Structural Engineering & Statics
where the load goes
Concept

Harmonic — where it appears

One sine wave in a sum that rebuilds a load or a moment along a member. Each wave can be solved on its own and the answers added, so a property that depends on wavelength, such as how much of a flange a stress reaches, can be found for any load from its mix of long and short waves.

Named by 2 essays across one field — 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 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

Named alongside it

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

Box girderContinuityEffective widthFlangeShear lagStress distributionEmpirical ruleFree bodyNet sectionPlane sectionsSaint-Venant's principleShear flow

All concepts