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Use the principle of virtual work to find approximate linear, quadratic, cubic, and quartic polynomial displacement solutions for a simply supported beam with length , Young’s modulus , moment of inertia , and a distributed load . Compare the approximate solutions with the exact solution for units, units, and units.
The virtual work method, also referred to as the method of virtual force or unit-load method, uses the law of conservation of energy to obtain the deflection and slope at a point in a structure. This method was developed in 1717 by John Bernoulli.
The Principle of Virtual Work Definitions: Virtual work is the work done by a real force acting through a virtual displace-ment or a virtual force acting through a real displacement. A virtual displacement is any displacement consistent with the constraints of the structure, i.e., that satisfy the boundary conditions at the supports.
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.
Principle of Virtual Work. The principle of virtual work states that if a body is in equilibrium, then the algebraic sum of the virtual work done by all the forces and couple moments acting on the body is zero for any virtual displacement of the body. Thus,
We see that (B) is the generalized form of (A’). The principle of virtual work states that for any compatible virtual displacement field imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work.
The principle of virtual work for deformable bodies says that the external virtual work applied to a structure must equal the internal virtual work that is caused within the structure: \begin{equation} \boxed{W_{v,e} = W_{v,i}} \label{eq:virtual-work} \tag{12} \end{equation}
The principle of virtual work is fundamental to the finite element method, which is used to solve problems described by systems of partial differential equations in many disciplines. Here is a typical example of how we can apply the principle of virtual work to find the deflections at some point in an elastic solid. Consider a beam carrying ...
A very similar method is to use the principle of virtual work. In this method, we imagine that we act upon the system in such a manner as to increase one of the coordinates. We imagine, for example, what would happen if we were to stretch one of the springs, or to increase the angle between two jointed rods, or the angle that the ladder makes ...
From the physical point of view, the principle of virtual work is an attempt to characterize unequivocally an equilibrium configuration of a mechanical system (as defined in statics (q.v.)) by observing how it reacts to a small kinematical perturbation, called a virtual displacement.