# Slope Deflection Method

## Slope Deflection Method

### Introduction of the slope deflection method

- Presented by Prof. George A. Maney in 1915
- Suitable for analysis of continuous beam (Statically indeterminate beam)and rigid jointed frame
- Using this method basic unknown like slopes and deflection of joints can be calculated

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.

- Due to given loadings end moments MFAB or (𝑀 ̅𝑎𝑏) and MFBA or(𝑀 ̅𝑏𝑎) develop without end rotations at ends.
- In fig. below Settlement, δ takes place without any rotations at ends. End moment developed are 6𝐸𝐼δ/𝑙2
- 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.
- 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.

- MAB= MFAB + 4𝐸𝐼/𝐿 ia + 2𝐸𝐼/𝐿 ia - 6𝐸𝐼δ/𝑙2
- MAB= MFAB + 𝟐𝑬𝑰/𝑳 (2ia +ib -𝟑𝜹/𝒍 )

- 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.**