ME2113-2010SEM1 Past year paper
-
Upload
unknown-uploader -
Category
Documents
-
view
116 -
download
0
description
Transcript of ME2113-2010SEM1 Past year paper
ME2113
NATIONAL UNIVERSITY OF SINGAPORE
ME2113 – MECHANICS OF MATERIALS I
(Semester I : AY2010/2011)
Time Allowed : 2 Hours
INSTRUCTIONS TO CANDIDATES: 1. This examination paper contains FOUR (4) questions and comprises SEVEN (7)
printed pages. 2. Answer ALL FOUR (4) questions. 3. All questions carry equal marks. 4. This is a CLOSED-BOOK EXAMINATION. 5. Answer Questions 1 and 2 in one booklet and Questions 3 and 4 in another booklet. 6. Programmable calculators are NOT allowed for this examination.
Page 2 ME2113
QUESTION 1 (a) A beam of flexural rigidity EI is loaded by a uniformly distributed load of intensity w
as shown in Figure 1a. Using a graphical method, show that the tangential deviation (tAB) of point A from the tangent at point B is given by (usual notations apply):
dxEIMxt B
AAB ∫=
Hints: Consider a segment CD of length dx on the beam deflection curve and construct the tangents at points C and D.
(10 marks)
(b) A beam AB of length 2L and flexural rigidity EI is fixed at A and is loaded by a uniformly distributed load of intensity w over the right-hand half of the beam as shown in Figure 1b. Draw the M/EI diagram along the length of the beam and determine the slope and deflection at point B using the moment-area method.
(15 marks)
Figure 1a
C D B A X
dx
Load intensity w per unit length
x
Y
Page 3 ME2113
The following information (usual notations apply) may be used: The area A0 under the parabolic curve PQ in Figure 1c is given by:
EIwLA6
3
0 −=
L/4
X
M/(EI)
L
-wL2/2EI
P
Q
Centroid
Figure 1c
Figure 1b
Load intensity w per unit length
A B L
Y
L
X
Page 4 ME2113
QUESTION 2 A beam AB simply supported at A and B is loaded by a trapezoidally-distributed load and a concentrated moment M0 = 10 kNm at the mid-span as shown in Figure 2a. The intensity of the distributed load varies linearly from 30 kN/m at support A to 60 kN/m at support B.
(a) Draw the shear force distribution along the length of the beam, showing the values of
shear force at x = 0, 0.8 m, and 1.6 m only (you need not determine the intermediate shear force values).
(7 marks)
(b) If the beam has a hollow circular cross-section with inner radius R1 = 90 mm and outer radius R2 = 100 mm as shown in Figure 2b, determine the maximum shear stress in the beam.
(18 marks) The following information (usual notations apply) may be used:
(i) The shear formula is given by: yAI
Fb
xyxy .1
=τ
(ii) For a semi-circular area of radius R, the centroidal distance y (as shown in
Figure 2c) is given by:
π34Ry =
Figure 2a
π34Ry =
Centroid of semi-circular area
R
Figure 2b
R2 = 100 mm R1 = 90 mm
Figure 2c
60 kN/m
30 kN/m
A B
0.8 m M0
0.8 m
Page 5 ME2113
QUESTION 3 (a) Show that the solid circular rod of radius r under combined bending moment M, and
torsion T, in Figure 3a will not yield according to Tresca criterion if,
where σy is the yield stress of the rod material.
(12 marks)
(b) Figure 3b shows a right-angled L-shaped rod ABC clamped perpendicularly to a wall at A. A force P=300N is applied at C at an angle θ as shown. The force P lies on a plane parallel to the wall. Using the expression in (3a) or otherwise, determine the range of θ that will not cause the rod to yield according to Tresca criterion. The rod has a circular cross-section of radius 20mm and a yield stress of 100MPa.
(13 marks)
Figure 3a
Figure 3b
P θ
1m
L
2m
A
B
C
T T
M M
Page 6 ME2113
QUESTION 4 An axial force P is applied to a simply-supported beam-column of length L at a distance of e directly above the centroid of its cross-section as shown in Figure 4. The maximum deflection occurs at B. Given that the beam-column has a square cross-section of dimension
, and that where , determine (a) the reaction forces at A and C,
(5 marks)
(b) the value of α and the deflection at B in terms of e and L, and (10 marks)
(c) the maximum compressive stress at where the maximum deflection occurs in terms of P, e and a, taking into consideration the beam deflection. (Hint : Sketch the free body diagram of a section of the deformed beam-column from one end to the where the maximum deflection occurs)
(10 marks)
Figure 4
P e A
C
B
αL (1−α)L
Page 7 ME2113
LIST OF EQUATIONS
Unless otherwise stated, the following expressions may be used without derivation. All symbols have their usual meaning.
1. Tresca yield criterion
{ } Yσσσσσσσ =−−− 133221 ,,max 2. Mises yield criterion
2213
232
221 ])()()[(
21
Yσσσσσσσ =−+−+−
3. Rankine criterion
{ } ultσσσσ =321 ,,max
4. Deflection of beam-columns
−≥−
−−−
−
−≤−
=cLx
xLPL
cLQxLkkLPk
cLkQ
cLxxPLQckx
kLPkkcQ
v
,)()()(sinsin
)(sin
,sinsinsin
where
EIPk =2 .
- END OF PAPER -