PolyPropertiesX(1)S... · submatrix , , , ,1 rows P 1 2 ux uy 1 0 1 0 1 0 0 #P #P augment P, 1 matrix rows P ,1 Translate , ,u "use homogeneous coordinates" y u x P rotatinga polygonaboutthe

PolyProperties

@description: calculate properties of generic polygons

@version: 2013.02.01

@author: Davide Carpi >>> [email protected]

@translation: Davide Carpi

@SMath: 0.95.4594 >>> http://en.smath.info

@license: CC BY-SA 3.0 >>> http://creativecommons.org/licenses/by-sa/3.0

just remember to give credit where it's due, and don't try to pass off other people's work as your own,

or your own as anyone else's.

@disclaimer: this script should be used with caution; the author assumes no responsibility for any damage resulting

from its use or misuse.

SEE THE LICENSE (human-readable summary)

DRAWING TIPS and RULES

A__POLYGON

A polygon is a closed chain of straight lines (the polygon sides);

the points where two sides meet are the polygon's vertices.

In this SMath Studio worksheet:

- a polygon is defined by it's vertices;

- definition of vertices can be either clockwise or anticlockwise;

- last vertex must be equal to the first;

- intersections points must be numered twice.

10 mar 2013 14:05:03 - PolyProperties

1 / 41

POLYGONS__(discontinuous__areas)

To have multiple polygons:

- draw all the vertices having same wise;

- close each polygon;

- don't forget to close the vertices chain in the first vertex.

NOTE:

- perimeter calculation cut off overlapped sides with opposite wise.

POLYGONS__(overlapped__areas)

With multiple polygons you can draw overlapped areas; like above:

- draw all the vertices having same wise;

- close each polygon;

- don't forget to close the vertices chain in the first vertex.


2 / 41

A__HOLLOW__POLYGON

To have a hollow polygon, draw the internal and the external vertices with opposite wise;

don't forget to close the vertices chain in the first vertex!

CURVED__SIDES

Using an adequate approximation you can draw curved sides (and therefore generic figures).


3 / 41

SHORTCUTS

Save time: use "SHORTCUTS" to draw common shapes and to make transformations (rotate, scale, translate, etc...)

On the right of the GEOMETRY area there is a complete list of all available SHORTCUTS


4 / 41

SHORTCUTS LIBRARY

Snippet: Operations with units 2012.01.00

TRANSFORMATIONS

rotation matrix; rotatation about the origin of the Cartesian coordinate system.

θ: anticlockwise rotation.

cos θsin θ

sin θcos θRot θ

reflection matrix; reflection about a line through the origin of the Cartesian coordinate system.

θ: angle of the reflection line through the origin with the x-axis.

cos θ2sin θ2

sin θ2cos θ2Ref θ

scaling matrix; scaling about the origin of the Cartesian coordinate system.

s.x: x-scaling factor;

s.y: y-scaling factor.

sy0

0sx

Sc

, sy

sx

shear matrix; shearing about the origin of the Cartesian coordinate system.

λ.x: shearing factor, parallel to x;

λ.y: shearing factor, parallel to y.

1λx

λy1

Sh

, λy

λx


5 / 41

geometric translation of a polygon.

P: polygon matrix;

u.x: x translation;

u.y: y translation.

submatrix , , , , 21rows P1

1uy

ux

010

001

#P

augment , matrix , 1rows P1P#P

"use homogeneous coordinates"Translate , , uy

ux

P

rotating a polygon about the origin of the Cartesian coordinate system.

P: polygon matrix;



100

0cos θsin θ

0sin θcos θ

#P


"use homogeneous coordinates"Rotate , θP

scaling a polygon about the origin of the Cartesian coordinate system.

P: polygon matrix;

s.x: x scaling;

s.y: y scaling.


100

0sy0

00sx

#P


"use homogeneous coordinates"Scale , , sy

sx

P


6 / 41

shearing a polygon about the origin of the Cartesian coordinate system.

P: polygon matrix;

λ.x: shearing factor, parallel to x;

λ.y: shearing factor, parallel to y.


100

01λx

0λy1

#P


"use homogeneous coordinates"Shear , , λy

λx

P

mirroring a polygon.

P: polygon matrix;

θ: angle of the reflection line through the origin with the x-axis or case-insensitiveaxes name ["x","y","xy"].


100

0cos Θ2sin Θ2

0sin Θ2cos Θ2

#P


if

else

θΘ

for

if

else

0

2kθa

Θ

1kθa

θ

..rows θa

1k

T

4

π

4

π

2

π

2

π00

"XY""xy""Y""y""X""x"

θa

"N.A."Θ

IsString θ

"use homogeneous coordinates"Mirror , θP


7 / 41

create a chunk of a polygon.

pgon: complete polygon;

TLV: Threshold Limit Value;

zone: (relative) semi-plane to keep ["y+","y-","x+","x-"].

Rot ∆θ#chunk#chunk

for

if

else

0

1jj

#TLV2j

#chunk

eval1k

#pgon

2k#pgon

21k#pgon

2k#pgon#TLV

1k#pgon

11k#pgon

1j#chunk

0#TLV21k

#pgon#TLV2k

#pgon

if

else

0

1jj

2k#pgon

2j#chunk

1k#pgon

1j#chunk

#TLV2k

#pgon

.. 1r1k

1j

rows pgonr

00#chunk

for

if

else

0

break

Rot ∆θpgon#pgon

3kΖTLV#TLV

2kΖ∆θ

zone1k

Ζ

..41k

12

π"x-"

12

π"x+"

1π"y-"

10"y+"

Ζ

chunk , , zoneTLVpgon


8 / 41

#chunk

eval21

#chunk2j

#chunk

eval11

#chunk1j

#chunk

Rot ∆θ#chunk#chunk10 mar 2013 14:05:03 - PolyProperties

9 / 41

POLYGONS

convex regular polygons.

s: number of sides/vertices;

c: circumradius;


#out

eval21

#out21s

#out

eval11

#out11s

#out

Rot θ#out#out

for

eval coss

kπ2c

21k#out

eval sins

kπ2c

11k#out

..s0k

max3

ss

00#outpgon , , θcs

hollow convex regular polygons.


c: external circumradius;

t: border thickness;


Rot θstack , eval row , 1#out#out

stack , reverse pgon , , 0tcspgon , , 0cs#outhPgon , , , θtcs

overlapped convex regular polygons.


ce: external circumradius;

ci: internal circumradius;



stack , pgon , , 0ci

spgon , , 0ce

s#outoPgon , , , θci

ce

s


10 / 41

star regular polygons.

p: number of sides/vertices[must be ≥ 3];

q: connected vertices spacing [{p/q} is the polygon Schläfli symbol];

c: circumradius;


NOTE: stellations of regular or star polygons are for drawing purposes only.

if

if

else

if

else

"stellation of a star polygon"

Rot θstr2num concat , substr , 5num2str #out"sys("

for

pgon , , 1kp

π2cround , 0

q'

pk

#out

..q'1k

"stellation of a regular polygon"

1q'

GCD

gcd , q'pGCD

"stellations"

#out

eval21

#out21p

#out

eval11

#out11p

#out

Rot θ#out#out

for

eval cosp

q'kπ2c

21k#out

eval sinp

q'kπ2c

11k#out

..p0k

00#out

1q'1gcd , q'p

3p

out

if

else

gcd , mod , babout

aout

0bgcd , ba

min1round , 0

2

p

q

q'

sPgon , , , θcqp


11 / 41

else

0

if

else

0

eval str2num concat , substr , 5num2str #out"sys("

for

Rot 1kp

π2#star

k#out

..GCD1k

sPgon , , , θcGCDGCD

p#star

0truncGCD

q'

GCD

q'

"stellation of a star polygon"10 mar 2013 14:05:03 - PolyProperties

12 / 41

edge of star polygons.

p: number of sides/vertices[must be ≥ 3];

q: connected vertices spacing [{p/q} is the polygon Schläfli symbol];

c: circumradius;


if

else

0

#out

eval21

#out21p2

#out

eval11

#out11p2

#out

Rot θ#out#out

for

eval cosp2

kπ2r

21k#out

eval sinp2

kπ2r

11k#out

if

else

ci

c

0trunc2

k

2

kr

.. p20k

sinp2

π2

q'

p2

π2q'

p

π

sin

q'

p2

π2q'

p

c

ci

00#out

3p

min1round , 0

2

p

q

q'

esPgon , , , θcqp


13 / 41

star-shaped polygons.

s: number of spikes [must be ≥ 3];

c.e: external circumradius;

c.i: internal circumradius;


if

else

0

#out

eval21

#out21s2

#out

eval11

#out11s2

#out

Rot θ#out#out

for

eval coss2

kπ2r

21k#out

eval sins2

kπ2r

11k#out

if

else

ci

ce

0trunc2

k

2

kr

.. s20k

00#out

3sssPgon , , , θci

ce

s

triangle.

B: x-length;

H: y-shift of top vertex;

a: x-shift of top vertex;


Rot θ

00

Ha

0B

00

triangle , , , θaHB


14 / 41

right triangle.

B: x-length;



Rot θ

00

H0

0B

00

rTriangle , , θHB

isosceles triangle.

B: x-length;



Rot θ

00

H2

B

0B

00

iTriangle , , θHB

equilateral triangle.

a: side length;


Rot θ

00

a2

3

2

a

0a

00

eTriangle , θa


15 / 41

rhombus.

p: x-diagonal;

q: y-diagonal;


Rot θ

02

p

2

q0

02

p

2

q0

02

p

rhombus , , θqp

square.

a: sides length;


Rot θ

00

a0

aa

0a

00

square , θa

rectangle.

B: width;

H: height;


Rot θ

00

H0

HB

0B

00

rectangle , , θHB


16 / 41

golden rectangle.

B: width;


Rot θ

00

H0

HB

0B

00

φ

BH

2

51φ

goldenRect , θB

rounded rectangle.

h: external height;

b: external width;

r: corner radius;


Rot θ#out

stack , , eval row , 1#outeval reverse10

01#out#out#out

stack , eval reverse10

01#out#out#out

if

else

2

h

2

b

#tmp

for

eval r2

hsin

pts

k

2

πr

21k#tmp

eval r2

bcos

pts

k

2

πr

11k#tmp

..pts0k

15pts

00#tmp

0r#outroundedRect , , , θrbh


17 / 41

parallelogram.

b: base;

a: x-shift of top base;

h: height;


Rot θ

00

ha

hab

0b

00

parallelogram , , , θhab

trapezoid.

a: top base;

b: bottom base;

c: x-shift of top base;

h: height;


Rot θ

00

ha

hac

0b

00

trapezoid , , , , θhcba

right trapezoid.

a: top base;

b: bottom base;

h: height;


Rot θ

00

h0

ha

0b

00

rTrapezoid , , , θhba


18 / 41

isosceles trapezoid.

a: top base;

b: bottom base;

h: height;


Rot θ

00

h2

ab

h2

aba

0b

00

iTrapezoid , , , θhba

L shape.

a: top base;

b: bottom base;

c: top height;

h: total height;


Rot θ

00

h0

ha

cha

chb

0b

00

L , , , , θhcba


19 / 41

T shape.

a: top base;

b: bottom base;

c: top height;

h: total height;


Rot θ#out


01#out#out#out

02

b

ch2

b

ch2

a

h2

a

#out

T , , , , θhcba

circle.

D: diameter.

pgon , , 02

D100circle D

semicircle.

D: diameter;


Rot θ#out

stack , eval row , 1#out#out#out

for

eval sinpts

kπ

2

D21k

#out

eval cospts

kπ

2

D11k

#out

..pts0k

100pts

00#outsemicircle , θD


20 / 41

segmental arch.

r.e: external radius;

r.i: external radius;

ω: inner angle;


Rot θ#P

stack , eval row , 1#P#P#P

stack , #P2#P1#P

for

eval sinpts

kω

2

ωπri21kpts

#P2

eval cospts

kω

2

ωπri11kpts

#P2

..0ptsk

50pts

00#P2

for

eval sinpts

kω

2

ωπre21k

#P1

eval cospts

kω

2

ωπre11k

#P1

..pts0k

50pts

00#P1arch , , , θωri

re


21 / 41

ellipse.

p: x-axis;

q: y-axis;


Rot θ#out


01#out#out#out

for

eval

2

a

11k#out

1b21k

#out

eval kpts

pa

11k#out

..pts0k

40pts

00#out

2

qb

2

pa

ellipse , , θqp


22 / 41

Hollow Structural Sections.

h: external height;

b: external width;

t: wall thickness;

r.e: external corner radius (typical=2t);

r.i: internal corner radius (typical=t);



stack , eval reverse roundrect , , , 0ri

t2bt2hroundrect , , , 0re

bh#outHSSbeam

, , , , , θri

re

tbh

Circular Hollow Sections.

De: external diameter;

t: border thickness.

hPgon , , , 0t2

De

75CHSbeam

, tDe

Elliptical Hollow Sections.

p: external x-axis;

q: external y-axis;

t: wall thickness;



stack , eval reverse ellipse , , θt2qt2pellipse , , θqp#outEHSbeam

, , , θtqp


23 / 41

european I-beam (UE:IPE/HE/HL/HD/HP;UK:UB/UC/UBP; US:W/HP; RU:HG; JP:H).

h: beam height;

b: flange width;

tw: web thickness;

tf: flange thickness;

r: web-flange junction radius;


Rot θ#out


01#out#out#out


01#out#out#out

stack , #out

tf2

h

2

b

2

h

2

b

#out

if

else

tf2

h

2

tw

#tmp

for

eval rtf2

hsin

pts

k

2

π

2

πr

21k#tmp

eval r2

tw

cospts

k

2

π

2

πr

11k#tmp

..pts0k

25pts

00#tmp

0r#outIbeam

, , , , , θrtf

tw

bh


24 / 41

european (un)equal legs angles.

h: height;

b: width;

t: thickness;

r1: legs junction radius;

r2: legs corner radius;


if

for

eval r2

tcospts

k

2

πr211k

#tmp

..pts0k

30pts

00#tmp

0r2

#P4

if

else

tt

#tmp

for

eval r1

tsinpts

k

2

ππ

2

3r121k

#tmp

eval r1

tcospts

k

2

ππ

2

3r111k

#tmp

..pts0k

30pts

00#tmp

0r1

#P3

if

else

tb

#tmp

for

eval r2

tsinpts

k

2

πr221k

#tmp

eval r2

bcospts

k

2

πr211k

#tmp

..pts0k

30pts

00#tmp

0r2

#P2

0b

00

h0

#P1

Lbeam

, , , , , θr2

r1

tbh


25 / 41

Rot θ#P

stack , , , , #P5#P4#P3#P2#P1#P

h0#P5

else

ht

#tmp

eval r2

hsinpts

k

2

πr221k

#tmp

european U-beam [DIN 1026-1:2000] (UPN) and parallel flange channels (UPE/PFC).

h: beam height [mm];

b: flange width;

tw: web thickness;

tf: flange thickness;

r1: web-flange junction radius;

r2: flange corner radius (UPE/PFC when r2=0);


if

00#tmp

0r2

#P2

2

hb

2

h0

#P1

eval tf

r1

2tw

r1

ubp2hd

if

else

0p

2

bu

if

else

100

5p

2

tw

b

u

100

8p

2

bu

300h

0r2

Ubeam

, , , , , , θr2

r1

tf

tw

bh


26 / 41

Rot θ#P

stack , , #P5eval reverse10

01#P#P#P

stack , , , #P4#P3#P2#P1#P

2

h0#P5

if

else

tf2

htw

#tmp

for

eval2

dsin

pts

k

2

π

2

πr121k

#tmp

eval r1

tw

cospts

k

2

π

2

πr111k

#tmp

..pts0k

25pts

00#tmp

0r1

#P4

tf2

hub#P3

else

tf2

hb

#tmp

for

eval r2

pr2

utf2

hsin

pts

k

2

πr221k

#tmp

eval r2

bcospts

k

2

πr211k

#tmp

..pts0k

25pts

00#tmp

PSEUDONYMS

CHSbeam

, tDe

ring , tDe


27 / 41

GEOMETRY

GEOMETRY

polygon coordinates - x y

mmUbeam

, , , , , , °156.513139.585240polygon

polygon

polygon

-0.125 0 0.125

0.125

0.0625

0

-0.0625

-0.125

x

y

Axes conventions


28 / 41

PLOT SETTINGS

CALCULATIONS OF CROSS-SECTION PROPERTIES

coordinates

eval col , 1polygonx

eval col , 2polygony

vertices (+1)

length xn

perimeter

=

1n

1i

2

iy

1iy

2

ix

1ixP

calculate total length of overlapped segments with opposite wise

eval

#links

for

if

else

0

eval sWiseoWise22

jy

1jy

2

jx

1jx#links#links

0oWise

for

if

else

if

else

0

1sWisesWise

ky

1jy

kx

1jx

1ky

jy

1kx

jx

1oWiseoWise

1ky

1jy

1kx

1jx

ky

jy

kx

jx

..n2jk

0oWisesWise

.. 2n1j

0#links∆P

∆PPP


29 / 41

area

=

1n

1ii

y1i

x1i

yi

x2

1A

centroid

=

1n

1ii

y1i

x1i

yi

x1i

xi

xA6

1Cx

=

1n

1ii

y1i

x1i

yi

x1i

yi

yA6

1Cy

use centroid as origin

dRound , 10Cx

Cx

dRound , 10Cy

Cy

Cx

xX

Cy

yY

second moments of area (any cross section allowed)

=

1n

1ii

Y1i

X1i

Yi

X2

1iY

1iY

iY

2

iY

12

1JXX

=

1n

1ii

Y1i

X1i

Yi

X2

1iX

1iX

iX

2

iX

12

1JYY


30 / 41

=

1n

1ii

Y1i

X1i

Yi

Xi

Y1i

X1i

Y1i

X2i

Yi

X21i

Yi

X24

1JXY

output adjustements due to clockwise/anticlockwise vertices definition

dRound , 10PP

AA

JXX

IX

JYY

IY

dRound , 10JXY

IXY

radii of gyration

A

IX

iX

A

IY

iY

elastic section modulus

max Y

IX

Wel;top


31 / 41

min Y

IX

Wel;bot

min X

IY

Wel;left

max X

IY

Wel;right

orientation of principal axes of inertia (u,v)

if

else

sign IXY4

π

2

π0I

YIX

sign JXX

atanIY

IX

IXY

2

2

1

0dRound , 10IY

IX

α

principal moments of inertia

2IXY

42

IY

IX2

1

2

IY

IX

Iu

2IXY

42

IY

IX2

1

2

IY

IX

Iv

radii of gyration about principal axes of inertia

A

Iu

iu

A

Iv

iv


32 / 41

plot with centroid as origin

Translate , , Cy

Cx

polygonpolygonC

ellipsoid of gyration

eval stack , , , , , 0iv00i

u0iu000ellipse , , 0eval i

u2eval i

v2ellipsoid

vertices numbering

function to numbering vertices of polygons.

values for opt:

"all" to enumerate all vertices;

### to enumerate vertices spaced at least the ### value;

any other text to disable numbering.

if

for

if

#colorv

#size4c

vn

num2str k3c

vn

2kpgon

2cvn

1kpgon

1cvn

opt#df

opt#dp

eval2

21kpgon

2kpgon

2

11kpgon

1kpgon#d

f

eval2

21kpgon

2kpgon

2

11kpgon

1kpgon#d

p

.. 1r2k

2c

#color#sizeconcat , , ")"num2str r"1("21pgon

11pgonv

n

IsString opt

rows pgonr

verticesnamesize

#size

verticescolor

#colornumbering , optpgon


33 / 41

else

if

else

00

vn

""3r

vn

concat , , ")"num2str r"1("31

vn

for

eval

#color5k

vn

#size4k

vn

num2str k3k

vn

..r1k

pgonvn

"all"opt

vn

else

0

1cc

#color5c

vn

numbering , verticesnumbering

polygonC

vertices

augment , submatrix , , , , 53rows vertices1verticesRot αsubmatrix , , , , 21rows vertices1verticesverticesuv


34 / 41

CROSS-SECTION__PROPERTIES polygon refered to centroid

-0.125 0 0.125 0.25

0.125

0.0625

0

-0.0625

-0.125

x

y

82

291(111)

110

G

perimeter

mm775.65P

area

2mm4230.01A

centroid (relative to global coordinates x,y)

mm21.6Cx

mm5.79Cy

second moments of area (relative to centroid)

4cm3373.7I

X

4cm471.96I

Y

4cm837.66I

XY

radii of gyration

cm8.93iX

cm3.34iY

elastic section modulus

3cm255.35W

el;top

3cm277.22W

el;bot

3cm89.62W

el;left

3cm51.55W

el;right


35 / 41

orientation of principal axes of inertia (u,v) orientation about principal axes of inertia

-0.125 0 0.125

0.125

0.0625

0

-0.0625

-0.125

x

y

82

291(111)

110

G

°15α

principal moments of inertia

4cm3598.15I

u

4cm247.51I

v

radii of gyration about principal axes of inertia

cm9.22iu

cm2.42iv

CALCULATION OF PLASTIC SECTION MODULUS (CREOSS-SECTION HAVE CONSTANT YELDING STRESS)

plastic neutral axis position

polygon area

if

else

0

eval

=

1r

1n2n

pgon11n

pgon21n

pgon1n

pgon2

1

3r

rows pgonrAPpgon

SMath geometry Unit of Measurement

UoM PUoM

check if "polygon" could have multiple areas

eval 0∆PisMultiPgon

search for a Plastic Neutral Axis (PNA).

pgon: polygon matrix;

axis: axis orthogonal to the PNA orientation ["x" or "y"];

mPgon: 1 if is a multiple polygon, 0 otherwise.


36 / 41

if

for

if

if

p#

plast

0round , 5fp#

Apgon

APchunk , , concat , "-"axisp

#pgon2A

pgonfp#

0round , 5UoM

ptmp

p#

evalk

Pchunk

p#

..11length Pchunk

k

sort col , columnchunk , , concat , "-"axisptarget

pgonPchunk

1targetp

targetptmp

plast

"search PNA regions"

0mPgon

0∆target

bisection , pmax

pmintarget

eval max col , columnpgonpmax

eval min col , columnpgonpmin

if

else

if

else

bisection , Ubound

p

bisection , pLbound

0fp

p

0fp

eval round , 5Apgon

APchunk , , concat , "-"axisppgon2A

pgonfp

2

Ubound

Lbound

p

bisection , Ubound

Lbound

AApgon

eval "y"axis1columnPNAhunter

, , mPgonaxispgon


37 / 41

∆target

target

else

0

for

if

else

0

p#

ptmp

if

else

break

if

else

0

1target

2target∆

target

plast2

target

ptarget

plast

p#

plast

0round , 5fp#

Apgon

APchunk , , concat , "-"axisp

#pgon2A

pgonfp#

0round , 5UoM

ptmp

p#

evalk

Pchunk

p#

..length Pchunk

2k

ptarget

ptmp

plast

sort col , columnchunk , , concat , "+"axisptarget

pgonPchunk

else

0

p#

ptmp

else

break

plast1

target

#last10 mar 2013 14:05:03 - PolyProperties

38 / 41

PNA x-position

eval PNAhunter

, , isMultiPgon"x"polygonPNA

eval1

PNAPNAx

eval2

PNAPNA∆x

PNA y-position

eval PNAhunter

, , isMultiPgon"y"polygonPNA

eval1

PNAPNAy

eval2

PNAPNA∆y

allowable region for PNAs

Translate , , Cy

1PNA

yCx

1PNA

xrectangle , , 0PNA

∆yPNA

∆xPNAs

BOX

PNA infos to plot

#P

for

eval stack ,PNA

colorPNA

namesizePNA

name2kPNAs

BOX1k

PNAsBOX

PNAcolor

PNAsymbolsize

PNAsymbol2k

PNAsBOX

1kPNAs

BOX

#P#P

..21k

centroidcolor

centroidnamesize

centroidname00

centroidcolor

centroidsymbolsize

centroidsymbol00

#P

PNAsplot

script has found discontinuous equal areas; green box/line show the PNAs region0msg1

PNAs and plastic section modulus calculated for equal areas0msg2

if

description msg2

0PNA∆y

0PNA∆x

msg


39 / 41

else

description msg2

plastic section modulus

PNAx;y

Cnp

Anp

PNAx;y

Cpp

App

eval

=

1r

1ii

y1i

x1i

yi

x1i

yi

yAnp

6

1Cnp

AP#chunkA

np

rows #chunkr

col , 2#chunky

col , 1#chunkx

chunk , , "y-"PNAx;y

#pgon#chunk

eval

=

1r

1ii

y1i

x1i

yi

x1i

yi

yApp

6

1Cpp

AP#chunkA

pp

rows #chunkr

col , 2#chunky

col , 1#chunkx

chunk , , "y+"PNAx;y

#pgon#chunk

if

else

eval pgon

eval Rot2

πpgon

"x"axis#pgonWpl

, , axisPNAx;y

pgon

plastic section modulus (x)

Wpl

, , "y"1

PNAy

polygonWpl;x

plastic section modulus (y)

Wpl

, , "x"1

PNAx

polygonWpl;y


40 / 41

plastic neutral axes (relative to global coordinates x,y)

1PNAx

PNAx

1PNAy

PNAy

plastic neutral axes (relative to centroid)

Cx

PNAx

PNAX

Cy

PNAy

PNAY

PLASTIC__SECTION__MODULUS plastice neutral axes center/region

-0.125 0 0.125

0.125

0.0625

0

-0.0625

-0.125

x

y

ZGZ

plastic neutral axes (relative to global coordinates x,y)

mm15.92PNAx

mm1.23PNAy

plastic neutral axes (relative to centroid)

mm5.68PNAX

mm4.56PNAY

plastic section modulus

3cm345.42W

pl;x

3cm111.7W

pl;y

note: "PNAs and plastic section modulus calculated for equal areas"


41 / 41

Documents

PolyPropertiesX(1)S... · submatrix , , , ,1 rows P 1 2 ux uy 1 0 1 0 1 0 0 #P #P augment P, 1 matrix rows P ,1 Translate , ,u "use homogeneous coordinates" y u x P rotatinga polygonaboutthe