Potential Energy

Potential energy V of a system in a state is defined to be an energy increment from a reference state,

(3.75)

in which

(3.76)

Here, the potential energy depends on one or more functions of one or more variables, representing the displacement field. In Euler's problem for instance, V is a so-called functional (i.e., a function of a function) of the form V[ w( x)], where w( x) is an arbitrary admissible function. The term admissible means that all the kinematic constraints of the problem are satisfied. In the case of a column (Table 3.1), admissible displacement fields or configurations are represented by continuous function w( x) having continuous first derivatives and satisfying the kinematic (or geometric) end conditions. This requirement on continuity is essential to exclude fracture of the column. In the case of the cantilever column (Table 3.1), the kinematic end conditions at the fixed end are w( o) = 0, w'( o) = 0. They represent kinematic (geometric) constraints, while the condition w"( l) = 0 is called static end condition since it merely states that the bending moment vanishes at the free end. At an equilibrium stale, the potential energy functional takes a stationary value, in other words, the first variation of the potential energy vanishes.

(3.77)

Thus the change in internal energy of the system is equal to...