Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
13
Types of Finite Elements
1-D (Line) Element
(Spring, truss, beam, pipe, etc.)
2-D (Plane) Element
(Membrane, plate, shell, etc.)
3-D (Solid) Element
(3-D fields - temperature, displacement, stress, flow velocity)
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
14
III. Spring Element
“
Everything important is simple
.
”
One Spring Element
Two nodes:
i, j
Nodal displacements:
u
i
, u
j
(in, m, mm)
Nodal forces:
f
i
, f
j
(lb, Newton)
Spring constant (stiffness):
k (lb/in, N/m, N/mm)
Spring force-displacement relationship:
F
k
= ∆
with
∆ =
−
u
u
j
i
k
F
=
/
∆
(> 0) is the force needed to produce a unit stretch.
k
i
j
u
j
u
i
f
i
f
j
x
∆
F
Nonlinear
Linear
k
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
15
We only consider linear problems in this introductory
course.
Consider the equilibrium of forces for the spring. At node i,
we have
f
F
k u
u
ku
ku
i
j
i
i
j
= −
= −
−
=
−
(
)
and at node j,
f
F
k u
u
ku
ku
j
j
i
i
j
= =
−
= −
+
(
)
In matrix form,
k
k
k
k
u
u
f
f
i
j
i
j
−
−
=
or,
ku
f
=
where
k = (element) stiffness matrix
u = (element nodal) displacement vector
f = (element nodal) force vector
Note that k is symmetric. Is k singular or nonsingular? That is,
can we solve the equation? If not, why?
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
16
Spring System
For element 1,
k
k
k
k
u
u
f
f
1
1
1
1
1
2
1
1
2
1
−
−
=
element 2,
k
k
k
k
u
u
f
f
2
2
2
2
2
3
1
2
2
2
−
−
=
where
f
i
m
is the (internal) force acting on local node i of element
m (i = 1, 2).
Assemble the stiffness matrix for the whole system:
Consider the equilibrium of forces at node 1,
F
f
1
1
1
=
at node 2,
F
f
f
2
2
1
1
2
=
+
and node 3,
F
f
3
2
2
=
k
1
u
1,
F
1
x
k
2
u
2,
F
2
u
3,
F
3
1
2
3
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
17
That is,
F
k u
k u
F
k u
k
k u
k u
F
k u
k u
1
1 1
1
2
2
1 1
1
2
2
2
3
3
2
2
2
3
=
−
= −
+
+
−
= −
+
(
)
In matrix form,
k
k
k
k
k
k
k
k
u
u
u
F
F
F
1
1
1
1
2
2
2
2
1
2
3
1
2
3
0
0
−
−
+
−
−
=
or
KU
F
=
K is the stiffness matrix (structure matrix) for the spring system.
An alternative way of assembling the whole stiffness matrix:
“Enlarging” the stiffness matrices for elements 1 and 2, we
have
k
k
k
k
u
u
u
f
f
1
1
1
1
1
2
3
1
1
2
1
0
0
0
0
0
0
−
−
=
0
0
0
0
0
0
2
2
2
2
1
2
3
1
2
2
2
k
k
k
k
u
u
u
f
f
−
−
=
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
18
Adding the two matrix equations (superposition), we have
k
k
k
k
k
k
k
k
u
u
u
f
f
f
f
1
1
1
1
2
2
2
2
1
2
3
1
1
2
1
1
2
2
2
0
0
−
−
+
−
−
=
+
This is the same equation we derived by using the force
equilibrium concept.
Boundary and load conditions:
Assuming,
u
F
F
P
1
2
3
0
=
=
=
and
we have
k
k
k
k
k
k
k
k
u
u
F
P
P
1
1
1
1
2
2
2
2
2
3
1
0
0
0
−
−
+
−
−
=
which reduces to
k
k
k
k
k
u
u
P
P
1
2
2
2
2
2
3
+
−
−
=
and
F
k u
1
1
2
= −
Unknowns are
U
=
u
u
2
3
and the reaction force
F
1
(if desired).
Lecture Notes: Introduction to Finite Element Method
Chapter 1. Introduction
© 1998 Yijun Liu, University of Cincinnati
19
Solving the equations, we obtain the displacements
u
u
P k
P k
P k
2
3
1
1
2
2
2
=
+
/
/
/
and the reaction force
F
P
1
2
= −
Checking the Results
•
Deformed shape of the structure
•
Balance of the external forces
•
Order of magnitudes of the numbers
Notes About the Spring Elements
•
Suitable for stiffness analysis
•
Not suitable for stress analysis of the spring itself
•
Can have spring elements with stiffness in the lateral
direction, spring elements for torsion, etc.