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m2l9

# m2l9 - Module 2 Analysis of Statically Indeterminate...

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Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Version 2 CE IIT, Kharagpur

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Lesson 9 The Force Method of Analysis: Beams (Continued) Version 2 CE IIT, Kharagpur
Instructional Objectives After reading this chapter the student will be able to 1. Calculate additional stresses developed in statically indeterminate structures due to support settlements. 2. Analyse continuous beams which are supported on yielding supports. 3. Sketch the deflected shape of the member. 4. Draw banding moment and shear force diagrams for indeterminate beams undergoing support settlements. 9.1 Introduction In the last lesson, the force method of analysis of statically indeterminate beams subjected to external loads was discussed. It is however, assumed in the analysis that the supports are unyielding and the temperature remains constant. In the design of indeterminate structure, it is required to make necessary provision for future unequal vertical settlement of supports or probable rotation of supports. It may be observed here that, in case of determinate structures no stresses are developed due to settlement of supports. The whole structure displaces as a rigid body (see Fig. 9.1). Hence, construction of determinate structures is easier than indeterminate structures. Version 2 CE IIT, Kharagpur

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Version 2 CE IIT, Kharagpur
The statically determinate structure changes their shape due to support settlement and this would in turn induce reactions and stresses in the system. Since, there is no external force system acting on the structures, these forces form a balanced force system by themselves and the structure would be in equilibrium. The effect of temperature changes, support settlement can also be easily included in the force method of analysis. In this lesson few problems, concerning the effect of support settlement are solved to illustrate the procedure. 9.2 Support Displacements Consider a two span continuous beam, which is statically indeterminate to second degree, as shown in Fig. 9.2. Assume the flexural rigidity of this beam to be constant throughout. In this example, the support B is assumed to have settled by an amount as shown in the figure. b Δ This problem was solved in the last lesson, when there was no support settlement (vide section 8.2). In section 8.2, choosing reaction at B and C as the redundant, the total deflection of the primary structure due to applied external loading and redundant and is written as, 1 R 2 R ( ) 2 12 1 11 1 1 R a R a L + + Δ = Δ ( 9 . 1 a ) ( ) 2 22 1 21 2 2 R a R a L + + Δ = Δ ( 9 . 1 b ) wherein, and are the redundants at 1 R 2 R B and C respectively, and , and are the deflections of the primary structure at () 1 L Δ 2 L Δ B and due to applied loading. In the present case, the support

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m2l9 - Module 2 Analysis of Statically Indeterminate...

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