ML20054D079

Forwards Addl Info in Response to Draft SER Re Intake Tunnel Structure.Response Will Be Incorporated Into Future FSAR Amend

ML20054D079
Person / Time
Site:	Perry
Issue date:	04/19/1982
From:	Davidson D CLEVELAND ELECTRIC ILLUMINATING CO.
To:	Schwencer A Office of Nuclear Reactor Regulation
References
NUDOCS 8204220317
Download: ML20054D079 (23)

Text

. - -

TliE CLEi!EL AN D ELECTRIC ILL U MIN ATIN G CO M P AN Y l

P O. 00X 5000 e CLEVELAND. OHIO 44101 e TELEPHONE (216) 622-9800 e ILLUMINATING DLDG e 55 PUBLIC SoVARE '

Da:wyniL Davidson ewing e Best Location t'n the Nation VICE PRESwENT SYS TE M ( NGINE E RING AN;.) CONS 1 RUCTION W o>

April 19, 1982 e e N

g , $-

~

e qV I Fr. A. Schwencer Chief, Licensing Branch No. 2 g9 ' @ .

Division of Licensing y 0V fy ;j e3 U. S. Nuclear Regulatory Commission <\

Warnington, D. C. 20555 Cu g \'

Perry Nuclear Power Plant Docket Nos. 50 11440; 50 111 44 Respanse to Draft SER Structural Engineering Bramh

Dear Mr. Schwencer:

This letter and its attachment is subnitted to provide additional information regarding the intake tunnel structure.

It is our intention to incorporate this response in a sur, sequent amendment to our Final Safety Analysis Report.

Verv Truly Yours, 5

t W Dalwyn d. Davidson Vice President System Engineering and Construction DRD: mlb cc: Jay Silberg John Stefano Max Gildner I

Q 00 s

/ I o

$$84558Sk7SSSh8

l Perry Nuclear Power Plant Intake and Discharge Tunnels Responses to Additional Information Requested by the NRC - March 30, 1982

1. With respect to the design of the 10 inches concrete tunnel lining provide justification and basis for taking exception on the limitations  ;

as imposed on Section 1.4.3 of ACI 322-78 code.

There is no code that has been developed that reflects the design consideration of tunnel supports and linings. The fact that the tunnel i lining is not a simple flexural or compression member makes application of existing codes for reinforced (ACI 318-77) and structural plain ,

concrete (ACI 322-72) inadvisable. The tunnel is not a detenninant structure because of its complex interaction with the surrounding rock '

mass and therefore the ACI code, as developed for beams, columns and slabs for buildings, is not appropriate for describing tunnels. The general failing of most building codes to consider tunnels has prompted the United States Department of Transportation, Urban Mass Transit Administration to fund the development of a more specialized " tunnel code". This research has been carried out by the Department of Civil

, Engineering at the University of Illinois. The results are currently in draft form only, subject to review by some industry representatives and the American Society of Civil Engineers' Tunnel Lining Design Committee, and not available for reference at this time.  !

The design of the concrete lining for the Perry tunnels did use the existing ACI codes for guidance, but recognizing the deficiences in the code with respect to tunnel lining design, the codes were not the sole source of justification for the design. In particular, the behavior of an unreinforced concrete-lined tunnel under seismic loads is not considered by ACI 322-72. The limitation of paragraph 1.4.3 is thus not appropriate for tunnels.

1

A two-fold approach to the design of the tunnel liniag under seismic loads was used in the case of the Perry tunnels. First, seismic  ;

loads were applied via computer analyses to the proposed tunnel lining.

• The tensile stresses generated by the seismic loads were then compared to the tensile strength of the concrete, a function of the modulus of I rupture. The'second approach to the consideration of the behavior of a j concrete-lined tunnel under seismic loads was to study the history of other existing tunnels which have been subjected to seismic loads.  !

This history was well-documented in STRAAM Engineers, Inc. report,

" Design of Concrete Final Lining," November 1979, with the resulting >

conclusion that all tunnels, whether reinforced concrete-lined, or unreinforced concrete-lined, behave well under seismic loads with  ;

damage being very minimal. l

[

i.  :

2. With respect to using allowable concrete tensile stress for tunnel lining design, it is believed that under the range of stress computed '

in your response, cracking will probably develop in the plain concrete lining. Please provide an assessment this anticipated cracking on the integrity of tunnel lining and structure. Also provide a consequence analysis of the tunnel structure assuming conservatively the collapse -

of some segment of concrete lining during an SSE and demonstrate that the structural function and integrity of the tunnel is maintained.

Cracking of the concrete is in itself not necessarily detrimental to the integrity of the tunnel lining. A tunnel lining is a very redundant structure owing to the interaction between the lining and the surrounding rock mass. Cracking, to some degree, is beneficial to the distribution of stress in a tunnel liner. Cracks allow the liner to deform slightly, the crown-invert dimension shortens while the spring-line dimension lengthens. This deformation releases the moments and associated tensile stress developed in the lining and allows the lining to carry the loads in compression for which concrete is more suitable.

The mechanical interlock between the irregular rock surface and the the cast-in-place concrete prevents the accumulation of longitudinal strains and consequently large cracks from developing. Cracks which do occur are generally well distributed.

The unlikely scenario of a segment of concrete lining collapsing into the tunnel will in any case have little effect on the function of the tunnels. The tunnel is self-supporting in a good quality shale.

Thus, the loss of concrete lining in the crown serves only to open up the space to the rockline of the original machine-driven rock tunnel.

The cross-sectional area of the tunnel is not reduced, therefore the flow quantity will be maintained. The roughness of the tunnel is increased due to the segment pieces lying in the invert and due to the lack of coherent lining in the crown. The velocity of flow is technically reduced by the increased roughness but practically the velocity remains nearly unchanged.

3. a) Provide information regarding the design basis for use of rein-forced tunnel lining as well as unreinforced tunnel lining which were built in the United States.

b) Provide an *-luation based on available data on the performance of unreinfvg.d tunnel lining under earthquake and other pertinent loads.

_2_

. 1 l

l a) In general, there is no simple basis for deciding to use reinforce-ment in a tunnel liner. The use of reinforcement in a concrete tunnel liner is based on a variety of considerations, the most important being the ground conditions surrounding the tunnel, the size of the tunnel, the shape of the tunnel, and the depth of the tunnel. In good quality rock some tunnels are left unlined.

Reinforcement is often found at tunnel intersections with other tunnels or shafts as in the case of the Perry tunnels. The unusual shape of the opening in these areas causes the stresses to be distributed unequally. Tunnels may show reinforced concrete at the portals. This is usually a result of hillside movements, not a consideration of ground support.

Attached is a list of 46 tunnels and typical statistics which are routinely recorded by the Bureau of Reclamation. Of the 46 tennels on the list, only one shows any significant reinforcement in the lini ng.

b) The performance of tunnels under seismic loads is documented in STRAAM Engineers, Inc., " Design of Concrete Final Lining", November, 1979.

4. We reviewed your responses on the seismic analysis of tunnel structures, the following additional clarifications are needed for our final evalua-ti on.

a) Your basis for using 1957 Golden Gate earthquake, instead of R.G.1.60 response spectra anchored to SSE as input motion.

b) Your basis for selecting the shear modulus of water as 31000 psi.

c) Your basis for selecting only six mode in the seismic analysis.

What are the cumulative participation factors for the modes selected?

d) The basis for use of 5% damping for both OBE and SSE.

e) Why the difference in system stiffness and responses of the tunnel structure when different methods of analysis (i.e.,

Time Marching vs. Modal) are used.

a) The time-history analysis was used only to obtain information that would help construct a model for extracting modes for a modal analysis and to provide a check on the modal analysis. It was not used for the final design analysis. In an effort to l reduce the computer costs for the time-history analysis, the Golden Gate record was used since only about 3 seconds of the record was required to match the horizontal criteria spectrum of the safe Shutdown Earthquake between 4 and 20 Hz as required by Regulatory Cuide 1.60.

l l l I

The NRC requirement published in the 1975 Standard Review Plan is that the time-history must provide a spectrum that will match or exceed the criteria spectrum throughout the range of frequen-cies. The liner frequencies of concern are in the 8 to 17 Hz range and the range of 4 to 20 Hz was studied on this basis.

b) The properties of water were based on two assumptions, the bulk modulus of elasticity, K, for ordinary temperature and pressures is 300,000 psi (Streeter, V.L. and Wylie, E.B., Fluid Mechanics, 6th ed., McGraw-Hill Book Company, New York,1975, p.17), and Poisson's ratio, / , is 0.45. The shear modulus, G, falls out of the relationship between these quantities G= 3 (1-2/) K = 31,000 psi 2 (1+ /)

c) The Rayleigh-Ritz approach was used to extract the modal shapes and associated frequencies of the tunnel liner under seismic loads.

In using the Raleigh-Ritz method a set of prescribed force patterns is used as input data. The resulting displacements considered were independent sets and contained the lower mode shapes of the system.

Twenty-four force patterns were used on the Perry tunnel liner analysis. The computations were terminated when six lower fre-quencies and mode shapes were obtained. Frequencies and mode shapes were limited to the lower range (< 20 Hz) of the criteria spectrum.

The maximum frequency suggested by the Reg. Guide 1.60 criteria spectrum is 35 Hz. The stress contribution above 20 Hz frequency was neglected since the magnitude was small.

Of the six modes analyzed, the responses were combined when frequencies of the modes were within 10".. As directed by Reg.

Guide 1.92 the absolute sum of the closely spaced modes were combined and the square root of these sums taken.

d) Damping used in the seismic computer analysis is taken as a function of the mass and stiffness matrices in the form:

[c] =

C4 [M] + B [K]

where x and B are related to the critical damping ratio by

• $=

An = + B 2 Wn 2 i

l l

l l

Since the predominant frequencies of earthquake acceleration records normally occur between 1 and 8 cps, values of oc and B were selected for the first trial so that 2n varied from 4 to 8 percent and averaged 5 percent.

e) The time-marching solution was not used as a basis for design.

The time-marching solution provided information on the response of the tunnel lining to a specific earthquake record, the Golden Gate S80E component. This general information was used to develop the model for the modal analysis only. The details of the time-marching solution were not refined as was the modal solution and the criteria spectrum was not used as input. These basic inconsistencies and the fact that the programs themselves were developed on the basis of different assumptions would of course result in different responses of the tunnel structure. The magnitude of that difference was not considered inconsistent.

5. We need additional information and reference materials pertaining to your longitudinal tunnel analysis procedures. Also assess and discuss your analysis approach with respect to the applicable requirement of SRP Section 3.7.3.

The longitudinal analyses of the tunnel lining is discussed in " Stress Analysis of Perry Nuclear Power Plant,10-ft Diameter Cooling and Emergency Service Water Tunnel System, Vol. II." by Agbabian Associates, July 1975. The appropriate sections are reproduced herein.

l l

l l

1 I

i 1

It is again noted that the dead load D, the grout load G, and the ground load H' are not included in these cases for tne shaf t analyses. l 1

For this reason, Case 1 is not a real !cading condition. Also, it will be noted  ;

that the shaf t stresses are ecual to cr lower than the tunnel s tresses for

- all' cases.

5 STRESSES PARALLEL TO THE LON3;TUDINAL AXES OF TUNNELS AND SHAFTS lt is pointed out in Section 4 of Volume I that stresses produced by both the static and dynamic loads are of greatest significance in pianes normal to the longitudinal axes of the tunnels. For this reason, detailed analyses have been provided in Volume I of typical transverse sections of the tunnels and extrapolated to the shafts in the previous section. Plane strain finite element models, which restrain deformations parallel to the longitudinal axes (z-direction) of the tunnels were used for these analyses. The re fo re ,

except for one seismic load condition, the stress (c,) parallel to the longi-

.. tudinal axes is given directly in the computer printout of stresses for the grout load G, the ground load H', and the temperature stresses T g and T'. The c stresses derived from these analyses are shown in Figures 10

4. z through 16. The longitudinal stresses resulting from the dead load are small 9-and have been neglected. Also, for the external and internal fluid pressures, J .

I the longitudinal stresses of the tunnel are low and approximately uniform l circumferentially. Stresses of +20 psi tensien result from an internal water pressure of 50 psi and, -60 psi compressive stresses result from an external water pressure of the sar.e magni tude. Plots of stresses are not

!, shown for the fluid pressure loads.

Stresses developed parallel to the longitudinal axes of the tunnels from seismic loads can be divice6 :nto two categories for presentation. First, the plane strain models used for the transverse modal analysis (see Figs. 41 and 42 of Volume 1) indicate longitudinal stresses c, which are provided in Figures 17 and 18 for the OBE and DSE, respectively. These stresses are of

. . low intensity. 30 psi or less, and can be characterized as being stresses I

13 t

.- - _ ., .. .- , = - . . . . - . ,

1 E

STRESSCS AT CONCRETE 121 SURFACCS HAVE BEEtt EXTRAPOLATED Lit!EARLY FROM ELEMEllT STRESSES

-16  !

l ELEMENT NO  ;- ; -

~ 7. 5 l  ! 106, go" 103

-12

{

45 J

[ \ D' g\b i 8

~

p I IM .-3 -

i. 70

-3 s 3

r 45 88 g

+1 90 i

+2 o +1 37.5 o 75-0 l

. +1 1 73 PLOT OF CIRCUMFERENTIAL NORMAL STRESSES:

. RADIAL VARIATl0tl SHOWN ON LEFT OllE-HALF

- 0F SECTiott AVERAGE STRESS ACROSS SECTION SHOWN ON RIGHT ONE-tiALF OF SECT 10tl ALL STRESSES ARE lti PSI:

+ = tells 10N

! - = COMPRESS 10tl l

FIGURE 10. LONGITUDINAL STRESSES P,ESULTING IN PERMANENT LINING FROM GROUT LOAD G

- 14

E 121 STRESSES AT CONCRETE SURFACES HAVE BEEN EXTRAPOLATED Lit 1EARLY FROM ELEMEllT STRESSES ,

-49 I

ELEMENT NO 4. ; ly, , ,

a.

^

~ 7.5 O s.I '

u 45

-27 3 -23 ..

4 106, .. 1" 0

! -45 103

-36 .

b f go, g\ _9

-_ i i 85

.E-T' L i

-12 -

j 70 -12 g

[ -12 1 :1 t'

45 88 f 3 90 i o +3

' _j 37.5 o-~-

7.5 s +5 4 l \ '

-2 p

+1 3 73 PLOT OF CIRCUMFERE!1TIAL 110RMAL STRESSES:

. RADIAL VARIAT10t1 SHOWN ON LEFT ONE-HALF OF SECTION AVERAGE STRESS ACROSS SECTION SHOWN 0:1 RIGHT ONE-HALF 0F SECTION ALL STRESSES ARE IN PSI:

+ = TEtiSIOtt

- = COMPP.ESSION FIGURE 11. LONGITUDINAL STRESSES RESULTING IN PERMANENT LINING FROM GROUND LOAD H' 15 1

s

l E

121 STRESSES AT C0!1 CRETE SURFACES HAVE BEEN EXTRAPOLATED LINEARLY FROM ELEMENT STRESSES ,

-587 '

i ELEMENT NO. (;; f ,,,

- -695 ,,- u.-u 75, ,w o M

-584 45 -

4

-641 C~Yg 106, u .g go f

-e90 ,4

-637 3

. 103 r 45

, f gD O 5". g 6-I s'c

- t, ,7 :

e- L 'i > -635 f./

t

\- -687 7.,

-583 45 88 -587 .

', s

. , t 90 i  ;,

-645 A

. 37.5 75 o

y,

. /4

.a . -687 ,

.. I. -699

-486 i /I

-5911 73 PLOT OF CIRCUMFERENTIAL NORMAL STRESSES:

. RADIAL VARIATION SHOWN ON LEFT ONE-HALF 0F SECTION 75 AVERAGE STRESS ACROSS SECT 1011 SHOWN ON l RIGHT ONE-HALF 0F SECTION ALL STRESSES ARE IN PSI:

+ = TENS 1014

- = COMPRESSION FIGURE 12. LONGITUDINAL STRESSES RESULTING IN PERMAt1ENT LINING FROM SEASONAL TEMPERATURE RISE T' (To = +570) o 16

I l E

STRESSES AT CONCRETE 121 SUP. FACES HAVE BEEN EXTRAPOLATED Lit 1EARLY FROM ELEr.ENT STRESSES

+ 220 ',

, y _

.- ~ n\ t f . ,. ,

ELEMEt1T l10. ,yd j

+255  %

118 ,

~

+222 , o 7.5 g

- +248 45 +237 106 ,

.. s" o

/

+235 103 .(

\b s

o, 9 W

., +253 f _ __ .p ..__ _

+219i j 85 S +236 .!

d 45 88 +226 7

~

I \ 90 ,

+262

+250

+ '30

.o. 7.5 -

s. .I s, +266 -s.

+190 ' :4 j g f 1 1

+234 , i 73 PLOT OF CIRCUMFERENTIAL NORMAL SikE55L5:

RADI AL VARI AT10tl SHOWN Ott LEFT ONE-flAll OF SECTION 7

AVERAGE STRESS ACROSS SECTION SHOWri ou RIGilT ONE-HALF 0F SECTION ALL STRESSES ARE lH PSI:

+ = TENSION

- = COMPRESS 10rl FIGURE 13 LONGITUDINAL' STRESSES RESULTING IN PERMANENT LINING FROM SEASONAL TEMPERATURE DROP T g (T = -230)

'(WITH PARTIAL ROCK RESTRAINT) 17

l C_

STRESSES AT CONCRETE 121 *

.. SURFACES HAVE BEEN  !

EXTRAPOLATED LINEARLY  !

I FROM ELEliENT STRESSES

+192:

ELEMENT NO.

e -724 l

+190 7.5 45 -266 p 106 , sOh

, f -722 / l 103  ! -266 45 ,

, go' J

~

" I 05 i 3

+189  ;

-270

~

g 70 ' .,

-729 45 oo

\

\ N

-259 No k-274

/ / . )u -

I l ' S~, ~~t\

j. -706

+188 __.

., i

-r i 5 7 '

73 PLOT OF CIRCUMFEREllTIAL NORMAL SlRESSC.,:

RADIAL VARIATION SHOWil ON LEFT OllL-ilALF OF SECTION

! AVERACF STRESS ACROSS SECTION 5110'.r: 0N RIGHT ONE-HALF 0F SECTION ALL STi(CSSES ARE IN PSI:

t = TENSION

- = CONPRESSION FIGURE 14. LONGITUDl!1AL STRESSES RESULTING IN PERMANENT LINING FROM RAPID TEMPERATURE RISE T' (T' = +570) i 18 i .-

l STRESSES AT CONCRETE 121 SURFACES HAVE BEEN EXTRAPOLATED LINEARLY FROM ELEMEllT STRESSES

-71 l ELEMENT I40.

+385 w 118

-70

[7 +384 45

- 7.5

+157

/

5, 9 103 +157 45 I g \b

'O,

, +387 -

lD 1 1

85 r- +158 _

70 g

_ _73 -

n, 45 83 , +153 g

\ 90 i

+378 l .g3 I

30

.' 7.5 L- ,

/ N +392 [

-72 L

f\ ,

-70 t 73 PLOT OF CIRCultFERENTI AL NORMAL STRESSES:

RADIAL VARIAI1011 SHouri ON LEFT ONE-ilALF OF SECTl0!1 b

AVERACE S TRESS ACROSS SECTI0li Sil0WN On ki GMT Otil-ilAI.F 0F SECT 10N

• ALL STRESSES ARL tu PSI:

+ = TENSIOlJ

- = CottPRESb l0H FIGURE 15 LONGITUDINAL STRESSES Rr5ULTlHG lti PERftANENT LitiltJG FROM RAPID TEMPERATURE DROP T' (T' = -230) (WITH FULL ROCK RESTRAltlT) 19 l

t ..

\

5 STRESSES AT C0!! CRETE 171 SURFACES HAVE SEEtt EXTRAPOLATED LitJEARLY FROM ELEMEliT STRESSES

-68 I

, A 22

\,

-58.[! W +253 45"/ if il 106

/ / +97 \

f f j  ! \b L l TC, 9 '

,/  !

45 ce, I

kN co .i

+100 N +108 30

/ 7.5L-s +271 -

' e

^

-4__  ;

\

-54

~- '

W

-54 1 73 PLOT OF CIRCUMFEREt1TIAL tl0kliAL STRES$ts:

RADI AL VARI ATI0tt SliOWri Ott i Er f OllE-itALF Ol' SECT 10:1 AVERAGE STRESS ACROSS SECTI0tt Sil0Wil Oil RIGHT Or;E-IIALF OF SECT 10ft ALL STRESSES ARE I l'S ! :

+ = tells l ut.1

- = COMI'RESSIOli FIGURE 16. L0t1GITUDif1AL STRESSES RESULT!!1G lti PERMAtJEtiT LitiltiG FROM RAPID TEMPERATURE DROP T' ]T' = 320)

(WITH PARTIAL ROCr) RESTRAltlT) 20 __

l ..

l

E 121 STRESSES AT CONCRETE SURFACES HAVE BEEF 4 EXTRAPOLATED LINEARLY FROM ELEMEllT STr. ESSES

'7 l 12L

- <J l ELEMEt4T NO.  ;

- g

+11 N

+13 _7}-45 +12 45

- \

106, , go"

/

105 45 o \,

103 9 \b- 1 . . . .:-i

\0 112 $ +15-M b

e 45 88

\

+11 '^

/

45*~ .W 45

^

\ s

• J Y%

+5 / ,_ l hD, 79 l M I +7 73 I PLOT OF L0tlGITUDir:AL STRESSES l 76 ALL STRES3ES Ac.E !!! PSI STRESSES CAli GE E Tota tells 1011 0R CC ' PRESS I0 FIGURE 17 LONGITUDillAL STRESSES IN PET.MAfE:;T Lit:ll:0 FOR OPET.AT'!;G SASIS EARTHQUAKE RESULTil1G FROM TRA!!SVERSE t'CDAL A!!ALYSIS 21

E STRESSES AT C0tlCRETE I2I SURFACES HAVE BEEll .

EXTRAPOLATED LINEARLY FROM ELEMEllT STRESSES

+13l 124

,~ le l

/

l

+22

+26 45 - Q45 +23 106 ,

, go" 109 \

/

45 103 0

\' *

, {+30 _

+23

{ .

- 45 l

94

\

r-45 88

+27 45

' o 45 /

\

+10 l 4 , 7g I 1 I+13 73 PLOT OF L0tlGITUDitlAL

, STRESSES 76 ALL STP. ESSES ARE It! PSI STRESSES CAN BE EITHER tells 10tl l OR COMPRESS 10!!

FIGURE 18. LO!1GITUD!!!AL STRESSES Irl PERMAllEffr LittitlG FOR SAFE SHUTDOWil EARTHQUAKE RESULTillG FR0ft TRAllSVERSE MODAL AtlALYS I S 22 1

- that result f rom the Poisson ef fect of seismic loads resisted by the tunnel in the transverse direction due to circumferential loading of the tunnel. Addi-tional stress, or a second category of longitudinal seismic stress, results f rom the fact that the tunnel section is translated laterally and deformed longitudinally as the earthquake stress waves engulf the tunnels and pass i through the shale surrounding the tunnels. Additional discussion of the phenomenology associated with this second category of stress is required and follows in the next paragraphs.

Stresses resulting parallel to the longitudinal axes of the tunnels f rom translation and deformation of the tunnel axes were calculated by the approach given by Keusel (Ref. 1). In this method the strain in the longi-tudinal direction of the tunnel induced by an earthquake stress wave having a wave length L, and a single wave ampli tude A, is given by:

c = c, + cb" L s n $ cos $ + 2 L

.. Here, c = Axial strain

- s cg = Bending strain 4 = Depth or width of structure

$ = Angle between the direction of wave propagation and longitudinal axis This equation shows that when 0 = 45 , c reaches a maximum value and wLen v = 0 , the bending strain c b is a maximum. In addition, i t 'indi cates that for a structure having a width W much smaller than the wave length L, the axial strain is predominant.

23 k.

t W y+e

i To solve this equation, the relationship of the length L and the amplitude A of an earthquake wave traveling in the medium in which the tunnel is located must be determined. Relationships between A and L for an earthquake wave traveling in sand and clay media are given by Kuesel and are shown in Figure 19 For Chagrin shale, the relationship between wave amplitude and wave length has been estimated based on the A versus L curves for clay and sand, and from the moduli of the three materials (Chagrin shale, clay, and sand). The A versus L and A/L'versus L curves estimated for Chagrin shale are given in Figure 20.

Having determined the A and L relationship of earthquakes in the Chagrin shale, the seismic stress in the longitudinal direction was computed in the following manner:

From Figure 20,

-6 (A/L) max

=

19 x 10 at L =

9000 ft u_

I" Since the, wave length, L, is much g'reater than the tunnel width, W, which L is only 10 ft, only the fi rst term in the equation is important and the r- equation may be reduced to

=

2nA .

c i c3 sin

• ces $

From this equation, it is evident that the strain c reaches a maximum s

when * = 45 and is equal to the following:

(c,) ,,x =

[^ sin 45 cos 45 =-f-Substi tuting the (A/L) , value

-6 -6

( c,) ,, = 19 x 10 x 3 1416 - 59 7 x 10 24 ,

4 e

28 3

I i CURVE l A = CL" A f, L IN FT l M A 24 2

L. r 0@

0 6 .,d d CURVE 8 l

@ s .'

• g (a) SOFT CLf*T e

\@ p) n '. 1 -

I a 5 G 2 3 E E

"U  ;;

--- 12 ot*55 pt;c

_ l g o t i.,

s g

g- 8 l

4 3

0 0 4 8 12 16 20 24 28 32 36 *Q -

' WAVE LEMOTH "L" (N 1,000 FT.

FIGURE 19 TRANSVERSE GROUl10 DISPLACEME!1T SPECTRUM (FROM REF. 1) e 4

25

+ . . .

o . . . . . . . . , -, , .

6 . J o

II I I i i i i I I I I I I I I I I 2%

i I 22 11 -

20 10 18 3 -

8 -

g8 -

16 f ~ l%

/ e

~

I2 6 -

hs

< /g e m _ 10 -

4 .-

8

~

6 3 (UPNO _-----_

Ciggg f

5B h g 4

2 -

p

/ /

2 1

p

' ' ' '- l J- ' ' - L- - - 3 L I ' I 8 I '

• O -I in in 20 O

C i 2 3 4 5 6 7 8 9 to 17 i3 a5 16 17 18 19

~

L (1000 IIi g593g FIGURE 20. TRAllSVERSE GR00t10 DISPLACEMEt!T SI'ECTl:Ull FOR CllAGI,Ill SilALE

The maximum strain can be either tensile or compressive, and the equiva-c, lent stress in the tunnel lining is given by:

9

= = +215 ps!

(s) max E(cs) max -

which is uniformly distributed throughout the entire section. In the above calculation the short term modulus of concrete was used. This value of stress is assumed to apply to the SSE condition. One-half of this value, +110 psi is assumed for the OBE.

( The maximum bending strain, c, b

f the tunnel can be demonstrated to be insignificant in the following manner:

For p=0 9 2 3

2n'AW cos o , 2r AW

.. b) max g2 g 2

i Substituting in this equation the A/L, and L values found above gives the i

following for V = 10 ft, e-(c b max = 0.42 x 10' 1

This is the maximum value of bending stress that could develop for the same I

A/L, but it does not develop simultaneously with the maximum value of c s

. as a different value of $ has been used in the two equations. Since this value is only +2 psi, it can be neglected, it should be noted that an absolute maximum for c b ccurs for a different value of A/L than considered above (i .e. , A/L 2

should be a maximum). However, in all cases c b is an rder of magnitude lower than e smax and can be neglected.

27 I

. ~ . -_ - . _ _ - . . .. . . - .- - ., .- _ -

0

• Comparison of the longitudinal stresses given in Figures 10 and 11 for the grout load G and the ground load H' with the stresses resulting f rom the fluid pressures indicates that the longitudinal static load stresses are quite low and vary from tensile stresses of less than +50 psi to compres- ,

sive stresses of about -100 psi. This statement neglects temperature stress effects. The temperature and seismic stresses, however, are suf ficient to cause transverse cracking. The designer i s , therefore, faced with four j _. alternatives. Recognizing that transverse temperature cracks will result i in a sllsht 1::-! increase in stress in the transverse section, but will not impai r the load-carrying capacity of the tunnel, three of the alternatives involve relieving the longitudinal temperature stresses through the use of transverse Joints or cracks. These alternatives are (1) provide' transverse constructicn join *s at regular intervals along the longitudinal axes of the tunnels; (2) provide control joints rather than construction joints; and (3) provide no joints but let transverse cracks develop at random. The fourth alternative is to control the temperature cracks by the use of longi.udinal temperature reinforcing steel. These alternatives are preroga-

)r-I tives of the designer, if the longitudinal stresses are to be carried by i

!. reinforcing steel, the stresses that must be considered are given in

, . Figures 21 through 27 for the seven loading conditions considered in this analysis.

Since the dead load, grout load, and ground loads do not act as transverse loads.on the shafts, stresses in the longitudinal direction of the shaf ts f rom static loads are less than estimated for the tunnels. It is conservative, and recommended that the same stress conditions be assumed in design of the shafts as computed for the tunnels.

1 s

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E STRESSES AT CCNCRETE 121 SURFACES HAVE BEEN EXTRAPOLATED LINEARLY FRCM dLEMENT STRESSES

-864 l 1

ELEMENT NO. [ l {,

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-818' 73 PLOT OF LO!!GITUDitlAL STRESSES:

RADIAL VARIATI0tt SHO'.Jil ON LEFT OrlE-HALF 0F SECT 10tl 75 AVERAGE STRESS ACROS3 5ECilott SliG'.JN ort RICiti J;.E-r!ALF 0F iECT . L'.

SAFE SHUTDOWN EARTHQUAKE CONDITION: ALL SinESSES ARE tu MI-U = 1.000 + 1.00H' + 1.00T + 1.00E' + - trnston

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. c.y s FIGURE 27 L0tlGITUDINAL STRESSES RESULTitlG FROM CASE 7 LOADS I 35

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