Hillerborg’s Model for Compressive Failure in Concrete Beams
Hillerborg’s (1990) model for compressive failure in concrete beams follows, in the formal aspects, the classical bending theory for concrete: a uniaxial stress-strain relation with plane cross sections remaining plane and no-tension for concrete. The essential difference is that he introduces softening and strain localization in compression to explain the size effect on ductility.
Fig. 10.5.10a shows the central section of the beam where the inelastic behavior is represented by a no-tension crack and a compressed zone (shaded in the figure) of width h into which the strain will localize. Hillerborg assumes that
h = rjx. (10.5.15)
as indicated in the figure, where x is the depth of the compressed zone, and rj is a constant (approximately equal to 0.8). Hillerborg further assumes linear softening expressed by a stress-displacement a(w) curve, where w here has the meaning of an inelastic displacement in compression, equivalent to the crack opening in tension. Fig. 10.5.10b shows the corresponding stress-strain curve. Note that the slope depends on the depth of the compression zone x, hence also on the size of the beam. wc is assumed to be a material property, and so is the compressive strength fc and the elastic modulus E.
According to these hypotheses, the beam depth (Fig 10.5.10c) is divided into three parts: over part LA no stress is transferred (except across the reinforcement), over part AB the concrete is compressed and elastic, and over BC the concrete undergoes crushing and strain localization. The strain is assumed to vary linearly as shown in Fig. 10.5.1 Od, so that
є = kz (10.5.16)
where re is the curvature. From this, together with the stress-strain curves of the steel and of the concrete already defined, the stress profile can be computed as a function of the position of the neutral line x and
the curvature re, as sketched in Fig. 10.5. lOe. Then x is computed from any given re from the equilibrium of forces, and next the bending moment is computed from the equilibrium of moments. In this way, the full moment-curvature diagram is obtained.
The essential feature is that, since the stress-strain curve of concrete is made to depend on the depth of the compression zone, the resulting moment-curvature diagrams are size-dependent. More specifically, they depend on the dimensionless size D* defined as
ic represents the characteristic material length for fracture in compression. Fig. 10.5.lOf illustrates Hillerborg’s results, which clearly display an increase of brittleness with the size and with an increasing steel ratio.
Certainly this model is crude, but offers one simple way of taking into account the softening behavior in compression to predict a size-dependent response of concrete in bending. This is not actually included in the codes, which take size-independent stress-strain curves for concrete in compression, such as the parabola-rectangle diagram of the CEB-F1P Model Code shown in Fig. 10.5.lOg. llillerborg suggests a simple way of using this kind of diagrams to include the size effect: the strain cut-off єи is made to depend on the depth of the compression zone as
Єи -= —
where k is a parameter, with dimensions of length, which includes the fracture properties in compression —roughly proportional to (c in (10.5.17)— and which might, eventually, depend on the geometrical details of the beam. However, further research is required, both on the experimental and theoretical sides, to settle on the best model that should go into the code provisions.