Beams and shapes

A beam's supports share its load like a lever, and its stiffness depends on its shape far more than on how much material it has: depth counts cubed. Students compare a strip flat and on its edge, an I-beam and a tube, work out support forces, and choose a section to a brief.

A ruler laid flat between two books sags under a pencil case. Turn it on its edge and it barely moves. It's the same ruler. Why?

Builds on: Levers (Engineering › Mechanisms) · Does balanced mean no forces? (Science › Year 7 Physics)

A beam is a member that carries a load across a gap, sitting on supports. Two questions matter: how much does each support hold, and how much does the beam sag?

The beams here are pine, 500 mm between supports. Each cross-section (the shape you'd see if you sawed through it) uses about the same amount of wood.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Flat or on edge?

A 30 × 3 mm strip is turned from flat onto its edge. Same wood, same load. Its sag…Drops to about a hundredthGets bigger: it might tip overStays the same: same woodHalves

TryTry it on the board and compare the result with your prediction.

Try: four shapes

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Switch the shape to the strip on its edge.

Use the board. This ticks when you get there.

About how many times less does it sag on its edge?

Try the I-beam and the tube. Which sags least?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Worked example: supports and stiffness

Worked exampleBefore each step, say what you think comes next.

$R_\text{left} = P \times \frac{b}{L}$ The load P is a away from the left support and b from the right. The far distance decides each share (as on a lever).
$10 \times \frac{0.25}{0.5} = 5 \text{ N each}$ In the middle, the supports share it equally.
$I = \frac{\text{width} \times \text{depth}^3}{12}$ Stiffness of a rectangle (in mm⁴): depth counts cubed.
$\text{flat: } \frac{30 \times 3^3}{12} = 67.5 \quad \text{edge: } \frac{3 \times 30^3}{12} = 6750$ Same wood, a hundred times the stiffness on its edge.

A 10 N load sits 0.1 m from the left support of a 0.5 m beam. How much does the left support hold, in newtons?

A beam is made twice as deep (same width). How many times stiffer is it?

ModifyChange the problem on purpose and see what happens.

Vary: move the load, choose the shape

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Move the load until the left support holds 8 N.

Use the board. This ticks when you get there.

The sag must be 0.5 mm or less, using the least wood (90 mm² or less). Choose the shape.

Use the board. This ticks when you get there.

PractiseA few on your own, to check you have it.

Beams

A 60 N load sits 1 m from the left support of a 4 m beam. How much does the left support hold, in newtons?

A plank is made three times as deep (same width). How many times stiffer is it?

Why are steel beams in buildings I-shaped?Most of the steel is at the top and bottom, far from the middle, where it adds most stiffnessI-shapes are easier to paintThe middle of a beam does all the workThey weigh more

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A shelf

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A shelf holds 25 N in the middle and may sag at most 0.5 mm. Choose the shape that meets the brief.

Use the board. This ticks when you get there.

In one or two sentences: which shape you chose, why it is stiffer, and one reason a builder might still choose another shape.

For teachers

Students compare four cross-sections with about the same wood (a strip flat and on edge, an I-beam, a tube), find depth counts cubed, work out how a beam's supports share a load, and choose a section for a shelf brief. About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can work out how much each support of a beam holds.
I can explain why a deeper beam is much stiffer, and why I-beams are shaped as they are.
I can choose a beam section to meet a sag limit.

Before class

A ruler between two books, flat then on its edge, is the whole lesson in ten seconds; then the board puts numbers on it.

The model: a simply supported pine beam (E = 10 000 N/mm²), 500 mm span, a point load; sag at the load point = P a² b² ÷ (3 E I L). Sections (mm): flat 30 × 3, edge 3 × 30, I-beam 30 deep with 20 × 1.5 flanges and a 1.5 web, square tube 20 × 20 × 1.2. Strength (breaking) isn't modelled, only sag. The picture exaggerates the sag.

Curriculum hooks: Design and Technologies, materials and technologies (properties of materials), engineering principles and systems; Maths, powers (cubes) and proportion.

Running the lesson

Predict: hand out rulers before the board.
Try: ask where the I-beam keeps its wood.
Worked example: link the support shares back to Levers (Mechanisms).
Make: ask why a builder might not choose the I-beam anyway (cost, joining, looks).

The hard part (on purpose)

Students expect the same material to behave the same way. Have them flex a ruler flat and on its edge.

When students get stuck

A student says the nearer support holds less. Ask: If the load sat right on a support, how much would the other one hold? Fix: The nearer support holds more.
A student says twice as deep is twice as stiff. Ask: What does the formula do to depth? Fix: Cubes it: 2³ = 8.

Stretch and support

Stretch: Stiffness of a rectangle is width × depth³ ÷ 12. Design a 90 mm² rectangle stiffer than the I-beam. Why don't builders use it?

Support: Give the four sections drawn to scale, and ask students to rank them before the board.

Where it leads

Next: the Capstone projects; Mechanisms › Levers for the support-share rule from the other side.

Real examples: floor joists on edge, steel I-beams, aluminium extrusions, bike frames (tubes).

Sources

TeachEngineering: K-12 engineering design challenges (benchmark; the briefs and wording are our own) (checked 12 October 2026)

What each step is for

flat or on edge?: Students expect the same material to sag the same way. Committing first makes the hundredfold difference land.
Try: four shapes: Switching shapes and reading the sag builds the idea that where the material is matters more than how much there is.
Worked example: supports and stiffness: Two rules: the supports share the load like a lever (the nearer support holds more), and stiffness depends on depth cubed (the I value).
Vary: move the load, choose the shape: First the support shares on the board, then a brief: the least wood that keeps the sag within a limit.
beams: Support shares, the depth-cubed rule, and why real beams are I-shaped.
a shelf: A heavier load narrows the choice to one shape: students use the facts line to decide and justify.

Now you can

Work out how much each support of a beam holds.
Explain why a deeper beam is much stiffer, and why I-beams are shaped as they are.
Choose a beam section to meet a sag limit.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.You reached the endBack to Structures

Source: content/engineering/structures/beams/lesson.md