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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 …… 此处隐藏：16975字，全部文档内容请下载后查看。喜欢就下载吧 ……