PRIST UNIVERSITY, Thanjavur

PRIST UNIVERSITY

(Estd. u/s 3 of UGC Act, 1956) Vallam, Thanjavur -613403

_________________________________________________________________M.TECH (STRUCTURAL ENGINEERING)

QUESTION BANK

Course Details:Course Code & Title

: 13255H13-THEORY OF PLASTICITY AND

ELASTICITYRegulation

: 2013 Regulation

Nature of the Programme: M.Tech (Structural Engineering)

H.O.D

Staff-In-Charge

13255H13 THEORY OF PLASTICITY AND ELASTICITY L T P C

3 1 0 4AIM and OBJECTIVES:Emphasis is placed on static problems with linear material and small deformation. Many basic 2-D problems (such as plane strain and plane stress) and 3-D problems.UNIT I ANALYSIS OF STRESS AND STRAIN 12Analysis of stress and strain, stress strain relationship. Generalized Hooks law. Plane stress and plane strain.UNIT II DEFORMATION 12Basic concepts of deformation of deformable bodies - Transformations of stresses and strains in Cartesian and polar co-ordinates - Equilibrium equations in two and three dimensions in Cartesian co-ordinates.UNIT III TORSION 12Torsion of non-circular section - methods of analysis - membrane analogy - torsion of thin rectangular section and hollow thin walled sections.UNIT IV THEORY OF ENERGY 12Energy methods Energy principles and theorem.UNIT V INTRODUCTION TO PROBLEMS IN PLASTICITY 12Physical assumption - criteria of yielding, yield surface, Flow rule (plastic stress strain relationship). Elastic plastic problems of beams in bending - plastic torsion.REFERENCES:1. Timoshenko, S. and Goodier T.N. "Theory of Elasticity", McGraw Hill Book Co., New York, II Edition 1988.2. Chwo P.C. and Pagano, N.J. "Elasticity Tensor, Dyadic an Engineering applications", D.Van Nestrand Co., In Co., 1967.3. Chenn, W.P. and Henry D.J. "Plasticity for Structural Engineers", Springer Verlag New York 1988.4. Mendelson, A., (2002), Plasticity: Theory & Applications, Mac Millan & Co., NewYork.

5. Sadhu Singh, (2004), Theory of Plasticity, Dhanpat Rai sons Private Limited,New Delhi. UNIT I

PART- A

1.Define generalized Hooks law.

2.Differentiate between external forces and internal forces.

3.Explain the different types of stress.

4.Write a short notes on principle of super position.

5. Differentiate between body forces and surface forces.

6.Define stress at a point

7.Define plane stress and give an example.

8.Define plane strain and give an example.

9.Find the minimum diameter of a steel wire which is using to raise a load of 400 N if the stress in the rod not to exceed 95N/ .

10.Define stress variants.

PART B1.The state of sress at a point B given by the following stress tensor.

= Mpa

a. Calculate the stress variants

b. Magnitude and direction of the principal stress

c. Spherical and deviator stress tensors.

2.The state of stress is given by =100, =200, =-100, =-200,=100 and

=300

Kpa, find the

a) The stress invariants

b) The principle stresses and

c) The direction Cosines of the principle planes.

3. The stress components at a point are given by the following array.

Mpa

Calculate the principle stress and principle planes.

4.Explain the plane stress problem with example and derive the basic equations.

5.The stress tensor at a point is given by

=

Determine the magnitude and direction of the principle stress.

6.Derive plain strain problem with examples and derive the basic equations.

UNIT II

PART- A

1.Differentiate between rigid and deformable bodies.

2.Write short notes on strain hardening.

3.Mention the stages of deformation.

4.Write short notes on Re crystallization.

5.Write down the equation of transformation of stress in three dimensional Cartesian

coordinates.

6. Write down the equation of transformations of engineering strains in three dimensional Cartesian coordinates.

7.Write down the 6x6 consecutive law for an Isotropic material.

8.What are the properties of direction cosines.

9.Write the methods to solve the two dimensional problems. PART- B1.Prove the following are Airys stress functions and examine the stress distribution represented by them. i) = A+B ii) = A iii) = A(-3)

2.Derive strain displacement relations in polar co ordinates.3. The state of stress at a particular point relative to the xyz co ordinate system is given by the stress matrix.

Mpa

Determine the normal stress and magnitude and direction of the shear stress on a surface intersecting the point and parallel to the plane given by the equation 2x-y+3z=9.4.Derive the differential equation of equilibrium in 3D Cartesian co ordinates.5.Derive the strain displacement relations in rectangular co ordinates.6. Derive the differential equation of equilibrium in Polar co ordinates.

UNIT- III PART A1.Write brief note on St.Venants Theory on Torsion.

2.Explain Prandtls stress function Theory on Torsion.

3.What is warping function.

4.Why circular shafts are preferred in Torsion

5.Explain why torsion of multi celled thin walled tube is statically indeterminate problem.

6.Write a short notes on shear centre.

7.Write the torsion equation of non-Circular section.

8.Explain briefly about Residual stress.

9.Explain membrane analogy for torsions.

PART- B

1.An elliptical bar is subjected to a twisting moment T. Derive the expressions for shear stress

and angle of twist at any point in the bar and hence the maximum shear stress. 2.A two cell tubular section whose wall thickness are as shown in figure. If the member is subjected to a Torque T, determine the shear flows and the angle of twist of the member per unit length.

3.a) Derive the expression for angle of twist and shear stress for a thin walled closed Non circular section due to twisting moment and hence determine the shear stress and angle of twist for a hollow thin walled aluminium tube of rectangular cross section as shown in figure. Subjected to a torque of 56.50 KN.m. Assume G=28 Gpa.

(8)

b) Derive the Torsion equation for narrow thin Rectangle Bar. (8)4.Derive the Governing equation and boundary condition in St.Venants warping function

approach for Torsion of non circular sections.5. Derive the Governing equation and boundary condition in Prandtls stress function approach for Torsion of non circular sections.6. Derive the expression for maximum shear stress and angle of twist for a equilateral triangular shaft subjected to torque.

UNIT- IV

PART A

1.Define the term Strain energy.

2.Define principle of virtual work.

3. State Castinglianos second theorem.

4.Explain unit dummy load method.

5.State the principle of stationary potential energy.

6.State the principle of complementary energy (or) least work. PART-B1.Applying the principle of minimum potential energy, determine the vertical and horizontal displacements of D due to the load for the following figure which shows a three bar truss.

The point D of which is subjected to P units of force. The members have equal cross sectional areas.

2.a) Calculate the vertical deflection of joint A by the principle of virtual work. Area of cross section of AB and AC is 5 and E=200Gpa.

(8)

b) Determine the support reaction for the propped cantilever shown in figure. (8)

3. State and derive Castinglianos Ist theorem.4. State and prove Maxwell-Betti theorem of reciprocal displacements.5. Determine the maximum deflection of simply supported beam subjected to uniformly distributed load using energy principles.6. State and explain principle of strain energy.

UNIT V

PART A

1.State maximum principle stress theory (Rankines Theory)

2.State maximum principle stain theory.

3.State total stain energy theory.

4.Trescas maximum shear stress theory.

5.Huber distortion energy theory.

6.What do you understand by yield criteria.

7.Define theory of plastic bending

8.Define Residual stresses.

9.Define Sand heap analogy.

10.Give the formula to determine the twisting couple at elasto plastic yield and full yielding condition.

PART B1.a) A circular shaft of inner radius 4cm and outer radius 10cm is subjected to a twisting couple so that the outer 2cm deep shaft yield plastically. Determine twisting couple applied to shaft, yield stress in shear for the shaft material is 145 N/. Also determine the couple for full yielding.

(8)b) The state of stress at a point is given by =700 Kg/, =1200 Kg/ and =350 Kg/. If the yield strength of material is 1250 Kg/. Determine the

uniaxial tensile stress, whether yielding will occur according to Trescas and Von-mises yielding condition or not.

(8)2.a) A solid circular shaft of 8 cm radius is subjected to twisting couple so that the outer 3cm deep shaft yield plastically. If the yield stress in shear for the shaft material is 150Mpa, determine the value of twisting couple applied and associated angle of twist take G=0.84x N/.Alsofind full yield twisting moment and . (8) b) The state of stress at a point is given by =800 Kg/, =1000 Kg/ and =600 Kg/. If the yield strength of material is 1500 Kg/. Determine whether yielding will occur according to Trescas and Von-mises condition. (8)3. A steel bolt is subjected to a twisting moment of 0.12KN.m and a bending moment of

0.2KN.m. If the yield stress of the material in tension is 250N/. Determine the diameter according to Von-mises yield criteria and Tersca,s yield criteria.

4. A rectangular beam 8cm wide and 10cm deep is 2m long and simply supported at the ends. The yield strength for the beam material is 250 N/. Determine the value of concentrated load applied at the beam mid-span if,

a) The outer most fibres just start yielding.

b) The outer face up to 3cm depth yields.

c) Whole of the beam yields. Assume linear stress strain idealised curve for the beam materials.

5. Expalin the elastic and plastic problem of beam in bending with an example.

6. Explain what are the factors affecting plastic deformation.

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