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HARMONIC FINITE-ELEMENT THERMOELASTIC ANALYSISOF SPACE FRAMES AND TRUSSESDan Givoli a & Omri Rand aa Department of Aerospace Engineering , Technion—Israel Institute of Technology , Haifa,32000, IsraelPublished online: 25 Apr 2007.

To cite this article: Dan Givoli & Omri Rand (1993) HARMONIC FINITE-ELEMENT THERMOELASTIC ANALYSIS OF SPACE FRAMESAND TRUSSES, Journal of Thermal Stresses, 16:3, 233-248

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HARMONIC FIMTE-ELEMENT THERMOELASTIC ANALYSIS OF SPACE FRAMES AND TRUSSES

Dan Givoli and Omri Rand Department of Aerospace Engineering

Technion-Israel Institute of Technology Haifa 32000, Israel

A numerical procedure is devised for the thermoelastic analysis of three-dimensional frame- or tms-type space structures exposed to solar radiation. Thin-walled frame or t m s members with cross sections of arbitrary shape are considered. Tension- compression, bending, shear, and torsional effects due to the temperature distribution induced by the solar radiation are all taken into account. The procedure proposed inuolues finite element discretization in the axial direction and a harmonic analysis in the circumferential direction of each member. This procedure is an extension of the one employed previously to obtain the temperature field in trusses. A multibay frame structure serves as a model to demonstrate the performance of the proposed method. The temperature, displacement, and stress fields in the frame are found in various cases.

INTRODUCTION

Thermal and thermoelastic numerical analyses have become an essential part of the design process of large frame- and truss-type space structures. Solar radiation and radiation from other sources induce a temperature field in the structure, which in turn generates an elastic displacement field. The latter is often very important to the designer, for two main reasons. First, the displacements must usually satisfy certain limitations dictated by the allowed working conditions of various instru- ments and antennas in the space vehicle. For example, a parabolic reflector may cease to be effective when undergoing large deflection [I, 21. Second, the strain energy absorbed may affect the rigid-body dynamical behavior of the structure. In fact, elastic vibrations due to momentary activation of engines may cause undesired changes in the paths of satellites leading to the loss of their stability [3, 41.

The thermal problems involved in the analysis of space structures are usually highly nonlinear due to the presence of thermal radiation emitted by the structure and to nonlinear material behavior. A further complication is introduced when part of the structure overshadows another part of it. In addition, the three-dimensional discretization of a large space structure would typically require a very large number of degrees of freedom if accurate results are desired. For an overview of the factors involved in this type of analysis see [5-101. Methods have been devised for the one-dimensional thermoelastic analysis of space trusses [ll-151 and for the two-

This work was supported by the Adler Fund for Space Research managed by the Israel Academy of Sciences. We also wish to acknowledge the assistance of Mrs. Cecilia Zolotnizki in programming.

Journal of Thermal Stresses, 16:U3-248, 1993 Copyright 63 1993 Taylor & Francis

0149-5739/93 $10.00 + .OO

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234 D. GIVOLI AND 0. RAND

dimensional (cross-sectional) analysis of frame and truss structures [16, 171. The review paper by Pinson [I81 should also be consulted.

In [I91 we employed a special numerical procedure for the three-dimensional thermal analysis of large space structures composed of thin-walled members. This method yields the temperature distribution in all the structural members in both the axial and the circumferential directions, while avoiding the need to solve a very large system of nonlinear algebraic equations. In the present paper, we extend the approach introduced in [19] for the thennoelastic analysis of multibeam space structures. Tension-compression, bending, shear, and torsional effects due to the temperature field in the structure are all taken into account in the analysis, and the resulting displacements and stresses are obtained.

Following is the outline of the paper. First we summarize the harmonic finite-element method used for the thermal analysis. Then we present the thermo- elastic analysis. W e show how an existing finite element code can be employed within the framework of the harmonic approach, via the use of effective external loads. We validate the scheme by applying it to some elementary numerical models. We also demonstrate the performance of the method via an example involving a multibay frame structure exposed to solar radiation. The analysis includes self- shadowing effects. Stresses and displacements are obtained for various kinematical parameters and thermal loading conditions. Finally we make some concluding remarks.

The Thermal Analysis

Consider a frame- o r truss-type structure where each member is made of an orthotropic material and has a uniform thin-walled closed cross section of an arbitrary shape. Figure 1 shows a typical member in the structure. The axial coordinate is denoted 5, and the circumferential coordinate along the midline of the thin cross section is denoted by s. The latter coordinate starts from an arbitrary point on the midline and measures arclength along this line. The maximal value of s, namely, the cross-sectional perimeter, is denoted by p. The thickness of the cross section is assumed uniform and is denoted by t . In the present model the two edges of the rod are each characterized by a single temperature, assumed to be identical for all the rods connected at the same joint. In other words, the joints are considered perfect conductors and lack any heat capacity. The exterior surface of the rod is exposed to solar radiation.

In the present model we neglect the following effects: the heat exchange through radiation between different truss members, the temperature variation through the thickness of the cross section, and the thermodynamic influence of the elastic strain rate on the temperature field. We also assume steady-state condi- tions. The governing equation for the temperature T(.$,s) in each structural member is

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ANALYSIS OF SPACE FRAMES AND TRUSSES

solar radiation

solar radiation

Fig. 1 A typical member in the structure, exposed to solar radiation.

The first two terms on the left side of Eq. (1) are conduction terms, whereas the third and fourth terms are the radiation and thermal load terms respectively. The solution T( 6, s ) of Eq. (1) is assumed to be periodic in s with period p. In Eq. (1) K* and K, are the thermal conductivities in the axial and circumferential directions respectively, u is the universal Stefan-Boltzmann constant, E is the surface emissivity of the member, q,,, is the absolute value of the solar radiation vector, a, is the surface absorptivity, Po is the "view factor" associated with the location and orientation of the rod with respect to the solar radiation vector, and P,(s) is the "view factor" depending on the direction of the normal n to the outer surface of the cross section at each point on the surface (see Fig. 1). Both "view factors" can have values between zero and one, and both are affected by self-shadowing when the structure is opaque: Po becomes zero in any region of the structure which is hidden from the incident heat flux, whereas &(s) is zero over the hidden half of the perimeter of every cross section.

The numerical procedure proposed for the solution of the thermal problem in the structure can be described by the following steps:

1. Make an initial guess for the temperatures at all the joints. 2. For each structural member separately, perform a finite element discretization

in the axial direction. That is, divide each rod into one-dimensional finite elements in the 6-direction, and apply the Galerkin finite element method to Eq. (1) with respect to 6 only, while leaving the variation with respect to the variable s continuous. This results in a system of ordinary differential equa- tions in the variable s for each member.

3. For each member, solve the system of ordinary differential equations in s by a harmonic-balance procedure.

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236 D. GIVOLl AND 0. RAND

4. Based of the results of the current iteration, update all the joint temperatures. 5. Check convergence. Stop if convergence is achieved; otherwise, return to step

2.

We now comment on these five steps. Step 1 is the initialization step in the iterative solution procedure. The number of iterations needed for the convergence of the method (cf. step 5 ) strongly depends on the distance of the initial guess from the exact solution. Once the temperatures of all the joints assume certain values, it is possible to treat each structural member separately, as is done in step 2. The specified temperatures at the hvo edges of each member serve as boundary conditions for the Galerkin procedure in the &direction.

The system of ordinary differential equations in s, which is the result of step 2, has the form (see [I911

The subscript ss stands for the second derivative with respect to s. In Eq. (21, d is the global solution vector containing the nodal temperature values, K is the thermal conductivity matrix (in the axial direction), R is the radiation vector that depends nonlinearly on d, and F is the thermal load vector. The expressions for K, R, and F are standard. On the-other hand, the matrix M, which is associated with the thermal conductivity in the circumferential direction, is defined by the nonstan- dard expression (see [191)

Here N,, is the number of elements in the member (in the &direction), d,Ndll is the assembly operator, me is the element matrix corresponding to the global matrix M, fie is the element domain, and N, and Nb are the element shape functions associated with nodes a and b. Explicit expressions for all the element matrices and vectors can easily be obtained for a given choice of finite element shape functions.

In step 3, the system of ordinary differential equations (2) is solved for each structural member, using a harmonic-balance procedure. To be more specific, consider the unknown vector d(s) and the right-hand-side vector F(s) appearing in Eq. (21, and let them both be represented approximately by the finite Fourier expansions

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ANALYSIS OF SPACE FRAMES AND TRUSSES 237

Here, do, d,,, d,,, Fo, F,,, F,, (n = 1, . .., N ) are real constant vectors and 4 is the circumferential angle defined by

4 = 2 ~ r s / p (7)

The Fourier coefficients coefficients F,, F,,, and (FFT) scheme.

do, d,,, and d,, in Eq. (5) are, of course, unknown. The F,, in Eq. (6) are found using a Fast Fourier Transform

The expansions (5 ) and (6 ) are now substituted into Eq. (2). The differential and algebraic operators appearing in Eq. (2) are applied to the finite series (5) and (6), and the resulting Fourier series are,also truncated after N harmonics. The harmonic balance between the two sides of Eq. (2) leads to a system of nonlinear algebraic equations for the unknown coefficients do, d,,, and d,,. The algebra involved in constructing these algebraic equations is considerable, and therefore it is performed symbolical& by the computer code itself, using the symbolic manipu- lator described in [20]. The resulting algebraic nonlinear system is then solved by a modified Newton iterative scheme (see, e.g., [14]).

In step 4, the temperatures at all the joints are updated. The scheme for updating these temperatures is based on the balance of heat flux entering each joint. In the steady-state case considered here it is clear that the net contribution of heat flux to a certain joint by all the rods connected to it must be zero. Therefore, the temperature at each joint is corrected according to the magnitude and sign of the net heat flux entering the joint, which is calculated from the numerical results of the current iteration. An explicit formula for the updated joint temperature in the general case is given in [19]. In the special case where all the rods connected to a certain joint are identical in their axial conductivity K ~ , in their cross-sectional geometry and in the element sizes used in their discretization, the updating scheme becomes extremely simple. In this case the new joint temperature is simply the average of the zeroth-order Fourier coefficients at the second node of all the rods connected to the joint.

Steps 2-4 are repeated until convergence, checked in step 5, is achieved. This happens when the differences between joint temperatures obtained in two succes- sive iterations are sufficiently small. Numerical experiments show that the iterative procedure outlined above converges rapidly if the initial guess for the joint temperatures is reasonably close to the exact solution. Otherwise, some modifica- tions are needed in the updating scheme in order to guarantee convergence [19].

THE THERMOELASTIC ANALYSIS

Throughout this section the variable T with denote the difference between the actual temperature at a point and the reference temperature in which the structure is undeformed.

After completing the thermal analysis and obtaining the temperature field T ( 5 , S) throughout the structure it is possible to perform the thermoelastic analysis and obtain the displacement and stress fields induced by this temperature distribu-

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238 D. GIVOLl AND 0. RAND

fmire element

.",. "; & :$ .5 2 " ":; 11

a, 8 h:. .:'f -r, .,@. * * C*E',,L. ,...

Fig. 2 Thc four effective load components acting at each finite element node in the structure.

tion. The approach adopted here is to devise a scheme that is highly compatible with the standard finite element method, so that the use of existing commercial finitc element codes is made possible. In the scheme described below we use C" beam finite elements with two nodes and six degrees of freedom per node (three displacements and three rotations). The element coordinate system is (5 , 77, 51, in which 5 is the axial coordinate and 11 and 5 are chosen to be the orthogonal principal inertia directions of the cross section.

First, the thermoelastic problem is reformulated as an elastic problem in which eflectiue extenzal forces and moments replace the given temperature field in the structure. Figure 2 defines the four effective load components acting at each node. These are the effective axial force PC, the effective bending moments M, and MI, and the effective torsional moment MS. Note that the effective axial force PC and the two effective bending moments M, and MI are applied to each node on the element leuel. The effective force and moments acting at a global node are computed by summing up the contributions from all the elements which share this node. On the other hand, it is more convenient to define the effective torque Mg as attributed to each node on the global leuel from the outset.

Following Boley and Weiner [21], the expressions for P6, M,, and MI based on the Bernoulli-Euler beam theory are

Here A is the cross-section area, a is the thermal expansion coefficient, and E is Young's modulus. The minus sign in Eq. (10) is due to the load direction convcntion shown in Fig. 2. In our case dA = tds (see Fig. 11, or, by using Eq. (71, dA = t p / ( 2 ~ ) d 6 . Assuming that a and E are constant over the cross section, we

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ANALYSIS OF SPACE FRAMES AND TRUSSES 239

get from Eqs. (8)-(10)

We now recall that the temperature T at a node is given, as the result of the thermal analysis, in the form of a Fourier series in the angle 4, as in Eq. (5). Thus we have the expansion

In addition, we assume that the coordinates q and 5 along the midline of the cross section can also be expanded in a truncated Fourier series in 4, i.e.,

The coefficients appearing in Eqs. (13) and (14) can be found for various cross- sectional shapes by using the formulae of the differential geometry of curves. Now we substitute Eqs. (12)-(14) into Eq. ( l l ) , while taking advantage of the orthogo- nality of the trigonometric functions. This finally yields

P, = a Etp To (15)

Note that the effective axial force P* is determined by the average temperature To alone, whereas the effective bending moments M, and Mc are determined in general by all the Fourier coefficients.

As an example we consider the special case of a circular thin-walled cross section of radius R and thickness t . In this case p = 27rR, = R, is, = R , and all the other Fourier coefficients of q and 5 on the cross-sectional midline vanish. Then Eqs. (15)-(17) give

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240 D. GIVOLI AND 0. RAND

We note that in a circular thin-walled cross section, the effective bending moments are determined by the first-order temperature Fourier coefficients T,, and T,,. Higher-order harmonics do not contribute to the effective bending moments. (This, however, does not mean that a single harmonic suffices in the harmonic-balance procedure outlined in the previous section. Due to the nonlinear character of Eq. (I), lower-order harmonics in the temperature field are affected by higher-order harmonics in the data.)

For a thin-walled cross section in the shape of a square with dimensions a x a, the Fourier coefficients of 7 and g in Eqs. (13) and (14) are easily calculated by using a Fast Fourier Transform (FFT) scheme. We find that the even-order Fourier coefficients vanish and that Q, = = - 5,, = f;, = 0.405a, q3 = - Q, = - lC3 =

-h3 = 0.045a, etc. The absolute value of the coefficients continues to decrease rapidly for the higher-order harmonics. These coefficients can be substituted in Eqs. (16) and (17) to yield the bending moments M , and Ml to within a desired order of accuracy.

Next we consider the torsional effects. To this end we first recall that in the three-dimensional Navier equation of thermoelasticity, the term involving the temperature field T plays the role of a body force FT, having the expression (see, e.g., [221)

FT = - 3 a K VT (21)

Here K is the bulk modulus, related to Young's modulus E and to Poisson's ratio v by 3 K = E/(1 - 2v). Now for a beam with a thin-walled cross section the temperature variation through the thickness is small, and therefore from (21) only the tangent component of the body force FT, namely,

is significant. Using Eqs. (22), (7), and (12) we obtain the expansion

It is reasonable to define the shear flow vector q (force per length of cross-sectional midline) in the cross section associated with a certain node in the structure as

Here h is the average size of the elements which share the node under considera- tion, and e 5 s the unit vector tangent to the midline. See Figure 3. From Eq. (24) it is easy to obtain the following expression for the effective torque (see Figs. 2 and 3):

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ANALYSIS OF SPACE FRAMES AND TRUSSES

Fig. 3 The shear flow q acting in the cross section associated with a certain node in the structure.

Here e; and es are the components of e V n the 7- and 5-directions respectively. As an example, we again consider the case of a circular thin-walled cross

section of radius R. We use Eqs. (131, (14), and (23) in Eq. (251, with p = 2nR, e; = -sin 4, e; = cos 4, 7)c, = R, JS, = R, and all the other Fourier coefficients of 7 and 5 being zero. This gives

The integral in Eq. (26) vanishes since FT in Eq. (23) has zero average. Thus, in a circular thin-walled cross section the effective torsional moment is zero, and so no torsional effects are present.

For a square cross section of dimension a, it can readily be shown that the quantity (e jq - e i 5 ) in the integrand in Eq. (25) has the constant value a/2. Thus, we again obtain Mg = 0, as in Eq. (26). However, with less symmetric cross sections (e.g., a rectangular cross section) torsional effects are present, and Mt can easily be calculated.

The use of the expressions obtained above for the effective external loads enables one to employ a standard existing finite element code for the thermoelastic analysis while implicitly using the harmonic-balance approach. The four effective load components are computed and applied at each node in the structure. The model is then analyzed by a standard finite element code incorporating three- dimensional elastic beam elements, which yields the displacements, axial stress, and transverse shear stresses throughout the structure. This analysis does not involve the temperature field in the structure; the thermal effects are all taken care of by the effective external loads.

The displacements and shear stresses throughout the structure can be read out directly from the output of the finite element code. On the other hand, the values of the axial stress uct computed by the code are not correct, since the code makes

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242 D. GlVOLl AND 0. RAND

use of the elastic constitutive equations which do not include thermal effects. The correct expression for the axial stress is (cf. [21]),

Here F, is the internal axial force, and m, and m, are the internal bending momenis, all usually given in the output of the finite eiement code. Also, I, and lc are the principal cross-sectional moments of inertia.

We return once more to the example of a circular thin-walled cross section of radius R. We use Eqs. (12)-(14) in Eq. (27) with I , = II = 1 , q,, = R, lS, = R, and all the other Fourier coefficients of q and 5 being zero, to obtain:

From Eqs. (18)-(20), (26), and (28)-(31) we conclude that, for a structure com- posed of members with circular thin-walled cross section, the thermoelastic response is determined entirely by the zeroth- and first-order harmonics of the temperature field distributed in the structure.

For a square cross section of dimension a, all the odd harmonics contribute to the expression of the axial stress given by Eq. (27). However, as already indicated, the size of the Fourier coefficients of and 5 decreases rapidly for the high-order harmonics. Therefore, only a few terms are needed to capture the complete variation of ueg with high accuracy.

NUMERICAL EXAMPLES

The numerical scheme was validated via a number of elementary examples for which analytic solutions are available. Examples involving a cubic truss slanted with respect to the global system of coordinates yielded satisfactory results. An addi- tional example that was tested is that of a cantilever beam of thin-walled circular cross section, deforming due to the given circumferential temperature distribution T ( 4 ) = cos 4. The parameters chosen are a = lo-', the length of the beam is 1 = 10, and the radius of the cross section is R = 0.05. The value of the lateral deflection at the end of the beam obtained by the numerical scheme is w = 0.01098.

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ANALYSIS OF SPACE FRAMES AND TRUSSES 243

We compare this result with the formula given in [23, p. 1071 for the case of a cantilever beam whose upper and lower surfaces are heated with temperatures TI and T,, respectively:

Since the original temperature distribution is sinusoidal, the temperature differ- ence between the upper and lower surfaces is T2 - T, = 1 - (- 1) = 2. Substituting the numerical data, formula (32) yields w = 0.01. The agreement between the numerical and analytic results is thus satisfactory.

To demonstrate the performance of the method, we apply it to the multibay frame structure illustrated in Fig. 4. The global Cartesian system of coordinates ( x , y, z) is also shown in the figure. All the beam members have a thin-walled circular cross section of radius R = 0.0045 m and thickness r = 0.001 m. The global dimensions of the structure are 2 m x 2 m x 5 m. All the joints prevent relative rotational deformation between different members (later we will consider another situation as well). 'The base of the structure, lying in the plane z = 0, is rigidly clamped. The thermal and elastic material properties are: a, = 0.018, K~ = K, = 10.1 W/m OK, UE = 9.1. 10-lo w / ~ ~ " K ~ , E = 44. 109N/m2, v = 0.3, and a =

7.3. ~ O - ~ ~ K - I . The reference temperature in which the structure is undeformed is 273°K. The

structure is exposed to solar radiation of magnitude q,, = 1300 W/m2, directed in the -x-direction. The structure material is opaque and self-shadowing effects are taken into account. Thus, half of the entire structure and also half of the outer perimeter of every cross section are hidden from the incident heat flux.

Fig. 4 The multibay frame structure used to demonstrate the performance of the proposed numerical scheme. The structure is exposed to solar radiation coming from the -x-direction.

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244 D. GIVOLI AND 0. RAND

z coo rd ina t e (ml

Fig. 5 Thc tcmperature distribution along the right column of beams at 4 = 0' and 4 = 90°, in the symmetric and asymmetric cases.

In both the thermal and elastic analyses, each member was divided in its axial direction into 7 finite elements, using a cosine-type distribution in which the clements are small near the joints and become larger in the central region of the beam. In the thermal harmonic-balance procedure used in th t circumferential direction, the first 12 harmonics were taken into account. In a convergence test similar to the one performed in [19], these parameters proved to yield accurate results.

As already mentioned half of the structure is in the shadow. One should note, however, that the 10 members lying in the x = 0 plane (arranged in two columns of five members each) are exactly on the boundary line between the exposed and hidden regions of the structure. Whether they are themselves exposed or hidden is thcrcfore extremely sensitive to small geometrical perturbations. Here we consider two cases: the symmetric case in which both columns of beams are exposed to the sun, and the asymmetric case in which only the left one (see Fig. 4) is exposed whereas the right column is hidden.

Figure 5 shows the temperature distribution along the right column of beams at two cross-sectional circumferential positions, namely, at 4 = 0' and 4 = 90". Results in both the symmetric and asymmetric cases are shown. In the latter case thc right column is entirely in the shade and hence the temperature is circumferen- tially uniform. In the symmetric case, the temperatures at the two circumferential locations differ by more than 10°K. The average temperature in the asymmetric case is smaller by about 70°K.

Figures 6a , b show the deformed structure in the symmetric and asymmetric cases, respectively. The significant difference between the deformation patterns in the two cases demonstrates the great sensitivity of the structure to small perturba-

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ANALYSIS OF SPACE FRAMES AND TRUSSES

Fig. 6 The deformed structure: ( a ) the symmetric case; (b ) the asym-

( b ) metric case.

tions in geometry and orientation. In practice, some asymmetry is bound to be present.

Figure 7 shows the distribution of the axial stress utt along the right column of beams, at 4 = 0' and 4 = 90°, in the symmetric and asymmetric cases. The stress in the joint connecting the first and second bays turns out to be the largest. The numerical values of the stresses are small, as typical for a structure in space, although the deformation of the structure may be regarded as critical in some applications.

Up to this point all the joints in the structure were frame-type joints; namely, they were given the property of preventing relative rotational deformation between different members. Now we assume that some of the joints along the column of beams that faces the solar radiation vector (see Fig. 4) fail to have rotational rigidity. In the finite element model these frame-type joints are replaced by truss-type joints that cannot transmit any moments. We start by releasing the rotational rigidity of the joint at the free end of the structure, and we gradually do the same thing with the succeeding joints along the same column. We do this in both the symmetric and asymmetric cases.

Figures 8a , b show the results of this numerical experiment. The size of the deflection along the right column of beams is shown for various values of the number of "released" joints. Figure 8a corresponds to the symmetric case whereas Fig. 8 b corresponds to the asymmetric case. Although the two figures are slightly different, the effect that the partial loss of rigidity has on the deformation is apparent in both cases. When four joints loose their rotational rigidity, the tip

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246 D. GIVOLI AND 0. RAND

"0 ? 7 a- * -

syrnrn,$ = 0" ;... Summ,.!??.=9@ ..

Asyrnrn. - - - - - - - - - - - - - - - -

, I

-__, m

I I I I

0.0 1.0 2.0 3.0 4.0 6.0

z coordinate (rn)

Fig. 7 Thc distribution of the axial stress qc along the right column of beams at q5 = 0" and q5 = 90", in the symmetric and asymmetric cases.

displacement increases by a factor of 3.7 in the symmetric case and by 3.2 in the asymmetric case.

CONCLUDING REMARKS

We have devised a numerical solution procedure for the thermoelastic analysis of large frame- and truss-type space structures, based on the harmonic finite-element approach. This procedure yields the elastic response of the structure to thermal loading while being capable of treating thin-walled cross sections of general shapes and temperature distribution in both the axial and circumferential directions of each structural member. The method has the advantage that it enables one to use an existing finite element code incorporating standard elastic beam elements, while at the same time being based directly on the harmonic finite-element thermal analysis which eliminates the need for a large number of temperature degrees of freedom.

We believe that the method presented here brings large scale computations involved in the thermoelastic analysis of space structures within practical reach. An important addition to this work would be the extension to the dynamic case and, in particular, to the thermoelastic analysis of space structures in periodic motion. It seems that an application of the harmonic balance procedure with respect to both the circumferential coordinate (s) and time ( t ) may be beneficial. We hope to report on progress in this direction in the future.

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ANALYSIS OF SPACE FRAMES AND TRUSSES

( b )

Fig. 8 The size of the deflection along the right column of beams for various values of the number of "released" joints: ( a ) the symmetric case; ( b ) the asymmetric case.

f 9 :- * a

z- No released ioints .... !.r.e!es.ed.joint .... ...... f

2 released ioints - - - - - - - - - - - - - - - - - - - - - 3 released joints - 4- T r e E s e d ioints -

E - c 9 .o 2- - 0 al = '?- 0 0 0 n /

2 -

2 I I 1 I I

0.0 1.0 2.0 3.0 4.0 6.0

z coordinate (m)

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248 D. GlVOLl AND 0. RAND

REFERENCES

1. J. S. Archer, High Performance Parabolic Antenna Reflectors, J. Spacecraft Rockets, vol. 17, pp. 20-26, 1980.

2. S. C. Clark and G. E. Allen, Thermo-Mechanical Design and Analysis System for the Hughes 76-in. Parabolic Antenna Reflector, in H. E. Collicot and P. E. Bauer, Eds., Spacecraff Thermal Control, Ilesign Operatiott, AlAA Publ., New York, 1983.

3. S. Nour-Omid, B. Nour-Omid, and C. C. Rankin, Large Rotation Transient Analysis of Flexible Space Structures, Proc. Second World Congress on Computational Mechanics, Stuttgart, Germany, 1990.

4. G. Shaviv, Ofek 1, Ofek 2 and TECHSAT Satellites, Seminar in Astrophysics, Dept. of Physics, Technion, Haifa, 1991.

5 P. Santini and A. Paolozzi, Heat Conduction in a Large Space Structure, Acta Astro., vol. 19, pp. 162-170, 1989.

6. J. T. Farmer, D. M. Wahls, and R. L. Wright, Thermal-Distortion Analys$ of an Antenna Strongback for Geostationary High-Frequency Microwave Applications, NASA Technical Paper 3016, NASA Langley Research Center, Hampton, Virginia, 1990.

7. W. A. Nash and T. J. Lardner, Parametric Investigation of Factors Influencing the Mechanical Behavior of Large Space Structures, AFOSR report no. TR-86-0858, 1985.

8. J. M. I-ledgepcth and R. K. Miller, Structural Concepts for Large Solar Concentrators, Acta Asrro., vol. 17, pp. 79-89, 1988.

9. S. C. Peskett and D. T. Gethin, Thermal Analysis of Spacecraft, in R. W. Lewis and K. Morgan, Eds., Numerical Methods in Thermal Problems, Vol. V I , pt. 1, Pineridge Press, Swansea, pp. 713-729, 1989.

10. H. O I ~ , M. Menking, E. Hornung, and E. Erben, Development of Large Orbital Structure Systems, 40th Congress of the IAF, paper no. IAF-89-340, Beijing, China, 1989.

11. E. A. Thornton and D. B. Paul, Thermal-Structural Analysis of Large Space Structures: An Assessment of Recent Advances, J. Spacecraff Rockets, vol. 22, pp. 385-393, 1985.

12. J. Mahaney and E. A. Thornton, Self-shadowing Effects on the Thermal-Structural Response of Orbiting Trusses, J. Spacecraff Rockers, vol. 24, pp. 342-348, 1987.

13. W. L. KO, Solution Accuracies of Finite Element Reentry Heat Transfer and Thermal Stress Analyses of Space Shuttle Orbiter, Internat. J. Numer. Methods Eng., vol. 25, pp. 517-543, 1988.

14. D. Givoli and 0. Rand, Thermoelastic Analysis of Space Structures in Periodic Motion, J. Spacecraft Rockers, vol. 28, pp. 457-464, 1991.

15. 0. Rand and D. Givoli, An Integrated Thermoelastic Analysis for Periodically Loaded Space Structures, in Proc. of 17th Congress of ICAS, Stockholm, Sweden, pp. 1529-1533, 1990.

16. J. Maheney, E. A. Thornton, and P. Dechaumphai, Integrated Thermal-Structural Analysis of Large Space Structures, Computational Aspects of Heat Transfer in Structures Symposium, NASA Langley Research Center, Hampton, VA, NASA CP-2216, pp. 179-198, 1981.

17. J. D. Lutz, D. H. Allen, and W. E. Haisler, Finite-Element Model for the Thermoelastic Analysis of Large Composite Space Structures, J. Spacecraft Rockets, vol. 24, pp. 430-436, 1987.

18. L. D. Pinson, Recent Advances in Structural Dynamics of Large Space Structures, Acta Astro., vol. 19, pp. 162-170, 1989.

19. 0. Rand and D. Givoli, Thermal Analysis of Space Trusses Including Thee-Dimensional Effects, Internat. J. Numer. Methods Heat Fluid Flow, vol. 2, pp. 115-125, 1992.

20. 0. Rand, Harmonic Variables-A New Approach to Nonlinear Periodic Problems, J. Comp. Math. Appl., vol. 15, pp. 953-961, 1988.

21. B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, John Wiley, New York, 1960. 22. 1. S. Sokolnikoff, Mathematical Theory of Elasricify, 2nd ed., McGraw-Hill, New York, 1956. 23. R. J. Roark and W. C. Young, Formulas for Stress and Strain, 5th ed., McGraw-Hill, New York,

1975.

Receiued March 25, 1992

Address correspondence t o D a n Givoli.

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