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&[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 :

1000 random 100000 / 0.00525

function inputs (hidden):

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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 : 1 m * - 0.675 m * - 1 m * 0.675 m * - 1 m * 0.675 m * 1 m * - 0.675 m * 1 m * - 0.675 m * - 52 mat

c_rebar_XY# a# 2 / c# - (- a# 2 / c# - (b# 2 / c# - (b# 2 / c# - (22 mat 48 . A.s35 mm 2^/ (n.plot^red pts.line a# 2 / c# - (- a# 2 / c# - (b# 2 / c# - (- b# 2 / c# - (- 22 mat 18 . 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 2853.4 - 9541.93 - 015 mat

test function call:

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PzMxMy NA rotation % height in compr Pz Mx My degree % kN kN.m kN.m 25 mat :

PzMxMy PzMxMy 10 ° * 5 % * biaxial stack (:

PzMxMy NA rotation % height in compr Pz Mx My degree % kN kN.m kN.m 105 22429.62 5616.01 1140.75 - 35 mat

Now that we have a function that produces input loads based on crossection responce, we will use the function to solve for crossection responce based on input loads. For example lets solve for "% height in compression" na#, given known neutral axis orientation (10 degrees):

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x y xii yii xiii yiii x y a x y b a.l b.l B.a B.b C.a C.b a.ii b.ii Clear 1

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 #

x y a x y biaxial 3 el 2999.9993601 + eval :

solve for Pz = -3000

x y b x y biaxial 4 el x y biaxial 5 el / 1 - eval :

solve for Mx/My = 1 (changing this can crash)

a.l a.ii B.a xiii xii - (* C.a yiii yii - (* + + :

b.l b.ii B.b xiii xii - (* C.b yiii yii - (* + + :

xiii B.b C.a * B.a C.b * - (xii * C.a b.ii * - C.b a.ii * + B.b C.a * B.a C.b * - / :

yiii B.b C.a * B.a C.b * - (yii * B.a b.ii * + B.b a.ii * - B.b C.a * B.a C.b * - / :

Step 1:

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xi 5 ° * : 0.0873

yi 5 % * :

xii 15 ° * :

yii 15 % * :

a.ii xii yii a eval : 11510.0015

b.ii xii yii b eval : 0.6756 -

B.a a.ii xi yii a - xii xi - / eval : 85096.8759

B.b b.ii xi yii b - xii xi - / eval : 12.6255 -

C.a a.ii xii yi a - yii yi - / eval : 1.4119 105^* -

C.b b.ii xii yi b - yii yi - / eval : 60.9608

xiii ° 4.7983

yiii % 12.4206

Step 2:

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xi xii : ° 15

yi yii : % 15

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 4.7983

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 12.4206

100000 random 10000 / 2.5246

a.ii xii yii a eval : 2084.2716

b.ii xii yii b eval : 0.6254

B.a a.ii xi yii a - xii xi - / eval : 75982.4856

B.b b.ii xi yii b - xii xi - / eval : 8.9466 -

C.a a.ii xii yi a - yii yi - / eval : 2.2883 105^* -

C.b b.ii xii yi b - yii yi - / eval : 41.3484

xiii ° 7.2051 -

yiii % 6.3752

Step 2:

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xi xii : ° 4.7983

yi yii : % 12.4206

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 16.2144

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 6.3752

a.ii xii yii a eval : 24272.0771

b.ii xii yii b eval : 3.3653 -

B.a a.ii xi yii a - xii xi - / eval : 14381.1368

B.b b.ii xi yii b - xii xi - / eval : 7.4106 -

C.a a.ii xii yi a - yii yi - / eval : 1.3445 105^* -

C.b b.ii xii yi b - yii yi - / eval : 38.8965

xiii ° 80.673

yiii % 36.4611

Step 2:

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xi xii : ° 16.2144

yi yii : % 6.3752

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 80.673

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 36.4611

a.ii xii yii a eval : 7675.7662 -

b.ii xii yii b eval : 0.997 -

B.a a.ii xi yii a - xii xi - / eval : 8329.7697

B.b b.ii xi yii b - xii xi - / eval : 2.4127 -

C.a a.ii xii yi a - yii yi - / eval : 1.0697 105^* -

C.b b.ii xii yi b - yii yi - / eval : 8.4446

xiii ° 28.341

yiii % 22.1726

Step 2:

]]>

xi xii : ° 80.673

yi yii : % 36.4611

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 28.341

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 22.1726

a.ii xii yii a eval : 7852.755

b.ii xii yii b eval : 0.5524 -

B.a a.ii xi yii a - xii xi - / eval : 351.7146

B.b b.ii xi yii b - xii xi - / eval : 0.6545 -

C.a a.ii xii yi a - yii yi - / eval : 1.2727 105^* -

C.b b.ii xii yi b - yii yi - / eval : 5.6625

xiii ° 10.1341

yiii % 28.2549

Step 2:

]]>

xi xii : ° 28.341

yi yii : % 22.1726

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 10.1341

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 28.2549

a.ii xii yii a eval : 13630.4844 -

b.ii xii yii b eval : 2.4677

B.a a.ii xi yii a - xii xi - / eval : 43150.7871

B.b b.ii xi yii b - xii xi - / eval : 8.4147 -

C.a a.ii xii yi a - yii yi - / eval : 1.2507 105^* -

C.b b.ii xii yi b - yii yi - / eval : 22.7298

xiii ° 9.1902

yiii % 16.7882

Step 2:

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xi xii : ° 10.1341

yi yii : % 28.2549

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 9.1902

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 16.7882

a.ii xii yii a eval : 1452.0819

b.ii xii yii b eval : 0.2455

B.a a.ii xi yii a - xii xi - / eval : 90778.2302

B.b b.ii xi yii b - xii xi - / eval : 12.0991 -

C.a a.ii xii yi a - yii yi - / eval : 1.4047 105^* -

C.b b.ii xii yi b - yii yi - / eval : 24.2333

xiii ° 1.2098

yiii % 8.821

Step 2:

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xi xii : ° 9.1902

yi yii : % 16.7882

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 1.2098

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 8.821

a.ii xii yii a eval : 485.3112

b.ii xii yii b eval : 4.1183

B.a a.ii xi yii a - xii xi - / eval : 1.3186 105^*

B.b b.ii xi yii b - xii xi - / eval : 39.0648 -

C.a a.ii xii yi a - yii yi - / eval : 1.9245 105^* -

C.b b.ii xii yi b - yii yi - / eval : 251.2102

xiii ° 0.8363 -

yiii % 6.6263

Step 2:

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xi xii : ° 1.2098

yi yii : % 8.821

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 5.9854

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 6.6263

a.ii xii yii a eval : 21635.111

b.ii xii yii b eval : 1.9302 -

B.a a.ii xi yii a - xii xi - / eval : 1.4126 105^*

B.b b.ii xi yii b - xii xi - / eval : 30.4709 -

C.a a.ii xii yi a - yii yi - / eval : 2.6211 105^* -

C.b b.ii xii yi b - yii yi - / eval : 43.1955

xiii ° 19.0131

yiii % 27.1344

Step 2:

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xi xii : ° 5.9854

yi yii : % 6.6263

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 19.0131

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 27.1344

a.ii xii yii a eval : 4014.3696 -

b.ii xii yii b eval : 0.293

B.a a.ii xi yii a - xii xi - / eval : 57822.5208

B.b b.ii xi yii b - xii xi - / eval : 26.2014 -

C.a a.ii xii yi a - yii yi - / eval : 1.3749 105^* -

C.b b.ii xii yi b - yii yi - / eval : 17.2762

xiii ° 18.3732

yiii % 23.7449

Step 2:

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xi xii : ° 19.0131

yi yii : % 27.1344

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 18.3732

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 23.7449

a.ii xii yii a eval : 240.3989

b.ii xii yii b eval : 0.0072

B.a a.ii xi yii a - xii xi - / eval : 46463.3913

B.b b.ii xi yii b - xii xi - / eval : 4.8309 -

C.a a.ii xii yi a - yii yi - / eval : 1.3963 105^* -

C.b b.ii xii yi b - yii yi - / eval : 10.2493

xiii ° 19.3737

yiii % 24.4981

Step 2:

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xi xii : ° 18.3732

yi yii : % 23.7449

xii 360 ° * 100000 random 10000 / * % * xiii 0 ° * < 360 ° * 100100000 random 10000 / - (* % * xiii 360 ° * > xiii cases cases : ° 19.3737

yii 100000 random 10000 / % * yiii 0 % * < 100100000 random 10000 / - (% * yiii 100 % * > yiii cases cases : % 24.4981

a.ii xii yii a eval : 82.8815 -

b.ii xii yii b eval : 0.0027

B.a a.ii xi yii a - xii xi - / eval : 45205.9283

B.b b.ii xi yii b - xii xi - / eval : 4.877 -

C.a a.ii xii yi a - yii yi - / eval : 1.4984 105^* -

C.b b.ii xii yi b - yii yi - / eval : 10.4135

xiii ° 19.2728

yiii % 24.3897

result:

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xiii yiii biaxial 3 el 3001.08 -

xiii yiii biaxial 4 el xiii yiii biaxial 5 el / 1