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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:

1. 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.
   

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   Figure 2 Some Features of the Yield Surface

2. Figure 2 shows a yielded state, point A, defined by stresses σ 1A
   1A and σ 2A. Suppose that strain increments Δε 1 and Δε 2 are
   imposedare imposed, causing the stresses to change to σ 1B and σ 2B at point
   Bpoint B. Plasticity theory says that some of the strain increment is an
   elastic an elastic increment and the remainder is plastic flow. The elastic part
   of part of the strain causes the change in stress. The plastic part causes no
   change no change in stress. This is why the behavior is referred to as elasticplasticelastic-plastic.
    For yield of an e-p-p material under uniaxial stress there is
   no is no stress change after yield. Hence, all of the strain after yield is
   plastic is plastic strain.(3)
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
   of the plastic strain. Essentially, the theory states that the direction of
   of plastic flow is normal to the yield surface. For example, consider
   consider uniaxial stress along the 1-axis. As shown in Figure 2, the stress
   stress path is OC, and yield occurs at point C. After yield, the stress stays
   stays constant, and hence all subsequent strain is plastic. The normal to
   to the yield surface at point C has 1-axis and 2-axis components in
   in the ratio 2:1. Hence, the plastic strains are in this ratio, and the
   the value of Poisson's ratio is 0.5 for plastic deformation. This agrees
   agrees with experimental results.

These ingredients are sufficient to develop an analysis method for

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the yielding of steel. In particular, the theory can be extended from the e-

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pp case to the case with strain hardening. There are many

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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 a larger U surface (ultimate strength). These surfaces both have the same
shapesame shape. If the stress point is inside the Y surface the material is elastic. If
the If the material is on the Y surface the material is elastic-plastic-strainhardeningstrain hardening.
As  As the material hardens the Y surface moves, as indicated in
the in the figure. When the stress point reaches the U surface, the material is
elasticis elastic-plastic, as in the e-p-p case. Among other things, the Mroz
theory Mroz theory specifies how the Y surface moves as the material strain
hardens.

Image Added

              Figure 3 Trilinear Behavior With Mroz Theory

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Extension to P-M Interaction

1.2.1 Concept

In a piece of steel under biaxial stress, the σ1 and σ 2 stresses interact
with interact with each other. Plasticity theory models this interaction. By analogy,
plasticity  plasticity theory can be extended to P-M interaction in a column, where
the 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 interaction surface for the column cross section.1.2.2

A Case Where The Analogy Works

Consider a short length of column with a cross section consisting

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of two steel fibers (in effect, an I section with one fiber for each

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flange and a web that can be ignored). This is shown in Figure 4(a). Each

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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

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a bending moment, M, as shown. The axial force is applied at

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the reference axis for the column, which is the axis through the

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cross section centroid. This is important because it means that when

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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

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no curvature and M alone causes curvature but no axial strain (where

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axial strain is measured at the reference axis). If the reference axis is not

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at the centroid, P and M interact even before yield.

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                                  Figure 4 Simple Steel Column

Each fiber has only uniaxial stress, but the column has P-M interaction.

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 It is easy to show that the P-M interaction surface is as shown in

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Figure 4(b). This is the yield surface for plasticity theory.

To see whether plasticity theory correctly predicts the behavior of

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the column, consider the behavior when the column is subjected to

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axial and bending effects. The loading and behavior are shown in Figure 5.

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 First, apply axial compression force equal to one half the yield force.

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 The load path is O-A in Figure 5(a). Then hold this force constant

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and increase the moment. The load path is A-B. At Point B Fiber 1 yields

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in compression, while Fiber 2 remains elastic. The moment capacity

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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

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axis suddenly shifts from the center of the section to the unyielded fiber.

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 Hence, any subsequent change in curvature is accompanied by a

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change in axial strain (always measured at the reference axis). This is shown

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in Figure 5(c).

The strains after yield are all plastic. That is, there is plastic bending

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of the cross section and plastic axial deformation. When the axial force

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is compression the plastic axial strain is compression, so that the

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column shortens as it yields in bending. If the column were in tension, it

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would extend as it yields in bending.

Image Added

                      Figure 5 Behavior of Simple Steel Column

Figure 5(c) shows the changes in curvature, Δψ , and axial strain, Δε ,

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 after yield. The change in axial strain is Δε = 0.5dΔψ . This is the

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ratio that plasticity theory predicts, based on the normal to the yield surface.

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 In this case, therefore, plasticity theory is correct.
If the bending moment is reversed, keeping the axial force constant,

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 Fiber 1 immediately unloads, and the cross section returns to an

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elastic state with the neutral axis at the center of the section. When

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the moment is fully reversed Fiber 2 yields in compression, while Fiber 1
remains elastic. This behavior is correctly predicted by

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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

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again compression. Hence, as the column is cycled plastically in bending

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it progressively shortens. After a number of cycles, the amount

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of shortening can be substantial.

This example is for a very simple cross section and for

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elastic perfectly-

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 plastic material. However, it indicates that plasticity

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theory can correctly account for P-M interaction. Analyses of more

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complex cross sections show that plasticity theory can make reasonably

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accurate predictions of cross section behavior. Hence, inelastic

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components based on plasticity theory can be used to model steel columns with P-

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M interaction, for both push-over and dynamic earthquake analyses.

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A Case Where the Analogy Does Not Work So Well

Next, consider a simple reinforced concrete section, consisting of

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two concrete fibers and two steel fibers as shown in Figure 6(a).

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Image Added

                                Figure 6 Simple Concrete Column

The steel fibers are elastic-perfectly-plastic. The concrete fibers are

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e-p-

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p in compression and have zero strength in tension. The P-M

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strength interaction surface for this section is shown in Figure 6(b).

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For plasticity theory, this is also the yield surface.

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 Consider the case with bending moment only, and zero axial force.

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The behavior is as follows

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:

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1. 1. The concrete fiber on the tension side cracks immediately. Hence,

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1. 1.  the neutral axis shifts towards the compression side. This poses

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1. 1. a problem for plasticity theory. Specifically, what bending and

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1. 1. axial stiffnesses should be used for elastic behavior before the

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1. 1. yield surface is reached?

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1. 1. As the moment is increased there is both curvature and

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1. 1. axial tension strain (measured at the reference axis). The

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1. 1. relationship between curvature and axial strain depends on the shift of

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1. 1. the neutral axis, which depends on the steel and concrete areas

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1. 1. and moduli. In the plasticity theory there is no P-M interaction in

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1. 1. the elastic range.

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1. 2. When the moment reaches the yield moment the steel fiber on

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1. 1. the tension side yields. The bending stiffness reduces to zero and

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1. 1. the neutral axis shifts to the compression fiber. Plasticity

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1. 1. theory captures this behavior.

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1. 1. The moment remains constant as the curvature increases. The

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1. 1. axial strain is tension. The relationship between axial strain

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1. 1. and curvature is Δε = 0.5dΔψ . Plasticity theory also captures

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1. 1. this behavior.

Hence, plasticity theory correctly predicts the behavior at Steps (3)

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and (4), after the yield surface is reached, but the theory has problems in

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the elastic range.

Next cycle the bending moment from positive to negative, still

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with zero axial force. The behavior is as follows

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:

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1. 1. When the bending moment is reduced the steel tension

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1. 1. fiber immediately unloads and becomes elastic. Plasticity

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1. 1. theory correctly predicts unloading.

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1. 1. As the moment is decreased the curvature decreases and there

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1. 1. is axial compression strain, which is opposite to Step (2). As before,

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1. 1.  plasticity theory does not capture this behavior. 
   2. (7) Immediately after the moment reaches zero the second

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1. 1. concrete fiber cracks. Both concrete fibers are now cracked. The

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1. 1. neutral axis moves to the center of the section, and the bending stiffness

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1. 1. is the stiffness of the steel only. Plasticity theory assumes

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1. 1. constant stiffnesses in the elastic range, and does not capture this behavior. 
   2. (8) When the moment reaches the strength of the steel fibers,

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1. 1. both fibers yield. Plasticity theory does not capture this behavior. 
   2. (9) The steel fiber that previously yielded in tension is now yielding

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1. 1. in compression. When the total strain in this fiber becomes zero

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1. 1. the crack closes in the concrete fiber and it regains stiffness.

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1. 1. The bending stiffness of the section increases and the neutral

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1. 1. axis shifts. Plasticity theory does not capture this behavior.
   2. (10) When the moment reaches the yield moment in the

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1. 1. opposite direction the steel fiber on the tension side yields. The

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1. 1. bending stiffness reduces to zero and the neutral axis shifts to

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1. 1. the compression fiber. Plasticity theory does captures this behavior,

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1. 1.  but by now it is too late.
   2. (11) The moment remains constant as the curvature increases, as in

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1. 1. Step (4). The axial strain is tension. Plasticity theory does capture

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1. 1. this behavior, but again it is too late.

In summary, plasticity theory does a mediocre job of modeling
modeling reinforced concrete for monotonically increasing loads, and a poor job
job for cyclic loads.

A major error for cyclic loads is that for axial forces below the balance
pointbalance point, plasticity theory predicts plastic strain in tension after the yield
surface yield surface is reached, for both bending directions. Hence, under cyclic
bending cyclic bending the theory predicts that the column will progressively increase
in increase in length. There can be axial growth in reinforced concrete members,
but  but plasticity theory overestimates the amount for cyclic loading. 

1.2.4 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

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