Beam Design Formulas with Shear and Moment Diagrams — All 32 Cases, Worked and Verified

Every engineer keeps a copy of the classic beam tables within reach — reactions, shear, moment, and deflection formulas for the standard support and loading conditions, best known from the American Wood Council's free Design Aid No. 6, "Beam Design Formulas with Shear and Moment Diagrams" (AWC DA6 — Beam Design Formulas with Shear and Moment Diagrams). This page is a worked companion to those tables, with two things the tables themselves can't show:

1. One identical beam runs through all 32 cases. Same section, same span, same load intensity — only the supports and load arrangement change. The formulas stop being abstract: you watch M_max fall from 12,800 lb·ft (cantilever) to 3,200 lb·ft (simple span) to 2,133 lb·ft (fixed ends) for the same 100 plf.
2. Every closed-form value is checked against an independent calculation. The tables below print each formula's number next to the same quantity computed by StructSuite's finite-element beam engine (the analysis core of the Wood & Steel Beam module), with the difference. The agreement is locked by an automated test — if either side ever drifts, the build fails.

The drawings, layout, and worked numbers here are our own; the case set and figure numbering follow AWC DA6 so you can cross-reference the original PDF line by line. Formulas are the standard closed-form statics results found in every mechanics text.

#The setup: one beam, 32 conditions — where each input comes from

How to read each card: two sketches, then the two diagrams. The general case comes first, labelled purely with the symbols the formulas use — R₁/R₂ reaction arrows at the supports, P and w on the loads, the a/b/c/L dimensions as drafting dimension lines. Below it, the same case with this page's numbers (1,000 lb, 100 plf, 4 ft…), so symbol and value line up one over the other. Then the shear and moment diagrams as the engine actually computes them (not schematic shapes — jumps, kinks, and sign reversals are the real solution) with the governing values marked by name (M_max, M₁, V_max…), the case's full formula set, and the comparison table: closed form vs engine, with the difference. Anchors: link to any case as #fig-N.

#Simple beams

One span, a pin and a roller — every other case on this page is a variation on this one. Watch two numbers as the loading changes: M_max = wL²/8 for the full uniform load (Figure 1) is the yardstick everything else is measured against, and the deflection coefficient collapses fast as load concentrates toward the supports.

#Cantilever beams

Fix one end, free the other. The moment arm doubles and the coefficient jumps from wL²/8 to wL²/2 — four times the simple-span moment for the same load — and deflection grows almost an order of magnitude. This is why the same section that works fine on a simple span is rarely adequate as a full-length cantilever.

#Propped cantilevers (fixed at one end, supported at the other)

Adding a simple support to a cantilever (or fixing one end of a simple beam) makes the beam statically indeterminate: the fixed end picks up 5/8 of a uniform load, the moment splits into a hogging peak at the wall (wL²/8) and a smaller sagging peak (9wL²/128), and deflection drops to less than half the simple span’s.

#Overhanging beams

Push the beam past its support and the hogging moment over that support (wa²/2 or Pa) fights the sagging moment between supports. A short overhang barely dents the span moment; load only the overhang and the far reaction goes negative — the beam tries to lift off its first support.

#Fixed-fixed beams

Full end restraint is the stiffest condition here: the uniform-load moment drops to wL²/12 at the walls (and wL²/24 at midspan), and deflection is a fifth of the simple span’s. The price is real end moments the connections must actually develop — which is why true fixity is rare in wood construction.

#Two-span continuous beams

Continuity trades midspan moment for a hogging peak over the center support (wL²/8 for both spans loaded) and an uplift surprise: load ONE span and the far-end reaction goes negative. The center support collects 10/8 of wL — the reaction the post and footing below must be designed for.

#Summary of results — one beam, every condition

All values below are the engine's, for the identical inputs above (Δ_max is the largest deflection anywhere on the member, overhang tips included).

Four comparisons worth internalizing:

• Support conditions are worth more than section size. The same 100 plf produces M_max = 12,800 lb·ft as a cantilever, 3,200 lb·ft simply supported, and 2,133 lb·ft with both ends fixed — a 6.0:1 spread with identical load and length. No practical section upgrade moves capacity that far.
• Deflection is even more dramatic. Under the uniform load the cantilever moves 3.83 in where the simple span moves 0.40 in and the fixed-fixed beam only 0.080 in — a 48× range. For the midspan point load the cantilever-to-simple ratio is 16× (6.39 in vs 0.399 in).
• Continuity concentrates load on the middle support. With both 16-ft spans loaded, the center support carries R₂ = 2,000 lb — 10/8 of wL, and 25% more than the 1,600-lb total a simple span would deliver to EACH of two separate posts. The post, plate crushing, and footing under R₂ are designed by this number, not by wL/2.
• Partial loading can pull a support DOWN. Load only one span of a continuous beam (Figure 26) and the far reaction is R₃ = -100 lb — negative. The same happens loading only an overhang (Figure 19). Pattern loading isn't an academic refinement; it changes the sign of real connections.

#Practical pitfalls when using these formulas

• Units eat more designs than statics does. The deflection formulas want consistent units: with w in lb/in, L in inches, E in psi and I in in⁴, Δ comes out in inches. Feed L in feet into wL⁴/8EI and the answer is off by 12⁴ ≈ 21,000×. This page evaluates everything in inches and pounds, then reports M in lb·ft — do the same, in one direction, once.
• Superposition is legal only while everything stays linear-elastic on the SAME structure. You may add Figure 1 + Figure 7 on the same simple span; you may not add a simple-span case to a fixed-end case, and you may not superpose onto a beam whose supports have changed (a prop added later is a different structure).
• "Fixed" in the table means a real moment connection. A stud wall top plate or a couple of toe-nails is a pin, whatever the framing looks like. Use the fixed-end cases (Figures 27–32) only where the detail can actually develop M = wL²/12 — in wood construction that is rare enough to treat as the exception.
• Pattern loading changes signs, not just magnitudes. Loading one span of the two-span beam (Figure 26) turns the far reaction NEGATIVE — a connection you sized for bearing now needs a tie-down. Any time a formula's reaction can go negative under partial load, detail for it.
• The maximum is not always where you expect. For a load at "any point" (Figure 8), M_max is under the load, not midspan; the deflection maximum sits at x = √(a(a + 2b)/3) (a > b) — near, but never at, the load. Read each formula's "at x = …" qualifier before you chase a midspan number.
• These are analysis values, not designs. Nothing here contains C_D, C_M, C_L or a single allowable stress. Take R/V/M/Δ from this page, then run the member through the Wood & Steel Beam module (or your own NDS/AISC checks) for capacity.

#How these numbers are verified

Each case's closed-form values and the finite-element results above come from one shared definition file, evaluated by StructSuite's production beam engine (Hermite-element stiffness solution with consistent nodal loads — reactions are exact to machine precision for the true load positions, and shear is enveloped with support-face statics exactly as the design checks do). The automated test tests/verification/awc-da6-cases.test.ts re-runs all 32 cases on every build and fails if any quantity drifts. The check cuts both ways: a mistyped formula disagrees with the engine, and an engine regression disagrees with the formulas. The largest difference anywhere in the tables above is 0.20% — finite-element discretization at the app's own mesh, on quantities the closed forms give exactly.

#Frequently Asked Questions

#What is the formula for the maximum moment in a simply supported beam?

For a uniformly distributed load, M_max = wL²/8 at midspan (Figure 1); for a concentrated load at midspan, M_max = PL/4 (Figure 7); for a concentrated load at any point, M_max = Pab/L under the load (Figure 8). On this page's 16-ft reference beam those give 3,200 lb·ft, 4,000 lb·ft, and 3,000 lb·ft respectively — each verified against the finite-element engine to within a small fraction of a percent.

#What is the deflection formula for a cantilever beam?

Δ_max = wL⁴/8EI at the free end for a uniform load (Figure 12), and Δ_max = PL³/3EI for a point load at the tip (Figure 13). Note the contrast with the simple span (5wL⁴/384EI): for the same w, L, and EI the cantilever deflects 48/5 ≈ 9.6 times more — on this page's beam, 3.83 in versus 0.40 in.

#Are these formulas ASD or LRFD?

Neither — they are statics. Reactions, shear, moment, and elastic deflection depend only on geometry, stiffness, and load, not on the design method. ASD vs LRFD enters afterwards, when factored (or unfactored) demands are compared to member capacities. That is also why one engine can serve both the NDS (ASD) wood checks and the AISC (LRFD) steel checks in the beam module: the analysis layer this page verifies is shared.

#Why do the engine results differ from the formulas at all?

Because the engine is a finite-element solution: the beam is meshed into ~2-inch Hermite elements, moment is recovered element by element, and deflection is interpolated between nodes. Reactions come out exact to machine precision; moment and deflection carry discretization differences that stay in the hundredths of a percent at the app's mesh — the largest anywhere on this page is 0.20%. An agreement much tighter than engineering inputs are ever known to (loads are rarely known to 1%) — the point of the comparison is independence, not decimals.

#Where do I get the original AWC Design Aid No. 6?

It's a free PDF from the American Wood Council: AWC DA6 — Beam Design Formulas with Shear and Moment Diagrams. Keep it in the truck; keep this page open next to your model. The figure numbers here match it one for one, so a reviewer can check either source against the other — and the resources directory collects it alongside the NDS, SDPWS, and the other documents a wood-design desk actually uses.