The joint that is not a pin

A truss is defined by an assumption. Its members are straight, its loads arrive only at the joints, and the joints are frictionless pins that transmit force and no moment at all. Grant those three and every member carries pure axial force, the analysis becomes two equations per joint, and a structure with fifty members can be solved with arithmetic.

Walk under a real truss and count the pins. There are none. The members are welded to a gusset plate, or bolted through one with four bolts in a rectangle, or — in a tubular truss — cut to a saddle profile and welded directly to the chord. Every one of those is a moment connection. The joint rotates as a unit, the members are bent by being forced to rotate with it, and the resulting bending is called secondary.

“Secondary” is not a statement about mechanism — the bending is entirely real and arrives with the first load. It is a statement about magnitude, and magnitude is a thing that can be measured rather than asserted.

Where the bending comes from #

Under load, a truss deflects. The chords change length, the diagonals change length, and the joints move to wherever those length changes put them. In a pin-jointed frame the members then simply rotate about the pins to suit, at no cost.

With rigid joints the members cannot rotate independently. Every member framing into a joint must rotate through the same angle as the joint itself, and a member forced to rotate at both ends by different amounts is a member in bending. The end moments are whatever it takes to enforce that compatibility:

$$M = \frac{EI}{L}\left(4\theta_1 + 2\theta_2 - 6\frac{\Delta}{L}\right)$$

which is the slope-deflection relation, and its important feature here is the factor $EI/L$. The stiffer the member and the shorter it is, the more moment it takes to bend it through the same angle — so a stocky truss with heavy members suffers more secondary bending than a light one, which is the opposite of the way most reserves of strength work.

How large it actually is #

The figures here solve the question rather than quoting a rule of thumb, and the answer has a clean shape.

Running the same geometry at a range of member slendernesses, and reading the worst bending stress as a fraction of the axial stress in the same member, gives a nearly exact inverse law: the ratio times the slenderness is a constant of about 4.4 across the whole range. Doubling the members’ length-to-depth ratio halves the secondary bending.

That inverse law is not a coincidence. The end moment goes as $EI\theta/L$; the bending stress goes as $Md/2I$; so the bending stress goes as $E\theta d/2L$, in which $I$ has cancelled entirely and only the shape ratio $d/L$ survives. The axial stress, meanwhile, is fixed by the load. Slenderness is therefore the whole story, and the second moment of area — which decides everything about a beam — decides nothing here.

Which free body produced the number #

The truss in these figures is a four-panel Pratt truss of 3 metre panels and a depth of 2.7 metres, loaded with 10 kN at each top-chord panel point, its members square hollow sections whose side is the panel length divided by the stated slenderness.

Two analyses are run on the same nodes and the same loads. The axial forces come from the pin-jointed solver, which assembles the two joint-equilibrium equations at each of the nine joints and solves the resulting system. The end moments come from the plane-frame stiffness solver, in which each member has six degrees of freedom and the joints are continuous.

The two are compared on a single member. At a member slenderness of eighteen the worst is member 0 — the bottom chord next to the support — carrying 16.7 kN of axial force and picking up an end moment of 0.20 kNm, which is a bending stress 24.3% of the axial one. At a slenderness of thirty the same figure reports 14.6%, and at nine it reports 48.1%; the products are 437, 438 and 433, which is the inverse law stated as three measurements rather than as an algebraic claim.

The cross-check that makes this trustworthy is that the two solvers agree on the thing they should agree on. Run them both on a simple three-member frame and the axial forces come out 11.111 kN against 11.012 kN, and −14.948 against −14.819 — a difference of about 0.9%, which is the axial shortening the frame solver includes and the truss solver does not. Two independent routes to the same number, differing by the term one of them omits.

Getting this wrong was easy. The first version of the figure read a field the frame solver does not return, so every bending moment came back as zero and the picture reported a tidy 0.0% — a well-formed drawing of nothing at all. The second version had the geometry in metres and the section properties in millimetres, which scrambled the ratio of axial to bending stiffness by a factor of a thousand and reported ratios in the thousands of per cent, with the trend against slenderness running backwards. Neither error is visible in the drawing; both are visible in the trend.

What the constant 4.4 turns out to be #

The inverse law was reported as a measurement — the ratio times the slenderness comes to 4.4 at three slendernesses — and a constant that is measured and not explained is a constant that might be an artefact. This one is not, and identifying it turns the law into something usable on a truss that has not been solved.

Follow the algebra through. The bending stress is $E\theta d/2L$, with $\theta$ the rotation the joint imposes; the axial stress is $E\varepsilon$. So

$$\frac{\sigma_b}{\sigma_a} = \frac{d}{2L}\cdot\frac{\theta}{\varepsilon}$$

and the product with the slenderness $L/d$ is simply $\theta/2\varepsilon$ — the joint rotations measured against the member strains, which is a property of the truss’s shape and of nothing else. Members do not appear in it. That is why it is constant across the sweep.

What shape? The truss in the figures spans 12 m at a depth of 2.7 m, and

$$\frac{S}{D} = \frac{12}{2.7} = 4.44$$

against a measured constant of 4.4. Put that back and the whole result is

$$\boxed{\;\frac{\sigma_b}{\sigma_a} \approx \frac{S/D}{L/d}\;}$$

the truss’s own aspect ratio divided by its members’. Checked against the three figures: 4.44/18 = 24.7% against 24.3 measured, 4.44/30 = 14.8% against 14.6, and 4.44/9 = 49.3% against 48.1. Three points is a fit rather than a proof, and the agreement is close enough to be worth stating and testing on a second geometry.

The second geometry has to move the ratio the first three did not. All three were the same truss with different members, so the constant 4.44 was measured once and reused three times; a real test changes the truss and leaves the members alone. Six panels of the same three metres at the same depth of 2.7 metres spans eighteen rather than twelve, which is a span-to-depth of 6.67, and the members stay at a slenderness of eighteen. The formula predicts 6.67/18, which is 37.0%.

37.8 against 37.0 is a second point at a geometry the constant was not fitted to, and it is a harder test than it looks, because the worst member has moved. In the twelve-metre truss it was member 0, the bottom chord beside the support; in the eighteen-metre one it is member 18, a diagonal. The formula does not know which member it is describing — it carries no member index at all — and it is still within a point of the answer when the answer relocates.

It holds, and it is the useful form of everything on this page. Secondary bending is two aspect ratios in competition: the shape of the truss, which sets how much rotation the joints impose, and the shape of the members, which sets how much stress that rotation costs.

Which trusses have to be checked #

That form gives a threshold instead of a rule of thumb. Secondary bending passes a fifth of the axial stress when

$$\frac{L}{d} < 5\,\frac{S}{D}$$

For the truss here, whose span-to-depth is 4.44, that is a member slenderness below 22 — and the figure drawn at 18 reports 24 per cent, which is the threshold doing its job.

Read the other way it says which configurations are safe and which are not, and the answer is not “stocky members” on its own.

A deep truss with slender members is safe by both terms at once. Take the truss in these figures and give it the members a roof truss actually gets — light angles or small hollow sections, sixty times as long as they are deep. The formula asks for 4.44/60, which is 7.4%, and at that magnitude the idealisation is doing no harm at all.

Seven per cent is the number that makes the idealisation respectable rather than merely convenient. Nothing about the analysis changed to produce it: the loads, the geometry, the solver and the joints are the ones that reported 48% three figures earlier, and the only thing that moved is how stocky the members are. A roof truss is drawn with light members because the loads are small, and the same decision that makes it cheap makes the pin-jointed model honest.

A shallow truss with stocky members is punished by both. A transfer truss one storey deep spanning fifteen — $S/D = 4$ — built from heavy sections at $L/d = 10$ lands at 40 per cent, and the members will have been chosen heavy because the loads are large, which makes them stocky, which is the same decision arriving twice.

That coupling is the part worth carrying. The two ratios are not independent variables a designer sets separately: a truss made shallow because the depth was not available is a truss whose members must be heavier, and heavier members in the same panel are stockier ones. The configuration that most needs the pin-jointed idealisation to be true is the configuration in which it is least true, and nothing in a member check says so.

Why it is usually ignored, and legitimately #

Three arguments justify the pin-jointed idealisation, and they are good arguments rather than excuses.

The axial forces are right. This is the important one. Secondary bending is a self-equilibrating system of moments caused by imposed deformation, and it does not change the axial forces appreciably — as the figures show, they agree to a per cent. So the pin-jointed analysis gets the primary load path exactly right, and the question is only whether an additional stress has been left out.

Steel yields. The secondary moments arise from imposed rotations, not from imposed loads. A section that yields locally sheds them: the rotation is accommodated plastically and the moment falls away, in the same manner and for the same reason as a plastic hinge redistributes. A stress that disappears when the material yields does not threaten the ultimate strength, and this is why codes permit the idealisation for statically loaded structures with reasonably slender members.

Real joints are not fully rigid either. A bolted gusset has slip and a welded tube joint has local flexibility in the chord face. The true behaviour lies between the pinned and rigid extremes, and the figures here bound it from the unfavourable side.

There is a fourth argument that is weaker than it looks and worth naming so it is not leaned on. It is often said that the secondary moments are small because the joints are “not really rigid”. The figures here are computed with fully rigid joints, which is the unfavourable bound, and a real semi-rigid joint gives something between that and the pinned case — so the argument is true but it is a reduction on a number that has to be computed first, not a reason to skip computing it.

The exception is fatigue. A stress that cycles does not care whether it can be shed by yielding, because the crack grows in the elastic range and grows fastest exactly where the geometry is worst — at the toe of the weld, where the secondary moment is largest. Bridge trusses under traffic are therefore designed with the secondary moments computed rather than dismissed, and joint detailing in a road bridge is a fatigue exercise before it is a strength one. The same reasoning applies to any structure whose loading cycles enough times to matter: crane gantries, machine supports, and anything carrying wind in a wind climate that makes it move.

Where the model stops #

Loads between the joints. The idealisation assumes load arrives only at panel points. A purlin sitting mid-panel puts a genuine primary bending moment in the chord, which has nothing to do with joint rigidity and is much larger. The two get confused; they are unrelated.

Eccentric setting-out. If the members’ centroidal axes do not meet at a point — a common consequence of detailing to fit bolt lines rather than to fit the analysis — the joint carries a moment equal to the axial force times the eccentricity, from the first load. That is not secondary bending and it does not shrink with slenderness; it is a primary moment created by the drawing office and it must be designed for. It is worth putting a number on, because it is the one hazard on this page that is created rather than inherited, and the number is not small.

Twenty-five per cent from 60 mm of setting-out is the same order as the secondary bending the whole essay has been measuring, and it arrives from a decision nobody records as a structural one. Sixty millimetres is a bolt gauge. The two hazards also behave in opposite ways: secondary bending falls as the members get slender, and eccentricity does not, because the moment is the axial force times a distance and neither term knows anything about $I$. A slender roof truss detailed to bolt lines can have the smaller of the two hazards under control and the larger one undrawn.

Which joint, not just which truss. The figures on this page are all Pratt trusses, in which a vertical and a diagonal meet each chord node at very different angles. A Warren truss, where every member is a diagonal, imposes a different rotation on each member at the same joint for the same deflection, so a Warren joint and a Pratt joint of the same size do not suffer equally. That is a geometric effect the two aspect ratios do not capture, and it is the reason the formula above is a shape to check against rather than a coefficient to design by.

Compression members. A member carrying compression and bending together is the awkward case, because buckling is sensitive to bending in a way tension is not: the moment amplifies as the axial load approaches the critical value, which is the second-order effect at the member scale. A 24% secondary bending stress in a tension member is a nuisance and in a stocky compression member is a design case.

Two dimensions, again. A real truss has out-of-plane joint stiffness too, and a chord continuous through several panels is restrained out of plane by a purlin at every second one. Lateral-torsional behaviour of the compression chord is a genuinely three-dimensional question that neither analysis here touches.

The figures cannot show the thing an inspector would most want to see, which is the distribution of the bending along each member. The shading reports one number per member — the largest end moment — and a member’s moment varies along its length, passing through zero somewhere near the middle. A drawing that shaded the whole member by its peak implies a uniformity that is not there, and the honest reading is that the shading marks where to look rather than what will be found.

The frame that gave up on diagonals #

The clearest way to see that joint rigidity is a real structural mechanism rather than a nuisance is to build something that depends on it entirely.

Remove the diagonals from a truss and the count says mechanism: nothing triangulates the panels, and a panel without a diagonal racks. Weld the joints and it stands anyway, because the joint rigidity that produced a 24% nuisance in a triangulated frame is now the only thing resisting the racking. That structure is a Vierendeel girder, and it works — expensively, because the members carry bending as their primary action, and beautifully, because the panels are open rectangles a person can walk through.

Between the fully triangulated truss and the Vierendeel there is a continuum, and a structure’s position on it is decided by how much of the shear each mechanism takes. The frame that leans is at one end of it; the truss in the figures above is very near the other, with about a quarter of its members’ stress coming from the mechanism it officially does not use.

The generalisation #

The pattern is general and worth naming: an idealisation that simplifies the load path usually simplifies the deformation too, and the deformation is where it lies.

The pin-jointed truss, the rigid-body free body, the plane section that stays plane, the point load applied at a point — all four ignore a deformation to gain an equation, and in all four the ignored deformation reappears as a stress the model cannot see. In the truss it is secondary bending. In the free body it is the local stress under the load. In plane sections it is shear lag and warping. In the point load it is bearing and web crippling.

What distinguishes the good idealisations from the bad ones is not whether they leave something out — they all do — but whether the thing left out is self-equilibrating. A self-equilibrating stress system has zero resultant, so it cannot change the overall load path, and it can be shed by any mechanism that accommodates the imposed deformation: yielding, cracking, slip, creep. That is why residual stresses from welding are tolerable, why settlement moments in a redundant frame are usually survivable, and why secondary bending in a truss is a footnote rather than a chapter.

The name is older than the understanding. Nineteenth-century bridge engineers built pin-connected trusses with literal pins precisely because the analysis assumed them, and the practice persisted in America long after riveted joints had proved cheaper and more durable — the pins wore, the eyebars rattled, and several notable failures traced back to a pin. The theoretical justification for riveting them solid arrived afterwards, in Manderla’s and Engesser’s work on secondary stresses in the 1880s, at which point the profession discovered that the joints it had been avoiding were fine.

There is a moral in the order of events. The pins were built because the analysis assumed them, which is the analysis dictating the structure rather than describing it — and the assumption was never a claim about how joints ought to be made, only a device for reducing the equations to a number a person could handle in an afternoon. Two generations of bridges carried a real hazard so that a fictional one could be avoided, and the fiction was in the model the whole time.

The ladder from here #

Later rungs on this anchor: eccentricity at the joint, which is the primary moment that looks like a secondary one. Chord continuity, and the fact that a chord running through several panels is a continuous beam whatever the joints do. Fatigue at the weld toe, where the secondary moment governs. Semi-rigid joint modelling, where the connection is given a rotational spring rather than an extreme. The tubular joint and its chord-face flexibility. And the vierendeel, which is a truss that has abandoned the diagonals altogether and asked the joint rigidity to carry everything — the case in which what is secondary here becomes the entire structural mechanism.