98
Finite Element Analysis and Design
10. The stepped bar shown in the figure is subjected to a force at the center. Use FEM to
determine the displacement at the center and reactions RL and RR.
Assume: E = 100GPa, area of cross sections of the three porti

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,

CHAP 6.
FINITE ELEMENTS FOR PLANE SOLIDS
1. Repeat Example 6.2 with the following element connectivity:
Element 1: 124
Element 2: 234
Does the different element connectivity change the results?
Solution:
(1) Element 1: Nodes 124
Using nodal coordinates, w

216
Finite Element Analysis and Design
4. The right end of a cantilevered beam is resting on an elastic foundation that can be
represented by as spring with spring constant k = 1,000 N/m. A force of 1,000 N acts
at the center of the beam as shown. Use the

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

214
Finite Element Analysis and Design
3. Use the Rayleigh-Ritz method to determine the deflection v(x), bending moment
M(x), and shear force Vy(x) for the beam shown in the figure. The bending moment
and shear force are calculated from the deflection as:

CHAP 3 Weighted Residual and Energy Methods
191
14. A vertical rod of elastic material is fixed at both ends with constant cross-sectional
area A, Youngs modulus E, and height of L under the distributed load f per unit
length. The vertical displacement u(

CHAP 3 Weighted Residual and Energy Methods
189
13. Consider a finite element with three nodes, as shown in the figure. When the solution
is approximated using u(x ) N 1(x )u1 N 2 (x )u2 N 3 (x )u3 , calculate the
interpolation functions N1(x ), N2 (x ),

CHAP 3 Weighted Residual and Energy Methods
197
16. Consider a tapered bar of circular cross section. The length of the bar is 1 m, and the
radius varies as r (x ) 0.050 0.040x , where r and x are in meters. Assume
Youngs modulus = 100 MPa. Both ends of t

CHAP 3 Weighted Residual and Energy Methods
187
3
12. Repeat Problem 11 by assuming w(x )
ci i (x ) c1x 2 c2x 3 c3x 4
i 1
Solution:
Using the second derivatives of the two trial functions, 1 2, 2 6x, 3 12x 2 , we
can calculate the coefficient matrix and

170
Finite Element Analysis and Design
4. One-dimensional heat conduction problem can be expressed by the following
differential equation:
k
d 2T
Q 0,
dx 2
0x L
where k is thermal conductivity, T (x ) temperature, and Q heat source per unit
length. Q , t

168
Finite Element Analysis and Design
3. Using the Galerkin method, solve the following differential equation with the
approximate solution in the form of u(x ) c1x c2x 2 . Compare the approximate
solution with the exact one by plotting them on a graph.

CHAP 3 Weighted Residual and Energy Methods
185
11. The boundary value problem for a cantilevered beam can be written as
d 4w
dx 4
p(x ) 0,
w(0)
0x 1
dw
d 2w
d 3w
(0) 0,
(1) 1,
(1) 1: boundary condtions
dx
dx 2
dx 3
p(x ) x . Assuming the approximate de

124
Finite Element Analysis and Design
22. Determine the normal stress in each member of the truss structure. All joints are
balljoint and the material is steel whose Youngs modulus is E = 210 GPa.
Y
A
b
8 kN
a
a = b = 10 cm
6m
C
X
B
Z
5m
3m
D
6m
Solution

116
Finite Element Analysis and Design
18. Use FEM to solve the plane truss shown below. Assume AE = 106 N, L = 1 m.
Determine the nodal displacements, forces in each element and the support reactions.
1
y
x
1
L
2
L
2
L
4
3
10,000 N
3
Solution:
Connectivi

118
Finite Element Analysis and Design
19. The plane truss shown in the figure has 2 elements and 3 nodes. Calculate the 44
element stiffness matrices. Show the row addresses clearly. Derive the final
equations (after applying boundary conditions) for the

92
Finite Element Analysis and Design
6. In the structure shown, rigid blocks are connected by linear springs. Imagine that
only horizontal displacements are allowed. Write the structural equilibrium equations
[K]cfw_Q cfw_F after applying displacement bo

CHAP 2 Uniaxial Bar and Truss Elements
101
12. A stepped bar is clamped at one end, and subjected to concentrated forces as shown.
Note: the node numbers are not in usual order!
3
5 kN
1m
2
1
2 kN
1m
Assume: E=100 GPa, Small area of cross section =1 cm2,

CHAP 6 Finite Elements for Plane Solids
307
3. Using two CST elements, solve the simple shear problem described in the figure and
determine whether the CST elements can represent the simple shear condition
accurately or not. Material properties are given