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A plasticity model for punching shear of laterally restrained slabs with compressive membrane action
Int. J. Mech. Sci. Vol. 35, No. 5, pp. 371-385, 1993 Printed in Great Britain.
0020-7403/93 $6.00 + .00 © 1993 Pergamon Press Ltd
A PLASTICITY MODEL FOR PUNCHING SHEAR OF LATERALLY RESTRAINED SLABS WITH COMPRESSIVE MEMBRANE ACTION J. S. KUANGand C. T. MORLEY Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 IPZ, U.K. (Received 3 June 1992; and in revised form 16 November 1992) Ai~tmet--A plastic theoretical model is presented for the punching shear failure of laterally restrained concrete slabs, in which a parabolic Mohr failure criterion for concrete is adopted. The proposed method allows for the effect of compressive membrane action and a membrane-modified flexural theory of elasto-plasticity is used to calculate the compressive membrane forces. The predictions by the proposed analysis show good agreement with a wide range of experimental test results.
NOTATION a ck DA d do dl fc f'c f, fy h Nrs na nr P R r S u Wo x
radius of circular slab parameter related to ratio of compressive to tensile strengths of concrete, defined by Eqn (2) internal-energy dissipation per unit length effective depth of slabs punch diameter outer diameter of punched cone uniaxial compressive strength of concrete cylinder strength of concrete uniaxial tensile strength of concrete yield strength of steel reinforcement slab thickness sum of radial compressive membrane forces acting on failure surface compressive membrane force acting at boundary radial compressive membrane force in slab ultimate punching shear load radius of edge ring beam function of generatrix of failure surface or horizontal coordinate stiffness parameter of laterally restrained slab relative displacement critical central deflection vertical coordinate angle between yield surface and displacement vector fl angle between relative displacement and vertical normal stress on an arbitrary plane oh,or3 principal stresses shear stress on an arbitrary plane 0 flexibility factor of laterally restrained slab ~o angle of internal friction 1. I N T R O D U C T I O N
Few problems in reinforced concrete structures have been studied more extensively than the punching shear failure of concrete slabs. However, as the complexities of punching shear behaviour have precluded the development of a satisfactory theoretical treatment, the present understanding of punching shear in concrete slabs is based largely on experimental studies of simple laboratory specimens, and the design prcMsions incorporated in the various codes of practice are a direct result of the empirical procedures derived from tests on such specimens. Moreover, it is widely recognized that the punching load-carrying capacity of concrete slabs is significantly increased when the slab edges are restrained against lateral movement ~1-9]. This restraint induces large arching forces within the slab between the supports-a phenomenon of compressive membrane action. It is this arching effect which is respons371
372
J.S. KUANGand C. T. MORLEY
ible for the enhanced strength of the slab. The development of compressive membrane action is considered to be an important aspect of concrete slab behaviour which has been ignored in the formulation of punching shear strength provisions in present design requirements. This may lead to a conservative prediction of the ultimate load capacity of a restrained slab as no direct account is taken of the enhancement due to the in-plane restraint in many types of reinforced concrete slab systems, such as bridge decks, flat-slab construction, offshore structures, marine platforms, protective shelters and nuclear containment vessels. Some research effort has been devoted to the theoretical investigation of enhanced punching strength of concrete slabs. Hewitt and Batchelor [3] presented a theoretical approach for evaluating the punching strength of restrained slabs in which the boundary restraining forces and moments were incorporated into Kinnunen and Nylander's model [10], which was developed for punching shear failure of unrestrained simply supported slabs. In the late 1970s, an extensive series of field tests on existing structures was carried out by the Ontario Ministry of Transportation and Communications in Canada. As a result of this research, an empirical and less conservative method of design which recognises compressive membrane action, provided certain boundary conditions are satisfied, was permitted by the Ontario highway bridge design code [11]. Kirkpatrick et al. [6] have recently investigated the punching strength of M-beam bridge decks. As part of this study, a semi-empirical formula for prediction was derived, in which it was assumed that bridge slabs were fully restrained laterally. This was based on the punching shear equation in a previously reported two-phase approach [12], with consideration of the enhancement in punching strength due to compressive membrane action accounted for by an equivalent percentage reinforcement parameter, the actual slab reinforcement being neglected. A more recent research effort has been devoted to the investigation of the enhanced punching strength of interior slab--column connections by Rankin and Long [7]. A semi-empirical method of predicting the enhancement in punching strength of full panel specimens has been proposed, which was also based on the previously reported two-phase approach [12]. However, there remains a real need to develop a physically meaningful and consistent theoretical approach which can properly and clearly present the important structural characteristics of punching shear in concrete slabs with compressive membrane action. The purpose of this paper is to present a rigid-plastic theoretical model for the punching shear failure of laterally restrained concrete slabs. The proposed analytical method allows for the effect of compressive membrane action, and the predictions show good agreement with a wide range of experimental test results.
2. CONSTITUTIVE MODEL FOR CONCRETE Parabolic Mohr failure criterion
In the analysis of concrete plasticity, concrete is regarded a rigid-perfectly plastic material [13]. Although much theoretical work has been done using the modified Mohr-Coulomb failure criterion for concrete, it has been recognised that Coulomb's friction hypothesis, which is a straight-line Mohr envelope with a frictional angle tp (Fig. 1), cannot provide an accurate approximation for the failure of concrete [14-16]. This is because the angle of internal friction of concrete is not a constant and decreases as the compressive stress increases. To represent the real mechanical characteristics of concrete, a parabolic failure criterion [17] is adopted, which is a second-order parabolic envelope with a varying friction angle of ~, touching Mohr's circles for simple compression and simple tension, as shown in Fig. 2. This parabolic Mohr failure criterion is expressed as: o
+ f t = 1,
(1)
where z and tr are shear and normal s t r e s s e s o n an arbitrary plane, fc and ft are concrete uniaxial compressive and tensile strengths and ck is a parameter related to the ratio of
Aplasticitymodelforpunchingshear
~
compression
373
"tension~ a
FIG.1. ModifiedMohr-Coulombfailurecriterionforconcrete.
tension c°mpressi°n~t"_
fc-
FIG.2. ParabolicMohrfailurecriterionforconcrete.
compressive to tensile strengths of concrete given by:
CR= X/1 + f-eft-- 1.
(2)
The shear and normal stresses in the parabolic Mohr failure envelope are given as functions of the parameter ct by: z = f~ ~ cot ~,
(3)
__C2 cot2~X).
(4)
/
From Mohr's circle the maximum and minimum principal stresses can then be found to be: trl = f t
1 -- -~-(cosec~t - 1) 2 ,
(5)
374
J.S. KUANGand C. T. MORLEY (33
~ l
plane strain e2 = 0
modified Mohr-Coulomb yield criterion
plane stress (~2=0 /
Parabolic Mohr yield criterion
/
FIG. 3. Yield loci in cases of plane stress and plane strain.
I
u
=- t FIG. 4. Deforming zone between two rigid parts.
a3
= f t [ 1 _ c~~-,~ c o s e c ~ + 1)2] .
(6)
The yield locus for the cases of plane stress and plane strain is shown in Fig. 3. It may seem paradoxical to use a plastic theory, which requires some ductility in the material, based on the tensile strengthft of the concrete, which marks the initiation of brittle rather than plastic behaviour (and is for that reason unreliable). However, this formulation is merely chosen for convenience: it is entirely possible to write the formulae in terms of the compressive strength f¢ and the ratio # offt to f¢. Thus Eqns (5) and (6) become: at = f c { # - [ ¼ - ½ ~
+ P)3 (cosecc~ - 1)2},
(7)
a3 =f~{# - [¼ - ½ x / ~
+ #)] (cosec c~ + t)1}.
(8)
In the limiting case where/~ ~ 0 (i.e. zero tensile strength): 0"1 ~
- - ~f¢
(cosec ~ -- 1)2,
(9)
0"3 ~
- - ~f¢ (cosecc¢ + 1) 2.
(10)
Yield line and internal eneroy dissipation The yield line between two blocks moving in one plane can be investigated by considering the rigid moving blocks I and II separated by a plane homogeneous strain field of thickness 6, as shown in Fig. 4. If the angle between the relative displacement u and the surface is ~O,
A plasticity model for punching shear
375
the strains in the deforming zone are:
~t = 0,
(11)
u
8n = ~ sin ~,,
(12)
u
Ynt = ~ cos O.
(13)
The principal strains can therefore be determined by:
'~= ½ 1.(~n + ~t) "0- %/(~n -- ~t) 2 + Vn2t"l =
U . ~(slnO _
1).
(14)
According to the normality flow rule of the theory of plasticity, using Eqns (5) and (6), with strains regarded as entirely plastic:
d~71 ~3 -da--3=el
sin • - 1 sin~+f
(15)
From Eqn (14): s3 ~1
sin ~ - 1 sinO + 1"
(16)
It is evident from the comparison between Eqns (15) and (16) that if and only if ~, = a can the stress state described by the parabolic Mohr yield condition (Fig. 2) produce the strain vector given by expression (14). Internal energy dissipation per unit length in the deforming zone for the parabolic Mohr material is then: DA = (0"1~ 1 q- 0"383)6 = uft 1 -I-
cot2g
sin~
(17)
This equation applies for any condition of plane strain, and for plane stress on the tension side of points A, A' in Fig. 3, i.e. for the ratio ex/e3 exceeding a certain limit. For other strain ratios in plane stress, the stress point will be at A or A' in Fig. 3, and the energy dissipation will be correspondingly different. It is seen from Eqn (17) that the energy dissipation is independent of the thickness 6, showing that a yield line is obtained in the limiting case when the value of 6 tends toward zero.
3. C O M P R E S S I V E
MEMBRANE
FORCES
Previous research by Bnestrup and Morley 118] has involved the development of a membrane-modified flexural theory of plasticity for compressive membrane action in circular reinforced concrete slabs, where an elasto-plastic analysis for arching action in restrained slabs has been presented. The elasto-plastic restrained slab was treated by the flow theory, lumping the support stiffness and in-plane slab stiffness into a single boundary spring, and assuming membrane action to start at an initial elastic deformation. Consider a clamped circular slab of depth h and radius a, subjected to concentrated load P at the centre, where a conical failure mechanism is assumed with the central deflection Wo, as shown in Fig. 5a. Deformations of the radial segment of the slab are shown in Fig. 5b. The rotation and middle surface extension are 0 and A, respectively. The geometrical condition of compatibility can be written as: (a - A, - Ab + e)2 + Wo 2 = a 2,
(18)
in which subscripts a and b indicate the positions at support and midspan, respectively, while e represents the radial displacement, defined by e = - N / S , where N is the membrane force and S the restraining stiffness. If Ab and A,b are the sectional areas of the restraining 35:S-C
376
J.S. KUANG and C. T. MORLEY P i
t-- .....
_l_
a I
(a) Conical failure mechanism a
Aa
h
7 ~
Ab
0bl
(b) Deformations of radial segment of slab FIG. 5. Conical failure mechanism for centrally loaded restrained circular slab. (a) Conical failure mechanism. (b) Deformationsof radial segment of slab.
ring beam and reinforcement in the beam respectively, the value of S can be determined by: 1
aR
a
0.8 EcAb + EsAsb
+ --
Ech'
(19)
where R is the radius of the edge beam and E~ and Es are the elastic moduli of concrete and steel respectively. It is suggested by Hognested 1-19] that an appropriate value of the elastic modulus of a concrete member subjected to bending and axial load is obtained from: Ec=4730~ ( N m m 2), (20) in which f'¢ is the cylinder strength of concrete. The corresponding kinematical condition of Eqn (18) is given for small A, etc., by: W0
.
Aa + As -- d = - - W o .
(21)
a
By introducing the dimensionless distance from the middle surface to the axis of rotation (Fig. 6a):
A hO
q = -~
(22)
and the dimensionless membrane force in mid-depth of the slab (Fig. 6b): n =
N
23)
Eqn (21) can then be rewritten as: dna - qa + qb + 2hq~ dwo
Wo h '
(24)
Aplasticitymodelforpunchingshear
377
f~
N
h ~ ~
~-~VIh
~l~
Ashf~
(a) (b) FIG.6. Strainratesandstressesofyieldingslabsection. where the value of t/is positive when the geometrical neutral axis is above the middle surface and negative when it is below; ~b is a flexibility factor, defined by:
af~
= 2hS"
(25)
By using the flow rule and considering the horizontal equilibrium of the slab and the kinematical condition [Eqn (24)], the following solutions for the radial membrane forces have been derived: na=kexp
-
no+
nr = na +
1-
~b + ~ - , ,
(26a) (26b)
in which the value of constant k is determined by: k=
n0+]0-]
exp
,
(27)
where wi is an initial elastic deflection at which membrane action starts, taken as 0.03 h empirically [18,20]. If No represents the compressive membrane force corresponding to rotation about mid-depth of the slab (Fig. 6b), then: No = ½hA - Ashfy,
(28)
in which Ash is the sectional area of steel reinforcement per unit length and the value of no in Eqns (26) and (27) is determined from Eqn (23) with N = No. When there is no flexural reinforcement in the slab, No = 1/2 hfc and then no = 0.5. The dimensionless compressive membrane force at the support of the slab can thus be found directly from the analytical expression, Eqn (26a). A plot of compressive membrane force against central deflection for different flexibility factors is shown in Fig. 7.
4. ULTIMATEPUNCHINGSHEARSTRENGTH Failure mechanism of laterally restrained slabs Consider a laterally restrained, centrally loaded circular slab, as shown in Fig. 8a. The failure mechanism of the slab is illustrated by Fig. 8b, where nl and nr represent the membrane restraining forces acting at the boundary and on the failure surface, respectively. In this treatment of the collapse of the slab, only translation and no rotation is allowed for the supports. It is assumed that the failure mechanism consists of the punching out of an axisymmetric, solid cone-like plug at the centre. The generatrix o f the failure surface is defined by a function r = r(x), and the relative velocity u is inclined at an angle/~ to the vertical, as indicated in Fig. 8b. For rigidly restrained slabs, it is recognized [1, 6, 21, 22] that the influence of the level of steel reinforcement on the ultimate capacity is small, which was also confirmed by the
378
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i i'* 0.3 0.4
i
[ / ," ."
0.2
/.,;:;/
0.1
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~
0.0 0
./.
"
I
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0.2
0.4
0.6
0.8
1
Central deflection, ~ / h FIG. 7. Membrane force--deflectioncurves for laterally restrainedcircular slab.
PI /,
D=2a
7Z
lla
77
(a) Partially restrained, centrally loaded circula? slab
d o
2 D
I
n a
~--
nr
F
n a
T
dl
2a (b) Failure mechanism F]o. 8. Laterallyrestrainedslab subjectedto concentratedloading.(a) Partially restrained,centrally loaded circular slab. (b) Failure mechanism.
A plasticitymodelfor punchingshear
379
results of model tests presented in Ref. [9]. The contribution of flexural reinforcement in the slab to the punching shear strength is thus neglected in the derivation. In punching tests, as mentioned in the previous research [9], many of the slab specimens reach a load plateau, and in some cases turn down to a lower load, before the sudden punching shear failure. The peak load might thus compare well with a membrane-modified flexural theory, even though final failure was by punching shear. Therefore, it is suggested that in the assumed punching failure mechanism, the values of compressive membrane forces n, and nr can be calculated using the membrane-action flexural theory presented in the previous section.
Upper-bound solution of plasticity For the assumed failure mechanism, the rate of external work done by the ultimate load associated with the relative displacement is:
WE = Pu cos/~
(29)
and the rate of internal work dissipated by the mechanism is: WA = ~ DA 2~rrv / ] - + r '2 dx + N . u sin r,
(30)
where N,~, the sum of radial compressive membrane forces acting on the failure surface, is determined by:
Nrs = f¢ I h nr 27rrdx.
(31)
3o
Substituting for nr from Eqn (26b) into Eqn (31) gives:
N,.=2~f~f'o[n.+~-~(l-r)]rdx 1 Wo
(32,
and: r' = tan(~ + fl).
(33)
Pucosfl = f~ DA27rrx/1 + r '2 dx + Nr~ usinfl.
(34)
The work equation is therefore:
The upper-bound solution for the punching-shear failure load, obtained by substituting for D^ and r' from Eqns (17) and (33), respectively, into Eqn (34), is then:
P = 2rift
f~or(r' -
[
/1 + r'tanF~21
tan ~) 1 + ~ \ r" ~- t-~nfl / J dx + Nn tan t,
(35)
or is expressed as: P - 2~ft f~ F (r, r') dx + N~ tan/~, where:
[
(36)
c2(I + r'tanB'~21
F ( r , r ' ) f r r' - tanfl +-~ \ rT-_-t-~n r /
j.
(37)
The function r(x) which minimizes the ultimate punching load can be found by the calculus of variations. The appropriate Euler's equation, which is for a functional of the form S~ F(r,r')dx, to minimize r ffi r(x) can be written as: aF
Or
d(0F) dx ~r'
•0"
(38)
380
J.S. KUANGand C. T. MORLEY
Because F(r, r') does not contain x, Euler's equation has the first integral:
F _ r ,OF Or-'-;= C,
(39)
where C is a constant. Assuming that the value of//is given, substituting Eqn (37) into Euler's equation (39) with consideration of the boundary conditions: do and r(0)=-~-
dl r(h)=-~
(40)
and ignoring the higher order terms gives the minimizing function of the generatrix:
r = C1eC~X_ --,tanfl
(41)
C2 where constants C~ and C2 can be determined numerically by the following equations: do 2
dt =
C1
tan fl C2
(42)
(~ tanfl~eC:h
(43)
tanfl C2
+ C2 }
Once the minimizing function of the generatrix has been found, the least upper bound on the failure load can then be obtained from Eqn (35): P = 2~f,
C 2 2c2h - 1) - ~2C1 2---~(e _ c2 I h
(eC2h _ 1)tanfl +
tan23
2C1 (eC~h _ l)tan fl + C2 tan2fl (e 2c~h - 1) + tan4fl_]
+ N,, tan r,
(44)
where the sum of radial compressive membrane forces acting on the failure surface, Nrs, is obtained by substituting for r(x) from Eqn (41) into Eqn (32): h
Nrs = 2rife(n. + ~1 h ) [ ~ - ~ ( e C ~ h - 1 ) ~ 2 2 tariff] 2C~ (e c~h -- 1)tanfl + C-~ h tan2fl] " --nfc-d-~l_2c2W°~C~(e2C:h--1) ----~-22
(45)
Consideration of aP/dfl = 0 gives the following equation:
nftc-2z I-2Ct (eC~h - 1)+ C2(e2C~h - 1)tanfl + ~22 2 tan 3fl ] L-~2
( lw0 h tanfl
r2Cl (eC2h-1)-- 2~htanfl] - 2nfo n, +
- 2rift L-C-~2 + nf~ ~
{-'2
i
(e c~h - I) - C--~tan fl
2 h / C2
tan ]3 + N,s = 0.
(46)
With the values of do, h, fc, ft and na known, the punching-shear failure load can be determined by the following procedure: (1) (2) (3) (4)
Choose a value for the boundary condition dl, the outer diameter of failure cone. Assume a value for the inclined angle ft. Determine the constants C2 from Eqn (43) and then C1 from Eqn (42). Substitute the values of r, C1 and C2 into Eqn (46). If the condition is not satisfied, change the value of/~ and repeat step (3).
A plasticity model for punching shear
381
(5) Calculate the value of N,, the sum of compressive membrane forces acting on the failure surface, using Eqn (45). (6) Find the value of punching failure load P from Eqn (44). The initial value of d1 is usually chosen as one smaller than that in the case of a simply supported slab and is then decreased until a minimum value of P is obtained by an iterative process. The value of d1 for a simply supported slab is determined by the following expression [ 171:
where ck is obtained from Eqn (2). 5. EXPERIMENTAL
Theoretical
VERIFICATION
results
In the theory of concrete plasticity, to allow for the lack of ductility in real concrete, the effectiveness on tensile strength is much lower than that on compressive strength. It is suggested [15] that a tensile strengthf, =fJlOO or smaller would give realistic results. This extremely low value indicates that the effectiveness factor for tensile strength of concrete is very small, and this is possibly due to the brittle nature of separation failure [13]. Throughout this investigation, the plastic effectiveness factor for concrete compressive strength and the ratiof,/f, are 4.22/A [13] and l/100, respectively, wheref, is in MPa and equal to 0.85 times the cylinder compressive strength. The value of h is taken as the effective depth of the slabs and the diameter of equivalent circular slabs is equal to 1.1 times the clear span of the corresponding square slabs. The critical deflection of slabs at failure is assumed to be half the slab depth, which is used to calculate the membrane forces. This value is borne out by an examination on the wide range of measured deflections at failure as cited by Park [23] for the results from a series of tests on laterally restrained concrete slabs. Some of the slabs tested by Snowdon [22] and Holowka et al. [24] are analysed using the proposed method. The predicted punching load and corresponding compressive membrane force acting on the surface of failure cone are calculated, for different values of dimensionless boundary force n, ranging from 0.075 to 0.375. When a slab is rigidly restrained and the critical deflection is taken as one-half of the slab depth, the value of n, is equal to 0.375. Figure 9 shows the variation of the dimensionless predicted punching load with compressive membrane force. As would be expected, there is a definite increase in the load-carrying capacity as the compressive membrane force increases, showing the strength enhancing effect of compressive membrane action on punching shear of laterally restrained slabs. Moreover, the plotted curves in Fig. 9 are convex, indicating that the rate of strength enhancement gradually decreases as the compressive membrane force increases. This agrees with the experimental results presented in Refs [73 and [9]. The relationship between the predicted outer diameter of the punched cone and compressive membrane force is shown in Fig. 10. It is interesting to note that there is a small reduction in outer plug diameter dl as the value of N,, increases. It is likely that the concrete is confined by the compressive membrane force and the propagation of inclined cracks in the slabs is then prevented. Comparison
with experimental
results
Figure 11 shows the correlation of the predictions by the proposed theory with the results of 86 tests carried out by the authors [9] and various researchers [l, 7,22,24,25]. It can be seen that the overall correlation with the test results is good, on the whole. The 86 tests analysed by the proposed method have a mean ratio of experimental to predicted ultimate loads of 1.020, with a standard deviation of 0.139 and a coefficient of variation equal to 13.62%. The correlations of the predictions by British and American code methods [26,27] are also presented in Figs 12 and 13. The mean ratio of the test to predicted punching loads by the British code method is 1.368 with a standard deviation of 0.275 and a coefficient of variation equal to 20.1%, while that by the American code method is 1.632 with a standard
382
J.S. KUANGand C. T. MORLEY P 0.19 • (d. + h)hfc Slab NO. - 33/34 (d= 107 nun, f,0=55.5 MPa do= 2 8 ~
0.18
i
0.17
i
0.16 ~
M
P a do= 283 nun, dl= 616 ram)
0.15
SlabstestedbySnowdon[22] ,
0.14 0.25 (a)
,
I
I
0.45
J
0.65
~ionless
I
,
1
0.85
,
1.05
1.25
compressive membmae foroe Nrs
ddtfc
P 0.19 g(d.+ h)hfc Slab No. - ALIA3 (d= 38.1 ram, I",=27.4 MPa do= 9 ~
0.18
~
0.17
"/"~/B2
f .~
(d= 31.8 ram, f~=27.7 MPa do= 90 nun, dl= 217 nun)
0.16
Slabs tested by Holowka et al. [24] 0.15 0.45
0.25
(b)
0.65
0.85
1.05
1.25
Dimensionless cornpressive membrane force Nrs dlhfc FIG. 9. Variation of predicted punching load with compressivemembrane force. 260 Slab No. - A1/A2 (do=90 ram, d= 38.1 mm, o-..-~..,k_..~ f== 27.4 MPa)
250 240
I
Slab No. - B 1 / B 2 (do=90 mm, d= 31.8 mm, f,= 27.7 MPa)
230 220 210
Slabs tested by Holowka ot al. [24] 200 40
I
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I
I
I
I
60
80
100
120
140
160
180
200
Compressivemembrane force Nrs (kN) FIG. 10. Relationshipbetweenouter diameter of failure cone and compressivemembrane force
A plasticity model for punching shear 600
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i
& Aoki 4" Snowdon
300O
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200 ~ _j
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@ Holowka
A :
~
m Rankin
&
• Author &
I00 =
0
"
•
0
100
200
300
400
500
600
Predicted punching load (kN) FIG. 11. Correlation of predictions by proposed method with 86 test results.
600-
,
/
0"i: 7
500-
.i-~
o
400=
~
300~
I-
4. ~
O Taylor
d 4. ~
AAoki
4
÷ Snov~don
/
@ Holowka
A
200~
M Rankin • Author
k
100
f°
0 0
100
200
300
400
500
600
Predicted punching load (kN) Fie. 12. Correlation of predictions by BS 8110 with 86 test results.
deviation of 0.269 and a coefficient of variation equal to 16.5%, where the safety factors have been removed. Thus, the overall correlation of the proposed theory with the experimental results is considered to be satisfactory in view of the wide range of variables involved and the variety of test sources. 6. CONCLUSIONS
An upper-bound plastic solution for the punching shear strength of laterally restrained concrete slabs has been presented, in which concrete was assumed to be rigid-plastic, but with yielding controlled by a parabolic Mohr failure criterion. The proposed model allows
384
J.S. KUANGand C. T. MORLEY 600++ z
500-
.,_ ,.
/ Taylor
400~
Aoki Snowdon
300 L
4a
'A A._
Z
°
Holowka
[] Rankin
/
Author
0
ml
0
100
200
300
400
500
600
Predicted punching load (kN) FIG. 13. Correlation of predictions by ACI-318 with 86 test results.
for the enhancement in punching strength due to the development of compressive membrane action, and the compressive membrane forces are calculated by a membranemodified flexural theory of elasto-plasticity. The predictions show good agreement with a wide range of experimental results. REFERENCES 1. R. TAYLORand B. HAYES,Some tests on the effect of edge restraint on punching shear in reinforced concrete slabs. Mag. Conc. Res. 17, 39 (1965). 2. Y. P. TONG and B. DEV. BATCHELOR,Compressive membrane enhancement in two-way bridge slabs. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 271. American Concrete Institute, Detroit (1971). 3. B.E. HEWITTand B. DEV. BATCHELOR,Punching shear strength of restrained slabs. J. Struct. Div. Am. Soc. Cir. Engrs 101 (ST9), 1837 (1975). 4. R. A. DORTON, M. HOLOWKAand J. P. C. KING, The Connestogo River Bridge--design and testing. Can. J. Cir. Engng 4(1), 18 (1977). 5. D. BEAL, Load capacity of concrete bridge decks. J. Struct. Div. Am. Soc. Cir. Engrs 108 (ST4), 814 (1982). 6. J. KIRKPATRICK,G. I. B. RANK1Nand A. E. LONG, Strength evaluation of M-beam bridge deck slabs. Struct. Engr 62B (3), 60 (1984). 7. G. I. B. RANKIN and A. E. LONG, Predicting the enhanced punching strength of interior slab-column connections. Proc. Instn Cir. Engrs Part 1 82, 1165 (1987). 8. P.C. PERDIKARISand S. BEIM, RC bridge decks under pulsating and moving load. J. Struct. Div. Am. Soc. Cir. Engrs 114 (ST3), 591 (1988). 9. J. S. KUANGand C. T. MORLEY,Punching shear behaviour of restrained reinforced concrete slabs. ACI Struct. J. 89(1), 13 (1992). 10. S. KINNUNENand H. NYLANDER,Punching of concrete slabs without shear reinforcement. Trans. Royal Inst. Tech., No. 158, Stockholm (1960). 11. ONTARIOMINISTRYOF TRANSPORTATIONANDCOMMUNICATIONS,The Ontario Highway Bridge Design Code. Toronto (1979). 12. A.E. LONG,A two-phase approach to the prediction of the punching strength of slabs. J. Am. Conc. Inst. 72 (2), 37 (1975). 13. M. P. NIELSEN,Limit Analysis and Concrete Plasticity. Prentice-Hall, New York (1984). 14. M. W. BR.,ESTRUP,Punching shear in concrete slabs. Plasticity in Reinforced Concrete, Introductory Report, p. 115. Colloquium of IABSE, Copenhagen (1979). 15. W. F. CHEN, Plasticity in Reinforced Concrete. McGraw-Hill, New York (1982). 16. M. W. BRF~.STRUP,Plastic analysis of structural concrete. Proc. 1 U TAM Symposium, p. 661. Evanston, IL (1983). 17. J.S. KUANG,An Upper-Bound Plastic Solution for Punching Shear Failure of Concrete Slabs, Technical Report No. CUED/Struct-D/TR. 136. Department of Engineering, University of Cambridge, U.K. (1991).
A plasticity model for punching shear
385
18. M. W. BR.ESTRUP and C. T. MORLEY, Dome effect in reinforced concrete slabs: elastic-plastic analysis. J. Struct. Div. Am. Soc, Cir. Engrs 106 (ST6), 1255 (1980). 19. E. HOGNESTED,A Study of Combined Bending and Axial Load in Reinforced Concrete Members, Bulletin No. 339. Engineering Experiment Station, University of Illinois (1951). 20. C. T. MORLEY,Some experiments on circular concrete slabs with lateral restraint. Paper presented at the International Conference on Engineering Mechanics, Warsaw (1974). 21. J. F. BROTCHIEand M. J. HOLLEY,Membrane action in slabs. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 345. American Concrete Institute, Detroit (1971). 22. L. C. SNOWDON,Some Tests on the Strength in Punching Shear of Restrained Concrete Slabs, unpublished report. Building Research Station, Wafford (1973). 23. R. PARK, Ultimate strength of rectangular concrete slabs under short-term uniform loading with edges restrained against lateral movement. Proc. lnstn Cir. Engrs 28, 125 (1964). 24. M. HOLOWKA,R. DORTONand P. CSAGOLY,PunchingShear Strength of Restrained CircularSlabs, unpublished research report. The Ontario Ministry of Transportation and Communications, Toronto (1979). 25. Y. AOKI and H. SEKI, Shearing strength and cracking in two-ways slabs subjected to concentrated load. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 103. American Concrete Institute, Detroit (1971). 26. BRITISHSTANDARDINSTITUTION,Code of Practicefor Design and Construction, BS 8110: Part 1. BSI, London (1985). 27. ACI COMMITTEE318, Building Code Requirementsfor Reinforced Concrete and Commentary (ACI 318-89 and ACI 318R-89). American Concrete Institute, Detroit (1989).
0020-7403/93 $6.00 + .00 © 1993 Pergamon Press Ltd
A PLASTICITY MODEL FOR PUNCHING SHEAR OF LATERALLY RESTRAINED SLABS WITH COMPRESSIVE MEMBRANE ACTION J. S. KUANGand C. T. MORLEY Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 IPZ, U.K. (Received 3 June 1992; and in revised form 16 November 1992) Ai~tmet--A plastic theoretical model is presented for the punching shear failure of laterally restrained concrete slabs, in which a parabolic Mohr failure criterion for concrete is adopted. The proposed method allows for the effect of compressive membrane action and a membrane-modified flexural theory of elasto-plasticity is used to calculate the compressive membrane forces. The predictions by the proposed analysis show good agreement with a wide range of experimental test results.
NOTATION a ck DA d do dl fc f'c f, fy h Nrs na nr P R r S u Wo x
radius of circular slab parameter related to ratio of compressive to tensile strengths of concrete, defined by Eqn (2) internal-energy dissipation per unit length effective depth of slabs punch diameter outer diameter of punched cone uniaxial compressive strength of concrete cylinder strength of concrete uniaxial tensile strength of concrete yield strength of steel reinforcement slab thickness sum of radial compressive membrane forces acting on failure surface compressive membrane force acting at boundary radial compressive membrane force in slab ultimate punching shear load radius of edge ring beam function of generatrix of failure surface or horizontal coordinate stiffness parameter of laterally restrained slab relative displacement critical central deflection vertical coordinate angle between yield surface and displacement vector fl angle between relative displacement and vertical normal stress on an arbitrary plane oh,or3 principal stresses shear stress on an arbitrary plane 0 flexibility factor of laterally restrained slab ~o angle of internal friction 1. I N T R O D U C T I O N
Few problems in reinforced concrete structures have been studied more extensively than the punching shear failure of concrete slabs. However, as the complexities of punching shear behaviour have precluded the development of a satisfactory theoretical treatment, the present understanding of punching shear in concrete slabs is based largely on experimental studies of simple laboratory specimens, and the design prcMsions incorporated in the various codes of practice are a direct result of the empirical procedures derived from tests on such specimens. Moreover, it is widely recognized that the punching load-carrying capacity of concrete slabs is significantly increased when the slab edges are restrained against lateral movement ~1-9]. This restraint induces large arching forces within the slab between the supports-a phenomenon of compressive membrane action. It is this arching effect which is respons371
372
J.S. KUANGand C. T. MORLEY
ible for the enhanced strength of the slab. The development of compressive membrane action is considered to be an important aspect of concrete slab behaviour which has been ignored in the formulation of punching shear strength provisions in present design requirements. This may lead to a conservative prediction of the ultimate load capacity of a restrained slab as no direct account is taken of the enhancement due to the in-plane restraint in many types of reinforced concrete slab systems, such as bridge decks, flat-slab construction, offshore structures, marine platforms, protective shelters and nuclear containment vessels. Some research effort has been devoted to the theoretical investigation of enhanced punching strength of concrete slabs. Hewitt and Batchelor [3] presented a theoretical approach for evaluating the punching strength of restrained slabs in which the boundary restraining forces and moments were incorporated into Kinnunen and Nylander's model [10], which was developed for punching shear failure of unrestrained simply supported slabs. In the late 1970s, an extensive series of field tests on existing structures was carried out by the Ontario Ministry of Transportation and Communications in Canada. As a result of this research, an empirical and less conservative method of design which recognises compressive membrane action, provided certain boundary conditions are satisfied, was permitted by the Ontario highway bridge design code [11]. Kirkpatrick et al. [6] have recently investigated the punching strength of M-beam bridge decks. As part of this study, a semi-empirical formula for prediction was derived, in which it was assumed that bridge slabs were fully restrained laterally. This was based on the punching shear equation in a previously reported two-phase approach [12], with consideration of the enhancement in punching strength due to compressive membrane action accounted for by an equivalent percentage reinforcement parameter, the actual slab reinforcement being neglected. A more recent research effort has been devoted to the investigation of the enhanced punching strength of interior slab--column connections by Rankin and Long [7]. A semi-empirical method of predicting the enhancement in punching strength of full panel specimens has been proposed, which was also based on the previously reported two-phase approach [12]. However, there remains a real need to develop a physically meaningful and consistent theoretical approach which can properly and clearly present the important structural characteristics of punching shear in concrete slabs with compressive membrane action. The purpose of this paper is to present a rigid-plastic theoretical model for the punching shear failure of laterally restrained concrete slabs. The proposed analytical method allows for the effect of compressive membrane action, and the predictions show good agreement with a wide range of experimental test results.
2. CONSTITUTIVE MODEL FOR CONCRETE Parabolic Mohr failure criterion
In the analysis of concrete plasticity, concrete is regarded a rigid-perfectly plastic material [13]. Although much theoretical work has been done using the modified Mohr-Coulomb failure criterion for concrete, it has been recognised that Coulomb's friction hypothesis, which is a straight-line Mohr envelope with a frictional angle tp (Fig. 1), cannot provide an accurate approximation for the failure of concrete [14-16]. This is because the angle of internal friction of concrete is not a constant and decreases as the compressive stress increases. To represent the real mechanical characteristics of concrete, a parabolic failure criterion [17] is adopted, which is a second-order parabolic envelope with a varying friction angle of ~, touching Mohr's circles for simple compression and simple tension, as shown in Fig. 2. This parabolic Mohr failure criterion is expressed as: o
+ f t = 1,
(1)
where z and tr are shear and normal s t r e s s e s o n an arbitrary plane, fc and ft are concrete uniaxial compressive and tensile strengths and ck is a parameter related to the ratio of
Aplasticitymodelforpunchingshear
~
compression
373
"tension~ a
FIG.1. ModifiedMohr-Coulombfailurecriterionforconcrete.
tension c°mpressi°n~t"_
fc-
FIG.2. ParabolicMohrfailurecriterionforconcrete.
compressive to tensile strengths of concrete given by:
CR= X/1 + f-eft-- 1.
(2)
The shear and normal stresses in the parabolic Mohr failure envelope are given as functions of the parameter ct by: z = f~ ~ cot ~,
(3)
__C2 cot2~X).
(4)
/
From Mohr's circle the maximum and minimum principal stresses can then be found to be: trl = f t
1 -- -~-(cosec~t - 1) 2 ,
(5)
374
J.S. KUANGand C. T. MORLEY (33
~ l
plane strain e2 = 0
modified Mohr-Coulomb yield criterion
plane stress (~2=0 /
Parabolic Mohr yield criterion
/
FIG. 3. Yield loci in cases of plane stress and plane strain.
I
u
=- t FIG. 4. Deforming zone between two rigid parts.
a3
= f t [ 1 _ c~~-,~ c o s e c ~ + 1)2] .
(6)
The yield locus for the cases of plane stress and plane strain is shown in Fig. 3. It may seem paradoxical to use a plastic theory, which requires some ductility in the material, based on the tensile strengthft of the concrete, which marks the initiation of brittle rather than plastic behaviour (and is for that reason unreliable). However, this formulation is merely chosen for convenience: it is entirely possible to write the formulae in terms of the compressive strength f¢ and the ratio # offt to f¢. Thus Eqns (5) and (6) become: at = f c { # - [ ¼ - ½ ~
+ P)3 (cosecc~ - 1)2},
(7)
a3 =f~{# - [¼ - ½ x / ~
+ #)] (cosec c~ + t)1}.
(8)
In the limiting case where/~ ~ 0 (i.e. zero tensile strength): 0"1 ~
- - ~f¢
(cosec ~ -- 1)2,
(9)
0"3 ~
- - ~f¢ (cosecc¢ + 1) 2.
(10)
Yield line and internal eneroy dissipation The yield line between two blocks moving in one plane can be investigated by considering the rigid moving blocks I and II separated by a plane homogeneous strain field of thickness 6, as shown in Fig. 4. If the angle between the relative displacement u and the surface is ~O,
A plasticity model for punching shear
375
the strains in the deforming zone are:
~t = 0,
(11)
u
8n = ~ sin ~,,
(12)
u
Ynt = ~ cos O.
(13)
The principal strains can therefore be determined by:
'~= ½ 1.(~n + ~t) "0- %/(~n -- ~t) 2 + Vn2t"l =
U . ~(slnO _
1).
(14)
According to the normality flow rule of the theory of plasticity, using Eqns (5) and (6), with strains regarded as entirely plastic:
d~71 ~3 -da--3=el
sin • - 1 sin~+f
(15)
From Eqn (14): s3 ~1
sin ~ - 1 sinO + 1"
(16)
It is evident from the comparison between Eqns (15) and (16) that if and only if ~, = a can the stress state described by the parabolic Mohr yield condition (Fig. 2) produce the strain vector given by expression (14). Internal energy dissipation per unit length in the deforming zone for the parabolic Mohr material is then: DA = (0"1~ 1 q- 0"383)6 = uft 1 -I-
cot2g
sin~
(17)
This equation applies for any condition of plane strain, and for plane stress on the tension side of points A, A' in Fig. 3, i.e. for the ratio ex/e3 exceeding a certain limit. For other strain ratios in plane stress, the stress point will be at A or A' in Fig. 3, and the energy dissipation will be correspondingly different. It is seen from Eqn (17) that the energy dissipation is independent of the thickness 6, showing that a yield line is obtained in the limiting case when the value of 6 tends toward zero.
3. C O M P R E S S I V E
MEMBRANE
FORCES
Previous research by Bnestrup and Morley 118] has involved the development of a membrane-modified flexural theory of plasticity for compressive membrane action in circular reinforced concrete slabs, where an elasto-plastic analysis for arching action in restrained slabs has been presented. The elasto-plastic restrained slab was treated by the flow theory, lumping the support stiffness and in-plane slab stiffness into a single boundary spring, and assuming membrane action to start at an initial elastic deformation. Consider a clamped circular slab of depth h and radius a, subjected to concentrated load P at the centre, where a conical failure mechanism is assumed with the central deflection Wo, as shown in Fig. 5a. Deformations of the radial segment of the slab are shown in Fig. 5b. The rotation and middle surface extension are 0 and A, respectively. The geometrical condition of compatibility can be written as: (a - A, - Ab + e)2 + Wo 2 = a 2,
(18)
in which subscripts a and b indicate the positions at support and midspan, respectively, while e represents the radial displacement, defined by e = - N / S , where N is the membrane force and S the restraining stiffness. If Ab and A,b are the sectional areas of the restraining 35:S-C
376
J.S. KUANG and C. T. MORLEY P i
t-- .....
_l_
a I
(a) Conical failure mechanism a
Aa
h
7 ~
Ab
0bl
(b) Deformations of radial segment of slab FIG. 5. Conical failure mechanism for centrally loaded restrained circular slab. (a) Conical failure mechanism. (b) Deformationsof radial segment of slab.
ring beam and reinforcement in the beam respectively, the value of S can be determined by: 1
aR
a
0.8 EcAb + EsAsb
+ --
Ech'
(19)
where R is the radius of the edge beam and E~ and Es are the elastic moduli of concrete and steel respectively. It is suggested by Hognested 1-19] that an appropriate value of the elastic modulus of a concrete member subjected to bending and axial load is obtained from: Ec=4730~ ( N m m 2), (20) in which f'¢ is the cylinder strength of concrete. The corresponding kinematical condition of Eqn (18) is given for small A, etc., by: W0
.
Aa + As -- d = - - W o .
(21)
a
By introducing the dimensionless distance from the middle surface to the axis of rotation (Fig. 6a):
A hO
q = -~
(22)
and the dimensionless membrane force in mid-depth of the slab (Fig. 6b): n =
N
23)
Eqn (21) can then be rewritten as: dna - qa + qb + 2hq~ dwo
Wo h '
(24)
Aplasticitymodelforpunchingshear
377
f~
N
h ~ ~
~-~VIh
~l~
Ashf~
(a) (b) FIG.6. Strainratesandstressesofyieldingslabsection. where the value of t/is positive when the geometrical neutral axis is above the middle surface and negative when it is below; ~b is a flexibility factor, defined by:
af~
= 2hS"
(25)
By using the flow rule and considering the horizontal equilibrium of the slab and the kinematical condition [Eqn (24)], the following solutions for the radial membrane forces have been derived: na=kexp
-
no+
nr = na +
1-
~b + ~ - , ,
(26a) (26b)
in which the value of constant k is determined by: k=
n0+]0-]
exp
,
(27)
where wi is an initial elastic deflection at which membrane action starts, taken as 0.03 h empirically [18,20]. If No represents the compressive membrane force corresponding to rotation about mid-depth of the slab (Fig. 6b), then: No = ½hA - Ashfy,
(28)
in which Ash is the sectional area of steel reinforcement per unit length and the value of no in Eqns (26) and (27) is determined from Eqn (23) with N = No. When there is no flexural reinforcement in the slab, No = 1/2 hfc and then no = 0.5. The dimensionless compressive membrane force at the support of the slab can thus be found directly from the analytical expression, Eqn (26a). A plot of compressive membrane force against central deflection for different flexibility factors is shown in Fig. 7.
4. ULTIMATEPUNCHINGSHEARSTRENGTH Failure mechanism of laterally restrained slabs Consider a laterally restrained, centrally loaded circular slab, as shown in Fig. 8a. The failure mechanism of the slab is illustrated by Fig. 8b, where nl and nr represent the membrane restraining forces acting at the boundary and on the failure surface, respectively. In this treatment of the collapse of the slab, only translation and no rotation is allowed for the supports. It is assumed that the failure mechanism consists of the punching out of an axisymmetric, solid cone-like plug at the centre. The generatrix o f the failure surface is defined by a function r = r(x), and the relative velocity u is inclined at an angle/~ to the vertical, as indicated in Fig. 8b. For rigidly restrained slabs, it is recognized [1, 6, 21, 22] that the influence of the level of steel reinforcement on the ultimate capacity is small, which was also confirmed by the
378
J.S. KUANGand C. T. MORLEY 0.5
i i'* 0.3 0.4
i
[ / ," ."
0.2
/.,;:;/
0.1
11!,
~
0.0 0
./.
"
I
I
1
I
0.2
0.4
0.6
0.8
1
Central deflection, ~ / h FIG. 7. Membrane force--deflectioncurves for laterally restrainedcircular slab.
PI /,
D=2a
7Z
lla
77
(a) Partially restrained, centrally loaded circula? slab
d o
2 D
I
n a
~--
nr
F
n a
T
dl
2a (b) Failure mechanism F]o. 8. Laterallyrestrainedslab subjectedto concentratedloading.(a) Partially restrained,centrally loaded circular slab. (b) Failure mechanism.
A plasticitymodelfor punchingshear
379
results of model tests presented in Ref. [9]. The contribution of flexural reinforcement in the slab to the punching shear strength is thus neglected in the derivation. In punching tests, as mentioned in the previous research [9], many of the slab specimens reach a load plateau, and in some cases turn down to a lower load, before the sudden punching shear failure. The peak load might thus compare well with a membrane-modified flexural theory, even though final failure was by punching shear. Therefore, it is suggested that in the assumed punching failure mechanism, the values of compressive membrane forces n, and nr can be calculated using the membrane-action flexural theory presented in the previous section.
Upper-bound solution of plasticity For the assumed failure mechanism, the rate of external work done by the ultimate load associated with the relative displacement is:
WE = Pu cos/~
(29)
and the rate of internal work dissipated by the mechanism is: WA = ~ DA 2~rrv / ] - + r '2 dx + N . u sin r,
(30)
where N,~, the sum of radial compressive membrane forces acting on the failure surface, is determined by:
Nrs = f¢ I h nr 27rrdx.
(31)
3o
Substituting for nr from Eqn (26b) into Eqn (31) gives:
N,.=2~f~f'o[n.+~-~(l-r)]rdx 1 Wo
(32,
and: r' = tan(~ + fl).
(33)
Pucosfl = f~ DA27rrx/1 + r '2 dx + Nr~ usinfl.
(34)
The work equation is therefore:
The upper-bound solution for the punching-shear failure load, obtained by substituting for D^ and r' from Eqns (17) and (33), respectively, into Eqn (34), is then:
P = 2rift
f~or(r' -
[
/1 + r'tanF~21
tan ~) 1 + ~ \ r" ~- t-~nfl / J dx + Nn tan t,
(35)
or is expressed as: P - 2~ft f~ F (r, r') dx + N~ tan/~, where:
[
(36)
c2(I + r'tanB'~21
F ( r , r ' ) f r r' - tanfl +-~ \ rT-_-t-~n r /
j.
(37)
The function r(x) which minimizes the ultimate punching load can be found by the calculus of variations. The appropriate Euler's equation, which is for a functional of the form S~ F(r,r')dx, to minimize r ffi r(x) can be written as: aF
Or
d(0F) dx ~r'
•0"
(38)
380
J.S. KUANGand C. T. MORLEY
Because F(r, r') does not contain x, Euler's equation has the first integral:
F _ r ,OF Or-'-;= C,
(39)
where C is a constant. Assuming that the value of//is given, substituting Eqn (37) into Euler's equation (39) with consideration of the boundary conditions: do and r(0)=-~-
dl r(h)=-~
(40)
and ignoring the higher order terms gives the minimizing function of the generatrix:
r = C1eC~X_ --,tanfl
(41)
C2 where constants C~ and C2 can be determined numerically by the following equations: do 2
dt =
C1
tan fl C2
(42)
(~ tanfl~eC:h
(43)
tanfl C2
+ C2 }
Once the minimizing function of the generatrix has been found, the least upper bound on the failure load can then be obtained from Eqn (35): P = 2~f,
C 2 2c2h - 1) - ~2C1 2---~(e _ c2 I h
(eC2h _ 1)tanfl +
tan23
2C1 (eC~h _ l)tan fl + C2 tan2fl (e 2c~h - 1) + tan4fl_]
+ N,, tan r,
(44)
where the sum of radial compressive membrane forces acting on the failure surface, Nrs, is obtained by substituting for r(x) from Eqn (41) into Eqn (32): h
Nrs = 2rife(n. + ~1 h ) [ ~ - ~ ( e C ~ h - 1 ) ~ 2 2 tariff] 2C~ (e c~h -- 1)tanfl + C-~ h tan2fl] " --nfc-d-~l_2c2W°~C~(e2C:h--1) ----~-22
(45)
Consideration of aP/dfl = 0 gives the following equation:
nftc-2z I-2Ct (eC~h - 1)+ C2(e2C~h - 1)tanfl + ~22 2 tan 3fl ] L-~2
( lw0 h tanfl
r2Cl (eC2h-1)-- 2~htanfl] - 2nfo n, +
- 2rift L-C-~2 + nf~ ~
{-'2
i
(e c~h - I) - C--~tan fl
2 h / C2
tan ]3 + N,s = 0.
(46)
With the values of do, h, fc, ft and na known, the punching-shear failure load can be determined by the following procedure: (1) (2) (3) (4)
Choose a value for the boundary condition dl, the outer diameter of failure cone. Assume a value for the inclined angle ft. Determine the constants C2 from Eqn (43) and then C1 from Eqn (42). Substitute the values of r, C1 and C2 into Eqn (46). If the condition is not satisfied, change the value of/~ and repeat step (3).
A plasticity model for punching shear
381
(5) Calculate the value of N,, the sum of compressive membrane forces acting on the failure surface, using Eqn (45). (6) Find the value of punching failure load P from Eqn (44). The initial value of d1 is usually chosen as one smaller than that in the case of a simply supported slab and is then decreased until a minimum value of P is obtained by an iterative process. The value of d1 for a simply supported slab is determined by the following expression [ 171:
where ck is obtained from Eqn (2). 5. EXPERIMENTAL
Theoretical
VERIFICATION
results
In the theory of concrete plasticity, to allow for the lack of ductility in real concrete, the effectiveness on tensile strength is much lower than that on compressive strength. It is suggested [15] that a tensile strengthf, =fJlOO or smaller would give realistic results. This extremely low value indicates that the effectiveness factor for tensile strength of concrete is very small, and this is possibly due to the brittle nature of separation failure [13]. Throughout this investigation, the plastic effectiveness factor for concrete compressive strength and the ratiof,/f, are 4.22/A [13] and l/100, respectively, wheref, is in MPa and equal to 0.85 times the cylinder compressive strength. The value of h is taken as the effective depth of the slabs and the diameter of equivalent circular slabs is equal to 1.1 times the clear span of the corresponding square slabs. The critical deflection of slabs at failure is assumed to be half the slab depth, which is used to calculate the membrane forces. This value is borne out by an examination on the wide range of measured deflections at failure as cited by Park [23] for the results from a series of tests on laterally restrained concrete slabs. Some of the slabs tested by Snowdon [22] and Holowka et al. [24] are analysed using the proposed method. The predicted punching load and corresponding compressive membrane force acting on the surface of failure cone are calculated, for different values of dimensionless boundary force n, ranging from 0.075 to 0.375. When a slab is rigidly restrained and the critical deflection is taken as one-half of the slab depth, the value of n, is equal to 0.375. Figure 9 shows the variation of the dimensionless predicted punching load with compressive membrane force. As would be expected, there is a definite increase in the load-carrying capacity as the compressive membrane force increases, showing the strength enhancing effect of compressive membrane action on punching shear of laterally restrained slabs. Moreover, the plotted curves in Fig. 9 are convex, indicating that the rate of strength enhancement gradually decreases as the compressive membrane force increases. This agrees with the experimental results presented in Refs [73 and [9]. The relationship between the predicted outer diameter of the punched cone and compressive membrane force is shown in Fig. 10. It is interesting to note that there is a small reduction in outer plug diameter dl as the value of N,, increases. It is likely that the concrete is confined by the compressive membrane force and the propagation of inclined cracks in the slabs is then prevented. Comparison
with experimental
results
Figure 11 shows the correlation of the predictions by the proposed theory with the results of 86 tests carried out by the authors [9] and various researchers [l, 7,22,24,25]. It can be seen that the overall correlation with the test results is good, on the whole. The 86 tests analysed by the proposed method have a mean ratio of experimental to predicted ultimate loads of 1.020, with a standard deviation of 0.139 and a coefficient of variation equal to 13.62%. The correlations of the predictions by British and American code methods [26,27] are also presented in Figs 12 and 13. The mean ratio of the test to predicted punching loads by the British code method is 1.368 with a standard deviation of 0.275 and a coefficient of variation equal to 20.1%, while that by the American code method is 1.632 with a standard
382
J.S. KUANGand C. T. MORLEY P 0.19 • (d. + h)hfc Slab NO. - 33/34 (d= 107 nun, f,0=55.5 MPa do= 2 8 ~
0.18
i
0.17
i
0.16 ~
M
P a do= 283 nun, dl= 616 ram)
0.15
SlabstestedbySnowdon[22] ,
0.14 0.25 (a)
,
I
I
0.45
J
0.65
~ionless
I
,
1
0.85
,
1.05
1.25
compressive membmae foroe Nrs
ddtfc
P 0.19 g(d.+ h)hfc Slab No. - ALIA3 (d= 38.1 ram, I",=27.4 MPa do= 9 ~
0.18
~
0.17
"/"~/B2
f .~
(d= 31.8 ram, f~=27.7 MPa do= 90 nun, dl= 217 nun)
0.16
Slabs tested by Holowka et al. [24] 0.15 0.45
0.25
(b)
0.65
0.85
1.05
1.25
Dimensionless cornpressive membrane force Nrs dlhfc FIG. 9. Variation of predicted punching load with compressivemembrane force. 260 Slab No. - A1/A2 (do=90 ram, d= 38.1 mm, o-..-~..,k_..~ f== 27.4 MPa)
250 240
I
Slab No. - B 1 / B 2 (do=90 mm, d= 31.8 mm, f,= 27.7 MPa)
230 220 210
Slabs tested by Holowka ot al. [24] 200 40
I
I
I
I
I
I
I
60
80
100
120
140
160
180
200
Compressivemembrane force Nrs (kN) FIG. 10. Relationshipbetweenouter diameter of failure cone and compressivemembrane force
A plasticity model for punching shear 600
383
/
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i
& Aoki 4" Snowdon
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~
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•
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100
200
300
400
500
600
Predicted punching load (kN) FIG. 11. Correlation of predictions by proposed method with 86 test results.
600-
,
/
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500-
.i-~
o
400=
~
300~
I-
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k
100
f°
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100
200
300
400
500
600
Predicted punching load (kN) Fie. 12. Correlation of predictions by BS 8110 with 86 test results.
deviation of 0.269 and a coefficient of variation equal to 16.5%, where the safety factors have been removed. Thus, the overall correlation of the proposed theory with the experimental results is considered to be satisfactory in view of the wide range of variables involved and the variety of test sources. 6. CONCLUSIONS
An upper-bound plastic solution for the punching shear strength of laterally restrained concrete slabs has been presented, in which concrete was assumed to be rigid-plastic, but with yielding controlled by a parabolic Mohr failure criterion. The proposed model allows
384
J.S. KUANGand C. T. MORLEY 600++ z
500-
.,_ ,.
/ Taylor
400~
Aoki Snowdon
300 L
4a
'A A._
Z
°
Holowka
[] Rankin
/
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0
ml
0
100
200
300
400
500
600
Predicted punching load (kN) FIG. 13. Correlation of predictions by ACI-318 with 86 test results.
for the enhancement in punching strength due to the development of compressive membrane action, and the compressive membrane forces are calculated by a membranemodified flexural theory of elasto-plasticity. The predictions show good agreement with a wide range of experimental results. REFERENCES 1. R. TAYLORand B. HAYES,Some tests on the effect of edge restraint on punching shear in reinforced concrete slabs. Mag. Conc. Res. 17, 39 (1965). 2. Y. P. TONG and B. DEV. BATCHELOR,Compressive membrane enhancement in two-way bridge slabs. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 271. American Concrete Institute, Detroit (1971). 3. B.E. HEWITTand B. DEV. BATCHELOR,Punching shear strength of restrained slabs. J. Struct. Div. Am. Soc. Cir. Engrs 101 (ST9), 1837 (1975). 4. R. A. DORTON, M. HOLOWKAand J. P. C. KING, The Connestogo River Bridge--design and testing. Can. J. Cir. Engng 4(1), 18 (1977). 5. D. BEAL, Load capacity of concrete bridge decks. J. Struct. Div. Am. Soc. Cir. Engrs 108 (ST4), 814 (1982). 6. J. KIRKPATRICK,G. I. B. RANK1Nand A. E. LONG, Strength evaluation of M-beam bridge deck slabs. Struct. Engr 62B (3), 60 (1984). 7. G. I. B. RANKIN and A. E. LONG, Predicting the enhanced punching strength of interior slab-column connections. Proc. Instn Cir. Engrs Part 1 82, 1165 (1987). 8. P.C. PERDIKARISand S. BEIM, RC bridge decks under pulsating and moving load. J. Struct. Div. Am. Soc. Cir. Engrs 114 (ST3), 591 (1988). 9. J. S. KUANGand C. T. MORLEY,Punching shear behaviour of restrained reinforced concrete slabs. ACI Struct. J. 89(1), 13 (1992). 10. S. KINNUNENand H. NYLANDER,Punching of concrete slabs without shear reinforcement. Trans. Royal Inst. Tech., No. 158, Stockholm (1960). 11. ONTARIOMINISTRYOF TRANSPORTATIONANDCOMMUNICATIONS,The Ontario Highway Bridge Design Code. Toronto (1979). 12. A.E. LONG,A two-phase approach to the prediction of the punching strength of slabs. J. Am. Conc. Inst. 72 (2), 37 (1975). 13. M. P. NIELSEN,Limit Analysis and Concrete Plasticity. Prentice-Hall, New York (1984). 14. M. W. BR.,ESTRUP,Punching shear in concrete slabs. Plasticity in Reinforced Concrete, Introductory Report, p. 115. Colloquium of IABSE, Copenhagen (1979). 15. W. F. CHEN, Plasticity in Reinforced Concrete. McGraw-Hill, New York (1982). 16. M. W. BRF~.STRUP,Plastic analysis of structural concrete. Proc. 1 U TAM Symposium, p. 661. Evanston, IL (1983). 17. J.S. KUANG,An Upper-Bound Plastic Solution for Punching Shear Failure of Concrete Slabs, Technical Report No. CUED/Struct-D/TR. 136. Department of Engineering, University of Cambridge, U.K. (1991).
A plasticity model for punching shear
385
18. M. W. BR.ESTRUP and C. T. MORLEY, Dome effect in reinforced concrete slabs: elastic-plastic analysis. J. Struct. Div. Am. Soc, Cir. Engrs 106 (ST6), 1255 (1980). 19. E. HOGNESTED,A Study of Combined Bending and Axial Load in Reinforced Concrete Members, Bulletin No. 339. Engineering Experiment Station, University of Illinois (1951). 20. C. T. MORLEY,Some experiments on circular concrete slabs with lateral restraint. Paper presented at the International Conference on Engineering Mechanics, Warsaw (1974). 21. J. F. BROTCHIEand M. J. HOLLEY,Membrane action in slabs. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 345. American Concrete Institute, Detroit (1971). 22. L. C. SNOWDON,Some Tests on the Strength in Punching Shear of Restrained Concrete Slabs, unpublished report. Building Research Station, Wafford (1973). 23. R. PARK, Ultimate strength of rectangular concrete slabs under short-term uniform loading with edges restrained against lateral movement. Proc. lnstn Cir. Engrs 28, 125 (1964). 24. M. HOLOWKA,R. DORTONand P. CSAGOLY,PunchingShear Strength of Restrained CircularSlabs, unpublished research report. The Ontario Ministry of Transportation and Communications, Toronto (1979). 25. Y. AOKI and H. SEKI, Shearing strength and cracking in two-ways slabs subjected to concentrated load. Cracking, Deflection and Ultimate Load of Concrete Slab Systems, SP-30, p. 103. American Concrete Institute, Detroit (1971). 26. BRITISHSTANDARDINSTITUTION,Code of Practicefor Design and Construction, BS 8110: Part 1. BSI, London (1985). 27. ACI COMMITTEE318, Building Code Requirementsfor Reinforced Concrete and Commentary (ACI 318-89 and ACI 318R-89). American Concrete Institute, Detroit (1989).