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Strength of Materials: Deflection by Integration

Deflection by Integration

If you know what the moment equation is that represents the particular beam of interest then you can take a double integral to determine what the deflection of the beam is. The first integration will represent the slope of the beam, while the second will represent the deflection.

Before I go any further on this subject though, you need to understand some basic calculus. For the most part all you will have to deal with is polynomial functions. Refer to equation 1 to see how to perform a basic polynomial integral.

Polynomial equation integral theory (1)

To reverse an integral you would take the derivative of the function; equation 2.

Polynomial equation derivative theory (2)

Another thing that has be considered when relating the moment equation to deflection is the Young's modulus of the material. The reason for this is because of Hook's law relates stress and strain, so effectively the Young's modulus is used to transform stress to strain so that a deflection can be determined. Equation 3 shows how to use calculus and Young's modulus, along with beam theory to determine the deflection and slope from the moment.

Beam Deflection moment slope and deflection relation thru calculus (3)

An example is shown below

Deflection by Integration example problem

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