Theorem of three moments: Difference between revisions

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In civil engineering and structural analysis Clapeyron's theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam.

Let A,B,C be the three consecutive points of support, and denote by l the length of AB by $l'$ the length of BC, by w and $w'$ the weight per unit of length in these segments. Then^[1] the bending moments $M_{A},\,M_{B},\,M_{C}$ at the three points are related by:

M_{A}l+2M_{B}(l+l')+M_{C}l'={\frac {1}{4}}wl^{3}+{\frac {1}{4}}w'(l')^{3}.

This equation can also be written as ^[2]

M_{A}l+2M_{B}(l+l')+M_{C}l'={\frac {6a_{1}x_{1}}{l}}+{\frac {6a_{2}x_{2}}{l'}}

where a₁ is the area on the bending moment diagram due to vertical loads on AB, a₂ is the area due to loads on BC, x₁ is the distance from A to the center of gravity for the b.m. diagram for AB, x₂ is the distance from C to the c.g. for the b.m. diagram for BC.

The second equation is more general as it does not require that the weight of each segment be distributed uniformly.

Notes

^ J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [1]
^ Srivastava and Gope: Strength of Materials, page 73 [2]

External links

CodeCogs: Continuous beams with more than one span

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[1] J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [1]

[2] Srivastava and Gope: Strength of Materials, page 73 [2]

[1]

[2]

@@ Line 9: / Line 9: @@
 :<math>M_A l + 2 M_B (l+l') +M_C l' = \frac{6 a_1 x_1}{l} + \frac{6 a_2 x_2}{l'}</math>
-where a<sub>1</sub> is the area on the [[Shear and moment diagram|bending moment diagram]] due to vertical loads on AB, a<sub>2</sub> is the area due to loads on BC, x<sub>1</sub> is the distance from A to the center of gravity for the b.m. diagram for AB, x<sub>2</sub> is the distance from C to the c.g. for the b.m. diagram for BC.
+where ''a''<sub>1</sub> is the area on the [[Shear and moment diagram|bending moment diagram]] due to vertical loads on AB, ''a''<sub>2</sub> is the area due to loads on BC, ''x''<sub>1</sub> is the distance from A to the center of gravity for the b.m. diagram for AB, ''x''<sub>2</sub> is the distance from C to the c.g. for the b.m. diagram for BC.
 The second equation is more general as it does not require that the weight of each segment be distributed uniformly.