1989-4 analisis of curved expandable space bar structures
DESCRIPTION
The paper present a new way tor solution oí non-linear analysis oí curved expandable space bar structures. Matricial methods are used combined wi th the eífects oí stabili ty functions aad eífects oí geometric changes. Several applications at cilindrical and spherical expanTRANSCRIPT
'.~
" ., .
. :"
ANALYSIS OF CURVED EXPANDABLE SPACE
BAR STRUCTURES.
by
.'
•• "1. -
JUAN P. VALCARCEL. Catedrático ETS Arquitectura La Coru~a. Spain.FELIX ESCRIG. Proí. Titular ETS Arquitectura Sevilla. Spa1n.
SUM.KARY
The paper present a new way tor solution oí non-linear analysis oícurved expandable space bar structures. Matricial methods are used combinedwi th the eífects oí stabili ty functions aad eífects oí geometric changes.Several applications at cilindrical and spherical expandable structures ar91ncluded.
---;-- ... ---r--_..... . -- "~':.. :..... ~~ - ""'r'" tt-. _- ,.
13
~
A.- INTRODUCTION.
Expandable bar structures are articulated structures that can mod1fytheir shape from completely folded, with a11 bar in a bundle, to expandeá,covering a great area. They are isostatics or cuasi-isostatics structuresbecause there must be a mechanism during the expansiono
Structural components are bars articulated at the ends with a joint atan inner point. Standard programs are unable to solve this problem and wewill define a computer program in order to solve that. Equilibrium matrixmethods are used.
Bar equilibrium is completely defined by the system oí forces Pl, P2,Nl, N2 . Reíerences (2) and (3) explain linear calculus oí thesestructures. However the efíect oí inner articulation make a bending on thebars and axial íorces modifies the value oí the bending moments oí thebars. Fig 1 shows the equilibrated system oí forces and reactions with nonlinear eífects ofaxial forces.
y
RZ2
z
,N~',,'.'
• l.,
'.."
. ',,-~
.,.~ ..
figure 1
STIFFNESS MATRIX IN LOCAL CooRDINATES.
P = K . Z- -E A
Nl = --- Ul11
3 E I 1Pl = -------- 01
;11 12 122
E AB2 = --- U2
12
3 E 1 1P2 = -------- 02
;211 2 122
';~'tl - _--'-"'"':2.-~...... ,_a'••4 ..... _ ......_~__ ~_ • .. , .........:.-.
In matrix form
E A11, 1 I --- O
1,
E A.'
lb I I O12
= II
p, I I O O
o O I I U:zI II I
3 E 11 1-------- O I I o,1,1,~122
3 E 1:2 1O -------- I I 02
121,21:z2
"~
p:z o o
o o u,
COMPATIBILITY MATRIX.- Orientation oí local axis like fig 2.
x axis ~ direction oí bar.y axis ~ direction of normal to plane oi scissor. 'tie can obtIl1.11 by
vectorial product oi vectors of bars.z axis ~ d1rectioo perpendicular of x and y axis.
figure 2
Direct10ns of three axis are io a general case:
-
'.
x axis ~
y axis ~
z axis ~
(cos a" cos ~ll cos ~,)
(cos ~Zl cos ~21 COS ~2)
(cos ~3, cos ~3, cos ~3)
:'1.;:'1.
.........'1 , """::'~ ~.-'-r-"', .:,:•. -.-: , ..... ".--; •.. ~",~",.:"".,,..._r,._·,
o
~
If the expandable structure is made by bundle madulus we can nat defineaplane ai scissqrs. In this case the definitian af local axis can be madefaro
x axis -l cas a , cas ~ • cas ~ )
cos j3 cas ocy axis -l (-
sen 't sen 't
cas a . cas 'tz axis -l (- -------------
sen't
o )
cas j3 . cas 't
sen 't, sen 't )
After defining lacal axis, the relationship between local and globalsystems are
U'2 - U'l
U l3 - U l
2
Y= OZ-Ol. L;:~/l
-03.11/1
w= ,!2-';1' lzl1-13.1111
U 1 =UZ =
11.'2
-._.-"_.- -
"1,..."
Ll __ ~ __--------------r1------ ~ ,~ _---- I 1
., ..;r--- ' 'e ' ' c. '~ ,!}~ :~~.q;e='=-=~~~.~ 11
Figure 3I J I XlI Ul I I YII 1 I ZII Uz I I X2
In matrix form Z = A . X I I = A I Y2I y I - 1 zzI I 1 X3I W I I Y3I I I Z3
The compatibility matrix A is--casal -COSj3l -COS'tl cosal cas~l COS'tl O O O
O O O -casal -COSf3l -COS'tl cosal COS,Bl COS'¡'I
-lzcasaz -12cosj32 -lzcos't2 -11cosa2 -11casf32 -llcos'tz------- ------- ------- casaz cos.lh cos'tz -------- ------- -------
1 1 1 1 1 1
-lzcasa3 -b~cosJb -12cos't3 -11cosa3 -11COSf33 -11cas't3------- ------- ------- cosa3 cos.lb COS't3 ------- ------- -------
1 1 1 1 1 1
';
.~, 'l'''' .. -~...- ..... :'":'"'. ...,...,...------,..~~'''''''''-''''''''''--'''--'''''-~-~-'-.~-' '''''~ ':; .""-".---
e
~
STIFFNESS MATRIX IN GLOBAL COORDIRATES.
Matrix equation 15 L = A~ . K . A . X = S . X
Stiffness matrix in global eoordinates wi11 be.
a+m.kl b+n.kl e+o.kl
d+p.kl e+q.kl
f+r.kl
-a-m.k3 -b~n.k3 -e-o.k3
-b-n.k3 -d-p.k3 -e-q.k3
-e-o.k3 -e-q.k3 -f-r.k3
m.k5
n.k5
o. k5
n.k5
p.k5
q.k5
0.k5
q. k5
r.k5
a+g+m b+h+n e+i+o 1-&-m.k4 -h-n.k4 -i-o.k4I
S= I d+j+p e+k+q l-h-n.k4 -j-p.k4 -k-q.k4I
f+l+r l-i-o.k4 -k-q.k4 -1-r.k4
g+m. k2 h+n.k2 ita. k2
j+p. k2 k+q.k2
l+r. k2
being
122 1,2 12 1, 11.1 2
kl= ---- k2= --- k3= --- k4= --- k5=p~ 12 1 1 12
E.A E.Aa = ----- eos2 al g = ----- eos2 a,
1, b~
E.A E.Ab = ----- eos a, cos .13, h = ----- cos a, cas .131
1, 12
E.A E.Ae = ----- cos a, cos ~, i = ----- cos al cos ~,
1, h~
E.A E.Ad =----- cos2 .13, j = ----- eos2 ~1
1, 12
E.A E.Ae = ----- cos .13, cos y, k = ----- COS ~1 COS ~,
1, 12
E.A E.Af = ----- cos2 ~, 1 = ----- cos~ ~1
1, 12
. ~ ¡......-. •• .._~~ •. III.\·_-·~ ....~.•.~_.'-:-~"",~.:_.,.-_. .<;...... ·••",.........--;·v .... _ . .,..... -.~.... .. ,.L1"":"':·....~--~~:--:-··~-~ ..·"""""':'.-:-------:~7~~~1.;-.- " -:':~:~ --;,~
&
..
ro =3E1,1
;, 1,2 1:;::::ces2 a2 +
3EIz1
'J2 1 , 2 122
ces.2 a8
3E1,1 3E121n = ---------- ces a, ces j3, + ---------- ces a, ces j3,
11' 1,~ 122 12 1,2 122
3E1,1 38121e = ---------- ces a, ces ~, + ---------- ces a, ces '11
1, 1,2 122 12 1, 2 122
3E1,1 3Eb1P = ---------- ces.2 j3, + ---------- ces2 j3,
1, 1,:2 1:;:2 ;2 1,2 b:2
3E1,1 3E121q = ---------- ces ~1 ces 'i, + ---------- ces j3, ces 't,
~, 1,2 122 '2 1,2 122
r =3E1,1
;, 1,::;: 1:;:2cas2 'tl +
3E121
;2 1,2 122
ces.2 'i,
C.- STRESS ANALYS1S.- Frem stlffness equatien in 1eca1 ceerdinates:
E.A E.AN, = ----- u, Ji:¡: = ----- U2
1, 12
3.8.1,.1 3.E.b.1p, =---------- P2 = ----------
~, 1,2.12:2 ;2 1 , 2 .122
Being
u, = -x, ces a, - y, ces j3, - z, ces 'il + X2 ces a, + Y2 ces j3, + Z2 ces 'i,
U2 = -X:;: ces a, - y:;: cas j3, - Z:2 cas 1, + X8 cas a, + Y8 cas t3, + Z8 ces 'i,
v = -x, <1:;:/1) ces a:;: - y, (1211) cas 132 - z, <12/1> ces 'h + X2 ces a2 +Y2 cas j32 + Z2 ces 'i2 - X3(1,/1)ces a2 - Y3(1,/l)ces j32- z3<l,/1)ces 'i2
w = -x, (1211) ces 0:3 - y, (1211) cas ~s - z, (12/1) cas 'i3 + x:;: cas a3 .;Y2 ces j33 + Z;z ces 'is - x3(1,/1)cas a3 - Y3(1,/l)ces j33- z3(l,/1)cas 'i3
Bending mementsP, 1, 12 N, 12 lb 1,
M., = -------- - ----- ó, - ----- Ó,
1 "1 1
: . " P2 1, 1:;: N, 12 lb 1,I M2 = -------- - ---7- Ó2 - ----- Ó2~
1 1 1
:- ,. -~----.-'''"7,.. ....-,....... ...-------." :.-::~'_..r.-:..~~~~ -~ 7!'.-......-::-:-::-:;..:~ •.-: .,,"To#-:.:'--:_+IV; ...~.~-.. ~·:"':7~.r-:.;-:-_'r~-"'·"· :.- -~. #~
•
:>
Shear stresses
p, 12 N,-82 P2 12 8,-82V,,= ----- + ----- O, V2'= ----- + ----- 02
1 1 1 1
P, 1, 11', -82 P2 1, B,-82V'2= ----- - ----- o, V22= ----- - ----- 02
1 1 1 1
CALCULUS OF STABILITY FUICTIOIS.
The process of ca1culus above explained needs the formulation ofstability functions because the effect of inner articulation makes that, ingeneral, axial forces are different in two spans. The way chosen is theintegration of differential equations of bar in two spans and apply theboundary conditlons. Axial forces can be compression, traction or null. Bycomblnation of two forces there are five differents cases.
a.- N, < O N2 < O
We suppose the bar in deflected position <fig 4).
p
• 1, --;i;¡1, -_._-.__.__=_:::~_=._~._=-:_._...
N'''~ L/lt~~ J:7 fh,li1-ék~.Eh. + N2-N, á - - - - - . I lI I N
2_ NI
Figure 4
beingN,-lb
q, =x,
q2 =N,-N2
N2
Bending moments in both spans are
P 12
M:, = -----1
x +N,-N2
ó x + B, y1
PI, 8,-82M:2 = ------ <l-x) + ------- Ó <l-x) + N2 y
1 1
:'.~, ...._~,..,.., - -'~l .~'~~":;7t""~"'?~."""""'::'~.~':..rl"""'.'~ •....-r--~.-.~-- .. -,' _ ~ --: - ...,...-, •• ;~>:-~ ~. "-.~.~, - _""'r>'''_'~~:_'''' .... ••., ~.-u-, .---- ~-~.--- - -.' .-:- r: ~;;.""~':-;r'"".,-;.- ~", o:. ~.
•o
solvig different1al equations and roaking
IN, Ip,~ =
El
and with the boundary conditions
pZ2 =Ilbl
El
x = Ox = 1x = 1,x = 1,
x = 11
we can obtain
y = OY = Oy, = óyz = Ódy, dY2
=dx dx
1 1 1 1( ----- + ----- ) - ---------- - ----------
Pl,2l z 2 p,2l, p22l2 p, tg p,l, P2 tg p2l2
Ó = -------- 3 ---------------------------------------------------------31 El
1,12 [p,
tg p,l,(l-q,1,) +
P2
tg p2h(1+q212) + <q,-q2)]
"
Deflection in lineal calculus 1s
Pl,21 2 2
v =31 El
we can see that in non lineal calcu1us
Pl,2l z 2
Ó = v.; = -------- ;31 El
Being ; the stability function
1 1 1 1( ----- + ----- ) - ---------- - ----------
p,2l, P2212 p, tg p, 1, P2 tg p2bI = 3 ---------------------------------------------------------
~ .._._......... 4"'_~_ +- "1',."
1,12 [p,
tg p,l,
P2<l-q, 1,) +
tg p2h
--" ....~~.
(1+q212) + <q,-q2)]
I~-'I""""'-'r-_:::~"'~~~.r-~.... "..-...:.~." .......
•~·•
•
ti
b. - N, ) O N:<: ) O
1 1 1 1- ( ----- + ----- ) + ---------- + ----------
p,21, p22 12 p, th p,1, P2 th p212; = 3 ---------------------------------------------------------
PI P21112 [ ------- (1-ql11) + ------- <l+q2b) + <ql-q2»)
th p,l, th p2b
c. - N, < O N:<: ) O
1 1 1 1( ----- - ----- ) - ---------- + ----------
p,21, p2212 PI tg PI 1, P2 th p2b1 = 3 ---------------------------------------------------------
d.- N, < O
1,12 l
N2 = O
p,
tg pl1,
P2(l-ql1,) +
th p2b(1+q212) + (q,-q2»)
+ --- + ---- - --------
3- ---------- + 12
p, tg P,11; =
e.- N, ) O
1,1 2
lh = O
3
p,:21,
p,
tg p,l,
12
1
1
1
1
12
p,="h:2
3 1
3 3- ----- + ---------- + 12
p,21, p, tg pd,; =
p,1,12 [ -------
tg p,11
1,
1
1+ --- +
1
1
12
p,2122
+ --------3 1
.'
.l;
BEHAVIOUR OF STABILITY FUNCTIO~S .
For the study oí the behaviour oi stabili ty functions we take areference in the critical load R' oí bar with the sama axial force in bot~
spans.
1'(2. E. 1}J' = ------
12
-,'-- ---'''- -:; .....;.,..,,-
~
"
In fig 5 we can observefor different relationship of
the behaviour oí severallengh in both spans.
stability functions
FunclonG& utcbllldcxlt 11- I l2- 1
-\.S
-1.1... 1
.... 1...,
.... 1.1 .. ..
.1 .. \..\.1
\.S
~" ,'':J'>
......... ...", ...,.....,.",>'" -la-.
I.'"'
)' -.1....,,, .... S·l
. '"');:" __ ¡.J,::' o!···1
..T,1.1t.,'
l2.3FLK1CIOflQIt ..tcbllldado 11- I
,<::<.<:):= ....\L~<. ...
.,.! )(-<.. ,.,:-.c.. ,,0( ......
.,,< ..("...... '> ....... l~~"'''''''>~<'''''''''''-:''''''''v>'''«~<.:<, 'J....<;~
-t.I .... '" ....... >......" ....
-u "'_', _',' ~.•..... l ........ ~"" .,... .... ~'.., _\.,
..... ',< , '>', J _\.,
...1 , .....,,::-..;. -.1
....S '<.' ....:',.,' " •. ;.T-.S ... , ' ....,,;... ,,"""" _•••• 1
.1 ',< .1•• '~, .1
., '.... .S.7 ........ -< ".l
.• ',< r)La ........... l0
1.1 L5 l.J
....
Figure 5
EXAMPLES.- From the corresponding analytlcalseveral kinds of curved expandable space grids,
program, we have studiedwith the following results.
.'
.,. ~~ C~-:-:---"'~~~.,· ---'.~'': , ....... ~l!''"':'"' .....,
o
0., •
"
';" .0
", '.' '
CYLINDRICAL TVlD-IiAY GRIDS. <Fig 6). - 10 achieve angular stiffness, cablewere diagonaly placed to improve the behaviour of textil cover. Most partoí them are compressed and are not 'necessary except as stirfers. The betterbehaviour is for the grids fixed at boundary or with stiffers connectingboundary joints of two 1ayers._. ! '
t2m
r9.82 m
1I 15m
11=14= 2,73 m j 12=13= 2,45 m i q= 50 kg/mr
LINEAR CALCULUS. <Lower boundary joints are fixed). 4 iterations.
Dma,,: = 30,5 cm P = 4939,9 kg
NON-LINEAR CALCULUS. 2 iterations.
Omax = 31, O cm P = 4967,9 kg
NON-LINEAR CALCULUS <great def1ections):
7~h iteration 60max = 1,75 cm P = 4967,9 kg
Stresses in bars.
I I IBar I O' (linear) I O' <non-linear) M <great defl.)
I II I
6 I 1866,8 I 1896,2 231,6II
46 I 1980,9 I 2011,5 113,2II
80 I 2380,4 I 2415,4 68,0I II I
114 I 2477.9 I 2501,7 66,5
I
.!-,-~~.~_,_ .. ,".--,--_ "C_'---:--~_-:-_''''_'''~'" ."l".c.-'""':-' • ..,. ~••~ -- -"-.~ '-e- .-- _ ..,
!'l
CILINDRICAL THREE-WAY GRIDS. <Fig 7>. - Calculus oí these structures havespecial characteristics as impose initial bending in several bars íorgeometric compat1bility during the expansiono Structural behaviour 1s verygood but their weight is relativily high.
" ..
....'
• 't.
11=Lt= 1,05 m12=1 3 = 0,95 m0:= 12,5"r= 4 m : h= 2 roq= 50 kg/JIt'Z
LINEAR CALCULUS. <Lower boundary joints are fixed>. 5 iterations.
Omax = 9,96 cm P = 843,03 kg.
NON-LINEAR CALCULUS. 2 iterations.
o",.x = 1O, 1 cm P = 843,03 kg
""!
:.1
. '~j
,;:'.' ..".l" j
NON-LINEAR CALCULUS <great deflectians>;
8~h iteración 6om~x = 0,071 cm P = 8-43,03 kg
stresses in bars.
I I IBar I O' <linear) I O' <non-linear> I M <great defl. >
II
11 I 1314,9 1 1317,2 I 12,1!I
41 I 545,3 897,8 1 13,2II
68 I 652,1 656,2 I 8,7II
93 I 1879,0 1892,5 I 6,5
n
SPHERICAL TWO-WAY GRIDS WITH BUNDLE MODULUS <F1g. 8),- Is a very1nterest1ng application to cover great areas. Structural behaviour 1s goodif boundary jo1nts are f1xed.
~I 12.95 m
" '(
11=1,751 mi 12=1,713 m13=1,713 mi 14=1,751 mq= 50 kg/m2 a = 4º
LINEAR CALCULUS, <Lower boundary joints are fixed). 9 iterations.
6msx = 9,28 cm P = 788,9 kg
NON-LINEAR CALCULUS. 3 iterations.
ÓO's>.: = 9.60 cm P = 788,9 kg
NON-LINEAR CALCULUS <great deflections):
13~h iteración 6ó m ax = 0,0228 cm
Stresses in bars.
",~
Bar
2
12
. 22
32
O' <linear)
1061,4
1041,8
372,2
1090,9
O' (non-11 near)
1087,4
1084,2
417,3
1009,2
60' <great defl.)
4.9
16,6
14,4
1,0
"
SPHERICAL THREE-WAY GRIDS VIT» SCISSOR MODULUS (F1g 9).- In thesestructures 1t 15 only poss1ble to guarantee tbe compatibility oí geometryin tolded and unfalded posit1ons. Tbe structure goes through 1ntermediatestages in which it has to be forced with an energy input. lf it 1scoropletely deployed has a very good structural behaviour, specially if tbelower jaints of boundary are fixed.
I
II
I
I
I ---- --- I_._~._.- '-'-30mI
q= 50 kg/nt2d= 30 m
LINEAR CALCULUS. (Lower boundary joints are fixed). 4 iterations.
0,,, ... ><: = 2,59 cm P = 1743,8 kg
NON-LINEAR CALCULUS. 3 1terations.
Om&><: = 3,07 cm P = 1743,8 kg
\_..:.-
NON-LINEAR CALCULUS (great deflections):
'8~n 1teration: 60mb"" = 0,044 cm
_... _- __ ._-.._...= ,~,,_~.,,"""""9 ,_... ..~ ,_..._..........~_·1';"·."":·n-~~_""'"'T'ó:.-"f'... ;•.-.-..... ...~ - 't", '.'- -•. ~"!o",\,,,,,,~,-~........ , -';:",,:... ~.-- ~~= "l ....... _.~::':. "_-.._r --.~r"':··"":",,·"'··"··- ~ -- -.- ,--""",-",
'!'l
Stresses in bars.
I IBar I ()' <linear) I ()' (non-linear) I ~()'{great defl.)
I1
2 I 160,7 I 161,5 I 6,3I II I
13 I 394,1 I 410,7 I 19,3I I I1 I
17 I 425,1 507,1 I 15,3II
49 I 846,2 936,2 I 36,2II
76 I 103,0 139,5 38,3
109 I 779,3 811,6 27,5
137 1 927,0 I 901,8 23,61I
184 I 2565,5 I 2522,1 113,8
REFERENCES.
(1)- P. Pifíero E."Estructures reticulées". L'Architecture d'Aujourd'hui.Vol 141. Dec. 1968. pp. 76-81
(2)- Escrig F. y P.Valcárcel J."lntroducción a la geometría de las estructuras espaciales desplegables de barras". Boletín Académico de la ETSA de LaCorufía. NQ 3. íeb 1986.
(3)- Escrig F. y P.Valcárcel J. "Analysis oí Expandable Space Bar Structures". lnt. Symposium on Kembrane Structures and Space Frames. lASS. Osaka1986,
(4)- Escrig F. y P. Valcárcel J. "Great Size Umbrellas solved w1th Expanda.ble Bar Structures". First International Conference on LightweightStructures in Architecture. Sydney 1986.
(5)- P.Valcárcel J. y Escrig F. "Bases para el cálculo no lineal de estructuras espaciales". Boletín Académico de la E.T.S.A. de La Coruna. NQ 7. sep1987.
(6)- Escrig F. y P.Valcárcel J. "Curved Expandable Space Grids". NonConventional Structures '87. Londres 1987.
_0 __-.'_".
L
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9
.....t,.
-'-----~-~-----.'-.',
I,1
,'
"
,00
'0...
00'
,oo','>
.\
JOSS-¡:°S,.
"
•
J..,.~........- '\'
,........
-
r\,7
---4
~.~
-.
.......
'o'
~
..'
,,'
.'."
'oo
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