Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

Structural Analysis

GATE : Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

The document Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev is a part of the GATE Course Structural Analysis.
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Strain Energy Method

  1. Strain energy stored due to axial load
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, P = Axial load
    dx = Elemental length
    AE = Axial rigidity
  2. Strain energy stored due to bending
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, MX = Bending moment at section x-x
    ds = Elemental length
    El = Flexural rigidity
    or E = Modulus of elasticity
    l = Moment of inertia
  3. Strain energy stored due to shear
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, q = Shear stress
    G = Modulus of rigidity
    dv = Elemental volume
  4. Strain energy stored due to shear force
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, AS = Area of shear
    S = Shear force
    G = Modulus of rigidity
    ds = Elemental length
  5. Strain energy stored due to torsion
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, T = Torque acting on circular bar
    dx = Elemental length
    G = Modulus of rigidity
    lP = Polar moment of inertia
  6. Strain energy stored in terms of maximum shear stress
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    where, image007 Maximum shear stress at the surface of rod under twisting.
    G = Modulus of rigidity
    V = Volume
  7. Strain energy stored in hollow circular shaft is
    Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
    Where, D = External dia of hollow circular shafts
    d = Internal dia of hollow circular shaft
    τmax = Maximum shear stress

Castigliano's first theorem

∂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.

Castiglianos Second Theorem

∂U/∂P = Δ, ∂U/∂M = θ

Betti's Law

∑Pmδmn = ∑Pnδnm

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

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'.

Maxwells Reciprocal Theorem

δ21 = δ12

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

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).

Standard Cases of Deflection

  1. For the portal frame shown in the figure below horizontal deflection at D due to load P, assuming all members have same flexural rigidity is given by
    δ = Ph2(2h + 3b)/3ElDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, δ = Deflection of D in the direction of load P.
  2. Semicircular arch whose one end is hinged and other supported on roller carried a load P as shown in the figure.
    δ = π/2.PR3/ElDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, δ = Deflection at B in the direction of load P.
  3. A quadrantal ring AB of radius r support a concentrated load P as shownDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, δV = Vertical deflection (deflection in the direction of load P) at end A
    δH = Horizontal deflection of end A.
  4. A portal frame as shown in figure carries a load P at ADisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, δH & δV is horizontal & vertical deflection at end A respectively.
  5. Figure shows two identical wires OA and OB each of area A and inclined at 45° from horizontal. A load P is supported at O
    δv = PL/AEDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, δV = Vertical deflection at 'O'.

Moment distribution method (Hardy Cross method)

  1. Stiffness: It is the force/moment required to be applied at a joint so as to produce unit deflection/rotation at that joint.
    k = F/Δ or M/θ
    Where, K = Stiffness
    F = Force required to produce deflection Δ
    M = Moment required to produce rotation θ.
  2. Stiffness of beam
    (i) Stiffness of member BA when farther end A is fixed.
    k = 4El/LDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevWhere, K = Stiffness of BA at joint B. When farther end is fixed.
    El = Flexural rigidity
    L = Length of the beam
    M = Moment at B.
    (ii) Stiffness of member BA when farther end A is hinged
    k = 4El/LDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, K = Stiffness of BA at joint B. When farther end is hinged
  3. Carry over factor
    Carry over factor = Carry over moment/Applied moment
    COF may greater than, equal to or less than 1.
  4. Standard Cases
    (i) COF = 1/2
    (ii) COF = 0Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev(iii) COF = a/bDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev
  5. Distribution Factor (D.F.)
    DF = stiffness of a member/Sum of stiffness of all members at that joint 
    or
    DF= Relative stiffness of a member/Sum of relative stiffness of all member at that jointDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere M1, M2, M3 and M4 are moment induced in member OA, OB, OC and OD respectively.
  6. Relative Stiffness
    (i) When farther end is fixed
    Relative stiffness for member = l/L
    (ii) When farther end is hinged
    Relative stiffness for member = 3l/4L
    Where, l = MOl and L = Length of BeamDisplacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevStiffness of OA = 4El/l
    Stiffness of OB
    Stiffness of OC = 34El/l
    Stiffness of OD = 0
  7. Fixed Convention
    +ve → Sagging
    –ve → Hogging
    and All clockwise moment → +ve
    and All Anti clockwise moment → –ve
    Span length is l

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

Slope deflection Method (G.A. Maney Method)

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:

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev

(ii) The slope deflection equation at the end B for member BA can be written as:

Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRevwhere, L = Length of beam, El = Flexural Rigidity Displacement Methods: Slope Deflection and Moment Distribution Methods Notes | EduRev 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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