# Slope Deflection Method - Ishwaranand

## Slope Deflection Method

### Introduction of the slope deflection method Image Source ~ Crafted With ©Ishwaranand - 2020 ~ Image by ©Ishwaranand

### 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 δ 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 ̅𝑎𝑏) and MFBA or(𝑀 ̅𝑏𝑎) develop without end rotations at ends.
2. In fig. below Settlement, δ 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 + 𝟐𝑬𝑰/𝑳 (2ia +ib  -𝟑𝜹/𝒍 )
A final moment at the end B
• MBA= MFBA + 4𝐸𝐼/𝐿 ib + 2𝐸𝐼/𝐿 ib - 6𝐸𝐼δ/𝑙2
• MBA= MFBA + 𝟐𝑬𝑰/𝑳 (2ib +ia  -𝟑𝜹/𝒍 )
Equation 1 and 2 is called a slope deflection equation.