Calculations

Calculations

• Tension resistance

  The design tension resistance of the gross section, and of the net section when one is given.

  Nt,Rd = min(A fy ÷ γM0, 0.9 A_net fu ÷ γM2)

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Shear resistance along the web

  The plastic shear resistance, and the shear buckling resistance of a slender web.

  Vpl,Rd = Av fy ÷ (√3 γM0); λ̄w = hw ÷ (86.4 tw ε); χw = η or 0.83 ÷ λ̄w

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Bending resistance with high shear

  The major-axis bending resistance reduced when the shear exceeds half the plastic shear resistance.

  ρ = (2 V_Ed ÷ Vpl,Rd − 1)²; W = Wpl,y − ρ Aw² ÷ (4 tw)

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Bending and axial force on the cross-section

  The utilization of the cross-section under axial force and bending about both axes.

  class 1–2: (My ÷ M_N,y)² + (Mz ÷ M_N,z)^β, β = max(5n, 1); class 3: σ ÷ (fy ÷ γM0)

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Buckling reduction factor

  The reduction factor χ for a slenderness and a buckling curve.

  Φ = 0.5 (1 + α (λ̄ − 0.2) + λ̄²); χ = 1 ÷ (Φ + √(Φ² − λ̄²)) ≤ 1

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Buckling resistance in compression

  The design buckling resistance for flexural buckling about both axes and torsional buckling.

  Ncr = π² E I ÷ L²; λ̄ = √(A fy ÷ Ncr); Nb,Rd = χ A fy ÷ γM1

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Lateral-torsional buckling resistance

  The design bending resistance of an unrestrained segment.

  Mcr = C1 (π² E Iz ÷ L²) √(Iw ÷ Iz + L² G It ÷ (π² E Iz)); λ̄LT = √(W fy ÷ Mcr); Mb,Rd = χLT W fy ÷ γM1

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Bending and compression on the member

  The left sides of the two member interaction equations (6.61) and (6.62).

  N ÷ (χy N_Rk ÷ γM1) + kyy My ÷ (χLT My,Rk ÷ γM1) + kyz Mz ÷ (Mz,Rk ÷ γM1), and the same with χz, kzy, kzz

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Equivalent uniform moment factor

  The factor Cm of Table B.3 read from a segment's moment diagram.

  straight diagram: max(0.6 + 0.4 ψ, 0.4); loaded span: 0.95 + 0.05 αh or 0.90 + 0.10 αh

  Steel member resistance to EN 1993-1-1 frame/frame-design-en.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Yield and tensile strength of the steel

  The fy and fu used for a member from its grade and thickness.

  table value; EN: the lower pair when the thickest part is over 40 mm

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  Exact

• A diagram value between stations

  Reads a force or moment at any distance along the member from the station values.

  a = aₖ + (x − xₖ) ÷ (xₖ₊₁ − xₖ) × (aₖ₊₁ − aₖ)

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• The diagram on one unbraced segment

  The stations inside a segment plus interpolated values at its two ends.

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Quarter-point moments of a segment

  The largest moment of a segment and the moments at its quarter, middle and three-quarter points.

  |M| at x₀ + 0.25 L, 0.5 L, 0.75 L; max |M| over the stations

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Buckling lengths and segments of a member

  The unbraced length, the number of segments and the effective lengths used in the checks.

  lb = min(L, override or ratio × L); segments = ⌈L ÷ lb − 1e-9⌉; Lcy = ky L; Lcz = kz lb; Lt = lb

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Demands and ratios, AISC 360

  Collects each check's demand, capacity and ratio for one member and combination.

  ratio = demand ÷ capacity; demand = largest station value

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Demands and ratios, EN 1993

  Collects each check's demand, resistance and ratio for one member and combination.

  ratio = demand ÷ resistance; demand = largest station value

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Elastic moduli used in the checks

  Chooses E and G for the capacity formulas.

  EN: E = 210 000, G = 81 000 MPa; AISC: the model's steel

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Deflection against the span limit

  The largest deflection off the chord over the service combinations, against L ÷ n.

  ratio = deflection ÷ (L ÷ limit)

  Steel member checks on a solved frame frame/frame-design.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• A combination with its loads scaled

  Makes a copy of the model whose governing combination has every factor multiplied by a load factor.

  f′ = f × λ

  Steel member checks on a solved frame frame/frame-breakit.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Load factor at first failure

  The factor on the governing combination at which the first member reaches a utilization of 1.

  largest utilization(λ) = 1, found by solving the frame again for each λ

  Steel member checks on a solved frame frame/frame-breakit.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Connection utilization

  The shear and axial force at a member end against the resistance the app states for the joint.

  V = √(Vy² + Vz²); utilization = max(V ÷ shear, |N| ÷ axial)

  Steel member checks on a solved frame frame/frame-breakit.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Shear factor and Gauss points

  The shear correction factor and the position of the integration points.

  k = 5 ÷ 6; ξ, η = ±1 ÷ √3

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Jacobian of the element

  The map between the element's own square coordinates and the plate's plane, its determinant and inverse.

  det J = x,ξ y,η − y,ξ x,η; J⁻¹ = adj(J) ÷ det J

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Plate axes, warp and area

  The plate's own x, y and normal, its corners in that plane, how far the corners leave the plane, and its area.

  n = unit(d₁ × d₂) of the diagonals; warp = max |z| of the corners from the mean plane; area = ½ |Σ (xᵢ yᵢ₊₁ − xᵢ₊₁ yᵢ)|

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Plane-stress material matrix

  The matrix that turns strains into forces per length, or curvatures into moments per length.

  D = f ÷ (1 − ν²) × [[1, ν, 0], [ν, 1, 0], [0, 0, (1 − ν) ÷ 2]]

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Element stiffness

  The 24 × 24 stiffness of one plate in its own axes and in the model's.

  K = ∫ Bᵀ D B dA by 2 × 2 Gauss; D_b = E t³ ÷ (12 (1 − ν²)); D_s = 5⁄6 G t; membrane: k_cc − k_ci k_ii⁻¹ k_ic

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Transverse shear at the tying points

  The shear strain rows at the four edge midpoints that the element interpolates from.

  γ̃_ξ at (0, ∓1); γ̃_η at (∓1, 0)

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Shear strains at a point

  The two transverse shear strains at an integration point from the tied values.

  [γ_x, γ_y] = J⁻¹ [γ̃_ξ, γ̃_η]; γ̃ interpolated linearly between opposite edges

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• 4 × 4 inverse

  Inverts the stiffness block of the incompatible modes.

  Gauss–Jordan with partial pivoting

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Element stiffness in global axes

  Turns the plate's stiffness from its own axes to the model's.

  K_g = Tᵀ K T, by 3 × 3 blocks

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Forces and moments in a plate

  Membrane forces, bending and twisting moments and transverse shears per unit length at a point of a plate.

  N = D_m ε; M = −D_b κ; Q = D_s γ

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Gauss point values to the corners

  Corner values of a result from its four integration point values.

  bilinear through the 2 × 2 points, read at ξ, η = ±√3

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Corner shares of a plate's area

  The part of a uniform pressure or of the plate's weight that goes to each corner.

  ∫ Nᵢ dA by 2 × 2 Gauss

  Plate element for slabs and walls frame/plate.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Poisson's ratio of a plate's material

  The ratio the plate element uses, from the material's two moduli.

  ν = E ÷ (2G) − 1

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Master point of a rigid floor

  The point a floor's in-plan movement is measured at, and each node's offset from it.

  x_m = mean of the nodes' x; y_m = mean of the nodes' y (or the point given)

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Rigid floor constraint

  Ties a floor node's two in-plan movements and its rotation about the vertical to the floor's master point.

  u_x = U_x − (y − y_m) R_z; u_y = U_y + (x − x_m) R_z; r_z = R_z

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Skyline Cholesky factorization

  Factorizes the stiffness of the whole model once for every load set.

  K = L Lᵀ; L_ii = √(K_ii − Σ L_ik²)

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Forward and back substitution

  Solves the factorized system for one load vector.

  L y = f; Lᵀ x = y

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Loads on plates

  Turns a plate's own weight and area loads into forces at its corners.

  corner force = (∫ Nᵢ dA) × p; own weight p = −factor × f × ρ × 1e-9 × g × t

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Node displacements from the solution

  Gives every node its six displacements, including those tied to a floor's master point.

  u_q = Σ c × x_e over the constraint map of q

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Plate forces into the reactions

  Adds each plate's corner forces to the support reactions and stores the plate's center results.

  f = K_g u, added at the corners; center results at ξ = η = 0

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Plate results at the nodes

  The moments and forces shown at each node of a slab or wall.

  inside: mean of the corner values of the plates at the node; on an edge: a + b x + c y fitted through nearby Gauss values, read at the node

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Least-squares plane through Gauss values

  The value at a node on a slab's edge, from a plane fitted through the integration point values around it.

  normal equations 3 × 3, first unknown by Cramer's rule

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Story drift

  How far each rigid floor moves relative to the floor below, at its center and at its worst node.

  drift = move(floor) − move(floor below) at the same point; ratio = |drift| ÷ story height

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Combined plate results

  Adds plate results and floor movements of several load sets, each times its factor.

  result = Σ fᵢ × resultᵢ

  Frame analysis with slabs, walls and rigid floors frame/plate-solve.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Cleaning an outline

  Removes repeated points and corners that lie on a straight run, and makes the outline counter-clockwise.

  drop a point when |p − q| ≤ 1e-6; drop a corner when |cross| ≤ 1e-9 × |b − a|²

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Is the outline square-cornered

  Finds the turn that makes every edge of the outline and its openings run along one of two square directions.

  turn = atan2 of the longest edge; each edge: min(|dx|, |dy|) ≤ 1e-6 × max(1, length)

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Grid mesh

  Lays grid lines through every corner and given point and divides each gap into equal parts no longer than the size asked for.

  n = ⌈gap ÷ size − 1e-9⌉; line_j = a + (b − a) × j ÷ n

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Point inside a polygon

  Decides whether a grid cell's center lies inside the outline or inside an opening.

  even–odd crossing count

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit

• Mesh of a four-corner outline

  A structured mesh between the opposite edges of a convex four-corner outline.

  P(s, t) = (1 − s)(1 − t) P₀ + s (1 − t) P₁ + s t P₂ + (1 − s) t P₃; s = i ÷ n, t = j ÷ m

  Meshing a slab or wall into plates frame/plate-mesh.js

  • CadCraft
  • BimCraft
  • DetailCraft
  64-bit