Worked examples
Worked engineering calculation examples
15 calcsheets across 8 disciplines, each set out the way an engineer would on paper: the inputs, the working, the checks and a results summary. Open one in Maths Explore, change an input and watch every result below it update — no account needed.
Electrical
Cable voltage drop and power loss
A single-phase 230 V final subcircuit feeds a 32 A load at the end of a long run. Current flows out on the active and back on the neutral, so the circuit…
Off-grid solar and battery sizing
Sizing the battery and PV array for a small off-grid cabin. Power (W) is a rate; energy (Wh) is what batteries store and loads consume — the schedule…
Facade
Facade panel and supporting mullion
A glazed panel spans between vertical mullions; each mullion spans floor to floor and hands the wind load to slab fixings. From a specified design wind…
Geotechnical
Earthworks quantities and haul planning
A basement excavation partly reused as site fill: the same soil occupies different volumes as bank (in the ground), loose (in the truck) and compacted (in…
Retaining wall stability
Sliding, overturning and bearing checks for a mass-concrete gravity wall retaining level granular backfill, per metre of wall. Active pressure from…
Hydraulics
Pumping water between two tanks
The motor power needed to pump water up to a header tank: velocity from the flow, friction from the Darcy–Weisbach equation with an explicit Swamee–Jain…
Mechanical
Bolt tightening and preload
The wrench torque that develops a target preload in a metric bolt, split into its thread and collar (under-head friction) parts, with the bolt stress…
Geared motor driving a winch
Sizing the electric motor of a winch that lifts a load through a single gear stage: the torque to accelerate the load and both shafts, the steady lifting…
Machine vibration and isolation
A fan on resilient mounts, modelled as a single degree of freedom: one mass on the summed mount stiffness with light damping. The question is how much of…
Vehicle braking on a slope
Stopping distance on a downhill grade, split into thinking and braking parts. Braking model: constant deceleration at the friction limit, a = g(μ·cosθ −…
Process
Batch mixing and dilution
Diluting a concentrate to a working-strength batch: how much concentrate and water make the target mass, checked by a mass balance on the active…
Structural
Reinforced concrete beam design to AS3600
Ultimate-limit-state bending design of a singly reinforced rectangular beam using the rectangular stress block of AS 3600:2018. Scope: bending strength…
Steel floor beam: strength and deflection
A simply supported steel beam carries a strip of office floor. The area loads become a line load, then bending strength and deflection are checked and two…
Thermal
Heat exchanger sizing
Sizing the area of a water-to-water counterflow exchanger by the LMTD method: the duty from the hot stream, the cold outlet from an energy balance, the…
How long ice lasts in an esky
Heat leaks into an esky by conduction through its foam walls and lid and melts the ice inside. Steady one-dimensional conduction through all six faces…