First Semester M.E. (Civil) Degree Examination, February 2014 (Str. Engg.) (2K8 Scheme) Structural Engineering/Prestressed Concrete/Earthquake Engineering SE 101/EQ 101 : THEORY OF ELASTICITY AND PLASTICITY Time : 3 Hours
Max. Marks : 100 Note : 1) Answer any one question from Part B and any four from Part A. 2) Additional data if required may be suitably assumed. PART – A
1. a) Derive Cauchy’s stress equations to solve the problems in continuum mechanics for three dimensional systems.
10
b) A rectangular bar of metal of cross-section 30 mm × 25 mm is subjected to an axial tensile force of 180 kN. Calculate the normal, shear and resultant stresses on plane whose normal has the following direction cosines i) l = m =
10
1 and n = 0 2
ii) l = m = n =
1 3
2. a) A body is subjected to external loading. Derive the equations of equilibrium in three dimensions considering the body forces. 10 b) The state of stress at a point in a body is given by the matrix shown below. ⎡2 2 1 ⎤ σij = ⎢⎢2 0 2 ⎥⎥ MPa ⎢⎣1 2 σz ⎥⎦
Determine σ z such that there is at least one plane passing through the point in such a way that the resultant stress on that plane is zero. Also determine the direction of that plane.
10 P.T.O.
JUP – 068
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*JUP068*
3. a) Explain the need for strain compatibility equations. Derive the expressions for compatibility in case of three dimensional strain field.
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b) The strain tensor at a point in a body is given below. Determine i) Octahedral normal and shearing strains ii) Deviator and spherical strain tensors. ⎡1 2 ε ij = ⎢2 3 ⎢ ⎢⎣5 4
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5⎤ 4 ⎥ × 10 − 4 ⎥ 5 ⎥⎦
4. a) Explain with examples plane stress and plane strain problems. Give the corresponding state of stress and strain.
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b) A rectangular strain rosette gives the following data. Find the principal stresses if E = 2 × 105 MPa and μ = 0.30 10 ε 0 = 670 × 10 −6
ε 45 = 330 × 10 −6 ε 90 = 150 × 10 −6
5. a) Derive Biharmonic expression for bodies involving plane strain conditions considering the body forces.
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b) For the beam shown in Fig. 5(b), the following stress functions are considered. Investigate the stress fields and comment. 10 i) φ = C1x 2 ii) φ = C1xy and iii) φ = C1y 3
*JUP068*
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JUP – 068
6. a) A long thick walled circular cylinder of internal radius R1 and external radius R2 is subjected to an internal pressure ‘P’. Given that the sum of radial and hoop stresses at any point in the wall is constant and also that no shear stresses are generated, show that the maximum hoop stress is given by K2 + 1 σθ = P 2 , where K = R2/R1. K −1
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b) A long closed cylinder has an internal diameter of 300 mm and an external diameter of 600 mm. It is subjected to an internal pressure of 100 MPa. Determine the maximum circumferential and axial stresses in the cylinder. Also sketch their distribution in the cylinder. 10 PART – B 7. a) List various theories of failure. Explain any two of them.
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b) The state of stress at a point is given below. ⎡ 105 52 .5 ⎤ ⎢52 . 5 180 ⎥ MPa ⎣ ⎦
If the yield strength of the material is 188 MPa obtained by uniaxial tensile test, verify whether yielding will occur according to Tresca’s or von-Mises yield criteria. 10 8. a) Write a note on the geometrical representation of yield criteria. b) Explain the following : i) Strain hardening behaviour ii) Strain softening behaviour.
First Semester M.E. (Civil) Degree Examination, February 2014 ... b) The state of stress at a point in a body is given by the matrix shown below. MPa. 1 2. 2 0 2.
5. a) Derive the governing equation for the continuous longitudinal vibrations of a prismatic ... rotates at 5 cm radius and total mass of vibrating system is 25 kg.
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Book Synopsis. This collection of papers by leading researchers in the field of finite, nonlinear elasticity concerns itself with the behavior of objects that deform when external forces or temperature gradients are applied. This process is extremely
