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# Quick Guide to Deflection of Beams - Calculation, Formula and Table

Updated: Dec 11, 2020

Deflection means bending or displacement of a structural member(beam) from its original position. When a beam is loaded, it bends. This displacement of all beam points in the y-direction is called the deflection of the beam.

Slope of the beam (θ) is the angle between the original and deflected beam at a particular point.

In this guide, we will show you the basics of finding the slope and deflection of a beam straight away. If you are looking to download the beam deflection formula table, here it is.

### What is Deflection of a Beam?

To understand the concept of deflection better, let’s take a case of a cantilever beam. The upward force P is applied at the free end. By virtue of which the beam bends. Deflection of a cantilever beam

The point which earlier was on the x-straight axis has moved up in the y-direction. This displacement of all beam points in the y-direction is called the deflection of the beam.

According to a structural analysis point of view,

• The deflection should not exceed the allowable limit.

• The deflection curve helps a lot during the analysis of indeterminate structures.

• Reactions and moments can be found easily drawing the deflection curve.

So, yes. It is same as the literal meaning of deflection. When you drive a vehicle, you must have had an eye on the speedometer, the hand of the speedometer deflects. This is what is deflection all about.

### What is the slope of a beam?

Slope(θ) is the angle between the original and deflected beam at a particular point. The slope at any section in a deflected beam is defined as the angle in radians which the tangent at the section makes with the original axis of the beam.

There are different methods to find slope and deflection of a beam:

## Double Integration Method

From the Euler-Bernoulli bending theory, at a point along a beam, we have:

1/R = M/EI

where: R is the radius of curvature of the point, M is the bending moment at that point, EI is the flexural rigidity of the member.

We also have dx = R dθ and so 1/R = dθ/dx. Again for small displacements, θ ≃ tan θ ≃ dy/dx and so:

1 / R = d²y / dx² = M / EI

### Formula used to find the slope and deflection of the beam • M is the Bending Moment at a particular section

• EI is the flexural rigidity of the member

• y represents the vertical deflection of the beam and x is the lateral direction.

• dy/dx represents the slope of the beam at that particular point.

Using this relation, vertical deflection and slope can be found quickly for determinate beams.

Not clear? Let’s calculate a bit of it; you can then master this concept.

## Double Integration method to find deflection and slope of a beam

Take a simply supported beam AB of length L, place a point load ‘P’ at a distance ‘a’ from the left support A. ### Step 1: Finding the reactions at support(s)

We can use the horizontal equilibrium (ΣH = 0), vertical equilibrium (ΣV = 0) and moment equation (ΣMx = 0) for the reaction calculation.

The reaction at A is Pb/L and B is Pa/L.

### Step 2: Writing bending moment pattern at different sections of beam

In this case, the bending moment from L = 0 to L = a follows one pattern and the bending moment from L = a to L = a + b, follows another pattern.