## 211 Inelastic Analysis 2111 An Overall View

Inelastic analyses can be generalized into two main approaches. The first approach is known as plastic hinge analysis. The analysis assumes that structural elements remain elastic except at critical regions where plastic hinges are allowed to form. The second approach is known as spread of plasticity analysis. This analysis follows explicitly the gradual spread of yielding throughout the structure. Material yielding in the member is modeled by discretization of members into several line elements and subdivision of the cross-sections into many "fibers." Although the spread of plasticity analysis can predict accurately the inelastic response of the structure, the plastic hinge analysis is considered to be computationally more efficient and less expensive to execute.

If geometric nonlinear effect is not considered, the plastic hinge analysis predicts the maximum load of the structure corresponding to the formation of a plastic collapse mechanism (Chen and Sohal 1994). First-order plastic analysis has considerable application in continuous beams and low-rise building frames where members are loaded primarily in flexure. For tall building frames and for frames with slender columns subjected to side sway, the interaction between yielding and instability may lead to collapse prior to the formation of a plastic mechanism (SSRC 1988). If an incremental analysis is carried out based on the updated deformed geometry of the structure, the analysis is termed second order. The need for a second-order analysis of steel frame is increasing in view of the modern codes and standards that give explicit permission for the engineer to compute load effects from a direct second-order analysis.

This section presents the virtual work principle to explain the fundamental theorems of plastic hinge analysis. Simple and approximate techniques of practical plastic analysis methods are then introduced.

The concept of hinge-by-hinge analysis is presented. The more advanced topics such as second-order elastic-plastic hinge, refined plastic hinge analysis, and spread of plasticity analysis are covered in Section 2.12.

### 2.11.2 Ductility

Plastic analysis is strictly applicable for materials that can undergo large deformation without fracture. Steel is one such material with an idealized stress-strain curve as shown in Figure 2.91. When steel is subjected to tensile force, it will elongate elastically until the yield stress is reached. This is followed by an increase in strain without much increase in stress. Fracture will occur at very large deformation. This material idealization is generally known as elastic-perfectly plastic behavior. For a compact section, the attainment of initial yielding does not result in failure of the section. The compact section will have reserved plastic strength that depends on the shape of the cross-section. The capability of the material to deform under constant load without decrease in strength is the ductility characteristic of the material.

### 2.11.3 Redistribution of Forces

The benefit of using a ductile material can be demonstrated from an example of a three-bar system shown in Figure 2.92. From the equilibrium condition of the system

Assuming elastic stress-strain law, the displacement and force relationship of the bars may be written as

Since L2 = L1/2 = L/2, Equation 2.201 can be written as

where T1 and T2 are the tensile forces in the rods, L1 and L2 are lengths of the rods, A is the cross-section area, and E is the elastic modulus. Solving Equations 2.201 and 2.202 for T2

The load at which the structure reaches the first yield (in Figure 2.92b) is determined by letting T2 = oyA. From Equations 2.203

Elastic

Perfectly plastic

Strain

FIGURE 2.91 Idealized stress-strain curve.

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