Direct integration of a beam

Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam.

For a beam with an applied weight ${\displaystyle w(x)}$, taking downward to be positive, the internal shear force is given by taking the negative integral of the weight:

The internal moment M(x) is the integral of the internal shear:

The angle of rotation from the horizontal, ${\displaystyle \theta }$, is the integral of the internal moment divided by the product of the Young's modulus and the area moment of inertia:

Integrating the angle of rotation obtains the vertical displacement ${\displaystyle \nu }$:

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