## Displacement Method of Analysis (Slope Deflection Method)

Slope deflection method is useful to analyse indeterminate structures like continues beams and plane frames.

The unknowns in this method are degree of freedom i.e., slope (**θ**) and deflection (**Δ**).

Combinedly these slopes and deflections are known as displacements. Thus, slope deflection method is a displacement method.

This is a classical method on which moment distribution method, **Kani’s method** and **stiffness matrix** method are based.

Analysis of a beam and frame means determination of bending moment and shear force throughout the length of the member i.e., determination of BMD and SFD for the member. BMD and SFD for a member of structure can be drawn if we know the internal end moments of a member.

**For example:**

If we have a fixed beam as shown below-

The BMD and SFD can be found if we know the internal end moments of member i.e., M_{AB} and M_{BA}.

- Thus, in the slope deflection method we establish a relationship between the degree of freedom (θ, Δ) and the member end moments. This relationship is called Slope Deflection Relationship.
- Finally, by using the equilibrium equation we find the slope deflection relationship to obtain the member moments.
- To find out the slope-deflection relationship, the method of superposition is used.

### Continuous Beam

Δ= deflection of joint B with respect of joint A

θ_{A} and θ_{B }= rotation of A and B

Internal member end moment at A = M_{AB}

Internal member end moment at B = M_{BA}

To find out the effect of external loading, rotation θ_{A}, θ_{B} and displacement ∂ on internal moments, we follow method of superposition.

### Steps in the determination of slope deflection relationship

Consider all the joints to be fixed. The member end moment due to external loads is M_{FAB} and M_{FBA} as shown in the figure below.

Allow the support B to settle with respect to A. The member end moments generated are as shown below.

Allow end A to rotate. The member end moments are as shown below.

Allow end B to rotate. The member end moments are as shown below.

Axial and Shear deformation is neglected.

*Clockwise moment is taken as +ve.*

*Clockwise rotation is taken as +ve.*

If ∂ gives clockwise rotation, it is taken as positive. For example, in the above discussion if end support B settles by amount ∂ with respect to A, it is taken as +ve.

Thus, from the principal of superposition, we have-

**M _{AB} = M_{FAB} + [2EI {2θ_{A} + θ_{B} – 3∂/l}]/l**

**M _{BA} = M_{FBA} + [2EI {2θ_{B} + θ_{A} – 3∂/l}]/l**

This equation A is called **Slope Deflection Equation**.

## Steps for Analysis in Slope Deflection Method

- Computation of fixed end moments
- Relate member end moments to joint displacement
- Formulate equilibrium equations
- Solve the equations
- Sketch SFD and BMD.

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