m3l18

m3l18 - Module 3 Analysis of Statically Indeterminate...

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Module 3 Analysis of Statically Indeterminate Structures by the Displacement Method Version 2 CE IIT, Kharagpur
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Lesson 18 The Moment- Distribution Method: Introduction Version 2 CE IIT, Kharagpur
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Instructional Objectives After reading this chapter the student will be able to 1. Calculate stiffness factors and distribution factors for various members in a continuous beam. 2. Define unbalanced moment at a rigid joint. 3. Compute distribution moment and carry-over moment. 4. Derive expressions for distribution moment, carry-over moments. 5. Analyse continuous beam by the moment-distribution method. 18.1 Introduction In the previous lesson we discussed the slope-deflection method. In slope- deflection analysis, the unknown displacements (rotations and translations) are related to the applied loading on the structure. The slope-deflection method results in a set of simultaneous equations of unknown displacements. The number of simultaneous equations will be equal to the number of unknowns to be evaluated. Thus one needs to solve these simultaneous equations to obtain displacements and beam end moments. Today, simultaneous equations could be solved very easily using a computer. Before the advent of electronic computing, this really posed a problem as the number of equations in the case of multistory building is quite large. The moment-distribution method proposed by Hardy Cross in 1932, actually solves these equations by the method of successive approximations. In this method, the results may be obtained to any desired degree of accuracy. Until recently, the moment-distribution method was very popular among engineers. It is very simple and is being used even today for preliminary analysis of small structures. It is still being taught in the classroom for the simplicity and physical insight it gives to the analyst even though stiffness method is being used more and more. Had the computers not emerged on the scene, the moment-distribution method could have turned out to be a very popular method. In this lesson, first moment-distribution method is developed for continuous beams with unyielding supports. 18.2 Basic Concepts In moment-distribution method, counterclockwise beam end moments are taken as positive. The counterclockwise beam end moments produce clockwise moments on the joint Consider a continuous beam ABCD as shown in Fig.18.1a. In this beam, ends A and D are fixed and hence, 0 = = D A θ .Thus, the deformation of this beam is completely defined by rotations B and C at joints B and C respectively. The required equation to evaluate B and C is obtained by considering equilibrium of joints B and C . Hence, Version 2 CE IIT, Kharagpur
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0 = B M 0 = + BC BA M M (18.1a) 0 = C M 0 = + CD CB M M (18.1b) According to slope-deflection equation, the beam end moments are written as ) 2 ( 2 B AB AB F BA BA L EI M M θ + = AB AB L EI 4 is known as stiffness factor for the beam AB and it is denoted by . is the fixed end moment at joint B of beam AB when joint B is fixed.
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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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m3l18 - Module 3 Analysis of Statically Indeterminate...

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