Choosing a Beam Section for a Crane Jib
A small workshop crane jib is 3.0 m long and must lift a 1.2 t load at its tip. Two candidate sections are available, both of the same mass per metre: a solid rectangular bar 60 mm x 120 mm, and a hollow square tube 100 mm x 100 mm with 8 mm walls. The jib is treated as a cantilever fixed at the mast. The engineer notes the solid bar is oriented with its 120 mm depth vertical. Yield strength of both steels is 350 MPa. The design brief requires a safety factor of at least 2.5 against yielding at the root. The client also wants the tip deflection kept small, and asks which section should be specified and why the lighter-looking tube is not automatically better.
1. Calculate the maximum bending stress at the root for each section and judge which meets the safety factor of 2.5.
Read the worked answer
Bending moment at root: M = F x L = (1.2 x 1000 x 9.81) x 3.0 = 11 772 x 3.0 = 35 316 N·m.
Solid bar: I = b d^3 / 12 = 0.060 x 0.120^3 / 12 = 8.64 x 10^-6 m^4. Section modulus Z = I / (d/2) = 8.64 x 10^-6 / 0.060 = 1.44 x 10^-4 m^3. Stress = M / Z = 35 316 / 1.44 x 10^-4 = 245 MPa.
Hollow tube: outer 0.100 m, inner 0.084 m. I = (0.100^4 - 0.084^4)/12 = (1.0 x 10^-4 - 4.98 x 10^-5)/12 = 4.18 x 10^-6 m^4. Z = I / 0.050 = 8.37 x 10^-5 m^3. Stress = 35 316 / 8.37 x 10^-5 = 422 MPa.
Judgement: the solid bar at 245 MPa gives a safety factor of 350/245 = 1.43, which fails the 2.5 requirement. The tube at 422 MPa exceeds yield outright, so it is worse. Neither passes; the solid bar is the better of the two but still needs a deeper section or a shorter jib.
2. The client argues the tube should be stronger because hollow sections are 'more efficient'. Explain, using the numbers, why this reasoning fails here, and what property of the section actually governs.
Read the worked answer
Hollow sections are efficient only when material is placed far from the neutral axis relative to the section's overall depth. Here the tube's overall depth is 100 mm versus the bar's 120 mm, and bending stress depends on the section modulus Z, which scales with depth cubed for a given shape. The bar's greater depth more than compensates for its solid core.
Compare Z values: bar 1.44 x 10^-4 m^3 against tube 8.37 x 10^-5 m^3, so the bar has about 1.7 times the section modulus at equal mass per metre. The governing property is therefore Z (or I if deflection governs), not whether the section is hollow. The judgement call is that 'hollow is better' is a rule of thumb that only holds at matched outer depth; at unequal depth it misleads.
3. If tip deflection, not yield, becomes the governing criterion, explain qualitatively how the ranking of the two sections could change, and state what additional data you would need.
Read the worked answer
Tip deflection for a cantilever is PL^3/(3EI), so it depends on I, not Z. Because the bar's I (8.64 x 10^-6 m^4) is about twice the tube's (4.18 x 10^-6 m^4), the bar would deflect roughly half as much for the same load. So under a deflection criterion the bar's advantage persists and may even widen, since deflection is linear in I while stress is linear in Z.
The ranking would only reverse if the tube's outer depth were increased (deflection scales with depth cubed), or if the bar were reoriented so its shallow axis carried the load. To quantify, I would need the allowable tip deflection, the elastic modulus E for both steels (assumed equal here), and confirmation of the effective cantilever length including any mast clamp flexibility.