# 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..

## Examples

In the following examples, clockwise moments are positive. The vertical reactions are not shown since they can be easily determined from statics.

 Concentrated load of magnitude P Linearly distributed load of maximum intensity q0 Uniformly distributed load of intensity q Couple of magnitude M0

The two cases with distributed loads can be derived from the case with concentrated load by integration. For example, when a uniformly distributed load of intensity ${\displaystyle q}$ is acting on a beam, then an infinitely small part ${\displaystyle dx}$ distance ${\displaystyle x}$ apart from the left end of this beam can be seen as being under a concentrated load of magnitude ${\displaystyle qdx}$. Then,

${\displaystyle M_{\mathrm {right} }^{\mathrm {fixed} }=\int _{0}^{L}{\frac {qdx\,x^{2}(L-x)}{L^{2}}}={\frac {qL^{2}}{12}}}$
${\displaystyle M_{\mathrm {left} }^{\mathrm {fixed} }=\int _{0}^{L}\left\{-{\frac {qdx\,x(L-x)^{2}}{L^{2}}}\right\}=-{\frac {qL^{2}}{12}}}$

Where the expressions within the integrals on the right hand sides are the fixed end moments caused by the concentrated load ${\displaystyle qdx}$.

For the case with linearly distributed load of maximum intensity ${\displaystyle q_{0}}$,

${\displaystyle M_{\mathrm {right} }^{\mathrm {fixed} }=\int _{0}^{L}q_{0}{\frac {x}{L}}dx{\frac {x^{2}(L-x)}{L^{2}}}={\frac {q_{0}L^{2}}{20}}}$
${\displaystyle M_{\mathrm {left} }^{\mathrm {fixed} }=\int _{0}^{L}\left\{-q_{0}{\frac {x}{L}}dx{\frac {x(L-x)^{2}}{L^{2}}}\right\}=-{\frac {q_{0}L^{2}}{30}}}$