Answer:
The converse of the Principle of Virtual Work states that if the virtual work is zero for all possible virtual displacements consistent with the constraints of a system, then the system is in equilibrium.
To prove the converse of the Principle of Virtual Work, we start with the assumption that the virtual work is zero for all possible virtual displacements consistent with the constraints of a system.
Let's consider a system in equilibrium, subjected to external forces and constraints. The virtual work done by external forces on the system is given by:
δW_ext = ∑ F_i * δr_i
where δW_ext represents the virtual work done by external forces, F_i represents the ith external force, and δr_i represents the virtual displacement in the direction of the external force.
Since the system is in equilibrium, the virtual work done by internal forces is also zero:
δW_int = ∑ F_i * δr_i = 0
where δW_int represents the virtual work done by internal forces.
Now, let's consider a virtual displacement consistent with the constraints of the system. This virtual displacement can be expressed as a combination of virtual displacements due to external forces (δr_ext) and virtual displacements due to internal forces (δr_int):
δr = δr_ext + δr_int
Substituting this expression into the equation for the virtual work done by external forces, we get:
δW_ext = ∑ F_i * (δr_ext + δr_int)
Expanding this equation, we have:
δW_ext = ∑ F_i * δr_ext + ∑ F_i * δr_int
Since the virtual work done by external forces is zero (as assumed), the equation becomes:
0 = ∑ F_i * δr_int
Since this equation holds for all possible virtual displacements consistent with the constraints of the system, it implies that the virtual work done by internal forces is also zero for all possible virtual displacements:
δW_int = ∑ F_i * δr_i = 0
Therefore, we can conclude that if the virtual work is zero for all possible virtual displacements consistent with the constraints of a system, then the system is in equilibrium. This proves the converse of the Principle of Virtual Work.
Explanation: