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Total potential energy

The bar chooses the displacement that makes the total potential energy smallest. The energy is the strain energy minus the work the loads can do. With the $L^2$ inner product $(f, g) = \int_0^L fg\,dx$ the expression is short.

\Pi(u) = \tfrac12\int_0^L EA(u')^2dx - \int_0^L F_x\,u\,dx - P\,u(L)

total potential energy

\Pi(u) = \tfrac12 B(u, u) - F(u)

short form

Symbols

$\Pi$	total potential energy	J
$B(u, u)$	twice the strain energy	J
$F(u)$	work of the loads	J

Example

A spring with stiffness $k$ and force $F$ : $\Pi(u) = \tfrac12 ku^2 - Fu$ . Minimum where $\Pi'(u) = ku - F = 0$ , so $u = F/k$ . The bar is the same idea with integrals.

You find the minimum of

\Pi

with the Gateaux derivative:

\delta\Pi(u, v) = 0

for all admissible

v

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Part of Finite Element Method: Bar elements and stiffness matrices.