Fixed end moment

The fixed end moments are reaction moments developed in a beam member under certain load conditions with both ends fixed.

A beam with both ends fixed is statically indeterminate to the 3rd degree, and any structural analysis method applicable on statically indeterminate beams can be used to calculate the fixed end moments.

apart from the left end of this beam can be seen as being under a concentrated load of magnitude

Then, Where the expressions within the integrals on the right hand sides are the fixed end moments caused by the concentrated load

For the case with linearly distributed load of maximum intensity