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, NY 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 sec- tion 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 con- tainment structures to seismic effects.
1. I n t r o d u c t i o n
A major design problem in construction of rein- forced concrete nuclear containment vessels and other similar heavy-walled structures is to transfer the large seismic membrane shear forces effectively across slight- ly open cracks in concrete by the combined mecha- nisms of interface shear transfer (aggregate interlock), dowel forces, and tensile forces in the bars crossing the cracks. Four-way 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 des- cribed llere is confined to two-way 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 defor- mation 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 SMiRT-5/USNRC Panel Session "Status of Research in Structural and Mechanical Engi- neering for Nuclear Power Plants", Berlin (West), 14 August, 1979.
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. Vari- ables 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 dis- cussed here.
85
86 R.N. White et al. /Strength and stiffness oj'reinJbrced concrete contain,nent
VESSEL
/ /
(
1
EARTHQUAKE
N v
l CRACKED ELEMENT element v
i - - " Nh --II------
Nv
Nh
Fig. 1. Nuclear containment vessel under combined internal pressurization and seismic forces.
2. Experimental program
2.1. Specimen description
The specimen shown in fig. 2 is a 48-inch square flat slab, 6 inches thick, with ttfickened corner regions for application of the shearing forces. The slab is rein- forced 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 cen- tered 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
Ty
4-- ~ "'-~.. .#
- k , ~ - ~ ÷ 6 " SPACING -?-+ 7 ' L ~ - - T W O #6 " r 6" °P^'ING
THICKENED ,an" |Y ~ "~ """ CORNER REGIONS ~ -~
6"
i o DETAILS OF BIAXIAL SPECIMEN
TENSILE CORNER ~\LOAD
COMPRESSIVE CORNER LOAD
~f. COMPRESSIVE CORNER LOAD
.DIAGONAL D2 ~ )
DIAGONAL DI
TENSILE CORNER LOAD
b. LOADING METHOD C SHEAR STRESSES ALONG X = O
Fig. 2. Specimen geometry and loading.
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 speci- men. The corner forces were generated by large hydraulic rams fastened to a massive prestressed con- crete 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.
Extensive measurements were made in the central 2-ft square region of the specimen. Dial gages and dis- placement 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 distor- tion 7, as illustrated in fig. 4. It should be noted that because all the tension-induced 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 87
Fig. 3. Experimental set-up.
88
Table 1 Test results
R.N. White et al. /Strength and stiffness of reinforced concrete containment
(1) (2) Ultimate o u (3) (4) Specimen Applied Strength ply - o N Failure no. a rebar ou(psi) x/fc (psi) c at cycle d
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 locat ions along the cracks were no 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 monoton ica l ly increasing shear to
failure, and 12 under fully reversing cyclic shear. All
were precracked by tensioning the reinforcing steel to
approximate ly 0.6Jy (36 ksi) in bo th directions.
Twelve specimens were under tension during shearing
and four had zero tension during the applicat ion o f
shear load. A summary of the testing program is given
in table 1, which also explains the load his tory and
the meaning of the ident i f ica t ion given to each speci- men.
T h e cyclic shear stress his tory is shown in fig. 5.
For most specimens, the initial shear load was -+ 125 psi for 10 cycles. This was increased, in 50 psi incre-
ments , to +-175 psi for another 10 cycles, unti l failure
resulted. During the first, second and ten th cycle o f
shear loading at each stress level, the shear load was
changed in small increments to permi t de te rmina t ion
o f the hysteret ic behavior o f the specimen. For the
o ther seven load cycles at each stress level, the shear
load was applied direct ly to peak values in bo th
directions.
R.N. White et al. I Strength and stiffness of reinforced concrete containment 89
V V \ j /
" ~ . / / d / \ - - ~,
I /
/ / / " \ \ ,
I~,1 + IAol E< Ad= 2 ; G. = 2 ( l+v)
r ,+ r2 ~ ) ' = - - 2 = o = 589.2 A d x 10 - 4
g = I__ = va . ~ = v a ( l + v ) . / ' ~
~, Aclqt'~" ' Go A d E c
Fig. 4. Calculation o f shear deformations.
3. Experimental results and discussion
3.1. Cracking patterns
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 through- out 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 mono- tonically 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 ten- sioning of the specimen at the beginning of the test. The formation of diagonal cracks is crucial in deter- mining the overall stiffness and strength of the speci- men. Shear cycling can produce significant additional diagonal cracking, which results in loss of integrity of the concrete, an increasing rate of degradation of stiff- ness and larger shear deformations.
3.2. Extensional stiffness
m o .
I .
UJ r r
I - -
n-
( / )
22s !
175
125
0
- 1 2 5
- 1 7 5
- 2 2 5
I0 CYCLES
, Qlfl
lj
NEXT SHEAR STRESS LEVEL
5 0 psi INCREMENTS
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:
1 c = average crack spacing of orthogonal cracks (in), fy = nominal ]¢ield strength of steel (ksi), p = steel reinforcement ratio in the direction of ten-
sion, 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
90 R.N. White et al. / Strength and stiffness of reinforced concrete containment
K
+ 175 I
o. SPECIMEN .9 (M)
TENSION ONLY
,] / r] ?/
I
/ / / /
/ /
/
/ 1 + 3 0 0 ps, (FAILURE)
\ / / / [ \/\ x~ Z
\
\ \ \ / \
/
SHEAR ] ~, / \ / • _,,s(, 1 / /~ /
(FAILURE)
1 [
575(,) I
TENSION ONLY [ I
\ \ ~1 SHEAR [ ~, [ -+125(1)
• 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 signifi- cant role in the ability of a specimen to transfer the applied shear forces across the crack surfaces success- fully. The extensional stiffness KN is proportional to
.9
2 #6 REBAR I #6 REBAR
ul I-
ra
4 8 12 16 2tO 24
INITIAL CRACK WIDTH, Cwo(in x 10 -3)
Fig. 7. Average initial crack width versus the applied tension in both directions.
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
nd2 (dis I - 74d2PEs (kip/in) (2) KN = -~-- \dcwo/ 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.
3. 3. Shear stiffness for monotonic shear loading
The variation of the average shearing deformation measured across the diagonals of the specimens loaded with monotonically increasing shear, plotted as a func- tion 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 91
425
525
~: 225
to 12E
25
°o 2~o
J ~.3 (M)
//
I// .....
~ sts (No.14, Nolo bars)
6' ' ' 0 120 180 AVERAGE SHEAR STRAIN, 7" (RAD x 10 -4)
3.0
2.0 >
W
1.0
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 defor- mation 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 spe- cimens 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
' 05 O 1.5 EO 25 SO 35 ~ SHEAR STRESS, v (MPo)
w¢ ~ . 1 2 UNCRACKED ~, MONOTONIC SHEAR {M)
O" ,IO - - SECANT ~< . . . . TANGENT
o8
(M)
w " , , , " \
\ \
o L~J i i , , ~ "r - - ~ - ~ - T , ', x,,,, 0 25 }25 225 525 425 525
SHEAR STRESS, v (psi)
Fig. 9. E f f ec t i ve shea r m o d u l u s versus a p p l i e d shea r s t ress
fo r m o n o t o n i c a l l y l o a d e d spec imens .
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(a-d). These show the effect of shear, axial tension, and cycling on the overall hysteretic behavior of the speci- mens. 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 por- tion 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
3,75
525 E ~ 275
225
tn 175
125 to
75
_ 210
/ ]
i / / - - CYCLE I Z ,'//~/ . . . . CYCLE I0
' 2'0 4'0 60 810 I()0 120 140
AVERAGE SHEAR STRAIN, 7" (RAD x I0 -4)
O. SPECIMEN .O(A)
425
3 7 5
3 2 5
- ~ 2 7 5
~ 225
~ 175
125
~ 75
- 2 0 0
CYCLE I - - - CYCLE I0
I
20 4'0 6'o
AVERAGE SHEAR STRAIN, 7" (RAD x 10 -4)
b. SPECIMEN .3 (A)
- - CYCLE I
810 l i -40 0 40 r20 160
i 200
AVERAGE SHEAR STRAIN, 7 , (RAD x I0 -4)
525
,--~275
225 w
c. SPEClMEN .6 (A )
- - CYCLE I . . . . CYCLE I0
/ / f
11 " , ,V ,25, . 1/,1 ,' ,'
i i / i /
I / t ~. i t i i
- 1 2 0 - 8 0 - 4 0 0 40 80 I 0 160 2 0 0 240
AVERAGE SHEAR STRAIN, 7(RAD x I0 - 4 )
d. SPECIMEN.9 (B)
i
2 8 0
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 o f reinforced concrete containment 93
.08 G O / /
/ /
SHEAR ~.45 G O / /
STREssl L/ " ,' ':OECREAS'"O (0'" I /
/ , ~ ~S'@ .02 G O to .05 G O / - - //
1 / i I I / I ]// I/I
i / / i i i i/ / / I
/ CONSTANT / /
SHEAR "~ ^A ~ / S RES --. . . ~
(psi) I - 7 - - // /
\
.01G O
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 loa- ding 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 stiff- ness 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 sub- stantially larger crack widths anu more severe degra- dation of the bond mechanism between the reinfor- cing 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 dis- tinct 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 inter- face 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 clo- sing of orthogonal cracks due to compressive strut action in the compressive direction. The interface shear transfer mechanism along the cracks is activa- ted, 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 asperi- ties 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 ap- plied biaxial tension stress level in the reinforcement
94 R.N. White et al. / Strength and stiffness o f reinforced concrete containment
I0
6
>= 4
0 0
8 5- 4.o f,/f, /
r.4'- "~" . ~ . f /f • • s y l ~ (
i [
I I
f
I I t
0.5
I
#6 REBARS o CYCLIC
• MONOTONIC i
I i
0.6 0.9 fs / f y
Fig. 12. Ult imate strength versus tension in reinforcing bars.
o I / ° ~ ' ~ / / - "(LOW R LIMIT) / / / 3 . 2 + . 0 0 7 - (ply o- N )
~1~/ / PRESENT RESEARCH • d~6 REBAR
4p #4 REBAR MATTOCK'S
o #3, ~4, #5, REBAR 800 I000 1200
pfy- o- N
is given in fig. 12. Tile shear stress parameter Ou/~/j" c is plotted against the dimensionless tensile stress param- eter Js / f y . A linear regression analysis gives the follo- wing straight lines:
For monotonic shear loading:
v u = [8.5 - 4.0 Us/Jy)J x/f 'c , (3)
for cyclic shear loading:
Vu = [7.4 3.7 (Js//y)] X/Jc, (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 com- pared 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 (pJy -- O N) , 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
Fig. 13. Comparison with Mattock's results and shear friction theory for the monotonical ly 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 con- crete confinement (no transverse steel is used) com- bine to produce these somewhat lower strengths.
Both mean value and lower limit straight lines are given in the interaction diagram of fig. 13. The pro- posed 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 95
providing the relationships 378 ~< ply <~ 744 and 38 ~< (pjy - ON) ~< 744 are satisfied. The quantities./y, ON, and fc are all in psi.
3. 6. Equilibrium model
The free-body diagram of the corner region of specimen 0.9(M), bounded by the main diagonal fail- ure 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 reinforce- ment, and no shearing forces in the concrete (no aggre- gate interlock), the equilibrium of forces in the weak direction (x) results in the following equation
V = Asxfy + Vd - Nx , (7)
where
Asx = cross-sectional area of bars in the x-direction, V = applied shear force at the corner,
Fig. 14. Free body equilibrium diagram at failure for specimen 0.9(M) (all forces are in kips).
Vd = dowel force in the double layer (y-direction), Nx = normal force applied in the single layer (x-
direction). 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. Simi- larly, in the single layer, equilibrium in the y-direction 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 develop- ment.
4. Conclusions
On the basis of the experimental research reported here, with specimens containing orthogonal reinforce- ment with pJy equal to either 378 or 744 psi in one direction and twice these amounts in the other direc- tion, the following conclusions may be drawn con- cerning 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 speci- mens) to about 5.0 x/f'c for 0.9fy tension in the ortho- gonal 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 high-level 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, compression-strut system will soon predo- minate, leading to shear distortions because of the prevailing opening mode in the diagonal cracks.
00 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 reinforce- ment 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 calcu- lated 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 simi- lar 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 ten- sioned 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 ten- sion and membrane shear must be correlated with on- going 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. Horizon- tal 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. Experi- mental evidence summarized ii1 this paper indicates the tfighly nonlinear nature of the shear stiffness of cracked walls. Dynamic analyses are required to eva- luate the displacements, accelerations, and stresses in the vessel for a variety of ground nrotions. In particu- lar, 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 dyna- mic response of structures is greatly reduced by dam- ping. 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 speci- mens 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 over- turning 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 9 7
finite element analysis can be used to predict the shear stress distribution in cracked cylindrical shells.
{f) Diagonal bars. Most current designs include dia- gonal bars, at least near the base of the vessel. The con- tribution of these bars needs to be evaluated. If more reinforcement results in a better distribution of cracks and more shrinkage cracking, what is the correspon- ding shear stiffness, as compared with elements with- out diagonal bars? How should the shear and normal forces be proportioned to the reinforcing in a 4-way reinforcing pattern?
(g) Prestressed elements with shear. Prestressed vessels have much lower steel percentages than non- prestressed 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 reinforce- ment is subjected to high-level 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.
Acknowledgement
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.
Cwo = average initial crack width in orthogonal cracks (in)
d = diameter of reinforcing bars, (in) D = length of diagonals in the central region,
in (34 in) E c = Young's modulus of concrete, ksi
(3500 ksi) E s = Young's modulus of steel, ksi (28 000
ksi) fs = applied tensile stress in the reinforce-
ment (ksi) f y ---- yield strength of reinforcement, ksi (61
ksi) fc = average cylinder compressive strength of
concrete, psi (3 800 psi) G o = shear modulus of uncracked concrete,
of cracked concrete (ksi) KN = effective extensional stiffness (kip/in) l c = average crack spacing of orthogonal cracks
(in) N x, Ny = applied normal tensile force in the x- and
y-direction (kips) v, V = shear stress, shear force applied to the
concrete (psi, kips) Vd = dowel force (kips) Vu = nominal shear stress at failure (psi) 3' = average shear strain of cracked concrete
(rad) A c = total compressive diagonal deformation
(in) A t = total tensile diagonal deformation (in) A d = average diagonal deformation (in) u = Poisson's ratio in concrete (0.16) p = steel reinforcement ratio (1.22% and
2.44% in x- and y-direction) ON = applied normal tension stress (psi)
Nomenclature
a
Asx, Asy
= dimension of the central square region of specimen, in (24 in)
= cross-sectional area of reinforcement in the x- and y-direction (in 2)
References
[1] P. Gergely and R.N. White, Nucl. Engrg. Des. 50 (1978) 41-47.
[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).
98 R.N. White et al. / Strength and stiffness o f reinforced concrete containment
[3] A.H. Mattock and N.M. Hawkins, Shear transfer in rein- forced concrete-recent 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) 55-75.
[5] R. Jimenez, P. Gergely and R.N. White, Shear transfer across cracks in reinforced concrete, Report 78-4, Dept. Struct. Engrg., Cornell University (August 1978).
