Page 35 - Mechanics of Microelectromechanical Systems

22 Chapter 1

and the complementary energy:

If the strain energy can be expressed as a function of solely the displacement
function, as:

then the variation in the strain energy can be written as:

which, by comparison to Eq. (1.63) yields:

Equation (1.67) is actually the mathematical expression of Castigliano’s first
theorem, stating that the load which is applied to an elastic body can be
calculated as the partial derivative of the strain energy stored in that body
taken with respect to the deformation set at the considered point about the
load’s direction.
When the complementary energy is a function of simply the loads acting
on the elastic body in the form:

the variation of this energy is:

By comparing Eqs. (1.69) and (1.64) results in:

which is the Castigliano’s second theorem , also known as the displacement
theorem, stating that an elastic deformation can be found by taking the partial
derivative of the complementary energy in terms of the load that is applied at
that point and about the considered direction.