Companion Solver [Alex/Davide]

569b0738-3810-4f63-ae0e-f95536a6b12d 40 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

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 :

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 10 ° * 5 % * biaxial stack (: 21 line

PzMxMy NA rotation % height in compr Pz Mx My degree % kN kN.m kN.m 105 811.34 392.47 - 557.59 - 35 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 Explorer wrt "target Pz"

rot 5 ° * :

target 3000 - :

_% target rot 108 -^Solver (: 0.128

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 5 12.786 3000 - 2722 - 1699 - 25 mat

rot 6 ° * 8 ° * 10 ° * 12 ° * 14 ° * 51 mat :

Collect/Tabulate/Document

target 4000 - :

User wrt [rot°, target]

Collect/Tabulate wrt [rot°, target] User header rot° h % Pz Mx My 15 mat : i 1 rot rows range_% i el target rot i el 108 -^Solver : b1 i el rot i el_% i el biaxial 1 el : b2 i el rot i el_% i el biaxial 2 el 3 round : b3 i el rot i el_% i el biaxial 3 el 0 round : b4 i el rot i el_% i el biaxial 4 el 0 round : b5 i el rot i el_% i el biaxial 5 el 0 round : 61 line for Biaxial header b1 b2 b3 b4 b5 augment stack : Biaxial 51 line :

User rot° h % Pz Mx My 6 15.387 4000 - 3279 - 1899 - 8 17.114 4000 - 3227 - 2357 - 10 18.463 4000 - 3180 - 2696 - 12 19.545 4000 - 3135 - 2967 - 14 20.426 4000 - 3092 - 3194 - 65 mat

sub User 26 range 21 range el :

sub 15.387 6 17.114 8 18.463 10 19.545 12 20.426 1452 mat

X sub 1 col : Y sub 2 col : x U X Y x cinterp : 31 line

Interpolated system wrt TARGET

x 19 :

x U 10.93662

10.93662 ° * 19 % * biaxial 10.9366 19 4000.9757 - 3159.0659 - 2829.5903 - 15 mat

sub sub 2 1 sys

Program to create the "Scatter xyz" Mathcad style X Y Z x0 x1 y0 y1 u v Sr count 1 : i 1 u range j 1 v range xx x0 i 1 - (u 1 - / x1 x0 - (* + : yy y0 j 1 - (v 1 - / y1 y0 - (* + : XYZ 1 count el xx yy X xx yy Y xx yy Z 31 mat : count count 1 + : 41 line for for XYZ transpose 31 line :

t0 1 time :

u rot : v_% : 12 mat

u v X u : u v Y v : u v Z u v biaxial 3 el : 31 line

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

x0 0 : x1 10 : y0 0 : y1 1 : u 10 : v 10 : 16 mat XYZ X Y Z x0 x1 y0 y1 u v Sr : data XYZ : xyz data Unest : 41 line

1 time t0 - min 0.4

x xyz 1 col : y xyz 2 col : z xyz 3 col : i 1 x rows range x i el x i el 2 round : y i el y i el 3 round : z i el z i el 0 round : 31 line for xyz x y z augment : 51 line

Notes: 1. The Smath 3D quick plot turns dead every times "xyz" is manipulated. You must re-insert from the menu. 2. As constructed, col 'z' is the corresponding "Pz biaxial". 3. There is no repeated value(s) in "Pz" 4. Easy to lookup in "Pz" for a value, and get the row ... thus giving back the biaxial pair [rot°, _%]. 5. At this point, determine the range of [rot°, _%] ... mesh at will as dictated by the "Pz".

xyz 0012030 0.111 5410 - 0 0.222 11085 - 0 0.333 16627 - 0 0.444 22170 - 0 0.556 27712 - 0 0.667 33456 - 0 0.778 39253 - 0 0.889 44986 - 0150676 - 1.11 01439 1.11 0.111 727 - 1.11 0.222 5307 - 1.11 0.333 12507 - 1.11 0.444 20027 - 1.11 0.556 27654 - 1.11 0.667 35306 - 1.11 0.778 42929 - 1.11 0.889 50311 - 1.11 155360 - 2.22 01433 2.22 0.111 491 - 2.22 0.222 4134 - 233 mat

xyz

image

u rot : v_% : 12 mat

biaxial is not analytical
Smath CreateMesh fails u v f u v u v biaxial 3 el 31 mat :

i 1 xyz rows range : Pz xyz 3 col : Pz i Pz augment : 31 line

v x search j 0 : i 1 v length range j j 1 + : v i el x ≡ vv v i el : break 21 line vv NO match : if 21 line for j vv 21 mat 31 line :

Pz 27654 - search Pz 50311 - search Pz π search 31 mat 3227654 - 21 mat 3850311 - 21 mat 200 NO match 21 mat 31 mat

Intuitively, maybe there is an infinite
combination of [rot°, _%] that produce a
given Pz ... No idea how to do  ... and if
that exist, the other problem is determine
some sort of technical valid mutuality ? xyz 32 row xyz 38 row 21 mat 3.33 0.111 805 - 13 mat 3.33 0.778 42442 - 13 mat 21 mat