ML19319D729: Difference between revisions
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SUPPLEMENT 6 h ANALYSIS AND DESIGN OF TENDON ANCHORAGE ZONES l | SUPPLEMENT 6 h ANALYSIS AND DESIGN OF TENDON ANCHORAGE ZONES l | ||
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TABLE OF CONTENTS (CONT'D.) | TABLE OF CONTENTS (CONT'D.) | ||
.. ..;. Page, | .. ..;. Page, | ||
,-. , 4.7 . LOADING CASES AND COMBINATIONS - l'( - . | ,-. , 4.7 . LOADING CASES AND COMBINATIONS - l'( - . | ||
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{Pxi, Pxj, Pxk, Pyt, Pyj , Pyk} (2) and the nodal displacements vector {6} can be expressed as, (6) = {ut , uj , uz , vi, v3 , va) (3) which are abovn in Figure 10. | {Pxi, Pxj, Pxk, Pyt, Pyj , Pyk} (2) and the nodal displacements vector {6} can be expressed as, (6) = {ut , uj , uz , vi, v3 , va) (3) which are abovn in Figure 10. | ||
Formulation of Element Stiffness Matrix Corresponding to the above assu=ptions, the displacements of a point at (x,y) are written as linear functions of the coordinates: | Formulation of Element Stiffness Matrix Corresponding to the above assu=ptions, the displacements of a point at (x,y) are written as linear functions of the coordinates: | ||
u = ui + C ix + C27 v = v1 + C3x + C4y (4) | u = ui + C ix + C27 v = v1 + C3x + C4y (4) | ||
The stiffness matrix [k} can be obtained by the following steps: | The stiffness matrix [k} can be obtained by the following steps: | ||
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: 12. Three Mile Island Nuclear Station Unit 1, Metropolitan Edison Company and Jersey Central Power and Light Company FSAR, Docket No. 50 - 289, Appendix D. | : 12. Three Mile Island Nuclear Station Unit 1, Metropolitan Edison Company and Jersey Central Power and Light Company FSAR, Docket No. 50 - 289, Appendix D. | ||
: 13. Johns-Manville Research and Engineering Center Retort E 455-T-266 filed in Robert E==ett Ginna Final Facility Description and Safety Analysis Report, Section 5, Appendix 5B. | : 13. Johns-Manville Research and Engineering Center Retort E 455-T-266 filed in Robert E==ett Ginna Final Facility Description and Safety Analysis Report, Section 5, Appendix 5B. | ||
: 14. J. Kosaka, Experimental Research of Unsteady-State Heat Transfer Concerned | : 14. J. Kosaka, Experimental Research of Unsteady-State Heat Transfer Concerned | ||
* l vith Nuclear Energy Facilities (Preliminary ReDort), Division of Nuclear l Energy Facilities, Taisei Construction Ltd. ,1969 l | * l vith Nuclear Energy Facilities (Preliminary ReDort), Division of Nuclear l Energy Facilities, Taisei Construction Ltd. ,1969 l | ||
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1 | 1 TABLE 2 BTETRESS ANCHOR ZONE SPALLING M=P {c - e - (c/h) (2h - 3e - c + * ]} | ||
TABLE 2 BTETRESS ANCHOR ZONE SPALLING M=P {c - e - (c/h) (2h - 3e - c + * ]} | |||
c = varies e = 16.5 in, h = 70 in. | c = varies e = 16.5 in, h = 70 in. | ||
M = P {c - 16.5 - (c/70) [(2 x 70 - 3 x 16.5) - c ( 1 - 2 x i6.5))) g C (in~) (c - 16.5 (c/70)2 _3( 1_ 2 x 16.5 ) M/P Mm (in. - k) 70 | M = P {c - 16.5 - (c/70) [(2 x 70 - 3 x 16.5) - c ( 1 - 2 x i6.5))) g C (in~) (c - 16.5 (c/70)2 _3( 1_ 2 x 16.5 ) M/P Mm (in. - k) 70 | ||
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* rece sc/- | * rece sc/- | ||
7 ._ | 7 ._ | ||
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. O *isa.a taveug. 4 7i | . O *isa.a taveug. 4 7i | ||
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"Jind " jai =tf CiHt H " | "Jind " jai =tf CiHt H " | ||
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Revision as of 22:37, 18 February 2020
| ML19319D729 | |
| Person / Time | |
|---|---|
| Site: | Crystal River, 05000303  |
| Issue date: | 08/10/1967 |
| From: | FLORIDA POWER CORP. |
| To: | |
| References | |
| NUDOCS 8003240765 | |
| Download: ML19319D729 (57) | |
Text
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( l l l 1 SUPPLEMENT 6 h ANALYSIS AND DESIGN OF TENDON ANCHORAGE ZONES l t i 1 ( i a 4 !. i I i I I i r l
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TABLE OF CONTDITS
'%d N
1 DESIGN DETAILS 1 1.1 TENDON ANCHORAGE HARDWARE 1 - l 1.2 ANCHORAGE ZONE REINFORCEENT 1 2 TENDON LOADS AND ALLOWABLE STRESSES 1 l 2.1 TENDON LOADS _ 1 2.2 ALLOWABLE STRESSES 1 ) 3 ANCHOR BLOCK STRESSES 3 3.1 BEARING STPISSES 3 3.2 EQUIVALENT FLUID PRESSURE METHOD 4 i 3.3 TRANSVERSE TENSION-PRISM ICfHOD 5 3.3.1 REINMRCEENT FOR TPRISVERSE FORCES 6 p v 3.h U'IBALANCED FORCES 7 3.h.1 VERTICAL UNBALANCED FORCES T 3.4.2 HORIZONTAL UNBALANCED FORCES 8 3.4.3 EQUILIBRIUM CONDITION IN THE ANCHOR BLOCK 8 4 CHECK ANALYSIS BY FINITE-ELEENT METHOD 9 h.1 INTRODUCTION 9 4.2 PLANE STRESS FINITE-ELEMENT COMPUTER PROGRAM 10 h.3 . GEOMETRY OF BUTTRESSES WITH TENDONS 12 h.h THERMAL LOADS AND TEMPERATURE PROFILES 12 4.5 Noh THERMAL LOADS 15 h.6 MATERIAL PROPERTIES 16 (~3
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t --,_ 3 r T - i -- 4 5 TABLE OF CONTENTS (CONT'D.)
.. ..;. Page,
,-. , 4.7 . LOADING CASES AND COMBINATIONS - l'( - .
\
! 4.8 STRESS ANALYSIS - 19 - ! 4.9 ANALYSIS OF RESULTS 19 i ,
' 4.10 -CONCLUSIONS 20
[ -5 REFERENCES i 21 l { 4 9 l 1 4 I 1 1 f. 1 4' i i !' i e i 1 A J- ( ,-- . \ i i ') l l ) ! l i .; a ? i , !- i-i' s as v-w e=e,n-mv-.,ww w w.,w-= w r e===-, --.w.m e-+,w -- .--+----w m- w,v .w+.r-- ww- ,, e # m we esww--e w-ww w n w ,rw w-we e =rr+_ ,, ,-v- c s ,% w w w gy- -pr-i
LIST OF TA3LES O V Table Number Title Page Table 1 Buttress Anchor Zone Transverse Stresses in Radial and Vertical Planes 22 Table 2 Buttress Anchor Zone Spalling 23 I _ 1
-lii-
t)- -LIST OF FIGURES V Figure No. Title Figure 1 Mat Anchorage Zone Reinforcement Details Figure 2 Buttress Anchorage Zone Reinforcement Details Figure 3' Typical Hoop Tendon Arrangement Figure h Buttress-Hoop Tendon Bearing Stresses Figure 5 Buttress-Radial Bursting Stresses Figure 6 Buttress-Vertical Bursting Stresses Figure T Hoop Tendon Typical Anchor Force Distribution Figure 8 Hoop Prestress Idealized Vertical Stress Flow in Buttress Figure _9 Idealized Model to. Spalling Figure 10 A Triangular Plane Stress Finite-Element Figure 11 Reactor Building With Buttresses F1-ure 12 Adiabatic Sections [')N
\_
ture 13 Model of Gridwork for Thermal Analysis Figure 14 Temperature Profiles Through Wall and Buttress for Various Thermal Load Conditions Figure 15 Iso-Thermal Curves for Winter Normal Operating Temperature Condition Figure L6 Iso-rhermal Curves for Winter Accident Temperature Condition at Time = 10,000 Sees. Figure 17 The Hoop Tendon Loads, Internal Pressure and the Equivalent Liner Pressure Figure 18 Finite-Elem:nt Grid Work for Buttress Analysis (hY # l l
/"'T 4
\/ l
-iv-1
,i N~
LIST OF FIGURES (CONT'D) N)
-Figure No. Title Figure 19 Minimum Principal Stresses for Maximum Prestress Forces and Winter Normal Operating Condition (Case A) (PSI)
Figure 20 Maximum Principal Stresses for Maximum Prestress Forces and Winter Normal Operating Condition (Case A) (PSI) Figure 21 Minimun Principal Stresses for Maximum Prestress Forces and Winter Accident Conaition (Case B) (PSI) Figure 22 Maximum Principal Stresses for Maximum Prestress Forces and Winter Accident Condition (Case B) (PSI)
^
Figure 23 Minimum Principal Stresses for Minimum Prestress Forces and Winter Normal Operating Condition (Case C) (PSI) Figure 2h Maximum Principal Stresses for Minimum Prestress Forces and Winter Normal Operating Condition (Case C) (PSI) Figure 25 FEnimum Principal Stresses for Minimum Prestress Forces and Winter Accident Condition (Case D) (PSI) Figure 26 Maximum Principal Stresses for Minimum Prestress Forces [ ) and Winter Accident Condition (Case D) (PSI) i l I
,)L .
_v-l
1 . DESIGN DETAILS v 1.1 TENDON ANCHORAGE HARDWARE The prestressing system employs the BBRV anchorage system utilizing 163-Tmm
. diameter wires conforming to the applicable portions of ASTM A421-65, Type BA.
The dimensions and capacities of the tendons used hereafter for the anchorage zone analysis are more fully described in PSAR Appendix 53, Figures 5B-9 thru 5B-ll. 1.2 ANCHORAGE ZONE REINFORC1 E The mild steel reinforcement in'tne tendon anchorage zones of the buttress and the base mat are shown on Figures 1 and 2. This reinforcement was conservatively designed by the methods enumerated hereafter in Section 3 of this report and is judged to be sufficient to accommodate all predicted stress states without jeopardizing the ability to satisfactorily place the concrete. 2 TENDON LOADS AND ALLOWABLE STRESSES 2.1 TENDON LOADS The minimum guaranteed ultimate tensile strength of the tendons is approximately 2,330 kips. Before the tendons are anchored (locked off) at 70 percent of the fs ultimate tendon force, which is equal to 1633.5 kips, they may be temporarily () overstressed to 1866.8 kips in order to reduce friction losses. The anchor zones are designed for anchor forces of 1633.5 kips per tendon. The anchor zones are also checked to resist safely temporary anchor force of 1866.8 kips per tendon. 2.2 ALLOWABLE STRESSES The allowable stresses for concrete and steel vill conform to those given in the PSAR Appendix 5B, Sections 6.3 and 6.4. Due to the specialized nature of anchor zone design the following conditions and stress criteria vill be used in the design. These stress criteria include adequate conservatism in their application.
- First, verify that the allowable concrete compression and tensile stressee are not exceeded due to bearing and bursting stresses respectively.
Second, check the tensile stresses in reinforcing steel when no credit is taken for concrete tensile capacity in the investigated anchor zones. Under these conditions, the allowable stresses will be:
.s c \
b ,s ,-
CONCRETE
- 1. Allowable bearing stress lh f
cp = 0.6f c 3/ A b'/Ab according to ACI/318-63,1 Paragraph 2605(C) f,; = 5.0 ks1.
- 2. Allowable compression stress in the structure beyond the anchorage zone f = 0.45f' c e f' = 5.0 ksi.
c
- 3. Allovable Tension Stress 3/ f' c
f' = 5.0 ksi, c REINFORCING STEEL Allowable tension for A615-Grade 40,sf = 20 ksi. . d O. 4
- tq 3 ANCHOR BIDCK STRESSES
. v' The investigation of the buttress anchorage zones for the hoop tendons is described in the following sections. The analytical work as presented, is based on the end anchor zone stresses and stress flow pattern studies documented by Y. Guyon2 and F. Leonhardt3 Their work has been substantiated through extensive strain measurements and stress flow pattern established by studies utilizing photoelastic techniques.
The design presented herein is quite conservative; nevertheless, it has been adopted as the basis for determination of required reinforcing steel in the end anchor zones to overcome tensile splitting stresses and to provide for spalling stresses near the concrete surface. 3.1 BEARING STRESSES Figures 3 and 4 show the typical hoop tendon arrangement and buttress anchor zone. Bearing area: = 242 _ 112 ( ) = 481 in. Concrete anchor surface: Af=332 _ 112 ( ) = 994 in.2 Allowable bearing stress: f cp = 0.6(5.0)( )l/3 = 3.82 ksi Use of 28-day concrete strength is conservative since initial tensioning vill occur significantly later in time. O) Bearing pressure for anchor force:
- a. P = 1633.5 kips, f b =1 '5 = 3.40 kai <f = 3.82 ksi
- b. P = 1866.8 kips, fb=
= 3.8S ksi The depth below the bearing plate, at which concrete stresses are identical to those allowed beyond the anchorage zone area, is computed on the con-servative assumption that the load e.preads at a 300 angle from the plane normal to the bearing plate surface.
Obtain the required concrete area to satisfy the above condition for: 7 P = 1633.5 kips, A g = 1633.5 = 727 in.2 2.25 If 0it is conservatively assumed that the anchor force spreads at an angle of 30 , the depth (h) below the bearing plate at which the concrete stress vill
)
r, (.) Y u..=
have been reduced to the allovable compression stress as shown above, can now be computed. See Figure hb. A g (24 + 2htan 30)2 ,(112(g)) 727 = (2h + b)2 - 95, h = k.0 in. 6 Special provisions must be made to confine the concrete within the 4.0 in. below the bearing plate. From then on, spiral reinforcement meeting the requirement for tensile splitting stresses vill be investigated as described in part 3.3. 3.2 EQUIVALENT FLUID PRESSURE METHOD Directly below the bearing plate the necessary reinforcement for confining the concrete is computed for an assumed equivalent fluid pressure "p". The equivalent fluid pressure is equal to the concrete :;cmpression stress multiplied by the Poissons Ratio, which is assumed to be 0.2. The concrete compression stress 2 in, below the bearing plate, which is about at the center of the critical h.0 in. depth, vill be used to determine the equivalent fluid pressure. The initial 1633.5 kips anchor force is considered. See Figure 4 Ag = [2h + h(tan 30)]2 _ [112 ( )] = 596 in.2 fc
- 96 5 = 2.74 ksi The equivalent fluid pressure vill be:
p = 2.7h (0.2) = 0.5h8 kai The reinforcement is arranged as shown on Figure 2. fne #6 spiral is 36 in, long with a 24 in. diameter and 3 in, pitch. Horizontal ties 2-#11 and vertical reinforcement 1-#11 plus 1-#9 are also available to resist this equivalent fluid pressure. The required steel area for the 24 in, diameter spiral vill be computed for an allovable steel stress of fs = 20 ksi. Required steel area: A 8
= 0.Sh8(h.0)(2h)=
20 2.63 in.2 (steel area req'c.. over length of 4 in. from backface of bearing plate) Q CD h-
Steel provided: b a '. Radially, Spiral 2-#6 = 'O.88'in. Ties 2-#11 = 3.12 4.00 in.2 >2.63 in.2
- b. Vertically, Spiral 2-#6 = 0.88 Vert. 1-#11 = 1.56 Vert. 1-#9 = 1.00 3.44 in.2 >2.63 in.2 3.3 TRANSVERSE TENSION-PRISM METHOD Transverse stresses are produced when the anchor force spreads out into the surrounding concrete. Directly behind the bearing plate the stress lines are parallel to the tendon axis. The angle change of the strese-flow-lines, where they bend outwards, produces in these regions compression in the concrete, which JS of no concern. Further away, the stress-flow-lines tend to bend back and teasile transverse stresses are created as a result of this angle change.
Figures 5 and 6 show the idealized compression stress-flow-lines in plane and the approximate location of transverse stresses. Further, it can be seen that the angle changes vill be small, and that the stress intensity along the compression stress-flow-lines reduces as they spread out into the concrete mass. From these considerations it can be concluded that the resulting tensile transverse stresses will be small and within the tensile capacity of the concrete. Deformed vertical, circumferential and radial reinforcement is provided throughout the buttress in addition to Q'v the spiral reinforcement at each bearing plate. The reinforcement will tie the concrete mass together and insures that thermal cracking of the concrete mass, if it should occur, is of no significance as far as structural integrity of the Reactor Building is concerned. Figure 6 shows that the critical condition, with respect to transverse tensile stresses in the vertical plane, exists when only individual tendons are stressed. When all tendons are stressed, the stress-flow-lines are more or less parallel. Figures 5 and 6 show, superimposed over the anticipated stress-flow-lines,
" equivalent beams" for which the transverse stresses can be computed. Com-putation for transverse stresses in radial planet, are performed in tabulated form in Table 1. Three cases are analyzed.
Case 1 Finds the maximum transverse radial stresses in symmetrical equivalent beam. Variation la investigates anticipated conditions during normal operation. Varfstions lb and le are made for extremely shallow and extremely iG
,J n.
u
deep " equivalent beams" to establish hov variations of beam depths affect the magnitude and location of the transverse stresses. Case 2 Investigates the effect of eccentric loads on equivalent beams. Guyon's
" Symmetrical Prism Method" is used2 According to this method the depth of the eccentric beam does not affect the magnitude of radial transverse stresses; only the dimension of the pris= drawn sy= metrical to the tendon center line determines their magnitude. Possible variations in eccentric beam depths are therefore of no concern.
All stresses were computed on the conservative assumption that the load spreads at 30 in the circumferential direction. The stress co=putations confirm the previously made prediction that transverse stresses are within the acceptable concrete stress range. Case 3 Finds the maximum transverse vertical stresses in sy= metrical equivalent beam. Variations 3a and 3b investigate anticipated conditions during normal operation. Vertical transverse forces counteract each other under operating conditions largely through vertical compression forces in the buttress concrete. See Figure 6b. Variation 3e investiget es anticipated conditions if only one anchor force acts. See Figure 6a. 3.3.1 REINFORCEMENT FOR ' ANSVERSE FORCES The allovable concrete tensile stresses in the radial and verti d planes are 9 not exceeded and no special transverse reinforcement is required for design loads. However, reinforcement is available in the form of buttress reinforce-ment in the hoop and meridional direction consisting of radial tie-reinforce-ment anchored into the shell and spiral reinforcement in the maximum bursting zone. The buttress is ruaforced sufficiently to resist all transverse tensile forces Zy (design) acting within the limits of the buttress without relying on concrete tensile resistance. Spiral: 3/h" $. 31/2" pitch, d = 24", length = 36" Maximum tensile elitting stress, Table 1, f b-y
= 0.133 ksi Tension in spiral:
t) D T=3.5(0.133)(f)=5.60,f K = = 12.7 ksi The spiral is designed for Zy (design) = 490 kips T = 15.h5 ksi design "2 30) " *
- f s(design) =
w
3.g UNBALANCED FORCES ,f-~s , ~
\' Figure 7 shows the tendon anchor forces for six buttresses'in the Reactor Building.
The solid line horizontal force vector in Figure 7 represents the tangential components of the anchor forces. The anchor forces are applied on opposite faces of the buttresses and counteract each other. However, since the vertical spacing of the anchors varies, it is not possible to balance all the forces. The unbalanced horizontal forces must be transferred from the buttress into the Reactor Building wall. The two dashed force vectors, Figure 7, adjacent and parallel to each anchor force vector are obtained by proportioning the anchor force according to its position with respect to two opposing anchors, similar to determining the reactions of a simply supported beam with an unsymmetrical concentrated load. The dashed horizontal force vector on the buttress center line indicates the unbalanced force between two opposing anchor for:es. How these horizontal forces are resisted is discussed later. Since the horizontal anchor forces on opposing sides are offset from each other, the forces must change direction in order to balance each other. Transverse . forces in the vertical direction are a result of this angle change. The vertical dashed force vectors shown in Figure 7 represent the unbalanced vertical forces behind each anchor zone.
,_ Vertical forces in Figure T vere computed for the simplified assumption that the anchor forces travel in straight lines from anchor plate to anchor plate t' ~';
after first moving 18 in. horizontally into the concrete and gradually bend over into the inclined position. See Figure 8. The check analysis employing the Finite-Element kethod did reveal a limited region of surface tensile stresses which presumably would produce cracking. However after concrete cracking has occared with the extent limited by sup-plemental reinforcing, a new state of equilibrium will exist. The following section discussing vertical and horizontal unbalanced forces verifies that a mechanism does exist to maintain equilibrium. 3.4.1 VERTICAL UNBALANCED FORCES During initial prestressing and during tendon inspection the balance of opposing tendon anchors might not exist and the tota' vertical transverse stress must be resisted. The maximum concrete transver e tensile stress for this condition was computed in Table 1, Case 3c, fdy = 103 psi. Figure 6a shows the anticipated typical vertical stress-flow-lines for such conditions. The d v I figure also shows the " equivalent beam" for which transverse stresses were determined according to Guyon's "Sy= metrical Prism Method.2 Sufficient reinforcing steel in form of a spiral has been provided for Case 3e without relying on concrete tensile resistance. In addition in the buttress, consisting of 6-#11, A = 9.36 in.2,,the canvertical resist areinforcement transverse splitting force of Z = 9.36 (0.95) (40)s= 356 kips, which is slightly more than the expected force Zy = 338 kips shown in Table 1. The reinforcement is placed mainly on the buttress surfaces in order to provide access for concrete placement. There is no need to distribute this reinforcement throughout the cross-section of the buttress or to place it where the vertical transverse forces actually act. Wherever the reinforcement is placed, it will not pre-vent the concrete from cra: king and will only tie the concrete together pre-venting the development of uncontrolled cracks. Consequently, surface re-inforcement is sufficient. It restrains the buttress in the vertical direction, and no failure mechanism caused by vertical transverse stresses can develop. 3.h.2 HORIZONTAL UNBALANCED FORCES Unbalanced horizontal forces shown in Figure 7 ut he transferred into the vall. I;uring initial prestressing and tendon inspection the counteracting effect of opposing anchor forces might be lost at any anchor zone position. All anchor zones will therefore be reinforced as the analysis requires for such conditions. Figure 5a shows the anticipated typical horizontal stress-flow-lines for such conditions. The figure also shows the " equivalent beams" for which transverse stresses were determined according to Guyon's "Sy= metrical Prism Method."2 The analysis for horizontal transverse forces is tabulated in Table 1. The critical loading condition exists when a tendon is stressed without being counterbalanced by opposing anchor forces. The maximum concrete transverse tensile stress for this condition is represented by Case 2a, fby = 133 psi. Sufficient reinforcing steel in form of a spiral has been proviced for Case 2a without relying on concrete tensile resistance. In addition, the radial tie > reinforcement in the buttress, consisting of 8-!11, A = 12.5 in.2, can resist , a transverse splitting force of Z = 12.5 (0.95) (ho) =s L75 kips which is J larger than the expected force Zy = 338 kips shown in Table 1. The tie C reinforcement restrains the buttress in radial direction, and no failure mechanism caused by radial transverse stresses can develop. 3.h.3 EQUILIBRIUM CONDITION IN THE ANCHOR BLOCK The reinforcement is checked based on the assumption of an existing spalling crack in the vertical direction. The forces and the extent of cracking
(3
\ s/ in the anchor block were based on a method developed by Gergely, Sozen, and Siess of arresting cracks due to transverse stresses.h The idealized =athe-matical model used for this approach, shown in Figure 9, is quite conservative.
The computation is shown in Table 2. It should be noted that zero moment exists at B' when c = 31 in. , and that the governing moment, Mn, prevails at c = hh in. , that is, far beyond the face of the vall. Concerning the visco-elastic behavior of concrete, it can be stated that the resulting redistribution of stresses developed by Gergely vill not signifi-cantly affect reinforcing requirements.5 The super-position of bursting stresses and nor=al stresses from vertical tendons results in a tri-axial state of stress which is less critical than during construction stages. No additional investigations for the final state of stress in the anchorage zones, therefore, appear necessary. The various sources of errors in a numerical tri-axial analysis (uncertainties of material properties, element definition, etc.) may lead to less accurate values than were obtained by two-dimensional methof.s. 4 CHECK ANALYSIS BY FINITE-ELEMENT METHOD
4.1 INTRODUCTION
The finite-element method has been e= ployed to check the design of tendon anchorage zones in the Reactor Building. The purpose of this section is to present the method of analysis, load conditions, and results for the buttress () and adjacent vall, and in conclusion to correlate these results with the design of the buttress reinforcement. The dominant loads are:
- a. Prestress forces
- b. Thermal loads
- c. Pressure forces due to loss of coolant accident (LOCA)
Although the stress analysis of the buttress and adjacent vall is a three dimensional problem, it can be reduced to a two dimensional problem, as the meridional usually constant. Thestress of the cylindrical finite-element methodportion of planeofstress the vesse3 isO*, I analysis which can account for complex loadings, arbitrary geometric configurations, and arbitrary boundary conditions, is an ideal tool for dealing with such a problem. Therefore, a plane stress finite-element computer programI was used in the buttress analysis.
.O ,-s, D
For the plane stress finite-element program the thermal loads were obtained from another computer program. The results fro = this program, for the steady state vinter normal operation and the transient state during vinter start-up, vinter shut-down, and vinter loss of coolant accident were used as input data in the finite-element program. As descriced later, a system of 768 direct stiffness simultaneous equations required solution. To save computer storage, the Gauss-Seidel iterative methodo, which requires the storage of only the non-zero elements of the stiffness matrix, was employed. Ir. order to expedite convergence, an over-relaxation factor 9 was also used in Reference 7; this factor usually rangt3 from 1.0 to 2.0. Cracking of the concrete for the vinter LOCA load condition is predicted. Areas of high tensile stresses have been defined and reinforcement capable of resisting these vill be provided. 4.2 PLANE STRESS FINITE-ELEIENT COMPUTER PROGRAM Lue to the fact that the meridional stresses of the cylindrical portion of a containment vessel are essentially uniform and constant, the horizontal cross section of the cylinder may be considered as a section subjected to plane stresses only. The finite-element method is a versatile method of analysis and is capable of providing statee of stress in the buttress for complex loadings , arbitrary geometric configurations, and srbitrary boundary conditions. In the plane stress finite-element analysis, the horizontal cross sectional plane of the cylindrical portion of the vessel is replaced by a finite number of elemtnts interconnected at a finite number of nodal points. Con-tinuity between elements of the system is maintained by requiring that within each element, " lines initially straight remain straight in their displaced positions." This requirement is satisfied if the strains within each element are assumed to be constant. Therefore, the stresses within each element are also constant. Distributed loads are replaced by equivalent nodal forces which act at the three nodes of the element. Based on these assumptions, it is possible to derive the stiffness matrix of a typical ' element, which is an expression for the nodal forces resulting from unit .
;j nodal displacements:
a {p} = [k] {6} (1) where {p} 1 the vector of six nodal forces which corresponds to the vector {6) of six nodal displacements (Figure 10), and [k] is the stiffness = atrix of the element.
f"%
\~~/ The nodal force vector {p} can be expressed as:
(P) = {Pxi, Pxj, Pxk, Pyt, Pyj , Pyk} (2) and the nodal displacements vector {6} can be expressed as, (6) = {ut , uj , uz , vi, v3 , va) (3) which are abovn in Figure 10. Formulation of Element Stiffness Matrix Corresponding to the above assu=ptions, the displacements of a point at (x,y) are written as linear functions of the coordinates: u = ui + C ix + C27 v = v1 + C3x + C4y (4) The stiffness matrix [k} can be obtained by the following steps:
- a. Substitute the nodal coordinavss into Equation (h) to obtain the displacement functions in terms of the nodal displacements.
7g b. Substitute results of (a) into the linear strain-displacement (,,/ equations to obtain the strains in terms of nodal displacements.
- c. Substitute the results of (b) into the stress-strain equation to obtain the stress in terms of nodal displacements.
- d. Find the nodal resultant forces which are statically equivalent to the stresses found in (c) so that the relationships between the nodal forces and the nodal displacements can be written in a matrix form as shown in Equation (1).
The explicit formulations of the matrices which co= pose the element stiffbess matrix can be found in Reference 7. W LJ l l O ' T - T-
Solution of the Tctal Stiffness Equations 3 After the stiffness matrices are fomulated for each of the elements, they O are assembled to form a total system stiffness matrix. If the boundary conditions and the loading conditions are properly specified, the total stiffness simultaneous equations are readily solvable to give the nodal displacements. Then, the nodal displacements are used to generate the stresses in each element and at each node. The directions and magnitudes of the principal stresses can also be computed by using the Mohr's circle technique. Iterative Method and Optimum Over-Relaxation Factor The Gauss-Seidel iterative procedure 0 is applied to solve the stiffness simultaneous equations. The method requires the storage of only the non-zero elements of the stiffness matrix during the iterative process. Since the stiffness matrix [k] is positive definite, the iterative method will always converge 10 The convergence rate is, however, often rather slow. An over-relaxation factord can be applied which greatly increases the convergence rate. The determination of proper over-relaxation factor is difficult and is based mostly on experience and trial and error. 4.3 GEOMETRY OF BUTTRESSES WITH TENDONS The pertinent geometry of the Reactor Building as it affects the buttress analysis is shown in Figure 11. In Figure lla, a sectional view illustrates the vessel with six buttresses. Figure 11b, shows the arrangement of the h, hoop tendons on a stretch out of the cylindrical v?.11. Figure lle, shows the dimensions of one buttress and the arrangement of the hoop and vertical tendons. h.4 THERMAL LOADS AND TEMPERATURE PROFILES If the temperature profile in a buttress is known, the difference between the temperature of any point and the reference temperature can be determined. t
'l The temperature difference, with the modulus of elasticity and thermal i expansion coefficient of the concrete, provides the equivalent constant Q stress for each finite-element. The constant thermal stress of each element can be converted to equivalent nodal resultant forces which can be ccnsidered as external forces applied at the nodal points of a non-heated element.
Temperature Profiles The temperature profiles and iso-thermal curves were calculated by a computer program developed by Brunnelll for the solution of steady state and transient state problems in one, two or three dimensional Cartesian or cylindric<t1 systems. The governing differential equations were solved by the =ethod of finite differences, using standard relaxation techniques.
V The convergence criterion T3 (step i) - T3 (step i-1) l 2 (, top 1) l 4 0.00001 (5) whe.e T3 is the tenperature at node j , was applied at every node of the difference lattice. To ensure stability of the transient solutions, the time increment was selected according to the method presented in Reference 11. The buttress and wall temperatures were determined for the following conditions:
- a. Normal operation in winter
- b. Start-up in winter-
- c. Shut-devn in winter
- d. Loss of coolant accident during vinter The vinter season was chosen because it produces the greatest thermal gradients in the shell of the Reactor Building.
p The temperature profiles shown in Figure lh were used in the analysis. These v profiles were obtained from Reference 12, for a nuclear generating station located in Northern U. S. A. , which has a more severe vinter climate than Crystal' River Unit No. 3. Thus the use of these profiles for Crystal River is considered to be conservative. For the start-up condition, it was assumed that the inside air temperature of the Reactor Building rose from 60 F to 110 F in one hour, and for the shut-down the reverse was used. The inside film coefficient was taken as 1.490 Btu /hr/sq ft/F for cases a, b, and e above. This was a modified film coefficient, which took into account the steel liner and the still air between the liner and the concrete: 1 h3 = (6) x x L steel air h k k i steel air n r Qh 3 t(O Equation 6 neglects the circumferential conduction and heat capacity in the steel liner and air gap. However, due to the relatively small thickness of the liner and the air gap when compared with the thickness of the concrete vall, these effects may be neglected. The preliminary analyses verified this h assumption. The following values, obtained by theoretical analysis 13, were used in the thermal analysis: inside film coefficient = 2.0 Btu /hr/sq ft/F specific heat, Cp con = 0.210 Btu /lb/F thermal conductivity, k = 10.0 Btu in./hr/sq ft/F con
's ,g ,,1 = 300.0 Btu in./hr/sq ft/F k
air = 1.836 Stu in./hr/sq ft/F The density of the concrete, P eon was assumed to be 140 lbs/cu ft. The above carried out values agrg: closely with the results from experimental research in Japan C p con = 0.200 Btu /lb/F k con = 11.2d Btu in./hr/sq ft/F The censity of the concrete Peon, was assumed to be 146 lb/cu ft. Tne thickness of the steel and air gap was: Oj X air = 1/32 inch Xsteel = 3/8 inch Using Reference 15, the local film coefficient for air at 20 F and a free stream velocity of 20 mph flowing past a concrete cylinder was calculated . to be 3 h = 0.56 Btu /hr/ft/F Q As a safe approximation the outside film coefficient was chosen as: ho = 1.0 Btu /hr/sq ft/F g
h O' V' The buttress model was ' replaced by a' mesh of 61' nodes, with _ adiabatic boundaries at the center line of the buttress and at the radial mid-section of the wall between two neighboring buttresses (Figures 12 and 13). Due to symmetry,'.there is n6 heat transferred across these boundaries. For Case d the liner temperature during LOCA was used as the inner boundary temperature in conjunction with a film coefficient of 5.880 Btu /hr/sq ft/F, which simulated the conduction through the air gap: hm * (I) air The transient state temperature analyses were performed at various time l steps. The temperature profiles for Cases a, b, c, and d are shown in Figure 1k. Because the most severe temperature gradients occur for cases a and d, only these two cases were considered in the stress analyses. Iso-thermal curves for cases a and d are shown in Figures 15 and 16.
~ The design pressure inside the Reactor Building during LOCA is equal to 55 psig and the maximum pressure was initially calculated to occur at approx-mately 200 seconds after initiation of the accident. The maximum liner temperature of 280 F was initially calculated to occur at approximately 650 seconds after the initiation of the accident. However, the maximum temperature gradient, which causes the most severe thermal stresses, was calculated to occur at 10,000 seconds after initiation of the accident.
Consequently, the determination of the most severe stress state, due to thermal. and pressure loads, requires that a number of stress analyses
- be
- conducted at various elapsed times after commencement of the accident.
A conservative estimate of the stresses was obtained, by superimposing j directly the maximum values for accident temperature and pressure. Consequently although subsequent calculstions alter the time at which I minimum pressure and liner temperature occur, these changes do not affect this analysis.
; h.5 NON THERMIL LOADS l In addition to the thermal loada, there are other loads which act on the
- _ structure
j a. Hoop Tendon Anchor Pressure l The tendon- force was assumed to' be distributed uniformly up the
- face.of the buttress. The resulting average force (=pa) was applied _to each of six nodes. The location of the nodes approxi-mately corresponded to the position of the tendon.
[ u C 1- , Q
,Q ~
. . - - - . . , ~ _ . - . ,
,-- ...-m--
- b. Hoop Tendon Pressure in a Radial Direction s
Radial pressures, due to the hoop tendens in the vall, were cuiculated as average stresses in pounds /in. l The two radial pressures applied (refer to Figure 17):
- 1. From point A to the tangent point T. Over this length the tendon spacing causing radial pressure (i.e. continuous past the buttress) is 39 in., giving a pressure of pt, and
- 2. For the length from point T to point B (midway between buttresses) the average tendon spacing causing radial pressure is 19-1/2 in. ,
giving a pressure of 2p g.
- c. Internal Pressure Due to LOCA Internal pressure acting on the inside face of the Reactor Building vall was considered as a combination of two separate effects:
- 1. Pressure due to LOCA. The maximum pressure due to LOCA was taken at 82.5 psig. (1.5 P)
- 2. Pressure due to expansion of liner caused by the increase in temperature during the LOCA condition (110 F to 280 F) based upon a limiting yield strength of 1.2 f .
y The above two loads were assumed to act si=ultaneously during the LOCA condition, giving an equivalent internal pressure of pg . The magnitude and combination of the above non-thermal loads is noted in Section 4.7. 4.6 MATERIAL PROPERTIES Prestressing Steel Wire
'l The type of prestressing vire to be used in the construction is a low re- s laxation (stabilized) type. Dimensions, properties, etc. , of the wire ;]
as used in the analysis were: CD
- a. Size of tendon = 163 wires
- b. Diameter of wires = Tcm
- c. Minimum guaranteed ultimate tensile stress = 240,000 psi
- d. Tenden force @ 70 percent GUTS = 1633.5 kips g
-a
r d
- e. Modulus of elasticity = 29 x 106 psi
- f. Coefficient of friction = 0.16
- g. Coefficient of vobble friction = 0.00030 Concrete The assumed properties of the concrete used in the analysis were:
- a. Compressive strength @ 28 days = 5000 psi
- b. Modul.us of elasticity = 4 x 100 psi
- c. Poissons ratio = 0.20 Liner The liner is fabricated from 3/8 in, thick steel plate manufactured to ASTM A-283 Grade C. The minimum yield stress for this material is 30,000 psi. In determining the equivalent liner pressure due to LOCA,' the yield stress was assumed _to be 36,000 psi.
4.7 LOADING CASES AND COMBINATIONS O The buttress and vall were analyzed for four loading cases. The first two cases considered the buttress and vall being subjected to the maximum prestress forces, i.e. , prestress forces allowing only those losses that would occur between prestressing and start-up (beginning of plant life). The latter two cases considered the buttress c.nd vall being subjected to the minimum prestress ) forces, i.e. , prestress forces after 40 year losses (end of plant life). The load cases were:
- a. Winter normal operation with maximum prestress force,
- b. Winter LOCA vith maximum prestress force.
- c. Winter normal operation with minimum prestress force.
d.- Winter LOCA with minimum prestress force. These cases were considered to be more critical than the summer normal operation /LOCA conditions. The summer condition results in a smaller temperature differential in the vall which in turn results in maximum compressive and tensile -stresses less than the vinter condition. Hence, the summer condition was not considered in the analyses, n ', \ The numerical value of the applied loads was: O Load Case a b c d t LT 11 0 150 110 150 t gp 110 280 110 280 t, 20 20 20 20 pa 6.25 6.73 5.52 6.03 pg 46.07 49.75 29.25 42.2 2p 92.13 99.50 78.5 84.4 t g pt 0. 0 96.1 0.0 96.1 where t LT = Temperature (F) of liner for thermal gradient in vall. tpg^ = Temperature (F) of liner for equivalent pressure of liner due to expansion during LOCA. t, = External air temperature (F) pa = Prestress force at bearing plate (kips / inch) pt = Radial prestress force from point A to tangent point T (lbs/ inch) A-T .7 2p = Radial prestress force from tangent point T to point B-(lbs/ inch) r-T-B 3 p1 = Equivalent internal pressure (LOCA pressure + liner expansion C pressure lbs/ inch) Figure 17 shows the application of these loads on the buttress and wall. Loads pg , pt, and 2pg for Case a incorporate losses due to elastic shortening of the concrete, friction, creep, shrinkage and steel relaxation losses that occur between prestressing and start-up. Case b prestress forces reflect increases from Case a, due to elastic elongation of the structure during LOCA. Tnus, Cases a and b reflect the maximum prestress force that vill exist in the structure, at the earliest time in the life of the plant, that a LOCA could occur. 0 l 1 U
V Loads pa, P t , and 2pt f r Cases e and d are similar to Cases a and b respectivein except that the losses due to creep, shrinkage and steel relaxation are those that are assu=ed to occur over 40 years to end of plant life. Case d prestress forces reflect an increase from Case c due to elastic elongation of the structure during LOCA. 4.6 STRESS ANALYSIS Due to the sy= metrical nature of the structure only one-twelfth of the cylinder, as shown in Figure 12 need be analy:rd. Preliminary analyses showed that beyond a certain point in the vall, the stress state, for all loading cases, was uniform. To conserve computer storage and give a more accurate solution, it was decided to reduce the size of the model to that shown in Figure 18. Nodal displacements of both boundaries were specified. The displacements were obtained from a computer program as developed by Kalnins10 using a shell of revolution theory. There were a total of 672 elements and 384 nodal points, so that 768 simultar eous equations required solution. For the computations, the Gauss-Seidel iterative solution procedure was used. 4.9 ANALYSIS OF RESULTS Figures 19 - 26 show isostress curves for the maximum and minimum principal stresses for the four loading conditions outlined in Section 4.7. It should Cs be noted that these curves represent the algebraic maximum and minimum values V of the principal stresses. They do not show the orientation of the stress traj ectories. Inspect'on of the iso-stress curves shows that, as expected, high compressive stresses exist directly beneath the bearing plate for all load conditions. Under vinter normal conditions, a region of tensile bursting stress occurs in the buttress at some distance removed from the bearing plate; during LOCA, however, these bursting stresses merge with those resulting from the pressure and temperatt.e loads. The variation of temperature through both the vall and buttress leads to sub-stantial hoop stress gradients. It should be noted that in the curves for LOCA, the direction of the stresses in the vall changes in the vicinity of the centerline. Of primary interest, however, is the high localized tensile stress that occurs at the buttress-wall re-entrant corner during LOCA, which results from both the thermal gradient as well as the local bending arising from the variation of resistance of the buttress and wall to internal pressure. nD h u 4.10 CONCLUSIONS Tne results indicate that for the most severe loading conditions, the maximum stresses in the buttress and adjacent wall are within safe limits, except for the localized area at the buttress vall corner where tensile stresses - greater than 6 / f'c occur. Reinforcement to stabilize cracking in this local region will be provided. All other results vould indicate that the design method employed was conservative. O s ( J
~)
C I -
-2 0-g
y,) 5 REFERENCES
- 1. Building Code Requirements for Reinforced Concrete, ACI 318-63, American Concrete Institute, Detroit, Michigan,1963.
- 2. Y. Guyon, Prestressed Concrete, Contractors Pecord Ltd. London and John Wiley and Sons Inc., New York, 1960.
- 3. F. Leonhardt, Prestressed Concrete Design and Construction, Wilhelm Ernst and Sohn, Berlin-Munich,19o4 4 ?. Gergely, M. A. Sozen and C. P. Siess, The Effect of Reinforcement on Anchorage Zone Cracks in Prestressed Concrete Members, Civil Engineering Studies, Structural Research Series No. 271, University of Illinois, Urbana, Illinois, July 1963.
5 Anchorage S9 stems in Prestressed Concrete Pressure Vessels; Anchorage Zone Problems, Oak Ridge National Laboratory, ORNL-TM-23Td, January,1969.
- 6. M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp, " Stiffness and Deflection Analyses of Complex Structures," J1. of Aero. Sci. , V. 23, No. 9, Sept. 1965, p. 805
! 7. E. L. Wilson, Finite-Element, Analysis of Two Dimensional Structures, Ph. D. Thesis , Univ. , e ' ?,L at Berkeley, June 19o3, d 8. M. G. Salvadori and M. -. Baron, p erical Methods in Engineering, Prentice-Hall, Englewood Cliffs , N. J. ,19o1.
9 F. G. Lehan, " Simultaneous Equations Solved by Over-Relaxization," Proc. ASCE, 2nd Conf. - on Electronic Computation, Pittsburgh, Pa. , USA, Sept. 1960,
- 10. S. H. Crandall, Engineering Analysis, McGraw-Hill Book Co. , N. Y. , USA,1956.
- 11. R. D. Brunnel, Compuer Program Heating and Heating 2, Atomic International Division, Report No. Al-b4-MEM0-177, North American Aviation, Inc. , Detroit, Michigan, USA, 1964.
- 12. Three Mile Island Nuclear Station Unit 1, Metropolitan Edison Company and Jersey Central Power and Light Company FSAR, Docket No. 50 - 289, Appendix D.
- 13. Johns-Manville Research and Engineering Center Retort E 455-T-266 filed in Robert E==ett Ginna Final Facility Description and Safety Analysis Report, Section 5, Appendix 5B.
- 14. J. Kosaka, Experimental Research of Unsteady-State Heat Transfer Concerned
- l vith Nuclear Energy Facilities (Preliminary ReDort), Division of Nuclear l Energy Facilities, Taisei Construction Ltd. ,1969 l
I
- 15. F. Kreith, ginciples of Heat Transfer, 2nd Ed. International, Scranton, Pa. ,- 1965.
p L/ 16. A. Kalnins, Computer Programs, Crystal _ River Unit No. 3, Florida Power ny Corporation, PSAR, Pages 5C-1 and SC-2. O
TABLE 1 Bl,TfRESS ANCHOR ZONE TRANSVERSE STRESSES IN RADIAL AND VERTICAL PLANES Case 2a' 2a 2a'/2a fy z Eb' 2b=2b'+ A=2a2b Zy=0.3P (..) (in) P fbz" f Zy(design)" ** (in) (in) (2az)tg300 (in2) (kips) P/A fbz by=fy (1-a'/a) (in) 0.3P (ksi) (ksi) (kips) (kips) [1] [2] [3] [4] [5] [6] Radial Plane la 24.0 52 0.h6 ' O.2h 0.81 24.0 48.3 2510 1633.5 0.651 0.150 26h 24.0 26 490 21.1 lb 0 92 0.0h 0 90 24.0 37.5 975 1633.5 1.675 0.067 39 2h.0 0.31 0.28 490 11.7 Ic 78 0.73 24.0 56.9 4430 1633.5 0.369 0.103 338 h90 28.5 2a 24.0 33 0.73 0.11 0.90 24.0 41.1 1355 1633.5 1.205 0.133 132 490 14.8 Ultimate Load - Radial Plane 2a 0.73 0 90 24.0 l24.0 l33 0.11 41.1 1355 2333.5 1 720 0.189 789 700 14.8 Vertical Planes hi 3a 24.0 39 See case 2a 3b 24.0 26 See Case Ib 3c 24.0 78 See case Ic [1] Guyon, " Prestressed Concrete", Figure 92 , [2] Guyon, " Prestressed Concrete", Figure 92 , [3] 1866.8 Kips and 233.3.; kips anchor forces is not critical. [4] Changes in tensile splitting stresses with vary ng "2a" dimension. j [5] Zy = total + ransverse force Leonhardt "Prestrer sed Concrete" p. 2T1.3 [6] For design purpose "a'/a" is conservatively assumed to be equal to zero. 1.- N e ob3 e e
1 TABLE 2 BTETRESS ANCHOR ZONE SPALLING M=P {c - e - (c/h) (2h - 3e - c + * ]} c = varies e = 16.5 in, h = 70 in. M = P {c - 16.5 - (c/70) [(2 x 70 - 3 x 16.5) - c ( 1 - 2 x i6.5))) g C (in~) (c - 16.5 (c/70)2 _3( 1_ 2 x 16.5 ) M/P Mm (in. - k) 70
-5 -11.5 c.0051 - 2.6h -11 95 10 - 6.5 0.020h - 5.28 - 8.24 16.5 0 0.0555 - 8.70 k.53 20 +35 0.0817 -10.55 - 3.03 25 + 8.5 0.1275 -13.20 - 1.35 zero at 31 in.
30 +13.> 0.1835 -15.85 - 0.20 hh +27 5 0.395 -23.25 + 0 95 1550 50 +33 5 0.511 -26.h0 + 0.70 (in. - kips) 70 +53.5 1.0 -37.00 0 y F Z = 3 in, T " h-z h = 70 in, s allowable = 105 As = 1,56 sq in. (#11 bars) W = 0.005 f, 2 105 /0.005/1 56. = 5.66 ksi , F K T = 701550
= 23.l A, = FT/ # s = 23.1/5.66 = h.08 sq in. .
As providedi #11 @ 12 in. + 1 #11. Loop = 8.19 sq in. n
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