MECH593 Finite Element Methods

1
MECH303 Advanced Stresses Analysis
Lecture 5 FEM of 1-D Problems Applications
2
Torsional Shaft
Review
Assumption Circular cross section
Shear stress
Shear strain
Deformation
3
Finite Element Equation for Torsional Shaft
4
Bending Beam
y
Review
M
M
x
Pure bending problems
Normal strain
Normal stress
Normal stress with bending moment
Moment-curvature relationship
Flexure formula
5
Bending Beam
y
Review
q(x)
x
Relationship between shear force, bending moment
and transverse load
Deflection
Sign convention
-

M
M
M
-

V
V
V
6
Governing Equation and Boundary Condition

• Governing Equation

0ltxltL

• Boundary Conditions -----

Essential BCs if v or is specified at the
boundary. Natural BCs if or
is specified at the boundary.
7
Weak Formulation for Beam Element

• Governing Equation

• Weighted-Integral Formulation for one element

• Weak Form from Integration-by-Parts ----- (1st
  time)

8
Weak Formulation

• Weak Form from Integration-by-Parts ----- (2nd
  time)

9
Weak Formulation

• Weak Form

q(x)
Q1
y(v)
Q3
Q2
Q4
x
x x2
x x1
L x2-x1
10
Ritz Method for Approximation
q(x)
Q1
y(v)
Q3
Q2
Q4
x
x x2
x x1
L x2-x1
where
Let w(x) fi (x), i 1, 2, 3, 4
11
Ritz Method for Approximation
12
Ritz Method for Approximation
13
Selection of Shape Function
The best situation is -----
Interpolation Properties
14
Derivation of Shape Function for Beam Element
Local Coordinates
How to select fi???
and
where
Let
Find coefficients to satisfy the interpolation
properties.
15
Derivation of Shape Function for Beam Element
How to select fi???
e.g.
Let
Similarly
16
Derivation of Shape Function for Beam Element
In the global coordinates
17
Element Equations of 4th Order 1-D Model
q(x)
u1
y(v)
u3
u2
u4
x
x x1
L x2-x1
x x2
f4
f1
1
1
f3
f2
xx2
xx1
18
Element Equations of 4th Order 1-D Model
q(x)
u1
y(v)
u3
u2
u4
x
x x2
x x1
L x2-x1
19
Finite Element Analysis of 1-D Problems -
Applications
Example 1.
Governing equation
Weak form for one element
where
20
Finite Element Analysis of 1-D Problems
Example 1.
Approximation function
f4
f1
f3
xx1
f2
xx2
21
Finite Element Analysis of 1-D Problems
Example 1.
Finite element model
Discretization
P2 , v2
P3 , v3
P4 , v4
P1 , v1
II
III
I
M1 , q1
M3 , q3
M4 , q4
M2 , q2
22
Matrix Assembly of Multiple Beam Elements
Element I
23
Matrix Assembly of Multiple Beam Elements
24
Solution Procedures
Apply known boundary conditions
25
Solution Procedures
26
Shear Resultant Bending Moment Diagram
27
Plane Flame
Frame combination of bar and beam
Q1 , v1
E, A, I, L
Q3 , v2
P1 , u1
P2 , u2
Q4 , q2
Q2 , q1
28
Finite Element Model of an Arbitrarily Oriented
Frame
y
q
x
y
q
x
29
Finite Element Model of an Arbitrarily Oriented
Frame
local
global
30
Plane Frame Analysis - Example
Hinge Joint
Rigid Joint
F
F
F
F
Beam II
Bar
Beam
Beam I
31
Plane Frame Analysis
Q3 , v2
Q4 , q2
P2 , u2
P1 , u1
Q2 , q1
Q1 , v1
32
Plane Frame Analysis
Q3 , v3
Q1 , v2
P1 , u2
P2 , u3
Q4 , q3
Q2 , q2
33
Plane Frame Analysis
34
Plane Frame Analysis