##### From Theory of Beam Columns: In-Plane Behavior and Design, Volume 1

## 5.1 INTRODUCTION

The basic quantity required in any column analysis is the value *EI*, which here can be considered as the slope of the relation curve between moment *M*, and curvature ?. Since the governing differential equation of a beam-column problem always involves this quantity in its coefficients as can be seen in Eq. (3.19). moment-curvature relation of column segments is to be studied before we undertake the theory of plastic stability of long compression members. In elastic range. this quantity is constant and therefore presents no difficulties. In plastic range, however, this quantity is not constant because the moment-curvature response is nonlinear and the instantaneous bending rigidity *EI* should be used.

A typical moment-curvature relationship for a steel wide flange cross-section under pure bending is shown diagrammatieatly in Fig. 5.1, which can be obtained from the stress-strain curve given in Fig. 2.5(b). Although the stress-strain relationship used is assumed to be elastic-perfectly plastic, the generalized stress-strain or moment-curvature relationship of the segment is seen to show an elastic-strain hardening characteristic. This is due to the fact that the section can be partially elastic and partially plastic as shown by the insets in the figure (see problem 5.2).

Figure 5.1: Moment-curvature relationship

The moment-curvature relationship depends on the existence of axial thrust *P* and, of course, on the shape of the cross section.

In this chapter, we will study the elastic-plastic response of column segments subjected to compression combined with a bending moment. In order...

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