1213sem2-me2114
DESCRIPTION
MechanicsTRANSCRIPT
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ME2114
NATIONAL UNIVERSITY OF SINGAPORE
ME2114 MECHANICS OF MATERIALS II
(Semester II : AY2012/2013)
Time Allowed : 2 Hours
Matriculation Number
___________________________________________________________________________ INSTRUCTIONS TO CANDIDATES: 1. Write your Matriculation Number in the box above.
2. This examination paper contains FOUR (4) questions and comprises TWENTY-
THREE (23) printed pages. 3. Answer 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
dxxuEA
21dx
EAN
21U
L
0
2L
0
2
2. Strain energy in slender rods due to bending
dxx
vEI21dx
EIM
21U
L
0
2
2
2L
0
2
3. Strain energy in slender rods due to torsion
dxGJT
21U
L
0
2 4. Stiffness matrix of a truss member
LEAksins,cosc
vuvu
scsscscsccscscsscscsccsc
k
vuvu
22
22
22
22
and where
localk
5. Stiffness matrix of a plane stress triangle element
BEBk ~~A4t T
local where
yxyxyxx0x0x00y0y0y
~B
and
)1(2E00
01
E1
E
01
E1
E
22
22
E
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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 Castiglianos 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 Youngs modulus, cross sectional area and second moment of area for the truss elements and beam are E, A and I, respectively.
(25 marks)
Figure 1
2L
L
L 2L
A E
P
D B
C
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QUESTION 2 The stiffness matrix of the plane stress triangle element in Figure 2a with a Youngs modulus of E and thickness t is
u v u v u v
[klocal] = 2LEt
0.375 0 0.375 -0.1875 0 0.1875 u
0 1.125 -0.1875 -1.125 0.1875 0 v
0.375 -0.1875 0.6563 0.375 -0.2813 -0.1875 u
-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)
Figure 2a
Figure 2b
Figure 2c
L
2L
a
b c
d
P 45o
L
2L
L
2L
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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 Tp until a plastic zone has occurred partially in the shaft. Derive an expression for the torque Tp and hence the torque Tu required to cause full plasticity.
(10 marks) Assuming the yield stress in shear =150 MPa determine:
(a) the fully plastic torque Tu ,
(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 cross-section 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 Youngs 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 Eulers buckling load Pcr is given by (usual notations apply):
(for a pinned-pinned column)
(for a pinned-fixed column)
22
LEI
2.5 m
W
0.8 m
Figure 3
0.8 m 0.8 m
A
Column Q
Column P
3.5 m
crP
2205.2
LEI crP
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