tw6 university of bolton school of engineering b.eng …

TW6

UNIVERSITY OF BOLTON

SCHOOL OF ENGINEERING

B.ENG (HONS) MECHANICAL ENGINEERING

EXAMINATION SEMESTER 1 - 2016/2017

MECHANICS OF MATERIALS AND MACHINES

MODULE NO: AME5002

Date: Monday 9th January 2017 Time: 2.00 – 4.00

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 11 Mechanical Engineering Examination Semester 1 2016/2017 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. (5 marks)

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

(3 marks)

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

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

labelling 𝜎1, 𝜎2 the principal stresses and the maximum shear stress 𝜏𝑚𝑎𝑥. Verify

the results found in part a).

(8 marks)

c) Illustrate on a sketch of the element:

(i) The orientation of the principal planes. (3 marks)

(ii) The orientation of the plane where the shear stress is maximum.

(3 marks)

Total 25 Marks

Please turn the page

𝝈𝒙 = 𝟔𝟎 𝑴𝑷𝒂

𝝈𝒚 = 𝟓𝟎 𝑴𝑷𝒂

𝝉𝒙𝒚 = 𝝉𝒚𝒙 = 𝟑𝟎 𝑴𝑷𝒂


Q2. A uniform horizontal cantilever, as shown in figure Q2, is supported at the free end with a spring of stiffness k with zero force at zero deflection. A downward point load F2 is applied at 1/3 of the length from the free end.

Given: E=270GPa, k=28MN/m, F2=250kN, L=3m

a) Calculate the bending moment related to x. (4 marks)

b) Derive an expression for the maximum deflection at the end A.

(9 marks)

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

deflection is not to exceed 5mm. (3 marks)

d) Determine the dimension of the cross section beam if it has a circular cross

section. (6 marks)

e) Determine the increase of deflection if the spring is released. (3 marks)

Total 25 Marks


Figure Q2


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

section with an external diameter of 150mm and a wall thickness of 25mm as

shown in figure Q3. If E=210GPa:

a) Calculate the Euler crushing load. (5 marks) b) Calculate the Rankine crushing load taking σc=550MPa and a=1/6500 and

compare with the Euler crushing load found in part a) (6marks)

c) Comment on the validity of Euler formula. (6marks)

d) For which length of the column would Euler and Rankine formulas give the

same crushing load (8 marks)

Total 25 Marks


Figure Q3

𝑥

𝑦


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 12MPa and an external pressure of

7MPa, 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. (7 marks)

c) The maximum shear stress at the inner and outer surfaces. (4 marks)

d) The circumferential strain (𝜀𝐶) and radial strain (𝜀𝑅) at the inner surface if the

longitudinal stress (𝜎𝐿) is 90 MPa compressive. (8 marks)

e) The final diameter of the cylinder. (3 marks)

Take E=250GPa and a poisson ratio of ѵ=0.3.

Total 25 Marks


400mm

800mm

Figure Q4

of 11

Mechanical Engineering Semester 1 Examination 2016/2017

Mechanics of Materials and Machines Module No. AME5002

Q5. A 1.8 by 2m observation platform is supported by a single 15x20cm wood post as shown

in figure 5.1. The maximum mass of the observer is 130kg. It is assumed that the observer can

stand within 15cm of the railing in any corner. If the observer stands in one of the four corners,

the distance from the post center will be 85cm and 75cm as shown in figure 5.2. The observer

weight will cause a bending moment about both the y and z axis that are in the plane of the

platform, as shown in the diagram. This results in asymmetric bending stress. For this situation,

calculate:

a) The asymmetric bending moment about the y and z axis separately. (5marks)

b) The moment of inertia about the y and z axis for the member cross section. (4marks)

c) The position of the neutral axis (NA) and plot it in (yz) axis. (7marks)

d) The maximum asymmetric bending stress, the normal axial compression stress and the total compression stress caused by the vertical weight of the observer. (9marks)

Total 25 Marks

Total 25 Marks Please turn the page

Figure 5.2: Observer Location

and Post Dimensions

2m

1.8m 15x20cm

Observer 130kg

Figure 5.1: Observation platform

Post cross-section

of 11

Mechanical Engineering Semester 1 Examination 2016/2017

Mechanics of Materials and Machines Module No. AME5002

Q6. A truck as shown in figure 6 is unloading a heavy machine having a mass of

500kg by a crane. The steel cable has a length of 7m and a stiffness of k=1MN/m

and, it was suddenly seized (jammed) at time t from a descending velocity

v=0.4m/s. It is anticipated that the heavy machine will undergo an “up-down-up”

vibration after such seizure. Determine the following:

a) The frequency of vibration of the machine that is seized from descending.

(3 marks)

b) The maximum tension in the cable induced by the vibrating machine.

(10 marks)

c) The maximum stress in the cable, if the stranded steel cable has a diameter of

20mm.

(5 marks)

d) The extension of the cable and its final length, if the elastic modulus E=210GPa.

(4 marks)

e) Would the cable break if the maximum allowable strength is 150 MPa.

(3 marks)

Total 25 Marks

END OF QUESTIONS

Elastic cable with k=1MN/m

V=0.4m/s

m=500kg

Figure 6

of 11 Mechanical Engineering Semester 1 Examination 2016/2017 Mechanics of Materials and Machines Module No. AME5002

FORMULA SHEET

Deflection:

Plane Stress:


Lame’s equation

Vibrations:

Free Vibrations:

𝑓 =1

𝑇 𝜔𝑛 = 2𝜋𝑓 = √

𝑘

M

Differential equation Homogeneous form:

𝑎�̈� + 𝑏�̇� + 𝑐𝑦 = 0 Characteristic equation:

𝑎𝜆2 + 𝑏𝜆 + 𝑐 = 0

i. If 𝑏2 − 4𝑎𝑐 > 0, 𝜆1 and 𝜆2 are distinct real numbers then the general solution of the differential equation is:

𝑦(𝑡) = 𝐴𝑒𝜆1𝑡 + 𝐵𝑒𝜆2𝑡 A and B are constants.

ii. If 𝑏2 − 4𝑎𝑐 = 0, 𝜆1 = 𝜆2 = 𝜆 then the general solution of the differential equation is:

𝑦(𝑡) = 𝑒𝜆𝑡(𝐴 + 𝐵𝑥)


A and B are constants.

iii. If 𝑏2 − 4𝑎𝑐 < 0, 𝜆1 and 𝜆2 are complex numbers then the general solution of the differential equation is:

𝑦(𝑡) = 𝑒𝛼𝑡[𝐴𝑐𝑜𝑠(𝛽𝑡) + 𝐵𝑠𝑖𝑛(𝛽𝑡)]

𝛼 =−𝑏

2𝑎 𝑎𝑛𝑑 𝛽 =

√4𝑎𝑐 − 𝑏2

2𝑎

A and B are constants.

Asymmetric Bending:

𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔 =𝑀𝑦 𝑧

𝐼𝑦−

𝑀𝑧 𝑦

𝐼𝑧

Stress

σ = Force/Area = F/A

Hook’s law

σ = E∙ε

ε = L/L


Struts:

END OF PAPER

tw6 university of bolton school of engineering b.eng …

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