Wednesday - June 7, 2023

Slope Deflection Method

Slope Deflection Method

Introduction of the slope deflection method

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Steps of slope deflection method

  • Moments at the ends of a member is first written in terms of unknown slopes and deflections of the end joints.
  • Considering joint equilibrium conditions, a set of the equation is formed and simultaneously solve to get unknown slopes and deflections.
  • Then end moments of individual members can be calculated.

Derivation for Slope deflection equation

  • Give AB do an intermediate span about a continuous beam subjected to an external weight system.
  • Let ia and ib be the slopes at the ends A and B.
  • Let Ξ΄ be the transverse downward deflection of the right end B with respect to the left end  A.
  • Let Mab and Mba be the final end moments at A and B.
  1. Due to given loadings end moments MFAB or (𝑀 Μ…π‘Žπ‘) and MFBA or(𝑀 Μ…π‘π‘Ž) develop without end rotations at ends.
  2. In fig. below Settlement, Ξ΄ takes place without any rotations at ends. End moment developed are 6𝐸𝐼δ/𝑙2
  3. Moment M’AB and M’BA give final rotations ia and ib to the beam AB. Where in fig. below Release the fixity at A. Maintain the fixity at B. Apply a moment 4𝐸𝐼/𝐿 ia at A to produce a slope ia at A. This will induce a moment 2𝐸𝐼/𝐿 ia at B.
  4. In fig. Release the fixity at B. Maintain the fixity at A. Apply a moment 4𝐸𝐼/𝐿 ib at B to produce a slope ib at B. This will induce a moment 2𝐸𝐼/𝐿 ib at A.

The final moment in the end A

  • MAB= MFAB + 4𝐸𝐼/𝐿 ia + 2𝐸𝐼/𝐿 ia – 6𝐸𝐼δ/𝑙2
  • MAB= MFAB + πŸπ‘¬π‘°/𝑳 (2ia +ib  -πŸ‘πœΉ/𝒍 )

A final moment at the end B

  • MBA= MFBA + 4𝐸𝐼/𝐿 ib + 2𝐸𝐼/𝐿 ib – 6𝐸𝐼δ/𝑙2
  • MBA= MFBA + πŸπ‘¬π‘°/𝑳 (2ib +ia  -πŸ‘πœΉ/𝒍 )

Equation 1 and 2 is called a slope deflection equation.

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Fortran ~ Bisection Method