PHYSICSBeam under load — bending moment, shear, deflection. The structural classics, instrumented.
─ The Static Beam
v0.3 · OPA Engineering Suite · drag the load. drag the piers. drop a UDL. watch the diagrams.
College X · CIVL 220 / STRC 220
Simply Supported Beam
Two supports · pin + roller · M=0 at supports
Beam: simply supported · L = 30 ft · EI = 1.0e7 k·in²Load: point P = 5.0 kip @ x = 0.50 L
→ drag the brass arrow to move the load · drag the brass-handled piers in continuous mode
Load type
Position
0.50 (mid)
Point P
5.0 kip
UDL w
1.00 k/ft
UDL preset
Span L
30 ft
EI stiffness
1.0e7 k·in²
Spans
Reactions—kip at supports
Pier moments—k·ft at interior
Max |V|—
Max +M—
Max −M—
δₘₐ₧ deflection—downward
L / δ ratio—deflection check
Code check—IBC service limit
About this lab
The Static Beam — v0.3 (merged)
A browser structural lab. Three classical configurations — simply supported, continuous over 2–5 spans, cantilever — each accepting two load types — a draggable point load or a uniform distributed load with snow / crowd / wind / dead presets. Shear, moment, and deflection diagrams redraw live as you drag the load, drag the piers, or change a slider.
Why this exists, and how it got here
This file is the merger of two earlier labs that overlapped: The Live Beam v0.2 (point-load only, normalized units, draggable load + draggable piers) and an interim Static Beam v0.1 (UDL + presets + dimensional units + code check). The two labs were teaching the same physics through different UI strengths. The v0.3 merge keeps the live drag-the-load + drag-the-piers + n-span Clapeyron chassis from Live Beam, layered with the UDL + presets + dimensional units + L/360 check from the interim build.
The name "Live Beam" was always about live UI (drag and the diagrams redraw), not live loads. That bothered Travis enough to ask for the true dynamic counterpart — the one that's actually about loads that move (crowd at 2 Hz, footfall, resonance) — and that one now lives at /labs/live-beam.html.
The three tabs in order
Simply Supported is the textbook reference. M = 0 at both supports; max moment under or near the load. Build the intuition here first.
Continuous (n spans) is the freshman surprise. A single load on one span lifts adjacent spans. Negative moment forms over interior piers. The action isn't where the load is — it's at the piers. Try 2/3/4/5 spans and drag the piers to see how much steel a smart layout saves.
Cantilever is the punchline. Length is the enemy of the cantilever. Tip deflection grows as the cube of the lever arm. The same equation governs the concert hall balcony, the stadium roof, and the wing.
Two load types in each tab
Point load — drag the brass arrow along the beam, or use the sliders. Best for building intuition about where the moment peak tracks.
Distributed (UDL) — the load that lives on every floor of every building under its own dead weight, plus everything sitting on top. Snow, crowd, wind on a wall, the floor system itself. Pick a preset (snow 20 psf, crowd 100 psf, wind 30 psf, dead 80 psf) or use the magnitude slider.
The hidden lesson
Engineering school spends a lot of time on stress — "does the beam break?" But the working-engineer reality is that deflection governs almost every real design. A floor strong enough to hold the load but sagging enough that the tenant notices is, to the tenant, broken. The L/360 service limit isn't safety — it's credibility. Read the L/δ ratio in the readouts. That's the number the user actually cares about.
What isn't here
Small-deflection Euler-Bernoulli beam theory. The lab doesn't model: large-deflection P-Δ effects, plastic hinging, shear deformation (Timoshenko correction), torsion, or moving loads (those live at the Live Beam, the dynamic one). For real design work use ASCE 7 (loads), AISC 360 (steel), ACI 318 (concrete), or AASHTO LRFD (bridges). Deflections are exaggerated on screen by ~10-100× for visibility; the readout numbers are the true dimensional values.
Suite context
Lab of the OPA Engineering Suite · dual of the Live Beam (dynamics) · sibling to The Horseshoe Vortex and The Wing. Filed under College X (Engineering), ELUSK Building 10. Cross-listed CIVL 220 (Statics & Beam Analysis), CIVL 320 (Indeterminate Structures · for the continuous tab), STRC 220 (Beam Design).
Math under the hood
What the lab is solving
Simply supported · point load P at position a
Rₐ = P(L−a)/L R₋ = Pa/L
Mₘₐ₧ = Pa(L−a)/L at x = a
δₘₐ₧ (center load) = PL³/(48EI)
Simply supported · uniform w
Rₐ = R₋ = wL/2
Mₘₐ₧ = wL²/8 at midspan
δₘₐ₧ = 5wL⁴/(384EI) at midspan
Continuous beam · n equal-ish spans · Clapeyron's three-moment
The lab builds an (n−1)×(n−1) tridiagonal system for the n−1 interior pier moments, with end moments M₀ = Mₙ = 0 (simple supports at each end). Solves via the Thomas algorithm in O(n) time. Same engine for point load and UDL; only the RHS (6Aā/L term) changes:
Point load P at distance a from left of a span of length L: 6Aā/L (from right) = P·a·(L²−a²)/L · (from left) = P·b·(L²−b²)/L where b = L−a.
UDL w over a span of length L: 6Aā/L = wL³/4 (symmetric, same from both ends).
Once interior moments are known, reactions are computed span-by-span from statics and combined at shared piers. V(x) and M(x) are evaluated by summing reaction contributions and the load to the left of x.
Cantilever · point load P at distance a from wall
Rₘₙₜ = P, Mₘₙₜ = −P·a
δₘₐ₧ (tip) = PL³/(3EI) for a = L
The cantilever deflects 16× as much as a simply-supported beam of the same length under the same center load (PL³/3EI vs PL³/48EI). That's the cost of having only one support.
Cantilever · uniform w
Mₘₐ₧ = wL²/2 at the wall
δₘₐ₧ = wL⁴/(8EI) at the tip
Service deflection limits (IBC / AISC)
L/360 — live load on floors with plaster ceiling (working-engineer default)
Hibbeler, Structural Analysis, 10th ed., Ch. 8–9 (statically determinate beams + force method for continuous). Clapeyron's three-moment equation: Hibbeler §10.3. AISC Steel Construction Manual, 15th ed., beam tables Part 3. Roark's Formulas for Stress and Strain, 8th ed., Ch. 8 (closed-form deflections). Numerical: Thomas algorithm for tridiagonal systems, Burden & Faires, Numerical Analysis.
Where the preset loads come from
Real loads, real codes
Each UDL preset is a working number, not the worst-case ASCE 7 governing combination. They're meant for an undergrad to feel the order of magnitude. The lab assumes a 10-ft tributary width when it converts psf to k/ft (so 100 psf × 10 ft = 1.0 k/ft).
Snow · 20 psf
Typical ground snow load for the U.S. South / mid-Atlantic per ASCE 7-22 Fig. 7.2-1. Buffalo or northern Michigan would use 50–80 psf; Nashville, 10 psf. Roof snow on a flat roof is roughly 70% of ground snow.
Crowd · 100 psf
ASCE 7-22 Table 4.3-1 live load: "assembly areas with fixed seats" is 75 psf; "lobbies and first-floor corridors" is 100 psf. 100 psf is the working concert/event live load. This is the load that talks to the Live Beam — same 100 psf, but moving in sync at 2 Hz, is what brought down the Hyatt Regency walkway in 1981.
Wind · 30 psf
Roughly the design wind pressure on a wall surface in a 110-mph wind zone (Florida coast, Gulf Coast). Hurricane Helene at Asheville peaked ~85 mph sustained — the design code already covered it. The structural failures there were soil failures, not beam failures.
Dead load · 80 psf
A typical built-up dead load for an office floor: 4-inch concrete slab on metal deck (50 psf) + steel framing + ceiling + MEP allowance + partition load = ~80 psf. Dead load is on the beam every day forever; the beam is sized for dead + live with appropriate combinations.
What about the 6-in cobble in a snowstorm?
The lab tops out at 10 k/ft (= 1,000 psf at 10-ft tributary width — ten times a normal office floor). Useful for: stadium seating + crowd moving in sync, pool-on-roof scenarios, ice load doubled, or "engineer's nightmare" testing. If your real design is over 10 k/ft you're either in a heavy-industrial building or you've made a units mistake.