Calculations

Calculations

• Pipe slope, cover and structure levels

  Length, slope and least cover of each pipe; rim and sump of each structure; and the list of rule problems.

  slope = (inv_up − inv_dn) ÷ L; crown(t) = inv_up + (inv_dn − inv_up)·t + D; cover = min over samples (ground − crown); sump = lowest invert − sump depth

  Gravity pipe network: slopes, cover and automatic inverts pipes.js

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• Automatic inverts

  Sets each pipe's upstream and downstream inverts from the ground, the minimum cover, slope and drop.

  up = min(ground − cover − D, lowest incoming invert − drop); dn = min(up − s·L, min over samples (up + (ground − cover − D − up) ÷ t))

  Gravity pipe network: slopes, cover and automatic inverts pipes.js

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• Rounding an invert down

  Rounds a computed invert down to two decimals.

  ⌊v × 100 + 1e-7⌋ ÷ 100

  Gravity pipe network: slopes, cover and automatic inverts pipes.js

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• Bench outlines, area and perimeter

  The closed outlines of the mined area at one level, each with its area and perimeter.

  point = origin + index × block size; area = Σ (x_i·y_j − x_j·y_i) ÷ 2; perimeter = Σ edge lengths

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Pit floor per column

  The bottom elevation of the lowest mined block in each column.

  floor = origin_z + k × dz

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Overall slope with a ramp

  The overall wall angle when every bench carries a ramp lane.

  OSA′ = atan(H ÷ (H ÷ tan(OSA) + w))

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Outline moved inward

  The bench outline moved inside by half the ramp width.

  vertex shift = (n₀ + n₁) × d ÷ (1 + n₀ · n₁)

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Haul ramp route, length and drop

  The ramp's 3D line down the pit, its length, its drop, and how many laps each bench needs.

  run = dz ÷ grade; z = z_top − dz × min(1, walked ÷ run); length = Σ 3D segment lengths; laps = run ÷ lane perimeter

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Pit cut into the ground and volume dug

  The ground surface with the pit cut in, and the volume between the ground and the pit floor.

  node z = min(ground, mean of the 4 columns round it); dug = Σ max(0, ground − floor) × dx × dy

  Open pit benches, haul ramp and pit surface pit-geom.js

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• Depth of cut and fill by cell

  The depth at every cell, the cut and fill volumes, and the deepest cut and highest fill.

  dz = design − ground at ((i + 0.5)·c, (j + 0.5)·c); fill = Σ dz·c² (dz > 0); cut = Σ −dz·c² (dz ≤ 0)

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Depth classes

  Equal steps of depth for shading the map, with a level band around zero.

  n = ⌈depth ÷ step − 1e-9⌉; level when |dz| < step ÷ 20

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Spot levels

  Ground level, design level and their difference on a grid, with the deepest cut and highest fill marked.

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Sloped pad as a plane

  The elevation of each pad corner on a plane through a point, rising in a direction at a grade.

  z = z₀ + grade × ((x − x₀)·cos θ + (y − y₀)·sin θ)

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Ramp strip

  The four corners of a ramp between two points, its length and its grade.

  L = hypot(Δx, Δy); grade = (z_b − z_a) ÷ L; corners = ends ± normal × width ÷ 2

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Free section

  Ground and design cut along a polyline, with cut and fill areas and the distance of each vertex.

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Stretches of a line inside the graded regions

  The distances along the section line where it runs inside a graded region.

  t = ((p − a) × e) ÷ (d × e); inside tested at the middle of each piece

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Section CSV

  Writes the section as distance, ground level and design level rows, then the cut and fill areas.

  value = round(v × 100) ÷ 100

  Cut and fill map, spot levels, sloped pads and free sections platform.js

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• Plot scales

  The list of scales and the millimeters on paper per project unit for each.

  mm per foot = 25.4 ÷ n (1 in = n ft); mm per meter = 1000 ÷ n (1:n)

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Scale that fits

  The largest standard scale at which the drawing's extents fit the sheet.

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Drawing window in project coordinates

  The part of the project that a sheet shows at a scale.

  half width = frame width ÷ 2 ÷ mm per unit; window = center ± half

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Project coordinates to paper

  Where each point of the plan lands on the sheet, in millimeters.

  x_mm = x·s + o_x, o_x = frame middle − c_x·s

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Scale bar length

  A round length about 50 mm long on paper for the scale bar.

  want = 50 ÷ s; length = largest of {1, 2, 5, 10} × 10^⌊log₁₀ want⌋ not over want

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Arcs as Bézier curves

  Turns a circular arc into cubic Bézier pieces for the PDF.

  n = ⌈|sweep| ÷ 90° − 1e-9⌉; d = sweep ÷ n; k = 4 ÷ 3 × tan(d ÷ 4); control = end ± k·r·tangent

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• A number in the PDF

  Formats every coordinate and width written to the page.

  text = v.toFixed(4) with trailing zeros removed

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Millimeters to PDF points

  The page transform that makes the content's units millimeters.

  72 ÷ 25.4 points per millimeter

  Plan sheets and plan-and-profile sheets as PDF plot.js

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• Stations per sheet

  The stretch of alignment each sheet shows and where each sheet starts and ends.

  fits = plan window width ÷ mm per unit; length = ⌊fits ÷ round⌋ × round; sheet k starts at start + k × length

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Design profile points on a sheet

  The stations at which the design profile is drawn within a sheet's range.

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Vertical exaggeration

  The largest standard exaggeration (10, 5, 4, 2 or 1) at which every sheet's elevations fit the profile window.

  fits when range × mm per unit × exaggeration ≤ available height; vertical scale = s × exaggeration

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Plan turned along the alignment

  Where each plan point lands on a plan-and-profile sheet, turned so the stretch runs left to right.

  q = R(−θ)·(p − p_a); x_mm = x₀ + q_x·s; y_mm = c_y + (q_y − y_mid)·s

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Profile grid and scales

  The station and elevation grid of the profile and the mapping from station and elevation to paper.

  X = x₀ + (sta − a)·s; Y = y₀ + (z − z₀)·s·exaggeration; z₀ = ⌊(z_mid − span ÷ 2) ÷ step⌋ × step

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Vertical scale label

  The vertical scale printed in the title strip.

  meters: 1:round(1000 ÷ s_v); feet: 1 in = (25.4 ÷ s_v) ft to 1 decimal

  Plan sheets and plan-and-profile sheets as PDF pp-sheets.js

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• Volumes done, remaining and percent complete

  For each survey: cut and fill done, what remains, over-cut, over-fill, stockpiles, digs and percent complete.

  cut done = Σ clamp(E − A, 0, E − D)·A_cell; fill done = Σ clamp(A − E, 0, D − E)·A_cell; percent = 100 × (cut done + fill done) ÷ planned

  Earthwork progress from repeat surveys progress.js

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• Forecast finish date

  The date the work reaches 100 percent at the pace of the last surveys.

  slope = Σ (t − t̄)(p − p̄) ÷ Σ (t − t̄)²; t_end = t_last + (100 − p_last) ÷ slope

  Earthwork progress from repeat surveys progress.js

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• Scheduled percent on a date

  The planned percent complete on a date, between the schedule's points.

  p = p₀ + (p₁ − p₀) × (t − t₀) ÷ (t₁ − t₀)

  Earthwork progress from repeat surveys progress.js

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• Gravity and unit constants

  The constants behind cant and unit conversion.

  g = 9.81; km/h per mph = 1.609344; meters per foot = 1200 ÷ 3937

  Railway cant, cant ramps and track quantities rail.js

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• Speed and length conversions

  Converts a speed to km/h, a project length to meters and millimeters to project units.

  km/h = mph × 1.609344; m = ft × 1200 ÷ 3937; project = mm ÷ 1000 (÷ meters per foot)

  Railway cant, cant ramps and track quantities rail.js

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• Equilibrium cant

  The cant at which a train at the design speed puts equal load on both rails.

  E = G × v² ÷ (g × R), v = V ÷ 3.6

  Railway cant, cant ramps and track quantities rail.js

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• Cant ramp shapes

  How the cant rises from 0 to full along a ramp for each transition kind.

  linear u; Bloss u²(3 − 2u); Helmert 2u² then 1 − 2(1 − u)²; cosine (1 − cos πu) ÷ 2; sine u − sin(2πu) ÷ 2π

  Railway cant, cant ramps and track quantities rail.js

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• Steepest slope of each ramp shape

  The factor that turns the average cant gradient of a ramp into its steepest gradient.

  linear 1; Bloss 1.5; Helmert 2; cosine π ÷ 2; sine 2

  Railway cant, cant ramps and track quantities rail.js

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• Rail heights from cant

  Each rail's height above the profile for an applied cant and a pivot rule.

  center: ∓E ÷ 2; inner rail fixed: 0 and E; low rail fixed: max(0, ±E)

  Railway cant, cant ramps and track quantities rail.js

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• Cant ramp gradient and rate

  The steepest cant gradient of each ramp in mm per meter and the rate of change of cant in mm per second.

  gradient = |E| ÷ length (m) × peak factor; rate = gradient × v; ramp on a straight: length = |E| ÷ gradient limit

  Railway cant, cant ramps and track quantities rail.js

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• Cant, deficiency and excess checks

  Compares the applied cant, the cant deficiency and the cant excess with the limits.

  deficiency = E_eq − E; excess = E − E_eq(slowest train); flagged when over the limit + 1e-9

  Railway cant, cant ramps and track quantities rail.js

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• Cant segments along the alignment

  The cant layout as constant and ramp segments, with each rail's height at both ends of each.

  Railway cant, cant ramps and track quantities rail.js

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• Cant at a station

  The applied cant and each rail's height at any station.

  u = (sta − ramp start) ÷ ramp length; E = cant × g(u) on the way in, cant × (1 − g(u)) on the way out

  Railway cant, cant ramps and track quantities rail.js

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• Track cross section

  The rails, sleeper, ballast and sub-ballast outline square to the alignment, and the ballast and sub-ballast areas.

  slope = (h_R − h_L) ÷ base; toe distance d = (z_shoulder − z₀ + |o|·fall) ÷ (1 ÷ ballast slope − fall)

  Railway cant, cant ramps and track quantities rail.js

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• Area of a section polygon

  The area of the ballast or sub-ballast outline in the cross section.

  A = Σ (o_i·z_j − o_j·z_i) ÷ 2

  Railway cant, cant ramps and track quantities rail.js

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• Track quantities

  Track and rail lengths, sleeper count and volume, and gross and net ballast and sub-ballast volumes.

  volume = Σ (A_i + A_(i+1)) ÷ 2 × Δsta; sleepers = ⌊length ÷ spacing + 1e-9⌋ + 1; net ballast = gross − sleepers × L × W × (depth − crib)

  Railway cant, cant ramps and track quantities rail.js

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• Gravity and the foot

  The constants the simulation uses.

  g = 9.80665 m/s²; 1 ft = 0.3048 m

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Grid from the surface

  Samples the ground surface into square cells in meters.

  n_x = round(W ÷ c), n_y = round(H ÷ c); c = max(W ÷ n_x, H ÷ n_y); z (m) = elevation × meters per unit

  Rain on the ground: overland flow, ponding and outflow rain.js

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• Low spots and what they hold

  Finds each depression, its area, the volume it holds before it spills, its greatest depth and its sill level.

  filled = max(ground, lowest way out); depth = filled − ground; volume = Σ depth × cell²

  Rain on the ground: overland flow, ponding and outflow rain.js

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