Structural Engineering & Statics
where the load goes
Concept

Continuous restraint — where it appears

A lateral restraint spread along a compression member rather than applied at points. Sheeting on a purlin, a deck on a flange and the U-frames along a through girder's chord are all examples, and smearing them into a stiffness per unit length turns the member into a strut on an elastic foundation, whose critical force stops depending on how long it is.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Also named here as u-frame — the same set of essays touches all of them, so they are one junction rather than several.

The restraint chooses the buckling length, and it is not the member's. A compression flange 12 m long held sideways not at points but everywhere, by a restraint of 0.35 N/mm per mm of length. Unrestrained it would buckle at 173 kN in a single half-wave, drawn faintly. Restrained it buckles at 1968 kN — 11.4 times as much — in two half-waves, because the sum n²π²EI/L² + kL²/n²π² has its minimum there and every other n is worse. The effective length that answer implies is 3555 mm, which is 0.30 of the member and is a property of the restraint rather than of the span.

Held everywhere, and it forgets its length

A brace at a point divides a member's buckling length. A restraint spread along the whole member does something else — the member chooses its own number of half-waves, and past a few of them the critical load stops depending on the length at all.

stability · Continuous restraint The same chord, buckling under a constant force and under a varying one. Two buckled shapes of the same 40 m compression chord on the same continuous restraint of 0.35 N/mm per mm, at the same scale. Under a constant force the buckle fills the member — 8 half-waves over 98 per cent of the length — and the critical force is 1881 kN, which is the length-free answer the closed form gives. Under the parabolic force a uniformly loaded deck delivers to it, the buckle LOCALISES: 5 half-waves over 56 per cent of it, gathered where the force is largest, and the peak force at buckling is 2113 kN. The shaded curve is the force distribution the second shape is buckling under.

The buckle that will not spread out

A compression chord on a continuous restraint chooses its own number of half-waves and forgets how long it is. Give it the force it actually carries — a parabola, largest at midspan — and the number barely moves while the shape changes completely, which is the half that decides where the restraint has to be.

stability · Continuous restraint The same restraint, spread and gathered. The buckled shape of a 24.0 m compression chord with the same smeared restraint, 0.35 N/mm per mm, delivered by U-frames at two spacings. With frames every 2.0 m the chord buckles in half-waves of about 5.1 m that ignore the frames — the smeared shape — at 1892 kN, matching the smeared answer. With frames every 4.0 m it buckles between them, with a node at every frame, at 1548 kN: 18 per cent below the smeared 1897 and at the Euler load of one bay. The dots are the frames.

A row of frames is not a foundation

The top chord of a half-through girder is held sideways by U-frames, and the standard calculation smears them into a continuous elastic foundation. That is right while the frames are close and quietly wrong once they are not. The crossover sits at seven-tenths of the buckle's own half-wavelength, whatever the frames' stiffness — and beyond it the chord buckles between frames at a load no stiffening of the frames can raise.

stability · Continuous restraint The bow the frames have to hold. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN critical at 1548 kN, drawn with its sideways movement exaggerated; its critical load is 1.66 times the chord force. Dashed, the worst of the bows tried: four half-waves, each 12.0 mm high, a five-hundredth of its own length. Solid, the chord once the compression has grown the bow; each arrow is the force the frame there must supply, the largest 13.6 kN — 1.47 per cent of the chord force.

What the frames hold is the bow

A row of U-frames is sized for a force, and the rules give that force as a percentage of the compression chord's force that nobody derives. Derived, it is not a property of the frames at all. It is the chord's initial bow, grown by how near the chord works to buckling — and it is least, not greatest, near the stiffness at which the chord stops using the frames. Stiff frames tend to a limit that is π²/500 of the chord force: the familiar two per cent, arriving from a bow of one five-hundredth.

stability · Continuous restraint The deck moves the frames' feet, and the chord goes with them. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, in plan, sideways movements drawn 6 times the scale of the length: each frame's foot pushed 7.5 mm inward by its cross-girder's rotation under the deck (arrows), the chord's movement under that push alone (solid), and under the worst bow alone, four half-waves at 12.0 mm (dashed). The chord follows the push, and under its compression goes a little further — 9.1 mm at its middle frames against the 7.5 mm the feet moved — because a chord 24 m long is far softer sideways than the frames holding it; the frames that carry most are the end ones, beside the chord's fixed ends.

The frames the deck pushes first

A half-through girder's U-frames are the ends of the cross-girders that carry its deck. Load the deck and every cross-girder sags, its ends rotate, and every frame's foot swings inward by millimetres before the compression chord has asked the frames for anything. It looks like a second imperfection on top of the chord's bow. It is a much weaker one: the frames all move together and the chord simply goes with them, so a 7.5 mm push adds 1.4 kN to frames that the bow loads with 13.6, and stiffer frames make it smaller, not larger. What the push leaves is mostly in the frames' own members, not in what they ask of the chord.

stability · Continuous restraint

Named alongside it

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

U-frameBracingBucklingElastic foundationImperfectionAmplificationCritical loadEffective lengthEigenvalueHalf-wavelengthMode shapeSecond-order

All concepts