--- title: "Beams and shapes" subtitle: "Structures · Beams · Lesson 4" --- # Beams and shapes **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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? :::teacher-note **In one sentence:** 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. **Students should be able to say:** 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. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict :::keep-together **A 30 × 3 mm strip is turned from flat onto its edge. Same wood, same load. Its sag…** ☐ Drops to about a hundredth ☐ Gets bigger: it might tip over ☐ Stays the same: same wood ☐ Halves My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 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. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Worked example: supports and stiffness :::keep-together ### Two rules 1. $R_\text{left} = P \times \frac{b}{L}$ — 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). 2. $10 \times \frac{0.25}{0.5} = 5 \text{ N each}$ — In the middle, the supports share it equally. 3. $I = \frac{\text{width} \times \text{depth}^3}{12}$ — Stiffness of a rectangle (in mm⁴): depth counts cubed. 4. $\text{flat: } \frac{30 \times 3^3}{12} = 67.5 \quad \text{edge: } \frac{3 \times 30^3}{12} = 6750$ — 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?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 10 × 0.4 ÷ 0.5 = 8 N (and the right holds 2 N). ::: :::keep-together ### Deeper **A beam is made twice as deep (same width). How many times stiffer is it?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 2³ = 8 times stiffer: it sags an eighth as much. ::: :::typical ## Vary: move the load, choose the shape **Shares.** **Least wood.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Beams :::keep-together ### 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?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 45 60 × 3 ÷ 4 = 45 N. ::: :::keep-together ### Deeper **A plank is made three times as deep (same width). How many times stiffer is it?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 27 3³ = 27. ::: :::keep-together ### 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 ☐ I-shapes are easier to paint ☐ The middle of a beam does all the work ☐ They weigh more ::: :::teacher-note **Answer:** 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. ::: ## Make: a shelf **A shelf.** **Evaluate.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A plank 20 mm deep is replaced by one 40 mm deep, the same width. How does its sag change?** {{lines-3}} ::: :::teacher-note **Answer:** Twice as deep is 2³ = 8 times stiffer, so it sags an eighth as much. :::