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Compurm& Snucrurrr Vol. 59. No. I. pp. 131-140. 1996 CopyrIght Q.I 1996 Elsevier Science Ltd Printed in Great Bntain. All rights reserved 0045.7949/96 fl5.00+ 0.00

ANALYSIS

OF PANTOGRAPH

FOLDABLE

STRUCTURES

A. Kaveht and A. Davaranl TBuilding and Housing Research Centre, P.O. Box 13145-1696, Tehran, Iran IIran University of Science and Technology, Tehran-16, Iran(Received 19 August 1994)

Abstract-An efficient method is developed for the analysis of scissor-link foldable structures. The stiffness matrix of a unit of such a structure, called a duplet, is derived and incorporated into a standard stiffness method. A computer program is developed and many examples are studied. The results are compared to the analysis using uniplet elements of Ref.[l]. Substantial improvement is obtained through the introduction of duplets in place of the use of uniplets.

1. INTRODUCTION

The need for mobile and reusable structures that are characterized by fast and easy erection procedures has existed for a long period. Such structures are often used in temporary construction industry, quick sheltering after natural disasters and in the aerospace industry. One of the most important foldable (deployable) structures is the scissor-link structure. The first such structure has been designed and constructed by Pinero[2]. Substantial contributions to the general understanding of geometric and kinematic behaviour of scissor-link structures is due to Escrig[3], and Escrig and Vaicarcel[4]. Further studies have been made by Zeigler[S], Derus[6], Nodskov[7], Gantes et al.[8], Rosenfeld et al.[9] and Shan[IO]. A pantograph is a foldable structure which consists of scissor-link units called duplets (see Fig. 1). Such a structure will be referred to as“p-structure” from now on. A duplet consists of two elements, called uniplets, which are capable of rotating about their intermediate pivot node, Fig. 2. Two uniplets which form a duplet are in fact beam elements with three nodes acting as pin-joints having only translational degrees of freedom. No torsion is produced in the members, however, axial forces and bending moments can be developed. A uniplet together with its degrees of freedom and the corresponding nodal forces are illustrated in Fig. 3. The stiffness matrix of a uniplet can be obtained by assembling first the stiffness matrix of two beam type elements, and then condensing and removing the rotational degrees of freedom of three nodes, Refs[I, lo]. The result is a 9 x 9 stiffness matrix for the uniplet element. This matrix is given in Appendix A. Once such a matrix is obtained, the analysis of p-structures follows a standard stiffness method. In this paper the stiffness matrix of a duplet is formulated using the stiffness matrices of its consti-

tuting uniplets. For this purpose the stiffness matrices of uniplets with the angle (p between them are assembled in a 15 x 15 matrix. Then the translational deg

rees of the freedom of the connecting pivot node are condensed, in order to obtain a 12 x 12 stiffness matrix for a duplet element. A comparison of the efficiency of the analysis of p-structures using duplet and uniplet elements is presented in subsequent sections.2. STIFFNESS MATRIX OF A DUPLET Consider a typical duplet element of a space pstructure, as shown in Fig. 4. The geometric properties of this element are illustrated in Fig. 4a and local and system coordinate are depicted in Fig 4b. The local coordinate system of uniplet l-S-2 is taken as xyz and that of uniplet 3-S-4 is considered as@.?. Both x and 2 axis are coincident with the axial longitudinal axes of the uniplets and the axes y-z and p-2 are parallel to the principal axes of the sections of uniplet 1-5-2 and 3-5-4, respectively. Obviously the plane x-y will coincide with plane a-j, and the angle between x and i in a counter clockwise direction is measured as 4. The nodal forces in local coordinate systems xyz and,?j% are illustrated in Fig. 5, where the nodal forces in node 5 are eliminated in both uniplets for clarity. Uniplet (l-5-2) will be specified by index 1 and (3-5-4) will be denoted by index 2. The forcedisplacement relationships for these uniplets can be written as

(1) (2)In these equations p, and d, are the force and displacement vectors of uniplet (l-5-2) in xyz coordinate131

132

A. Kaveh

and A. Davaran

Fig.

I. A typical

p-structure

system, and&, d2 are force and displacement vectors for uniplet (3-5-4) in system coordinate .@Z. These vectors are

and .Izj are parallel and therefore the following formation matrices R and T can be used:fiz=T.pz;;i,=T.d,

trans-

(5)

fiz and I, can be obtained, similar to eqn (3) by exchanging the indices of (2. 5,1) to those of (4, 5, 3) and exchanging the coordinate system .u_rr to .Gj2. The stiffness matrices Kd and Kt corresponding to uniplets 1 and 2 in local coordinate systems have the following patterns:

where d, and p2 are the force and displacement vectors of uniplet 2 in XJJZcoordinate system corresponding to uniplet 1. R and T are orthogonal transformation matrices. Using eqn (2) and eqn (5)

p>=T’+T.d>=K;.d, K’”= T’g2T.”

(7) (8)

where each block S,, and S,, of the above matrices are 3 x 3 submatrices and their entries are evaluated in Appendix A. In order to assemble the stiffness matrix of these elements, forces and displacements of uniplet 2 (3-54) should be transferred to .XJ: coordinate system. XJJ

Now eqns (I), (7) and (8) can be used to assemble the stiffness matrix. For this purpose the nodal numbers of uniplets are exchanged, such that the pivot node is numbered after the other two end nodes. This operation can be carried out with exchanging the corresponding blocks in vectors and

uniplets

node

Fig. 2. The structure of a duplet.

Analysis of pantograph foldable structures

133

Fig. 3. A uniplet and the corresponding degrees of freedom.

stiffness matrices. For simplicity the same

notations are used after the exchange operation. The assembled stiffness matrix before condensation is 15 x 15 and has the following form:

Therefore

eqn (9) becomes

p&&k].Expanding eqn (12) results in K,,.d,+K,,.dS=p,, K&.d,+K,,.d,=p5. Combining eqns (14) and (15) leads to d,= KG’(P~ - K1z. dd, pd=(K,,-K,z.K22’.KIZ).dd+K,2.K221.p5.

(13)

pd=Kd.iid,where

(9)

(14) (15)

(10)

(16)(17)

and (11) where[S,,jl and[S,,], are 3 x 3 matrices corresponding to the mid node of the uniplet 1 and 2, respectively. The stiffness matrix in eqn (10) is partitioned in a different form in order to separate the block corresponding to node 5 as follows: K22=SSS. (12)

In eqn (17) the second term contains the fixed end action in non-pivotal nodes of the duplet. In practice the pivot node is rarely loaded and one can ignore this term. However, in general, fixed end action of the nodes of a duplet can be obtained using simple statics: Pd=&.&t K,= K,, - K,, . K,’ . K;, . (18) (19)

&=[

-fl,--

~-;:-I:

In the above equations, the relation between load and displacement of a duplet in the xyz coordinate system is established. In these relations p,, and dd are the force vector and displacement vector of all nodes of duplet, respectively.

(20)

134

A. Kaveh and A. Davaran

(4

(b)z

X

AFig. 5.

Y

Fig. 4. A typical duplet of a foldable structure and considered coordinate systems. rest of the calculations can be carried out using standard stiffness method. It should be noted that for derivation of the 12 x 12 stiffness matrix of a duplet, a 3 x 3 matrix K,, should be inverted. This can be done easily because of the sparsity of such a matrix. The process of this inversion is omitted for the sake of brevity. The final result is represented in Appendix B and the list of the computer program for the stiffness matrix of a duplet is provided in Appendix C. Application of duplets in place of uniplets results in a considerable reduction of the number of structural nodes. Consequently the total degrees of freedom and the bandwidth of the stiffness matrix of the structure is reduced. Consequently reduction in storage and the computer time is achieved, Kaveh[I 1). For the assembling of the stiffness matrix, one 12 x 12 matrix is employed in place of two 9 x 9

Plz

Once the stiffness matrix of a duplet is obtained in the local coordinate system of one of the uniplets, the

1bFig. 6. A planar foldable p-structure.

Analysis Table

of pantograph of foldable of

foldable p-structures

structures using uniplets and duplets TimeS

135

1. ComparativeType of element uniplet duplet

analysis

MaxNumber deflection Example

nodes 32 20 14s 8190 50

Bandwidth18

mm0.125 0.125

1 2

12 38 22 63 3663 36111

1.37 1.21 4.50

uniplet duplet uniplet duplet uniplet duplet uniplet duplet

16.650 16.650 12.700 12.700 85.286 85.286 26.527 26.527

1.938.24 2.69 15.38 4.23 51.53 11.21

3

4

166 90242 130

5

60

matrices. The simplicity of configuration processing and

data generation is another advantage of using duplets. 3.NUMERICAL RESULTS

and I,= I,= 290 mm4. Example 1: a planar foldable p-structure is considered as shown in Fig. 6. The loads applied at three nodes are P= 100 N and a= 200 mm and b= 100 mm. Other properties of the structure are as mentioned in the above. Example 2: this is another planar p-structure as shown in Fig. 7. The corresponding data is the same as Example 1 except the grid is 8 x 8. Example 3: in this example a space p-structure is considered as shown in Fig. 1, which is a 4 x 4 grid in plan with a= 200 mm and h= 300 mm. The height is taken as h= 100 mm and P= 100 N. The grid contains only the bottom layer and four vertical bars at supports. For providing the statical stability (rigidity) of the structure in xy plane, the nodes on the axes of symmetry at the bottom layer are constrained in the .x and JJ directions. For complete foldability“a” should be taken equal to“b”. Example 4: a space p-structure is considered as shown in Fig. 8. The mechanical properties of the structure are the same as in Example 3 except the grid

In this section the results of the analysis of five p-structures using uniplets and duplets are presented. Examples 1 and 2 are chosen from planar structures and examples 3, 4 and 5 are taken from space foldable p-structures. In all the examples the structures are analysed using both uniplets and duplets. The nodal ordering considered in these examples correspond to stiffness matrices with minimum bandwidth. The computer time for the analysis on a PC 486DX50 are measured from the moment of entering the data up to the end of the solution of equations for evaluating the nodal displacements. The results obtained for the examples are presented in Table I. In all the examples the following data are used:

E=210,000Nmm-‘,

A=28mm*

Fig. 7. A planar foldable p-structure

with 8 x 8 grid.

136

A. Kaveh and A. Davaran *

Fig. 8. A space foldable p-structure.

is 8 x 4 and h= 200 mm. The nodes on the axis of symmetry in x and y directions are fixed in order to provide the statical stability of the structure. The applied loads are taken as P= 100 N. Example 5: a foldable p-structure is taken from Ref.[12], as illustrated in Fig. 9. The dimensions of the grid are the same in both directions, i.e. a= b= 200 mm. The height is taken as h= 100 mm. The mechanical properties are the same as the previous examples, and P= 100 N, which are applied at nine middle nodes of the bottom layer of the structure. The axes of symmetry of the bottom layer of the structure are constrained laterally in horizontal directions.

4. CONCLUDING

REMARKS

The stiffness matrix of a duplet element with four nodes is formulated. The results of using this element is compared with those of uniplets. The number of nodes for a structure modelled with duplets is less than that of uniplets, e.g. for space structures the number of nodes can be reduced approximately by one third.

Reduction of the total degrees of freedom affects the bandwidth of the stiffness matrix of the structure, increasing the efficiency of its solution. Table 1 shows that the saving obtained using the new element becomes considerable as the number of nodes of the structure increases. For foldable structures

Fig. 9. A space foldable p-structure (taken from Ref.[12]).

Analysis of pantograph foldable structures with cable elements, due to the reanalysis required,the saving becomes more significant. 7.REFERENCES

137

6. D. L. Derus,

8.

I. W. Shan, Computer analysis of foldable structures. Compuler. Struct. 42, 903-912 (1992). 2. E. P. Pinero, Materia-estructa-forma. Hogar Y Arquilect. 40, 25-30 (1962). 3. F. Escrig, Expandable space structures. Int. J. Space Struct. 1, 79-91 (1985). 4. F. Escrig and J. P. Valcarcel, Geometry of expandable structures. Znt. J. Space Struct. 8, 71-84 (1993). 5. T. R. Zeigler, Collapsible self-supporting structures. U.S. patent no. 3.968.808 (filed 6 November, 1976) (1976).

9.

10. 11. 12.

Collapsible articulated wall structures. U.S. patent no. 4.580.375, 1986 (filed 20 September, 1985) (1986). P. Nodskov, Collapsible exhibit panel, U.S. patent no. 4.580.375 (filed 20 September, 1983) (1986). C. J. Gantes, R. D. Logcher, J. J. Connor and Y. Rosenfeld, Depolyability conductions for curved and flat, polygonal and topological deployable structures. Int. J. Space Struct. 8, 97-106 (1993). Y. Rosenfeld, Y. Ben-Ami and R. D. Logcher, A prototype“clicking” scissor link deployable structures. Int. J. Space Struct. 8, 85-93 (1993). W. Shan, Foldable structure. Ph.D. thesis, University of Surrey (1990). A. Kaveh, Structural Mechanics: Graphs and Matrix Mefhods, RSP, 2nd edn. Wiley, Exeter (1995). H. Nooshin and P. Disney, Elements of formian. Comput. Struct. 41, 1183-1215 (1991).

APPENDIX A

The stiffness matrix of a uniplet element is as follows:

Kl, 0 K,,0 K,, 00

K,=

0 0 K,,

I I K,,j -_--r--------___--~-_----___-_-I 0; 6, ! I 0 I 0 K,, I I

I I I I I

I I I I 1

1 sym.

;

p,= K,.d“I

OKjO 0Ki ___________6_3_c____-------%-/_-_____._-____ 0 0; K,, 0 0 0 1K,, 0 0 K,, 0 0i 0

K,,0

0

/ 0

K,,0

K,, 1 0

K,,/ 0

K,19 I9

where: K11= -K41= EA.m Kzz= 3E1,. m3n/(m+n) Ksz= 3E1, (-m’n-m*n’)/(m+n) Kg2= 3EIz. m’n’/(m+n) K33= 3EIy.m3n/(m+n)&3= 3EIy (-m3n-m2n2)/(m+n) Kg3= 3EIy. m%*/(m+n) I<44= EA.(m+n) K74= - K77= - EA.n

Kss= Kg5= I<66= Kg6= Ksg= K99=

3E1, (m3n+mn3+2m2n2)/(m+n) 3E1, (-mn’-m*n*)/(m+n) 3EIy. (m3n+mn3+2m2n2)/(m+n) 3EIy. (-mn3-m2n2)/(m+n) 3EIz. mn3/(m+n) 3EIy. mn3/(m+n) 1 n=- 1, m=L, L,

APPENDIX B

The element stiffness matrix of a duplet:

; pd=K,.d,,

where Sij is a submatrix of order 3x3; ij= 1,...,4, j 2 i.

138

A.

Kavehand A. DavaranAPPENDIX C

FORTRAN

77 subprogram for the formation ofa duplet stifl%ess matrix:

subroutine duplet(fi,e,al,xl,x2,yyl,zzl,a2,x3,x4,yy2,zz2,tp) common/stf/sll(3,3),sl2~3,3),s13(3,3),sl4~3,3~ * s22(3,3),~23(3,3),~24(3,3) * s33(3

,3),s34(3,3) s44(3,3) *, implicit double precision (a-h) implicit double precision (o-z) double precision fi,xl,x2,x3,x4,xmn2,xm3,xn3,eal,eyyl,ezzl,yyl,zzl double precision yyZ,zz2,eyy2,ezz2 double precision c,s,cs,c2,s2,zl,z2,z3,zq,el,el,e2,e5,e9 double precision ul,wl,u2,w2 eal=e*al eyyl=e*yyl ezzl=e*zzl ---~-------------------xm=l.O/xl xn=l.O/x2 xmn=xm+xn xmn2=xm*xm*xn*xn xm3=xm*xm*xm*xn xn3=xn*xn*xn*xm --------__-----~---_---rll=eal*xm r41=-rll r22=3.0*ezzl*xm3/xmn r52=3.0*ezzl*(-xm3-xmn2)/xmn r82=3.0*ezzl*xmn2/xmn r33=3.0*eyyl*xm3/xmn r63=3.0*eyyl*(-xm3-xmn2)/xmn r93=3.0*eyyl*xmn2/xmn r44=eal*xmn r74=-eal*xn r77=-r74 r55=3.0*ezzl*(xm3+xn3+2.O*xmn2)/xmn r85=3.0*ezzl*(-xn3-xmn2)/xmn r66=3.0*eyyl*(xm3+xn3+2.O*xmn2)/xmn r96=3.0*eyyl*(-xn3-xmn2)/xmn r88=3.0*ezzl*xn3/xmn r99=3.0*eyyl*xn3/xmn -------------~---------------------if(tp.eq.l.O)goto 10 ea2=e*a2 ezz2=e*zz2 eyy2=e*yy2 -__----__----__---------__---~_---~_ xm=l.O/x3 xn=l.O/x4 xmn=xm+xn xmn2=xm*xm*xn*xn xm3=xm*xm*xm*xn xn3=xn*xn*xn*xm --~__--~_---~---~-~__---_--~~--~--~~ tll=ea2*xm t41=-t11 t22=3,0*ezz2*xm3/xmn t52=3.0*ezz2*(-xm3-xmn2)/xmn t82=3.0*ezz2*xmn2/xmn t33=3.0*eyy2*xm3/xmn t63=3.0*eyy2*(-xm3-x2)/xmn t93=3.0*eyy2*xmn2/xmn t44=ea2*xmn t74=-ea2*xn

C

C C

C

C

Analysis ofpantograph foldable structures t77=-t74 t55=3.0*ezz2*(xm3+xn3+2.O*xmn2)/xmn t85=3.0*ezz2*(-xn3-xmn2)/xmn t66=3.0*eyy2*(xm3+xn3+2.O*xmn2)/xmn t96=3.0*eyy2*(-xn3-xmn2)/xmn t88=3.0*ezz2*xn3/xmn t99=3.0*eyy2*xn3/xmnC C __---_-__----_-_----~-~---~-----~-----

139

got0 20__------_------_------------~-----~~--

10 tll=rll t41=r41 t22=r22 t52=r52 t82=r82 t33=r33 t63=r63 t93=r93 t44=r44 t74=r74 t77=r77 t55=r55 tSS=r85 t66=r66 t96=r96 t88=r88 t99=rPPC ___________----_------~~~----~~~

20 c=cos(fi) s=sin(fi) c2=c*c s2=s*s cs=c*sC

zl=r44+c2*t44+s2*t55 z2=-cs*(t44-t55) z3=rSS+s2*t44+c2*t55 z4=r66+t66 det=zl*z3-z2*z2 el=z3/det e2=-z2/det eS=zl/det eP=l.O/z4C C --___---__----__----____________________~~

Duplet Stiffness Submatrices sll(l,l)=rll-r41*r41*el s11(1,2)=-r41*e2*r52 s11(2,1)=s11(1,2) s11(2,2)=r22-r52*r52*e5 s11(3,3)=r33-r63*r63*e9___-____-__----_______~--~~~~-~~~~~~---~~~

:

C

s12(1,1)=-r41*el*r74 s12(1,2)=-r41*e2*r85 s12(2,1)=-r52*e2*r74 s12(2,2)=r82-r52*eS*r85 s12(3,3)=r93-r63*e9*rP6C -_~--___-__---__----_-~~~~~--~~--___--____

ul=t41-t52 wl=C2*ul+t52 zl=S2*ul+t52C -_~--___------__----______________________

si3(1,1)=-r41*el*wl+ r41*e2*cs*ul s13(1,2)=r41*el*u1*cs - r41*e2*zl s13(2,1)=-r52*e2*wl+ r52*eS*cs*ul s13(2,2)=r52*e2*cs*ul - r52*e5*zl s13(3,3)=-r63*e9*t63C _____________________~~-~~~~~~~-----------

u2=t74-t85 w2=c2*u2+t85

140

A. Kavehand A. Davaran z2=s2*u2+t85 __________--____-----_------~~------~----~ s14(1,1)=-r41*el*w2+ r41*e2*cs*u2 s14(1,2)=r41*el*cs*u2 - r41*eZ*z2 s14(2,1)=-r52*e2*w2+ r52*e5*cs*u2 s14(2,2)=r52*e2*cs*u2 - r52*e5*z2 s14(3,3)=-r63*e9*t96 ___________-______---_-------___------_--s22(1,1)=r77 - r74*r74*el s22(1,2)=-r74*e2*r85 s22(2,1)=s22(1,2) s22(2,2)=r88 - r85*r85*e5 s22

(3,3)=r99 - r96*r96*e9 ---___------_----------------__-------__-s23(1,1)=-r74*el*wl+ r74*e2*cs*ul s23(1,2)=r74*el*cs*ul - r74*e2*zl s23(2,1)=-r85*e2*wl+ r85*e5*cs*ul s23(2,2)=r85*e2*cs*ul - r85*e5*zl s23(3,3)=-r96*e9*t63 ------------_-_--------------_--------____ s24(1,1)=-r74*el*w2+ r74*e2*cs*u2 s24(1,2)=r74*el*cs*u2 - r74*e2*z2 s24(2,1)=-r85*e2*w2+ r85*e5*cs*u2 s24(2,2)=r85*e2*cs*u2 - r85*e5*z2 s24(3,3)=-r96*e9*t96 --------------------------------------_---s33(1,1)=c2*tll+s2*t22-wl*wl*el+2.*cs*ul*wl*e2-cs*cs*ul*ul*e5 s33(1,2)=-cs*(tll-t22)+cs*ul*wl*el-(cs*cs*ul*ul+zl*wl)*e2+ *zl*cs*ul*eS s33(2,1)=s33(1,2) s33(2,2)=s2*tll+c2*t22-cs*cs*ul*ul*el+2.O*cs*ul~zl*e2-zl*zl*e5 s33(3,3)=t33-t63*t63*e9 --------------------~-----------------__--s34(1,1)=s2*t82-w2*wl*el+cs*(w2*ul+wl*u2)*e2-cs*cs*u2*ul*e5 s34(1,2)=cs*t82+cs*u2*wl*el-(cs*cs*u2*ul+z2*wl)*e2+z2*cs*ul*e5 s34(2,1)=cs*t82+w2*cs*ul*el-(w2*zl+cs*cs*u2*ul)*e2+cs*u2~zl*e5 s34(2,2)=c2*t82-cs*cs*u2*ul*el+cs*(u2*zl+ul*z2)*e2-z2*zl*e5 s34(3,3)=t93-t63*eg*t96 ---_-------_-----------------------------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ s44(1,2)=-cs*(t77-t88)+w2*cs*u2*el-(w2*z2+cs*cs*u2*u2)*e2+ *cs*u2*z2*e5 s44(2,1)=s44(1,2)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ s44(3,3)=tQQ-t96*t96*eQ return end

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