Abstract. 1 Introduction

Size: px

Start display at page:

Download "Abstract. 1 Introduction"

Basil Boone
5 years ago
Views:

1 Buckling force for deployable pantographic columns I. Raskin, J. Roorda Department of Civil Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L SGI Abstract The method of calculating the buckling load for a uniform pantographic column is presented. The deployable column is substituted with a simpler model - a composite column consisting of links having infinite bending and finite axial stiffness. The links are connected by hinges and rotational springs. The parameters of the model are calculated based on the condition of stiffness equivalence of the two structures. The stiffness matrix of the axially loaded composite column is derived from the expression of potential energy under the condition of small lateral displacements. Some entries of the matrix are quadratic functions of the load. The solution of the nonlinear eigenvalue problem gives the buckling load. A number of different columns are analyzed. The results show that for constant height and constant degree of deployment the buckling load increases as the number of pantographic units in the column grows. The buckling load decreases as a particular column deploys. Certain conclusions are made concerning the applicability of the procedure. 1 Introduction The pantographic column is representative of deployable structures. The basic unit of the column is a scissors-like element. It is formed from two beams, with hinges at the ends, connected at their intermediate points by a shear resisting element which does not restrict their relative rotation. When placed along a straight line, several units form a planar pantographic column. Such systems were studied by Escrig & Valcarcel\ McNulty*, Zanardo*, Gantes et al/, Kwan

2 306 Mobile and Rapidly Assembled Structures & Pellegrino^ and other researchers. This paper presents a method of finding critical compressive load for a uniform pantographic column. 2 Analytical model of a deployable column The behaviour of a pantographic column under lateral loading (force P% in Fig. la) resembles that of a solid column (Fig. Ib), so that an equivalent bending stiffness, (EI)p (index "p" stands for "pantographic"), based on the equal tip deflections of the two structures, can be obtained. When the number of pantographic units in the column is sufficiently large this stiffness does not depend on the number of units, in the same manner as the bending stiffness of the solid column is only a characteristic of the cross-section, not of the length of the structure. However, when an axial load (force PI in Fig. la) is applied to the deployable column, the equivalent axial stiffness, (EA)p, obtained by using the same approach, is inversely proportional to the squared number of units (Appendix). The equivalent properties are given by: (El) = 6 El sin y, (EA) =, ^ ^ ' ^ /? acos'y(4n'-l) ' where a - half-length of each bar; El - bending stiffness of each bar; y - degree of deployment (when y = 0 the column is fully folded); n - number of units in the column. These equivalent stiffnesses of the pantographic column with equal units are derived under an assumption that deformations of the structure are caused only (EQp, (EA)p c, (EA). i-th element 99%%^^^% (a) (b) (c) Figure 1: Pantographic (a), equivalent solid (b) and composite (c) columns.

3 Mobile and Rapidly Assembled Structures 307 by the bending of bars. For the purpose of stability analysis it was decided to model the pantographic column as another structure that is stiffnesswise equivalent The composite column shown in Fig. Ic consists of straight elements with infinite bending and finite axial stiffnesses. They are successively hinged to each other and to the base with rotational springs resisting relative rotations at hinges. The length of each element is equal to the height of a pantographic unit in the original deployable column. Units and elements in both columns are numbered from top to bottom. The parameters of each element - A, (%, pi - are its length, the stiffness coefficient of the rotational spring at its bottom and its axial stiffness coefficient, respectively. Since we are modeling a deployable column made of equal units, all elements in the composite column have the same length /. Also, to make the bending stiffness uniform along the column, all oci are taken to be the same and the index is dropped. The equivalent bending and axial stiffnesses of such a structure, (EI)c and (EA)c (index "c" stands for "composite"), obtained in the same manner as those for the pantographic column, are: -^- (2) where L = n / - total length of the column. The next step is to assign values to /, a and pi. The first two follow from equating the height of the pantographic unit to the element length and the equivalent bending stiffness of the original structure to that of the model:,. 6 El sin y 3 El / = 2asmy, <x = = - (3) / a The situation with the axial stiffness coefficients pi is more complicated. Based on the internal force distribution shown in Fig. Alb for the deployable column under the axial loading, vertical displacement of the top nodes of an i-th pantographic unit can be found as: Similarly, for unit i+1 : v. =^Icos'y^(4k'-4ik + l) (4) 3 El k=i *-4(i + l)k + l] (5) ^ 3 El Since the top nodes of unit i+1 are the bottom nodes of unit i, the vertical deformation of the i-th unit in the axially loaded column is obtained by subtracting eqn (5) from eqn (4):.,+, J C/l The elemental axial stiffness coefficient: ] (6)

4 308 Mobile and Rapidly Assembled Structures 3 El 1, z (7) Vi-Vj+i a cos Y [2n(n + l)-2i(i + l) + l] At this point we are able to calculate all parameters of the composite model and can proceed to investigate its stability. 3 Buckling analysis A composite column, with all dimensions and stiffness parameters specified, subjected to axial compressive force P applied at the top, is shown in Fig 2. The potential energy of this system in a buckled configuration, V, equals the difference between the strain energy of the axially compressed elements and deformed rotational springs, U, and the work done by the load, W: (8) i=l L ^ ^ J i=l where Q\ - rotation of the i-th element (6n+i = 0); 5j = P cos 6i/pi - elemental compressive deformation. Substituting 8; into eqn (8) and using the series expansion cos0j «1 - Q?/2 for small 0i we have:?-p*/pi] (9) Differentiating eqn (9) two times with respect to the angles we obtain the stiffness matrix of the structure: Figure 2: Initial and buckled configurations of composite column.

5 Mobile and Rapidly Assembled Structures 309 -P/, for,,,,. =-a, if i-i=l; - = 0, otherwise. 36,98 j.. Hence, the stiffness matrix is tridiagonal with the elements on the main diagonal being quadratic functions of the load. To find the critical load we have to solve the nonlinear eigenvalue problem. For solving this problem the combination of bisection and sign counting suggested by Wittrick & Williams* was employed. After the critical load is found and substituted into the stiffness matrix, the matrix becomes degenerate. The standard procedure can be used then to calculate linear eigenvalues and eigenvectors. At least one of the eigenvalues is zero, and the corresponding eigenvector gives the mode shape for the critical load. 4 Test of procedure Since the results of buckling analysis of pantographic columns are not available in the literature, it was decided to perform some analytical tests on axially rigid columns to gain at least partial confidence in the procedure. The first structure (Roorda^) is the column composed of two rigid elements of the same length / connected to each other and to the base by hinges and the same rotational springs a. The critical force for this column is found to be equal 0.382a// which is in absolute agreement with the known result. The next test was done to check the assumption that, as the number of elements in the composite column increases, the critical force approaches that of Euler for a solid column. The elements are axially rigid and the solid column has a bending stiffness equal to the equivalent stiffness of the composite one. The parameters of the solid column, (EI)c and L, were chosen to make l dd The elemental length and spring coefficient, / and a, were calculated using the relations from eqn (2). The graph for the critical load versus 1/n for n from 5 to 100 is shown in Fig Results of buckling analyses Next we modeled deployable columns of constant length L and degree of deployment y equal to 70, but with different numbers of units n varying from 5 to 100. The equivalent bending and axial stiffnesses of these columns differ very slightly with n. It should be noted that the amount of material is the same for all columns.

6 310 Mobile and Rapidly Assembled Structures , /n Figure 3: Buckling load for axially rigid column versus number of elements. The plot of the nondimensional critical force PcrL^/EI versus the reciprocal of the number of units is shown in Fig. 4. Here, as before, El is the bending stiffness of bars the pantographic units are made of. The increase of the critical force with the number of units, which may somewhat contradict the intuitive expectations, can be attributed to redistribution of the axial stiffness along the column. As can be seen from the graph a growing number of units brings the discrete model closer to the column with a continuously varying axial stiffness. The limiting critical load corresponding to n approaching infinity (=16.8) is the critical force for such column. In thefinalexercise, one of the previously analyzed columns was taken (the one consisting of 10 units) and considered at different degrees of deployment. Angle y was changing from 63 to 83 (Fig. 5). Although the equivalent bending stiffness of the deployable column is proportional to sin y, and therefore increases as the column deploys, the column length also increases. Since in the classical case the critical load is inversely proportional to the squared length, the observed drop in the buckling load is sensible. Another reason justifying decrease of the buckling load is the growth of the equivalent axial stiffness during deployment. 6 Conclusions We described the method of calculating the buckling load for a uniform pantographic column. The analyses of the composite columns representing the

7 Mobile and Rapidly Assembled Structures 311 Figure 4: Buckling load for pantographic column of constant height and degree of deployment versus number of units (" a)/el Figure 5: Buckling load for pantographic column versus degree of deployment.

8 312 Mobile and Rapidly Assembled Structures original structure show that for constant height and constant degree of deployment the buckling load increases as the number of pantographic units in the column grows. The buckling load decreases during deployment of a particular column. The question about the limits of applicability of the proposed technique is still open. The expressions for the equivalent stiffnesses of the deploy able column were derived based on the assumption of linear behaviour of the structure and small displacements. This may not be true for the units close to the top, where the corresponding elements in the composite column are axialry soft, especially for small degrees of deployment. This may lead to a situation in which these elements undergo axial deformations comparable to their original length before the load reaches its buckling value. On the other hand, for configurations close to full deployment, the finite axial stiffness of the bars, which was neglected in the equivalent stiffness derivation, becomes important. This is the reason for the equivalent bending stiffness being insensitive to an apparent singularity at y = 90. References 1. Wittrick, W.H. & Williams, F.W. A general algorithm for computing natural frequencies of elastic structures, Quarterly Journal of Mechanics and Applied Mathematics, 1971, Vol. 24, Pt. 3, pp Roorda, J. Buckling of Elastic Structures, University of Waterloo Press, Waterloo, Ontario, Escrig, F. & Valcarcel, J.P. Analysis of expandable space bar structures (ed. K. Heki), Vol. 3, pp , Proceedings of I ASS Symposium on Shells, Membranes and Space Frames, Osaka, Japan, McNulty, O. Foldable space structures (ed. K. Heki), Vol. 3, pp , Proceedings oflass Symposium on Shells, Membranes and Space Frames, Osaka, Japan, Zanardo, A. Two-dimensional articulated systems developable on a single or double curvature surface, Meccanica, 1986, Vol. 21, pp Gantes, C., Connor, J.J. & Logcher, R.D. Equivalent continuum model for deployable flat lattice structures, Journal of Aerospace Engineering, 1994, Vol. 7, no. 1, pp Kwan, A.S, & Pellegrino, S. Matrix formulation of macro-elements for deployable structures, Computers & Structures, 1994, Vol. 50, no. 2, pp

9 Mobile and Rapidly Assembled Structures 313 Appendix. Equivalent stiffnesses of pantographic column Forces acting on the i-th unit of the pantographic column, separately for lateral and axial loadings (P% and PI in Fig. la, respectively), are shown in Fig. Al. Note that in both cases the internal forces depend linearly on the distance from the top, which corresponds to the situation of the lateral load, and differs from that of the vertical load, applied to a solid column. Assume that the contribution of axial deformations of the bars to the nodal displacements is negligible compared to that due to bending. The moments are linear along the bars with the maximum at their midpoints and zero at the ends. For the lateral loading the maximum moments are: -Mm=-McE (Al) - l)pztany Pi/2 B _?2/2 (i-l)picoty i-l)picoty ipicoty ipztanyt lipztany Pi/21 lpi/2 (a) (b) Figure Al: Forces acting on i-th unit of pantographic column subjected to lateral (a) and axial (b) loads. Using the principle of virtual work one can easily find the horizontal deflection of the top of the pantographic column. This deflection is then equated to that of a solid column with the same length and overall bending stiffness and under the same load: Using L = 2 a n sin y we obtain: _ For large n: I<2i-D*

10 314 Mobile and Rapidly Assembled Structures lim = lim = -, hence (ED = 6 El sin y (A4) ~ Even columns with only a few units have equivalent bending stiffnesses quite close to this value. For example, for n = 4, the numerical coefficient in the expression for (EI)p is already 6.1. Going along the same lines for the case of axial loading: r\ MCA = MCD =Pi -(l-2i)cosy = -McB = -M^ (A5) Vertical top deflections of the deployable and the solid columns are equated: Pa I 2 X n i _. o ID i, TI j *'"7If "**&<"-'> =^; which gives the following equivalent axial stiffness: 18 El sin y %z coss (4.2-1) M

Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods of Structural Analysis

uke University epartment of Civil and Environmental Engineering CEE 42L. Matrix Structural Analysis Henri P. Gavin Fall, 22 Review of Strain Energy Methods and Introduction to Stiffness Matrix Methods