# Calculating the deflection of a simply supported beam from first principles

When calculating the deflection of a simply supported beam for any loading combination we use two theorems:

- The change in slope of a beam between two points is the area of the M/EI diagram between those two points
- The deflection of point B on the beam from the tangent to the M/EI diagram at point A is equal to the moment of the M/EI diagram between A and B from B.

So, for a simply supported beam first work out the area of the M/EI diagram between the reactions. This tells you the change in slope between the two ends (one end will have a negative slope, the other positive). Now working at R2 calculate the bending moment of the M/EI diagram and this will give you the deflection of R2 from the tangent at R1. The slope at R1 is equal to this figure/span.

Now if you work along the beam calculating the slope using 1. above you will come to the point where the slope becomes zero (assuming no reverse loadings). This is the point of maximum deflection. At this point work out the moment of the M/EI diagram to the left of this point and this will tell you what the deflection is from the tangent at R1. Correct for the slope at R1 and you get the maximum deflection.

As EI is a constant you can use the BM diagram in the calculation above to get a generic deflection/EI. Substituting the actual E and I values for any member will give you the actual deflection. Rather than trying to work out exact figures, you will find that you get quite acceptable results if you split the span into 20 or 50 points and work out the areas and moments etc for each slice treated as a trapezium.

As a simple example consider a beam spanning 4m with a 10kN point load in the centre. The numbers below are all in kN and metres and are to be divided by the EI in each case.

- Max B.M. = 10kNm (WL/4)
- Area of M/EI diagram = 10x4/2 = 20 = change of slope between R1 and R2
- By inspection the system is symmetrical so slopes are -10 and 10 and maximum deflection is at 2m
- M/EI diagram between R1 and 2m is a triangle, base 2m, height 10(/EI). Centroid will be at 2/3m from R2.
- So moment of M/EI diagram between R1 and 2m from R2 is (2x10/2)x2/3 = 6.66. This is the deflection at 2m from the tangent to the slope at R1.
- The slope at R1 is -10 so at 2m the tangent line is at -20. Add the figure just calculated and you get a net deflection of 13.33(/EI).

This agrees with the standard formula of PL^{3}/48EI which is
10x4^{3}/48EI = (640/48)/EI = 13.33/EI

The above method is used to calculate deflection in our SuperBeam, ProSteel and EuroBeam programs. In EuroBeam we use the mid-span deflection as a proxy for maximum deflection - the two will be within 3% of each other whatever the loading pattern.