1. 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.

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

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

4. 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.

5. 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”

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

7. 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

8. 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