2 marks Questions and Answers

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Thomasina Howard
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1 1. Define the term strain energy. A: Strain Energy of the elastic body is defined as the internal work done by the external load in deforming or straining the body. 2. Define the terms: Resilience and Modulus of resilience. A: Resilience: Strain energy per unit volume of the material is known as strain energy density or resilience. Modulus of resilience: When the stress f is equal to proof stress, f p at the elastic limit, the corresponding resilience is known as proof resilience, u p = f p 2 2E The proof resilience is known as modulus of resilience. It is the property of the material. It s unit is N-m/m 3 = N/m 2 1

2 3. Write down the expression for strain energy stored in a bar of cross sectional area A and length l subjected to a tensile load W. A: Strain energy due to axial tensile load W is U = W2 L 2AE 4. Write the expression for strain energy due to bending. A: Strain energy due to bending is, U L 0 2 M 2EI dx 2

3 5. State Castigliano s first and second theorem for strain energy. A: Castigliano s first theorem: For linearly elastic structure, the Castigliano s first theorem may be defined as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action. δ i = U W i 3

4 Castigliano s second theorem: The energy U i can be express in terms of spring stiffnesses k 11, k 12 (or k 21 ), & k 22 and deflections δ1 and δ2; then it can be shown that U 1 U i i 2 W W 1 2 This is Castigliano s second theorem. 4

5 6. What is the strain energy due to axial force of a tapering bar of circular section? A: 2 marks Questions and Answers D P d P U = 2P2 L πedd 5

6 7. What is the strain energy due to axial force of a tapering bar of rectangular section? L t A: x P b B P U = P 2 L 2 B b te log e B b 6

7 8. Calculate the strain energy stored in a cantilever beam subjected to a point load W at the free end. M Px U L 0 2 M 2EI dx L 0 2 P x 2EI 2 dx 2 P L 6EI P L U 6 EI 7

8 9. State Maxwell s reciprocal theorem. A: In any beam (or) truss, the deflection at any point C due to load W at any point B is the same as the deflection at B due to the same load W applied at C. W A B C D δ C = W A B C D δ B δ C = δ B 8

9 10. State the principle of virtual work. A: The partial derivative of internal energy with respect to a load applied at a point where the deflection is zero is called principle of virtual work. And hence, U W = 0. 9

10 11. Define fixed beam? A: A fixed beam is a beam whose end supports are such that the end slopes remain zero (or unaltered) and is also called a built-in or encaster beam. 12. What are the disadvantages of fixed beam? A: (i). It is practically difficult to maintain the two ends of the beam at exactly at the same level. Any subsidence of one of the supports, however small it may be, will set up considerable stresses. (ii). Temperatute variations also produce large stresses in a fixed beam. 13. Define theorem of three moments. A: If AB and BC are any two consecutive spans of a continuous beam subjected to an external loading, the support moments M A, M B and M C at the supports A, B and C are given by the relation, M A (l 1 ) + 2M B l 1 + l 2 + M C l 2 = 6a 1x 1 + 6a 2x 2 l 1 l 2 Is known as theorem of three moments. 14. What is meant by prop? A: Simply supported at the free end of cantilever beam is known as prop. 10

11 15. Find the fixed end moments of a fixed beam subjected to a point load at the center. A: A = A M l = marks Questions and Answers l Wl 4 A l/2 W + Wl 4 l/2 B M = Wl 8 = M A = M B M Free BMD - Fixed BMD Wl M Resultant BMD 11

12 16. For the fixed beam subjected to an eccentric point load W, write end moments. M A = Wab2 l 2 M B = Wba2 l 2 a A W l b B Wab 2 Wab l 2 l Wba l 2 Resultant BMD 12

13 17. A fixed beam of span L is subjected to UDL of W/m throughout the span. What are end moments? A: W kn/m A = A A l m 2 wl2 l wl 2 /8 3 8 = M l + B M = wl2 12 M A = M B = Wl2 12 M - M 13

14 18. What are the advantages and limitations of theorem of three moments? A: Advantages: It is useful to find support moments of a indeterminate beams like continuous beam, fixed beam and propped cantilever beam. The effects of sinking of support will be taken into consideration for a continuous beam. It is also useful if moment of inertia is different for different spans of continuous beams. Limitations: To use this theorem, minimum two spans are required. 14

15 19. What is the value of prop reaction in a propped cantilever of span L, when it is subjected to a UDL of w/m over the entire length? A: A A L=0 2 marks Questions and Answers L m W kn/m B R B wl 2 /8 By using clayperon s theorem of three moments for the spans A A and AB, one can find W kn/m M A = wl2 8 A B M A =0, wl 2 /8 R A L m R B R B l + wl2 8 wl2 2 = 0 R B = 3 8 wl 15

16 20. What is the value of prop reaction in a propped cantilever of span L, when it is subjected to a point load of W at the centre? A: A A L= wl 2 marks Questions and Answers L m W wl 4 B R B By using clayperon s theorem of three moments for the spans A A and AB, one can find W M A = 3 16 wl 3 16 wl A R A L m B R B M A =0, R B l wl wl 2 = 0 R B = 5 16 w 16

2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C

2. Determine the deflection at C of the beam given in fig below. Use principal of virtual work. W L/2 B A L C CE-1259, Strength of Materials UNIT I STRESS, STRAIN DEFORMATION OF SOLIDS Part -A 1. Define strain energy density. 2. State Maxwell s reciprocal theorem. 3. Define proof resilience. 4. State Castigliano