Engineering checks

Questions of engineering, asked of the exact geometry and kept as checks: a section's area and stiffness, a tank's fill level, where a hull floats and whether it stays upright, a stress, and what happens over a motion.

These build on the measures, checks and outputs of Measuring and assertions: read that first.

Below a plane, sections and levels

Three questions about a plane through closed bodies, answered on the exact geometry (the part below the plane is cut off by the kernel and integrated; the section's moments come from the curves in which the plane cuts the bodies). The plane is given as measure.symmetry takes it (xy, yz, zx, a datum plane, {"point", "normal"}) or as a flat face by its name ("F1:face(1)", with its outward normal), moved by offset along its normal; the default is the xy plane. Below is the side the normal points away from.

[
 ["feature.block", {"size": [4000, 1300, 1000], "position": [-2000, -650, 0], "name": "Tank"}],
 ["measure.below", {"of": "Tank", "offset": "300 mm"}],
 ["measure.levelFor", {"of": "Tank", "volume": "1.56 * (1 m)^3"}],
 ["measure.section", {"of": "Tank", "plane": "yz"}]
]

The first answers volume 1.56e9 (mm³), centroid [0, 0, 150]; the second, the fill level for 1.56 m³ of liquid, offset 300 (the kernel finds it by Newton's method, steps saying how many levels it measured); the third, the cross-section square to x: area 1.3e6, with xAxis y and yAxis z, ix = 1300 × 1000³/12 (about the axis along y) and iy = 1000 × 1300³/12.

• offset is a length, volume a volume, both expressions of the variables; xAxis sets the section's x axis (square to the normal; by default the world axis most square to it).
• Errors you can act on: outside (0, total) (no plane leaves that volume below), a sheet has no inside (close it first), did not converge after 64 steps.
• Bodies that overlap are counted once each.
• Floating objects: a body of mass m in a liquid of density ρ settles where levelFor of m/ρ puts the plane. A Jig computes what it needs from the centroid and the section's moments; the developer documentation has a worked example (design/sections-and-levels.md).

A section's torsion and shear centre

measure.section also twists the section: torsion is its St Venant torsion constant J (the length unit to the fourth power), shearCentre the point a sideways load must pass through for the beam not to twist (a point in space; shearOffset is it from the centroid along xAxis and yAxis), and warping the warping constant Cw (the unit to the sixth power). They hold for any section: closed tubes, open channels, sections with cutouts or holes, several pieces. 3DCraft solves the warping of the section by finite elements and refines until J and the shear centre change by less than torsionTolerance (default 1e-4, relative); torsionSolve says whether it got there (converged), how much J changed in the last round, and how many elements it took (at most maxElements, default 60000). A solve that stops at the cap says converged: false; it never claims more. torsion: false leaves the torsion out (it costs from a few tens of milliseconds to a few tenths of a second).

[
 ["feature.block", {"name": "Outer", "size": ["500 mm", "100 mm", "150 mm"]}],
 ["feature.block", {"name": "Inner", "size": ["502 mm", "94 mm", "142 mm"], "position": ["-1 mm", "3 mm", "4 mm"]}],
 ["feature.boolean", {"name": "Box", "op": "subtract", "target": "Outer", "tool": "Inner"}],
 ["measure.section", {"of": "Box", "plane": "yz", "offset": "250 mm"}]
]

The welded box beam, walls of 3 and 4 mm: torsion about 5.58e6 mm⁴, 1.4% above Bredt's thin-walled formula (which is an approximation; the solve is not), and shearCentre at the centroid, as the symmetry says. Cut a slot in one wall and the section through it is open: J drops by three orders of magnitude and the shear centre moves away from the slot.

Sections and levels as checks

As assertions: volumeBelow, centroidBelow (with axis), sectionArea, sectionCentroid (with axis), sectionInertia (with about: x, y, xy, major or minor), sectionTorsion (J), sectionShearCentre (with axis: a coordinate of the shear centre; a solve that stops at its cap is notProven), sectionExtent (with axis: x, y, z or [x, y, z]; how far the section reaches along it, the length of a hull's waterline at its level) and levelFor (with volume), each with the same of, plane, offset (not for levelFor) and xAxis (not for sectionExtent):

[
 ["assert.add", {"name": "stiff enough", "measure": "sectionInertia", "of": "Tank", "plane": "yz", "about": "minor", "min": "1e11"}],
 ["assert.add", {"name": "fill level", "measure": "levelFor", "of": "Tank", "volume": "1.56 * (1 m)^3", "max": "350 mm"}]
]

Floating

measure.floating finds the plane a floating body settles at when its centre of gravity is known too: the level and the tilt at which the volume below is volume (or mass / density, kg and kg/m³) and the centroid below lies straight under the centre of gravity. plane is the upright waterplane (default xy, up +z); heel and trim tilt the start.

[
 ["feature.block", {"size": [1000, 200, 200], "position": [-500, -100, 0], "name": "Pontoon"}],
 ["measure.floating", {"of": "Pontoon", "mass": 20, "density": 1000, "centerOfGravity": [0, 0, 50]}],
 ["measure.floating", {"of": "Pontoon", "mass": 20, "density": 1000, "centerOfGravity": [0, 0, 90]}]
]

The first answers one equilibrium, upright at offset 100, stable, with its two metacentric heights under principal (gm 33.3 mm about the length, 833 mm about the width). The second has its centre of gravity too high: upright is an equilibrium but unstable, and the two it falls to follow, heeled 32.3° to each side (the box's angle of loll); floats is the index of the first stable one. Each equilibrium gives heel, trim and tilt in degrees, normal, centroid, arm (how far the centre of gravity is off the vertical through the centroid) and what measure.below gives there. The search starts from the upright plane (or heel, trim), finds the nearest equilibrium by Newton's method and, where that one is unstable, the ones the body falls to from it; other equilibria are not searched. neutral says that the first one turns freely in some direction (a sphere with its centre of gravity at its centre).

Floating, as assertions and outputs

What measure.floating finds can be asserted too, so a design that floats upright keeps doing so as it changes: floatingOffset (the waterplane's offset along its normal), floatingHeel and floatingTrim (its angles from the upright plane; write the bounds with a unit, "3 deg"), floatingGM and floatingGML (the metacentric heights for a heel, a turn about the plane's x axis, and for a trim, about its y axis) and floatingStable (is: stable both ways; notProven when it is neutral within measurement), all of the equilibrium reached from the start. They take the parameters of measure.floating, and every value may be a formula of the variables and the outputs, so the centre of gravity can be computed by the design:

[
 ["feature.block", {"size": [2000, 600, 400], "position": [-1000, -300, 0], "name": "Barge"}],
 ["variable.set", {"name": "KG", "expr": "180 mm"}],
 ["variable.set", {"name": "xg", "expr": "64 mm"}],
 ["variable.output", {"name": "zG", "expr": "KG"}],
 ["assert.add", {"name": "upright", "measure": "floatingGM", "of": "Barge", "mass": 300, "density": 1000, "centerOfGravity": ["xg", 0, "zG"], "min": "0 mm"}],
 ["assert.add", {"name": "trim", "measure": "floatingTrim", "of": "Barge", "mass": 300, "density": 1000, "centerOfGravity": ["xg", 0, "zG"], "min": "-3 deg", "max": "3 deg"}],
 ["variable.measure", {"name": "GMf", "measure": "floatingGM", "of": "Barge", "mass": 300, "density": 1000, "centerOfGravity": ["xg", 0, "zG"]}]
]

The barge settles 2.86° down towards +x, where its centre of gravity lies (a trim of tan⁻¹ 0.05, the wall-sided closed form) with a GM of 66.75 mm, so both pass. With KG at 300 mm its GM is negative and upright fails, and so does trim: the barge then trims by 3.16°, past its bound of 3°. A search for the equilibrium that does not settle makes such an assertion notProven, with the reason, never passed, and an output of it unknown; a displacement more than the body holds is an error (it sinks). Each assertion solves the equilibrium itself, and again only when what it depends on changes.

Stability as outputs: a hull's GM

Some answers are formulas of measures. A boat's stability, the metacentric height GM = KB + BM − KG, comes from where it floats (the draught T), the height of the centre of what is under water (KB), and the waterplane's moment (I, so BM = I / V). As outputs (Measuring and assertions › Outputs), a box hull's GM is worked out after every change, and checked:

[
 ["variable.set", {"name": "KG", "expr": "180 mm"}],
 ["variable.set", {"name": "V", "expr": "0.3 * (1 m)^3"}],
 ["feature.block", {"size": ["2000 mm", "600 mm", "400 mm"], "position": ["-1000 mm", "-300 mm", "0 mm"], "name": "Hull"}],
 ["variable.measure", {"name": "T", "measure": "levelFor", "of": "Hull", "volume": "V"}],
 ["variable.measure", {"name": "KB", "measure": "centroidBelow", "of": "Hull", "offset": "T", "axis": "z"}],
 ["variable.measure", {"name": "I", "measure": "sectionInertia", "of": "Hull", "offset": "T", "about": "x", "xAxis": "x"}],
 ["variable.output", {"name": "BM", "expr": "I / V"}],
 ["variable.output", {"name": "GM", "expr": "KB + BM - KG"}],
 ["assert.add", {"name": "stable", "measure": "value", "expr": "GM", "min": "0 mm"}],
 ["output.list", {}]
]

The hull floats at T = 250 mm (the level below which it holds 0.3 m³), KB = 125 mm, I = 2000 × 600³/12 mm⁴, so BM = 120 mm and GM = 65 mm: the assertion stable passes. Set KG to 300 mm and GM is −55 mm, and stable fails; only the assertion is measured again, since the waterline does not depend on KG.

Forces, stresses and moments

Forces, stresses and moments have units of their own:

• g and kg for a mass, s for a time;
• N and kN for a force; Pa, kPa, MPa and GPa for a pressure or a stress; J for an energy;
• a force unit followed by a length unit for a moment: 20 N m, 5 kN mm.

They are derived from mass, length and time (a newton is kg·m/s²), so a load over an area is a stress: 1000 N / (100 * (1 mm)^2) is 10 MPa, and 1 MPa is 1 N / (1 mm)^2. Unlike mm or deg, these symbols are not reserved: a count may still be called N and a spacing s. They are units only right after a number, a parenthesis or pi (N * 5 N is N times five newtons).

A formula with units in it has the dimension they give: a load in newtons over an area is a stress. So a stress needs no unit in its name:

[
 ["variable.set", {"name": "load", "expr": "1000 N"}],
 ["feature.block", {"name": "Bar", "size": ["100 mm", "10 mm", "10 mm"]}],
 ["variable.measure", {"name": "area", "measure": "sectionArea", "of": "Bar", "plane": "yz", "offset": "50 mm"}],
 ["variable.output", {"name": "stress", "expr": "load / area", "label": "Bending stress", "unit": "MPa"}],
 ["variable.output", {"name": "moment", "expr": "load * 100 mm", "label": "Bending moment"}],
 ["assert.add", {"name": "below yield", "measure": "value", "expr": "stress", "max": "235 MPa"}],
 ["output.list", {}]
]

output.list (and the Parameters tab) shows Bending stress 10 MPa and Bending moment 100 N m. An output without a unit is shown in its dimension's: a force in N, a pressure or a stress in MPa, a moment (or an energy) in N m, a mass in kg, anything else in the document's length unit to its power. A unit is a unit of the expressions or a product and quotient of them: N/mm², kN m, kN*mm, kg/m^3, mm/m. It must have the output's dimension: "unit": "mm" on a stress is refused, with both dimensions named, and a unit that stops fitting after an edit (the load written without its unit) makes the output unknown with the reason. An assertion's bounds take the same units ("max": "235 MPa"); a plain number bounding a stress is in MPa, a force in N, a mass in kg and a density in kg/m³ (as a floating measure's density). A measured mass is a mass: mass + 1 is a unit mismatch, write mass + 1 kg. Mind that a unit belongs to the number just before it: 1000 N / 100 mm^2 divides by (100 mm)², 10 000 mm², and is 0.1 MPa; write 1000 N / (100 * (1 mm)^2). A unit's symbol alone, an undefined N, is a variable: write 1 N. The set filter thickness (Selection and named sets) is a flat face's plate thickness, not this measure's thinnest place.

Over a motion

A design that moves (its timeline t, see The window) is checked over the whole motion with motion.check: every assertion is measured at evenly spaced moments of t (samples), on a copy, and the result says per assertion whether it held at the samples, the first sample it failed at (firstFailure.t, with after, the last sample before it that passed: the failure begins between the two), and its least and greatest value at the samples with their t. It proves what held at the samples, no more: a collision shorter than the spacing of the samples can be missed, so the result says sampled: true; take more samples, or a narrower t around a suspect moment, to look closer. Any measure works: overlap with is false gives the first sample that collides, clearance the least gap at the samples and where it is. measure checks an assertion written in place without keeping it.

[
 ["variable.set", {"name": "t", "expr": "0"}],
 ["feature.block", {"name": "Wall", "size": ["10 mm", "10 mm", "10 mm"], "position": ["-10 mm", "0 mm", "0 mm"]}],
 ["feature.block", {"name": "Slider", "size": ["10 mm", "10 mm", "10 mm"],
   "position": ["15 mm + 10 mm * cos(360 deg * t)", "0 mm", "0 mm"]}],
 ["assert.add", {"name": "gap", "measure": "clearance", "from": "Wall", "to": "Slider", "min": "6 mm"}],
 ["motion.check", {"samples": 33}],
 ["motion.check", {"measure": {"measure": "overlap", "of": "Wall", "to": "Slider", "is": false}}]
]

The first check fails at t = 0.4375 with the least clearance 5 mm at t = 0.5 (the slider swings to within 5 mm of the wall); the second passes. A collision shorter than the spacing of the samples can be missed: give more samples where it matters.

Next: Making Jigs.

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