Fixed end moment

In the following examples, clockwise moments are positive.

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,

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

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

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 ${\displaystyle q_{0}}$,

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

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

• Moment distribution method
• Statically Indeterminate
• Slope deflection method
• Matrix method

• Yang, Chang-hyeon (2001-01-10). Structural Analysis (in Korean) (4th ed.). Seoul: Cheong Moon Gak Publishers. ISBN 89-7088-709-1. Archived from the original on 2007-10-08. Retrieved 2007-09-03.