Analysis of a 3D Frame

8/12/2019 Analysis of a 3D Frame

1/94

3D Frame Analysis

Analysis of a 3D frame subject to distributed loads, point loads and moments

Principal Axes

Beam principal axes are defined in the same way as in the FEA program Strand7:

i.e. the Z axis is in effect defined as the vertical axis in the model, and the ! plane is hori"ontal.

Units

Any consistent #nits may $e #sed.

Solvers

%hree alternative solvers are availa$le from the dropdown $ox on the &'#tp#t& sheet:

%he Algli$ Sparse Solver is $y far the fastest, $#t may not converge for some pro$lems.

%he compiled solver is recommended for cases where the sparse solver does not perform

%he (BA solver is provided for cases where the necessary dll files for the compiled and sp

For instr#ctions on installing the dll file see:

End Releases

%r#ss mem$ers transmitting axial load only may $e defined $y setting )*, )+ and to "ero.

utput

-licing the &/ecalc#late& $#tton generates node res#lts 0deflections and reactions1 and forces an

Beam res#lts relative to $eam principal axes are generated $y clicing the &/ecalc#late $eam res#

)ntermediate res#lts may $e generated for any n#m$er of points.

After generating $eam res#lts, res#lts for selected $eams may $e plotted on the 2lot Beam& sheet.

i3 is the unit vector directed from Node 1 to Node 2.

i2 is the unit vector arising from i2= Z i3where Zis the unit vector in the global Z diri1 completes the right-handed system such that i1 i2= i3

%his proced#re in effect creates the i!Axis parallel to the ! plane, and the i"Axis in the plane pa

)f the i3 axis is parallel to the Z axis then the i!axis is parallel to the ! axis in the positive direction.

Beam principal axes may $e rotated a$o#t the i3axis $y an angle specified in the $eam connectio

2ositive rotation is clocwise when looing in the positive i3direction.

http:44newtonexcel$ach.wordpress.com4+5*+4**4*64frame8now8with8added8algli$4

Beam end may $e released for translation or rotation. 9ote all end releases are relative to the #lo
http://newtonexcelbach.wordpress.com/2012/11/16/frame4-now-with-added-alglib/http://newtonexcelbach.wordpress.com/2012/11/16/frame4-now-with-added-alglib/


2/94

well.

arse solvers are not installed

deflections at $eam ends.

lts& $#tton.

ction

rallel to the Z axis3

s range.

alaxes.


3/94

3D Frame Analysis

Analysis of a 3D frame subject to distributed loads, point loads and moments

$eam Property %ypes

2roperty 9#m$er Area )* )+

* 5.; 5.5*5*667 5.5*66667 5.5+


4/94

9ode -oordinates =

Shear Area* Shear Area+ Shear ?od#l#s 9#m$er !

5.55E@55 5.55E@55 =,555,555 * 5.55 5.55

5.55E@55 5.55E@55 =,555,555 + ;.55 5.55

=,555,555 5.55 5.55

;.55 5.55

; 5.55 ;.55

6 ;.55 ;.55

7 5.55 ;.55

= ;.55 ;.55


5/94

Enter &F& for fixed restraint, or displacement val#e for prescri$ed displacement 0rot

9ode /estraints

Z 9#m$er ! Z ?

5.55 * F F F F5.55 + F F F F

.55 ; F F F F

.55 6 F F F F

5.55

5.55

.55

;.55


6/94

ation in radians1

Beam -onnections **

?! ?Z Beam 9#m$er 2roperty %ype 9ode * 9ode +

F F * * * F F + * +

F F * ; 7

F F * 6 =

; +

6 + 7 =

7 + =

= 7

< =

*5 7

** + ;


7/94


8/94

!+ Z+ ?+ ?!+ ?Z+

*75

*75

*75;

*756

+;;=

+;; /es#lts are o#tp#t for each listed point in '#t2oints MMMMMMM

> '#t 8 '#tp#t index: * 0defa#lt1 8 Shear, ?oment, Slope and eflection at each '#tpoint3 + 8 Force MMMMMMM

im SS/es01 As o#$le, SS/es+01 As o#$le, SS/es01 As o#$le, 9#msegments As ong, 9#

im i As ong, G As ong, As ong, oaden As o#$le, %otoad As o#$le, %ot?om As o#$le MMMMMMM

im N As o#$le, Ndash As o#$le, N9eg As o#$le, As o#$le, * As o#$le, + As MMMMMMM

im StartSlope As o#$le, EndSlope As o#$le, Startef As o#$le, Slope-hange As o#$le, 9 MMMMMMMim S* As o#$le, S+ As o#$le, S*9ode As ong, S+9ode As ong, Span As o#$le, Segm MMMMMMM

im S*ef As o#$le, S+ef As o#$le, S*/Force As o#$le, S+/Force As o#$le, ef-hange MMMMMMM

im 9#m/ows As ong, 9#m-ols As ong, 9#mB-ols As ong, 9#mSeg-ols As ong, 2oads+0MMMMMMM


68/94

im A As o#$le, EA As o#$le, BeamAng As o#$le, Beam-os As o#$le, BeamSin As o#$ MMMMMMM

im 2%rans As o#$le, 2Ax As o#$le, NA As o#$le, NAdash As o#$le, NA9eg As o#$le, 2 MMMMMMM

im End+% As o#$le, End+Ax As o#$le, End*efa0* %o *, * %o +1 As o#$le, End*efAng MMMMMMM

-onst %ol As o#$le O 5.5555555555555*

)f %ype9ame0Beam-oords1 O &/ange& %hen Beam-oords O Beam-oords.(al#e+ MMMMMMM

)f %ype9ame0BeamSect1 O &/ange& %hen BeamSect O BeamSect.(al#e+ MMMMMMM

)f )s?issing0'#t-ols1 O False %hen MMMMMMM )f %ype9ame0'#t-ols1 O &/ange& %hen '#t-ols O '#t-ols.(al#e+ MMMMMMM

9#m'#t-ols O PBo#nd0'#t-ols, +1 MMMMMMM

Else MMMMMMM

9#m'#t-ols O = MMMMMMM

End )f MMMMMMM

9#mB-ols O PBo#nd0BeamSect, +1

O Beam-oords0+, *1 8 Beam-oords0*, *1 MMMMMMM

! O Beam-oords0+, +1 8 Beam-oords0*, +1 MMMMMMM

Beamength O 0 Q + @ ! Q +1 Q 5.; MMMMMMM

Beam0*, *1 O Beamength MMMMMMM

BeamAng O A%n+0, !1 MMMMMMM > )f BeamAng R 5 %hen MMMMMMM

Beam-os O -os0BeamAng1 MMMMMMM

BeamSin O Sin0BeamAng1 MMMMMMM

> End )f MMMMMMM

MMMMMMM

)f )s?issing0EndefA1 O %r#e %hen MMMMMMM

End-ols O * MMMMMMM

Else MMMMMMM

)f %ype9ame0EndefA1 O &/ange& %hen EndefA O EndefA.(al#e+ MMMMMMM

End-ols O PBo#nd0EndefA, +1 MMMMMMM

End )f MMMMMMM S* O 5 MMMMMMM

S+ O Beamength MMMMMMM

Span O S+ 8 S* MMMMMMM

/eim '#t2oints0* %o 9#mSegs @ *, * %o *1

'#t2oints0*, *1 O 5

Segength O Span 4 9#mSegs MMMMMMM

9#m'#t O 9#mSegs @ *

For i O + %o 9#m'#t

'#t2oints0i, *1 O '#t2oints0i 8 *, *1 @ Segength

9ext i

)f )s?issing0oads1 O False %hen

oads O /ange+Array0oads, 9#m/ows, 9#m-ols, , , %r#e, +1

)f )s?issing09#mloads1 O %r#e %hen 9#mloads O 9#m/ows

/eim 2reserve oads0* %o 9#m/ows, * %o %rans start MMMMMMM

oads0i, 71 O 8oads0i, 1 M BeamSin > %rans end MMMMMMM

oads0i, =1 O oads0i, 1 M Beam-os > ong start MMMMMMM

oads0i, ong end MMMMMMM Else MMMMMMM


69/94

oads0i, 61 O oads0i, 1 M Beam-os > %rans start MMMMMMM

oads0i, 71 O oads0i, 1 M Beam-os > %rans end MMMMMMM

oads0i, =1 O 8oads0i, 1 M BeamSin > ong start MMMMMMM

oads0i, ong end MMMMMMM

End )f MMMMMMM

9ext i MMMMMMM

Else 9#mloads O 5

End )f


2oads O /ange+Array02oads, 9#m/ows, 2oad-ols, , , False1

)f )s?issing09#m2loads1 O %r#e %hen 9#m2loads O 9#m/ows

For i O * %o 9#m/ows MMMMMMM

)f BeamAng R 5 %hen MMMMMMM

2Ax O 802oads0i, +1 M Beam-os @ 2oads0i, 1 M BeamSin1 MMMMMMM

2%rans O 2oads0i, 1 M Beam-os 8 2oads0i, +1 M BeamSin MMMMMMM

Else MMMMMMM

2Ax O 82oads0i, +1 MMMMMMM 2%rans O 2oads0i, 1 MMMMMMM

End )f MMMMMMM

2oads0i, +1 O 2%rans >2%rans MMMMMMM

2oads0i, 1 O 2Ax > 2Ax MMMMMMM

9ext i MMMMMMM

Else

9#m2loads O 5

End )f

/eim 2oads+0* %o 9#m2loads, * %o ;1

> MMM MMMMMMM

> )nsert additional nodes if reD#ired at segment ends and s#pports.

> )nsert segment ends

Segments O -om$ineArray0Beam, '#t2oints, 9#m, , , %r#e1

9#m9odes O 9#m

9#msegments O 9#m 8 *

)f 9#mB-ols R + %hen MMMMMMM

/eim SS/es0* %o 9#m9odes, * %o =1 MMMMMMM /eim SS/es+0* %o 9#m9odes, * %o =1 MMMMMMM

Else

> )ncl#de shear slope and deflection and total slope and deflection

/eim SS/es0* %o 9#m9odes, * %o **1 MMMMMMM

/eim SS/es+0* %o 9#m9odes, * %o **1 MMMMMMM

End )f MMMMMMM

S*9ode O *

S+9ode O 9#m9odes

/eim Segments+0* %o 9#m9odes, * %o 9#mB-ols @ *1

O *

For i O * %o 9#m9odes


70/94

Segments+0i, *1 O Segments0i, *1

)f Segments+0i, *1 R 5 %hen

)f 0Segments+0i, *1 8 Beam0, *11 4 Segments+0i, *1 %ol %hen O @ *

End )f

Segments+0i, +1 O BeamSect0, *1 MMMMMMM

)f 9#mB-ols * %hen Segments+0i, 1 O BeamSect0, +1 MMMMMMM

)f 9#mB-ols + %hen Segments+0i, 1 O BeamSect0, 1 MMMMMMM

9ext i

> MMM

>Solve ass#ming "ero rotation at node *

> SS/es col#mns: SS/es+ col#mns: MMMMMMM

> *: *: Shear force MMMMMMM

> +: Shear force +: Bending moment MMMMMMM

> : Bending moment : Axial force MMMMMMM

> : Slope : MMMMMMM

> ;: eflection d#e to $ending ;: ! MMMMMMM

> 6: Axial force 6: /Z MMMMMMM

> 7: Axial deflection 7: %rans MMMMMMM

> =: Shear slope =: Ax MMMMMMM >


71/94

SS/es0i, +1 O SS/es0i, +1 @ N M * @ Ndash M * Q + 4 + @ N9eg M + 8 Ndash M + Q

SS/es0i, 1 O SS/es0i, 1 @ N M * Q + 4 + @ Ndash M * Q 4 6 @ N9eg M + Q + 4 + 8 N

SS/es0i, 1 O SS/es0i, 1 @ 0N M * Q 4 6 @ Ndash M * Q 4 + @ N9eg M + Q 4 6 8

SS/es0i, ;1 O SS/es0i, ;1 @ 0N M * Q 4 + @ Ndash M * Q ; 4 *+5 @ N9eg M + Q 4

SS/es0i, 61 O SS/es0i, 61 @ NA M * @ NAdash M * Q + 4 + @ NA9eg M + 8 NAdash M MMMMMMM

SS/es0i, 71 O SS/es0i, 71 @ NA M * Q + 4 + @ NAdash M * Q 4 6 @ NA9eg M + Q + 4 MMMMMMM

9ext G

For G O * %o 9#m2loads

* O 2oads0G, *1

* O x 8 *

K O 2oads0G, +1

2A O 2oads0G, 1 MMMMMMM

>)f * R 5 %hen * O 5

)f * 5 %hen

SS/es0i, +1 O SS/es0i, +1 @ K

SS/es0i, 1 O SS/es0i, 1 @ K M *

SS/es0i, 1 O SS/es0i, 1 @ 0K M * Q + 4 +1

SS/es0i, ;1 O SS/es0i, ;1 @ 0K M * Q 4 61 SS/es0i, 61 O SS/es0i, 61 @ 2A MMMMMMM

SS/es0i, 71 O SS/es0i, 71 @ 2A M * MMMMMMM

End )f MMMMMMM

9ext G

)f 9#mB-ols * %hen MMMMMMM

>AdG#st shear slope and deflection for A

A O Segments+0i, 1

)f A 5 %hen

SS/es0i, =1 O SS/es0i, +1 4 A

SS/es0i,


72/94

SS/es0i, *1 O x

9ext i

> AdG#st slope and deflection for E)

MMMMMMM

For i O + %o 9#m9odes

E) O Segments+0i, +1 MMMMMMM

x O Segments+0i, *1 O x 8 Segments+0i 8 *, *1

)f i O + %hen StartSlope O SS/es0i 8 *, 1 Else StartSlope O EndSlope

)f i O + %hen Startef O SS/es0i 8 *, ;1 Else Startef O Endef

EndSlope O StartSlope @ 0SS/es0i, 1 8 SS/es0i 8 *, 11 4 E)

Endef O Startef @ StartSlope M @ 0SS/es0i, ;1 8 SS/es0i 8 *, ;1 8 0SS/es0i 8 *, 1 M 11

SS/es0i 8 *, 1 O StartSlope

SS/es0i 8 *, ;1 O Startef

9ext i

i O i 8 * SS/es0i, 1 O EndSlope

SS/es0i, ;1 O Endef

> inBeam+ O SS/es MMMMMMM

>Exit F#nction MMMMMMM

)f 9#mB-ols * %hen MMMMMMM


SS/es0i, ;1 O SS/es0i, ;1 @ SS/es0i, AdG#st deflection MMMMMMM

End+% O 80EndefA0+, +1 8 EndefA0*, +11 M BeamSin @ 0EndefA0+, 1 8 EndefA0*, 11 M Bea MMMMMMM

End+Ax O 0EndefA0+, +1 8 EndefA0*, +11 M Beam-os @ 0EndefA0+, 1 8 EndefA0*, 11 M Be MMMMMMM

ef-hange O 0SS/es0S+9ode, ;1 8 End+%1 MMMMMMM

)f ef-hange R 5 %hen MMMMMMM

Slope-hange O ef-hange 4 Span MMMMMMM

For i O + %o 9#m9odes MMMMMMM

SS/es0i, 1 O SS/es0i, 1 8 Slope-hange MMMMMMM

SS/es0i, ;1 O SS/es0i, ;1 8 Slope-hange M SS/es0i, *1 MMMMMMM

9ext i MMMMMMM

End )f MMMMMMM

O * MMMMMMM

/eim SS/es+0* %o 9#m9odes, * %o MMMMMMM


73/94

SS/es+0i, 61 O 0SS/es0i, ;1 M Beam-os @ SS/es0i, 71 M BeamSin1 > ! MMMMMMM

SS/es+0i, 71 O SS/es0, 1 MMMMMMM

SS/es+0i, =1 O SS/es0, ;1 MMMMMMM

SS/es+0i, Add initial deflection of start node MMMMMMM

)f EndefA0*, +1 R 5 'r EndefA0*, 1 R 5 %hen MMMMMMM inc O 0EndefA0+, +1 M 5 @ EndefA0*, +11 4 Span MMMMMMM

!)nc O 0EndefA0+, 1 M 5 @ EndefA0*, 11 4 Span MMMMMMM

End*efa0*, *1 O EndefA0*, +1 MMMMMMM

End*efa0*, +1 O EndefA0*, 1 MMMMMMM

End*efAng O A%n+0EndefA0*, +1, EndefA0*, 11 @ 2i 4 + MMMMMMM

End*/tn O 00EndefA0*, +1 Q + @ EndefA0*, 1 Q +1 Q 5.; M -os0BeamAng 8 End*efAng11 M MMMMMMM

MMMMMMM

For i O * %o 9#m9odes MMMMMMM

SS/es+0i, =1 O SS/es+0i, =1 @ End*/tn MMMMMMM

SS/es+0i,


74/94

8a*

8a+ a

8a 8a;

8a=

a; a74+

8a a64+

a*

a+ 8a

a a;

a=

a; a7

8a a6

8a*

-a3 a+

-a! -a

8a=a a.:!

-a+ a0:!

a*

a3 -a+

a! a

a=

a a.

-a+ a0

Frame8sheardef

F#nction inBeam0ength As o#$le, BeamSect As (ariant, 9#mSegs As ong,'ptional By(al EndefA As (ariant, 'ptional By(al 9#mloads As (ari

'ptional '#t-ols As (ariant, 'ptional '#t As ong O *1 As (ariant

> Find deflected shape for $eam #nder specified transverse loads #sing local axis sy

> /es#lts are o#tp#t for each listed point in '#t2oints

> '#t 8 '#tp#t index: * 0defa#lt1 8 Shear, ?om, /otation, eflection

> '#t O +: 8 Force and moment im$alance and deflection and rotation difference

im SS/es01 As o#$le, SS/es+01 As o#$le, SS/es01 As o#$le, 9#msegment

im i As ong, G As ong, As ong, ei As o#$le, Segments+ As (ariant, 9#m A

im N As o#$le, Ndash As o#$le, N9eg As o#$le, As o#$le, * As o#

im StartSlope As o#$le, EndSlope As o#$le, Startef As o#$le, 2oad-ols Aim S* As o#$le, S+ As o#$le, S*9ode As ong, S+9ode As ong, Segment

im End-ols As ong

im 9#m/ows As ong, n#mcols As ong, 9#mB-ols As ong, 2oads+01 As o#


75/94

im A As o#$le, ea As o#$le, Beam0* %o *, * %o *1 As o#$le

im 9#m'#t-ols As ong

-onst %ol As o#$le O 5.5555555555555*

)f %ype9ame0BeamSect1 O &/ange& %hen BeamSect O BeamSect.(al#e+

)f )s?issing0'#t-ols1 O False %hen

)f %ype9ame0'#t-ols1 O &/ange& %hen '#t-ols O '#t-ols.(al#e+ 9#m'#t-ols O PBo#nd0'#t-ols, +1

Else

9#m'#t-ols O =

End )f

9#mB-ols O PBo#nd0BeamSect, +1

Beam0*, *1 O ength

)f )s?issing0EndefA1 O %r#e %hen

End-ols O *

Else )f %ype9ame0EndefA1 O &/ange& %hen EndefA O EndefA.(al#e+

End-ols O PBo#nd0EndefA, +1

End )f

S* O 5

S+ O ength

/eim '#t2oints0* %o 9#mSegs @ *, * %o *1

'#t2oints0*, *1 O 5

Segength O ength 4 9#mSegs

9#m'#t O 9#mSegs @ *

For i O + %o 9#m'#t

'#t2oints0i, *1 O '#t2oints0i 8 *, *1 @ Segength

9ext i


oads O /ange+Array0oads, 9#m/ows, n#mcols, , , %r#e, +1

)f )s?issing09#mloads1 O %r#e %hen 9#mloads O 9#m/ows


76/94

Else 9#mloads O 5

End )f


2oads O /ange+Array02oads, 9#m/ows, 2oad-ols, , , False1

)f )s?issing09#m2loads1 O %r#e %hen 9#m2loads O 9#m/ows

Else

9#m2loads O 5

End )f

/eim 2oads+0* %o 9#m2loads, * %o ;1

> )nsert additional nodes if reD#ired at segment ends and s#pports.

> )nsert segment ends

Segments O -om$ineArray0Beam, '#t2oints, 9#m, , , %r#e1

9#m9odes O 9#m

9#msegments O 9#m 8 *

)f 9#mB-ols R %hen

/eim SS/es0* %o 9#m9odes, * %o *+1

Else

> )ncl#de shear slope and deflection and total slope and deflection

/eim SS/es0* %o 9#m9odes, * %o *1

End )f

S*9ode O *

S+9ode O 9#m9odes

/eim Segments+0* %o 9#m9odes, * %o 9#mB-ols @ *1

O *



77/94

Segments+0i, *1 O Segments0i, *1

)f Segments+0i, *1 R 5 %hen

)f 0Segments+0i, *1 8 Beam0, *11 4 Segments+0i, *1 %ol %hen O @ *

End )f

Segments+0i, +1 O BeamSect0, +1

)f 9#mB-ols + %hen Segments+0i, 1 O BeamSect0, 1

)f 9#mB-ols %hen Segments+0i, 1 O BeamSect0, 1

9ext i

> MMM

>Solve ass#ming "ero rotation at node *

> SS/es col#mns:

> *:

> +: Shear force

> : Bending moment

> : Slope

> ;: eflection d#e to $ending

> 6: Shear slope

> 7: Shear deflection

SS/es0*, 1 O 5

SS/es0*, ;1 O 5

SS/es0*, 71 O 5

)f 2oads0*, *1 O 5 %hen SS/es0*, +1 O 2oads0*, +1

)f 2oads0*, 1 O 5 %hen SS/es0*, 1 O 2oads0*, 1


5 O Segments+0i 8 *, *1

O Segments+0i, *1

O 8 5

For G O * %o 9#mloads

* O oads0G, *1

+ O oads0G, +1

N O oads0G, 1 N9eg O 8oads0G, 1

)f + R * %hen

Ndash O 0oads0G, 1 8 oads0G, 11 4 0+ 8 *1

Else

Ndash O 5

End )f

* O 8 *

)f * R 5 %hen * O 5 + O 8 +

)f + R 5 %hen + O 5


78/94

SS/es0i, +1 O SS/es0i, +1 @ N M * @ Ndash M * Q + 4 + @ N9eg M + 8

SS/es0i, 1 O SS/es0i, 1 @ N M * Q + 4 + @ Ndash M * Q 4 6 @ N9eg M

SS/es0i, 1 O SS/es0i, 1 @ 0N M * Q 4 6 @ Ndash M * Q 4 + @ N9eg

SS/es0i, ;1 O SS/es0i, ;1 @ 0N M * Q 4 + @ Ndash M * Q ; 4 *+5 @ N9

9ext G

For G O * %o 9#m2loads

* O 2oads0G, *1

* O 8 *

K O 2oads0G, +1

>)f * R 5 %hen * O 5

)f * 5 %hen

SS/es0i, +1 O SS/es0i, +1 @ K

SS/es0i, 1 O SS/es0i, 1 @ K M *

SS/es0i, 1 O SS/es0i, 1 @ 0K M * Q + 4 +1

SS/es0i, ;1 O SS/es0i, ;1 @ 0K M * Q 4 61 End )f

9ext G

)f 9#mB-ols + %hen

>AdG#st shear slope and deflection for A

A O Segments+0i, 1

)f A 5 %hen

SS/es0i, =1 O SS/es0i, +1 4 A

SS/es0i,


79/94

SS/es0i, *1 O

9ext i

> AdG#st slope and deflection for E)

ei O BeamSect0*, +1


O Segments+0i, *1 O 8 Segments+0i 8 *, *1

)f i O + %hen StartSlope O SS/es0i 8 *, 1 Else StartSlope O EndSlope

)f i O + %hen Startef O SS/es0i 8 *, ;1 Else Startef O Endef

EndSlope O StartSlope @ 0SS/es0i, 1 8 SS/es0i 8 *, 11 4 ei

Endef O Startef @ StartSlope M @ 0SS/es0i, ;1 8 SS/es0i 8 *, ;1 8 0SS/es

SS/es0i 8 *, 1 O StartSlope

SS/es0i 8 *, ;1 O Startef

9ext i

i O i 8 * SS/es0i, 1 O EndSlope

SS/es0i, ;1 O Endef

)f 9#mB-ols + %hen


SS/es0i, ;1 O SS/es0i, ;1 @ SS/es0i,


80/94

)f )s?issing0EndefA1 O False %hen

SS/es0*, 1 O EndefA0*, 1 SS/es0*, ;1 O EndefA0*, +1

)f SS/es0*, 1 R 5 'r SS/es0*, ;1 R 5 %hen


O Segments+0i, *1

SS/es0i, 1 O SS/es0i, 1 @ EndefA0*, 1

SS/es0i, ;1 O SS/es0i, ;1 @ EndefA0*, +1 @ EndefA0*, 1 M

9ext i

End )f

End )f

)f 9#m'#t-ols O = %hen

inBeam O SS/es Else

/eim SS/es0* %o 9#m9odes, * %o 9#m'#t-ols1


For G O * %o 9#m'#t-ols

O '#t-ols0*, G1

SS/es0i, G1 O SS/es+0i, 1

9ext G

9ext i

inBeam O SS/es

End )f

End F#nction


81/94

xl 4 ell yl 4 ell "l 4 ell

08xl M yl M -g 8 ell M "l M Sg1 4 en en M -g 4 0ell M ell1 08yl M "l M -g @ ell M xl M Sg1 4 en

0xl M yl M Sg 8 ell M "l M -g1 4 en 8en M Sg 4 0ell M ell1 0yl M "l M Sg @ ell M xl M -g1 4 en

5 * 5

8-g 5 Sg

Sg 5 -g

xl 4 ell yl 4 ell "l 4 ell


82/94


83/94


84/94


85/94


86/94

i 8 *, 1 M 11 4 ei


87/94


88/94

le


89/94


90/94


91/94


92/94


93/94


94/94

Analysis of a 3D Frame

Documents