_______________________________ _____________________________________________ _____________________________ ______________________________ _______________ INSTRUCTIONS TO CANDIDATES:
1.
Write your Matriculation Number in the box above.
2.
This examination paper contains FOUR (4) questions and comprises TWENTYTHREE (23) printed pages.
3.
Answer ALL ALL FOUR (4) questions.
4.
Write your answers in the space provided in this question booklet.
5.
This is a CLOSED-BOOK EXAMINATION. Question Number
Marks Obtained
Maximum Marks
1
25
2
25
3
25
4
25
Total
100
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LIST OF EQUATIONS
1.
Strain energy in slender rods due to axial loads
u U dx EA 2 0 EA 20 x L
1 N 2
2.
L
2
dx
Strain energy in slender rods due to bending L
U
3.
1
1 M2
2 0 EI
2 v dx EI 2 2 0 x 1
L
2
dx
Strain energy in slender rods due to torsion L
U
4.
1 T2
dx
v
u
2 0 GJ
Stiffness matrix of a truss member u
v
c2 cs c 2 cs u cs s2 cs s 2 v k local k 2 c cs c 2 cs u 2 2 cs s v cs s where c cos , s sin 5.
and k
EA L
Stiffness matrix of a plane stress triangle element
k local
t 4A
B~ EB~ T
where ~ B
y 0 y 0 y 0 0 x 0 x 0 x x y x y x y
and
E 1 2 E E 1 2 0
E 1 2 E 1 2 0
0 E 2(1 ) 0
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QUESTION 1 Figure 1 shows a solid beam ABC supported by two truss elements BD and BE. A force P is applied at C. Using Castigliano’s theorem, determine the reaction force from the simple support at E and the vertical displacement of C. Take into consideration strain energy due to axial forces only for truss elements BD and BE and strain energy due to bending moments only for beam ABC. The Young’s modulus, cross sectional area and second moment of area for the truss elements and beam are E , A and I , respectively. (25 marks)
C
P
2L
B
D
L A E
L
2L Figure 1
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QUESTION 2 The stiffness matrix of the plane stress triangle element in Figure 2a with a Young’s modulus of E and thickness t is
[klocal] =
u
v
u
v
u
v
0.375
0
0.375
-0.1875
0
0.1875
u
0
1.125
-0.1875
-1.125
0.1875
0
v
Et
0.375
-0.1875
0.6563
0.375
-0.2813
-0.1875
u
L2
-0.1875
-1.125
0.375
1.2188
-0.1875
-0.0938
v
0
0.1875
-0.2813
-0.1875
0.2813
0
u
0.1875
0
-0.1875
-0.0938
0
0.0938
v
Show that the stiffness matrix of the element in Figure 2b is the same as that for the element in Figure 2a. Hence, solve for the displacement of node C in Figure 2c which shows the finite element mesh of a plate clamped along the top and left edges carrying a load P. ~ (Hint : Compare the matrix [B] as defined in the given list of equations for both the elements in Figures 2a and 2b) (25 marks)
2L
L
L
2L
Figure 2a
Figure 2b
d
a
L c b
2L o
45 Figure 2c
P
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QUESTION 3 A solid shaft which has a radius of 30 mm is made of an elastic perfectly plastic material. It is loaded by a slowly increasing torque T p until a plastic zone has occurred partially in the shaft. Derive an expression for the torque T p and hence the torque T u required to cause full plasticity. (10 marks) Assuming the yield stress in shear τ γ =150 MPa determine: (a)
the fully plastic torque T u ,
(b)
the residual shearing stress at the outer surface after the torque is removed and plot the residual stress distribution indicating the residual stress values at the outer surface and center of the shaft cross-section. (15 marks)
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QUESTION 4 A rigid horizontal bar pivoted at the left-hand end, supported by two columns P and Q, is loaded at point A by a load of W (see Figure 3). The columns have a hollow circular crosssection with inner and outer diameters of 80 mm and 100 mm respectively. Column Q is pinned at both ends and column P is pinned at one end and fixed at the other. The columns are fabricated from a material with a Young’s modulus of 180 GPa. Using a safety factor of 1.5 determine which column will buckle first and hence the maximum allowable value of W. (25 marks) The Euler’s buckling load P cr is given by (usual notations apply): 2
