Structural Engineering & Statics
where the load goes
Concept

Stress distribution — where it appears

How stress varies across a section or through a body, as distinct from its peak value. Two distributions with the same resultant load a structure quite differently, which is why a check on a resultant can pass where a check on the distribution fails.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The lag follows the shear, so it is worst at the supports. Effective width along the span of a simply supported beam under a uniform load, summed over 25 odd harmonics. Each harmonic has its own half-wavelength L/n and its own, smaller, effective width — 0.815 for the first, 0.400 for the third, 0.269 for the fifth — and near a support the short harmonics carry a bigger share of what little moment there is. So the working fraction is 0.603 at the support against 0.839 at mid-span, a difference of 23.6 percentage points on the same flange. The single sinusoid's answer, 0.815, is drawn as the flat line, and it is only right at mid-span. This is the behaviour a code reproduces by shortening L_e near a support, and the reason it does is that shear lag is driven by w‴, which is the shear force.

The flange works least where the shear is largest

Shear lag is driven by the shear force rather than by the moment, so the effective width of a wide flange is not a property of the beam. It is a function of position along it, worst at the supports, and a single number quoted for a whole span is right at mid-span and nowhere else.

sections · Effective width Cut the throat, and the face carries a moment and a tension at once. A crane hook of trapezoid section, 50 to 120 mm radius and 40 to 15 mm wide, carrying 100 kN, drawn beside the section at the cut and the stress across it. The load hangs on a line through the centre of curvature, so cutting the throat and taking everything below the cut as the free body leaves a face carrying a direct tension of 100 kN and a moment of N·R = 7.970 kN·m about the section's own centroid, which sits a full R = 79.70 mm from the load line. The stress is a hyperbola, zero at r = 75.04 mm rather than at the centroid 4.65 mm outside it, reaching 497.5 N/mm² of tension at the inner fibre and 281.3 of compression at the outer. The straight-beam formula, drawn dashed, reports 323.4 N/mm² for the bending part against the true 445.5, and leaves the 51.9 N/mm² of direct tension out altogether — between them, 54% under the real peak, at the fibre where a hook actually breaks. This is why a hook is trapezoidal: both effects are worst inside, so the material goes there.

The wide side goes inside

A crane hook's section is a trapezoid with its broad face towards the centre of curvature, and that is not a casting convenience. Turn the same section round — same area, same depth, same moment — and the stress at the fibre that breaks rises by thirty-nine per cent. The shape is doing two things at once, and only one of them is in a straight beam's arithmetic.

sections · Curved beam The coefficient a web gets depends on where its neutral axis is. The plate buckling coefficient for an internal element against ψ = σ₂/σ₁, the ratio of the stresses at the two edges of the panel. Uniform compression is ψ = 1 and k = 4; a gradient running from compression to zero is ψ = 0 and k = 7.81; pure bending is ψ = −1 and k = 23.92, six times the value a column's flange gets. The 1800 × 12 mm web drawn here starts at ψ = −1.000 and k = 23.92 and ends at ψ = −0.910 and k = 21.63, because losing width from the compressed half drops the neutral axis and deepens the compression zone. The curve is steepest exactly where a bending web sits, so a small movement of the neutral axis costs 2.29 of coefficient.

Classified by a gradient it does not have

A web in bending is the one plate whose buckling coefficient cannot be looked up. It depends on the stress gradient, the gradient depends on where the neutral axis is, and the neutral axis depends on how much of the web the coefficient has just taken away — so the answer is a fixed point, and the calculation everyone does is its first term.

stability · Plate buckling A curved I-beam's flanges curl. The cross-section of an I-beam bent to a radius of 3.0 m, its flanges 300 mm wide and 15 mm thick, with the flanges' radial movement exaggerated 47 times; dashed, where the flanges would be if they did not bend across their width. It is bent the way that opens the curve, so the outer flange is in tension and the inner in compression. A tensioned flange round a curve is pulled toward the centre of curvature and a compressed one pushed away from it, so here both are pressed toward the web, and each outstand bends like a cantilever from the web, its tip moving 0.80 mm. Bent the other way, both would curl away from it by the same amount. The bars are the stress along the beam across each flange: the full value at the web, falling toward the tips to 0.72 of it, so that the flange works as if it were 89 per cent as wide.

The flange that curls away from its stress

Bend an I-beam into a curve and each flange, carrying its stress round the bend, is pressed sideways by that stress. The outstands bend like cantilevers from the web, and in bending they move to a different radius and shed the very stress that pushed them. At a tight radius only a strip beside the web works. But the flange's loss of width arrives slowly, and a stress nobody computes for a straight beam arrives at once: the flange bending across its own width, harder than it is stressed along the beam.

sections · Curved beam The flange over the support works over a third of its width. The effective width of the flange as a share of its overhang along two continuous 20 m spans with a flange overhang of 3.0 m each side of the web, the support bearing over 1.0 m, under a uniform load, left out where the moment is under three tenths of its peak (solid), with EN 1993-1-5's factors over the regions they apply to (dashed) and the simply supported span's mid-span value (dotted). At the sagging peak, 7.6 m from the end, the flange works over 0.81 of its width; over the support, over 0.33. EN 1993-1-5 gives 0.83 in the span and 0.34 over the support, from effective lengths of 17.0 and 10.0 m; a single span of 20 m has 0.88 at mid-span.

The reaction that is made of short waves

Run a wide-flanged girder continuously over a support and its worst moment moves to the support. That moment is made by a reaction, a reaction is made of short waves, and a short wave puts its stress next to the web. Over the support of two 20 m spans with a 3 m overhang the flange works over a third of its width, against four fifths at the sagging peak, so the stress at the web there is 4.2 times the sagging peak for a moment only 1.7 times as large — and how much flange works depends on what the support bears on.

sections · Effective width A girder softer to short waves sheds a little of its support moment. For two continuous 20 m spans of a girder whose flanges are 6.0 m wide and 30 mm thick, 1.20 m apart on 24 mm of web — the flanges 97% of its stiffness — under a uniform load, each interior support 1.0 m long: the bending moment along the girder per unit load, sagging up — for a prismatic girder (dashed), which is what a global analysis gives with one effective width per span, and for the girder whose every wave of moment bends it with that wave's own effective flange (solid). The interior reaction falls by 0.88%, and because the support moment is the difference of two much larger moments, that becomes 4.6% of the support moment, 47.0 to 44.8; the sagging peak rises by 2.9%, 28.1 to 28.9.

The reaction moves one per cent and the moment five

Over the support of a wide-flanged continuous girder the flange works over a third of its width, so the girder there should be softer and shed moment into its spans. It does — but it is the reaction that the softness changes, by less than one per cent, and the support moment, a small difference of two large ones, moves by five. The stress spike over the support follows the reaction rather than the moment and eases by two. A prismatic analysis is right about the moments to a few per cent, and the spike stays a problem of the section.

sections · Effective width

Named alongside it

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

Effective widthBox girderContinuityFlangeHarmonicShear lagBending stressCurvatureCurved beamFree bodyNeutral axisSection modulus

All concepts