Structural Engineering & Statics
where the load goes
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The load must go somewhere — page 2

Essays 25 to 48 of 298 on this thread, in the same order.
A bolt group under an eccentric load. A 3 by 2 bolt group carrying 100 kN at 150 mm from its centroid, with the resultant force on each bolt drawn to scale, by the elastic vector method. The load is shared equally and the torque is not, so the worst bolt carries 50.37 kN against 16.67 kN of direct shear alone — 3.02 times as much. Connections

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.

 A bolt group under an eccentric load. A 3 by 2 bolt group carrying 100 kN at 150 mm from its centroid, with the resultant force on each bolt drawn to scale, by the elastic vector method. The load is shared equally and the torque is not, so the worst bolt carries 50.37 kN against 16.67 kN of direct shear alone — 3.02 times as much. Connections

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.

 Prying action in a tee stub. A tee stub pulled by its web with 100 kN per bolt. The 20 mm flange is in the one-hinge regime, so the prying force at the flange tip is 50.63 kN and the bolt carries 150.63 kN — 1.51 times what was applied. The flange stops prying entirely at 26.97 mm thick, and collapses on its own at 110 kN. Connections

The force the bolt never saw applied

Pull a tee stub with a hundred kilonewtons and its bolt carries a hundred and fifty. The extra comes from the flange bending and pressing its own edge against the thing it is bolted to, and no free body of the connection as a point contains it.

 Block shear: the metal between the holes. Three 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%. 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.

 A preloaded joint, before and after it slips. Two preloaded bolts at 137 kN each, on one friction face at μ = 0.5. The joint carries 137 kN by friction with the bolts in tension and not in shear at all; past that it slips into bearing and carries 188 kN with the bolts now in shear. Two different mechanisms, one joint. Connections

The joint that carries nothing until it slips

Tighten the bolts hard enough and the plates are clamped together with a force nothing applied. The joint then carries shear by friction, the bolts are in tension and not in shear at all, and the load path has nothing in common with the joint it looks identical to.

 An angle bolted through one leg. A 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. Connections

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.

 A joint is springs in series. The five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 25227.71 kN·m per radian. Connections

A joint made of springs in series

A connection's stiffness is not a property of the connection. It is the series combination of the flexibilities of everything the force passes through — so the softest component decides, and stiffening any of the others changes almost nothing.

 How a moment crosses a gap. A moment end plate with three bolt rows. The moment is carried as a couple: tension in the rows, compression through bearing at the bottom flange. The plastic distribution reaches 172.8 kN·m and the elastic one 142.29 kN·m, a factor of 1.21 — and the compression at the bottom flange is 540 kN either way, which is the check that gets forgotten because it is not a bolt. Connections

Making a moment cross a gap

A moment is not a thing that can be handed across a joint. It has to be turned into a pair of forces, carried separately, and reassembled — and the whole design is a question of how the tension is shared between bolt rows that are not equally able to take it.

 A base plate, and when the bolts start working. A 500 × 400 mm plate carrying 600 kN and 90 kN·m, so the resultant sits 150 mm from the centre against a kern of 83.33 mm. The plate is in partial contact: bearing over 300 mm at a peak of 10 N/mm², with the holding-down bolts carrying 0 kN. The plate lifts at 50 kN·m and crushes at 120 kN·m, and the bolts are not needed until 150 kN·m. Connections

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.

 20 kN applied at once and held, on a structure of 0.500 s period. Displacement against time for a single-degree-of-freedom structure of natural period 0.500 s and 2.0% damping, under 20 kN applied at once and held. The static deflection under the same peak force is 12.67 mm and the peak response is 24.56 mm — a factor of 1.94. Dynamics

Twice the deflection, for the same load

A weight placed gently on a beam deflects it by one amount. The same weight let go from rest, a millimetre above the same beam, deflects it by twice as much — and the factor of two is exact, for every structure ever built.

 How much of the mass each mode carries, over eight modes. The effective modal mass of each of the eight modes of a frame of eight storeys, as a percentage of the total. Mode 1 carries 85.6% and mode 2 9.1%; two modes are needed to reach 90% of the mass. The masses sum to exactly the total, which is a property of the eigenvectors rather than a normalisation applied afterwards. Dynamics

Most of the mass moves together

A sixteen-storey frame has sixteen modes, and the first one carries eighty-four per cent of the mass. That is why a hand calculation on one mode gets the base shear right to within a tenth — and why the same calculation gets the acceleration at roof level wrong by a factor of two.

 What a point on a response spectrum is: three structures, three integrations, three points. Three oscillators of periods 0.3, 0.8, 1.8 s, each integrated through the whole of the same ground motion, and the peak of each one plotted against its own period on the curve at the right. The peaks are 12.54, 46.2, 108.95 mm. The complete spectrum is that done 44 times. Nothing in the curve is a property of the earthquake alone: every point on it carries a period and a damping ratio that belong to a structure. Dynamics

The spectrum is not a load

A response spectrum looks like a load curve and is not one. Every point on it is the peak of a complete time integration of one particular structure, and the curve is what you get by doing that again for every structure there could be.

 What a drop height is worth, as a factor on the answer for a weight placed slowly. The peak displacement as a multiple of the static deflection, against the height a weight is dropped from divided by the deflection that weight causes when it is placed. The curve is 1 + √(1 + 2h/δ), which is conservation of energy and nothing else: the weight does work over the height it falls PLUS the distance the structure then gives, and the structure stores work only over the second. At a ratio of 40 the factor is 10.00, and at zero it is exactly 2 — the marked point, where a dropped weight becomes a placed one. Dynamics

The weight that was dropped

A half-tonne load lowered onto a beam produces 5 kN. The same load dropped one metre onto the same beam produces 160 kN — and onto a beam ten times softer, 54 kN. The stiff structure is the one that suffers, which is the opposite of nearly every other rule about structures.

 Weight is the only thing holding it down. A body 2.5 m wide and 6 m tall weighing 120 kN, under a wind pressure of 1 kN/m². The wind delivers 48 kN and an overturning moment of 144 kNm about the leeward toe; the weight restores 150 kNm, a factor of 1.04. The resultant lands 1.20 m from the centre against a middle third of ±0.42 m, so the base is lifting over 2.35 m of its width. Equilibrium

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.

 Where a beam's load comes from. A 8 × 6 m panel carrying 5 kN/m², divided at 45° from the corners. The long beams take a trapezoid of 15.0 m² each and the short beams a triangle of 9.0 m²; the four areas sum to 48.0 m², which is the panel, so no load has been invented or lost. The line loads quoted are the uniform equivalents; the real distributions peak at 15.00 kN/m at midspan. Equilibrium

The load a beam is given is a decision

Every beam calculation so far has started with a load per metre, handed over as though it were a property of the beam. It is not. It is the answer to a prior question nobody draws, and two defensible answers to it differ by sixty per cent on the same floor.

 A torque diagram is a shear diagram about a different axis. A torque of 40 kNm applied 2 m along a member of 6 m held against twist at both ends. The two ends take 26.7 and 13.3 kNm, in inverse proportion to their distances, because the two halves are springs in parallel and torsional stiffness is GJ over length. The diagram steps at the applied torque and closes at the far end, exactly as a shear diagram does — the only difference is which axis the arrows turn about. Internal forces

The internal force with no diagram

A cut through a member reveals four things, and this collection has drawn diagrams for three of them. The fourth is a torque, it obeys exactly the same rules, and whether it exists at all can depend on a decision the designer is free to make.

 Two triangles that cross zero, and a block that does not. Stress across a 300 × 700 mm section at each stage, compression positive. The prestress alone gives -6.33 MPa at the top and 20.61 at the bottom; at transfer, with only self-weight on it, the top is at -2.47 MPa and in service the section runs from 7.61 to 3.82 MPa — compression everywhere. The same beam with no prestress reaches -12.67 MPa at the bottom fibre, which is 4.2 times what the concrete can hold. Internal forces

The load put on backwards

Every other structure in this collection waits for its load and then resists it. A prestressed one is given a load first — chosen, permanent, and pointing the wrong way — so that when the real one arrives the two nearly cancel and the material never has to do the thing it is bad at.

 The middle third, computed. The kern of a 400 × 600 mm rectangle, computed by asking, for every direction, how far the resultant can move before the far face would be pulled. It reaches ±100.0 mm vertically and ±66.7 mm horizontally, which are h/6 and b/6 exactly, and the region between is a rhombus rather than the ellipse the two numbers suggest — a resultant on a diagonal has less room than either axis allows. Sections and stress

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.

 A two-way slab is a one-way slab as soon as it is not square. The share of the load carried by the strips spanning the short way, against the ratio of the sides. The two families of strips cross at the centre and must deflect equally there, and a strip's deflection goes as the fourth power of its span — so at a ratio of 1.33 the short strips already take 76% and at 2 they take 94%. The panel drawn here is 6 × 8 m, a ratio of 1.33, and its short strips take 76.0%. Two-way action is worth having at a ratio of one and worth almost nothing by two. Structural form

The slab that spans both ways

A panel supported on four sides sends its load in two directions at once, and the share is decided by a fourth power — so a panel a third longer than it is wide has already stopped being a two-way slab in any useful sense. What it does at collapse is a different calculation with a different answer.

 The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 12 kN/m of wet concrete and 18 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 292 MPa; propped, the finished composite section takes everything and reaches 186 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two. Structural form

The structure that was never complete

Every analysis in this collection is of a finished structure loaded once. Real ones are built in pieces, and each piece carries whatever was present at the moment it became structural — so the stress in a member depends on when it arrived, which appears nowhere on any drawing.

 Five loads behind one wall, and the water is the biggest. The horizontal pressure on a 6 m wall retaining soil at 18 kN/m³ with a friction angle of 30°, a surcharge of 10 kPa and the water table 2 m down, drawn once as the profile the wall feels and then once per term. The terms are surcharge 20.0 kN/m at 3.00 m, soil above water 12.0 kN/m at 4.67 m, soil at the water table 48.0 kN/m at 2.00 m, submerged soil 27.2 kN/m at 1.33 m, water 78.5 kN/m at 1.33 m, and they sum to 185.7 kN/m — matched to 7e-8 by integrating the drawn profile numerically. The largest single term is the water, at 78.5 kN/m: water has no shear strength, so its coefficient is exactly one where the soil's is 0.333, and it acts on top of the soil's effective stress rather than instead of it. The combined resultant sits at 1.90 m above the base, 0.317 of the height rather than the third point at 2.00 m that a pure triangle would give. Equilibrium

The load that depends on what carries it

Every other load in this collection is a number the structure is given. Retained soil is not — it pushes with a fraction of its own weight, and the fraction is decided by how far the wall moves. Six millimetres of retreat on a six-metre wall takes a third off the load, and being held still puts it back.

 The moment does not stop at the end of the beam. A portal frame of 8 m by 4 m with fixed bases, carrying 20 kN/m on the beam. The bending moment is drawn on the tension side of every member, and it runs round the corner without a break: 65.2 kNm arrives at the end of the beam and 65.2 kNm leaves down the column, which is the same number, since joint rotational equilibrium is one of the equations the frame solve satisfied. Midspan carries 94.8 kNm, and the two add to 160.0 — the 160.0 kNm of a simply supported span, to 0.0e+0 kNm. The corner takes 61% of the wL²/12 a fully built-in beam would have carried, because the columns are springs rather than walls: the beam-to-column stiffness ratio is 1.27. The beam's moment crosses zero 0.92 m from the corner and the column's 1.33 m above its base. Internal forces

The moment that goes round the corner

At a rigid knee the bending moment does not stop at the end of the beam. It turns and runs down the column, and in the same instant the beam's shear becomes the column's axial force — while the block of steel that has to carry the turn appears on no member diagram anywhere.

 Two beams tied together, and the deeper one takes 89% of the load. Two simply supported beams of 6 m, one twice as deep as the other, tied together at midspan so that they have to move as one. A load of 100 kN stands on the tie. Point stiffness is 48EI/L³, so the deeper beam is 8 times as stiff — depth cubed, nothing else — and the load divides in that ratio: 11.1 kN into the shallow beam and 88.9 kN into the deep one, 11% against 89%. Both midspan points move 5.00 mm, which is the whole of the argument: the geometry of the load never entered it. The deflection is drawn 78 times full size — the real sag is 5.00 mm on a 6 m span, about 1 in 1200. Internal forces

The stiffest path takes the load

When two members share a force the split can be argued about. When they share a displacement it cannot — stiffness settles it, and nothing about the load or the plan drawing gets a vote. The consequence is that stiffening a lightly loaded member raises its stress, and the way to unload something is to soften it.

 The ground pushes back hardest where the beam has gone down furthest. A strip 16.4 m long and 1 m wide on ground of subgrade modulus 50 × 10³ kN/m³, carrying 1000 kN at its centre. The beam settles 3.90 mm under the load and the ground pushes back in proportion — the arrows are k times the settlement above them, peaking at 195 kN per metre — so the pressure diagram is the settlement bowl and not an assumed distribution. The characteristic length 1/β is 2.56 m: the bowl crosses zero at 6.04 m, which is 3π/4 of it, and beyond that the arrows reverse because the beam has lifted off. By 8.05 m — one π/β — the disturbance is 4.3% of what it was, which is why the moment 641 kNm and the peak pressure 195 kN/m contain no length at all. The settlement is drawn 217 times full size — the real bowl is 3.90 mm deep over 16.4 m, about 1 in 4207 — and at true scale the beam would be a straight line. Internal forces

The beam that sits on the ground

Every other beam in this collection is held at points. A footing is held everywhere, by something that pushes back in proportion to how far it is pushed — and that single change hands the structure a length it did not choose. Two or three of those lengths from the column, nothing knows the load happened.

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