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m1l5 - 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 5 Virtual Work Version 2 CE IIT, Kharagpur
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Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows 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. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F F system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,....... , , 2 1 n ,..... , 2 , 1 n u u u ,...... , , 2 1 n F F F ,....... , , 2 1 ij σ and real internal strains ij ε inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F δ ,...... , , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement Version 2 CE IIT, Kharagpur
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configuration n u u u δ , ,......... , 2 1 . The virtual loading system produces virtual internal stresses ij δσ and virtual internal strains ij δε inside the beam. Now, apply the second system of forces on the beam which has been deformed by first system of forces. Then, the external loads and internal stresses i F ij σ do virtual work by moving along i u and ij . The product i i u F is known as the external virtual work. It may be noted that the above product does not represent the conventional work since each component is caused due to different source i.e.
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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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m1l5 - Module 1 Energy Methods in Structural Analysis...

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