Through **Champion Study Plan for GATE Civil Engineering (CE) 2022**, we are providing very useful basic notes and other important resources on every topic of each subject. These topic-wise notes are useful for the preparation of various upcoming exams like GATE Civil Engineering (CE) 2022/ IES/ BARC/ ISRO/ VIZAG/ DMRC/ SSC-JE/ State Engineering Services examinations and other important upcoming competitive exams.

Read here the fundamental notes on the "**Slope Deflection Method**" topic of the "**Structural Analysis**" subject.

## Slope Deflection Method

**Introduction**

- In the stiffness method, displacements (rather than forces) are taken as unknown quantities. For this reason, the method is also called the displacement method.
- The unknown displacements are obtained by solving equations of equilibrium (rather than equations of compatibility) that contain coefficients in the form of stiffnesses.

In this method, if the slopes at the ends and the relative displacement of the ends are known, the end moment can be found in terms of slopes, deflection, stiffness and length of the members.

In- the slope-deflection method the rotations of the joints are treated as unknowns. For any 1 member bounded by two joints, the end moments can be expressed in terms of rotations.

In this method all joints are considered rigid; i.e the angle between members at the joints are considered not-to change in value as loads are applied, as shown in fig 1.

### Assumptions in the Slope Deflection Method

This method is based on the following simplified assumptions.

- All the joints of the frame are rigid, i.e, the angle between the members at the joints do not change, when the members of the frame are loaded.
- Distortion, due to axial and shear stresses, being very small, are neglected.

**Degree of Freedom**

The number of joints rotation and independent joint translation in a structure is called the degrees of freedom. Two types for degrees of freedom.

- In Rotation- For beam or frame is equal to D
_{r}.

Where,

Dr = degree of freedom.

j = no. of joints including supports.

F = no. of fixed support.

- In translation- For frame is equal to the number of independent joint translation which can be given in a frame. Each joint has two joint translation, the total number of possible joint translation = 2j. Since on other hand, each fixed or hinged support prevents two of these translations, and each roller or connecting member prevent one these translations, the total number of the available translational restraints is

The slop defection method is applicable for beams and frames. It is useful for the analysis of highly statically indeterminate structures which have a low degree of kinematical indeterminacy. For example, the frame shown below.

The frame (a) is nine times statically indeterminate. On the other hand only tow unknown rotations, θb and θc i.e Kinematically indeterminate to a second degree- if the slope deflection is used.

The frame (b) is once indeterminate.

**Sign Conventions**

Joint rotation & Fixed and moments are considered positive when occurring in a clockwise direction.

**Fixed End Moments**

**Slope Deflection Equation**

In this method, joints are considered rigid. It means joints rotate as a whole and the angles between the tangents to the elastic curve meeting at that joint do not change due to rotation. The basic unknown is joint displacement (θ and Δ).

To find θ and Δ, joints equilibrium conditions and shear equations are established. The forces (moments) are found using force-displacement relations. Which are called slope deflection equations.

**Slope Deflection Equation**

(i) The slope deflection equation at the end A for member AB can be written as:

(ii) The slope deflection equation at the end B for member BA can be written as:

where,

**L = Length of beam, El = Flexural Rigidity**

** are fixed end moments at A & B respectively.**

**M _{AB} & M_{BA} are **final

**moments at A & B respectively.**

**θ _{A} and θ_{B} are **rotation

**of joint A & B respectively.**

**Δ = Settlement of support**

- Sign Convention
- +M → Clockwise
- -M → Anti-clockwise
- +θ → Clockwise
- -θ → Anti-clockwise
- Δ → +ve, if it produces clockwise rotation to the member & vice-versa.

The number of joint equilibrium conditions will be equal to a number of ‘θ’ components & number of shear equations will be equal to a number of ‘Δ’ Components.

**Application of Slope-Deflection Equations to Statically Indeterminate Beams.**

The procedure is the same whether it is applied to beams or frames. It may be

summarized as follows:

1. Identify all kinematic degrees of freedom for the given problem. This can be done by drawing the deflection shape of the structure. All degrees of freedom are treated as unknowns in the slope-deflection method.

2. Determine the fixed end moments at each end of the span to the applied load. The table given at the end of this lesson may be used for this purpose.

3. Express all internal end moments in terms of fixed end moments and near end, and far-end joint rotations by slope-deflection equations.

4. Write down one equilibrium equation for each unknown joint rotation. For example, at support in a continuous beam, the sum of all moments corresponding to an unknown joint rotation at that support must be zero. Write down as many equilibrium equations as there are unknown joint

rotations.

5. Solve the above set of equilibrium equations for joint rotations.

6. Now substituting these joint rotations in the slope-deflection equations evaluate the end moments.

7. Determine all rotations.

**Example 1**

A continuous beam ABC is carrying uniformly distributed load of 2 kN/m in addition to a concentrated load of 20 kN as shown in Figure below. Draw bending moment and shear force diagrams. Assume EI to be constant.

curve of the beam is drawn in figure in order to identify degrees of freedom. By fixing the support or restraining the support against rotation, the fixed-fixed beams area obtained as shown in figure.

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