Calculations

Calculations

• One step of iterative refinement

  Solves the leftover out-of-balance once more and adds the correction to the displacements.

  r = loads − member forces; x₂ = x + K⁻¹ r, kept if max |r₂| < max |r|

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Sum of the reactions

  The total force and moment of all support reactions, for the equilibrium check printed in the report.

  ΣF = Σ R; ΣM = Σ (M + r × R) about the global origin

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Sum of the applied loads

  The total force and moment of every load of a set about the global origin.

  ΣF = Σ F; ΣM = Σ (M + r × F); distributed loads by 4-point Gauss

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Linear combination of results

  Adds computed load sets, each times a factor, into one result.

  u = Σ fᵢ uᵢ, the same for end forces, reactions, loads and totals

  Frame analysis: loads, releases, end forces and reactions frame/frame-solve.js

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• Internal forces at a cut

  N, Vy, Vz, T, My and Mz at a distance x from the member's start.

  S = −(F_i + ∫₀ˣ q); M = −(M_i + r × F_i + Σ r × q)

  Forces and deflection along a member frame/frame-solve.js

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• Deflection along the member

  The member's displacement between its ends, from its curvature.

  v″ = M_z ÷ (E I_z); w″ = −M_y ÷ (E I_y); u′ = N ÷ (E A); trapezoids, twice

  Forces and deflection along a member frame/frame-solve.js

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• Station values and deflection off the chord

  The values kept at each station, and how far the member bows from the straight line between its ends.

  rel = √(v_rel² + w_rel²); v_rel = v − v_end × x ÷ L

  Forces and deflection along a member frame/frame-solve.js

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• Largest and smallest values along a member

  The maximum and minimum of each internal force, where they occur, and the largest deflection with its span ratio.

  max and min over the stations; ratio = L ÷ deflection

  Forces and deflection along a member frame/frame-solve.js

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• Root fillet constants

  The area of one root fillet over r², and how far its centroid is from the corner over r.

  area ÷ r² = 1 − π ÷ 4; centroid ÷ r = 0.2234

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Section properties for the checks

  Plastic and elastic moduli, radii of gyration, the warping constant and the slenderness ratios of an I-shape.

  Zx = bf tf (d − tf) + tw hc² ÷ 4 + 4 A_f (hc ÷ 2 − 0.2234 r); Cw = Iy ho² ÷ 4; rts = √(√(Iy Cw) ÷ Sx)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Slender-element limits

  Decides whether a flange or web is too slender for the checks that are built.

  bf ÷ 2tf > 0.56 √(E ÷ Fy); h ÷ tw > 1.49 √(E ÷ Fy); bending: 1.0 √(E ÷ Fy), 3.76 √(E ÷ Fy)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Tension strength

  The design strength in tension: yielding of the gross section, and rupture when a net area is given.

  φPn = min(0.90 Fy Ag, 0.75 Fu Ae)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Compression strength

  The design strength in compression for flexural buckling about both axes and torsional buckling.

  Fe = π² E ÷ (KL ÷ r)²; Fcr = 0.658^(Fy ÷ Fe) Fy or 0.877 Fe; φPn = 0.9 Fcr Ag

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Moment gradient factor Cb

  The factor that raises the lateral-torsional buckling strength when the moment is not uniform.

  Cb = 12.5 Mmax ÷ (2.5 Mmax + 3 M_A + 4 M_B + 3 M_C)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Major-axis bending strength

  The design bending strength with lateral-torsional buckling and flange local buckling.

  Lp = 1.76 ry √(E ÷ Fy); Lr by (F2-6); Mn by (F2-1) to (F2-4); φMn = 0.9 Mn

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Minor-axis bending strength

  The design bending strength about the weak axis.

  Mp = min(Fy Zy, 1.6 Fy Sy); noncompact flange: straight line to 0.7 Fy Sy; φMn = 0.9 Mn

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Web shear coefficient

  The reduction of shear strength for a slender web or flange.

  Cv = 1; 1.10 √(kv E ÷ Fy) ÷ (h ÷ t); 1.51 kv E ÷ ((h ÷ t)² Fy)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Shear strength along the web

  The design shear strength in the direction of the web.

  φVn = φ 0.6 Fy d tw Cv; φ = 1.0 when h ÷ tw ≤ 2.24 √(E ÷ Fy), else 0.9

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Shear strength along the flanges

  The design shear strength in the direction of the flanges.

  φVn = 0.9 × 0.6 Fy (2 bf tf) Cv, kv = 1.2

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Bending and axial force together

  The utilization of a member under axial force and bending about both axes.

  Pr ÷ Pc ≥ 0.2: Pr ÷ Pc + 8⁄9 (Mrx ÷ Mcx + Mry ÷ Mcy); else Pr ÷ 2Pc + (Mrx ÷ Mcx + Mry ÷ Mcy)

  Steel member strength to AISC 360-16 frame/frame-design-aisc.js

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• Section properties for the checks

  Plastic and elastic moduli, shear areas, the warping constant and radii of gyration of an I or H section.

  Wpl,y = b tf (h − tf) + tw hc² ÷ 4 + 4 A_f (hc ÷ 2 − 0.2234 r); Avz = A − 2 b tf + (tw + 2r) tf; Iw = Iz (h − tf)² ÷ 4

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

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• Cross-section class

  The class of the flange and web in compression and in bending.

  ε = √(235 ÷ fy); flange c ÷ t ≤ 9ε, 10ε, 14ε; web ≤ 72ε, 83ε, 124ε (bending) or 33ε, 38ε, 42ε (compression)

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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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