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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 2
(EMECH_8_Y2)
Summer 2008
Mechanics of Materials
(Time: 3 Hours)
Instructions Answer FIVE questions. Question ONE is compulsory, answer FOUR other questions.
Examiners: Mr. W. Corr Mr. P. Clarke Prof. M. Gilchrist
Keep your answers to Q1 short and in accordance with the instruction: “sketch”, “demonstrate”, “explain”, etc.
Q1. (a) Sketch a 3D stress element showing positive direct and shear stresses on all faces. (2 marks) (b) List the four elastic constants for a material. If a material is homogenous isotropic, how many are independent? What does this mean? (2 marks)
(c) Sketch 2D and 3D Mohr’s circles for the stress situation in a thin-walled spherical pressure vessel. (2 marks)
(d) Consider two solid columns of equal length, material and end conditions: one square of side d, the other circular of diameter d. Which will buckle first? Why? (2 marks)
(e) Natural cork has Poisson’s ratio v = 0. Why use this material in wine bottles? (2 marks)
(f) Distinguish between statically determinate and statically indeterminate systems. (2 marks)
(g) Natural rubber has Poisson’s ratio v = 0.5. Why use this material in 0-ring seals? (2 marks)
(h) Describe the failure mechanism when a piece of chalk is twisted until it breaks. (2 marks)
(i) Why and where might built-in beams be pinned? How might the section between pins be analysed? (2 marks)
(j) Sketch a free-body diagram for one half of a simply supported beam carrying a uniformly distributed load. (2 marks)
Q2. (a) Show that a simplified shaft design based on the maximum distortion stress theory of failure can be expressed as
3 2
1 2 2 4
= ^ M + T
S
d N
π y
(5 marks)
(b) A flywheel of mass 500kg is mounted on a shaft of diameter 75mm and midway between self-aligning bearings 600mm apart.
What will be: (i) the maximum tensile stress, (4 marks) (ii) the maximum compressive stress, (4 marks) (iii) the maximum shear stress, (4 marks) if the shaft transmits a power of 20kW at 5 rev/s?
(c) Sketch two stress elements to illustrate these results. (3 marks)
Q5. A thin rectangular plate of dimensions 75 mm x 200 mm is formed by welding two triangular plates (see Figure Q5). The plate is subjected to a compressive stress of 3. MPa in the long direction and a tensile stress of 10.0 MPa in the short direction. (a) Determine the normal stress σw acting perpendicular to the line of the weld and the shear stress τw acting parallel to the weld. (8 Marks) (b) (i) Calculate also the principal stresses and maximum shear stress in the plate. (4 Marks) (c) (i) Make a rough sketch of Mohr's Circle of stress for the plate. (4 Marks) (ii) Locate the weld on your circle. (4 marks)
Q6. (a) Define the slenderness ratio for a column and show that the critical slenderness ratio can be expressed in the form
Syt
E
k
Le c
π^2
where the usual notation applies. (Start from the Euler buckling load) (6 marks)
(b) A vertical post AB is embedded in a concrete foundation and held at the top by two cables (see figure). The post is a hollow steel tube with modulus of elasticity 200 GPa, outer diameter 40mm, and thickness 5 mm. The cables are tightened equally by turnbuckles. If a factor of safety of 3.0 against Euler buckling in the plane of the figure is desired, what is the maximum allowable tensile force Τ allow in the cables? (14 marks)
Q7. (a) State Mohr’s Moment-Area Theorems and illustrate their application to solving for:
- Slopes at bearings
- Deflections at the gear mesh For a simply-supported midspan spur gear drive. (6 marks)
(b) The shaft of Figure Q7 is made of steel with Young’s Modulus 200 GPa. The bearings at A and B exert only vertical reactions on the shaft. (i) Sketch the BM diagram, indicating values at the pulleys. (4 marks) (ii) Determine the midspan deflection in mm. (6 marks) (iii) Determine the slope at A in degrees. (4 marks)
Figure Q
