Force is easy to picture and pressure is easy to misjudge. The gap between the two is entirely about area, and once you see that force alone tells you almost nothing about what a load will do, a surprising number of engineering judgements become straightforward. This guide builds from F = m × a up to hydraulic force and structural stress, with units checked at every step.
Force, weight and gravity
Newton's second law is the starting point:
F = m × a
Set a = g and the same equation gives weight, so a 500 kg mass exerts 500 × 9.81 = 4,905 N near sea level. Use g = 9.81 m/s² in metric work and 32.174 ft/s² in imperial; both appear on calculators and datasheets, and mixing them produces a 15% error. Note that the force calculator reports weight separately from mass, because they are different quantities and only one of them changes with location.
Pressure: the same force, spread thinner
Pressure is force per unit area:
P = F ÷ A
This single division explains an everyday observation. Press down with 5 N on a table top of 0.5 × 0.5 m and the pressure is 5 ÷ 0.25 = 200 Pa — nothing. Press the same 5 N on a 0.5 mm diameter needle point, area = π × 0.00025² = 1.96 × 10⁻⁷ m², and the pressure is 25.5 MPa.
That is a factor of about 128,000 from the same finger push, and it is the entire explanation for why needles puncture, why a coin in a paywall machine jams, and why a sharp knife cuts when a blunt one does not. The force required to break a material depends on the stress — pressure — not on the force. Sharpen the edge and the force needed to break the material falls.
Units: Pa, kPa, bar, psi and atm
Five units cover nearly all practical work, and the conversions are worth having in your head rather than looking up:
- 1 bar = 100 kPa = 14.504 psi
- 1 psi = 6.895 kPa (exactly 4,894.757 Pa)
- 1 atm = 101.325 kPa = 14.696 psi
- 1 MPa = 1,000,000 Pa = 10 bar = 145.04 psi
The reason imperial units are so awkward is historical: psi comes from pound-force per square inch, and the pound-force is a force unit defined by gravity's effect on a pound-mass. The pressure calculator handles all of them, which is worth knowing before you convert by eye.
Gauge versus absolute pressure
This distinction causes real equipment failures, so it is worth stating plainly:
P_absolute = P_gauge + 101.325 kPa
Every pressure gauge you own reads gauge pressure, because it measures the difference between the system's pressure and the atmosphere acting on its sensor. A gauge reading zero is not zero pressure — it is one atmosphere still pushing. A system at 250 kPa gauge is therefore at 351.3 kPa absolute.
Three things must use absolute pressure. Any gas-law calculation needs it, because the volume of a gas depends on the total molecular collision pressure, not on the portion above atmosphere. Vacuum work needs it, since a vacuum is a pressure below atmosphere, which has no meaning as a negative gauge reading. And any specification that says "absolute" or "at sea level" needs it, because a gauge reading is only meaningful at the altitude where it was taken — altitude changes the reference.
This is why a vacuum pump rated at "−85 kPa" is a good pump: −85 kPa gauge is 16.3 kPa absolute, and the achievable ultimate pressure is bounded below by the vapour pressure of the pump oil, not by how hard it pulls. The dilution guide covers the other place where precise concentrations demand care.
Worked example 1: a bearing plate
A 50 kN load sits on a 200 × 200 mm bearing plate. The area is 0.2 × 0.2 = 0.04 m², so the pressure is 50,000 ÷ 0.04 = 1,250,000 Pa = 1.25 MPa.
Now the useful part. Structural steel has a yield strength around 250 MPa. Using that yield directly as a limit gives a factor of 200, which is not a design — it ignores stress concentrations, weld defects, fatigue, load direction and the fact that steel is not perfectly uniform. Apply a safety factor of 2.0 and the allowable is 125 MPa, still 100 times the actual stress, so the plate is enormously strong for this job. If the plate were cast iron with an allowable of about 40 MPa, it would also pass. The load is simply not the limiting factor, and knowing that early saves a great deal of unnecessary work.
The same 50 kN distributed over a smaller footprint changes everything: on a 50 × 50 mm pad the area is 0.0025 m² and the stress is 20 MPa — 16 times higher from a footprint one sixteenth the area. Dimensioning a bearing pad is almost always about keeping this number under the allowable, not about the total load.
Worked example 2: a hydraulic cylinder
A double-acting cylinder with a 30 mm bore is supplied at 20 bar. The pressure is 20 bar = 2,000,000 Pa. The piston area is the full circle, A = π × (0.030 ÷ 2)² = π × 0.015² = 7.069 × 10⁻⁴ m².
The thrust is F = P × A = 2,000,000 × 7.069 × 10⁻⁴ = 1,414 N. Round numbers are nice but this one is honest: 2 × 10⁶ × π × 2.25 × 10⁻⁴ = 1,413.7 N.
Two things are worth noting. The force comes from the pressure, and the pressure came from the pump, which produced it against whatever load the piston presented — not from the motor's speed. And a 30 mm bore at 20 bar is a modest cylinder: doubling the pressure doubles the thrust, and doubling the bore quadruples it, because area scales with the square of diameter.
Why hydraulics multiply force
Pascal's principle states that pressure applied to a confined fluid is transmitted undiminished throughout the fluid. Since F = P × A, pushing a small piston creates a pressure that acts on every other piston in the system, and the force on the large piston is that pressure times the large area.
So a 5 cm piston pushing with 500 N creates a pressure of 500 ÷ 1.963 × 10⁻³ = 254.6 kPa. Applied to a 30 cm piston of area 7.069 × 10⁻² m², that yields 254,600 × 0.07069 = 18,000 N — 36 times the input force, because the area ratio is (30 ÷ 5)² = 36.
Here is the part that makes it a mechanism rather than a free lunch: the large piston moves one thirty-sixth as far. The volume pushed through the small piston must equal the volume gained by the large one, so travel divides by exactly the factor force multiplies. Total work is unchanged. A hydraulic jack is not a machine that creates energy; it is a machine that trades distance for force with near-perfect efficiency, which is exactly what a gear ratio does, in linear form rather than rotational.
Gas pressure: why temperature must be in Kelvin
For a gas, pressure follows the ideal gas law, and the form that matters in practice is the ideal gas law rearranged for pressure:
P × V = n × R × T
with R = 8.314 J/(mol·K). Check it: 1 mol of gas at 300 K in 0.02 m³ gives P = 1 × 8.314 × 300 ÷ 0.02 = 124,700 Pa, about 1.23 atmospheres.
The one non-negotiable rule is that T must be absolute temperature in kelvin, not Celsius. A temperature is a measure of molecular energy, and energy is proportional to absolute temperature — a gas at 27 °C is 300 K, not 27. Substituting 27 into the formula gives an answer 11 times too small, and it is the single most common error in this area.
The reason is visible in the arithmetic: 0 °C is 273.15 K, not 0. Doubling the Kelvin temperature doubles the pressure, which means pressure rises roughly 0.34% per degree Celsius at constant volume. A tyre's pressure warning is not fussiness — a 1 atm tyre's absolute pressure climbs about 14% between 0 °C and 40 °C if the volume is fixed.
Stress, allowable stress and safety factor
For a structural member, stress is the same idea as pressure — force over area — with a geometric correction for the shape:
σ = F ÷ A for direct tension or compression, and for a bending member the section modulus takes over, since stress varies across the depth
Design does not use the material's strength directly. It uses an allowable stress, which is the strength divided by a safety factor:
σ_allow = σ_yield ÷ SF
For 250 MPa yield steel with SF = 2.0, the allowable is 125 MPa. The margin is not superstition: it absorbs unexpected loads, weld defects, stress concentrations around holes and notches, material variability, fatigue and wear, and the fact that loads are usually estimated rather than known. As the Newton's three laws guide sets out, static equilibrium assumes forces you have actually accounted for, and structures meet forces you have not.
Brittle materials deserve more margin. Cast iron, glass, ceramics and concrete have no yield stage — they give no warning, they simply fail. Common practice is SF = 3 to 4 for cast iron, and considerably more for pressure vessels, lifting equipment and anything where failure injures someone.
Working the earlier example backwards shows what the allowable really does. A 50 kN load with an allowable of 125 MPa needs at least 50,000 ÷ 125,000,000 = 4 × 10⁻⁴ m², which is 4 cm² — a 20 × 20 mm square. The 200 × 200 mm plate offers 2,500 times more material than the minimum, which is the practical meaning of "this load is not a problem for steel".
Habits that keep you out of trouble
Convert units before computing, not after. Mixing kPa and Pa silently gives answers off by a thousand, and the result still looks plausible.
Ask which area, every time. Nominal contact area, bolt-hole net area and bearing area are different numbers, and the smallest one is usually the one that governs.
Decide gauge or absolute deliberately. Write it down at the start of any gas or vacuum problem, and default to absolute for those.
Use kelvin for gases. No exceptions, and add 273.15 if you have only Celsius.
Keep a safety factor, and record it. An unrecorded factor of safety is indistinguishable from no factor of safety.
Frequently asked questions
Why is pressure defined as force divided by area rather than force alone?
Because the same force spread over different areas produces completely different effects. A 5 N push on a table top spread over 0.25 m² gives 20 Pa, while the same push on a needle point of 0.5 mm diameter concentrates into 25.5 MPa — roughly a thousand times more. Material failure depends on how concentrated the load is, not on how large it is in total.
What is the difference between gauge and absolute pressure?
Absolute pressure is measured from a perfect vacuum and is the pressure you use in thermodynamics, gas laws and vacuum work. Gauge pressure is what a gauge reads, and it is atmospheric pressure subtracted away. Absolute equals gauge plus 101.325 kPa. A gauge showing 0 is not zero pressure; it is one atmosphere of pressure still acting.
How does a hydraulic jack multiply force?
Pascal's principle says pressure applied to a confined fluid is transmitted undiminished to every part of it. Since force equals pressure times area, a small piston can drive a piston of nine times the area to produce nine times the force. The cost is that the large piston moves one ninth as far, so force multiplies while travel divides, and total work is still conserved.
What safety factor should I use for a structural design?
For static loads on ductile materials like structural steel, a factor against yield of about 1.5 to 2.0 is conventional, and 2.0 is a sensible default for general work. For cast iron, which is brittle and has no warning before failure, use 3 to 4, and for pressure vessels, lifting gear and anything with a consequence of failure, use considerably more.