Structural Engineering & Statics
where the load goes
Depth

Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
Three details, and no material anywhere on the plot. Stress range against cycles to failure for three detail categorys — 160, 90, 36 N/mm² at two million cycles. The lines are parallel because they share a slope of three, and the spread between them is a factor of 4.4 in stress and therefore 88 in life. Nothing on this plot depends on the strength of the steel: the same detail in a grade twice as strong lies on the same line. At a stress range of 70 N/mm² the lives are 160: 6.8e+7, 90: 4.3e+6, 36: 2.7e+5 cycles. The knee in each line is the constant-amplitude limit, past which the slope becomes five.

Fatigue

  1. 1 The load that never came near failing anything
  2. 2 Designed to be found in time
  3. 3 The cycles that do not count
  4. 4 Which reversal closes the loop
  5. 5 The loops a crack grows on
  6. +3 more
8 essays · materials The area is the rotation, and its first moment is the movement. A 6 m cantilever under a tip load of 10, with the M/EI diagram beneath it. The shaded area is 180.00, which by the first theorem is the change of slope along the whole member. Its centroid is at 2.000 m, and the first moment about the tip is 720.00 — which by the second theorem is the deviation from the tangent, and for a cantilever that tangent is horizontal, so it is the deflection itself. Integrating the curvature twice instead gives 720.00.

Moment-area

  1. 1 The area of a diagram is a rotation
  2. 2 The fixed-end moment is a column stress
  3. 3 The elastic centre is not on the frame
  4. 4 The centre that hangs in the air
  5. 5 The axes that have to be turned first
  6. +2 more
7 essays · deflection The deflection that arrives years late. The multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.00, after five years by 3.29, and it approaches 3.38. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.

Creep

  1. 1 The deflection that arrives three years late
  2. 2 The stress that leaks away
  3. 3 The creep that belongs to the member
  4. 4 The slab that dries from one face
  5. 5 The support that had no moment when it was cast
  6. +1 more
6 essays · materials The reaction lies inside the cone, so the block stands. A block of 100 on a plane at 15°, against a coefficient of friction of 0.35. Resolving across and along the plane gives a normal force of 96.6 and a friction demand of 25.9, against a capacity of μN = 33.8 — a ratio of 0.77. Added together the two make one contact reaction leaning 15.0° from the normal, and the admissible reactions fill a cone of half-angle arctan μ = 19.3°. Equilibrium is possible exactly when the demanded reaction lies inside that cone, which here it does. The weight enters neither the cone nor the lean: a block of any weight on this slope leans its reaction by the same 15.0°, which is why the angle of repose is a material property and the size of a heap of sand is not.

Friction

  1. 1 The force that is whatever it needs to be
  2. 2 The area that is not in the equation
  3. 3 The force that is capped on purpose
  4. 4 Seventy-five per cent each way
  5. 5 The pier that moves in jumps
  6. +1 more
6 essays · equilibrium A Pratt truss of 6 panels. A Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing.

Truss

  1. 1 The triangle that cannot fold, and everything built out of it
  2. 2 The joint that is not a pin
  3. 3 One drawing solves the whole truss
  4. 4 Halving the panel buys a shorter strut
  5. 5 The chord is a continuous beam
  6. +1 more
6 essays · structures 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.

Bolt group

  1. 1 The bolt that carries more than its share
  2. 2 The bolt group has no neutral axis
  3. 3 The bracket pushed from the wrong side
  4. 4 The better method flatters the worse layout
  5. 5 The bolt that was never fitted
5 essays · connections The answer arrives in instalments. The hogging moment at support 1 of a three-span beam, cycle by cycle. It starts at the fixed-end moment of 53.3 kNm — the value with every joint clamped — and settles at 64.0 kNm against an exact 64.0. The error falls by about a factor of four per cycle: 14.00, 3.50, 1.95, 0.59 kNm after one, two, three and four. Two cycles is an engineering answer and nobody had to invert anything.

Moment distribution

  1. 1 Solved by passing it around
  2. 2 Why it converges, and how fast
  3. 3 Every joint balanced, and the frame still leaning
  4. 4 Told what the far end is doing
  5. 5 The table that cannot be read halfway
5 essays · deflection Buildings that sway alike need almost no gap between them. The separation two adjacent buildings need, against the ratio of their periods. The obvious answer is the sum of what each can do — 340 mm — and it is wrong, because the two peaks do not occur at the same instant. The right combination is the one modal responses use, √(u₁² + u₂² − 2ρu₁u₂), with ρ the cross-correlation coefficient of the two responses. ρ depends on the period ratio and behaves the opposite way to intuition: at a ratio of one the two buildings sway together, ρ = 1, and the gap collapses to the difference of the two, 100 mm. At the 0.57 drawn ρ is 0.029 and the gap is 248 mm — 27 per cent less than the sum, and 99 per cent of the square root of the sum of squares.

Pounding

  1. 1 The gap between two buildings
  2. 2 The floor that arrives at a column
  3. 3 The force that belongs to the model
  4. 4 The damper that ends up as a joint
  5. 5 The force a filler can promise
5 essays · dynamics 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.

Prying

  1. 1 The force the bolt never saw applied
  2. 2 The thickness that decides who fails
  3. 3 How much of the plate is bending
  4. 4 The rows have to share one fold
  5. 5 The bolt that was already stretched
5 essays · connections Evenly spaced modes, so one of them is always where the feet are. The first 6 modes of a 120 m stay under 3.50 MN. A taut string's frequencies are an arithmetic progression — every one of them 1.006 Hz above the last — where a beam's go as the square of the mode number and spread out. That difference is the whole of why a cable is a lively member and a beam is not: a beam has a first mode and then a gap, and a cable has a mode every 1.01 Hz for ever. The shaded band is ordinary walking, 1.6 to 2.4 Hz, and mode 2 sits inside it. Nothing about the tension can move a mode out of the band without moving another one in.

Cable dynamics

  1. 1 The force read off a frequency
  2. 2 The damper that is too near the end
  3. 3 The line of ties that stops short
  4. 4 The stay shaken along its own length
4 essays · dynamics The section that is checked is not the section that was chosen. A 457 mm beam coped 50 mm deep over 120 mm to frame into a girder. What is left is a tee with a section modulus of 3.836e+5 mm³ against the whole section's 1.438e+6 — 27 per cent. The moment at the end of the cope is the reaction on a lever arm of 130 mm: 23.4 kNm, giving 61 N/mm² and a flexural utilisation of 0.17. The web now has a free edge along the cope, so its buckling coefficient collapses from 4 to 0.425 — a factor of 9.4 — and the re-entrant corner has a stress concentration of 5.5 on a 10 mm radius.

Coped beam

  1. 1 The section that is checked is not the one chosen
  2. 2 The end that is only a plate
  3. 3 The repair that fixes the wrong check
  4. 4 The plate that arrives after the load
4 essays · connections Three forces must meet at a point. A body held by two supports and one load. The reaction at the roller is vertical and the load's direction is given, so their lines of action fix a meeting point — and the pin reaction has to point at it.

Graphic statics

  1. 1 Three forces must meet at a point, and a drawing can find it
  2. 2 The line that pairs four forces
  3. 3 The pole decides the drawing, not the answer
  4. 4 The pin that is not a point
4 essays · equilibrium Three modes of a five-storey frame. The first three mode shapes of a five-storey shear frame, from the eigenvalue problem rather than sketched. Mode 1 has a period of 0.6 s, no node and carries 88.0% of the mass; Mode 2 has a period of 0.21 s, one node and carries 8.7% of the mass; Mode 3 has a period of 0.13 s, two nodes and carries 2.4% of the mass. The nth mode crosses the axis n−1 times, which is a theorem rather than a drawing convention.

Mode shapes

  1. 1 A structure has more than one period
  2. 2 The modes that were left out
  3. 3 The twist the combination rule invents
  4. 4 The damping that belongs to no mode
4 essays · dynamics 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.

Overturning

  1. 1 Weight is the only thing resisting it
  2. 2 Whether it tips or slides
  3. 3 Half as far between the legs
  4. 4 It tips inside its own hull
4 essays · equilibrium A 8 mm plate, and the width it can be. The elastic critical stress of a plate in compression against its width, with the yield stress drawn across it. Below 370 mm the plate reaches yield before it buckles; above it the plate ripples first, and the fraction of the width still carrying load falls away — at 700 mm only 47 per cent of it is still working.

Plate buckling

  1. 1 The plate that ripples, and the width that is left
  2. 2 The coefficient that is not four
  3. 3 Four was never a fact about plates
  4. 4 Classified by a gradient it does not have
4 essays · stability A restoring moment that gets smaller the further it leans. Restoring moment against rotation for a block 0.90 m wide and 4.20 m tall, both normalised — the moment by its value at first uplift, the rotation by the angle α = 12.1° at which the block topples. It is mgR·sin(α − θ), which is a weight times a geometry with no material property in it at all, and it has two features an elastic system does not. There is a jump at the origin: the moment is whatever the ground demands until uplift and then it is mgR sinα, so the law is discontinuous where a spring's is steepest. And the slope is negative — the further it leans the less it pushes back — so the equilibrium at θ = 0 is stable only because of that jump, and there is no stiffness to divide into a mass. The straight line is what a spring of the same first-uplift strength would have done.

Rocking

  1. 1 The block that is safer for being bigger
  2. 2 The only damping is the landing
  3. 3 A wall that is allowed to lift
  4. 4 The tendon that forgets its prestress
4 essays · dynamics 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
  2. 2 The damping that is radiated
  3. 3 The frequency below both checks
  4. 4 The damping that comes through the sides
4 essays · dynamics 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
  2. 2 The radius rule, and where it fails
  3. 3 The eccentricity at right angles to the drawing
  4. 4 Two fasteners that never arrive together
4 essays · connections 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.

Base plate

  1. 1 Where the structure meets the ground, and when the bolts start working
  2. 2 The plate that is only as big as it is thick
  3. 3 The compression that stays under the flange
3 essays · connections The hour that is really a temperature. The retention factors for carbon steel against temperature: the yield stress and the elastic modulus. The modulus falls away first — at 500°C the steel has kept 78% of its strength and 60% of its stiffness — so a member's failure mode can change during a fire. A member working at 60% of its cold capacity runs out of strength at 558°C, and out of the stiffness for the same ratio at 500°C, 58 degrees earlier. There is nothing about time in any of it: a fire rating is a temperature the member must not reach, converted into the minutes a particular fire takes to get it there.

Fire

  1. 1 The hour that is really a temperature
  2. 2 The temperature is a shape
  3. 3 The fire that goes out
3 essays · materials

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