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:

${\displaystyle V(x)=-\int w(x)\,dx}$

The internal moment ${\displaystyle M(x)}$ is the integral of the internal shear:

${\displaystyle M(x)=\int V(x)\,dx}$ = ${\displaystyle -\int \left[\int w(x)\,dx\right]dx}$

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:

${\displaystyle \theta (x)={\frac {1}{EI}}\int M(x)\,dx}$

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

${\displaystyle \nu (x)=\int \theta (x)\,dx}$