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CivilKits/Calculators/Structural/Loading analysis/Two-span partial UDLs

structural / loading analysis

Two-span continuous beam with partial UDLs.

Place a uniform-load patch anywhere on either span. Compare the middle-support hogging, three reactions, and positive span moments as the patch locations or span stiffnesses change.

continuous beam / load patch

Move the loading without losing support continuity.

Each of the two spans has its own patch start a, loaded length c, intensity w, and relative flexural stiffness EI. Both outer supports are simple. The beam remains continuous over the unyielding centre support B, so a patch in one span also changes the reactions in the other.

Three-moment method

For each span, the patch resultant W = wc acts at a + c/2. First construct its simply supported moment diagram Mss(x). Integrate that diagram to obtain its moment area and first moments. The three-moment equation combines the left-span first moment about A and the right-span first moment about C, each weighted by its span length and EI. The outer end moments are zero.

PATCH MOMENT AND CONTINUITYMss(x) = Rssx − w⟨x−a⟩²/2 + w⟨x−a−c⟩²/2
MB = −3[J1/(L1EI1) + K2/(L2EI2)]/(L1/EI1 + L2/EI2)

Here ⟨z⟩ means max(z, 0), J1 = ∫Mss,1(x)x dx, and K2 = ∫Mss,2(x)(L2−x) dx, over their respective spans. Span reactions follow from equilibrium after MB is known. The program checks each patch boundary and any zero-shear point inside a patch for the maximum sagging moment. Signed bending uses sagging positive and hogging negative; a negative reaction flags uplift in the idealised model.

Assumptions and limits

  • Two prismatic spans, simple outer supports, continuous middle support, and zero support settlement.
  • One downward uniform-load patch per span. A zero-intensity patch represents an unloaded span.
  • Constant EI within each span, linear elasticity, small deflection, and no shear deformation.
  • No support width, redistribution, moving-load envelope, cracking adjustment, deflection result, vibration, lateral stability, or member design.