Introduction to the Finite Element Method
This section presents the general steps included in the finite element method.
Typically, for the structural stress analysis, it is required to determine the stresses and
deformation (strains) throughout the structure which is in equilibrium and is subjected to
applied loads.
The finite element method involves modeling of the structure using small units
(finite elements). A displacement function is associated with each finite element. The
followings are the steps used in finite element method. This will be followed by
illustration of the application of these steps on springs and plane stress cases.
Step 1. Discretize and Select Element Types
Divide the structure into an equivalent system of finite elements with associated nodes.
The simplest line elements, Fig.1.a has two nodes, one at each end. The basic twodimensional elements, Fig. 1.b are loaded by forces in their own plane (plane stress).
They are triangular or quadrilateral elements. The common three dimensional elements,
Fig.1.c, are tetrahedral and hexahedral (brick) elements. They are used to perform three
dimensional stress analysis in 3-D solid bodies.
Step 2. Select a Displacement Function
Choose a displacement function within each element using the nodal values of the
element. Linear, quadratic, and polynomials are frequently used functions.
Step 3. Define the Stress/Strain Relationships
= du/dx
 = E
Step 4. Derive the Element Stiffness Matrix and Equations
The stiffness matrix and element equations relating nodal forces and displacements are
obtained using force equilibrium conditions or the principle of minimum potential energy.
Step 5. Assemble the Element Equations to obtain the Global Equations
Step 6. Solve for the Unknown Displacements
Step 7. Solve for the Element Strains and Stresses
Step 8. Interpret the Results
The final goal is to interpret and analyze the results for use in the design process.
Stiffness (Displacement) Method in Finite Element Analysis
The linear spring is introduced to provide a simple yet generally instructive tool to
illustrate the basic concepts of stiffness method.
The principle of minimum potential energy is introduced and applied derive the spring
element equations, and use it to solve a spring assemblage problem. This principle will be
illustrated using the simplest of elements (those with small numbers of degrees of
freedom) so that it will become a more easily understood concept when necessarily
applied to elements with large numbers of degrees of freedom in subsequent discussions.
Derivation of the Stiffness Matrix for a Spring Element
The symbol k is called the spring constant or stiffness of the spring.
Analogies to actual spring constants arise in numerous engineering problems.A prismatic
uniaxial bar has a spring constant k = AE/L, where A represents the cross-sectional area
of the bar, E is the modulus of elasticity, and L is the bar length. Similarly, a prismatic
circular-cross-section bar in torsion has a spring constant k=JG/L, where J is the polar
moment of inertia and G is the shear modulus of the material. For one-dimensional heat
conduction, k=AKxx/L, where Kxx is the thermal conductivity of the material and onedimensional fluid flow through a porous media, k=AKxx /L, where Kxx is the permeability
coefficient of the material.
Consider the linear spring element shown in Figure1. The reference points 1&2 are called
the nodes of the spring element. The local nodal forces are d1x and d2x for the spring
element. These nodal displacements are called the degrees of freedom at each node.
k
1
2
x
f1x , d1x
L
f2x , d2x
Figure 1 Linear spring element with positive nodal
displacement and force conventions
A relationship between nodal forces and nodal displacements for a spring element need
to be developed. This relationship is the stiffness matrix. The nodal force matrix is related
to the nodal displacement matrix as follows:
 f1x   k11 k12   d1x 
 
 
 f 2 x  k 21 k 22  d 2 x 
(1)
Where the elements kij of the matrix in Eq. (1) are to be determined.
Step 1 Select Element Type
Consider the linear spring (Fig.2) subjected to resulting nodal tensile forces T. The
original distance between nodes before deformation is denoted by L.
Step 2 Select a Displacement Function
A displacement function u is assumed. Here a linear displacement variation along
the x axis of the spring is assumed. Therefore
k
1
2
T
x
d1x
d2x
L
x
1
2
Figure 2 Linear spring subjected to tensile forces
u  a1  a2 x (2)
In general, the total number of coefficients a is equal to the total number of degrees
of freedom associated with the element. Here the total number of degrees of freedom is
two – an axial displacement at each of the two nodes of the element. In matrix form, Eq.
(2) becomes
a 
u  1 x 1  (3)
a 2 
Evaluating u at each node and solving for a1 and a 2 from Eq.(2) as follows:
u (0)  d1x  a1 (4)
u ( L)  d 2 x  a2 L  d1x (5)
Or, solving Eq. (5) for a 2 ,
a2 
d 2 x  d1 x
(6)
L
On substituting Eqs.(4)and (6) into Eq. (2), we have
u(
d 2 x  d1x
) x  d1x (7)
L
In matrix form, we express Eq. (7) as
  [1 
x  d1 x 
]  (8)
L d 2 x 
x
L
or
u  [ N1
Here
N1  1 
d 
N 2 ] 1x  (9)
d 2 x 
x
L
and
N2 
x
(1)
L
,
are called the shape functions because the N i s express the shape of assumed
displacement function over the domain of element when the ith element degree of freedom
has unit value and all other degrees of freedom are zero.
In addition, they are often called interpolation functions because we are interpolating to
find the value of a function between given nodal values.
Step 3 Define the Strain/Displacement and Stress/Strain Relationships
The tensile forces T produces a total elongation (deformation)  of the spring. For
the linear spring, T and  are related through Hooke’s law by
T  k (11)
where  is the deformation of the spring,
  d 2 x  d1x (12)
Step 4 Derive the Element Stiffness Matrix and Equations
Nodal forces are
f1x  T
f 2 x  T (13)
Using Eqs. (11), (12), and (13), we have
T   f 1x  k (d 2 x  d1x )
(14)
T   f 1x  k (d 2 x  d1x )
f1x  k (d 2 x  d1x )
(15)
f 2 x  k (d 2 x  d1x )
Now expressing Eqs.(15) in a single matrix equation yields
 f 1x   k
 
 f 2 x   k
 k   d1x 
  (16)
k  d 2 x 
k
 Stiffness Matrix is k  
 k
 k
(17)
k 
Step 5 Assemble the Element Equations to Obtain the Global Equations and
Introduce Boundary Conditions
N
K  [K ]   k
e 1
(e)
N
and
F  [F ]   f
(e)
(18)
e 1
Where k and f are now element stiffness and force matrices expresses in global
reference frame.
Step 6 Solve for the Nodal Displacements
The displacements are then determined by imposing boundary conditions and
solving a system of equations, F  K d simultaneously.
Step 7 Solve for the Element Forces
Finally, the element forces are determined by back-substitution, applied to each
element.