Stiffness and Strength Tailoring in Uniform Space-Filling Truss Structures

NASA Technical Paper 3210 April 1992 Stiffness and Strength Tailoring in Uniform Space-Filling Truss Structures Mark S. Lake

Summar This paper presents a deterministic procedure for tailoring the continuum stiness and strength of uniform space-lling truss structures through the appropriate selection of truss geometr and member sies (i.e., eural and aial stinesses and length). The trusses considered herein are generated b uniform replication of a characteristic truss cell. The repeating cells are categoried b one of a set of possible geometric smmetr groups derived using crstallographic techniques. The elastic smmetr associated with each geometric smmetr group is identied to help select an appropriate truss geometr for a given application. Stiness and strength tailoring of a given truss geometr is enabled through eplicit epressions relating the continuum stinesses and failure stresses of the truss to the stinesses and failure loads of its members. These epressions are derived using an eisting equivalent continuum analsis technique and a newl developed analtical failure theor for trusses. Several eamples are presented to illustrate the application of these techniques and to demonstrate the usefulness of the information gained from this analsis. Introduction In the future, the primar structures of man large orbiting spacecraft will be lightweight trusses. Although numerous studies have been performed to determine the feasibilit and structural characteristics of these trusses (e.g., refs. 1 through 3), little work has been done to establish deterministic procedures for their design. The selection of appropriate truss designs is inuenced b both structural optimiation and spacecraft operational considerations. Currentl, structural optimiation of these trusses is a predominantl heuristic process involving trial and error procedures. This paper presents a deterministic procedure for truss geometr selection and member design based on tailoring the continuum stiness and strength characteristics of the truss. Analsis of the truss stiness and strength characteristics is performed using an equivalent continuum analog (ref. 4). This approach is preferred because it oers better insight into structural behavior than the conventional numerical analsis techniques oer. The trusses considered herein are generated b uniform rotational and/or translational replication of a characteristic cell, as shown in gure 1, and the are thus called uniform space- lling trusses. In most cases, the repeating truss cell and the resulting truss structure inherentl possess some geometric smmetr. The presence of geometric smmetr implies elastic smmetr that reduces the number of independent equivalent elastic constants characteriing the truss. In this stud, the crstallographic techniques are used to dene the possible geometric smmetr groups associated with repeating cells that generate uniform trusses. In addition, the number of independent elastic constants associated with each geometric smmetr group is identied to help select an appropriate truss geometr for a given application. The independent elastic constants characteriing a truss can be tailored to specic values b selecting appropriate member stinesses. In the present stud, this stiness tailoring is accomplished using eplicit relationships between the equivalent continuum stinesses of a truss and the aial stinesses of its members. Also, the continuum strength characteristics of a truss are tailored using a strength tensor that is written eplicitl in terms of the local elastic buckling loads of the truss members. To illustrate the application of these techniques, a commonl used truss geometr is analed to determine member sies that produce optimum isotropic and orthotropic (i.e., one direction of high stiness and strength) designs. All derivations presented have been performed smbolicall using a computeried mathematics routine (ref. 5), and results have been converted into a numerical form when necessar. The advantage in using smbolic algebra is that eplicit relationships can be determined between the design parameters and the continuum elastic behavior of the truss. These eplicit relationships signicantl enhance the utilit of the stiness and strength tailoring procedures presented. 1

Smbols A A c A n A o C ijkl C 0 ijkl cross-sectional area of members in regular octahedral truss cross-sectional area of members in cubic lattice of Warren truss cross-sectional area of members in nth group cross-sectional area of members in octahedral lattice of Warren truss continuum elastic stinesses (tensor form) transformed continuum elastic stinesses (C 0 1111 ) n continuum unidirectional stiness for nth group of parallel members c mn E Eeq (Eeq) iso (Eeq) Geq L l n r n S ijkl s mn T ij v n continuum elastic stinesses (matri form) Young's modulus of truss material equivalent continuum Young's modulus equivalent Young's modulus of isotropic Warren truss equivalent -direction Young's modulus equivalent continuum shear modulus characteristic dimension of truss repeating cell length of members in nth group radius of gration of members in nth group continuum elastic compliances (tensor form) continuum elastic compliances (matri form) coordinate transformation tensor volume fraction of nth group of parallel members ; ; Cartesian coordinates 0 c n " ij strain tensor member longitudinal direction length ratio of repeating truss cell in direction ratio of cross-sectional areas of members in Warren truss ratio of cross-sectional areas of members in nth group to that of rst group " 0 ij transformed strain tensor ("crit) n eq eq ij ult critical aial strain for nth group of members equivalent continuum Poisson's ratio densit of truss material equivalent continuum densit stress tensor continuum compression strength 2

( ult) -direction compression strength ( ult) iso compression strength of isotropic Warren truss i direction cosine with the ith coordinate ais ; ' spherical coordinates [ kl ] n strength tensor Truss Geometr Selection The design of a truss is often governed b considerations other than the structural performance (e.g., as shown in ref. 6). For eample, operational concerns such as the arrangement and integration of spacecraft subsstems onto a truss might dictate a particular geometr for the truss repeating cell. For applications in which operational concerns do not dominate, selecting a truss geometr b matching its inherent elastic behavior with the structural requirements of the spacecraft is prudent. Even in situations in which operational concerns prevail, enough latitude probabl eists in the selection of a truss geometr so that structural considerations can be incorporated. This section categories the elastic characteristics of most uniform space-lling truss structures b eamining their geometric smmetr. The uniform truss structures considered herein are similar to crstalline lattices because the both can be generated b replicating a characteristic repeating cell that tpicall possesses geometric smmetr. Of interest are smmetr with respect to specic rotations about one or more aes and smmetr with respect to reection about one or more planes. Smmetr in the truss geometr (i.e., lattice arrangement and member designs) implies smmetr in the elastic characteristics of the truss. This implied elastic smmetr reduces the number of independent equivalent elastic constants characteriing the continuum behavior of the truss, and it thus simplies the task of stiness and strength tailoring. Rotational Smmetr Groups Crstallographic studies (refs. 7 and 8) have shown that the rotational and reectional smmetries in reticulated, or discrete, structures are limited to a set of 32 possible combinations that are commonl called crstallographic smmetr groups. Love (ref. 9) determined that the elastic behavior of most crstallographic smmetr groups can be derived b considering onl rotational smmetr. For brevit, the few cases in which reectional smmetr is important are not considered herein. B neglecting reectional smmetr, the 32 crstallographic smmetr groups reduce to the 10 rotational smmetr groups shown in gure 2. Each smmetr group in gure 2 is identied b a specic combination of aes about which rotational smmetr eists. The orientations of these aes are shown relative to a Cartesian coordinate sstem, and the order of rotational smmetr is given b one of four graphical smbols: a cusped oval, a triangle, a square, or a heagon. These smmetr smbols are related to the order of smmetr in the ke. This order of smmetr is dened as n-gonal where the rotation angle is 2=n and n is either 2, 3, 4, or 6. Notice that in smmetr groups i and j, the trigonal smmetr aes lie along lines connecting the center of a cube with its corners, thus structures of these smmetr groups are often referred to as cubic structures. Smmetr groups that possess more than one ais of rotational smmetr are called multiaial. The three rotational smmetr aes presented for each of the multiaial groups are not the onl smmetr aes for those groups. A complete set can be generated b appling the smmetr operation of each ais to the others. For eample, in smmetr group d, appling trigonal smmetr about the -ais identies four additional digonal smmetr aes separated b 60 in the - plane. 3

An truss structure that possesses aes of rotational smmetr can be categoried b one of the 10 rotational smmetr groups in gure 2. This classication is accomplished b identifing all rotational smmetr aes within the structure and then b selecting a Cartesian coordinate sstem relative to these aes which matches one of the given smmetr groups. Once the smmetr group of the truss is identied, its inherent elastic behavior is determined using the methods that follow. Elastic Characteristics of Rotational Smmetr Groups A uniform truss structure can be represented b an equivalent homogeneous anisotropic continuum characteried b 21 empirical elastic constants. These elastic constants appear as stinesses c mn or C ijkl in the constitutive equations given in equation (1a) in matri form and equation (1b) in tensor form: 8 >< >: 11 22 33 23 13 12 9 >= >; = 2 6 4 c11 c12 c13 c14 c15 c16 c12 c22 c23 c24 c25 c26 c13 c23 c33 c34 c35 c36 c14 c24 c34 c44 c45 c46 c15 c25 c35 c45 c55 c56 c16 c26 c36 c46 c56 c66 38 >< 7 5>: "11 "22 "33 2"23 2"13 2"12 9 >= >; (1a) ij = C ijkl " kl (1b) When the truss possesses geometric smmetr, elastic smmetr is implied, which reduces the number of independent continuum elastic constants. A continuum that possesses geometric smmetr with respect to a rotational or a reective transformation (characteried b T ij ) also possesses smmetr in its elastic constants (see, for eample, ref. 10). Therefore, the transformed stiness tensor C 0 ijkl must be identical to the original tensor C ijkl. Hence, C 0 ijkl = C mnopt im T jn T ko T lp = C ijkl (2) The number of independent elastic constants associated with each smmetr group, presented in gure 2, is determined using equation (2). A transformation tensor T ij is determined for the specied rotation about each smmetr ais and substituted into equation (2) to give 21 conditions on the stinesses C ijkl. Some of these conditions are identicall satised, whereas others can be satised onl b the elimination or restriction of certain elastic constants. This process is repeated for all rotational smmetr aes in the given smmetr group, and the resulting reduced set of elastic constants denes the continuum elastic characteristics of an truss structure that is a member of that smmetr group. For eample, the independent elastic constants characteriing trusses of smmetr group a are determined b enforcing elastic smmetr with respect to a rotation of 180 about the -ais. The transformation matri for this rotation is T ij = 2 6 4,1 0 0 7 0,1 05 (3) 0 0 1 4 3

Substituting equation (3) into equation (2) gives the following result: C ijkl = C ijkl (4a) if an even number (or none) of the indices is 3 and C ijkl =,C ijkl (4b) if an odd number of the indices is 3. Satisfing equation (4b) requires the following to be true (note that, because of smmetr in C ijkl, man possible permutations of the subscripts have been omitted): C1123 = C1113 = C2223 = C2113 = C3323 = C3313 = C2312 = C1312 =0 (5) Emploing the usual conversion from tensor to matri form (ref. 10), the following equivalent conditions eist for the components of the stiness matri: c14 = c15 = c24 = c25 = c34 = c35 = c46 = c56 =0 (6) Similar calculations can be made for the remaining smmetr groups in gure 2. Without presenting the details, the conditions on continuum stinesses as well as the number of independent elastic constants for each smmetr group are presented in table I. A similar derivation shows that the conditions presented in table I must also be obeed b the components of the continuum compliance tensor. An obvious conclusion from table I is that the presence of an smmetr in a truss lattice signicantl reduces the number of independent elastic constants characteriing its continuum behavior. This result greatl simplies the task of tailoring the stiness and strength of most trusses. Remember that the conditions on the elastic constants presented in table I are valid onl for the coordinate aes presented in gure 2. For eample, smmetr groups b; f; g; h; i; and j are indicated to have ero shear coupling stinesses (e.g., c14 ;c 15 ; and c 16) in the given coordinate sstem, but the might have nonero coupling stinesses in an alternate coordinate sstem. As eplained b Rosen and Shu (ref. 11), and seen in table I, none of the permissible geometric smmetr groups possesses sucient smmetr to ensure isotropic elastic behavior. However, this research shows that isotrop can be obtained b tailoring the relative stinesses of dierent truss members. The information in table I should help select appropriate truss geometries for particular truss applications and determine additional stiness tailoring requirements for the selected truss geometr. For eample, if the primar loads in a truss are epected to occur in onl one direction, considering geometries that have less smmetr and which can easil be tailored to have signicantl higher stinesses and strengths in that direction (i.e., an orthotropic design) is more ecient. However, for a structure that ma have to sustain loads in multiple directions or one for which the loading conditions are not well-dened, considering truss geometries that possess more smmetr and which can be tailored to behave isotropicall ma be best. Stiness and Strength Tailoring Once a truss geometr has been selected, its independent elastic constants are identied using table I. The values of these constants can be adjusted for a particular application b tailoring the relative aial stinesses of the members comprising the truss. Likewise, changing the relative elastic buckling loads of dierent members alters the equivalent continuum strengths of the truss. Changing onl the dimensions and member stinesses of a truss which do not violate its geometric 5

smmetr causes it to remain in the same rotational smmetr group; thus, the conditions on its continuum stinesses given in table I remain valid. Alternativel, changing dimensions and member stinesses of a truss which violate its geometric smmetr changes its rotational smmetr group, thus altering the number of independent elastic constants characteriing its behavior. Stiness and strength tailoring will be demonstrated for a truss in which geometric smmetr is maintained and one in which geometric smmetr is altered. Equivalent Continuum Elastic Constants Once a candidate truss for stiness tailoring is selected, its continuum stinesses are calculated in terms of the aial stinesses of its members. The approach used in this stud for calculating these stinesses was developed b Nafeh and Hef (ref. 12); this approach is similar to a three-dimensional generaliation of classical laminated plate theor (ref. 13) in which groups of parallel members within the truss are analogous to individual lamina. Because truss members carr onl aial loads, each group of parallel members forms a unidirectional elastic continuum that has no transverse or shearing stinesses. The truss assemblage stinesses are obtained b summing the stinesses of each of the groups of parallel members. This superposition of stinesses implies that the continuum displacement eld within a truss is single-valued, which is consistent with the fact that truss members connected at a common point must have the same displacement at that point. Note that this is not the case for trusses with cross-laced members that can slide relative to one another; therefore, such designs should not be analed using the techniques of this stud. Each group of parallel members is characteried b one nonero equivalent stiness that is in the local 0 direction (the member longitudinal direction). This equivalent unidirectional stiness is determined in equation (7) for the nth group of members: (C 0 1111 ) n = Ev n (7) where E is the Young's modulus of the truss material in the members and v n is the volume fraction of the group of members (i.e., the ratio of the total volume of material in the members to the total volume of the truss). The continuum stinesses for a truss are calculated b transforming the unidirectional stinesses for each of its groups of parallel members into a global coordinate sstem using equation (2) and b summing the results, as indicated b C ijkl = X n (C 0 1111 ) n (T 1iT1jT 1k T 1l) n (8) Elements of the rst row of the transformation tensor T1i are simpl the direction cosines between the longitudinal ais of the members and the ith coordinate ais. Therefore, equation (8) can be rewritten as X C ijkl = (C1111 0 ) n ( i j k l ) n (9) n where i is the ith direction cosine of the members. The continuum stinesses dened b equation (9) are eplicit functions of the member etensional stinesses. These functions enable the desired continuum stiness characteristics to be translated into member aial stiness tailoring rules. Equation (9) produces additional restrictions on the continuum stinesses of uniform trusses which should be noted. Emploing the usual conversion from the matri form of the elastic 6

constants to the tensor form (ref. 10), the values for the transverse and shear stinesses c12 and c66 are c12 = C1122 = X n (C 0 1111 ) n (2 1 2 2 ) n (10) c66 = C1212 = X n (C 0 1111 ) n (2 1 2 2 ) n (11) Thus, c12 = c66 (12) Similarl, c13 = c55 c23 = c44 c45 = c36 c25 = c46 c14 = c56 (13) Remember that these identities must be valid for an uniform space-lling truss, regardless of its geometr, and therefore these identities should be added to those alread presented in table I for all smmetr groups. Thus, under these assumptions, a generall anisotropic space-lling truss structure has onl 15 independent elastic constants rather than the 21 that are normal for a generall anisotropic solid. Trusses that are tailored to behave as isotropic continua can be characteried b two elastic constants, an equivalent continuum Young's modulus Eeq and an equivalent continuum Poisson's ratio eq. Writing the stinesses in equation (12) in terms of these equivalent constants gives the following condition: eqeeq (1 + eq)(1, 2eq) = Eeq 2(1 + eq) (14) Solving equation (14) for eq gives the result that eq is equal to 1/ 4. Therefore, an uniform three-dimensional space-lling truss structure that is globall isotropic must have an equivalent Poisson's ratio equal to 1/ 4, and, thus, it has onl one remaining independent elastic constant, which is its equivalent Young's modulus. Using a similar procedure, the two-dimensional space- lling trusses that behave isotropicall must have an equivalent Poisson's ratio of 1/ 3. Equivalent Stiness-to-Densit Ratio Stiness-to-densit ratios are commonl used as indicators of the ecienc of materials. Likewise, equivalent stiness-to-densit ratios are useful indicators of the ecienc of uniform trusses. Most equivalent truss stiness-to-densit ratios are dependent on the design of the truss. However, an equivalent stiness-to-densit ratio that is onl a function of the modulus-to-densit ratio of the parent material will be shown to eist. In equation (15), a sum of equivalent continuum stinesses for a truss is shown to be equal to the sum of the uniaial stinesses of its individual groups of members. Notice that the direction cosine terms drop out because the sum of the squares of the three direction cosines for an member is equal to one. 7

c11 + c22 + c33 +2c23 +2c13 +2c12 = C1111 + C2222 + C3333 +2C2233 +2C1133 +2C1122 = X n (C 0 1111 ) n (4 1 + 4 2 + 4 3 +22 2 2 3 +22 1 2 3 +22 1 2 2 ) n = X n (C 0 1111 ) n (2 1 + 2 2 + 2 3 )2 n = X n (C 0 1111 ) n (15) The equivalent densit of a space-lling truss is determined b multipling the densit of the parent material b the sum of the volume fractions of all groups of parallel members. Considering equation (7), this relationship can be written as eq = X n v n = E X n (C 0 1111 ) n (16) Dividing equation (15) b equation (16) gives the following equivalent stiness-to-densit ratio: c11 + c22 + c33 +2c23 +2c13 +2c12 eq = E (17) Equation (17) is a unique relationship because it provides a direct correlation between an equivalent continuum stiness-to-densit ratio of the truss and the modulus-to-densit ratio of the parent material in the truss members. Once the parent material is dened for a truss, equation (17) provides a direct relationship between the equivalent anisotropic stiness of a truss and its equivalent densit. This relationship can be used in a number of was. For eample, changes in the continuum stinesses because of stiness tailoring of the truss members can be directl translated into a proportional change in the equivalent densit of the truss. Similarl, requiring the sum of the continuum stinesses in the numerator of equation (17) to be constant during stiness tailoring results in the equivalent densit remaining constant. This requirement allows the eects of material redistribution within a truss lattice to be convenientl studied. Equation (17) can be simplied for trusses that are tailored to be globall isotropic. Without presenting details, equation (17) reduces to the following equation b writing the equivalent continuum stinesses in terms of an equivalent Young's modulus and Poisson's ratio (equal to 1/ 4): Eeq = 1 E (18) 6 eq The signicance of equation (18) is that all uniform space-lling trusses that are globall isotropic must have the same equivalent modulus-to-densit ratio regardless of their geometries or member sies. Furthermore, this modulus-to-densit ratio must be eactl 1/ 6 of the modulus-to-densit ratio of the parent material. Equivalent Continuum Strength Tensor The continuum strength of a truss structure is dened herein as the maimum continuum stress that the truss can sustain before an of its members buckle elasticall. This failure mode, which is a local phenomenon within the truss lattice, will have one of two eects on the continuum behavior of the truss. If redundant members eist and load is redistributed, local buckling will cause a change in the continuum stinesses of the truss. However, if no load redistribution takes 8

place, local buckling will precipitate a catastrophic failure of the truss lattice. These continuum eects are analogous, respectivel, to ielding and ultimate failure in a material. Because the local failure mode in trusses can be determined analticall, a purel analtical failure theor for trusses can be constructed. In this section, a tensor that describes the strength of a truss will be constructed, and failure analsis using this strength tensor will be discussed. Having a tensor that represents the strength of a truss is advantageous because it allows strength to be readil determined in alternate reference frames or under multiaial stress states. Material strength is not a tensor quantit, and, thus, analsis of failure in materials under multiaial stress can be accomplished onl with approimate, semiempirical theories such as that proposed b von Mises (e.g., as eplained in ref. 14). A strength tensor is constructed for trusses b converting the applied stresses into strains using the compliance equations given in equations (19) and b analing these strains to determine if the aial compression strain in an truss member has eceeded its critical elastic buckling limit: 9 2 38 9 "11 s11 s12 s13 s14 s15 s16 11 "22 s12 s22 s23 s24 s25 s26 8>< >= 22 "33 s13 s23 s33 s34 s35 s36 = 6 >< >= 33 7 (19a) >: 2"23 2"13 2"12 >; 6 4 s14 s24 s34 s44 s45 s46 s15 s25 s35 s45 s55 s56 s16 s26 s36 s46 s56 s66 7 5>: 23 13 12 >; " ij = S ijkl kl (19b) Note that the compliance matri in equation (19a) is simpl the inverse of the stiness matri given in equation (1a). Therefore, the equivalent continuum compliances for a truss can be determined from the equivalent continuum stinesses derived previousl. The continuum strains, dened in tensor form in equation (19b), can be transformed into a new coordinate sstem described b the linear transformation tensor T ij. The resulting transformed strains " 0 ij are " 0 ij = T iot jp " op = T io T jp S opkl kl (20) The aial strain in an member of the truss is determined b dening an alternate coordinate sstem with one of its aes aligned along the longitudinal direction of the member and evaluating the normal strain along that ais. Assuming that the -ais of the alternate coordinate sstem is aligned this wa, the aial strain in the member is given as " 0 11 = T 1iT1jS ijkl kl = i j S ijkl kl (21) where, as dened before, i is the ith direction cosine of the member. Failure occurs in a member if its aial strain eceeds a critical value determined for elastic buckling. For the present stud, the truss members are assumed to be slender and therefore to buckle as Euler columns (ref. 15); thus, the critical strain for the nth group of members is dened as ("crit) n =, 2 rn l n 2 (22) 9

where r n is the radius of gration and l n is the length of the members in the nth group. The minus sign in equation (22) indicates that the critical strain is compressive. A fail-safe criterion can be constructed from equations (21) and (22) b requiring the aial strains in all members to be less than the critical value. This fail-safe criterion can be written as 2 6 4 ( i j ) n S ijkl, 2 rn l n 3 7 2 5 kl =[ kl ] n kl 1 (23) The bracketed term in equation (23) can either be thought of as a third-order tensor representing the strength of the truss or as a collection of second-order tensors, each representing the strength of a group of parallel members within the truss. The product of this strength tensor and the second-order applied stress tensor kl isavector of constants, one for each of the groups of parallel members. For elastic failure to occur, an one of these constants must be 1. Thus, the critical stress at which failure occurs is the minimum stress at which one or more of these constants is equal to 1. Equation (23) represents a purel analtical failure theor for space-lling trusses which can be used with equal ease to anale strength under multiaial or uniaial loading. Similarl, strength in alternate coordinate sstems can be readil handled b simpl transforming the collection of second-order strength tensors kl in the same wa that a stress or strain tensor would be transformed. Equation (23) can be used, as described, to determine the strength of a given truss design. Additionall, this equation is useful for tailoring the strength of a truss design because it is an eplicit relationship between the strength of individual members (i.e., r n =l n ) and the continuum strength of the truss. Strength tailoring is accomplished b varing the strength of individual members to eect a desired change in the continuum strength of the truss. Note that because the continuum compliances of the truss appear in equation (23), strength tailoring is not independent of stiness tailoring. Consequentl, tailoring the continuum stinesses of a truss also will change its continuum strength characteristics. In the remaining sections of this paper, eamples of stiness and strength tailoring of uniform trusses are presented. Truss geometries are selected for analtical simplicit, thus allowing emphasis to be placed on developing an understanding of the analsis techniques. Eamples of Stiness and Strength Tailoring in Trusses Equations (9), (17), and (23) provide the basis for analsis of the continuum stiness, densit, and strength of uniform space-lling truss structures. B providing eplicit relationships between these continuum quantities and truss design parameters, these equations are eective tools that enable ecient tailoring of the truss stiness and strength characteristics. In this section, these equations are applied to the analsis of two commonl used truss geometries and to the tailoring of designs that have continuum isotropic and orthotropic behaviors. Regular Octahedral Truss The octahedral truss (also known as the tetrahedral truss, ref. 2, or the octet truss) is a common geometr that derives its name from its members that connect to form octahedrons and tetrahedrons. For the present stud, a regular octahedral truss is considered which has all identical members. A repeating cell from this truss is shown in gure 3. The cell contains a regular octahedron at its center (g. 3(a)) and tetrahedrons connected to each of the eight faces of the octahedron (g. 3(b)). Space is lled b translational replication of this cell in each of the three coordinate directions. 10

Because all members are identical, the octahedral truss has digonal smmetr aes along the lines = ; = ; and = ; trigonal smmetr aes along the lines = = ;, = = ; =, = ; and = =,; and quadragonal smmetr aes along the -, -, and -aes. This combination of smmetr aes indicates that the regular octahedral truss is a member of rotational smmetr group j. Calculation of continuum stiness and densit. In table I, the behavior of the regular octahedral truss is characteried b the three independent elastic constants c11;c12; and c66. Equation (12) further reduces this number to two. However, these constants lack the relationship c66 =(c11, c12)=2; thus, the regular octahedral truss is not globall isotropic. Values for the elastic constants can be determined from equations (7) and (9). Si dierent groups of parallel members eist in the octahedral truss, and all members are identical and assumed to have a cross-sectional area of A. With the half-height of the regular octahedron dened to be L, as shown in gure 3, the length of each of the members is p 2L. Then, the equivalent unidirectional stiness for each of the si groups of parallel members is (C1111 0 ) n = p EA 2L 2 (24) Substituting equation (24) into equation (9) along with the appropriate direction cosines for the dierent member groups, gives the result presented in equation (25) for the equivalent continuum stiness matri of the octahedral truss: [c mn ]= EA 2 p 2L 2 2 6 4 2 1 1 0 0 0 1 2 1 0 0 0 1 1 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 Notice that the continuum stinesses obe the restrictions in table I and equation (12). Because all members in the regular octahedral truss are identical, the relative magnitudes of the continuum stinesses for the octahedral truss are constrained b the proportions given in the matri of equation (25). Therefore, changing the aial stiness of the truss members can onl uniforml change all continuum stinesses. The equivalent densit of the octahedral truss can be calculated b substituting the stinesses from equation (25) into equation (17). Rearranging and simplifing gives 3 7 5 (25) eq = 3p 2A L 2 (26) Calculation of continuum strength. Before appling equation (23) to calculate the continuum strength of the octahedral truss, the tensor form of the continuum compliances must be determined from the stiness matri given in equation (25). This process is done b inverting the stiness matri to get the compliance matri and then emploing the usual conversion from matri form to tensor form on the individual compliances (ref. 10). The onl remaining unknown truss parameter is the radius of gration of its members. 11

Suppose that the strength of the octahedral truss under a continuum uniaial compression is required. Assuming this stress to have magnitude ult and to be applied along a vector given b the spherical coordinates and ' (as shown in g. 4), the applied continuum stress tensor can be written as [ kl ]=, ult 2 6 4 (sin 2 cos 2 ') (sin 2 sin ' cos ') (sin cos cos ') (sin 2 sin ' cos ') (sin 2 sin 2 7 ') (sin cos sin ') 5 (27) (sin cos cos ') (sin cos sin ') (cos 2 ) 3 The compression strength is determined b substituting equation (27) into equation (23). After simplication, equation (23) reduces to a set of si scalar equations (n = 1 to 6), one for each group of parallel members in the truss. Each of these equations can be solved for the value of ult which is necessar to cause Euler buckling in the corresponding member. The minimum value of ult determined from these si equations is the lowest uniaial compression stress at which local buckling occurs within the truss lattice. This value is dened as the uniaial compression strength for the given set of and '. A three-dimensional plot of the uniaial compression strength of the octahedral truss is presented in gure 4 for a range of and ' from 0 to 90. Because of smmetr, the strength in all other quadrants is identical. A factor of 2 variation eists in the compression strength of the lattice, and, not surprisingl, the directions of minimum strength are coincident with the directions of the members of the truss. Maimum strength occurs for loading along the three coordinate aes and along the line = =. The value of the minimum strength is ult = EA2 r 2 2 p 2L 4 (28) Because all members are identical, changing the strength of the members would change the vertical scale of the strength plot given in gure 4, but it would not change its shape. Introducing member-specic properties will alter the equivalent continuum stiness and strength; however, this would destro the geometric smmetr of the lattice and introduce additional independent stinesses. In the following section, a truss based on the octahedral lattice is designed for isotropic stiness and nearl isotropic strength. Isotropic Warren Truss The lattice of the regular octahedral truss is modied b adding members that connect all si vertices of each octahedron to the geometric center of the octahedron, as shown in gure 5(a). The resulting arrangement of new members forms a cubic lattice within the octahedral lattice, with the edges of the cube ling parallel to the three coordinate aes and each cube containing a regular tetrahedron, as shown in gure 5(b). The members of the cubic lattice are of length L, whereas the members of the original octahedral lattice are of length p 2L. This truss geometr is often referred to as the Warren truss because its lattice arrangement is similar to that of a common two-dimensional truss of the same name. Similar to the regular octahedral truss, the Warren truss is a member of smmetr group j, and it has two independent elastic constants c11 and c12. However, unlike the octahedral truss, the Warren truss has two dierent members whose relative stinesses and strengths can be tailored to aect the continuum behavior of the truss without violating its geometric and elastic smmetr. In this section, it is demonstrated that the continuum strength and stiness properties of the lattice can be tailored b redistributing material within the truss lattice. The material is transferred from the octahedral lattice members 12

to the cubic lattice members so that the continuum stinesses become isotropic. Also, the relative strengths of the members are tailored to reduce variations in continuum compression strength. Continuum stiness tailoring. The Warren truss is composed of nine dierent groups of parallel members. Three groups correspond to the cubic lattice, and si groups correspond to the octahedral lattice. The continuum stinesses for the Warren truss can be determined b adding the contributions because of the cubic lattice members to the result presented in equation (25) for the octahedral lattice. The cross-sectional areas of the members in the cubic lattice and the octahedral lattice are dened to be A c and A o, respectivel. Thus, the equivalent uniaial stinesses of the three groups of parallel cubic lattice members are given b (C 0 1111 ) n = EA c L 2 (29) Substituting equation (29) into equation (9), along with the appropriate direction cosines, and adding the result to that presented in equation (25) gives [c mn ]= EA o 2 p 2L 2 2 6 4 2+2 p 2 c 1 1 0 0 0 1 2+2 p 2 c 1 0 0 0 1 1 2+2 p 2 c 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 where c is dened as A c =A o. If c is equal to 0, the cross-sectional area of the cubic lattice members is 0, and equation (30) is identical to equation (25). As before, an equivalent densit can be calculated using equation (17) and the stinesses presented in equation (30). The result is 3 7 5 (30) eq = (3p 2+3 c )A o L 2 (31) To stud the eects of redistribution of material within the truss, the total amount of material must remain constant. For convenience, the densit of the Warren truss is required to be the same as that of the regular octahedral truss b setting equation (26) equal to equation (31). The result is A A o = p (32) 1+ c = 2 where A is the cross-sectional area of the members in the regular octahedral truss that was analed previousl. Equation (32) denes the relation between the cross-sectional areas of the cubic and octahedral lattice members within the Warren truss; this relation must be valid to keep the equivalent densit of the Warren truss equal to that of the regular octahedral truss. Substituting equation (32) into equation (30) gives eplicit equations for the continuum stinesses of the Warren truss in terms of the member area ratio c.to better understand the eects of redistribution of material, the stiness components in equation (30) are translated into equivalent Young's modulus, Poisson's ratio, and shear modulus, as follows: Eeq = (c 11 +2c12)(c11, c12) c11 + c12 13 = 4EA(1 + 2p 2 c ) 2 p 2L 2 (3+2 p 2 c ) (33)

c12 eq = c11 + c12 = 1 3+2 p 2 c (34) Geq = c66 = p EA p 2 2L 2 (35) (1 + c = 2) These stiness components are plotted in gure 6 as functions of the area ratio c. For c = 0, no material has been redistributed from the octahedral lattice to the cubic lattice, and the stinesses represent those of the octahedral truss. As c is increased, material is moved from the octahedral lattice to the cubic lattice, and this process is accompanied b an increase in the equivalent Young's modulus and decreases in the equivalent Poisson's ratio and the equivalent shear modulus. As seen from equations (34) and (35), when c becomes large, both the Poisson's ratio and the shear modulus approach 0. This eect is consistent with the fact that the cubic lattice of members is not a kinematicall stable truss b itself. Because of this, considering designs with ver large values of c is unreasonable. For the Warren truss to be globall isotropic, the stinesses must satisf the following condition: Eeq Geq = (36) 2(1 + eq) Substituting the epressions from equations (33) to (35) into equation (36) shows that c must be equal to 1=(2 p 2) for isotrop. Substituting this value of c into equation (32) gives a value of 4A/5 for the cross-sectional area of the members in the octahedral lattice and, consequentl, avalue of p 2A=5 for the cross-sectional area of the members in the cubic lattice. Thus, if 1/ 5 of the material that was originall in the members of the octahedral truss is redistributed into the members of the cubic lattice, the resulting truss behaves isotropicall. The isotropic values for the equivalent Young's modulus, Poisson's ratio, and shear modulus are (Eeq) iso = p EA 2L 2 (eq) iso = 1 4 (Geq) iso = p 2EA 5L 2 (37) Notice that the equivalent isotropic Poisson's ratio is 1/ 4, which is the value that was predicted earlier for globall isotropic trusses. Also, calculating the ratio of the equivalent isotropic Young's modulus (eq. (37)) to the equivalent densit (eq. (26)) gives the result predicted in equation (18) for globall isotropic trusses. Continuum strength tailoring. Appling the same procedure used for the octahedral truss, the continuum strength of the isotropic Warren truss can be determined and the eects on continuum strength of varing the strength of the truss members can be evaluated. For comparison, the same continuum stress tensor given in equation (27) is also applied to the Warren truss. Two cases are analed. In the rst case, all members in the truss are assumed to have the same radius of gration, and in the second case, all members are assumed to have the same buckling load. The rst case is representative of a truss with thin-walled members of equal cross-sectional diameter. The second case illustrates the eects of tailoring individual member buckling strengths on the continuum strength of the truss. For the rst case, the radius of gration of all members is r, and the lengths of the members are L for the cubic lattice and p 2L for the octahedral lattice. These values, the continuum compliances determined from equation (30), and the appropriate direction cosines are substituted into equation (23). The result is a set of nine scalar equations, one for each group of parallel 14

members in the truss, from which the minimum value of ult is determined for the given set of and '. A three-dimensional plot of the uniaial compression strength of the isotropic Warren truss is presented in gure 7 for the same range of and ' as in gure 4. The shape of the strength plot is similar to that of the octahedral truss, and, despite the redistribution of material from the octahedral lattice, the values and the directions of the minimum and maimum strength are the same as those for the octahedral truss. The directions and maimum strength are coincident with the directions of the cubic lattice members, and the directions of minimum strength are coincident with the directions of the octahedral lattice members. Requiring that all members have the same radius of gration causes the cubic lattice members to have twice the buckling load of the octahedral lattice members because of the dierence in their lengths. This eect causes a factor of 2 variation in the continuum strength. Variation in truss strength might not be a concern for man design applications; however, if it is desirable to have a truss that behaves isotropicall in stiness, it is probabl also desirable for the truss to behave isotropicall in strength. B tailoring the buckling loads of the cubic lattice members to be the same as those of the octahedral lattice, the variations in continuum strength can be signicantl reduced. For this case, the radius of gration of the cubic lattice members is reduced to r=p 2 so that the buckling loads of all members are the same. A plot of the resulting continuum compression strength is presented in gure 8. Although some variation still eists in the continuum strength, the magnitude of the variation has been signicantl reduced. The use of three-dimensional strength plots is particularl helpful for developing strength tailoring rules because these plots provide visualiation of the correlation between member orientations and continuum strength variations. Without this correlation, developing strength tailoring rationale for the members would be dicult. The eample presented is fairl simple because of the isotropic stiness behavior and geometric smmetr of the Warren truss. Therefore, the correlation between variations in continuum strength and the orientation of members is fairl obvious. However, for trusses with less geometric smmetr or more comple applied stress tensors, this correlation might not be apparent without the use of a threedimensional strength plot. Orthotropic Warren Truss Man applications eist for large truss structures with orthotropic, rather than isotropic, continuum properties. For orthotropic applications, the requirements on continuum stiness and strength are much higher in one direction than in others. For eample, man applications involve beam-like trusses that primaril carr bending and torsional loads. In these cases, the longitudinal (along the length of the beam) stiness and strength requirements are much higher than the transverse stiness and strength requirements. Therefore, using a truss with orthotropic continuum properties is probabl more ecient than using one with isotropic properties. Table I shows that trusses of smmetr groups i and j are not candidates for orthotropic design because their stinesses (and strengths) must be the same in all three coordinate directions. Trusses of all other smmetr groups are candidates for orthotropic tailoring because their properties in the direction can dier from those in either the or the direction. The truss presented in gure 9 is a variation of the Warren truss design that is a member of smmetr group f and is, thus, a possible candidate for orthotropic design. The lattice arrangement of this truss is identical to that of the Warren truss ecept the length of the repeating cell in the direction diers from that in either the or the directions b the proportion. This section will show the results of appling stiness and strength tailoring techniques to generate orthotropic designs that have high stinesses and strengths in the direction but which have the same equivalent densit as that of the isotropic Warren truss. 15

Calculation of continuum stinesses. The orthotropic Warren truss shown in gure 9 has four dierent members. The cross-sectional areas for members of groups 1 and 2 are dened as 1A and 2A, respectivel, where 1 and 2 are variable area ratios and A is the cross-sectional area assumed earlier for the members in the octahedral truss. The equivalent uniaial stinesses for groups of these members are determined using equation (7), and the results are given in equations (38) and (39): (C1111 0 ) 1 = 1EA L 2 (38) (C 0 1111 ) 2 = 2EA(1 + 2 ) 1=2 2L 2 (39) For simplicit, members of groups 3 and 4 are assumed to be the same as those in the isotropic Warren truss. Therefore, the cross-sectional area of members of group 3 is p 2A=5, and the cross-sectional area of members of group 4 is 4A=5. The equivalent uniaial stinesses are the same for member groups 1 and 2, and the value of this stiness is given in equation (40): (C 0 1111 ) 3 =(C0 1111 ) 4 = p 2EA 5L 2 (40) Substituting these uniaial stinesses and the appropriate transformation tensors into equation (9) and simplifing gives the following values for the nonero continuum stinesses: " p c11 = c22 = EA 2 2 L 2 + 5 # 2 (1 + 2 ) 3=2 (41) c12 = c66 = p 2EA c13 = c23 = c44 = c55 = EA 5L 2 (42) 2 2 L 2 (1 + 2 ) 3=2 c33 = EA L 2 1 + 24 2 (1 + 2 ) 3=2 Note that these stinesses obe the conditions presented in table I and equations (12) and (13) for trusses of smmetr group f. Equations (41) through (44) are eplicit functions of the three remaining design parameters ; 1; and 2. Therefore, these equations can be used directl to determine how variations in the design parameters aect the orthotropic characteristics of the truss. An equivalent densit can be calculated for the orthotropic Warren truss b substituting the stinesses from equations (41) through (44) into equation (17). The result is eq = A L 2 " 6 p 2 5 + 1 + 2(1 + 2 ) 1=2 2 # (43) (44) (45) 16

Setting equation (45) equal to equation (26) ensures that the equivalent densit of the orthotropic Warren truss is the same as that of the regular octahedral truss and the isotropic Warren truss. The resulting epression can be rearranged to give the following condition on the area ratio 2: 2 = (3p 2, 1), 6 p 2=5 2(1 + 2 ) 1=2 (46) Equation (46) reduces the set of independent design parameters to the repeating cell length ratio and the cross-sectional area ratio 1. An equivalent -direction Young's modulus can be determined for the orthotropic Warren truss b inverting the s33 component of the compliance matri as follows: Performing this calculation gives the result (Eeq) = (Eeq) = 1 s33 p p 2EA 15 1= 2+183,5(1=p 2, 6=5) 2 L 2 (15, 51=p 2+12+6 3 ) (47) (48) To determine the improvement in stiness in the direction, the modulus given in equation (48) is divided b the Young's modulus of the isotropic Warren truss given in equation (37). The resulting normalied -direction Young's modulus is (Eeq) (Eeq) iso = 30 1= p 2+363,10(1=p 2, 6=5) 2 15, 51=p 2+12+6 3 (49) A three-dimensional plot of the normalied -direction Young's modulus is presented in gure 10 for ranges of and 1. The isotropic Warren truss is characteried b 1 = p 2=5 and = 1; this point on the plot corresponds to a normalied modulus equal to 1. As 1 increases, for a ed value of, the material transfers from members of group 2 to members of group 1 (see g. 9). This material transfer causes an increase in the modulus because the group 1 members are oriented parallel to the direction. As increases, for a ed value of 1, the number of group 3 and group 4 members in a given volume decreases. To maintain constant densit, material is redistributed among group 1 and group 2 members, thus also causing an increase in the modulus. Calculation of continuum -direction strength. The strength of the orthotropic Warren truss is calculated for a uniform continuum compression applied in the direction. This applied stress tensor is given in equation (50) and is substituted into equation (23): [ kl ]= 2 6 4 0 0 0 0 0 0 0 0,( ult) 3 7 5 (50) Because their alignment is parallel to the direction, members in group 1 buckle at lower continuum stresses than the remaining members in the truss. (This result was veried through 17

additional analsis not presented herein.) Thus, considering onl buckling in group 1 members, equation (23) can be reduced to equation (51), where r1 and l1 are the radius of gration and length of members in group 1: ( ult) = 2 r 2 1 l 2 1 s 33 Dening the radius of gration of these members to be r and their length to be L (see g. 9) and substituting the result from equation (47) gives the following epression for the -direction compression strength of the orthotropic Warren truss: ( ult) = 2 r 2 (51) 2 L 2 (E eq) (52) The -direction compression strength of the isotropic Warren truss can be determined from gure 7 ( =0 ), and this value can be used to normalie equation (52). The result is ( ult) ( ult) iso = (E eq) 2 (Eeq) iso (53) Unlike the modulus, the factor of 2 in the denominator of equation (53) causes the -direction strength to decrease with increasing. However, it is apparent that both modulus and strength have the same variation with 1. A three-dimensional plot of the normalied -direction compression strength is presented in gure 11 for comparison with the modulus plot in gure 10. Because both modulus and strength increase as 1 increases, selecting the largest practical value for 1 is best. As an eample, if the cross-sectional areas of all members within the truss are constrained so p that the dier b no more than a factor of 5, the maimum allowable value for 1 would be 2. Assuming this value for 1 gives the following for all the member cross-sectional areas: A1 = p 2A A2 = (10, 6)A 5(2 + 2 2 ) 1=2 A3 = p 2A=5 A4 =4A=5 (54) A plot of the normalied -direction strength and modulus is presented in gure 12, assuming 1 is equal to p 2. As eplained, etending the length of the Warren truss cell in the direction (increasing ) increases the stiness while decreasing the strength of the truss. Therefore, the optimum length for the truss cell depends on the relative importance of continuum strength and continuum stiness in the design. Concluding Remarks A deterministic procedure has been presented for tailoring the continuum stiness and strength of uniform space-lling truss structures through the appropriate selection of truss geometr and member sies (i.e., eural and aial stinesses and length). A ke aspect of this procedure is smbolic manipulation of the equivalent continuum constitutive equations to produce eplicit relationships between truss member sies and continuum strength and stiness. To help select an appropriate truss geometr for a given application, a nite set of possible geometric smmetr groups which characterie uniform trusses has been presented, and the implied elastic smmetr associated with each geometric smmetr group has been identied. Equivalent continuum stiness has been determined using an eisting technique assuming that the displacement eld within a truss is single-valued and the members within a truss carr onl aial load. Based on these assumptions, generall anisotropic trusses are shown 18