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Nonlinear analysis of normal- and high-strength concrete slabs
H. MARZOUK AND Z.W. CHEN Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John's, Nfld.,
Canada A IB 3x5
Received June 26, 1992
Revised manuscript accepted January 4, 1993
Concrete slabs supported on four edges and loaded axially and transversely are used in many civil engineering applica- tions. High-strength concrete slabs are commonly used for marine structures and offshore platforms. The catastrophic nature of the failure exhibited by reinforced concrete slabs when subjected to concentrated loads has been a major concern for engineers over many years. Therefore, there is a great need to develop accurate numerical models suitable for normal-strength or high-strength concrete in order to reflect properly its structural behaviour.
Proper simulation of the post-cracking behaviour o f concrete has a significant effect on the nonlinear finite element response of such slabs. Cracking and post-cracking behaviour of concrete which includes aggregate interlock, dowel action, and tension-stiffening effects is especially crucial for any nonlinear concrete analysis. The post-cracking behaviour and the fracture energy properties of high-strength concrete are different from those of normal-strength concrete. This can be realized by comparing the experimental testing results of plain normal- and high-strength concrete. The experimental results of testing plain high-strength concrete in direct tension indicated that the total area under the stress - crack width curve in tension is different from that of normal-strength concrete.
A suitable softening and tension-stiffening model is recommended for high-strength concrete; other existing models suitable for normal-strength concrete are discussed. The proposed post-cracking behaviour models are implemented in a nonlinear finite element program in order to check the validity of such models by comparing the actual experimental data with the finite element results. Finally, a parametric study was conducted to provide more insight into the behaviour of high-strength concrete slabs subjected to combined uniaxial in-plane loads and lateral loads. The effects of the magnitude of in-plane load and the sequence of loading on the structural behaviour of such slabs are examined.
Key words: high-strength concrete, slabs, punching shear, fracture energy, tension-softening, tension-stiffening, parametric study.
Les dalles de beton avec support en bordure, soumises a des charges axiales et transversales, ont bon nombre d'appli- cations dans le domaine du genie civil. Les dalles de beton a haute resistance sont frequemment utilisees dans la cons- truction de structures marines et de plate-formes au large des c6tes. La nature catastrophique des defaillances constatees dans les dalles de beton arme soumises a des charges concentrees preoccupe les ingenieurs depuis bon nombre d'annees. Par consequent, il y a tout lieu de developper des modeles numeriques precis qui permettent de predire adequatement le comportement structural de beton a resistance normale et a haute resistance.
Une simulation adequate du comportement du beton aprks fissuration a un effet important sur la reponse non lineaire des elements finis des dalles de beton. Le comportement en fissuration et apres fissuration du beton, incluant le couplage des granulats, l'action des goujons et les effets du raidissement de traction, sont d'un interEt crucial pour toute analyse non lineaire du beton. Le comportement apres fissuration et les proprietes energetiques de rupture du beton a haute resistance different de ceux du beton a resistance normale. Cette constatation est possible en comparant les resultats des essais experimentaux des deux types de beton. Les resultats experimentaux des essais du beton a haute resistance en traction directe indiquent que la surface totale sous la courbe contrainte - largeur de fissure en traction differe de celle du beton a resistance normale.
Un modele adequat d'adoucissement et de raidissement de traction est recommande pour le beton a haute resistance; les autres modeles qui conviennent au beton a resistance normale sont discutes. Les modeles de comportement apres fissuration proposes sont mis en oeuvre dans le cadre d'un programme d'elements finis non lineaire afin d'en verifier la validite en comparant les donnees experimentales aux resultats de la methode des elements finis. Enfin, une etude parametrique a ete realisee afin de mieux comprendre le comportement des dalles de beton a haute resistance soumises a la fois a des charges laterales et uniaxiales au plan de I'ossature porteuse. Les effets de l'ampleur des charges au plan de I'ossature porteuse et de la sequence de chargement sur le comportement structural de ces dalles sont examines.
Mots clPs : beton a haute resistance, dalles, force de cisaillement, energie de rupture, adoucissement de traction, raidissement de traction, etude parametrique.
[Traduit par la redaction]
Can . J . Civ. Eng. 20. 696-707 (1993)
Introduction High-strength concrete is widely used for offshore con-
crete platforms, marine structures, tall buildings, and long- span bridges. High-strength concrete slab plates represent the main structural element of a concrete offshore structure.
NOTE: Written discussion of this paper is welcomed and will be received by the Editor until December 31, 1993 (address inside front cover).
Several nonlinear finite element models are available to model normal-strength concrete behaviour. However, the research work related to high-strength concrete is only limited to change some of the mechanical properties like f,', 8, f,, and E without considering the major difference in post-cracking behaviour of the two materials. Based on the material testing program conducted at Memorial University of Newfoundland (Marzouk and Hussein 1990, 1991a; Chen 1993), it has been found that a fundamental difference exists
Printed in Canada / lmpr~mc au Cal~ada
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, ........................................ ,,.!.Special spring elements i ...
Transverse point load
---- A '
3 x 3 F.E. mesh for one quarter slab i 8
Reinforcement
- 750 mm -4 4
i in transverse direction i ..........................................
FIG. I . Finite element mesh and model used for the analysis: (a) 8-node quadrilateral shear flexible shell element with the reduced 2 x 2 Gaussian integration over the plane and nine-point Simpson integration through the thickness; (b) modelling of one quarter slab.
between the post-cracking behaviour of the two materials, in particular the effect it has on tension stiffening and fracture energy. The difference between high-strength and normal-strength concrete has a profound effect on the struc- tural behaviour of the slab structural element. The flexural strength, punching capacity, and load-deflection character- istics of high-strength concrete slabs are different from those of normal-strength concrete slabs, as detailed by Marzouk and Hussein (1991a, 1991 6) .
This paper focuses on the difference in post-cracking behaviour between high-strength concrete and normal- strength concrete. The tension-softening behaviour of normal-strength concrete is quite different from that of high- strength concrete. Therefore, the post-cracking behaviour of high-strength concrete must be properly reflected through a tension-stiffening and shear degradation model suitable for nonlinear finite element analysis.
Material model A plasticity-based incremental elastic-plastic concrete
model using a simple form of yield surface written in terms of the first two stress invariants is used for both materials. The model adopts the classical concepts of plasticity theory: a strain rate decomposition into elastic and inelastic strain rates; elasticity; yield; flow; and hardening. The post- cracking behaviour is assumed based on the brittle fracture concept of Hilleborg et al. (1976) and Crisfield (1986) where the fracture energy is required to form a unit area of crack surface. This model significantly simplifies the actual behav- iour of associated flow and isotropic hardening. The adopted material model is described in full detail by ABAQUS (1990).
Cracking dominates the material behaviour in the case of tensile state of stress. A crack detection plasticity surface is used to determine the location of the crack and the orien-
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'Jl V1 Q)
M + 'Jl
Normalized strain, et/eto
FIG. 2. Normalized average net stress vs. normalized strain.
tation of the crack. The analysis is based on a smeared crack model, where constitutive calculations are performed inde- pendently at each integration point of the finite element model. Once concrete has cracked, three phenomena are normally manifested, namely, aggregate interlock, dowel action, and tension stiffening. Aggregate interlock or "shear friction" accounts for the transfer of shear force across a crack, when shear displacements occur parallel to the direc- tion of the cracks. As the crack width increases, the con- tact area of the concrete on the two sides of the crack decreases. Thus, the shear forces transferred by the aggregate interlock or the shear friction mechanism decrease. In reinforced concrete structures, reinforcement crossing the cracks tends to prevent the cracked pieces from moving apart and provides significant clamping forces which enhance aggregate interlock mechanisms. Moreover, any movement of cracked pieces of concrete parallel to the crack causes the reinforcement crossing the crack to transfer shear forces by dowel action. On the other hand, due to the bond effect, concrete is still capable of carrying tensile stress after the formation of primary cracks. As the load increases, more secondary cracks appear and tensile stresses in the concrete are releated gradually. The cracking behaviour of concrete is modelled through a smeared crack approach in this study.
Finite element program The concrete material model described earlier is provided
in a general purpose finite element code, ABAQUS (1990). This code provides various types of elements for one-, two-, and three-dimensional problems such as plane stress, plane strain, three-dimensional solid elements, straight or curved beams, and shell elements. In the shell element library, curved geometries as well as axisymmetric shells are pro- vided, which contain both "thin" shells - shell elements in which the Kirchhoff constrains are imposed algebraically, and "shear flexible" shells - shell elements in which transverse shear deformation is allowed. The essential idea of shear flexible shells is that the displacement of a point in the shell reference surface and the components of a vector which is approximately normal to the reference surface are
interpolated independently. This allows for nonlocking shear and membrane deformation. The performance of these shell elements is further improved by introducing the concept of selectively reduced integration. Thus, the application of such shear flexible shell elements has been extended into the range of thick shell problems.
The program is suitable for nonlinear problems, since an automatic control of time stepping is provided. When a step and certain tolerance are defined, the program automatically selects the increments to model the step. In a nonlinear analysis, the automatic incrementation is commonly used due to the unknown nature of the response. This is based on the convergence rate of the early iteration process at each increment. The automatic incrementation assumes that con- vergence will occur relatively smoothly as the iterations progress, so that the number of iterations needed to achieve convergence to the specified tolerances can reasonaly be predicted from the rate at which the residuals are reduced from one iteration to the next during the early iterations. However, the convergence within an increment is often erratic when the adopted material model involves oriented failure due to concrete cracking, crushing, and rebar yielding. Therefore, the automatic subdivision algorithm tends to lead to unreasonably small increments.
The modified Riks algorithm is used to obtain static equi- librium. The ultimate collapse can be genuinely defined to reveal the collapse mode and the ductility or deformation capacity of the structure. This is usually difficult to achieve with a load control solution strategy. The Riks algorithm is based on attempting to step along the equilibrium path (the load-displacement response curve) by prescribing the path (arc) length along the curve to be traversed in each increment, with the load magnitude included as unknown.
One quarter of the slab is modelled with a 3 x 3 mesh using an 8-node quadrilateral shear flexible element (thick shell) with six degrees of freedom at each node. A 2 x 2 reduced Gaussian integration rule is used over the element plane and nine Simpson-type integration points are used through the thickness of the concrete slab. The finite element model and the used mesh are shown in Fig. 1.
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Normalized average strain, et/eto
FIG. 3. Tension-softening model of high-strength concrete up to four times tensile strain at the peak load.
Post-cracking behaviour Post-cracking behaviour of concrete is of vital importance
when nonlinear finite element analysis is used for accurate predictions of deflection, crack width, bond transfer, and shear transfer. Several researchers conducted tests on the cracking behaviour of plain concrete in direct tension, such as Gopalaratnam and Shah (1985), Guo and Zhang (1987), Baiant and Oh (1983), and Yankelevsky and Reinhardt (1987). According to the brittle fracture concept of post- cracking behaviour of concrete (Hilleborg et al. 1976), the fracture energy required to form a unit area of crack surface, GF, can be calculated by measuring the tensile stress as a function of the crack opening displacement:
In general, the fracture energy can be calculated from the area under the net stress versus average displacement of a plain concrete specimen under direct tensile load. Also, the width of the fracture-process zone, w,, is used by Baiant and Oh (1983) to define the average crack strain. The energy dissipated in the opening of a crack in a tension specimen, GF, is directly related to the strain energy density, Awi, as illustrated by Massicotte et al. (1990b). The area under the ascending and descending branches of the stress-strain curve of plain concrete in tension is as follows:
Several research workers simplified the stress-strain relation of plain concrete under tensile load and the calculation of Awi, such as Scanlon (1971), Lin and Scordelis (1975), Baiant and Oh (1983), and Massicotte et al. (1990b).
For normal-strength concrete, it has been found that a bilinear simplified curve such as that proposed by Baiant and Oh (1983) or a trilinear curve as proposed by Massicotte et al. (1990b) with a linear ascending branch and a bilinear softening branch for concrete beyond cracking can provide good results for the prediction of normal-strength concrete
slabs. However, when such models are used for high- strength concrete slabs the results were not very accurate.
Massicotte et al. (1990b) reported that the detailed analysis of 52 test results revealed that the total area under the ascending and descending branches of the stress-strain curve of the plain normal-strength concrete under tensile load is equal to ten times the area under the ascending branch of the same curve. The experimental evidence by Marzouk and Chen (1993), from testing plain high-strength concrete (including silica fume in the mix) under tensile load, indicated that the stress-strain characteristics are quite different from those of normal concrete. The total area, Awi, under the stress-strain curve is only about five times the area under the ascending branch of the same curve. Also, the crack strain in tension is recorded at about 120 micro- strain. The recorded normalized average net stress versus the normalized strain for high-strength concrete in direct tension is shown in Fig. 2. The complete details of the test setup and instrumentation are provided by Marzouk and Chen (1993). The ascending branch of that curve can be fitted perfectly to a parabolic equation as follows:
where a = 2 and b = - 1. For the descending branch, it has been found that an equation similar to the expression used by Guo and Zhang (1987) provides an excellent fit.
where a = 2.84 and /3 = 1.6655. The fitted equations ([3] and [4]) can represent the test results as shown in Figs. 3 and 4. Therefore, the idealized tension behaviour of high- strength concrete, based on direct tension tests of high- strength concrete, can be proposed as follows:
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1 Test result (ascending) + Test result (descending) - Fitted curve
0.8
0.6
0.4
0.2
Normalized average strain, E ~ / E ~ ~
FIG. 4. Complete tension-softening model of high-strength concrete.
- E, is the average crack opening strain and is the concrete tensile strain at the peak load. According t o the detailed analysis of 52 test results by Massicotte et al. (1990b), the cracking energy is significant up to an ultimate tensile strain of 16 times cIo. Therefore, the strain energy density, Awi , of plain high-strength concrete can be provided by integrating the area under the curve represented by the following equation:
load.
Tension-stiffening model It has been well established that the tension-stiffening
effect of reinforced concrete is dependent on the percentage of steel, diameter of steel reinforcement, distribution of rein- forcement and bond stresses. Also, it has been found that tension stiffening is more pronounced in the reinforced con- crete element with low percentage of reinforcement. The strain energy density, A,, of reinforced high-strength concrete must be estimated from the strain energy density, Awi , of plain high-strength concrete. Prakhya and Morley (1 990) recommended that a tension-softening curve could be used with some modifications to the descending branch of the stress-strain curve of plain concrete t o reflect the tension-stiffening behaviour of a reinforced concrete element.
The same concept can be adopted for high-strength con- crete by changing the descending branch of the tensile
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Normalized average strain, et/cto
FIG. 6. Complete tension-stiffening model of high-strength concrete.
stress-strain curve. This can be easily accomplished by changing the value of a in [4] to convert the tension- softening expression into a tension-stiffening curve. The change of a values will change the total area under the curve from A w i to A,, as illustrated in Figs. 5 and 6 . The initial portion of the curve up to a normalized average strain of E [ / E , ~ = 4 is shown in Fig. 5. The normalized strain value of E [ / E [ ~ of 16 is used as the cut-off point for the integra- tion process. It is evident from Figs. 5 and 6 that there is hardly any change in the ascending portion of the curve.
A parametric study was carried out to establish the area under the stress-strain curve, in order to convert the soften- ing behaviour ( A w i ) into a stiffening behaviour (A,). The results of over 20 high-strength concrete slabs tested at Memorial University of Newfoundland were utilized in a parametric study to determine the magnitude of tension stiffening of high-strength concrete slabs (Chen 1993). It has been found that for slabs with high to moderate steel rein- forcement ratios, the tension-stiffening values can be rep- resented as follows:
[7] A, = 1.20Awi to 2.0Awi
For slabs with low steel reinforcement ratios, the tension stiffening values are
For the analyzed high-strength concrete slabs, the strain energy density of reinforced concrete (A,) was chosen to be 1.5 the strain energy density of plain concrete ( A w i ) . Therefore, the fitted curve given by [6] can be modified by changing the constant (a) from 2.84 to 1.3862 in order to provide the proper tension-stiffening model for high-strength concrete slabs as follows:
The proposed model is then implemented into the finite element program. The proposed tension-stiffening model given by [9] may cause numerical problems for certain computer programs. It has been found that a simplified bilinear stress-strain curve can also provide a good result, provided the area under the stress-strain curve is only about five times the area under the ascending portion and if the appropriate steel reinforcement ratios modification factors represented by [7] or [8] are implemented.
In order to illustrate the post-cracking behaviour effects on high-strength concrete slab behaviour, slab HS5 tested by Marzouk and Hussein (1991a) is further examined. Different tension-stiffening values are assumed based on the proposed model and a simplified bilinear model. Figure 7 represents different tension-stiffening values based on the recommended model (model 1) expressed by [ 9 ] . The dif- ferent values were obtained by modifying the area under the softening curve ( A w i ) by factors of 2.6, 1.7, 1.50, and 1.20. In Fig. 8, a simplified bilinear tension-stiffening curve similar to the one recommended by Baiant and Oh (1983) will be referred as model 2. The simplified model is used instead of [9] to illustrate the post-cracking effect of high- strength concrete. The simplified expression can be used when the recommended expression poses a numerical prob- lem with the computer program. Extremely high tension- stiffening values of A , = 5.3Awi was assumed, as shown in Fig. 8, to demonstrate the effects of tension-stiffening values on the behaviour of high-strength concrete slabs.
The concrete shear modulus of the cracked plane is reduced by a linear type of function as follows:
' 16Ei11 f ;x dx where Go is the initial shear modulus of concrete in its + 1 1 . 3 8 6 3 ( ~ - 1)1.6655 + x uncracked state, ( is the linear degradation function of shear
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Model 1
_-- 4 ,,," anlt
- * Test re--.. 3 .,' (1) - Model 1, A, = 1.2&. " :2--.,.. (2) -.- Model 1, A, = 1.5&.
3 100 ;' (3) - Model 1, A, = 1.7&. (4) - - Model 1, A, = 2.6&. 1
Central deflection, D (mm)
F I G . 7. Tension-stiffening effect on the load-deflection curve of slab HS5 (model 1; eq. [9]).
*
Test result (1) - Model 2, = 5.803-4, (A, = (2) -.- Model 2, c,", = 7.40E4, (A, = (3) - - Model 2, c,", = 8.403-4, (A, =
(4) - Model 2, €2 = 14.OE4, (A, = (5) -.- Model 2, EL = 16.OE4, (A, =
Central deflection, D (mm)
F I G . 8. Tension-stiffening effect on the load-deflection curve of slab HS5 utilizing a simplified bilinear model (model 2 ) .
modulus. E,, and E,U, are the total average shear-effective crack opening strain and ultimate shear-effective tensile strain at which concrete shear stiffness is reduced to zero. E,, is assumed to be equal to 0.005 and 0.001 for normal- and high-strength concrete respectively. The concrete shear stiffness is assumed to have been damaged for the cracks that subsequently close.
[13] ,$ = ,$"Ow for El < 0
where ,$"OSe is the reduction factor of shear modulus related to the cracks that subsequently close during the calculation. This was assumed equal to 0.8 for both normal- and high- strength concrete.
Validation of the post-cracking model Slabs loaded transversely
In order to check the validity of the proposed post- cracking behaviour of normal- and high-strength concrete, the model was tested to predict the experimental behaviour of previously tested slabs loaded transversely (Marzouk and Hussein 1991a). In order to illustrate the test setup, slab HS17 is shown in Fig. 9 during testing and in Fig. 1 0 after the test was completed. The new model was tested and the results are reported by Chen (1993). In this work, only the results of two normal-strength concrete slabs, N S l and NS2, and two high-strength concrete slabs, HS3 and HS17, are presented to validate the accuracy of the finite element model, as given in Table 1. It is evident that the predicted
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FIG. 9. Test setup for slab HS17.
TABLE 1. Results of finite element analysis versus experimental measurements on four tested slabs
Normal-strength High-strength
Property NS 1 NS2 HS3 HS17
Slab thickness (m) Reinforcement ratio,
P (Yo) f, (MPa) f,' (MPa) .A (MPa) Poisson's ratio, p Type of failure
1.093 420 67 3.48 0.24
Ductile Punching Parabolic/ power function
1.473 496 42 3.51 0.2
Punching
0.944 496 30 2.5 1 0.2
Ductile Punching Bilinear
1.474 496 69 3.59 0.24
Punching
Tension stiffening Bilinear Parabolic/ power function
Recorded failure load (kN)
Finite element failure load (kN)
Recorded ultimate deflection (mm)
Finite element ultimate deflection
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F I G . 10. Slab HS17 after testing.
behaviour of both normal- and high-strength concrete slabs is adequate, compared with the test results, as shown in Figs. 11-14, when appropriate post-cracking behaviour is considered. However, it is important to point out that an 8-node quadrilateral flexible shear element with six degrees of freedom per node was used in the analysis. Therefore, it is expected that the results of finite element analysis for the pure punching shear failure types will be less accurate than the results of flexural failure and ductile-punching failure.
Slabs loaded axially and transversely The behaviour of normal-strength concrete slabs loaded
axially and transversely was extensively examined at the University of Alberta recently. The research included both experimental (Aghayere and MacGregor 1990) and numerical (Massicotte et al. 1990a, 1990b) investigations. Their study was conducted on normal-strength concrete slab panels with uniaxial compressive strength, f,', of 35 MPa. The results of these investigations will be used to compare with the predicted behaviour of high-strength concrete slabs.
In order to investigate numerically the effect of the in- plane loading on the response of high-strength concrete slab supported on four edges and free to rotate and unrestrained in the in-plane direction, slab HS17 is selected for the parametric study. The uniaxial compressive strength of slab HS17 concrete is 67 MPa; the reinforcement steel ratio, p , is 1.093%; and the overall dimensions are 1950 x 1950 x 150 mm. It can be seen that the lateral load-carrying capacity of slab HS17 is improved with the application of the in-plane load, as shown by comparing Figs. 14 and 15. From Fig. 15, it can be observed that the lateral load-carrying capacity was further improved with the increase of in-plane load magnitude. The applied in-plane load relative magnitude, I,, varied between 0.10 and 0.40, where I,,, is
in which P i s the total in-plane load, a and h are the width and the thickness of the sIab respectively, and f,' is the ultimate uniaxial compressive concrete strength. This phe- nomenon indicates that the compressive membrane force,
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Central deflection, D (mm)
FIG. 1 1 . Comparison of finite element model with test results for slab NSI.
Central deflection, D (mm)
FIG. 12. Comparison for finite element model with test results for slab NS2.
introduced by axial in-plane load, enhanced the lateral load capacity in a similar fashion to the prestressing of concrete members. However, ductility was significantly reduced due to the application of in-plane load, but remained basically the same with further increase of in-plane load magnitude. Similar results were reported for normal-strength stocky panels by Massicotte et al. (1990a). Slab HS17 failed because of concrete crushing (brittle failure) before the extensive yielding due to the initial compressive strain introduced by the axial in-plane load was developed.
The effect of the loading sequence on slab HS17 response is demonstrated in Fig. 16. In the first case, the lateral load, Q, was applied up to a load of 440 kN, close to the ultimate load capacity of the slab under lateral load alone (Q =
Central deflection, D (mm)
FIG. 13. Comparison of finite element model with test results for slab HS3.
* Test result d
e - Model 1, A, = 1 . 5 A ~ 0
3 100
Central deflection, D (mm)
FIG. 14. Comparison of finite element model with test results for slab HS17.
512.0 kN). Then the lateral load was kept constant while the in-plane load was being applied, up to a value of I , = 0.11. In the second case, the in-plane load was applied to I , = 0.11 and kept constant, while the lateral load was increased until it reached the value of Q = 456.0 kN. This was close to the failure load in the first case, but ductility was significantly reduced. The best performance of the slab was achieved in the third case, where the lateral and in-plane loads were applied proportionally. This produced a lateraI value of Q = 772.0 kN and I , = 0.19. The load capacity was improved by 75%; however, the deflection was also the largest among the three cases analyzed. The slab ductility was consistently reduced due to in-plane load, regardless of loading sequence, as shown in Fig. 17. Extensive cracking
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3 0 - 200 Model 1
+ I, = 0.11; Q = -456 kN Im=0.19;Q=-528kN
* I, = 0.40; Q = -624 kN
Central deflection, D (mm) FIG. 15. Effect of in-plane load on the load-deflection curve
of slab HS17.
Model 1 * Q = -440 kN; I, = 0.11
100 + I, = 0.11; Q=-456 kN 0 Q = -772 kN; I, = 0.19
Central deflection, D (mm)
FIG. 16. Effect of loading sequence on the load-deflection curve of slab HS17.
of such a slab was prevented by the application of the in- plane load. Therefore, a more brittle failure was evident. In general, a reduction of about 43% in load capacity was associated with prior application of either P or Q in both axial and lateral loads. This indicates the slab response could be very sensitive to the loading sequence.
Massicotte et al. (1990a) reported similar effects on the loading sequence of normal-strength, thick concrete slabs. It has been suggested that when lateral load is applied first, lateral deflection is significantly larger than in the other two cases. This is due to extensive cracking and the impact of the second-order effects.
Conclusions The post-cracking behaviour and the fracture energy of
high-strength concrete are different from those of normal- strength concrete. This fundamental difference must be
Model 1 * Q=-772 kN;I,,=0.19
100 1 o Q=-512 !+I only i Central deflection, D (mm)
FIG. 17. Effect of I,,, at proportional loading on the load- deflection curve of slab HS17.
reflected in any nonlinear finite element analysis. The difference between the two materials is not only related to the material properties such as f:, f;, p, and E but other differences must be considered. The post-cracking behaviour and the fracture energy must be reflected in the analysis. The tension-softening and consequently the tension- stiffening of each material must be properly represented in the nonlinear analysis. The difference is clearly indicated by the stress-strain behaviour of plain concrete under direct tensile load, the area under the curve (Awi), and the ratio of the area under the total area to the ascending portion under the curve. The total area (A,,,i) under the stress-strain curve is about five times that of the area under the ascend- ing branch for high-strength concrete and about ten times that of normal-strength concrete.
A tension-softening and stiffening model is recommended for the analysis of high-strength concrete slabs. The model is based on the experimental measurements of strain- softening behaviour of high-strength concrete and on a parametric study conducted over 20 high-strength concrete slabs. The recommended tension-stiffening model can be represented as follows:
The proposed model was implemented in the finite ele- ment program and the predicted results are provided. For simplification, a bilinear post-cracking response can also be assumed with the same area under the curve (A,).
An incremental elastic-plastic concrete model imple- mented in the finite element program, formulated in the con- tent of the 8-node quadrilateral shear-flexible (thick) shell element with all six degrees of freedom per node accounted for, is capable of predicting the behaviour of concrete slabs
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with appropriate modelling of the tension stiffening for normal- and high-strength concrete. The program is used to check the test results of slabs loaded laterally. A para- metric study was conducted to provide more insight into the behaviour of high-strength concrete panels subjected to com- bined uniaxial in-plane loads and lateral loads. The results of this numerical study can be summarized as follows:
1. Membrane compressive stress introduced by in-plane load severely reduces slab ductility, resulting in a brittle failure.
2. The ultimate load capacity of the slab is improved with the increase of the in-plane load magnitude for the analyzed slab.
3. Loading history has a significant effect on slab behav- iour. Prior application of a small amount of in-plane load led to approximately the same load capacity as achieved with the prior application of lateral load; however, ductility was significantly reduced. Both ultimate strength and ductility of the analyzed slab are improved when lateral and in-plane axial loads are applied proportionately.
ABAQUS. 1990. User's manual, Version 4-8. Hibbit, Karlsson and Sorensen Inc., Providence, R.I.
Aghayere, A.O., and MacGregor, J.G. 1990. Tests of reinforced concrete plates under combined in-plane and transverse loads. Journal of the American Concrete Institute, 87(6): 615-622.
Baiant, Z.P., and Oh, B.H. 1983. Crack bond theory for fracture of concrete. Materials and Structures, 16(93): 155-177.
Chen, Z.W. 1993. Nonlinear analysis of high-strength concrete structures. M.Sc. thesis, Memorial University of Newfoundland, St. John's, Nfld.
Crisfield, M.A. 1986. Snap-through and snap-back response in concrete structures and the danger of under integration. Inter- national Journal for Numerical Methods in Engineering, 22: 751-767.
Gopalaratnam, V.S., and Shah, S.P. 1985. Softening response of plain concrete in direct tension. Journal of the American Con- crete Institute, 82(3): 310-323.
Guo, Z.H., and Zhang, X.Q. 1987. Investigation of complete stress deformation curves for concrete in tension. American Concrete Institute Materials Journal, 84(4): 278-285.
Hilleborg, A., Modeer, N.M., and Peterson, P.E. 1976. AnaIysis of crack formation and crack growth in concrete by means of fracture mechanics and finite element. Cement and Concrete Research, 6: 773-782.
Lin, C.H., and Scordelis, A.C. 1975. Nonlinear analysis of R.C. shells of general form. American Society of Civil Engineering Proceedings, 101(3): 523-538.
Marzouk, H.M., and Chen, Z.W. 1993. Tension softening behav- iour of high-strength concrete made with silica fume and fly ash. Proceedings of the Third International Symposium on Utilization of High-Strength Concrete, Lillehammer, Norway, Vol. 2, pp. 1154-1160.
Marzouk, H.M., and Hussein, A. 1990. Properties of high-strength concrete at low temperature. American Concrete Institute Mate- rials Journal, 87(2): 167-171.
Marzouk, H.M., and Hussein, A. 1991~ . Experimental investiga- tion on the behaviour of high-strength concrete slabs. American Concrete Institute Structural Journal, 88(6): 701-713.
Marzouk, H.M., and Hussein, A. 19916. Punching shear analysis of reinforced high-strength concrete slabs. Canadian Journal of Civil Engineering, 18(6): 954-963.
Massicotte, B., MacGregor, J.G., and Elwi, A.E. 1990a. Behaviour of concrete panels subjected to axial and lateral loads. ASCE Journal of Structural Engineering, 116(9): 2324-2343.
Massicotte, B., MacGregor, J.G., and Elwi, A.E. 19906. Tension- stiffening model for planar reinforced concrete members. ASCE Journal of Structural Engineering, 116(11): 3039-3058.
Prakhya, G.K.V., and Morley, C.T. 1990. Tension-stiffening and moment-curvature relations of reinforced concrete elements. American Concrete Institute Structural Journal, 87(5): 597-605.
Scanlon, A. 1971. Time dependent deflections of reinforced con- crete slabs. Ph.D. thesis, University of Alberta, Edmonton, Alta.
Yankelevsky, D.Z., and Reinhardt, H.W. 1987. Response of plain concrete to cyclic tension. American Concrete Institute Materials Journal, 84(5): 365-373.
List of symbols material-dependent constants area under the ascending and descending branch of reinforced high-strength concrete tensile stress-strain curve area under the ascending and descending branch of plain high-strength concrete tensile stress-strain curve secant modulus of elasticity concrete tensile stress ultimate concrete tensile strength shear modulus of concrete fracture energy required to form a unit area of crack surface initial shear modulus of concrete axial in-plane load along two opposite slab edges lateral load at slab centre crack width when f, reaches zero width of the crack fracture-process tension-stiffening constant material-dependent constant concrete tensile strain concrete cracking strain average crack opening strain average shear-effective crack opening strain ultimate shear-effective crack opening strain ultimate tensile-effective crack opening strain (simplified bilinear tension-stiffening model) linear shear degradation parameter shear degradation parameter steel reinforcement ratio
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