Calculations

Calculations

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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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  64-bit

• 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

  • CivilCraft
  Exact

• 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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• Rain depth of a storm

  The total depth of rain over the storm's duration.

  depth (mm) = Σ i × Δt ÷ 60

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

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• Design storm by alternating blocks

  Builds a storm from an intensity-duration curve with the heaviest block in the middle.

  depth_k = idf(k·Δt) × k·Δt ÷ 60 − depth_(k−1); i_k = depth_k × 60 ÷ Δt

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

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• Time step

  The length of each simulation step.

  dt = min(dt_max, α × Δx ÷ √(g × max(h_max, 0.01)))

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

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• Flow across a cell face

  The unit flow between two neighboring cells for one time step.

  q = (q − g·h_f·dt·Δη ÷ Δx) ÷ (1 + g·dt·n²·|q| ÷ h_f^(7/3)); |q| ≤ h_f·√(g·h_f)

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

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• Flow leaving at an open edge

  The unit flow out of the grid at an open edge or beside a hole in the ground.

  q = h^(5/3) × √S ÷ n, S = max(edge slope, bed slope to the cell behind); q ≤ h·√(g·h)

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

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• Keeping depths from going negative

  Scales the outflows of a cell so it cannot give away more water than it has.

  when Σ out × dt ÷ Δx > available: out ← out × available ÷ (Σ out × dt ÷ Δx)

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

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• New depth, deepest water, fastest flow and first-wet time

  Updates each cell's depth and records its greatest depth, greatest speed and the time it first got wet.

  h ← h + rain − losses + (dt ÷ Δx) × (q_W − q_E + q_S − q_N); speed = |q| ÷ h

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

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  32-bit in part

• Where the records are kept

  Sets up the arrays that hold the per-cell maximum depth, maximum speed and first-wet time.

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

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  32-bit in part

• Drains

  Takes water out of a drain's cell up to its capacity each step.

  take = min(capacity × dt, depth × cell area)

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

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  64-bit

• Volumes of rain, loss and outflow

  Adds each step's rain, losses and outflow to the running totals.

  out += Σ q × Δx × dt; rain += Σ depth of rain × cell area

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

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• Mass balance and the results

  The stored volume, the mass balance error, the pond volumes at the end and the peak outflow.

  error = (rain − loss − out − drained − stored) ÷ rain

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

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  64-bit

• Section prepared for the solver

  Splits a cross section at the banks, roughness changes and ineffective areas, and adds a point at each split.

  z(o) = z_p + (z_q − z_p) × (o − o_p) ÷ (o_q − o_p)

  River water surface profiles by the standard step method river-solve.js

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  64-bit

• Area, wetted perimeter, conveyance and velocity coefficient

  The hydraulic properties of a section at a water level.

  A = Σ (d_a + d_b) ÷ 2 × Δo; P = Σ hypot(Δo, Δz); K = (k ÷ n) × A × (A ÷ P)^(2/3); n_c = (Σ P_i·n_i^1.5 ÷ P)^(2/3); α = max(1, A²·Σ(K_i³ ÷ A_i²) ÷ K³)

  River water surface profiles by the standard step method river-solve.js

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• Root finding

  Solves the energy balance for the water level between two bracketing levels.

  stop when |bracket ÷ 2| ≤ 2ε|x| + tol ÷ 2, tol = 1e-6

  River water surface profiles by the standard step method river-solve.js

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• Minimum search

  Refines the level of least specific energy.

  River water surface profiles by the standard step method river-solve.js

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• Specific energy

  The energy level at a section for a water level and a flow.

  E = WS + α·Q² ÷ (2g·A²)

  River water surface profiles by the standard step method river-solve.js

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• Critical water surface

  The water level of least specific energy at a section.

  River water surface profiles by the standard step method river-solve.js

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• Normal depth boundary

  The water level at which the section carries the flow at a given energy slope.

  K(WS) = Q ÷ √S

  River water surface profiles by the standard step method river-solve.js

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  64-bit

• Velocity head and flow split

  The velocity head, energy level and the flow in the left overbank, channel and right overbank.

  h_v = α·Q² ÷ (2g·A²); Q_part = Q × K_part ÷ K

  River water surface profiles by the standard step method river-solve.js

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  64-bit

• Energy loss between two sections

  The friction loss and the contraction or expansion loss over a reach.

  L = (L_lob·Q_lob + L_ch·Q_ch + L_rob·Q_rob) ÷ Q; S_f = ((Q₁ + Q₂) ÷ (K₁ + K₂))²; h_e = L·S_f + C·|h_v2 − h_v1|

  River water surface profiles by the standard step method river-solve.js

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  64-bit

• Results at a section

  Depth, velocities, Froude numbers, energy slope and specific force for the results table.

  Fr = V ÷ √(g·A ÷ T); S_f = (Q ÷ K)²; specific force = Q² ÷ (g·A) + A·ȳ

  River water surface profiles by the standard step method river-solve.js

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  64-bit

• Subcritical run

  Steps upstream from the downstream boundary, solving each section's water level.

  River water surface profiles by the standard step method river-solve.js

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• Supercritical run

  Steps downstream from the upstream boundary, solving each section's water level below critical depth.

  River water surface profiles by the standard step method river-solve.js

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• Mixed regime

  Keeps, at each section, the subcritical or supercritical answer with the larger specific force, and marks hydraulic jumps.

  River water surface profiles by the standard step method river-solve.js

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• Decimals in the results table

  The number of decimals each result is shown with in the table, the text report and the CSV.

  River water surface profiles by the standard step method river-report.js

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• Table rows

  Formats each section's results for the table and the CSV.

  text = v.toFixed(d)

  River water surface profiles by the standard step method river-report.js

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• Water surface at a plan point

  The water level at any point from the results at the sections either side.

  rs = centerline length − distance along; WS = WS_a + (WS_b − WS_a) × (rs − rs_a) ÷ (rs_b − rs_a)

  Flood extent on the ground surface river-extent.js

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  64-bit

• Behind a levee

  Whether a point is kept dry by a levee the water does not top.

  offset and crest interpolated between the two sections; dry when beyond the levee and WS < crest

  Flood extent on the ground surface river-extent.js

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  64-bit