NASA Technical Paper 3659 Buckling Behavior of Long Symmetrically Laminated Plates Sujected to Shear and Linearly Varying Axial Edge Loads Michael P. Nemeth Langley Research Center Hampton, Virginia National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-2199 July 1997
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Astract A parametric study of the uckling ehavior of infinitely long symmetrically laminated anisotropic plates that are sujected to linearly varying edge loads, uniform shear loads, or cominations of these loads is presented. The study focuses on the effects of the shape of linearly varying edge load distriution, plate orthotropy, and plate flexural anisotropy on plate uckling ehavior. In addition, the study examines the interaction of linearly varying edge loads and uniform shear loads with plate flexural anisotropy and orthotropy. Results otained y using a special purpose nondimensional analysis that is well suited for parametric studies of clamped and simply supported plates are presented for [±θ] s thin graphite-epoxy laminates that are representative of spacecraft structural components. Also, numerous generic ucklingdesign charts are presented for a wide range of nondimensional parameters that are applicale to a road class of laminate constructions. These charts show explicitly the effects of flexural orthotropy and flexural anisotropy on plate uckling ehavior for linearly varying edge loads, uniform shear loads, or cominations of these loads. The most important finding of the present study is that specially orthotropic and flexurally anisotropic plates that are sujected to an axial edge load distriution that is tension dominated can support shear loads that are larger in magnitude than the shear uckling load. Introduction Buckling ehavior of laminated plates that are sujected to comined loads is an important consideration in the preliminary design of aircraft and launch vehicles. The sizing of many structural sucomponents of these vehicles is often determined y staility constraints. One sucomponent that is of practical importance in structural design is the long rectangular plate. These plates commonly appear as sucomponents of stiffened panels used for wing structures and as semimonocoque shell segments used for fuselage and launch vehicle structures. Buckling results for infinitely long plates are important ecause they often provide a useful conservative estimate of the ehavior of finite-length rectangular plates, and they provide information that is useful in explaining the ehavior of these finite-length plates. Moreover, knowledge of the ehavior of infinitely long plates can provide insight into the uckling ehavior of more complex structures such as stiffened panels. An important type of long plate that appears as a sucomponent of advanced composite structures is the symmetrically laminated plate. In the present paper, the term symmetrically laminated refers to plates in which every lamina aove the plate midplane has a corresponding lamina located at the same distance elow the plate midplane, with the same thickness, material properties, and fier orientation. Symmetrically laminated plates remain flat during the manufacturing process and exhiit flat preuckling deformation states. These characteristics and the amenaility of these plates to structural tailoring provide symmetrically laminated plates with a significant potential for reducing structural weight of aircraft and launch vehicles. Thus, understanding the uckling ehavior of symmetrically laminated plates is an important part of the search for ways to exploit plate orthotropy and anisotropy to reduce structural weight. In many practical cases, symmetrically laminated plates exhiit specially orthotropic ehavior. However, in some cases, such as [±45] s laminates, these plates exhiit anisotropy in the form of material-induced coupling etween pure ending and twisting deformations. This coupling is referred to herein as flexural anisotropy, and it generally yields uckling modes that are skewed in appearance. The effects of flexural orthotropy and flexural anisotropy on the uckling ehavior of long rectangular plates that are sujected to single and comined loading conditions are ecoming etter understood. For example, recent in-depth parametric studies that show the effects of anisotropy on the uckling ehavior of long plates that are sujected to compression, shear, pure inplane ending, and various cominations of these loads have een presented in references 1 through 5. The results presented in these references indicate that the importance of flexural anisotropy on the uckling resistance of long plates varies with the magnitude and type of the comined loading condition. However, none of these studies supply results for plates loaded y uniform shear and a general linear distriution of axial load across the plate width. Both the uniform axial compression and the pure in-plane ending loads are special cases of the general linear distriution of axial edge loads. Results for this class of loadings are useful in the design of aircraft spar wes and panels that are located off the neutral axis of a fuselage or launch vehicle that is sujected to overall ending and torsion loads. Moreover, the importance of neglecting flexural anisotropy in a uckling analysis is practically unknown for this class of loadings.
One ojective of the present paper is to present uckling results for specially orthotropic plates that are sujected to uniform shear, a general linear distriution of axial load across the plate width, and cominations of these loads in terms of useful nondimensional design parameters. Other ojectives are to identify the effects of flexural anisotropy on the uckling ehavior of long symmetrically laminated plates that are sujected to the same loading conditions and to present some previously unknown results that show some unusual ehavior. Results are presented for plates with the two long edges clamped or simply supported and that are free to move in their plane. Several generic uckling-design curves that are applicale to a wide range of laminate constructions are also presented in terms of the nondimensional parameters descried in references 1, 2, 5, and 6. Symols A m, B m D 11, D 12, D 22, D 66 D 16, D 26 E 1, E 2, G 12 K K s ( n 1 ) cr ( n xy1 ) cr K s, K γ =δ= γ =δ= K x K y c ( n x1 ) cr ( n y1 ) cr displacement amplitudes (see eq. (22)), in. plate width (see fig. 1), in. orthotropic plate-ending stiffnesses, in-l anisotropic plate-ending stiffnesses, in-l lamina moduli, psi nondimensional uckling coefficient associated with critical value of an eccentric in-plane ending load (see eq. (21) and fig. 1(a)) nondimensional uckling coefficient associated with critical value of a uniform shear load (see eq. (2) and fig. 1(a)) shear and in-plane ending uckling coefficients, defined y equations (2) and (21), respectively, in which anisotropy is neglected in the analysis nondimensional uckling coefficient associated with critical value of a uniform axial compression load (see eq. (18) and fig. 1(a)) nondimensional uckling coefficient associated with critical value of a uniform transverse compression load (see eq. (19) and fig. 1(a)) L 1, L 2, L 3, L 4 c n x1, n y1, n xy1, n 1 c n x2, n y2, n xy2, n 2 N N N xc N x, N y, N xy c N x1, N y1, N xy1, N 1 c N x2, N y2, N xy2, N 2 p, p cr w N ( ξη, ) nondimensional load factors defined y equations (14) through (17), respectively nondimensional memrane stress resultants of system of destailizing loads defined y equations (1) through (13), respectively nondimensional memrane stress resultants of system of sucritical loads defined y equations (1) through (13), respectively numer of terms in series representation of out-of-plane displacement field at uckling (see eq. (22)) intensity of eccentric in-plane ending load distriution defined y equation (5), l/in. intensity of constant-valued tension or compression load distriution defined y equation (5), l/in. longitudinal, transverse, and shear memrane stress resultants, respectively (see eqs. (5), (7), and (8)), l/in. memrane stress resultants of system of destailizing loads (see eqs. (6) through (9)), l/in. memrane stress resultants of system of sucritical loads (see eqs. (6) through (9)), l/in. nondimensional loading parameter (see eqs. (14) through (17)) and corresponding value at uckling (see eqs. (18) through (21)), respectively out-of-plane displacement field at uckling defined y equation (22), in. x, y plate rectangular coordinate system (see fig. 1), in. α, β, γ, δ ε, ε 1 nondimensional parameters defined y equations (1), (2), (3), and (4), respectively in-plane ending load distriution parameters (see fig. 1 and eq. (5)) 2
η = y/, ξ = x/ θ nondimensional plate coordinates fier angle of a lamina (see fig. 1), deg half-wavelength of uckling mode (see fig. 1), in. / uckle aspect ratio (see fig. 1) lamina major Poisson s ratio ν 12 Φ m ( η) Analysis Description asis functions used to represent uckling mode (see eq. (22)) In preparing generic design charts for uckling of a single flat plate, a special purpose analysis is often preferred over a general purpose analysis code, such as a finite element code, ecause of the cost and effort usually involved in generating a large numer of results with a general purpose code. The results presented herein were otained y using such a special purpose analysis. The analysis details are lengthy; hence, only a rief description of the analysis is presented. Symmetrically laminated plates can have many different constructions ecause of the wide variety of material systems, fier orientations, and stacking sequences that can e used to construct a laminate. A way of coping with the vast diversity of laminate constructions is to use convenient nondimensional parameters. The uckling analysis used in the present paper is ased on classical plate theory and the classical Rayleigh-Ritz method and was derived explicitly in terms of the nondimensional parameters defined in references 1, 2, 5, and 6. This approach was motivated y the need for generic (independent of laminate construction) parametric results for composite plate uckling ehavior that are expressed in terms of the minimum numer of independent parameters needed to fully characterize the ehavior and that indicate the overall trends and sensitivity of the results to changes in the parameters. The nondimensional parameters used in the present paper are given y α β -- D 11 1/4 = -------- D 22 D 12 + 2D 66 = ----------------------------- ( D 11 D 22 ) 1/2 (1) (2) δ D 26 = ----------------------------- 3 ( D 11 D 22 ) 1/4 where is the plate width and is the half-wavelength of the uckle pattern of an infinitely long plate (see fig. 1). The suscripted D-terms are the ending stiffnesses of classical laminated plate theory. The parameters α and β characterize the flexural orthotropy, and the parameters γ and δ characterize the flexural anisotropy. The loading cominations included in the analysis are uniform transverse tension or compression, uniform shear, and a general linear distriution of axial load across the plate width, as depicted in figure 1. The longitudinal stress resultant N x is partitioned in the analysis into a uniform tension or compression part and a linearly varying part corresponding to eccentric in-plane ending loads. This partitioning is given y N x = N xc N [ ε + ( ε 1 ε )η] where N xc denotes the intensity of the constant-valued tension or compression part of the load, and the term containing N defines the intensity of the eccentric inplane ending load distriution. The symols ε and ε 1 define the distriution of the in-plane ending load, and the symol η is the nondimensional coordinate given y η = y/ (see fig. 1). The analysis is ased on a general formulation that includes comined destailizing loads that are proportional to a positive-valued loading parameter p that is increased until uckling occurs and independent sucritical comined loads that remain fixed at a specified load level elow the value of the uckling load. Herein, the term sucritical load is defined as any load that does not cause uckling to occur. In practice, the sucritical loads are applied to a plate prior to the destailizing loads with an intensity elow that which will cause the plate to uckle. Then, with the sucritical loads fixed, the destailizing loads are applied y increasing the magnitude of the loading parameter until uckling occurs. This approach permits certain types of comined load interaction to e investigated in a direct and convenient manner. (4) (5) The distinction etween the destailizing and sucritical loading systems is implemented in the uckling analysis y partitioning the preuckling stress resultants as follows: N xc = c c N x1 + N x2 (6) γ = D 16 ----------------------------- 3 ( D 11D22 ) 1/4 (3) N y = N y1 + N y2 N xy = N xy1 + N xy2 (7) (8) 3
N = N 1 + N 2 (9) where the stress resultants with the suscript 1 are the destailizing loads, and those with the suscript 2 are the sucritical loads. The sign convention used herein for positive values of these stress resultants is shown in figure 1. In particular, positive values of the general linear edge stress distriution parameters N 1, N 2, ε, and ε 1 correspond to compression loading. Negative values of N 1 and N 2, or negative values of either ε or ε 1 yield linearly varying stress distriutions that include tension. The two normal stress resultants of the c system of destailizing loads, N x1 and N y1, are defined as positive valued for compression loads. This convention results in positive eigenvalues eing used to indicate instaility caused y uniform compression loads. The uckling analysis includes several nondimensional stress resultants associated with equations (6) through (9). These dimensionless stress resultants are given y c n xj = c 2 N xj ------------------------------------ π 2 ( D 11 D 22 ) 1/2 N yj 2 n yj = -------------- π 2 D 22 N xyj 2 n xyj = ------------------------------------ π 2 3 ( D 11 D 22 ) 1/4 N j 2 n j = ------------------------------------ π 2 ( D 11 D 22 ) 1/2 (1) (11) (12) (13) where the suscript j takes on the values of 1 and 2. In addition, the destailizing loads are expressed in terms of the loading parameter p in the analysis y c n x 1 n y 1 n xy1 = = = L 1 p L 2 p L 3 p (14) (15) (16) n 1 = L 4 p (17) where L 1 through L 4 are load factors that determine the specific form (relative magnitude of the load components) of a given system of destailizing loads. Typically, the dominant load factor is assigned a value of 1, and all others are given as positive or negative fractions. Nondimensional uckling coefficients used herein are given y the values of the dimensionless stress resultants of the system of destailizing loads at the onset of uckling; that is, K x K s K c ( n x1 ) cr K y ( n y1 ) cr ( n xy1 ) cr ( n 1 ) cr c ( N x1 ) cr 2 = ------------------------------------ π 2 ( D 11 D 22 ) 1/2 = L 1 p cr ( N y1 ) cr 2 = ----------------------- π 2 = L 2 p cr D 22 ( N xy1 ) cr 2 = ------------------------------------ π 2 3 ( D 11 D 22 ) 1/4 = L 3 p cr ( N 1 ) cr 2 = ------------------------------------ π 2 ( D 11 D 22 ) 1/2 = L 4 p cr (18) (19) (2) (21) where p cr is the magnitude of the loading parameter at uckling. Positive values of the coefficients K x and K y correspond to uniform compression loads, and the coefficient K s corresponds to uniform positive shear. The direction of a positive shear stress resultant acting on a plate is shown in figure 1. The coefficient K corresponds to the specific in-plane ending load distriution defined y the selected values of the parameters ε and ε 1. The mathematical expression used in the variational analysis to represent the general off-center and skewed uckle pattern is given y N w N ( ξη, ) = ( A m sin πξ + B m cos πξ)φ m ( η) m=1 (22) where ξ = x/ and η = y/ are nondimensional coordinates, w N is the out-of-plane displacement field, and A m and B m are the unknown displacement amplitudes. In accordance with the Rayleigh-Ritz method, the asis functions Φ m ( η) are required to satisfy the kinematic oundary conditions on the plate edges at η = and 1. For the simply supported plates, the asis functions used in the analysis are given y Φ m ( η) = sin [ mπη] (23) for values of m = 1, 2, 3,..., N. Similarly, for the clamped plates, the asis functions are given y Φ m ( η) = cos [( m 1)πη] cos [( m + 1)πη] (24) 4
For oth oundary conditions, the two long edges of a plate are free to move in the x-y plane. Algeraic equations governing the uckling ehavior of infinitely long plates are otained y sustituting the series expansion for the uckling mode given y equation (22) into the second variation of the total potential energy and then y computing the integrals appearing in the second variation in closed form. The resulting equations constitute a generalized eigenvalue prolem that depends on the aspect ratio of the uckle pattern / (see fig. 1) and the nondimensional parameters and nondimensional stress resultants defined herein. The smallest eigenvalue of the prolem corresponds to uckling and is found y specifying a value of / and y solving the corresponding generalized eigenvalue prolem for its smallest eigenvalue. This process is repeated for successive values of / until the overall smallest eigenvalue is found. Results otained y using the analysis descried herein for uniform compression, uniform shear, pure inplane ending (given y ε = 1 and ε 1 = 1), and various cominations of these loads have een compared with other results for isotropic, orthotropic, and anisotropic plates otained y using other analysis methods. These comparisons are discussed in references 1 and 2, and in every case the results descried herein were found to e in good agreement with those otained from other analyses. Results otained for isotropic and specially orthotropic plates that are sujected to a general linear distriution of axial load across the plate width were also compared with results presented in references 7 through 13. In every case, the agreement was good. Results and Discussion Results are presented for clamped and simply supported plates loaded y a general linear distriution of axial load across the plate width, uniform shear load, and cominations of these loads. For convenience, plates loaded y a general linear distriution of axial load across the plate width are referred to herein as plates loaded y linearly varying edge loads. To otain the various edge load distriutions used herein, ε 1 = 1 was specified and ε was varied. Sketches that show the linearly varying edge loads for several values of ε are shown in figure 2. For loading cases that involve shear, a distinction is made etween positive and negative shear loads whenever flexural anisotropy is present. A positive shear load corresponds to the shear load shown in figure 1. Although the analysis presented herein previously includes the means for applying comined loads y using sucritical loads, the comined loads considered in the present paper were applied as primary destailizing loads. Results are presented first for the familiar [±θ] s angle-ply plates that are loaded y linearly varying edge loads or y comined loads. (Several results for corresponding [±θ] s angle-ply plates that are sujected to uniform uniaxial compression, uniform shear, or pure in-plane ending loads have een presented in refs. 1 and 2.) These thin laminates are representative of spacecraft structural components and are made of a typical graphite-epoxy material with a longitudinal modulus E 1 = 127.8 GPa (18.5 1 6 psi), a transverse modulus E 2 = 11. GPa (1.6 1 6 psi), an in-plane shear modulus G 12 = 5.7 GPa (.832 1 6 psi), a major Poisson s ratio ν 12 =.35, and a nominal ply thickness of.127 mm (.5 in.). Generic results are presented next, in terms of the nondimensional parameters descried herein, for ranges of parameters that are applicale to a road class of laminate constructions. The term generic is used herein to emphasize that these uckling results are very general ecause they are presented in a form that is independent of the details of laminate construction; i.e., stacking sequence and ply materials. The ranges of the nondimensional parameters used herein are given y.1 β 3., γ.6, and δ.6. The results presented in references 1 and 2 indicate that γ = δ =.6 corresponds to a highly anisotropic plate. For isotropic plates, β = 1 and γ = δ =. Moreover, for symmetrically laminated plates without flexural anisotropy, γ = δ =. (These plates are referred to herein as specially orthotropic plates.) Values of these nondimensional parameters that correspond to several practical laminates (and several material systems) are given in references 1 and 2. To simplify the presentation of the fundamental generic ehavioral trends, results are presented herein only for plates in which γ and δ have equal values (e.g., [±45] s laminates). However, these ehavioral trends are expected to e applicale to laminates with nearly equal values of γ and δ such as a [+35/ 15] s laminate made of the typical graphite-epoxy material descried herein. For this laminate, β = 1.95, γ =.52, and δ =.51. Furthermore, results showing the effects of α, or equivalently (D 11 /D 22 ) 1/4, on the uckling coefficients are not presented. References 1 and 2 have shown that variations in this parameter affect the critical value of the uckle aspect ratio / ut not the uckling coefficient (i.e., the uckling coefficient remains constant) of plates that are sujected to uniform tension or compression loads, uniform shear loads, and pure in-plane ending loads. This trend was found in the present study to e valid also for the plates loaded y linearly varying edge loads and uniform shear loads considered herein. For clarity, the shear and in-plane ending uckling coefficients, defined y equations (2) and (21), respectively, are expressed as K s and K when the generic results are γ=δ= γ =δ= 5
descried for plates in which flexural anisotropy is neglected in the uckling calculations. Plates Loaded y Linearly Varying Edge Loads or Shear Results are presented in figures 2 and 3 for simply supported and clamped [±45] s plates, respectively, that are sujected to linearly varying edge loads that correspond to values of ε = 2, 1.5, 1,.5,,.5, and 1. In these figures, the minimum value of the loading parameter p, found y solving the generalized eigenvalue prolem for a given value of /, is shown y the solid lines for values of / 2 (flexural anisotropy is included in the analysis). The overall minimum value of the loading parameter for each curve is indicated y an unfilled circle, and these minimum values of the loading parameter correspond to the value of the uckling coefficient for each curve. The corresponding values of / are the critical values of the uckle aspect ratio. The results presented in figures 2 and 3 show the effect of the load distriution shape and oundary conditions on the uckling coefficient and the corresponding critical value of /. As the amount of tension load in the load distriution increases (ε decreases), the uckling coefficient increases sustantially, and the critical value of / decreases. Moreover, the results show that the clamped plates exhiit larger uckling coefficients and smaller critical values of / than corresponding simply supported plates. As the amount of tension load in a linearly varying edge load distriution increases, the amount of the plate width that is in compression decreases. As a result, the uckles appearing in the plate are typically confined to the narrower compression region. The width d of this narrower compression region of a plate is otained from the equation that defines the corresponding neutral axis of the in-plane ending load component (defined y N x = ); that is, d = /1 ( ε ) where ε <. For plates with large negative values of ε, the critical value of / may e much less than a value of 1. For these cases, a more accurate expression of the critical uckle aspect ratio is given y /d. Results that indicate the effect of the edge load distriution shape on the uckling coefficients for simply supported and clamped [±θ] s plates with θ =, 3, 45, 6, and 9 (see fig. 1) are shown in figure 4. The solid and dashed lines correspond to uckling coefficients for clamped and simply supported plates, respectively, in which the flexural anisotropy is neglected in the analysis. These results show that the clamped plates always have higher uckling coefficients than the simply supported plates and that the uckling coefficients for the clamped plates are more sensitive to variations in the load distriution parameter ε (indicated y the slope of the curves). The results also show very large increases in uckling coefficient as the amount of tension in the edge load distriution increases (ε decreases) for all values of θ. For a given value of ε, the largest uckling coefficient is exhiited y the plates with θ = 45, followed y the plates with θ = 3 and 6, and then y the plates with θ = and 9. Moreover, the results indicate that the plates with θ = 3 and 6 have the same uckling coefficients as do the plates with θ = and 9. The importance of neglecting the anisotropy in the calculation of the uckling coefficients given in figure 4 for [±θ] s plates is indicated in figure 5. In figure 5, the ratio of the anisotropic-plate uckling coefficient K to the corresponding specially orthotropic-plate uckling coefficient K is given as a function of the load γ=δ= distriution parameter ε and the fier angle θ. The solid and dashed lines shown in the figure correspond to results for clamped and simply supported plates, respectively. The results indicate that the simply supported plates generally exhiit larger reductions in the uckling coefficient ratio ecause of anisotropy and that the simply supported plates are more sensitive to the load distriution parameter ε than are the clamped plates. In particular, the results predict that the simply supported plates are only slightly sensitive to variations in ε and that the clamped plates exhiit practically no sensitivity to variations in ε. The largest reductions in the uckling coefficient ratio are predicted for the plates with θ = 45, followed y the plates with θ = 6 and θ = 3, respectively. The simply supported plates with θ = 45 and with ε = 2 and ε = 1 (uniform compression) have values of approximately.76 and.74, respectively, for the uckling coefficient ratio. Generic effects of plate orthotropy. Generic uckling results for specially orthotropic (γ = δ = ) simply supported and clamped plates that are sujected to linearly varying edge loads are presented in figures 6 through 1. The solid and dashed lines in the figures correspond to results for clamped and simply supported plates, respectively. The results presented in figure 6 show the uckling coefficient as a function of the orthotropy parameter β for selected values of the load distriution parameter ε = 2, 1.5, and 1. Similar results are presented in figure 7 for values of ε =.5,,.5, and 1. The results presented in figures 8 through 1 show the uckling coefficient as a function of the load distriution parameter ε for discrete values of the orthotropy parameter β =.5, 1, 1.5, 2, 2.5, and 3. In figures 8 through 1, results are presented for 2 ε 1, 1 ε, and ε 1, respectively, ecause of the large variation in the uckling coefficient with ε. 6
The generic results presented in figures 6 through 1 show that the uckling coefficient increases sustantially as the orthotropy parameter β increases. In contrast, the uckling coefficient decreases sustantially as the load distriution parameter ε increases and the amount of compression in the load distriution increases. In addition, the results presented in figures 6 and 7 indicate that the clamped plates are more sensitive to variations in the orthotropy parameter β (indicated y the slope of the curves) than the simply supported plates for the full range of load distriution parameters considered. Moreover, the results presented in figures 8 through 1 indicate that for a given value of β, the clamped plates are typically more sensitive to variations in the load distriution parameter ε than are the simply supported plates. Some generic uckling results for specially orthotropic simply supported and clamped shear-loaded plates that have een presented in references 1, 2, and 5 are presented in figure 11 for completeness of the present study and for convenience. The solid and dashed lines in the figure show the uckling coefficient as a function of the orthotropy parameter β for clamped and simply supported plates, respectively. The results indicate that the shear uckling coefficient increases sustantially as the orthotropy parameter β increases and that the clamped plates are typically more sensitive to variations in β than are the simply supported plates. This trend for the shearloaded plates is the same as the corresponding trend predicted for the plates that are loaded y linearly varying edge loads. Generic effects of plate anisotropy. Results are presented in figures 12 through 16 for simply supported and clamped plates that are sujected to linearly varying edge loads. In figures 12 through 16, the ratio of the anisotropic-plate uckling coefficient K to the corresponding specially orthotropic-plate uckling coefficient K (see figs. 6 through 1) is given for equal values of the anisotropy parameters (γ = δ) ranging from.1 γ =δ= to.6. In figures 12 and 13, the generic effects of plate anisotropy on the uckling coefficient ratio are given for simply supported and clamped plates, respectively, as a function of the orthotropy parameter β. For each value of γ = δ given in figures 12 and 13, two curves are presented. The solid and dashed lines correspond to values of the load distriution parameter ε = 1 (uniform compression) and ε = 2 (the maximum amount of tension in the load distriutions considered herein), respectively. In figures 14 through 16, the uckling coefficient ratio is given as a function of the load distriution parameter ε for discrete values of the orthotropy parameter β = 3, 1.5, and.5, respectively. The solid and dashed lines in figures 14 through 16 correspond to results for clamped and simply supported plates, respectively. Figures 12 through 16 show that the anisotropicplate uckling coefficient is always less than the corresponding orthotropic-plate uckling coefficient for all values of parameters considered. In addition, these results predict that the effects of neglecting anisotropy are typically more pronounced for the simply supported plates than for the clamped plates, ut only y a small amount. Moreover, for the full range of anisotropy considered, figures 12 and 13 show a trend of monotonic increase in the uckling coefficient ratio as the orthotropy parameter β increases. The results in figures 12 through 16 also predict that the effects of neglecting plate flexural anisotropy in the uckling analysis of simply supported plates ecome slightly less pronounced as the load distriution parameter ε decreases and the amount of tension in the edge load distriution increases. Moreover, this effect is practically negligile in the corresponding clamped plates. This ehavioral trend, in which the importance of anisotropy is reduced as the amount of tension in the edge load distriution increases, is similar to a ehavioral trend given in reference 5. There, the importance of anisotropy on the uckling load of a plate that is sujected to destailizing uniform axial compression is shown to e reduced as the amount of sucritical transverse tension load N y2 that is applied to the plate is increased. However, for this case, the clamped plates exhiit more sensitivity to the transverse tension load than do the simply supported plates. Results that show the importance of flexural anisotropy on the uckling ehavior of shear-loaded simply supported and clamped plates have een presented in reference 1 and are presented in figure 17 in a different form for convenience. In this figure, the ratio of the anisotropic-plate uckling coefficient K s to the corresponding specially orthotropic-plate uckling coefficient K s (see fig. 11) is given as a function of the γ =δ= orthotropy parameter β for equal values of the anisotropy parameters (γ = δ) ranging from.2 to.6. Two groups of curves that correspond to positive and negative shear loads are shown in the figure. For each value of γ = δ given in figure 17, two curves are presented in each group. The solid and dashed lines correspond to results for clamped and simply supported plates, respectively. The results presented in figure 17 for simply supported and clamped plates that are sujected to positive shear loads indicate that the anisotropic-plate uckling coefficient is always less than the corresponding orthotropic-plate uckling coefficient. However, this trend is reversed for negative shear loads. The results in figure 17 also show that the effects of neglecting plate 7
anisotropy ecome smaller as the orthotropy parameter β increases and the anisotropy parameters γ and δ decrease. Furthermore, these results show that the effects of neglecting plate anisotropy are only slightly more pronounced for simply supported plates than for clamped plates. Comparing the results presented in figures 12, 13, and 17 indicates that the reductions in uckling coefficient caused y neglecting anisotropy are, for the most part, more pronounced for the shear-loaded plates than for the plates that are sujected to linearly varying edge loads. Plates Loaded y Shear and Linearly Varying Edge Loads Buckling interaction curves otained y neglecting the plate anisotropy in the uckling calculations for simply supported [±45] s plates that are sujected to uniform shear and linearly varying edge loads are presented in figure 18. Negative values of K indicated on the figure correspond to results in which the sign of N 1 is reversed, and negative values of K s correspond to negative shear loadings. Several curves that indicate the staility oundaries corresponding to values of the load distriution parameter ε = 2, 1.5, 1,.5,,.5, and 1 are presented in figure 18. Each curve shown in the figure is symmetric aout the line K s =, and the curve for ε = 1 (pure in-plane ending) is also symmetric aout the line K =. In addition, all the curves pass through the same two points on the line K = ; i.e., the points that correspond to positive and negative shear uckling. The curves shown in figure 18 for values of ε =,.5, and 1 are open, paraola-like curves that extend indefinitely in the negative K -direction. For these comined loadings, shear loads that are greater in magnitude than the shear uckling load can e sustained only when the linearly varying edge loads are tensile loads. In contrast, specially orthotropic plates that are sujected to pure in-plane ending (ε = 1) can never sustain shear loads that are greater in magnitude than the shear uckling load. However, the curves shown in figure 18 for ε = 2, 1.5, and.5 are significantly different from the conventional uckling interaction curves found in the literature and shown in figure 18 for plates that are sujected to uniform shear and uniform axial compression (ε = 1) or pure in-plane ending loads. Specifically, the results predict that shear loads that are larger in magnitude than the shear uckling load can e supported y an unuckled plate when a tension-dominated linearly varying edge load distriution is applied to the plate first. It is important to oserve that the loading with ε =.5 is tension dominated for negative values of K ut not for positive values. In general, when trying to determine whether a linearly varying edge load is tension dominated, oth positive and negative values of K should e considered. The aility to carry shear loads greater in magnitude than the positive and negative shear uckling loads is attriuted to the fact that the stailizing effect of the tension part of the linearly varying edge load is greater than the destailizing effect of the compression part. Additional uckling interaction curves that correspond to the curves shown in figure 18 are presented in figure 19 for simply supported [±45] s plates that are sujected to shear and linearly varying edge loads. The curves shown in figure 19 include the effects of flexural anisotropy which are manifested y skewing and translation of the curves presented in figure 18 in the K -K s plane. Thus, the results indicate that for a given value of K, the anisotropic plates can carry a negative shear load that is much greater in magnitude than the corresponding positive shear load. Moreover, the results predict that the anisotropic plates with a tension-dominated linearly varying edge load distriution (i.e., ε = 2, 1.5, and.5) can also support shear loads (positive or negative) that are larger in magnitude than the shear uckling load and that this effect is much more pronounced for plates that are loaded in negative shear. This greater negativeshear load capacity for the plates is attriuted to the greater shear uckling resistance of these plates under negative shear loads. That is, flexurally anisotropic plates generally exhiit two unequal plate ending stiffnesses along the directions of the diagonal compression and tension generated y the shear load. For a negative shear load, the higher plate ending stiffness acts in the direction of the diagonal compression generated y the load. The importance of neglecting anisotropy in the calculation of uckling interaction curves for simply supported [±θ] s plates that are loaded y shear and a tension-dominated linearly varying edge load distriution (ε = 2) is indicated in figure 2. Curves are shown in this figure for values of θ = 3, 45, and 6. The solid and dashed lines are uckling interaction curves in which the plate anisotropy is neglected (specially orthotropic) and included, respectively. Figure 2 indicates that the specially orthotropic plates with θ = 45 have larger (in magnitude) uckling coefficients than the corresponding plates with θ = 3 and 6. Moreover, the uckling interaction curves for the specially orthotropic plates with θ = 3 and 6 are identical. For the anisotropic plates, however, these two trends are not valid. That is, the uckling coefficients for the anisotropic plates with θ = 45 are not always larger in magnitude than the uckling coefficients for the corresponding plates with θ = 3 and 6. Moreover, the uckling interaction curves for the anisotropic plates with θ = 3 and 6 are different. The results also predict the capaility of carrying shear loads that are greater in magnitude than the shear uckling loads (positive and negative) for 8
all the specially orthotropic and anisotropic plates considered. Generic effects of plate orthotropy. Generic uckling interaction curves for specially orthotropic (γ = δ= ) simply supported plates that have a value of β = 3 and that are sujected to shear and linearly varying edge loads are presented in figure 21. In particular, several curves that indicate the staility oundaries that correspond to values of the load distriution parameter ε = 2, 1.5, 1,.5,,.5, and 1 are presented in the figure. The generic uckling interaction curves shown in figure 21 exhiit the same characteristics as the corresponding curves presented in figure 18 for the [±45] s plates in which the effects of anisotropy are neglected. That is, each curve shown in the figure is symmetric aout the line given y a zero value of K s, and the γ =δ= curve for ε = 1 (pure in-plane ending) is also symmetric aout the line given y a zero value of K. In γ =δ= addition, all the curves pass through the same two points on the line given y a zero value of K, i.e., the γ =δ= points that correspond to positive and negative shear uckling. Like the results presented in figure 18 for the [±45] s plates, the generic curves shown in figure 21 for values of ε =,.5, and 1 are open, paraolalike curves that extend indefinitely in the negative K -direction. Thus, for these three loadings, shear γ =δ= loads that are greater in magnitude than the shear uckling load can e sustained only when the linearly varying edge loads are tensile loads. However, the plates that are sujected to pure in-plane ending (ε = 1) can never sustain shear loads greater in magnitude than the shear uckling load. Furthermore, the generic results also predict an aility to carry shear loads that are larger in magnitude than the corresponding shear uckling load for plate loadings with ε = 2, 1.5, and.5 (i.e., tensiondominated load distriutions). This aility to carry shear loads greater in magnitude than the positive and negative shear uckling loads is again attriuted to the fact that the stailizing effect of the tension part of the linearly varying edge load is greater than the destailizing effect of the compression part. Generic uckling interaction design curves for specially orthotropic (γ =δ=) simply supported and clamped plates that are sujected to shear and linearly varying edge loads are presented in figures 22 through 28 for values of ε = 2, 1.5, 1,.5,,.5, and 1, respectively. The solid lines and dashed lines in the figures correspond to results for clamped and simply supported plates, respectively, and curves are given for values of the orthotropy parameter β =.5, 1, 1.5, 2, 2.5, and 3. Moreover, curves are shown for positive shear loading only. Results for negative shear loading are otained y noting that uckling interaction curves for specially orthotropic plates that are loaded y shear are symmetric aout the K -axis. Points on the curves correspond to γ =δ= constant values of the stiffness-weighted load ratio N 1 ----------- D 22 1/4 --------, as illustrated in figure 29 y the line N xy1 D 11 emanating from the origin of the plot. An important characteristic of all the uckling interaction curves presented in figures 22 through 28 is that, for a given stiffnessweighted load ratio, the magnitude of the uckling coefficients increases sustantially as the orthotropy parameter β increases. The results presented in figure 22 for the plates with ε = 2 indicate that the maximum value of K s γ =δ= N 1 occurs within the range 2.86 ----------- D 22 -------- 1/4 and N xy1 D 11 4.44 N 1 3. ----------- D 22 -------- 1/4 for the simply supported N xy1 D 11 4.13 and clamped plates, respectively, as the orthotropy parameter β increases from.5 to 3.. Similarly, the results presented in figure 23 for the plates with ε = 1.5 indicate that the maximum value of K s occurs γ=δ= N 1 within the range 2.64 ----------- D 22 -------- 1/4 and N xy1 D 11 3.97 N 1 2.58 ----------- D 22 -------- 1/4 for the simply supported N xy1 D 11 3.64 and clamped plates, respectively. Moreover, the results presented in figure 25 for the plates with ε =.5 indicate that the maximum value of K s occurs γ=δ= N 1 within the range 8.88 ----------- D 22 -------- 1/4 and N xy1 D 11 5.72 N 1 8.31 ----------- D 22 -------- 1/4 for the simply supported N xy1 D 11 6. and clamped plates, respectively. Furthermore, the results presented in figure 24 for the plates with ε = 1 indicate that the maximum value of K s occurs γ=δ= when K =. γ =δ= The results presented in figures 24, 26, 27, and 28 for the plates with ε = 1,,.5, and 1, respectively, indicate that the uckling interaction curves for the simply 9
supported plates generally have more curvature than the curves for the corresponding clamped plates. Thus, the simply supported plates are generally more sensitive to variations in the stiffness-weighted load ratio. Generic effects of plate anisotropy. Generic uckling interaction curves for simply supported anisotropic plates with β = 3 and γ = δ =.6 that are sujected to shear and linearly varying edge loads are presented in figure 3. More specifically, curves that indicate the staility oundaries for ε = 2, 1.5, 1,.5,,.5, and 1 are presented. The generic uckling interaction curves shown in figure 3 exhiit the same characteristics as the curves presented in figure 19 for the corresponding [±45] s anisotropic plates (these plates have β = 2.28 and γ = δ =.517). That is, the effects of anisotropy are manifested y skewing and translation in the K -K s plane of the curves for the corresponding specially orthotropic plates presented in figure 21. Thus, as was seen for the [±45] s anisotropic plates, the generic results for the anisotropic plates also predict that, for a given value of K, negative shear loads can e carried that are much greater in magnitude than the corresponding positive shear load. Moreover, the anisotropic plates with tensiondominated axial-edge load distriutions (ε = 2, 1.5, and.5) can also support shear loads that are larger in magnitude than the shear uckling load. This effect is much more pronounced for plates that are loaded y negative shear. This difference in ehavior exhiited y anisotropic and specially orthotropic plates loaded y negative and positive shear is attriuted to directional dependence of the shear uckling resistance of anisotropic plates previously descried herein. Buckling interaction curves that show the generic effects of plate anisotropy are presented in figure 31 for simply supported and clamped plates with β = 3 that are sujected to shear and a linearly varying edge load corresponding to ε = 2 (tension dominated). Several curves that indicate the staility oundaries for equal values of the anisotropy parameters (γ = δ) ranging from to.6 are shown in the figure. The solid and dashed lines in figure 31 correspond to results for clamped and simply supported plates, respectively. These results indicate that the curve corresponding to a specially orthotropic plate (γ = δ = ) ecomes more skewed and is translated more in the negative K s -direction as the values of the anisotropy parameters increase. These effects of anisotropy appear to e nearly the same for the clamped and simply supported plates. The results presented in figure 31 also show that for some values of the stiffness-weighted load ratio N 1 ----------- D 22 1/4 --------, N xy1 D 11 the uckling interaction curves for plates with γ = δ = and γ = δ intersect. Thus, for these values of the stiffness-weighted load ratio, neglecting anisotropy has no effect on the critical value of the loading parameter p cr (see eqs. (18) through (21)). Moreover, the results indicate that as the stiffnessweighted load ratio varies, so does the importance of the plate anisotropy. An indication of the importance of plate anisotropy, with respect to the stiffness-weighted load ratio, is otained y introducing the angle that the line emanating from the origin of the uckling interaction curve plot shown in figure 29 makes with the K s -axis as an independent variale. This angle is referred to herein as the stiffness-weighted load ratio angle and is denoted y the symol Ψ. Then, for a given stiffness-weighted load ratio (constant value of Ψ), the importance of anisotropy on the uckling p cr coefficients is expressed y the ratio ---------------------- where p cr γ =δ= p cr ---------------------- p cr γ =δ= K = ---------------------- = K γ = δ = Results are presented in figure 32 that show the importance of plate anisotropy on the uckling ehavior of simply supported and clamped plates with β = 3 that are sujected to shear and a linearly varying edge load corresponding to ε = 2. The results for the simply supported plates with γ = δ =.6 correspond to those presented in figure 3. Curves that indicate the uckling p cr coefficient ratio ---------------------- for equal values of the anisotropy parameters (γ = δ) ranging from to.6 are shown p cr γ =δ= in figure 32 as a function of the stiffness-weighted load ratio angle Ψ. The solid and dashed lines in the figure correspond to results for clamped and simply supported plates, respectively. The results presented in the figure show a variation in the uckling coefficient ratio of approximately ±.5, with the greatest variations occurring at values of Ψ 7, 1 Ψ 2, and 29 Ψ 36. Moreover, the results indicate that the importance of plate anisotropy is only slightly more pronounced for the simply supported plates than for the clamped plates. Concluding Remarks K s ---------------------. K s γ = δ = A parametric study of the uckling ehavior of infinitely long symmetrically laminated anisotropic plates that are sujected to linearly varying edge loads, uniform shear loads, or cominations of these loads has een presented. A special purpose nondimensional analysis that is well suited for parametric studies of clamped and simply 1
supported plates has een descried, and its main features have een discussed. The results presented herein have focused on the effects of the shape of the linearly varying edge load distriution, plate flexural orthotropy, and plate flexural anisotropy on the uckling ehavior. In addition, results have een presented that focus on the interaction of linearly varying edge loads and uniform shear loads with plate flexural anisotropy and orthotropy. In particular, results have een presented for [±θ] s thin graphite-epoxy laminates that are representative of spacecraft structural components. Also, numerous generic uckling results have een presented that are applicale to a road class of laminate constructions that show explicitly the effects of flexural orthotropy and flexural anisotropy on plate uckling ehavior under these comined loads. These generic results can e used to extend the capaility of existing design guides for plate uckling. An important finding of the present study is that the uckling coefficients increase significantly as the D 12 + 2D 66 orthotropy parameter β = ----------------------------- increases or ( D 11 D 22 ) 1/2 as the load distriution parameter ε decreases (the loading distriution ecomes tension dominated). In contrast, the uckling coefficients decrease significantly D 16 as the anisotropy parameters γ = ----------------------------- and 3 ( D 11D22 ) 1/4 D 26 δ = ----------------------------- with equal values (γ = δ) increase, for 3 ( D 11 D 22 ) 1/4 all linearly varying edge loads considered and for the plates that are sujected to positive shear loads. For negative shear loads, the trend is generally reversed. The effect of linearly varying edge load distriution shape (determined y ε ) on the importance of plate anisotropy is generally small, with uniform compression-loaded plates exhiiting the largest reductions in uckling resistance. The results presented herein also show that the effects of plate anisotropy are slightly more pronounced for simply supported plates than for clamped plates when the plates are sujected to either linearly varying edge loads, uniform shear loads, or cominations of these loads. The most important finding of the present study is that specially orthotropic and flexurally anisotropic plates that are sujected to a tension-dominated axial edge load distriution (e.g., ε = 2, 1.5, and.5) can support shear loads that are larger in magnitude than the shear uckling load. This aility to carry a shear load greater in magnitude than the corresponding shear uckling load is attriuted to the fact that the stailizing effect of the tension part of the linearly varying edge load is greater than the destailizing effect of the compression part. Moreover, this unusual ehavior is much more pronounced for anisotropic plates than for specially orthotropic plates that are loaded in negative shear. This trend is reversed for plates that are loaded in positive shear. This difference in ehavior exhiited y anisotropic plates and specially orthotropic plates loaded y negative shear and the trend reversal for positive shear loads is attriuted to directional dependence of the shear uckling resistance of anisotropic plates. NASA Langley Research Center Hampton, VA 23681-2199 January 27, 1997 References 1. Nemeth, M. P.: Buckling Behavior of Long Symmetrically Laminated Plates Sujected to Comined Loadings. NASA TP-3195, 1992. 2. Nemeth, M. P.: Buckling of Symmetrically Laminated Plates With Compression, Shear, and In-Plane Bending. AIAA J., vol. 3, no. 12, Dec. 1992, pp. 2959 2965. 3. Zeggane, Madjid; and Sridharan, Srinivasan: Buckling Under Comined Loading of Shear Deformale Laminated Anisotropic Plates Using Infinite Strips. Int. J. Numer. Methods in Eng., vol. 31, May 1991, pp. 1319 1331. 4. Zeggane, M.; and Sridharan, S.: Staility Analysis of Long Laminate Composite Plates Using Reissner-Mindlin Infinite Strips. Comput. & Struct., vol. 4, no. 4, 1991, pp. 133 142. 5. Nemeth, Michael P.: Buckling Behavior of Long Anisotropic Plates Sujected to Comined Loads. NASA TP-3568, 1995. 6. Nemeth, Michael P.: Importance of Anisotropy of Buckling of Compression-Loaded Symmetric Composite Plates. AIAA J., vol. 24, no. 11, Nov. 1986, pp. 1831 1835. 7. Rockey, K. C.: Shear Buckling of Thin-Walled Sections. Thin- Walled Structures, A. H. Chilver, ed., John Wiley & Sons, Inc., 1967, pp. 261 262. 8. Oyio, G. A.: The Use of Affine Transformations in the Analysis of Staility and Virations of Orthotropic Plates. Ph.D. Thesis, Rensselaer Polytechnic Inst., 1981. 9. Brunnelle, E. J.; and Oyio, G. A.: Generic Buckling Curves for Specially Orthotropic Rectangular Plates. AIAA J., vol. 21, no. 8, Aug. 1983, pp. 115 1156. 1. Timoshenko, Stephen P.; and Gere, James M.: Theory of Elastic Staility, Second ed. McGraw-Hill Book Co., 1961, pp. 373 379. 11. Bulson, P. S.: The Staility of Flat Plates. American Elsevier Pul. Co. Inc., 1969, pp. 63 77. 12. Bleich, Friedrich: Buckling Strength of Metal Structures. McGraw-Hill Co., Inc., 1952, pp. 399 413. 13. Galamos, Theodore V., ed.: Guide to Staility Design Criteria for Metal Structures. John Wiley & Sons, Inc., 1988, p. 13. 11