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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 w {\displaystyle w} , taking downward to be positive, the internal shear force is given by taking the negative integral of the weight:
The internal moment M {\displaystyle M} is the integral of the internal shear:
The angle of rotation from the horizontal, θ {\displaystyle \theta } , is the integral of the internal moment divided by the product of the Young's modulus and the area moment of inertia:
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