m3l15

m3l15 - Module 3 Analysis of Statically Indeterminate...

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Unformatted text preview: Module 3 Analysis of Statically Indeterminate Structures by the Displacement Method Version 2 CE IIT, Kharagpur 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 AB as shown in Fig.15.1.The support B is at a higher elevation compared to A by an amount . Hence, the member axis has rotated by an amount from the original direction as shown in the figure. Let L be the span of the beam and flexural rigidity of the beam EI , 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 A , B and (settlement) Version 2 CE IIT, Kharagpur The given beam with initial support settlement may be thought of as superposition of two simple cases as shown in Fig.15.1 (b) and in Fig. 15.1(c). In Fig.15.1b, the kinematically determinate beam is shown with the applied load. For this case, the fixed end moments are calculated by force method. Let A and B be the end rotations of the elastic curve with respect to rotated beam axis AB (see Fig.15.1c) that are caused by end moments and . Assuming that rotations and displacements shown in Fig.15.1c are so small that ' AB M ' BA M l = = tan (15.1) Also, using the moment area theorem, A and B are written as EI L M EI L M AB AB A A 6 ' 3 ' = = (15.2a) Version 2 CE IIT, Kharagpur EI...
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This note was uploaded on 06/30/2009 for the course CE 358 taught by Professor Trifunac during the Fall '07 term at USC.

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m3l15 - Module 3 Analysis of Statically Indeterminate...

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