Beam sag: flat or on edge?

• Designing columns, beams and supports needs the strength and elastic behaviour of the materials used. Structural engineering explains why bridge beams have an I-shaped section and why a sand heap or a hill takes a pyramid shape.
• Crane rope for 10 tonnes (1 metric ton = 1000 kg): the rope must stay within its elastic limit, so A ≥ Mg/σy. With mild steel σy ≈ 300 × 10⁶ N m⁻², A ≥ (10⁴ × 9.8)/(300 × 10⁶) = 3.3 × 10⁻⁴ m², a radius of about 1 cm.
• A safety margin of about ten times the load raises the recommended radius to about 3 cm. A single wire that thick would be a rigid rod, so crane ropes are many thin wires braided together, for easier manufacture, flexibility and strength.
• Rest a bar (span l, breadth b, depth d) on supports near its two ends and hang a load W from its middle: the middle drops by δ = Wl³/(4bd³Y).
• To reduce sagging use a material of large Y, keep the span short, and increase the depth rather than the breadth: δ ∝ d⁻³ but only ∝ b⁻¹.
• A deep, thin bar can buckle sideways when the load is not exactly in place, as with moving traffic. The I-section is the compromise: a large load-bearing surface and enough depth to resist bending, with less weight and cost for the same strength.
• A pillar with rounded ends supports less load than one whose ends are spread out (distributed).
• Maximum height of a mountain: the base feels a shear stress of about hρg. Setting it equal to a typical rock's elastic limit, 30 × 10⁷ N m⁻², with ρ = 3 × 10³ kg m⁻³ and g = 10 m s⁻², gives h = 10 km, more than the height of Mt. Everest.

Teaching this idea? Put the simulation on your own page or LMS with this code.