Calculations

Calculations

• 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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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• 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

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• Cutting an outline into triangles

  Ear clipping of an outline that is neither square-cornered nor four-cornered.

  an ear: cross(a, b, c) > 1e-12 and no other corner inside

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

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• Triangles into quadrilaterals

  Cuts each triangle into three quadrilaterals and each of those into k × k plates.

  k = ⌈longest edge ÷ 2 ÷ size − 1e-9⌉; nodes by bilinear blend

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

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• Mesh quality

  The largest side ratio and the smallest corner angle of the plates.

  aspect = longest side ÷ shortest side; angle = acos(u·v ÷ (|u| |v|))

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

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• Mesh nodes onto model nodes

  Uses an existing node of the frame for a mesh node that lies within the merge tolerance of it, else adds a new node.

  |existing − new| ≤ tol (1 mm by default)

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

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• Wall plates

  Meshes a vertical wall along a line in plan between two heights.

  node = from + e_x × u, z = z₀ + v; e_x = (to − from) ÷ L

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

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• Is a plate a proper quadrilateral

  Refuses a plate whose corners are in a line or folded over.

  each corner: (e₁ × e₂) · n ÷ |n| > 1e-3 × |e₁| × |e₂|

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

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• Nodes of a rigid floor by height

  Collects the nodes that belong to a floor given by its height.

  |z_node − z_floor| ≤ tol (1 mm by default)

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

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• Bar positions of a beam

  Places the top, bottom and side bars of a beam section from their count and cover.

  y_i = −b ÷ 2 + c + (b − 2c) × i ÷ (n − 1); side bars evenly between the rows

  Concrete section by strain compatibility frame/rc-section.js

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• Bar positions of a column cage

  Places the bars around the perimeter of a column, corners counted once.

  y_i = −y_c + 2 y_c × i ÷ (n − 1); y_c = b ÷ 2 − cover

  Concrete section by strain compatibility frame/rc-section.js

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• Bars gathered into layers

  The depth of each bar from the compression face, with bars at one depth added into one layer.

  d = depth ÷ 2 − sign × position, rounded to 1e-6 mm

  Concrete section by strain compatibility frame/rc-section.js

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• Axial force and moment for a neutral axis depth

  The section's axial force and its moment about mid-depth when the neutral axis is at depth c.

  a = min(β₁ c, h); C_c = α₁ f′c b a; ε_s = ε_cu (c − d) ÷ c; f_s = clamp(E_s ε_s, ±f_y); P = C_c + Σ F; M = Σ F × (h ÷ 2 − d)

  Concrete section by strain compatibility frame/rc-section.js

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• Neutral axis for a given axial force

  The neutral axis depth at which the section's axial force equals the target.

  bisection on c between 1e-6 × depth and 1000 × depth

  Concrete section by strain compatibility frame/rc-section.js

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• Stress block factor β₁

  The depth of the equivalent stress block as a share of the neutral axis depth.

  0.85 for f′c ≤ 28; 0.85 − 0.05 (f′c − 28) ÷ 7; 0.65 for f′c ≥ 55

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Strength reduction factor

  The factor φ for axial force and bending from the strain in the extreme tension steel.

  0.90 for ε_t ≥ ε_ty + 0.003; 0.65 (0.75 spiral) for ε_t ≤ ε_ty; a straight line between

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Square root of the concrete strength in shear

  The √f′c used in the shear formulas.

  min(√f′c, 8.3)

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Bending strength of a beam section

  The nominal and design moment of a section at zero axial force, with the minimum steel and ductility tests.

  c from P(c) = 0; Mn = M(c); φMn; As,min = max(0.25 √f′c ÷ fy, 1.4 ÷ fy) b d

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Steel required for a moment

  The steel area a singly reinforced section needs for a factored moment.

  Rn = Mu ÷ (φ b d²); As = (0.85 f′c ÷ fy) (1 − √(1 − 2 Rn ÷ 0.85 f′c)) b d

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• One-way shear strength

  The concrete and stirrup contributions to shear strength, the minimum stirrups and the spacing limit.

  Vc by Table 22.5.5.1 (a), (b) or (c); λs = min(1, √(2 ÷ (1 + 0.004 d))); Vs = Av fyt d ÷ s; φVn = 0.75 (Vc + Vs)

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Axial force and moment diagram

  The nominal and design strength outline of a column section for bending about one axis.

  (Mn, Pn) at 121 depths c from 20 h down to 0.001 h, plus the balanced, tension-controlled and P = 0 points; Po = 0.85 f′c (Ag − Ast) + fy Ast

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Utilization on the diagram

  How far a factored axial force and moment are along the ray from the origin to the diagram.

  ratio = √(Mu² + Pu²) ÷ t, t = where the ray meets the outline

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Threshold torsion

  The torque above which torsion would have to be designed for.

  φ Tth = 0.75 × 0.083 λ √f′c Acp² ÷ pcp

  Concrete member strength to ACI 318-19 frame/rc-aci318.js

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• Strengths of a section, worked out once

  The bending strengths for both axes and signs, the two strength diagrams, the effective shear depths and the threshold torsion of a section.

  shear depth d = centroid of the bars beyond mid-depth, or 0.8 × depth when there are none

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

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