Structural Engineering & Statics
where the load goes
Connections

The metal between the holes, which comes out as a block

A bolted end connection can fail without a single bolt breaking and without the plate reaching its tensile strength anywhere. A block of metal simply comes out, bounded by two surfaces with two different strengths on them.

Assumes The connection is not a point, and every diagram on this site says it is and What a cut reveals, and why it was there all along.

17 min read 8 figures The load must go somewhereWhich failure arrives first

Here is a failure that no member calculation on this site can express, and that is the point of showing it first among the plate failures.

A gusset plate is bolted to a bracing member with three bolts in a line. The member pulls. Nothing about the plate is bending; there is no neutral axis in it, no section modulus, no curvature. The plate is in tension, which is the simplest state a piece of steel can be in.

And it fails by a block of metal coming out of it.

Block shear: the metal between the holesThree bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 421.7 kN, of which the shear plane carries 70.43%.pullshear plane, 180 mmtension plane, 40 mmcapacity 421.7 kN0.6 fu Anv = 322.5 kN · 0.6 fy Agv = 297 kN · fu Ant = 124.7 kNthe yield value governs the shear plane
Fig. 1 The block, shaded, and the two surfaces that bound it. Along the bolt line a shear plane 180 mm long; across the end bolt a tension plane 40 mm long. The plate does not run out of tensile strength and the bolts do not break. The block simply separates, at 421.7 kN, and the capacity is the sum of two different strengths acting on two different planes.

Why it is not any of the checks that were made

A bolted end connection has an established checklist, and it is a good one as far as it goes.

Bolt shear. Each bolt has a shear capacity; the group must not exceed the sum.

Bearing. The bolt presses against the side of its hole; the plate must not crush or tear out to the edge.

Net section tension. The plate has holes in it; the reduced section must carry the load in tension.

Every one of those is a check at a point or across a single line. Block shear is neither, because the failure surface is a path — it runs along one direction for 180 mm and then turns and runs across for 40 mm — and the strength is different on the two legs of the path.

The consequence is the one that makes this mode dangerous rather than merely additional: neither leg of the path looks critical on its own. The tension plane is 40 mm long and would fail, alone, at 124.7 kN. The shear plane is 180 mm long and would fail, alone, at 297 kN. Neither is close to the applied load. It is only when the two are made to fail together, in the shape the block requires, that a number as low as 421.7 kN appears — and no check that considers one plane at a time will produce it.

The two strengths, and why they are different

Look at what has to happen for the block to come out.

The material along the two long sides has to shear. Steel’s shear strength is about 0.6f0.6f of its direct strength, which is what a von Mises criterion gives: the material yields when the distortion energy reaches a limit, and pure shear reaches it at 1/3=0.5771/\sqrt{3} = 0.577 of the direct yield stress.

The material across the end has to pull apart. That is direct tension, at the full strength.

So the capacity is

P=0.6 f Ashear+f AtensionP = 0.6\,f\,A_{\text{shear}} + f\,A_{\text{tension}}

and the question is which ff goes in each term. The convention — and it is a convention arrived at from tests rather than a derivation — is that the tension plane is taken at the ultimate strength on the net area, while the shear plane is taken as the lesser of the ultimate strength on its net area and the yield strength on its gross area.

That mixture looks arbitrary and is not. It is an attempt to describe a sequence.

Two planes that do not fail together

The mixture exists because the two planes do not reach their limits at the same displacement, and the block’s capacity is what they can supply simultaneously.

The tension plane is short. To separate, it needs the material across it to reach ultimate strain, which is a large strain, and over 40 mm that is a large displacement — several millimetres.

The shear planes are long, and shear yields at a small strain. By the time the tension plane has stretched enough to rupture, the shear planes have long since yielded and are sliding.

So the honest picture is that the shear planes yield first and then the tension plane ruptures, and the load at that instant is the yield strength on the sheared area plus the ultimate strength on the torn area. Which is exactly what the mixed formula says, and it is why using ultimate strength on both — the arithmetically tidier choice — overestimates what the connection can supply at one moment.

For the connection above, the shear rupture value is 322.5 kN and the shear yield value is 297 kN, so yielding governs and the sum is 421.7 kN. The shear planes contribute 70% of it.

Block shear: the metal between the holesFive bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 320 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 652.7 kN, of which the shear plane carries 80.89%.pullshear plane, 320 mmtension plane, 40 mmcapacity 652.7 kN0.6 fu Anv = 570.18 kN · 0.6 fy Agv = 528 kN · fu Ant = 124.7 kNthe yield value governs the shear plane
Fig. 2 Five bolts instead of three. The shear planes lengthen by 140 mm and the tension plane does not change at all, so the capacity rises to 652.7 kN and the shear share rises to 81%. Adding bolts to a line makes this failure mode more shear-dominated, which changes which strength governs and which detailing rule matters.

Which dimension buys what

The two planes are controlled by different dimensions, and knowing which is which is most of the practical content of this mode.

The shear plane length is the end distance plus the bolt pitches: e1+(n−1)p1e_1 + (n-1)p_1. Lengthening it means more bolts or wider spacing, both of which make the connection longer.

The tension plane length is the edge distance, plus the gauge if there are two lines of bolts. Lengthening it means a wider plate.

Compare what each buys. Going from three bolts to five raises the capacity from 421.7 to 652.7 kN — 55% for two extra bolts and 140 mm of extra length. Going from 40 mm of edge distance to 80 mm raises it from 421.7 to 593.7 kN — 41% for 40 mm of plate width.

Block shear: the metal between the holesThree bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 180 mm long and a tension plane 80 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 593.7 kN, of which the shear plane carries 50.03%.pullshear plane, 180 mmtension plane, 80 mmcapacity 593.7 kN0.6 fu Anv = 322.5 kN · 0.6 fy Agv = 297 kN · fu Ant = 296.7 kNthe yield value governs the shear plane
Fig. 3 The same three bolts with the edge distance doubled to 80 mm. The tension plane doubles, the shear planes do not move, and the capacity goes to 593.7 kN with the shear share falling to 50%. Forty millimetres of extra plate width bought more than two extra bolts and 140 mm of extra length did.

The edge distance is by far the better exchange rate, and it is the dimension most likely to have been set by a minimum rather than chosen. That is worth knowing because the instinct, on discovering that a connection fails block shear, is to add bolts; and adding bolts is the expensive lever.

There is a second reason to prefer widening. The shear term is 70% of the capacity at three bolts and 81% at five, so adding bolts pushes the connection further into a mode governed by shear yield — a strength that does not benefit from a higher grade of steel in the way the ultimate strength does. Widening moves the balance the other way.

The three checks it sits between

It helps to see block shear alongside the two plate checks it is most easily confused with, because all three are about holes in a plate and all three have different answers.

Bearing and tear-out against end distanceA 20 mm bolt in a 10 mm plate. Below 165 mm of end distance the bolt tears a channel out to the end and the capacity is proportional to that distance; above it the plate crushes in front of the bolt and the end distance stops mattering. At 40 mm the capacity is 52.12 kN and the mode is tear-out.020406080100120140050100150200end distance, mmbearing capacity, kN40 mm → 52.12 kNthe corner is at 165 mm, past this plotplate crushesbolt tears out
Fig. 4 Bearing and tear-out at a single hole: the check that asks what happens immediately in front of one bolt. Below 165 mm of end distance the bolt tears a channel to the end of the plate and the capacity is proportional to that distance; above it the plate crushes. At the 40 mm end distance of the connection above, one bolt bears at 52.1 kN, so three bear at 156 kN — which is a different number from 421.7 and answers a different question.

Bearing is local to one hole, and its failure is a channel of metal shoved forward by one bolt. Block shear takes the whole group and the metal between and around all of it. The two are related — a very long tear-out from the last bolt is on its way to being a block — but the areas are different and so are the strengths.

The net section, and the path the tear takesA 200 mm plate with two holes staggered by 0 mm at a gauge of 60 mm. The straight path through one hole leaves 156 mm; the diagonal path through both leaves 156 mm after the s²/4g correction adds 0 mm back. The shorter of the two decides, at 78% of the gross section.g = 60net width 156 mm of 200the critical path crosses two holes, with s²/4g = 0 mm added back
Fig. 5 Net section tension: the check that asks what is left of the plate across its width. The holes remove their own diameters and the remaining 156 mm of a 200 mm plate carries the load. This failure is a straight line across the member, at right angles to the load. Block shear’s failure is a path partly along it, which is why the two produce unrelated numbers.

Net section tension is a cut across the member. Block shear’s surface runs mostly along it. A connection can be comfortable on the net section and fail as a block, because the material the net section check counts — the full width of the plate — is not the material that has to shear for the block to come out.

The three checks are therefore not three approximations to one answer. They are three separate mechanisms, and a connection has whichever capacity is smallest.

Block shear: the metal between the holesTwo bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 110 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 306.2 kN, of which the shear plane carries 59.27%.pullshear plane, 110 mmtension plane, 40 mmcapacity 306.2 kN0.6 fu Anv = 198.66 kN · 0.6 fy Agv = 181.5 kN · fu Ant = 124.7 kNthe yield value governs the shear plane
Fig. 6 Two bolts rather than three. The shear planes shorten by 70 mm, the tension plane is unchanged, and the capacity falls to 306.2 kN with the shear share down to 59%. Every bolt removed takes a fixed amount off the shear term and nothing off the tension term, so the balance between the two moves with the bolt count.

Where it hides

Block shear is easy to miss and there is a structural reason for that, beyond the plane-by-plane checking above.

It is a failure of the connected part rather than of the fastener or of the member, and connected parts are frequently detailed rather than designed: a gusset plate’s size is chosen to fit the bolts on with the minimum edge distances, and then nobody returns to it. The member has been sized, the bolts have been sized, and the plate is a consequence of both.

The configurations where it governs share a shape:

  • coped beam ends, where the top flange has been cut away so the beam can pass under another, leaving a short web to bolt through and a very short tension plane above the top bolt;
  • gusset plates at bracing connections, where the plate is sized to the bolt pattern;
  • angles connected through one leg, where the tension plane is the edge distance on a leg that is already small — and which are losing capacity to shear lag at the same time and for a related reason;
  • short end connections generally, where the block is small and both planes are small with it.

A coped beam end is the classic case and the one to remember, because the cope removes the material that would otherwise have made the tension plane long. It is also the case where the reason for the cope has nothing to do with the connection: the flange was cut away so that a beam could pass beneath another one, a clearance decision made by somebody looking at a section through the floor, and the structural consequence lands on a plane 40 mm long that appears on no drawing as a dimension anybody chose.

Why the formula has a minimum in it

The awkward-looking min⁡\min between shear rupture and shear yield repays a paragraph, because it is not a safety device and it is not indecision.

Both quantities are real limits on the same plane and they are limits on different areas. Shear rupture is the ultimate strength acting on the net shear area — the length of the plane less the material removed by the holes it passes through. Shear yield is the yield strength acting on the gross shear area, the whole length, because yielding does not care that there are holes: the material between them yields, and so would the material in them if there were any.

For the connection at the top of this page those two come out at 322.5 and 297 kN, close enough that either could govern with a small change of geometry. Which one does is decided by the ratio of hole area to gross area on that plane, and therefore by the bolt pitch: a tightly pitched group removes proportionally more of the plane and pushes rupture down, while a widely pitched one leaves more metal and lets yield govern.

So the minimum is a statement about which of two competing descriptions of the same plane is the binding one, and it moves with the detailing. It is exactly the same structure as the lower of the squash load and the Euler load deciding a column: two mechanisms, one plane, whichever arrives first.

What it looks like when it happens

Block shear failures are visually unmistakable in a way that most steel failures are not, and it is worth knowing what to look for.

There is no necking, because the plate never reached a uniform tensile limit. There is no buckling. What there is, is a rectangle of plate — the block — still bolted to the member it was connected to, having come out of the plate it was connected through. The bolts are usually undamaged and still in their holes. The member is usually undamaged. The gusset plate has a rectangular bite taken out of one end.

That signature is diagnostic and it is a useful thing to carry, because it says immediately which of the checks was the one that was not made. A sheared bolt says bolt shear. A plate necked across its width says net section. An ovalled hole says bearing. A missing rectangle says nobody added the two planes together.

Block shear: the metal between the holesThree bolts in a 10 mm plate end connection. The shaded block tears out along a shear plane 165 mm long and a tension plane 40 mm long. Shear yields first, and the capacity is the sum of two different strengths on two different planes: 396.95 kN, of which the shear plane carries 68.59%.pullshear plane, 165 mmtension plane, 40 mmcapacity 396.95 kN0.6 fu Anv = 283.8 kN · 0.6 fy Agv = 272.25 kN · fu Ant = 124.7 kNthe yield value governs the shear plane
Fig. 7 The same three bolts with the end distance cut from 40 mm to 25. The capacity falls to 396.9 kN — 6% — because the end distance appears once, at the end of the shear plane. Compare that with the 41% the edge distance was worth: the two dimensions look interchangeable on a drawing and are not.
An angle bolted through one legA 100 × 75 × 10 angle connected through its 100 mm leg with three bolts at 75 mm pitch. The centroid sits 19.77 mm from the connected face over a connection 150 mm long, so U = 1 − 19.77/150 = 0.87 and 13.18% of the net area is not working.x̄ = 19.77connected legoutstanding legLc = 150net areaU = 0.87 of it worksU = 1 − x̄ / Lc = 0.87both halves are geometry — where the centroid sits, and how long the connection is
Fig. 8 And the other thing a short connection does to the member it is on. Block shear removes a piece of the plate; shear lag leaves part of the member unstressed because the force has not had room to spread across it. Both are consequences of the connection being short, and neither is visible in the member’s own section properties.

The ductility this quietly requires

There is an assumption underneath the sum that is worth surfacing, because it is the same one that runs under every plastic argument on this site.

Adding a shear term and a tension term treats them as though both are available at once. They are, but only because the material can deform enough to keep the shear planes carrying their yielded load while the tension plane goes on stretching to rupture. In a brittle material there is no such overlap: whichever plane reaches its limit first fails, the rest follows, and the sum is not available.

So the block shear equation is a lower-bound plasticity argument in the same sense as the thrust line in a masonry arch or the plastic hinge in a beam. It proposes a mechanism, checks that every element of the mechanism can supply its share, and relies on ductility to let the structure find that mechanism.

Which is why the equation is written for structural steel and why it should be treated with suspicion for anything else — high-strength bolts in thin plate, steel at low temperature, or any detail where a flaw could turn the tension plane brittle before the shear planes have finished yielding.

What to take from it

Some failures are shapes, not stresses. Nothing in beam theory, section theory or buckling can express a block coming out of a plate, because all of those describe a stress at a point in a continuous field and this is a surface separating.

Two planes, two strengths, one sum. Shear on the long sides at 0.6f0.6f, tension across the end at ff, and the choice between yield and ultimate on each is a statement about which one gets there first rather than about which is more conservative.

Neither plane looks critical alone, and that is why the check is separate. 124.7 kN on the tension plane and 297 kN on the shear planes, in a connection that fails at 421.7. Every individual check passes.

The cheap lever is the edge distance. Widening the plate is worth more per millimetre than lengthening it, and it moves the failure away from the shear-dominated end where the grade of steel stops helping.

And the mode is a property of the connected part, which is the piece nobody designed. The member was sized, the bolts were sized, and the gusset was drawn to fit the bolts on. That is the ordinary sequence and it is the sequence in which this failure survives: every element that somebody chose has been checked, and the failure surface runs entirely through the one that was merely drawn.

Where this goes next

100 kNe = 150centroidworst bolt 50.37 kNSix bolts · direct shear 16.67 kN eachelastic vector method The connection is not a point, and every diagram on this site says it is Every free body drawn here has joined its members at points. Real structures fail at the joints far more often than in the members, and the reason is that a joint is exactly the region the theory behind every other page explicitly excludes. s = 50g = 60net width 166.42 mm of 200the critical path crosses two holes, with s²/4g = 10.42 mm added back The tear that goes diagonally, and the correction that has no derivation Stagger the holes so that no straight line crosses more than one and the plate does not get its strength back. The tear runs at an angle instead, and the arithmetic that makes it come out right is a century-old piece of curve-fitting nobody has improved on. 020406080100120140160180200050100150200end distance, mmbearing capacity, kN40 mm → 52.12 kNcorner at 165 mmplate crushesbolt tears out The hole that goes oval, and the one that tears to the edge A bolt pressing on the side of its hole either crushes the plate in front of it or shoves a channel of metal out to the end. Which one happens is decided entirely by a distance that is usually set by a minimum in a table. neutral axispeak 0.4stressflow, q = VQ ÷ Imean stress 0.19 — the value a shear divided by an area would givepeak 2.15× that, and in the place bending ignoresthe flow is continuous; the stress jumps wherever the width does The shear nobody draws A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains. 00.5%1%2%2%050100150200250300350strainstress, N/mm²mild steelcast iron The property that appears in none of the equations Ductility is in no design formula on this site. Every method on this site depends on it — and a brittle structure does not merely fail early, it makes the analysis wrong.

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Block shearConnectionDuctilityLoad pathNet sectionShear planeShear yieldUltimate strength