What is refuted here, and by what
Most subjects leave a reader ignorant. This one leaves them taught, which is a harder starting position, because a sentence that has to be dislodged first is doing damage that silence would not. Almost everybody who reads this site already knows that a stronger steel is a better steel, that Euler's formula gives a column's capacity, that a truss joint is a pin, and that an extra support is a free margin of safety. Three of those are false and the fourth is a simplification whose direction of error nobody was told.
So the wrong explanations are treated as content rather than as omissions. Each one below is stated in the strongest form it is usually met in, given a verdict, and settled by a number that came out of a solver on this site rather than out of a better textbook. The verdict is what the computation returned.
The four verdicts are not interchangeable, and the fourth is the one this subject needs. A great deal of structural engineering is taught as a simplification that errs safe — and then repeated for long enough that the label falls off. Those are not misconceptions and they are not harmless: each is chosen for a case in which it is conservative, and each has a direction in which it stops being so. Naming that direction is the only thing that turns a habit back into a decision.
1154 claims in all, in four groups. Each group runs in the order the essays do, and the list continues across the pages linked at the foot of each one.
False
The claim is wrong, and something on this site computes by how much. These are the ones worth the most, because a reader carrying one of them is not merely missing something. 630 claims in this group.
An influence line is a bending moment diagram for a moving load.
What decides it: They are plotted against different axes and answer different questions. A moment diagram fixes the load and varies the station; an influence line fixes the station and varies the load position. The curve here is built by literally moving a unit load to each of three hundred positions and re-solving the beam every time — so it is a summary of three hundred load cases at one station, not one load case at three hundred stations.
Tested in The worst place to stand, at the figure it turns on · the essays about influence line.
More steel is more strength, so a heavier section is a stronger one.
What decides it: Four profiles enclosing exactly the same area, with ∫y²dA computed for each from the rectangles that make it up. Identical weight, identical cost, stiffnesses differing by a large factor — which is why a steel catalogue holds hundreds of shapes rather than a range of solid bars.
Tested in The same steel in a different shape, and a factor of forty, at the figure it turns on · the essays about section shape.
A brace has to be strong: it holds the column, so it carries a share of the column's load.
What decides it: Every point on the curve is an eigenvalue — discretise the column, assemble its elastic and geometric stiffness, add the spring, find the smallest load at which the pair goes singular. It climbs with brace stiffness and then goes flat at 4π²EI/L² to five figures, because the second mode has a node exactly where the brace is and the brace cannot obstruct it. What is required is stiffness, the requirement is exact, and past the knee more of it buys nothing at all.
Tested in The brace that need not be strong, at the figure it turns on · the essays about effective length.
A beam strong enough to carry its load is stiff enough to use. Check the stress and the deflection takes care of itself.
What decides it: The two limits do not scale together — strength as the square of the span, deflection as the fourth power — so they cross rather than track. Both are plotted against span for one section, and for an ordinary steel floor beam the crossing lands at roughly six to eight metres. Below it strength governs; above it deflection does, and increasingly emphatically.
Tested in Stiffness is not strength, and usually it is the one that governs, at the figure it turns on · the essays about serviceability.
Specifying a higher grade of steel makes the member stiffer as well as stronger.
What decides it: The two stress–strain curves drawn together. Steels whose yield strengths differ by 67 per cent are the same line for the first 0.13 per cent of strain — not similar, the same — because every steel ever rolled has a modulus near 210,000 N/mm². There is no strength anywhere in 5wL⁴/384EI, and none in π²EI/L² either.
Tested in The one number a stronger steel does not change, at the figure it turns on · the essays about elastic modulus.
A bigger hole concentrates stress more than a small one.
What decides it: Kirsch's exact solution contains one length, the radius, so every dimensionless quantity in it can depend only on a/r — which is 1 at the hole's edge whatever a is. A ten-millimetre hole and a metre hole in wide plates both multiply the applied stress by exactly three. What a large hole costs is net section, which is a separate effect and equally computable.
Tested in The hole that multiplies the stress by three, at the figure it turns on · the essays about stress concentration.
A member whose stresses never approach yield cannot fail from loading.
What decides it: Stress range against cycles for three detail categories, in a steel whose grade is not stated because it makes no difference. At a working range of 70 N/mm² the best detail lasts 6.8 × 10⁷ cycles and the worst 2.7 × 10⁵ — a factor of 250 in life, decided by the geometry of a joint rather than by any margin against strength.
Tested in The load that never came near failing anything, at the figure it turns on · the essays about fatigue.
A bolt in tension carries the tension applied to it. Add thirty per cent for prying and the connection is covered.
What decides it: The allowance is wrong in both directions and by more than it is worth. A 20 mm flange under 100 kN per bolt carries 150.6 kN — a ratio of 1.51, not 1.30. A 27 mm flange under the same load carries exactly 100.0, because prying has vanished entirely. And below 19.07 mm the flange has become a mechanism, at which point the bolt is not the thing that decides and no percentage is meaningful.
Tested in The force the bolt never saw applied, at the figure it turns on · the essays about prying.
Stagger the bolt holes and the net section is the gross section less one hole, because no straight line crosses two.
What decides it: The tear is not obliged to be straight. In a 200 mm plate with two 22 mm holes at 60 mm gauge, the straight path leaves 178 mm and the diagonal path through both leaves 166.4 mm even after the s²/4g correction adds 10.4 mm back. The diagonal governs until the stagger reaches 72.7 mm, which is more than the gauge itself.
Tested in The tear that goes diagonally, and the correction that has no derivation, at the figure it turns on · the essays about net section.
The enhancement for a transverse weld is an empirical bonus from test data.
What decides it: It falls out of the directional check with no data in it at all. A weld loaded along its length puts everything into τ∥ and the criterion reads √3·τ. Loaded across, the force resolves onto the 45° throat as equal σ⊥ and τ⊥ of F/(√2·a·l) each, and the criterion reads √2·F/(a·l). The ratio of the two capacities is √3/√2 = 1.224745, computed rather than fitted.
Tested in The weld that is stronger across than along, at the figure it turns on · the essays about weld strength.
Stress in a weld group falls off with distance from the centroid, so the outermost weld is the most highly stressed.
What decides it: Only the torsional component depends on radius. The direct component is the same vector at every point on the group, so the two add at one end and partly cancel at the other — and on two horizontal runs, points at exactly equal radius carry 0.93 and 0.56 of the same units. Radius cannot distinguish them because radius is not what separates them.
Tested in The corner that is not the worst point, at the figure it turns on · the essays about weld group.
The slip resistance depends on the bolt grade, since it comes from the bolts.
What decides it: It depends on the preload and the surface. Halving the friction coefficient from 0.5 to 0.2 — which is the difference between blasted steel and an ordinary mill-scale surface — takes the same two bolts from 137 kN to 54.8 kN. The bolt is unchanged and the joint has lost 60% of its slip resistance to how the plate was prepared.
Tested in The joint that carries nothing until it slips, at the figure it turns on · the essays about Slip-critical.
Shear lag is another way of accounting for the holes, so taking the worse of the two is enough.
What decides it: They are different mechanisms and they multiply. Net section removes material that is not there; shear lag discounts material that is there and is not fully stressed. On the same angle the two give 0.867 and 0.868, and the effective area is 0.752 of gross rather than 0.867.
Tested in The angle that uses half of itself, at the figure it turns on · the essays about shear lag.
A connection is either pinned or rigid, and which one is a property of the connection.
What decides it: The boundaries are multiples of the beam's own EI/L, so the same joint changes class when the beam does. A flush end plate at 12,000 kN·m/rad is semi-rigid on a beam with EI/L = 14,000 and would be rigid on one four times stiffer at the joint. The classification is a statement about the pair, and neither member of the pair can answer it alone.
Tested in Neither pinned nor rigid, which is every real connection, at the figure it turns on · the essays about joint stiffness.
To stiffen a connection, stiffen the connection — thicker plate, bigger bolts, more weld.
What decides it: The components are in series, so the flexibilities add and the softest dominates. Doubling the stiffness of the column flange in bending, which is 39.6% of the total flexibility, raises the joint's stiffness by a factor of 1.247. Doubling the bolts in tension, at 8.9%, raises it by 1.046 — under five per cent for twice the bolt.
Tested in A joint made of springs in series, at the figure it turns on · the essays about component method.
A connection is close enough to pinned if it was detailed as a pinned connection.
What decides it: The classification depends on the beam. The same web cleats at 1,800 kN·m/rad are pinned against a beam with EI/L = 14,000 and would not be against one four times more flexible, where the pinned boundary falls to 1,750. Nothing about the connection would have changed.
Tested in The redistribution nobody chose, at the figure it turns on · the essays about joint classification.
Once the moment exceeds the middle-third limit the base plate lifts and the holding-down bolts go into tension.
What decides it: It lifts and the bolts carry nothing. Between e = L/6 and e = L/2 the contact length simply shrinks so the triangular bearing block's resultant sits under the applied resultant, and compression alone satisfies equilibrium. For a 500 mm plate under 600 kN the bolts stay at zero until 150 kN·m, three times the moment at which the plate first lifted.
Tested in Where the structure meets the ground, and when the bolts start working, at the figure it turns on · the essays about base plate.
A load is a load. Its magnitude is what matters, and how quickly it is applied is a detail.
What decides it: The same 20 kN on the same structure gives 12.67 mm if it is applied slowly, 25.33 mm if it is applied at once, and 7.83 mm if it is applied at once and removed a tenth of a period later. Three answers, one load, a spread of more than three to one — and the only thing that differs between them is the history.
Tested in Twice the deflection, for the same load, at the figure it turns on · the essays about dynamic amplification.
Finding a natural frequency needs a dynamic analysis, so it is extra work.
What decides it: Rayleigh's quotient over the deflected shape a beam already takes under its own weight gives 4.915 Hz against an exact 4.909 Hz — 0.12% high, and high rather than low for a reason that is a theorem. The deflection it needs was computed for the span/360 check. The frequency is one line of arithmetic over numbers that already exist.
Tested in The period nobody chose, at the figure it turns on · the essays about natural period.
The higher modes are overtones of the first, so a building's periods are one, a half, a third of the fundamental.
What decides it: They are eigenvalues of a matrix, not harmonics of a string. An eight-storey uniform frame has periods of 0.932, 0.314, 0.193 and 0.143 s — ratios of 1 : 2.97 : 4.83 : 6.53, which is nothing like 1 : 2 : 3 : 4 and is not a series at all. The spacing depends on how the mass and stiffness are distributed, and changing one storey changes every period in the list.
Tested in A structure has more than one period, at the figure it turns on · the essays about mode shapes.
A structure vibrating in one mode will be dragged into the others, since they share the same material.
What decides it: Modes are orthogonal through the mass matrix: the product of any two distinct mode shapes, weighted by the floor masses, is zero to machine precision. That is what allows an N-storey building to be analysed as N independent single-degree-of-freedom problems, and it is the single most useful fact in structural dynamics.
Tested in A structure has more than one period, at the figure it turns on · the essays about mode shapes.
Effective modal masses are a normalisation convention, so their sum is arbitrary.
What decides it: They sum to the total mass exactly, for every structure, in every set of units, to machine precision — 100.000% across all eight modes of the frame drawn here. It is a completeness property of the eigenvectors, and it is what makes "how many modes is enough" a question with a numerical answer instead of a judgement.
Tested in Most of the mass moves together, at the figure it turns on · the essays about modal mass.
Floor vibration is a strength problem in disguise — a floor that moves that much must be overstressed.
What decides it: The peak acceleration on the worst case here is 0.11 m/s², which on a 12-tonne modal mass is a dynamic force of about 1.3 kN — a fifth of the weight of the person causing it, on a floor designed for tonnes. Nothing is close to being overstressed. The complaint is entirely about perception, and perception responds to acceleration rather than to stress or deflection.
Tested in The floor that is strong and unusable, at the figure it turns on · the essays about floor vibration.
A response spectrum describes an earthquake, so two structures at the same site experience the same thing.
What decides it: A point on the curve is a property of a structure and a record together. The record drawn here gives 5.50 m/s² to a structure of 0.3 s period and 1.33 m/s² to one of 1.8 s — a factor of four between two buildings on the same ground in the same event. The curve is an envelope over structures, not a description of the shaking.
Tested in The spectrum is not a load, at the figure it turns on · the essays about response spectrum.
Halving a structure's strength doubles its displacement under an earthquake.
What decides it: Measured over five records at periods from 0.5 to 1.5 s, a structure given a quarter of the elastic strength reaches 1.08 times the elastic displacement, not four times. The strength was divided by four and the displacement moved by eight per cent. Below about half a second the rule breaks and the ratio rises to 2.77 at 0.15 s, which is the direction that matters.
Tested in The earthquake asks for a displacement, at the figure it turns on · the essays about ductility demand.
The design wind is the strongest wind, so a structure checked for a fifty-year storm is checked for everything.
What decides it: The critical speed for vortex shedding on the chimney drawn here is 6.0 m/s — a breeze, met hundreds of times a year, and far below any storm. The cross-wind amplitude there is 181 mm on a 1.2 m cylinder, and in the fifty-year storm it is essentially zero because the shedding has moved to a frequency the structure ignores. The worst case is not the largest load.
Tested in The wind that brings its own frequency, at the figure it turns on · the essays about vortex shedding.
A crowd is a load, so more people means more force and the response scales with the number.
What decides it: Below the critical number the response is small however many people are on the bridge; above it, the motion grows exponentially from nothing. For a deck of 120 tonnes modal mass at 0.5 Hz with 0.6% damping, the threshold is 30 people — and 29 people produce a bridge that feels solid while 31 produce one that has to be closed.
Tested in The bridge that was pushed by its own sway, at the figure it turns on · the essays about crowd synchronisation.
More damping in the absorber is better, since damping is what removes energy.
What decides it: There is an optimum and both sides of it are worse. On a 3% absorber the peak response is 55.1 with no absorber damping, 7.5 at the optimum 10.5%, and 20.3 at 50%. A heavily damped absorber is locked to the structure and moves with it, so its spring never stretches and its dashpot never works.
Tested in The mass that helps by being late, at the figure it turns on · the essays about tuned mass damper.
Dropping a load from twice the height doubles the force.
What decides it: The factor is 1 + √(1 + 2h/δ), so it grows as the square root of the height rather than in proportion to it. Dropped on a beam whose static deflection is 2 mm, a load from 100 mm gives a factor of 11.05 and from 1000 mm — ten times the height — gives 32.64, which is not ten times but three.
Tested in The weight that was dropped, at the figure it turns on · the essays about impact factor.
A stronger, stiffer structure is safer under an impact.
What decides it: A 500 kg load dropped one metre produces 160 kN on a structure that deflects 2 mm under it, 54 kN on one that deflects 20 mm, and 21 kN on one that deflects 200 mm. Same load, same drop, same energy — and the force is set almost entirely by how far the structure gives while absorbing it.
Tested in The weight that was dropped, at the figure it turns on · the essays about impact factor.
More damping in a mount is always safer.
What decides it: Past root two it is the reverse. At a ratio of 3, transmissibility is 0.130 with 5% damping and 0.193 with 20% — half as much again gets through the better-damped mount. Damping helps only while passing through resonance; at running speed it is a leak path.
Tested in The machine that shakes the building, at the figure it turns on · the essays about vibration isolation.
The Tacoma Narrows bridge failed by resonance: the wind gusted at the bridge's natural frequency.
What decides it: A steady wind has no frequency to resonate with, and the wind that day was steady. The torsional motion that destroyed the deck was self-excited — the deck's own rotation altered the flow so as to add energy to the rotation, which is an instability with a critical speed rather than a response with a magnification factor. The distinguishing test is in the pictures: a resonant response is bounded and needs a matching frequency; this one is unbounded and needs only enough speed.
Tested in The motion that feeds itself, at the figure it turns on · the essays about aeroelastic instability.
A structure that is strong enough to carry a wind load is safe against being blown over by it.
What decides it: Strength and stability are decided by different quantities and the second contains no material property at all. The hoarding below is a trivial strength problem — 48 kN spread over eight metres of frame — and its factor against overturning is 1.04. Doubling the strength of every member changes that number by nothing whatever.
Tested in Weight is the only thing resisting it, at the figure it turns on · the essays about overturning.
Nothing has happened to a base until the structure is on the point of tipping over.
What decides it: The far edge stops pressing down at an eccentricity of B/6, which for the same body arrives at a height smaller by exactly √3 — 3.54 m against 6.12 m here. Past that point the contact shortens, the peak pressure rises without limit, and the factor of safety quoted against overturning says nothing about either.
Tested in Weight is the only thing resisting it, at the figure it turns on · the essays about overturning.
The load on a beam is a property of the floor above it, so any correct method of working it out gives the same answer.
What decides it: The same 8 × 6 m panel at 5 kN/m² gives its long beams 15.00 kN/m if the slab is taken as spanning one way and 9.38 kN/m if it is divided at 45°. Both divisions hand over the whole 240 kN and neither is wrong; they are different decisions about a slab, and the beams differ by 60%.
Tested in The load a beam is given is a decision, at the figure it turns on · the essays about tributary area.
A plane frame analysis is a complete analysis of a plane frame.
What decides it: It satisfies three of the six equations. The other three — vertical moments about the two in-plane axes and the horizontal force out of the page — are satisfied by restraints the drawing does not contain, and a frame that has none of them is a mechanism in a direction the analysis never examined.
Tested in Six equations, and the drawing shows three, at the figure it turns on · the essays about Three-dimensional equilibrium.
Closing a section adds a little torsional stiffness, in the way that adding material usually does.
What decides it: It multiplies it. A 300 × 500 × 12 box has a torsion constant of 6.11 × 10⁸ mm⁴; slit along its length, with every millimetre of steel still present, it has 8.94 × 10⁵ — a factor of 683, and the shear stress rises by 45 times for the same torque.
Tested in The internal force with no diagram, at the figure it turns on · the essays about torsion.
Prestressing a beam is a way of making it stronger, so the critical condition is when it is fully loaded.
What decides it: The hard case is the empty beam. At transfer the 12 m beam below has −2.47 MPa at its top fibre with only its own weight on it, and 7.61 MPa of compression there once the full load arrives. The prestress is a load, and a load without its opposing load is a load acting alone.
Tested in The load put on backwards, at the figure it turns on · the essays about prestress.
A beam deflects in the direction it is loaded.
What decides it: Only where the section has an axis of symmetry in the plane of loading. An equal angle loaded vertically moves 59.2% as far sideways as it moves down, and a Z-purlin moves 166% — further across than along the load. The deflection runs perpendicular to the neutral axis, and the neutral axis is tilted by the product of inertia.
Tested in Loaded straight down, and it moves sideways, at the figure it turns on · the essays about principal axes.
A resultant inside the middle third in each direction keeps the section in compression.
What decides it: The kern of a rectangle is a rhombus, not the rectangle or the ellipse the two half-widths suggest. On the diagonal it reaches 56.7 mm where an ellipse through the same axis points would allow 78.4 — so a resultant inside the limit in both directions separately can still be outside the kern.
Tested in The middle third, at the figure it turns on · the essays about kern.
A shell that fails at a third of its calculated buckling load shows that the calculation is wrong.
What decides it: The calculation is right about the perfect shell and the shell is not perfect. The deficit follows a power law with an exponent below one — 0.662 fitted for an unstable symmetric system and 0.488 for an asymmetric one — so an imperfection of a hundredth costs 7% on one system, 17% on another, and two thirds on a cylinder. Every one of those numbers comes from the same critical load.
Tested in A third of what the theory promised, at the figure it turns on · the essays about imperfection sensitivity.
A structure with a larger factor of safety against buckling is less sensitive to how well it is built.
What decides it: Sensitivity is a property of the post-buckling path, not of the margin. Three systems with identical critical loads are drawn below: one climbs past it, one falls symmetrically, and one falls in one direction only. The same 0.02 radian imperfection costs them nothing, 11% and 23% respectively.
Tested in A third of what the theory promised, at the figure it turns on · the essays about imperfection sensitivity.
Thermal effects are a serviceability matter, not a strength one.
What decides it: A restrained member in compression is a column carrying a load nobody applied. The temperature rise that reaches its Euler load is π²i²/αL², with no material property in it: 53.3 °C for a 30 m member of this section, and 13.3 °C for a 60 m one. Nothing has been loaded and the member has buckled.
Tested in The movement nobody applied, at the figure it turns on · the essays about thermal movement.
Moment distribution is an approximate method, superseded by exact matrix analysis.
What decides it: It converges to the exact answer and it does so quickly: on a three-span beam the support moment goes 53.33, 66.67, 67.50, 64.22, 64.01, 64.00 kNm, against a stiffness-matrix solution of 64.000 obtained by a route sharing no arithmetic with it. Stopping early is approximate; the method is not.
Tested in Solved by passing it around, at the figure it turns on · the essays about moment distribution.
Robustness can be quoted as a number for a structure.
What decides it: It depends entirely on which member is lost. Removing a midspan bottom chord asks a survivor for 2.03 times what it was carrying; removing a diagonal asks for 1.07; removing an end diagonal asks for nothing at all, because there is no structure left to ask.
Tested in The structure that survives losing a member, at the figure it turns on · the essays about robustness.
The yield lines of a rectangular slab run at 45° from its corners.
What decides it: They do for a square panel and not otherwise. Searched over six hundred positions, the lowest collapse load of a 6 × 8 m panel puts the intersection 3.413 m from the short edge rather than the 3.000 a 45° line would give — an angle of 41.3°, and the difference is the mechanism choosing the lowest upper bound rather than the tidiest picture.
Tested in The slab that spans both ways, at the figure it turns on · the essays about Two-way spanning.
Two lateral systems acting together are two springs in parallel, so their stiffnesses add.
What decides it: They interact. A 20-storey wall alone reaches 146.5 mm at the top and the frame alone 140.0; springs in parallel would give 71.6 mm and the pair actually gives 58.0 — 24% stiffer than the sum, because each is stiff exactly where the other is not and the interaction force between them changes sign at level 16.
Tested in How a tall building stands still, at the figure it turns on · the essays about lateral system.
An outrigger belongs at the top of the building, where the wall's rotation is largest.
What decides it: Its job is to apply a restraining moment to the wall over as much height as possible, and one at roof level has no height left to act over. Sweeping every level, the best is level 11 of 20 — 55% of the height — giving 17.0 mm against 34.7 mm at roof level and 58.0 mm with none.
Tested in How a tall building stands still, at the figure it turns on · the essays about lateral system.
Two identical structures carrying identical loads have identical stresses.
What decides it: Not if they were built differently. The same 12 m composite beam under the same 30 kN/m reaches 291.7 MPa in its bottom flange if the slab was cast unpropped and 186.2 MPa if it was propped — 57% more, from a decision about temporary works that appears on no structural drawing.
Tested in The structure that was never complete, at the figure it turns on · the essays about construction sequence.
The friction force at a contact is μN.
What decides it: μN is a ceiling, not a value. Walking the ladder's one free parameter from end to end gives a continuum of states, every one satisfying ΣFx, ΣFy and ΣM to within 1e-13, with the floor's friction anywhere from 344.5 to 390.7 N. Statics prefers none of them, and only two of them have F equal to μN at either contact.
Tested in The force that is whatever it needs to be, at the figure it turns on · the essays about friction.
Water behind a wall is a drainage matter rather than a structural one.
What decides it: Of the five terms behind a 6 m wall with the table 2 m down, the water is the largest at 78.5 kN/m of 185.7. Soil at 18 kN/m³ and 30° pushes like a fluid of 6.00 kN/m³ against water's 9.81. The same wall drained has factors of 2.53 against overturning and 1.26 against sliding; with the drain blocked they are 2.08 and 0.91, and 0.91 is a wall on its way down the slope.
Tested in The load that depends on what carries it, at the figure it turns on · the essays about lateral pressure.
A load standing on two members divides between them according to the geometry of what they support — the area, or the length, or half each when there is nothing else to go on.
What decides it: A 6 m beam crossing a 9 m beam with 100 kN standing exactly where they cross has no area to divide. The plan drawing can only say half each; compatibility of the two midspan deflections says 77.1 kN and 22.9 kN, which is 9³/(6³ + 9³) exactly. The tributary answer is wrong by 54% on the beam that matters, and the free body that settles it is the crossing point, not the panel.
Tested in The stiffest path takes the load, at the figure it turns on · the essays about Load-sharing.
A member that is working too hard should be made stiffer, and a deeper section is a safer section.
What decides it: Stiffness from depth goes as d³ and section modulus only as d², so doubling a member's second moment of area raises its capacity by 1.587 while raising its force by up to 2. The two cross at a share of 0.260, which is the cube root of two minus one. The shallow beam of the hero figure carries 11% and ends up 13% more stressed for having been deepened.
Tested in The stiffest path takes the load, at the figure it turns on · the essays about Load-sharing.
A longer raft carries more bending moment, because the moment in a beam grows with its span.
What decides it: The peak moment under a column on a flexible bed is P/4β, which contains no length of beam anywhere. Spreading the same 1000 kN uniformly over 16 m and taking PL/8 returns 3.12 times the true moment — and the same calculation understates the peak contact pressure by exactly the reciprocal, 0.32, whose product with 3.12 is 1.000.
Tested in The beam that sits on the ground, at the figure it turns on · the essays about elastic foundation.
Shear deflection is a small correction, so two beams at the same span-to-depth ratio can be treated alike.
What decides it: The share depends on I/As, not on slenderness alone. At L/d = 8 every rectangle there is gives 4.64%, because I/As is d²/10 for all of them — and a 600 × 200 I-section at the same slenderness gives 13.06%, because its shear area is the web alone and κ falls from 0.833 to 0.513. The rectangle reaches a tenth of its deflection in shear at L/d = 5.29 and the I-section is still there at L/d = 9.30.
Tested in The deflection that is not bending, at the figure it turns on · the essays about shear deflection.
The members carrying the most force are the ones to strengthen when a truss deflects too far.
What decides it: In the six-panel Pratt truss drawn here the vertical at mid-span carries the whole 10 kN panel load and contributes exactly 0.000 to the mid-span deflection, because the unit load applied at the node beneath it leaves f = 0.000 in that member. Stiffening the busiest member in the picture would change the 631.06 of movement by nothing at all, while the top chord beside it is 14.80% of the whole.
Tested in Which member moved the roof, at the figure it turns on · the essays about truss deflection.
A cable roof reaches its limit when the tension in a cable approaches the strength of the cable.
What decides it: The limit is a loss of geometry. In the 30 m net below the hogging family's tension falls as the roof deflects and reaches zero at 407 mm, under 7.84 kN/m² — at which point half the surface has stopped working, while the sagging family has only climbed from 400 to 925 kN. Nothing is anywhere near failing and the structure has run out.
Tested in The stiffness that comes from the shape, at the figure it turns on · the essays about cable stiffness.
A masonry dome that is cracking is a dome that was built too thin, and the remedy is more stone.
What decides it: The stress in a dome carrying nothing but its own weight is γR/(1 + cos φ₀), which contains no thickness at all — the load is γt and the resisting area is t, and the two cancel exactly. The figure sizes the same 30 m dome at 50, 100, 200 and 400 mm: the meridional force rises 25 → 50 → 100 → 200 kN/m and every one of the four points reads 0.500 MPa. Thickening a dome adds hoop tension to carry and changes nothing about the stress, which is why the repair that worked at St Peter's was iron round the base rather than stone on the crown.
Tested in The surface that carries by being curved, at the figure it turns on · the essays about shell action.
Settlement of a support adds to the hogging moment over it, so the floor above a transfer needs more top steel.
What decides it: The induced moment over a settling support is a sagging one. It does not add to the 240 kNm of hogging, it eats it: 432 kNm of induced moment leaves 192 kNm of sagging at that support, and the face in tension over the transferred column is the one with no reinforcement in it.
Tested in The column that stops, at the figure it turns on · the essays about transfer structure.
Any system of forces can be replaced by a single resultant force somewhere.
What decides it: Only if the system is coplanar, concurrent or parallel. Two forces of 10 and 5 units, neither meeting nor parallel, reduce to a force of 11.1803 with a couple of 13.4164 left over — and that couple is parallel to the force, so no shift of the reduction point can remove it. The pitch, 1.2, is a property of the system.
Tested in Moving a force, and what it costs, at the figure it turns on · the essays about force couple.