Structural Engineering & Statics
where the load goes
Concept

Secondary moment — where it appears

A bending moment produced by the geometry of a structure rather than by an applied load — an eccentricity, a deflection, a member that is not where it was drawn. It is absent from a line-model analysis by construction and it has to be found in the setting-out or in a measurement.

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

A reaction with no load, and the moment it bends the beam with. The prestress moments in a 2-span beam. The primary moment is −P·e, the tendon acting on its own section, and it reaches 540 kNm over the middle support. The secondary moment is what is left when the primary is taken off the total, and it is 306 kNm — 57% of the primary, with the same sign, so it does not cancel anything. It comes from the middle support refusing to let the beam lift: 51.0 kN pressing down there and 25.5 kN lifting at each end, a reaction set that sums to -2e-13 because nothing external was applied. Its diagram is straight between supports to 2.0e-13% of its own peak, which it has to be: reactions are point forces and a point force puts no curvature in a span.

The prestress that pushes back

On a simply supported beam a tendon is an internal matter and changes no reaction. Put the same beam on three supports and the tendon lifts it off the middle one, the support refuses, and the force it takes to hold the beam down is a reaction produced with no load applied at all.

internal-forces · Secondary prestress The moment a joint makes out of geometry. Bending stress in the chord divided by the axial stress already in it, against how far the diagonal's working line misses the node. A truss analysed as pin-jointed is drawn with its members as lines through their own centroids meeting at a point, because that is what makes the joint a pin with nothing but forces in it. A fabricated joint is not obliged to oblige, and the resultant then has a moment about the node — 16 kNm at the 60 mm drawn. It divides between the members meeting there in proportion to their rotational stiffnesses, 4EI/L or 3EI/L, which is the moment-distribution rule applied to an unbalanced moment that came from geometry rather than from load. The chord takes 48 per cent of it and carries 25 per cent of its axial stress again in bending. Ten per cent arrives at 28 mm, which on a chord of any depth is a detailing decision rather than a mistake.

The joint that is not where it was drawn

A truss is drawn as lines through the centroids of its members meeting at points. A fabricated joint is under no obligation to oblige, and when the working lines miss each other the resultant has a moment about the node that has to go somewhere.

connections · Noding eccentricity A tubular K joint with a 12.6 mm gap. A 168.3 × 8.0 mm chord with two 114.3 × 6.3 mm braces at 45°, their toes 12.6 mm apart on the chord's crown. The braces' working lines, carried down through the chord, meet 3.0 mm below the chord's axis — an eccentricity of 0.02 chord diameters, inside the band of −93 to +42 mm in which the joint rules let its moment be neglected. Nobody chose that number. It follows from the gap, the brace diameter and the angle, and the gap was chosen so both toes could be welded. The working lines would meet on the chord's axis at a gap of 6.7 mm, which is a joint no rule permits.

The joint whose lines may not meet

In a welded tubular truss the noding eccentricity is not a fabrication error. It is fixed by the brace diameter, the brace angle and the gap between the toes, and the gap is chosen for the welder. For a third of ordinary proportions the one gap at which the working lines would meet lies in a band no fabrication rule permits — so the concurrent joint the truss was analysed with is the one joint that cannot be built.

connections · Noding eccentricity The room left between two planes of braces. The surface of a 168.3 mm chord unrolled flat — along the chord across, round it up — with the footprints where 114.3 mm braces at 45 degrees are welded to it: two from a K joint in one plane (solid), two from a K joint in a plane 90 degrees round the chord (dashed), both nodes at the same section, each with a 12.6 mm gap between its own toes. The narrowest gap between a brace of one plane and a brace of the other is 6.5 mm, against the 12.6 mm two welds need: too little to weld. Each footprint spans 85.6 degrees of the chord's circumference.

Two planes of braces and one circumference

A tower leg or a triangular girder's chord receives K joints from two planes at once, and each can be checked as a planar joint: its gap, its eccentricity, its band. What the planar checks cannot see is that the two planes' braces are welded to the same tube. A 114 mm brace takes up 86 degrees of a 168 mm chord's circumference whatever its gap, so a square tower's two planes leave 6.5 mm between them where the welds need 12.6, and a triangular girder's braces land on each other. And two eccentricities at the edge of their band give the chord √2 or √3 times the moment the band was written for.

connections · Noding eccentricity The order of building decides which supports keep the secondary moment. The secondary moment at each interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built and stressed from one end, from both ends towards a closure bay, from the middle outwards, and all at once. From one end: 161, 175, 160, 219 kN·m; from both ends: 161, 172, 172, 161 kN·m; from the middle out: 215, 120, 120, 215 kN·m; all at once: 344, 258, 258, 344 kN·m.

The beam built towards its own middle

A continuous prestressed beam built one bay at a time from one end keeps about half the secondary moment a beam stressed at once would carry, and most of it at the end where it was finished. Built from both ends towards a closure bay, it is symmetric, and the closure's supports keep only a little more than the rest: 172 kN·m against 161. Built from the middle outwards, the first interior supports — the ones the load hogs hardest — keep 215, 63 per cent of the at-once value, for any length of beam. The order of construction is a design variable. Creep then takes back most of the choice.

internal-forces · Secondary prestress A jack set for the first day is wrong by the last. The secondary moment over the first interior support of a five-span beam of 16 + 16 + 16 + 16 + 16 m prestressed with 2,400 kN on a parabolic tendon, built from both ends towards a closure bay, against the creep coefficient: not jacked; jacked at closure by the whole of what the order took, 183 kN·m there; jacked by 0.60 of it, so as to reach the at-once moment at a creep coefficient of 2.0; and by 0.89, so that the shortfall on the day and the overshoot at the end are equal; dotted, stressed at once, 344. On the day and at 2.0: not jacked, 161 and 302; jacked by the whole, 344 and 372; for the end, 271 and 344; balanced, 324 and 365.

The jack that creep carries past the mark

A continuous beam built from both ends towards a closure bay keeps less secondary moment at its supports than the same beam stressed all at once, and creep gives back three-quarters of the difference over the years. Bridge builders do not always wait: before the closure is cast, the supports can be lowered on jacks so that the beam starts with the moments it would have had. Jacked by the whole difference, it does start there — and creep, which goes on returning the staged deficit while it relaxes the jacked correction, carries it past them, by a seventh of what was jacked. Jacked by 0.89 of the difference, it starts a ninth short and ends a ninth over. And it takes a jack at every interior support: with fewer, the moment goes to the wrong places.

internal-forces · Secondary prestress

Named alongside it

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

EccentricityContinuityDetailingPrestressWorking lineBeam-columnConstruction sequenceCreepHollow sectionAxial forceCarry-overConcordant profile

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