Slope Deflection Examples

# Slope Deflection Examples - Slope Deflection Examples Fixed...

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Slope Deflection Examples: Fixed End Moments For a member AB with a length L and any given load the fixed end moments are given by: () 2 2 2 A BB FEM g g L =⋅ A 2 2 2 BA B A FEM g g L =− Where: g B and g A are the moments of the bending moment diagrammes of the statically determinate beam about B and A respectively. Example: Determine the fixed end moments of a beam with a point load. Simply supported beam with bending moment diagramme. Centroid in accordance with standard tables. 2 2 2 A FEM g g L A 2 2 2 23 AB LW a b b L a b a L FEM LL L  ++ ⋅⋅   2 22 2 3 AB Wab b L a L FEM L ⋅+ = 2 3 AB Wab b a b a FEM L + = 2 2 2 AB Wab FEM L = In a similar way FEM BA may be calculated. Slope-deflection Page 1 of 20 7/23/2003

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Use of slope-deflection equations: Example 1: Determine the bending moment diagramme of the following statically indeterminate beam. The unknowns are as follows: θ A , θ B , θ C = 0, ψ AB = 0, ψ BC = 0 We require two equations to solve the two unknown rotations: 00 AA B MM =∴ = () 2 23 A BA B A B EI MF L θθ ψ =⋅ + + A B E M 21 2 48 AB A B EI M ⋅⋅ + + 0 4 0,5 5 AB A B ME I E I θ =⋅+ ⋅⋅+ 0 A M = EI A + 0,5 EI B + 5 = 0 ( 1 ) BB A B C M + = 2 BA A B AB BA EI L =+ + E M 2 BA A B EI M 0 4 0,5 1 5 BA A B IE I ⋅+ ⋅− 2 BC B C BC BC EI L + + E M 2 25 2 61 BC B EI M + 6 2 0,66667 15 BC B M + M BA + M BC = 0 0,5 EI A + 1,66667 EI B + 1 0 = 0 ( 2 ) Solve the unknowns: A = - 2,35294/EI B = - 5,29412/EI Calculate the values of the moments: M BA = 0,5 x –2,35294 – 5,29412 – 5 = -11,471 kN.m Slope-deflection Page 2 of 20 7/23/2003
M BC = 0,6667 x – 5,29412 + 15 = + 11,471 kN.m () 2 23 CB B C BC CB EI MF L θθ ψ =+ + E M 2 25 61 CB B EI M θ ⋅− 6 2 M CB = 0,3333 x –5,29412 – 15 = -16,675 kN.m Draw the bending moment diagramme. The Modified Slope-Deflection Equation with a Hinge at A: We would like to eliminate θ A from the equation as we know that M AB = 0. 2 A BA B A B EI L =⋅ + + A B E M Solve for θ A . 3 22 2 2 A BB A FEM L EI θψ =− + A B 2 BA A B AB BA EI L + E M Replace θ A in this equation. 3 21 2 B BA B AB BA AB EI L  + +−   E M F E M 31 2 BA B AB BA AB EI E M L + F E M This equation may be used to reduce the number of unknown rotations. Solve the previous problem using the modified slope-deflection equation. Slope-deflection Page 3 of 20 7/23/2003

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The unknowns are as follows: θ A use the modified slope-deflection equation, θ B , θ C = 0, ψ AB = 0, ψ BC = 0 The total number of unknowns is reduced to θ B 00 BB A B C MM M =∴ + = () 31 2 BA B AB BA AB EI MF E M L θψ =− + F E M 3 1 04 11 04 48 2 BA B EI M θ ⋅⋅    8 0,75 7,5 BA A ME I =⋅ 2 23 BC B C BC BC EI L θθ ψ + + E M 2 25 2 61 BC B EI M + 6 2 0,66667 15 BC B M + M BA + M BC = 0 1,41667 EI B + 7 , 5 = 0 ( 1 ) B = - 5,29412/EI Bending moments are as calculated previously.
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Slope Deflection Examples - Slope Deflection Examples Fixed...

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