# 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 unknowns like slopes and deflection of joints can be calculated

### Steps ofΒ slope deflection method

- Moments at the ends of a member are 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 an intermediate span about a continuous beam subjected to an external weight system.
- Let ia and ib be the slopes at the ends of A and B.
- Let Ξ΄ be the transverse downward deflection of the right end B concerning the left endΒ A.
- Let Mab and Mba be the end moments at A and B.

- Due to given loadings end moments MFAB or (π Μ ππ) and MFBA or(π Μ ππ) develop without end rotations at ends.
- In Fig. below Settlement, Ξ΄ takes place without any rotations at the ends. The end moment developed is 6πΈπΌΞ΄/π2
- Moment MβAB and MβBA 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 4πΈπΌ/πΏ ia at A to produce a slope at A. This will induce a moment 2πΈπΌ/πΏ ia at B.
- In fig. Release the fixity at B. Maintain the fixity at A. Apply a moment 4πΈπΌ/πΏ ib at B to produce a slope ib at B. This will induce a moment 2πΈπΌ/πΏ ib at A.

The final moment at the end A

- MAB= MFAB + 4πΈπΌ/πΏ ia + 2πΈπΌ/πΏ ia β 6πΈπΌΞ΄/π2
- MAB= MFAB + ππ¬π°/π³ (2ia +ibΒ -ππΉ/π )

A final moment at the end of B

- MBA= MFBA + 4πΈπΌ/πΏ ib + 2πΈπΌ/πΏ ib β 6πΈπΌΞ΄/π2
- MBA= MFBA + ππ¬π°/π³ (2ib +iaΒ -ππΉ/π )

**Equation 1 and 2 is called slope deflection equation.**