Structural Engineering & Statics
where the load goes
Concept

Equivalent load — where it appears

A set of applied forces that produces exactly the internal forces of something that is not a force at all. A curved tendon is replaced by a distributed load of Pκ along its length and a pair at its anchorages, which turns a prestress into a quantity that can be traded against the loads.

Named by 7 essays across 3 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 tendon is a load, pointing the other way. A 14 m beam with a parabolic tendon dropping 260 mm to midspan, stressed to 1440 kN after losses. Its curvature pushes the beam up along its whole length with an intensity of 8Pe/L² = 15.28 kN/m, against an applied 17.63 kN/m — so 2.34 kN/m is left to bend anything, and the beam carries 57.4 kNm where an unstressed one carries 432 kNm. What the section then feels is 6.40 MPa of uniform compression and very little else.

The load that comes from changing direction

A force that travels in a straight line asks nothing of anything. Bend its path and it asks for a transverse load of F over R along every millimetre of the curve, and that load is real, is nowhere on the load schedule, and is the same statement behind a prestressing tendon, a hoop force, an arch thrust and a web that buckles with nothing applied to it.

internal-forces · Deviation force Four details, and no material anywhere on the plot. Stress range against cycles to failure for four detail categorys — 160, 112, 71, 36 N/mm² at two million cycles. The lines are parallel because they share a slope of three, and the spread between them is a factor of 4.4 in stress and therefore 88 in life. Nothing on this plot depends on the strength of the steel: the same detail in a grade twice as strong lies on the same line. At a stress range of 62 N/mm² the lives are 160: unlimited, 112: 2.1e+7, 71: 3.0e+6, 36: 3.9e+5 cycles. The knee in each line is the constant-amplitude limit, past which the slope becomes five.

The detail decides and the steel does not

A fatigue check contains no material strength anywhere. The same detail in a steel twice as strong lies on exactly the same line, because a fatigue life is decided by the geometry of a weld and by the stress range it sees — and two per cent of the traffic does most of the damage, because life goes as the inverse cube of the range.

connections · Detail category 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 1440 kNm over the middle support. The secondary moment is what is left when the primary is taken off the total, and it is 720 kNm — 50% 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: 102.9 kN pressing down there and 51.4 kN lifting at each end, a reaction set that sums to 0e+0 because nothing external was applied. Its diagram is straight between supports to 3.6e-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 tendon that can be moved

Lift a continuous beam's tendon at its interior support without changing its drape and nothing about the beam's total moment changes. The primary falls, the secondary rises by exactly as much, and the pressure line stays where it was — which turns a parasitic effect into a quantity a designer can place.

internal-forces · Secondary prestress Two load sets with the same resultant and different work. The two ways of putting a uniform load of 10 kN/m onto a beam element 6.00 m long. Both put 30.0 kN at each node, so both have the same resultant and the same moment about any point — they are equivalent for a rigid body. The consistent set adds a couple of 30.0 kN·m at each end, in opposite senses, which is what makes it do the same virtual work over the element's shape functions as the real load does. The couples cancel in the resultant, which is exactly why the resultant cannot see them, and they are the whole difference between an exact answer and one that is a third out.

Equivalent in work, not in resultant

Two force systems with the same resultant and the same moment about every point are interchangeable — for a rigid body. A finite element is not a rigid body, and substituting one for the other on a beam element leaves the tip of a cantilever a third too low with no warning of any kind.

equilibrium · Force couple A downward pressure that pulls the corners up. The nodal forces that do the same work as a uniform pressure on one eight-node serendipity quadrilateral, as fractions of the whole load, each node's share being the integral of its shape function over the element. The shares are −1/12, −1/12, −1/12, −1/12, 1/3, 1/3, 1/3, 1/3 at its eight nodes, adding up to one. The four corner forces are negative: the pressure pushes down and the work-equivalent forces at the corners pull up.

The corners the pressure pulls up

Replace a uniform pressure on a beam element by its work-equivalent nodal loads and the ends acquire couples the resultant cannot see. Do the same on an eight-node plate element and the corners acquire forces pointing the wrong way: a pressure pushing down is represented by the corners being pulled up, a twelfth of the load each. It is correct, it is the vector that makes the solution the best one available, and it is the reason a printout of nodal forces is not a picture of where a load goes.

equilibrium · Force couple The shear is read from the pressure line, not the tendon. The first 9.6 m of a two-span beam of 16 + 16 m, 400 × 900 mm, prestressed with 2,400 kN on a parabolic tendon and carrying 45 kN/m, with depths drawn from the top face (−450 mm) to the bottom (+450 mm): the tendon's centre (dashed) and the pressure line, where the resultant of the prestress actually acts once the supports have pushed back (solid). 0.8 m from the support the tendon falls 87.00 mm per metre, which is the slope the rule P·sin θ reads, worth 209 kN; the pressure line falls 76.39 mm per metre, worth 183 kN. The difference is the secondary shear, 26 kN, which the supports put back.

The relief that belongs to the pressure line

An inclined tendon carries part of a prestressed beam's shear, and the rule is to subtract its vertical component, P times the slope of the tendon. In a continuous beam the tendon also provokes reactions, and the reactions make a shear of their own — a constant in each span, too small to look at. It is not small where it matters. Beside the end support of a two-span beam it doubles the shear left for the links, and moving the tendon without changing the beam at all can make the tendon's slope report that there is none.

internal-forces · Secondary prestress

Named alongside it

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

PrestressConcordant profileContinuityFree bodyIndeterminacyPressure lineEccentricityFinite-elementLoad balancingReactionResultantSecondary prestress

All concepts