Page Comparison

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).

Image Modified

...

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 elasticplasticelastic-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.

Image Added                                     

Figure 2

...

Some Features of the Yield Surface

1. 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, Δ_ε1 and Δ_ε2 are 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)
2. 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 OCO-C, 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

...

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 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 figurein 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

...

Extension to P-M Interaction

1.2.1 Concept

In a piece of steel under biaxial stress, the σ1 σ₁ and σ₂ 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 of
two of two steel fibers (in effect, an I section with one fiber for each flange
and flange and a web that can be ignored). This is shown in Figure 4(a). Each fiber
is fiber is elastic-perfectly-plastic with area A and yield stress σYσ_Y.

A short length of the column is loaded with an axial force, P, and a
bending a bending moment, M, as shown. The axial force is applied at the
reference the reference axis for the column, which is the axis through the cross
section cross section centroid. This is important because it means that when the
column 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 nono curvature and M alone causes curvature but no axial strain (where axial
axial strain is measured at the reference axis). If the reference axis is not at
at the centroid, P and M interact even before yield.

Image Added

Figure 4 Simple Steel Column

Each fiber has only uniaxial stress, but the column has P-M interaction.
It  It is easy to show that the P-M interaction surface is as shown in Figure
4Figure 4(b). This is the yield surface for plasticity theory.

To see whether plasticity theory correctly predicts the behavior of the
columnthe column, consider the behavior when the column is subjected to axial
and axial and bending effects. The loading and behavior are shown in Figure 5.
First First, apply axial compression force equal to one half the yield force.
The  The load path is O-A in Figure 5(a). Then hold this force constant and
increase and increase the moment. The load path is A-B. At Point B Fiber 1 yields in
compressionin compression, while Fiber 2 remains elastic. The moment capacity has
now 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 axis suddenly shifts from the center of the section to the unyielded fiber.
Hence Hence, any subsequent change in curvature is accompanied by a change
in change in axial strain (always measured at the reference axis). This is shown in
Figure in Figure 5(c).

The strains after yield are all plastic. That is, there is plastic bending of
the of the cross section and plastic axial deformation. When the axial force is
compression is compression the plastic axial strain is compression, so that the columncolumn shortens as it yields in bending. If the column were in tension, it would
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 ,
after  after yield. The change in axial strain is Δε = 0.5dΔψ . This is the ratio
that ratio that plasticity theory predicts, based on the normal to the yield surface.
In  In this case, therefore, plasticity theory is correct.
If the bending moment is reversed, keeping the axial force constant,
Fiber  Fiber 1 immediately unloads, and the cross section returns to an elastic
state elastic state with the neutral axis at the center of the section. When the
moment the moment is fully reversed Fiber 2 yields in compression, while Fiber 1
remains elastic. This behavior is correctly predicted by plasticity
theoryplasticity 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
compressionagain compression. Hence, as the column is cycled plastically in bending it
progressively it progressively shortens. After a number of cycles, the amount of
shortening of shortening can be substantial.

This example is for a very simple cross section and for elasticperfectlyelastic perfectly-
plastic  plastic material. However, it indicates that plasticity theorytheory can correctly account for P-M interaction. Analyses of more complex
complex cross sections show that plasticity theory can make reasonably accurate
accurate predictions of cross section behavior. Hence, inelastic components
components based on plasticity theory can be used to model steel columns with P-M
M interaction, for both push-over and dynamic earthquake analyses.1.2.3

A Case Where the Analogy Does Not Work So Well

Next, consider a simple reinforced concrete section, consisting of two
concrete two concrete fibers and two steel fibers as shown in Figure 6(a).

Image Added

Figure 6 Simple Concrete Column

The steel fibers are elastic-perfectly-plastic. The concrete fibers are epe-p-
p in compression and have zero strength in tension. The P-M strength
interaction strength interaction surface for this section is shown in Figure 6(b). For
plasticity For plasticity theory, this is also the yield surface.
Consider  Consider the case with bending moment only, and zero axial force. The
behavior The behavior is as follows.
(1) The :

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

...

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

...

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

...

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

...

1. yield surface is reached?

...

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

...

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

...

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

...

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

...

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

...

1. the elastic range.

...

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

...

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

...

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

...

1. theory captures this behavior.

...

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

...

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

...

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

...

1. this behavior.

Hence, plasticity theory correctly predicts the behavior at Steps (3) and
and (4), after the yield surface is reached, but the theory has problems in the
elastic the elastic range.

Next cycle the bending moment from positive to negative, still with
zero with zero axial force. The behavior is as follows.:(5)

When the bending moment is reduced the steel tension fiber
fiber immediately unloads and becomes elastic. Plasticity theory
theory correctly predicts unloading.

1. (Step 6)As the moment is decreased the curvature decreases and there

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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

...

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 the effects of axial force on bending strength. Interaction between bending
and bending and axial deformations tends to be a secondary concern. In a typical
columntypical 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 many other complications and approximations in the modeling of inelastic
behavior inelastic behavior in columns, the fact that plasticity theory can overestimate the
amount 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
columna column, and if the extension of the column could have a significant
effect significant effect on the behavior of the structure, you should examine thethe calculated axial extensions (for example using the Hysteresis Loops
Loops task) and satisfy yourself that these deformations are not large enough
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
Fiber cross sections account for P-M-M interaction, but they use uniaxial
uniaxial stress-strain relationships and hysteresis loops for the fibers, and hence
hence do not make use of plasticity theory.

One case where axial deformations are definitely important is for shear
wallsshear walls. If a shear wall is wide, as it cracks and yields there can be quite
large quite large axial extensions. P-M hinges may not be sufficiently accurate for
modeling for modeling inelastic behavior in shear walls. This is why PERFORM
uses only fiber cross sections for inelastic shear walls.1.3

P-M-M Interaction

1.3.1 General

So far this chapter has considered only biaxial P-M interaction. For a
column a column element in PERFORM there can be triaxial P-M-M interaction.
The  The principles are exactly the same, the only difference being that the
yield 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  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 in shear hinges. Since the mechanism of inelastic shear in reinforced
concrete reinforced concrete is not plastic, plasticity theory really does not apply. However,
it  it should give reasonable results for most practical purposes.

1.3.2 P-M-M Yield Surfaces

PERFORM uses a P-M-M yield surface that is similar to that described
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
This yield surface requires only a few parameters to define its shape, yet
yet gives you substantial control over the details of this shape. 

 When you specify the parameters for a yield surface in PERFORM you
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.

Image Added

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) :

...

Image Added

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:

                             Image Added

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.

Image Added

Figure 8 Concrete Type P-M-M Yield Surface

The equations of the yield surface are essentially as follows.

In each P-M plane:

Image Added

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.

1.3.3 Strain Hardening

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

1.3.4 Plastic Flow

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