Version 2 CE IIT, Kharagpur Lesson 29 The Direct Stiffness Method: Beams (Continued)
Version 2 CE IIT, Kharagpur Instructional Objectives After reading this chapter the student will be able to 1. Compute moments developed in the continuous beam due to support settlements. 2. Compute moments developed in statically indeterminate beams due to temperature changes. 3. Analyse continuous beam subjected to temperature changes and support settlements. 29.1 Introduction In the last two lessons, the analysis of continuous beam by direct stiffness matrix method is discussed. It is assumed in the analysis that the supports are unyielding and the temperature is maintained constant. However, support settlements can never be prevented altogether and hence it is necessary to make provisions in design for future unequal vertical settlements of supports and probable rotations of fixed supports. The effect of temperature changes and support settlements can easily be incorporated in the direct stiffness method and is discussed in this lesson. Both temperature changes and support settlements induce fixed end actions in the restrained beams. These fixed end forces are handled in the same way as those due to loads on the members in the analysis. In other words, the global load vector is formulated by considering fixed end actions due to both support settlements and external loads. At the end, a few problems are solved to illustrate the procedure. 29.2 Support settlements Consider continuous beam ABCas shown in Fig. 29.1a. Assume that the flexural rigidity of the continuous beam is constant throughout. Let the support Bsettles by an amount Δas shown in the figure. The fixed end actions due to loads are shown in Fig. 29.1b. The support settlements also induce fixed end actions and are shown in Fig. 29.1c. In Fig. 29.1d, the equivalent joint loads are shown. Since the beam is restrained against displacement in Fig. 29.1b and Fig. 29.1c, the displacements produced in the beam by the joint loads in Fig. 29.1d must be equal to the displacement produced in the beam by the actual loads in Fig. 29.1a. Thus to incorporate the effect of support settlement in the analysis it is required to modify the load vector by considering the negative of the fixed end actions acting on the restrained beam.
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