DO NOT DETACH ANY PAGE(S) FROM THE EXAMINATION...

Surname: Given Name: Student Number:

UNIVERSITY OF TORONTO

FACULTY OF APPLIED SCIENCE AND ENGINEERING

WINTER TERM FINAL EXAMINATION

MIE222H1 S - Mechanics of Solids I

April 18, 2017 Time Allowed: 2 '/2 hours

Examiner - Professor S. A. Meguid

DO NOT DETACH ANY PAGE(S)

FROM THE EXAMINATION BOOKLET

INSTRUCTIONS

This paper contains FIVE (5) questions and comprises TWENTY-TWO (22) pages.

Attempt ALL questions

Please answer each question in the space provided.

Candidates must support their arguments with the appropriate expressions/figures

Only non-programmable and non-communicating calculators are allowed.

Pages 20 to 22 contain information you may find useful.

This is a CLOSED BOOK examination.

of 22


QUESTION #1: (20 Marks)

Starting from point A, the built-in support, calculate the slope and the deflection of the beam AC at point B using Macaulay's singularity functions method. You may assume that the flexural rigidity of the beam E1 0.65 IVfNm2.

W0= 5 kNlm

ly

s';: I

4 4 5m 5m

A B C

Fig. Qi: Beam with applied uniformly distributed load and built-in support

Slope atB4OB

Deflection at B, YB = mJ

of 22


QUESTION # 1 SOLUTION:

Please remember to insert final answer(s) in the provided box

of 22


QUESTION # 1 SOLUTION (continued):


of 22



The thick cylinder shown in Fig. Q2(a) is made of high strength steel with an internal diameter of 200 mn-i and an external diameter of 300 mm. It is subjected to an internal pressure Pi = 60 MPa. The cylinder is assumed to have closed ends, as depicted in Fig. Q2(b).

Determine the stress state acting on the element abcd located on the outer surface of the cylinder as a result of P1; see Fig. Q2(b) and the enlarged Fig. Q2(c). (8 Marks)

If the surface stress element abcd is rotated anticlockwise by 30°, as shown in Fig. Q2(c), find the new stress state acting on the newly rotated surface element a 'b 'c 'd'.

(6 Marks)

Verify your answer by accurately thawing Mohr' s circle to scale on the provided graph paper on Page 9. Show the positions of the surface elements abcd and a 'b 'c 'd' on your circle. (6 Marks)

Fig. Q2(a) Cross-section cut of thick-walled closed cylinder

101

r;i

Fig. Q2(b) Closed end thick walled cylinder Fig. Q2(c) Enlarged view of outer with outer surface stress elements surface stress elements

of 22


Element abed Element a'b'c'd'

Hoop Stress, u00 = MPa Hoop Stress, o'' = MPa

Longitudinal Stress, r = MPa Longitudinal Stress, o'' = MPa

Shear Stress, To = MPa Shear Stress, r'' = MPa

QUESTION #2 SOLUTION:

of 22


QUESTION #2 SOLUTION (continued):


of 22



Fig. Q3(a) shows a simply supported beam that carries a central concentrated load of 500 kN

together with a uniformly distributed load of 300 kN/m applied across the 3 m span of the

beam. Given the cross-section dimensions shown in Fig. Q3(b),

Calculate the shear stress distribution of a section 1 in from the left-hand support using Table Q3 provided on , and (15 Marks)

Draw the shear stress distribution acting in the cross-section in the provided space in Fig. Q3(b). (5 Marks)

500 kN 300 kN/m

1.5m 1.5m

Fig. Q3(a) Simply supported beam dimensions and loading details

Draw Shear Diagram Below S4 ---.-.---------.-------- T --------

- S4

S3 --------------------------------f --------f --------- E E

S2

Si

--------- [ 8

-- S 2

'--S

25 mm 25 ram

100 mm 4

Fig. Q3(b) Enlarged beam cross-section

Page 10 of 22

Section cut

Area (mm2) (mm) Q(mm3) q = VQ (N\

-i--- ) t (mm) r (MPa)

Si

S2

S3

S4

Table Q3: Shear stress calculation table




of 22


Question # 4: (20 Marks)

A 60 mm diameter shaft of 1.2 m in length is clamped at one end and is free at the other end. A

40 mm diameter concentric bore (hole) is machined out from the free end extending inwards.

The free end is subjected to a torque T. For this particular design, the maximum allowable angle

of twist is 2.5° and the tensile yield stress of the shaft material is 400 MPa, and its modulus of

rigidity is 85 GPa.

Determine the maximum allowable length of the bore LB and the maximum allowable

value of the applied torque T, if a factor of safety of 2 is assigned to the shaft. (14 Marks)

Determine the maximum permissible angle of rotation, if the bore length LB extends to

the full length of the shaft, and yielding is not permitted. (6 Marks)

III

Fig. Q4 Shaft with a concentric hole at one end

Maximum allowable Length of Bore, LB = mm

Maximum allowable Torque, T = Nm

Maximum Angle of Twist, 0 =

of 22




of 22


Question # 5: (20 Marks)

A material is subjected to two mutually perpendicular direct stresses of 80 MPa tensile and

50 MPa compressive, together with a shear stress of 30 MPa. The shear stress couple acting on

planes carrying the 80 MPa stress is clockwise in effect. Assuming that Young's Modulus of the

material E = 200 GPa and Poisson's ratio v = 0.3,

Calculate the magnitude and nature (tensile or compressive) of the principal strains, (6 Marks)

Calculate the magnitude of the maximum and minimum shear strains in the plane of

the given system, (4 Marks)

Calculate the direction of the planes on which the principal strains act, and (4 Marks)

Using the graph paper provided on Page 19, confirm your answer by means of

Mobr's circle in terms of strains. Use Mohr's circle to determine the magnitude of the

normal and shear strains acting on a plane inclined 20° counter clockwise to the plane

on which the strain component of the 50 MPa stress acts. (6 Marks)

Principal Strain,

Principal Strain, E2 = IIE

Maximum Shear Strain, Ym

Minimum Shear Strain, Ymin = IIE

Principal Stress Angles, O =

of 22


QUESTION # 5 SOLUTION:


of 22




of 22


QUESTION 9 5 SOLUTION (continued):


of 22


USEFUL FORMULA - DO NOT DETACH

e =(a — v(cy +a))

1 1 Yxz=jtxz '

E G=2(1) c EE

F My

Tr VQ J It

P=Tw

A =

7td 2 ird4 = j = 2irr3t

4 64 32

bh3 A=bh, 12

= I + Ad

qtt

T=2qA

TLds çb

4A2G

r

t

T = (dM/dx) Q/It

of 22


cr + L7yy o_xx - oyy

= 2 +

2 cos 28 + sin 28

Tx1y = -

a, sin 20 + cos 28

ozz = v(a. + ok,)

+

(x+ay) 2

2 - ~2 f

- a 2

tmax P 2 ) +) tan20 =

cr -

UY

M ab(3H2 - h2)

12

arr A-B/r2

a bH 2 Mf P

Pr Pr r00 =--, zz 2r

a09 = A + B / r2

tan20 = - a y 2rX),

M

Mpp

fp

6zz-

P1d - P2d

2 '- 1

°max

°max - °min - UY

-

2 2

[(aa)2+ 2 2]2oy 2

EI-=V(x) dx 2 dx 3

= q(x)

of 22


M=EI- dr2

dv rM - = j- dx + C1 El

V = $J

dxjdx + C1 x + C2

f (x) = (x - a)

1 0 x~a;n>O f fl(x)=(x—a)= x>a ; n>0

Type of Loading Singularity Function Pictorial Representation

Concentrated Moment M(x)M0(x—a)° I x

(doublet function) 0 a 2M0 *1

Concentrated Force

(impulse or M(x) FO (x - a)1 x Dirac delta function) 0 a

Uniformly distributed w0 M(x)=<x—a>2

tW

fflflh1t load (step function) o

a

w

1 d _ dwldx Linearly Varying Load

- M(x) -- (x - a)3 6 dx

0 _______________________________

a un

of 22

PLEASE DO NOT

TURN OVER THIS

EXAMINATION

PAPER UNTIL YOU

ARE ASKED TO DO SO

DO NOT DETACH ANY PAGE(S)

FROM THE EXAMINATION BOOKLET

DO NOT DETACH ANY PAGE(S) FROM THE EXAMINATION...

Documents

Transcript of DO NOT DETACH ANY PAGE(S) FROM THE EXAMINATION...