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CHAPTER 2
The Basic FEA Procedure
2.1 Introduction
This chapter discusses the spring element, especially for the purpose of introducing
various concepts involved in use of the FEA technique. A spring element is not very
useful in the analysis of real engineering structures; however, it represents a structure in
an ideal form for an FEA analysis. Spring element doesn’t require discretization (division
into smaller elements) and follows the basic equation F = ku. We will use it solely for the
purpose of developing an understanding of FEA concepts and procedure.
2.2 Overview
Finite Element Analysis (FEA), also known as finite element method (FEM) is based on
the concept that a structure can be simulated by the mechanical behavior of a spring in
which the applied force is proportional to the displacement of the spring and the
relationship F = ku is satisfied. In FEA, structures are modeled by a CAD program and
represented by nodes and elements. The mechanical behavior of each of these elements
is similar to a mechanical spring, obeying the equation, F = ku. Generally, a structure is
divided into several hundred elements, generating a very large number of equations that
can only be solved with the help of a computer.
The term ‘finite element’ stems from the procedure in which a structure is divided into
small but finite size elements (as opposed to an infinite size, generally used in
mathematical integration). The endpoints or corner points of the element are called nodes.
Each element possesses its own geometric and elastic properties. Spring, Truss, and
Beams elements, called line elements, are usually divided into small sections with nodes
at each end. The cross-section shape doesn’t affect the behavior of a line element; only
the cross-sectional constants are relevant and used in calculations. Thus, a square or a
circular cross-section of a truss member will yield exactly the same results as long as the
cross-sectional area is the same. Plane and solid elements require more than two nodes
and can have over 8 nodes for a 3 dimensional element.
Chapter 2 – The Basic FEA Procedure
A line element has an exact theoretical solution, e.g., truss and beam elements are
governed by their respective theories of deflection and the equations of deflection can be
found in an engineering text or handbook. However, engineering structures that have
stress concentration points e.g., structures with holes and other discontinuities do not
have a theoretical solution, and the exact stress distribution can only be found by an
experimental method. However, the finite element method can provide an acceptable
solution more efficiently. Problems of this type call for use of elements other than the line
elements mentioned earlier, and the real power of the finite element is manifested. In
order to develop an understanding of the FEA procedure, we will first deal with the
spring element. In this chapter, spring structures will be used as building blocks for
developing an understanding of the finite element analysis procedure. Both spring and
truss elements give an easier modeling overview of the finite element analysis procedure,
due to the fact that each spring and truss element, regardless of length, is an ideally sized
element and do not need any further division. Therefore, in the following sections spring
structures will be used to illustrate the finite element analysis procedure.
2.3 Understanding
Computer
and
FEA software interaction -
Using the Spring Element as an example
In the following example, a three-element structure is analyzed by finite element method.
The analysis procedure presented here will be exactly the same as that used for a complex
structural problem, except, in the following example, all calculations will be carried out
by hand so that each step of the analysis can be clearly understood. All derivations and
equations are written in a form, which can be handled by a computer, since all finite
element analyses are done on a computer. The finite element equations are derived using
Direct Equilibrium method. The example illustrates the interaction between computer and
the FEA software used for solution.
Example 2.1
Two springs are connected in series with spring constant k
1
, and k
2
(lb./in) and a force F
(lb.) is applied. Find the deflection at nodes 2, and 3.
k
1
k
2
F
1
2
3
Figure 2.1
ME 273 Lecture Notes © by R. B. Agarwal
Finite Element Analysis
2
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Chapter 2 – The Basic FEA Procedure
Solution:
For finite element analysis of this structure, the following steps are necessary:
Step 1: Derive the element equation for each spring element.
Step 2: Assemble the element equations into a common equation, knows as the global
or Master equation.
Step 3: Solve the global equation for deflection at nodes 1 through 3.
Step 1: Derive the element equation for each spring element.
First, a general equation is derived for an element e that can be used for any spring
element and expressed in terms of its own forces, spring constant and node deflections, as
illustrated in figure 2.2.
u
i
u
j
f
i
f
j
e
Figure 2.2
Element ‘e’ can be thought of as any element in the structure with nodes i and j, forces f
i
and f
j
, deflections u
i
and u
j
, and the spring constant k
e
. Node forces f
i
and f
j
are internal
forces and are generated by the deflections u
i
and u
j
at nodes i and j, respectively.
For a linear spring f = ku, and
f
i
= k
e
(u
j
– u
i
) = -k
e
(u
i
-u
j
) = - k
e
u
i
+ k
e
u
j
For equilibrium,
f
j
= -f
i
= k
e
(u
i
-u
j
) = k
e
u
i
- k
e
u
j
Or
-f
i
= k
e
u
i
- k
e
u
j
-
f
j
= - k
e
u
i
+ k
e
u
j
Writing these equations in a matrix form, we get
i
i
e
e
j
j
e
e
f
u
k
k
f
u
k
k
−
−
=
−
−
ME 273 Lecture Notes © by R. B. Agarwal
Finite Element Analysis
2
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Chapter 2 – The Basic FEA Procedure
The above matrix equation is a general form of an equation of a spring elements, and can
be used to derive element equations for any spring element in this example, and in
general, it is valid for any linear spring element. Thus, equations for each elements can be
written as follows:
Element 1:
k
1
f
1
f
2
1
2
Where, the superscript on the force matrix indicates the corresponding element.
Element 2:
k
2
f
2
f
3
2
3
−
−
=
−
−
2
1
1
1
1
1
)
1
(
2
1
u
u
k
k
k
k
f
f
−
−
=
−
−
3
2
2
2
2
2
)
2
(
3
2
u
u
k
k
k
k
f
f
Thus,
f
1
(1)
= -k
1
(u
1
– u
2
) f
2
(1)
= k
1
(u
1
-u
2
)
f
2
(2)
= -k
2
(u
2
– u
3
) f
3
(2)
= k
2
(u
2
-u
3
)
This completes the procedure for step 1.
Note that f
3
= F (lb.). This will be substituted in step 2. The above equations represent
individual elements only and not the entire structure.
ME 273 Lecture Notes © by R. B. Agarwal
Finite Element Analysis
2
-4
Chapter 2 – The Basic FEA Procedure
Step 2 : Assemble the element equations into a global equation
.
The basis for combining or assembling the element equation into a global equation is the
equilibrium condition at each node. When the equilibrium condition is satisfied by
summing all forces at each node, a set of linear equations is created which links each
element force, spring constant, and deflections. In general, let the external forces at each
node be F
1
, F
2
, and F
3
, as shown in figure 2.3. Using the equilibrium equation, we can
find the element equations, as follows.
Node 2
:
∑
F = 0 = f
2
(1)
+ f
2
(2)
+ F
2
F
2
Or
F
2
= -f
2
(1)
– f
2
(2)
=
= -k
1
(u
1
– u
2
) + k
2
(u
2
– u
3
)
f
2
(1)
f
2
(2)
= -k
1
u
1
– k
1
u
2
+ k
2
u
2
– k
2
u
3
Node 2
F
3
Node 3
:
∑
F = 0 , f
3
(2)
+ F
3
= 0
f
3
(2)
Or
F
3
= -f
3
(2)
= -k
2
(u
2
- u
3
)
Node 3
1
1
1
1
1
1
2
1 1
1
1:
0
(
)
Node
F
f
F
or F
f
k u
u
k u
k u
= = +
= − =
−
=
−
∑
2
Node 1
F
1
f
1
Figure 2.3
The superscript “e” in force f
n
(e)
indicates the contribution made by the element number
e, and the subscript “n” indicates the node “n” at which forces are summed.
Rewriting the equations, we get,
k
1
u
1
– k
1
u
2
= F1
- k
1
u
1
+ k
1
u
2
+ k
2
u
2
– k
2
u
3
= F2
(2.1)
- k
2
u
2
+ k
2
u
3
= F3
ME 273 Lecture Notes © by R. B. Agarwal
Finite Element Analysis
2
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