university of bolton school of engineering beng …€¦ · examination semester 1 2015/2016...

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UNIVERSITY OF BOLTON

SCHOOL OF ENGINEERING

BENG (HONS) MECHANICAL ENGINEERING

EXAMINATION SEMESTER 1 - 2015/2016

MECHANICS OF MATERIALS AND MACHINES

MODULE NO: AME5002

Date: Monday, 11 January 2016 Time: 2.00 – 4.00 p.m.

INSTRUCTIONS TO CANDIDATES: There are SIX questions.

Answer ANY FOUR questions only.

All questions carry equal marks.

Marks for parts of questions are shown in brackets.

Electronic calculators may be used provided that data and program storage memory is cleared prior to the examination.

CANDIDATES REQUIRE: Formula Sheet (attached).

of 10 School of Engineering BEng (Hons) Mechanical Engineering Examination Semester 1 2015/2016 Mechanics of Materials and Machines Module No. AME5002

Q1. A plane element of a material is subjected to a two dimensional stress

system as shown in figure Q1.

a) Determine via calculation:

(i) The magnitude of the principal stresses. (6 marks)

(ii) The angular position of the principal planes in relation to the X-axis

(4 marks)

(iii) The magnitude of the maximum shear stress. (3 marks)

b) Illustrate on a sketch of the element, the orientation of the principal planes

relative to the direction of σx and the orientation of the maximum shear

planes. (4 marks)

c) Sketch a Mohr’s Stress Circle from the information provided in figure Q1,

labelling σx, σy, the principal stresses and the maximum shear stress.

(8 marks)

Total 25 Marks

Total 25 Marks

PLEASE TURN THE PAGE

𝜏𝑥𝑦 = 𝜏𝑦𝑥 = 70 𝑀𝑃𝑎

𝜎𝑥 = 100 𝑀𝑃𝑎

𝜎𝑦 = 80 𝑀𝑃𝑎

Figure Q1


Q2. A simply supported beam with point load of 250 kN in the middle has a length

of 7m as shown in figure Q2.

Ignoring the mass of the beam the formula for calculating the maximum

deflection 𝑦𝑚𝑎𝑥 of the beam in figure Q2 is: 𝑦𝑚𝑎𝑥 = −𝐹𝐿3

48𝐸𝐼.

a) Calculate the flexural rigidity (EI) of the beam if the maximum allowable

deflection is not to exceed 5mm.

(3 marks)

b) Determine the dimension of the cross section beam if it is a square cross

section.

Take E=270GPa and ymax =5mm

(7 marks)

c) Determine the dimension of the cross section beam if it is a circular cross

section.

Take E=270GPa and ymax =5mm (7marks)

d) Now the beam can be treated as a simply supported beam having a mass of 25000kg placed at the middle and it deflects 5mm under the weight. Ignoring the mass of the beam calculate the frequency of transverse oscillations and comment on your result.

(8 marks)

Total 25 Marks

PLEASE TURNTHE PAGE

x

Y

F=250 kN

L=7m

Figure Q2


Q3. A steel pin-ended strut is 5m long and has a uniform rectangular hollow cross

section of 250mm x 375mm with a wall thickness of 30mm as shown in figure

Q3. When E=250GPa, the Yield Stress σy=470MPa and the Rankine

Constant is 1/6500, determine:

a) The slenderness ratio, 𝐿 𝑘⁄ . (8 marks)

b) The Euler Buckling Stress, 𝜎𝐸. (5 marks)

c) The Rankine-Gordon Buckling Stress, 𝜎𝑅. (6 marks)

d) Comment on the validity of the Euler bucking Stress. (6 marks)

Total 25 Marks


375 mm

260 mm 𝑥

𝑦

Figure Q3


Q4. A long, closed ended cylindrical pressure vessel has an outer diameter of

800mm and an inner diameter of 400mm as shown in figure Q4. If the vessel

is subjected to an internal pressure of 110 MPa and an external pressure of

70 MPa, determine the following:

a) The radial stress (𝜎𝑅) at the inner and outer surfaces. (3 marks)

b) The circumferential stress (𝜎𝐶) at the inner and outer surfaces.

(10 marks)

c) The circumferential strain (𝜀𝐶) at the inner surface if the longitudinal stress (𝜎𝐿)

is 90 MPa compressive. (12 marks)

Take E=250GPa and ѵ=0.3.

Total 25 Marks


400mm

800mm

Figure Q4


Q5. A machine of mass 2000kg is supported by four identical elastic springs and

set oscillating. It is observed that the amplitude reduces to 15% of its initial

value after 4 oscillations. It takes 2 seconds to do them.

Calculate the following:

a) The natural frequency of undamped vibrations (in Hertz).

(3 marks)

b) The effective stiffness of all four springs together. (4 marks)

c) The critical damping coefficient that will prevent oscillatory motion.

(4 marks)

d) The damping ratio. (7 marks)

e) The damping coefficient. (3 marks)

f) The frequency of damped vibrations. (4 marks)

Figure Q5

Total 25 Marks


Q6. The cross section of a cantilever section shown in figure Q6 is 3m long and is

loaded at its free end with 10KN. Given that the second moments of area

about the section are:

Ixx= 105

mm4

and I y y = 2 x 105

mm4, calculate:

a) The product second moment of area: (Ixy). (5 marks) b) The angular position of the principal plane in relation to the X-axis.

(3 marks)

c) The second moments of area (Iu) and (Iv) on the principal axes. (5 marks)

d) Find and state the angular position of the neutral axis of the section in relation to the principal axis u-u. (12 marks)

Total 25 Marks

Figure Q6

END OF QUESTIONS

10kN

FORMULA SHEET

Deflection:

Complex Stress:

Thick Cylinder:

Lame Equations Strain format

Struts:

Euler:

Rankine-Gordon:

of 10 ESS - B.Eng (Hons) Mechanical Engineering Examination 2015/2016 Mechanics of Materials and Machines Module No. AME5002

Vibrations:

Free Vibrations:

𝑓 =1

𝑇 𝜔𝑛 = 2𝜋𝑓 = √

𝑘

M

Damped Vibrations:

Asymmetric Bending:

𝑙𝑛 𝑥1𝑥2 =

2𝜋𝑚𝜁

1 − 𝜁2 , m is the number of oscillations

𝑓𝑑 =𝜔𝑑2𝜋 𝑐𝑐 = √4M𝑘 𝜁 =

𝑐

𝑐𝑐=𝑐

2𝑘 𝜔𝑛

𝜔𝑑 = 𝜔𝑛 1 − 𝜁2

university of bolton school of engineering beng …€¦ · examination semester 1 2015/2016...

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