Calculations

Calculations

• 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
  • BimCraft
  • DetailCraft
  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
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  • DetailCraft
  64-bit

• Deflected shape

  The members' deflected shape, enlarged so it can be seen.

  drawn point = point + scale × displacement; scale = 0.015 × model size ÷ largest displacement

  Analysis results drawn on the model frame/frame-draw.js

  • CadCraft
  • BimCraft
  • DetailCraft
  32-bit, display only

• Force diagram beside a member

  A filled diagram of one internal force drawn square to each member, with its largest value labeled.

  height = value × 0.08 × model size ÷ largest value

  Analysis results drawn on the model frame/frame-draw.js

  • CadCraft
  • BimCraft
  • DetailCraft
  32-bit, display only

• Plate contours

  Each plate as two triangles colored by its corner values, with the largest values labeled.

  color = ramp(|value| ÷ largest |value|); kN·m/m = N·mm/mm ÷ 1000

  Analysis results drawn on the model frame/plate-draw.js

  • CadCraft
  • BimCraft
  • DetailCraft
  32-bit, display only

• Points written to a file

  Each point's northing, easting and elevation as text.

  text = v.toFixed(3); 0 when |v| < 0.0005

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
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  64-bit

• Points read from a file

  Turns the northing, easting and elevation text of a row into numbers.

  x = Number(E); y = Number(N); z = Number(Z) or none

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
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  64-bit

• Difference between a design and an as-built point

  The offsets in x, y and height, the horizontal and 3D distance, and whether the point is within tolerance.

  dh = √(dx² + dy²); d3 = √(dh² + dz²); out when dh > tol or |dz| > tolZ

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
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  64-bit

• Matching points by nearness

  Pairs each design point with the closest unused as-built point within a radius.

  pairs sorted by horizontal distance, taken closest first; radius = 10 × tol by default

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
  • BimCraft
  64-bit

• Summary of the comparison

  The counts, the largest horizontal and height differences and the root mean square of the horizontal differences.

  rms = √(Σ dh² ÷ n)

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
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  64-bit

• Comparison table

  The rows of the comparison as text for the page and the CSV.

  text = v.toFixed(3)

  Layout point files and as-built comparison layout/layout-points.js

  • CadCraft
  • BimCraft
  64-bit

• Day of the year

  The day number of a date.

  N = days before the month + day (+1 after February in a leap year)

  Sun position for shadows m3d/sun.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  Exact

• Equation of time and declination

  How far sun time runs ahead of clock time, and the sun's angle above the equator, for the date.

  γ = 2π ÷ (365 or 366) × (N − 1 + (hour − 12) ÷ 24); eqtime = 229.18 × (0.000075 + 0.001868 cos γ − 0.032077 sin γ − 0.014615 cos 2γ − 0.040849 sin 2γ)

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Hour angle and altitude

  The sun's height above the horizon.

  tst = hour × 60 + eqtime + 4 × lon − 60 × tz; H = tst ÷ 4 − 180°; cos Z = sin φ sin δ + cos φ cos δ cos H; altitude = 90° − Z

  Sun position for shadows m3d/sun.js

  • CadCraft
  • BimCraft
  • ModelCraft
  • CivilCraft
  64-bit

• Azimuth

  The sun's compass direction, in degrees clockwise from north.

  azimuth = atan2(sin H, cos H sin φ − tan δ cos φ) + 180°, wrapped to 0…360

  Sun position for shadows m3d/sun.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  64-bit

• Direction toward the sun

  The unit vector shadows are cast from, in model axes (north is +Y, east is +X).

  dir = (sin a cos alt, cos a cos alt, sin alt)

  Sun position for shadows m3d/sun.js

  • CadCraft
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  • ModelCraft
  • CivilCraft
  64-bit

• Unit vector

  A vector scaled to length 1.

  n = a ÷ |a|; [0, 0, 0] when |a| ≤ 1e-15

  3D vectors, planes and transforms m3d/vec.js

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

• Polygon normal

  The unit normal of a polygon given by its corner points.

  N = Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁) taken per axis, then normalized

  3D vectors, planes and transforms m3d/vec.js

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

• Polygon area in 3D

  The area of a flat polygon in space.

  area = |Σ (pᵢ − pᵢ₊₁) × (pᵢ + pᵢ₊₁)| ÷ 2

  3D vectors, planes and transforms m3d/vec.js

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

• Plane through a point

  A plane as a unit normal and an offset, with the normal's sign made canonical.

  n·p = d; first component of n with |n| > 1e-9 is made positive

  3D vectors, planes and transforms m3d/vec.js

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

• Same plane test

  Whether two planes are the same plane.

  |n₁·n₂| > 1 − 1e-9 and |d₁ × sign(n₁·n₂) − d₂| ≤ 1e-6

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
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  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Closest points of a line and a segment

  The nearest points between a ray's line and an edge.

  den = (d·d)(u·u) − (d·u)²; t = (B·E − C·D) ÷ den; s = (A·E − B·D) ÷ den, clamped to [0, 1]

  3D vectors, planes and transforms m3d/vec.js

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

• Closest points of two lines

  The nearest points between two infinite lines, or nothing when they are parallel.

  den = A·C − B²; parallel when |den| < 1e-14 × A × C

  3D vectors, planes and transforms m3d/vec.js

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

• Ray and plane

  The point where a ray's line meets a plane.

  t = (d_plane − n·o) ÷ (n·dir); p = o + t × dir

  3D vectors, planes and transforms m3d/vec.js

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

• Ray and triangle

  The distance along a ray to a triangle, or nothing when it misses.

  Möller–Trumbore: det = e₁·(d × e₂); u, v, t from three scalar triple products ÷ det

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
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  • ModelCraft
  • TakeoffCraft
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  64-bit

• Rotation about an axis through a point

  The 4×4 matrix that turns geometry by an angle about an axis through a given point.

  M = T(about) × R(axis, θ) × T(−about), R from Rodrigues' formula

  3D vectors, planes and transforms m3d/vec.js

  • CadCraft
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  • ModelCraft
  • TakeoffCraft
  • CivilCraft
  64-bit

• Transform a point

  A point moved by a 4×4 matrix.

  p′ = (M × [x, y, z, 1]) ÷ w

  3D vectors, planes and transforms m3d/vec.js

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

• Inverse of a 4×4 matrix

  The matrix that undoes a transform, used to move points into a group's own coordinates.

  M⁻¹ = adj(M) ÷ det(M), with det from twelve 2×2 minors

  3D vectors, planes and transforms m3d/vec.js

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

• Unit constants

  The Boltzmann constant, Coulomb constant, acceleration factor and pressure factor for reduced and real units.

  real: kB = 0.0019872041 kcal/mol/K; ke = 332.0637; accel = 418.4; pressure = 68568.4 atm per kcal/mol/ų

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Complementary error function

  erfc(x), used for the real-space part of the Ewald sum.

  erfc(x) ≈ t × exp(−x² + P₉(t)), t = 1 ÷ (1 + |x| ÷ 2); 2 − that for x < 0

  Molecular dynamics molecular/md.js

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  64-bit

• Pair parameters and the cutoff shift

  σ and ε for each pair of atom types and the energy subtracted so the pair energy is zero at the cutoff.

  σᵢⱼ = (σᵢ + σⱼ) ÷ 2; εᵢⱼ = √(εᵢ εⱼ); shift = 4 εᵢⱼ (s² − s), s = (σᵢⱼ ÷ r_c)⁶

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Kinetic energy

  The kinetic energy of the system in its energy unit.

  KE = ½ Σ mᵢ (vx² + vy² + vz²) ÷ accel

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Temperature

  The instantaneous temperature from the kinetic energy.

  T = 2 KE ÷ (dof × kB), dof = max(1, 3N − 3 − constraints)

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Pair forces and energies

  The Lennard-Jones and real-space Coulomb force, energy and virial of one pair of atoms.

  U = 4ε(s⁶² − s⁶) − shift, s⁶ = (σ² ÷ r²)³; Coulomb: ke qᵢqⱼ erfc(αr) ÷ r; excluded pairs: −ke qᵢqⱼ erf(αr) ÷ r

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Bond stretch

  The energy, force and virial of each harmonic bond.

  U = ½ k (r − r₀)²; F = −k (r − r₀) ÷ r along the bond

  Molecular dynamics molecular/md.js

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  64-bit

• Bond angles

  The energy, forces and virial of each harmonic angle.

  θ = acos(a·b ÷ (|a||b|)); U = ½ k (θ − θ₀)²; dU/dcos = −k (θ − θ₀) ÷ sin θ

  Molecular dynamics molecular/md.js

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  64-bit

• Dihedral angles

  The energy and forces of each periodic dihedral.

  φ = atan2(|b₂| b₁·n₂, n₁·n₂); U = k_d (1 + cos(nφ − δ))

  Molecular dynamics molecular/md.js

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  64-bit

• Ewald reciprocal sum and self term

  The long-range part of the Coulomb energy and force.

  E = (2π ke ÷ V) Σ_k exp(−k² ÷ 4α²) ÷ k² × |S(k)|², |n|² ≤ kmax²; self = −ke α ÷ √π × Σ q²

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Pressure

  The virial pressure of the box.

  P = (2 KE + W) ÷ (3V) × unit factor

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Time stepping, thermostats and barostat

  Advances the system by velocity Verlet, with optional temperature and pressure coupling.

  v += ½ dt a; x += dt v; forces; v += ½ dt a; Berendsen λ = √(1 + dt÷τ (T₀÷T − 1)); μ = ∛(max(0.5, 1 − dt÷τ_p κ (P₀ − P)))

  Molecular dynamics molecular/md.js

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  64-bit

• Radial distribution function

  g(r): how many atoms sit at each distance, relative to an ideal gas of the same density.

  g(b) = count(b) ÷ (frames × shell × N_A × (N_B, or N_A − 1 for one type) ÷ V), shell = 4⁄3 π ((b+1)³ − b³) dr³

  Molecular dynamics molecular/md.js

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  64-bit

• Mean-square displacement and diffusion

  The mean-square displacement over time and the diffusion coefficient from its slope.

  D = (n Σtm − Σt Σm) ÷ (n Σt² − (Σt)²) ÷ 6 over the chosen part of the samples

  Molecular dynamics molecular/md.js

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• Mass density

  The density of the box in g/cm³ (real units).

  ρ = Σm ÷ 0.602214076 ÷ V (amu per ų to g/cm³)

  Molecular dynamics molecular/md.js

  • CadCraft
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  64-bit

• Fast Fourier transform

  The in-place radix-2 transform used on each axis of the charge mesh.

  Molecular dynamics molecular/pme.js

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  64-bit

• Cardinal B-spline

  The weight of a charge on a mesh point.

  M₂(u) = 1 − |u − 1|; Mₙ(u) = u ÷ (n−1) × Mₙ₋₁(u) + (n−u) ÷ (n−1) × Mₙ₋₁(u−1)

  Molecular dynamics molecular/pme.js

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  64-bit

• B-spline moduli

  The correction 1 ÷ |b(m)|² applied to each mesh wave vector.

  Molecular dynamics molecular/pme.js

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  64-bit

• Particle-mesh Ewald energy and forces

  The reciprocal Coulomb energy and the force on every charge from a mesh instead of the plain sum.

  E = ½ Σ_m G(m) |Q̂(m)|², G = ke ÷ (πV) × exp(−π² m² ÷ α²) ÷ m² × B(m)

  Molecular dynamics molecular/pme.js

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  64-bit

• Rigid bonds

  Moves atoms after a step so each constrained bond has its set length, and corrects the velocities.

  g = (d² − r²) ÷ (2 (1÷mᵢ + 1÷mⱼ) r·s); stop when |d² − r²| ≤ 2 × tol × d²

  Molecular dynamics molecular/pme.js

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• Mean and standard error of a run

  The mean of the samples and its standard error by block averaging.

  mean = Σx ÷ n; error = √(Σ(blockᵢ − m)² ÷ (nb − 1) ÷ nb), nb = min(5, n) blocks of ⌊n ÷ nb⌋ samples

  Molecular dynamics molecular/props.js

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  64-bit