m1l4

m1l4 - 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 4 Theorem of Least Work Version 2 CE IIT, Kharagpur
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Instructional Objectives After reading this lesson, the reader will be able to: 1. State and prove theorem of Least Work. 2. Analyse statically indeterminate structure. 3. State and prove Maxwell-Betti’s Reciprocal theorem. 4.1 Introduction In the last chapter the Castigliano’s theorems were discussed. In this chapter theorem of least work and reciprocal theorems are presented along with few selected problems. We know that for the statically determinate structure, the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load at the point of application of load. This theorem when applied to the statically indeterminate structure results in the theorem of least work. 4.2 Theorem of Least Work According to this theorem, the partial derivative of strain energy of a statically indeterminate structure with respect to statically indeterminate action should vanish as it is the function of such redundant forces to prevent any displacement at its point of application. The forces developed in a redundant framework are such that the total internal strain energy is a minimum. This can be proved as follows. Consider a beam that is fixed at left end and roller supported at right end as shown in Fig. 4.1a. Let be the forces acting at distances from the left end of the 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. 4.1a. This is a statically indeterminate structure and choosing n u u u ,..., , 2 1 n P P P ,.... , , 2 1 a R as the redundant reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the principle of superposition, this may be treated as the superposition of two cases, viz, a cantilever beam with loads and a cantilever beam with redundant force n P P P ,.... , , 2 1 a R (see Fig. 4.2a and Fig. 4.2b) Version 2 CE IIT, Kharagpur
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Version 2 CE IIT, Kharagpur
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In the first case (4.2a), obtain deflection below A due to applied loads .
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m1l4 - Module 1 Energy Methods in Structural Analysis...

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