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Which of the following is equation of theorem of three moment?
MBl1+2MAl1+l2+MCl2=−6a1x1l1−6a2x2l2
MCl1+2MBl1+l2+MAl2=−6a1x1l1−6a2x2l2
MAl1+2MBl1+l2+MCl2=−6a1x1l1−6a2x2l2
None of these
MAl1+2MBl1+l2+MCl2=−6a1x1l1−6a2x2l2
The Theorem of Three Moments (Clapeyron's Theorem) is a fundamental analytical tool in structural mechanics used to solve for support moments in continuous beams. It establishes a linear relationship between the bending moments at three consecutive supports (A, B, and C) and the applied external loads on the spans.
The Theorem of Three Moments (Clapeyron's Theorem) is a fundamental analytical tool in structural mechanics used to solve for support moments in continuous beams. It establishes a linear relationship between the bending moments at three consecutive supports (A, B, and C) and the applied external loads on the spans.
MAl1+2MB(l1+l2)+MCl2=−l16a1x1−l26a2x2 — General expression for three consecutive supports A, B, and C.
lax — Represents the area moment of the free bending moment diagram about the outer support.
The theorem is derived from the principle of least work or slope-deflection compatibility at interior supports. For a continuous beam of constant flexural rigidity (EI), it assumes that the supports are at the same level. The expression represents the equilibrium of deformation at the central support, balancing the effects of internal continuity (moments) and external loading (moment area properties).
Used for indeterminate continuous beams.
Assumes constant flexural rigidity (EI).
Requires support settlements to be zero for the standard form.
Bending moments at ends of a simply supported beam are zero.
Simplifies analysis of continuous beams compared to slope-deflection method.
Reduces the number of simultaneous equations required.
Difficult to account for varying moment of inertia (I).
Not directly applicable to frames without modification.
Analysis of multi-span continuous beams.
Calculation of redundant moments in statically indeterminate structures.
The indices a1,a2 represent the area of the free bending moment diagram for spans l1 and l2 respectively.
Variables x1 and x2 are the distances of the centroids of these areas from the outer supports (A and C).
C is correct — The standard Clapeyron's equation for three consecutive supports A, B, and C is MAl1+2MB(l1+l2)+MCl2=−l16a1x1−l26a2x2.
Always remember the order: outer support A, central support B, outer support C; the coefficient '2' is always associated with the central support moment (MB).
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