What is refuted here — page 11

A bolt group carrying a shear force shares it equally: divide the load by the number of bolts.

What decides it: True only when the load passes through the group's centroid, which almost no real bracket arranges. At 150 mm of eccentricity the same six bolts carry 50.4, 50.4, 36.4, 36.4, 34.8 and 1.5 kN — a spread of thirty-four to one — against an equal share of 16.7. The direct shear is shared equally; the torque is not, and adding the two as vectors is the whole calculation.

Tested in The bolt that carries more than its share, at the figure it turns on · the bolt group ladder.

A thicker flange helps because it is stronger.

What decides it: It helps because of where the moment goes, not because of a capacity. The thickness at which prying disappears is 2·√(T·m / p·fy) — set by the applied load and the geometry, and containing no bolt property at all. At 100 kN, m = 45, p = 90 and fy = 275 that is 26.97 mm, and a bolt twice as strong moves it by nothing.

Tested in The force the bolt never saw applied, at the figure it turns on · the prying ladder.

A bolted end connection is checked by three things: bolt shear, bearing on the plate, and tension on the net section. Pass all three and it is adequate.

What decides it: Those three are checks at a point or across a line, and this failure is neither. A three-bolt end connection whose shear plane and tension plane each look comfortable fails at 421.7 kN as a block, with the shear plane contributing 70% of that and the tension plane 30% — and no single-plane check produces the number, because the capacity is a sum across two surfaces at once.

Tested in The metal between the holes, which comes out as a block, at the figure it turns on · the block shear ladder.

Bolted connections are sized by the shear capacity of the bolts.

What decides it: An M20 bolt in single shear carries roughly 94 kN. The same bolt in a 10 mm plate at the minimum end distance of 26.4 mm bears at 34.4 kN — a third of it — and the plate is the limit by a wide margin. It takes 132 mm of end distance for bearing to reach 172 kN and 165 mm before the curve goes flat at 215.

Tested in The hole that goes oval, and the one that tears to the edge, at the figure it turns on · the bearing ladder.

A fillet weld has a strength per millimetre of throat, and the capacity is that strength times the length.

What decides it: Only if the direction is stated with it. The same 4 mm throat over 100 mm carries 116.8 kN loaded along its length and 143.1 kN loaded across it — a ratio of 1.2247, which is √3/√2 to five figures and is a consequence of the yield criterion rather than a measured allowance.

Tested in The weld that is stronger across than along, at the figure it turns on · the weld strength ladder.

A preloaded joint carries its load through friction, so it never reaches bearing.

What decides it: Only at serviceability. Slip resistance at μ = 0.5 is 137 kN against a bearing capacity of 188 kN, so the joint is expected to slip before it fails and the ultimate check is a bearing check on a joint that has already moved. Design it as though slipping is the end and the ultimate limit state has no calculation behind it at all.

Tested in The joint that carries nothing until it slips, at the figure it turns on · the Slip-critical ladder.

A tension member's capacity is its net area times its strength, whatever the connection.

What decides it: The connection has to be able to reach the whole section, and over a short one it cannot. A 100 × 75 × 10 angle bolted through its long leg with two bolts at 75 mm pitch has U = 0.736, so 26% of the net area is not working. With three bolts U is 0.868 and with six it is 0.947 — the member never changed, and neither did its net area.

Tested in The angle that uses half of itself, at the figure it turns on · the shear lag ladder.

Real connections are close enough to one idealisation or the other that the distinction is academic.

What decides it: Of three ordinary connections plotted against a 6 m beam of EI = 84,000 kN·m², one is pinned and two are semi-rigid — and neither of the two is rigid. The extended end plate, the stiffest connection in ordinary use, reaches 46,000 kN·m/rad against a rigid boundary of 112,000. The idealisation the analysis used is available to none of them.

Tested in Neither pinned nor rigid, which is every real connection, at the figure it turns on · the joint stiffness ladder.

A connection's stiffness is a property of the connection, so it can be tabulated per detail.

What decides it: Two of the five components — the column web in shear and in compression, 33.5% of the flexibility between them — belong to the column rather than to the connection. The same end plate on a heavier column is a stiffer joint, and no table indexed by connection detail can say so.

Tested in A joint made of springs in series, at the figure it turns on · the component method ladder.

Designing a beam as simply supported is conservative, because any end restraint that exists is extra capacity.

What decides it: Conservative for the beam and not for what the beam is attached to. Web cleats on a 6 m beam of EI = 84,000 deliver 6.0% of the fixed-end moment; a flush end plate delivers 30.0%. That moment is real, it goes into the column, and the column was designed for an axial load and a nominal eccentricity by an analysis that reported the connection as a pin.

Tested in The redistribution nobody chose, at the figure it turns on · the joint classification ladder.

A moment connection transmits moment, so its capacity is a moment capacity that can be compared directly with the beam's.

What decides it: The moment is carried as a couple, and the compression half of that couple is a force nothing in the moment check looks at. Three bolt rows at 180 kN each give 172.8 kN·m of moment and a compression of 540 kN at the bottom flange — which the beam flange, the column web and the welds all have to carry, and which appears in no bolt schedule.

Tested in Making a moment cross a gap, at the figure it turns on · the moment connection ladder.

A short, sharp load is the dangerous case, so a blast is worse than a steady push.

What decides it: Only above a duration of half a natural period. Below it the structure has not finished responding when the load leaves, and the peak falls away: a pulse lasting a tenth of a period produces 0.62 times the static deflection, which is a fifth of what the same force produces if it stays. A blast that is short compared with the structure is felt as an impulse, and the quantity that matters becomes the area under the force curve rather than its height.

Tested in Twice the deflection, for the same load, at the figure it turns on · the dynamic amplification ladder.

A stiffer structure is a faster one, so stiffening it is how its frequency is raised.

What decides it: True only if the mass is untouched, and it usually is not. The frequency goes as the square root of stiffness over mass, so doubling the stiffness raises it by 1.414 — measured, not argued — while the concrete or steel that doubles it also adds mass, which pulls the other way. Adding 50% to the mass alone multiplies the frequency by 0.8165, which is 1/√1.5 exactly.

Tested in The period nobody chose, at the figure it turns on · the natural period ladder.

Resonance is dangerous because the response becomes infinite.

What decides it: It becomes infinite only with no damping at all, which nothing has. At 2% of critical the response settles at 25 times the static deflection, at 5% it settles at 10, and the number is one over twice the damping ratio in every case. Resonance is dangerous because the multiplier is set by the least reliably known property the structure has — not because it is unbounded.

Tested in The only thing that stops it, at the figure it turns on · the damping ladder.

The first mode carries most of the mass, so the higher modes can be ignored.

What decides it: Ignored for what? On a sixteen-storey frame the first mode gives 92.6% of the base shear, so for a force it is very nearly the whole answer. The same mode gives 52% of the roof acceleration, because the higher modes have short periods where the spectral acceleration is five times larger — so for anything bolted to an upper floor, the modes carrying 12% of the mass supply half the demand.

Tested in Most of the mass moves together, at the figure it turns on · the modal mass ladder.

A stiffer floor is a better floor, so raising the frequency always helps.

What decides it: The response is not monotone in frequency. On the floor drawn here a 3 Hz version reaches a response factor of 3.4, a 4 Hz version reaches 11.3, and a 5 Hz version reaches 2.4. Stiffening from 3 Hz to 4 Hz makes the floor three times worse, because 4 Hz is where the second harmonic of a two-step-per-second pace sits.

Tested in The floor that is strong and unusable, at the figure it turns on · the floor vibration ladder.

Vortex shedding is resonance, so the response can be found by multiplying the shedding force by a magnification factor.

What decides it: Only outside the lock-in band. Inside it — from 5.1 to 7.8 m/s here — the shedding abandons the Strouhal line and follows the structure's own frequency instead, because the structure's motion organises the wake. The excitation is then a function of the response, which no magnification factor can express, and the amplitude is bounded by nonlinearity rather than by damping alone.

Tested in The wind that brings its own frequency, at the figure it turns on · the vortex shedding ladder.

Mounting a machine on springs reduces the vibration it transmits.

What decides it: Only above a frequency ratio of √2. Below it the mount amplifies: at a ratio of 1.2 with 5% damping the transmissibility is 2.21, so the floor receives more than twice the force it would have received with the machine bolted down. Every curve passes through exactly 1.000 at √2, for every damping ratio there is.

Tested in The machine that shakes the building, at the figure it turns on · the vibration isolation ladder.

A stiffer structure is safer against wind-induced instability.

What decides it: The galloping criterion contains the section's shape, its mass per metre and its damping — and the structure's frequency appears only through the damping term. Doubling the stiffness at constant mass raises the critical speed by 1.41 through the frequency, while doubling the mass alone doubles it. Changing the section's SHAPE can remove the instability altogether, at any speed.

Tested in The motion that feeds itself, at the figure it turns on · the aeroelastic instability ladder.

A tributary division is exact, because it is only geometry.

What decides it: The geometry closes exactly — the four regions sum to 48.0 m² against a panel of 48.0 — and the division itself is an assumption about where a two-way slab hands its load over. The elastic answer is not 45° lines: at this panel's proportions the short strips take 76% of the load rather than the 62.5% the tributary rule implies.

Tested in The load a beam is given is a decision, at the figure it turns on · the tributary area ladder.

A four-legged table is a stiffer and safer version of a three-legged one.

What decides it: It is also indeterminate, and the fourth leg buys nothing that equilibrium can see. The equilibrium matrix has rank three either way; with four legs there is a state of self-stress — one diagonal pair pushed down and the other lifted — satisfying every equation with no load at all. A floor 4 mm out of level turns 25 kN in each leg into 37 and 13.

Tested in Six equations, and the drawing shows three, at the figure it turns on · the Three-dimensional equilibrium ladder.

A torque in a member is a load like any other and has to be carried wherever it appears.

What decides it: Only where equilibrium demands it. Where a torque exists because two members are joined and rotate together, it is shared in proportion to stiffness, and softening the torsional member makes it go away: the same spandrel attracts 70% of the joint's moment as a closed box and 0.3% once it is slit, and nothing falls down in either case.

Tested in The internal force with no diagram, at the figure it turns on · the torsion ladder.

A prestressing tendon works by pulling the ends of the beam together.

What decides it: The axial force is the smaller half of it: 5.71 MPa of uniform compression here. What carries the beam is the tendon's curvature, which pushes upward along the whole span with an intensity of 8Pe/L² — 14.67 kN/m against an applied 17.25, leaving 2.58 kN/m to bend anything at all.

Tested in The load put on backwards, at the figure it turns on · the prestress ladder.

The second moment of area of a section is a number.

What decides it: It is a number for each direction, and it varies with the angle of the axis. For the equal angle below it runs from 0.734 to 2.866 × 10⁶ mm⁴ — a factor of 3.90 — and the value the drawing suggests, 1.800 × 10⁶, is neither the maximum nor the minimum but the value at an angle nothing physical happens at.

Tested in Loaded straight down, and it moves sideways, at the figure it turns on · the principal axes ladder.

The middle third is a rule about masonry.

What decides it: It is a property of a rectangle, arrived at as Z/A = b/6, and it appears wherever a section cannot be pulled: a base plate on grout, a footing on soil, a prestressing tendon near a beam end, a wall, an arch ring. Change the section and the number changes with it — an I-section's kern reaches 58.9% of its depth against a rectangle's 33.3%.

Tested in The middle third, at the figure it turns on · the kern ladder.

A structure that buckles has reached a load at which an alternative deflected shape becomes available.

What decides it: That is a bifurcation, and it is one of two ways to run out. A shallow frame stays exactly symmetric through its own collapse: at 133.4 kN its apex has moved 63.8 mm of a 150 mm rise, nothing has branched, and the load it can carry begins to fall. The failure is a maximum on a single path, not a meeting of two.

Tested in The roof that jumps, at the figure it turns on · the Snap-through ladder.

Snap-through is a hazard for any shallow arch or dome.

What decides it: Only for a very shallow one. The bar stress at the limit point is E(h/a)²/3, so it reaches yield at a rise-to-half-span ratio of √(3fy/E) — 7.1% for ordinary steel. Above that the members squash or buckle before the geometry runs out, and the snap-through calculation describes a state the structure never reaches.

Tested in The roof that jumps, at the figure it turns on · the Snap-through ladder.

A longer member is more affected by temperature, so a short one is safer.

What decides it: True for movement and false for stress. Fully held, every length carries the same 75.6 MPa at 30 °C. Held by a real spring of 100 kN/mm, the LONGER member is the more restrained one — 28.4% of full restraint at 5 m and 82.6% at 60 m — because it generates more movement for the same spring to absorb.

Tested in The movement nobody applied, at the figure it turns on · the thermal movement ladder.

The point of the method was to save arithmetic.

What decides it: It saves a great deal, and what it really supplies is a physical account of where a moment goes. Each cycle is a joint being released and rotating, each share is a stiffness ratio the engineer chose the sizes to produce, and the answer accumulates in a table that can be read as a story rather than inspected as an output.

Tested in Solved by passing it around, at the figure it turns on · the moment distribution ladder.

A redundant structure is a robust one.

What decides it: Redundancy is necessary and not sufficient. The frame below is four times redundant and four of its twenty-five members still leave a mechanism when removed — the two end diagonals and the two end bottom-chord panels — because redundancy is a global count and collapse is a local event.

Tested in The structure that survives losing a member, at the figure it turns on · the robustness ladder.

A slab supported on four sides shares its load between the two directions.

What decides it: Only while it is nearly square. The two families of strips must deflect equally where they cross, and a strip's deflection goes as the fourth power of its span, so the short strips take Ly⁴/(Lx⁴+Ly⁴): 50% at 1:1, 76% at 3:4, and 94% at 1:2. By a ratio of two it is a one-way slab with a decorative second direction.

Tested in The slab that spans both ways, at the figure it turns on · the Two-way spanning ladder.

The bending moments in a finished structure follow from its finished geometry and its loads.

What decides it: Only for the loads applied after it was finished. Two spans erected simply supported under 12 kN/m and made continuous before the remaining 18 arrived carry 324 kNm over the support rather than the 540 the finished analysis gives, and 378 at midspan rather than 270. Both are in equilibrium with the same total load.

Tested in The structure that was never complete, at the figure it turns on · the construction sequence ladder.

The lateral pressure coefficient is a property of the soil, so it is looked up from the friction angle and the load follows.

What decides it: The friction angle fixes the two ends of a curve and nothing between them. At 30° those ends are Ka = 0.333 and Kp = 3.000, a factor of nine, and which one applies is decided by the displacement of the wall itself: 6.0 mm of retreat on a 6 m wall brings the coefficient to 0.342, and a wall not permitted to move sits at K0 = 0.500 — 1.50 times what an active design was checked for.

Tested in The load that depends on what carries it, at the figure it turns on · the lateral pressure ladder.

A rigid connection has to be designed for the moment and the shear that the member diagrams show at the joint.

What decides it: The largest force at a knee is on neither diagram. The beam hands its 65.2 kNm over as a flange couple of 142.7 kN and the column takes only 24.5 kN of it away as shear, so 118.2 kN crosses the panel — 1.48 times the larger member shear under gravity load and 2.87 times under sway. Cutting the panel out as its own free body is the only way to see that number.

Tested in The moment that goes round the corner, at the figure it turns on · the corner moment ladder.

A raft cannot be designed until the subgrade modulus is known, because the answer depends on the ground.

What decides it: It depends on the fourth root of the ground. Sound rock to very soft clay is a factor of 1000 in k and a factor of 5.62 in the characteristic length — 0.81 m against 4.56 m — because 1000^¼ = 5.62. The design moment P/4β moves by that same 5.62 rather than by 1000. The number a site investigation is least sure of is the one this answer is least sensitive to.

Tested in The beam that sits on the ground, at the figure it turns on · the elastic foundation ladder.

Torsion is a shear problem: a torque produces shear stress, and the check is that stress against a shear capacity.

What decides it: Only where the section is free to warp. Hold a 305 × 165 I-section against warping at one end and twist it by 0.5 kN·m, and the largest stress there is longitudinal, not shear: 67.0 N/mm² of flange bending against 35.6 N/mm² of Saint-Venant shear. It lands at the flange tip, adds directly to the bending stress already there, and no shear check can see it.

Tested in The section that cannot stay flat, at the figure it turns on · the warping ladder.

Everything an internal cut reveals is one of six things — three forces and three moments.

What decides it: A restrained open section carries a seventh. The bimoment is four flange-tip forces whose resultant force is zero and whose resultant moment about all three axes is zero, so a free body can satisfy every equation of equilibrium and still have −0.897 kN·m² of it. Its units are kN·m², which is why there is no axis on any diagram it could be plotted against.

Tested in The section that cannot stay flat, at the figure it turns on · the warping ladder.

The bending stress in a flange is uniform across its width, because every fibre of it sits at the same distance from the neutral axis.

What decides it: Plane sections says so and plane sections is a kinematic assumption, not a law. Longitudinal stress enters a flange only through shear along the web junction, so the far parts of it lag: on a 3 m overhang at a 20 m span the free edge carries 0.765 of the web's stress, the working width is 2.531 m of 3 m, and the peak stress is 1.185 times what a uniform block would report. The free body that settles it is a strip of flange between the free edge and a plane at distance y, on which the only longitudinal force available is the shear on that plane.

Tested in The flange that is not all there, at the figure it turns on · the effective width ladder.

The bending formula follows from plane sections staying plane, so wherever that assumption holds the stress is linear across the depth and σ = My/I applies.

What decides it: Plane sections is necessary and not sufficient. In a curved bar the faces do stay plane, and the formula is still wrong: the inner fibre of the hook drawn here carries 222.8 N/mm² of bending stress against My/I's 161.7, a factor of 1.378, and the gap grows as the curvature tightens — 1.034 at a radius of ten depths, 1.447 at one. The missing hypothesis is that the fibres were the same length to begin with.

Tested in The bar that was bent before it was loaded, at the figure it turns on · the curved beam ladder.

A column buckles at π²EI/L², with I the smaller of the section's two second moments.

What decides it: That expression is the lowest of two modes, and an open section has three. The third is rotation about the longitudinal axis, resisted by G·J and by warping, and it is governed by a polar radius rather than by either second moment. Four sections of ordinary proportions at 3000 mm show the gap: the I-section buckles at 1454 kN, exactly its Euler load, while the tee buckles at 349 kN against an Euler value of 727 kN — a factor of 2.08 between the formula and the answer.

Tested in The column that twists instead of bending, at the figure it turns on · the Flexural-torsional ladder.

A column's effective length is set by its end conditions: pinned gives 1.0, fixed 0.5, fixed at one end and pinned at the other 0.7, and a cantilever 2.0.

What decides it: Those four are the eigenvalues of an isolated member with idealised ends, and no column in a building has them. Assemble the storey instead — elastic stiffness minus lambda times geometric stiffness, solved for the lowest root — and the same column comes out at k = 0.774 when its head is held against drift and k = 1.317 when it is free to drift, with nothing changed but one horizontal restraint that carries no vertical load. The test is whether the storey can sway, and it is a question about the frame that no end condition on the member can answer.

Tested in Held, and not held, at the figure it turns on · the sway stability ladder.

A beam's deflection is found by integrating the bending moment diagram twice.

What decides it: That integral is one of two terms. The other is ∫V·v/GAs, computed by the same virtual work over the same diagrams, and it is 16.3% of the movement of a 300 × 600 rectangle spanning four times its depth. It does not appear as a percentage on the same curve — it has its own shape, with one degree of curvature less than the bending shape has.

Tested in The deflection that is not bending, at the figure it turns on · the shear deflection ladder.

A truss is a beam with holes in it, so its deflection scales with span to the fourth power.

What decides it: Swept from 4 to 16 panels at a fixed depth the deflection runs 159.5 to 24900.8, a fitted log-log exponent of 3.65 rather than 4. The shortfall is structural rather than numerical: the chord share climbs 0.608 to 0.952 over the same sweep, so the sum is a mixture of two different scalings whose proportions are still changing.

Tested in Which member moved the roof, at the figure it turns on · the truss deflection ladder.

A structure's stiffness is a property of its material and its section, so the way to make something stiffer is a better material or more of it further out.

What decides it: True of everything that bends and false of everything that hangs. The initial stiffness of a cable is 8T₀/L exactly, in which no property of the material appears at all — 0.0, 33.3, 133.3 and 533.3 kN/m for four identical 30 m strands differing only in the tension put into them before the load arrived. The slack one leaves the origin flat and sags 1.059 m under the load that puts 0.276 m into the tightest.

Tested in The stiffness that comes from the shape, at the figure it turns on · the cable stiffness ladder.

A transfer beam is sized by the moment it has to carry, like any other beam.

What decides it: The moment asks for 1.94 m of depth and the answer is 2.55 m. The free body that settles it is not the transfer member at all but the floors above it: at the strength depth they pick up 1.80 times the bending they were designed for, purely from the settlement of the column standing on the beam.

Tested in The column that stops, at the figure it turns on · the transfer structure ladder.

A force can be slid along its line of action without changing anything.

What decides it: Only for a rigid body. Every reaction, every equilibrium equation and every resultant is unchanged; the internal forces are not, and they differ over a length of about the body's own depth. The same substitution applied to a spread load leaves the reactions agreeing to the last figure and the peak moment out by a third.

Tested in Moving a force, and what it costs, at the figure it turns on · the force couple ladder.

A beam should be cambered for its total load, so that it finishes flat.

What decides it: That rule leaves the beam hogged for every stage before the last, and its worst hog — 37.9 mm on the 12 m member here, one part in 317 of the span — occurs with nothing on it at all. Cambering against the wet concrete instead leaves 12.7 mm of sag at the end, one part in 945, and a floor that is flat on the day it is poured.

Tested in Built to the wrong shape on purpose, at the figure it turns on · the camber ladder.

Residual stress reduces a column's capacity because it uses up part of the yield stress.

What decides it: It is self-equilibrating, so it adds nothing to the section's total force and takes nothing from its squash load — a stub column reaches fy. What it removes is STIFFNESS, by yielding the tips at a lower applied stress, and stiffness is what buckling is about. At 80% of yield the elastic core is 67% of the flange width and the minor axis has 30% of its stiffness left.

Tested in The column that had yielded before it was loaded, at the figure it turns on · the inelastic buckling ladder.

A slab's capacity is what its yield-line calculation gives.

What decides it: Only if its ends are free to move apart. A restrained 200 mm strip spanning 4 m carries 116.6 per unit width against a yield-line load of 30.0 — 3.89 times as much — and it reaches that at a deflection of 48 mm, which is a quarter of its own thickness and one part in 83 of the span.

Tested in The force nobody put in the model, at the figure it turns on · the membrane action ladder.

The largest shear stress in a beam is at the neutral axis.

What decides it: The largest shear stress ON A HORIZONTAL PLANE is. The largest shear on ANY plane is the radius of Mohr's circle, and at the extreme fibre — where the horizontal shear stress is zero — it is exactly half the bending stress, which here is 43.0 against the neutral axis's 42.4. Mild steel's Lüders lines at 45° from a flange are that plane made visible.

Tested in The worst stress is not where the worst bending is, at the figure it turns on · the principal stress ladder.

The chords of a Vierendeel girder carry the bending moment, and the posts carry the shear.

What decides it: The chords carry both. At mid-span of the girder drawn here the chord axial force is 155 kN and each chord also carries 33.2 kNm of local moment from the panel shear, and the two add at the panel ends and cancel at the middles. 68% of the girder's deflection is that second term.

Tested in The truss with no diagonals, at the figure it turns on · the vierendeel ladder.

The stiffening girder of a suspension bridge carries part of the load, in proportion to its share of the stiffness.

What decides it: It takes 17% of a mid-span point load here, and it removes 60% of the deflection and 83% of the kink under it. What it is doing is distribution, not carrying: it spreads the load over 183 m — a characteristic length √(EI/H) — so what reaches the cable is spread rather than concentrated.

Tested in The deck is not there to carry the load, at the figure it turns on · the stiffening girder ladder.

A cable is the efficient structure, so the stiffer the deck the worse the bridge.

What decides it: Stiffness moves load off the cable and onto the girder, and at μ = L√(H/EI) below one the girder takes 89% of it and is simply a beam. But the same parameter decides the deflection and the kink, and a bridge with no girder at all deflects 1.87 units against 1.13 and has a slope discontinuity of 0.0083 radians under the load.

Tested in The deck is not there to carry the load, at the figure it turns on · the stiffening girder ladder.

A bigger column solves a punching problem, because the perimeter grows with it.

What decides it: The perimeter grows with the column, but most of it is not column. At two effective depths from the face the four corner arcs together are a full circle of radius 450 mm — 2,827 mm of perimeter that is there whatever the column does, and 64% of the whole. Fitted over columns from 200 to 1,400 mm the resistance rises as the 0.43 power of the column size, so doubling the column buys 35%.

Tested in A check made on a perimeter, not on a section, at the figure it turns on · the punching shear ladder.

Turning a section over cannot change its strength, because none of its properties change.

What decides it: None of its properties change and which of them governs does. For a moment of one sign the tee's flange is in compression and the web tip decides; reversed, the flange decides. The two elastic moduli are 220.1 and 75.3 × 10³ mm³, so the same tee under a sagging moment and a hogging one differs by a factor of 2.92.

Tested in Two strengths, depending which way up, at the figure it turns on · the asymmetric section ladder.

A column's slenderness is a property of the column, so a single number describes it.

What decides it: There is a radius of gyration about every axis, and the column buckles about the one with the smallest. For the I-section drawn here the two differ by a factor of well over two, so a member with a slenderness of 45 about one axis has 100 about the other and only the second number has ever mattered.

Tested in The one length a section takes into a column, at the figure it turns on · the radius of gyration ladder.

A flexible diaphragm distributes a lateral load to the walls by tributary area.

What decides it: Tributary area is not a limit of the model. Softening the plate takes the middle wall of three from 33.3% of the load toward 62.5%, and passes through the tributary 50% at one particular finite stiffness ratio on the way. The tributary rule is the answer for a plate that is soft AND discontinuous over the wall, which is an extra assumption nobody states.

Tested in The floor is a beam lying down, at the figure it turns on · the diaphragm ladder.

A beam that deflects too far is made stiffer by deepening it.

What decides it: Only the part of the deflection that is the beam's. The secondary beam here deflects 22.9 mm, of which 12.7 is its own bending and 10.2 is the primary beam beneath it going down. Doubling the secondary's second moment takes the total from 22.9 to 16.5 — a 28% improvement bought with twice the steel, because 44% of the movement was never the secondary beam's to fix.

Tested in The deflection that belongs to the support, at the figure it turns on · the support flexibility ladder.

Torsion can be avoided by detailing the connection as a pin.

What decides it: It is exactly right for the compatibility case and exactly wrong for the other one. A canopy cantilevering 2.2 m off the same spandrel delivers 29 kNm per metre of it, and the 116 kNm that arrives at the column contains no stiffness at all — releasing the connection does not reduce it, it removes the only load path there was.

Tested in The torsion that goes away if you let it, at the figure it turns on · the compatibility torsion ladder.

Prestress is internal, so it produces no reactions.

What decides it: True of a determinate beam and false of every other. The two-span beam here is lifted off its middle support by the tendon; holding it down takes 51 kN pressing down there and 25.5 kN lifting at each end. That set sums to zero — it is self-equilibrating, which is what internal means — and it bends the beam by 306 kNm over the support regardless.

Tested in The prestress that pushes back, at the figure it turns on · the secondary prestress ladder.