Slope Deflection Method

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-

Slope Deflection Method

The BMD and SFD can be found if we know the internal end moments of member i.e., MAB and MBA.

Slope Deflection Method
  • 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

Continuous Beam

Δ= deflection of joint B with respect of joint A

θA and θB = rotation of A and B

Internal member end moment at A = MAB

Internal member end moment at B = MBA

To find out the effect of external loading, rotation θA, θB 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 MFAB and MFBA 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-

MAB = MFAB + [2EI {2θA + θB – 3∂/l}]/l

MBA = MFBA + [2EI {2θB + θA – 3∂/l}]/l

This equation A is called Slope Deflection Equation.

Steps for Analysis in Slope Deflection Method

  1. Computation of fixed end moments
  2. Relate member end moments to joint displacement
  3. Formulate equilibrium equations
  4. Solve the equations
  5. Sketch SFD and BMD.

Right Understanding

Right Understanding We all know that the Human Desire is to be in continuous happiness which is the need of I (self). But do you know, from where Continuous happiness will come? No, right! So continuous happiness is to be in Right Understanding, Right Feeling, and Right Thought that is Activity of I (Self). Do…

Continue Reading Right Understanding

Where We Are

Where We Are (Self-Evolution) We exist as human being. We want to live a fulfilling life. We have some desires and we have some programs for the fulfilment of it. We need to understand our basic aspiration and program for its fulfillment correctly and comprehensively. Only then, we can ensure fulfillment. We should explore ourselves…

Continue Reading Where We Are

Highway Construction

Highway Construction Embankment Construction Materials and General Requirements The materials used in embankments, subgrades, earthen, shoulders, and miscellaneous backfills shall be soil, moorum, gravel, a mixture of these. Clay having liquid limit exceeding 70 and plasticity index exceeding 45; shall be considered unsuitable for embankment. Sub-grade and top 500mm portion of the embankment just below…

Continue Reading Highway Construction

Special Concretes

Special Concretes Concrete is most vital material in modern construction. In addition to normal concrete, other varieties in use are, high strength and high-performance concrete, self-compacting, lightweight, high density, fiber reinforced, polymer, colored concrete, etc. The making of concrete is an art as well as a science. Special types of concrete are those with out-of-the-ordinary…

Continue Reading Special Concretes

Marketing Practices

Marketing Practices Success in the world of business, no matter how you earn it, you have to rule on the marketplace. Although luck plays a role in the outcome of the market strategies. In the business decisions, there should be the understanding of market otherwise the failure will take place by the marked decisions. While…

Continue Reading Marketing Practices

Risk Analysis

Risk Analysis The risk that remains after the implementation of controls is called the residual risk. All systems will have residual risk because it is virtually impossible to completely eliminate risk to an IT system. In other words, we can say that there are two main parts of the security risk analysis known as Quantitative…

Continue Reading Risk Analysis

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *