Calculations

Calculations

• 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
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  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
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• 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
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• Length of a segment or arc

  The true length of one straight or curved piece of a run.

  θ = 4 · atan(b); r = c ÷ (2 · sin(θ ÷ 2)); length = r × |θ|

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• Area under an arc

  The area between an arc and its chord.

  area = r² × (θ − sin θ) ÷ 2

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• Area of an outline

  The area inside a closed outline by the shoelace formula, arcs included.

  A = ½ Σ (xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ) + Σ arc areas

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• Perimeter

  The length around a closed outline.

  P = Σ segment lengths, the closing segment included

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• Net area with cut-outs

  Gross area, cut-outs, net area and perimeter of an area mark.

  net = gross − Σ cut-outs

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• Centroid

  The center of area of an outline, where its label goes.

  C_x = Σ (xᵢ + xᵢ₊₁) · fᵢ ÷ 6A, fᵢ = xᵢ · yᵢ₊₁ − xᵢ₊₁ · yᵢ

  Takeoff lengths and areas geom.js

  • TakeoffCraft
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• 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
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• 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
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• 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
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• Knot span

  Which span of the knot vector a parameter falls in.

  i such that U[i] ≤ u < U[i + 1], by bisection

  B-spline basis functions nurbs/basis.js

  • ModelCraft
  • PartCraft
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• Basis functions

  The p + 1 basis functions that are not zero at a parameter.

  N(i, p)(u) by the Cox-de Boor recurrence (A2.2)

  B-spline basis functions nurbs/basis.js

  • ModelCraft
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• Basis function derivatives

  The basis functions and their derivatives up to a chosen order.

  N⁽ᵏ⁾(i, p)(u) for k = 0 … d (A2.3)

  B-spline basis functions nurbs/basis.js

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• Line and line

  The point where two lines cross.

  t = ((b₁ − a₁) × d₂) ÷ (d₁ × d₂); parallel when |d₁ × d₂| < 1e-12 × |d₁| × |d₂|

  Line, arc and circle intersections geom.js

  • CadCraft
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• Line and circle

  The points where a line meets a circle or arc.

  A·t² + B·t + C = 0; t = (−B ± √(B² − 4AC)) ÷ 2A

  Line, arc and circle intersections geom.js

  • CadCraft
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• Circle and circle

  The points where two circles or arcs meet.

  x = (d² + r₁² − r₂²) ÷ 2d; h = √(r₁² − x²)

  Line, arc and circle intersections geom.js

  • CadCraft
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• Is the point on the object

  Whether a computed crossing lies on the drawn part of each object.

  −tol ≤ t·L ≤ L + tol, with tol = 1e-7; arcs: the angle within 1e-7 of the sweep

  Line, arc and circle intersections geom.js

  • CadCraft
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• Plan extents and edge tolerance

  The outside box of the level's walls and how close to it a wall must lie to count as an outside edge.

  T = 1.2 × max(150 mm, thickest wall)

  Automatic plan dimensions autodim.js

  • BimCraft
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• Dimension line offset

  How far the dimension line sits from the two points it measures.

  offset = round((target − a)·n × 10) ÷ 10, n = (−dy, dx) ÷ |b − a|

  Automatic plan dimensions autodim.js

  • BimCraft
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• Reference station on a wall

  The saved position of a dimension's end along its wall, 0 at one end and 1 at the other.

  station = round(station × 10⁶) ÷ 10⁶

  Automatic plan dimensions autodim.js

  • BimCraft
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• Is this dimension already there

  Compares two references so the same dimension is not made twice.

  Automatic plan dimensions autodim.js

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• Measured segment along an axis

  Makes one dimension between two points, measured along X or along Y.

  Automatic plan dimensions autodim.js

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• Stops along an outside edge

  The wall ends, opening edges and meeting walls that a row of dimensions runs between.

  opening stations = offset ÷ L and (offset + width) ÷ L

  Automatic plan dimensions autodim.js

  • BimCraft
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• Inverse of a 4 × 4 matrix

  The inverse of the model's coordination matrix.

  M⁻¹ = adj(M) ÷ det(M)

  Issue viewpoints in BCF files bcf.js

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  • ModelCraft
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• Viewer point to model point

  A camera position or a point on a section plane in the model's own coordinates, in meters with Z up.

  (x, y, z) = C⁻¹ v; IFC = (x, −z, y)

  Issue viewpoints in BCF files bcf.js

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• Viewer direction to model direction

  A camera direction, up vector or section plane normal in the model's own axes.

  Issue viewpoints in BCF files bcf.js

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• Model point to viewer point

  A point read from an issue file, placed in the viewer.

  v = C (x, z, −y)

  Issue viewpoints in BCF files bcf.js

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• Model direction to viewer direction

  A direction read from an issue file, in the viewer's axes.

  Issue viewpoints in BCF files bcf.js

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• Numbers written to the file

  How each coordinate, direction, field of view and scale is written in a viewpoint.

  round(v × 10⁶) ÷ 10⁶

  Issue viewpoints in BCF files bcf.js

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• Numbers read from the file

  Reads the X, Y and Z of a point or direction.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
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• Camera read from a viewpoint

  The camera of an issue as BimCraft keeps it.

  Issue viewpoints in BCF files bcf.js

  • BimCraft
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• Which stated quantity is used

  Picks an element's volume, area or length from the quantities the file states.

  Element quantities for carbon and cost carbon-calc.js

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• Volume of a mesh

  The volume of an element from its triangles, used when the file states none.

  V = |Σ a·(b × c)| ÷ 6 × (unit ÷ 1000)³

  Element quantities for carbon and cost carbon-calc.js

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• Share of the volume per material

  How an element's volume is divided between the materials it is made of.

  layer share = tᵢ ÷ Σt; constituent share = fᵢ ÷ Σf; otherwise 1 ÷ n

  Element quantities for carbon and cost carbon-calc.js

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• Floor area of the building

  The floor area the carbon total is divided by when the person has not typed one.

  area = Σ space floor areas, else Σ floor slab areas

  Element quantities for carbon and cost carbon-calc.js

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• Quantities per material part

  The volume, area and length handed on for each material of an element.

  part volume = element volume × share

  Element quantities for carbon and cost carbon-calc.js

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• Middle of a room

  The point the interior elevation cuts through.

  middle = (min + max) ÷ 2 on each axis

  Interior elevations and their dimensions comforts-views.js

  • BimCraft
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• Frame of an interior elevation

  The width and height of the wall shown, and how deep the view looks.

  x₀, x₁ = min, max of p·right; depth from the room's middle to max(p·dir) + 50 mm

  Interior elevations and their dimensions comforts-views.js

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• Dimension values on elevations and sections

  The widths, heights and level spacings written on a view.

  width = x₁ − x₀; height = y₁ − y₀; level spacing = yᵢ₊₁ − yᵢ

  Interior elevations and their dimensions comforts-views.js

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• Where a door or window is tagged

  The plan point, the height range and the outward side of each opening.

  middle station = offset + width ÷ 2; top = sill + height

  Interior elevations and their dimensions comforts-views.js

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• Views placed at scale on the sheet

  Takes view millimeters to paper millimeters at the sheet's scale.

  k = 1 ÷ scale ratio; paper = origin + view × k

  Interior elevations and their dimensions comforts-views.js

  • BimCraft
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• Where a drag puts an item

  The position and turn saved when a placed item is dragged or turned by its grip.

  turn = round(atan2(dy, dx) × 180 ÷ π ÷ step) × step, step 15° or 1° with Shift

  Interior elevations and their dimensions comforts-ui.js

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• Room height on the interior elevations

  The floor-to-floor height the elevations are framed to, and the height of each tagged item.

  height = next level − this level, else the level's own height

  Interior elevations and their dimensions comforts-ui.js

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• Project origin and survey point

  Where the site's survey point is in the file's coordinates, with its elevation and map grid position.

  survey point = site placement origin × file unit in mm; elevation in m = RefElevation × unit ÷ 1000

  Coordinate readouts and survey point coords.js

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• Level a height is on

  Picks the level whose floor is at or just below a height.

  the highest level with elevation ≤ z + 1 mm

  Coordinate readouts and survey point coords.js

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• Foot and US survey foot

  The two sizes of foot a map's unit is compared with.

  ft = 0.3048 m; US ft = 1200 ÷ 3937 m

  Coordinate readouts and survey point coords.js

  • BimCraft
  Exact

• Map grid coordinates as text

  A point's eastings and northings in the map's own unit.

  value = meters ÷ map unit in meters, written with 3 decimals

  Coordinate readouts and survey point coords.js

  • BimCraft
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