Nuclear Engineering and Design 59 (1980) 8598 © NorthHolland Publishing Company
STRENGTH AND STIFFNESS OF REINFORCED CONCRETE CONTAINMENTS SUBJECTED TO SEISMIC LOADING: RESEARCH RESULTS AND NEEDS * Richard N. WHITE, Philip C. PERDIKARIS and Peter GERGELY Department of Structural Engineering, Cornell University, Ithaca, N Y 14853, USA
The primary purpose of this paper is to present results of an experimental investigation on the strength and stiffness of reinforced concrete subjected to combined biaxial tension and simulated seismic forces. The test specimens represent a section of a wall of a containment structure carrying combined pressurization and seismic loading. Shear stiffness and strength, and their degradation with shear cycling, are given, along with simple expressions for predicting strength and extensional stiffness. The secondary purpose of the paper is to discuss research needs for improved prediction of the response of containment structures to seismic effects.
Possible degradation of both strength and stiffness after many cycles of reversing membrane shear must also be determined. The complexities of this behavior preclude any direct analytical approach to predicting either strength or stiffness in the general case of fully reversing shear. The experiments described in this paper were directed to resolving the many questions involved in combined shear and biaxial tension, and to lay a foundation for improved licensing procedures for containments. Variables included the level of axial tension applied to the reinforcement (0, 0.3Jy, 0.6)'y, and 0.9fy), the type of shear loading (monotonic to failure or fully reversing cyclic shear), and the level of shear stress. Reinforcement ratios were maintained constant in this phase of the research program at 2.44% in one orthogonal direction, and 1.22% in the other. Ongoing research at CorneU University will investigate the effect of varying the steel percentages. The stress conditions in the cracked wall of a pressurized containment are simulated on a square specimen in the form of a flat slab (fig. 1) subjected to the required tensile forces in the bars and shearing forces in the concrete. The original goals of the research program and some descriptions of the equipment and the testing program are given by Gergely and White [1]. The punching strength of biaxially tensioned slabs is also being investigated, although that research is not discussed here.
1. I n t r o d u c t i o n
A major design problem in construction of reinforced concrete nuclear containment vessels and other similar heavywalled structures is to transfer the large seismic membrane shear forces effectively across slightly open cracks in concrete by the combined mechanisms of interface shear transfer (aggregate interlock), dowel forces, and tensile forces in the bars crossing the cracks. Fourway reinforcing schemes are often used, with diagonals in both directions, even though it is recognized that the vertical and horizontal steel in the containment will mobilize considerable shear strength and stiffness. However, the research described llere is confined to twoway reinforcing schemes with no diagonals. Subsequent research at Cornell University will consider the presence of diagonal steel. In addition to the strength considerations involved in combined biaxial tension and membrane shear, it is necessary to determine the deformations (slips at cracks, crack opening and closing values, and shear distortion) that the structure will undergo. The deformation behavior has a significant influence on dynamic response, and also on the amount of deformations experienced by the steel liner plate in the containment. * Paper presented at the SMiRT5/USNRC Panel Session "Status of Research in Structural and Mechanical Engineering for Nuclear Power Plants", Berlin (West), 14 August, 1979. 85
86
R.N. White et al. /Strength and stiffness oj'reinJbrced concrete contain,nent Ty
VESSEL Nv
//
l element
(
1
Nh II
CRACKEDELEMENT v
Nh
i


"
4
~ "'~..
k ,
EARTHQUAKE
~
?+ 7
Nv THICKENED CORNER REGIONS ~

.#
÷ 6 " SPACING
~
' L~TWO
#6 " r 6" °P^'ING
,an" Y ~
~
"~ """
6"
Fig. 1. Nuclear containment vessel under combined internal pressurization and seismic forces. i o DETAILS OF BIAXIAL SPECIMEN
2. Experimental program TENSILE CORNER ~\LOAD
2.1. Specimen description
The specimen shown in fig. 2 is a 48inch square flat slab, 6 inches thick, with ttfickened corner regions for application of the shearing forces. The slab is reinforced with one layer of no. 6 reinforcing bars in one direction and two layers of no. 6 bars in the other direction. All reinforcement is Grade 60 and is centered in the thickness of the specimen. Sections through the central uniform thickness portion of the specimen have a shearing area of 288 in 2 . 2.2. Experimental setup and measurements
Tension in the reinforcing bars was applied by hydraulic rams acting against steel pipe reaction frames built around the specimen in both orthogonal directions (fig. 3). The frames are independent of each other and of the shear loading equipment, and float freely with the slab as it undergoes distortions. The sets of rams in each direction are connected to a common hydraulic pressure pump to ensure equal loads on each of the two sets of orthogonal steel. Shear loads were applied by alternately pulling and pushing on the thickened corners of the specimen. The corner forces were generated by large hydraulic rams fastened to a massive prestressed concrete test frame (fig. 3). Some secondary reinforcing steel (no. 4 bars) was placed in the corner regions to help transfer the tensile corner loads into the central region of the specimen without premature separation of the corners.
~fCOMPRESSIVE . CORNER LOAD .DIAGONAL D2
~
)
DIAGONAL DI
COMPRESSIVE CORNER LOAD
b. LOADING METHOD
TENSILE CORNER LOAD C SHEAR STRESSES ALONG X = O
Fig. 2. Specimen geometry and loading.
Extensive measurements were made in the central 2ft square region of the specimen. Dial gages and displacement transducers measured deformations. Other quantities measured included local values of crack width changes and crack slip along orthogonal cracks in both directions, and extensional stiffness of the specimen in both directions. Reinforcing bar strains were measured at locations outside the specimen, using wire electrical resistance strain gates. The key deformation pattern the shear distortion of the central part of the specimen  was calculated by taking the average of the tensile and compressive diagonal deformations. An effective shear modulus was then calculated from this average shearing distortion 7, as illustrated in fig. 4. It should be noted that because all the tensioninduced ortnogonal cracks did not propagate completely to the sides of the speci
R.N. White et al. / Strength and stiffness o f reinforced concrete containment
Fig. 3. Experimental setup.
87
R.N. White et al. /Strength and stiffness of reinforced concrete containment
88 Table 1 Test results (1) Specimen no. a
0.0(A) 0.0(B) 0.0(C) 0.0(M) 0.3(A) 0.3(B) 0.3(C) 0.3(M)
(2) Applied rebar tension b
No tension
0.3fy
Ultimate Strength ou(psi)
x/fc
(psi) c
(4) Failure at cycle d
+425.0 +425.0 + 475.0 +485.0
7.8 7.3 7.6 8.7
744 744 744 744
1st 3rd 1st 
375.0 + 375.0 400.0 +450.0
6.3 6.0 6.6 6.6
521 521 521 521
4th 6th 1st 
5.8 4.8 4.9 6.2
298 298 298 298
1st 10th 7th 
3.8 4.6 4.3 5.0
75 75 75 75
8th 1st 4th 
0.6(A) 0.6(B) 0.6(C) 0.6(M)
0.6fy
+ 325.0 275.0 +300.0 + 375.0
0.9(A) 0.9(B) 0.9(C) 0.9(M)
0.9fy
+ 225.0 275.0 +250.0 +300.0
ou
(3)
ply  o N
a (A) or (B): CYCLIC (10 cycles at each shear stress level; start at 125 psi and continue at 50 psi increments). (C) : CYCLIC (monotonic loading up to the average failure load of (A) and (B); if no failure occurs continue cycling). (M) : MONOTONIC (monotonic loading up to failure). b Rebar tension of 0.0fy, 0.3fy, 0.6fy and 0.9fy is the same in both directions (x and y). c p = 1.22% (weak direction) is used; o N = normal tension. d Cycle number is within a group of 10 cycles at the indicated stress level.
men, the local values o f crack width changes and slips at specific locations along the cracks were n o t always consistent, and they are used here only as qualitative data to help explain the overall specimen behavior.
2.3. Loading history Sixteen specimens were tested in Phase I o f the research; 4 under m o n o t o n i c a l l y increasing shear to failure, and 12 under fully reversing cyclic shear. All were precracked by tensioning the reinforcing steel to a p p r o x i m a t e l y 0.6Jy (36 ksi) in b o t h directions. Twelve specimens were under tension during shearing and four had zero tension during the application o f shear load. A summary o f the testing program is given
in table 1, which also explains the load history and the meaning o f the identification given to each specimen. T h e cyclic shear stress history is shown in fig. 5. For most specimens, the initial shear load was +125 psi for 10 cycles. This was increased, in 50 psi increments, to +175 psi for another 10 cycles, until failure resulted. During the first, second and t e n t h cycle o f shear loading at each stress level, the shear load was changed in small increments to p e r m i t d e t e r m i n a t i o n o f the hysteretic behavior o f the specimen. F o r the other seven load cycles at each stress level, the shear load was applied directly to peak values in b o t h directions.
R.N. White et al. I Strength and stiffness of reinforced concrete containment
\
V
j/
V
" ~ .
//
d/
\
I
3.1. Cracking patterns
/
I~,1 Ad=
\\,
2
r,+r2
)' =   2
g
E<
IAol
+
; G. = 2 (l+v) =
= I__ = va
~,
~
o
= 589.2 A d x 10  4
. ~
Aclqt'~" '
Go
3. Experimental results and discussion
~,
 
/
//"
89
= va(l+v)./'~ A d Ec
Fig. 4. Calculation o f shear deformations.
After cracking by tensioning the steel, the stress level was adjusted to one of four values: 0, 0.3fy, 0.6fy, or 0.9fy, and held constant at that level throughout the remainder of the test. Additional cracks were formed from diagonal tension stresses during the application of shear load. The orthogonal cracking due to bar tension and the additional diagonal cracking produced by shear stress are shown for specimens 0.9(M) and 0.9(B) in fig. 6. Note the much more severe cracking for the cyclically loaded specimen 0.9(B), than for the monotonically loaded specimen 0.9(M). Diagonal cracking initiates at shear stresses as low as 125 psi, which is about half the expected level. This could be due to excessive internal cracking caused by the biaxial tensioning of the specimen at the beginning of the test. The formation of diagonal cracks is crucial in determining the overall stiffness and strength of the specimen. Shear cycling can produce significant additional diagonal cracking, which results in loss of integrity of the concrete, an increasing rate of degradation of stiffness and larger shear deformations.
3.2. Extensional stiffness
NEXT SHEAR STRESS L E V E L
22s ! I0 CYCLES 175 m o. I.
UJ rr
I
, Qlfl
5 0 psi INCREMENTS
125 0
n
125 (/)
175
lj
225
Fig. 5. Cyclic shear stress history o f specimens A or B.
Tensioning the bars in both directions at preset stress levels imposes different initial widths at the orthogonal cracks. Dial gages were used to measure the axial deformations on the concrete surfaces in both directions. A linear regression analysis of the data resulted in the following expression for average crack widths: Cwo
:
(lcfy~ (2.7 + 10.8(fs/fy)) 10 3 (in), \ PEs!
(1)
where 1 c = average crack spacing of orthogonal cracks (in), fy = nominal ]¢ield strength of steel (ksi), p = steel reinforcement ratio in the direction of tension, E s = Young's modulus of steel reinforcement (ksi), fs = applied tensile stress in reinforcement (ksi).
This expression may be used to obtain a lower limit
R.N. White et al. / Strength and stiffness of reinforced concrete containment
90
TENSIONONLY [
/
I
///
\ \
TENSION ONLY
[
,]
K
\
[
///
\/\
x~Z \
\\\/
/
/ + 175
I
o. SPECIMEN .9 (M)
/
r]
/
?/ I
\
SHEAR ] ~ , / \ /
/
~1 SHEAR ~, [ +125(1)
•_,,s(, 1 //~ /
1 [
(FAILURE)
/
ps, (FAILURE)
1+300
575(,)I
• SPECIMEN .9 (B)
Fig. 6. Cracking patterns (diagonal cracking shown is the additional cracking at each specified shear stress level).
on the effective axial stiffness (also called clamping stiffness in shear specimens) in each direction for the embedded reinforcement. This stiffness plays a significant role in the ability of a specimen to transfer the applied shear forces across the crack surfaces successfully. The extensional stiffness KN is proportional to
I
#6
nd2 (dis I  74d2PEs KN = ~ \dcwo/
(kip/in)
(2)
Ic
where d = diameter of reinforcing bar (in). The variation of the initial crack width Cwo with the applied tension fs in both directions for the no. 6 bars used in these experiments is given in fig. 7.
.9
2 #6 REBAR
the, steel ratio p and is given by the slope of the straight line of a plot o f f s versus Cwo in eq. (1). The result is
REBAR
ul I
3. 3. Shear stiffness for monotonic shear loading
ra
4
8
12
16
2tO
24
INITIAL CRACK WIDTH, Cwo(in x 103)
Fig. 7. Average initial crack width versus the applied tension in both directions.
The variation of the average shearing deformation measured across the diagonals of the specimens loaded with monotonically increasing shear, plotted as a function of the applied shear stress, is given in fig. 8. The corresponding curve from a large scale specimen with tension of 0.9fy, as determined from a parallel research program at the Portland Cement Association Labora
R.N. White et al. / Strength and stiffness o f reinforced concrete containment ' 425
J
~.3 (M)
O
1.5 EO 25 SHEARSTRESS,v (MPo)
~ w¢ ~ ~,
//
.
1UNCRACKED
I//
2.0
.....
>
....
o8 (M)
w
°o
SECANT TANGENT
1.0
12E
25
35
MONOTONICSHEAR {M)
W
~: 225
SO
2
O" ,IO ~<
525
to
05
3.0
91
",,,"\ \\
~
sts ( N o . 1 4 ,
6'0
120 '
Nolo
bars)
180 '
o L~J 0 25
2~o
AVERAGE SHEAR STRAIN, 7" (RAD x 104)
Fig. 8. Average shear strain versus applied shear stress for the monotonic specimens.
tories [2], is also shown by the dashed lines. Two types of shear stiffness may be defined for these curves  a local tangent stiffness, which is the slope of the curve, and a secant stiffness, which is the slope of a line from the origin to any point on the curve. The latter value is representative of the total shearing deformation experienced at any stress level. The tangent shear stiffness measured above about 100 psi shear stress is very similar for the Cornell University and PCA specimens with 0.9fy tension, but there is a distinct shift of the Cornell data to higher levels of total shearing deformation at lower shear stresses. The reasons for these differences are not apparent and are under study at this time; one possibility is the effect of shrinkage cracks in the smaller specimen. It is also evident from the comparison that a more ductile behavior near failure loads is present in the Cornell specimen. This increased flexibility near failure is probably explained by the difference in bar size between the two test series (no. 6 for Cornell specimens and no. 14 and no. 18 for the PCA specimens) and by the fact that the failure crack passed through the measuring grid in the Cornell test, whereas it was outside the gage length in the PCA test. The test results show that there is a decrease in shear stiffness and an increase in deformation as the reinforcing bar tension level increases from 0 to the maximum value of 0.9jy. This is evident in fig. 9 where the shear stiffness behavior of specimens loaded
i
i }25
,
,
~
"r 
~
 ~ T 525
225 SHEARSTRESS,v (psi)
,
425
',x,,,,
525
Fig. 9. E f f e c t i v e s h e a r m o d u l u s v e r s u s a p p l i e d s h e a r stress for monotonically loaded specimens.
in monotonic shear is summarized. Effective tangent or secant shear moduli of less than 10% of that for uncracked concrete were calculated for each of the monotonically loaded specimens.
3. 4. Shear stiffness for cyclic shear loading The positive portions of the shear deformation versus shear load hysteretic loops for selected cycles of loading for specimens with applied bar tensions ranging from 0 to 0.9jy are given in figs. 10(ad). These show the effect of shear, axial tension, and cycling on the overall hysteretic behavior of the specimens. The first cycle at each shear stress level is indicated by solid lines and the tenth cycle by dashed lines. The main characteristics of these curves are the very low stiffnesses at shear stresses less than about 50 psi, and the sudden increase of stiffness at higher shear stresses, as the concrete surfaces along each crack come into bearing contact and begin to transfer shear stress by interface locking action. The unloading portion of each loop is very stiff. This behavior may be idealized with a bilinear representation, as shown in fig. 11 for axial tensions of 0 and 0.9fy. This figure also gives the effective tangent shear modulus G as a percentage of Go for uncracked concrete. For the situation of zero tension during shearing, and shear stresses less than 50 psi, values of G equal to 0.05G0 were calculated. This value decreased to
92
R.N. White et al. / Strength and stiffness o f reinforced concrete containment 425
425
325
3,75 525
~275
/ ]
E
~
275 225
tn 175
~ 225 i/ / Z,'//~/
....
CYCLE I CYCLEI0
~
175 125
125 to
~
75
_
210
CYCLE I CYCLE I0

375
' 2'0
4'0
60
810
I()0
120
140
I
75
20
0
20
6'o
4'0
AVERAGE SHEAR STRAIN, 7" (RAD x I0 4)
AVERAGE SHEAR STRAIN, 7" (RAD x 10 4)
O. SPECIMEN .O(A)
b. SPECIMEN .3 (A)  
40
0
CYCLE
810
40
I
l
i
i 160
r20
200
AVERAGE SHEAR STRAIN, 7, (RAD x I 0 4) c. SPEClMEN . 6 ( A )  ....
525
CYCLE I CYCLE I 0
,~275
w
225 //f
,25,
.
11
,'
1/,1
i/
i
120
80
40
0
40
t
~.
i
80
I 0
160
" ,,V ,'
i/
t
I/
i
i
200
240
i
280
AVERAGE SHEAR STRAIN, 7(RAD x I 0  4 ) d. S P E C I M E N . 9 (B)
Fig. 10. Average shear strain versus applied shear stress for the cyclically loaded specimens (only the positive portions of the hyste retic loops are shown.
R.N. White et al. / Strength and stiffness of reinforced concrete containment .08
93
GO
/
/ / / SHEAR
~ . 4 5 GO
/
SHEAR
/
STREssl L/ " ,'':OECREAS'"O (0'" I /
/ ~ ~S'@
1/
/
.02 GO to .05 GO
RES
(psi)
. .
I
7
/
.~
//
/
/
//

\
.01G O
I
I
iii
S
/ /
,
iI
/
i//
CONSTANT "~ ^A ~
/
]//
i/
I/I
I
/
O. NO TENSION
b.
TENSION =
.9fy
Fig. l 1. Bilinear idealization of the hysteretic loops and the corresponding effective tangent moduli for specimens with 0 and 0.9fy biaxial tension.
0.02Go from the comoined effects of cycling and increasing the peak shear stress level. The stiffer loading portion of the curves also show a decrease in stiffness from about 45% that of uncracked concrete to about 8% as failure is approached. Specimen 0.9(A) with 0.9fy applied tension had an extremely low shear stiffness of less than 0.01Go at low shear stresses, with the stiffer portion of the response exhibiting an approximately constant stiffness of 0.03 to 0.04Go. This is similar to that for the monotonically loaded specimen with the same bar tension. The increase in flexibility, particularly near failure, is apparent for this specimen, which had substantially larger crack widths anu more severe degradation of the bond mechanism between the reinforcing steel and the concrete. The shear stiffness degrades with an increasing number of shear loading cycles, and a 10 to 20% increase in shearing deformation occurs. Three distinct stages of behavior may be identified for the cycled specimens: (a) Stage 1 (low stiffness; shear stress less than 50 psi). After several load reversals and formation of diagonal cracks, large crack openings require large deformations to develop the shear transfer mechanism. As the number of cycles increases, dowel action becomes a more important mechanism, with the interface shear transfer along the cracks becoming less important because of the progressive damage to the concrete surfaces. Tile increased dowel forces then cause additional deterioration of bond because of
local crushing around tile bars at the cracks, and dowel action then becomes less effective and more flexible. (b) Stage 2 (high stiffness; shear stress greater than 50 psi). At this stage the diagonal tension mechanism begins to predominate, causing opening of diagonal cracks normal to the tensile direction and some closing of orthogonal cracks due to compressive strut action in the compressive direction. The interface shear transfer mechanism along the cracks is activated, particularly when the crack widths are small, and the tangent shear stiffness shows a dramatic increase. With cycling, the interface shear transfer and friction become less effective, resulting in degradation of the stiffness and a cumulative gain in shear distortions. In specimens with reinforcing bar tension as high as 0.9jy this behavior is not so important, because the initial crack widths are so great that dowel action is tile major shear transfer mechanism. (c) Stage 3 (Unloading from peak shear stress). Unloading leads to some reduction in deformation, but appreciable residual deformation can remain at zero sitear stress because of the interlocking of asperities along tire crack surfaces and the wedging action of concrete at the surface of the reinforcement. Tile tangent shear stiffness remains nearly the same as in the second stage.
3.5. Ultimate strength An interaction curve for shear strength versus applied biaxial tension stress level in the reinforcement
R.N. White et al. / Strength and stiffness o f reinforced concrete containment
94
14:
I0
8 5 4.o f,/f,
7/
12
LOWER LIMIT
/
I0
6 r.4'•
•
fs /f y l "~" . ~ .
>= 4
i [ I I
f
0
0
~
%
#6 REBARS o CYCLIC • MONOTONIC i
0.5
0.6
IV~AN ~ / , / / / /
6
o I/°~'~/// //
0.9
fs / f y
oo'l"'
Fig. 12. Ultimate strength versus tension in reinforcing bars.
is given in fig. 12. Tile shear stress parameter Ou/~/j"c is plotted against the dimensionless tensile stress parameter Js/fy. A linear regression analysis gives the following straight lines: For monotonic shear loading: (3)
for cyclic shear loading: Vu = [7.4
3.7 (Js//y)] X/Jc,
"(LOW R LIMIT) 3 . 2 + . 0 0 7 (ply 
o N )
PRESENT RESEARCH 4 ~1~// ~  ' / ' SHEAR FRICTION • d~6 REBAR 2t/'/'~j (F = 1,4) 4p #4 REBAR MATTOCK'S o #3, ~4, #5, REBAR 800 I000 1200
I i
vu = [8.5  4.0 Us/Jy)J x/f'c,
(MEAN)3.6 +.007 (plycr N)
8
(
I
I I t
871 I
(4)
where Jc is in psi, and tile steel percentages are 1.22% in one direction and 2.44% in the other, with Grade 60 steel. Thus, tile initially cracked specimens with zero tension in tile reinforcement during the application of shear carried a shear stress of about 8.5x/jc and 7.4x/jc, respectively, for the two cases of monotonic and cyclic shear stress. When applied bar tension is increased to 0.9.Iv, the shear strength decreases to about 5x/f'c and 4%/f'c for monotonic and cyclic shear loading, respectively. Strength is decreased by some 15% by the action of cyclic loading, as compared to the monotonically loaded specimen strengths. These same results, along with several results obtained on very recent Cornel1 University tests with lower percentages of steel and monotonic loading, are plotted in fig. 13 in terms of the stress parameter ( p J y  ON), where p is the steel ratio in tile more lightly reinforced direction, and ON is the applied normal tension stress in the same direction. Results
obo
pfy oN Fig. 13. Comparison with Mattock's results and shear friction theory for the monotonically loaded specimens.
from experiments conducted by Mattock [3,4] for monotonically shear specimens are included, along with the straight line from shear friction theory with a friction coefficient of 1.4. As observed by Mattock, strength increases with increasing steel ratio p or with decreasing axial tension. An approximately 50% variation in strength is observed with applied tension ranging from 0 to 0.9fy. For low values of (Ply  oN), less than about 200, the present experiments correlate well with Mattock's results, and the shear friction theory is on the conservative side. As the applied tension decreases or the reinforcement ratio increases, strengths of the Cornell specimens are substantially lower than those predicted from Mattock's expressions. For the latter case, where the bar tensions are smaller than 0.6Jy, the shear friction theory does not give conservative results. The effects of the bidirectional cracking, the biaxial tension, and the absence of any effective concrete confinement (no transverse steel is used) combine to produce these somewhat lower strengths. Both mean value and lower limit straight lines are given in the interaction diagram of fig. 13. The proposed simple expressions to predict the strength for uncycled specimens are: Lower limit: o u = [3.2 + 0.007 (pJy
oN)] ~/]c "~ 0'13Jc
(5)
R.N. White et al. / Strength and stiffness of reinforced concrete containment
mean value: Vu = [3.6 + 0.007 (Ply  oN)l X/Jc ~< 0 . 1 3 £ ,
(6)
providing the relationships 378 ~
The freebody diagram of the corner region of specimen 0.9(M), bounded by the main diagonal failure crack, is given in fig. 14. The diagonal crack is inclined 45 degrees to the direction of reinforcement. The external forces are the nominal shear forces at failure and the tensile loads in the bars, determined from load cells and electrical wire gages, respectively. The dowel forces in the topmost horizontal and leftmost vertical bars in fig. 14 are neglected because they have short development lengths and the concrete splits at low loads. Assuming dowel action in the remaining bars, simultaneous yielding in the reinforcement, and no shearing forces in the concrete (no aggregate interlock), the equilibrium of forces in the weak direction (x) results in the following equation V
=
Asxfy + Vd

Nx
(7)
,
where Asx V
= crosssectional area of bars in the xdirection, = applied shear force at the corner,
2 #6
DOWEL
5 ~ 3 ,.6, 2 4 . 2 ~
,
530
0OWEL 268
W ll;
3S A
Iv
c~'~
2 4.2
11 32 ..... +424.2
4
~26.8 ~
48.1 ': ~ ......... '~N,24 2 ~ ,.a~/J ; ~ : ~ ;  7 T~  '  ~ " T ' ' ~ " •
v ] 48.4
l48.4
48.4
48.4
Fig. 14. Free body equilibrium diagram at failure for specimen 0.9(M) (all forces are in kips).
95
Vd Nx
= dowel force in the double layer (ydirection), = normal force applied in the single layer (xdirection). Substitution of the applied forces into eq. (7) gives Vd
=43.2 3 X0.44 X 61 +0.9 X3 X0.44 ×61 = 35.2 kips.
This means that a dowel force of 5.9 kips per bar in the double layer exists at the diagonal crack. Similarly, in the single layer, equilibrium in the ydirection gives a dowel force of 9.2 kips per bar. Since, for the specimens with less tension than 0.9Jy in the bars, the opening of the failure diagonal crack width is smaller, aggregate interlock contribution of the concrete has to be taken into account in addition to the dowel and tensile forces in the bars. A more complete equilibrium model is now under development.
4. Conclusions On the basis of the experimental research reported here, with specimens containing orthogonal reinforcement with pJy equal to either 378 or 744 psi in one direction and twice these amounts in the other direction, the following conclusions may be drawn concerning the shear transfer mechanism of initially cracked reinforced concrete: (a) The monotonic shear capacity decreases approximately linearly with increasing biaxial tension: from 8.5x/j'c for zero tension (but precracked specimens) to about 5.0 x/f'c for 0.9fy tension in the orthogonal steel during the application of the shear. (b) Fully reversing cyclic shear reduces the shear capacity by about 15 to 20%, with the reduction decreasing as the level of biaxial tension increases, (c) Orthogonal cracking reduces the effective shear modulus of the concrete to less tban 10% of that for uncracked concrete, for monotonic shear loading. (d) Repeated application of reversing shear stress causes progressive degradation of the shear stiffness and a decrease in the area of the hysteresis loop. (e) An accumulated increase in shear deformations of about 10 to 20% can occur due to cycling. The deformations increase at a faster rate near failure and for high stresses in the reinforcement.
96
R.N. White et aL / Strength and stiffness of reinforced concrete containment
(f) Although the shear stiffness is low,/'or biaxially tensioned specimens subjected to highlevel cyclic shear, the effect of shear deformations and slips at cracks on the linear stresses and distortions depends on the frequency content of the excitation and on tire distribution of shear stresses around the circumference of the cylindrical vessel. These effects have not yet been studied extensively. (g) Formation of inclined (diagonal) cracks at sllear stress levels as low as 125 psi, superimposed on the already existing orthogonal cracks produced by biaxial tension, is probably the major factor in the irreversible loss in stiffness and strength. A diagonal tension, compressionstrut system will soon predominate, leading to shear distortions because of the prevailing opening mode in the diagonal cracks. 0 0 Although analysis of all experimental data is not yet complete, it appears that the shear capacity is always governed by yielding of the reinforcement in the region near the corner that is loaded in tension from the equivalent shear loading. (i) Assuming simultaneous yielding in the reinforcement in both directions at the peak load and no shearing forces transmitted by the concrete, dowel forces of about 5.9 kips per bar in the double layer and 9.2 kips per bar in the single layer were calculated for a specimen tensioned to 0.9fy. At lower bar tension levels, the reinforcement can probably sustain higt~er dowel forces because the concrete around the bar is less damaged by high bond stresses. (j) The effect of diagonal steel to be determined after additional tests on specimens with fourway reinforcing patterns are completed. (k) This research should be coordinated with similar work at PCA as soon as possible. This is discussed further in the next part of this paper, which is a separate section on research needs.
5. Research needs Extensive research on uniaxially and biaxially tensioned specimens subjected to reversing shear loads has resulted in a reasonably complete understanding of their behavior. Stiffness and strength characteristics have been established for a variety of conditions. Nevertheless, several problems must be studied before the seismic behavior of reinforced concrete
shells, with and without diagonal bars, can be fully evaluated. The the most significant questions remaining are: (a) Shear transfer and dowel action in large bars. The Cornell research results on combined biaxial tension and membrane shear must be correlated with ongoing research on larger specimens with no 14 and no. 18 reinforcing bars at the PCA Laboratories in Skokie, IL. The two projects are complementary in that the large number of tests on small specimens at Cornel can be used to evaluate many parameters while the few tests at PCA can focus on the detailed behaviour of large specimens with limited parameters. (b) Shear transfer at construction joints. Horizontal construction joints occur at about every 6 ft in the wall and their shear transfer characteristics need to be studied. Various types of surface preparation and the effects of nearby reinforcing can be investigated, using the biaxial tension setup described earlier. (c) Dynamic response o f containment walls. Experimental evidence summarized ii1 this paper indicates the tfighly nonlinear nature of the shear stiffness of cracked walls. Dynamic analyses are required to evaluate the displacements, accelerations, and stresses in the vessel for a variety of ground nrotions. In particular, the total strain imposed on the liner plate is of great interest. In addition, analysis can help to develop equivalent linear models, for design and for guidance in planning the load history to be used in testing. (d) Effect o f vibration on shear transfer. The dynamic response of structures is greatly reduced by damping. Relatively large amounts of hysteretic damping are present at the open cracks. Thus, dynamic testing is required to evaluate the hysteretic and frictional damping as a function of frequency. Existing shaking tables in the U.S. cannot apply shear forces high enough for the testing of sufficiently large size specimens and therefore eccentric mass shakers or blast loading of models will be required. (e) Shear distribution in the vessel. The elastic shear stress distribution in a cylinder is sinusoidal. However, the nonlinear stiffness at cracks and the decrease of crack width in one half of the shell, due to the overturning effect, could alter the shear distribution. For example, the shear may increase in the neighborhood of the plane of the loading and this radial shear poses another design problem: splitting of concrete at large bars subjected to radial shears. Complex nonlinear
R.N. White et al. / Strength and stiffness o f reinforced concrete containment
finite element analysis can be used to predict the shear stress distribution in cracked cylindrical shells. {f) Diagonal bars. Most current designs include diagonal bars, at least near the base of the vessel. The contribution of these bars needs to be evaluated. If more reinforcement results in a better distribution of cracks and more shrinkage cracking, what is the corresponding shear stiffness, as compared with elements without diagonal bars? How should the shear and normal forces be proportioned to the reinforcing in a 4way reinforcing pattern? (g) Prestressed elements with shear. Prestressed vessels have much lower steel percentages than nonprestressed shells. A few tests should be made to study the shear transfer mechanism of specimens with limited prestressing. The test setup used in the biaxial tension work is suitable for this investigation. (h) Stress and strain limits in bars. The reinforcement is subjected to highlevel tension, cyclic shear, and cyclic local bending at cracks. What are the limits of stress and strain for various load histories and load combinations? (i) Biaxial tension and punching shear. The work on punching shear strength in the presence of biaxial tension needs to be continued to ascertain the effects of additional factors, such as dowel action, quality of concrete, and cover.
Cwo d D Ec Es fs fy
fc Go
KN lc
N x, Ny v, V Vd Vu 3'
Acknowledgement Ac The research program described in this paper is supported by the Nuclear Regulatory Commission (Site Safety Research Branch). It is part of a broader research at Cornell University aiming at various aspects of the shear transfer phenomenon in cracked reinforced concrete. Any findings, views, concluding remarks or recommendations expressed herein are those of the authors and do not necessarily reflect the views of NRC.
At Ad
u p ON
97
= average initial crack width in orthogonal cracks (in) = diameter of reinforcing bars, (in) = length of diagonals in the central region, in (34 in) = Young's modulus of concrete, ksi (3500 ksi) = Young's modulus of steel, ksi (28 000 ksi) = applied tensile stress in the reinforcement (ksi)  yield strength of reinforcement, ksi (61 ksi) = average cylinder compressive strength of concrete, psi (3 800 psi) = shear modulus of uncracked concrete, ksi (1 500 ksi) = effective secant or tangent shear modulus of cracked concrete (ksi) = effective extensional stiffness (kip/in) = average crack spacing of orthogonal cracks (in) = applied normal tensile force in the x and ydirection (kips) = shear stress, shear force applied to the concrete (psi, kips) = dowel force (kips) = nominal shear stress at failure (psi) = average shear strain of cracked concrete (rad) = total compressive diagonal deformation (in) = total tensile diagonal deformation (in) = average diagonal deformation (in) = Poisson's ratio in concrete (0.16) = steel reinforcement ratio (1.22% and 2.44% in x and ydirection) = applied normal tension stress (psi)
References Nomenclature a
Asx, Asy
= dimension of the central square region of specimen, in (24 in) = crosssectional area of reinforcement in the x and ydirection (in 2)
[1] P. Gergely and R.N. White, Nucl. Engrg. Des. 50 (1978) 4147. [2] R.G. Oesterle and H.G. Russell, Shear tests on reinforced concrete membrane elements, ASCE National Convention on Civil Engineering and Nuclear Power, Vol. I, Boston, MA (April 1979).
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[3] A.H. Mattock and N.M. Hawkins, Shear transfer in reinforced concreterecent research, PCI Journal, Vol. 17 (March/April 1972). [4] A.H. Mattock, L. Johal and H.C. Chow, Shear transfer in reinforced concrete with moment or tension acting
across the shear plane, PCI Journal, Vol. 21 (Jan/Feb. 1976) 5575. [5] R. Jimenez, P. Gergely and R.N. White, Shear transfer across cracks in reinforced concrete, Report 784, Dept. Struct. Engrg., Cornell University (August 1978).