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# 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.
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 the ends. The end moment developed is 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 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 at 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 of 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 slope deflection equation.

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