Structural Engineering & Statics
where the load goes
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Shear flow — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 0.41 against a mean of 0.19 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero.

    The shear nobody draws

    A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains.

    part 1 · sections
  2. A short timber beam is a shear problem, and a steel one never is. Utilisation of the bending and shear checks on one beam, against span-to-depth. The two cross where the ratio equals f_m ÷ f_v exactly — no load, no width and no span survives the cancellation — which for this timber is 6.7 and for steel is 1.73. So a timber beam shallower than about six times its depth is governed by shear parallel to the grain, and a steel beam would have to be shorter than twice its own depth before the same thing happened, which is not a beam. The third check is bearing across the grain, which does not move with the span at all: on the beam drawn it is at 0.40, and it is the one that governs.

    The shear that decides a timber beam

    A steel beam is never governed by shear, because its bending strength is only 1.73 times its shear strength and no beam is that short. Timber's ratio is 6.7 along the grain and 23 across it, so shear governs at proportions people build every day.

    part 2 · sections
  3. A crack starts at the corner and runs along the grain. The end of a timber beam 400 mm deep and 140 mm wide, notched on its underside so that 200 mm stands over the support, with the notch's corner 50 mm from the line of the reaction, drawn to scale. A crack starts at the corner and runs along the grain towards mid-span. With a shear strength of 3.5 N/mm² the un-notched end would carry 130.7 kN; Eurocode 5's factor kᵥ of 0.40 on the reduced depth leaves 25.8 kN, 20% of it.

    The notch that costs more than its depth

    Cut a timber beam's underside away at the support, so that half its depth sits on the bearing, and a stress check on what is left says the end has lost half its shear strength. The beam says otherwise. A crack starts at the notch's corner and runs along the grain, and whether it runs is a question of energy, not stress — so the answer has a size in it. A 400 mm glulam beam notched to half its depth keeps a quarter of its shear resistance, an 800 mm one keeps 18 per cent, and the deeper the beam the less a notch leaves. Sloping the notch buys much of it back, and moving the corner away from the support costs it again.

    part 3 · sections

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