Items related to Inelastic Analysis of Structures
Synopsis
The modeling of mechanical properties of materials and structures is a complex and wide-ranging subject. In some applications, it is sufficient to assume that the material remains elastic, i.e. that the deformation process is fully reversible and the stress is a unique function of strain. However, such a simplified assumption is appropriate only within a limited range, and in general must be replaced by a more realistic approach that takes into account the inelastic processes such as plastic yielding or cracking.
This book presents a comprehensive treatment of the most important areas of plasticity and of time-dependent inelastic behavior (viscoplasticity of metals, and creep and shrinkage of concrete). It covers structural aspects such as:
* incremental analysis
* limit analysis
* shakedown analysis
* optimal design
* beam structures subjected to bending and torsion
* yield line theory of plates
* slip line theory
* size effect in structures
* creep and shrinkage effects in concrete structures.
The following aspects of the advanced material modeling are presented:
* yield surfaces for metals and plastic-frictional materials
* hardening and softening
* stress-return algorithms
* large-strain formulations
* thermodynamic framework
* microplane models
* localization of plastic strain.
Inelastic Analysis of Structures is a textbook for basic and advanced courses on plasticity, with a slight emphasis on structural engineering applications, but with a wealth of material for geotechnical, mechanical, aerospace, naval, petroleum and nuclear engineers. The text is constructed in a very didactical way, while the mathematics has been kept rigorous.
"synopsis" may belong to another edition of this title.
About the Author
Milan Jirasek and Zdenek P. Bazant are the authors of Inelastic Analysis of Structures, published by Wiley.
From the Back Cover
The modeling of mechanical properties of materials and structures is a complex and wide-ranging subject. In some applications, it is sufficient to assume that the material remains elastic, i.e. that the deformation process is fully reversible and the stress is a unique function of strain. However, such a simplified assumption is appropriate only within a limited range, and in general must be replaced by a more realistic approach that takes into account the inelastic processes such as plastic yielding or cracking.
This book presents a comprehensive treatment of the most important areas of plasticity and of time-dependent inelastic behavior (viscoplasticity of metals, and creep and shrinkage of concrete). It covers structural aspects such as:
* incremental analysis
* limit analysis
* shakedown analysis
* optimal design
* beam structures subjected to bending and torsion
* yield line theory of plates
* slip line theory
* size effect in structures
* creep and shrinkage effects in concrete structures.
The following aspects of the advanced material modeling are presented:
* yield surfaces for metals and plastic-frictional materials
* hardening and softening
* stress-return algorithms
* large-strain formulations
* thermodynamic framework
* microplane models
* localization of plastic strain.
Inelastic Analysis of Structures is a textbook for basic and advanced courses on plasticity, with a slight emphasis on structural engineering applications, but with a wealth of material for geotechnical, mechanical, aerospace, naval, petroleum and nuclear engineers. The text is constructed in a very didactical way, while the mathematics has been kept rigorous.
From the Inside Flap
The modeling of mechanical properties of materials and structures is a complex and wide-ranging subject. In some applications, it is sufficient to assume that the material remains elastic, i.e. that the deformation process is fully reversible and the stress is a unique function of strain. However, such a simplified assumption is appropriate only within a limited range, and in general must be replaced by a more realistic approach that takes into account the inelastic processes such as plastic yielding or cracking.
This book presents a comprehensive treatment of the most important areas of plasticity and of time-dependent inelastic behavior (viscoplasticity of metals, and creep and shrinkage of concrete). It covers structural aspects such as:
* incremental analysis
* limit analysis
* shakedown analysis
* optimal design
* beam structures subjected to bending and torsion
* yield line theory of plates
* slip line theory
* size effect in structures
* creep and shrinkage effects in concrete structures.
The following aspects of the advanced material modeling are presented:
* yield surfaces for metals and plastic-frictional materials
* hardening and softening
* stress-return algorithms
* large-strain formulations
* thermodynamic framework
* microplane models
* localization of plastic strain.
Inelastic Analysis of Structures is a textbook for basic and advanced courses on plasticity, with a slight emphasis on structural engineering applications, but with a wealth of material for geotechnical, mechanical, aerospace, naval, petroleum and nuclear engineers. The text is constructed in a very didactical way, while the mathematics has been kept rigorous.
Excerpt. © Reprinted by permission. All rights reserved.
Inelastic Analysis of Structures
By Milan Jirasek Zdenek P. BazantJohn Wiley & Sons
Copyright © 2002 Milan Jirasek Zdenek P. BazantAll right reserved.
ISBN: 978-0-471-98716-1
Chapter One
Combined Plastic Bending and Compression or Tension
13.1 GENERALIZED PLASTIC HINGE
So far we have assumed that a plastic hinge forms if the bending moment in a critical cross section reaches the plastic limit value, [M.sub.0]. We have tacitly neglected the effect of the other internal forces on the formation of the yield hinge. This effect, however, can be appreciable, for example in multi-storey frames or frames with a large horizontal thrust where the axial forces are large. So, we will now refine our analysis and consider the interaction of bending moment M and normal force N. The resulting fully plasticized cross section will be referred to as a generalized plastic hinge. The effect of the shear force is usually less important, and is also more difficult to incorporate. We will postpone this problem to Part III, because it is beyond the scope of analysis under uniaxial stress.
A typical evolution of the strain and stress profiles during the formation of a generalized plastic hinge is depicted in Figure 13.1. As usual, it may be assumed that the cross sections remain planar at all stages of the loading process. Consequently, the variation of normal strains over the cross section is linear. The corresponding stress distribution is linear only in the elastic range (Figure 13.1(a)). As the loading process continues, yielding starts at the top or bottom fibers, and the plastic zone propagates into the interior of the cross section (Figure 13.1(b)). During the elastoplastic stage, the cross section has an elastic core with linear stress variation, and one or two plastic zones with constant stress equal to the positive or negative yield stress. For very large curvatures, the elastic core becomes negligibly small, and the stress distribution approaches a piecewise constant distribution, with one part of the cross section yielding in tension and the remaining part yielding in compression.
The plasticization process of course takes place in the neighboring cross sections as well, and the plastic zone occupies a certain volume (Figure 13.2(b)). However, for the purpose of modeling, we can lump the plastic hinge into one single cross section, the same as we did while analyzing the hinge under pure bending. The total plastic deformation in the idealized hinge is replaced by a rotation, [theta], and longitudinal displacement, [e.sub.p] (Figure 13.2(c)). From kinematic considerations, it follows that the plastic extension at an arbitrary point of the plasticized cross section can be expressed by a linear function
[[bar.e].sub.p](z) = [e.sub.p] + [theta]z (13.1)
where z is the centroidal coordinate perpendicular to the bending axis. The plastic extension at the centroid, [[bar.e].sub.p](0) = [e.sub.p], corresponds to the 'gap' between the centroids of the end cross sections of the two elastic parts connected by the generalized yield hinge in Figure 13.2(c). The bar over e helps to distinguish the distribution of the plastic extension over the cross section, [[bar.e].sub.p](z), which is a function of the distance from the centroidal axis, from the variable [e.sub.p], which characterizes the deformation of the generalized plastic hinge and is equal to the value of [[bar.e].sub.p] at z = 0.
Consider a given distribution of plastic extension, represented by the solid line in Figure 13.3(a). It might seem natural to assume that the fibers for which [[bar.e].sub.p](z) > 0 yield in tension, and those for which [[bar.e].sub.p](z) < 0 yield in compression; see Figure 13.4(b). This assumption corresponds to the so-called deformation theory of plasticity, in which the stress state is entirely determined by the current strain state, independently of the previous history of strain evolution (Hencky, 1924). Such a model is conceptually simple and convenient, but not always sufficiently realistic. The actual stress in an elastoplastic material is uniquely determined by the strain only as long as the loading process is monotonic. If the strain increment changes its sign, the material unloads elastically and eventually starts yielding in the opposite direction (Figure 13.4(a)). The deformation of a plastic hinge does not have any elastic component, and so the stress jumps to the opposite yield stress immediately after the plastic extension increment has changed its sign (Figure 13.4(b)). Clearly, the stress can be negative even at a positive extension. If we strictly adhere to the flow theory of plasticity we have to determine the sign of stress from the sign of plastic extension rate (increment) rather than from the sign of the plastic extension itself. This means that the position of the neutral axis separating the tensile zone from the compressive zone is determined by the condition [[??].sub.p](z) = 0 rather than [[bar.e].sub.p](z) = 0; see Figure 13.3(c).
Depending on the position of the neutral axis, a generalized plastic hinge can yield different combinations of internal forces M and N. The set of all such combinations is the so-called plastic limit envelope in the space of internal forces (also called the yield surface, yield locus, failure envelope, or interaction curve). States outside this envelope can never be reached. Similarly to the plastic hinge under pure bending, it is assumed that states inside the plastic limit envelope are entirely elastic, although this is not exactly true. One could define the elastic limit envelope that corresponds to states at which the first point of a cross section begins yielding (similar to the limit elastic moment in the case of pure bending). The states between the elastic and plastic limit envelopes are elastoplastic, but the corresponding plastic deformations that precede the formation of a full plastic hinge are usually neglected, especially when looking for the final collapse mechanism rather than for the details of the response in the elastoplastic range.
Example 13.1: Construct the elastic limit envelope and the plastic limit envelope for a rectangular cross section of width b and depth h.
Solution: 1) We start with the plastic limit envelope. Let us specify the position of the neutral axis by the coordinate [z.sub.n] = [eta]h/2 measured from the cross section centroid (Figure 13.5). Instead of integrating over the cross section, we can replace the positive stress by its resultant
[N.sup.+] = [A.sup.+] [[sigma].sub.0] = 1/2 (1 - [eta])bh][sigma].sub.0] (13.2)
and the negative stress by its resultant
[N.sup.-] = [A.sup.-] [[sigma].sub.0] = 1/2 (1 + [eta])bh][sigma].sub.0] (13.3)
as indicated in Figure 13.5. The arms of the resultants of positive and negative stress, i.e. the distances of the centroids of the areas under tension and under compression from the centroid of the entire cross section, are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.5)
The force and moment resultants at a plastic limit state are now obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.7)
The parameter [eta] describing the position of the neutral axis can vary within the limits. The values [eta] = -1, [eta] = 0 and [eta] = 1 correspond to the axial tension, pure bending and axial compression, respectively. For [eta] = -1, the normal force assumes its maximum possible value, N = [sigma].sub.0]bh [equivalent to] [N.sub.0] = plastic normal force, while the bending moment is zero. For [eta] = 0, the bending moment assumes its maximum possible value, M = [sigma].sub.0]b[h.sup.2]/4 [equivalent to] [M.sub.0] = plastic moment, while the normal force is zero. Finally, for [eta] = 1, the normal force assumes its minimum possible value, N = -[[sigma].sub.0]bh = -[N.sub.0], while the bending moment is zero. Note that the plastic normal force [N.sub.0] is identical to the plastic axial force [S.sub.0] we introduced in truss analysis; likewise, N and S are only two alternative symbols for the normal, or axial, force.
Relations (13.6) and (13.7), rewritten as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.8)
parametrically describe the plastic limit envelope. Eliminating the parameter [eta], we get an explicit equation for the envelope,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.9)
which obviously defines a parabolic arc.
Deriving the previous results, we have tacitly assumed that the bottom face of cross section is under tension and the top face under compression. In other words, equation (13.9) only applies if the bending moment is positive. To obtain a complete description of the plastic limit envelope, we have to consider the opposite case of tension at the top face and compression at the bottom. Reverting the signs of stresses and, consequently, of the internal forces, we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.10)
from which
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.11)
This equation describes a parabolic arc that is a mirror image of the previous one. The complete plastic limit envelope is shown as the solid curve in Figure 13.6.
Relations (13.9) and (13.11), separately describing the upper and lower branches of the plastic limit envelope, can be replaced by a single equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.12)
Note that the interior of the plastic limit envelope is characterized by f(N, M) < 0, and its exterior by f(N, M) > 0. A function f with this property is called the yield function.
2) The elastic limit envelope can easily be determined using the well-known expressions for stresses in the extreme points of the cross section,
[[sigma].sub.max] = N/A + M/[W.sub.el] (13.13)
[[sigma].sub.min = N/A + M/[W.sub.el] (13.14) where A = bh is the cross sectional area, and [W.sub.el] = b[h.sup.2]/6 is the elastic cross sectional modulus. The section remains elastic as long as |[sigma]| [less than or equal to] [[sigma].sub.0], i.e. the values of internal forces in the elastic domain satisfy the condition
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.15)
Note that (13.15) represents four inequalities that must be satisfied simultaneously. Using the relations [[sigma].sub.0]A = [N.sub.0] and [[sigma].sub.0][W.sub.el] = [M.sub.el] we can rewrite the condition as
|N/[N.sub.0] M/[M.sub.el]| [less than or equal to] 1 (13.16)
The previous description of the elastic domain applies to cross sections of an arbitrary shape, not only to rectangular ones. According to (13.16), the elastic domain is a parallelogram whose vertices correspond to the elastic limits under axial tension or compression, and under pure bending. Under axial tension or compression, the elastic limit concides with the plastic limit. Under pure bending, the ratio [M.sub.0]/[M.sub.el] = [alpha] depends on the shape of the cross section. For rectangular cross sections, [alpha] = 1.5, and the corresponding elastic limit envelope is marked in Figure 13.6 by the dashed lines.
Let us now explore the relation between the plastic deformations and the changes of internal forces. A generalized plastic hinge forms at a certain cross section when the internal forces reach the plastic limit envelope. In contrast to the plastic hinge under pure bending, the values of the internal forces do not have to remain constant as the hinge deforms. During plastic flow, the neutral axis can change its location while the cross section remains entirely plasticized. Consequently, the internal forces transmitted by the hinge move along the plastic limit envelope.
As explained before, the location of the neutral axis is determined, according to the deformation theory of plasticity, by the current values of the generalized strains e and [theta] or, according to the flow theory of plasticity, by their rates [??] and [??]. The neutral axis divides the cross section into the part yielding in tension, [A.sup.+], and the part yielding in compression, [A.sup.-]; see Figure 13.7. For a general cross section, the internal forces can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.18)
where
[S.sup.+] = [[integral].sub.[A.sup.+]] z dA (13.19)
is the static moment of the part under tension with respect to the centroidal axis, and
[S.sup.-] = [[integral].sub.[A.sup.-]] z dA (13.20)
is the static moment of the part under compression. Let [z.sub.n] denote the current position of the neutral axis. If the neutral axis currently located at [z.sub.n] moves by an infinitesimal distance d[z.sub.n], the corresponding increments of the areas and static moments can be expressed as
d[A.sup.+] = -[b.sub.n]d[z.sub.n] (13.21)
d[A.sup.-] = [b.sub.n]d[z.sub.n] (13.22)
d[S.sup.+] = [z.sub.n]d[A.sup.+] = -[b.sub.n][z.sub.n]d[z.sub.n] (13.23)
d[S.sup.-] = [z.sub.n]d[A.sup.-] = -[b.sub.n][z.sub.n]d[z.sub.n] (13.24)
where [b.sub.n] is the width of the cross section at the current level of the neutral axis; see Figure 13.7. Consequently, the internal forces change by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13.26)
The fact that the slope of the plastic limit envelope is equal to the distance of the neutral axis from the centroid, i.e.
dM/dN = [z.sub.n] (13.27)
has very important implications:
1. Convexity: As the position of the neutral axis changes from the bottom of the cross section ([z.sub.n] = [z.sub.max] > 0) to the top ([z.sub.n] = [z.sub.min] < 0), the corresponding point on the plastic limit envelope travels along the upper branch from the left corner to the right corner, and along the lower branch from the right to the left; see Figure 13.8. During this process, the slope dM/dN = [z.sub.n] monotonically decreases. Consequently, for the upper branch we have [d.sup.2]M/d[N.sup.2] < 0, and for the lower branch [d.sup.2]M/d[N.sup.2] > 0. This implies that the interior of the plastic limit envelope is a convex domain.
(Continues...)
Excerpted from Inelastic Analysis of Structuresby Milan Jirasek Zdenek P. Bazant Copyright © 2002 by Milan Jirasek Zdenek P. Bazant. Excerpted by permission.
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