Continuous beam analysis
A linear elastic analysis of a straight beam of any material: 1 to 6 spans with optional overhangs, pins or fixed supports, line loads, point loads and concentrated moments per load type, the ASCE 7-22 load combinations or one load set, and pattern live load. It gives the shear, moment and deflection envelopes and the reactions; size the member in its material calculation (steel beam, wood beam, concrete beam) with these forces.
Inputs of the beam analysis #
- Stiffness: an AISC shape (Shapes Database v16.0, Ix about the strong axis) or E and I entered. E = 29,000 ksi is prefilled (steel, AISC 360-22); enter the modulus of your material (wood: the NDS E or Emin you choose; concrete: Ec of ACI 318-19 19.2.2) and the gross, effective or cracked I you judge appropriate. EI is constant along the beam. With a shape, its self-weight can be added as dead load.
- Spans and supports: the left support and each span with the support at its right end; a support is a pin (roller), a fixed end, or none (a free end, for example a cantilever: left fixed, one span with no right support). The left and right overhangs are free ends beyond the outer supports (0 = none).
- Loads: line loads (uniform, partial or linearly varying), point loads and concentrated moments, each with its load type (D, L, Lr, S, R, W, E) and where it acts: a span, an overhang, or the beam measured from its left end. From and To are measured from the start of that span or overhang; From = To = 0 loads all of it. Loads act downward when positive; moments are clockwise positive.
- Load combinations: the ASCE 7-22 2.3.1 (LRFD) or 2.4.1 (ASD) combinations of the load types present, chosen by the method switch (W and E in both directions), or one load set: the unfactored sum of the loads entered.
- Pattern live load: ASCE 7-22 4.3.3 partial loading. When on, the live load (L) is placed on all segments (spans and overhangs), on each segment alone, on alternate segments (1st, 3rd, ... and 2nd, 4th, ...) and on each pair of adjacent segments, and every pattern is enveloped with the other loads at full value.
- Deflection limits: span / n for live load and for total load; IBC 2021 Table 1604.3 floor values (360, 240) are prefilled and 0 turns a check off. For an overhang or cantilever the span is twice its length.
Results and checks of the beam analysis #
| Item | Basis |
|---|---|
| Analysis basis | Euler-Bernoulli beam elements with consistent load vectors (exact for these load shapes); first-order; no shear deformation |
| Pattern live load | ASCE 7-22 4.3.3; a note states the patterns used, or that pattern loading is off |
| Reactions and envelope forces | Largest and smallest reaction at each support; largest positive and negative moment and largest shear, with their location and combination |
| Live load deflection | Each of L, Lr, S and R alone (service, with the live load patterns), against span / n |
| Total load deflection | The ASD gravity combinations (or the load set), service, against span / n |
The drawings (on screen, in the report and as a PDF / DXF drawing sheet) show the beam elevation (spans and overhangs, pin / roller and fixed supports, the loads by load type with their values, span and load-position dimensions, the shape's mark; with E and I entered the member depth is schematic) and, at the elevation's scale and aligned under it, the envelopes of shear, moment (drawn on the tension side) and service deflection over every case analysed, with the largest values of each span labelled. Moments are positive sagging; shears are positive with the left part pushed up. The sheet notes give the stiffness, the analysis basis and the combinations; member design, connections and bearing are not part of the calculation. A long beam on a letter sheet is drawn at 1/16" = 1'-0" at the smallest; a beam much over 65 ft reads best on an 11 × 17 sheet.
Limitations #
- Constant EI along the beam; no shear deformation, no second-order effects, no support settlement or springs.
- No member strength checks: the analysis gives forces and deflections only.
- Pattern loading applies to the live load L only (not to Lr, S or R), with the patterns listed above rather than every possible arrangement.
- Ev, the overstrength factor Ω0 and the 0.5L option of ASCE 7-22 2.3.1 are not modeled.
Verification examples #
Closed-form beam formulas (hand calculation), one unfactored load set, E = 29,000 ksi, I = 1,000 in⁴, w = 1 kip/ft, L = 20 ft unless stated. The closed forms below are exact and the app matches them within 0.1 %. The published beam cases of the AISC Steel Construction Manual, 16th Ed., Table 3-22 and the continuous-beam coefficients of Table 3-21c follow.
| Example | What it checks | Closed form | ILX Calcs |
|---|---|---|---|
| BA-1 | Simple span | wL²/8 = 50.0 kip-ft; 5wL⁴/384EI = 0.1241 in | 50.0; 0.1241 |
| BA-2 | Fixed-fixed beam | wL²/12 = 33.33, wL²/24 = 16.67 kip-ft; wL⁴/384EI = 0.0248 in | 33.33, 16.67; 0.0248 |
| BA-3 | Propped cantilever | wL²/8 = 50.0 kip-ft; 3wL/8 = 7.5 kips | 50.0; 7.5 |
| BA-4 | Two equal spans | wL²/8 = 50.0 kip-ft; 1.25wL = 25.0 kips; 9wL²/128 = 28.13 kip-ft | 50.0; 25.0; 28.13 |
| BA-5 | Two spans, pattern live load | 49wL²/512 = 38.28 kip-ft; R = 7wL/16 = 8.75 and −wL/16 = −1.25 kips | 38.28; 8.75, −1.25 |
| BA-6 | Cantilever, tip load 10 kips, 10 ft | PL = 100 kip-ft; PL³/3EI = 0.1986 in | 100.0; 0.1986 |
| BA-7 | 20 ft span with a 5 ft overhang | −wa²/2 = −12.5 kip-ft; R = 15.625, 9.375 kips | −12.5; 15.625, 9.375 |
| BA-8 | End moment 10 kip-ft (sign convention) | R = ±M0/L = ±0.5 kips; M = +10 kip-ft | ±0.5; 10.0 |
Published: AISC Steel Construction Manual, 16th Ed., Table 3-22 "Shears, Moments, and Deflections" (P = 10 kips, w = 1 kip/ft, l = 20 ft per span, same E and I). The continuous-beam cases print coefficients to three significant figures (deflections to two), which explains the small differences.
| Example | Table 3-22 case | Published | ILX Calcs |
|---|---|---|---|
| AISC-7 | 7, simple beam, load at center (p. 3-217) | M = Pl/4 = 50.0 kip-ft; Δ = Pl³/48EI = 0.0993 in | 50.0; 0.0993 |
| AISC-8 | 8, load at any point, a = 15 ft (p. 3-217) | M = Pab/l = 37.5 kip-ft; R = 2.5 / 7.5 kips; Δmax = 0.0694 in | 37.5; 2.5 / 7.5; 0.0694 |
| AISC-9 | 9, two symmetric loads, a = 5 ft (p. 3-217) | M = Pa = 50.0 kip-ft; Δ = 0.1366 in | 50.0; 0.1366 |
| AISC-16 | 16, fixed both ends, load at center (p. 3-220) | ±Pl/8 = 25.0 kip-ft; Δ = Pl³/192EI = 0.0248 in | ±25.0; 0.0248 |
| AISC-17 | 17, fixed both ends, a = 5 ft (p. 3-220) | M1 = −28.13, M2 = −9.38, Ma = 14.06 kip-ft; R1 = 8.44 kips | same |
| AISC-37 | 37, three spans, one end span unloaded (p. 3-229) | R = 7.66 / 24.0 / 9.0 / −0.66 kips; MB = −46.8, MC = −13.32, M+ = 29.4 kip-ft; Δ = 0.0563 in | 7.67 / 24.0 / 9.0 / −0.67; −46.67, −13.33, 29.39; 0.0563 |
| AISC-39 | 39, three spans, all loaded (p. 3-229) | R = 8.0 / 22.0 kips; MB = −40.0, M+ = 32.0 / 10.0 kip-ft; Δ = 0.0658 in | 8.0 / 22.0; −40.0, 32.0 / 10.0; 0.0656 |
| AISC-41 | 41, four spans, first and third loaded (p. 3-230) | R = 8.92 / 11.44 / 9.28 / −1.08 kips; MB = −21.44, MC = −14.28, M+ = 39.84 kip-ft; Δ = 0.0925 in | 8.93 / 11.43 / 9.29 / −1.07; −21.43, −14.29, 39.86; 0.0924 |
| AISC-42 | 42, four spans, all loaded (p. 3-230) | R = 7.86 / 22.8 / 18.56 kips; MB = −42.8, MC = −28.56, M+ = 30.88 / 14.56 kip-ft; Δ = 0.0620 in | 7.86 / 22.86 / 18.57; −42.86, −28.57, 30.86 / 14.54; 0.0616 |
AISC Steel Construction Manual, 16th Ed., Table 3-21c (continuous beams, equal spans, equally loaded, p. 3-214), spans of 20 ft, w = 1 kip/ft or P = 10 kips per load point; the coefficients are printed to two or three digits:
| Case | Published coefficients | Published | ILX Calcs |
|---|---|---|---|
| Three spans, uniform | −0.10 wL², +0.08 wL², R = 0.4 / 1.1 wL | −40.0, 32.0 kip-ft; 8.0, 22.0 kips | −40.0, 32.0; 8.0, 22.0 |
| Four spans, uniform | −0.107 / −0.071 wL², +0.077 wL², R = 11/28, 26/28 wL | −42.8 / −28.4, 30.8 kip-ft; 7.86, 18.57 kips | −42.86 / −28.57, 30.86; 7.86, 18.57 |
| Two spans, load at each center | −0.188 PL, +0.156 PL, R = 0.31 / 1.38 P | −37.6, 31.2 kip-ft; 3.1, 13.8 kips | −37.5, 31.25; 3.125, 13.75 |
| Three spans, load at each center | −0.15 PL, +0.175 PL, R = 0.35 / 1.15 P | −30.0, 35.0 kip-ft; 3.5, 11.5 kips | −30.0, 35.0; 3.5, 11.5 |
| Two spans, loads at the third points | −0.333 PL, +0.222 PL, R = 0.67 / 2.66 P | −66.6, 44.4 kip-ft; 6.7, 26.6 kips | −66.67, 44.44; 6.67, 26.67 |