Calculations

Calculations

• Stiffness matrix of a shear building

  Builds the tridiagonal floor stiffness matrix from one stiffness per story.

  K[i][i] = k[i] + k[i+1]; K[i][i+1] = K[i+1][i] = −k[i+1]

  Earthquake time-history of a building hazard.js

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• Periods, mode shapes and effective mass

  The natural periods, the mode shapes scaled to 1 at the top floor, the participation factor and the effective mass of each mode.

  S = K[i][j] ÷ √(mᵢ mⱼ); ω = √max(λ, 0); T = 2π ÷ ω; Γ = Σ m φ ÷ Σ m φ²; effective mass = (Σ m φ)² ÷ Σ m φ²

  Earthquake time-history of a building hazard.js

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• Story stiffness for a chosen first period

  The single story stiffness that gives the building the first period the person typed.

  k = (2π ÷ T₁)² ÷ ω₁², ω₁ = first frequency with k = 1

  Earthquake time-history of a building hazard.js

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• Made ground acceleration record

  A Ricker pulse or filtered noise record scaled to the peak ground acceleration asked for.

  Ricker: a = (1 − 2x²) × e^(−x²), x = π f (t − 1.5 ÷ f); then a × pga ÷ max|a|

  Earthquake time-history of a building hazard.js

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• Newmark time stepping

  The floor displacements, velocities and accelerations at every time step of the record.

  K* = K + c₁ M + c₂ C, c₁ = 1 ÷ (β Δt²), c₂ = γ ÷ (β Δt), β = 1/4, γ = 1/2; C = a₀ M + a₁ K

  Earthquake time-history of a building hazard.js

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• Peak drift, floor acceleration, base shear and roof displacement

  The largest values over the run that the person reads.

  drift = uᵢ − uᵢ₋₁; floor acceleration = ü + a_g; base shear = k₀ u₁ (shear building) or Σ m (ü + a_g) (full matrix)

  Earthquake time-history of a building hazard.js

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• Sway under static floor forces

  Floor displacements and story drifts under static sideways forces, used for the wind sway.

  shear building: driftᵢ = (Σ F above) ÷ kᵢ, u = running sum of drifts; full matrix: K u = F

  Earthquake time-history of a building hazard.js

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• Floor heights from the frame's nodes

  Groups the frame's nodes into floors by height and takes the lowest group as the base.

  nodes whose heights differ by ≤ tol join one group; floor height = mean of the group

  Earthquake time-history of a building hazard-frame.js

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• Inverse of the flexibility table

  Turns the floors' flexibility table into their stiffness matrix.

  Earthquake time-history of a building hazard-frame.js

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• Floor stiffness of a frame

  The stiffness the frame gives each floor sideways, and one stiffness per story for the drift check.

  F[i][j] = (mean sideways move of floor i under 1000 N on floor j) ÷ 1000 ÷ 1000; K = F⁻¹; k_story = V ÷ drift under forces ∝ height

  Earthquake time-history of a building hazard-frame.js

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• Floor masses from the frame

  The mass on each floor: half of each member's weight at each end, plus a load per square meter over the floor's plan box.

  member mass = ρ × A × L × 1e-9; floor area = (x extent ÷ 1000) × (y extent ÷ 1000); m = max(1, self + area × load)

  Earthquake time-history of a building hazard-frame.js

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• Reading a property value

  Turns an IFC property or quantity value into the number or text the index, the totals and the comparison use.

  number kept = round(x × 10⁶) ÷ 10⁶

  Quantity totals and schedules from IFC properties ifc-props.js

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• Rounding a total

  Cuts a sum to 12 significant digits so that binary noise such as 0.1 + 0.2 does not show.

  tidy(n) = Number(n.toPrecision(12))

  Quantity totals and schedules from IFC properties ifc-qto.js

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• Totals by group

  Counts elements and sums the chosen number fields for each combination of group values and for the whole selection.

  sum = Σ vᵢ over the elements with a number; counted = how many had one

  Quantity totals and schedules from IFC properties ifc-qto.js

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• Spaces schedule totals

  The total floor area and volume of the listed rooms.

  Quantity totals and schedules from IFC properties ifc-qto.js

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• Is a property value the same in both versions

  Decides whether a number or text property changed between two versions of a model.

  same when |a − b| ≤ 1e-9 × max(1, |a|, |b|)

  Quantity totals and schedules from IFC properties ifc-compare.js

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• Number shown in Properties

  How a property or quantity value from the file is written out in the Properties panel.

  shown = round(x × 10⁴) ÷ 10⁴

  Quantity totals and schedules from IFC properties ifc-model.js

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• Placement of an item relative to its story

  The location and X direction written to the file for a placed item.

  (lx, ly) = W⁻¹ × (x ÷ unit, y ÷ unit, story z); direction = W⁻¹ rotation × (cos r, sin r)

  Placing and pasting elements in an IFC model ifc-place.js

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• Position and rotation read back from a placement

  The x, y and rotation of each placed item as listed in the equipment schedule and used by copy and paste.

  x = W[12] × unit; y = W[13] × unit; rot = round(atan2(W[1], W[0]) in degrees × 1000) ÷ 1000

  Placing and pasting elements in an IFC model ifc-place.js

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• An item's body in the file's unit

  Writes a library item's mesh or box as IFC geometry in the file's length unit.

  coordinate in file units = millimeters ÷ (millimeters per file unit)

  Placing and pasting elements in an IFC model ifc-place.js

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• Plan outline as polylines

  Turns the lines, arcs, circles, ellipses and bulged polylines of an item's plan into the straight segments written as its footprint.

  arc: n = max(4, ⌈sweep ÷ 10°⌉) segments; bulge: θ = 4 atan(b), r = chord ÷ (2 sin(θ ÷ 2))

  Placing and pasting elements in an IFC model ifc-place.js

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• Viewer point to model millimeters and back

  Converts a point between the viewer's Y-up meters and the model's Z-up millimeters.

  model = (x × 1000, −z × 1000, y × 1000) of C⁻¹ × v; viewer = C × (x ÷ 1000, z ÷ 1000, −y ÷ 1000)

  Placing and pasting elements in an IFC model ifc-place.js

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• Move and turn of pasted elements

  The matrix that carries copied elements to where they are pasted, with a turn about a point and a change of story.

  M = T(about + d) × Rz(rot) × T(−about); dz = elevation of target story − elevation of source story

  Placing and pasting elements in an IFC model ifc-elements.js

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• Placement written for a pasted element

  The location, Z axis and X axis of the new placement, from its matrix.

  location = round to 6 decimals; axis = column ÷ its length, rounded to 9 decimals

  Placing and pasting elements in an IFC model ifc-elements.js

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• Is this mesh a straight prism

  Checks that the mesh is closed and consistently wound, and picks candidate directions for the prism's axis.

  eps = tol × bounding-box diagonal, tol = 1e-6

  Turning a face mesh back into an extrusion ifc-solids.js

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• Depth, frame and outline of the prism

  The prism's depth, its origin and axes, and its outline and holes in the cap's own two axes.

  h = v · n; depth = max h − min h; (u, v) = ((p − origin) · x, (p − origin) · y)

  Turning a face mesh back into an extrusion ifc-solids.js

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• Removing collinear outline points

  Drops outline corners that lie on the line between their neighbors.

  drop b when |(b − a) × (c − b)| ≤ tol × max(1, |c − a|)

  Turning a face mesh back into an extrusion ifc-solids.js

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• Volume of a closed mesh

  The signed volume of a closed triangle mesh, used to check a conversion.

  V = Σ A · (B × C) ÷ 6

  Turning a face mesh back into an extrusion ifc-solids.js

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• Area of the prism's outline

  The outline's area less its holes; times the depth it is the prism's volume.

  area = |shoelace(outer)| − Σ |shoelace(hole)|

  Turning a face mesh back into an extrusion ifc-solids.js

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• Writing the prism into the file

  The IFC lines of the outline, its placement and the extrusion depth.

  each number written as toFixed(6)

  Turning a face mesh back into an extrusion ifc-solids.js

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• Degrees from degrees, minutes, seconds and millionths

  Reads an IFC compound angle as decimal degrees.

  deg = d + m ÷ 60 + (s + μ ÷ 10⁶) ÷ 3600

  Where an IFC model point is on the earth ifc-geo.js

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• File point to map grid

  Easting, northing and height in meters of a point of the file.

  [E, N, H] = (matrix × p) × meters per map unit

  Where an IFC model point is on the earth ifc-geo.js

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• Longitude and latitude from the site's reference point

  An approximate WGS 84 position of a file point from its offset east and north of the site's point.

  w = √(1 − e² sin² φ); N = a ÷ w; M = a (1 − e²) ÷ w³; lon = lon₀ + east ÷ ((N + h) cos φ); lat = lat₀ + north ÷ (M + h)

  Where an IFC model point is on the earth ifc-geo.js

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• Degrees to degrees, minutes and seconds

  Formats a latitude or longitude with its hemisphere letter.

  d = ⌊|deg|⌋; m = ⌊(|deg| − d) × 60⌋; s = (|deg| − d − m ÷ 60) × 3600, shown with 2 decimals

  Where an IFC model point is on the earth ifc-geo.js

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• Viewer point to the file's own coordinates

  Takes a point picked in the viewer back to the file's Z-up coordinates in its own length unit.

  y = C⁻¹ × (place⁻¹ × p); file = (y_x, −y_z, y_y) ÷ meters per file unit

  Where an IFC model point is on the earth ifc-geo.js

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• Section of a run

  The outside diameter, wall and bend radius of a pipe from the size table, or the width and height of a duct, in millimeters.

  mm = inches × 25.4

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Run corners with the slope applied

  The corners of a run with its fall along the horizontal length from the first point.

  z = z₀ − slope × Σ horizontal leg length

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Bend radius of an elbow

  The center-line radius an elbow is built with.

  pipe: the table's radius; round duct: 1.5 × diameter; rectangular duct: its width for a turn in plan, its height for a turn up or down

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Elbow geometry at a corner

  The two tangent points, center and angle of the elbow where a run turns.

  θ = acos(d₁ · d₂); T = R × tan(θ ÷ 2); center = p₀ + n × R

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Where runs meet

  Finds run ends that touch another run's end or lie on another run's line, to make an elbow, reducer or tee there.

  end to end when |P − Q| ≤ max(tol_a, tol_b); tee when the foot of P on a leg is within the tolerance

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Elbow, reducer and tee allowances

  How much of each leg an elbow, a reducer or a tee takes.

  reducer length (duct) = max(150, max(|Δw|, |Δh|) ÷ 2 ÷ tan 15°); tee half length = max(25, 0.5 × branch + 0.25 × host)

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Straight pieces between fittings

  The start and end of every straight piece along each leg, after trims, tees, valves and weld cuts.

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Saving a run's definition

  The points, sizes, slope and flow kept in the run's property set.

  point = round to 3 decimals of a millimeter

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Length and surface written on each piece

  The length of a straight pipe or duct piece and a duct's outside surface, written as IFC quantities.

  L = |b − a|; perimeter = π w (round), π h + 2 (w − h) (oval), 2 (w + h) (rectangle); surface = perimeter × L

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Elbow written as a revolved section

  The axis and angle of the solid of revolution that an elbow is written as.

  axis at R to the side of the section; angle = θ

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Flat-oval outline

  The outline of a flat-oval duct with semicircular ends.

  r = H ÷ 2; c = W ÷ 2 − r; points at c + r cos a, r sin a

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Connecting ports that touch

  Joins a port to a port of another part that lies at the same place.

  connected when |p − q| < 0.6 mm

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Position along a run

  How far along a run a plan point lies, and how far off the line.

  t = clamp((p − a) · d, 0, L); s = Σ earlier legs + t

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Length of a run

  The length of a run's center line through its corners, with the slope applied.

  L = Σ |Vᵢ − Vᵢ₋₁|

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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• Pipe size for a flow and a velocity limit

  The smallest pipe of a material and class whose velocity stays at or under the limit.

  V = Q ÷ (π D² ÷ 4); first size with V ≤ V_max

  Pipe and duct runs: layout, lengths and sizing mep-model.js

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