Structural Engineering & Statics
where the load goes
Concept

Axial force — where it appears

The force acting along a member's own axis, positive in tension and negative in compression. It is the only internal force a two-force member carries, it is what a truss converts bending into, and it is the one internal force whose capacity contains no lever arm.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Two cantilevers, or one wall, and the beams decide which. The deflected shape of a coupled pair of 6 m walls, drawn against the two limits it lies between. Release the coupling beams entirely and the pair is two independent cantilevers, deflecting 111 mm. Make them rigid and it is one composite wall of the full width, deflecting 16 mm — 6.8 times stiffer, because the lever arm between the wall centroids is 8.40 m and everything inside either wall is smaller than that. Real beams of 600 × 350 mm over a 2.4 m opening land at 23 mm and carry 63% of the base overturning as an axial couple rather than as wall bending. The degree of coupling never reaches one, because a beam of finite depth cannot suppress the walls' curvature entirely.

Two walls that agreed to be one

A pair of shear walls with a row of doors between them is the commonest lateral system there is, and it has two readings that differ by a factor of seven. What decides which one applies is a beam 600 mm deep over a 2.4 m opening — and most of the overturning ends up as an axial couple that no bending diagram contains.

internal-forces · Wall coupling A force may be moved anywhere, at the price of a couple. A 80 kN force applied 250 mm off the centreline of a body, and the same force applied ON the centreline together with a couple of 20 kNm. The two systems are equivalent: they have the same resultant force and the same moment about every point in space, so no equilibrium equation written about the body can tell them apart. What they are not is the same loading — the stresses inside the body differ, and they differ over a distance of about the body's own depth. The offset is drawn to a scale that keeps the arrow on the body; the number beside it is the real one.

The moment the beam left behind

A beam reaction is drawn arriving on a column's centreline. It arrives on a cleat a hundred millimetres out from the face, and the difference is a couple that goes into the column and has to be shared between the lengths above and below it. Nothing about it appears in a frame model whose members meet at nodes.

internal-forces · Column eccentricity The corner columns take what the middle ones did not. Axial stress in the columns across one flange face of a 36 by 36 m framed tube, at the base. Plane sections says the flat line: every column on the face at the same distance from the neutral axis, therefore at the same stress. The solved distribution is the curve — 78.5 N/mm² at the corner against 13.4 in the middle, a ratio of 5.88. The middle columns lag because the only route the axial force has into them is the in-plane shear of the spandrel frame, bay by bay from the corner. The face is carrying its resultant on an effective width of 45 per cent, and the tube deflects as though its second moment were 64 per cent of the gross.

The columns that lean

A framed tube carries its wind load by bending the spandrel beams between its columns, and it does it badly — the corner columns take nearly six times what the middle ones do. Tilt the columns instead, so the perimeter is triangulated, and the same shear is carried axially. The concentration falls to 1.21 and the tube recovers most of the stiffness the plan said it had.

structures · Diagrid The worst force in a pile is not at the top of it. A 0.6 m pile 24 m long through ground that is settling, carrying 800 kN at its head. Above the neutral plane the soil moves down past the shaft and the friction acts downward, so the axial force grows with depth; below it the friction acts upward in the ordinary way and the force falls again to the 300 kN the base takes. The maximum is 1282.87 kN at 13.78 m — 1.60 times the load applied, and it is at a depth where nothing is applied, nothing is connected and nothing can be inspected. A pile section chosen for the head load is under-sized by that factor over the middle third of its length.

The ground that hangs on instead of holding up

A pile is driven through fill that has not finished settling. The fill goes down past the shaft, the friction along that length turns round, and the pile is now carrying the soil rather than the other way about. The worst force is not at the head and not at the toe, and nothing at that depth can be seen.

internal-forces · Downdrag 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 Squeeze an I-section and it loses its corner. Moment against curvature for an I-section of 3,000 mm² and 200 mm depth in mild steel, the moment over its unloaded plastic moment and the curvature over its unloaded first-yield curvature, at axial compressions of 0, 30, 60 per cent of its squash load (dots: first yield). At 0 per cent it first yields at 0.92 and reaches 98 per cent of its plastic moment, 1.00, at 1.2 times the unloaded first-yield curvature; at 30 per cent it first yields at 0.64 and reaches 98 per cent of its plastic moment, 0.75, at 2.3 times the unloaded first-yield curvature; at 60 per cent it first yields at 0.37 and reaches 98 per cent of its plastic moment, 0.44, at 6.4 times the unloaded first-yield curvature. Unloaded, the I-section turns its corner almost at once; squeezed, its web is yielding in compression before any moment arrives and its flanges have to be strained far past yield to carry what is left.

The hinge a squeezed section has to work for

Unloaded, a rectangle carries half again its first-yield moment once it is fully plastic and an I-section only a little more, and the I-section gets there almost at once while the rectangle has to bend four times further. Put a column's axial load on them and both rankings move. The rectangle's reserve grows as 1.5(1 + n), to nearly three times its first-yield moment under heavy compression; the I-section's barely moves. And the I-section loses the thing that made it the beam everyone uses: under three-fifths of its squash load it needs more than five times the curvature to develop its plastic moment that it needs as a beam, because the compression has yielded its web before any bending began.

materials · Moment-curvature Squeezed while it bends, the hinge rides down its plastic moment. Moment against curvature for an I-section of 3,000 mm² and 200 mm depth in elastic–perfectly plastic steel, the moment over its unloaded plastic moment and the curvature over its unloaded first-yield curvature: at a constant 10% of its squash load, at a constant 60%, and with the compression rising from the first to the second as it bends (dashed: the plastic moment at the compression the rising path has reached). At constant loads it settles at 0.96 and 0.44. The rising path carries 0.86 at twice the first-yield curvature, 0.70 at 6.0 times it and 0.43 at 12.0: it forms its hinge early, at the lower load, and then loses moment as fast as the plastic moment falls with the load.

The hinge squeezed while it turns

The leeward column of a frame overturning in an earthquake gains compression as its base hinge rotates. Followed fibre by fibre, with every fibre's plastic strain remembered, the hinge does neither of the things the constant-load curves suggest. It forms early, at the low load it started with, and then rides down the section's plastic moment as the load rises: an I-section whose compression goes from a tenth to six-tenths of its squash load holds 0.86 of its unloaded plastic moment at twice its first-yield curvature and 0.43 at twelve times it. The steel never softens, and the hinge still has a falling branch, set by statics. The relieved column on the other side gains what this one loses, less the curvature of the interaction curve.

materials · Moment-curvature

Named alongside it

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

Beam-columnDuctilityLoad pathStiffnessCompatibilityDriftEccentricityLateral systemMoment-curvatureMoment distributionOverturningPlastic hinge

All concepts