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Unformatted text preview: Chapter 5, Problem 6 A cantilever carries a concentrated load P as shown in Figure P5.6. Using Castigliano ’ s theorem, determine the vertical deflection v A at the free end A . M Qx M Qx P x AC CB a = = + − ( ) A P L C a x B Q Thus v M dx A EI i M Q i = ∫ 1 ∂ ∂ = + + − ∫ ∫ 1 EI a a L Qx x dx Qx P x a x dx [ ( )( ) [ ( )]( ) Set Q and integrate: = 0, v P x a xdx L aL a A EI a L P EI = − = − + ∫ 1 6 3 2 3 2 3 ( ) ( ) ↓ Chapter 5, Problem 7 A cantilevered spring of constant flexural rigidity E I is loaded as depicted in Figure P5.7. Applying Castigliano ’ s theorem, determine the vertical deflection at point B . Assumption: The strain energy is attributable to bending alone. M Px M PL PR BC CA = = + sin θ Thus δ θ ∂ ∂ ∂ ∂ π B EI BC M P CA M P L M dx M Rd BC CA = + ∫ ∫ 1 2 [ ] = + + + P EI L RL R L R 12 3 2 2 4 6 24 3 ( ) π π 3 Chapter 5, Problem 9 A continuous beam is subjected to a bending moment M o at support C (Figure P5.9). Applying Castigliano ’ s theorem, find the reaction at each support. Consider R as redundant. A C B R B M x L L/2 x ’ R C M L R A = + 2 A R A M R x M x M v M d AB A CB M L R A i M R A i A = = + − = = ∫ ( ) ' , 2 1 2 ∂ ∂ x Thus v R x xdx x M dx A A M L R x L L A = + + − ∫ ∫ ( ) ( ) ' ] ' 2 2 2 = After integrating R M L = ↑ 2 3 A Then R = ↑ C M L 4 3 For the entire beam, F R y B M L = = ↓ ∑ 2 : Chapter 5, Problem 22 A curved frame of a structure is fixed at one end and simply supported at another, where a horizontal load P applies (Figure P5.22). Determine the roller reaction F at the end B , using Castigliano ’ s theorem. Assumption: The effect of bending moment is considered only. M FR PR M dx v EI M F = + − = = ∫ sin ( cos ), θ θ δ ∂ ∂ 1 1 Therefore ∫ − + = π θ θ θ θ δ 1 ) sin )]( cos 1 ( sin [ Rd R PR FR EI v = + = = − = π π π FR EI PR EI P P F 3 3 2 2 4 0, ↑ 4 Chapter 5, Problem 30 A simply supported beam carries a distributed load of intensity w = w o sin π x/L as shown in Figure P5.30. Using the principle of virtual work, determine the expression for shown in Figure P5....
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 Winter '10
 Kim
 Shear Stress, Trigraph, Tensile strength, MPa

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