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, taking downward to be positive, the internal shear force is given by taking the negative integral of the weight:
The internal moment is the integral of the internal shear:
The angle of rotation from the horizontal,, 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 :

Integrating

Each time an integration is carried out, a constant of integration needs to be obtained. These constants are determined by using either the forces at supports, or at free ends.

Sample calculations

Take the beam shown at right supported by a fixed pin at the left and a roller at the right. There are no applied moments, the weight is a constant 10 kN, and - due to symmetry - each support applies a 75 kN vertical force to the beam. Taking x as the distance from the pin,
Integrating,
where represents the applied loads. For these calculations, the only load having an effect on the beam is the 75 kN load applied by the pin, applied at x=0, giving
Integrating the internal shear,
Assuming an EI value of 1 kNmm
Because of the vertical supports at each end of the beam, the displacement at x = 0 and x = 15m is zero. Substituting and, we can solve for constants =-1406.25 and =0, yielding
For the given EI value, the maximum displacement, at x=7.5m, is approximately 500 times the length of the beam. For a more realistic situation, such as a uniform load of 1 kN and an EI value of 5,000 kN·m², the displacement would be approximately 1 cm.