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# 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.
== 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
{\displaystyle q}
is acting on a beam, then an infinitely small part
d
x
{\displaystyle dx}
distance
x
{\displaystyle x}
apart from the left end of this beam can be seen as being under a concentrated load of magnitude
q
d
x
{\displaystyle qdx}
. Then,
M
r
i
g
h
t
f
i
x
e
d
=
∫
0
L
q
d
x
x
2
(
L
−
x
)
L
2
=
q
L
2
12
{\displaystyle M_{\mathrm {right} }^{\mathrm {fixed} }=\int _{0}^{L}{\frac {qdx\,x^{2}(L-x)}{L^{2}}}={\frac {qL^{2}}{12}}}
M
l
e
f
t
f
i
x
e
d
=
∫
0
L
{
−
q
d
x
x
2
(
L
−
x
)
L
2
}
=
−
q
L
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
q
d
x
{\displaystyle qdx}
.
For the case with linearly distributed load of maximum intensity
q
0
{\displaystyle q_{0}}
,
M
r
i
g
h
t
f
i
x
e
d
=
∫
0
L
q
0
x
L
d
x
x
2
(
L
−
x
)
L
2
=
q
0
L
2
20
{\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}}}
M
l
e
f
t
f
i
x
e
d
=
∫
0
L
{
−
q
0
x
L
d
x
x
(
L
−
x
)
2
L
2
}
=
−
q
0
L
2
30
{\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}}}
== See also ==
Moment distribution method
Statically Indeterminate
Slope deflection method
Matrix method
== References ==
- [[Professional Practice/Construction Management/Contract Administration]]
- [[Building Construction/Specifications and Detailing/Construction Specifications]]
- [[Professional Practice/Codes & Standards/National Building Code of India/Part 06 - Structural Design/Section 1 - Loads Forces and Effects]]
- [[History and Theory/World History/Modern Architecture]]
- [[Urban and Planning/Urban Design and Planning]]
- [[Urban and Planning/Urban Design]]
- [[Professional Practice/Client Management]]
- [[Wiki-Architecture/Architectural Concepts and History]]
- [[Professional Practice/Codes & Standards/Standards Index]]
- [[Professional Practice/Codes & Standards/National Building Code of India/Part 11 - Sustainability]]
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.