h a===ine                                                                                                                                                     ""
_throuth the head and bed ioints, as shown in Fie. 3(bL It is a jois                                                                                              !
                      '" e and the blocks are not affected. For ungrouted Slockwork, failure
                                                                                                                                                                                       *                          @50                              sp,.- to occurs at the block-mortar interfaces by tensile bond failure along the                                                                                                                                                       ,,, tsa head joints and shear bond failure along the bed joints. 'Ihe failure of                                                                                                                                                      .fs. to grouted assemblages is smular, but with shearing of the cores at the                                                                                                                               o    so loo sso zoo 250 Mo horizontal section of minimum grout area (maximum q.). The failure of the grout is actually a tensde failure resulting from the shear forces (5,6).
W M8EE                       '''9#
                     "therefore, the splitting tensde strength of the grout, a,, was chosen as                                                                                                    FIG. Se-ENect of O'out Tensus Stengui en Tonese Spens #M W Paramel most accurately wy.. .. ting this type of failure. The tensile strength of__
to med Joines (Felius, asede n) 2S4                                                                                       -
235

              .       W on the assemblage knolon capedty, espedaHy for lower values of                                                                                  ususie                           -
: 9. (larger areas of grout).
n          go           a      ao                     ,
: 3. The net-no.grtas-area ratio of the block In the horhontal hh                                                        800              '         '            '        '
                     %, has an appreciable effect on the tensde capacity. A larger value of                                                            '""*"                                              ,         !
g means a mduced area of grout, thereby causing a reduction in its                                                     k          ****                                                   ^~
horizontal shear resistance and, consequently, a lower average tensile                                                 8          ''"
      -              strength of the grouted assemblage. However, for ungrouted masonry, iiiso    -
m.se to or the unlikely case in which.the shear bond strength between the mor-                           '
E          **         .as                                              ;
tar and the block is greater than the shear resisting capacity (tensile                                                                                          'SiU strength) of the grout, larger values of gn will result in increased assem-bkge tensile strengths,
{                 '"oocY m
j     .
                                                                                                                                                                                                     "E i
i      ..
                                                      -                                                                                         80 0 -
E Eqs. 4 and 6 express the lension capacity of either ungrouted or grouted                                                                                                                      I rnas nry under tensile stresses parallel to the bed joints for Failure Modes                                            W I and II, respectively. The noverning formula is the one giving the lower as             .
a rapacity. Thus, the mode of fadure could be deternuned for vanous                                !
strWgth~and geometric characteristics of the components. This is shown                                                  $        '
IR 88? Iof r'Presentative constant values of grout stwngth and net-to-                           !
h                           -unsame masomw groas-area ratios of the block. From this it is suggested that the mode                                                                             -enourso masomer of fahure and the capacity under tension paraBel to the bed joints are                                                   $
affected most by the following factors-                                                                                  g             -         -      -       -      '
3      o              200o        aooo          scoo
__1. The first factor is the aspect ratio of the block, 2a/b, which is a smh characteristic of the assemblage 'Ihe possibility of Failure Mode F 7 o                alues of's        ,
here F use Mode goveris                  p lSo         .       .        ,              ,
increasing the aspect ratios results in larger tensile capacities because the
                                             'W'es- L8'                                                                    failure path will be longer, thereby providing greater resistance along re /'m==tse                                             h y
I                                                                                      the horizontal joints. 'Ihis is particularly true for grouted masonry where
                                            - h- too g,1.2s Mn-*?s                                                     y                     the core area has a significant effect.                                               '
8        ''****
8                                     3                        2. The second factor is the block-tensile-strength-to-mortarc. ,.: cad-E                     shear 4x d-strength ratio,'which is a strength characteristic of the as-                 ,
S LOO                                                    --     ~                       semblage. Figs. 7 and 8 show that higher strength ratios increase the 2                               S'                                              .!
possibility of Failure Mode II occurring, especiaily for ungrouted ma-
                                                                .d ''                                              '
nis'                                                                                   sonry where the mortar is the only media joining the blocks. (In Fig. 8
* block tensile strength is taken as 10% of block compressive strength.)                   ;
o                                       These figures also indicate that tor grouted masonry, Fadure Mode 1 a
                                                                      . * *g                                                (splitting along vertical plane passing between the columns of groutDs
[50
                                                                           '',,                                    !
* ine expecrea mode for most r>ractical cases in which the a/b ratio is greater _
than one and the grout and block strengths are about the same. Com+
b O25-                                                       -
Parisons between the experimental results (2,4) and Eq. 4 for the tensile strenSth of masonry parallel to the bed joints (Failure Mode I) are pre-
                                                                -iseaoure """r                                              sented in Fig. 9. In the opinion of the writers, the proposed expression
                                                .       .       7'"*",""''*""                                               provides a good estimate of the measured strength, o      2        4        6              8      to      12 stocx TENSILE STRENGTH To MORTAR BOND                                                  Diaoonat Tsusste STneNoTH oF Masouny t
sm Ramen,**'"
When the angle between the principal tensile stress and the bed joint l
FIG. 7,-annuence g e -              .
is 45*, it has been suggested (8) that the assemblage tensile strength can enese,        p.,,u,gn          Q Q                               en Mode of Fouure tanderTen-            be considered as a measure of the shear strength of masonry elements such as beams and shear walls.
(
8                                                        '
237 l

w
                                                                                                ,,,,,,,,                                       is            .          .            .
         ..              ,.            to        ..      se   a,e                       i.       ..      3. 4.
                      $           ,=,,, ,"=_,.= ".88g"
{           . m -= =.u.=                                           *12-
                                                                                                                                                                                            ~
                          '"- .-.-e==n
                                    ===ca u.,
E
                                                                                       = ;- ;;;;;,                                         7e I                                                              -     l,
                     ,,,                      o.=                ma     y=-                                                                                                                 .                       ,
                                                                                                               . ,.                        5 LM                                                                     !
                     !                                                 E                   .                                               t'                       a-an                a 3
g y '"-
m            .
                                         / m .oero ..
t g
g m.o a. e. ..         .
i                      go.
D F
g  g== -                                      3                         .
j
[
                     *-              N 8 ''8"                                                                                              " 12     -
o-UNGfp0UTED taasotmY
                                                                -os    ,I_
g to-            .- ea e 'aa                                     m    oamm..an.
                                                                                       ..,-7
                                                                                                               -os                         ti            e-onouTED uasOserr
                                    % " **88                               so  .
                                                                                            " 88 fu' w
M l.1
* as0TE:EacH PoweTis aN metast Olr af -
l                  - oro                          g                      go                                                           LEast TwtE tests.
l         **
g O lo-                                             -
                                                                                                                                                                                                                     )
6 o       da 6 6 .aa 6                .a            o      6 6 6 a 6 ..
enour setsTTess TENsELE                 ,pel eROUT SpuTTafG TDeLE sTRDETgped         i                                                                                                h O        055          10             5    ZD FIG. SM m '- . of Meeeured and Calculated Values of Masonry Tenelle                                                          SPLITTfdG TENSILE STIENGTH ratio.
Strength Perellel to Bed Jointe (Fellure Mode 1): (s) Nominal 6 in. Block (143 msn                                                            ag/' Hit                                           -
Actuel);(6) ISO nwn Block (7.48in. Actual)
Tm Cepecay and Tennes Stengthe                 [
In Principal Directione Current Investigation.-For orientations of principal tension in direc-                                                                                                                         !
tions other than parallel and normal to the bed joints, the most probable                                                                                                           .
mode of failure is a mixed shear (debonding at the interfaces)-tension                                                                  I (splitting of the units and the mortar joints) mode of failure. This is due to the fact that the failure crack does not always follow the most direct f, . 2 '(q,,, + q.) aa ., + l        )ana 3                        4>
path, especially for grouted masonry (2,5). The tension capacity under 3
tensi!e stresses oriented at 45" from the bed ioints is assumed here to be                             + (1 qg)          J + (1 - tin) Geg      ' " . . " ' ' * " ' " " * " ' * " * *
* a runcuon on the average strengths in the two orthogonal directions (nor-                                              >                     -
_ man ana perallel to the bed inintM as follows-for Failure Mode 11 (debonding failure). The minimum of the strengths
                                     + f,,                                                                                 caiculated from the above two equations governs. The mode of failure fw=f              2       .................................................(7)                          as well as the strength is thus predicted using these pmposed formulas.
Fig.11 presents the average diagonal tensile strength of ungrouted Utilizing the expenmental results of splitting tests of masonry disks                              masonry, as calculated from the proposed formulas, versus the bl (2), a strong correlation was found between the diagonal tension capac-                                 tensile-strength-to-mortar-bond-strength ratio, a /a . It is s own ,                         ,
ity and the average strengths. As shown in Fig.10, a simple linear re-there are two distinct modes of failure possible under diagonal tension                      ,
lationship of the form given in Eq. 8 predicts streneths within 5% for a stresses.
_large range of grout to block tensue strength ratios.                                           ~
i" 4 (f,,, + f,,)                                                                                    1. A tension failure is distinguished by a crack passing mainly thmugh                    l fu=3               2       .................................................(8)                           the blocks and the mortar joints. This failure mode wd, l tend to donunate                  p in blocks of lower tensile strength, mortars of high shear bond strength,                   i Substituting the values of f,,, from Eq. 2 and f, from either Eq. 4 or                                and when the overlap of running bond (indicated by the s/b ratio) is Eq. 6 into Eq. 8 yields the following two equations (i.e., two different failure modes related to f,,):
                                                                ~
high"
: 2. A shear-tension failure (debonding failure) is distinguished by a 2    '/1               i stepped crack occurring mostly along the mortar bed and head joints.
                                            + tin I an  . + 1mem + (1 - rin)a,,
                                   - vi                                                                                     This mode of failure will. tend to occur in blocks of higher tensd fw = 3 l(2                      )          2                         ..................(9)           .
mortars of lower bond strength, and with units of low s/b ra                  .              .
for Failure Mode I (splitting tension failure) and                                                  '
be seen in Fig.11 that the aspect ratio of the block has a significant effect L                                    -
i .
g 1

         ,-
* soo                                                              }
       ,                            g                -      -             -
                                                                                 - noo
                                                                                                                                        ,W.s af                                                                                                                                       ,
tes cr   .w-i its ,, - ro
                                                                            .r.
s        # t ,e
: n. -ars                                                                             .
p                                           .
                                          ,, .avo              l
                                                                                 .an                                            iso-
                                                                                                                                                      ,w/ - cant a          .
too mo 3.. too               l:
                                          , , , ,                     M -'8                                                 t; e.-w-                              l
                                                                                                        ,                                                    , - to *
                                                     ;         I                                        -
l Ts-                                                                                   3                                 3. -ars       -ars
                                                   -l          l                 -asoj                                       E co-                           '*"*''                              }
n=a.Ium-me.                                                          W                                '=~'**
s              :
N                   Fa***E       '*
* E                                                                                                        i aso.                       l                                                                      _remeu rastume. re s                   .oso E                          !                                                                      _.--,m..
                                                                                -ors                                                                  o a so. _e.m.p.-'
(,3. rcasa l        = =-n===                                       ,
                                                                                                       '                                     m-           ra-ic.      .                          ;
                                                                                                                                        ,,,f,,,,- o s                       .ans
                                   <h               l         !                                        f                                 rossa resume I Mm-TEh8m o
                                             .      .         1   .     .
i i ,.wic ..<. -i a     4       e      e     o       iz stocx Toots steam To uoeran sono                                                                    ;        ;      ;      ;      6   ,2 snam narc 'w/'en                                                                  etocx Tasu steam To montan sonc                                 '
sinom nare.,m/,
Sect Cherecterletice of F _            ^ on Diagonal Tenelle Strength g                        of Cosnponente on Diagonal Teneas Strengih of arouted useonry on the tension capacity, especially for cases of high strength ratios, a       m/                                                                                    .
a.
Fig.12 contains the calculated average diagonal tension strengths of                      Staff rd-Smith et al. (11) suggested that the diagonal tension strength grouted masonry from Eqs. 9 and 10 for different block-tensile-strength-                   of brickwork equals either the tensile strength of the mortar or the brick,
,                to-mortar-bond-strength ratios, am /a,.,,,, using two grout strengths (rel-                whichever is less. The experimental results (2) associated with this re-l ative to the block strength). For an aspect ratio of 2, (a/b = 1.0) and a                   search pmgram indicated a reduction of diagonal 'ension strength of l
ratio of a,,/a m equal to 0.5, the demarcation between the tension mode                       nly abat 13 for a reduction in mortar strenge of akut E Es of failure and the shear-tension mode is identified as point A                              indicates that, for concrete block masonry, no proportionality exists be-For lower values of a,/, , tension failum will control (Eq' 9)$ whereas                tween diagonal tension strength and the mortar tensile strength.
t' shear-tension failure will govern (Eq.10) for higher values of am /a                           From an extensive investigation of block masonry piers, Balachandran ,
l               In other cases of high aspect ratios (a/b > 2) or higher grout strenh                l (1) also concluded that increasing the mortar compressive or tensile (e era: a m), or both, the tension failure mode will control.                               strengths does not appreciably increase the shear strength. In cases where i
Comments on Other Proposals.-Sinha and Hendry (10) suggested                           debonding is the governing mode of failure, the tensile strengths of nei-I               the following empirical formula to relate the diagonal tensile strength of           ;      ther the block nor the mortar seem to have much effect on the diagonal solid brick masonry, fu, to its compressive strength f' :                                   tension capacity of the assemblage. This is attributed to the fact that.thg l                                                                                                           bondine strength of the mortar is not predominately a function of its_
f'             [w = 2 N        .......................................".,,,,,,,,,,(13)                      comriressive or sensde strength (5,0).
I                             u       -          .                                                             For grouted block maso            , Balachandran (1) used the method of least Using a scon-- yu. .m.1 relationship to relate the geometnc and strength characteristics to the prism compressive strength, it is possible to get                     squares to nlate fa and               . vakes          b W whal data h a close agreement with the strengths predicted by Eq. II by using Eq. 9.               4       straight-line relatimship as follows:
However, when the diagonal tension failure is a combined shear-tension               '
                                                                                                            !""       g(If7                                                                   12 7              failure (Eq.10), Eq.11 overestimates the capacity (5). This can be at-tributed to the fact that the tension and shear bond strengths of mortar                     Because this is a best fit equation, it can provide reasonable agreement are not directly related to the compression strength (2,6). It is important                  for a limited range of conditions. However, as Balachandran pointed to mentmn that Eq.11 was developed for brick masonrf where the as-                           out, it is not possible to attribute any physical relationship between the l              Pect ratio cf the bricks (2a/b) is g,reater than 3 and, thus, the tensile            ,
failure of a grouted prism in axial compression and a grouted pier in mode of failure would tend to govern.                                                         diagonal tension. Therefore, the correlation between the diagonal ten-I                                                                                                         '
240                                                                                              241 i

r
           ,      sim stegth of ray and its compressive strength is not sound,                                     2. Dere are two possible modes of failure under tension parallel tp
       .          Particularly for grouted =.a,s y where the interschon of the mmponent                         the bed joints. These cre a tension mode described by a failure crack materials is mm complex.
passing through the head joints cnd the block's face shells, and a shear-tension mode described by stepped cracking at the block-mortar inter-lup u-* or Ikocet Cons seense                                                                 faces. The components' geometric and strength characteristics determme which one of the two is the governing mode.
  ~
For the calculations used in the previous sections, the values of the            I
: 3. The block's strength and its geometric characteristics sigmficantly net-to-gmss-area ratios were-taken at the critical sections for different           l         influence the tensile strength of masonry assemblages. Therefore, ga                         .
orientations of the tensile stresses. As can be seen from the proposed                        Proaches that consider the mortar type to be the main parameter af-                          j equations, these parameters have a sigruficant influence on the pre.                          Iectmg masonry tensile strength would seem to be unreanstic and would meno to lead to a sutistantial underestimation of masonrv mnantv. es-                           '
dicted tensile strengths of the masonry assemblages. Therefore, it is ap.
Propnate that some comments be made:                                                         -pecially under tenstle stresses parallel to the bed points.
: 4. 1here is no reason to expect that any direct correlaMon should exist i
: 1. For failure due to tensile stresses normal to the bed joints, the             !         between the diagonal tensile strength of masonry and its compressive                         l' strength of grouted masonry is substantially reduced as the net-to-gross-                      strength, because the factors affecting the two characteristics are quite area ratio y., increases. Therefore, the existence of flared top sections                      different. Relating the diagonal tensile strength to the tension capacity of the block (see the sketch in Fig. 2) results in a reduced tensile strength.                 nanal and parallel to the bed joints mms to be more malisdc.
The cause is that the grout area is decreased in the top section, which becomes the critical area governing the capacity of the assemblage. In                         Acomptroosem a cases where the net-to-gross-area ratio, y. , is increased from 0.6 to 0.75 was handed h@ operaung grants imm & Natual due to the flared shape, the capacity would be decreased by about 25%               ,               us usea for medium grout strength [m, = 300 psi (2.1 MN/m')]                                1          Science and Engineering Research Council of Canada and the Masonry
: 2. For Failure Mode I, due to tensile stresses rallel to the bed ' ~ ts                     Research Foundation of Canads. Also, associated with the experimental the practice of tapering the face shells to increase their thickness to rds Part of this pmgram, we wish to thank the Ontario Concrete Block As-the middle web (producing pear-shaped cores) may appreciably increase                          s ctatim for providm, g the concrete blocks and the Ontario Masoruy the tensile capacity, because it increases the area of the block normal to                     Catractors Associaten for providing the mason,s time.
the applied stresses. For the nominal 6 in. (190 mm) blocks used in On-tario, this increase of the face shell thickness helped, by increasing y,e                     Aressout I.-Rerenences from 0.46 to 0.7, to increase the overall espacity by about 30%.                                  1. Balachandran, K., "An Investigation of the Strength of Concrete Mawnry i
: 3. For Failure Mode II, due to tensile stresses parallel to the bed joints,                      Shear wall Structures," thesis presented to the University of Florida, at the flared shape of the top of the block reduces the area of gmut in the                             Gainsville, Fla., in 1974, in partial fulfillment of the i@...u.ts for the critical section, thereby causing a reduction in the horizontal shear re.                            Doctor of Philosophy.
sistance and, consequently, in the average tensile s'rength of grouted                           2. Drysdale, R. G., and Hamid, A. A., " Influence of Block Properties on the                  ,
essemblages.                                                                        '
Rexural Strength of Concrete Masonry," Procmfings of the Scoenth Australs-sian Conference on the Mechanics of Structures eruf Materints, Nedlands, West Australia, May,1980, pp.179-184.
Cosectussoses                                                                      i             3. Drysdale, R. G., and Hamid, A. A., "In-Plane Tensile Strength of Concrete
                                                                                                    ,                Masonry
* Canafian fournal of Civil Engineering Vol. 9, No. 3, Sept.,1982,
                   'Ihe equations presented herein to describe the tensile failure criteria        .
4,              yh G., Hamid, A. A., and Heidebrecht, A. C., " Tensile Strength or ungrouted and gmuted concrete block masonry provide a rational                 '
of Concrete Masonry," Journal of the Structural Drvision, ASCE, Vol.105, No.
approach to assessing the influence of the many strength and geometric                               ST7, July,1979, pp.1261-1276.
parameters. From a study of these equations the following additional              !              5. Hamid, A. A., "Behaviour Characteristics of Concrete Masonry," thesis pre-conclusions may be made:                                                          ,                  sented to McMaster University, at Hamilton, Ontario, Canada, in 1978, in l
partial fulfillment of the requirements for the degree of Doctor of Philosophy.
                                                                                                         $ 6. Hamid, A. A., Drysdale, R. G., and Heidebrecht, A. C., " Shear Strength of
: 1. The tensile strength of concrete block masonry in a specified di-          4                   Concrete Masonry loints," fournal of ihr Structural Division, ASCE, Vol.105, rection is not only a function of the strength characteristics of the com.                            No. ST7, July,1979, pp.1227-1240.
ponent matenals, but also a function of their geometric characteristics.           '
: 7. " Handbook on Reinforced Grouted Brick Masonry Construction," Brick In-The block's geometric parameters, such as the aspect ratio and the net-            :                  stitute of America,8th ed., I.os Angeles, Calif. '974.
to-gross-area ratio in the horizontal and vertical directions, have signif-        f              8. Johnson, F. B., and 'Ihompson, J. N., "Devch+dw.t of Diametncal Testing i                  Procedures to Provide a Measure of Strength e _acteristics of Masoruy As-icant effects on the tensile strength of both ungrouted and grouted              l                    ,gmygage,,~ Designing, Engineering and Construs. ion reith Masonry Products, F.
masonry.
B. Johanson, ed., Gulf Publishing Co., Houston, Tex.,1969, pp. 51-57.
242                                                   '
243 9

  ' ,.# .                                                                                                                                                                                   ~
    *. 9. Jones R. M , edunics of Composite Materisis,1st ed., McGraw Hill, Wash-
* ULTIMATE STRENGTH OF DAMAGED TUBULAR
: 10. Sirb, B. P., and Hendry, A. W., "RackingTests on Storey Height Shear                                                    CRACING MEMBERS Wall Structures with Openings Subjected to . -,..              , Desi       En-
           , gineering and Constrwetson anth Masonry Products, F."B. Johanson,gning, ed.,      Gulf                                                                                       '
Publishmg Co., Houston, Tex.,1969, pp.192-199.
By Charles F. Ellinas'                                l
: 11. Stafford. Smith, B., Carter, C., and Choudhery, J. R., "The Diagonal Tensile                                                                                                     '
Strength of Brickwork," ne Structural Engineer, london England, Vol. 48,                                                                                  deceksped 2 es-No. 4, June,1970, pp. 219 225.                                                                       f                          pact          tutsular tiram.cokunn awe-         ,
bers wie overen tiending and kical denting damage. De analysis is tissed on
: 12. Vanderkeyl, R., "Behaviour Characteristics of Brick Masonry," thesis sub-                       a member with a reduced effecove section to account for the effects of such mitted to McMaster University at Hamilton, Ontario, Canada, in 1979, in                       damage. Comparisons show the awhod 2 preide &see kiwer W ' &
partial fulfillment of the mquirements for the degree of Master of Engineering.               tions of available theoretical end expenmental results. Finany, ew n .nenks
-                                                                                                            of damage are emanuned in a detailed paramwtree study, the results of which can t= usefulin direct design apphcations.
APPE9 mix II.--Notatioes The following symbols are usal in this paper:                                             hemooucinoen                                                                            I^
2a      =    nominallength of block:                                                           Circular tubular sections that form the main bracing members in off-                  ,
A,      =    gross cross-sectional area of block;                                           shore jacket platforms are often prone to Incahwd darnage caused snainly A.     =
net area of block cross section in bed joint plane;                            by collisions with supply vessels and by the impact of falling objects.
6   =     nominal height of block;                                                   l   Available design codes for these components (1,3) do not provide any
* f'     =
compressive strength of masonry;                                            j   information on how the effects of such damage on member strength can f.,   = diagonal tensile strength of masonry;                                          ;   be estimated.
         /ea
                 =
                       ''"''''''''"''"'-""'r"'-'''"'''''"'
tensile strength of masonry parallel to bed joint where tension            i resulted in a considerable research effort, both expenmental and theo-                   ,
i is governing failure mode;                                                      retical, over the last few years (2,5-11). As a consequence, the mechan-f,a      =
tensile strength of masonry parallel to bed joint where com,                    ics of damage is now better un<.lerstood and, in certain cases, the weak-bined shear-tension is governing failure mode;                                  ening effect on member strength can be estimated with reasonable f   =    thickness of block;                                                             confidence.
9a     =
net-to-gross-area ratio of block;                                                  Depending on the type and intensity of collision and the member ge-9.s     =    net-to-gross-area ratio of block in a vertical cross section cross-             ometry and material characteristics, bracing tubulars will absorb the im-ing face shells just beside intermediate web;                                   Pact energy by developing deformations of the following forms:
                 =
        %             ratio of mortared area of head joint to gross area; e.e.,    =
shear bond strength of mortar bed joints;                                          1. Local denting of the tube wall without overall bending of the rr,em-e, s    =
splitting tensile strength cf block;                                            ber. This form of damage takes place mainly in short or thin-walled e m .,   = tensile bond strength of mortar; and                                                members. It also occurs close to the ends of tubular members, and can a,,     =
splitting tensile strength of grout.                                            be the result of impulsive collision loads.
: 2. Overall bending without denting of the tube wall. This is likely to occur with long members having small diameter-to thickness ratios, where collision loads are applied more slowly and closer to mid-span.
: 3. Combined overall bending and denting deformations. This is the                      j most likely form of damage that can occur in tubular memberw subjected                   p to lateral impact forces.
p'   i i
In the case of overall bending damage without denting, a good cor-                         ,
relation with test results has been obtained by the use of an clastic-plas-
                                                                                                         'llonarary Research Asst., Civ. Engrg. Dept., Univ. Coll. london, Imndon, England; Engrg. Specialist, J. P. Kenny & Partners, l.ondori, England.
Note.-Discussion open until July 1,1964. To extend the closing date one month,             i a written request must be filed with the ASCE Manager of Technical and Profes-
                                                                                                ;      sional Publications. The manuscript for thisymper was submitted for review and               '
possible publication on January 24, 1983. This paper is part of the fournal of Senactural Engineering, Vol.110, No. 2, February, lyse. CASCE, ISSN 0733-9445/
84/3002 0245/501.00. Paper No.18591.
,                                                                      244
'                                                                                                                                                   245                                             .

y                     f
* I                                                                                         .
4                           )e                                                                                                                                            14889                                   JULY 1979                                 'ST7 4                                                                                                                                                                                                      .
JOURNAL OF TILE d                                                                                                                                                               ,,
STRUCTURAL DIVISION TENSILE STRENGTH OF CONCREFE MASONRY
                 'l-                                                                                                                                                                                                By Robert G. Drysdale,' Ahumed A. Hannid,'
                   -(                        ,
and Arther C. Heldebrecht," M. ASCE I                   .
I                   t i
r                                                                                                                                                                   lernocoenow f                                                                                                                                                                    The tensile strength of masonry is an important parameter in the behavior              j a                                                                                                                                                                   of structural masonry elements such as shear walls, where horizontal forces j                                                                                                                                                                  will produce tension or shear stresses, or both. A possible failure mode of g                                                                                                                                                                   mascary shear walls is diagonal tension failure that is mainly governed by the l                                                                                                                                                 tensile strength characteristics of the combined masonry materials.
8                                                                                                                                                    Numerous studies on the diagonal tensile strength of masonry have been done, but mostly for brick masonry. Many investigators (I,3,9,17.19) considered the diagonal tension capacity to be a ccostant value directly proportional to the square root of the compressive strength of masonry. "Ihis assumption is not justified when masonry is considered to be a composite material having g                 ,                                                                                                                                                  more than one tensile strength characteristic (7,10). It is reported (22) that to
                         +                !                                                                                                                                                  be able to predict the strength capacity of masonry under a multiaxial state
                           -1I                                                                                                                                                               of stress, the directional variation of the trissonry tensile strength should be
                     .)    '
considered.                                                                               ,
T ae effect of grouting on the behavior of concrete masonry under different r              ,                                                                                                                                                   tensile stress orientations has not been adequately documented and this may
                         'f              ,                                                                                                                                                   account for the assignment of similar allowable stress values for ungrouted               '
                         ]'              ;                                                                                                                                                   and grouted masonry in typical masonry design codes (13,16).This design approach ignores the influence of the directional continuity Drovided by the trout that
                        ;1                                                                                                                                                                   could be envisioned to significantly af fect tie capacity, especially under tensile l                                                                                                                                                                                      ,
                        ,                                                                                                                                                                    stresses parallel to the core direction (norma.1 to the bed joint).
j This paper describes an experimental study that has been camed out to g                                                                                                                                                                    investigate the strength characteristics of ungrouted and grouted concrete masonry        l Note.-Discussion open until December I,1979. To cAtend the Closing date one month, h                                                                                                                                                                    a written request must be filed with the Editor of Technical Publications. ASCE. This e                                                                                                                                                                    paper is part of the copyrighted Journal of the Structural Division. Proceedings of the 9,                                                                                                                                                                   American Society of Civil Engineers, Vol.105, No. 517. July,1979. Manuscript was          ;
submitted for review for possible publication on August 24,1978.                          ;
                                                                                                                                                                                                ' Assoc. Prof., Dept. of Civ. Engrg. and Engrg. Mechanics, McMaster Univ., Hamillos, Ontare Canada.
                                                                                                                                                                                                ' Post-Doctoral Fellow Dept. of Civ. Engrg. and Engrg. Mechanics, McMaster Univ.,
                        -i                                                                                                                                                                   Hamilton, Ontario, Canada.
                                                                                                                                                                                                ' Prof., Dept. of Cav. Engrg. and Engrg. Mechames, McMaster Univ., Hamilton, Ontario,
                  ,,                                                                                                                                                                         r'amam 8m 5                                                                                                                                                                                                                1261

(L           -
l                                                                                           .<
                     . 1262                                      JUI.Y 1979                                   ST7    ST7                               TENSILE ST"ENGTH                               1253 U                assembl:ges under diffirent tensile stress orient:tions. The results show the                 w:re 440 psi (3,000 kN/m') cnd 250 psi (I,700 kN/m'), respectively. The W              effects of grout strength, mortar type, and bed joint reinforcement on the tension            coefficients of variation for the compressive strength, the Ocxural tensile strength, capacity. The failure mechanisms for different conditions are examir.ed and                   and the splitting tensile strength were 4.3%,8.5% and 11.5%, respectively.
the current design provisions are evaluated.                                                      Mortars.-The proportion of the two types of mortar used are listed in Table I. The proportions were actually controlled by weight rather than by volume.                 ,
                   . Mateniats                                                                                      Both ricasures are shown. The sand was sieved to meet the specifications in l
ASTM C-144. The water contents were established by the mason's requirements
                   ;         The ranges of parameters have been limited to a single type and strength                  for suitable workability but then maintained for all subsequent batches. The l     of concrete block, two types of mortar, and three types of grout. Specific                    mortars were thrown out after a 1/2-hr period to avoid variations resulting properties are examined in the following sections.                                            from retempering. The control specimens were 2-in. (50-mm) cubes were tested Concrete Blocks.-The autoclaved concrete masonry units used in this inves-                under .nxial compression (ASTM C-476) at approximately the same age as the corresponding assemblages.
TABLE 1 Morter Mis Proportions                                         Groot.-The proportions of the three different types of grout used are presented in Table 2. Note that the weak grout, GW, and the strong grout, GS, do not              '
initial        satisfy the proportions specified in ASTM C-476. Batching was controlled by flow, as        weight The water-ccment ratios were established to give about a 10-in. (250 mm) y o,,,,                Proportions by Volume (Weight)                                    slump assuring a Guid grout that could be poured in the cores without separation l
e per.
type          Cement        time                  Sand     Water        contage         of its components. The two types of control specimens used were air cured                 i l
(1)            (2)           (3)                   (4)       (5)          (6)          3-in. x 6-in. 06-mm x 150-mm) cylinders, and block-molded prisms using paper              i i
S              I           0.5                   4.0                     115          towels as a porous separator so that water could be absorbed by the blocks.
'                                                                                                                                                                                                                l j                                               (0.21)                 (4.24)    (0.9)                       The prism dimensions were 2-1/2 in. x 5-5/8 in. x 7-5/8 in. (64 mm x I40                   ,
N              I           l.25                  6.75                                 mm x 190 mm), which give nearly the same surface area to volume ratio as I                                               (0.53)                 (7.16)    (I.46)         118          a block's core. The control specimens were tested under axial compression and splitting tension p) at the time of testing the corresponding assemblages.
TABLE 2.-Grout Mis Proportions Tm Soms Proportions by Volume (Weight)                               Johanson and Thompson (10) developed a method of testing to establish the              i
          ]
tensile strength of brickwork using 15-in. (380-mm) diam circular disks loaded
* gr u          Coment         Ume                   Sand    aggr      to   Water         in c mpression along the diameter. The typical mode of failure was by splitting           !
(1)            (2) along the loaded diameter as a result of the mduced transverse tensile stresses           :
              ,                                                (3)                  (4)        (5)          (6) that are approximately constant for about 65% of the diameter. The test is                 !
:                     GW                I                                5 similar to the indirect method of determining the tensile strength of concrete I
GM                I          0.1 where a cylinder is split under loads applied along its diameter. For clastic (0.44)                (3.55) behavior, the tensile strength of the specimen is calculated as
            ,                   GS                I                                I           I                             2P g                                                                       (1.1l}      (0.9)       (0.45)      f, =         ...........,,.,,,,,.,,,,,,,,,,,,,,,(1) w                                                                                                                  1rA                                                                                   ,
I.             tigation were the standard hollow 6-in. (ISO-mm) blocks with two pear-shaped                 in which f, - the tensile strength; P = the failure load; and A = the area i          cores.Their physical and mechanical properties were based on five test repetitions           along the splitting plane.
p(             each.                                                                                            The splitting test, as a technique for determining the tensile strength, h'as
(                  The gross area per block was 87.9 sq in. (567 cm') and the net area was                  several advantages over other techniques such as the direct tension or the flexoral          -
found to be 52.1 sq in. (336 cm'), which results in a net to gross area ratio                tension test. The test procedure is simple and less susceptible to testing errors l
of 0.59. The specific gravity was 2.00 with a coefficient of variation of I.5%.              and it leads to less variation in the results compared to either the direct tensile           '
An initial rate of absorption (IRA) of 36.3 g/ min /30 sq in. with a coefficient             test or the flexural test (4,10,14). The test results are innuereced less by stress
          .l             of variation of 8.4% was measured [American Society for Testing and Materials                concentrations compared to the situation for direct tension tests and the rate (ASTM) C-67]. The compressive strength of half blocks, hard capped and tested                of loading has littie effect on the test results (4). Also, since splitting failure natwise (similar to ASTM C-140 but with half blocks), was 2.850 psi (19,600                  occurs through the central portion of the specimen, surface imperfections and kN/m') and the flexural tensile (ASTM C-78) and splitting tensile strengths                  multidirectional drying of the test specimens will have less effect. In addition l
l                                                                                                         -

l                                                                                                 ,.
l 1264                                   JULY 1979                                'J7         ST7                               TENSCLE STRENGTH                               1265 ts the aforementioned adv:ntag:.s, the splitting test fzcilitit:s testing identical        of 24 br, groot was poured into the cores and puddled using a steel rod.
specimens ur der tensile stresses at different orientations from the mortar bed The disks were tested under splitting loads having three different orientaticevs t
joints and the nearly uniform tensile stresses in the failure zone most closely              with respect to the bed joints. The maximum principal tension stresses are
!                     reproduce the conditions for principal tension stresses for in-plane loading of
'                                                                                                                 perpendicular to the loading plane and are defined as being oriented at an angle, walls.
       ~
: 8. from the bed joint direction. Using 8 = 90' allows measurement of the i
               '         In this investigation, the circular shape was modified to the hexagonal shape            tensile strength normal to the bed joint (tensile bond strength of the mortar l
shown in Fig. I because it was n: ore convenient to prepare. A finite element in the case of ungrouted specimens). For 8 = 45*, measurement of the diagonal i      pronram (7) was used to analyze the stress distribution along the loaded plane             tensile strength is obtained and at 8 = 0', the tensile strength parallel to the of the suggested hexagonal shape. The longitudinal and transverse stresses along           bed joint is found. The program was composed of three groups of tests, each j       -     ;       the loaded plane of the hexagonal disk from the finite element analysis were               concerned with one parameter.
close to those for a thin circular plate analyzed using elasticity theory. Thus,              in Group I, the effect of the mortar type on the tension capacity of ungrouted l
Eq. I can be used to accurately calculate the tensile strength assuming a                  masonry for orientations of O*, 45', and 90' from the bed joint was studied.
              ,       homogenous elastic material.                            .
Types S mortar and N mortar were used. Four repetitions for each test type Using the finite element method, Stafford-Smith, et al. (18) analyzed nonhomo-          were adopted.                                                                            '
l                     genous circular brickwork disks under splitting loads oriented at 45' from the For Group 2, the effect of the grout strength on the tension capacity of masonry assemblages under principal tension stresses at 0*,45', and 90' from               l'
                                                                      -       i        samas,                    the bed joints was studied. The three different grout mixes were used and
                                                                      '7               ''''
                                          '$ ~                                                                   Type S mortar was adopted for all of these specimens.                                      i
                                                                                 \
t                                             /
* 4       \
7,y
                                                                                       -s The effect of providing joint reinforcement in each bed joint on the cracking ,
load and the tension capacity was investigated in Group 3. Orientations of 45' i
l i
4 1
* d
                                                                        *N                                       and 90' from the bed joint for ungrouted disks and of 45' from the bed joint               j I
for grouted disks were studied. Only Types GM grout and S mortar were used.                i
:stau                               Mechanical gage points were mounted on both faces of the disks using a i     *
                                                                                       ,m,m,                    4-in. (100-mm) gage length, so that the deformations in the transverse direction N             . .
ff*d?d'
                                                                                       ,                        along the loaded plane and in the vertical direction at the center of the disks g".7jk N
c
                                                                                  //   Emes could be measured. A typical arrangement of the gage points is shown in Fig.
: 1. Strain measurements were taken at regular load increments up to about 90%
s se -                                  of the failure load. Strips of I /2-in. (13-mm) plywood along with I/2-in. (13-mm)         :
I                                        Ewo                        rRONT                                     thick steel plates provided the bearing points at the top and the bottom. The              !
ELEvaTroN                 ELEVaTK)N strips had a 2-in. (50-mm) width that assured a distribution of the applied load RG.1 Test Sotop for Splitting Tension Test (1 in. - 25.4 mm)                   over a width equal to 1/16 of the height of the disk. The dimensions and
          '                                                                                                     the test setup are shown in Fig.1. A finite element analysis (7) indicated that            !
this load distribution results in a deviation of about 2% from the stresses calculated l          bed joints. The results demonstrated that when the brick is stiffer than the on the besis oflineloads. This smalldeviation is ignored in the strength calculation.
      ,   !          mortar, concentrations of high principal tensile stresses are induced in the mortar        He crack patterns at the onset of failure were recorded.
layers, ne localized stresses may be of particular interest for a detailed                                                                                                            ,
i examination of the interaction of the constituent materials. However, in this              Test Resats g          investigation, the effects of the constituent materials are_ detected only in the     l                                                                                               I geraaed properties of the composite. It was felt that_this_ approach would be Modes of Failure.-The general mode of failure was that of splitting along more likely to lead to the practical application of the tat results (21).                  the central portion of the disk due to the induced transverse tensile stresses.
Expeansewrat Devans ano Testuse Pnoceoune                                                  All specimens with the bed joints perpendicular to the loaded plane (0 - 0*)
failed in the tension mode with the failure line passing through the intercepted 3
face shells of the blocks and along the mortar-block interfaces at the head L
A total of 63 f J-sized ungrouted and groute'd masonry disks of the type              joints, in the most direct line between the load points. The same mode of shown in Fig. I were constructed in running bond with full bed and head joints.            failure occurred for both ungrouted and grouted specimens. In the latter case, The mortarjoints were tooled on both faces of the specimens. All disks were               the failure plane crossed the block face shells at the interface between the constructed by the same experienced mason using a good standard of workman-center webs and the grout, which left both the center web and the grout core ship. Upon completion of each specimen, two blocks were placed on top to                  Intact. Typical failures for both angrooted and grooted disks under tensile stress j                  provide some weight on the bed joints of the upper courses. After a minimum          l i    parallel to the 1,ed joint are shown in Fig. 2.

_V
                                                                        ~
       .b           *                                                                                                                                                                                                  .
                  . 1266                                    JULY 1979                                      ST7         ST7                                 TENSILE STRENGTH                                       1267 When the loading plane was parallel to the bed joints (0 = 90'), the failure                    disks occurred along the bottom of the bed joint which was the weakest plane line was along the mortar-block interface and for grooted disks, the grout failed                  because it had the minimum grouted areas (see Fig. 3). As will be considered.
by splitting tension along the same line. For the ungrouted disks the failuri                      in more detaillater, the grout capacity is the dominant factor for tension normal to the bed joint.
h f,,, .' 4
                                         <         f:.!fbh M        d      *                                                      ..
:                                                                                                  .8 p
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(a)                                             t6)
       ,            FIG. 2mTypicalFailures for S;: 'n:= Oriented at e = 0':(a) Grouted:(6) Ungrouted i
S Morter:(6) Type N Morter
                                                                              .E  UE                      '
j l                                                                                                  !
* Eet       .
r                                                                                                                                                                      '
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                              'D '
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I*I                                                     t6>                   FIG. 5-Typlent Failures for Grouted Specimens Oriented et e - 45':(e) Week Grout:
(6) Medium Grout; (c) Strong Grout FIG. 3.-Typleal Failures for S;: ' :n Oriented at 8 = 90*: (e) Ungrouted: (6)
Graned                                                                                                                                                                                      ,
evident. For high strength mortar (Type S) the failure mode of ungrouted disks took place along tl.e top of the bed joint corresponding to the minimum contact                approached the ideal tension failure with the fracture crack extending through area and the last block to come in contact with the mortar. Because the blocks                 the block and the mortarjoints in the most direct path between the load points
:                 have flared webs and face shells at their top surfaces, failure of the grouted             ;    [see Fig. 4(a)]. For a low strength mortar (Type N) the failure crack did not

     .~                            .
i
                        !       f.   *
                     ,                                                                                                                                                                                                                                                            s p
1268                                                          JULY 1979                                                                  ST7 t
* ST7                                     TENSILE STRENGTH                      1[89 i
       '                                                                                                                                                                                    pass through the block all along the loaded plane, but sometimes followed the'-
* mortar-block interface along part of the failure plane as shown in Fig. 4(b).
TABLE 3-Spiltting Test Results for Masonry Dielle                                                                       This latter case was *a mixed shear (slip at the block-mortar interface) tension m a=
(splitting of the block, mortar, or grout. or all three) mode of failure. For 6
T a.a. so.aem 1. v                on. aim aa, s.
    *~$      .*                                                                                 %,,,,,                      fr n e.s J aen. m s ne. p= .e                 =*
200
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sp==.n e e,.puen p= .g inch ceas een*      7.ne.n*
n.
It       Mn n.
it   ha.es n.
st     Mn a,
p                            m                m                    (2>           m            m     m             m            m         m        (=         (m                                g e 1 I       caer       e.              3.ono                               les                        ut j
         "il                                           Type s                 (tz.Mr                                us             ne         sia        u4 se so         es 5 , iso.                            e
  *            )j           i
                             !                         m.nar                                                        314        (S.2%)*        106    (4 99f         M     (18 94f                            g$                                               ,,gg HS
       ''l  ,
l U.grees.d.                 3.180 122 tel                          96 10 43                                       gg                                  *
* I Type N                  (9 74f                              tot i         $
nar 100        808        tal        $1         43                            ,,
y $
94       0 4%f           106   (6.3%f         M     (810%f                             sg i        l                2      Type GW y         t.        3. 00          1.710                114 1
Type s lie                               114                 107 yo            [
(11 31 /        (1965 /      (40Sf    12I W
j lit         138       144       807        106 wz                                                   '.E 6
se.nar                                                      ll?        Q 3%f           149    (S 6%i*      102      O 193 8                          6- O 0 .04.,p Type Gh4 gr       l.                                                                                                                                                                                          >g Type s 3.0e0 (12.Mf 3.290 (7.'*f GSesf 3o0    122 In 823 141 in        15:
114 H1         sla                             g8
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                                                                                                                                                                                                                                                         \g         m t-i                                   Type        p                                                  l
                ''                                                  t.          3.cos (t:1sf S.240 HI Isf deo                               I                   I                                                                N M Mal m m                -05 Type s                                             (f osf   les            H3          167       isa       lis        IM                                              ,_. ,  ggg,,,,,,3        g 3
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                                                       ,e.
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                                                    .,v                         3.             use          n.                                l Type GM               (12.3%)*         (7.95 f     G SSf                               les       ISS go    e. Type s                                                                         143    (' 4S f                                                       ''
nar                                                                                                                                                                                                1
                      )   *{              *C     preseeve o'                          45*          90=
                   ,                                                                                  If t.ectes for Typ. s on.nar d ths beechs. f e Type M .e.nar.
Ti.e. e e. ,e. . serv.sth  .v     of e.. ear.d c.hes. (ses.d                                                                                                                                          ANGLE e                          ,
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no. 7-Effect of Orientation of Bed Joints on Splitting Tenelle Strength i             t                                                                                                                                                                                             Ea 8 ll i               ;i                                       1  1 f    -
c    g.                 ,.
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                                                                                                                                       ;r    w 3                        i 3                             da -= -[                                    e
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FIG. 3.--Verlation of Stresses Normal to Bed Joint wereue Orientation of Bed Joint i
FIG. S.-Typical Feilures for Reinforced Specimens: (s) Ungrouted, e - 45*: (6)                                                                    grouted disks, a similar mixed mode of failure was also observed as shown Ungrouted, e = 0*; (c) Grouted, e - 45e                                                                                                      l    in Fig. 5.
                                                                                                                                                                                            ,The results of the Group 3 tests showed that joint reinforcement had no effect on the crack patterns. However, it helped to ;nvent the sudden collapse

1,        *                                                                                                                                                                                             .
t                                                                                                                                                                                                             .
1270                                  JULY 1979                                  ST7           ST7                                TENS!LE STRENGTH                                      1271 exhibited by the nonreinforced specimens. Fig. 6 shows typical reinforced disks                are muchJigherJhan the tension bond stren sh of the martar It is interesting
                   !           after failure,                                                                                 to observe that grouting had no effect on the capacity for tensile stresses parallol
.                                  Effect of Orientatica of Bed Jelet.-The tensile strength,f,,is calculated front            to the bed joints (9 = 0') regardless of the grout strength. As shown in Fig.
Eq.1, on the basis of the gross area for botti ungrouted and grouted specimens.                2, the failure crack occurred along the blocks and the head joints and did The results for the v'ngrouted, and grouted specimens are listed in Table 3 for                not cross the grouted cores. It seems that the tension bond between the blocks Group Nos. I and 2 respectively. The results of the corresponding control                      and the grouted cores has a negligible contribution.
      .                        specimens are also presented.' Fig. 7 shows the influence of the orientations                     It is worth noting in Fig. 7 that the maximum tensile capacity for the grouted j            of the principal tensile stresses with respect to the bed joints (measured by                 specimens was achieved for 6 = 45' (the diagonal tension case) where the angle 6) on the splitting tensile strength of the masonry disks. The obvious                  influence of the relatively small mortar bond strength is not as significant. The variation of the tensils strength with the orientation of the bed joint was as                capacity in this case is mainly influenced by the contribution of the tensile
                 ,             expected and this illustrates why the basic strength characteristics in tension               strengths of the block and the grout.
should be evaluated in the principal material directions. Similar observations                   Use of Net Aree versus Gross Area of Bleek.-The tensile strengths verses were made by Johanson and Thompson (10) from tests on brickwork disks                         the orientations of the maximum tensile stress with respect to the bed joint under splitting loads.
For ungrouted disks the minimum capacity was obtained when the tensile                                            mu.'
                                                                                                                                =              '*        '"       "               =
stress was normal to the bed joints (0 = 90'). In this case, the capacity was                                                                                          .
I only influenced by the mortar bond strength. Under tensile stresses parallel                 jj                                                          "u.mourro T
to the bed joints (0 = 0*), both the block splitting tensil: strength and the
                                                                                                                                                        ,. ' y .    -      t  ,,
j "*-
N\                  ,
tensile bond strength of the head joints contributed to the capacity. For 0 =                 g 45', the capacity was governed by either the block and mortar teasile strengths                                                          ro                                      \.
                                                                                                                                      /y", ,y',j f;                                                j                                         ..
                             . for a tension mode of failure, or the block tensile strength and the mortar                   !;;                                               g'             eemos: anc :       \              i tension and shear bond strength for a mixed shear-tension failure. Because                    3
                                                                                                                                                       .                  . y Islan                                                                                 .\ 'gg the tensile strength of the block is much greater than the tensile bond strength of the mortar, a higher contribution of the block increases the tension capacity of the specimen. Examination of the normal stresses applied on the bed joint,                *
                                                                                                                                          /[;          .             , o, g    [ ,,,_
3
                                                                                                                                                                               ,9   o                                 ,
j for various orientations of the bed joint, as shown in Fig. 8, with the tension              E.                  e-o - .         .
l capacities in Fig. 7, indicate a correlation between higher normal compression               !"                  Z _I            I             f st-                                             ;
and high tension capacity for ungrouted masonry. The normal compressive                      I                                                 j                      o                o         i stresses applied on the bedjoints seem to increase the resistance of the assemblage                     ' '** "*I.. U "*'" ""'"               g 5          T 8       o
            .                   to the debonding failure at the block-mortar interfaces that forces splitting to                            ' " * " " "                       w
                                                                                                                                                                                      %'Q',m',".e',ur"vmos'"c"a e-w rs i                             e"nour I                   occur in the block and thus increases the tension capacity of the assemblages.                    *
                                                                                                                                     ,,y,,,",, ,g,,,1 gg                            o-
                                                                                                                                                                                                     .g            . ,o.
I   .                      For grouted specimens, the anistropic inhavior is also evident in Fig. 7, but
            ,                   is less pronounced than for the ungrouted specimens. This behavior indicates                FIG. 9-Effect of Grout Strength on                FIG.10-Tenelle Strrythe for Type S that the grouted cores provide some continuity, which tends to give masonry                 Splitting Tene31e Strength                       _Morter versus Orientt. ion of Bed Joints j
i                    a more isotropic behavior.                                      .
I                         Effect of Morter Strength. 'Le test results for the ungrouted disks indicate            are displayed in Fig.10 using the net area for ungrouted masonry and the 4                       that the mortar type had little effect on the tensile strength. A decrease of              gross area for grouted masonry. The use of the net area for the calculation I
about 60% in the mortar compressive strength reduced the average assemblage                of the stresses for units with voids is a widely adopted practice (1,2,13,20).
j                     capacity by approx 11%. The relatively insignificant effect of the mortar strength         De current results indicate that calculating the tensile stresses for ungrouted on the tensile strength can be attributed to the fact that the mortar bond strength        masonry on the basis of the net area for 8 = 0* and 45' may lead to higher is not predominantly a function of its compressive or tensile strength (8). The           tension strengths than for similar grouted assemblages. It seems that basing mortar bot.d strength is mainly influenced by both the plastic behavior of the            the tensile strengths far grouted masonry on the tensile strengths calculated mortar, measured by its initial flow and water retentivity, and the block's surface       on the basis of the net are for ungrouted masonry could lead to unconservative condition, measured by its roughness and initial rate of absorption,                      values except for tensile stresses normal to the bed joints (0 = 90*). De fracture Effect of Grout Strength.-Fiz. 9 shows that aroutina rianificantiv affected            surfaces showed that for 8 - 0' and 45', the areas along the splitting planes
                               ,)he tensile strength of masonry disks. The grout's contribution towards increasinz_ .      were different from the horizontal cross-sectional net area of the blocks.
Therefore, it seems that there is no adequate single definition of net area for the tensilewhen contribution    strength    varied the principal     with tensile    the orientation stresses were normal of     thebed to the   bed iMauioint.11the had      a maximum calculation             _ stresses of tensile l        for ungrouted masonry subject to tension (6 - 90') since the arouted cores contributed by their tensile strenaths that l           at differect orientations from the bed joints.
k

             >              l. a 1272                                   JULY 1979                                  ST7     ,    ST7                             TENSILE STRENGTH
            .9                                                                                                                                                                                                    1273
   .                                   Effect of Joint Retsforcement.-Eleven disks, having Dur-0.Wal Steel (No.                   observed experimentally that the small amount of horizontal steel embedded-k                       9 Gage Wire) as the joint reinforcement in each bed joint, were tested under                  in the bed joints of ungrouted concrete masonry shear walls had no significant j                     ' tensile stresses oriented at 0 = 0' and 45' from the bed joints to study the                  effect on the shear (diagonal tension) capacity of the walls. From the failure l                       effect ofjoint reinforcement on the tensile capacity of masonry. The test results             modes in the current experimental program it was obvious that the joint
     -R                             are listed for Group 3 in Table 3 and are plotted in Fig.11 along with the                    reinforcement helped to prevent the sudden brittle failure exhibited by the I                       results for similar unreinforced specimens.                                                   nonreinforced disks.
     ' 'h No tests were done for the ' reinforced case with the maximum principal tensile
                          ;         stress normal to the bed joint (0 = 90'). Because it was lot.ical to assume                   Review or Coor PaowmoNs that the reinforcing steel would have no effect on the tensile strength in this direction, the reinforced case was considered to coincide with the previously                    Typical masonry design codes (13,16) assign different allowable tensile stresses tested unreinforced cases designated as points'I and 2 in Fig. II.                            depending on the orientation of the stress, the mortar type, and whether the 7g                          For the case of tensile stresses parallel to the bed joints (8 = 0'), only                 cores are filled or not. (The proposed American Concrete Institute Standard, "Bai! ding Code Requirements for Concrete Masonry Structures," which is 3          ,
200 L'
approved for discussion, gives allowable tension stress values similar to those P=oo8%                                   from Ref.13.) For the latter factor, if the grout is at least as strong as the
                                                      .                                  TYPC GM GHouT
                                                                                         " 8 " ""                                 unit, the SIlowable stresses are the same or greater than for units without voids.
                                                       $                                                                                                                                                                   l rs                                                        The following comments are based on the observed behavior:
                                                       ?
(                                           ;. 15d     -     ._-.~.,,,,N                                                                                                                                        i M
io                            I. The allowable tensile stresses for units with voids (ungrouted masonry) t; N                                    do not have consistent factors of safety for different orientations of the bed          r N                              joints. An example of this is shown in Fig.10. Also, the data presented in l7, ,, {     -----
Fig. 7 indicate that, for the type of block used, the mortar type or strength q*                        g'N, \       PoWT2 does not have a pronounced Cffect on the tensile strength of ungrouted masonry.
s,
                                                                                           \y                                    It is suggested that the tensile bond strength of the mortar, which is the dominant parameter, depends not only on the mortar properties but also on the physical ht                                         N -as                                                                                                                  {
                                                                                                   \                             properties of the block (i.e., surface roughness and initial rate of absorption).       :
                     !                                 Es                                                                           2. For grouted masonry, design values for tension normal to the bed joints           i
          ;.f        !                                 d    3n_                                        ,                         are similar to those for solid masonry and depend only on the mortar type.
romT'                                                                                                                        !
g                                                                        As shown in Fig.10, ignoring the contribution provided by the continuity of
               '                                                                                                                                                                                                         i g                         UNGRourED GROUTED
                                                                                             *-*                                 the grout, which has much higher tensile strength than the tensile bond strength
                                                       .a           UNREWoRCED c         3 y                                                          RE woRcED      C     O SW                                   of the mortar, resuits in a substantial underestimation or the increase in strength.
Fig. 9 indicates that for 9 - 90* increased tensile strength of the grout causes        '
         ;]                                                    ,,                 j3                  ,o.                    ,   a significant increase in the capacity of concrete masonry and as such should n                                                                     aungg                                           be considered as a major parameter in the assignment of design stresses.
7*                                                                           ,                                            3. For tensile stresses parallel to the bed joints (0 = 0'), lower allowable FIG.11 Effect of Bed Joint Reinforcement on Splitting Tensile Strength stresses are assigned to hollow units compared to grout filled units. In Fig.            i p                                                                                                                   9, it can be seen that grouting has no effect on the capacity. Therefore, when ungrouted specimens were tested with joint reinforcement. This' decision was               the strength of ungrouted masonry is based on net area, as shown in Fig.10,              t N                        based on the behavior of the unreinforced specimens, which indicated little                its allowable strength should be higher than for grouted masonry, which is h         ;
influence of the grout on tensile capacity parallel to the bed joints (see Fig.            based on gross area. Similarly, units without voids should have a higher allowable ID                          9). Therefore, Point 3 in Fig.11 is actually the strength for ungrouted reinforced         strergth than grouted units for tension stresses parallel to the bed joints. From 1                        specimens and it is also assumed to represent the grouted case. Point 4 represents         the failure modes shown in Fig. 2, it is apparent that the tensile strength of           '
the cracking stress, which is essentially the same as the strength of the unreinforced     the block should be the most significant parameter in determining the tensile I' .                                 specimens. In this case, the reinforcement increased the capacity by about 25%.            strength in this direction.
             -                          The results show that, under a diagonal tension loading (0 = 45'), reinforcing              4. Current code provisions do not assign design values for diagonal tensiors, 78                         the bed joints with Dur-O-Wal steel had no significant effect on the splitting         i except as can be interpreted from allowable shear stresses. The test results
            %                        strength. Because of the low steel percentage used, the component of the steel             show th.it the tensile capacity in a diagonal direction is not the same as in i                        tension force after cracking and assuming that the steel had reached its yic1d             the principal material directions. The low values assigned for allowable shear stress, was less than the capacity of the unreinforced specimens. Jolley (II)              stress would seem to indicate that only debonding failure along the bed joints I

dt        .    .                                         .                                              i JULY 1979                                  ST7     ST7                                    TENSILE STRENGTH                                  1275 1274 w s considered. This s2bstantially enderestimatis the t;nsile capacity of masonry                W;11 Struct1res," thesis presented ta the Uziversity of Florida, at Gainsville, Fla..
where a combined failure mode can be created by shear and compression stresses.            2. '3iume, J. A., and Proulx, J.,n       ShearN4, m in paMalBnck Grouted   fu!Mment MasonryofWall theElementV' requirements fo l
* Western States Clay Products Association San Francisco, Calif.,1968.
Summany ano Cowetumons                                                                      3. Borchelt. J. G., " Analysis of Shear Walls Subject to Axial Compression and In-Ptane Shear." Neredings of Second International Brick Masonry Conference, Stoke-on.
The behavioral characteristics of masonry under tensile stresses at different               Trent. Englaed. Apr.,1970, pp. 263-265.
     <                  orientations from the bd joint direction were presented and examined. The                  4. Borchett, J. G., and Brown, R. H., "An Indirect Tensile Test foi Masonry Units,"
e             raluation, American Society for Testing and Materials, Vol.
effects of grout strenph, mortar type, and bed joint reinforcement on the                            ('f9fg,sdas           3 assemblage tensile strength were investigated.                                             5. Cowan, II. J., "The Strength of Plain, Reinforced and Prestressed Concrete under The following conclusions are drawn from the experimental results presented                  the Action of Combined Stresses, with Particular reference to the Combined Bending in this paper:                                                                                 andersion of Rectangular Sections," Magasine of Concrete ReseareA, No.14 Dec.,
1953, pp. 75-86.
: 6. Goodier, J. N., " Comparison of Rectangular Block and the Bending Beams by I. The tensile strength of cht ungrouted or grouted masonry varies with
            '                                                                                                           N                                  di 8 Forces
* Transactions. American Society of the stress orientation. This is attributed to the anistropic nature of masonry.                 y M",*;*[ $5 "["[l "g 5           I932, p          '
: 2. For constant physical properties of a block, the mortar type had little             7. Ifamid A. A., "Behaviour Characteristics of Concrete Masonry," thesis presented influence on the tensile strength of ungrouted masonry. The effect is not                      to McMaster University, at 11amilton, Ontario, Canada, in 1978, in partial fulfillment proportional to the strength of the mortar.                                                    of the requirements for the degree of Doctor of Philosophy.                             l
: 8. llandbook on Reinforced Grouted Brick Masonry Construction, 8th ed., Brick Institute
: 3. Calculating the strength of ungrouted masonry for tension parallel to the               of Cahfornia. I os Angeles, Cabf.,1974.
bed joints on the basis of the net area of the horizontal cross section does              9. Itatziriikolas, M., " Shear Behaviour o( Masonry Walls Subdivided by Floor Slabs "
not relate to the failure plane that is normal to the bed jomt. A more appropriate            Nccedings of rne first Canadian Masonry Symposium, Calgary, Alberta, Canada, approach might be to use either the gross area or the face shell area of the                  June,1976, pp. 304-323.
: 10. Johanson, F. B., and Thompson, J. N.," Development of DiametralTesting Procedures block.                                                                                             evide a Measure f Strength Characteristics of Masonry Assemblages " Designing,
: 4. Grouting contributes differently to the tensile strength of bleck masonry.
                                                                                                                       '[,gineering and Construction with Masonry Nducts. F. 3. Johanson, ed ,        .
It has a maximum contribution when the tensile stresses are in the direction                  Pubtithing Co., llouston, Tex., May,1969, pp. 51-57,                                    6 of the grouted cores (i.e., normal to bed joint). It has no contribution when          11. Jo!!ey, R., " Shear Strength: A Predictive Technique for Masonry Walls," thesis I
these stresses are normal to the direction of the grouted cores (i.e., parallel               presented to Brigham Young University at Provo, Utah in 1976,in partial fulfillment of the requirements for the degree of Doctor of Philosophy.                              ;
i              to bed joint). Therefore, allowable stresses based on gross area should take           12. Jones. R. M., Chapt. 2. Mechanics of Composite Materials,1st ed., McGraw-Hill
[                                    .
this behav.ior into account.                                                                  Book Co., Inc., New York, N.Y.,1975.                                                    j a  l                  5. The continuity provided by the grouted cores in the vertical direction has       13. " Masonry Design and Construction for Buildings," CSA Standards, S3041977, a significant effect on the strength of concrete masonry under tension normal                Canadian Standards Association, Rexdale, Ontario, Canada,1977.
to bed joints. This factor should be considered in assigning allowable stresses        14. M rsv. E., "An Investigation of Mortar Properties influencing Brickwork Strength."         ,
       >                                                                                                              thesis presented to the University of Edinburgh, at Edinburgh, Scotland, in 1968 for grouted masonry.                                                                         in partial fulfillment of the requirement for the degree of Doctor of Philoso h
* 6. It is suggested that the current philosophy for assigning allowable tensile     15. Neville, A. M., Chapt. 8. Nperties of Concrete, 2nd ed., Pitman Publish ng," Bath
* stresses for concrete masonry may not be based on a sound scientific assessment      ,
Engla:id,1973.
s                   of the behavior and that in order to realize the full potential of concrete masonry,   16. " Recommended Practice for Engineering Brick Masonry," Structural Clay Products "i te, McLean, Va., 8%9.
this should be reviewed. The writers are currently investigating a promising           37* 3,''h
    .[
g analytical model that is fairly simple and accounts for the strength and geometric Stru Eure's th Op ngs Sulhe t to Pre mpre s a                  s sin     ngin ng n characteristics of the components that contribute differently depending on the n,                                                                                                               Construction with Masonry Nducts. F. B. Johanson, ed., Culf Publishing Co.,
ki                  governing mode of failure.                                                                  Ilouston, Tex., May,1969, pp.192-199.
I                                                                                                        18. Stafford-Smith, B., Carter, C., and Chowdhury, J. R., "ne Diajonal Tensile Strength of Brickwork," De Structural Engiacer, l.ondon, England, Vol. 48, No. 4. June, S                    Acunomecoments                                                                              1970, pp. 219-225.
p m a1 Resul This research was funded through Operating Grants from the National Research             y",'"o     hi ""d      **
                                                                                                                                       , ,, f,",,d,    ',    ,m *P  ,       ,,,      oj the Streng h of Brick*
Council of Canada. The writers appreciate the contribution of mason's time                 Stoke-on-Trent. England, Apr.,1970, pp.149-155.
and equipment made available through the Ontario Masonry Industry Promotion           20. Umform svitding Code Chapt. 24, International Conference of Building Officials, Whittier, Calif.,1973 ed.
Fund and we also thank General Concrete Ltd. for providing the concrete blocks.
                                                                                                              , 21. Wright,'P. J. F., " Comments on an Indirect Tensile Test on Concrete Cylinders,"
                                                                                                              ,      Magazine of Concrete Research, Vol. 7, No. 20 July,1955, pp. 87-96.
Amnoix l.-Reremencea                                                                l 22. Yokel. F., and Fattal, G., " Failure Hypothesis for Masonry Shear Walls," Journal of the Structural DiFision, ASCE, Vol.102, No. ST3. Proc. Paper l1992. Mar.,1976'
: 1. Balanchandran, K., "An Investigation of the Strength of Concrete Masonry Shear pp.515-532.

                                           ~ _ _ _ _ _ _ .
                                                                                                                                                                                                         .         o i                . .                                                                                                                                                       ,
                                                                                                                                                                                                        }-
4-                                                                                                              Journal of the
         *1 STRUCTURAL DIVISION                            .                        .             :
Pmceedings of the American Say of Civil Engineers                                                     ';
c ;:i                   .
:) ~ :'
          .s                                  e*                        *
                                                                                      ,                                                                                                                j-T5                                                                                                                                                                                          n E
* FINITE ELEMENT ANALYSIS OF HEINM)ltCEI) CONCHETE SLA118                                               ).-
                                                                                                                                                                                                    . . i.,-
1                                                                                                  By Jan C. Jofriet' and Gregory M. McNelee 8                                      * * *; *l,
                                                                                                                                                                                                . t. i '.#.o l
T.))
N                                                                         *                                                                                 .
                                                                                                                                                                                               ',.h. .Y        g 1         2
          .h '                                                                                                               INTitODUCTION
* kAh*    a:c: ,'  '
[t i           I                                                                                 Inthe analysis of reinforced concrete slabs, it is important to consider the effect that cracking of the concrete in the tensile zones has on the slab load-
                                                                                                                                                                                                ]k'
: l. .
            )
l 1,
deflection response. Cracking of certain regions reduces the overall flexural rigidity of the stab, causing an increase in deflection. Extensivecracking may
                                                                                                                                                                                               ; ; y~  .rr 3,-                                                                                                                                                                                     4 ,0*.f.,
well effect the serviceabilityof thestructure. Thelmportanceof serviceability
 ;          ) i.                                                                          as a design criterion is becoming more and more apparent' with the ever-                   '
                                                                                                                                                                                              . },t<* -4      .
;           c
* Increasing use of higher strength materials in reinforced concrete and ulti-i mate design procedures.                                                                               ' 4,j'h:W 3 .
         '*                                                                                    Cracking of certain regions losadirect influence on slab analyuls because                             D.['. :. .
r
'          'J                                                                   .         the rigidity of acracked section, as opposed to an uncracked section, depends                               '.''
greatly on the percentage of reinforcing steel. Crackingwillbeginat moments                                f equal to approximately one quarter of the yleid moment capacity of the cross
                                                                                                                                                                                                    %*g              .
section for stecipercentages onthe order of I %. For small amounts of steel, ontheorder of 0.25 %, the cracking moment may be equal to the ultimate mo-
                                                                                                                                                                                                . k.
                                                                                                                                                                                              ,7 frf h[
ment. *Ihe extent of cracking over the area of a slab at service load depends 3                                                                             mainly on the span to depth ratto am! the concrete strength.In the came of                           '
* 7 ''..p
                                                                                                                                                                                               , d'
  ' .j                                                                                    extensive cracking a variable of mesh reinforcement will result in wklely                                 .
varying rigidities throughout the slab which in turn results in a moment field                            .d that differs greatly from that used for the design.In many cases the design                                   ; ,,
        ,2
* values are obtained using the assumption that the slab is isotropic even for                                  .       f,
{,                         g I rectangular slabs.                                                                                      !
* j[.U' 1         6                 g i                                   A slab analysis that includes the effects of cracked regions must include:                       ''g" y (1) A means of taking account of the orientation of the cracks with respect to                                  -
                                                                                                                                                                                                ; f "i
             !,                          1 the coordinate system of the stab; (2) a reasonably good estimate of the ri-                                              ,
j                                                                                                                                                                                  8 gidity of a cracked region at moment levels greater than the cracking mo-_                                      ..
jl                                                                                  Note.-Dascussionopenuntil August 1,1971. To extend the closing date one month, a          ,
e l'. pea 8          -
written request must be filed with the Executive Director, ASCE. This paper is part
       ' f.,'j of the copyrigMed Journal of the Structural Division. Proceedings of the American                        [(+(~
j
              '                                                                            Society of Civil Engiurers. Vol. 97. No b*r3. March,1971. Mansmeript was subnaltted
              .l ;
for review for possible publication on June 12,1970.
: 4. . , 
                                                                                                                                                                                                    .'P'<.-
;            ,                                                                                 8 Research Asst., Solid Mech. Div.. Univ. of Waterloo. Waterloo. Ontario. Canada. .
              , j.                                                                             s Asst. Prof., Sottd Mech. Div., Univ. of Waterloo, Waterloo Ontario, t'===d=                                       (
d
(,
                                                                                                                                                                                           .         h. w. .
                                                                                                                                      *fss                                                             ,.g-go.
: j.                                                           -
swa N         w
               '                                                                       S                                                                                               _

q r*                 , ,, ,
                                                                                                                                                                                                                                                               .m yl,186                                                    March,1971                                               ST S                Sr S                      ItEDfFORCED CONC 1 TETE SLADS                                     787             -
v-                 . "                                .
          ,            saent; and (3) an actimat) of the effect on rigidity of steel trientation eith                                        assuming no nonlinear geometric;l behavior. Substitutise of Eqs. 3 and 4 lato                          *
                    , re pect ts th2 crackdirection. A suggested procedars (Ir cach of these prob-                                            11:ada t2                                                                                                          i          *
     *!                less is contained herein.                                                                                                                          D D M A . . . . . . . . -. . . . . . . . . , . . . . . . . . (5)                       *
      .',                   A classical heading analysis of a slab with varying rigidity throughout is                                              Ei n   " OA [g                                                                                   ,
                                                                                                                                                                                                                                                               , L, , ,
Si' Effleult. The finite dtNerence and flatte element methods can deal with such a                                        De external work done by virtual nodal displacements can be writtee as
        ,              problem if the slab is discretised isAo flatte regions andthe rigidity kept con.                                                                                                                                                     '
i                stant la specific directicas throu@ each ladividual region. Decause the                                                      Wu ,= 648 F . . . . . . . . . . . . . . .. .. . . .. . . . . . . . . . . . . . . (6)                         .
3       ' .'IIItse spectelement        method to flexural         enables rigidntles,   it waseach     region thosen         to be of la favor       modelled       differencewith re-itnewhich the finiteseparately                                  F = the nodal forces. For equilibriu n the laternal work must equal                                      M en rna ww an Hause 6A are v unt quanuties, it foHows that
      ." d             scheme.                                                                                                                                                                                                                                 .;
Successful application of the finite element method has been made to both l                                             F = f AIV D Il sfA A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)                     b material and geometrical nonlinear analyses of plates in flexure, e.g., the                                                                                                                                                             *j
      }'d analyses are peculiar to metallic plates and specifically to metals that satisfy j
* ccrks of Marcal et al. (20), Armen et al. (3), and Whang (24). Ilowever. these ! or sim ply F = K A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)                                           .
the von allses criterion and the Prandtl-Reuse type relations. A successful e                                        in which K = the stiffness matrix of the element.                                                                   0N         .<
                 .
* ittempt to model the elastic-plastic behavice of reinforced concrete slabes I
      ..l cas made by McNetce (21) using the square yleid criterion. Ilowever, the l                                                                                  SELECTION OF DENDING ELEMENT
                                                                                                                                                                                                                                                            'Q 3? effect of cracking was not studied and consequently slab behavior at the ser- l                                                                                                                                                                  [,ts..
, J 3
viceability level was not modeled properly.                                                                              Since publication of the first Ignding element in 1959 (22) many models of he present study does notinclude post t leld      y behavinr but coseccentrait u oss                           varying cosuplexity have beces proimised (5,11,U,10,I1), it is sip to the user las                         , . .hg.,
the nonlinear behavior due solely to changes in slab rigidity during a mono-                                   g select the best element for his purpose. In most of the available literature                                       'f.- j.
         !              tonic increase in applied load. The step-by-step analysis performed employs
        ? i;' a bilinear moment curvature relationship for each sobregion or the convergence characteristics of an element are demonstrated by plotting element.
                                                                                                                                                                                                                                                            *h.. #
i                                                                                                                                            the accuracy of the solutions versus the number of subdivisions for a few
* i!                                                                                                                                   simple plate bending problema Ilowever. a more appropriate comparison is                                            e. i l                                                                                                                         l the accuracy versus the numberof degrees-of-freedom,therefore,the number FINITE ELEMENT METHOD                                                                                                                                                                                ' y. 4
,'             ~
o
         ;                                                                                                                              g of equations to be solved resulting from the particular subdivision. Such a i                                                                                                                              : comparts n was made for 10 bending elements using the published results of                                       ':9 In the finite element method (FEM) the continuum is divided into finite                                        finite ele      t analyses and enct solutions (23) of sejuare plates having Imth cited regions called elements. Withineach element the continuum behavior is                                                                                                                                                             g,n, clamped and simNy supputed bauMes M). h applimiload casistal M
,              .*       tpproximated. The formulationof the element behavior is effected through the                                         a unif rmly distributed load and a point load at the center. Fig.1 shows one                                  %g g
'                       use of energy principles applicable to the continuous medium but applied to                                          such comparison. Of the elements indicated by solidlines the quadrilateral 7*             sach discrete element. The addition of the force-dluplacement relationships elements developed by Clough-Felippa (Q19) and by Fraeljs de Veubeke (CQ) 3 I        I              far the elements leads to a finite number of simultaneous linear algebrale                                                                                                                                                     , /,g m est.          e   agh-FeHppa element W has far cwner nodes only                                            pj, j
3
          )
IIrce-displacement relationships for the structure upon which matrix opera-                                         with three degreta-of freedom each whereas the CQ element (11) has in ad-n he m 1 I ec on theory of plate bending, neglecting shear deforma-dition four midside nodes with one degree-of-freedom each. The presence of                                  ,(pf.g
                                                                                                                                                                                                                                                           ]
the midside nodes results in a significant increase in bandwidth which in-5'
         ;j.             tions, the internal virtual work for an element may be writ *en as                                             e creases computer storage requirements. The Clough-Felippa element was                                           y.4 IL*g. = Mfk8 M sfA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 ) j therefore                  selected as the most suitable model.                                                                   -** T    .
1 4
J
                                                                                                      ,                                          The convergence curves in Fig. I marked P and Q refer to the NRC tri-                                         h I               in which A = area of an element; 8e = virtual curvatures due to virtual nndal l angular element (10); it can be seen that it performs better than the Q19                                                                           . "N
    . 'i                 displacements; and 11 = moments. The moments are related to the curvatures ' element. Ilowever, because the curvatures are included in the degrees-of-h               by                                                                                                              . freedom, difficulties arise if adjacent elements are of different thicknesses.
[ In such cases, curvatures are in reality discontinuous. The NRC element is
                                                                                                                                                                                                                                                            * *(gj 1                                                                                                                                                                                                                                              iO '
        .j j                      11 = D x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 )              incapable of modelling this because the curvatures are fundamental degrees.                                     'T h ~ 1A which D = the flexural plate stiffness properties. From an assumed shape of-freedom that must be made continuous for convergence.                                              ,
1
               ;         function relating transverse displacements to nodal displacements, a rela-                                              The Clough-Felippa element is a fully compatible element and is described Ig tionship between the curvatures, e, and the nodal displacements, A, can be                                          in detall elsewhere (8). Fig. 2 shows how it is built up from four triangles.                                     !
                      ' derlVed i
Each of these triangles in turn consists of three subtrian'gles represented by                                      W i                                                                                                                                          acomplete 10-term polynomlatin se,the transverse displacement. Curvatures                                         ei j{.,*   ;
x = Ua........................................                                                             therefore vary linearly within each subregion. An evaluation has been made                                          A
           !I and, the refore A = B 6 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)                       of the Ql9 element for use on a IBM 360 series computer for problems in-                                          .M
                                                                                                                                                                                                                                                                , sv r
}.                                                                                                                                                                                                                                                                       .
4
                                                                                                                                                                                                                                                                 ?J....
u

mi                                                                                                                                                                                                                                          .s
      !i.788a         '-
* hlarch,1971                                            ' ST 3       3T 3                       REINFOftCED CONCItETE 8 LADS                                   iss                          *-
3P                                                                                                                                                                                                                                               *
[                                                                                                                         volving various (terment n:pect ratios (14,15).
                   ' . 20
                                                                                                                                   'Ihc nece:sary transfir:iatism, methixts of calcutilng the rigidity sit c
* l".I''d.
*1
   *} l ' '                                                 E0f 8     cracked region, rnit the influenc3 on rigidity of the track orientation is de-scribed subsequently herein.            *
  .'}
8' -
:.j j                                                                             .
O                          o -- - ,;;;p                                                                           TIIANSFOltMATION OF FLEXUltAL IllGIDITY MAT 111X                                                 ,
l           .
k Y                                                                                                                                                                                                                                     . ! ,at.lt- ' -
From Eqs. 7 and 8 lt follows that the stiffness matrix of an element may Jj                   at ,,o       0,8j              p/                                    m nart um, ctan accas                          be expressed by
   .'.i                                    2                                                ,,, c e,,,
                                                                                                  ,     ,, gg,,,,         .                                                                                                                     ,
K = [ 11' D B dA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (D)             .;.
 ]
tcCTs                                                                       in which D = a 3 by 3 matrix of flexural plate rigidity properties with respect                  -
K'
.j.                       -So                           .
50 80 0 500 8000
                                                                                                      .                         to the coordinate systein, x, y, of the element.
Dealing with acracked regloin of a concreto slab the clastic propertics urs
(.j,fh(.*  y ,,,
 -11
   '3                                                         NO.0F EQUATIONS                                                   not the same in all directions. In Lbc came where the cracking direction coln-                                    .
I                                          FIG.1.-CONVEftGENCE CUltVES                                                                                                   -y                                         .
                                                                                                                                                                                                                                            ,[{ Nf I
i                            LY                                                                                                                                                                                                       s[. . .
  '.                                                                                  x* ,X I+X2+X
* fg[l!
4            3                                 s +34
  ,i:                                                     ,                                       4 g;
e y_ y,+y2+Y S+Y4                                                        m                          e                                             . y-
  .fi                                                DE        I,                                 4 g
A a                                      .(.w.g;
                                                                                                                                                                                                                                              .~...
 .,l,,
6 I gW.E l
          $                                    i                                                                                                                             75As                                                         a
   .i ,                         e x                                                                                                                                             ;. g, .
                                                                                                                                                                                            'A g                                      g b'e       ,.:
I                         cusoniLATEnAL EttMENT DivfDED INTO 4 LccT-il TRIANGLE $
ay AREA @
i                          RDNFORCEMENT                                                                       h. *.    -
                                                                                              = AREA @                                        FIC. 3.-CitACKED ltEGION OF ItEINEVitCED CONCitETE SLAB
                                       .                      '3(5)                          = AREA @
cides with either the x or y direction the problem la simply one of special                               ** .r
                                                                                                                                                                                                                                            , 4-[ e
: j.     '
g
                                                           @ o @ SIT) orthotropy, for wh;ch Huber (23) has suggested the following rigidities for a reinforced concreto stab:                                                                                        j.4 U 8.8   "g                               i D .s " Us. " Pe @s.sD           ;                         i
        ;                                                                                       NUMBERS OF                                               , I Us.s " g                                           '
g gi)                                       QUADRILATERAL ELDIENT                                                                                                                               .~             a
   )l i                                                                                                                                 D,,, = 1 .2 s."                ; D , = D.,, = D , = D., = 0 . . . . . . (10)
X_
                                                                                                                                                                                                                                                  ,    .j
     ,j l TeANGbLAR ELEMENT OlVIDED INTO 3 $Us TRIANGLES                                                                     "    "         "                       "             #                                           '''*''
l'                    Lcci - 12 ELEutNT HAS S No0ES. I- 4 INCL                                                          l j                      LcCT- 11 ELEMEh? HAS S N00ES, 1.2.3.S.,AND 6 tu g c CaSO where the cracking direction doco not colncido willi cither the                                ,-hD x or y direction the problem is one of general orthotropy resulting in a fully                                      'i
 'l  {
j ,.                                    FIG. 2.-QUADillLATEttAL ELEhlENT                                                    populated D matriz. The terms of this matrix can no longer be derived di-                                        j{-[.
G.
ie
                                                                                                                                                                                                                                                  . e J.
k*
j                                                                                                            .
              ,. .                                                 .. .                                                                                                                                                                             lr
                                                                                                                                                                                                 ..     . . . .               ...-m

   .' .. Tse" " e                                                                                           klarch, it'll                                                ST 4          3T 3                        REINFORCED CONCRETE St.AD$                                      TVI                                 e i *, "
:-l ! .                                                                                                                                                                                                                                                                                                       .. t :s J j .Irectly from the materialand sectl0nalproperties when viewed tiong the x and                                                                                                fross which it follows that                                                                                                        ~AM Fig. 3 hows e cracked region. The normal, C, ta the tracking direction                                                   ,              D"4 C,~8 D' C, a f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (21)                                         fp
   ,l                         .anakes angle a with the x axis. The a and i axes are now the princigxtl axes of                                                                         The inverse of C, is the transpose of C ; thus Eq. 21 becoenos                 ,
     /                         orthotropy with the moment-curvature relationship given by                                                                                                                                                                                                                                 o' D=     , D* 3C .... .................... ..... ...... (22) h,
   ,,j                                                                  .tf,             D,       D    0                                                                                Consequentlya simple cemgruent transformation of D' gives the D matrix with                                                       ,.,
l l
    ,j e
11 t    "     D , D:        0 h              ...................(11)                      {             TAst.E 1.-oNE-WAY STABS WIT 11 GENERAL ORTHOTROPIC PROPERTiggs                                                           - ) , ;,
                                                                                                                                                                                                                                                                                                                    . : . .G    '
                                                                         '\#ag            0       0   Dat        2 , , ,',~                                                                                                                          e,la degreen                                                 [
i                                                                                                                                                                                                                                                                                                                   *
                                                                       ~ ~                               ~     ""
                                                                                                                                                                                                                                                                                                                            'M
                         *                                                              ~
l    'i                                                                                                                                                                                         Jtab property            Mode                                                                      '
l in which .11, and .11t = the bending moments in the n and i directions; JLfat *
                                                                                                                                                                                                                                                                                                    *          **               'N d
   ', j                            the twisting moment; and se = the transverse displacement. The terms in thu                                                                                                           21 25   0.021      0.02s1       0.0003      0.140    0.217 D r tatrix are calculated using Eq.10 by replacing X with n and y with f.
Average dettrettime x gn"                                                                                                              .Yp 1                                                                                                                                                                                                                                                                                                        .9gI. R i                                                 The relationship in Eq.11 may be rewritten in matrix algebra form as
* wtecipal snomien* x ge          21    0.264      0.249        0.250       0.26s    0.253
                                                                                                                                                                                                                                                                                                                    ,NW
   '?                                                                                                                                                                                                                      22    0.253      0.267        0.2ss       0.286    0.250 hl' = D ' z ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)                                                        23   0.247      0.270        0.301       0.2st    0.248 4                                         ,
The moments hl in the x and y directionsare related tothe moments M' in the
      ]                           c and i directions by the transformation Pri ctpal ronient stirec.
tion measured from 21 22 0.0 0.0 s.02 12.15 15.52 to.51 15.G7 20.01 0.0 0.0 i.HN MV q
: z. axis                         23   0.0       13.22       20.64       20.94      0.0
* d ?"' O T'                                                                             "cd         s'     - 2 es"                                                                                                                                                                                                             4. (. '.
Reactions k ed                        0.0469     0.1306        0.2285      0.244s  0.0551 s 8 c a 2 es        M.......................(13)
                                                                                                                                                                                 ,                                            1 0.05 t r. 0.0239      -0.0332     -0.03 t a 0.0453                                    M
                                                                                                                                                                                                                                                                                                                        ';p,,s
   ~l-                                                           . hl' =                                                                                                                                                      2
        ?
                                                                                   ##       * #I #: *8 -
3  0.05*9     0.0498        0.0446      0.0361   0.0412 0.0453
                                                                                                                                                                                                                                                                                                                   ' 'g.,
s I                                                                                                                                                                                                                      4  0.0516      0.0360       0.0104      0.0097                                                    A S   0.0469     0.0001     -0.0003     -0.00s9    0.0591                                   "N $
   ,j                                in Which e and s refer to cos a and sin a respectively; or simply
* see Fig. 4; D.
* EI./(t - #1 = 1; 0, = EI,/(t - 9) = 0.t.
84f
                                                                                                                                                                                                                                                                                                                          ;[j
    .e gg . = C: hl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)                                                                                                                                                      , , .
f l Cmilarly for the curvatures j                            p       10     15 to 25         30 35          40 45                                         .. ..$h.;
        .l'                                                                     ,
c'           s'      - es_                                                                      s                                              y~                                                                               . ',;>. Z 5 ,
j                                                               s'=          s'           c'        cs             x . . . . . . . . . . . . . . . . . . . . . . ( 15)                                .4__--_
{                            44.                                                  t-J                                                                       "2 es - 2 es c8 - s'                                                                                                                                t:                                                                                    JW!r j .;
                                                                                                                                                                                                              >S---                 @-                           438     8                                                  g) j j tr K' = C E........................................
t (16)
                                                                                                                                                                                                              >2--
42e, f"lj
        ,i Substituting Eqs.14 and 16 into Eq.12 yields                                                                                                                                                                                                               -,specRTs                                            lg. .,e
         . .                                                                                t                                                                                        ,
                                                                                                                                                                                                                 '                                                   a                                                     :M,. 
                  !                                                     C, h! = D' r................................... (17)                                                                          ,7       n g      ,   ,     }t     3      3 ose 3 ,
s-
      .I I                                                                                                                                                                                                  a-            oss                                               s  . i
                                                                                                                                                                                                                                                                                                                          ) ?
jj In the x, y coordt te system                                                                                                                                                                                                                                                                                   'h 5
hl = D s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18)                                  ,I ~ a              I,. 0 0833          t, . 0 00633                                                                  ;4
:                                                                                                                                                                                                                                                                                                                .. .,4 j                      [13 obtained, from which it follows that                                                                                                                                                                                                                                   -
:4 FIO. 4.-RECTANGULAR ONE-WAY ORTIIOTROPic SLAB                                                                    i ** 1 J
C, D s = D ' C, a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (19)                                                                                                                                                          ' 0:' -
A
         .J
* re8Pect to the x, y coordinate system required in the analysis,it can now be Premultiplying both sides of Eq.19 with the square t'natrix (C D)~8 yields                                                                      incorporated directly into Eq. 9 for the stiffness matrix.'Ihe coefficients of                                                    *. i.$,,
       '!                                                               x = (C,D)~8 D ' C, c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 0)                         D are given explicitly in Ref.16 expressed in terms of the coefficients of D'.                                                      j;g p
         .                                                                                                                                                                                                                                                                                                                 . ~ ,.;
                                                                                                                                                                                                      *                                                                                                                       ?; .'
19

              . ' 74 ,.       .                      Klarch,1971                               ST 3             ST 3                          REINFOltCE JGCEISU[3 C&dD                                                     'u m                      .y M They cgres eith those given by Hearnon in Ref.12                                                    lioned earlier.13 both methods a bilinear moment-curvature relationship 13                                                           ''N; To dementrate the effent of generalcrthotropic propertieson the beh.svlor                  seed. Itceby'3 first nu.thent define 2 the ft:xiral ris'idity, ft , Izfore tracking                                           .           ,,
     .g 3- [' of C pl. ate, the method descr,1 bed hereta ca3 rpplied ta                            the as I
cimply supported                                                                                                                                *N
         ;      one-way stab with a center span line load (Fig. 4). The first analysis is for                                                                                                                                                               9 j.j the special orthotropic case with a moment of inertia about the y axis 10 R, = E, I, . . . . . . . . . . . . . * * * * * * * * * * * * * * * * * * * * * * * * (2 4)
                                                                                                                                                                                                                                                       'q '
i       times that about the x axis. The principal axes of orthotropy are then rotated                  la which E, = Young's modulus as specilled in the ACI code (1) and 1, e the                                                        .
1                  22.5 *, 45*, 67.5*, and finally 90*. Table I contains the values and di-             anoment of inertia cf the croes section including the eHect of the reinforcement.                                                          p; p
* y,t.through l rections of the principal moments, cpater deflections, and the distribution of
: f.      reactions. By inspection it is evident that even a small deviation of the axis                                                                                                                                                                ...
TAllLE 2.-MOMENTS OF INEltTI A.1,, OF CitACKED SECTIONe
     'y
     '. ! of orthotropy from lhe x axis has a 'significant effect on the behavior of tho              ,
         - i plate.                                                                                                                                                                                                                                           .
i
                                                                                                                'compresetve                                                                     #'
8                     #'8                              '.*
etre h ut                                                                                               4
[          -
FLEXIJRAL RIGIDITY OF CRACKED REINFOllCED
                                                                                                                               ,    u%                                 fcr la l{ 8                                g                                       '.Q.{
      .1                                         CONCRETE REGION                                       ,
                                                                                                                               ",      ness,f.                 p         ,,,,p,              the tourth            the tourth                            ,
                                                                                                                                                                                                                                                                      ,}
i
                                                                                                                       , quay, h he                                  inch             Powr per              power per                        ,*4 I           The determhnation of the rigidity of a cracked concrete region normal to                             te                                                                     inch                   inch                          .f g
        )       the crack is difficult. The problems of the nonhomogeneous characteristics of            i                (1)            (2)                  (3)             (4)                (5)                    (H                          ( hy,y.
the material are for a cracked region compounded by a variation of sectional                                                                                                                                                         '
* d'                                                                                                        -
0.004              3 3                                       2.s9 properties from section to section, depending on whether the section is taken                           3,000               0 6              0.001              5.01               3.21                   4.39 5.62 b           ,y j i ct a crack or between. Furthermore the effect of tension stiffening, which is                                                    e              0.010              e.64               4.01                                              .
3       the effect on the overall rigidity of the uncracked parts of the slab between                                               s              0.004              s.25               5.63                   1.39                                g *- ]
j       cracks, reduces as the bending moments increase. AlsotheYoung's modulus                                                                    0.001            12.33                s.26                 11.22                                             .
of concrete is dependent on the stress level.
e s                               1 2
                                                                                                                                                                                                                         *~
50
   .J               hlany methods have been proposed for dealing with this problem with re-             i                                 10               0.001            26.3s               16.95                 22.as                          NW
        }, spect tothecalculation of deflection of reinforced concrete members. A sum
* 34.58               21.04                 29.21 4        mary has been given in a report by ACI Committee 435 (2).
to 32 C.010 0.004            29.02               20.52                 2s.s0               *         '. a .,
i          The simplest methwlistoese the cracked transformed section as the flex-                                                                 0.001           46.92               30.26                  40.10 I       ttral rigidity of the cracked zones. However, because no account is taken of 12 12               0.010           61.50               as.s9                  52.0s                         ; %' ,
   ] t;nsion    ber  of stiffening this methodgrossly overestimates the deformation. A num-experimental    investigations into the effect of cracking on the flexural                      4,000              0               0.004              2.s1               1.93                   2.58 .
e               0.001              4.53               2.53                   3.96 5
rigidity of concrete beams have been performed. The most recent studies in-                                                                                                      1 59                    5.10                    *  ,' ?*.p'.
6               0.010              5.96                                                                       <'..
    ~'
            . Clude those by Yu anet Winter (26), Branson (7), and Beeby (6).                                                               e               0.004              1.31               4.95                   s.s0                    ,
                                                                                                                                                                                                                                                           .PQ The Yu and Winter's formula gives an expression for deflection in terms               ,                                  s              0.001           11.55                 1.28                 10.11                               ,4 *g a               0.010           15.22                 9.2s                 13.02           g 4      .of thedeflection calcueted on the basis of a cracked transformed section with                                                                     i                             ' '
              ' O modification factor to allow for a certain degree of tension stiffening. The                                             $                h[0             [ 51                14 93                    0                                        '!
4*
        ;       use of.this method for application inthe finite element analysis is limited be*                                            10               0.010           31.06               19.01                  26.56                          5 h,,
5
        ;       case a measure of the actual flexural rigidity rather than the deflection is                                               is                0.004          26.53                1s.05                 23.94                           .. ydg 1 . r: quired in Eq. 9.                                                                                                            12                0.001           41.91              28.64                  36.es                                 y.,,.
The Branson method does give an expression for the effective moment of                                                 12                0.010           55.23               34.01                  41.23                               ,.
l
{ ' inertia of a region after cracking. Dranson suggests that                                      l
* See Eq. 29 E, a 29 x 108 pel; E, = 51.600 M; f, = 1.5 M.                                                                        ?
Idf "       f    It+     I*          Ier .. . .... . ... . .... .. . (23)                                                                                                                 a       ,
i
           ' in which .ti and.11
                                      = the limiting cracking moment and bending raoment, re-The flexural rigidity after cracking is given as                                                                                        g c                                                                                          p* ,g;y".        ,,,,,,,,,,,,,.,                   , , , , . , , . . . . . . . . . . . (2 5)                                     J, . . )
Cpectively; I g a the moment of inertia of the gross concrete section; and                                                                                                                                                     ,            p
      ],        I,, = the moment of inertia of the cracked transformed section.                                     In which E} = a reduced modulus of elasticity of 0.57.x E, and f}, = the                                                                M y.
        ;1          Deeby examined deflection data obtained from a large number of beam                      i moment of inertia of the cracked transformed section calculated using E}.                                                                           -
i      .t       tats performed by the Dritish Cement and Concrete Association amt by                                    Deeby's secomi method defines the rigidity beforo cracking as                                                                       Y
        ; ',' others. On the basis of these results he recommends the use of one of two                                                                                                                                                                     . . ,
i
        ' , mithods, which are claimed to yletd superior results to those methods men.
R,=E,it ************************************                                                                                        I.
l
        ,                                                                                                                                                                                                                                                   '+ A i
                                                                                                                                                                                                                                                              *I l       i       .
                                                           ~
                                                                                                                                                                                      - - -                                                                   *s*;.
4
                                                                                                                                                            .                 .o  .         .4-      . . . e.  . e. M , , . .e .sa=E .*.v .
                                                                                                                                                                                                                                                ' 'a                     '
l

  ,].
ygj "
* hlarch,1971                                           ST 3           3T 3                   REINFORCED CONCRETE SLABS                                          'i9S           -
     '3
                       'lkie rigidity sit;r cracking is then det:rmined by the slope of the moment-has not been reported exinively becruse thb is, as a rt, of littl) intir:st                          .             *
        )*                                                                                      g                                    either to the deitgner cr ID most riscarchers. For c low steel percentrge,                                       .*..
      .f
* curvat and e =rs       lineIcr);
31,/(Ee   p.as:ing        throughyleid Jfs = the maximum             31momente Af,,ofathe   e cross Af,/(E,     I ) cndI which section*           through31           = case is the usual  Als in slabs, the reinforcr.nent contributes Ilttle to the                                    .,
q*                   The,Branson method and Beeby's first method were selected as the most                                   sioment of resistance of an uncracked section. Anuncracked. region therefore                          .             .
           * -          suitable for use in the finite element analysis and an evaluation of these me-                               may be treated as isotropic, regardless of orientation and orthotropy of re-
      .i                thods was carried outinorder to select the best une. Both methods were used                                  inforcement. The region is assumed to crack initially normal to the major                                    , ,,
3*              la a number of analyses of beams and slabs for which computed deflections                                    principal moment. This in fact has been confirmed experimentally by Len-
:'                were comixared with experimental Tesults.                                                                    schow and Sosen (19). Presumably this means that in orthotropically rein-                                       4, ..
The Deeby formut.t is in a form that can be incorporated directly into the                              forced slabs the eventual yield lines do not follow the direction of the initial
          !.            finite element analysis because R, in Eq. 24 and R, in Eq. 25 are tangent ri-                         ,
cracks.                                                                                                          '9.
giditics. Before cracking, an element is assigned the rigidity if,,, but after e
     'i                                                                                                                                  %e finite element analyuls presented herein la Intended to deal wil5 slahd                                   W-d         . the rigidity R,. The Branson formula however does not yield directly the                                     in the service laid range and it is assumed that the crack orientation is.nor-                                    t.
M                 tangent flexural rigidity of a region after cracking but rather the secant                                  sial to the principal moment direction.                                                           .                        .
rigidity.                                                                                                        In the prcylous ucction it has been shown how the rigidity of a concreto                     .
                                                                                                                                                                                                                                                 "3(('
1 can be shown (16) that the Uranson method also leads to a bilinear mn-ment curvature reistionship where inthe uncracked clastic region the rigidity reldon ran be cannputed given the trannforsed moment of hierfla. In the cano                                  h i                is equal tothat given in Eq. 24 and in the cracked elastic region it may be ap-N cracking normal to one of the layers os reinforcing steel the procedure is simple. The depth of the neutral axis from the compression face, c , is Q.y. o d-                  proximated by                                                                                                              r                                                                                                         I' en 1
                                                                                                                                                = d VZ t* n + (tm)" - p nj . . . . . . . . . . . . . . . . . . . . . . . . . (32)
                                                                                                                                                                                                                                               'D ; i9':
       ]:1                       ye . U~ ~ II #" I" .. .. .. .. .... .... ..... ..... . ... . (27)                                                                                                                                              4, ,
1                               rQ-Icr                           .                                                        In which d = the effective depth; A = the percentage reinforcing steel normal
    .].                   In which r = a f.ictor that is related to the reinforcing steel percentage, p.                            to the crack; and n = the modular ratin. The Young's modulus of the ennereto                                 ,
From a least-squares fit, r is approximately                                                              is reduced as given hi Eq. 25. The cracked transformed moment of inertla,                                     i
     ''f                         r = 3.8 - 32 0 p + 1,450 p* . . . . . . . . . . . . . . . . . . . . . . . . . . (28)               lir, then ts                                                                                                    g"}:
Table 2 presents some representative values of 1,, which is defined as                                        79 ,      e,s,+wpdGf-    c.)'............                  . .......... (33)
                                                                                                                                                                                                                                    ~
                                                                                                                                                                                                                                                 ~ $[
     ^i                          1, =         ,..................................(2M                                               The first part of Eq. 33 constit ites the contritntion from the concrete above
     '!                                   C the neutral axis. The second is the contribution from the reinforcing steel.                                 * '' b 3                 The values have beencomputed for a variety of slab thicknesses and percent-
* If cracking occurs atan arbitrary angle, e, to the reinforcement direction, ages of reinforcing steel. It is apparent from an examination of these values                            the contribution from the reinforcing steel to the rigidity of the reglon be-                                     . s.s
          !               that the calculated rigidity of a cracked region will always be greater with                             comes more complex. Considerable work on the effects of steelorientationon 7 >
ll                . the Deel>y method than with the Branson method. The Beeby method was sub-                                 the rigidity of slabs has been carried out by Lenschow and Sosen (19). From                               **
h                sequently selected for the finite element analysis used herein.                                    ' , this work it is possible to arrive at expressions for equivalent areas of rein-                              .!
     -j                                                                                                                            forcing steel normal to the cracks. This has been done in Appendix 1.                                         t'        ,M
    -l j                              CRACK ORIENTATION AND ITS INFLUENCE ON RlGIDITY                                                   Three cases must be considered separately. They are regions that have
          .!'                                                                                                                      cracked in: (1) One direction only; (2) two mutually orthogonal directions on
                                                                                                                                                                                                                                                     ,-[. %
l                   'the orientation of cracks im reinforced concrete slabs has been the sub- *, the same side of the stab; and (3) two mutually orthogonal directions on op-i                                                                                                                        Posite sides of the slab. In case 1 the equivalent area of steel normal to the                                            -
,    l i-ject of extensive studies in co'nnection with the yield criterion of reinforced concrete (17,18,19,25). For an isotropically reinforced stab, yield lines form                          cracks may be empressed by                                                                                     . g(
          ?'  '            normal to the major principal moment. For an orthotropically reinforced                                            as ,
slab however, this is not normally the case (18,19). The direction angle, Se og f                                                                   ..................,. 3..                                                  .
the normal to the yleld line is deterpined by                                                        , in which As = the area of steel in the x direction; a = the angle between the                                     : b i,;
4        .
11, = m , . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * . *     * * . . (3 0)      normal to the cracks 2nd the x axis; and p = the rctio of steel areas in the y                                    M .~
[
and x directions (Fig. 3). In case 2 the equivalent area of steel in the major                       .
s." I l't                and -88 818 =h................*.*****************.(1) 88 princW ment hection a may be written as
                                                                                                                                                       -                                                               .                            .I .
                                                                                                                                                                                                        ~
h                In which .11, = the nUrmal bending moment and-m, = the resisting normal                                         A s, = A s cos' a + p sin
* a +            (1 + p) cos' a sin          8 a       (35)
    .4 '                                                                                                                                                                                                               "
j ! ' ' moment.In the avillable literature on slab tests, the orientation of initial cracks                              The equivalent area of steel in the minor principal moment direction f is 3
                                                                                                                                                                                                                                                     ,(..
t                                                                                                                                                                                                                                         . w. 1 ic                                                                                                                                      .                                                                                                    t!.''
          ,                                                                                                                      .                                                                                                                   %.h 1
              *-                                                                                                                                                                                                                                 NI

;       ,, i.                    .. e top                                      maarek,1973                                          er a     srs                          a Iwroncan conca Ts SLABS                                                         Tof
)                                ,
l i,-                                             g                                                                            esaparison of the esperimental and consputodeenier defleettene for two lov.
                                                                                                                                                                                                                                                                     ,I
,   y*,                          Ad
* A ss Fst af,       ............................... (36)                                  els of load tre presented in Fig. 6.                                                                                          -
1                                                                                                                  >
!    .4 ' in which r., = ratto of steel tensile strain ta direction a to steel tensile strain D
                                                                                                                                                                                                                   ~
800
: ljf la direction i1.V. = major principal moneent; and A1 = minor principal mo-                                                                                ,a                      pg
: b                     ment, la case 3 the equivalent steel area la the major principal moment di-                                                                                                         g I rection is the same as for case 2 with the exception of the ratio of straines,                                                      ,    ,,
                                                                                                                                                                                                                                                             *.8.'.           .
                        'r.g . It must be replaced by a ratio r',g_which may be approxinsated by                                                       .
   ]
}i rk = - 2 .5 rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (37) .
gwensawraL                                            'i. . .
..i                                                                                                                        i                 soo .
] .y                         With the modular ratto m known and the appropriate amounts of steel given                                   $                                                                            ,                                               ,.
bej
; .j                    by Eq. 34 or Eq. 35 and Eq. 36, the valos of 1,, c.sn be determined using Eqs.                                   g t,$                    ,                  ..d,-Q
                                                                                                                                                                                                                                                                               ,. T i      ,                 32 and 33.                                                                                                      ,_ goo   .
r                         ,
[ ._ sc,r=Q=                    .                              ;;g4 z,                                           .
o                                                                                                          -
                                                                                                                                                                                              ,,.J        ..            .
SOLUTION PROCEDURE                                                              $4an     .                                              is       s  s
* r                                       ,jq,7 k(*y,.*w lw h
      )                                                                                                                                                        f'.5100 c          ps i                                                                              i 8''i+f The slab with a known distribution of reinforcing steel is divided into ele.                  I E 29s lo8psj                          *" sua                                                                       T
                                                                                                                                                                                                                                                             !@.4 ments. For each element or region a stiffness matrix is derived using a ri-                                         200               Ec.4 s sio'pai                                                                                             -
j g                      gndity m. atrix based on uncracked section geometry. The centroid of each                                                             , .ocos,                                                                                      '
                                                                                                                                                                                                                                                             . , , ,jd i                        rrglon is constrieredinall calculations to represent the average properties of                                                                                     '                     '
that region. The elements are thenassembled into the total slab structure and i                              *          '                                '
{ *: .g
                 . boundary conditions applied.
j l H !"                       Aunit. load is applied and scaled down until only one region cracks. This is                                                           DEFLECTION AT CENTER Bf BICHES alD I                              ~j."y       '
f- gave rned by the magnitude of the maximusa principal moment. The effects of                               l                                                                                                                                         ,(<M l                                                                                                                                    FIO. 8.-14AD DEFLECTION CUltVES FOR CElfrER OF ONE-WAY ELAB
                                                                                                                                                                                                                                                             .; M l
stiel and crack orientation on the rigidity matrix relative to the element co-L    .l ', trdinate system are determined next. The stiffness matrix is then changed                                                                                                        *
                                                                                                                                                                                                                                                             . af) t i
d
                    . tppropriately for this cracked region. The unit load is applied again and re.                                                                                                                                                          , 'i. h' .,
l                     . cults scaled to the next crack observation. This procedure is repeated until                                                     10         Expetacfint. /ERNEM
! ]
th3 desired load level is reached. The analysis allows cracking in two ortho-                                                                                                                                                       . N; -
l                    ' gonal directions if the minor principal moment in an element reaches the                                                                                                        fen emmetse tracktr.g moment value.                                                                                                          g _
j It is assumed that once a crack forms at right angles to a principal mo-ment direction, its orientation is unaltered during any increase of load. This                    b EXPERa ettat                                                                                   .
                                                                                                                                                                                                                                                             ,y Q',
4,j' . ,
Eg i
7                   has been vertiled la the analyses presented. The greatest change la principal
 . O*                    moment direction was less than 5*.
                                                                                                                                                    ''                                                                                                       5 b: ,;7 f
4 anni P
                                                                                                                                                                                                                                                             '[
]-
EXAhlPLES AND COntPARISON WITH EXPERIMENTAL RESULTS                                                              a 4  '~
q ,, ,,,                                    y                                                            .j
!       l                   ' To demonstrate the capability of the method for reinforced concrete mem-                      ,                             ,               ,                     [,,
* I       l             , bers and in particular to establish which of the two methods, Bransonor Dee-                                                                 ,.ocos3                                                     *g
                                                                                                                                                                                                                                    '^'#-
                                                                                                                                                                                                                                            !
* f,g ,
                      , by, represents experissental results best, the following six analyses were                                                                                                                                  . .                              . M '.
q performed:                                                                                                                                            i         i           i         e i                                                                     .
e 02         04       os                                                                   '.,.
j                                                                                                                                                                                             <>e       to                                                                   .
* 1. A one-way slab,12 in, wide by 1.75 in., thick, a span of 34 in., and 0.8                                                               DEFLECTION pf INCMS                                                            .
h                    ''e reinforcing steel, tested by McNelce as a control specimen for concrete                                                                                                                                                               .g O, 7h 4'                  clab tests 121). The division of the slab into elements and the load-deflecilon                             FIG. 8.-LOAD. DEFLECTION CUltVES FOtt CElrTER OF3EAnt NO. les                                                                 '
Curves from the two analyses together with the experimentalload-deflection I                                                                                                                                                             M,f turve are shown in Fig. 5.                                                                              3. A corner supported two-way slab,36 in, square by 1.75 in. thick with an t
: 2. A beam INo.106) 7 in, wide by 15.25 in. deep, a span of 180 in., and                         isotrople mesh of 0.85 % ret.iforcingsteel, tested under scentral point load by 0.e3 % reinforcing steettested by the Cement and Concrete Association (4). A , McNelce (21). Fig. 7 shows the finite element grid and the numbering of the
                                                                                                                                                                                                                                                                  .h'
                                                                                                                                                                                                                                                                    *M.
i        l. .
4    ,1 '                                                                                                                                                                                                                                                          U 3. .
!       j j.
* 1 i.;                                       .
it
\       l                                                                                                                                                                                                                                                    ..%
i

March,1971 ST 3          !rt 3                             REINFORCED CONCRE' E SLABS                                                         TIP-             yt.;is;.
  ' d ;,798,                                                                                                                                                                                                                                                                                                          *
        ;I                           "                                                                                                                               and Sonen la their Investigation into n yleid crit:rton for retriforced concrete                                                        ,,,
{l modes; Figs. 8, 9,10, and 11 show the load deflection curves from expert.                                                                                                                                                                                                                     * '.
stabs (19). All slabs cn 413. thick and r'inforced w13 approximately 1 % of                                                                       ,
       ;'                                                                                                                                                   '        steel. Table 3 presents the comparable results.
   '' ..1> !, . mental and computed results for four points on the stab.                                                                                                                                                                                                                                        ,
      .,'8i e
I                                .
4 Smu.                             SteFORT
       .t i.                                                                                                                                                                                                                                                                                                    r Ni                                              2                                       -              M aMMe                    [* ~
3 o0   .
    '9,
* 6                                        48 32 %
e                                                                                                                 -
6i                                                                                                                                           '
47                                                                         FEM. (BEE 8Yl-j                                                                                       S                                                                                2000   -
F.EM leRMODIG                                    ( ,(,l .
M.;                                                                                       4                                       4s %                                 "$                                                                                                                            ...-     -
2400    -
                                                                                                                                                                                                                                                                                                          . g f.: * -
E
[                                                                                       3    -            -
43                                    a. 2000      -                                EXPERIDENTAL                                                         ,
1                                         m
[                                                                                                                                                                                                                                                          !y
                                                                                                                                                                                                                                                                                                           , N 'y 2                                       44                                   g ,,oo       .
     ']
I                                       43' -~*                             3 1200-
                                                                                                   -(,roafH'T e,*
       )
       .                                                     i.ers,                                                                                                                                                                                                                                       1,r     w'2 I                                                     a . i 3i-                                       -tmo P                                                                eno    .
                                                                                                                                                                                                                                                                                                          ?
fje S600 p*si                                                                                                         400 -
                                                                                                                                                                                                                                                                                                          $;M      .l; t.:f i
Ege29aIO psi                                                                                                                                                                                                      '
Ege415 s O'e s t                                                              h                                                         '        '        I         I 16        20 I
24 28 I
32 7/fI
                                                                                                                                                                                                                                                                                                          . g > y. y ,
4         8        12 i                                                    , .o :5                                                                                                                                                                                                         2
* t. ,.;,)
    .',.?
p *O 0005                            .                                                                                                       DEF1.ECT10N K NODE 4 IN NQE3 O                                                                   , dW l               FIO. 9.-LO%D-DEFI.ECTION CUltVES AT NODE 4 OF TWO-WAY SLAB                                                                    , ; c.s,.N q                                                                                                                                                                                                                                                                                                  sr
        +                                              ,
se.
v-                                                                                                                                                       .         ., g,. .v      ;
e
                                                         ;_,                      as.                                          _ , , ,c                                                                                                                                                                    ,u.<.
       .                                                                                                                                                                                                                 EXITRedEN1AL                                                                        g' **g
* 3600-                                                                                                                **b
      ., j FIG. 7.-CORNER SUPPORTED TWO-WAY SLAB                                                                                                                                                                F EM. (BEEBY)
                                                                                                                                                                                                                                                                                                           'g.(
       **                                                                                                                                                                                    3200  -
3600                                                                                                                                       g2000-                                                        F.E.M. (BRANSO4                                                 .yj,3 s
i ; ''                                                                                                         F.E.M.tBEE8Y)                                                        2400-                                                                                                         . 4*d 3200-                                                                                                                                                                                                                                                                 '
                                                                                                                                                               '                     3                                                                                                                                  I
                                                -                                                                                                                                            2tXX) -
h2000                                                                                                                                                                                                                                                                             ;
                                                                                                                                                                                                                                                                                                            .- '' [g
    '!.                   3 2400 EXPERMENTAL F.EM.lERANSONI
    -l- ,
a 2o=                 -
_ i2m-                                                                                                                   . .
                                                                                                                                                                                                                                                                                                               ~
i                                                                                                                                                                                      800   -
h8600
;                                                                                                                                                                                                                                                                                                              ? :.
s                       a                         ?                                                                              r                                                     4m 9,-
i .,                                stoo   -     <
* I        f         f          1            I                     f I                                   goo                                                                                                                                                                                                                                                                 ,/
28
                                                                                  '                                                                                                                        4        a         82          16          20                     24 2
* g;  l) ,,
ll; 4M                                                                                                                                                               DEFLECTION K NODE 7 H INOES a10
                                                                                                                                                                                                                                                                                                                 .m, w t                  t        t             i           i         t           t       t                            ,
et                                                                                                                                                                                                                                                                                                      $. '
e        2            16           20       24          as       32     36 4                                                                                                                     FIO.10.-LOAD-DEFLECTION CURVES AT NODE 7 OF TWO-WAY SLAB
[t DEFLECDON AT NODE 2 IN NOES a 10 g
                                                                                                                                                                                                                                                                                                       ,        Ipg In all but the one-way slab by McNetce and slab C1 of the Lanschow and                                                               j,1Q l.l                                                                                                                                                       l Soten tests, the results using the Decby method are in bcffer agreement with FIG. 8.-LOAD-DEFLECTION CURVES AT NODE 2 OF TWO-WAY SLAB                                                                      experlmsntal ones than those obtained by the Branson method. In the case of                                                             d' S    '' 15 e
the one-vay slab the experimental deflections are greater than either of the
       ],j  ,
In :. idition to the aforementioned analyses, rigidities calculated by Eqs. 25 two analyilcal ones. Th's may be due to the fact that flat bars were used for                                                            . y. ;;
and 27 m ere compared directly with those obtained experimentally by Lenschow                                                                                                                                                                                                                  ep l*
        ,                                                                                                                                                                                                                                                                                                         o a                                                                                                                     9y
                                                                                                                                                                                                                                              ,,,,                                                            ..'g.. 9 8                                                                                                                                 . . W .4TIP/

!, e                #
e*      e i f.                                                       March,1971                                            ST 3        ST 3                        REIMFORCED CONCRETE SLAlm f
r ret s                 ich (ould have led to cxe:ssive bond slip. With respect to                         and citir cracking Isy the Deeby method cra given 13 Eqs, 24 and 25,                                .       :4. ?
,)!              g                               noted that the only w y it Cff:ra with clabs C2 (nd C3 is respectivity.
ilccause it is recessary to tak2 into a; count the crlontition of the track]                          id .
 .,              0 higher concrete strength.                                                                                       with respect to the coordinate system of the finito clement, the matrix trans-
  .7                            '
4                  -
tablished in Eq. 22.
e g, N
                .EX          k        A I        E'T-C          AU CURVES                                                                 Procedures are developed that allow a close approximation to be made of q    '                                                                                                                             the effect on rigidity of steel orientation. Eqs. 34,35, and 36 give equivalent
                 . g Lge Ref.19)
ExperimentalE te. E,1,,Branson bleth- Eefe, Beeby blethod, od, in inch-pounds        in inch-pounds                steel areas normal to the et icks for direct use in the calculation of                                  .; f..,
th
* 3                                           'lach-pounde f ront Ref.19.             $n.e   x10-*                   mto-*            i transformed moment of inertia of a cracked region.                                                             s ['3, ,
n10-                                                                    This study shows how the finiteclement method can bo used to                  analyse re-j                        g                           II                           I3)                     (1)                                                                    **               *             "
7                                                                                                                                             *   """        "        " E *"                        *
                                                                                                                                                                                                                                           . .k4 '
by-step procedure that incorporates the change in rigidity of each region,                 *         '
Di                            s.2                        4,3s                     s.2s Rio                           5I                                                  5.62                   cracked or uncrackci, under increasing load. Because the analysis is prl.                                            4 marily intended to study serviceability of reinforced concreto slabs in tho C1                            4*T                         4 44                    6.50 service load range, pustylcid behavior is not included.The analyuls could iso Ac.p(
1 s.16                                                                                                                        f C,                            8.7                         4.35                                           extended into the posty1 eld range using the method previously employed by
  '}
C3                            g.3                         4.23                    5.cs                   McNelec (21).
                                                                                                                                                                                                                                           ;fg&
t A (hilte element analysis mich as that devcinped herein can be a valuable                       . I @*
tool for slab research.11 is the opinlent of the writers that mimurical unintri-yd
                                                                . . _ . _ _ .                      ~
mentation supplemented by physical experiments, chosen judiciously,is not                            "y
      },'
 ,j                                          3600 ExPDWMDf7AL.                                                        only a more economicalapproach to slab research but will allow the important.?
F E M. (BEEBq design parameters to be isolated quickly and examined more thoroughly.It                                  jh. 4
 ' .I' ,                     .               3200    -
was for this reason that the present study was comiucted,                                                 u,p
(                                                                          FE.M (BRAfee J
             ' ' -                      m 200o i
                                                                                                                                                                                                                                   .       <. yeh 4                                        2400 '-
ACKNOWLEDGMENTS                                                            t #
F 2,
: o. 2 coo     -
                                                                                                                                             *this work was carried out with the financialassistance of the NationalRe-
                                                                                                                                                                                                                                               '* [
L f,
  $,                                    g goo search Council of Canada. The writers express their sincere appreciation to their Dritish colleagues forthe suggestions and information which lead to this
* 2h.       *[
a l'!                                     200     .
study. In particular to K. O. Kemp, University College, and Deeby and Taylor                             R&
4 800.                                                                                      of the Cement and Concrete Association.                                                              .t            3 400                                                                                                                                                                                                    c 4                                                      4          e         i2         16       to   24
* J.                                                                                                   2                                       APPENDIX !.-DERIVATION OF EQUIVALENT STEEL AREAS                                               ; Qe.-
DEFLECTION M NODE 28 IN BOIES a D l[
      .=
g
* f * ':
FIG.11.-LOAD-DEFLECTION CURVES AT NODE 28 OF TWO-WAY SLAB                                                                                                                                                                         '
: f.                                                                                                                                       Fig. 3 shows a reinforced concrete plate region with an orthogonal steel parisons the Beeby method appears to give                  6 arrangement in the x and y directions. The steelarea in the x dtrection is A ,                       ,g  h.
D be te e                a       ft          Eld 1 y I a reinforced concrete region than the in the y direction p A,. If this reinforced concrete region is subheted to                                  t .f I.
      !            Branson method.                                                                                                        a direct tensile force N, in the a direction at an angle a with the x direction, j!                                                                                                                                                                                                                                           %*,'l Lenschow and Sozen have shown thatafter cracking, in the elastic range, the
      .                                                                                                                            i l{,                                               

==SUMMARY==
AND CONCLUSIO!!S                                                           tenstle strain in the steelin the n direction (, may' be expressed by                         '            O' 1
it is proposed herein that the rigidity of a cracked region of a reinforced                                                 #8 *
                                                                                                                                                         ' N     ~        *              '"           cos' a + p sin
* a
     ;, c nerete slab be based on a bilinear moment-curvature relationship. The                                                    I                     ~Es '                                    '                                                '*A
                                                                                                                                                                                                                                                   .i;h, l          by and the Branson methods were used in the derivation of the rigidit                                                in which e' = eg + yet (cotan a - p tan a)/(1 + p); (g = steel strain in 3 .                 analytical results withboth methods compared with experimental results                                                                        shear strain in steel. Adopting Eq. 38 established by l               appears that the Beeby method yields better results. The rigidilles before                                           direction f; and y,g    =
l,I..
* r           .                                                                                                                                                                                                                                    .:*
                                                                          ,M   .                                                                                                                                                                     .
S*

               '. 868 ' **
* liarek,1971 '                                         ST 3                                                                                                                                               t !
                                                                                                                      ,                  gr 3                                 RE N N RETEBI.AB3                                                         003            -*:               ,
     .h janschow End Soren it is pos;lhis to deriv 3 apprenimate expressions for                                                       '
be compded to be                                                                       *'
1              eqelvilent tr';as of reinfIre: ment with respect tothe crack direction. If a cast                                                tis e
      ;1             f tre the principst directions, the shear strain in the steel Yet e is sero and                                                                                               #=                                                                    '
al = ag. The mornsat strain sesy then be written as
                                                                                                                                                       's *                                                                     * 'i"" e
                                                                                                                                                                                                                                                   ,.,(44)                    '
       ~}                                                                                                                                                      EA cos* e + p sin
* e + (t * "} * *
      -I                         t=           "
                                                     . a s(1 + p) esse , og ,e ,                                I                               *                           *                                                               ~
l ).p
                                         .E, A ,                                           ,  cos' e + p sin
* a            (39) has for esse 2 the equivalent steel area in the a direction is
{I ;.l*
,jg                  E Creeks have only formed la one aHrection, rg will be much smallerthan 8 e and therefore can be neglected. Eq.-39 then becontes A ,. = A , ccis* e, + p sin
* a + {I + p) cos'            rg e sla' e' ****** (49}
     .' '
* N.                   .
In.the transverse direction a relationship between cg and Ng le Mquired.                                                     .h)..       .'
.      v                        '' " E ,A ,(cosi e + p sin
* a) * * * * * * * * * * * * * * * * * * * *
* I         From Eqs. 45 and 48
        "                                                                                                                                                                                                                                                 .              d'* M ,
I                                                                                                                                                                                                                                                    .        s g .f M' A , (cos' e + p sin
* e) = A,                                                                                                  eg = 'a. ,,,,. ,,.............................. (50)
                                                                         . . . . . . . . . . . . . . . . . . . . . . . . . (41)                                                                                                                                      M,y then Eq. 40 can be rewritten as y"                                                                               h h;
                                                                                                                                         -tt                                                                                                            D'l
      ;j              .
c =      ,,y;,,...................................(42)                                                                     u.a .        ,
cos'  e  4  y sina    .r . o , g3 .,y  = ,,-,d = r-        -
r.t 9g j f                 in'which Aa = an equivalent area of steel in the a direction computed by                                                                                                                                                                           .. s Tensile force Nt is approximately related to N. by                                                                                  g j g ,, ,7 i
me:ns of Eq. 41. The principal bending moment in the a direcit..n leads to a -
l 3            t;nstle force in the steel, N ,and the corresponding curvature in the n direc.                                                        =             ......................................(53)                                                          ) .T' i
       .l. ' tion leads to a tensile steel strain, t . The equivalent steel area, A u, may                                                              8         8 therefore be used directly for the calculation of the steel contribution to the                                                                                                                                                                                  .
g transformed moment of inertia in the a direction in case I l.e., for regions                                              i in which M and M, = the major and minor princl nali snonients. Substitution i                                                                                                           . ,M
          ,         that are cracked in ong direction only.                                                                              of Eq. 52 into Eq. fit leads to                                                                                       .
f.r Is case 2, where cracking occurs in teto orthogonal directions at the same                                                                                          #*
i facs .of the stab. the strain in the transverse direction, tg , can no longer be                                                                                                                                                    (53) 4
                  , neglected. An approximate relationship between steel strains in the x and y t     't "
sAs cos* a + p slM e + (1 + n) cos' r, sin 8 e' "I
jf, Ma g                        N' ot Yst directions is                                                                                                                              _                                                                  .
h.'I W
gr
[
j . I' ,; ,.
ts"    [y       #fr" Fry         8 y..........           .......... ....... (43)                   i and an expression for the equivalent steel area in the i direction of                                                                     g Y
Ast " A s           cos' s + p sini     e + (g , ,) eo,a e sin 8 e' F"I ,M                   M.fa (54)     ..hak j
rw                       .            i
                                                                                                                                                                                                                                                                                 *7
               ,f,laBec:use the x and a      and I are       byprincipst directions they are related to the steel strains
!                                       y directions y                                                                                . 4 ei.Or t, = e, cos* e + Eg sin
* e                                                                               ***                   * * * ** '*
* M            ****************************
i
                                                                            ....................... (44)
* l-                     fy"t a sin' e + ti cos' e 4
If e, and tg are rehted by
                                                                                                                            ,                      In case 3, where cracking occurs in two orthogonal directions at opposite faces of the slab,the strain in the transverse direction, af, le a compressive k..t g."g 
! .                 ,,                                                                                                                   strain. It can be related to the tensile strain, Eg, by t.=rte tg           ....s.............................. (45)                                                                                                                                                                              ,
                 ' it follows from Eqs.'43,44 and 45 that g, yg g( ,               , ,      ,      ,
his ratto r,, will depend on the strain distribution through the section and
                                                                                                                                                                                                                                  ,,    ,,,,1,,,. (56)
T.%'
         ;,'                             rr        s e - sin 8 e                                                                       can be caiculated from the prevalling stress state. Fw case 3 the equivalent                                                    ,5
        )f,                    f or " cos,'cos   e - r,y sin' a * * * * * * * * * * * * * * * * * . . . . . . . . ( 46)                  steel area in the a direction is therefore g..*
s.'8'  !. ubstitution of Eq. 45 into Eq. 39 yields                                                                         I A es = A s cos' a + p sin
* a + (1 + y) cos' a sin'                                        .... (57)               h.hNg
                                                                                                                                                                                                                                                                         . . w. .
jf-j                   en.          =
                                       .E, A ,
                                                       .f.a.
rg a    (1 + p) cos' e sin' e. cos* a + p sin
* e (47)                                                                                                                                                  .
in which r,',g = the product of rit and the ratio r. le found from Eq. 46. The
[h.9
                                                                                                                                  )
equivalent area of steel inthe transverse direction is as given by Eq. 35. The                                                      ;
     .j                                                                                                                                                                                                                                                                         .       -
l w.
                                                                                                                                                                                                                                                                          * .a i.\-           .
i gp
                                                                                                                                                                                                            ~'--

5!.,e <                       .
WR                                                                                                 ft 3
* REDfFORCED CONCRETE SLAgg                                                                  006                                  -
lesPelin it'll                                              sf 3 -                                                                                                                                                .,
           $         .       valeeff th? r'tio rjf does not have a great Effect on Asa nor on the rigidity g                             '
st.Johansen.K.W **Vield Linelheory.'*Cenisse anlConcsses Ameciation.l.sadun.1962.
i' 0 region. Its offeet on the overall rigidity of a slab is very small indeed. It is ( la. Kamp. K. 04"The Yicid Criterien for Ortheorepnelly Reselesend Coussues Elmhn "Inessansson-                                                                        ,7p      ,
       "t therefore reasonable to adopt se average value for rgg. Assemlag an average                                         alJournald hW                                        #
f,,,,,,,,g p,,,,,, ,g,, g;,,g,g g ,
          ?,                 depth 16 lhe neut rat axis of 0.4 d, a r===a==hta value for er , would be - 2.5.Thee le                       19.1cmhew R aml                                  A         c                                                                             ,
                                                                                                                                                                                  ,,;g,, n.,g,,g 5,,,4r,,Se,in-turuf Mrsnesse virs No. Jf f. Uniscenioy of Ecl. 57 r,*,g may be repLeeed by . 2.5 r.g. This value has lieen used forall                                                                ,,, ,,,4
                                                                                                                                                 "['' , gp ,,g g,jg,,,,
analyses presented herela.                                                                                                                        gg,,-Elessic. Plastic Analysis of Het Pimees by the Finies FJuesut 1.%
Nc,1,od. A             can S.s , .i Moon.n.c.ungin es Winier - M-d= nd E -gr s*-                                                                 c.
I.
I
       .v                                                    #
I          seen Eposition. New York, N.Y. Dec 1963.                                                                                                         -
fl                                                                                                                             [ 31.McNcise. G. M "Llantic-Plassic IIcehns of Plaecs and Stahs by the Finies mment Medmd.,,                                                              ,a,A ,
i                                                       APPENDEX H.-REFERENCES                                                       incsis preiemcd to the University of London et 1.ondon, England. in 1967. In pestial fulfillment                                                ,;
d
* elthe recluifessee Ws Ier the elegree of Doctor of Plulosophy.
g.Papenfess. S. W.. "l.ateral Plase Deficction by Seaffness Matri Motheds whh Applwedse                                                                     -'.
I se a Maritisce." thesih Pecscsiecil in ties t Issi,ctsisy nf Wasteiserseen, at Scutelc.Washenytms.in 19W.                      .
Qk          .-
[.
1                                                                                                                                                                                                                                                                                                        ,..-
in p,osial lulfillnesait ut she evigentcsiecnis for alic slcrecc i.i Masscr est %scese.
I. American Concrcie lentisuee. Committee JIs. AmiWmr Cde N vasermenti A,r ReinA,n nf Can. g H.Tinunhenka, S. P. anal Woenuwsky-Kricsst. S Daarf al Ms8r8 8mf Sar#s,2nd IJstima,                                                                                e,..
r8' J8881983.                                                                                    6          McCnew-Hin Book Company, New York.1959.
l
,          v,             , 2."Anienean Coswrese Institute Comneissee 4J5. " Deflections of Reinforccd Concrcse                                                Fleseral                                                                                                                          *Mi ping, N **Elado.Pbmaic Orthineispic Places and Shcus" rg .lw e/#4r Srmpasione se Memhcri"J..a.in.sf o/sAr. imerscen Concrrrr initisarr. Vol. 63. No. 6. June 19ei6, pp. 6U 67),                                                                                                                                                       I ' ,, . *f
 ,     ;[                                                                                                                                t
                                                                                                                                                 'g,,,g,,,,,f f.,,,,,,, g y,,,,,,,, y,,n,dv m ('ivil t'nrinrrrms. Vanderhusle Unive'nity. Mmhedle.
I                .         , 3. Armen.11 Parke A. aml 1. crime. II.,"A I-inite Licnient Mctiseul fur Ilic Plantie Hcolmr Anal.
  '                                                                                                                                              Tsusicsicc,N.sv P86'f.
* I ) ' ''
Den of Structurcs," Presimfmri Jef Cem/carniv cm Merrn Methenfs in Serem-rmref Mn Assairs, f,.                      Wright.Paitcriosi Air Force Base Oluo,1963,
: 15. Weed. R. fl Plassic sal Eleutic Design of Stahs," Thesacs anil tiedmen. Lesulon,1968.
3,, y,, w, w, ,,"g wi,,,,, G.,alnemensensees and I.eng. Time Deflections of Reinfaced Cenesses
{* *
  ,         .j , , . d. Bane, G. D., ReaJ. J. B., Sceby. A. W. and Taylor.11. P. J.,"An Investigation of the Ciach
                                . Control Characecristics of various Typen of Har in Meniforced Caesrcsc licains," Neinsern g,,,,, ,,,g,, w, Ling 3.nnds " Juanesi n/ 84r .4meverwn (*mererre feesirere, Vol. 57, Nn. l. July.                                          A (!
       *14
                                ' Re.rt .h i1. Ceneens and Conctcte Annociation, Lusadosi. Due.,1966.
g,g,ppy w,
           , I*
s                                                                                                                                       ,                g%;
                   '           3. uuele3. G. P., Cheung, Y. L., Ironn. B. M. aml ZicnLicuies. O. C., " Triangular Elements in l                                                                                                                                                                       .2 i'         j                       Plate lesmimg Conforming and Non-Conforming Salesions." Prearenfinsi Osm/,.vnce un Me.              ,
           'i                     seri Mrshad. vs Structa,,of Mn konici, Wright.Patect=ni Air l'orse Nase, Ohio.1963.                                                                                                                                                                                 M, 4
;                  ;           6. Bech3 . A. W "Skat.1crne Dclorsmassose.uf McinforscJ Concrcie Mcmhcri."Cemcat andCon.                                                               APl'ENI)lX lil.-NOTATION etese hooeiation. Tag knavalMrport No. TN.-l 4rtT. London March,1963.
       . ,                , 7. Bran =>n. D. E., "In.iansaneous and Time. Dependent Deficctions of Sinople and Coe'inuous                                                                                                                                                                                   %.
         ..                       RcinforecJ Concrete Beami,'' .lfw&enso flir4=wr Neinsere Neert No. 7. Rurcas of Public
(\
: 8. I     y.              . and l'eloppa. C. A.,"A Itefined Quadrilateral Element for Analysis of Place            N I'N'*I"E 'I"                                                              P*              '
        .4
           }                      9cnomg." Pr.werJmen 2nd Con /creuser on Mentrix.Methuifs m Strescessrel Ma hanirs. Wright.
* Pasterson Air Force Base Ohio,1963.
A = area of element;                                                                                                                          Q
(           ?                  9.Clough. R. W. and Tocher, J. L "Finise Eleniens Seiffness Mairices for the Analysis of Place                  As = area of reinforcing steel per taalt width of stair,                                                                                        t l
BenJmg." ProcccJmes Conferrmy un Marris Mrra.ds in .V#rimtermi Mn-Annks. Wright.                            A,, = equivalent As IR 88 dIf'CII'"i                                                                                                         -
j,          !                     Patterion Air Forec Banc. Ohio.1965.                                                                        A,, e equivalent A, in i direction;
           !                 10.Comper. G. R liosto. E Limihers.G. M Olsen. M. D "A lligh Precision Triangular Plate.                             B = matrix relating curvatures to nodal displacements;                                                                                      j'?r,
{                     BenJeg Elemens," .4rroneurkuf Neport LN.Jfd. National Rcncarch Council of Canada, Dcca                                                                                                                                                                       , %,.
3"                                                                                                            C, = matrix of direction cosines for moments;                                                                                                   '
1l                                                                                                                                  C8 = matrix of direction cosines for curvatures;                                                                                               '" M I
                .            II.Fracij de VeubcLe, B.,"A Conforming Finite Elenient for Plate Bending ** farerns#6morJune.            t                                                                                                                                                        *C
                                                                                                        '                                         e = cos a;                                                                                                                                  1.,
{ l.                  nolof .%fs2. aoeJ Sormerares. Vol. 4, 8968.                                                         s fn = depth to neutral axis from compression face;
           'J                12.Itcarnon. R. F. S., "An leerudestion so Applied Ani.ostorie Elasticity." Omfurd Universisy' .                                                                                                                                                                 Q:q p,,,,, 3 9g.                       i                                                                            D = general matrix of flexural plate rigidities;                                                                    j ,                       ;',4 f'        13.llotmes. M. ami Domnham R. J.,"Emperisnental Seiffness sad Yicld Critens for Reinforccd                         D' = D matrix for the special orthotropic case;                                                                                                it*F.
3      J    ,          Con.rcie Stah " #wifJmg Scornrr. Vol. 4.19ee.pp. 31 42.                                                   Dgj = coefficients in matrix D;
            ;                14.Joinct. J C "The Enese Elemens Method in Sendias-Scicetion. Evaluanian and Application                            af = effective depth of stab;                                                                                                               o ++'.
            ;                     of a QuaJedateral Plate Displacenient Muilel." she.is nehmissed so the Universie) of Waection,               E, = Young's modulus of concrete;                                                                                                                      .,4
         '!                       as Onsatio. CanaJa,in 19ee in panial fulfillaient of the teilmirements for the degrce of Masse' j            E{. = reduced Young's nuidulms of concrete; b" " ""                                                                                                 *
         .                   15.       m       0            esec G. M. and Green. K "De Clough.I'clippa Llenient in Slah Anal mis,"
                              . Mm.rr .bs 4T.Solmi Medannes Disi. ion University of Waterloo.Onsatio.19M.
3 F = nMal forces,                                                                                                                                   %Y   ;
l                                                                                                                       FEM = finite element method;
            ;[               16.Mnct.J. C. and McNeice, G. M "Fimie Llement Analysis of Reinforecil Comercie Stahn with                                                                                                                                                                      '. :
             *;                   Progrewne Caxting," Nersure No. .'ef. Salmi Mcebanics Division, Universie) of Waterl.io,                     f *. = compressive strength of concrete;
            ;1 .                  Watciloo. Uneario. June.19M.                                                                                  fg = modulus of repture of concrete,                                                                                                          . ,M
              ~I
: l.                                                                                                                                                     .                                                                                                                                 1; .c 4
I6

                 , 1. m .e, _                     - ._ . _ .- .,
[ dom
               .          .                                                                                           o Deflection Of Two Way
                                 -                           Reinforced Concrete Floor Systems:
C State-Of-The-Art Report                                           .
By ACI Committee 435                                                                          ,
G. M. Sabnis*                           A. H. Nilson*
Chairman, Subcommittee 5 Chairman W. C. Alsmeyer*                     R. S. Fling
* F. Roll
* J. R. Libby                 C. G. Salmon D. E. Brar .on B. L. Meyers*                A. Scanlon*
D. R. Buei *.ner
* M. Saeed Mirza*              A. C. Scordelis R. A. Cris.
N. Norby Nielsen             A. F. Shaikh            ,
L. M. Dallam M. V. Pregnoff               S. Zundelevich i
Synopsis _:              This state of the art paper sumnarizes practical method for calculating the deflection of two-way slab systems, including flat slabs, flat plates, beam-supported slabs, and wall-supported slabs.         A selected list of references is included.                .
Keywords:           bea=s (supports _); eracking (fracturing); creep properties; deflection; flat coverete plates; flat concrete slabs; flexural strength;
_ floors; modulus of elasticity; reinforced concrete; reviews; structural analysis; two-vay slabs.
l-                   .
  '                                 t
* Members of subcommittee 5 which prepared this report.
                                                                                 -       43521 g
         ''*'**'****"*Pe Me+,*              ,,,%      _   _     ,      _

                                                                                                                                                 =
to obtcin a suitablo force-displacement relationship
        -                   batwson tha nodel forcos cnd the corresponding disp 1cce-
                          . osnts at the nodal degrees of freedom.                                                  A further complica-tion, in applying the method to reinforced concreto, is the derivation of a suitable set of constitutive relations to model the slab behavior under various loading conditions.
The response is generally not linearly elastic under all loading conditions.
When the constitutive relations become non-linear, and cracking and time-dependent effects of creep and shrinkage are to be taken into account, the finite element analysis of reinforced concrete slab systems becomes much more complex (see below). Iterative or incremental tech-niques (or a combination of these) have been used to ob-tain a solution (16,18,19,23,31).
CHAPTER 7 - EFFECTS OF CRACKING The effec; of cracking in slabs, as in beams, is to reduce the flexural stiffness of the member                                                 and thus to increase the deflection at any load. Prior to cracking, deflection calculatio'ns are generally based on the moment of inertia of the gross concrete cross section neglecting the contribution of the reinforcement; no significant error is introduced. Upon cracking, the moment of inertia is reduced, but the tensile concrete between cracks con-tinues to have a stiffening influence, and use of the cracked transformed moment of inertia for deflection cal-                                                       '
culation leads to serious overestimation of deflection.
For simple-span beams, Branson has recommended use of an average effective moment of inertia (6) for the entire length of a span; fM 1 3
I
* fM/ 39 Ier i I               (7'I)
I e "(F                      s              I'5 R           ,              s where I I           = moment of inertic of the gross uncracked con-crete section I
cr
                                                 = moment of inertia of the cracked transforried concrete section                                                -
l                                        Mer     = cracking moment of the reinforced concrete l                                                         beam M,      = maximum value of bending moment in the span This equation is included in the 1971 ACI Code. For con-l                             tinuous spans the Code permits use of the simple average i
of 1,,for the negative and positive bending regions.
i Although Eq. (7.1) was derived on the basis of tests of simple span rectangular and T beams, its use for two-                                                           ,'
l way slabs is permitted under the 1971 ACI Code. In
                                                                                                                            ~ --
                                                  '                                                 ~
3 y                       ?       .                      ,
              - . , - -                    [                   .,    . _ _ . , , . _ , _ . _ .             f. ___    _         _        ,, _

  "~~ ~
centrast to the aany beca tests which provide documenta-                         3
                     ~
tien for the uso of Eq. (7.1) for predicting beam defice-gions, there is little experimental dato to demonstrate its applicability to slobs. The Code contains the stato-'                               -
ment that other values of I, may be used if they result in deflection prediction in reasonable agreement with the results of tests.
In two way slabs, the ratio of M     g ,/M, is typically quite high. Also, the effect of cracking on deflections                          i is likely to be less than for beams having the same ratio of N g,/M, because the maximum moment occurs over only a fraction of the total width of the critical sections.
Tests of flat plate and flat slab floors have shown (30) that most cracking is in the negative bending regions close to the columns. At service load levels such sicbs may b.e mostly uncracked.        Ref.11 contains the recommenda-                 '
tion that the average gross moment of inertia be used for two-way joist and waffle slabs, while for solid slabs, it                       8 is recommended that an average ratio of Mer/Ma be used In Ref. I it is and    that I, hat,be suggested      t      computed by Eq. (7.1).
for flat plates,   the average ratio of Mg ,/M,be based on positive and negative moment region values for the long direction column strip, while for two-way slabs it be based on positive and negative moment region values for the short direction middle strip.
Various approaches have been used to incorporate the                ''
influence of cracking in the finite element analysis of slabs.      Jofriet and McNeice (18) used empirical bilinear moment curvature relationships, the first stage represent-ing the stiffness before cracking occurs and the second stage representing the effects of the cracking in reducing the stiffness. Cracks were assumed to propogate in a direction determined by the orientation of the maximum                         :
principal' moment in an element at first cracking. The
;                         element was then assumed to be orthotropic with respect            -
to the principal moment axes. Using effective steel areas with respect to these axes, the appropriate moment-curvature relationships were obtained and transformed to the slab coordinate systems. An incremental loading technique was used, where the element stiffnesses were modified as cracking was detected under the increasing load.
An alteenative to the modified EI approach is to                     ,
consider the slab to consist of a series of layers allow-ing g variation in material properties through the thick-ness. Youssef et al (31) used a layered finite element technique for nonlinear analysis of reinforced and pre-stressed concrete slabs.         Nonlinearities due to cracking were accounted for by using a piecewise linear incremental e
                         ~
& - -, . r. : ,n yr. m.+ & -, - -l~ ? --v .~;- - ~--~~~:-~ ~~ +- - - -
                                                                                                 --J-   -      --
i                                                                                              -
t

          ,                        precodura based on acterial nonlinearitics alens. Tha
            .                -     computed lood-dsflection characteristics, the principal strains, the machanisms of failure and the failuro loads showed good ogreement with the experimenta1 data (31) from                                                  i three post-tensioned prestressed concrete slabs tested to destruction. Hand, Pecknold and Schnobrich (16) used a layered nonlinear finite element procedure for determining the deflection history of plates loaded to failure. They used a twenty-degree-of-freedom shallow shell layered ele-
        .                          ment. Nonlinear behavior was introduced through the mate-
                               ' rial properties. Geometric nonlinearities were not con-sidered. An iterative procedure, often referred to as the
                                   " incremental-variable elasticity" procedure, was used to
                                  ~ perform the nonlinear analysis. Each iteration was treated as a linear problem. Several numerical examples were pre-sented to confirm the adequacy of the mathematical model and to establish the validity of the assumed material prop-                                               -
erties. Scanlon and Murray (23,24) used a layered rectan-gular plate bending element to model the response of slabs subject to cracking and time dependent strains. To account for the effects of tension stiffening between cracks the modulus of elasticity was considered to be strain-dependent
,                                  and decreases in step-wise fashion as cracking progresses.
The variations of the basic finite element approach 4                                 just described, while they permit accounting for the ef-
* feet of progressive cracking, are of interest mainly as research tools, and are probably tod complex and expensive i                           to apply in design practice.                          It has been suggested that a                           ,
three-step program might be developed for practical use in which (a) a finite element analysis is performed based on properties of the uncracked slab to determine the elastic distribution of moments, (b) from the elastic distribution of moments, the stiffness of each element is recalculated using an appropriate modification of Eq. (7.1) for calcu-l:
lating I,, and finally (c) the finite element analysis is repeated using the modified stiffnesses.                                                '
i CHAPTER 8 - LONG TIME DEFLECTIONS Long-time deflections due to creep and shrinkage are influenced by many variables, including load intensity, mix components, mix proportions, age of slab at first loading, curing conditions, presence of compressive rein-
,.i                                 forcement, relative humidity, p.nd temperature.                                           While time-dependent deflection of slabs has noz been studied exten-e 9                               sively, it is generally known that time-dependent deflec ti'ons may be about two to three times initial elastic P
deflections (32), and often result in unsatisfactory service load performance.
For beams and one-way slabs it has been the practice to estir. ate additional long-time deflection as a simple M
W h
* e                       3
               ,. .,             . . ~ . - - . e , .m
                                                                         ,. 4. .a.-. . .,                   .   ,. ,... _ .g ... _ ... r. . .- ._.    .

             .-                                                               nstnr. u xR<i'mu . msv ,
        ,                                  1952, p. 869.
: 14.          Filencnko-Borodich,M),TheoryofElasticity, Dover Publications , Inc. , Ne,w York,1965.
: 15.          Furr, W. L., " Numerical Method for Approximate Analy-sis of Building Slabs", Journal of ACI, Vol. 31, No.
6, Dec.1959, p. 511.                                ,
: 16.          Hand, R. A., Pecknold, D. A., and Schnobrich, W. C.,
A Layered Finite Element Non-Linear Analysis of Rein-forced Concrete Plates and Shells, Structural Re-search Series No. 389, Univ. of Illinois, August 1972.
l
: 17.          Jensen, V. P., Solutions for Certain Rectangular Slabs
:                                    Continuous Over Flexible Supports, Univ. of Illinois Engg. Expt. Sta. Bull. No. 303, 1938.                                      -
: 18.          Jofreit, J. C. and McNeice, G. M., " Finite Element 3
Analysis of Reinforced Concrete Slabs", Journal of                          l Structural Div., ASCE, Vol. 97, No. ST3, Mar. 1971,                          ,
: p. 787.                                                                      l
: 19.           Lin, C. S. , Nonlinear Analysis of Reinforced Concrete i                                     Slabs and Shells, Report No. 73-7, Div. of Structural Engg. and Structural Mechanics, Univ. of California
      ;                                     at Berkeley, Apr. 1973.
: 20.          Marcus, H., Die Theorie Elasticher Gewebe und Ihre                    '
     .                                     Anwendung Aug die Berechnung Biegsamer Plattan, Julius Springer, Berlin, 1924.
: 21.          Marsh, C. F. , Reinforced Concrete, D. Van Nostrand g                                     Co., New York, 1904, p. 283.
: 22.           Peabody, D., " Continuous Frame Analysis of Flat Slabs", Journal of Boston Soc. of Civil Engs., Vol.
XXXV, No. 1. Jan. 1948, p. 1.
:                       23.           Scanlon, A. , Time Dependent Deflections od Reinforced
      ,                                     Concrete Slabs, Structural Engg. Report No. 38, Dept.
of Civil Engg. , Univ. of Alberta, Dec.1971.
If l'
: 24.           Scanlon, A. and Murray, D. W., "An Analysis to Deter-mine the Effects of Cracking in Reinforced Concrete I
      !                                     Slabs", Proc. Conference on Finite Element Methods in li                               ,
Civil Engg., McGill Univ., 1972, p. 841.
l                                 s
   *1 We e           *       ,
e           et           *    *t f- - % yq : - .-: ,c. :$.;&y. . 5. .~. - .: - ,; . - .& -.. c.; - , ... n. m N-, . . - -
                                                                                           'in. m &-..Nr.. . u ~. - .
           .        _ _. _                _._..2            .._ _ _         _               _         ._-

y,,,,.      m, _, .e
    -.-      ,e,n, m e
             ,,o,   ,e               t,.         e e~...         .ee.t e,. -
D*****'**    -
       ~
Deflections of Reinforced Concrete Flexural Members                                                                                                :
Reported by ACI Committee 435                                                                                                    :
DAN E. BRANSON ch.,.
JACK R SENJAMIN                                                    CHAtt!5 M. HERD AttEN N BROWNFIELD                                                 HAROLD J. JOS5E
     -                                            CAttOS D SuttOCK                                                   GEOaGE t. tARGE W. GENE COtttY                                                     MARVIN A. LAR5ON LAWe!NCE N. DALLAM                                                 DONALD R. PEltCE eU55Ett 5. FLING                                                   MICHAEL V. PREGNOFF J.A.HANSON                                                         A.C.SCOaDELt1 DAVID WAT5ftlN Discusses the principal factors affectin                                                 d long time deflections of reinforced concrete flexural members, g short time an5everal methods for computing deflec
* fions are reviewed and a study made of the accuracy of these rnethods for pre.
dicting initial and time. dependent deflections.
Key words: beams structural); compressive strength; concrete; continuity; cracting; creep; deflect (ions; lightweight concrete; live loads; loading; modulus of elasticity; modulus of rupture; moment; moment of inertia; reinforced concrete;                                                                       -
reinforcement; shrinlage; sttein; warping.
CONTENTS                                                                                           C Cha pt er 1-Introduction . . . . . . . . . s . ...................................4352 101 Object and scope of the re,wi          ~
102 Notation Cha pter 2-Behavior of Concrete. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435-4                                   ,
201 Behavior under short. time loading
     ~
202 Behavior under long time loading Cha pter 3-Cilcula tion of Deflections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 35 6 301 Introductory remarks 302 Description of deflection response                                                                                    --
303 Deflections under short. time loading 303.1 Calculation procedures                                                                                                                    I       <
I
,                                        304 Deflecticas under long. time loading                                                                                                                  '
                                                                                                                                                                                                   )
304.1 Effect of concrete cree 304.2 Shrinkage warping of bams .                                                                                                                       !
,                                            304.2.1 Equivalent tensile force method                                                                                                               )
304.2.2 Empirical methods                                                                                           -                                 .
304 2.3 Comparison with test results                                                                                                                  1 304 3 Calculation procedures                                              .
I Chapter 4-Comparison with Test Results and Discus.sion ............... 43517                                                                             :
i 401 Comparison with test results                                                                                                                           l 402 Discussion                                                                                                         435-21 Ch a p t e r 5-S u m ma ry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 5 21                            l
                                 . Chapter 6-Reference, A ppendix l-Sample Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 23
                                  . Appendix 2-Data Parameters. Comparison with Test Results ........... 435 26 1
l 435-1                                                                                                    l l
l l-       .        -. , -       .               .-
,r          - - .; . .-. - w- J y .e         y--                  ,v a.r--l.W.-n..m----,-,-..--~-----,.~---------                                                           .

I                        Ing clong th) span length of the beam, the sfiectiva flexural rigidity
: +.
              -          is n:t o constant. On2 could evalusta the zones (Fig.1) fir which the th; rstic:1 cr:cking m: mint (it which th2 c:ncrete falls in tensi:n) had been exceeded and then calculatt the corrasponding trcnsfarmed section moments of inertia along the beam length based on the appropriate cracked or uncracked sections. With El known along the span length, Eq.
(9) could be used to evaluate deflections. This procedure would under-estimate the flexural rigidity of the beam, however, since it would neglect the participation of tens!!e concrete between the cracks. Concrete tensile strength may be even more important with regard to slab deflections
                 .        since large portions of a slab are frequently uncracked.
4 303.1 Calculation procedures Due to the complex nature of the flexural behavior of reinforced concrete beams and the desirability of having a simplified method for calculating deflections due to short-time loads, various procedures have been proposed. These procedures will be described in order of their                                                       -
chronological appearance in the literature.
In 1914 Maney*2 proposed a formula:
a = k3L MEl-= = k3L I f + afe . _ . _                             ..  (12)
                                                                                      \ E.6 where km is a constant depending on loading and end conditions and M.. is the maximum bending moment in the span (ks= 5/48 and M = WL/8 for a uniformly loaded simple beam, etc.). Eq. (12) is based on a cracked transformed section throughout the length of the span.
In 1924 Swaints presented a useful expression for El based on a                                         ,
cracked section as follows:
                                                                    - g,-   = u ."k)d but r,= f,/E, and M = f,A.jd which when substituted into Eq. (13) gives:
El = E.A.(1 - k)jdt        .                          .  .(14)
In 1931 Myrlea,'' reporting as chairman of ACI Committee 307, sug-gested the adoption of Eq. (12) which is based on a cracked section,                                                                  I for the calculation of deflections.
In 1940 Murashev" proposed that the cracked transformed section be used for calculating I, but that an increased effective steel modulus of elasticity,2,, be used to account for the participation of tensile concrete I                        between cracks:
t, = E./, where e d 1.0 . .                              . .(IS) g = 1 - (2/3) (M,,/M): ,             .           . . ,     (1g) in which M is the moment at service load, and L is the cracking mo.
                          .                                                      fes
                                                 < ... #^                        f'a
                                            ' .a.st
                                                          /       .4             y sinales             STntes           I" * '' " ''** ****
(steet strela eseumed tete senatent)          beam subjected to long-tel                       tal               (el                           time creep WIEK m.m           w.msrammmem *-                                             ~.m- gy ,mm .. 3                                            . m_7 7 -    ~;..v-- ,r.,q  +~...e     .
q .      ,.y_._.,.,,           .
       ~

      ".                                                              m:nt. A ditdled descripti:n (in English) cf Murash:v's d:riv:ti:n was                                                                                                               .
                                   ~
                       ,                       ,                      reportsd by Pregn:ff.*
Fcr pr:ctied purposes, th2 cricking m:m:nt c:n be computed as:                                                                                     .
ht ;= f**I*                                                            07)                                        .
1 We where f,a is the modulus of rupture of the concrete and I, and w refer                                                                                                              .
to the gross concrete section (neglecting steel reinforcement).                                                                                                                     .
In 1947 the Portland Cement Association,5* in a widely followed bul-1etin, recommended the use of the uncracked moment of inertia of the
                                            .                         gross section, ignoring the effect of the steel. Comparisons were made
 !                                                                    with experimental data indicating that at service loads this procedure gives satisfactory results. This approach gives reasonably correct values for beams which are not too shallow and are not too severely cracked.
However, with the use of shallower beams and high strength steels, ae-                                                                                                                    ,
l ccmpanied by higher workir.g strenes and greater cracking, this ap-                                                                                                  ~
.                                                                     proach will underestimate the deflections.                                                                                                                                                1 In 1960 Yu and Winter,*s reporting on an extensive study of existing                                                                                                               i beam deflection test data, suggested two methods which could be used
* for calculating short-time deflections:
Nethod A-The cracked transformed section I at midspan is used as a                                                                                                            l constant value throughout the length of the span for simple spans, and                                                                                                           ,
the average moment of inertia for the positive and negative moment reeions is used for continuous beams.
Method B-To take into account the participation of the tens!!e concrete                                                                                                  '
between the cracks, the short-time deflections computed by Method A are                                                                                                     .
                                        .                                      multiplied by the following correction factor:                                                                                                                              ,
( > - . -p )                                                            n.)                                   .     ,
in which M             is the moment under working loads and Ma is defined as:
N = 0.1(f,*)s/sD(D - kd)                                                            09)
The derivation of Eq. (18) and (19) followed an elastic theory approach with the factor 0.1 having been determined empirically. The moment M., was a pure bending moment in the derivation, and the factor 0.1 was determined on the basis that M.,is the maximum moment in the                                                                                                                     -
span for the beams tested. Method B is applied directly to deflection calculations with the use of the following effective moment of inertia                                                                                                              '
in the same way that the cracked transformed section Ils used in Method A:                                                                                                                                '
T,ir = r d ( 1 - 6' ," )                                                            (30)
                                                                                                                                                                                                                                       ~
Comparisons with test data indicated that Method B provides some-                                                                                                                 l what better results than Method A.                                                                                                                                                  .
In 1961 the European Concrete Committee (CEB)ss recommended that                                                                                                                  l deflections be computed in two parts as shown in Fig. 4:                                                                                                                            ,
                                                                    ,             Per J
                                                              $                               N: = moment corresponding to the part of the load that produces first cracking (see Eq. 07)1 K: = Ecle .         .           .. .         ..                    . (21) where E,= 18.000V), kg per sq cm, which when converted to cylinder strength in psi locomes E, = 73,5006*, pel
                                                                              .h is obtained using Na and K Part H                                                                                                                   r Mu = M - Ms. where M is the moment corresponding to the totalload -
Kn = K.Aeds(1 - 2q) 0 - %g)                                                           (22)
                                                                           *Privene eemmunseetles to the eemmittee unpubushed.             .
f*~~~-*-              '.'    h ~              -~e r < 3*M ~ eso v-.,*a *-. m -e-me , , . , . =v ,s.,s y =- -                                     .
                                                                                                                                                      +e~ y --=..           wes. ,-.. wee = ~~w           k .u-s~..m.<4 - 7 .ps          *
                                                                                                                                                                                                                                             -s-==*..-~
y-   w   -.wu.-    --   +,y   -m              <, --      %     e.-y,,.y+r     em- y-r,       e  ,e-,-,---w-r--       ,,,g-ywmm-
                                                                                                                                              , . - -           ,w---,rgg-r                   ,,ye--r.-                    ,,s> -w -

     ,,                                            mate equiv:lents in ultim:12 strength station of (1-2q) and (1 a64).
respectively.
i
                 ,      .                          An is obtained using M and Ku                                                                   ,
K'n =0.75Ku A'n is obtained using Na and K'n Solution A = As + A'u ns An .                     .-      .
(23)
This approach provides for a consideration of loading stages but does not account for variations in flexural rigidity along the beam. It does con-sider both initial loading (before the beam has cracked) behavior and reloading (after the beam has cracked) behavior. No special reference is made to continuous beams in the CEB report.
The 1963 ACI Code,8 Section 909, specifies that where short time deflections are to be computed, the moment of inertia should be based on the gross section when pf, is equal to or less than 500 and on the cracked transformed section when pf, is greater. This is an attempt to guard against underestimating deflections (using gross section I) for shallow beams, among others, associated with pf, values above 500. For continuous spans the ACI Code states that the moment of inertia may be taken as the average of the values obtained for the positive and negative moment regions. Under ultimate strength design, Section 1507, the Code requires that deflections must be checked at the service load level by the provisions of Section 909, whenever p, (p- p'), or (p -ps) ex-ceeds 0.18f//f, or whenever the specified yield strength, f,, exceeds 40,000 psi.
In ultimate strength design by Whitney's method," a balanced steel percentage is given by Eq. (24):                                                                                                         -
p : = 0.46/,'/f,                 ..
(24)
Investigators" have felt that a deflection warning should be sour.ded when the ratio p for singly reinforced beams, (p-p') for doubly rein-forced beams, and (p,- p,) for T-beams exceeds 0.18f//f,. This ratio is close to the balanced steel ratio by elastic theory and less than one-
:,                                            half the balanced design ratio by ultimate strength theory (or a 2-plus
         *                                                                                                           ~
load factor). For singly reinforced beams this marginal steel percentage is p = 0.18f//f, and pf, = 0.18f/= 450 and 540 when f/ = 2500 and 3000 psi, respectively. Hence the ACI value of pf,= 500 was selected for these and higher strength concretes as a governing criterion.
In a 1963 study by Branson,88 an empirical expression was presented for the effective moment of inertia at any particular cross section of a
                                                                                                  *n
                                                                                     .a..r         *n                ..
                               .                                                                                     /
                                      .                                                           .               /
7
                                                                                               ,.        / No.' ** . aan
                                                                                                                           *%*un 3                       /              No Firauret Rigisty d                   Kg=seus en eene owns.
Fig. 4-European Concrete Ibg ;          /
fj No"# fe'"s Naes saatsal Na Committee recommended                                                        , g , 'E procedure for calculating               e
                                                                                        /                                                            '
short.fime deflection Jtden        i from Reference 20)                        Defiesns , a
                                                                                                                                                                                          ~
o
                                   .              h                                     #
D     8*      gN     '.T       * ="           ..              e 9
                                                                                                        . . . _ _            . _ _ _ _ _ . _           _ , . _ , _ _ _ . ~ _ _ . _ _ _ . _ . . .

be m ts a functi:n cf th2 bending m: mint, section properties, cnd con-creta str:ngth (in a f:rm that includes tha effect cf extint cf cr:cking):
I,tr = ([ Ie+{ 1- (h)* ] I,                              ..                   (25) where M,is determined from Eq. (1*l).The appropriate power of 4 was determined numerically from a sizable number of test results that in-cluded both rectangular beams (simple and continuous) and T-beams                                                                                  .
(simple).
An expression for the average effective moment of inertia over the                                                                           .
entire length of a simply supported, uniformly loaded, rectangular or T-beam was determined as:                                                                                                                            +
                                                                                                  ~
f,st = ( [" )* I. +
                                                                                    ,             ,  1-([")']Jn-   ,-                                (28)                                      1 Eq.' (25) and (26) apply only when M or M..,is greater than or equal to M.,; otherwise 1,n= J,. In a supplemental study (Reference 32) the                                                                   -
average of the positive and negative moment region values in Eq. (26) was recommended for continuous beams.                                                                                                             '
Eq. (25) and (26) are presented in a form that is thought to be con-venient for engineers. since these equations are bounded by the well-known limits of I, and I as a beam is loaded to a state corresponding                                                                             ,
to first cracking (M or M.,,,= M,) and to a severely cracked state (M                                                                             i or M .,>> M.), respectively. The uncracked transformed section I                                                                                  -
might be more accurately used instead of the gross section I in these equations, especially for heavily reinforced members.
304-Deflections under long. time loading                                                                                                       '
304.1 Effect of concrete creep Time-dependent deflections or reinforced concrete flexural members,                                                                           i resulting solely from effects of sustained load (creep deflections), are usually greater then, and often two to three times as great as, deflec-tions resulting from all other effects combined during the life of a structure subjected predominantly to sustained loads. Thus, creep deflee-tions are of primary interest and should always be considered in addi-                                                                             -
tion to those resulting from short-time loads and shrinkage.
in addition to the difficulty of computing the creep-time history of a particular concrete under constant, uniformly-distributed sustained stress, a reinforced concrete flexural member is subjected to a nonuni.
form stress distribution and often a variable load history. An accurate analysis of the effects of a variable stress history even for uniformly i
loaded specimens, requires creep-time curves and a knowledge of the loading history.The rate-of-creep method"' or the superposition method" can be used to calculate deflections when detailed creep and load data are available.
Usually such a detailed analysis is not feasilple. and a shorter. more
                                    .,        approximate method is used. One such method is the sustained modulus
                                              . method which refers to concrete under a constant sustained stress. In l                                             this case a reduced or effective modulus called the sustained modulus of i                                             elasticity is used for computing initial-plus-creep deflections (see Ref.
erences 15 and 33.for example):
r, -         ''***'"'       =        "'**'2*'         =
E' rwsw + rren,           (1 + Celt,.aw          1+C which is Eq. (8).                                                                               ._ ~
When the sustained modulus of elasticity is used in computing initial-plus-creep deflections for uncracked sections. the computed creep deflee-
                        ,                     tions are simply equal to the initial deflections multiplied by the creep coefficient. Many of the methods that have been proposed for calculating
* e--     *        .                 .
                                  ; t-                                                                                                                              ; , ,,
        ' * ** ~ ;        -                              . 2 .- ~ ~ - - ~ ~     ~~     -----s-~~--~~--                        -...-.n.                    -,.g              e.,,.;,. , ... .
_n     .       __- . - _ _ _ - _                   _       -- .                -

       **                                20. Keeton, '' Study of Creep in Cancrete" Phases 1 through 5, Tschzies! Re-                                 I
                    -                        NA R333-1 2, U.S. Nr 21 Civil Engineering Labor-tzry, Part Hu neme,
: 21. H:nsen, T. C., cnd MLttock, A. H., "Thi InRuence of Size end Shipe af Member on the Shrinkage and Creep of Concrete," ACI JOURNAL, Proceedings V.
63, No. 2, Feb.1966, pp. 267-290.
: 22. Maney, G. A., " Relation Between Deformations and DeRections in Rein-forced Concrete Beams," Proceedings ASTM, V. 14 1914 Part II.
: 23. Swain, G. F., Structure! Engineering, Strengtk of Materials, McGraw-Hill Book Co., New York,1924.
: 24. Myrlea, T. D., author-chairman, "DeRection of Reinforced Concrete Mem-                                   .
bers," Progress Report of ACI Committee 307, ACI JOURNAL, Proceedings V. 27, 1931,p.351.                                                                                                     g
: 25. Murashev, V. E., "Iheory of Appearance and O tion of Rigidity of Reinforced Concrete Members,"pening                 of Cracks,Computa-Stroireinaya Promishlenost (Moscow). No. II,1940.
: 26. "DeRection of Reinforced Concrete Members" Bulletin ST-70, Portland Ce-ment Association,1947.
: 27. Yu, Wei-Wen, aisd Winter, George " Instantaneous and Long-Time DeRec-tions of Reinforced Concrete Beams Under Working Loads," ACI JOURNAL, Pro-eeedings V. 57, No.1, July 1960, pp. 29-50, for discussion see pages 1105 through 1371.
: 28. Levi, Franco, " Work of the European Concrete Committee," ACI JOURNAL,                                   I Proceedings V. 57, No. 9, Mar.1961, pp.1041-1070, for discussion see pages 1811                                          '
through 1822.
I
: 29. Whitney, Charles S., "Plastle Theory of Reinforced Concrete Design" Transactions ASCE, V.107,1942, p. 251.
: 30. ACI-ASCE Committee 327 " Ultimate Strength Design," Proceedings ASCE, Paper No. 809, Oct.1955. The report without certain test data appear in ACI JOURNAL, Proceedings V. 52, No. 7, Jan.1956, pp. 505-524, for discussion see pages 1333-1354.                              .
: 31. Branson, Dan E. " Instantaneous and Time-Dependent DeRections of Simple and Continuous Reinforced Concrete Beams," Report No. 7, Alabama Highway Research Report, Bureau of Public Roads, Aug. 1963,(1965).                                                        *
: 32. Bewtra, S. K "A Stud of Different Methods for Predicting Short-Time and Long-Time Dehections o[ Reinforced Concrete Beams," MS thesis, Univer-sity of Iowa, Aug.1964.                                                                                           I
: 33. Pauw Adrian, and Meyers, Bernard, "Effect of Creep and Shrinkage on the Behavior of Reinforced Concrete Members," Symposium on Creep of Con-crete, SP-9, American Concrete Institute, Detroit,1965, pp.129-158.
: 34. Ferguson, Phil M., discussion of " Warping of Reinforced Concrete Due to Shrinkage," by A. L. Miller, ACI JOURNAL, Proceedings V. 54, No. 6, Dec.1958, pp.1393-1402.
: 35. Miller, Alfred L, " Warping of Reinforced Concrete Due to Shrinkage,"
ACI JoURNA1., Proceedings V. 54, No. II, May 1958, pp. 939-950, for discussion see
     ,                                 pp.1393-1402.
: 36. Washa, G. W., and Fluck, P. G., "Ihe Effect of Compressive Reinforcement                                  I on the Plastic Flow of Reinforced Concrete Beams," ACI JOURNAL, Proceedings
     .                                 V. 49, No. 8 Oct.1952, pp. 89-108, for discussion see pages 108-1 through 108-8.                                  ,
d                                         37. "CRSI Design Handbook," Concrete Reinforcing Steel Institute, Chicago, Revised 1961.
: 38. ACI Committee 317, Reinforced Concrete Desigs Handbook-Working Stress 3
Design, SP-3, 3rd Edition. American Concrete Institute, Detroit, 1965, 271 pp.
(,                                        39. Washa, G. W., and Fluck, P. G., " Plastic Flow (Creep) of Reinforced Con-
 ':                                    crete Continuous Beams," ACI JOURNAL, Proceedings V. 52, No. 5, Jan.1956, pp. 549-561, for discussion see pages 1367 through 1872.
* I l'6 :                               .
l l
1
                                                                                                                                                               )
i
: 7. 7 Mrf ,.H--gf .yg e n . -- . ' . r . . .., - ,_. -, . . .                     .
                                                                                                          .. ....,: . .,*-,. . ., ,.,;... . b. . . _ c f. . _ i
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