Theorem of three moments
Encyclopedia
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
the length of BC, by w and 
the weight per unit of length in these segments. Then the bending moments 
at the three points are related by:
This equation can also be written as
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.
Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...
and structural analysis
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
Clapeyron
Benoit Paul Émile Clapeyron
Benoît Paul Émile Clapeyron was a French engineer and physicist, one of the founders of thermodynamics.-Life:...
'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
the length of BC, by w and the weight per unit of length in these segments. Then the bending moments at the three points are related by:This equation can also be written as
where a1 is the area on the bending moment diagram
Shear and moment diagram
Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of an element. Using these diagrams the type and size of a member of a given material can...
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.