Plasticity theory is often used to model column elements with P-M interaction. This note provides a simplified explanation of the essential features of plasticity theory. This noter also shows that it can be reasonable to apply this theory to steel columns, but generally not to concrete columns.
PERFORM includes inelastic hinge components that have P-M interaction and are based on plasticity theory. Before using these components, you should be clear on the limitations of plasticity theory and hinge components.
This note also describes the yield surfaces that are used for P-M interaction in column elements.
Yield of Metals
Figure 1(a) shows a piece of steel plate subjected to biaxial stress. Assume that the behavior is elastic-perfectly-plastic (e-p-p), and that the yield stress in simple (uniaxial) tension is σ Y . The uniaxial stress-strain relationship is shown in Figure 1(b).
Figure 1 Steel Plate With Biaxial Stress
The well known von Mises theory says that for biaxial stress the material has a yield surface as shown in Figure 1(c). If the stress point is inside the yield surface the material is elastic. If the stress point is on the yield surface the material is yielded, and its behavior is elasticplastic. This means that it is partly elastic and partly plastic, as explained below. Stress points outside the yield surface are not allowed.
The yield surface thus defines the strength of the material under biaxial stress. Plasticity theory defines the behavior of the material after it reaches the yield surface (i.e. after it yields). The ingredients of the theory are essentially as follows:
- As long as the stress point stays on the yield surface, the material stays in a yielded state. However, the stress point does not remain in one place. The stresses can change after yield, even though the material is e-p-p, which means that the stress point can move around the surface. The stress does not change after yield for an ep- p material under for uniaxial stress, and hence biaxial stress is fundamentally different from uniaxial stress.
Figure 2 shows a yielded state, point A, defined by stresses σ 1A
and σ 2A . Suppose that strain increments Δε 1 and Δε 2 are
imposed, causing the stresses to change to σ 1B and σ 2B at point
B. Plasticity theory says that some of the strain increment is an
elastic increment and the remainder is plastic flow. The elastic part
of the strain causes the change in stress. The plastic part causes no
change in stress. This is why the behavior is referred to as elasticplastic.
For yield of an e-p-p material under uniaxial stress there is
no stress change after yield. Hence, all of the strain after yield is
plastic strain.
(3) Plasticity theory also defines the direction of plastic flow. That is,
it defines the ratio between the 1-axis and 2-axis components of
the plastic strain. Essentially, the theory states that the direction of
plastic flow is normal to the yield surface. For example, consider
uniaxial stress along the 1-axis. As shown in Figure 2, the stress
path is OC, and yield occurs at point C. After yield, the stress stays
constant, and hence all subsequent strain is plastic. The normal to
the yield surface at point C has 1-axis and 2-axis components in
the ratio 2:1. Hence, the plastic strains are in this ratio, and the
value of Poisson's ratio is 0.5 for plastic deformation. This agrees
with experimental results.
These ingredients are sufficient to develop an analysis method for the
yielding of steel. In particular, the theory can be extended from the e-pp
case to the case with strain hardening. There are many hardening
theories. PERFORM uses the Mroz theory. For the case of trilinear
behavior the Mroz theory is illustrated in Figure 3.