Structural Engineering & Statics
where the load goes
Concept

Locked in stress — where it appears

Stress carried by a section from a stage of loading that reached it before the section had its final shape or properties. It adds to later stresses and cannot be recovered by dividing the total load by the final section modulus, so a propped and an unpropped member differ although nothing about the finished drawing does.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 12 kN/m of wet concrete and 18 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 292 MPa; propped, the finished composite section takes everything and reaches 186 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.

The structure that was never complete

Every analysis in this collection is of a finished structure loaded once. Real ones are built in pieces, and each piece carries whatever was present at the moment it became structural — so the stress in a member depends on when it arrived, which appears nowhere on any drawing.

structures · Construction sequence The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 6 kN/m of wet concrete and 9 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 146 MPa; propped, the finished composite section takes everything and reaches 93 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.

The section that changed while it was being loaded

A stress is computed from a moment and a section modulus. When part of the moment arrived while the section was a different shape, there is no single section modulus to divide by — the stresses add and the properties do not, and two identical finished beams can differ by half again in stress with nothing on the drawing to say which is which.

sections · Staged section Same deck, same load, and two pier forces. A deck bearing on a pier of 20 kN/mm with μ = 0.03, taken to the same final state two ways: the deck moves 4 mm over the pier, and the bearing's load rises from 2000 to 4000 kN. Moved first, while the bearing carries 2000 kN, the pier force reaches the limit of 60 kN and the bearing slides for the rest of the movement; the load arriving afterwards raises the limit and changes nothing, and the pier is left carrying 60 kN. Loaded first, the limit is 120 kN before the deck moves, the bearing grips throughout, and the pier carries 80 kN. Both states are at the same displacement under the same load, and both satisfy equilibrium and the friction bound; the order is the only difference, and it appears in neither.

The order the loads arrived in

Statics allows a contact with friction a whole range of forces and has no way to choose between them. A real structure does choose, and what it chooses by is the order in which things happened to it — so the force in a pier under a sliding bearing is a record of its history, not a function of its loads.

equilibrium · Friction After a yield, the prestress left does not remember the prestress put in. The prestress left in the tendon of a 2.0 × 8.0 m rocking wall of 400 kN once it has rocked to each rotation and come back upright, for a tendon yielding at 1,370 kN and stiffening by 20 kN a millimetre as the base opens, prestressed to 300 kN, 600 kN and 900 kN. Each keeps all its prestress until it yields — at 53.5 mrad, 38.5 mrad and 23.5 mrad — and then loses 20 kN for every further milliradian, so past 53.5 mrad the three lines are one: the yield force less the stiffness times the stretch. Every one of them is slack at upright past 68.5 mrad. With bars of 400 kN the wall re-centres only while 400 kN remains, dashed, and it keeps that only up to 48.5 mrad, whatever it was prestressed to.

The tendon that forgets its prestress

A rocking wall comes home because a tendon pulls it, and the tendon must not yield — yet the rotations that test the wall are exactly the ones that stretch it. Past its yield, the force a tendon keeps is its yield force less its stiffness times the stretch, whatever it was prestressed to, so the guarantee a design needs is a limit on rotation, and more prestress only reaches that limit sooner.

dynamics · Rocking The roller's pressure, not the soil's. Horizontal pressure down the back of a wall, for backfill of 20 kN/m³ and φ′ = 30° behind a wall 3.0 m high, compacted by a roller of 60 kN per metre of its width. Dashed, the soil's own active pressure, Kₐ·γz (Kₐ = 0.33), and at-rest pressure, K₀·γz (K₀ = 0.50). Solid, Ingold's compaction envelope: rising as the passive limit γz/Kₐ to 27.6 kPa at 0.46 m, then constant at √(2pγ/π) = 27.6 kPa all the way down, since the active pressure would not reach it until 4.15 m. Its thrust is 76.6 kN per metre of wall against the active 30.0, and its moment about the base 106.3 kN·m against 30.0.

The pressure the roller leaves behind

A retaining wall is designed for the pressure of the soil it holds, a triangle growing with depth. The soil behind most walls was not tipped there. It was placed in layers and rolled, and each pass of the roller pushed the soil sideways against the wall harder than the soil's own weight ever could, and left most of that push locked in when it moved on. Behind a 3 m wall compacted by an ordinary vibrating roller the locked-in pressure is a constant 28 kPa from half a metre down, the thrust two and a half times the active triangle's and the moment at the base three and a half.

equilibrium · Lateral pressure

Named alongside it

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

Construction sequenceSuperpositionComposite actionCreepEnergy dissipationPrestressActive pressureAt rest pressureBackfillBearingCamberCompaction

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