Structure Analysis : Slope-Deflection Method Study Notes

By Deepanshu Rastogi|Updated : January 3rd, 2022

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     

Slope Deflection Method

Introduction

  • In the stiffness method, displacements (rather than forces) are taken as the 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.

byjusexamprep

byjusexamprep

 

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 Dr.

byjusexamprep

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

byjusexamprep

byjusexamprep

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.

byjusexamprep

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.

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

Fixed End Moments

byjusexamprep

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:

image045

image046

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

image047

where,

L = Length of beam, El = Flexural Rigidity

image048 are fixed end moments at A & B respectively.

MAB & MBA are final moments at A & B respectively.

θA and θB 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. 

byjusexamprep

byjusexamprep

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.

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

byjusexamprep

 

You can avail of BYJU’S Exam Prep Online classroom program for all AE & JE Exams:

BYJU’S Exam Prep Online Classroom Program for AE & JE Exams (12+ Structured LIVE Courses)

You can avail of BYJU’S Exam Prep Test series specially designed for all AE & JE Exams:

BYJU’S Exam Prep Test Series AE & JE Get Unlimited Access to all (160+ Mock Tests)

Thanks

Team BYJU’S Exam Prep

Download  BYJU’S Exam Prep APP, for the best Exam Preparation, Free Mock tests, Live Classes.

Comments

write a comment

AE & JE Exams

AE & JEAAINBCCUP PoliceRRB JESSC JEAPPSCMPPSCBPSC AEUKPSC JECGPSCUPPSCRVUNLUPSSSCSDEPSPCLPPSCGPSCTNPSCDFCCILUPRVUNLPSPCLRSMSSB JEOthersPracticeMock TestCourse

Follow us for latest updates