174
Finite Element Analysis and Design
6. Consider the following differential equation:
d 2u
u x 0,
dx 2
u(0) 0
du
dx
0x 1
1
x 1
The solution is approximated as
u(x ) c1x c2x 2
Calculate the unknown coefficients using Galerkin method. Compare u(x) and
du
64
Finite Element Analysis and Design
28. For a plane stress problem, the strain components in the xy plane at a point P are
computed as
xx yy .125 102 , xy .25 102
(a) Compute the state of stress at this point if Youngs modulus E = 21011 Pa and
Poissons
208
Finite Element Analysis and Design
21. Consider the tapered bar in Problem 16. Use the Rayleigh-Ritz method to solve the
same
problem.
Assume
the
displacement
in
the
form
of
u(x ) (x 1)(c1x c2x 2 ) .
Solution:
The assumed displacements satisfy the ess
CHAP 2.
UNIAXIAL BAR AND TRUSS ELEMENTS DIRECT
METHOD
1. Three rigid bodies, 2, 3 and 4, are connected by four springs as shown in the figure.
A horizontal force of 1,000 N is applied on Body 4 as shown in the figure. Find the
displacements of the three b
202
Finite Element Analysis and Design
18. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to
determine the displacement field u(x), axial force distribution P(x), and reactions RL
and RR.
Assume: E = 100GPa, area of cro
108
Finite Element Analysis and Design
15. For a twodimensional truss shown in the figure, determine displacements of the
nodes and normal stresses developed in the members. Use E = 30106 N/cm2 and the
diameter of the circular cross section is 0.25 cm.
N3
EML 4500 Finite Element Analysis and Design, Fall 2009
8th Period (3:00-3:50 pm) MWF, TUR L005
http:/cpdlt.mae.ufl.edu/roy/eml4500.htm
1. Catalog Description: Credits: 3; Stress-strain analysis and design of machine elements; finite element
analysis.
2. P
CHAP 4 Finite Element Analysis for Beams and Frames
235
16. Consider a cantilevered beam with spring support at the end, as shown in the figure.
Assume E = 100 ksi, I = 1.0 in4, L = 10 in, k = 200 lb/in, beam height h = 10 in, and
no gravity. The beam is
296
Finite Element Analysis and Design
11. Find the heat transfer per unit area through the composite wall in the figure. Assume
one-dimensional heat flow and there is no heat flow between B and C. The thermal
conductivities are kA = 0.04 W/m/oC, kB = 0.1
CHAP 5 Finite Elements for Heat Transfer Problems
291
9. A well-mixed fluid is heated by a long iron plate of conductivity k = 12 W/m/oC and
thickness t = 0.12m. Heat is generated uniformly in the plate at the rate Qg = 5,000
W/m3. If the surface convecti
CHAP 5 Finite Elements for Heat Transfer Problems
289
8. Consider heat conduction in a uniaxial rod surrounded by a fluid. The right end of
the rod is attached to a wall and is at temperature TR. One half of the bar is insulated
as indicated. The free str
CHAP 5 Finite Elements for Heat Transfer Problems
283
4. Determine the temperature distribution (nodal temperatures) of the structure shown in
the figure using two equallength, linear finite elements with the cross-sectional area
of 1 m2. The thermal cond
256
Finite Element Analysis and Design
28. The frame shown in the figure is subjected to some forces at Nodes 2 and 3. The
resulting displacements are given in the table below. Sketch the axial force, shear
force, and bending moment diagrams for Element 3
CHAP 4 Finite Element Analysis for Beams and Frames
247
24. The frame shown in the figure is clamped at the left end and supported on a hinged
roller at the right end. The radius of circular cross-section r = 0.05 m. An axial force
P and a couple C act at
CHAP 4 Finite Element Analysis for Beams and Frames
213
2. The deflection of the simply supported beam shown in the figure is assumed as
v(x ) cx (x 1) , where c is a constant. A force is applied at the center of the
beam. Use the following properties: EI
CHAP 4 Finite Element Analysis for Beams and Frames
239
18. A linearly varying distributed load is applied to the beam finite element of length L.
The maximum value of the load at the right side is q0. Calculate work equivalent
nodal forces and couples.
q
220
Finite Element Analysis and Design
7. A simply supported beam of length L is under a uniformly distributed load p. When
two equal-length beam elements are used, the finite element analysis yields the
following nodal DOFs:
pL3
5pL4
pL3
.
cfw_Qs T cfw_
224
Finite Element Analysis and Design
10. Let a uniform cantilevered beam of length L be supported at the loaded end so that
this end cannot rotate, as shown in the figure. For the given moment of inertia I,
Youngs modulus E, and applied tip load P, calc
298
Finite Element Analysis and Design
12. Consider a wall built up of concrete and thermal insulation. The outdoor temperature
is To = 17C and the temperature inside is Ti = 20C. The wall is subdivided into
three elements. The thermal conductivity for co
CHAP 6 Finite Elements for Plane Solids
305
2. Solve Example 6.2 using one of finite element programs in Appendix.
Solution:
We will solve the problem using MATLAB Toolbox in Appendix D. The program list is
shown below:
Edof=[ 1 1 2 3 4 5 6; 2 1 2 5 6 7 8
310
Finite Element Analysis and Design
5. A structure shown in the figure is approximated with one triangular element. Plane
strain assumption is used.
(a) Calculate the straindisplacement matrix [B].
(b) When nodal displacements are given by cfw_u1, v1,
Truss-like Structures
Bar Elements
A truss-like structure is made up of many bar members of constant cross-sectional area connected
together by pin-joints that are frictionless. For finite element analysis each bar-like member in
the structure is assumed
EML 5526 Finite Element Analysis and Applications
Assignment
Problem 1)
Create a connectivity table that includes element angles, compute element stiffness matrices and assemble
the global matrix to solve for the nodal displacements and stresses in each e
EML 5526 Finite Element Analysis and Applications
Assignment
Problem 1:
For the beam shown in the figure, determine the following using the finite element
method. Show all the steps in the method (including element equations, assembly and
application of b
EML 5526 Finite Element Analysis and Applications
Homework #1
Problem 1)
Rewrite the following equations using index notation.
(a)
(b)
The equilibrium equations for static 2D elastic structures:
xx xy
0
x
y
yx yy
0
x
y
The convection diffusion equation:
CHAP 4 FINITE ELEMENT ANALYSIS OF
BEAMS AND FRAMES
FINITE ELEMENT ANALYSIS AND DESIGN
Nam-Ho Kim and Bhavani Sankar
Instructor: Subrata Roy
1
INTRODUCTION
We learned Direct Stiffness Method in Chapter 2
Limited to simple elements such as 1D bars
In Cha
CHAP 6 Finite Elements for Plane Solids
331
18. A quadrilateral element in the figure is mapped into the parent element.
(a) A point P has a coordinate (x, y) = (, y) in the physical element and (s, t) = (,
t) in the parent element. Find y and t coordinat