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Unformatted text preview: Section 5 WORK AND ENERGY PRINCIPLES The concepts of work and energy are fundamental to the analysis of structures. They provide our most helpful tools and allow us to simplify the processes of satisfying the three basic requirements of equilibrium, stressstrain and compatibility. Without these principles it would be very difficult to solve many problems with any degree of assurance; they form the basis of modern analysis including the finite element techniques implemented in a variety of software. Engineers, mathematicians and physicists try to find grand unifying principles to explain what they do. For mechanics, including dynamics, materials and structures, work and energy relationships tie together forces, displacements, velocities and accelerations. For structures we normally focus on the relationships between forces and displacements; this makes work the most obvious place to look to find a fundamental principle. Consider a particle that can translate in threedimensional space. Let us suppose that several forces act on it while it undergoes a small displacement du. The situation is illustrated by the following figure. F 1 F 2 F 3 x y O z Figure 5.1 Particle Translating in Space The work done by any of these forces during the displacement is i dW F du = r r g . Over a finite displacement from to a b u u r r , the work that this force does is given by the integral. i where is the component of F in the direction of the displacement b b a a u u i i iu u u iu W F du F du F = = r r r r g r 1999, 2000 Robert O. Meitz 41 Lecture Notes Aerospace Structures Page 42 Note that the nature of the force is not defined in any particular way. All that has been said is that the force exists and that its point of application has moved by some amount. The work done by the force depends only on the history of the force as the displacement takes place as implied by the integral expressions. With several forces acting as shown in Figure 1, the total work is the sum of the work done by the several forces. 1 1 b a n n u i i u i i W W F du = = = = r r r r g The force system acting on a particle is statically equivalent to a single force called the resultant and may be replaced by it in any expression. If the forces on the particle are in equilibrium, the resultant force is zero and the net work done by the force system is also zero. This will be true no matter what direction the displacement may take. This observation makes it clear that we can use the concept of work to establish whether a force system is in equilibrium. Specifically, if we discover that the net work is zero for any possible displacement, then the forces on the object are in equilibrium. The precise formulation of any work principle will depend on the nature of the problem being considered....
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 Fall '06
 Meitz

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