# 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 of 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 δ be the transverse downward deflection of the right end B concerning the left end A.
- Let Mab and Mba be the end moments at A and B.

- Due to given loadings end moments MFAB or (? ̅??) and MFBA or(? ̅??) develop without end rotations at ends.
- In Fig. below Settlement, δ takes place without any rotations at the ends. The end moment developed is 6??δ/?2
- 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 4??/? ia at A to produce a slope at A. This will induce a moment 2??/? ia at B.
- In fig. Release the fixity at B. Maintain the fixity at A. Apply a moment 4??/? ib at B to produce a slope ib at B. This will induce a moment 2??/? ib at A.

The final moment at the end A

- MAB= MFAB + 4??/? ia + 2??/? ia – 6??δ/?2
- MAB= MFAB + ???/? (2ia +ib -??/? )

A final moment at the end of B

- MBA= MFBA + 4??/? ib + 2??/? ib – 6??δ/?2
- MBA= MFBA + ???/? (2ib +ia -??/? )

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