The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney. The slope deflection method was widely used for more than a decade until the moment distribution method was developed. In the book, "The Theory and Practice of Modern Framed Structures", written by J.B Johnson, C.W. Bryan and F.E. Turneaure, it is stated that this method was first developed "by Professor Otto Mohr in Germany, and later developed independently by Professor G.A. Maney". According to this book, professor Otto Mohr introduced this method for the first time in his book, "Evaluation of Trusses with Rigid Node Connections" or "Die Berechnung der Fachwerke mit Starren Knotenverbindungen".
Introduction By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined. Deformation of member is due to the bending moment.
Slope deflection equations The slope deflection equations can also be written using the stiffness factor K = I a b L a b {\displaystyle K={\frac {I_{ab}}{L_{ab}}}} and the chord rotation ψ = Δ L a b {\displaystyle \psi ={\frac {\Delta }{L_{ab}}}} :
Derivation of slope deflection equations When a simple beam of length L a b {\displaystyle L_{ab}} and flexural rigidity E a b I a b {\displaystyle E_{ab}I_{ab}} is loaded at each end with clockwise moments M a b {\displaystyle M_{ab}} and M b a {\displaystyle M_{ba}} , member end rotations occur in the same direction. These rotation angles can be calculated using the unit force method or Darcy's Law.
θ a − Δ L a b = L a b 3 E a b I a b M a b − L a b 6 E a b I a b M b a {\displaystyle \theta _{a}-{\frac {\Delta }{L_{ab}}}={\frac {L_{ab}}{3E_{ab}I_{ab}}}M_{ab}-{\frac {L_{ab}}{6E_{ab}I_{ab}}}M_{ba}}
θ b − Δ L a b = − L a b 6 E a b I a b M a b + L a b 3 E a b I a b M b a {\displaystyle \theta _{b}-{\frac {\Delta }{L_{ab}}}=-{\frac {L_{ab}}{6E_{ab}I_{ab}}}M_{ab}+{\frac {L_{ab}}{3E_{ab}I_{ab}}}M_{ba}}
Rearranging these equations, the slope deflection equations are derived.
Equilibrium conditions
Joint equilibrium Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,
Σ ( M f + M m e m b e r ) = Σ M j o i n t {\displaystyle \Sigma \left(M^{f}+M_{member}\right)=\Sigma M_{joint}}
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