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Double Integration Method For Beam Deflections

Deflection problems generally occur in classical mechanics. In mechanical engineering education, deflections are explained with simple beams. Also, lots of theories that explain the deflections or other physical results are explained via simple beams. 

One of these theories is the deflection of simple beams on different loading conditions. But in here, we explain the general principle of beam deflections, which is also called as ‘double integration method’. 

What Is the Double Integration Method?

Basic example of a simply supported beam(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg. 254).

Actually, all the problems related to beam deflections can be examined with one differential equation;

As you see above, there is a second-order differential equation here. This means that you need to integrate this equation two times to obtain a general solution to it. If you apply the boundary conditions for this general solution, you will obtain the deflection equation for your beam system. 

Mathematical manipulations are very important here. 

The ‘E’ is the elasticity modulus of the beam material. And ‘I’ is the area moment of inertia of the cross-section of the beam. 

‘M’ is the moment. The moment must be calculated or obtained correctly to obtain the correct deflection equation. The moment occurrence depends on your loads and the boundary conditions of your beam system. So, determining the moment equation of your system depends on your engineering ability. 


So, the logic of the double integration method is very simple like that. You need to take a look at the different types of loading conditions and different types of boundary conditions for different beam systems to completely understand the double integration method. 

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