ABSTRACT
A powerful method for calculating deflection is the method of virtual work. Consider the beam (bridge) of Figure A-1. We would like to know the deflection of the beam at midspan due to a truck load Q located as shown. I locate myself at midspan before the truck reaches the bridge. I am the virtual load, q. Comes now the truck. As it reaches the location shown, I feel myself being lowered by the deflection of the bridge at midspan due to the truck load Q. I do virtual work because of deflection, D , of the bridge by the truck. My virtual work is qD . But this virtual work acts on the bridge which stores my virtual work as (potential) virtual energy. The stored virtual energy is mdq, where m is the moment at any point on the bridge due to the virtual load, and dq is the change in curvature of the bridge at that point due to the truck load. But dq = Mdx/EI. So the virtual stored energy is mMdx/EI. Equating virtual work to virtual stored energy,
qD =foL mMdx/EI where q = virtual load, (Let q = 1.) m = moment in the beam due to load q, M = moment in the beam due to load Q, E = modulus of elasticity of material, I = moment of inertia of the cross section of
the beam about its horizontal neutral surface,
L = length of the beam.
