Structural Engineering & Statics
where the load goes
Concept

Span-to-depth — where it appears

A member's span divided by its structural depth, which is the single ratio that fixes how stiff it will be. Choosing it is choosing the deflection, long before any material property is decided.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

Two curves climbing together, and the one that catches up first. A 6 m member tapering from 200 to 600 mm, with the moment it carries and the moment it can carry drawn on the same scale below it. The demand rises linearly and the capacity as the square of the depth, so the gap between them closes and then opens again. It is narrowest at 3.00 m from the free end, where the member is 400 mm deep and 79% used, against 70% at the root where the moment is largest.

The section that changes along the span

A prismatic beam is checked where the moment is largest, and everyone knows where that is. A tapered one is not, because the capacity is moving too — and for a cantilever with a load at its tip the governing station is exactly where the depth has doubled, with no length, no load and no material in the answer.

sections · Tapered member Whether the floor shares the load out by stiffness or by area. The share of a uniform storey force taken by each of three equally stiff walls, against the stiffness of the floor plate that spans between them. A plate far stiffer than the walls translates almost rigidly, every wall deflects the same and the share is the ratio of stiffnesses — 60% to the middle wall here, the same as everyone else. A plate far softer than the walls behaves as a continuous beam over them and the middle wall takes 33%. Neither end is the tributary-area answer of 50%, which assumes a plate that is both soft and discontinuous over the wall, and which nothing here ever reaches.

The floor is a beam lying down

A floor plate spans horizontally between the walls that resist a lateral load, carries a distributed inertia load, and has chords, a web and a span-to-depth ratio like any other beam. Its stiffness decides whether the walls share the load by their stiffness or by the area of floor nearest them — and the familiar tributary answer turns out to be neither limit.

structures · Diaphragm Strength at an angle, and the straight line that is not it. Compressive strength against the angle between the load and the grain. Hankinson's formula — f₀f₉₀ ÷ (f₀sin²α + f₉₀cos²α) — is an interpolation rather than a failure theory, and what makes it worth having is how far it sits from the straight line anyone would otherwise draw between 21 and 2.5 N/mm². At forty-five degrees it gives 4.5 N/mm² against the line's 11.8: 38 per cent of it, and 21 per cent of the strength along the grain. The curve drops away in the first twenty degrees because the weak direction starts governing as soon as it has any component at all, which is the same arithmetic as a section's weak axis and the reason a skewed bearing detail is a real loss rather than a small one.

The material that has a direction

Every material in this collection so far has had one modulus and one strength. Timber has three of each, differing by more than an order of magnitude, and the consequence is not a correction to steel design — it is a different set of checks with a different one governing.

materials · Anisotropy The chords take the shear the web is credited with. A cantilever of 6 m tapering from 400 to 1200 mm, under a 120 kN tip load, with the shear divided between the web and the two inclined chords. The chord force is M/z and it is not horizontal, so its vertical component is real: V_web = V − (M/z)·dz/dx, which is Résal's result of 1899. At the root the web is left with 33 per cent of the applied shear — exactly d₀/d₁ for a straight taper, with no length, load or material in it. Turn the same member round and the figure becomes 300 per cent: the chords stop helping and start adding, at the section where the moment is largest as well. And the limit is worth having — a member whose depth is proportional to its moment leaves the web nothing to do at all, which is the triangular cantilever every crane jib is.

The shear the chords take

Every shear check in this collection has assumed the two chords of a beam are parallel, so that the whole of the shear crosses the web. Taper the member and that stops being true — and the sign of the correction is decided by which end the haunch is at.

internal-forces · Inclined chord A fan, and where its forces go. Half a cable-stayed bridge: a tower 70 m above a deck, 12 stays reaching out over 200 m, and a uniform 200 kN/m on the deck. Each stay is drawn at a weight proportional to the force in it, from 3427 kN at the innermost to 10090 kN at the outermost — the outer stay carries the same vertical share and is far flatter, so it carries far more. The deck's shading is its own accumulated compression, 61905 kN at the tower, which is 1.55 times the load being lifted and is the horizontal half of every stay force added up. Nothing in this drawing is a catenary: every stay is straight and every one of them is a spring.

The cable that is a spring

A suspension bridge's cable is a funicular — it takes the shape the load gives it, and the deck's job is to make the load one that shape is right for. A stayed deck is not that at all. Its cables are straight, each reaches the deck at one point, and every one of them behaves as an inclined spring.

structures · Cable-stayed The check that depends on a date. Total deflection and the deflection occurring after the brittle finishes are built, for one 12 m beam, against the day those finishes go up. The total barely moves — the beam ends up where it ends up. The increment falls from 32 mm at a week to 14 mm at a year, because creep is fast at first and slow later and a partition built early inherits nearly all of it: 44% of the final creep has already happened by day 28. The span/500 limit is 24 mm and the span/250 limit is 48; this beam passes the first only after day 25. Camber subtracts from both terms of the difference and therefore changes the upper curve and not the lower one, which is the reason a cambered beam can satisfy every total-deflection check and still crack the wall.

The limit that depends on a date

Total deflection can nearly always be met, and on a long span it is met with camber. The limit that actually decides the member is the other one — the deflection occurring after the brittle finishes are built — and camber does nothing for it whatever, because it is subtracted from both terms of a difference. The same beam passes or fails on the day the partitions went up.

deflection · Incremental deflection Every level added is a longer span, not a shorter one. Steel per square metre of floor against the number of levels in the hierarchy, for a 12 m bay with a deck that can span 3.0 m. A bending level's weight per unit area is 3ρqrL/8σ — it contains the SPAN and not the spacing — so breaking a floor into more levels cannot make the members lighter by making them closer together. It adds one more system, and the last system always spans the whole bay: a second level costs 52 per cent more steel than one, and a third 107 per cent. The structural zone grows with it, 730 mm to 1346 mm. Hierarchy is not an economy, it is a way of reaching, and it is paid for in both currencies at once.

Every level is a longer span

A floor is a hierarchy — deck to joists to beams to girders — and the reason usually given is that breaking a long span into short ones saves material. A bending level's weight per square metre contains its span and not its spacing, so it does not.

structures · Hierarchy A short timber beam is a shear problem, and a steel one never is. Utilisation of the bending and shear checks on one beam, against span-to-depth. The two cross where the ratio equals f_m ÷ f_v exactly — no load, no width and no span survives the cancellation — which for this timber is 6.7 and for steel is 1.73. So a timber beam shallower than about six times its depth is governed by shear parallel to the grain, and a steel beam would have to be shorter than twice its own depth before the same thing happened, which is not a beam. The third check is bearing across the grain, which does not move with the span at all: on the beam drawn it is at 0.40, and it is the one that governs.

The shear that decides a timber beam

A steel beam is never governed by shear, because its bending strength is only 1.73 times its shear strength and no beam is that short. Timber's ratio is 6.7 along the grain and 23 across it, so shear governs at proportions people build every day.

sections · Shear flow The shear deflection is not a correction. The share of a sandwich panel's deflection that is shear rather than bending, against how slender the panel is. A solid beam at a span-to-depth ratio of 20 spends about a per cent of its deflection on shear; this panel spends 25% at the same ratio, because its core is 2800 times softer in shear than its faces are in tension. At the 37 of the panel drawn it is 9%. The curve falls as the square of the span because bending grows as the fourth power and shear as the second, so the term that is negligible for a long panel is the whole answer for a short one.

The stiffness that belongs to the span

A section's flexural rigidity is a property of the section, and a member's is not. Once shear deformation is counted the effective stiffness contains the span, so the same panel is a different member at three metres and at six — and for a sandwich the correction is not a correction.

deflection · Shear deflection Soft in shear, the wall gives up moment to the span. The bending moment along a 300 × 600 rectangle fixed at its left end and propped at its right, under a uniform load, per unit load and span, at span-to-depth ratios of 4, 2, 1, and for the same beam treated as rigid in shear, dashed. Rigid in shear the wall carries 0.125 wL² and the prop 0.375 of the load. Counting the shear it deforms by, the wall moment falls to 0.119 wL² at 4, 0.105 wL² at 2, 0.070 wL² at 1, and the prop's share rises to 0.381, 0.395, 0.430. The load has not changed and the beam is no weaker: its forces have moved, because in a redundant beam they come from how it deforms.

The shear that moves the moments

In a statically determinate beam, shear deformation adds movement and changes no force. In a redundant one the forces come from how the beam deforms, so a stubby beam soft in shear carries less moment at its wall and more in its span — and the carry-over factor, the one half every hand method passes along, falls to nothing at φ = 2 and then changes sign.

deflection · Shear deflection One quantity, three names. The middle braced bay's share of the storey force (solid) and one end bay's (dashed), for a 48 by 12 m plate, 200 mm of concrete, spanning two 24 m bays between three braced bays of 100,000 kN/m each, under 40 kN/m, against a factor on the plate's stiffness as built, on a logarithmic scale. The share moves continuously from 61 per cent, a continuous plate on unyielding supports, to 33 per cent, the rigid plate. Left of 0.02 (shaded) ASCE 7 calls the plate flexible and assigns tributary shares, 50 per cent to the middle; the plate there actually gives it 59 per cent at the boundary. Right of 1.54 (shaded) EN 1998-1 calls it rigid and assigns stiffness shares, 33 per cent; at that boundary the plate gives 37. The plate as built, factor 1, gives 38 per cent to the middle and 31 to each end.

Rigid by one code and not by the other

A floor plate between shear walls shares its storey force out in proportions that move continuously with its stiffness, from a continuous beam's to a rigid body's. The codes put it in one of two boxes and use a different model in each. ASCE 7 and EN 1998-1 draw their boxes' edges a hundred times apart in plate stiffness, each at a point where its own licensed model already misstates the middle wall by about a tenth — and an ordinary 200 mm slab sits in the gap between them.

structures · Diaphragm

Named alongside it

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

Shear deflectionFree bodyDeflectionLoad pathShear modulusTributary areaAnisotropyChord forceConstruction sequenceDiaphragmEquilibriumFlexural rigidity

All concepts