Structural Engineering & Statics
where the load goes
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Which failure arrives first — page 9

Essays 193 to 216 of 223 on this thread, in the same order.
The studs are evenly spaced and the demand is not. The force per unit length the shear connection carries along half of a 12 m composite beam, from Newmark's solution. It is largest at the support — 282 N/mm — falls to nothing at mid-span, and averages 156: the end studs are asked for 1.81 times the mean. Studs are nevertheless placed at a uniform spacing, and the justification is the one the variable-angle truss uses for its stirrups — a ductile connector sheds what it cannot carry to its neighbours, so the uniform distribution is a plastic redistribution and not a description of the elastic state. Internal forces

The connection is busiest where the beam is not

A composite beam's studs are spaced evenly along it and the demand on them is not even at all. It peaks at the supports, where the bending stress is nothing, and falls to zero at mid-span, where the section is working hardest — so the connection is designed from a diagram nobody looks at.

 The second moment of area is a function of direction. Second moment of area of an equal angle against the angle of the axis it is taken about, with the product of inertia beneath it. The maximum is 5.943 × 10⁶ mm⁴ and the minimum 1.523 × 10⁶, a ratio of 3.90, and they occur where the product of inertia passes through zero — at 45.0° from the drawn axis. The value the drawing suggests, 3.733 × 10⁶, is neither of them. Sections and stress

The axis a column buckles about

A strut buckles about the axis with the smallest second moment of area, and for a section with no axis of symmetry that axis is neither of the two on the drawing. An angle used as a strut is 2.45 times weaker than the number a designer reads off its own dimensions.

 The shear centre of a channel. A channel of 100 by 250, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 38.0 outside the web to leave the section untwisted — a point in the air, outside the material entirely. Sections and stress

The eccentricity a purlin cannot avoid

A channel's shear centre is outside the material, so a load applied anywhere on the section misses it. The distance is fixed by the proportions rather than by the detailing, it is 38 mm on an ordinary purlin, and the torque it produces is not an error anybody made.

 The lag follows the shear, so it is worst at the supports. Effective width along the span of a simply supported beam under a uniform load, summed over 25 odd harmonics. Each harmonic has its own half-wavelength L/n and its own, smaller, effective width — 0.815 for the first, 0.400 for the third, 0.269 for the fifth — and near a support the short harmonics carry a bigger share of what little moment there is. So the working fraction is 0.603 at the support against 0.839 at mid-span, a difference of 23.6 percentage points on the same flange. The single sinusoid's answer, 0.815, is drawn as the flat line, and it is only right at mid-span. This is the behaviour a code reproduces by shortening L_e near a support, and the reason it does is that shear lag is driven by w‴, which is the shear force. Sections and stress

The flange works least where the shear is largest

Shear lag is driven by the shear force rather than by the moment, so the effective width of a wide flange is not a property of the beam. It is a function of position along it, worst at the supports, and a single number quoted for a whole span is right at mid-span and nowhere else.

 One point, every plane through it, one circle. A point carrying 180 N/mm² across one face, 90 across the other and 40 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 60.2 centred at 135.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 195.2 and 74.8, on planes 20.8° from the face the 180 acts on; the largest shear on any plane is 60.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 170.6. Sections and stress

The circle nobody draws

A plane stress state has three principal stresses and the third is zero. When the two on the drawing share a sign, the largest shear in the state involves the one that is not there — and the circle a designer has drawn is not the circle that governs.

 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. Sections and stress

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.

 A 10 mm plate, and the width it can be. The elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 462 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 900 mm only 46 per cent of it is still working. Stability

The coefficient that is not four

A plate's buckling stress carries a coefficient that looks like a constant and is not. It is 4 for an internal element, 0.43 for an outstand and 23.9 for a panel in shear — and the width a 10 mm plate may be runs from 152 mm to 1,130 across that range.

 The load being amplified is not the load doing the amplifying. The sway amplifier 1/(1 − ΣP/P_cr) against the storey's total gravity load, as columns are added that carry load and provide no lateral stiffness. The critical load of the storey is fixed at 8203 kN by the bracing that exists, and every leaning column moves the structure along the axis without changing it. At the storey drawn the amplifier is 1.57, so the second-order sway moment is 57% on top of the first-order one — and none of that 70% of the load which is causing it appears in any stability calculation done column by column. The storey stays stable across the whole of this axis. That is the practical reason a gravity-only column is drawn on the frame model rather than designed on its own: it is not being checked, it is being counted. Stability

Counted, not checked

A column with pinned ends and no bracing cannot buckle on its own, so nothing about it fails a stability check. It still carries load, and load with no stiffness attached lowers the buckling load of everything around it — which is why a gravity-only column is put on the frame model and never designed by itself.

 A base plate, and when the bolts start working. A 550 × 450 mm plate carrying 900 kN and 140 kN·m, so the resultant sits 155.56 mm from the centre against a kern of 91.67 mm. The plate is in partial contact: bearing over 358.33 mm at a peak of 11.16 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 82.5 kN·m and crushes at 192.95 kN·m, and the bolts are not needed until 247.5 kN·m. Stability

The pinned base that is not pinned

A column base drawn as a pin is a plate bearing on grout, and a plate in contact over its whole length resists rotation whether anybody wanted it to or not. The stiffness it delivers depends on the axial load, so the assumption is one a frame can leave and re-enter as its loads change.

 Warping stiffens a short member and nothing at all a long one. The stiffening 1/[1 − tanh(κ)/κ] against kL, both axes logarithmic, over kL from 0.05 to 200. At the low end the curve is a straight line of slope −2, because for small kL the bracket is κ²/3 and the stiffening is 3/kL²: it reaches 1201 at kL = 0.05, falls to 1.005 at the top, and every open section ever rolled sits somewhere on it. The same three plates arranged three ways are marked: the 533 by 190 mm I-section at kL 2.76 and ×1.562, the tee at kL 37 and ×1.027, the angle at kL 34 and ×1.030. A tee's warping constant is 288 times smaller than the I-section's and an angle's 236 times, because their plates meet at a point and there is no pair of flanges to bend against each other — so they have no warping resistance to offer at all, and that is the reason an angle is a poor thing to twist. Stability

The restraint that beats the gradient

A moment-gradient factor is worth up to 2.7 on a beam's critical moment and is tabulated everywhere. Holding the ends against warping is worth more, is achieved by a detail rather than by a load case, and appears in no table at all.

 A channel has three critical loads, not one. The three critical loads of a channel in compression, against its length, with the load it actually buckles at drawn over them. At 3500 mm the flexural loads are 13970 kN about the major axis and 2306 kN about the minor, while twisting about the shear centre takes 1555 kN. The lowest root is 1484 kN, and the column twists. The shear centre sits 109.7 mm from the centroid, so the modes cannot happen separately: the lowest root of the coupled problem is 4.5 per cent below the lowest of the three, and the Wagner coefficient β is 0.60. The governing mode changes at 5813 mm: below that length the column twists, above it, it bends — because the torsional resistance keeps a term that does not grow when the member is shortened, and the flexural loads have none. Stability

The third root of the cubic

A column has three buckling loads and an Euler calculation finds two of them. The third is a twist about the shear centre, and for a section whose shear centre is not at its centroid the three cannot happen separately — so the answer is the lowest root of a cubic and can be a third below anything the two familiar modes report.

 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. Deflection

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.

 The crack length at which the strength stops mattering. Failure stress against crack length for a toughness of 100 MPa√m, with three steel grades drawn. The falling curve is fracture — Kc divided by Y times the root of pi a — and it does not know what the yield stress is. The horizontal lines are the grades. At 275 N/mm² the two cross at a crack 33.5 mm long; At 355 N/mm² the two cross at a crack 20.1 mm long; At 460 N/mm² the two cross at a crack 12.0 mm long. The stronger grade's transition is the shorter one — raising the yield stress does not raise the strength of a cracked member, it only shortens the crack that takes it away. At a working stress of 120 N/mm² the critical crack is 176.0 mm. Materials

Designed to be found in time

A fatigue design that promises a detail will not crack is making a claim about a hundred years of traffic. A damage-tolerant one assumes it will crack, and sets the inspection interval from how long a crack takes to grow from the smallest size anybody can find to the largest the section can survive.

 Which of them stops moving. Three load cases on the same rectangle, each a constant moment plus a temperature profile cycled from nothing to a peak and back, over sixteen cycles. At 20% of the plastic moment with a 40°C profile it never yields at all; At 50% of the plastic moment with a 150°C profile it shakes down; At 90% of the plastic moment with a 260°C profile it ratchets, at 51.0% of the first-yield curvature per cycle. The ratcheting case never collapses and never returns: it simply arrives somewhere further round every cycle, which is a serviceability failure that no collapse calculation contains. Materials

The map with three regions

A structure carrying a constant load and a cycling temperature has three possible fates and only one of them is a collapse. It can stay elastic, it can yield once and then stop, or it can gain a little more deformation every cycle for ever — and the third has no failure load at all.

 Yielding one way makes it easier to yield the other. Mild steel taken to a strain of 1.20% and then pushed back the other way. The stress falls by 550 N/mm² before it yields again, against a yield stress of 275 — the elastic range is twice the yield stress and not once it, which is the Bauschinger effect and is a consequence of the yield surface sliding rather than growing. Materials

Yielding one way, and then the other

A material that has yielded in tension yields earlier in compression than it did the first time, and by an amount that is exactly what makes its elastic range twice its yield stress rather than once it. That is a property no monotonic test reports and every reversing structure depends on.

 Three specimens cannot see the tail. The factor k applied to the sample's own scatter when a characteristic value is estimated from n specimens. With the scatter known in advance it is z·sqrt(1 + 1/n) and barely moves; with the scatter estimated from the same n results it is the Student t quantile instead, and it runs from 7.73 at two specimens to 1.73 at 30. At n = 4 the characteristic strength comes out at 23.8 N/mm² against 27.9 for a population known exactly — 15% lower, for a material that is identical. A small test programme does not report a worse estimate of the strength; it reports a worse strength. Materials

The strength that belongs to the test programme

A characteristic strength is a fractile of a distribution, and a distribution estimated from four specimens is not the same object as one known exactly. The same material tested four times reports a strength 15 per cent below what it reports when its scatter is known — and nothing about the material differs.

 The stress that leaks away. A restrained shrinkage strain of 320 microstrain in concrete of modulus 34000 N/mm². Ignoring creep it produces 10.88 N/mm², which is above the tensile strength of 3.8 and predicts that every restrained concrete member ever cast has cracked. Counting creep by the superposition integral leaves 1.72 N/mm² after 55 years, and the one-line age-adjusted shortcut at the usually quoted ageing coefficient of 0.8 leaves 3.05. The two disagree — this creep function implies an ageing coefficient of 1.67, not 0.8 — and both are below the tensile strength, so the conclusion turns on counting creep at all rather than on how it is counted. Materials

The stress that leaks away

Creep makes a load's deflection grow and an imposed strain's stress shrink, and the second is why restrained concrete does not crack as often as an elastic calculation says. The same material property runs both ways, and which way it runs depends on whether the structure was given a force or a movement.

 Two per cent of the traffic and most of the damage. A 120-year traffic spectrum on one detail of category 71, with each band's share of the cycles and its share of the damage. The two bars have almost nothing to do with one another, and the reason is the slope of three: life goes as the inverse cube of the stress range, so a cycle twice as large does eight times the damage and a cycle a third as large does a twenty-seventh of it. The a full train band is 2% of the crossings and 58% of the damage; the smallest band is 35% of the crossings and, being under the cut-off, does none at all. The equivalent constant range that would do the same damage in the same number of cycles is 25.5 N/mm², which is the one number a designer is usually handed — and it is a cube-weighted average, so it is nearer the heaviest vehicle than to the average one. Materials

The cycles that do not count

A fatigue spectrum has to be reduced to one number, and the reduction is a cube-weighted average rather than an ordinary one. Two per cent of the traffic does most of the damage, a third of it does none at all, and the equivalent range that comes out is nearer the heaviest vehicle than the average one.

 Throat stress round a fillet weld group. A l shape weld group carrying 150 kN at 200 mm from its centroid. The peak throat stress is 2.17 kN per mm of throat, at (139.53, 100); the worst point at maximum radius from the centroid carries 1.99. Checking by radius is wrong here by 9.17%, and points at identical radius differ by a factor of 1. Connections

The radius rule, and where it fails

A weld group under an eccentric load is checked at the point furthest from its centroid, on the reasoning that the stress from the twist grows with the radius. That reasoning ignores the direction the two stresses point in, and for one common shape it misses the peak by nine per cent.

 Prying against flange thickness. The ratio of bolt force to applied force, for a tee stub carrying 140 kN per bolt, as the flange thickness varies. Prying disappears above 31.91 mm and the flange has become a mechanism below 22.75 mm, where the shaded region begins and the bolt has stopped being the thing that decides. Connections

The thickness that decides who fails

A bolt in a tee stub carries more than the load applied to it, because the flange bends and levers against its own edge. How much more, and whether the bolt or the flange is the thing that gives way, are both decided by one dimension — and the two regimes it separates fail in completely different ways.

 A preloaded joint, before and after it slips. Four preloaded bolts at 172 kN each, on one friction face at μ = 0.5. The joint carries 344 kN by friction with the bolts in tension and not in shear at all; past that it slips into bearing and carries 362 kN with the bolts now in shear. Two different mechanisms, one joint. Connections

The hole made bigger so the steel would fit

A preloaded joint carries load by friction, and the friction is reduced by the shape of the hole the bolt passes through — not by how much steel the hole removes, but by a coefficient in a table. An oversize hole costs fifteen per cent of the resistance; a long slot costs thirty-seven. Both are provided because the steel would not otherwise line up.

 Block shear: the metal between the holes. Three bolts in a 9 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 55 mm long. Shear ruptures first, and the capacity is the sum of two different strengths on two different planes: 524.79 kN, of which the shear plane carries 63.03%. Connections

The end that is only a plate

Cut one flange off a beam's end and what is left is a tee. Cut both and what is left is a plate with holes in it — no flanges, no section modulus worth the name, and none of the checks the beam was selected by. Three plate checks replace them, and the one that governs depends on dimensions that appear in no section table.

 Where the drift went. Storey drift at the target displacement, for the same frame with and without a soft ground storey at 50 per cent of the others' stiffness and strength. The regular frame spreads 219 mm over every storey; the soft one reaches 266 mm and puts 5.58 per cent of it into the ground storey against 0.53 next to it — a concentration of 5.9 against 1.7. The roof goes 22 per cent further, and where that extra displacement lands is the whole of the difference between the two buildings. Dynamics

Weaker in one place, and better on every average

Take an eight-storey frame and make its ground storey half as stiff and half as strong. Its ductility demand falls, its first mode carries more of the mass, and its period lengthens into a gentler part of the spectrum. Three global numbers all improve, and the building is the one that collapses.

 A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.697 N/mm² against a resistance of 0.658. Internal forces

Turn the column, and the slab passes

A flat slab that is comfortable under gravity fails its punching check the moment a moment arrives at the column, and nothing about the load has changed. The fix is not more concrete. It is the column's plan shape and, at equal area, which way round it is turned — worth more than adding half again as much column.

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