Method of Virtual Work - Structural Analysis. Virtual work - Wikipedia, the free encyclopedia. Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. The principle of virtual work states that for any compatible virtual displacement Virtual Displacement & Virtual Work If a set of particles or a body is in equilibrium under the action of forces then there is no motion and consequently there is no. The principle of virtual work can be derived form the equations of equilibrium and vice versa. The condition of equilibrium for a. Brief explanation of the principle of virtual work and a description of the process to calculate deflections in structures using the method of virtual work. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies. It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians. He was able to solve problems for both rigid bodies as well as fluids. Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1. Varignon's second volume of Nouvelle m. This formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. His idea was to convert a dynamical problem into static problem by introducing inertial force. A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic, essentially the D'Alembert's principle, was given in his M. The principle of virtual work, which is the form of the principle of least action applied to these systems, states that the path actually followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero (to first order). The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, and is termed the calculus of variations. Consider a point particle that moves along a path which is described by a function r(t) from point A, where r(t = t. B, where r(t = t. It is possible that the particle moves from A to B along a nearby path described by r(t) + . The components of the variation, . This can be generalized to an arbitrary mechanical system defined by the generalized coordinatesqi , i = 1, .., n. In which case, the variation of the trajectory qi (t) is defined by the virtual displacements . When considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero. Introduction. The work done by the force F is given by the integral. W=. It is important to notice that the value of the work W depends on the trajectory r(t). Now consider particle P that moves from point A to point B again, but this time it moves along the nearby trajectory that differs from r(t) by the variation . Suppose the force F(r(t)+. The work done by the force is given by the integral. W. Hence, we can define n generalized coordinates qi (t) (i = 1, 2, .., n), and express r(t) and . That is,r(t)=r(q. In other words, both linear momentum and angular momentum of the system are conserved. The principle of virtual work states that the virtual work of the applied forces is zero for all virtual movements of the system from static equilibrium. This principle can be generalised such that three dimensional rotations are included: the virtual work of the applied forces and applied moments is zero for all virtual movements of the system from static equilibrium. It is sufficient to use only f coordinates to give a complete description of the motion of the system, so fgeneralised coordinatesqk , k = 1, 2, .., f are defined such that the virtual movements can be expressed in terms of these generalised coordinates. 2.4.4 The principle of Virtual Work. The principle of virtual work forms the basis for the finite element method in the mechanics of solids and so will be. ArXiv:physics/0510204v2 Therefore, the generalized forces Qk are zero, that is. There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis. The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint. Lanczos. The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero. Newton's laws state that at equilibrium the applied forces are equal and opposite to the reaction, or constraint forces. This means the virtual work of the constraint forces must be zero as well. Law of the Lever. The lever is operated by applying an input force FA at a point A located by the coordinate vector r. A on the bar. The lever then exerts an output force FB at the point B located by r. B. The rotation of the lever about the fulcrum P is defined by the rotation angle . These forces are given by. FA=FA. If the opposite is true that the distance from the fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever reduces the magnitude of the input force. This is the law of the lever, which was proven by Archimedes using geometric reasoning. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next. For this analysis, we consider a gear train that has one degree- of- freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear. Transmission of motion and force by gear wheels, compound train. The size of the gears and the sequence in which they engage define the ratio of the angular velocity . Let R be the speed ratio, then. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train. Let the angle . This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque. Dynamic equilibrium for rigid bodies. The result is D'Alembert's form of the principle of virtual work, which is used to derive the equations of motion for a mechanical system of rigid bodies. The expression compatible displacements means that the particles remain in contact and displace together so that the work done by pairs of action/reaction inter- particle forces cancel out. Various forms of this principle have been credited to Johann (Jean) Bernoulli (1. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity Vi and angular velocities . Such a system of rigid bodies is said to have one degree of freedom. Consider a single rigid body which moves under the action of a resultant for F and torque T, with one degree of freedom defined by the generalized coordinate q. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force Q* associated with the generalized coordinate q is given by. Q. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that. This condition yields m equations,Fj+Qj. Let's define two unrelated states for the body: The superscript * emphasizes that the two states are unrelated. Other than the above stated conditions, there is no need to specify if any of the states are real or virtual. Imagine now that the forces and stresses in the . The principle of virtual work then states: External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains. It includes the principle of virtual work for rigid bodies as a special case where the internal virtual work is zero. Proof of Equivalence between the Principle of Virtual Work and the Equilibrium Equation. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero. Now recap. We have shown through the above derivation that. They are valid irrespective of material behaviour. Principle of virtual displacements. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify: The virtual work equation then becomes the principle of virtual displacements. Conversely, (f) can be reached, albeit in a non- trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on St. The virtual work principle is also valid for large real displacements; however, Eq.(f) would then be written using more complex measures of stresses and strains. Principle of virtual forces. It has another name: the principle of complementary virtual work. Alternative forms. According to D'Alembert's principle, inclusion of inertial forces as additional body forces will give the virtual work equation applicable to dynamical systems. More generalized principles can be derived by: allowing variations of all quantities. Lagrange multipliers to impose boundary conditions and/or to relax the conditions specified in the two states. These are described in some of the references. Among the many energy principles in structural mechanics, the virtual work principle deserves a special place due to its generality that leads to powerful applications in structural analysis, solid mechanics, and finite element method in structural mechanics. See also. Shames, Solid Mechanics: A Variational Approach, Mc. Graw- Hill, 1. 97. Danilo Capecchi, History of Virtual Work Laws, Springer- Verlag, Italy, 2. Ren. Levinson, Dynamics: theory and applications, Mc. Graw- Hill, New York, 1. Usher, A. A History of Mechanical Inventions. Harvard University Press (reprinted by Dover Publications 1. ISBN 9. 78- 0- 4. Retrieved 7 April 2. Levinson, Dynamics, Theory and Applications, Mc. Graw- Hill, NY, 2. Bibliography. ISBN 0- 1.
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