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.
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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.
Timing
Predict and Try 8 min · Worked example 10 min · Vary 8 min · Practise and Make 14 min
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
What you will see
A question to ask
The fix
A student says the nearer support holds less.
If the load sat right on a support, how much would the other one hold?
The nearer support holds more.
A student says twice as deep is twice as stiff.
What does the formula do to depth?
Cubes it: 2³ = 8.
Answers to the checks
flat or on edge?: A 30 × 3 mm strip is turned from flat onto its edge. Same wood, same load. Its sag… About a hundredth. Stiffness grows with the depth cubed: 10 times deeper is 1000 times stiffer, and a tenth as wide is a tenth of that: 100 times.
On its edge: Switch the shape to the strip on its edge.
It sags 0.39 mm instead of 38.6 mm.
How many times?: About how many times less does it sag on its edge? 100 38.6 ÷ 0.39 ≈ 100.
I-beam and tube: Try the I-beam and the tube. Which sags least?
The I-beam: it puts most of its wood at the top and bottom, as far from the middle as it can.
Two rules: 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? 8 10 × 0.4 ÷ 0.5 = 8 N (and the right holds 2 N).
Deeper: A beam is made twice as deep (same width). How many times stiffer is it? 8 2³ = 8 times stiffer: it sags an eighth as much.
Shares: Move the load until the left support holds 8 N.
0.1 m from the left support: 10 × 0.4 ÷ 0.5 = 8 N.
Least wood: The sag must be 0.5 mm or less, using the least wood (90 mm² or less). Choose the shape.
The strip on its edge: 0.39 mm with 90 mm² of wood. The tube also passes the sag, but uses 90.2 mm².
Shares: 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? 45 60 × 3 ÷ 4 = 45 N.
Deeper: A plank is made three times as deep (same width). How many times stiffer is it? 27 3³ = 27.
I-beams: 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 stiffness Stiffness comes from material far from the middle. An I-beam puts its flanges there and keeps a thin web between them.
A shelf: A shelf holds 25 N in the middle and may sag at most 0.5 mm. Choose the shape that meets the brief.
Only the I-beam: 0.44 mm. The strip on its edge sags 0.96 mm and the tube 1.22 mm.
Solutions
One possible solution for each task. Other correct programs are fine.
Vary: move the load, choose the shape › Shares
Open task: any program that does what the step asks.
Vary: move the load, choose the shape › Least wood
Open task: any program that does what the step asks.
a shelf › A shelf
Open task: any program that does what the step asks.
a shelf › Evaluate
Open task: any program that does what the step asks.
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.
Exit ticket
A plank 20 mm deep is replaced by one 40 mm deep, the same width. How does its sag change?
Twice as deep is 2³ = 8 times stiffer, so it sags an eighth as much.
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.
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).