3. The cantilever beam AB shown has an extension BCD attached to its free end. A...

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3. The cantilever beam AB shown has an extension BCD attached to its free end. A force P acts at the end of the extension. a) Find the ratio a/L so that the vertical deflection of point B will be zero. b) Find the ratio a/L so that the angle of rotation at point B will be zero. c) For part b), determine the equation of the deflection curve for the cantilever beam AB. PV 4. Solve Problem 3 using the method of superposition. B C Beam and Loading 1 3 0 Appendix D Beam Deflections and Slopes Maximum Deflection Slope at End Elastic Curve --- Ju b Ymax Bx ymas Ymax Bx PL 3EI wL 8EI ML 2EI PL 48EI For a > b: Pb(L-b/2 9V/3EIL at x = 5wL 384EI ML 9V/3EI L-b 3 PL 2EI wL 6EI ML EI PL 16EI 0 + Pb(L-b) 6EIL 8A = + Pa(L-a). 6EIL 24EI ML 6EI ML 3EI Equation of Elastic Curve y 6EI (x-3Lx) y w (x - 4Lx + 6Lx) 24EI For x = L: P y= 48EI y = M 2EI For x < a: Pb y = [x - (L-b)x] 6EIL (4x - 3Lx) For x = a: y = Pab 3EIL w 24EI (x-2Lx + Lx) M 6EIL -(x - Lx) 3. The cantilever beam AB shown has an extension BCD attached to its free end. A force P acts at the end of the extension. a) Find the ratio a/L so that the vertical deflection of point B will be zero. b) Find the ratio a/L so that the angle of rotation at point B will be zero. c) For part b), determine the equation of the deflection curve for the cantilever beam AB. PV 4. Solve Problem 3 using the method of superposition. B C Beam and Loading 1 3 0 Appendix D Beam Deflections and Slopes Maximum Deflection Slope at End Elastic Curve --- Ju b Ymax Bx ymas Ymax Bx PL 3EI wL 8EI ML 2EI PL 48EI For a > b: Pb(L-b/2 9V/3EIL at x = 5wL 384EI ML 9V/3EI L-b 3 PL 2EI wL 6EI ML EI PL 16EI 0 + Pb(L-b) 6EIL 8A = + Pa(L-a). 6EIL 24EI ML 6EI ML 3EI Equation of Elastic Curve y 6EI (x-3Lx) y w (x - 4Lx + 6Lx) 24EI For x = L: P y= 48EI y = M 2EI For x < a: Pb y = [x - (L-b)x] 6EIL (4x - 3Lx) For x = a: y = Pab 3EIL w 24EI (x-2Lx + Lx) M 6EIL -(x - Lx)