Beam depression formula has d³ in it: why are deep beams used, and why are I-beams and hollow shafts stronger?

7 answers & replies

• [+][–]Tushar PawarDropperSample· 15d ago

  2. Don't guess, use the proportionality. Everything else is constant, so

  $\delta \propto \frac{1}{b\,d^3}$

  Beam A: $4\times6^3 = 4\times216 = 864$. Beam B: $6\times4^3 = 6\times64 = 384$.

  $\frac{\delta_B}{\delta_A} = \frac{864}{384} = 2.25$

  So the depression becomes 2.25 times, not 1.5. Cross-section area is the same (24 cm²), but beam A has the depth in the right place.

  Report

  • [+][–]Ishaan Bose11thSample· 15d agoAuthor

    bruh 2.25. the answer was sitting there and I guessed. thanks. also the neutral layer idea is the first time "obviously" was actually explained

    Report
• [+][–]Kabir Joshi11thSample· 15d ago

  Also the formula has $Y$ in the denominator, so material matters too: $\delta \propto 1/Y$. For the same beam size and load, a steel beam ($Y \approx 2.0\times10^{11}$ N m⁻²) sags about 2.9 times less than an aluminium one ($Y \approx 0.7\times10^{11}$ N m⁻²), going by NCERT's table. So you can't fix a floppy design just by switching material, shape matters a lot, and shape is exactly the $b\,d^3$ part of the formula.

  Report

  • [+][–]Tushar PawarDropperSample· 15d ago

    Nice add, Kabir. And to put numbers on the shape part, δ ∝ 1/(b d³): double the depth and the sag is $1/2^3 = 1/8$. Double the breadth and it's only 1/2, because b is to the power 1. 👍

    Report
• [+][–]Zoya Khan12thSample· 15d ago

  Why depth matters so much, in words. When a beam bends, the top surface is compressed and the bottom surface is stretched, while the layers around the middle (the neutral layer) are hardly strained. The deeper the beam, the further the top and bottom layers sit from that middle, so they take a stronger share of the work resisting the bending. Material far from the neutral layer is the "useful" material.

  That's the intuition for $d^3$. Breadth only adds more columns of material side by side, so it counts once.

  Report
• [+][–]Lakshmi TripathiDropperSample· 15d ago

  I-beam: the top and bottom flanges carry most of the compression and tension, and a thin vertical web joins them. The centre doesn't do much work, so you remove it and save a large amount of material and weight, without losing much resistance to bending. That is why the shape of an I.

  Hollow shaft: NCERT says that for the same mass, a hollow cylinder is more rigid and stronger against twisting (torsion) than a solid one, because the material is placed away from the axis. And a hollow tube is also lighter for the same strength, so it's used in machine shafts.

  Report
• [+][–]Nisha Verma12thSample· 15d ago

  Exam tip: these are almost always ratio questions. Write the formula in the form $\delta \propto \dfrac{1}{bd^3}$ (keep $W$, $l$, $Y$ out of it) and plug the numbers. Also check units are consistent; sometimes they give mm for one and cm for another.

  And ask whether the question says "load at centre" and "beam supported at two ends". The formula you wrote is only for that case.

  Report