Plasticity theory for P-M interaction

By Dr. Graham H. Powell, Professor Emeritus, UC Berkeley

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 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

Figure 2 shows a yielded state, point A, defined by stresses σ_1A and σ_2A. Suppose that strain increments Δ_ε1 and Δ_ε2 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 elastic-plastic. 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.
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 O-C, 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-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 Theory

Extension to P-M Interaction

Concept

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 σ_Y.

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 Column

Each 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 Column

Figure 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 Column

The 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:

The concrete fiber on the tension side cracks immediately. Hence, the neutral axis shifts towards the compression side. This poses a problem for plasticity theory. Specifically, what bending and axial stiffnesses should be used for elastic behavior before the yield surface is reached?
As the moment is increased there is both curvature and axial tension strain (measured at the reference axis). The relationship between curvature and axial strain depends on the shift of the neutral axis, which depends on the steel and concrete areas and moduli. In the plasticity theory there is no P-M interaction in the elastic range.
When the moment reaches the yield moment the steel fiber on the tension side yields. The bending stiffness reduces to zero and the neutral axis shifts to the compression fiber. Plasticity theory captures this behavior.
The moment remains constant as the curvature increases. The axial strain is tension. The relationship between axial strain and curvature is Δε = 0.5dΔψ . Plasticity theory also captures this behavior.

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.

(Step 6)As the moment is decreased the curvature decreases and there is axial compression strain, which is opposite to Step (2). As before, plasticity theory does not capture this behavior.
(Step 7) Immediately after the moment reaches zero the second concrete fiber cracks. Both concrete fibers are now cracked. The neutral axis moves to the center of the section, and the bending stiffness is the stiffness of the steel only. Plasticity theory assumes constant stiffnesses in the elastic range, and does not capture this behavior.
(Step 8) When the moment reaches the strength of the steel fibers, both fibers yield. Plasticity theory does not capture this behavior.
(Step 9) The steel fiber that previously yielded in tension is now yielding in compression. When the total strain in this fiber becomes zero the crack closes in the concrete fiber and it regains stiffness. The bending stiffness of the section increases and the neutral axis shifts. Plasticity theory does not capture this behavior.
(Step 10) When the moment reaches the yield moment in the opposite direction the steel fiber on the tension side yields. The bending stiffness reduces to zero and the neutral axis shifts to the compression fiber. Plasticity theory does captures this behavior, but by now it is too late.
(Step 11) The moment remains constant as the curvature increases, as in Step (4). The axial strain is tension. Plasticity theory does capture this behavior, but again it is too late.

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 Interaction

General

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 Surface

The 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 P_Y0 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 = P_Y0.

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 (1.1) defines the M values at which yield occurs, in both the P-M₂ and P-M₃ planes (put f_PM = 1 and solve for M). Call these values M_YP2 and M_YP3. The yield function in the M₂-M₃ plane
is then:

El Tawil and Deierlein suggest values for the exponents α and γ.

Concrete Yield Surface

Figure 8 shows the yield surface for a concrete section.

Figure 8 Concrete Type P-M-M Yield Surface

The equations of the yield surface are essentially as follows.

In each P-M plane:

where f_PM = yield function value, = 1.0 for yield, P = axial force, P_B = axial force at balance point (assumed to be the same in both P-M planes), M = bending moment, P_Y0 = yield force at M = 0 , and M_YB = yield moment at P = P_B .

Different values for the exponent α and the yield force P_Y0 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, but PERFORM requires a value larger than 1.0.

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 f_PM = 1 and solve for M). The yield function in the M₂-M₃ plane is then given by Equation (1.2).

Again, El Tawil and Deierlein suggest values for the exponents α and γ.

Strain Hardening

For trilinear behavior PERFORM uses the Mroz hardening theory, as described earlier in this chapter.

Plastic Flow

PERFORM assumes plastic flow normal to the yield surface. Generally this means that as a P-M-M hinge yields in bending it also extends or shortens. As considered in this chapter, this may not be an accurate model of actual behavior. It may be noted that El Tawil and Deierlein assume no axial plastic deformation in the strain hardening range, and assume normal plastic flow only when the outer yield surface is reached. This has not been done in PERFORM, mainly because non-normal flow implies a non-symmetrical stiffness matrix, which can cause both theoretical and computational problems.