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.
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.
Step 3: Putting the bending moment value in double integration formula
Now putting the double integration formula and then integrating twice
Step 4: Solving the integration constants using boundary conditions
In this step we will apply the boundary conditions (at x = 0, y = 0 and at x = L, y = 0).
Step 5: Find deflection at any point
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
Step 6: Finding maximum deflection
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.
Step 7: Finding slope
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
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.
Moment area method
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.
First-moment area theorem: Used to find the slope at any point of a beam.
Second-moment area theorem: Used to find the deflection at any point of a beam.
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.
Principle of superposition
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.
Beam Deflection Formula Table
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
Deflection of Beam Example Problem
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.
Deflection of Beam Problem: Try this!
Try this question.