Structural Engineering & Statics
where the load goes
Concept

Shear area — where it appears

The part of a section that carries the shear, which is the web on an I-section and about five sixths of a rectangle. A section has two of them and they are different numbers: one is the area that reaches the shear yield stress, and the other is the area an energy integral gives for a deflection.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

Nothing happens, and then everything happens. The moment capacity left to a section already carrying shear, against the shear as a fraction of what the web can take. The web holds 26.1% of this section's plastic modulus and the flanges hold the rest, and only the web's share is reduced — by the factor √(1 − v²) that von Mises leaves it. So the curve is flat for most of its length: the first per cent of moment is not lost until v = 0.27, half the shear capacity costs 3.5%, and 15% is not reached until v = 0.9. The tangent at v = 1 is vertical, which is why the last tenth of the shear range costs more than the first eight.

Both at once, and neither matters until it does

A section carrying shear has less moment capacity, and the reduction is the web's share of the plastic modulus times one minus the root of one minus the shear ratio squared. On a rolled beam that share is a quarter, so half the shear capacity costs three and a half per cent — and then the last tenth costs more than the first eight.

sections · Shear moment interaction A section has two areas and the tables give one of them. Peak shear stress divided by the mean, for four sections of exactly the same gross area and depth. The mean is V/A and is the number a first calculation uses; the peak is what the material actually sees, and the ratio between them is a property of shape alone. A rectangle's is 1.5 — the parabola's peak over its average — and it is one of the few numbers in this subject that is exactly derivable and universally ignored. An I-section's is near 1.98, and the reason is on the second bar: 97% of the shear is inside a web that is 56% of the area. So the flanges carry the moment and almost none of the shear, and the web carries the shear and almost none of the moment — which is why a shear check on an I-section uses the web area and a moment check uses the whole section, and why the two checks are about two different pieces of steel.

The section has two areas

A shear force divided by the area of the section is not the shear stress anywhere in it. A rectangle's peak is exactly one and a half times that number and an I-section's web carries nearly all of the shear over a fifth of the area, which is why a moment check and a shear check on the same member are checks on two different pieces of steel.

sections · Shear area A transverse load with nothing applied. Web slenderness against web thickness, with the limit the flange's own curvature sets. A flange carrying 6213 kN and curved to a radius of 592 m needs 10.5 N per millimetre of radial force to stay on its curve, and the only thing available to supply it is the web. Nothing has been applied to the girder: the load comes from the deflected shape, which is why a straight beam has none of it and a beam at a plastic hinge has a great deal. Setting the radial force against the web's own plate-buckling resistance gives, in four lines, h_w/t_w ≤ k·(E/f_yf)·√(A_w/A_fc) — the form the codes use, arrived at without them. The constants differ: an elastic flange strain gives k = 1.34 and the rule uses 0.3, a factor of 4.5, and the gap is the curvature assumed. k goes as the inverse square root of the flange strain, so 0.3 is a flange strained to 3.4% — which is what a plastic hinge does to it. The rule is not conservative; it is written about a different beam.

The web that is crushed from inside

A plate girder's compression flange is curved by the beam's own deflection, and a curved force needs a transverse load to stay on its curve. The only thing available to supply it is the web. So a deep girder can buckle its web vertically with nothing applied to it at all, and the rule that prevents it is the only clause in the codes about a load no load case contains.

stability · Flange induced The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 120 mm to frame into a girder. What is left is a tee with a section modulus of 3.836e+5 mm³ against the whole section's 1.438e+6 — 27 per cent. The moment at the end of the cope is the reaction on a lever arm of 130 mm: 23.4 kNm, giving 61 N/mm² and a flexural utilisation of 0.17. The web now has a free edge along the cope, so its buckling coefficient collapses from 4 to 0.425 — a factor of 9.4 — and the re-entrant corner has a stress concentration of 5.5 on a 10 mm radius.

The section that is checked is not the one chosen

A beam framing into a girder has its top flange cut away so the two can sit at the same level. What is left is a tee with a quarter of the section modulus, a web with a free edge, and a re-entrant corner — and the beam was selected on a table entry that describes none of it.

connections · Coped beam The shear deflection is not a correction. The share of a sandwich panel's deflection that is shear rather than bending, against how slender the panel is. A solid beam at a span-to-depth ratio of 20 spends about a per cent of its deflection on shear; this panel spends 25% at the same ratio, because its core is 2800 times softer in shear than its faces are in tension. At the 37 of the panel drawn it is 9%. The curve falls as the square of the span because bending grows as the fourth power and shear as the second, so the term that is negligible for a long panel is the whole answer for a short one.

The stiffness that belongs to the span

A section's flexural rigidity is a property of the section, and a member's is not. Once shear deformation is counted the effective stiffness contains the span, so the same panel is a different member at three metres and at six — and for a sandwich the correction is not a correction.

deflection · Shear deflection Block shear: the metal between the holes. Three bolts in a 9 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 55 mm long. Shear ruptures first, and the capacity is the sum of two different strengths on two different planes: 524.79 kN, of which the shear plane carries 63.03%.

The end that is only a plate

Cut one flange off a beam's end and what is left is a tee. Cut both and what is left is a plate with holes in it — no flanges, no section modulus worth the name, and none of the checks the beam was selected by. Three plate checks replace them, and the one that governs depends on dimensions that appear in no section table.

connections · Coped beam Each repair reaches the checks its plate touches, and no others. The four checks on a 457 mm beam coped 50 mm deep over 200 mm, carrying a reaction of 300 kN through three bolts, as utilisations, for the end as coped and with three repairs: an 8 mm doubler on the web, a 100 × 10 mm plate along the free edge, and both. As coped the utilisations are flexure 0.46, shear 0.41, tear-out 0.57 and local buckling 0.46. The doubler thickens the web, which is most of what a coped tee is, and lowers all four; the edge plate gives the tee back a flange and lowers only flexure and buckling, leaving shear and tear-out exactly where they were. What governs: as coped, tear-out at 0.57; doubler, tear-out at 0.30; edge stiffener, tear-out at 0.57; both, tear-out at 0.30.

The repair that fixes the wrong check

A coped beam end that fails its checks is usually repaired by welding a plate along the edge the cope left, because the cope removed a flange and the plate puts one back. On ordinary proportions that repair restores the check that was not failing. The check that governs is carried by the web, and only a plate on the web reaches it.

connections · Coped beam Soft in shear, the wall gives up moment to the span. The bending moment along a 300 × 600 rectangle fixed at its left end and propped at its right, under a uniform load, per unit load and span, at span-to-depth ratios of 4, 2, 1, and for the same beam treated as rigid in shear, dashed. Rigid in shear the wall carries 0.125 wL² and the prop 0.375 of the load. Counting the shear it deforms by, the wall moment falls to 0.119 wL² at 4, 0.105 wL² at 2, 0.070 wL² at 1, and the prop's share rises to 0.381, 0.395, 0.430. The load has not changed and the beam is no weaker: its forces have moved, because in a redundant beam they come from how it deforms.

The shear that moves the moments

In a statically determinate beam, shear deformation adds movement and changes no force. In a redundant one the forces come from how the beam deforms, so a stubby beam soft in shear carries less moment at its wall and more in its span — and the carry-over factor, the one half every hand method passes along, falls to nothing at φ = 2 and then changes sign.

deflection · Shear deflection A plate welded to a loaded beam carries only the load that comes after it. The bending stress across the depth of a 457 mm beam coped 200 mm long and 50 mm deep, carrying a reaction of 300 kN, at the end of the cope, repaired with a doubler 8 mm thick on the web. With 60 per cent of the reaction already on the beam when the plate is welded, the original steel carries that share on its coped section and the rest on the repaired one. At the web's free edge it reaches 137 N/mm², against 164 as coped and 97 if the plate had been there from the start. The plate carries only the later load, 39 N/mm² at its outer edge. The repair has removed 40 per cent of the stress it would have removed on an unloaded beam.

The plate that arrives after the load

A repair plate welded to a beam already in use carries only the load that arrives after its weld has cooled. The elastic checks remember that order — a doubler welded under 60 per cent of the reaction removes 40 per cent of the stress it would remove from an unloaded beam — while tear-out, a plastic mechanism, gets the whole of the repair. Propping the beam is what hands the plate the load already there.

connections · Coped beam

Named alongside it

The objects these essays reach for when they reach for this one.

Local bucklingLoad pathNet sectionPlate bucklingBlock shearConnectionFree bodySection modulusShear deflectionWebDeflectionDetailing

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