Eccentricity — where it appears
Named by 45 essays across 8 fields — each of them below, with the objects they name alongside it.
The point that is not in the section
A channel loaded down its web twists. To stop it, the load must be applied through a point outside the steel entirely — in the air beside the section, where nothing can be attached.
The load that makes itself worse
A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load.
Two ways to fail, and the curve between them
A column carrying both compression and bending has two capacities and a rule for sharing them out. The rule is a straight line, the truth is a curve, and for a rectangle the straight line gives away a quarter of the plastic moment at half the squash load.
The connection is not a point, and every diagram so far 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.
The bolt that carries more than its share
Six bolts, one hundred kilonewtons, and a worst bolt carrying fifty. The load is shared equally and the torque is not, and the second one is invisible on any drawing where the connection is a point.
The corner that is not the worst point
Check the point furthest from the centroid. It is the standard rule for a weld group under an eccentric load, it is exactly right for some shapes, and for others it misses the peak by sixteen per cent — or picks one of four points it cannot tell apart whose stresses differ by two thirds.
The angle that uses half of itself
Bolt an angle through one leg and the other leg is not fully working. The correction is one over a length — both halves of it are geometry, neither involves a material, and a two-bolt connection throws away a quarter of the section.
Where the structure meets the ground, and when the bolts start working
Push a base plate off its middle third and it lifts off the foundation. The holding-down bolts then carry exactly nothing, and go on carrying nothing until the plate has crushed the concrete underneath it.
Weight is the only thing resisting it
A structure that is strong enough everywhere can still be blown over, and nothing in its material properties has any part in whether it is. The whole answer is a weight and a width — and the failure begins long before anything tips, at the moment one edge stops pressing down.
The middle third
A material that cannot be pulled imposes a condition on where the load may land, and the condition is a region rather than a point. For a rectangle it is the famous middle third; for every other section it is a shape nobody quotes, and one ordinary section's is nearly twice as generous as the rule allows.
Moving a force, and what it costs
Every free body on this site begins by putting a force somewhere convenient. That move is free along the force's own line, costs a couple across it, and in three dimensions leaves behind something no choice of point can remove.
The corner that moves most
A lateral force is shared out in proportion to stiffness only if it passes through the centre of rigidity, which is not the centre of the plan and not the centre of mass. The distance between the two is a torque, and the wall that pays for it is the one furthest away and carrying least.
The prestress that pushes back
On a simply supported beam a tendon is an internal matter and changes no reaction. Put the same beam on three supports and the tendon lifts it off the middle one, the support refuses, and the force it takes to hold the beam down is a reaction produced with no load applied at all.
What is left after it ripples
A thin plate that buckles locally has not failed. It has stopped taking load in its middle and gone on taking it near its edges, so the member is now made of a different section from the one that was drawn — and the new one has its centroid somewhere else, which turns a concentric load into an eccentric one.
The member with only one direction
If a body is in equilibrium under forces applied at exactly two points, those forces are equal, opposite and along the line joining the points. It is three conclusions from two equations, it is the shortest real theorem in statics, and nearly everything that follows depends on it without saying so.
Four inequalities and a wedge
A prestressed section has to satisfy two stress limits when the force is largest and the load smallest, and two more when the force has relaxed and the load has arrived. Each is linear in one over the force — which turns a search for a prestress into a region on a page, and turns an impossible section into an empty one.
The section that will not keep its shape
A box girder is closed, so torsion costs it almost nothing. What an eccentric load actually does to it is something a torsion calculation contains no term for — the rectangle becomes a parallelogram, in its own plane, along the whole length of the span.
The column that fails years later
A concrete column under sustained load goes on straining at constant stress, so its deflection grows — and because the second-order moment is the load times that deflection, the demand grows with it. There is a load below which the two settle and one above which they never do.
The weight that makes it safer
Every load in this collection makes a structure worse. A pinnacle does not. A masonry pier fails when the line of compression leaves the stonework, and adding weight at the top rotates that line back towards the vertical without adding anything the pier cannot carry — so the stone is not being strengthened, it is being aimed.
It does not buckle, it runs out of width
Every stability failure in this collection is a member that could have carried tension deciding to go sideways instead. Masonry cannot carry tension, and its failure under an eccentric load is not a bifurcation at all — the bearing area simply shrinks until it runs out. The capacity is exactly linear in the eccentricity, Euler's load is ten times anything allowed, and no material property appears until the very end.
The load that is really a lean
No frame is ever plumb. The columns are out of upright by something like a three-hundredth, and every tonne of gravity load standing on that lean has a horizontal component. The force that represents it is not a safety allowance — it is an exact statics substitution for a geometry nobody drew.
The moment the beam left behind
A beam reaction is drawn arriving on a column's centreline. It arrives on a cleat a hundred millimetres out from the face, and the difference is a couple that goes into the column and has to be shared between the lengths above and below it. Nothing about it appears in a frame model whose members meet at nodes.
The strength thrown away on purpose
Masonry, concrete and soil are all analysed as though they had no tensile strength whatever. Each of them has some. The decision to set it to zero is the single most consequential modelling assumption in the subject, it is safe for one kind of check and unsafe for another, and almost nothing that uses it says which.
The joint that is crooked by construction
Lap two plates and fasten them and the two load paths are offset by the thickness of a plate. The joint carries a moment nobody applied, the peak stress is four times the mean, and the rotation that is supposed to straighten it out saves eleven per cent — because the rescue works for thin sheet with a long lap and a bolted structural joint is neither.
The joint that is not where it was drawn
A truss is drawn as lines through the centroids of its members meeting at points. A fabricated joint is under no obligation to oblige, and when the working lines miss each other the resultant has a moment about the node that has to go somewhere.
The shape that came out of the shop
A weld cools by seven hundred degrees while the plate holds it, so it yields in tension and stays that way. What is left is a locked-in force of three hundred kilonewtons applied where the weld is, and if that is not on the centroid the member leaves the shop bent.
Whether it tips or slides
A free body pushed sideways has two ways of leaving, and which one it takes is decided before any load is known. The condition is a width divided by a height set against a coefficient of friction, and the weight, the wind pressure and the depth of the body all cancel out of it.
The tendon that can be moved
Lift a continuous beam's tendon at its interior support without changing its drape and nothing about the beam's total moment changes. The primary falls, the secondary rises by exactly as much, and the pressure line stays where it was — which turns a parasitic effect into a quantity a designer can place.
The eccentricity a purlin cannot avoid
A channel's shear centre is outside the material, so a load applied anywhere on the section misses it. The distance is fixed by the proportions rather than by the detailing, it is 38 mm on an ordinary purlin, and the torque it produces is not an error anybody made.
The pinned base that is not pinned
A column base drawn as a pin is a plate bearing on grout, and a plate in contact over its whole length resists rotation whether anybody wanted it to or not. The stiffness it delivers depends on the axial load, so the assumption is one a frame can leave and re-enter as its loads change.
The radius rule, and where it fails
A weld group under an eccentric load is checked at the point furthest from its centroid, on the reasoning that the stress from the twist grows with the radius. That reasoning ignores the direction the two stresses point in, and for one common shape it misses the peak by nine per cent.
Turn the column, and the slab passes
A flat slab that is comfortable under gravity fails its punching check the moment a moment arrives at the column, and nothing about the load has changed. The fix is not more concrete. It is the column's plan shape and, at equal area, which way round it is turned — worth more than adding half again as much column.
Three actions on one web
A load applied off the centreline of a box girder is three actions at once, and every textbook decomposes it into them. What the decomposition does not say is that all three land on the same piece of plate — added on one web and subtracted on the other, so the two walls of one section differ by a factor of ten.
The weld that is stronger where it is pulled
A fillet weld pulled across its axis is √1.5 stronger than the same weld pulled along it. On a group under an eccentric load the direction of the pull varies from point to point, so the capacity does too — and the gain that follows is worth twenty per cent at one eccentricity and nothing at all at another.
The eccentricity at right angles to the drawing
A bracket's load stands off the plane of its welds as well as being offset within it, and the second eccentricity produces a completely different object — bending about an axis through the group rather than torsion about a point in it. The weld line has no compression zone to argue about, so its neutral axis is its own centroid, and the whole length works.
Half as far between the legs
A body standing on feet, legs or pads has for its base the polygon its supports enclose, and how far its weight can be pushed before it tips depends on which way it is pushed. A three-legged stand pushed toward the gap between two legs has exactly half the reach it has pushed toward one of them.
The compression that stays under the flange
Put a moment on a base plate and a pressure-block calculation pushes the compression out to the plate's edge, where the lever arm to the holding-down bolts is longest. A plate that is not rigid cannot put it there. The compression stays within a few tens of millimetres of the compressed flange, the lever arm shrinks, and the bolts carry twice what the pressure block said.
The throat that misses the load
A partial-penetration butt weld's throat faces the load squarely, so a millimetre of it carries more than a fillet's, and in every grade what limits it is a cap on normal stress rather than the directional criterion. Made from one side, the throat sits off the plates' line of action. At half penetration it keeps a quarter of that capacity, and the stress the eccentricity adds lands on the root, which is a crack.
The bolt group has no neutral axis
A bolt group carrying a moment about an axis in its own plane has some bolts in tension and something in compression somewhere else. Where that somewhere else is decides the answer, nothing in the group decides it, and the two defensible choices give the worst bolt 50 kilonewtons and 21.
The twist the combination rule invents
Two modes close together respond together, and the square root of the sum of squares assumes they do not. The error has the sign of the two modal contributions: where they agree, as they do in base shear, the rule comes up short, and where they oppose, as they always do in torque, it comes up long — by a factor of five for a floor whose stiffness sits twenty centimetres off its mass, a torque the building does not have.
The bracket pushed from the wrong side
A six-bolt bracket checked for a load straight down carries 198.5 kN. Push the same load through the same point at 138 degrees and it carries 135.8 — the direction changes the torque as well as the shear, and the worst direction is one no drawing shows. The capacity of a group whose load can turn is a closed curve, the bolt that governs it changes as the load turns, and what the curve rewards is not a larger polar moment but a smaller distance to the furthest bolt.
The better method flatters the worse layout
The instantaneous centre method finds capacity the elastic method cannot see, and how much it finds depends on the layout it is applied to. On a six-bolt rectangle under a load of unknown direction it is worth 24 per cent; on a ring of six with a smaller polar moment it is worth 14. The two methods agree on which layout to choose and disagree by a factor of five about the margin — and the reason is that a layout the elastic method already likes is one with nothing left to redistribute.
The bolt that was never fitted
A six-bolt bracket found with five bolts in it has lost a sixth of its fasteners and between four and thirty-four per cent of its capacity, depending which one is missing. The share is the smallest of the three things that changed: the centroid moves away from the gap, which lengthens the load's own lever arm, and the polar moment falls by more than the count does. The bolt whose absence costs most is not the bolt that governed the check.
The strongest layout leans on one bolt
Search a plate for the six bolt positions that carry most and the answer carries 234 kN — and loses 36 per cent of it if one particular bolt is missing. Move that one bolt fifty millimetres, into the corner the optimum had just left, and the group carries 232 kN and loses 25 per cent whichever bolt goes. Robustness here costs one per cent of strength, and a search for strength alone will never find it.
The joint whose lines may not meet
In a welded tubular truss the noding eccentricity is not a fabrication error. It is fixed by the brace diameter, the brace angle and the gap between the toes, and the gap is chosen for the welder. For a third of ordinary proportions the one gap at which the working lines would meet lies in a band no fabrication rule permits — so the concurrent joint the truss was analysed with is the one joint that cannot be built.
Named alongside it
The objects these essays reach for when they reach for this one.
Free bodyBolt groupConnectionKernTorsionLoad pathSecond-orderBearing pressureDetailingElastic methodEquilibriumImperfection