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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Autumn Examinations 2009/
Module Title: Mechanical Materials (3D)
Module Code: MECH 8012
School: School of Mechanical and Process Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering – Stage 3
Programme Code: EMECH_8_Y
External Examiner(s): Prof. Robin Clarke, Mr. John J. Hayes Internal Examiner(s): Mr. S.F. O Leary
Instructions: Answer THREE questions. All questions carry equal marks.
Duration: 2 Hours
Sitting: Autumn 2010
Requirements for this examination: Graph paper and Logarithmic Books to be provided.
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
Q1. (a) Derive from first principles the following expressions for the hoop stress, H , and radial
stress r , set up at any radius r of a disc of external radius R 2 and central hole of radius R 1 rotating with a radial velocity w :
^ ^2
2 2 12 22 12 2
2 ( 3 ) 8 r r
ew R R RR
r
^ ^2
2 2 12 22 12 2
2
H ew 8 ( 3 ) R R RrR ( 1 3 ) r
Where e is the density and is the Poisson’s Ratio of the disc material.
Hence, determine expressions for the magnitudes and locations of the maximum hoop and radial stresses. (26 marks)
(b) A steel rotor disc is of uniform thickness 30 mm and has an outer diameter of 500 mm and a central hole of 100 mm diameter. If the disc rotates at 5,000 revolutions per minute, determine the magnitude and locations of the maximum hoop and radial stresses developed. (8 marks) Given for the Steel,
Poisson’s Ratio = = 0.
Density = e = 7460 kg/m^3
Q4. The three dimensional strain field at a critical design point in a machine element with respect to a Cartesian coordinate XYZ system has been determined to be the following:
Direct Strains: εx = 229 x 10-6^ contractional
εy = 168 x 10-6^ extensional
εz = 50 x 10-6^ contractional
Shearing Strains: γxy = - 9.1 x 10-3^ degrees γxz = + 12.5 x 10-3^ degrees γyz + = 22.8 x 10-3^ degrees
Calculate the magnitudes of the maximum principal stresses and strains.
Hence, determine, assuming a factor of safety of 3, whether the machine element has failed according to the Maximum Shear Strain Energy Theory of Elastic Failure. Given material properties: Modulus of Elasticity = 205 GN/m^2 Poisson’s Ratio = 0. Yield Stress = 240 MN/m^2 (34 marks)
Q2. (a) A thick cylinder of 100 mm internal radius and 150 mm external radius is subjected to an internal pressure of 160 MN/m^2 and an external pressure of 50 MN/m^2. Determine the hoop and radial stresses at the inside and outside of the cylinder together with the longitudinal stress if the cylinder is assumed to have closed ends. (17 marks)
(b) Given that the cylinder is subjected to a loading cycle consistent with the loading of part (a) and that the cylinder is made of a ductile steel with the following properties: Static Tensile Ultimate Stress u = 450 MPa Fatigue Strength Reduction Factor = 0. Determine the fatigue life of the cylinder using the Maximum Shear Stress Theory of Elastic Behaviour together with the Modified Goodman Criterion. (17 marks)
Given:- NCR = Nf ln( )
ln( ) ,
1
e
f
e
f CR b N
f b N
and Table 1 Q
Fatigue Criterion
Ductile Steels (σu≤ 1750 Mpa)
Brittle (hard) steels (σu≤ 1750 Mpa) σf Nf σe Ne σf Nf σe Ne Modified Goodman
0.9 σu 103 1/2 Kσu 106 0.9 σu 103 1/3 Kσu 108
Soderberg 0.9 σu 103 1/2 Kσu 106 0.9 σu 103 1/3 Kσu 108 Gerber 0.9 σu 103 1/2 Kσu 106 0.9 σu 103 1/3 Kσu 108 SAE σu + 350 x 10^6 1 1/2 Kσu 106 σu 1 1/3 Kσu 108
Table 1 Q2: End Points for S-N Diagram
