# Application of Double Integration and Superposition Methods to Restrained Beams

## Superposition Method

There are 12 cases listed in the method of superposition for beam deflection.

- Cantilever beam with...
- concentrated load at the free end.
- concentrated load anywhere on the beam.
- uniform load over the entire span.
- triangular load with zero at the free end
- moment load at the free end.

- Simply supported beam with...
- concentrated load at the midspan.
- concentrated load anywhere on the beam span.
- uniform load over the entire span.
- triangular load which is zero at one end and full at the other end.
- triangular load with zero at both ends and full at the midspan.
- moment load at the right support.
- moment load at the left support.

See beam deflection by superposition method for details.

## Double Integration Method

Moment at any exploratory section

$EIy'' = M$

Slope of the beam at any point

$\displaystyle EIy' = \int M + C_1$

Deflection of beam at any point

$\displaystyle EIy = \int \int M + C_1x + C_2$

See the deflection of beam by double integration method for details.