∂U/∂Δ = P & ∂U/∂θ = M
where, U = Total strain energy
Δ = Displacement in the direction of force P.
θ = Rotation in the direction of moment M.
∂U/∂P = Δ, ∂U/∂M = θ
∑Pmδmn = ∑Pnδnm
where, Pm = Load applied in the direction m.
Pn = Load applied in the direction n.
δmn = Deflection in the direction 'm' due to load applied in the direction 'n'.
δnm = Deflection in the direction 'n' due to load applied in the direction 'm'.
δ21 = δ12
where, δ21 = deflection in the direction (2) due to applied load in the direction (1).
δ12 = Deflection in the direction (1) due to applied load in the direction (2).
In this method, joints are considered rigid. It means joints rotate as a whole and the angles between the tangents to the elastic curve meeting at that joint do not change due to rotation. The basic unknown are joint displacement (θ and Δ).
To find θ and Δ, joints equilibrium conditions and shear equations are established. The forces (moments) are found using force displacement relations. Which are called slope deflection equations.
Slope Deflection Equation
(i) The slope deflection equation at the end A for member AB can be written as:
(ii) The slope deflection equation at the end B for member BA can be written as:
where, L = Length of beam, El = Flexural Rigidity are fixed end moments at A & B respectively. MAB & MBA are final moments at A & B respectively. θA and θB are rotation of joint A & B respectively.
Δ = Settlement of support
Sign Convention
+M → Clockwise
-M → Anti-clockwise
+θ → Clockwise
-θ → Anti-clockwise
Δ → +ve, if it produces clockwise rotation to the member & vice-versa.
The number of joint equilibrium conditions will be equal to number of ‘θ’ components & number of shear equations will be equal to number of ‘Δ’ Components.
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