INTRODUCTION AND LITERATURE REVIEW
A- CWSLC; B-CWSMC; C-CWSSC
4.4 MATERIAL MODELLING
translations in the nodal X, Y and Z directions. Fig. 4.4 shows SOLID46 element.
Fig. 4.4 SOLID46 Element
Fig. 4.5 Typical Stress–strain curve of concrete. [Chung et al. 2002]
in Fig. 4.5. The point A represents the peak stress fcc and strain εcc . B corresponds to the stress and strain of 0.85 fcc and ε0.85 respectively, while C represents the stress and strain of 0.3 fcc and ε0.3. The co-ordinate of C lies at the extension line obtained by connecting the points A and B.
4.4.1.1 Multi-linear Isotropic Hardening Stress-Strain Curve
The uniaxial stress-strain behaviour of concrete was proposed by many researchers in the form of different empirical formulae. In the present numerical analysis, the uniaxial behavior of concrete was modeled by the numerical expression proposed by Desayi and Krishnan [1965] incorporating the modification proposed by Gere and Timoshenko [1997]. It is described by a piece wise linear stress-strain curve, starting at the origin with positive stress and strain values. The slope of the first segment of the curve must correspond to the elastic modulus of the material. All segments other than the first one should have slope less than elastic modulus of the material. All segments should have positive slope. The ANSYS program requires the uniaxial stress-strain relationship for concrete in compression only. The multi-linear isotropic stress-strain implementation requires that the first point (0.3fcc ) of the curve be defined by the user. Stress strain properties of concrete up to failure is presented in Table 1 (Appendix A) and the TH-951_06610408
multi-linear isotropic stress-strain curve for concrete has been shown in Fig. A.1 (Appendix A).
4.4.1.2 Failure Criteria for Concrete
The concrete material model available in ANSYS is capable of predicting failure modes like cracking and crushing failure. The two input strength parameters i.e., ultimate uniaxial tensile and compressive strength are needed to define the failure surface for the concrete. The 3-D solid element (SOLID65) is capable of incorporating cracking and crushing. Further, the ability of concrete to undergo plasticity with the William and
Fig. 4.6 Failure surface of concrete
Warnke [1974] failure surface besides creep behaviour had also been utilized. The presence of a crack at an integration point is represented through modification of the stress-strain relations by introducing a plane of weakness in a direction normal to the crack face. If the material at an integration point fails in uniaxial, biaxial, or triaxial compression, the material is assumed to crush at that point. In SOLID65, crushing is defined as the complete deterioration of the structural integrity of the material (e.g.
material spalling). Further, crushing implies that the material strength have degraded to such an extent that the contribution to the stiffness of an element at the integration point TH-951_06610408
in question can be ignored.
A failure surface for concrete is shown in Fig. 4.6. The most significant nonzero principal stresses are in the X and Y directions, represented by σxp and
σyp respectively. Three failure surfaces are shown as projections on the σxp −σypplane. The mode of failure is a function of the sign of σzp (principal stress in the Z direction). For example, if σxp and σypare both negative (compressive) and σzp is slightly positive (tensile), cracking would be predicted in a direction perpendicular to theσzp. However, if σzp is zero or slightly negative, the material is assumed to crush [ANSYS 2003].
4.4.2 Steel Reinforcement
Unlike concrete, steel is very uniform and as such generally the specification of a simple stress-strain relation is adequate to define it numerically. Typical stress-strain curves for reinforcing steel bars used in concrete construction are obtained from standard tensile tests. Practically speaking, steel exhibits the same stress-strain curve in both compression and tension. The stress-strain relation shows a linear elastic portion at the initial stage, followed by a yield plateau until a strain hardening region is reached, in which stress again increases with strain before eventually dropping off as microscopic fracture occurs.
The length of the yield plateau is dependent on the tensile strength of steel. The yield plateau of high strength, high-carbon steels is generally much shorter than relatively low- strength, low-carbon steels. Typical uniaxial stress-strain curve are as shown in Fig. 4.7 for various grades of steel.
Modulus of elasticity of steel (ES), which corresponds to the slope of the initial linear elastic portion was adopted as 2×10 N/mm as per IS-456 [2000] in the present analysis. 5 2 An abrupt change in the stress-strain behaviour is observed for the material beyond the yield point. Moreover, upon unloading the material at any point beyond the yield point, a permanent deformation is introduced. This kind of behaviour can be idealized by a TH-951_06610408
Fig. 4.7 Uniaxial stress-strain curves for steel
bilinear stress-strain relationship with two slopes, the second slope being called the tangential modulus (ET). After reaching the yield point, the slope could be less than, equal to, or greater than zero. In the present analysis (ET) has been assumed as1000N/mm . One typical idealized stress-strain curve for steel is shown in Fig. 4.8. In 2 addition to modulus of elasticity, the value of Poisson’s ratio is also provided as the input data for modeling of steel in FE analysis. This value was assumed as 0.3 in the present analysis.
Fig. 4.8 Typical idealized stress-strain curve for steel
4.4.3 FRP Composite
FRP composite is a brittle material. The stress-strain curve is a purely straight line up to the failure strength. The various moduli which are needed for the modeling of FRP composite in FE analysis with ANSYS have been adopted from the properties of FRP
0 100 200 300 400 500 600
0 0.04 0.08 0.12 0.16 0.2
strain (mm/mm)
Stress (MPa) Fe 250
Fe 415 Fe 500
2 10 (for all steel)5
Es = ×
TH-951_06610408
composite as shown in Table 4.1. Composites are orthotropic materials. There exist numerous strength theories for prediction of failure of orthotropic materials. These are the maximum stress theory, maximum strain theory, Tsai Hill theory and Tsai Wu theory.
Tsai Hill theory [Tsai, 1966] is regarded as superior. The advantage of this theory is that it gives a single criterion for the detection of failure as compared to the other stress and strain based criterion.
Table 4.1 Properties of FRP Materials
Type of FRP
Ultimate Strength (N/mm2)
Ex
(MPa) Ey
(MPa) Ez
(MPa) xy yz zx
Gxy
(MPa) Gyz
(Mpa) Gzx
(Mpa)
Fibre Orientat- ion CFRP 670 46300 46300 23309 0.3 0.25 0.25 16189 9713.3 9713.3 00 GFRP 350 11670 11670 7787.7 0.3 0.25 0.25 10385 5182.7 5182.7 ±450