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

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.

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.

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,

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.

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:

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
 

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.

Take a simply supported beam AB of length L, place a point load ‘P’ at a distance ‘a’ from the left support A.

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.

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.

Now putting the double integration formula and then integrating twice

In this step we will apply the boundary conditions (at x = 0, y = 0 and at x = L, y = 0).

 Now we can find deflection at any point(x) on the beam using the final equation given in step 4.

If we want to find the deflection at half of the span, then putting x = 0.5 L

The maximum deflection will not result at the centre. At the point of maximum deflection, the slop should be zero. Equating the slope to zero, we can find the value of x.

To find the rotation or slope, the equation of dy/dx(Refer step 3 for the equation) should be written again. Now putting the value of x (depending on the point where slope is to be found) the slope value can be determined.

In this case, to find slope of point A, x = 0 and for slope of point B, x = L.

Macaulay's method is a subcase of double integration method, in Macaulay’s method there is only one bending moment equation written across the whole span of beam, instead of writing different moment equations for different sections of the beam.

This method is one of the most effective and mostly used methods for getting the bending deflections in beams as well as frames. Here in the method, the area of the bending moment diagrams is used for determining the slope and or deflections at specific points along the axis of the beam. There are two theorems, one is to get the slope, and another is for finding deflection.

The first theorem is used to calculate the change in the slope between two points on the beam.

The second theorem is used to compute the vertical displacement/tangential deflection between a point on the curve and a line tangent to the curve at the second point.
 

The deflection produced in a beam by combined loads is the same as the summation of deflections produced when they are acted upon the beam individually.

Some tricky problems to find deflection can be solved using the principle of superposition.

It is always advised to remember some of the slope and deflection formula for some standard load cases. This will save you from a lot more calculation, so confusion in examinations.
 

Some of the more important load cases are presented below.

Download Beam Deflection Formula Table PDF below

Calculating deflection of beams is so simple, isn't it? But 80% of you got this one question wrong from one of the APSEd Tests, I will tell you how you can tackle such tricky questions.
 

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