BUCKLING OF LONG COMPRESSION-LOADED ANISOTROPIC PLATES RESTRAINED AGAINST INPLANE LATERAL AND SHEAR DEFORMATIONS

Transcription

1 AIAA BUCKLING OF LONG COMPRESSION-LOADED ANISOTROPIC PLATES RESTRAINED AGAINST INPLANE LATERAL AND SHEAR DEFORMATIONS Mihael P. Nemeth * Mehanis and Durability Branh, NASA Langley Researh Center Hampton, Virginia Abstrat An approah for synthesizing bukling results and behavior for thin balaned and unbalaned symmetri laminates that are subjeted to uniform axial ompression loads and elastially restrained against inplane expansion, ontration, and shear deformation is presented. This approah uses a nondimensional analysis for infinitely long, flexurally anisotropi plates (oupling between bending and twisting) that are subjeted to ombined mehanial loads and is based on nondimensional parameters. In addition, nondimensional loading parameters are derived that aount for the effets of the elasti inplane deformation restraints, membrane orthotropy, and membrane anisotropy on the indued prebukling stress state. The loading parameters are used to determine bukling oeffiients that inlude the effets of flexural orthotropy and flexural anisotropy. Many results are presented, for some seleted laminates, that are intended to failitate a strutural designer s transition to the use of the generi bukling design urves that are presented and disussed in the paper. Several bukling response urves are presented that provide physial insight into the behavior for ombined loads, in addition to providing useful design data. An example is presented that demonstrates the use of the generi design urves, whih are appliable to a wide range of laminate onstrutions. The analysis approah and generi results indiate the effets and harateristis of laminate orthotropy and anisotropy in a very general and unifying manner. * Assistant Head. Assoiate Fellow, AIAA. Copyright 3 by the Amerian Institute of Aeronautis and Astronautis, In. No opyright is asserted in the United States under Title 7, U. S. Code. The U. S. Government has a royaltyfree liense to exerise all rights under the opyright laimed herein for Governmental purposes. All other rights are reserved by the opyright owner. Introdution Strutural tailoring of laminated-omposite plates to enhane their bukling resistane is an important element in the development of new, advaned aerospae vehiles. One strutural omponent that is often examined in design of airraft and spaeraft is the long retangular plate. Plates of this type ommonly appear as elements of stiffened panels that are used for wing strutures, and as semimonooque shell segments that are used for fuselage and launh vehile strutures. Thus, establishing a broad understanding of the bukling behavior of long plates is essential to the advanement of strutural-tailoring tehnology for aerospae vehiles. An important type of long plate that appears as an element of advaned omposite strutures is the symmetrially laminated plate. In the present study, the term, "symmetrially laminated," refers to plates in whih every lamina above the plate midplane has a orresponding lamina loated at the same distane below the plate midplane, with the same thikness, material properties, and fiber orientation. Symmetrially laminated plates are, for the most part, flat after the manufaturing proess and exhibit flat prebukling deformation states, whih is desirable for many appliations. More importantly, the amenability of these plates to strutural tailoring provides symmetrially laminated plates with a signifiant potential for reduing the weight of aerospae vehiles or for meeting speial performane requirements. Thus, understanding the bukling behavior of symmetrially laminated plates is an important part of the searh for ways to exploit plate orthotropy and anisotropy to redue strutural weight or to fulfill a speial design requirement. In many pratial ases, symmetrially laminated plates exhibit speially orthotropi behavior. However, in many ases, these plates exhibit anisotropy in the form of material-indued oupling between pure bending and twisting deformations. This oupling is referred to herein as flexural anisotropy and it generally yields bukling modes that are skewed in appearane (see Fig. ). Symmetrially laminated plates that are unbalaned are also being investigated for speial-purpose uses in aerospae strutures. In the present study, the term, "unbalaned laminate," refers to symmetri laminates in whih eah ply with a positive-valued fiber orientation is not balaned by a orresponding ply with a negative-valued fiber orientation. Unbalaned laminated plates exhibit anisotropy in the form of material-indued oupling between pure inplane dilatation and inplane shear deformations, in addition to flexural anisotropy. This oupling is referred to herein as membrane anisotropy and it generally yields ombined inplane stress states for simple loadings like uniform edge ompression when inplane displaement onstraints are imposed on one or more edges of a plate. For example, when the two unloaded, opposite edges of an unbalaned, symmetrially laminated plate that is ompression loaded, suh as a [+45 // Amerian Institute of Aeronautis and Astronautis

2 9 laminate, are totally restrained against expansion and ontration and inplane shearing deformations, inplane shear stresses are developed in addition to the biaxial ompression stresses that are typially present in balaned laminates (see Fig. ). These kinematially indued shear stresses an be relatively large in magnitude, ompared to the diret ompressive stresses, and as a result an affet greatly the bukling behavior of the plate and yield bukling modes that are skewed in appearane. The effets of flexural orthotropy and flexural anisotropy on the bukling behavior of long retangular plates that are subjeted to single and ombined loading onditions are beoming better understood. For example, in-depth parametri studies that show the effets of flexural orthotropy and flexural anisotropy on the bukling behavior of long plates that are subjeted to ompression, shear, pure inplane bending, and various ombinations of these loads have been presented in Refs. through 3. The results presented in these referenes detail the ways in whih the importane of flexural anisotropy, on the bukling resistane of long plates, varies with the magnitude and type of the ombined loading ondition. Similar results for plates loaded by uniform shear and a general linear distribution of axial load aross the plate width have also been presented in Ref. 4. Several studies have also addressed the behavior of retangular plates that are restrained against inplane movement- an important researh area beause inplane movement is typially restrited in aerospae strutures by adjaent panels and stiffeners. In partiular, Harris 5 examined the effets of lateral inplane restraint on the behavior of ompression-loaded, speially orthotropi plates, and Obraztsov and Vasil ev 6 examined the same effets on ompression-loaded, balaned angle-ply laminates. Sherbourne and Pandey 7 examined the behavior of balaned and unbalaned, symmetri laminates subjeted to uniaxial ompression loads and with either lateral inplane movement restrained or both lateral movement and inplane shear deformation restrained. Their study highlights the effets of fiber orientation and plate aspet ratio for seleted laminate staking sequenes made of a typial graphite-epoxy material system, and explored the possibility of tailoring laminates to have a negative Poisson s ratio, that results in improved bukling resistane (Insight into how laminate staking sequene affets Poisson s ratio is found in Refs. 8-.). Similar studies that fous on bukling-load optimization of ompression-loaded retangular laminates are presented in Refs. and. These two studies examine the effets of unloaded edges that are either rigidly or elastially restrained agained inplane, lateral movement, and inlude the effets of transverse-shear flexibility. Results are presented for balaned, symmetri laminates made of a typial graphite-epoxy material, for several plate aspet 3, 4 ratios and five different boundary onditions. Bedair and Walker, et. al. 5, have also presented studies that inlude the effets of restrained inplane movement, along with the presene of nonuniform applied ompression loads. Speifially, the results in Refs. 3 and 4 fous on the behavior of finite-length isotropi plates with elasti inplane restraints. The results presented in Ref. 5 fous on bukling-load optimization of retangular, graphite-epoxy laminates that are balaned and symmetri. Studies that address the effets of membrane orthotropy and membrane anisotropy on the bukling behavior of long retangular plates that are restrained against axial thermal expansion or ontration and subjeted to uniform heating or ooling and mehanial loads have been presented in Refs. 6 and 7. Likewise, similar results for plates that are either fully restrained or elastially restrained against thermal expansion and ontration and subjeted to uniform heating or ooling have been presented in Refs. 8 and 9, respetively. These studies are omprehensive and have provided a better understanding of the load interation effets of balaned and unbalaned, symmetrially laminated plates that are subjeted to mehanial loads and restrained against thermal expansion or ontration. As evidened by the previous studies disussed, the effets of membrane orthotropy and anisotropy, flexural orthotropy and anisotropy, and the restraint of inplane movement on the bukling behavior of retangular plates that are subjeted to mehanial and thermal loads are beoming better understood. However, omprehensive review of these studies indiates that there remains a need for in-depth studies that address, in a broad way, the effet of a ompliant, elasti restraining medium on the bukling of ompression-loaded, symmetrially laminated plates that are restrained against lateral expansion and ontration and inplane shearing deformations. This researh area is important beause it represents a lass of problems that must be well understood in order to determine potential benefits and pitfalls of strutural tailoring. Thus, the objetive of the present study is to present an analytial approah that indiates the effets of a ompliant, elasti restraining medium on the bukling behavior of ompression-loaded, balaned and unbalaned, symmetrially laminated plates in a very general manner. Towards that objetive, a bukling analysis is presented first that follows the analysis presented in Ref. 9. To ahieve this objetive, the bukling analysis is formulated in terms of nondimensional bukling oeffiients and load fators that depend on the inplane ompliane oeffiients for a given plate and the relative ompliane of a restraining medium. Results are then presented for infi- Amerian Institute of Aeronautis and Astronautis

3 nitely long plates with the two long, unloaded edges lamped or simply supported and elastially restrained against inplane movement. These results inlude nondimenional bukling loads for seleted laminates that are made from one of nine different material systems, and generi results that are appliable to a vast range of laminate onstrutions, inluding hybrid laminates. Bukling results for infinitely long plates are important beause they often provide a pratial estimate of the behavior of finite-length retangular plates (lower bounds to festoon bukling urves), and they provide information that is useful in explaining the behavior of these finite-length plates. Moreover, knowledge of the behavior of infinitely long plates an provide insight into the bukling behavior of more omplex strutures suh as stiffened panels. Finally, an example is presented that illustrates the use of the generi bukling-design urves presented herein and in Ref. 8, and highlights the effets of a restraining medium on the bukling behavior. Analysis Desription In preparing generi design harts for bukling of a single flat thin plate, a speial-purpose analysis is often preferred over a general-purpose analysis ode, suh as a finite-element ode, beause of the ost and effort that is usually involved in generating a large number of results with a general-purpose ode. The results presented in the present paper were obtained by using suh a speial-purpose bukling analysis that is based on lassial laminated-plate theory. The analysis details are lengthy; hene, only a brief desription of the bukling analysis is presented herein. First, the bukling analysis for long plates that are subjeted to a general set of mehanial loads is desribed. Then, the mehanial loads that are indued in ompression-loaded plates by elastially restraining the inplane lateral and shear deformations are derived. Bukling Analysis Symmetrially laminated plates an have many different onstrutions beause of the wide variety of material systems, fiber orientations, and staking sequenes that an be seleted to onstrut a laminate. A way of oping with the large number of hoies for laminate onstrutions is to use onvenient nondimensional parameters in order to understand overall behavioral trends and sensitivities of the strutural behavior to perturbations in laminate onstrution. The bukling analysis used in the present paper is based on lassial laminatedplate theory and the lassial Rayleigh-Ritz method, and is derived expliitly in terms of the nondimensional parameters defined in Refs. -4 and 6-. This approah was motivated by the need for generi (independent of a speifi laminate onstrution) parametri results for omposite-plate bukling behavior that are expressed in terms of the minimum number of independent parameters needed to fully haraterize the behavior, and that indiate the overall trends and sensitivity of the results to hanges in the parameters. The nondimensional parameters that were used to formulate the bukling analysis are given by δ = (4) (D D 3 ) /4 where b is the plate width and λ is the half-wave length of the bukle pattern of an infinitely long plate (see Fig. ). The subsripted D-terms are the bending stiffnesses of lassial laminated-plate theory. The parameters α and β haraterize the flexural orthotropy, and the parameters γ and δ haraterize the flexural anisotropy. The mehanial loading onditions that are inluded in the bukling analysis are uniform transverse tension or ompression, uniform shear, and a general linear distribution of axial load aross the plate width, as depited in Fig.. Typially, an axial stress resultant distribution is partitioned into a uniform part and a pure bending part. However, this representation is not unique. The longitudinal stress resultant N x is partitioned in the analysis into a uniform tension or ompression part and a linearly varying part that orresponds to eentri inplane bending loads. This partitioning is given by N x =N x N b[ε +(ε ε )η] (5) where N x denotes the intensity of the onstant-valued tension or ompression part of the load, and the term ontaining N b defines the intensity of the eentri inplane bending load distribution. The symbols ε and ε define the distribution of the inplane bending load, and the symbol η is the nondimensional oordinate given by η = y/b. This partiular way of partitioning the longitudinal stress resultant was used for onveniene by eliminating the need to alulate the uniform and pure bending parts of an axial stress resultant distribution prior to performing a bukling analysis. The analysis is based on a general formulation that inludes ombined destabilizing loads that are proportional to a positive-valued loading parameter p that is inreased until bukling ours, and independent subritα = b λ D /4 β = D +D 66 (D ) / γ = D 6 (D 3 ) /4 D 6 () () (3) 3 Amerian Institute of Aeronautis and Astronautis

4 ial ombined loads that remain fixed at a speified load level below the value of the bukling load. Herein, the term "subritial load" is defined as any load that does not ause bukling to our. In pratie, the subritial loads are applied to a plate prior to, and independent of, the destabilizing loads with an intensity below that whih will ause the plate to bukle. Then, with the subritial loads fixed, the ative, destabilizing loads are applied by inreasing the magnitude of the loading parameter until bukling ours. This approah permits ertain types of ombined-load interation to be investigated in a diret and onvenient manner. For example, in analyzing the stability of an airraft fuselage, the nondestabilizing transverse tension load in a fuselage panel that is aused by abin pressurization an be onsidered to remain onstant and, as a result, it an be represented as a passive, subritial load. The ombined shear, ompression, and inplane bending loads that are aused by flight maneuvers an vary and ause bukling and, as a result, they an be represented as ative, destabilizing loads. The distintion between the ative, destabilizing and passive subritial loading systems is implemented in the bukling analysis by partitioning the prebukling stress resultants as follows N x = +N x N y = +N y N xy = +N xy N b =N b +N b (6) (7) (8) (9) where the stress resultants with the subsript are the destabilizing loads, and those with the subsript are the subritial loads. The sign onvention used herein for positive values of these stress resultants is shown in Fig.. In partiular, positive values of the general linear edge stress distribution parameters N b, N b, ε, and ε orrespond to ompression loads. Negative values of N b and N b, or negative values of either ε or ε, yield linearly varying stress distributions that inlude tension. Depitions of a variety of inplane bending load distributions are given in Ref. 4. The two normal-stress resultants of the system of destabilizing loads, and, are defined to be positive-valued for ompression loads. This onvention results in positive eigenvalues being used to indiate instability due to uniform ompression loads. The bukling analysis inludes several nondimensional stress resultants assoiated with Eqs. (6) through (9). These dimensionless stress resultants are given by n N xj b xj = π (D ) / () n yj = N yj b () π N xyj b n xyj = () π (D D 3 ) /4 N n bj = bj b (3) π (D ) / where the subsript j takes on the values of and. In addition, the destabilizing loads are expressed in terms of the loading parameter p in the analysis by (4) (5) (6) (7) where L through L 4 are load fators that determine the speifi form (relative ontributions of the load omponents) of a given system of destabilizing loads. Typially, the dominant load fator is assigned a value of and all others are given as positive or negative frations. Nondimensional bukling oeffiients that are used herein are given by the values of the dimensionless stress resultants of the system of destabilizing loads at the onset of bukling; i.e., K x n x n x =L p n y =L p n xy =L 3 p n b =L 4 p r = r b π (D ) / =L p r K y n y r = r b =L p r K s n xy r = r b π (D D 3 ) =L /4 3 p r N b K b n b r = r b π (D ) =L / 4 p r (8) (9) () () where the quantities enlosed in the parentheses with the subsript r are ritial values that orrespond to bukling and p r is the magnitude of the loading parameter at bukling. Positive values of the oeffiients K x and K y orrespond to uniform ompression loads, and the oeffiient K s orresponds to uniform positive shear. The diretion of a positive shear-stress resultant that ats on a plate is shown in Fig.. The oeffiient K b orresponds to the speifi inplane bending load distribution defined by the seleted values of the parameters ε and ε (see Fig. ). 4 Amerian Institute of Aeronautis and Astronautis

5 The mathematial expression used in the variational analysis to represent the general off-enter and skewed bukle pattern is given by w N (ξ,η)= N Σ m= (A m sinπξ +B m osπξ)φ m (η) () where ξ =x/λ and η =y/b are nondimensional oordinates, w N is the out-of-plane displaement field, and A m and B m are the unknown displaement amplitudes. In aordane with the Rayleigh-Ritz method, the basis funtions Φ m (η) are required to satisfy the kinemati boundary onditions on the plate edges at η = and. For the simply supported plates, the basis funtions used in the analysis are given by Φ m (η) = sin mπη for values of m =,, 3,..., N. Similarly, for the lamped plates, the basis funtions are given by Φ m (η) = os(m )πη os(m+)πη (3) (4) For both boundary onditions, the two long edges of a plate are free to move in-plane, unless noted otherwise. Algebrai equations that govern the bukling behavior of infinitely long plates are obtained by substituting the series expansion for the bukling mode given by Eq. () into a nondimensionalized form of the seond variation of the total potential energy and then omputing the integrals appearing in the nondimensional seond variation in losed form. The resulting equations onstitute a generalized eigenvalue problem that depends on the aspet ratio of the bukle pattern λ/b (see Fig. ) and the nondimensional parameters and nondimensional stress resultants defined herein. The smallest eigenvalue of the problem orresponds to bukling and is found by speifying a value of λ/b and solving the orresponding generalized eigenvalue problem for its smallest eigenvalue. This proess is repeated for suessive values of λ/b until the overall smallest eigenvalue is found. Results that were obtained from the analysis desribed herein for uniform ompression, uniform shear, pure inplane bending (given by ε = - and ε = ), and various ombinations of these mehanial loads have been ompared with other results for isotropi, orthotropi, and anisotropi plates that were obtained by using other analysis methods. These omparisons are disussed in Refs. -3, and in every ase the results desribed herein were found to be in good agreement with those obtained from other analyses. Likewise, results were obtained for isotropi and speially orthotropi plates that are subjeted to a general linear distribution of axial load aross the plate width and ompared with results that were obtained by seven different authors (see Ref. 4). In every ase, the agreement was good. Prebukling Stress Resultants In general, ompression-loaded plates that are symmetrially laminated, but unbalaned, beome subjeted to a ombined inplane stress state when the lateral, inplane expansion and ontration and inplane shearing deformations are restrained at the plate edges (see Fig. ). As the magnitude of the ompression load inreases, the indued loads inrease proportionally, whih an ause premature bukling, ompared to the bukling resistane of the orresponding unrestrained plate. These indued mehanial loads are determined in the present study by using the inverted membrane onstitutive equations that are based on lassial laminated-plate theory; that is, ε x ε y γ xy = a a a 6 a a 6 a 6 a 6 N x N y N xy (5a) where ε x, ε y, and γ xy are the prebukling, inplane strains and the subsripted a-terms are the plate membrane ompliane oeffiients. An alternate form of this equation (see Ref., p. 79) that is also used in the present study, that utilizes the overall laminate properties, is given by ε x ε y γ xy = E x ν yx E y ν xy E x Ey η xy,x E x η x,xy G xy η y,xy G xy η xy,y E y G xy N x /h N y /h N xy /h (5b) where h is the laminate thikness, E x and E y are the laminate moduli, G xy is the laminate shear moduli, ν xy and ν yx are the major and minor Poisson s ratios, respetively, η x,xy and η y,xy are the oeffiients of mutual influene of the first kind, and η xy,x and η xy,y are the oeffiients of mutual influene of the seond kind. Relationships between the various onstitutive terms are obtained by noting that the oeffiient matrix of Eq. (5b) is symmetri. Following Ref., the oeffiients of mutual influene are referred to herein as shearextension oupling oeffiients. The effets of the elasti boundary restraints depited in Fig. a are obtained by noting that the indued stress resultants are proportional to the strains aused by 5 Amerian Institute of Aeronautis and Astronautis

6 expansions or ontrations and shearing deformations of the plate and the resistane provided by the restraining medium. Typially, the elasti resistane of the restraining medium is simulated with linear springs and expressed in terms of the orresponding spring stiffnesses. In the present study, the elasti resistane of a homogeneous restraining medium is desribed approximately in terms of two overall ompliane oeffiients of the restraining medium, denoted by C and C 3. In partiular, the ation of the elasti restraining medium is represented in a simple manner by and ε y = - C N y (6) γ xy = - C 3 N xy (7) for a uniform, positive-valued set of strains in a plate. The negative sign in Eq. (6) indiates that a positive, expansional strain in the y-diretion is reated by a ompressive stress in the y-diretion, that orresponds to a negative value for N y. In other words, for a plate that is free to deform under axial loading, a restraining medium would require a ompressive restoring fore to suppress the deformation. Similarly, the negative sign in Eq. (7) indiates that a positive shearing strain indues, or is reated by, a negative shearing-stress resultant. In the present study, the plates are presumed to be supported and loaded suh that nonuniformities in the prebukling stress field are negligible. Expressions for the indued stress resultants are obtained by substituting Eqs. (6) and (7) into Eq. (5a) and then solving two of the resulting equations for N y and N xy in terms of the applied stress resultant N x. This step gives Beause of these definitions, R and R 3 are referred to herein as ompliane ratios. By using Eqs. (3) and (3), Eqs. (8) and (9) beome and (3) (33) Next, beause all the subritial loads used in the bukling analysis are zero-valued for this problem, N x = -, N y = -, and N xy = (see Figs. and ). This substitution yields and a N y =N 6 a 6 a +R 3 x +R +R 3 a 6 a N xy =N a 6 a 6 +R x +R +R 3 a 6 a =N 6 a 6 a +R 3 x +R +R 3 a 6 a = N a 6 a 6 +R x +R +R 3 a 6 (34) (35) Equations similar to equations (34) and (35), that express the indued stress resultants in terms of the overall laminate material properties, are obtained by substituting Eqs. (6), (7), (3), and (3) into Eq. (5b) and then solving two of the resulting equations for N y and N xy in terms of the applied stress resultant N x. This proedure gives = +R 3 ν yx + η xy,y η x,xy +R +R 3 η xy,y η y,xy (36) and a N y =N 6 a 6 a +C 3 x +C +C 3 a 6 (8) and = +R η x,xy + ν yx η y,xy +R +R 3 η xy,y η y,xy (37) (9) To simplify these two equations further and to provide a simple way to estimate the influene of a restraining medium, the ompliane oeffiients of the restraining medium are defined as relative proportions of the plate ompliane oeffiients; that is, and a N xy =N a 6 a 6 +C x +C +C 3 a 6 C = R (3) C 3 (3) Although the fous of the present study in on ompression loaded plates ( >), Eqs. (34) through (37) learly indiate that bukling an our under axial tension loads for some laminate onstrutions. Equations (34) through (37) ontain three speial ases of interest. First, when a plate is rigidly restrained suh that ε y = γ xy =, the ompliane ratios R and R 3 are zero valued. For this ase, Eqs. (34) through (37) give = A A = ν yx + η xy,y η x,xy η xy,y η y,xy (38) 6 Amerian Institute of Aeronautis and Astronautis

7 and = A 6 A = η x,xy + ν yx η y,xy η xy,y η y,xy (39) where A, A, and A 6 are membrane stiffnesses of lassial laminated-plate theory. Seond, when a plate is elastially restrained against lateral expansion or ontration and unrestrained against inplane shearing deformations suh that =, the ompliane ratio R 3. This value for R 3 means that the restraining medium is muh more ompliant in shear than the plate. For this ase, Eqs. (34) through (37) give = and L = L 3 = D / = a 6 a 6 a +R 3 +R +R 3 a 6 D /4 = a a 6 a 6 +R +R +R 3 a 6 D D / /4 (44) (45) where = ν yx +R ν yx = a a = A A A A A A 66 A 6 (4) (4) For balaned laminates, a 6 = a 6 =, Eq. (45) gives L 3 =, and Eq. (44) beomes L = D / = ν yx +R D / (46) Equation (4) agrees with the result given in Ref. 7 for the even simpler ase when R =. Third, when a plate is elastially restrained against inplane shearing deformations and unrestrained against lateral expansion or ontration suh that =, the ompliane ratio R. This value for R means that the restraining medium is muh more ompliant in lateral expansion than the plate. For this ase, Eqs. (34) through (37) give = and where = η x,xy +R 3 η x,xy = a 6 a = A A A A A A A (4) (43) Equations (34) through (37) define a ombined loading state that is indued by elastially restraining the inplane lateral and shearing deformations of a plate. The bukling problem is posed by determining the load fators L and L 3 that appear in Eqs. (5) and (6). For an applied ompression load, the load fator L = by definition. The values for the other two load fators that are needed to ompletely define the prebukling stress state in the nondimensional bukling analysis are obtained by dividing Eqs. (5) and (6) by Eq. (4), with L =, and by using Eqs. () - (). This step yields where ν yx = A A. For an isotropi plate, Eq. (46) redues to L = = ν +R (47) where ν is Poisson s ratio of a homogeneous, isotropi material. With L = and L and L 3 defined by Eqs. (44) and (45), the ritial value of the mehanial loading parameter p r an be alulated by using the nondimensional bukling analysis. Note that p r = p r β, γ, δ, L,L 3 for a given set of bending boundary onditions (e.g., simply supported and lamped edges). Finally, it is important to mention that the approah used herein to define the prebukling stress state also applies for a more sophistiated plate theory, like a first-order transverse-shear deformation theory, beause the inplane stiffness and ompliane oeffiients are idential to those of lassial laminated-plate theory. For this theory, p r would depend also upon additional nondimensional parameters that haraterize the transverseshear flexibility. Thus, the only differene in the results for the two plate bending theories is the atual value of p r that is used in Eqs. (8)-(), for a given problem. It is also important to point out that p r for a long plate does not depend on the bukle aspet ratio parameter α. This fat has been shown and disussed in Refs Amerian Institute of Aeronautis and Astronautis

8 Results for Seleted Laminates Results are presented in this setion that illustrate the behavioral trends for several seleted symmetrially laminated plates that are loaded by uniform axial ompression. Nine different material systems are onsidered that inlude boron-aluminum, S-glass-epoxy, a typial boron-epoxy, AS4/35-6 graphite-epoxy, AS4/35 graphite-epoxy, IM7/56 graphite-bismaleimide, Kevlar 49-epoxy, IM7/PETI-5, and P-/35 pith-epoxy materials (see Table ). The plates are either rigidly restrained, elastially restrained against only lateral, inplane movement, elastially restrained against only inplane shear deformation, or are elastially restrained against both lateral inplane movement and inplane shearing deformations. First, results are presented in Table and Figs. 3- for several balaned, symmetri laminates that exhibit relatively small degrees of flexural anisotropy; that is, [(±45/ axially stiff laminates, [(±45/ 9 laterally stiff laminates, [(±45//9 quasi-isotropi laminates, and [(±θ angle-ply laminates. Then, results are presented for /, /9, and //9 unbalaned laminates in Tables 3-4 and Figs. - that exhibit signifiant degrees of membrane and flexural anisotropies. In addition, results are presented in Figs. -6 for [+θ 3 /5 unbalaned laminates that exhibit a wide range of values for the overall laminate Poisson s ratio, inluding negative values. All the results are based on lassial laminated-plate theory and the nominal ply thikness used in the alulations was.5 in. Results for Balaned Laminates For balaned laminates, there are no inplane shearing deformations beause A 6 = A 6 = a 6 = a 6 =, and as a result, the ompliane ratio R 3 is immaterial and the indued shearing stress resultant is zero valued. With the use of Eq. (4), Eq. (36) simplifies to N +R y = ν N yx = A x A (48) where R <. Thus, graphs or tables of ν yx for balaned, symmetri laminates yield the indued lateral stress resultant for an infinite number of different ompliane ratios. Results are presented in Table that show the load ratio +R, or equivalently, ν yx for [(±45/, [(±45/9, and [(±45//9 laminates made from one of the nine different material systems given in Table. The results are independent of the number of laminate plies and indiate that the laterally stiff [(±45/9 laminates exhibit the largest values of ν yx for a given material system; the smallest values are exhibited by the axially stiff [(±45/ laminates. Moreover, for the [(±45/9 and [(±45/ laminates, the largest value of ν yx is obtained for the P-/35 pith-epoxy and the boron-aluminum materials, respetively. For the [(±45//9 quasi-isotropi laminates, the largest value of ν yx is obtained for the Kevlar 49-epoxy material. For all the laminates and material systems, < < for all allowable values of the ompliane ratio R and the laminates experiene a state of uniform biaxial ompression prior to bukling. Nondimensional bukling loads are shown in Fig. 3 as a funtion of the number of laminate plies for the axially stiff [(±45/ laminates made of the IM7/56 material. The nondimensional bukling load is given by r b where D is defined in terms of the lamina material properties (see Table ) and plate thikness h by E D = L E T h 3 ν LT ν TL (49) (5) This bending stiffness is used herein to permit the bukling performane of laminates that are made of the same material, but with different ply orientations, to be ompared diretly. Two groups of urves are shown in the figure; the dashed and solid urves orrespond to results for lamped and simply supported plates, respetively. Four urves appear within eah group that orrespond to values of the ompliane ratio given by R = (rigidly restrained),.,.5, and. The results for R = orrespond to an unrestrained plate for whih = =. The results in Fig. 3 show a monotoni redution in the nondimensional bukling load as the number of plies (8m) inreases. However, beause D inreases with the ube of the plate thikness h, the atual bukling load inreases by the fator m 3 as the number of plies inreases. The results in Fig. 3 also show a redution in the bukling load as the ompliane ratio dereases, for both simply supported and lamped plates, as expeted. Speifially, the unrestrained plates, that experiene uniaxial ompression, exhibit the highest bukling loads and the orresponding rigidly restrained plates, that experiene biaxial ompression, exhibit the lowest bukling loads. The orresponding bukling interation urves for 8 Amerian Institute of Aeronautis and Astronautis

9 the axially stiff [(±45/ laminates of Fig. 3 with m = (8 plies) and m = 6 (48 plies) are shown in Fig. 4. The dashed and solid urves orrespond to results for lamped and simply supported plates, respetively. In addition, the horizontal, flat portions of the urves orrespond to wide-olumn bukling modes for infinitely long plates. Points on the urves that orrespond to values of R = (rigidly restrained),.,.5,,, and (unrestrained) are indiated by six different symbols. These points are loated by noting that the slope of a line that emanates from the origin in the figure, is given by = ν yx +R (5) For these axially stiff laminates, ν yx =. 93. Additionally, the 8-ply plates have a muh higher degree of flexural anisotropy (γ =.8, δ =.) than the 48-ply plates (γ =., δ =.). These results indiate that the rigidly restrained plates (R = ) experiene the largest amount of transverse ompression, as expeted. Bukling interation urves for [(±45/ ), [(±45/ 9 ), and [(±45//9) 6-ply laminates made of the IM7/56 material are presented in Fig. 5 for simply supported (solid lines) and lamped (dashed lines) boundary onditions. Points on the urves for values of R = (rigidly restrained),.,.5,,, and (unrestrained) are also indiated by six different symbols. These points are also loated by noting that the slope of a line that emanates from the origin in the figure, is given by Eq. (5). Values of ν yx for these laminates, to be used with Eq. (5), are given in Table. Like for Fig. 4, the horizontal, flat portions of the urves orrespond to wide-olumn bukling modes. The results in Fig. 5 show the basi effets of ply orientation on the bukling resistane of the 6-ply laminates, as the laminate onfiguration hanges from axially stiff to quasi-isotropi to transversely stiff. In general, the transversely stiff [(±45/9 ), plate exhibits the greatest bukling resistane for states of biaxial ompression. In ontrast, the axially stiff [(±45/ ) plate exhibits the lowest bukling resistane for states of biaxial ompression. For a state of uniaxial ompression, the quasi-isotropi laminate exhibits the greatest bukling resistane. For all ases, the lamped plates are more bukling resistant than the simply supported plates, as expeted. Moreover, the simply supported plates exhibit wide-olumn modes for the smaller values of the ompliane ratio R designated by the symbols, whereas none of the lamped plates exhibit wide-olumn modes. A omparison of the strutural effiieny of 6-ply [(±45//9) quasi-isotropi plates with simply supported edges is presented in Fig. 6. Eah urve in this figure orresponds to one of the nine material systems defined in Table. A thik, solid gray urve is also shown for plates made of aluminum with an elasti modulus E = 6 psi, a Poisson s ratio ν =.33, and a density ρ Al =. lb/in 3. In this figure, the bukling loads are normalized by the bending stiffness D Al, whih is obtained by substituting the properties for the aluminum material into Eq. (5). Moreover, the nondimensional bukling loads are weighted by the density ratio ρ Al /ρ, where ρ is the density of the material (see Table ) that orresponds to a given urve in the figure. Thus, plates with higher bukling resistane per unit mass are represented by urves that are farther from the origin of the graph. Also, points on the urves that orrespond to values of R = (rigidly restrained),.,.5,,, and (unrestrained) are indiated by symbols. The results in Fig. 6 show that all the materials out perform the aluminum material exept the Kevlar 49-epoxy and S-glass-epoxy materials. The best performane is exhibited by the P-/35 pith-epoxy material, followed by the boron-aluminum material. The worst performane is exhibited by the S-glass-epoxy material, followed by the Kevlar 49-epoxy material. The symbols shown in the figure indiate a very pronouned effet of lateral edge restraint, whih varies somewhat with material system. Results are presented in Fig. 7 for [(±θ balaned, angle-ply laminates that dramatize the effets of fiber orientation and material system on the indued lateral load. Eah urve in this figure also orresponds to one of the nine material systems defined in Table, and is independent of the number of laminate plies. Moreover, the results are appliable to an infinite range of ompliane ratios given by R <. The results in Fig. 7 show the largest variation in the load ratio / in the approximate range of 5 deg < θ < 8 deg. The greatest variations in the load ratio are exhibited by the laminates made of the P-/35 pith-epoxy material, followed by those made of the IM7/PETI-5 material. In ontrast, the smallest variations are exhibited by the laminates made of the boron-aluminum material, followed by those made of the S-glass-epoxy material. It is important to note that the indued lateral load for many of the laminate onfigurations exeeds the magnitude of the applied load; the largest being about 3.3 times the applied load. The effets of fiber orientation and lateral edge restraint on the bukling resistane of 4-ply, highly anisotropi [±θ laminates made of the IM7/56 material given in Table are shown in Fig. 8. In partiular, the nondimensional bukling load defined by Eq. (49) is giv- 9 Amerian Institute of Aeronautis and Astronautis

10 en as a funtion of the fiber angle θ. Two groups of urves are shown in the figure. The dashed and solid urves orrespond to results for lamped and simply supported plates, respetively. In addition, four urves appear within eah group that orrespond to values of the ompliane ratio given by R = (rigidly restrained),.,.5, and (unrestrained, = = ). The results in Fig. 8 show a big effet of fiber orientation and lateral edge restraint. Unlike the orresponding results for the load ratio shown in Fig. 7, the largest variation in bukling load is in the approximate range of 5 deg < θ < 7 deg. Generally, the results indiate that the lamped plates are more bukling resistant than the simply supported plates, as expeted, but for several values of θ, the unrestrained, simply supported plates are more bukling resistant than the orresponding lamped plates with R = (rigidly restrained) and R =.. This somewhat surprising result illustrate a detrimental effet of the biaxial ompression state that is indued by severely restraining the lateral movement of the lamped-plate edges. Thus, negleting the effets of inplane restraint in a preliminary-design bukling analysis ould lead to an erroneous representation of the true response and negative margins of safety. The results in Fig. 8 also show signifiant differenes in the shapes of the urves for the orresponding lamped and simply supported plates with R =,., and.5 and in the range of approximately 5 deg < θ < 65 deg. Insight into these differenes is obtained by examining the orreponding bukling interation urves that are presented in Figs. 9 and for the simply supported and lamped [±θ laminates, respetively, made of the IM7/56 material. Five urves are shown in Fig. 9, and in Fig., that orrespond to values of θ = 5, 3, 45, 6, and 75 deg. Also, points on the urves that orrespond to values of R = (rigidly restrained),.,.5,,, and (unrestrained) are indiated by six different symbols. Comparison of the symbols shown in Fig. 9 for the simply supported plates indiate wide-olumn bukling modes for values of θ = 3, 45, and 6 deg. For the remaining values of θ, the bukling modes are not wideolumn modes. In addition, none of the modes for the lamped plates with the values of R indiated by the symbols are wide-olumn modes. Thus, the differene in the shape of the urves in Fig. 8, for the lamped and simply supported plates with lateral edge restraint, appear to be assoiated with the differene in mode shapes. Results for Unbalaned Laminates For unbalaned laminates, inplane shearing deformations will develop under uniaxial ompression loading, unless rigidly restrained, beause the shearextensional oupling oeffiients are nonzero; that is, a 6 and a 6 (see Eq. (5a). The indued loads assoiated with restraining lateral movement and inplane shearing deformations are given by Eqs. (34) and (35), or by Eqs. (36) and (37). Two speial ases that will be addressed subsequently are the ases in whih only the lateral inplane movement of a plate is restrained (R 3 ) and the ase in whih only the shear deformation of a plate is restrained (R ). The indued loads for the first ase are defined by Eqs. (4) and (4), and those for the seond ase are defined by Eqs. (4) and (43). Results are presented in Table 3 that show the load ratio +R, or equivalently, ν yx for /, /9, and //9 unbalaned laminates made from one of the nine different material systems given in Table and for the ase in whih only the lateral inplane movement of a plate is restrained (R 3 ); that is, inplane shear deformations are unrestrained and =. The results are independent of the number of laminate plies and indiate that the /9 laminates exhibit the largest values of ν yx for a given material system; the smallest values are exhibited by the / laminates. Moreover, for the /9 laminates, the largest value of ν yx is obtained for the Kevlar 49-epoxy material. For the //9 and / laminates, the largest value of ν yx is obtained for the boron-aluminum material. For all the laminates and material systems, < < for all allowable values of the ompliane ratio R and the laminates experiene a state of uniform biaxial ompression. Results are presented in Table 4 that show the load N ratio +R xy 3, or equivalently, η x,xy for /, /9, and //9 laminates made from one of the nine different material systems and for the ase in whih only the inplane shear deformation of a plate is restrained (R ); that is, inplane lateral movements are unrestrained and =. These results, all negative, are also independent of the number of laminate plies and indiate that the /9 laminates exhibit the largest magnitudes of η x,xy for a given material system. The smallest magnitudes of η x,xy are exhibited by the / laminates. Moreover, for the /9 and //9 laminates, the largest magnitude of η x,xy is obtained for the P-/35 pithepoxy material. For the / laminate, the largest magnitude of η x,xy is obtained for the S-glass-epoxy material. For all the laminates and material systems, and for all allowable values of the ompliane ratio R 3, the Amerian Institute of Aeronautis and Astronautis

11 laminates experiene a state of uniform axial ompression and negative shear (see Fig. b for the positiveshear diretion). The presene of negative shear is rationalized by the fat that the 45-deg plies tend to rotate ounterlokwise in the x-y plane when the plates are subjeted to the uniaxial ompression load and shear deformations are not restrained. Thus, the restoring fore of the restraining medium must at to rotate these plies in the opposite diretion, whih implies the presene of negative shear stresses. Although not addressed speifially in the present study, the results in Table 4 indiate the possibility of bukling under uniaxial tension loads when only the inplane shear deformation of an unbalaned laminate are restrained. Graphs of the load ratios / and / are presented in Figs. and, respetively, for / laminates made of the IM7/56 material defined in Table. These results are independent of the number of laminate plies and are given as a funtion of the ompliane ratio R by the blak urves, for values of R 3 = (rigidly restrained against inplane shear deformation),.,.5,,, and (unrestrained against inplane shear deformation). The dashed gray urves in the figures orrespond to similar results in whih the ompliane ratios are equal; that is, R. The irular symbol on the ordinate of the graphs orrespond to results in whih the plates are rigidly restrained against all inplane movement. The results in Figs. and define a very broad spetrum of ombined biaxial ompression and negative shear loads that an result from restraining the inplane expansion, ontration, and shear deformation of the plates, for a very general variety of restraint ombinations. For all restraint senaros represented in the figures, the indued loads diminish as either of the restraint onditions is relaxed. Moreover, the magnitudes of the indued loads never exeed % of the magnitude of the applied axial ompression for this family of laminates. The effets of hanging material system on the load ratios / and / are presented in Figs. 3 and 4, respetively, for the / laminates (m =,,...). In partiular, urves that show the load ratios as a funtion of the ompliane ratio R are presented, where R = R. Values of R = and R = orrespond to plates that are rigidly restrained and unrestained against inplane deformation, respetively. Eah urve shown in these two figures orresponds to one of the nine material systems defined by Table. Suprizingly, the results in Fig. 3 show that the boron-aluminum and the P-/ 35 pith-epoxy materials exhibit the largest and smallest values of / for the range of R shown. Also, the magnitudes of / never exeed 5% of the magnitude of the applied axial ompression for these laminates and tend to diminish rapidly with inreasing values of the ompliane ratio R. In ontrast, the results in Fig. 4 show that both the boron-aluminum and P-/ 35 pith-epoxy materials exhibit the smallest magnitudes of / for the range of R shown. The other materials exhibit, for the most part, the larger, nearly equal magnitudes of the indued negative-shear load. The magnitudes of / never exeed % of the magnitude of the applied axial ompression for these laminates and also tend to diminish rapidly with inreasing values of the ompliane ratio R. The effets of laminate staking sequene and ompliane ratio R (R = R) on the load ratios / and / are illustrated in Fig. 5 for the /, /9, and //9 laminates made of the IM7/56 material. The solid and dashed urves orrespond to results for / and /, respetively. Moreover, the gray solid and dashed urves orrepond to results for the //9 laminates. Like the previous two figures, the results show a deline in the magnitudes of the load ratio with inreasing values for the ompliane ratio R. The deline is the most pronouned for the /9 laminates and the least pronouned for the / laminates. These results also show that plaing more laminate fibers perpendiular to the axis of the applied load dramatially amplifies the magnitudes of the indued ompression and shear loads. The largest magnitudes of the load ratios are on the order of 6% to 7% of the applied load, for the rigidly restrained /9 laminates. Nondimensional bukling loads, defined by Eq. (49), are shown in Fig. 6 as a funtion of the number of laminate plies for the /9 laminates made of the IM7/56 material. Two groups of urves are shown in the figure; the dashed and solid urves orrespond to results for lamped and simply supported plates, respetively. Four urves also appear within eah group that orrespond to different values of the ompliane ratio R, where R = R. Speifially, the values are R = (rigidly restrained),.,.5, and. The results for R = orrespond to an unrestrained plate for whih = =. Unlike the results shown in Fig. 3 for the [(±45/ balaned laminates (note that R 3 = and R = R for the balaned laminates), the results in Fig. 6 for the /9 unbalaned laminates show a monotoni inrease in the nondimensional (and atual) bukling load as the number of plies inreases, and show a reversal in the response trend for the lamped plates with R =. Speifially, the unrestrained lamped plates exhibit Amerian Institute of Aeronautis and Astronautis