TABLE OF CONTENTS So far we only analyzed elastic beams. When a beam, or another structure composed of thin rods, reaches yield stress at some cross section, then this section behaves like a hinge. Thus the beam transforms into a mechanism whose parts can rotate around this hinge, causing the collapse of the system (Fig. 8-75). However, when a statically indeterminate beam, made of a ductile material, reaches yield stress at a certain cross section, it may still carry loads. This is studied by the theory of limit analysis based on the theory of plasticity. A few examples will help in understanding this theory. Let us consider the three-bar symmetric truss shown in Fig. 8-76. It is assumed that each bar has the same cross-sectional area A and is made of the same material with Young's modulus E in the elastic range. First we calculate the forces and the deformations of each bar. We note that the truss is statically indeterminate and we remove bar 2-4, replacing its action by the unknown force X (Fig. 8-77). From the equilibrium equation of node 4 we find that the force N in each remaining bar is
In order to determine X we use the conditional equation
which states that the vertical displacement of the system 1-4-3 at node 4 equals the extension of the removed bar 2-4. However,
whereas
where the...
