What is refuted here, and by what

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 influence line ladder.

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 section shape ladder.

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 effective length ladder.

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 serviceability ladder.

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 elastic modulus ladder.

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 stress concentration ladder.

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 fatigue ladder.

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 prying ladder.

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 net section ladder.

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 weld strength ladder.

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 weld group ladder.

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 slip critical ladder.

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 shear lag ladder.

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 joint stiffness ladder.

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 component method ladder.

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 joint classification ladder.

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 base plate ladder.

A masonry arch stands because its shape is right — because it matches the funicular of the loads it carries.

What decides it: Heyman's safe theorem asks for far less: it is enough that *some* line of compression can be drawn inside the stonework, and there are infinitely many candidates. The family is drawn for one arch under one load, with the minimum- and maximum-thrust members both fitting inside the ring — which is why a cracked arch is usually a working one, and why the question 'what is this arch doing?' is both unanswerable and unnecessary.

Tested in The line that must stay inside, at the figure it turns on · the arch ladder.

The bending stress in a beam is My/I.

What decides it: It is, and only because a cross-section plane before bending is still plane afterwards — a kinematic assumption, not a law and not derived from one. Strain is drawn across a cut face at span-to-depth ratios of 8, 4, 2 and 1 against the straight line the theory assumes: they coincide at the first and the real distribution is nothing like a straight line at the last. Everything in the subject that fails does so where that sentence stops holding.

Tested in Plane sections stay plane, and what the assumption costs, at the figure it turns on · the plane sections ladder.

The Euler load is the load a slender column can carry.

What decides it: Euler's load is the load at which a *perfectly straight* column becomes indifferent to being bent, and no column is perfectly straight. Started with an initial bow, the deflection grows continuously from zero load: there is no bifurcation, and the Euler load is an asymptote the column never reaches. The Southwell plot then recovers that unreachable load from readings taken nowhere near it.

Tested in The column that was never straight, at the figure it turns on · the buckling ladder.

The instantaneous centre method is a more accurate version of the elastic vector method, so the difference between them is an error.

What decides it: They are different assumptions about what the plate does, not two attempts at one number. Swept over eccentricity for a 3 × 2 group, the gap is 16% at 40 mm, 13% at 150 mm and 12% at 300 mm — a curve, not the single factor it is usually quoted as — and just off zero eccentricity the instantaneous centre comes out 1.2% below, because it anchors deformation at the furthest bolt while a concentric elastic group has every bolt at exactly its capacity.

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

The s²/4g term is an approximation to the true diagonal net area.

What decides it: It is not an approximation to a length at all. The diagonal path is geometrically longer than the straight one by a factor of sec θ, which at s = 50 and g = 60 is 1.302 — 18.1 mm of extra length where the rule adds back 10.4, and adding back the extra length would be a correction in the wrong direction besides. The rule stands in for the diagonal leg carrying a mixture of tension and shear rather than for its length, and it is a fit to test data with no derivation behind it.

Tested in The tear that goes diagonally, and the correction that has no derivation, at the figure it turns on · the net section ladder.

Bearing on the plate is a check on the material's crushing strength.

What decides it: Only past a corner it usually is not past. Below an end distance of 2.5·3·(d+2) = 165 mm the failure is not crushing at all: the bolt shoves a channel of metal out to the end of the plate, and the capacity is proportional to the end distance rather than to any material property of the region in front of the bolt.

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

For a weld group under an eccentric load, check the point furthest from the group's centroid.

What decides it: True for a group symmetric about the load's axis and false otherwise. Measured on five shapes: a C, a plain vertical run and a deep C put the peak exactly at the furthest point; an L-shaped group puts it 15.6% higher at a point nearer the centroid; and two horizontal runs put four sampled points at identical radius carrying stresses that differ by a factor of 1.656.

Tested in The corner that is not the worst point, at the figure it turns on · the weld group ladder.

Tightening a bolt harder makes the connection stronger.

What decides it: It makes the joint stiffer and it changes the mechanism; it does not raise the ultimate capacity. Two M20 bolts preloaded to 137 kN each carry 137 kN of slip resistance at μ = 0.5, and 188 kN in bearing once slipped. The ultimate capacity is the second number and preload has not moved it. What preload bought is that the first 137 kN arrive with no movement at all.

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

Bolt rows share the tension in proportion to their distance from the compression centre.

What decides it: That is the elastic distribution and it is one of two. With three equal rows it gives 142.3 kN·m; letting every row reach its own limit gives 172.8 — a factor of 1.214 for no extra steel. Which is available depends on whether the top row can keep carrying its load while the lower rows catch up, and that is a property of which component governs that row rather than of the arithmetic.

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

The middle-third rule is the design limit for a base plate under moment.

What decides it: It marks a lift-off, not a capacity. The regime it opens ends when the peak bearing pressure reaches the concrete's limit, and for this plate that is at e = 200 mm and M = 120 kN·m — well before the bolts are needed at 150. The plate crushes with the bolts still idle, so the governing boundary is neither the kern nor the bolt.

Tested in Where the structure meets the ground, and when the bolts start working, at the figure it turns on · the base plate ladder.

If m + r = 2j, a pin-jointed frame is statically determinate and stable.

What decides it: The count asks whether there are enough equations and never whether they are different from one another. Arrangements satisfying it exactly are drawn folding up, and the honest test is the rank of the equilibrium matrix — with the mechanism sitting, visibly, in its null space.

Tested in The count that does not see it, at the figure it turns on · the determinacy ladder.

A member is safe when the stress in it is below the yield stress of its material.

What decides it: Failure load against slenderness, with both modes on one plot. Squashing happens at σ_y·A and does not care about length; Euler buckling happens at π²EI/L² and cares about nothing else. The governing capacity is the lower, and the yield stress does not appear in the second at all — so a slender column of high-strength steel and one of ordinary steel buckle at exactly the same load, with no stress anywhere near a limit.

Tested in Strong enough and still falls over, at the figure it turns on · the buckling ladder.

An extra support is a free margin of safety — more supports, more capacity.

What decides it: Equilibrium cannot divide a load between two routes to the ground; the missing condition is compatibility, one equation per redundancy, and it brings EI into a calculation statics never needed. Load then goes where the stiffness is — and the same structure acquires a sensitivity it did not have, because a support that moves produces a full moment field with no load applied at all.

Tested in One support too many, and what it costs to know, at the figure it turns on · the indeterminacy ladder.

A connection is designed to carry the forces the frame analysis reports at that node.

What decides it: The frame analysis reports forces at a point, and the connection is a region across which those forces have to be transferred by a specific mechanism at specific offsets. A bolt group solved at an eccentricity of 150 mm has its worst bolt at 50.4 kN where the direct share is 16.7 — three times what the node forces suggest, and the extra is entirely a consequence of the connection having size.

Tested in The connection is not a point, and every diagram on this site says it is, at the figure it turns on · the connection design ladder.

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.

Truss joints are pins, so truss members carry axial force and nothing else.

What decides it: The same truss solved twice — pin-jointed, and with the joints continuous as welding actually makes them. The axial forces agree to about a per cent, which is why the idealisation is a good one; the bending it omits reaches a quarter of the axial stress in the worst member. It scales with EI/L, so a stocky truss of heavy members suffers more of it than a light one — which is the opposite of the way most reserves of strength work, and is the direction in which the simplification stops erring safe.

Tested in The joint that is not a pin, at the figure it turns on · the truss ladder.