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.
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
  4. 4 The fixed base the bolts decide
  5. 5 The base that is rigid until the bed lets go
  6. +3 more
8 essays · connections 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
  6. +3 more
8 essays · connections 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. +3 more
8 essays · materials 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 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
  5. 5 Every space a point, every joint a polygon
  6. +3 more
8 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
  5. 5 The top floor the bearings shake
  6. +3 more
8 essays · dynamics 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
  6. +3 more
8 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
  6. +3 more
8 essays · connections Every member's share of the movement, and they are not the members expected. A Pratt truss of six panels at a depth of 0.85, carrying 10 kN at each top node, with the movement of the bottom chord at mid-span attributed member by member. The unit-load sum δ = ΣF·f·L/EA gives 631.06 at EA = 1: 47.2% from four top chords, 28.5% from six bottom chords, 22.3% from six diagonals, 2.0% from five verticals. The single worst member is a top chord at mid-span at 14.8% of the whole. Each member is drawn at the width of its own share. The same deflection from a stiffness solution that shares none of this arithmetic is 631.06, a relative residual of 3.6e-15.

Truss deflection

  1. 1 Which member moved the roof
  2. 2 The member that is not worth stiffening
  3. 3 The drawing that is right except for a rotation
  4. 4 The truss that is stiff by accident
  5. 5 The calculus answer, rounded, is the worst one
  6. +3 more
8 essays · deflection 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
  5. 5 The rock that sends the waves back
  6. +3 more
8 essays · dynamics A brace on the wrong flange never gets there, however stiff it is. The critical moment of an 8 m beam against the stiffness of a single midspan brace, drawn three times for the three heights the brace could sit at. On the compression flange it climbs from 143 kNm to the two-half-wave plateau of 447 — the beam braced into two 4.0 m beams — and reaches 99% of it at 447 kN/m. At the shear centre it needs 2252 kN/m, 5.0 times as much. On the tension flange it never arrives at all: at the stiffness that would have done the job on the other flange it has bought a factor of 1.068, and a stiffer brace in the same place buys the same nothing. Past the plateau the beam stops using the brace, which is where the idea of an ideal stiffness comes from.

Beam bracing

  1. 1 The brace on the wrong flange
  2. 2 Twice the moment, four times the brace
  3. 3 The stiffness a row of braces shares out
  4. 4 Three braces where the moment is
  5. 5 The braces that stay where the framing puts them
  6. +2 more
7 essays · stability 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 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
  5. 5 The ballast that helps it over
  6. +2 more
7 essays · equilibrium Four inequalities, and the wedge between them. The Magnel diagram: every limit on a prestressed section, plotted as a bound on 1/P against the eccentricity. Two of the four come from transfer, when the force is largest and the only moment is the beam's own weight, and two from service, when 20% of the force has been lost and the moment is 640 kNm. Each is linear in 1/P, which is the substitution that makes the problem a picture rather than a search. The shaded region is every force-and-eccentricity pair the section will accept: it is a wedge opening to the right, so the cheapest prestress is always at the largest eccentricity the cover allows — 1029 kN at e = 400 mm here. The section's kern is 241 mm, and every useful answer is outside it.

Prestress limits

  1. 1 Four inequalities and a wedge
  2. 2 The flange the prestress cannot use
  3. 3 The strands that are sleeved at the ends
  4. 4 The beam that leaves the bed
  5. 5 The beam at the bottom of the stack
  6. +2 more
7 essays · sections The shear centre of a channel. A channel of 80 by 200, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 31.7 outside the web to leave the section untwisted — a point in the air, outside the material entirely.

Shear centre

  1. 1 The point that is not in the section
  2. 2 The eccentricity a purlin cannot avoid
  3. 3 The plate that closes the circuit
  4. 4 The web that moves the centre and not the twist
  5. 5 The web that leans and lifts the centre
  6. +2 more
7 essays · sections 3% of the mass, hung on a spring, against the peak it removes. The magnification of a structure with 1.0% damping, with and without a tuned mass damper of 3.0% of its mass, tuned to 0.9709 of its frequency with 10.5% damping of its own. The bare peak is 50; with the absorber the single peak becomes two of 7.34, a reduction to 15% — a factor of 6.8. The marked points at frequency ratios 0.923 and 1.044 are the fixed points: the response there is the same whatever damping the absorber is given, which is what makes the optimum a question with an answer. The absorber's own stroke at the worse of the two peaks is 29.57 times the structure's static deflection, and that stroke is what decides whether it fits.

Tuned mass damper

  1. 1 The mass that helps by being late
  2. 2 The actuator that arrives late
  3. 3 The actuator that pushes against the mass
  4. 4 The law that throws the tuning away
  5. 5 Half a spring is neither tuning nor freedom
  6. +2 more
7 essays · dynamics An eccentric load is three load cases, and only two of them are checked. A line load of 40 N/mm at 1.5 m from the axis of a 3.0 by 2.0 m box, replaced by the three cases it is equivalent to. Bending is the load on the axis. The torque 60 kNm per metre then splits into a set of edge forces that drives Bredt's shear flow and distorts nothing, and a set with the flange forces reversed — 10.0 kN/m up one web and down the other, 15.0 kN/m across the flanges — which carries no torque at all and squashes the rectangle into the rhombus drawn behind it. Its generalised load is exactly half the torque, so a box girder spends half of an eccentric load's torsion on changing its own shape, and no torsion calculation contains that half.

Box distortion

  1. 1 The section that will not keep its shape
  2. 2 One diaphragm is nearly none
  3. 3 The diaphragm that is a few angles
  4. 4 The deck whose frame is its crossbeams
  5. 5 The stiff frame among soft ones
  6. +1 more
6 essays · sections 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
  5. 5 The deck that forgets its own rhythm
  6. +1 more
6 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
  5. 5 The hole at the tip of the crack
  6. +1 more
6 essays · connections Six ways for one dowel to fail, and the capacity is the smallest. Johansen's single-shear mechanisms for a 12 mm dowel through 40 and 40 mm members, each drawn as the shape it is: the dowel straight and the timber crushing, the dowel rotating rigidly, one plastic hinge, then two. The capacity under each is that mechanism's own, and the joint's strength is the smallest — 5.02 kN by mode c, which is the dowel rotates rigidly and both members crush. That is the kinematic theorem of plasticity: every mechanism gives an upper bound and the true collapse is the lowest of them. The crushed timber is shaded, and the circles are plastic hinges in the steel.

Dowel yield

  1. 1 The smallest of six failures
  2. 2 The pull that Johansen left out
  3. 3 The screw that is pulled instead of bent
  4. 4 Twenty screws that peak together
  5. 5 The row that remembers its overload
  6. +1 more
6 essays · connections

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