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Two-span continuous beam with independent UDLs.

Calculate the internal-support hogging moment, three vertical reactions, and positive sagging moments for two constant-EI spans.

read the result

Use continuity at the middle support.

With simple end supports, zero support settlement, and constant EI, the three-moment relation gives the internal support moment. Each span is then treated as a beam segment with known end moments, and the reactions and interior positive-moment location follow from equilibrium and zero shear.

Implemented equations and signs

Let L1, L2 be the span lengths and w1, w2 the downward full-span UDLs. The signed middle-support moment is MB = -(w1L13 + w2L23)/[8(L1 + L2)]; the headline displays its hogging magnitude. The end reactions are RA = w1L1/2 + MB/L1 and RC = w2L2/2 + MB/L2. The internal reaction is the sum of the forces delivered at B by the two spans: RB = (w1L1 - RA) + (w2L2/2 - MB/L2). Sagging candidates are obtained from M(x) = Mleft + Rleftx - wx2/2 at x = 0, L, and x = Rleft/w when that point lies inside the span.

Classic load-case variants covered

Independent span UDL inputs cover equal spans with equal loading, unequal spans, unequal UDLs, one loaded span with the other unloaded by setting one UDL to zero, and the limiting equal-span symmetric case. The page covers full-span UDLs only; a point load, partial UDL, support settlement, or an additional span needs a different stiffness or three-moment model.

Keep it as a first-order elastic screen.

  • Both end supports are simple vertical supports; the middle support is continuous and has zero vertical settlement.
  • Both spans have constant EI, so EI cancels from the force and moment equations. The input form requires positive span lengths and accepts non-negative UDLs.
  • The displayed sagging locations are measured from the left end of each respective span; the middle-support moment is shown as a positive hogging magnitude.
  • Deflection, shear deformation, redistribution, cracking, vibration, lateral stability, member design, and code load combinations are not checked.