The concepts discussed in the preceding sections also underlie the constitutive models of concrete behaviour incorporated in most packages currently used for the numerical analy-sis of concrete structures. Although emphaanaly-sis is placed on the description of the post-peak material characteristics which essentially describe the behaviour of a discontinuous mate-rial, since they reflect the effect of cracking on deformation, the material modelling relies on continuum mechanics theories. Within the context continuum mechanics, a theory (e.g., endochronic, damage, plasticity, etc.) or combination of theories (plastic fracturing, visco-plasticity, etc.) considered as the most appropriate for the description of the phenomenologi-cal features of the behaviour of concrete under load is selected for formulating analytiphenomenologi-cal expressions describing the stress–strain and strength characteristics of concrete under
z
Calculation of average stress σα
Fs = Asfy = 56.55 × 417 ≈ 23600 N
Figure 1.13 Assessment of average stress in compressive zone based on the measured values obtained from testing the beam in Figure 1.12. (From Kotsovos M. D., 1987, ACI Structural Journal, Proceedings, 84(3), 266–273.)
generalised (triaxial) stress states. The formulation of analytical expressions is followed by calibration through the use of experimental data. Such typical data obtained from tests on concrete under axisymmetric (Kotsovos and Newman 1980) and biaxial (Kupfer et al.
1969) states of stress are shown in Figures 1.14 through 1.16 and Figures 1.17 and 1.18, respectively.
Figure 1.14 presents stress–strain curves obtained from tests on concrete cylinders under triaxial axisymmetric compression, that is under the combined action of an axial com-pressive stress, σa, and a confining pressure, σc, such that σa > σc assuming compression as positive. The cylinders were first subjected to a hydrostatic pressure (σa = σc) which was increased to a predefined value; then, σc was maintained constant during the subsequent application of the displacement controlled σa which was increased monotonically until the cylinder suffered significant loss of load-carrying capacity.
The figure shows that all curves exhibit similar trends of behaviour which are indepen-dent of the applied σc. Both axial stress–axial strain and axial stress–lateral strain curves comprise ascending and gradually descending branches. It is important to note, however, that, when the σa approaches its peak value, the rate of increase of the lateral strain (i.e., the strain in the direction of σc (<σa) becomes significantly larger than the rate of the axial strain (i.e., the strain in the direction of σa).
Figure 1.14 Stress–strain curves for a typical concrete in triaxial axisymmetric compression (σa > σc).
–10
Figure 1.15 Stress–strain curves for a typical concrete in triaxial axisymmetric extension (σa < σc).
Figure 1.15 presents stress–strain curves obtained from tests on concrete cylinders under triaxial axisymmetric extension (i.e., under the combined action of σa and σc such that σa < σc). As for the case of the triaxial compression tests, the cylinders were first subjected to a hydrostatic pressure (σa = σc) increasing to a predefined value; then, σc was maintained constant during the subsequent application of a displacement-controlled axial stress coun-teracting the vertical component of the hydrostatic pressure until the specimen suffered a complete loss of load-carrying capacity.
The figure shows that all curves exhibit similar trends of behaviour which are indepen-dent of the applied σc, but, unlike the stress–strain curves in Figure 1.14, these curves have only ascending branches; when the axial stress reaches a critical value, the cylinder suffers a complete and immediate loss of load-carrying capacity. However, as for the curves obtained
–1 0 1 2 3 4 5
–1 σa/fc
0 1
√2σc/fc2 3 4
σa = σc
σa > σc
σa < σc σc σα
Figure 1.16 Strength envelope of concrete under axisymmetric states of stress.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
–4 –2 0 2 4
ε1 ε2 ε3 Tensile
σ2 = 0.5σ 1 σa/fc
σ1
Compressive
Strain × 10–3
Figure 1.17 Stress–strain curves for a typical concrete in biaxial compression.
from the triaxial compression tests, the rate of increase of the strain in the direction of the smaller stress, σa is significantly larger than that of the strain in the direction of the larger stress, σc when the axial stress approaches its peak value.
Figure 1.16 shows the variations with confining stress, σc, of the maximum and minimum values of the axial stress, σa, sustained by concrete for the cases of triaxial axisymmetric compression (σa > σc) and extension (σa < σc), respectively. The figure shows that, for the case of triaxial compression, σa increases sharply with σc, even for a small increase of the latter;
similarly, for the case of triaxial extension, σc increases sharply with σa, even for a small increase of the latter. On the other hand, the presence of small tensile stress is sufficient even to reduce to zero the load-carrying capacity of the material in the orthogonal direction.
The stress–strain curves in Figure 1.17 have been obtained from tests on square concrete plates under a plane state of stress σ1, σ2 such that σ2 = 0.5σ1 > 0. As for the case of the stress–strain curves of concrete under axisymmetric compression, the stress–strain curves under biaxial compression comprise an ascending and a gradually descending branch with the descending branch, which describes the out-of-plane deformational response (ε3), exhib-iting a significantly smaller slope than the slopes of the in-plane descending branches (ε1, ε2).
As regards the strength of concrete under biaxial (plane) stress conditions, Figure 1.18 shows that, while the presence of a compressive stress up to fc in any of the two principal directions (σ1, σ2) leads to an up to 25% increase of the compressive strength of concrete in the orthogonal direction, the presence of a small tensile stress (smaller than the uniaxial tensile strength of concrete) rapidly diminishes the compressive strength of concrete in the orthogonal direction to zero.
The calibration of the analytical formulations of the constitutive models proposed to date appears to place emphasis only on the use of stress–strain data (such as those presented above) describing the deformational response of concrete in the direction of the maximum principal compressive stress. (It should be noted that the directions of the axes of symmetry of the specimens tested for establishing the stress–strain behaviour of concrete are the direc-tions of the principal stresses.) On the other hand, the shapes of the stress–strain curves in the directions orthogonal to the direction of the maximum principal compressive stress (i.e., the directions of the intermediate and minimum principal stresses) are dictated by the con-tinuum mechanics theory adopted for the formulation of the analytical expressions.
Although the post-peak material characteristics as described by the constitutive models proposed to date also describe the effect of cracking on deformation, only the formulation of
–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
–0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 σ2/fc
σ1/fc σ1
σ2
Figure 1.18 Strength envelope of concrete under biaxial states of stress.
the deformational response of concrete in compression is considered to adequately describe this effect. Under a state of stress with at least one of the principal stress components being tensile, the numerical description of the effect of cracking on deformation is complemented with the use of numerical techniques (such as e.g., the smeared-crack approach [Ngo and Scordelis 1967] with fixed [Cervera and Chiumenti 2006] or free-rotating [Jirásek and Zimmermann 1998] crack axes and the discrete-crack approach [Saouma and Ingraffea 1981]) which allow for the effect of cracking on deformation through the implementation of modifications in the geometry of the structure analysed. Such techniques place emphasis on the shear resistance considered to be provided by friction developing across the crack inter-faces through the introduction of shear-resistance parameters such as the shear-retention factor (Scotta et al. 2001).
1.2.2 Inconsistencies of concepts underlying constitutive modelling
RC structures subjected to in-plane loading, such as, for example, frames, structural walls and so on are often analysed assuming plane-stress conditions. Even if in-plane loading were possible (since there always exist unintended eccentricities of the applied load), the assump-tion of plane-stress condiassump-tions is unrealistic; this is because the development of out-of-plane actions is inevitable due to variations in the transverse expansion resulting from the non-uniform distribution of the internal stresses. As discussed in Section 2.4.2, such variations in transverse expansion inevitably result in the development of small transverse stresses, for purposes of transverse deformation compatibility, the effect of which is considerable in strength, as indicated in Figure 1.16. Ignoring the development of transverse stresses on account of their small magnitude leads to misinterpretations of the available experimental information and assumptions which divert attention from the true causes of observed and measured structural response.
A typical misinterpretation links the causes of the ‘size effect’ phenomenon (i.e., the dependence of the behavioural characteristics of certain concrete members, such as, beams without transverse reinforcement, on the actual member dimensions) with intrinsic material properties (Bazant and Oh 1963; Hillerborg 1985; Gustafsson and Hillerborg 1988). On the other hand, it has been suggested by Kotsovos and Pavlovic (1994, 1997) that size effects are due to out-of-plane actions resulting from non-symmetrical cracking caused by unintended eccentricities of the applied load and/or the heterogeneous nature of concrete. In fact, it has been shown that realistic predictions of the size effect on the load-carrying capacity of RC beams without transverse reinforcement can be obtained by numerical analysis which allows for the formation of non-symmetrical cracking (Kotsovos and Pavlovic 1997).
As discussed in Section 1.1.2.2, the post-peak stress–strain behaviour of concrete under any state of stress is significantly affected by visible cracking which first occurs when the peak-stress level is reached. As it will be fully discussed in the following chapter, visible cracking predominantly affects the strains measured in the direction orthogonal to that of the crack plane, that is, the strains in the direction of σc in Figures 1.14 and 1.15 and strains ε3 in Figure 1.17. This effect is reflected on the rate of increase of these strains which exceeds the rate of increase of the strains in the other directions by an amount significantly larger than the amount that could be described by a continuum mechanics theory. As a result, constitutive models developed on the basis of a continuum mechanics theory cannot provide a realistic description of the post-peak deformational behaviour of concrete as a material.
The inability of constitutive models based on continuum mechanics concepts to provide a realistic description of the post-peak concrete characteristics is more pronounced for the case of states of stress with at least one tensile principal stress component. In such cases, the description of cracking on deformation is complemented through the use of numerical
techniques (e.g., smeared- or discrete-crack approach) which modify the geometry of the structure so as to account for the effect of cracking in excess of that which is accounted for by the adopted constitutive model.
Moreover, although the large rate of increase of the strains orthogonal to the crack plane implies void formation due to the lack of contact of the crack faces, it is often further assumed that some resistance to the shearing movement of the crack faces is possible to develop due to friction. This resistance is usually described by assuming that ‘cracked’ con-crete retains a portion of the shear rigidity of ‘uncracked’ concon-crete.
As a result of the above inconsistencies characterizing the constitutive models, the appli-cability of the analysis packages which incorporate them appears to be limited only to par-ticular structural elements. In fact, there has been no evidence presented to date on the ability of the analysis packages to provide realistic predictions of the behaviour of a wide range of structural configurations without a suitable modification of the adopted constitu-tive model of concrete behaviour. The above implies that the ability of the analysis packages for realistic predictions is linked with the use of a constitutive model dependent on the type of structural element analysed; however, there has not as yet been any criterion suggested for selecting a ‘suitable’ constitutive model. This apparent lack of generality of the analysis packages, as a result of the lack of objective criteria for adopting a particular constitutive model, represents a major drawback of the use of numerical methods for the analysis of RC structures. Moreover, even if such criteria existed, linking the ability of realistic predictions of structural behaviour to the use of a particular constitutive model does not appear to be rational, since such a link implies that concrete possesses some sort of intelligence allowing it to adapt its behaviour to the needs of particular structures.