Calculations

Calculations

• Polyline vertices read

  Reads the vertices of a lightweight polyline.

  DWG object values written and read dwg/objects-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• North direction of older geolocation data

  Turns the north angle of a version 1 geolocation object into a direction vector.

  north = (sin a, cos a)

  DWG object values written and read dwg/objects-read.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  • DetailCraft
  • PartCraft
  64-bit

• Fit point parameters and knots

  The parameter of each fit point along the spline, and the knot vector built from them.

  u₀ = 0; uᵢ = uᵢ₋₁ + |Qᵢ − Qᵢ₋₁|; knots = [0,0,0,0, u₁ … uₙ₋₂, L,L,L,L]

  Spline through fit points dwg/spline-fit.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Cubic basis functions at a parameter

  The four B-spline basis values that weigh the control points at one fit point.

  Cox–de Boor recurrence, degree 3

  Spline through fit points dwg/spline-fit.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Start condition

  The rows that pin the first control point and set the spline's start tangent or a natural start.

  P₀ = Q₀; P₁ − P₀ = T₀ × u₄ ÷ 3 or P₀ ÷ u₄ − P₁ (1 ÷ u₄ + 1 ÷ u₅) + P₂ ÷ u₅ = 0

  Spline through fit points dwg/spline-fit.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• End condition

  The rows that pin the last control point and set the spline's end tangent or a natural end.

  Pₘ₋₁ = Qₙ₋₁; Pₘ₋₁ − Pₘ₋₂ = T₁ × (L − uₘ₋₁) ÷ 3 or the zero second derivative row

  Spline through fit points dwg/spline-fit.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Solve for the control points

  Solves the banded system for the x, y and z of every control point.

  A · P = B

  Spline through fit points dwg/spline-fit.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• A load split onto the member's axes

  Turns one point or distributed load in a global or local direction into pieces along the member's own three axes.

  piece = load × (direction · local axis); a, b clamped to [0, L]

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Inverse of the released block

  Inverts the small stiffness block of a member's released end actions.

  Gauss–Jordan with partial pivoting

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Member stiffness with released ends

  Removes pinned or sliding end actions from a member's 12 × 12 stiffness.

  k_c = k_kk − k_kr · k_rr⁻¹ · k_rk

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Fixed-end loads with released ends

  Moves the load a released end cannot carry onto the member's other end actions.

  p_c = p_k − k_kr · k_rr⁻¹ · p_r

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Member stiffness in global axes

  Turns a member's stiffness from its own axes to the model's.

  k_g = Tᵀ · k · T, by 3 × 3 blocks

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Directions nothing stiffens

  Finds translations and rotation directions at a node that no member resists, so they can be held instead of stopping the run.

  translation: diagonal + spring ≤ 1e-14 × largest diagonal; rotation: eigenvalue ≤ 1e-10 × largest rotational diagonal

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Eigenvalues of a symmetric 3 × 3 matrix

  The principal stiffnesses and directions of the rotation block at a node.

  cyclic Jacobi rotations; t = sign(θ) ÷ (|θ| + √(θ² + 1))

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
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  64-bit

• Loads of a case or combination

  Adds up node forces and member loads of every case in a set, each times its factor, including the members' own weight.

  own weight w = −factor × f × ρ × 1e-9 × A × g (N/mm along −Z)

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Member end forces and reactions

  The six forces at each end of every member and the forces at the supports, from the solved displacements.

  f = k_c · u_local − p_c; reaction = Σ Tᵀ f − loads applied at the node

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• One step of iterative refinement

  Solves the leftover out-of-balance once more and adds the correction to the displacements.

  r = loads − member forces; x₂ = x + K⁻¹ r, kept if max |r₂| < max |r|

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Sum of the reactions

  The total force and moment of all support reactions, for the equilibrium check printed in the report.

  ΣF = Σ R; ΣM = Σ (M + r × R) about the global origin

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Sum of the applied loads

  The total force and moment of every load of a set about the global origin.

  ΣF = Σ F; ΣM = Σ (M + r × F); distributed loads by 4-point Gauss

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Linear combination of results

  Adds computed load sets, each times a factor, into one result.

  u = Σ fᵢ uᵢ, the same for end forces, reactions, loads and totals

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Cleaning an outline

  Removes repeated points and corners that lie on a straight run, and makes the outline counter-clockwise.

  drop a point when |p − q| ≤ 1e-6; drop a corner when |cross| ≤ 1e-9 × |b − a|²

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
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  64-bit

• Is the outline square-cornered

  Finds the turn that makes every edge of the outline and its openings run along one of two square directions.

  turn = atan2 of the longest edge; each edge: min(|dx|, |dy|) ≤ 1e-6 × max(1, length)

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Grid mesh

  Lays grid lines through every corner and given point and divides each gap into equal parts no longer than the size asked for.

  n = ⌈gap ÷ size − 1e-9⌉; line_j = a + (b − a) × j ÷ n

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Point inside a polygon

  Decides whether a grid cell's center lies inside the outline or inside an opening.

  even–odd crossing count

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Mesh of a four-corner outline

  A structured mesh between the opposite edges of a convex four-corner outline.

  P(s, t) = (1 − s)(1 − t) P₀ + s (1 − t) P₁ + s t P₂ + (1 − s) t P₃; s = i ÷ n, t = j ÷ m

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Cutting an outline into triangles

  Ear clipping of an outline that is neither square-cornered nor four-cornered.

  an ear: cross(a, b, c) > 1e-12 and no other corner inside

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Triangles into quadrilaterals

  Cuts each triangle into three quadrilaterals and each of those into k × k plates.

  k = ⌈longest edge ÷ 2 ÷ size − 1e-9⌉; nodes by bilinear blend

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Mesh quality

  The largest side ratio and the smallest corner angle of the plates.

  aspect = longest side ÷ shortest side; angle = acos(u·v ÷ (|u| |v|))

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Mesh nodes onto model nodes

  Uses an existing node of the frame for a mesh node that lies within the merge tolerance of it, else adds a new node.

  |existing − new| ≤ tol (1 mm by default)

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Wall plates

  Meshes a vertical wall along a line in plan between two heights.

  node = from + e_x × u, z = z₀ + v; e_x = (to − from) ÷ L

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Is a plate a proper quadrilateral

  Refuses a plate whose corners are in a line or folded over.

  each corner: (e₁ × e₂) · n ÷ |n| > 1e-3 × |e₁| × |e₂|

  Meshing a slab or wall into plates frame/plate-model.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Nodes of a rigid floor by height

  Collects the nodes that belong to a floor given by its height.

  |z_node − z_floor| ≤ tol (1 mm by default)

  Meshing a slab or wall into plates frame/plate-model.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• A number as report text

  Writes a value with a fixed number of decimals.

  text = v.toFixed(d); 0 when |v| < 0.5 × 10⁻ᵈ

  Numbers in the frame analysis report frame/frame-report.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Force in kN

  A force in newtons printed in kilonewtons.

  kN = N ÷ 1000, 2 decimals; kN·m = N·mm ÷ 10⁶, 2 decimals

  Numbers in the frame analysis report frame/frame-report.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Reactions, largest displacement and the envelope

  Gathers each set's reactions, the node that moves most, each member's extremes, and the largest value of each force over all sets.

  displacement = √(ux² + uy² + uz²); envelope = the value with the largest magnitude over the sets

  Numbers in the frame analysis report frame/frame-report.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Deflection as a span ratio

  Prints a member's deflection as L over a whole number.

  'L/' + round(L ÷ deflection)

  Numbers in the frame analysis report frame/frame-report.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Demand and capacity as text

  Prints a design check's demand and capacity in kN or kN·m.

  kN = round(N ÷ 10) ÷ 100; kN·m = round(N·mm ÷ 10⁴) ÷ 100; ratios to 3 decimals

  Numbers in the frame analysis report frame/frame-design.js

  • CadCraft
  • BimCraft
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  64-bit

• Utilization as a percentage

  Prints a utilization ratio as a percentage with one decimal.

  round(ratio × 1000) ÷ 10, 1 decimal

  Numbers in the frame analysis report frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• A number in the concrete and seismic tables

  Writes a value with a chosen number of decimals.

  round(v × 10ᵈ) ÷ 10ᵈ, then d decimals

  Numbers in the frame analysis report frame/frame-concrete.js

  • CadCraft
  • BimCraft
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  64-bit

• Values in the shared problem list

  The found and required values attached to a failed concrete check or drift.

  utilization rounded to 0.0001; drift rounded to 0.01 mm

  Numbers in the frame analysis report frame/frame-concrete.js

  • CadCraft
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  64-bit

• Slab and wall rows of the report

  For each slab or wall: its largest deflection, its largest positive and negative moments, twisting moment and shear.

  max and min of the plates' center values; Q = √(Qx² + Qy²); w = largest node displacement; kN·m/m = N·mm/mm ÷ 1000, 2 decimals

  Numbers in the frame analysis report frame/plate-draw.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Values in a slab problem

  The required and provided steel, or the shear and its strength, attached to a failed slab check.

  steel rounded to 0.1 mm²/m; shear to 0.1 kN/m

  Numbers in the frame analysis report frame/slab-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Add, subtract, multiply, divide

  The four arithmetic functions on integers and reals.

  all integers: 32-bit result; any real: 64-bit result; integer ÷ integer truncates toward zero

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• FIX: a real to an integer

  Drops the fraction of a number.

  int(trunc(x))

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• Square root, exponential, logarithm and power

  SQRT, EXP, LOG and EXPT.

  Math.sqrt, Math.exp, Math.log, Math.pow

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• EQUAL with a fuzz

  Whether two numbers or lists of numbers are equal within a given amount.

  |a − b| ≤ fuzz

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• RTOS: a number as text

  Formats a length in decimal or in the drawing's other length formats.

  decimal: v.toFixed(prec); 0 when |v| < 0.5 × 10⁻ᵖʳᵉᶜ

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• ANGTOS: an angle as text

  Formats an angle in degrees, degrees-minutes-seconds, grads, radians or surveyor's bearings.

  degrees = a × 180 ÷ π; grads = a × 200 ÷ π; minutes = (deg − whole) × 60

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• ANGTOF: text to an angle

  Reads an angle written in any of the angle formats and returns radians.

  degrees × π ÷ 180; d + m ÷ 60 + s ÷ 3600; grads × π ÷ 200

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit

• Unit factors for CVUNIT

  The factor of each unit to its base unit: meter, radian, kilogram or square meter.

  inch 0.0254; foot 0.3048; yard 0.9144; mile 1609.344; pound 0.45359237; sq inch 0.00064516; sq foot 0.09290304; acre 4046.8564224

  LISP arithmetic, conversions and geometry lisp/builtins.js

  • CadCraft
  64-bit