# Analysis of Elasto-Plastic Frames

DOI: 10.4018/978-1-4666-6379-4.ch008

## Abstract

This chapter, as well as Chapter 4, deals with the algorithms for the analysis of frames; specifically, it shows how the models for elasto-plastic elements presented in Chapter 7 can be used in a structural analysis. In the first section, a particularly efficient algorithm is presented: the hinge-by-hinge method; in this case, the analysis of an elasto-plastic frame can be treated as a sequence of linear problems; this analysis also allows for an estimation of the ultimate resistance forces of the structure; on the other hand, the hinge-by-hinge procedure can only be used in very particular cases. A general procedure for the analysis of any kind of elasto-plastic frames under quasi-static forces is presented in Sections 8.2 and 8.3; this method is based on an algorithm called elastic predictor-plastic corrector that is a key concept for most of the inelastic structural analyses, even for the damage and fracture models that are described in the next chapters. The same algorithm can also be used for the dynamic analysis of elasto-plastic frames as discussed in Section 8.3.4.
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## 8.1 The Hinge-By-Hinge Method For Perfectly Plastic Frames

### 8.1.1 Formulation of the Problem

Consider a perfectly plastic two dimensional frame, the generalized deformation and generalized stress measures are thus given by (3.1.3) and (3.2.2); the elements of the frame are slender, the plastic elongations are neglected and nonlinear geometrical effects are not considered.

The structure is subjected to the following loading system: first, a set of distributed loads is applied for a time t that is arbitrarily defined as zero; it is assumed that these loads do not produce plastic deformations of the frame elements and, as it is usual, that they remain constant for the subsequent steps; these forces generate the matrix . Then, a second set of concentrated loads on the frame nodes is applied; it is assumed that these forces can be written as: ; where is a matrix of reference forces and l(t) is a loading parameter; the reference forces matrix may have non zero terms only in the locations corresponding to the free displacements and these terms are constant during the entire analysis. The term l(t) represents a monotonically increasing function of time; that is: for a t2> t1 then l(t2) > l(t1); the loading parameter takes the value zero for t = 0; next, it increases until collapse of the structure is achieved. All these external forces produce reactions that will be grouped into a matrix denoted .

Therefore, the problem to be solved can be defined as:

• 1.

Compute: the free displacements, reaction forces, generalized deformations, plastic rotations and stresses.

• 2.

With the Following Data: the initial configuration of the structure, the restricted displacements, the reference forces and those applied over the elements and, finally, the material and cross-section properties (E, I, A, My).

• 3.

Such that they verify:

• a.

The linear kinematic equation:

• b.

The linear equilibrium equation:

• c.

The elasticity law:

(8.1.1)

• d.

The plastic rotations evolution laws:

• e.

The yield functions:;

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