FINITE ELEMENT METHOD
DEVELOPMENT OF TRUSS EQUATION
1. Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates
Figure 1. Bar subjected to tensile forces T; positive nodal displacements and forces
are all in the local x direction
The bar element is assumed to have constant cross-sectional area A, modulus of elasticity E, and initial
length L. The nodal degrees of freedom are local axial displacements (longitudinal displacements
directed along the length of the bar) represented byd1x and d2x at the ends of the element as shown in
Figure 1.
Hooke’s law equation, 𝜎𝑥 = 𝐸𝜀𝑥
𝑑𝑢
The strain/displacement relationship, 𝜀𝑥 = 𝑑𝑥
Force equilibrium, 𝐴𝜎𝑥 = 𝑇 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑑 𝑑𝑢
The differential equation governing the linear-elastic bar behavior as, (𝐴𝐸 )=0
𝑑𝑥 𝑑𝑥
The following assumptions are used in deriving the bar element stiffness matrix:
a. The bar cannot sustain shear force or bending moment, that is, f1y = 0, f2y = 0, m1=0, and m2=0
b. Any effect of transverse displacement is ignored.
c. Hooke’s law applies; that is, axial stress x is related to axial strain x by x = Ex
d. No intermediate applied loads
The steps to derive the stiffness matrix for the bar element,
a. Step 1 - Select the element type
Represent the bar by labeling nodes at each end and in general by labeling the element
number (Figure 3–2).
b. Step 2 - Select a Displacement Function
A linear displacement function u is assumed,
The total number of coefficients a1 always equal to the total number of degrees of freedom
associated with the element.
Applying the boundary conditions and solving for the unknown coefficients gives,
and
or in another form,
MEITY WULANDARI - 3116202014 1
FINITE ELEMENT METHOD
with shape functions given by
c. Step 3 - Define the Strain/Displacement and Stress/Strain Relationships
The strain/displacement relationship is,
The stress/strain relationship is,
d. Step 4 - Derive the Element Stiffness Matrix and Equations
The element stiffness matrix is derived as follows. From elementary mechanics, we have
Substituting the stress-displacement relationship, we obtain
Also, by the nodal force sign convention defined in element figure, is,
So,
Expressing above equation together in matrix form, we have
Now, because f = kd we have,
Where, AE/L for a bar element is analogous to the spring constant k for a spring element.
e. Step 5- Assemble Element Equations to Obtain Global or Total Equations
The global stiffness and force matrices and global equations are assembled using the nodal force
equilibrium equations, and force/deformation and compatibility equations. Where k and f are the
element stiffness and force matrices expressed in global coordinates.
f. Step 6 - Solve for the Nodal Displacements
Determine the displacements by imposing boundary conditions and simultaneously solving a
system of equations, F = kd
MEITY WULANDARI - 3116202014 2
FINITE ELEMENT METHOD
g. Step 7 - Solve for the Element Forces
Determine the strains and stresses in each element by back-substitution of the displacements as
follow,
DEVELOPMENT OF BEAM EQUATION
Figure 2. Beam element with positive nodal displacements, rotations, forces, and
Moments
1. Beam Stiffness
The beam is of length L with axial local coordinate x and transverse local coordinate The local transverse
nodal displacements are given by diy and the rotations by i The local nodal forces are given by fiy and the
bending moments by mi as shown. We initially neglect all axial effects.
At all nodes, the following sign conventions are used:
a. Moments are positive in the counterclockwise direction.
b. Rotations are positive in the counterclockwise direction.
c. Forces are positive in the positive y direction.
d. Displacements are positive in the positive y direction.
The steps to develop the stiffness matrix and equations for a beam element
a. Step 1 - to develop the stiffness
Represent the beam by labeling nodes at each end and in general by labeling the element number
(Figure 2).
b. Select a Displacement Function
The transverse displacement variation through the element length is assumed,
Express v as a function of the nodal degrees of the nodal degrees of freedom d1y, d2y, 1 and 2 as
follows,
MEITY WULANDARI - 3116202014 3
FINITE ELEMENT METHOD
Where = dv/dx for the assumed small rotation .
In matrix form is
N1, N2, N3, and N4 are called the shape functions for a beam element. These cubic shape (or
interpolation) functions are known as Hermite cubic interpolation (or cubic spline) functions. For
the beam element, N1=1 when evaluated at node 1 and N1 = 0 when evaluated at node 2. Because
N2 is associated with 1 , (dN2/dx) = 1 when evaluated at node 1. Shape functions N3 and N4 have
analogous results for node 2.
c. Step 3 - Define the Strain/Displacement and Stress/Strain Relationships
The following axial strain/displacement relationship to be valid is assumed,
The axial displacement to the transverse displacemet by,
From elementary beam theory, we obtain,
The bending moment and shear force are related to the transverse displacement function. Because
we will use these relationships in the derivation of the beam element stiffness matrix, we now
present them as,
d. Step 4 - Derive the Element Stiffness Matrix and Equations
First, derive the element stiffness matrix and equations using a direct equilibrium approach to obtain,
MEITY WULANDARI - 3116202014 4
FINITE ELEMENT METHOD
where the minus signs in the second and third are the result of opposite nodal and beam theory positive
bending moment conventions at node 1 and opposite nodal and beam theory positive shear force
conventions at node 2 as seen by comparing, relate the nodal forces to the nodal displacements. In
matrix form become,
where the stiffness matrix is then
Where, k relates transverse forces and bending moments to transverse displacements and rotations,
whereas axial effects have been neglected.
MEITY WULANDARI - 3116202014 5
