Lesson 15 The Slope-Deflection Method: Beams (Continued) Version 2 CE IIT, Kharagpur
Instructional Objectives After reading this chapter the student will be able to 1. Derive slope-deflection equations for the case beam with yielding supports. 2. Estimate the reactions induced in the beam due to support settlements. 3. Analyse the beam undergoing support settlements and subjected to external loads. 4. Write joint equilibrium equations in terms of moments. 5. Relate moments to joint rotations and support settlements. 15.1 Introduction In the last lesson, slope-deflection equations were derived without considering the rotation of the beam axis. In this lesson, slope-deflection equations are derived considering the rotation of beam axis. In statically indeterminate structures, the beam axis rotates due to support yielding and this would in turn induce reactions and stresses in the structure. Hence, in this case the beam end moments are related to rotations, applied loads and beam axes rotation. After deriving the slope-deflection equation in section 15.2, few problems are solved to illustrate the procedure. Consider a beam ABas shown in Fig.15.1.The support Bis at a higher elevation compared to Aby an amountΔ. Hence, the member axis has rotated by an amount ψfrom the original direction as shown in the figure. Let Lbe the span of the beam and flexural rigidity of the beamEI, is assumed to be constant for the beam. The chord has rotated in the counterclockwise direction with respect to its original direction. The counterclockwise moment and rotations are assumed to be positive. As stated earlier, the slopes and rotations are derived by superposing the end moments developed due to (1) Externally applied moments on beams. (2) Displacements Aθ,Bθand Δ(settlement) Version 2 CE IIT, Kharagpur
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