Continuous Beam Calculator

How the calculator works

The unknowns are the rotations at each support. A fixed end has no rotation unknown. Each span contributes the slope-deflection stiffness 4EI/L and 2EI/L plus the fixed-end moments ∓wL²/12 of its uniform load. The page then enforces moment equilibrium at every free joint and solves the linear system. Once the end moments are known, it gets moment, shear and deflection anywhere in a span by statics. Deflection is the fixed-end uniform-load shape wx²(L−x)²/24EI plus the Hermite cubic driven by the end rotations.

Dead load acts on all spans. Live load runs through all 2ⁿ on/off patterns, and the page samples 200 points per span for each pattern and keeps the maximum and minimum. That captures the positive moment with alternate spans loaded, the negative moment with the adjacent spans loaded, and uplift at end supports of short spans next to long ones. The moment and shear envelopes use the load factors. Deflections are recomputed with unfactored loads, by integrating the curvature M/EI twice within each span.

HL-93 moving loads. With HL-93 on, the page first builds influence lines: it solves the beam once for a unit load at every sample point, giving moment, shear and reactions everywhere for each load position. It then moves the design truck across the beam in both directions with the rear spacing at 14, 18, 22, 26 and 30 ft, and moves the design tandem the same way, keeping the worst value at every point. The vehicle envelope is multiplied by (1 + IM) and added to the lane-load pattern envelope, which is AASHTO 3.6.1.3.1 cases (a) and (b). For the negative moment and reaction at each interior support it also runs case (c): 90% of two design trucks with 14 ft rear spacing and at least 50 ft between them, plus 90% of the lane load. Live-load deflection is the larger of the design truck alone and 25% of the truck plus the lane load, both with IM (3.6.1.3.2).

Check values (equal spans, uniform load on all spans)

Worked example — two spans, live load pattern

Limits

• Moving loads are stepped at two sample spacings (span/100) and interpolated between influence-line points, so vehicle results can differ from an exact analysis by a few tenths of a percent, most visibly in shear.
• HL-93 is applied per lane. Live-load distribution factors, multiple-presence factors, fatigue and the permit or state legal vehicles are not included. There are no other point loads.
• Prismatic beam: one EI for all spans, and no shear deformation.
• No support settlement, no cantilever overhangs, and no internal hinges.
• This is a line-beam analysis. It does not include composite-section staging, creep or restraint moments, and it uses one load factor for all dead load (no separate DC/DW or γ_p min/max).

References: Hibbeler, R.C., Structural Analysis, slope-deflection and stiffness methods (Ch. 11–15). AISC Steel Construction Manual, 16th ed., Table 3-23 (continuous-beam cases 28–36), used as the check values above. AASHTO LRFD Bridge Design Specifications, 9th ed., Art. 3.6.1.2.2–3.6.1.2.4 (design truck, tandem and lane load), 3.6.1.3.1 (application, including the 90% two-truck case), 3.6.1.3.2 (deflection loading), Table 3.6.2.1-1 (IM = 33%), Table 3.4.1-1 (Strength I), Art. 2.5.2.6.2 (deflection criteria).