m4l23

# M4l23 - Module 4 Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version 2 CE IIT Kharagpur Lesson 23 The Direct

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Module 4 Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version 2 CE IIT, Kharagpur

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Lesson 23 The Direct Stiffness Method: An Introduction Version 2 CE IIT, Kharagpur
Instructional Objectives: After reading this chapter the student will be able to 1. Differentiate between the direct stiffness method and the displacement method. 2. Formulate flexibility matrix of member. 3. Define stiffness matrix. 4. Construct stiffness matrix of a member. 5. Analyse simple structures by the direct stiffness matrix. 23.1 Introduction All known methods of structural analysis are classified into two distinct groups:- (i) force method of analysis and (ii) displacement method of analysis. In module 2, the force method of analysis or the method of consistent deformation is discussed. An introduction to the displacement method of analysis is given in module 3, where in slope-deflection method and moment- distribution method are discussed. In this module the direct stiffness method is discussed. In the displacement method of analysis the equilibrium equations are written by expressing the unknown joint displacements in terms of loads by using load- displacement relations. The unknown joint displacements (the degrees of freedom of the structure) are calculated by solving equilibrium equations. The slope-deflection and moment-distribution methods were extensively used before the high speed computing era. After the revolution in computer industry, only direct stiffness method is used. The displacement method follows essentially the same steps for both statically determinate and indeterminate structures. In displacement /stiffness method of analysis, once the structural model is defined, the unknowns (joint rotations and translations) are automatically chosen unlike the force method of analysis. Hence, displacement method of analysis is preferred to computer implementation. The method follows a rather a set procedure. The direct stiffness method is closely related to slope-deflection equations. The general method of analyzing indeterminate structures by displacement method may be traced to Navier (1785-1836). For example consider a four member truss as shown in Fig.23.1.The given truss is statically indeterminate to second degree as there are four bar forces but we have only two equations of equilibrium. Denote each member by a number, for example (1), (2), (3) and (4). Let i α be the angle, the i- th member makes with the horizontal. Under the Version 2 CE IIT, Kharagpur

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action of external loads and , the joint E displaces to E’ . Let u and v be its vertical and horizontal displacements. Navier solved this problem as follows. x P y P In the displacement method of analysis u and v are the only two unknowns for this structure. The elongation of individual truss members can be expressed in terms of these two unknown joint displacements. Next, calculate bar forces in the members by using force–displacement relation. Now at E , two equilibrium equations can be written viz., 0 = x F and 0 = y F by summing all forces in x and y directions. The unknown displacements may be calculated by solving
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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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M4l23 - Module 4 Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version 2 CE IIT Kharagpur Lesson 23 The Direct

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