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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.

AnswerALLFOUR (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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PAGE 3 ME2114

LIST OF EQUATIONS

1. Strain energy in slender rods due to axial loads

dxx

uEA2

1dxEA

N

2

1U

L

0

2L

0

2

2.

Strain energy in slender rods due to bending

dxx

vEI

2

1dx

EI

M

2

1U

L

0

2

2

2L

0

2

3. Strain energy in slender rods due to torsion

dxGJ

T

2

1U

L

0

2

4. Stiffness matrix of a truss member

L

EAksins,cosc

v

u

v

u

scsscs

csccsc

scsscs

csccsc

k

vuvu

22

22

22

22

andwhere

localk

5.

Stiffness matrix of a plane stress triangle element

BEBk ~~A4

t Tlocal

where

yxyxyx

x0x0x0

0y0y0y~B

and

)1(2

E00

01

E

1

E

01

E

1

E

22

22

E

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PAGE 4 ME2114

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 momentsonly for beam ABC. The Youngs modulus, cross sectional area and second moment of area

for the truss elements and beam areE,AandI, respectively.

(25 marks)

Figure 1

2L

L

L 2L

A

E

P

DB

C

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PAGE 9 ME2114

QUESTION 2

The stiffness matrix of the plane stress triangle element in Figure 2a with a Youngs modulus

ofEand thickness tis

u v u v u v

[klocal]= 2L

Et

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 elementin 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

P45

o

L

2L

L

2L

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PAGE 14 ME2114

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 Tpuntil 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 MPadetermine:

(a) the fully plastic torqueTu ,

(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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PAGE 19 ME2114

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 columnsare 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 loadPcris given by (usual notations apply):

(for a pinned-pinned column)

(for a pinned-fixed column)

2

2

L

EI

2.5 m

W

0.8 m

Figure 3

0.8 m 0.8 m

A

Column Q

Column P

3.5 m

crP

2

205.2

L

EIcr

P

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PAGE 23 ME2114

- END OF PAPER -

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Pg.

1

NATIONAL

UNIVERSITY

OF SINGAPORE

Department

of

Mechanical Engineering

Module Code: ME2114 Module Title: Mechanics

of

Materials II

Solution to Question No. _

c

Sh

em

- s.lreSS distribution

For elastic-plastic shaft, the torque in the shaft is given by:

T

=

S

r 2nr )dr

.

r +

LR y nr

2

dr

Elastic)

Plastic)

c

R

2

r

[r

r]

r

2

T

=

o nr dr + Jc y nr dr

2Jr I 3 fR 2

T

=

-ry

r dr

2Jrry

r dr

c 0 C

1 [ 4

21 [ 3 3

=

r y c

2c

-ry

R

3

c

=

R

3

c3 J

2Jrry 3 12

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-----------------------

Pg 2

a) For fully plastic shaft, C = 0 and the torque in the shaft is given by:

R3

_ =

nTyR3

=

2n

x 150 x 30

3

= 8 48x

10

6

Nmm

u = 2nT y

3

12

3 3

b) For elastic unloading we have:

TyR

3

[

2n

JR 4 _4x15 =2 MPa

R

3 =

-

3

~ =

3

y

TR 1[

R4

2

I.e. Residual stress at r

=

30

mm

is 150-200)

=

-50 MPa

Residual stress distribution:

150 MPa

-50 MPa

Set By :

_ J

Tay _

Date:

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Pg

.

t

NATIONAL UNIVERSITY OF SINGAPORE

Department of Mechanical Engineering

Module Code:

t \b-21

t / ~ . . 2 i l ' f M o d u l e

Title:.

M_ O -M...:.....JL..,(==--

_

Solution to Question No.

f

0-8

Co. /

-p

o ~ g

.> (

c

0

r ~

tPrtw

~ l i h l ~ W >

: i

f l/13

P6 +; ff= 3 W

, )

FI-f,.,en+ . t

Yee

- o / t . , ~ C . i

f:;

A)E)e

(2.-)

'.e. t

f-7

I

I f

r L ~

I f = d-

nrD

);h YV)

L6i =::

3 fDo -rn

t'Y)

~ e T - j

i

d1n,. , t.'en;

5 .,

J)

1-

.

o g

..... e.g o" R n

--frY

,-.

cv e-

-e.g :;::; o ~ 8 IY

3)

e

f

::: 2 . . ; o , ~ fY

SetBy:

____________________________

Date: ______________

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----------------------------

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NATIONAL UNIVERSITY OF SINGAPORE

Department of Mechanical Engineering

Module Code:

Module Title:

Solution to Question No. _

/J.5)hO

Ear.

t l

OJ/. p;:;:

2

~ ~ = 1f2-6::-

Xz,.oJ= O.:S2.t., E:r

l r)l. L

d-

-1 -= I j . ~ K ..J

SetBy:

________________________

__

Date: ___________________

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- - - - - - - - - - - - - - - - - - - - -

NATIONAL UNIVERSITY OF

SINGAPORE

Department of Mechanical Engineering

Module Code:

Module Title:

Solution to uestion

No.

___

_

E ~ s ( ) ) ~ C

b

\ l ~

f& fa::::.

o

4-r1- w =

'--

2

-fc.. \l

~ .

w::

6 /6

.

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C

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f

f:::

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-{C"v

(

8

)

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e W:::'

+ .

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OWl

n ~ S

f15

7)

7 ) ,

--H -e 10/JeY \lot

w e

~

IN

s ~ o , I J ~ ~ ~

H ~ V \

c..e.

-fcx fJR..olo-;'V),

r

",c,1e

s DiV1

J. Cb tAWlVl

6C

WI"

l

b &1 cr: k

fc yB C .

SetBy: ________________________

_ _

Date: