Bend a paperclip once and it holds its new shape. Bend it back and forth a dozen times and it snaps in your fingers, even though you never once pulled as hard as it can be pulled. A ruler squeezed end to end skips the squashing and bows sideways with a sudden bang. A diving board flexes under your weight and springs back, but a bit more weight would leave it permanently bent.
None of that is mysterious once you have the picture a mechanical engineer carries. Every solid is a very stiff spring. Every load makes it stretch or squash or twist by an amount you can predict, and the way it fails depends on what kind of load, how often, and where the stress piles up. Statics told you how big the forces are. This course is about what they do to the stuff.
Who takes it: mechanical, aerospace, civil and materials engineers, usually in the second year (sometimes called Strength of Materials or Solid Mechanics).
What it is for: predicting how much a part deforms and when it will fail, so you can choose its shape and size.
Comes after: Statics (34/10), and through it Physics I: Mechanics (34/06) and Linear Algebra (34/04).
Leads to: Machine Design (34/19), which turns it into shafts, bolts and gears. It also underlies Materials Science & Manufacturing (34/13) and Vibrations (34/17).
What this piece will and won't give you: you won't be able to certify a beam. You will see why a beam is shaped like an I, why a long thin column fails differently from a short fat one, and why a small hole can matter more than a big load.
Easier first read: Everyday Physics: The Answers Hiding in Plain Sight (Physics 5). Read Statics: Why Things Stand Still (34/10) first.
Stress and strain: divide out the size
A thick rod and a thin rod made of the same steel and pulled with the same force behave very differently, and a long rod stretches more than a short one. To compare materials rather than particular objects, engineers divide the size out.
Stress is force per unit area: how hard the material is working. Strain is stretch per unit length: how much it has deformed, as a fraction. A rod that grows from 1,000 mm to 1,001 mm has strain 0.001, or 0.1%.
For small stretches, stress and strain are proportional. Robert Hooke published the idea in 1678 as ut tensio, sic vis, "as the extension, so the force". The constant of proportionality for a material is its Young's modulus E, named after Thomas Young. It is the material's stiffness, independent of shape:
Steel's E is about 200 GPa, aluminium's about 69 GPa, and pine's roughly 10 GPa (values vary by alloy and species). Read it as: for the same stress, aluminium stretches about three times as much as steel. Try it.
With 10 kN on a 10 mm steel rod 1 m long: area 78.5 mm², stress 127 MPa, strain 0.000637, stretch 0.64 mm. Halve the diameter and the stress quadruples, because the area falls with the square of the diameter. Swap to aluminium and the stretch nearly triples (200 ÷ 69) while the stress stays put. That is why stiffness-critical parts (not just strength-critical ones) are chosen by E.
The stress–strain curve is a material's personality
Pull a bar in a test machine, record stress against strain, and you get a curve that is a fingerprint of the material. It has a few landmarks worth knowing by heart.
First comes a steep straight line: the elastic range. Let go and the bar returns exactly to its original length. The slope is E. At the yield point the line gives way: atoms begin to slip past each other, and release no longer restores the old shape, so some stretch is permanent. In a ductile metal like structural steel the curve then rises further (the metal work-hardens) to the ultimate strength, the highest stress it ever carries. After that the bar thins in one place (necking) and finally breaks. A brittle material such as glass or cast iron stays elastic almost to the end and snaps with little or no warning.
Engineers prefer ductile materials for structures because ductility gives notice: a steel beam sags visibly and absorbs energy long before it breaks. The yield point, not the ultimate strength, is usually the number a designer works from.
Illustrative curves: a steel-like metal (E = 200 GPa, yield 250 MPa, ultimate 400 MPa) and a glass-like brittle solid (E = 70 GPa, snaps at 50 MPa). They are idealised in shape, not measured.
The elastic range of the steel-like metal ends at strain 0.00125, since 250 MPa ÷ 200,000 MPa = 0.00125. That is just 1.25 mm on a metre of bar, a sliver of the total 25% it can stretch before breaking. Use the zoom button to see that sliver. Nearly everything a structure does in service happens inside it, which is why it is designed to stay there.
Beams: stiffness grows with depth cubed
When a beam sags, its top edge is squashed and its bottom edge is stretched. Somewhere between them is a layer that does neither, the neutral axis. Material near the middle is barely working. Material at the top and bottom is working hardest and is also the farthest from the middle, which gives it the most leverage against bending.
The shape's resistance to bending is captured in one number, the second moment of area I. For a rectangle of width b and depth d, I = b·d³ ÷ 12. The cube is the whole story. For a beam resting on two supports with a load in the middle:
Double the depth of a beam, keeping the width, and its stiffness rises 2³ = 8 times. Compare a plank on edge with the same plank laid flat: the edge-up plank is twice as deep (a factor of 8) but half as wide (a factor of ½), so it is 4 times stiffer. Doubling the span makes the sag 2³ = 8 times bigger. Depth and span, not weight, rule the stiffness.
That also explains the I-beam. If material near the middle isn't doing much, move it to the flanges and leave a thin web to hold them apart. The shape gives up some stiffness for a lot less steel.
A solid steel 50 × 100 mm beam on edge, 2 m span, 5 kN at the middle, sags 1.0 mm. Lay it flat and it sags 4 mm. Make the same outline an I-section (flanges and web each a tenth of the size) and it uses 28% of the steel but keeps 54% of the stiffness, sagging 1.85 mm. The beam is shaped like an I because that puts the steel where the bending lives.
Buckling: slender columns bow before they crush
You would expect a column to fail when the compression exceeds the material's strength. A short, stout one does. A long thin one never gets there. Press on the ends of a plastic ruler and at some load it suddenly bows out sideways, far below what it could carry if it stayed straight. That is buckling: an instability, not a strength failure.
Leonhard Euler worked out the load at which a perfectly straight slender column bows, in the 1750s:
The load depends on stiffness (E I), not strength, and falls with the square of the length: double the length and the load a column can take falls to a quarter. The factor K depends on how the ends are held: 1 for pinned ends, 0.5 if both are clamped (four times stronger), 2 for a flagpole fixed at the bottom only (a quarter as strong). A column's crushing load, on the other hand, is just yield stress times area and doesn't care how long it is.
A 20 mm steel rod, 1 m long and pinned at both ends, buckles at 15.5 kN, though it would take 78.5 kN to crush it. At 2 m it buckles at only 3.9 kN. At 0.2 m the buckling load is far above the crushing load, so it crushes instead. The formula is for ideal columns; real ones have imperfections and do worse, and real designs use empirical corrections with generous margins. In structures, compression members are the ones engineers watch for exactly this reason, and why a long pushed member is deeper or braced.
Torsion: a twisted shaft is a bar in shear
Stretching is not the only way to load a bar. Push the top of a deck of cards sideways and the cards slide over each other: that is shear, a sliding stress (force parallel to the surface, divided by the area). A bolt holding two plates that try to slide past each other is in shear. The shear modulus G plays the role for sliding that E plays for stretching; for steel it is about 80 GPa.
Twist a shaft and every thin slice slides a little relative to its neighbour, and the sliding is biggest at the surface. The surface stress is torque times radius divided by the shaft's polar second moment J (for a solid round shaft J = πD⁴ ÷ 32):
Because the centre of a shaft barely strains, you can bore it out and keep most of its torsional stiffness. That is why drive shafts, bicycle cranks and aircraft tubes are hollow.
A solid 20 mm steel shaft carrying 100 N·m has a surface shear stress of 63.7 MPa and twists 4.6° over a metre. Bore it out to 16 mm (80% of the outer diameter) and it keeps 59% of the twisting stiffness with 36% of the steel. The surface stress rises to 108 MPa, so you do pay something, which is why designers also check it. The yield strength in shear is roughly 0.58 times the tensile yield (about 144 MPa for 250 MPa steel).
Holes, cracks and repetition: why parts break below their strength
Stress flows through a part like water through a pipe. Where the pipe narrows or bends, the flow bunches up. A hole in a plate in tension makes the stress lines crowd around it, so the stress at the hole's edge is higher than the average, by a stress concentration factor K. For a small round hole in a wide plate, K = 3. For an elliptical hole whose width across the pull is a times its height, K = 1 + 2a. A sharper, longer notch across the pull means a bigger K, and a crack is the sharpest notch of all.
At 80 MPa average stress (well under yield) a round hole brings the edge to 240 MPa, a factor of safety of about 1.04. At 100 MPa it goes to 300 MPa and the metal there yields. A 5 : 1 slot puts K at 11. The factor of safety is just capacity divided by what you expect to load, with a deliberate margin for the unknowns. Aircraft designers typically work with slimmer margins than builders of bridges, because every kilogram of margin is paid for in fuel, so they have less room for a surprise like a concentration they missed.
Repetition makes it worse. A load applied again and again, far below the yield stress, can start a tiny crack at a stress concentration and grow it a little each cycle until what remains cannot carry the load. That is fatigue, and it is the paperclip's story at engineering scale. The de Havilland Comet, an early jet airliner, suffered two mid-air break-ups over the Mediterranean in 1954 (10 January off Elba, 35 people killed; 8 April near Naples, 21 killed). At Farnborough, investigators sealed a whole fuselage in a water tank and pressurised it again and again to imitate flights. It burst open after 3,057 pressure cycles (1,221 real flights plus 1,836 simulated), from a crack at the corner of the forward escape-hatch window. On the Elba wreckage the fatigue crack began at a rivet hole by the opening for an aerial. The story is often told as "square windows"; sharp corners were stress concentrators, but they were one of several in that airframe. The lesson that stayed: repeated pressurisation plus a stress concentration can break even a strong skin, so pressurised airframes must be designed and tested for fatigue.
What a graduate just sees
- Strength is stress (force over area), not force. Halving the diameter quadruples the stress.
- Every solid is a spring, and Young's modulus is its stiffness: steel stretches a third as much as aluminium for the same stress.
- Most of the working life of a structure is spent in the first fraction of a percent of strain, deliberately.
- Ductile means warning; brittle means none.
- A deep beam is stiff because stiffness grows with the cube of the depth. Put material where it is farthest from the middle.
- A long thin thing in compression is a buckling problem, not a crushing problem, and doubling its length cuts its capacity to a quarter.
- Hollow tubes are nearly as stiff in twist as solid bars at a fraction of the weight.
- Sharp corners, holes and cracks multiply stress; round them. Repetition finds them.
Where it shows up next
- Statics (34/10) feeds it: the reaction or member force found there becomes the F and T here.
- Machine Design (34/19): sizing a shaft, a bolt or a gear tooth is these formulas with factors of safety and fatigue data added.
- Materials Science & Manufacturing (34/13): why steel yields where it does, why glass is brittle, and how heat treatment moves the curve you played with.
- Vibrations (34/17): a beam is a spring, and its stiffness sets the pitch at which it rings.
- In the world: aircraft skins and wing spars, bridge girders, bicycle frames, tall chimneys, engine crankshafts, the reason window corners on ships and planes are rounded.
If you want to go deeper
- 📘 Standard textbook: R. C. Hibbeler, Mechanics of Materials (Pearson). Widely assigned, clear on method, with a great many worked problems.
- 📘 Standard textbook: F. P. Beer, E. R. Johnston Jr., J. T. DeWolf and D. F. Mazurek, Mechanics of Materials (McGraw Hill). The other common choice.
- 📘 Standard textbook: J. M. Gere and B. J. Goodno, Mechanics of Materials (Cengage). Known for patient explanations.
- 📘 Gentler: J. E. Gordon, Structures: Or Why Things Don't Fall Down, and its companion The New Science of Strong Materials. Written for non-engineers, and mostly equation-free.
- 🌐 Free course: MIT OpenCourseWare 2.001 Mechanics & Materials I, with lecture notes and problem sets.
Stress is force over area, strain is stretch over length, and Young's modulus is the exchange rate between them. A beam's stiffness rises with the cube of its depth, so it is shaped like an I. A slender column bows rather than crushes, at a load that falls with the square of its length. A twisted shaft is a bar in shear, and a hollow one is almost as good as a solid one. And every hole, corner and crack concentrates stress, which repeated loads then turn into fatigue.
Try this: bend a paperclip back and forth and count the bends until it breaks. Then look at any bridge, crane or bicycle and find where the metal is placed far from the middle, and where the corners are rounded.
Sources. The formulas (σ = F/A, ε = ΔL/L, Hooke's law, I = bd³/12, centre-load sag PL³/48EI, Euler's Pcr = π²EI/(KL)², τ = Tr/J, twist TL/GJ, the Inglis ellipse K = 1 + 2a/b) are standard textbook results. Hooke published the anagram's solution ut tensio, sic vis in 1678 (Wikipedia, "Hooke's law"). Typical E values: Wikipedia, "Young's modulus" (steel about 200 GPa, aluminium about 68–69 GPa, wood roughly 9–12 GPa). Euler's column formula dates from 1757 (Wikipedia, "Buckling"). Comet: BOAC Flight 781, 10 January 1954, 35 dead; South African Airways Flight 201, 8 April 1954, 21 dead; water-tank test fuselage G-ALYU failed on 24 June 1954 after 3,057 cycles (1,221 actual flights plus 1,836 simulated), from Wikipedia, "De Havilland Comet"; the fatigue origin on G-ALYP at a rivet hole by the aerial (ADF) window, and the square-window caveat, from Aerossurance, "Common Comet Misconceptions". Every number in the interactives was computed in node (steel rod: 127.3 MPa, 0.637 mm; beam: 1.00, 4.00 and 1.85 mm; column: 15.5, 3.88 and 78.5 kN; shaft: 63.7 and 107.8 MPa, 4.56°; notch: K = 3, 11). The two stress–strain curves are idealised and the G, yield and ultimate values are round textbook figures.
Go deeper. Gordon's two books were first published by Penguin in 1978 (Structures) and earlier (The New Science of Strong Materials).
- 🌐 MIT OpenCourseWare 2.001 Mechanics & Materials I
- 🌐 Aerossurance: Common Comet Misconceptions
- 📘 J. E. Gordon, Structures: Or Why Things Don't Fall Down