Fixed end moment explained

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.

Examples

In the following examples, clockwise moments are positive.

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

q is acting on a beam, then an infinitely small part dx distance x apart from the left end of this beam can be seen as being under a concentrated load of magnitude qdx . Then,M ^{fixed} _{right} = \int ^{L} _{0} (q dx x ^{2} (L - x))/(L ^{2}) = (q L ^{2})/(12) M ^{fixed} _{left} = \int ^{L} _{0} \left \{ - (q dx x ^{2} (L - x))/(L ^{2}) \right \} = - (q L ^{2})/(12) Where the expressions within the integrals on the right hand sides are the fixed end moments caused by the concentrated load qdx .

For the case with linearly distributed load of maximum intensity

q _{0} ,M ^{fixed} _{right} = \int ^{L} _{0} q _{0} (x)/(L) dx (x ^{2} (L - x))/(L ^{2}) = (q _{0} L ^{2})/(20) M ^{fixed} _{left} = \int ^{L} _{0} \left \{ - q _{0} (x)/(L) dx (x (L - x) ^{2})/(L ^{2}) \right \} = - (q _{0} L ^{2})/(30)

See also

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

References

• Book: Yang, Chang-hyeon. Structural Analysis. 4th. 2001-01-10. Cheong Moon Gak Publishers. Korean. Seoul. 89-7088-709-1. 2007-09-03. https://web.archive.org/web/20071008135424/http://www.cmgbook.co.kr/category/sub_detail.html?no=1017. 2007-10-08. dead.