Calculations

Calculations

• Slab reinforcement and shear

  The steel required and provided per meter in each layer, and the one-way shear check of a slab.

  d = h − cover − bar ÷ 2 (outer layer); provided = (π d_b² ÷ 4) × 1000 ÷ s; need = max(required, ρ_min × 1000 × h); ratio = need ÷ provided

  Concrete member and slab checks on a solved frame frame/slab-design.js

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• Concrete stiffness for the analysis

  The elastic modulus, shear modulus and density given to concrete members.

  E = 4700 √f′c; G = E ÷ 2.4; ρ = 2400 kg/m³

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Cracked stiffness factor

  The share of the gross stiffness used for cracked members.

  0.70 columns; 0.35 beams; 1 when not cracked

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Area of a bar

  The cross-section area of a reinforcing bar from its diameter.

  A = π d² ÷ 4

  Concrete member and slab checks on a solved frame frame/frame-concrete.js

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• Approximate period

  The building's approximate fundamental period from its height and structural system.

  Ta = Ct × (hn ÷ 1000)^x, hn in mm

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Upper limit factor on the period

  The factor Cu that caps a period taken from analysis.

  table of SD1 against Cu, straight lines between rows

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Seismic response coefficient

  The coefficient Cs that multiplies the building's weight to give the base shear.

  Cs = SDS ÷ (R ÷ Ie), at most SD1 ÷ (T R ÷ Ie) or SD1 TL ÷ (T² R ÷ Ie), at least max(0.044 SDS Ie, 0.01) and 0.5 S1 ÷ (R ÷ Ie) where S1 ≥ 0.6

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Height exponent

  The exponent k of the vertical distribution of the story forces.

  k = 1 for T ≤ 0.5 s; 2 for T ≥ 2.5 s; 1 + (T − 0.5) ÷ 2 between

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Base shear and story forces

  The base shear, the force on each floor and the shear in each story.

  V = Cs W; Cvx = wx hx^k ÷ Σ wi hi^k; Fx = Cvx V; Vx = Σ F from the top down

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Story force on a rigid floor

  Puts a story force on a floor at its master point, with the accidental torsion.

  Mz = −((x₀ − x_m) Fy − (y₀ − y_m) Fx) ± 0.05 × plan size × F

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Design drift against the allowable

  The design story drift and its ratio to the allowable drift.

  δ = Cd δ_e ÷ Ie; allowed = 0.020, 0.015 or 0.010 × story height; ratio = δ ÷ allowed

  Seismic loads and story drift to ASCE 7-22 frame/loads-asce7.js

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• Suggested floor levels

  The heights at which level members meet, offered as the building's floors.

  z = (z_a + z_b) ÷ 2 of a level member; levels within 1 mm are one; stored z rounded to 0.1 mm

  Seismic loads and story drift to ASCE 7-22 frame/frame-concrete.js

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• Floor weights into newtons

  Turns the floor weights typed in kN into the N the procedure works in.

  w (N) = w (kN) × 1000

  Seismic loads and story drift to ASCE 7-22 frame/frame-concrete.js

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• Period of the analyzed frame

  The fundamental period in each direction from the floor weights and the frame's displacements under the seismic forces.

  T = 2π √(Σ wᵢ δᵢ² ÷ (g Σ fᵢ δᵢ))

  Seismic loads and story drift to ASCE 7-22 frame/frame-concrete.js

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• Weight of each floor from the loads

  The vertical load of chosen load cases gathered at each floor, as its seismic weight.

  node load = node forces + member equivalent loads + plate corner loads; each node's load goes to the floor at or above it

  Seismic loads and story drift to ASCE 7-22 frame/frame-concrete.js

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Table values in millimeters

  Converts each published row to millimeters, mm² and kg/m.

  mm = in × 25.4; mm² = in² × 25.4 × 25.4; kg/m = lb/ft × 1.48816394; r = (k_des − t_f) × 25.4

  Steel section dimensions, areas and outlines shapes/sections.js

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• Exact section area

  The cross-section area worked out from the dimensions, with fillets and rounded corners included.

  I: 2·b_f·t_f + (d − 2t_f)·t_w + (4 − π)·r²; RHS: 2t(h + b − 2t) − (4 − π)(r_o² − r_i²); CHS: π(D − t)·t

  Steel section dimensions, areas and outlines shapes/sections.js

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• Section outline

  The outline of a section as loops of points, centered on its bounding box.

  arc point = (cx + r·cos a, cy + r·sin a), a = a₀ + (a₁ − a₀)·i ÷ n

  Steel section dimensions, areas and outlines shapes/sections.js

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• Rounded rectangle loop

  The outer or inner loop of a rectangular hollow section with rounded corners.

  Steel section dimensions, areas and outlines shapes/sections.js

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• Loop area

  The signed area of an outline loop.

  A = ½ × Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

  Steel section dimensions, areas and outlines shapes/sections.js

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• Loop perimeter

  The length around an outline loop.

  L = Σ √(Δx² + Δy²)

  Steel section dimensions, areas and outlines shapes/sections.js

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• Parameter levels

  The evenly spaced values of a parameter between its two bounds.

  vᵢ = from + (to − from) × i ÷ (steps − 1)

  Design study variants and ranking study/study.js

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• Rounding of parameter values

  Rounds a parameter value before it is put in a variant.

  v → v to 12 significant digits

  Design study variants and ranking study/study.js

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• Variant list

  Every combination of parameter levels, or a seeded random sample between the bounds.

  random: v = from + (to − from) × u, u = 32-bit integer ÷ 2³²

  Design study variants and ranking study/study.js

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• Meeting the limits

  Whether a variant's results meet every limit of the study.

  min goal: value ≤ limit; max goal: value ≥ limit

  Design study variants and ranking study/study.js

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• Trade-off front

  The variants that no other variant beats on every goal at once.

  a dominates b when a is no worse on every goal and better on at least one

  Design study variants and ranking study/study.js

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• Weighted ranking

  A score for each feasible variant from its goals and the weights a person gives.

  x = (value − lo) ÷ (hi − lo); score = Σ w × (x for max goals, 1 − x for min goals)

  Design study variants and ranking study/study.js

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• Load factor to the limit

  The factor by which every load can be multiplied before the design's utilization reaches 1.

  find λ with util(λ) = 1: double λ until util ≥ 1, then bisect with mid = √(lo × hi) until hi − lo ≤ 10⁻⁴ × hi

  Load reserve and escape route checks study/breakit.js

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• Escape route widths

  For each room, the route out whose narrowest door is widest, and its margin against the required width.

  width of a route = min of its door widths; margin = width − required

  Load reserve and escape route checks study/breakit.js

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• Part reserve from a stress study

  The load factor to first yield and to buckling of a solved part.

  λ = minimum safety factor ÷ wanted safety factor; utilization = 1 ÷ λ

  Load reserve and escape route checks study/breakit.js

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• Matrix product

  Combines two transforms into one.

  o[c][r] = Σₖ a[k][r] × b[c][k]

  USD transforms and placement usd/geom.js

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• Matrix inverse

  The transform that undoes another.

  inverse = cofactor matrix ÷ determinant

  USD transforms and placement usd/geom.js

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• Transforming a point

  Applies a transform to a point.

  p′ = (M × [x, y, z, 1]) ÷ w

  USD transforms and placement usd/geom.js

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• Rotation about an axis

  The matrix of a rotation by an angle in degrees about X, Y or Z.

  a = deg × π ÷ 180; entries cos a, ± sin a

  USD transforms and placement usd/geom.js

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• Quaternion to rotation matrix

  The rotation matrix of an orientation stored as a quaternion.

  q ÷ |q|; R₀₀ = 1 − 2(y² + z²), R₀₁ = 2(xy + wz), …

  USD transforms and placement usd/geom.js

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• Value at a time

  An animated value at a given time, between its stored samples.

  v = a + (b − a) × (t − t₀) ÷ (t₁ − t₀)

  USD transforms and placement usd/geom.js

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• An object's own transform

  Builds the transform of one object from its list of translate, rotate, scale and matrix operations.

  M = op₁ × op₂ × … in the listed order; an inverted operation uses its inverse

  USD transforms and placement usd/geom.js

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