Three-Moment Equation - Examples & Applications

Example

A two-span continuous beam ABCABC has equal spans L=6 mL=6\text{ m}, constant EIEI, and simple end supports at A and C. Each span carries a uniform load w=20 kN/mw=20\text{ kN/m}. Determine the interior support moment MBM_B.

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Example

A continuous beam ABCABC has LAB=4 mL_{AB}=4\text{ m} and LBC=6 mL_{BC}=6\text{ m}, constant EIEI, and simple end supports. Span AB carries a central point load P=40 kNP=40\text{ kN}; span BC is unloaded. Determine MBM_B.

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Example

For the same two-span beam, suppose the middle support B settles downward by a known amount ΔB\Delta_B while A and C remain fixed vertically. Explain how the settlement enters the three-moment equation.

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Key Takeaways
  • The three-moment equation links three consecutive support moments through load, stiffness, span, and settlement terms.
  • Free bending-moment diagrams are taken from the corresponding simply supported spans.
  • Sign conventions for support moments and settlements must remain consistent throughout the calculation.
  • After support moments are found, use span equilibrium to recover reactions, shear, and bending-moment diagrams.