Cantilever Beam (End Load)

stressmass-cost

Verified build 5 relations · 4 identities proven · 1 modeling step · 3 parity samples

Fix a beam into a wall, hang a load off the free end: this is the cantilever — balcony, diving board, wing spar, atomic-force-microscope probe, bookshelf bracket. It is the simplest structure that exhibits the full chain every structures course is built on: load → internal moment → stress → deflection → margin against failure.

The reason it’s a reference THING here is the material cascade. Pick a material in the widget and three independent properties fan out through different relations:

Young’s modulus $E$ drives the tip deflection $\delta = \dfrac{PL^3}{3EI}$ — and only the deflection.
Yield strength $\sigma_y$ drives the safety factor $\mathrm{SF} = \sigma_y/\sigma$ — and only the safety factor.
Density $\rho$ drives the mass — and only the mass.

Try it: switch from A36 structural steel to Ti-6Al-4V titanium. The “stronger, fancier” material deflects almost twice as much (E falls from 200 to 110 GPa) while the safety factor more than triples (σ_y jumps from 250 to 869 MPa) and the mass nearly halves. Stiffness is not strength. If you want less deflection, titanium is the wrong upgrade — change the section instead: stress falls with $h^2$ and deflection with $h^3$ , which is why I-beams put material far from the neutral axis.

The materials page plots this independence for the whole database, Ashby-style.

Try it

Material

T3, bare flat sheet 0.010-0.128 in. thick, AMS 4037 / AMS-QQ-A-250/4 (MIL-HDBK-5J Table 3.2.3.0(b1), p. 3-71)

Bound properties of 2024-T3 aluminum sheet (bare)
E	10.5 Msi	typical	mil-hdbk-5j
sigma_y	47 ksi	design min.	mil-hdbk-5j
rho	0.1 lb/inch**3	typical	mil-hdbk-5j

3 materials in the database are not listed here: no published value in our cited sources for every property this THING needs.

Materials modeled here: 2024-T3 aluminum sheet (bare) 304 stainless steel 6061-T6 aluminum 7075-T6 aluminum AISI 1045 medium-carbon steel AISI 4340 low-alloy steel (Ni-Cr-Mo) ASTM A36 structural steel (hot-rolled) C26000 Cartridge Brass (70/30) Nylon 6/6 (PA66), unfilled Ti-6Al-4V

Governing relations

I = \frac{b h^3}{12}

Assumes: solid rectangular cross-section, bending about the strong axis

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).

\delta = \frac{P L^3}{3 E I}

Assumes: linear elastic material; small deflections; Euler–Bernoulli kinematics (shear deflection neglected); load applied exactly at the free end; fixed support is perfectly rigid · Valid while: Tip deflection exceeds L/10 — the small-deflection assumption is breaking down, and the true behavior is stiffer than this formula predicts. This is a short, deep beam (L < 10h): shear deflection, neglected by Euler–Bernoulli theory, is no longer small.

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).

\sigma = \frac{M_{max}\,c}{I} = \frac{P L (h/2)}{I}

Assumes: maximum bending stress at the wall, at the outer fiber; stress concentration at the support ignored · Valid while: Bending stress exceeds the yield strength — the material has yielded and every linear-elastic number here stops being the truth.

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).

\mathrm{SF} = \frac{\sigma_y}{\sigma}

Assumes: safety factor against first yield at the outer fiber (not against collapse)

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).

m = \rho\, b\, h\, L

Assumes: prismatic beam, uniform density

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

M_{x} = P \left(L - x\right)

1. Cut the beam at a distance x from the wall and balance the free body beyond the cut: the internal bending moment must carry the load P acting on the remaining arm (L − x). It is largest at the wall, M_max = PL. — statics: method of sections modeling step

\delta = \frac{L^{3} P}{3 E I}

2. Euler–Bernoulli theory says curvature is proportional to moment, EI·v″ = M(x). Integrating twice with the clamped-end conditions v(0) = v′(0) = 0 and evaluating at the tip gives the classic result. (The build's test suite re-derives this integral with a computer algebra system as an independent physics check.) — integrate the moment–curvature relation

\sigma = \frac{L P c}{I}

3. Bending stress varies linearly through the depth, peaking at the outer fiber, a distance c = h/2 from the neutral axis — the flexure formula σ = Mc/I evaluated where the moment is largest, M(0) = PL at the wall. — flexure formula

\sigma = \frac{6 L P}{b h^{2}}

4. Substitute I = bh³/12 and c = h/2. Notice what this says: doubling the height h cuts the stress by 4× — section shape is a stronger lever than material strength. — substitute the rectangular section

SF = \frac{\sigma_{y}}{\sigma}

5. The safety factor compares the material's yield strength to the working stress. E is nowhere in this line — and σ_y is nowhere in the deflection line. Stiffness and strength are independent axes; that's the whole lesson of this THING. — definition of safety factor against yield

How it fails

The widget’s safety factor guards one failure mode — first yield at the outer fiber, at the wall. Real cantilevers have several more exits:

Yielding at the support. The moment peaks at the wall, so that’s where it lets go first. The sharp corner where beam meets support concentrates stress beyond the nominal $Mc/I$ — real designs add fillets; fatigue cracks start exactly there.
Fatigue. A cantilever that vibrates (and they all do — it’s a tuning fork) accumulates bending cycles at the wall. Fatigue strength is far below static yield for most metals, and the relations on this page say nothing about it.
Lateral-torsional buckling. A tall, narrow section ( $h \gg b$ ) loaded on its strong axis can twist and flop sideways before it yields — the failure is geometric, governed by stiffness, not strength.
Excessive deflection as “failure”. A machine spindle or optical mount can be ruined by a deflection a thousand times smaller than what yields it. Serviceability limits often govern.
Shear at the support for very short, deep beams — the regime the L > 10h validity warning flags, where Euler–Bernoulli theory itself stops applying.
Brittle materials break, not yield. For gray cast iron (in the materials list) there is no yield plateau; rate the section against the ultimate strength instead, with a generous factor — and note its tensile and compressive strengths differ by ~3.6×.

Fixed-Fixed Beam (UDL)
A beam built rigidly into a wall at BOTH ends under a uniform load. Two equilibrium equations, four unknown reactions — indeterminate to the second degree — so two compatibility conditions (zero slope and zero deflection at a released end) close the system, and the build solves the coupled 4×4 group exactly. The fixing moment at each wall governs, and none of the reactions cares about the material.
- stress
- mass-cost
Propped Cantilever (UDL)
A cantilever with a prop under its free end: one redundant support turns a determinate beam into a statically indeterminate one. Equilibrium alone cannot find the three reactions — compatibility (the prop deflects to zero) supplies the missing equation, and the build solves the coupled 3×3 system exactly.
- stress
- mass-cost
Simply Supported Beam (Center Load + UDL)
The floor joist under you right now: pinned at both ends, carrying a point load and a distributed load at once. Because the governing equation is linear, the two answers simply add — superposition, the single most-used trick in structural analysis, made visible.
- stress
- mass-cost
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
Push uniform pressure on a flat circular plate — a tank head, a porthole, a valve cover — and how hard it deflects and where it cracks depend entirely on the RIM. Bolt it down (clamped) and it is stiff and hottest at the edge; rest it on a ring (simply supported) and it sags four times as far and is hottest at the center. This is the page where Poisson's ratio moves a STRESS: the simply-supported stress carries ν, the clamped-edge stress carries no material property at all.
- stress
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
Bend a bar that is already curved and the neutral axis walks off the centroid, toward the inside of the curve — the inner fibers run hotter than the straight-beam Mc/I ever predicts. This is the Winkler formula behind a crane hook, a C-clamp, and a press frame: σ = M·c/(A·e·r), the tiny eccentricity e = r_c − r_n doing all the work, plus the direct P/A a hook's load adds on top.
- stress
Transverse Shear in Beams (τ = VQ/Ib, Shear Flow, Fastener Spacing)
A beam does not only bend — the shear force V drags its layers past one another, and that longitudinal shear is what a built-up beam's nails or bolts actually carry. The stress is a parabola (peak 3V/2A at the neutral axis, zero at the surfaces), and the shear flow q = VQ/I sets the fastener spacing. Statics and geometry only: no stiffness enters at all.
- stress

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- delta L
Impact Loading (Falling Mass, Energy Method)
- delta b
- delta d
- delta h
- delta L
- m_beam m
Symmetric Two-Bar Truss
- delta d
- delta L
Thermal Assembly (Two-Segment Bar Between Rigid Walls)
- delta L_1
- delta L_2
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
- delta a
- delta t
- sigma q
- sigma sigma_allow
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- delta r_i
- delta r_o
- delta w
Fixed-Fixed Beam (UDL)
- delta b
- delta h
- delta L
Propped Cantilever (UDL)
- delta b
- delta h
- delta L

+ 24 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — §5.5 (flexure formula), §9.3 (cantilever deflections by integration), App. E (section properties).