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 MetalsFigure 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 σY . The uniaxial stress-strain relationship is shown in Figure 1(b). Figure 1 Steel Plate With Biaxial StressThe 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 elastic-plastic. 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 e-p-p material under for uniaxial stress, and hence biaxial stress is fundamentally different from uniaxial stress. Figure 2 Some Features of the Yield Surface
These ingredients are sufficient to develop an analysis method for the yielding of steel. In particular, the theory can be extended from the e-p-p 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. There are two yield surfaces, namely a Y surface (initial yield) and a larger U surface (ultimate strength). These surfaces both have the same shape. If the stress point is inside the Y surface the material is elastic. If the material is on the Y surface the material is elastic-plastic-strain hardening. As the material hardens the Y surface moves, as indicated in the figure. When the stress point reaches the U surface, the material is elastic-plastic, as in the e-p-p case. Among other things, the Mroz theory specifies how the Y surface moves as the material strain hardens. Figure 3 Trilinear Behavior With Mroz TheoryExtension to P-M InteractionConcept In a piece of steel under biaxial stress, the σ1 and σ2 stresses interact with each other. Plasticity theory models this interaction. By analogy, plasticity theory can be extended to P-M interaction in a column, where the axial force, P, and the bending moment, M, interact with each other. For the e-p-p case the yield surface is now the P-M strength interaction surface for the column cross section. A Case Where The Analogy Works Consider a short length of column with a cross section consisting of two steel fibers (in effect, an I section with one fiber for each flange and a web that can be ignored). This is shown in Figure 4(a). Each fiber is elastic-perfectly-plastic with area A and yield stress σY. A short length of the column is loaded with an axial force, P, and a bending moment, M, as shown. The axial force is applied at the reference axis for the column, which is the axis through the cross section centroid. This is important because it means that when the column is elastic there is no interaction between P and M. With the reference axis at the centroid, P alone causes axial strain but no curvature and M alone causes curvature but no axial strain (where axial strain is measured at the reference axis). If the reference axis is not at the centroid, P and M interact even before yield. Figure 4 Simple Steel ColumnEach fiber has only uniaxial stress, but the column has P-M interaction. It is easy to show that the P-M interaction surface is as shown in Figure 4(b). This is the yield surface for plasticity theory. To see whether plasticity theory correctly predicts the behavior of the column, consider the behavior when the column is subjected to axial and bending effects. The loading and behavior are shown in Figure 5. First, apply axial compression force equal to one half the yield force. The load path is O-A in Figure 5(a). Then hold this force constant and increase the moment. The load path is A-B. At Point B Fiber 1 yields in compression, while Fiber 2 remains elastic. The moment capacity has now been reached, and the moment-curvature relationship is e-p-p, as shown in Figure 5(b). However, when one fiber yields the neutral axis suddenly shifts from the center of the section to the unyielded fiber. Hence, any subsequent change in curvature is accompanied by a change in axial strain (always measured at the reference axis). This is shown in Figure 5(c). The strains after yield are all plastic. That is, there is plastic bending of the cross section and plastic axial deformation. When the axial force is compression the plastic axial strain is compression, so that the column shortens as it yields in bending. If the column were in tension, it would extend as it yields in bending. Figure 5 Behavior of Simple Steel ColumnFigure 5(c) shows the changes in curvature, Δψ , and axial strain, Δε , after yield. The change in axial strain is Δε = 0.5dΔψ . This is the ratio that plasticity theory predicts, based on the normal to the yield surface. In this case, therefore, plasticity theory is correct. If the bending moment is reversed, keeping the axial force constant, Fiber 1 immediately unloads, and the cross section returns to an elastic state with the neutral axis at the center of the section. When the moment is fully reversed Fiber 2 yields in compression, while Fiber 1 remains elastic. This behavior is correctly predicted by plasticity theory. Hence, for this column the theory is also correct for cyclic load. After yield in the opposite direction, the plastic axial strain is again compression. Hence, as the column is cycled plastically in bending it progressively shortens. After a number of cycles, the amount of shortening can be substantial. This example is for a very simple cross section and for elastic perfectly- plastic material. However, it indicates that plasticity theory can correctly account for P-M interaction. Analyses of more complex cross sections show that plasticity theory can make reasonably accurate predictions of cross section behavior. Hence, inelastic components based on plasticity theory can be used to model steel columns with P-M interaction, for both push-over and dynamic earthquake analyses. A Case Where the Analogy Does Not Work So Well Next, consider a simple reinforced concrete section, consisting of two concrete fibers and two steel fibers as shown in Figure 6(a). Figure 6 Simple Concrete ColumnThe steel fibers are elastic-perfectly-plastic. The concrete fibers are e-p-p in compression and have zero strength in tension. The P-M strength interaction surface for this section is shown in Figure 6(b). For plasticity theory, this is also the yield surface. Consider the case with bending moment only, and zero axial force. The behavior is as follows:
Hence, plasticity theory correctly predicts the behavior at Steps (3) and (4), after the yield surface is reached, but the theory has problems in the elastic range. Next cycle the bending moment from positive to negative, still with zero axial force. The behavior is as follows: When the bending moment is reduced the steel tension fiber immediately unloads and becomes elastic. Plasticity theory correctly predicts unloading.
In summary, plasticity theory does a mediocre job of modeling reinforced concrete for monotonically increasing loads, and a poor job for cyclic loads. A major error for cyclic loads is that for axial forces below the balance point, plasticity theory predicts plastic strain in tension after the yield surface is reached, for both bending directions. Hence, under cyclic bending the theory predicts that the column will progressively increase in length. There can be axial growth in reinforced concrete members, but plasticity theory overestimates the amount for cyclic loading. Are These Errors Fatal? The major reason for considering interaction is to account for the effects of axial force on bending strength. Interaction between bending and axial deformations tends to be a secondary concern. In a typical column, the column will extend or shorten as it yields in bending, but the amount of axial deformation is probably not large. Given the many other complications and approximations in the modeling of inelastic behavior in columns, the fact that plasticity theory can overestimate the amount of axial deformation may not be very important. This is a decision that you must make. If you use P-M-M hinges in a column, and if the extension of the column could have a significant effect on the behavior of the structure, you should examine the calculated axial extensions (for example using the Hysteresis Loops task) and satisfy yourself that these deformations are not large enough to affect the accuracy of the results for design purposes. If you must calculate axial deformation effects more accurately, consider using fiber cross sections rather than P-M-M hinges. Fiber cross sections account for P-M-M interaction, but they use uniaxial stress-strain relationships and hysteresis loops for the fibers, and hence do not make use of plasticity theory. One case where axial deformations are definitely important is for shear walls. If a shear wall is wide, as it cracks and yields there can be quite large axial extensions. P-M hinges may not be sufficiently accurate for modeling inelastic behavior in shear walls. This is why PERFORM uses only fiber cross sections for inelastic shear walls. P-M-M InteractionGeneral So far this chapter has considered only biaxial P-M interaction. For a column element in PERFORM there can be triaxial P-M-M interaction. The principles are exactly the same, the only difference being that the yield surface is 3D rather than a 2D. In plasticity theory there are major changes required to go from uniaxial plasticity to biaxial plasticity. There are no major changes in going from biaxial to triaxial, or higher. PERFORM also uses plasticity theory for V-V shear interaction in shear hinges. Since the mechanism of inelastic shear in reinforced concrete is not plastic, plasticity theory really does not apply. However, it should give reasonable results for most practical purposes. P-M-M Yield Surfaces PERFORM uses a P-M-M yield surface that is similar to that described in the following pair of papers : Nonlinear Analysis of Mixed Steel-Concrete Frames, Parts I and II, by S. El-Tawil and G. Deierlein, Journal of Structural Engineering, Vol. 126, No. 6, June 2001. This yield surface requires only a few parameters to define its shape, yet gives you substantial control over the details of this shape. When you specify the parameters for a yield surface in PERFORM you can plot the surface to see the effect of the parameters on its shape. Steel Yield Surface Figure 7 shows the yield surface for a steel section. Figure 7 Steel Type P-M-M Yield SurfaceThe equations of the yield surface are essentially as follows: In each P-M plane (P-M2 and P-M3) : Different values for the exponent α and the yield force PY0 can be specified for tension and compression. Different values for the exponent α can also be used in the P-M2 and P-M3 planes. El Tawil and Deierlein use β = 1, which causes a sharp peak at P = PY0. PERFORM requires a value larger than 1.0 for β, with a suggested value of 1.1. This has little effect on the yield surface for smaller P values, yet avoids the sharp peak. For any value of P, Equation (1.3) defines the M values at which yield occurs, in both the P-M2 and P-M3 planes (put fPM = 1 and solve for M). The yield function in the M2-M3 plane is then given by Equation (1.2). |