The bracket pushed from the wrong side

A bolt group carrying an eccentric load shares the force equally and the torque unequally, and the bolt furthest from the centroid, in the corner where the two shares point the same way, carries most. That essay’s bracket carried a vertical load at a fixed eccentricity, which is how brackets are drawn: the load has a direction on the drawing, the eccentricity is measured at right angles to it, and the check is one calculation.

Many brackets are not loaded that way. A davit swings. A crane runway bracket takes the wheel load vertically and the surge laterally, in any proportion, which is an envelope rather than a load. A sign bracket takes wind from whichever way it blows. A lifting lug is pulled at whatever angle the sling happens to make. For all of these the direction of the load is a variable, and a check made in one direction is a check of one point on something that is not a point.

The direction changes the torque, not only the shear #

The elastic method needs two things from the load: its size, to share equally among the bolts, and its torque about the group’s centroid, to share in proportion to distance.

The torque is the load times the perpendicular distance from the centroid to the load’s line of action. If the load acts through a fixed point on the bracket — the eye of a lug, the tip of an arm — that perpendicular distance depends on which way the load points. A load aimed straight at the centroid, along the line from the load point through it, has no torque at all. A load at right angles to that line has the largest torque the point can produce: the load times the full distance from the centroid to the point.

So as the direction turns, both parts of every bolt’s force change: the equal share turns with the load, and the torsional share grows and shrinks and reverses. The bolt whose two shares happen to line up is a different bolt for different directions, and the direction in which some bolt’s two shares line up best is the weakest direction for the group.

A bracket checked straight down #

Take six bolts in three rows of two, 75 mm apart each way, each able to carry 100 kN in shear. The load acts through a point 150 mm across and 150 mm up from the centroid of the group — an arm reaching up and out.

Straight down, the perpendicular distance from the centroid to the load’s line is 150 mm, so the torque is $150V$. The polar moment of the group is $J = \sum r^2 = 4(37.5^2 + 75^2) + 2 \times 37.5^2 = 30{,}938$ mm². The lower right bolt sits at $(37.5, -75)$ mm, and the clockwise torque pushes it left and down: $\tfrac{150 \times 75}{30{,}938} = 0.364V$ sideways and $\tfrac{150 \times 37.5}{30{,}938} = 0.182V$ down. Add its equal share, $V/6 = 0.167V$ down, and its force is $\sqrt{0.364^2 + 0.348^2}\,V = 0.504V$. The group carries $100/0.504 = 198.5$ kN.

That is the calculation a drawing with a vertical arrow on it would produce, and nothing about it is wrong.

The same bracket at 138 degrees #

Now aim the same load through the same point at 138 degrees — up and to the left, the direction a sling pulling back toward the wall would take.

The load’s line now passes 212 mm from the centroid rather than 150, because the direction is nearly at right angles to the line from the centroid to the load point, which runs at 45 degrees. The torque is $211.8V$ anticlockwise. The upper right bolt at $(37.5, 75)$ mm is pushed by that torque at right angles to its own radius — up and to the left at 153 degrees — by $\tfrac{211.8 \times 83.85}{30{,}938} = 0.574V$. The equal share points along the load at 138 degrees, $0.167V$. The two are fifteen degrees apart, so they very nearly add: the bolt carries $0.736V$, and the group $100/0.736 = 135.8$ kN.

A 46 per cent overstatement from checking the direction on the drawing. Nothing about the bolts, the plate or the load’s size has changed. The worst direction is simply the one in which the largest torque the point can make and the direction in which the furthest bolt’s torsional share points come closest to agreeing, and for this geometry that is 138 degrees — between the 135 degrees at which the torque is largest and the 153 degrees at which the corner bolt is pushed.

Which bolt governs, as the load turns #

Sweep the direction through a full turn, and at each step note which bolt reaches its resistance first.

The answer is not one bolt. Bolt 6, the corner furthest along the diagonal the load point lies on, governs for half the turn — every direction in which the torque is anticlockwise. For the other half, clockwise, the governing bolt is whichever of the two remaining corners best lines up its torsional share with the load, and that changes at the directions where the load passes near the centroid.

That has a practical edge. A designer who has identified bolt 4 as critical from the vertical check, and specified a larger bolt or a tighter hole there, has strengthened a bolt that governs for a quarter of the directions and is not the one that fails in the worst of them.

The capacity as a curve #

Put the whole sweep on one axis: the group’s capacity against the direction of the load, for three layouts of six bolts.

It has a shape worth reading. Two spikes at 45 and 225 degrees, where the load aims at or directly away from the centroid, the torque vanishes and every bolt carries a sixth of the load: 600 kN, the group’s full shear capacity. Two troughs half a turn apart, near 135 and 315 degrees, where the torque is largest. And the curve repeats every half turn, because reversing a load reverses every bolt force and changes no magnitude.

The capacity that matters for a load that can turn is the bottom of the lower trough, and the check that finds it is a minimisation over direction, not a calculation in one.

Moving the load point to directly across from the centroid — the ordinary bracket — puts the troughs near vertical, which is why the vertical check is usually close. It is close, not right: the tall rectangle’s weakest direction is 258 degrees rather than 270, and its capacity there is 154.5 kN rather than the vertical check’s 157.6. The error is two per cent. It is 46 per cent when the load point is on a diagonal, and a bracket arm that reaches up as well as out puts it there.

Two mirror images that each win half the turn #

Turn the rectangle on its side — two rows of three instead of three rows of two — and the polar moment is identical, 30,938 mm², because it is the same set of distances.

For a load straight down through a point 200 mm across, the tall rectangle is 10 per cent stronger: its corner bolts’ radii are mostly vertical, so their torsional shares are mostly horizontal and meet the vertical equal share at right angles, while the wide rectangle’s corner radii are mostly horizontal and their torsional shares add straight onto it. That is the usual advice, and it is right for that direction.

Over the whole turn, though, the two layouts trade places. The tall rectangle is the stronger in 178 of the 360 directions and the wide one in 176 — each wins wherever its long dimension lies across the load. Their weakest capacities differ, 154.5 against 143.3 kN, because the load point being across rather than above makes one set of directions matter more. A layout chosen for the drawn direction is chosen for half the directions, and whether that half contains the worst one depends on where the load point is.

Two component checks, added two ways #

The usual way a load of uncertain direction is handled in practice is not a sweep. It is to split the load into components — a vertical one and a horizontal one — check the group for each, and combine the two utilisations. How they are combined turns out to decide whether the answer is safe.

For the diagonal load point, the group carries 198.5 kN straight up or down and 178.4 kN straight across. A load at 138 degrees has a vertical component of $0.669V$ and a horizontal one of $0.743V$.

Added linearly, the utilisations are $0.669V/198.5 + 0.743V/178.4$, and setting that to one gives $V = 132.7$ kN — two per cent below the exact 135.8, and safe.

Added as a square root of the sum of squares, as though the two components were independent, they give $V = 186.6$ kN — 37 per cent above the exact value, and not safe.

The linear rule is not merely safe here; it is safe in every direction, and the reason is one line. The elastic method is linear in the load, so each bolt’s force under the combined load is the vector sum of its forces under the two components. The length of a vector sum is never more than the sum of the lengths, and the largest of a set of sums is never more than the sum of the largest of each — so the worst bolt under the combination carries no more than the worst bolt under one component plus the worst under the other, even when those are different bolts. Swept over the whole turn for this bracket, the linear rule is never above the exact capacity and touches it only where the load is purely vertical or purely horizontal.

The quadratic rule has no such guarantee, because it assumes the two worst bolts are at right angles to each other’s contributions, and at the corner bolt they are nearly parallel. Across the turn it overstates the capacity by up to 38 per cent. The combination rule that looks more refined is the one that fails, and it fails in exactly the directions where the torque and the direct shear line up.

Where turning loads come from, and how much of the curve they use #

Not every load that can change direction can take every direction, and the curve is useful precisely because it can be read over a range.

A crane runway bracket carries the wheel load vertically, plus a lateral surge of the order of a tenth of it, in either direction. The resultant stays within about six degrees of vertical, so it uses a narrow band of the curve around 270 degrees — the band the ordinary vertical check sits in the middle of. For a bracket whose load point is directly across from the bolts that band contains the weakest direction, and the two per cent that separates 258 from 270 degrees is the whole error.

A davit or a slewing arm can point its load anywhere in the plane, and uses the whole curve.

A lifting lug is pulled along its sling, whose angle depends on the rigging. A lug designed for a vertical lift and used with a sling at 45 degrees to the vertical is being read at a different point on its curve, and for a lug whose eye is offset from its bolts both up and across, that point can be a trough.

Wind on a sign or a canopy arrives from any horizontal direction, but in the plane of the bolts only one component of it acts — the curve is read over the full turn, weighted by how the wind’s in-plane component varies with its compass direction.

In each case the check is the same operation: find the part of the curve the load can reach, and take its lowest point — which is the worst-position search an influence line makes along a beam, made here round a circle.

What the envelope rewards #

The third layout in every figure is a ring: six bolts equally spaced on a circle of radius 70.4 mm. Its polar moment is 29,737 mm², 4 per cent less than either rectangle. By the usual reasoning — a larger polar moment spreads torque more thinly — it should be the weakest of the three.

At a load point on the diagonal it is the strongest in 306 of the 360 directions, and its weakest capacity is 150.3 kN against the rectangles’ 135.8.

The reason is visible in the arithmetic of the worst bolt. Whatever the direction, the worst bolt’s force cannot exceed its equal share plus the largest torsional share any bolt can have, and those line up at worst exactly. The largest torque a load through a point at distance $d$ from the centroid can produce is $Vd$, and the largest torsional share is at the furthest bolt, $Vd\,r_{\max}/J$. So

$$C_{\min} \;\ge\; \frac{R_b}{\,1/n + d\,r_{\max}/J\,}.$$

For the tall rectangle with the load point on the diagonal, $d = 212.1$ mm and $r_{\max} = 83.9$ mm: the bound is $100/(0.167 + 0.575) = 134.8$ kN, against the exact 135.8. For the ring, $r_{\max} = 70.4$ mm: $100/(0.167 + 0.502) = 149.5$, against 150.3. On the diagonal the bound is within one per cent, because some direction lines the two shares up almost perfectly.

The quantity that matters is $r_{\max}/J$, not $J$. A rectangle gets its polar moment from its corners, and its corners are the furthest bolts, so every increase in $J$ comes with an increase in $r_{\max}$. A ring puts every bolt at the same distance, so for its polar moment its furthest bolt is as close as it can be. For a load of known direction the rectangle can be oriented to take advantage of its shape; for a load that can turn, there is no orientation to take advantage of, and the layout with no bad direction wins.

It is worth being clear why this bound does not fall into the trap the radius rule for weld groups falls into. That rule checks the furthest point for one load direction and can miss a nearer point whose two stress components line up better. The bound here does not choose a point or a direction: it adds the largest possible equal share to the largest possible torsional share as though they were aligned, which no bolt can exceed in any direction. It gives up a little accuracy for a guarantee.

The bound is also a design rule in its own right. It needs no sweep, no angle and no decision about which bolt is critical: the load point’s distance from the centroid, the group’s polar moment and its furthest bolt. Where it passes, every direction passes.

Where the model stops #

The elastic method. Every capacity here assumes bolt forces proportional to distance from the centroid, which is the conservative of the two methods. The instantaneous centre method gives larger capacities, and for a load that turns its centre of rotation moves around the group as the direction changes. The locus it produces is larger and differently shaped, and whether its weakest direction is the same one is not established here.

A bolt’s resistance independent of direction. The 100 kN is treated as the same whichever way the bolt is pushed. A bolt bearing on a plate near an edge is weaker toward the edge, because the tear-out path is shorter and the tear can run diagonally, so a bolt’s resistance is itself a function of direction — and the corner bolts that govern are the ones nearest two edges.

Static loading. A load that turns repeatedly reverses every bolt’s force twice a turn. For bearing-type bolts in clearance holes that means slip back and forth across the hole, which is why brackets under reversing loads are designed as preloaded, slip-resistant joints, where the resistance is friction and the check is against slip rather than bearing — and why oversized holes are a decision about the direction of the load as much as about fit-up.

A rigid plate. The bracket plate is assumed not to deform, as it is in the weld group’s version of the same calculation — and a bracket that is both bolted and welded does not add the two groups at all, because the two fasteners never arrive together. A thin plate under a large torque bends between bolts, and the force distribution moves toward the bolts nearest the load point.

A load in the plane of the bolts. A lug pulled at an angle out of the plane adds tension to the bolts, which is a different problem with a neutral axis of its own, and the combination of that with the in-plane locus here is a surface rather than a curve.

Still open: the centre of rotation that walks round the group #

The elastic method’s capacity locus has a lower bound in closed form because the method is linear: every bolt force scales with the load, and the worst direction is a matter of alignment. The instantaneous centre method is not linear. For each direction the group rotates about a point found by iteration, bolts nearer the load deform less than those further away, and the point itself moves as the load turns — along a closed path around the outside of the group. Whether that path, and the capacity it produces, still favours the layout with the smallest furthest bolt, or whether the method’s reliance on the furthest bolt’s deformation capacity rewards a rectangle’s long lever arm instead, is the question after this one.