biaxial(α,na)

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biaxial(α,na)

Calculation of biaxial demand (P, Mx, My) based on assumed responce

biaxial(α#,na#)
α# - orientation of neutral axis in degrees
na# - location of neutral axis as % rotated height in compression

variables c_outer_XY# (concrete outline) and c_rebar_XY# (rebar)
shall be defined prior to function call α# na# biaxial formulas used P# A Area Calculation Closing the polygon P#c P# P# 1 row stack eval : n# P#c rows eval : x#.i P#c i# 1 el eval : x#.i1 P#c i# 1 + 1 el eval : y#.i P#c i# 2 el eval : y#.i1 P#c i# 1 + 2 el eval : 12 / - (y#.i y#.i1 + (x#.i1 x#.i - (* i# 1 n# 1 - sum * eval 91 line : P# M.x 1st Moment of Area Closing the polygon P#c P# P# 1 row stack eval : n# P#c rows eval : x#.i P#c i# 1 el eval : x#.i1 P#c i# 1 + 1 el eval : y#.i P#c i# 2 el eval : y#.i1 P#c i# 1 + 2 el eval : 16 / - (x#.i1 x#.i - (y#.i 2^y#.i y#.i1 * + y#.i1 2^+ (* i# 1 n# 1 - sum * eval 91 line : P# M.y 1st Moment of Area Closing the polygon P#c P# P# 1 row stack eval : n# P#c rows eval : x#.i P#c i# 1 el eval : x#.i1 P#c i# 1 + 1 el eval : y#.i P#c i# 2 el eval : y#.i1 P#c i# 1 + 2 el eval : 16 / (y#.i1 y#.i - (x#.i 2^x#.i x#.i1 * + x#.i1 2^+ (* i# 1 n# 1 - sum * eval 91 line : θ Rot θ cos θ sin θ sin - θ cos 22 mat : 41 line check correctness of input na# na# 1000 random 100000 / Max : Locate Neutral Axis:Locate Neutral Axis rotate the section: c_outer_XY'# c_outer_XY# α# Rot * eval : rotate the rebar: c_rebar_temp# c_rebar_XY# 1 c_rebar_XY# rows range 12 range el eval : c_rebar_temp# c_rebar_temp# α# Rot * eval : c_rebar_XY'# c_rebar_temp# c_rebar_XY# 1 c_rebar_XY# rows range 3 c_rebar_XY# cols range el augment eval : locate Y' coordinate of NA: Y'.MIN# c_outer_XY'# 1 c_outer_XY'# rows range 2 el Min eval : Y'.MAX# c_outer_XY'# 1 c_outer_XY'# rows range 2 el Max eval : Y'.NA# Y'.MAX# na# Y'.MAX# Y'.MIN# - (* - eval : locate Y' coordinate of comp. extreme fiber: Y'.ext# Y'.MAX# eval : locate Y' coordinate of comp. block start: Y'.a# Y'.MAX# β.1 Y'.MAX# Y'.NA# - (* - eval : define rectangular stress block portion of the concrete section: #pgon c_outer_XY'# eval : #TLV Y'.a# eval : r #pgon rows eval : j 1 : k 1 r 1 - (range #pgon k 2 el #TLV ≥ #chunk j 1 el #pgon k 1 el eval : #chunk j 2 el #pgon k 2 el eval : j j 1 + : 31 line 0 if #pgon k 2 el #TLV - (#pgon k 1 + 2 el #TLV - (* 0 < #chunk j 1 el #pgon k 1 + 1 el #pgon k 1 el - (#TLV #pgon k 2 el - (* #pgon k 1 + 2 el #pgon k 2 el - / #pgon k 1 el + eval : #chunk j 2 el #TLV eval : j j 1 + : 31 line 0 if 21 line for c_outer_comp_XY'# #chunk : Calculate stress block properties (XY coordinates): c_outer_comp_XY# c_outer_comp_XY'# α# - Rot * eval : A.c_outer_comp# c_outer_comp_XY# A eval : xbar.c_outer_comp# c_outer_comp_XY# M.y A.c_outer_comp# / eval : ybar.c_outer_comp# c_outer_comp_XY# M.x A.c_outer_comp# / eval : summary of rebar force and moment: c_concr_ΣP# A.c_outer_comp# α.1 * φ.c * f'.c * eval : c_concr_ΣMx# c_concr_ΣP# ybar.c_outer_comp# * eval : c_concr_ΣMy# c_concr_ΣP# xbar.c_outer_comp# * eval : Calculate rebar properties: rebar coordinates: c_rebar_X# c_rebar_XY# 1 col eval : c_rebar_Y# c_rebar_XY# 2 col eval : rebar strain and stress: c_rebar_Y'# c_rebar_XY'# 2 col eval : c_rebar_ε c_rebar_Y'# Y'.NA# - Y'.ext# Y'.NA# - / ε.cu * eval : c_rebar_σ# φ.s f.y * c_rebar_ε sign * eval c_rebar_ε abs ε.sy ≥ φ.s c_rebar_ε * E.s * eval cases vectorize eval : rebar force: c_rebar_A# c_rebar_XY# 4 col eval : c_rebar_A# c_rebar_A# 1 n.plot /^mm 2^* vectorize eval : c_rebar_P# c_rebar_A# c_rebar_σ# * vectorize eval : rebar moment: c_rebar_Mx# c_rebar_P# c_rebar_Y# * vectorize eval : c_rebar_My# c_rebar_P# c_rebar_X# * vectorize eval : summary of rebar force and moment: c_rebar_ΣMx# c_rebar_Mx# sum eval : c_rebar_ΣMy# c_rebar_My# sum eval : c_rebar_ΣP# c_rebar_P# sum eval : Design Resistance - Pr, Mrx and Mry: P.rz# c_concr_ΣP# c_rebar_ΣP# + eval - : M.rx# c_concr_ΣMx# c_rebar_ΣMx# + eval - : M.ry# c_concr_ΣMy# c_rebar_ΣMy# + eval - : Output in kN and m; 1st column is α#, 2nd column is na#, 3rd is Pz, 4th is Mx, 5th is My α# ° / na# % / P.rz# kN / M.rx# kN m * / M.ry# kN m * / 15 mat eval 601 line :

Functions

Material Properties & Cover

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Concrete:

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f'.c 35 MPa * :

γ.c 2350 :

γ.c 9.814 1000 / * kN m 3^/ * kN m 3^/ 23.063

w.c 24 kN m 3^/ * :

E.c 3000 f'.c MPa / sqrt * 6900 + (γ.c 2300 / (1.5^* MPa * : GPa 25.4563

f.cr 0.4 f'.c MPa * sqrt * 3.2 MPa * Min : MPa 2.366

α.1 0.85 0.0015 f'.c * 1 MPa / * - : 0.798

β.1 0.97 0.0025 f'.c * 1 MPa / * - : 0.883

α.c_ΔT 10106 -^* :

coefficient of thermal expansion

Concrete cover to reinforcing steel unless noted otherwise:

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Cover for concrete cast in contact with soil:

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cover.0 60 mm * :

cover.1 100 mm * :

Concrete failure strain:

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Steel yield strain:

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ε.cu 0.0035 :

ε.sy 0.002 :

Steel:

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f.y 400 MPa * :

E.s 200 GPa * :

f.pu 1860 MPa * :

E.p 200 GPa * :

A.s15 200 mm 2^* :

A.s20 300 mm 2^* :

A.s25 500 mm 2^* :

A.s30 700 mm 2^* :

A.s35 1000 mm 2^* :

A.s45 1500 mm 2^* :

A.s55 2500 mm 2^* :

d.b15 16 mm * :

d.b20 19.5 mm * :

d.b25 25.2 mm * :

d.b30 29.9 mm * :

d.b35 35.7 mm * :

d.b45 43.7 mm * :

d.b55 56.4 mm * :

Resistance Factors:

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φ.c 0.75 :

φ.s 0.9 :

matrix [x1, x2 | y1, y2] - start and end coordinates
no - number of points
format - plotting type for 2D plot, say "."
size - font size of a point
color - color, say "red" xy12# no# format# size# color# pts.line pts# no# 5 matrix : x1# x2# y1# y2# 22 mat xy12# : dx# x2# x1# - no# 1 - / : dy# y2# y1# - no# 1 - / : xi# x1# : yi# y1# : i 1 no# range pts# i 1 el xi# : pts# i 2 el yi# : pts# i 3 el format# : pts# i 4 el size# : pts# i 5 el color# : xi# xi# dx# + : yi# yi# dy# + : 71 line for pts# 81 line :

matrice di rotazione; rotazione rispetto all'origine degli assi cartesiani.
 θ: angolo di rotazione antiorario. rotation matrix; rotatation about the origin of the Cartesian coordinate system.
 θ: anticlockwise rotation. θ Rot θ cos θ sin θ sin - θ cos 22 mat :

rettangolo.
 B: larghezza;
 H: altezza;
 θ: angolo di rotazione antiorario. rectangle.
 B: width;
 H: height;
 θ: anticlockwise rotation. B H θ rectangle 00 B 0 B H 0 H 0052 mat θ Rot * :

trasla un poligono.
 P: matrice del poligono;
 u.x: spostamento in x;
 u.y: spostamento in y. geometric translation of a polygon.
 P: polygon matrix;
 u.x: x translation;
 u.y: y translation. P u.x u.y Translate use homogeneous coordinates #P P 1 P rows 1 matrix + (augment : #P 100010 u.x u.y 133 mat * 1 P rows 12 submatrix 31 line :

ruota un poligono rispetto all'origine degli assi cartesiani.
 P: matrice del poligono;
 θ: angolo di rotazione antiorario. rotating a polygon about the origin of the Cartesian coordinate system.
 P: polygon matrix;
 θ: anticlockwise rotation. P θ Rotate use homogeneous coordinates #P P 1 P rows 1 matrix + (augment : #P θ cos θ sin - 0 θ sin θ cos 000133 mat * 1 P rows 12 submatrix 31 line :

Sign convention:

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Cross-section of column is a z-plane. Positive x and y are to the right and upward. Positive moments cause compression at positive x locations and positive y locations. Compression is positive

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Define section geometry assuming section centroid at the zero coordinates:

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a# 2000 mm * :

b# 1350 mm * :

c# 80 mm * :

n.plot 0.4 :

(power of size scaling for plotting rebar)

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c_outer_XY# a# b# 0 rectangle a# 2 / - b# 2 / - Translate 11 sys eval : m - 0.675 m * - m 0.675 m * - m 0.675 m * m - 0.675 m * m - 0.675 m * - 52 mat

c_rebar_XY# a# 2 / c# - (- a# 2 / c# - (b# 2 / c# - (b# 2 / c# - (22 mat 2 . A.s35 mm 2^/ (n.plot^red pts.line a# 2 / c# - (- a# 2 / c# - (b# 2 / c# - (- b# 2 / c# - (- 22 mat 2 . A.s35 mm 2^/ (n.plot^red pts.line 21 sys eval :

c_rebar_XY# c_rebar_XY# sys2mat eval :

n.rebar_entries# c_rebar_XY# rows eval :

i## 1 n.rebar_entries# range i## 1 ≡ c_rebar_temp# c_rebar_XY# i## el eval : c_rebar_temp# c_rebar_temp# c_rebar_XY# i## el stack eval : if for

c_rebar_XY# c_rebar_temp# eval :

0 ° * 7.3 % * biaxial 0 7.3 3158.51 - 2869.08 - 015 mat

Test function call

PzMxMy NA rotation % height in compr Pz Mx My degree % kN kN.m kN.m 25 mat : PzMxMy PzMxMy 5 ° * 5 % * biaxial stack (: 21 line

PzMxMy NA rotation % height in compr Pz Mx My degree % kN kN.m kN.m 55 627.1 514.86 - 652.32 - 35 mat

pilot the X, Y directions rot 46 4.25 range : h 46 4.25 range : 21 sys

Pz r 1 : ii 1 h rows range c 1 : j 1 rot rows range M r c el rot ° * (ii el h % * (j el biaxial 3 el eval : c c 1 + : 21 line for r r 1 + : 31 line for M 1 round vectorize 31 line :

2D function Table counter loop

CWD C:\SmathFile CurrentDirectory :

Pz CWD Alex_jmG wfile 1

New CWD Alex-jmG rfile :

The bilinear matrix New x/y rot stack h transpose Pz stack augment :

New x/y 4 4.25 4.5 4.75 5 5.25 5.5 5.75 64 940.8 834.2 730.3 628.2 527 426.2 325.3 223.9 121.7 4.25 959.1 855.1 754 654.8 556.7 459.1 361.7 263.8 165.2 4.5 975.1 873.5 774.9 678.3 582.9 488.3 393.8 299.1 203.8 4.75 989.2 889.8 793.4 699.1 606.3 514.2 422.4 330.5 238.2 5 1001.8 904.3 809.9 717.8 627.1 537.3 448 358.6 268.9 5.25 1013.1 917.3 824.7 734.5 645.8 558.2 471384 296.7 5.5 1023.1 928.9 838 749.5 662.7 577 491.8 406.9 321.7 5.75 1032.2 939.4 850.1 763.2 678 594.1 510.7 427.7 344.5 6 1040.4 948.9 861 775.6 692 609.6 527.9 446.6 365.3 1010 mat

x y M bilinear A M 112 M cols submatrix : B M 2 M rows 11 submatrix : C M 2 M rows 2 M cols submatrix : j 1 A length 2 range CC j el B C 1 C rows j j submatrix y linterp : for A transpose CC x linterp 51 line :

Interpolation in reverse order 55 New bilinear 5.5 4.5 New bilinear 4.5 5.5 New bilinear 31 mat 627.1 393.8 83831 mat

5 ° * 5 % * biaxial 3 el 4.5 ° * 5.5 % * biaxial 3 el 5.5 ° * 4.5 % * biaxial 3 el 31 mat 627.1 393.8 83831 mat

4.5 ° * 5.5 % * biaxial 4.5 5.5 393.8 662.1 - 798.8 - 15 mat

Companion Solver [Alex/Davide]

Alex, biaxial companion solver [adapted] target rot ε Solver x1 1 % * (- : x2 1 % * : ---------------------- f1 rot x1 biaxial 3 el target - : f2 rot x2 biaxial 3 el target - : ---------------------- df f2 f1 - x2 x1 - / f1 UnitsOf x1 UnitsOf / try : i 110000 range x2 x1 f1 df / - : f2 rot x2 biaxial 3 el target - : df f2 f1 - x2 x1 - / df try : f2 target / abs ε < break Done if x1 x2 : f1 f2 : 61 line for x1 101 line :

Command Solver wrt "rot, target Pz"

rot 4.5 ° * :

target 393.8 :

_% target rot 108 -^Solver (: 0.055

Collect Command Explorer

b1 rot_% biaxial 1 el : b2 rot_% biaxial 2 el 3 round : b3 rot_% biaxial 3 el 0 round : b4 rot_% biaxial 4 el 0 round : b5 rot_% biaxial 5 el 0 round : biax rot° h % Pz Mx My 15 mat : Biaxial biax b1 b2 b3 b4 b5 augment stack : 71 line

Biaxial rot° h % Pz Mx My 4.5 5.5 394662 - 799 - 25 mat

Given rot = 2.36, solving for Pz = -59852 generates "Division by zero" ... puzzling NA rotation % height in compr Pz Mx My angle Mrx / Mry degree % kN kN.m kN.m degree 2.36 34.64 59851.66 - 63006 64989.97 45.89 36 mat

5 ° * 5 % * biaxial 3 el 4.5 ° * 5.5 % * biaxial 3 el 5.5 ° * 4.5 % * biaxial 3 el 31 mat 627.1 393.8 83831 mat ≡