The bracket with no worst bolt

Every bolt group the earlier essays on this bracket examined was loaded in one of two ways. The bolt that carries more than its share and the essays that followed it put the load in the plane of the bolts, where it is shared as shear and twists the group about its centroid. The bolt group has no neutral axis took the other eccentricity, the distance in front of the column face at which a bracket actually carries its load, and found that it pulls the top bolts in tension against a compression somewhere below that nothing in the group locates. That essay added the two at one bolt, for a load straight down, and noticed that the bolt was the same bolt either way.

It stopped at the case that looked like a detail. A load that is not straight down — a raking strut, a hanger at an angle, a crane girder’s surge added to its weight — still twists the group and still pulls it off the column. But it does not load the same bolt most in both. The bolt the shear loads most and the bolt the tension loads most come apart, and the group’s check is then a question about which of the two reaches its interaction first, which neither calculation on its own can answer.

Straight down, one corner is worst at everything #

The bracket throughout is six M20 grade 8.8 bolts in three rows of two at 75 mm, the one every earlier essay on this group used, on a plate 200 mm wide with 75 mm from the bottom row to the plate’s lower edge. Its load acts 150 mm to the side of the group’s centroid and 150 mm in front of the column face. Each bolt resists 94 kN in shear and 136 kN in tension, and is checked on an interaction between the two.

With the load straight down, the shear on each bolt is the elastic method’s: an equal share of the load, plus a torsional component perpendicular to the line from the centroid, largest at the four corners and pointing downward on the side the load is on. The two right-hand corner bolts carry the most, 90 kN each at the group’s capacity. The tension is the edge model’s: the plate pivots on its bottom edge, and each row is pulled in proportion to its height above it, so the top row carries the most, 38 kN per bolt. One bolt is in both sets — the top right — and it governs.

That coincidence is what made the combined check look routine. It is not a property of bolt groups; it is a property of this load’s direction. The torsion puts the largest shear on the corners farthest from the centroid on the load’s side, and the pivot puts the largest tension on the row farthest from the compression edge, and for a vertical load both of those are the corner furthest up on the load side.

Tilt the load and the corners come apart #

Incline the load by 60° towards the side it is offset to and both distributions move. The horizontal component adds a direct shear to every bolt in the direction it acts, which the torsion’s shears on the right-hand column now mostly oppose and on the left-hand column mostly reinforce at the top — so the top right bolt still carries the most shear, 84 kN, but only just. And the same horizontal component, acting 150 mm in front of the column, pivots the plate about its right-hand edge as well as its bottom one, which pulls the left-hand column harder. The most tension is now on the top left bolt, 84 kN, while the top right carries 51.

Neither is worst at both. The top left carries 74 kN of shear and 84 of tension; the top right 84 of shear and 51 of tension. On the ellipse the top left reaches its limit first, at a group load of 227 kN, and the top right is at 0.97 of its own. The bolt that governs is the one that is worst at the thing its interaction is least tolerant of — here, tension — and a check that looks only at the bolt with the most shear, which is what an in-plane calculation delivers, has looked at the wrong bolt.

Where a bracket’s load stops being vertical #

The inclined load is not an exotic case. A bracket carrying a beam’s reaction is loaded straight down only while the beam carries gravity alone and bears level. A crane girder seated on a column bracket delivers its wheel loads and, with them, a horizontal surge from the trolley braking, of the order of a tenth of the lifted load, which leans the resultant by several degrees in whichever direction the trolley was moving. A raking prop or a knee brace framing into a bracket delivers a load along its own axis, whatever the angle the setting-out gave it. A canopy tie or a hanger at an angle delivers its force along the tie. And a bracket carrying a beam that expands in the sun receives the friction at its bearing as a horizontal component that reverses with the season — a contact force that can point any way it likes up to its limit, and does.

In each case the in-plane calculation is usually done correctly, because the horizontal component is visible on the drawing as a separate load, and the out-of-plane calculation is usually done correctly for the vertical component, because the bracket’s outstand is visible too. What goes missing is the horizontal component’s own lever arm in front of the column, which pivots the plate about its side edge and pulls one column of bolts harder than the other. That term is small when the horizontal component is small, and it is the term that decides which bolt governs.

The shortcut that treats two bolts as one #

The ordinary defence against not knowing which bolt governs is to take the largest shear in the group and the largest tension in the group and check them together, as if one bolt carried both. It is conservative by construction, since no bolt can carry more of either than the group’s largest. The question is how conservative, and where.

Plotted against the direction of the load, the true capacity and the shortcut coincide wherever one bolt is worst at both, and separate everywhere else. They coincide straight down. They separate as the load tilts either way, and they are furthest apart at the angle where the governing bolt changes — 55.4° on the ellipse, where the true capacity is 221 kN and the shortcut 201.5: the shortcut under-reads the group by a factor of 1.10 exactly at the switch. That is where two bolts are equally critical, and it is the case the shortcut treats worst, because it credits neither with being less loaded in the thing the other is loaded in.

The sweep shows two further things the vertical calculation hides. The capacity has a cusp just beside straight down, at −5.5°, where the top right and bottom right bolts exchange the lead; straight down sits a percent below that peak. And the weakest direction is not straight down but 15° towards the load side, where the group carries 173 kN against 179 — a 4 per cent loss that an inclination the load can easily acquire from a sloping member or a bearing that is not level.

Two bolts at their limit together #

At 55° the group is at the edge of its switch, and every bolt’s position on its own interaction shows what “no single worst bolt” means. The top two bolts are both at their limit to two figures; the middle pair is at about two thirds and the bottom pair at about half. A group in that state has a capacity that depends on two bolts, and any small change — a slightly different edge distance, a slightly larger pull-off, a slightly different interaction — decides which of them fails without changing the capacity much at all.

That has a consequence for a bolt that is weaker than the rest. The bolt that was never fitted asked what a group loses when one of its bolts is missing, and found the answer depends on which, and the strongest layout leans on one bolt found that a layout searched for strength can depend on a single one. At a switch the question has two answers at once, and a group that is robust against losing its worst-in-shear bolt may be fragile against losing its worst-in-tension one. The inspection that finds a loose bolt on a bracket needs to know which kind of worst it is.

Where each bolt governs #

The switch angle is not fixed. It moves with the distance in front of the column at which the load acts, because that distance scales every bolt’s tension and leaves every bolt’s shear alone.

Mapped over both — direction across, pull-off distance up — the group divides into three regions, each governed by one corner bolt, and the boundaries between them lean. With the load at the column face there is no tension at all and the governing bolt is decided by shear alone: the bottom right for loads leaning back, the top right for loads leaning forward, all the way to horizontal. As the load moves out from the face, tension grows, and it moves both boundaries. On the side the load leans toward, the top-left region opens and pushes its boundary down to smaller angles, because the horizontal component’s lever pulls the left-hand column. On the far side the top-right region grows back across straight down into loads that lean away, because the vertical component’s lever pulls the top row, and that makes the top right bolt more critical than the bottom right one the shear alone had favoured.

A fifth of the map — the dots — is within three per cent of a tie. On a bracket designed by checking one bolt, a fifth of the plausible load cases would have a second bolt nearly as close to failure as the one checked, and on those cases the choice of which bolt to check was nearly arbitrary. The bolt that governs a bracket depends on how far in front of the column its load is, not only on where the load points, which is the thing the in-plane and out-of-plane calculations, done separately and added at one bolt, cannot see.

The ellipse and the straight line, for a group #

Two interactions are in use for a bolt in combined shear and tension — the bolt-sized version of the question two ways to fail asked of a whole cross-section. One is an ellipse, $(V/F_v)^2 + (T/F_t)^2 \le 1$, the shape tests on single bolts fall along. The other, Eurocode’s, is a straight line, $V/F_v + T/(1.4F_t) \le 1$ with $T \le F_t$, which lies inside the ellipse everywhere except at the two ends where they meet. For one bolt the disagreement is known exactly: the ratio of the ellipse’s capacity to the line’s along a ray of fixed $T/V$ is $(s + t/1.4)/\sqrt{s^2+t^2}$ in units of the two resistances, which is one in pure shear and in pure tension and rises to $\sqrt{1 + 1/1.96} = 1.23$ where the tension, in proportion to its resistance, is 0.71 of the shear.

A group inherits its governing bolt’s ratio, so the disagreement between the two codes for a bracket is not a fixed 23 per cent or any other number: it is whatever the governing bolt’s mix of shear and tension makes it. With the load 50 mm from the column face the governing bolt is mostly sheared, and for loads within 60° of vertical the two rules differ by only 4 to 9 per cent, rising to 20 for a horizontal load. With it 150 mm out, the governing bolt’s mix sits near the worst proportion for nearly every direction and the rules differ by 13 to 23 per cent. With it 300 mm out they differ by 14 to 23 per cent for loads within 60° of vertical, and come together, to 3 per cent, only for a horizontal load, which pulls one column of bolts far harder than it shears any of them.

There is a second disagreement hidden in the first. The two rules weigh tension against shear differently, so they do not agree about which bolt reaches its limit first. For this bracket the straight line switches from the top right bolt to the top left at 50.5°, the ellipse at 55.4°, and for loads between the two angles the codes disagree not only about the capacity but about which bolt fails.

The pull-off distance, and the tension taking over #

The last figure holds the direction and moves the load out from the column. Straight down, the capacity falls gently from 187 kN with the load at the face to 147 with it 400 mm out, because the governing bolt is carrying mostly shear and the added tension enters its ellipse as a square. At 60° it falls from 254 to 125, by half, because at that inclination the horizontal component’s lever about the side edge pulls the left-hand column harder than the vertical component pulls the top row, and the governing bolt moves from a shear-dominated one to a tension-dominated one on the way.

The shortcut tracks the true capacity closely at both ends and separates in between, by up to a factor of 1.09 at 130 mm out, which is where the two top bolts share the burden most evenly: one mostly sheared, one mostly pulled, both near their limit. The distance at which a bracket’s load acts is usually fixed by what it carries — a beam’s bearing, a hanger’s pin — rather than chosen, and the figure is a reminder that the connection’s behaviour changes character over the range those details span.

The arithmetic at the top left bolt #

At 60° and 227 kN the load has a vertical component of $227\cos 60° = 113.5$ kN and a horizontal one of $227 \sin 60° = 196.6$ kN. The polar second moment of the six bolts about their centroid is $J = 4(37.5^2 + 75^2) + 2(37.5^2) = 30{,}938$ mm², and the torque about the centroid, from the vertical component at 150 mm, is $113.5 \times 150 = 17{,}025$ kN·mm.

At the top left bolt, at $(-37.5, 75)$, the direct shear is $196.6/6 = 32.8$ kN across and $113.5/6 = 18.9$ kN down. The torsional shear is $T r/J$ perpendicular to the radius, with components $17{,}025 \times 75/30{,}938 = 41.3$ kN across and $17{,}025 \times 37.5/30{,}938 = 20.6$ kN up. The resultant is $\sqrt{74.1^2 + 1.7^2} = 74$ kN — the direct and torsional shears nearly align across the plate there.

The tension comes from two levers of 150 mm. The vertical component pivots the plate on its bottom edge, and the top row sits 225 mm above it against a sum of squared heights of $2(75^2 + 150^2 + 225^2) = 157{,}500$ mm², giving $113.5 \times 150 \times 225/157{,}500 = 24.3$ kN. The horizontal component pivots it on the right-hand edge, and the left column sits 137.5 mm from it against $3(62.5^2 + 137.5^2) = 68{,}438$ mm², giving $196.6 \times 150 \times 137.5/68{,}438 = 59.3$ kN. Together, 83.6 kN.

On the ellipse, $(74/94)^2 + (83.6/136)^2 = 0.620 + 0.378 = 1.00$. The same arithmetic at the top right bolt gives 84 kN of shear and 51 of tension, and $(84/94)^2 + (51/136)^2 = 0.94$, or 0.97 as the square root the figure reports.

Elastic shear, a pivoting plate, and a rigid bracket #

The shear is the elastic method’s. The group is taken to rotate about its centroid with every bolt’s shear proportional to its distance, which the better method flatters the worse layout found under-reads the in-plane capacity of this layout by a quarter compared with the instantaneous centre. Whether the instantaneous centre’s gain survives the tension is a separate question: a bolt already carrying tension has less shear deformation to give before it fractures.

The tension is the edge model added about two edges. The plate is taken to pivot on its bottom edge under the vertical component and on its side edge under the horizontal one, each row or column pulled in proportion to its distance from that edge, and the two tensions added. A real plate bears on a region near the corner the two edges share rather than on two lines, and the bearing model that the neutral-axis essay solved for gives smaller tensions; the edge model is the one bracket guidance mostly uses, and the ranking of the bolts is more robust than their values.

The plate is rigid and there is no prying. A flexible plate adds prying forces to the tension bolts, largest where the plate bends most, which would load the top row more and could move the switch.

What the pictures cannot show #

That the group’s capacity is a function of direction and distance, and that a design is checked at a point on it. Every figure here is one bracket; the shape of the map — three regions with leaning boundaries and a band of near-ties along each — is general, but where the boundaries fall depends on the layout, the edge distances and the ratio of the bolt’s tension resistance to its shear resistance. A bracket loaded by a single member in a known direction can be checked at that point exactly. A bracket whose load’s direction depends on a load case — wind reversing, a crane moving — has to be checked along a curve on the map, and the curve is likely to cross a boundary.

Still open: the group that is checked where it is weakest #

The bracket pushed from the wrong side found that a group whose load can turn in its own plane has a capacity that is a closed curve, and a worst direction that no drawing shows. With the pull-off added, the curve becomes a surface over direction and distance, and its lowest point is a different bolt’s limit in different places. Whether the search for the worst case can still be reduced to a lower bound in closed form — as the in-plane case was, by the distance to the furthest bolt — or whether the interaction’s curvature makes every bracket a search, is the question that decides whether a bracket carrying loads from several directions can be designed by hand at all.