Calculations

Calculations

• Convert between two units

  A value in one unit as a value in another unit of the same kind.

  result = v × (n₁ × d₂) ÷ (d₁ × n₂)

  Unit conversion unit-table.js Every app
  Exact

• Typed text to a unit

  What was typed, with its unit, as a value in the unit the field wants.

  3 yd typed into a feet field = convert(3, yd, ft) = 9

  Unit conversion unit-table.js Every app
  Exact

• A value to the base unit

  A number in any unit as meters, square meters, cubic meters, radians or rise over run.

  base = v × n ÷ d (v × π × n ÷ d for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• A base value to a unit

  A stored base value as a number in the unit shown.

  value = base × d ÷ n (base ÷ π × d ÷ n for degrees and gons)

  Unit conversion unit-table.js Every app
  64-bit

• Which side of an edge

  Whether three points turn counter-clockwise, clockwise, or lie on one line.

  sign of (b − a) × (c − a)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Is an edge Delaunay

  Whether the point across an edge lies inside the circle through the triangle on this side.

  sign of the in-circle determinant of a, b, c, d

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  Exact

• Triangle under a point

  The triangle a point falls in and its three barycentric weights.

  w₁ = ((y_b − y_c)(x − x_c) + (x_c − x_b)(y − y_c)) ÷ d, w₃ = 1 − w₁ − w₂

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Elevation at a point

  The surface elevation at an easting and northing.

  z = w₁·z₁ + w₂·z₂ + w₃·z₃

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Slope of a triangle

  Rise over run and the downhill direction of one triangle.

  n = u × v; slope = √((n_x ÷ n_z)² + (n_y ÷ n_z)²)

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Plan and surface area

  The surface's area in plan and along the slope, with its elevation range.

  area in plan = Σ |n_z| ÷ 2; surface area = Σ |u × v| ÷ 2

  Surface triangulation tin.js

  • BimCraft
  • CivilCraft
  64-bit

• Member stiffness matrix

  The 12 by 12 stiffness of one beam-column in its own axes.

  EA ÷ L, GJ ÷ L, 12EI ÷ L³, 6EI ÷ L², 4EI ÷ L, 2EI ÷ L

  3D frame analysis frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Equivalent nodal loads

  A load along a member as forces and moments at its two ends.

  p = ∫ N(ξ) · w(ξ) dx, by 4-point Gauss-Legendre quadrature

  3D frame analysis frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Banded Cholesky factorisation

  Factorises the stiffness matrix once, for every load set.

  K = L · Lᵀ; pivot s = k_jj − Σ l²; mechanism when s ≤ 1e-11 × k_jj

  3D frame analysis frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Forward and back substitution

  Displacements for one load set from the factorised matrix.

  L · y = f, then Lᵀ · u = y

  3D frame analysis frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Add, subtract, multiply, divide

  The four operations on values that carry units.

  a + b and a − b need the same dimensions; a × b and a ÷ b combine them

  Calc sheets calc.js Every app
  64-bit

• Comparisons

  Pass or fail checks such as M <= 40 kN m.

  a ≤ b is true when a ≤ b + 1e-12 × max(1, |a|, |b|)

  Calc sheets calc.js Every app
  64-bit

• Number shown

  A result as text in the sheet.

  6 significant digits; exponent form from 1e9 up and below 1e-4

  Calc sheets calc.js Every app
  64-bit

• Reproject points

  Every point of a file from one coordinate system to another.

  (x, y) in the source system → (x, y) in the target system, point by point

  Map coordinate reprojection reproject-batch.js

  • BimCraft
  • CivilCraft
  64-bit

• Map origin from latitude and longitude

  Eastings and northings of a typed latitude and longitude.

  (lon, lat) → (E, N), rounded to 0.000001 of the map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Model to map conversion

  Where a model sits on the map: origin, rotation and scale.

  X axis = (cos a, sin a) rounded to 1e-12; scale = model unit ÷ map unit

  Map coordinate reprojection crs-georef.js

  • BimCraft
  • CivilCraft
  64-bit

• Sabine constant

  The constant both reverberation formulas multiply the volume by.

  C = 0.161 s/m

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Sound absorbed by the air

  The attenuation of a pure tone in air, in dB per meter, at the room's temperature, humidity and pressure.

  α = 8.686 f² [1.84e-11 (p_r ÷ p) √(T ÷ T₀) + (T ÷ T₀)^−2.5 (0.01275 e^(−2239.1 ÷ T) ÷ (f_rO + f² ÷ f_rO) + 0.1068 e^(−3352 ÷ T) ÷ (f_rN + f² ÷ f_rN))]

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Air absorption per meter

  Turns the dB figure into the energy absorption coefficient the formulas use.

  m = α ÷ 4.343

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Reverberation time per band

  The Sabine and Eyring times of a room in each of six octave bands, and the mid-frequency mean.

  A = Σ Sᵢαᵢ + extra; ᾱ = min(0.999, A ÷ S); Sabine = C V ÷ (A + 4mV); Eyring = C V ÷ (−S ln(1 − ᾱ) + 4mV)

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Reverberation over the target

  The finding raised when a room's mid-frequency time is over the person's target, with the absorption that would meet it.

  T = round(T_mid × 100) ÷ 100; need = round(max(0, V C ÷ target − (A₅₀₀ + A₁₀₀₀) ÷ 2))

  Room reverberation time acoustics.js

  • BimCraft
  64-bit

• Distance from a point to a room outline

  How far an opening's middle is from the room's outline, to decide which room the opening belongs to.

  d = min over edges of |P − (A + t (B − A))|, t clamped to 0…1

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room perimeter

  The length of the room's outline, used for the wall area when the walls' finishes are not read from the model.

  P = Σ |pᵢ₊₁ − pᵢ|

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Is a point inside a room

  Decides whether a ceiling, a floor or a piece of furniture lies in a room.

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room height and opening areas

  The height the room is given and the area of the doors and windows in its walls.

  H = height ÷ 1000; opening area = (width ÷ 1000) × (height ÷ 1000)

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Wall area per finish

  The area of wall each finish covers in a room, less its doors and windows.

  wall area = Σ edge length × H; net = max(0, gross − glass − doors); each finish × net ÷ gross

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Room volume and the figures reported

  The volume passed to the formulas and the area, height and volume listed for each room.

  V = floor area × H; aspect = longest ÷ max(shortest, 0.1)

  Room reverberation time acoustics-model.js

  • BimCraft
  64-bit

• Fast Fourier transform

  The transform behind the band filters and the convolution.

  X_k = Σ x_n e^(∓2πi kn ÷ N), N a power of two

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit

• Octave band weight

  How much of a frequency belongs to one octave band.

  w = cos(π ÷ 2 × |log₂(f ÷ f_band)|) within one octave, else 0

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit

• Zero-phase band filter

  Keeps one octave band of a signal.

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit

• Decaying reverberant tail

  The diffuse part of the response: noise that falls 60 dB in each band's reverberation time.

  tail(t) = Σ_bands noise_b(t) × e^(−k t), k = 6.907755 ÷ T_b; × (t ÷ onset)² up to onset = max(0.002, 2 × mean free path ÷ 343)

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit

• First reflections and the tail's level

  Places the six first reflections and sets the level of the tail against the direct sound.

  R = S a ÷ (1 − a); reflection gain = √(1 − a) × d₀ ÷ d at sample round((d − d₀) ÷ 343 × sr); tail energy = 16π d₀² ÷ R less the reflections; r_c = √(R ÷ 16π)

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  32-bit in part

• Reverberation time measured from a response

  The reverberation time of an impulse response in one band, by Schroeder's backward integration.

  E(t) = Σ over τ ≥ t of x²; L = 10 log₁₀(E(t) ÷ E(0)); T = 60 ÷ −slope of L between −5 and −35 dB

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  64-bit

• Convolution

  A dry sound passed through the room's impulse response.

  y = IFFT(FFT(x) × FFT(h)) ÷ N

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  32-bit in part

• Level matching

  Brings the sound in the room to the loudness of the dry sound, so switching between them changes the room and not the volume.

  k = rms(dry) ÷ rms(wet over the dry sound's length)

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  32-bit in part

• WAV file samples

  The numbers written to the downloaded sound file.

  s = round(clamp(x × k, −1, 1) × 32767), k = 1 ÷ peak when peak > 1

  Room impulse response and listening acoustics-ir.js

  • BimCraft
  32-bit in part

• Length unit of the file

  The factor that takes the file's lengths to millimeters.

  toMm = prefix × 1000, or factor × base × 1000 for a converted unit

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Axis placement

  The three axes and the origin of one IFC placement.

  X = norm(X − Z (X·Z)); Y = Z × X

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Placement chain

  An element's position and axes in the model, through every placement it is relative to.

  M = M_parent ∘ M_local

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Rounded rectangle outline

  The outline of a hollow rectangular section with rounded corners.

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Circle outline

  The outline of a round bar or tube.

  pᵢ = r (cos 2πi ÷ 64, sin 2πi ÷ 64)

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• I-shape outline

  The outline of an I section with its root fillets.

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Is this I-shape a rolled section

  Matches an I-shape's name and sizes against the section table, to decide whether it is treated as rolled or welded.

  |a − b| ≤ max(0.5 mm, 0.01 × b) for d, bf, tw, tf

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Member axis, length and section

  The line a member is analyzed on and the section properties it carries.

  axis from the section centroid c₀ to c₀ + depth × ext; skipped when |ext·U| or |ext·V| > 0.001

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Concrete wall as a line, thickness and height

  Reads a concrete wall's plan outline as a center line and a thickness for the plate model.

  turn the outline by −θ of its longest edge; thickness = y₁ − y₀; kept when |area| ≥ 0.9 × box area

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit

• Closest points of a line and a segment

  Where a member's own line passes nearest another member's axis, and how far apart they are there.

  s = (b e − c d) ÷ (a c − b²); t = (a e − b d) ÷ (a c − b²)

  Frame model from a building model analysis-bim.js

  • BimCraft
  64-bit