The catalog

Axial, Thermal & Impact

• Composite Bar (Core + Sleeve)

  A solid core inside a concentric sleeve, bonded between rigid end plates and pushed by a centric axial load. The two materials must stretch together, so the load splits in proportion to each member's axial stiffness A·E — and the build solves that coupled 2×2 share exactly. Swap the sleeve's metal and watch the load migrate to the stiffer member.

  • stress
  • mass-cost
• Impact Loading (Falling Mass, Energy Method)

  Drop a mass onto an elastic member and the peak stress is not W/A — it is n times the static stress, where the impact factor n = 1 + √(1 + 2h/δ_st). A suddenly-applied load (h = 0) already doubles the stress; a real drop multiplies it many times over. Stiffer members take HIGHER impact stress, because a smaller static deflection means a larger n.

  • dynamics
  • stress
• Symmetric Two-Bar Truss

  Two identical pin-jointed bars share a load at a common joint — the member force is P/(2cos α), which blows up as the truss flattens toward horizontal. Statically determinate by construction: equilibrium alone fixes the forces, no compatibility needed. Flatten it and watch the forces (and the joint deflection) diverge; in compression each bar must also clear Euler buckling.

  • statics
  • stress
  • stability
  • mass-cost
• Thermal Assembly (Two-Segment Bar Between Rigid Walls)

  Two different materials joined end to end and pinned between rigid walls, then heated or cooled uniformly. Neither segment can expand, so an internal force builds up — solved exactly from the coupled equilibrium-and-compatibility pair. Swap a segment's metal and watch the thermal stress move, because a stiff, high-expansion metal pushes hardest.

  • stress

Beams & Plates

• Cantilever Beam (End Load)

  A beam fixed at one end, loaded at the other — the fruit-fly of structures. One widget shows why stiffness (E) and strength (σ_y) are independent axes: swap steel for titanium and deflection goes UP while the safety factor also goes up.

  • 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
• 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
• 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

Torsion & Combined Loading

• Fixed-Fixed Torsion Shaft (Interior Torque)

  A solid circular shaft built into a wall at BOTH ends, with a torque applied at an interior station. Equilibrium gives one equation for the two wall reaction torques; the missing equation is compatibility — the twist at the load point is single-valued — and the build solves the coupled 2×2 system exactly. The larger reaction lands on the SHORTER segment, and the material cancels out.

  • stress
  • mass-cost
• Rectangular Shaft in Torsion (Saint-Venant)

  Twist a solid rectangular bar and the shear stress does something the round shaft never does: it peaks at the MIDDLE of the long side and drops to exactly zero at the corners. Two cited coefficients c1, c2 — functions only of the side ratio a/b — set the peak stress and the twist, and an equal-area round shaft beats it on both counts. Why square shafts are a bad deal.

  • stress
  • mass-cost
• Shaft in Torsion (Solid, Circular)

  The power-transmission workhorse: twist a solid circular bar and shear stress winds around it. Three material properties drive three different outputs — stiffness (G) sets the twist, strength (σ_y) sets the margin, and the stress itself doesn't care what the shaft is made of at all.

  • stress
  • torque-power
  • mass-cost
• Shaft under Combined Bending + Torsion

  Real shafts never get to choose: the belt that twists them also bends them. Bending and torsion land on the same surface element, Mohr's circle finds the worst plane, and two failure criteria — Tresca and von Mises — disagree by up to 15 % about how bad it is.

  • stress
  • mass-cost
• Thin-Walled Tube in Torsion (Bredt)

  Why driveshafts, airframes, and bike frames are closed tubes: in torsion, what matters is not how much metal you have but how much AREA the wall encloses. Bredt's shear flow makes any closed section solvable with two knobs — and the isoperimetric inequality polices which sections can exist at all.

  • stress
  • mass-cost

Columns & Stability

• Eccentric Column (Secant Formula)

  Load a column even slightly off-axis and the clean buckling story dissolves: it bows from the first newton, stress grows faster than load, and the Euler limit survives only as the asymptote the deflection chases. Because nothing here is linear, the safety factor must be taken on the LOAD — the page solves that transcendental equation live, by bracketed root-finding.

  • stability
  • stress
  • mass-cost
• Euler Column (Buckling)

  Push down on a slender strut and it fails sideways — at a load set entirely by stiffness and geometry. Yield strength is nowhere in Euler's formula: the "strong" steel column buckles at exactly the same load as the mild one. Strength only decides where the formula stops being true.

  • stability
  • stress
  • mass-cost

Pressure & Rotating Bodies

• Compound Cylinder (Shrink Fit)

  Where the monobloc wall gave up: shrink a jacket over a liner and the interference squeezes the bore into hoop compression before the pressure ever arrives. Service tension must spend that compression first — and at the balanced fit with the interface at √(r_i·r_o), the elastic pressure ceiling approaches DOUBLE the one no solid wall could pass.

  • stress
  • mass-cost
• Rotating Disk with a Central Bore

  Drill the smallest possible shaft hole through a spinning disk and the peak stress exactly doubles — not "roughly increases": doubles, in the limit of a vanishing bore. The solid flywheel's optimistic numbers meet the hole every real rotor needs.

  • stress
  • energy-storage
  • mass-cost
• Thick-Walled Cylinder (Lamé)

  Where the thin-wall pressure vessel hands off: the exact elastic field for any wall thickness. The stress piles up at the bore and decays as 1/r² — and the bore shear always exceeds the pressure, so past p = σ_y/2 no amount of thickness can contain it elastically.

  • stress
  • mass-cost
• Thin-Walled Pressure Vessel (Cylinder)

  A pressurized tube with closed ends — scuba tank, boiler, rocket stage. The hoop stress is exactly twice the longitudinal stress, which is why sausages split lengthwise; and because the relations are undirected, the same widget runs backwards as a design tool: pick a safety factor and the wall thickness falls out.

  • stress
  • mass-cost

Gears & Drives

• Belt Drive (Flat Belt / Capstan)

  Friction compounding like interest: every degree of wrap multiplies the tension a belt can hold, e^μθ in total — the same exponential that lets a sailor check a ship with two turns of rope. At speed, centrifugal relief steals tension back, so every belt has a power ceiling.

  • torque-power
• Planetary (Epicyclic) Gearset

  Three coaxial members — sun, ring, planet carrier — share one gear mesh law. With two degrees of freedom, it has no single "ratio": fix a different member and the same hardware becomes a different transmission.

  • kinematics
  • torque-power
• Power Screw (Square Thread)

  An inclined plane wrapped around a cylinder: torque in, brute linear force out. Friction is an honest knob here — it decides the torque, eats the efficiency, and (the good part) decides whether the load politely stays put or back-drives the screw when you let go.

  • torque-power
• Spur Gear Pair (Lewis Bending)

  Two meshing spur gears carry power through one tangential tooth force. The Lewis equation turns that force into a root-bending stress, and a cited form-factor table sets how much a tooth of N teeth can take. Same load, same module — yet the pinion, with fewer teeth, always works harder.

  • stress
  • torque-power

Shafts & Bearings

• Rolling-Contact Bearing Life (Load–Life + Weibull Reliability)

  A bearing's catalog rating C₁₀ is not a strength — it is the load at which 90% of a population survives one million revolutions. Halve the load and the fatigue life jumps eightfold (ball) or more (roller); ask for higher reliability than 90% and the life you can count on drops sharply. This page turns the two catalog numbers into a life in hours, and the reliability you demand into the life you actually get.

  • dynamics
  • reliability
• Shaft Critical Speed (Rayleigh + Dunkerley)

  Spin a shaft fast enough and it whips: at its critical speed the rotor whirls in resonance with its own static sag, ω_c = √(g/δ_st). Gravity sets the sag but cancels out of the answer — the critical speed is pure stiffness over inertia. Dunkerley's estimate folds in the shaft's own mass and is provably never higher than Rayleigh's.

  • dynamics
  • stress
• Stepped Shaft — Shoulder-Fillet Stress Concentration

  A shoulder where a shaft steps from a large diameter D to a small diameter d concentrates stress in the fillet of radius r. The peak stress is the nominal stress times a geometric factor K_t read from a cited chart — pure geometry: the material changes the safety factor but never K_t.

  • stress

Joints, Springs & Clutches

• Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)

  The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.

  • torque-power
• Bolted Joint with Gasket (External Tensile Load)

  A preloaded bolt clamping a gasketed joint, then pulled by an external tensile load. The bolt and the members act as two springs in parallel, so the external load does NOT all go to the bolt — it splits by stiffness. The build solves the coupled bolt/member force system exactly and refuses the moment the members go slack.

  • stress
• Helical Compression Spring

  A torsion bar wound into a package: push on the coil and the wire twists. G sets the rate, σ_y sets the margin, and the geometry trades them against three envelopes — coil bind, buckling, and a spring index you can actually wind.

  • stiffness
  • stress
  • mass-cost

• DC Motor (Permanent Magnet)

  The machine that turns current into torque — and its whole personality is one straight line. At fixed voltage a PM DC motor trades speed for torque along T = T_stall(1 − ω/ω₀): two datasheet numbers pin every operating point, and the peak power hides at half the no-load speed.

  • torque-power
  • kinematics
• Flywheel (Solid Rotating Disk)

  The machine that stores work as spin — and loads itself doing it. Centrifugal self-loading grows with ρω²R², so the energy a flywheel can hold per kilogram is capped not by its size but by one material index: strength over density.

  • energy-storage
  • stress
  • mass-cost
• Four-Bar Linkage (Position)

  Four pinned links — ground, crank, coupler, rocker — and the oldest mechanism in the book. Spin the crank and the rocker answers through pure geometry. Every position has TWO valid assemblies (open and crossed): the first THING in the catalog where one input has two honest answers, and the widget lets you pick the circuit.

  • kinematics
• Slider-Crank (Exact Kinematics and Gas Torque)

  Crank, connecting rod, piston — the four-bar linkage with one pivot pushed to infinity, and the heart of every reciprocating engine and pump. Spin the crank at a fixed speed and the piston's position, velocity, and acceleration follow by pure geometry and two derivatives; push on the piston with a gas force and the connecting-rod obliquity turns it into a crank torque that swings from zero to a peak and back every revolution. The classic two-term r/l approximation rides alongside the exact form so you can watch it drift.

  • kinematics
  • torque-power
• Torsional Oscillator (Disk on a Shaft)

  A disk on an elastic shaft is a torsion pendulum: twist it and let go and it rings at one natural frequency, ω_n = √(k_t/J_d). The pitch is set entirely by the shaft's stiffness and the disk's inertia — not by how hard you twist it — while the stress it survives is set by the amplitude.

  • dynamics
  • stress
  • torque-power