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The bar problem: strong form

A bar fixed at one end and pulled by a distributed load $F_x$ and a point load $P$ . Three relations give everything: strain $\varepsilon = u'$ , Hooke's law $\sigma = E\varepsilon$ and equilibrium $N' + F_x = 0$ . Together they give the differential equation with one displacement condition (essential) and one force condition (natural).

-(EA\,u')' = F_x

strong form for

0 < x < L

u(0) = 0,\quad EA\,u'(L) = P

essential and natural boundary condition

N = EA\,u'

normal force

Symbols

$u$	displacement	m
$EA$	axial stiffness	N
$F_x$	distributed load	N/m
$P$	point load at the end	N

Example

Constant $EA$ , uniform load $q$ , point load $P$ :

$u(x) = \frac{1}{EA}\big((P + qL)x - \tfrac12 qx^2\big)$ and $N(x) = P + q(L - x)$ .

Displacement conditions are essential (built into the function space). Force conditions are natural (enter through the boundary term in the weak form).

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