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m1l3 - Module 1 Energy Methods in Structural Analysis...

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Module 1 Energy Methods in Structural Analysis Version 2 CE IIT, Kharagpur

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Lesson 3 Castigliano’s Theorems Version 2 CE IIT, Kharagpur
Instructional Objectives After reading this lesson, the reader will be able to; 1. State and prove first theorem of Castigliano. 2. Calculate deflections along the direction of applied load of a statically determinate structure at the point of application of load. 3. Calculate deflections of a statically determinate structure in any direction at a point where the load is not acting by fictious (imaginary) load method. 4. State and prove Castigliano’s second theorem. 3.1 Introduction In the previous chapter concepts of strain energy and complementary strain energy were discussed. Castigliano’s first theorem is being used in structural analysis for finding deflection of an elastic structure based on strain energy of the structure. The Castigliano’s theorem can be applied when the supports of the structure are unyielding and the temperature of the structure is constant. 3.2 Castigliano’s First Theorem For linearly elastic structure, where external forces only cause deformations, the complementary energy is equal to the strain energy. For such structures, the Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. Version 2 CE IIT, Kharagpur

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Let be the forces acting at from the left end on a simply supported beam of span n P P P ,.... , , 2 1 n x x x ,...... , , 2 1 L . Let be the displacements at the loading points respectively as shown in Fig. 3.1. Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by (vide eqn. 1.8 of lesson 1) n u u u ,..., , 2 1 n P P P ,.... , , 2 1 n n u P u P u P W 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.1) Version 2 CE IIT, Kharagpur
Work done by the external forces is stored in the structure as strain energy in a conservative system. Hence, the strain energy of the structure is, n n u P u P u P U 2 1 .......... 2 1 2 1 2 2 1 1 + + + = (3.2) Displacement below point is due to the action of acting at distances respectively from left support .

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m1l3 - Module 1 Energy Methods in Structural Analysis...

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