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m2l8 - 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 8 The Force Method of Analysis: Beams Version 2 CE IIT, Kharagpur
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Instructional Objectives After reading this chapter the student will be able to 1. Solve statically indeterminate beams of degree more than one. 2. To solve the problem in matrix notation. 3. To compute reactions at all the supports. 4. To compute internal resisting bending moment at any section of the continuous beam. 8.1 Introduction In the last lesson, a general introduction to the force method of analysis is given. Only, beams, which are statically indeterminate to first degree, were considered. If the structure is statically indeterminate to a degree more than one, then the approach presented in the previous example needs to be organized properly. In the present lesson, a general procedure for analyzing statically indeterminate beams is discussed. 8.2 Formalization of Procedure Towards this end, consider a two-span continuous beam as shown in Fig. 8.1a. The flexural rigidity of this continuous beam is assumed to be constant and is taken as EI . Since, the beam is statically indeterminate to second degree, it is required to identify two redundant reaction components, which need be released to make the beam statically determinate. Version 2 CE IIT, Kharagpur
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Version 2 CE IIT, Kharagpur
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The redundant reactions at A and B are denoted by and respectively. The released structure (statically determinate structure) with applied loading is shown in Fig. 8.1b. The deflection of primary structure at 1 R 2 R B and C due to applied loading is denoted by ( ) 1 L Δ and ( ) 2 L Δ respectively. Throughout this module notation is used to denote deflection at redundant due to applied loads on the determinate structure. () i L Δ th i EI PL EI wL L 12 7 8 3 4 1 = Δ ( 8 . 1 a ) EI PL EI wL L 16 27 24 7 3 4 2 = Δ ( 8 . 1 b ) In fact, the subscript 1 and represent, locations of redundant reactions released. In the present case 2 ( ) 1 R R A = and ( ) 2 R R B = respectively. In the present and subsequent lessons of this module, the deflections and the reactions are taken to be positive in the upward direction. However, it should be kept in mind that the positive sense of the redundant can be chosen arbitrarily. The deflection of the point of application of the redundant should likewise be considered positive when acting in the same sense. For writing compatibility equations at B and , it is required to know deflection of the released structure at C B and due to external loading and due to redundants.
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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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m2l8 - Module 2 Analysis of Statically Indeterminate...

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