Our task is to estimate the axial displacement u at any section x 1-D ELASTICITY Our first practical application of FE will be an elastic rod subjected to a distributed axial load F(x) along its length. The rod may also have a variable stiffness. x dx EA(x) F(x) Our task is to estimate the axial displacement u at any section x We need a differential equation to describe this system, together with boundary conditions. Once we have these, we can then attempt to solve the problem by FE.

Consider a thin strip of the rod of length Area A 1) EQUILIBRIUM 2) CONSTITUTIVE 3) STRAIN/ DISPLACEMENT “small strain”

DIFFERENTIAL EQUATION These three equation can be simplified as follows: GOVERNING EQUATION Now we will attempt to solve the equation by FE

with one spatial dimension. The 1-d Rod Element This is the simplest 1-d finite element suitable for solving differential equations with one spatial dimension. Original problem 1 2 3 4 5 Element number Node number Finite element discretization x L u1 u2 A typical 1-d element Let the nodal displacements be u1 and u2. Assume a linear variation of from one end to the other.

Consider the following TRIAL SOLUTION across the element: 1 where c1 and c2 are the “undetermined parameters”, and y1 (x) = 1 and y2 (x) = x are the “trial functions” The Trial Solution must satisfy the boundary conditions corresponding to the displacement at each node of the element, thus: 2 3 Eliminate c1 and c2 from equations and to give: 2 1 3 or where “Shape Functions” and “Undetermined parameters”

FINITE ELEMENT SOLUTION Governing differential Equation Trial solution Matrix version Substitute the Trial Solution into the governing equation Residual

Galerkin weighting functions Weight the Residual using Galerkin’s method Assume EA and F are constant over each element. They can vary from one element to the next of course! Galerkin weighting functions After integration by parts and the elimination of “boundary terms”, this can be written as, or in matrix form as

This is the “ELEMENT STIFFNESS RELATIONSHIP” where Element stiffness matrix Element nodal displacements Element nodal forces The Galerkin Finite Element Method has turned the DIFFERENTIAL EQUATION into the MATRIX EQUATION