Lessons · Engineering · Quick reference
Engineering quick reference
83 topics, one line each, in the order Hone teaches them.
Hone is a place to practise a career, one idea a day. This sheet is the whole Engineering track at a glance: every idea it covers, in the order they are taught, one line each. It is a map rather than a lesson. Read opens the full explanation of an idea; Practise gives you a question on it. Both are free, and reading needs no account at all.
In beta. This sheet was written for Hone and has not yet been checked by a licensed engineer. Practice material, not professional advice. What that means.
From units to a working design · Units and numbers
SI and US units, and converting between themA unit is part of the number. Converting is multiplying by a fraction that equals one, written so the old unit cancels. Read: Two systems of units, and carrying a number between them · Practise SI and US units, and converting between them
significant figuresA calculated number cannot be more precise than the roughest measurement that went into it, and its written digits should say so. Read: Significant figures: how much of the number you actually know · Practise significant figures
checking an equation by its unitsReplace every symbol in an equation with its units and cancel. If what is left is not the unit the answer needs, the equation or a conversion is wrong. Read: Let the units check the equation for you · Practise checking an equation by its units
force, mass and weightWeight is a force: W = m × g, with g = 9.81 m/s² in SI and 32.2 ft/s² in US customary. Read: Mass is how much stuff; weight is the pull on it · Practise force, mass and weight
density and specific weightDensity ρ is mass per volume, in kg/m³. Specific weight γ = ρ × g is weight per volume, in N/m³. Read: Density and specific weight: how heavy a volume is · Practise density and specific weight
From units to a working design · Statics
vectors and their componentsA force is a vector: it has a direction as well as a size. Any vector can be replaced by a horizontal component F cos θ and a vertical component F sin θ, and components along the same axis simply add. Read: A force has a direction, so split it into two you can add · Practise vectors and their components
the free-body diagramCut the body free of everything touching it, and at every cut draw the force the removed thing was putting on the body. Nothing else goes on the page. Read: The free-body diagram: the body alone, and every force on it · Practise the free-body diagram
equilibrium of a particleWhen a small body is at rest, the forces on it sum to zero in every direction: ΣF_x = 0 and ΣF_y = 0. Read: Equilibrium of a particle: the forces add to nothing · Practise equilibrium of a particle
moments and the leverMoment is force times the perpendicular distance from the pivot to the line of the force: M = F × d, in N·m. Read: A moment: how hard a force is trying to turn something · Practise moments and the lever
equilibrium of a rigid bodyA body that is not moving has ΣF_x = 0, ΣF_y = 0 and ΣM = 0 about any point. Three equations, so three unknowns can be found. Read: Equilibrium of a rigid body: forces balance and turning balances · Practise equilibrium of a rigid body
reactions of a simple beamFor a beam on two supports, a moment sum about one support gives the other reaction, and the vertical force sum gives the first. Read: What each end of a simple beam carries · Practise reactions of a simple beam
a truss by the method of jointsAt a pin joint the member forces and the load add to zero, so a joint with two unknown members is solved by ΣF_x = 0 and ΣF_y = 0. Assume tension; a negative answer is compression. Read: A truss, one joint at a time · Practise a truss by the method of joints
From units to a working design · Strength of materials
stress: force over areaStress is force divided by area, σ = F / A. In N/mm² it is megapascals, and that is the unit steel is rated in. Read: Stress: the force, spread over the area that carries it · Practise stress: force over area
strain: how much it stretchesStrain is the change in length divided by the original length, ε = δ / L. It has no unit. Read: Strain: how much it stretched, for its length · Practise strain: how much it stretches
Hooke's law and Young's modulusBelow yield, σ = E × ε. E is Young's modulus, the stiffness of the material: about 200 GPa for steel, 70 GPa for aluminium. Read: Hooke's law: stress and strain are proportional, and E is the ratio · Practise Hooke's law and Young's modulus
axial deformation, PL over AEA rod of length L and area A under axial load P, in a material of modulus E, stretches δ = P L / (A E). Read: How much a rod stretches: PL over AE · Practise axial deformation, PL over AE
factor of safetyFactor of safety is the material's strength divided by the working stress, FS = σ_strength / σ_working. Turned round, the allowable stress is strength over the factor. Read: Factor of safety: how far the material is from its limit · Practise factor of safety
shear stress in a pinA pin loaded across its axis carries shear stress τ = V / A over its cross-section. In double shear two sections share the load, so the stress halves. Read: Shear in a pin: the force trying to slice it · Practise shear stress in a pin
thermal expansionA bar warmed by ΔT grows by δ = α L ΔT. If it cannot grow, that same strain becomes a stress σ = E α ΔT. Read: Thermal expansion: heat makes it longer, and holding it makes it stressed · Practise thermal expansion
From units to a working design · Circuits
Ohm's lawVoltage pushes, resistance resists, current is what results: V = I × R. Read: Ohm's law: push, resistance, and what flows · Practise Ohm's law
electrical powerPower is volts times amps, P = V × I, in watts. With Ohm's law it is also I² R and V² / R. Read: Electrical power: how fast the circuit turns energy into heat or work · Practise electrical power
series and parallel resistanceResistors in series add: R = R_1 + R_2. In parallel the reciprocals add: 1/R = 1/R_1 + 1/R_2, and the result is smaller than the smallest branch. Read: Series adds; parallel shares · Practise series and parallel resistance
Kirchhoff's voltage lawGoing once around a closed loop, the rises through sources and the drops across resistors sum to zero: ΣV = 0. Read: Kirchhoff's voltage law: round any loop, the volts add to zero · Practise Kirchhoff's voltage law
the voltage dividerTwo resistors in series across V_in give V_out = V_in × R_2 / (R_1 + R_2) across R_2, the one you read across. Read: The voltage divider: two resistors share the volts by their size · Practise the voltage divider
energy in a capacitorA capacitor C charged to V holds charge Q = C V and energy E = ½ C V². Read: A capacitor stores energy, and the energy goes as the square of the volts · Practise energy in a capacitor
RMS of a sine waveFor a sine wave, V_rms = V_peak / √2, about 0.707 of the peak. Power and heating use rms, so meters and ratings are in rms. Read: RMS: the DC voltage that would heat the same wire the same amount · Practise RMS of a sine wave
From units to a working design · Fluids and thermo
pressure at depthIn a fluid at rest, pressure rises with depth as p = ρ g h. Only depth, density and g matter; the shape of the container does not. Read: Pressure at depth: the weight of the column above you · Practise pressure at depth
Pascal's principle and the hydraulic jackPressure in a confined fluid is the same at every point, so F_1 / A_1 = F_2 / A_2. A small push on a small piston becomes a big push on a big one. Read: Pascal's principle: the same pressure everywhere, so area multiplies force · Practise Pascal's principle and the hydraulic jack
continuity: flow is area times speedFlow rate is area times speed, Q = A × v, and along a pipe with no leaks A_1 v_1 = A_2 v_2. A narrower pipe means faster fluid. Read: Continuity: what flows in must flow out · Practise continuity: flow is area times speed
Bernoulli between two pointsBetween two points along a flow with no losses, p + ½ ρ v² + ρ g z is the same at both. Faster means lower pressure; higher means lower pressure; and a tank drains at v = sqrt(2 g h). Read: Bernoulli: pressure, speed and height trade with each other · Practise Bernoulli between two points
the ideal gas lawP V = n R T, with P in pascals, V in cubic metres, n in moles, T in kelvin and R = 8.314 J/(mol·K). Read: The ideal gas law: pressure, volume, amount and temperature in one line · Practise the ideal gas law
sensible heat, Q = m c ΔTHeating a mass m of a material with specific heat c by ΔT takes Q = m c ΔT. For water, c is about 4186 J/(kg·K). Read: Sensible heat: how much energy to change a temperature · Practise sensible heat, Q = m c ΔT
efficiency of a heat engineNo engine running between a hot source at T_h and a cold sink at T_c can beat η = 1 − T_c / T_h, with both temperatures in kelvin. Read: The ceiling on a heat engine · Practise efficiency of a heat engine
work and powerWork is force times the distance moved along the force, W = F × d, in joules. Power is work over time, P = W / t, in watts. One horsepower is 746 W. Read: Work is force through a distance; power is how fast · Practise work and power
From units to a working design · Engineering economics and on the job
simple and compound interestSimple interest is I = P × i × n. Compound interest grows the whole sum each period: F = P × (1 + i)ⁿ. Read: Interest: simple grows in a line, compound grows on itself · Practise simple and compound interest
present and future valueA sum P today becomes F = P (1 + i)ⁿ in n periods; a sum F due in n periods is worth P = F / (1 + i)ⁿ today. Read: Present and future value: money moved through time · Practise present and future value
payback periodPayback period is the cost divided by the net saving per year, when the saving is the same every year. Read: Payback: how long until the saving has repaid the cost · Practise payback period
costing a bill of materialsA bill of materials, the BOM, is the list of every part a job needs. Each line is quantity times unit price; the material cost is the sum; labour and overhead are added on top. Read: Costing a bill of materials · Practise costing a bill of materials
tolerance stacksStack parts in a line and the nominals add, and so do the tolerances. Worst case is the plain sum; the statistical estimate is the root of the sum of squares. Read: Tolerance stacks: when the parts all sit at their limits · Practise tolerance stacks
a unit-conversion chain from a specA conversion chain is one multiplication per unit, each factor written as a fraction equal to one, so the old units cancel on paper before any number is touched. Read: A datasheet in one system, a design in another · Practise a unit-conversion chain from a spec
From units to a working design · Becoming licensed
the licence many engineers never need, and why that is not about youNearly every jurisdiction recognises some form of industrial exemption: an engineer employed by a manufacturer, a utility or a public agency, whose engineering work is incidental to that employer's own products and is not offered to the public, can generally work unlicensed, with the employer carrying the liability instead of a sealed drawing. Read: Most engineers are not licensed, and the reason has nothing to do with them · Practise the licence many engineers never need, and why that is not about you
what the licence actually lets you doA licensed professional engineer may seal engineering documents, and may offer engineering services to the public. The seal is the mechanism: it is how a document is submitted for public approval, and it is a personal assumption of responsibility for the work rather than a mark of seniority. Read: What the licence actually lets you do, which is take responsibility in writing · Practise what the licence actually lets you do
the degree, and the accreditation that changes the yearsThe usual route starts with an accredited engineering degree. What matters is the accreditation the programme holds rather than the reputation of the school, and the kind of accreditation changes how much experience a board will require: an engineering technology or related science degree generally carries a longer experience requirement than an engineering one. Read: The degree, and the accreditation that quietly changes how many years you owe · Practise the degree, and the accreditation that changes the years
the first examination, and why it is sat earlyThe Fundamentals of Engineering examination covers the broad material common to all engineering rather than any one discipline, and is normally sat near graduation. Passing it is the first formal step on the road, and it is not a licence. Read: The first examination, and why it is sat while you are still a student · Practise the first examination, and why it is sat early
progressive experience, and who it has to be underBetween the two examinations a board requires a period of progressive engineering experience, generally under the supervision of a licensed professional engineer. How long depends on the state and on the accreditation of your degree, and what counts as progressive is defined rather than assumed. Read: Progressive experience, and who it has to be under · Practise progressive experience, and who it has to be under
the second examination, which is your discipline'sThe Principles and Practice examination is discipline-specific: civil, mechanical, electrical and the rest are separate papers. The branch you chose is therefore not only a curriculum decision, it is the paper you will eventually sit. Read: The second examination, which is the one your branch decided · Practise the second examination, which is your discipline's
the board that issues, and moving between statesA state board grants the licence. The national body administers the examinations and keeps a record of your qualifications, and moving to another state generally runs through that record by comity rather than by starting again. Read: The board issues, the national body does not, and moving runs on a record · Practise the board that issues, and moving between states
From units to a structure that stands up · Loads and how they travel
Dead, live and the difference it makesA dead load is the weight of the structure itself and anything permanently fixed to it. A live load is everything that moves: people, furniture, stored goods, vehicles. They are kept apart because they are known with different certainty -- you can weigh a slab, you can only estimate a crowd -- and the codes treat the uncertain one more cautiously. Read: What is always there, and what comes and goes · Practise Dead, live and the difference it makes
Following a load to the groundA load path is the continuous route a force takes from where it is applied to the foundation: slab to beam to column to pad to soil. Every element on the route carries everything above it. A path that stops somewhere is not a design, it is a collapse waiting for the load. Read: Every load reaches the ground or the building falls down · Practise Following a load to the ground
Which area a member actually carriesTributary area is the piece of floor whose load a given beam or column actually takes: halfway to the neighbour on each side. Multiply that area by the load per square metre and you have what the member carries, before any arithmetic about the member itself. Read: A member carries the area that drains to it · Practise Which area a member actually carries
Why loads are combined before they are checkedA member is not checked against dead load, then live load. It is checked against combinations in which each load is multiplied by a factor that reflects how well it is known and how likely the loads are to peak together. The combination, not the raw load, is what the member must survive. Read: Loads are combined before anything is checked · Practise Why loads are combined before they are checked
From units to a structure that stands up · Members that carry them
Shear and moment along a beamShear is the force trying to slice the beam across; bending moment is the effect trying to bend it. Both vary along the span, and the design is governed by their peaks. For a simply supported beam under a uniform load, shear is largest at the supports and moment is largest at midspan. Read: Two diagrams that say where a beam is working hardest · Practise Shear and moment along a beam
How far it moves, and why that governsDeflection is how far a member moves under load. A beam can be nowhere near failure and still be unusable: floors bounce, doors jam, plaster cracks, and people believe a building is unsafe because it moves. Deflection is checked separately from strength and frequently governs. Read: Strong enough and still unacceptable · Practise How far it moves, and why that governs
Concrete works in compression, steel in tensionConcrete is strong in compression and weak in tension -- roughly a tenth as strong. Reinforced concrete works by putting steel where the tension is and letting concrete do the compression. That is why reinforcement sits near the bottom of a simply supported beam and near the top over a support. Read: Concrete takes compression, steel takes tension · Practise Concrete works in compression, steel in tension
Choosing a section from its propertiesA steel section is chosen by its properties, not its look. The section modulus relates bending moment to stress, so the required modulus is the moment divided by the allowable stress. Pick the lightest section in the table whose modulus is at least that. Read: Choosing a section from the numbers in the table · Practise Choosing a section from its properties
From units to a structure that stands up · The ground and the site
What the ground will holdSoil has a bearing capacity: a pressure above which it fails or settles unacceptably. A foundation spreads a column's load over enough area to bring the pressure below it. Required area is the load divided by the allowable bearing pressure. Read: What the ground underneath will actually hold · Practise What the ground will hold
Settlement, and why differential is the problemAll foundations settle. A building that settles uniformly by twenty millimetres is usually undamaged. One where a corner settles twenty and the middle settles five is distorted, and it is the difference -- differential settlement -- that cracks walls and jams doors. Read: Everything settles; the damage is in the difference · Practise Settlement, and why differential is the problem
Levelling: backsight, foresight, reduced levelLevelling transfers a known height across a site. You read a staff on a point of known level (the backsight) to find the height of the instrument, then read the staff at the unknown point (the foresight) and subtract. Instrument height equals known level plus backsight; the new level is instrument height minus foresight. Read: Backsight, foresight, and the level you actually want · Practise Levelling: backsight, foresight, reduced level
Runoff from a catchmentThe rational method estimates peak runoff as a coefficient times rainfall intensity times catchment area. The coefficient says what fraction runs off rather than soaking in: near one for asphalt, far lower for grass. Pave a field and you have not changed the rain, you have changed the coefficient. Read: How much water arrives from a piece of ground · Practise Runoff from a catchment
From units to a machine that keeps running · Turning and transmitting
Torque, speed and power are one relationshipPower is torque times rotational speed. Two of the three fix the third, so a motor rated in kilowatts tells you the torque only once you know the speed it turns at. Slow the same power down and the torque rises. Read: Torque, speed and power are one relationship · Practise Torque, speed and power are one relationship
Gear ratio trades speed for torqueA gear ratio multiplies torque and divides speed by the same number, or the reverse. It creates no power -- the product stays the same, less whatever is lost to friction. A 5:1 reduction gives five times the torque at a fifth of the speed. Read: A gearbox buys torque with speed · Practise Gear ratio trades speed for torque
A shaft in torsionA shaft transmitting torque carries shear stress that is zero at the centre and highest at the surface. That is why a hollow shaft is nearly as strong as a solid one of the same outside diameter: the material near the axis was barely working. Read: A shaft twisting under load · Practise A shaft in torsion
Bearing life falls off with the cube of loadFor a ball bearing, rated life goes with the ratio of dynamic capacity to applied load, cubed. Halving the load multiplies life by eight. Doubling it divides life by eight. Nothing else in routine machine design is that steep. Read: Bearing life falls off with the cube of the load · Practise Bearing life falls off with the cube of load
From units to a machine that keeps running · Holding together
A bolted joint is held by preload, not by the boltA properly tightened bolt stretches and clamps the parts together. The external load is then mostly carried by friction between the clamped faces, and the bolt sees only a small share of it. A loose bolt has no clamp, so the load goes straight into the bolt in shear and it fails. Read: A bolted joint is held by clamp, not by the bolt · Practise A bolted joint is held by preload, not by the bolt
Spring rate, deflection and the force you getA spring's rate is force per unit deflection, and for an ordinary spring it is constant: force equals rate times deflection. Rate is a property of the spring; force is what you get once you decide how far to compress it. Read: Rate, deflection, force · Practise Spring rate, deflection and the force you get
Things break at loads they carried yesterdayA part loaded repeatedly can fail at a stress far below the one that would break it once. Fatigue is driven by the range of stress and the number of cycles, and cracks start where stress concentrates: a sharp corner, a keyway, a hole, a weld toe. Read: Things break at loads they carried yesterday · Practise Things break at loads they carried yesterday
Tolerance and fit: clearance, transition, interferenceA fit is the relationship between a hole and the shaft that goes in it. Clearance means the shaft is always smaller and it turns or slides. Interference means the shaft is always larger and it must be pressed or shrunk in. Transition may be either, depending where in tolerance the two parts land. Read: Clearance, transition, interference · Practise Tolerance and fit: clearance, transition, interference
From units to a machine that keeps running · Moving heat and fluid
Head, flow and the power a pump needsHydraulic power is density times gravity times flow rate times head. Divide by efficiency to get the shaft power the pump actually needs. Head is the height the pump must lift against, plus everything the pipework wastes. Read: Head, flow and the power it takes · Practise Head, flow and the power a pump needs
Cavitation: the margin that stops itCavitation is vapour bubbles forming at the pump inlet and collapsing violently inside it. It is prevented by keeping the available net positive suction head (NPSH) above what the pump requires. Available comes from the installation; required comes from the pump. Read: The margin that stops a pump destroying itself · Practise Cavitation: the margin that stops it
Moving heat between two streamsThe heat a stream gives up or takes in is its mass flow times specific heat capacity times the temperature change. In a heat exchanger, what one stream loses the other gains, minus losses -- so knowing three quantities on one side and two on the other fixes the rest. Read: Moving heat from one stream into another · Practise Moving heat between two streams
Efficiency, and the ceiling nothing beatsNo heat engine working between two temperatures can be more efficient than one minus the ratio of the absolute temperatures, cold over hot. That is the Carnot limit. Real engines fall well below it, but nothing exceeds it, and the temperatures must be absolute. Read: The ceiling no engine beats · Practise Efficiency, and the ceiling nothing beats
From units to a circuit that behaves · Alternating quantities
Why alternating quantities are drawn as arrowsAn alternating voltage or current is a sine wave with a size and a timing. A phasor is an arrow that carries both: its length is the size, its angle is how far ahead or behind it runs. Adding two sine waves by arithmetic is hard; adding two arrows is geometry. Read: Why alternating quantities are drawn as arrows · Practise Why alternating quantities are drawn as arrows
Impedance: resistance that depends on frequencyImpedance is opposition to alternating current, and unlike resistance it depends on frequency. A capacitor's opposition falls as frequency rises; an inductor's grows. Resistance does neither. Impedance combines resistance and reactance as arrows at a right angle. Read: Resistance that depends on frequency · Practise Impedance: resistance that depends on frequency
Real, reactive and apparent powerApparent power is volts times amps. Real power is what actually does work. Power factor is the ratio between them. When current lags voltage -- as it does with motors -- the supply carries current that does no work but still heats the cable and still has to be paid for. Read: Real power, apparent power, and the gap between · Practise Real, reactive and apparent power
Three phases, and the root-three that followsA three-phase supply carries three alternating voltages a third of a cycle apart. Because they are out of step, the voltage between two lines is not twice the voltage to neutral but root three times it -- about 1.732. The same factor turns up in three-phase power. Read: Three supplies, one root three · Practise Three phases, and the root-three that follows
From units to a circuit that behaves · Machines and supplies
Turns ratio, and what it does to currentA transformer changes voltage in proportion to the ratio of its turns, and changes current in the opposite proportion. Power in equals power out apart from losses, so stepping voltage up by ten divides current by about ten. Read: Turns ratio, and what it does to current · Practise Turns ratio, and what it does to current
Slip, torque and starting currentAn induction motor's rotor always turns slightly slower than the rotating field; that difference is slip, and it is what produces torque. At the instant of starting, slip is total and the motor draws several times its running current. Read: Slip, torque, and the current at the moment of starting · Practise Slip, torque and starting current
Sizing protection to the cable, not the loadA fuse or breaker exists to stop the cable overheating. Its rating must be at or below what the cable can carry continuously, and at or above the load's normal current. The appliance protects itself; the cable cannot. Read: Protection guards the cable, not the appliance · Practise Sizing protection to the cable, not the load
Earthing and the fault pathEarthing gives a fault current a low-resistance path back to the source so that protection sees a large current and disconnects quickly. Without that path, exposed metal can sit at a dangerous voltage while the breaker, seeing only a small current, never operates. Read: A fault needs somewhere to go, fast · Practise Earthing and the fault path
From units to a circuit that behaves · Signals and devices
A device that conducts one wayA diode allows current in one direction and blocks it in the other, dropping a small roughly constant voltage when it conducts. That one-way behaviour is what turns alternating current into direct current and what protects circuits from reversed supplies. Read: A component that conducts one way · Practise A device that conducts one way
Using a transistor as a switchUsed as a switch, a transistor is either fully off or fully on. A small current or voltage at the control terminal decides which. That is how a logic output that can supply milliamps operates a relay, a motor or a lamp that needs amps. Read: A small current controlling a large one · Practise Using a transistor as a switch
The two rules that solve most op-amp circuitsWith negative feedback, an ideal op-amp behaves as if no current flows into its inputs, and as if it does whatever it must to keep the two inputs at the same voltage. Those two rules solve nearly every standard circuit without touching the device's internals. Read: Two rules that solve most op-amp circuits · Practise The two rules that solve most op-amp circuits
Sampling fast enough to be honestTo capture a signal without inventing content, it must be sampled at more than twice its highest frequency. Sample too slowly and high frequencies do not disappear -- they reappear disguised as low ones, which is aliasing, and nothing downstream can undo it. Read: Sampling fast enough to be honest · Practise Sampling fast enough to be honest