The Three-Moment Equation, introduced by Clapeyron, is a powerful tool for analyzing statically indeterminate continuous beams. It directly relates the internal bending moments at three consecutive supports (or joints) to the loads applied on the two spans connecting those supports.

Originally formulated by French engineer Émile Clapeyron in 1857, the Theorem of Three Moments (or Clapeyron's Theorem) expresses the relationship between the bending moments at three consecutive supports of a continuous beam. It is fundamentally derived by enforcing the geometric compatibility condition that the slope of the elastic curve must be continuous (the same on just the left and just the right) over any interior support.

The standard Three-Moment Equation requires three supports to formulate one equation. If a beam ends at a fixed support (e.g., Support $A$), there is no adjacent span to the left to form a triplet (Left-Middle-Right).

To solve this, we introduce an imaginary span (or dummy span) extending outward from the fixed support.

• The length of this imaginary span is zero ($L_0 = 0$).
• The moment of inertia of this imaginary span is infinite ($I_0 = \infty$), making it perfectly rigid so no rotation occurs at the fixed end.
• No loads are placed on this imaginary span, so its area term ($A_0$) is zero.

By applying the Three-Moment Equation to the newly created triplet (Imaginary Support - Actual Fixed Support - First Interior Support), we generate the necessary extra equation to solve for the unknown bending moment acting at the fixed support.

The terms $\frac{6A\bar{x}}{L}$ are standard load terms derived from the moment-area theorems. Common cases include:

• Uniform distributed load $w$ over entire span $L$: $\frac{6A\bar{x}}{L} = \frac{wL^3}{4}$
• Point load $P$ at distance $a$ from the left and $b$ from the right support:
  • Left side term (using distance from left support): $\frac{Pab(L+a)}{L}$
  • Right side term (using distance from right support): $\frac{Pab(L+b)}{L}$

A frequent error when applying the Three-Moment Equation involves confusing the left and right spans. Ensure you correctly designate $L_1$ for the left span and $L_2$ for the right span relative to the middle support $B$. When the loads are asymmetric, the load terms $\frac{6A_1 \bar{x}_1}{L_1}$ and $\frac{6A_2 \bar{x}_2}{L_2}$ differ, meaning $L_1$ and $L_2$ cannot simply be swapped without altering the results.

In the design of multi-span continuous highway bridges, the Three-Moment Equation offers an efficient method to rapidly determine the internal moment envelope over intermediate piers due to self-weight and moving traffic loads. It is especially useful for quickly checking alternative span arrangements during preliminary design before detailed finite element analysis is performed.

The basic Clapeyron's equation assumes rigid, unyielding supports. If supports settle, the resulting relative displacements induce additional bending moments. The equation is modified by adding a settlement term to the right side:

$M_1 L_1 + 2M_2 (L_1 + L_2) + M_3 L_2 = -6 \left[ \frac{A_1 \bar{x}_1}{L_1} + \frac{A_2 \bar{x}_2}{L_2} \right] + 6EI \left[ \frac{h_1 - h_2}{L_1} + \frac{h_3 - h_2}{L_2} \right]$

Where $h_1, h_2, h_3$ are the vertical downward settlements of supports 1, 2, and 3, respectively.