m1l6

m1l6 - Module 1 Energy Methods in Structural Analysis...

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Version 2 CE IIT, Kharagpur Module 1 Energy Methods in Structural Analysis
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Version 2 CE IIT, Kharagpur Lesson 6 Engesser’s Theorem and Truss Deflections by Virtual Work Principles
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Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove Crotti-Engesser theorem. 2. Derive simple expressions for calculating deflections in trusses subjected to mechanical loading using unit-load method. 3. Derive equations for calculating deflections in trusses subjected to temperature loads. 4. Compute deflections in trusses using unit-load method due to fabrication errors. 6.1 Introduction In the previous lesson, we discussed the principle of virtual work and principle of virtual displacement. Also, we derived unit – load method from the principle of virtual work and unit displacement method from the principle of virtual displacement. In this lesson, the unit load method is employed to calculate displacements of trusses due to external loading. Initially the Engesser’s theorem, which is more general than the Castigliano’s theorem, is discussed. In the end, few examples are solved to demonstrate the power of virtual work. 6.2 Crotti-Engesser Theorem The Crotti-Engesser theorem states that the first partial derivative of the complementary strain energy ( ) * U expressed in terms of applied forces is equal to the corresponding displacement. j F * 1 n jk k j k j U aF u F = = = (6.1) For the case of indeterminate structures this may be stated as, 0 * = j F U ( 6 . 2 ) Note that Engesser’s theorem is valid for both linear and non-linear structures. When the complementary strain energy is equal to the strain energy (i.e. in case of linear structures) the equation (6.1) is nothing but the statement of Castigliano’s first theorem in terms of complementary strain energy. Version 2 CE IIT, Kharagpur
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In the above figure the strain energy (area OACO) is not equal to complementary strain energy (area OABO) = = u du F U OACO Area 0 (6.3) Differentiating strain energy with respect to displacement, F du dU = ( 6 . 4 ) This is the statement of Castigliano’s second theorem. Now the complementary energy is equal to the area enclosed by OABO. = F dF u U 0 * ( 6 . 5 ) Differentiating complementary strain energy with respect to force , F u dF dU = * ( 6 . 6 ) Version 2 CE IIT, Kharagpur
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This gives deflection in the direction of load. When the load displacement relationship is linear, the above equation coincides with the Castigliano’s first theorem given in equation (3.8).
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m1l6 - Module 1 Energy Methods in Structural Analysis...

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