Structural Engineering & Statics
where the load goes
Depth

Ladders — page 13

A field says what an essay is about. A ladder says what else there is to say about it — the distinct arguments that stand against one idea, from the one that introduces it to the one that assumes all the others.
A basement is a boat. A 20 by 30 m substructure dug 6 m into ground whose water table stands 2 m down. The head on the underside of the base slab is 4.0 m, so the pressure there is 39.2 kN/m² over the whole plan — 23.5 MN of it, pushing upward. Nothing about the structure changes that number. What resists it is weight: 18.7 MN of concrete and whatever is built above, giving a factor of 0.80. The structure floats if the water reaches 2.82 m below the ground, and a base slab alone would have to be 1.64 m thick to hold it down.

Uplift

  1. 1 A basement is a boat
1 rung · equilibrium How much of a force a mount lets through, at three damping ratios. The force transmitted to the support divided by the force applied, against the ratio of the forcing frequency to the structure's own, at 2%, 5%, 20% of critical damping. Every curve passes through exactly 1 at a frequency ratio of root two, whatever the damping: below that ratio a mount amplifies what it was installed to isolate, and above it more damping lets more through.

Vibration isolation

  1. 1 The machine that shakes the building
1 rung · dynamics The shear goes round the corner instead of across it. A 6-panel Vierendeel girder, 12 m by 1500 mm, under 100 kN at mid-span. There is no diagonal in it, so each panel's 50 kN of shear is carried as bending in the chords: the curves drawn along them are the chord moments, and every one passes through zero at the middle of its own panel. The local moment is the panel shear times the panel length over four, 25.0 kNm, and it adds to an axial force of 200 kN from the global moment at the same point. The girder deflects 6.20 mm against 3.18 mm for the same members triangulated — 1.95 times — and 68% of that movement is chord bending that a diagonal would have removed entirely.

Vierendeel

  1. 1 The truss with no diagonals
1 rung · structures The deflection at x = 4, by virtual work. Three diagrams: the moment from the real load, the moment from a unit load placed where the answer is wanted, and their product. The area under the third, divided by EI, is the deflection — 213.33 here. No standard case was consulted, so the method works for any load pattern at all.

Virtual work

  1. 1 One deflection, without solving everything
1 rung · deflection Lock-in: the frequency the wind sheds at, and what it does to the chimney. A 1.2 m cylinder at 0.9 Hz. Vortices leave it at St·U/D, so the critical speed at which they match the structure is 6 m/s — a breeze, met many times a year rather than once in fifty. The upper panel shows the shedding frequency locking on to the structure across a band from 5.1 to 7.8 m/s; the lower shows the amplitude that results. At a Scruton number of 11.2 the peak amplitude is 180.94 mm, which is 15% of the diameter.

Vortex shedding

  1. 1 The wind that brings its own frequency
1 rung · dynamics Two cantilevers, or one wall, and the beams decide which. The deflected shape of a coupled pair of 6 m walls, drawn against the two limits it lies between. Release the coupling beams entirely and the pair is two independent cantilevers, deflecting 111 mm. Make them rigid and it is one composite wall of the full width, deflecting 16 mm — thirty times stiffer, because the lever arm between the wall centroids is 8.40 m and everything inside either wall is smaller than that. Real beams of 600 × 350 mm over a 2.4 m opening land at 23 mm and carry 63% of the base overturning as an axial couple rather than as wall bending. The degree of coupling never reaches one, because a beam of finite depth cannot suppress the walls' curvature entirely.

Wall coupling

  1. 1 Two walls that agreed to be one
1 rung · internal-forces A straight line, and the comfortable case is already two thirds down it. The capacity of a 215 mm masonry wall as a fraction of its squash load, against the eccentricity of the resultant. Nothing in this figure is a buckling calculation. A material that cannot be pulled bears on a strip of width 3(t/2 − e) under a triangular stress block, so the capacity is exactly 1.5f(t − 2e) — a straight line, zero when the resultant reaches the face, and already at 67% at the edge of the kern. The middle third is treated everywhere as the comfortable case; a wall loaded there has given away a third of its capacity before slenderness has been mentioned. The lower line is the same wall with slenderness in it, which enters as an ADDITIONAL eccentricity of 17.4 mm rather than as a reduced stress — h_ef²/2400t, for h_ef = 3000 mm. Euler's load for this wall is 9.6 times what the eccentricity rule allows, which is why no masonry calculation contains it.

Wall slenderness

  1. 1 It does not buckle, it runs out of width
1 rung · stability The flanges go opposite ways, and the pair of them is the bimoment. A 305 by 165 mm I-section held against warping and twisted by 0.5 kN·m, with the section on the left and the two flanges seen in plan on the right. At the built-in end each flange bends in its own plane, one way at the top and the other at the bottom, through 29.3 mm at the free end — drawn 20 times its true size against the 6 m length. The pair of flange shears is 1.69 kN each, and 1.69 × 295 mm is 0.500 kN·m — the whole torque at that section, carried by two forces neither of which is a torque. The pair of flange moments is 3.04 kN·m each, and 3.04 × 295 mm is 0.897 kN·m², which is the bimoment. It puts 67.0 N/mm² into two diagonally opposite flange tips and takes the same out of the other two, so its net force and its net moment about every axis are zero — which is exactly why no member diagram has a place for it.

Warping

  1. 1 The section that cannot stay flat
1 rung · sections A hole in a web is a Vierendeel panel. A 400 × 300 rectangular opening in a 533 deep beam, 15% along a 9 m span carrying 20 per metre — where the moment is 103 kNm and the shear 63 kN. The moment is a couple on the two tees, 301 kN on a lever arm of 343 mm, which is 70.3 N/mm² of uniform stress. The shear has nowhere to go but through the tees, so each carries 32 kN over the opening and bends in double curvature: a Vierendeel moment of 6.3 kNm and 167.6 N/mm² on top. So 70% of the stress at the corner exists because the hole has a LENGTH, and only 30% of the section's second moment has gone.

Web opening

  1. 1 The hole that costs nothing, and everything
1 rung · sections Throat stress round a fillet weld group. A c shape weld group carrying 100 kN at 150 mm from its centroid. The peak throat stress is 0.88 kN per mm of throat, at (79.5, -100); the worst point at maximum radius from the centroid carries 0.88. Checking by radius is right here, and points at identical radius differ by a factor of 1.

Weld group

  1. 1 The corner that is not the worst point
1 rung · connections A fillet weld is stronger across than along. Capacity of a 4 mm throat over 100 mm, against the angle between the weld's axis and the load. Loaded along its length it carries 116.83 kN; loaded across it, 143.09 kN. The ratio is 1.22, which is √3/√2 exactly, and it comes out of the failure criterion rather than out of a test.

Weld strength

  1. 1 The weld that is stronger across than along
1 rung · connections The wind pushes on one face and pulls on three. A 30 × 20 m building in plan, with the measured pressure coefficient on each face and the arrows drawn in the direction the pressure acts. Only the windward face is pushed; the other three are sucked, and the side faces are sucked hardest of all at c_p = -0.7. The horizontal resultant is 842 kN at a velocity pressure of 0.9 kPa, and the arithmetic of it is the whole point: the leeward suction pulls the building downwind, so it ADDS, supplying 38% of the answer, while the two side faces cancel each other exactly and supply none of it. The coefficients are wind tunnel data; what is computed is the free body they are applied to.

Wind pressure

  1. 1 Most of it is suction
1 rung · equilibrium One criterion inside the other, touching at six points. The two yield criteria in principal stress space with the third principal stress zero, both normalised by the yield stress. Von Mises is the ellipse — σ₁² − σ₁σ₂ + σ₂² = f_y², which is a circle seen at an angle — and Tresca is the hexagon inscribed in it, touching at the six points where one principal stress is zero or the two are equal. Everywhere else Tresca is the smaller, by up to 15.5 per cent, and the widest gap is at pure shear, where σ₁ = −σ₂ and the two answers are 205 and 178 N/mm². The ratio there is exactly 2/√3, computed rather than quoted, and it is the whole reason a web is checked against f_y over root three.

Yield criterion

  1. 1 The shear strength nobody measured
1 rung · materials

All essays