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First Semester M.E. (Civil) Degree Examination, March 2013 STRUCTURAL ENGINEERING/PRESTRESSED CONCRETE/ EARTHQUAKE ENGINEERING 2K8SE101 : 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 expressions for stress components on an arbitrary plane in three dimensions. Also obtain : i) Resultant stress ii) Normal stress and iii) Shear stress on the plane. 10 b) A rope of length L is hung from the ceiling. The density of the material of the rope is ρ . Find : i) the stress in the rope at the free end L and ii) the value of maximum tension and its location.

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2. a) What are stress invariants ? The state of stress at a point in x, y, z coordinate system is given below. ⎡10 ⎢ 9 ⎢ ⎢⎣ 0

8 −6 0

0⎤ 0 ⎥ MPa ⎥ 4 ⎥⎦

Considering another set of coordinate axes, x′y′z′ in which z′ coincides with z and x′ is rotated by 30° anticlockwise from x-axis, determine the stress components in the new co-ordinates system. 10

P.T.O.

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b) If the state of stress at a point is given as below. Determine the value of τxy in order that the stress distribution is in equilibrium if μ is the Poison’s ratio. 10 σ x = y 2 + μ(x 2 − y 2 ), σ y = x 2 + μ(y 2 − x 2 ), σ z = (x 2 + y 2 ) τ xy = f (x, y ), τ yz = τ zx = 0 .

3. a) Write a note on the practical significance of compatibility equations. If C and C1 are some constants, under what conditions the following strain system compatible ?

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ε x = C(x 2 − y 2 ) ε y = Axy ε xy = C1xy

b) A rectangular strain rosette gives the data as below. ε 0 = 670 micrometres/m, ε 45 = 330 micrometres/m and ε 90 = 150

micrometres/m Find the principal stresses σ 1 and σ 2 if E = 2×105 MPa and v = 0.3.

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4. a) Derive the compatibility equation in terms of stresses for plane stress problems including the body forces.

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b) Distinguish between plane stress and plane strain problems. Give the corresponding stress – strain relations.

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5. a) Using the stress function method, obtain the solution for a beam subjected to pure bending.

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b) Show that the following stress function satisfies the boundary condition in a beam of rectangular cross-section of width 2h and depth d under a total shear force W. ⎡ W ⎤ φ = −⎢ xy 2 (3 d − 2 y )⎥ 3 ⎣ 2hd ⎦

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6. a) A thick cylinder of internal diameter Di and external diameter Do is subjected to internal pressure only. Sketch the distribution of axial and hoop stresses in the cylinder. Comment on the maximum hoop stress. 10 b) A steel tube, which has an outside diameter of 10 cm and inside diameter of 5 cm, is subjected to an internal pressure of 14 MPa and an external pressure of 5.5 MPa. Calculate the maximum hoop stress in the tube. 10 PART – B 7. a) Explain : i) Elastic-perfectly plastic material and ii) Elastic-linear strain hardening material. Sketch the stress-strain behaviour of these materials.

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b) List various theories of failure. Explain any two of them.

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8. a) The state of stress at a point is given below. ⎡ 105 σ ij = ⎢ ⎣52 .5

52 .5 ⎤ MPa 180 ⎥⎦

If the yield strength of the material is 187.5 MPa obtained by uniaxial tensile test, verify whether yielding will occur according to Tresca’s or Von-Mises yield criteria.

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b) Write a note on the geometrical representation of yield criteria. __________________

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