May 2013

NASA/TM–2013-217999

An Approximate Solution and Master Curves 

for Buckling of Symmetrically Laminated 

Composite Cylinders 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA scientific and technical information (STI) 

program plays a key part in helping NASA maintain 

this important role. 

The NASA STI program operates under the 

auspices of the Agency Chief Information Officer. 

It collects, organizes, provides for archiving, and 

disseminates NASA’s STI. The NASA STI 

program provides access to the NASA Aeronautics 

and Space Database and its public interface, the 

NASA Technical Report Server, thus providing one 

of the largest collections of aeronautical and space 

science STI in the world. Results are published in 

both non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types: 

 TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

Programs and include extensive data or 

theoretical analysis. Includes compilations of 

significant scientific and technical data and 

information deemed to be of continuing 

reference value. NASA counterpart of peerreviewed formal professional papers, but 

having less stringent limitations on manuscript 

length and extent of graphic presentations. 

 TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., quick release reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis. 

 CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees. 

 CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, seminars, 

or other meetings sponsored or cosponsored by NASA. 

 SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest. 

 TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing information desk and personal search 

support, and enabling data exchange services. 

For more information about the NASA STI 

program, see the following: 

 Access the NASA STI program home page 

at http://www.sti.nasa.gov

 E-mail your question to help@sti.nasa.gov

 Fax your question to the NASA STI 

Information Desk at 443-757-5803 

 Phone the NASA STI Information Desk at 

443-757-5802

 Write to: 

 STI Information Desk 

 NASA Center for AeroSpace Information 

 7115 Standard Drive 

 Hanover, MD 21076-1320
National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

May 2013

NASA/TM–2013-217999

An Approximate Solution and Master Curves 

for Buckling of Symmetrically Laminated 

Composite Cylinders 

Michael P. Nemeth 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1

Summary

 Nondimensional linear-bifurcation buckling equations for balanced, symmetrically laminated 

cylinders with negligible shell-wall anisotropies and subjected to uniform axial compression 

loads are presented. These equations are solved exactly for the practical case of simply supported 

ends. Nondimensional quantities are used to characterize the buckling behavior that consist of a 

stiffness-weighted length-to-radius parameter, a stiffness-weighted shell-thinness parameter, a 

shell-wall nonhomogeneity parameter, two orthotropy parameters, and a nondimensional 

buckling load. Ranges for the nondimensional parameters are established that encompass a wide 

range of laminated-wall constructions and numerous generic plots of nondimensional buckling 

load versus a stiffness-weighted length-to-radius ratio are presented for various combinations of 

the other parameters. These plots are expected to include many practical cases of interest to 

designers. Additionally, these plots show how the parameter values affect the distribution and size 

of the festoons forming each response curve and how they affect the attenuation of each response 

curve to the corresponding solution for an infinitely long cylinder. 

 To aid in preliminary design studies, approximate formulas for the nondimensional buckling 

load are derived, and validated against the corresponding exact solution, that give the attenuated 

buckling response of an infinitely long cylinder in terms of the nondimensional parameters 

presented herein. A relatively small number of “master curves” are identified that give a 

nondimensional measure of the buckling load of an infinitely long cylinder as a function of the 

orthotropy and wall inhomogeneity parameters. These curves reduce greatly the complexity of the 

design-variable space as compared to representations that use dimensional quantities as design 

variables. As a result of their inherent simplicity, these “master curves” are anticipated to be 

useful in the ongoing development of buckling-design technology.

Symbols

A, B nondimensional modal amplitudes defined by 

equations (13)-(15)

A11, A12, A22, A66 orthotropic-cylinder membrane stiffnesses, lb/in.

cx nondimensional coefficient defined by equations (44)

D11, D12, D22, D66 orthotropic-cylinder bending stiffnesses, in-lb

Dm( ), Dc( ), Db( ) nondimensional linear differential operators defined by 

equations (3)-(5)

E, G Young’s modulus of elasticity and shear modulus of an

isotropic material, respectively, psi

 reference value of Young’s modulus of elasticity (see

equation (44a)), psi

E
2

EL, ET, GLT lamina moduli (see Table 1), psi

h cylinder thickness (see figure 1), in.

I nondimensional wall nonhomogeneity measure defined by 

equation (19)

L cylinder length (see figure 1), in.

m ∈ {1, 2, 3, ...} number of axial half waves in buckle pattern (see equation (13))

mcr value of m at buckling

n ∈ {0, 1, 2, 3, ...} number of circumferential waves in buckle pattern (see equation

(13) and figure 2)

ncr value of n at buckling

N, Ncr circumferential wave number defined by equation (20c) and 

value of N at buckling, respectively

Nx, compressive axial prebuckling stress resultant and 

corresponding value at buckling, respectively

, nondimensional loading parameter defined by equation (12) 

and corresponding value at buckling, respectively

R cylinder radius (see figure 1), in.

x, y cylinder coordinate system (see figure 1), in.

X, Xcr quantity defined by equation (28) and value at buckling,

respectively 

αb, αm nondimensional stiffness-weighted length-to-radius ratios

defined by equations (8) and (6), respectively

β nondimensional orthotropy parameter defined by equation (9)

δF nondimensional buckling-stress function (see equations (1)

and (2))

δW nondimensional radial-displacement field at buckling (see

equations (1) and (2))

nondimensional cylinder circumferential coordinate

θ lamina fiber angle (see figure 1), degrees

Nx

cr

p pcr

η = y

R
3

λ, λcr stiffness-weighted axial-buckle aspect ratio defined by 

equation (23) and value at buckling, respectively

Λ, Λcr nondimensional stiffness-weighted axial half-wave length of a

buckle pattern defined by equation (18) and value at buckling,

respectively

μ nondimensional orthotropy parameter defined by equations (7)

νm laminate membrane Poisson's ratio defined by equation (10)

νLT lamina major Poisson's ratio (see Table 1)

nondimensional cylinder axial coordinate

ρ stiffness-weighted radius-to-thickness ratio defined by 

equation (11)

value of the uniform axial stress at buckling, psi

Introduction

 As evidenced by recent studies, efforts to develop shell-buckling design technology for 

laminated-composite cylinders are still underway.1, 2 Despite these and other efforts, the 

technology still has not reached the level of maturity needed to obtain the full weight-saving 

potential of shell structures, especially those made of laminated-composite materials. The major 

impediment is a lack of validated analysis and design methods. As a result, NASA SP 8007,3

developed in the late 1960s, is still the primary resource used by American industry for the design 

of buckling-resistant cylindrical shells. The procedure used in NASA SP 8007 has become a 

paradigm that accounts for the effects of initial geometric imperfections and edge support 

conditions by applying empirical “knockdown” factors to relatively simple classical linearbifurcation buckling solutions. Although this class of solutions are very approximate, they 

generally provide insight into how variations in design parameters affect performance. Thus, it is 

useful to have relatively simple approximate analyses and formulas for predicting response trends 

that can be used as part of a building-block design process.

 Recent examples of design-technology development for buckling of laminated-composite 

cylinders have been presented in references 1 and 2. Specifically, baseline design formulas are 

presented for balanced, symmetrically laminated cylinders with walls that are effectively 

homogeneous and specially orthotropic. This approach was motivated by existing designs in 

which the allowable range of structural dimensions are such that the laminate is composed of a 

relatively large number of plies that are distributed uniformly through the thickness. Thus, it 

follows that modelling a cylinder wall as a homogeneous and specially orthotropic material is not 

a serious limitation in a preliminary design process, for a significant number of practical 

applications. Moreover, this approach is a logical first step in a building-block approach for more 

ξ = x

L

σx

cr

= Nx

cr

h
4

general, laminate constructions with plies at the outer faces that are substantially different from 

those near the mid-plane. Most importantly, this approach yields significant simplification of the 

buckling equations and facilitates the development of simple design formulas in terms of useful 

nondimensional parameters, as evidenced by the results presented in references 1 and 2. Of 

particular importance are the “master curves” presented in these references that represent the 

behavior of a wide range of practical laminate constructions in a very concise manner. 

Specifically, the “master curves” presented in references 1 and 2 are plots of normalized buckling 

stress versus normalized buckling strain that are presented as a function of laminate composition. 

By including strain in these plots, both structural stiffness and load-capacity requirements can be 

considered in the design process. Moreover, curves such as these permit common behavior trends 

to be identified for a wide variety of laminated-cylinder configurations. The formulas and results 

presented in references 1 and 2 represent a significant first-approximation advancement in design 

technology for compression-loaded laminated-composite cylinders.

 The objective of the present study is to extend work presented in references 1 and 2 to include 

the cylinder-wall heterogeneity that cannot be neglected in a buckling analysis without 

introducing significant errors. Herein, cylinder-wall heterogeneity refers to laminate 

constructions in which there exists a significant through-the-thickness variation of the layer 

properties or orientations. Of particular interest herein are symmetric laminates with plies at the 

outer faces that are substantially different from those near the mid-plane. For convenience, this 

class of heterogeneous laminates are referred to herein as “sandwich-like” laminates. Use of the 

term “sandwich-like” herein is intended to connote the arrangement of plies and does not imply 

the existence of significant transverse shearing flexibility commonly associated with sandwich 

plates. To accomplish this objective, a set of nondimensional linear-bifurcation buckling 

equations and parameters are presented for balanced, symmetrically laminated cylinders 

subjected to uniform axial compression loads and in a membrane-stress prebuckling state. 

Anisotropy of the shell wall is presumed to be negligible. Then, a linear bifurcation buckling 

analysis is presented for the classical simply supported boundary conditions that are typically 

used in the preliminary design of cylinders. Next, approximate formulas for the buckling loads of 

infinitely long cylinders are derived. Results that establish the range of the nondimensional 

parameters appearing in the buckling analysis and simplified formulas are then presented. With 

the parameter ranges established, plots of nondimensional buckling load versus a stiffnessweighted length-to-radius ratio are presented for various combinations of the remaining 

parameters governing the behavior and for a very broad range of laminate constructions. Finally, 

relatively fewer “master curves” are presented for infinitely long cylinders that give a 

nondimensional measure of buckling as a function of the orthotropy parameters and the wall 

inhomogeneity measure. In addition, approximate formulas for the critical stresses are given that 

utilize the information in these curves. Thus, these “master curves” are anticipated to be useful in 

the ongoing development of buckling-design technology.

Bifurcation-Buckling Equations

 To accomplish the objective of the present study, it is convenient to use the nondimensional 

parameters and equations presented in reference 4. When specialized to right circular cylindrical 

shells, such as that shown in figure 1, these equations correspond to Donnell-type, linear-
5

bifurcation buckling equations for symmetric laminates. In particular, the compatibility equation 

is given by 

 (1)

and the transverse equilibrium equation is given by

 (2)

where Dm( ), Db( ), and Dc( ) are nondimensional linear differential operators. The symbol δW 

denotes the nondimensional radial-displacement buckling mode and δF denotes the corresponding 

stress function. For symmetric laminates with negligible coupling between inplane stretching 

and shearing deformations, and between pure bending and twisting deformations, these operators 

are given by 

 (3)

 (4)

 (5)

The nondimensional coordinates in these equations are defined as and , where x is 

the axial coordinate, y is the circumferential arc-length coordinate, L is the cylinder length, R is 

the cylinder radius, 0 ≤ ξ ≤ 1, and 0 ≤ η ≤ 2π. The nondimensional parameters appearing in these 

equations are given by

 (6)

 (7)

 (8)

 (9)

Dm δF = 12 Dc δW

Db δW + 12 Dc δF = − pπ2 ∂

2

δW

∂ξ 2

Dm δF = 1

αm

2

∂

4

δF

∂ξ 4 + 2μ ∂

4

δF

∂ξ 2

∂η2 + αm

2 ∂

4

δF

∂η4

Dc δW = ρ ∂

2

δW

∂ξ 2

Db δW = 1

αb

2

∂

4

δW

∂ξ 4 + 2β ∂

4

δW

∂ξ 2

∂η2 + αb

2 ∂

4

δW

∂η4

ξ = x

L η = y

R

αm = L

R

A22

A11

1

4

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

αb = L

R

D22

D11

1

4

β = D12 + 2D66

D11D22
6

 (10)

 (11)

where A11, A12, A22, A66, D11, D12, D22, and D66 are the stiffnesses of classical laminated-shell 

theory (see reference 5), and the corresponding stiffnesses A16, A26, D16, and D26 are presumed 

negligible. The parameter defined by equation (11) is a stiffness-weighted shell-wall thinness 

metric. The quantity is the nondimensional loading parameter defined by

 (12)

where Nx is the magnitude of the uniform compressive axial stress resultant, that is presumed to 

exist prior to buckling, in classical linear bifurcation analyses for cylinders. 

Buckling Analysis

 For the purpose of preliminary design, cylinders with simply supported edges that restrict 

circumferential displacements are often used because they faciliate a simple analytical solution. 

These boundary conditions are usually referred to as the classical boundary conditions for simply 

supported shells and are denoted as “S2” boundary conditions in the recent book by Jones (see p. 

477).6

 The nondimensional buckling modes for these boundary conditions are given by

 (13)

where 0 ≤ ξ ≤ 1, 0 ≤ η ≤ 2π, m ∈ {1, 2, 3, ...} is the number of axial half-waves, n ∈ {0, 1, 2, ...} 

is the number of full circumferential waves (see figure 2), and A is the indeterminate modal 

amplitude. A typical buckling mode is shown in figure 3a that exhibits one full axial wave and 

four full circumferential waves. The corresponding contour plot of the radial displacement is 

shown in figure 3b. Buckling modes with n = 0 and n ≠ 0 are referred to herein as 

axisymmetric and asymmetric modes, respectively. 

 Substituting equation (13) into equation (1) indicates that the stress function can be represented 

by 

 (14)

where B is related to the amplitude A. Specifically, substituting equations (13) and (14) into 

equation (1) yields

νm = A12

A11A22

ρ = R

h

1 − νm

2 A11A22 h

4

144D11D22

1

4

p

p = NxR2

π2D11D22

δW = A sin mπξ cos nη

δF = B sin mπξ cos nη
7

 (15)

Substituting equation (15) into equation (14) and the result into equation (2), along with equation 

(13), and setting the coefficient of the trigonometric functions equal to zero yields the following 

equation for the eigenvalues 

 (16)

For a given set of material properties and cylinder dimensions, the nondimensional parameters 

appearing in equation (15) are fixed values. The nondimensional buckling load, , is then given 

by the smallest value of that is found for m ∈ {1, 2, 3, ...} and n ∈ {0, 1, 2, ...}. The 

corresponding wave numbers are denoted by mcr and ncr . 

Approximate Buckling Formulas

 To obtain simple, approximate buckling formulas, similar to that given in reference 1, equation 

(16) is differentiated as if is a continuous function and m and n are real variables. 

Although not strictly correct, this approach gives useful results for certain ranges of the 

nondimensional parameters appearing in equation (16). To obtain a minimum set, 

 (17)

is enforced. In general, the partial derivatives appearing in equation (17) are very complicated 

functions of the variables m and n. To remedy this situation, it is convenient to introduce a 

nondimensional axial half-wave length Λ of a buckle pattern defined by 

 (18)

and a positive-valued wall nonhomogeneity measure given by the following ratio of extensional 

and bending stiffnesses

 (19)

B = − 2 3 ρ m2

π2

n4αm

2

+ 2μn2m2π2+ m4

π4

αm

2

A

p = p m, n

pπ2= n4

αb

2

m2π2 + 2βn2

+

π2

m2

αb

2 +

12ρ

2

n

4

αm

2

m2π2 + 2μn2

+

m2

π2

αm

2

pcr

p

p = p m, n

dp = ∂p

∂m

dm + ∂p

∂n

dn = 0

Λ = L

πmR

2 A22

A11

1

2

I = A11D22

D11A22

1

2
8

For a homogeneous wall, I = 1. Because m is the number of axial half waves in a given 

bifurcation eigenvector, it follows that the quantity L/m is the axial half-wave length of the 

eigenvector. Substituting equations (18) and (19) into equations (6) and (8) yields 

 (20a)

 (20b)

As the cylinder length increases to infinity, Λ behaves like a continuous variable and 

minimization of by differentiation with respect to Λ represents solutions for an infinitely long 

cylinder that buckles into a local shell mode and not into an overall column mode. Likewise, it is 

convenient to introduce

 (20c)

By using equations (20), equation (16) becomes

 (21)

Now, it is observed that application of the chain rule of differentiation to equation (17) reveals that 

minimization of with respect to Λ and N is equivalent to minimization with respect to m 

and n, respectively. Enforcing and simplifying gives

 (22)

Likewise, enforcing gives 

 (23)

for values of N > 0, which yields

 (24)

αm

πm

2

= Λ

αb

πm

2

= ΛI

p

N = n2

pπ2= N2ΛI + 2βN +

1

ΛI

+

12ρ

2

ΛN2+ 2μN +

1

Λ

p

∂p

∂Λ = 0

IN2− 1

IΛ2

1

Λ

+ 2μ N + Λ N2 2− 12ρ

2

N2− 1

Λ2 = 0

∂p

∂N = 0

β + IΛ N 1

Λ

+ 2μN + ΛN2 2− 12ρ

2

μ + ΛN = 0

1

Λ

+ 2μN + ΛN2 2= 12ρ

2 μ +ΛN

β + IΛ N
9

Substituting equation (24) into equation (22) gives

 (25a)

for values of N > 0. For the case of N = 0, minimization of equation (21) with respect to nonzero 

values of Λ is given by equation (22), which reduces to

 (25b)

Here, it is important to note that values of Λ and N that satisfy equations (24) and (25), denoted 

by Λcr and Ncr, respectively, yield the critical value of , denoted by , when substituted into 

equation (21). 

 Solutions that minimize the right-hand side of equation (21) are obtained by examining the 

cases of N = 0 and N > 0 separately. For the case of N = 0, equation (25b) yields

 (26)

Substituting equation (26) for Λ and N = 0 into equation (21) gives the result

 (27)

where it is implied that Ncr = 0. This solution corresponds to short-wave-length, axisymmetric 

buckle patterns.

 Solutions of equations (24) and (25a) for N > 0 are found in the present study by defining a 

positive-valued quantity X given by

 (28)

so that equation (25a) becomes

 (29)

equation (24) becomes

 (30)

I2Λ2N2− 1 μ + ΛN − I β + IΛN Λ2N2− 1 = 0

− 1

IΛ2 + 12ρ

2

= 0

p pcr

Λcr = 1

2 3I ρ

pcr = 4 3 ρ

π2

I

X = IΛN

β − Iμ X2

+ 1 − I

2

X + I μ − Iβ = 0

I

X

+ 2μ + X

I

2

N2= 12ρ

2

I

Iμ + X

β + X
10

and equation (21) becomes

 (31)

This transformation allows N to be expressed in terms of the nondimensional parameters and the 

new variable X. In particular, equation (30) is solved for N to get

 (32)

Equation (32) is now used to eliminate N from equation (31), which yields a function of X; 

that is, equation (31) becomes

 (33)

The critical values of are obtained by substituting the positive-valued solutions to equation 

(29), denoted by X = Xcr, into equation (33). In particular, the solutions to equation (29) are given 

by

 (34a)

for and I ≠ 1. The corresponding values of and Ncr are given by

 (34b)

 (34c)

Additionally, equation (28) yields

 (34d)

pπ2= X + 2β +

1

X

N +

12ρ

2

X

I + 2μ +

I

X

N

N = 2ρX 3I

Iμ + X

β + X I2

+ 2μIX + X2 − 1

p

p = 2 3Iρ

π2

Iμ + X

β + X

X2+ 2βX + 1

X2

+ 2μIX + I2 +

β + X

Iμ + X

p

Xcr = I

2

− 1

2 β − Iμ

1 ± 1 +

4I β − Iμ Iβ − μ

I

2

− 1

2

β − Iμ ≠ 0 pcr

pcr = 2 3Iρ

π2

Iμ + Xcr

β + Xcr

Xcr

2

+ 2βXcr + 1

Xcr

2

+ 2μIXcr + I2 +

β + Xcr

Iμ + Xcr

Ncr = 2ρXcr 3I

Iμ + Xcr

β + Xcr

Xcr

2

+ 2μIXcr + I2 − 1

Λcr = Xcr

INcr
11

These results correspond to short-wave-length, non-axisymmetric buckle patterns. It is important 

to note that as Xcr + Iμ → 0, → ∞.

 For, the case of N > 0 and I = 1 and either or , the solution to 

equation (29) is given by Xcr = 1 and equation (34b) reduces to

 (35a)

which is identical to the corresponding result presented in reference 1. Additionally, equation 

(34c) reduces to

 (35b)

and equation (34d) becomes

 (35c)

For this case, → ∞ as μ → -1 and the corresponding axisymmetric buckling solution given 

by equations (26) and (27) govern the response. For cases with and I ≠ 1, equation 

(29) reduces to which yields physically inadmissible solutions for positive values of β, 

since X > 0 by definition. In addition, equations (34b)-(34d) become singular. Solutions for this 

case are obtained by re-examining equations (21)-(23). First, equation (23) yields 

 (36)

which is satisfied by selecting

 (37)

Substituting equation (37) into equation (22) gives

 (38)

pcr

β − Iμ ≠ 0 β − Iμ = 0

pcr = 4 3ρ

π2

1 + β

1 + μ

Ncr = 3 ρ

1 + β 1 + μ

Λcr = 1 + β 1 + μ

3 ρ

pcr

β − Iμ = 0

X = − β,

I 1

Λ

+ 2μN + ΛN2 2− 12ρ

2

μ + ΛN = 0

1

Λ

+ 2μN + ΛN2 2= 12ρ

2

I

I − 1

I

1

Λ2 = 0
12

Enforcing I ≠ 1 requires Λ-1 to go to zero, which means that the buckle patterns have axial half 

waves that are on the order of the cylinder length. Next, substituting Λ-1 = 0 back into equation 

(37) gives

 (39)

For this equation, and equation (21), it is presumed that N → 0 such that N2

Λ and NΛ are finite 

valued. As a result, equation (21) with β = Iμ yields the result 

 (40)

 The results producing equation (40) indicate that long-wave-length buckle patterns exist. In 

addition, it is important to note that the analysis presented by equation (22)-(35) is not valid for 

very long cylinders that exhibit buckle patterns with axial half waves that are on the order of the 

cylinder length. For the general case of long-wave-length buckle patterns, the parameter Λ is 

presumed to approach infinity and the parameter N is presumed to approach zero such that N2

Λ 

and NΛ remain finite. Applying these presumptions to equation (21) gives 

 (41)

Minimization of this equation with respect to N2

Λ gives 

 (42)

which is identical to equation (40), with no restrictions on the values of I, β, or μ. For this case, 

the buckle pattern is axisymmetric; that is, Ncr = 0. Note that the appears in the numerator of 

equation (42) and in the denomenator of equation (27) for the short-wavelength axisymmetic 

buckle pattern.

Results and Discussion

 In the present section, results that indicate the possible ranges of the nondimensional 

parameters and coefficients defined by equations (6)-(11) and (19) are presented first. Based on 

these ranges, generic buckling results obtained by minimization of equation (16) with respect to 

m and n are presented entirely in terms of the nondimensional parameters defined herein. These 

generic results are based on very wide parameter ranges and, as a result, encompass a wide range 

of geometries and laminate constructions and also illustrate a wide spectrum of behavioral trends. 

2μN + ΛN2

= ± 2 3

I ρ

pcr = 4 3I ρ

π2

pπ2= N2ΛI +

12ρ

2

N2Λ

pcr = 4 3I ρ

π2

I
13

Next, “master curves” for infinitely long cylinders are presented. In particular, two nearly 

identical sets of results are presented that were obtained by minimization of equation (16) with 

respect to m and n for extremely long cylinders and by using the approximate formulas derived 

herein. 

Parameter and Coefficient Ranges

 The nondimensional parameters and coefficients defined herein have been presented for a wide 

variety of balanced, symmetric laminates in reference 7 for the nine material systems given in 

Table 1. Two laminate families of particular interest that are discussed in this reference are the 

[(±45/0/90)p]S and [(0/90/±45)p]S quasi-isotropic laminates, and the [(+θ/−θ)p]S and [(−θ/+θ)p]S 

angle-ply laminates. The results presented in reference 7 for the [(+θ/−θ)p]S and the [(−θ/+θ)p]S 

angle-ply laminates indicate the following ranges: 

0.279 ≤ β ≤ 2.76 

-0.788 ≤ μ ≤ 5.43

0.341 ≤ (A22/A11)

1/4 ≤ 2.93

I = 1

for the nine material systems, regardless of the value of the stacking sequence index p. Likewise, 

the results for the [(±45/0/90)p]S quasi-isotropic laminates indicate:

1.03 ≤ β ≤ 2.22

0.926 ≤ (D22/D11)

1/4 ≤ 1.00

for the nine material systems and for p = 1,2, ..., 8. Moreover, for the [(0/90/±45)p]S quasiisotropic laminates the results indicate: 

0.266 ≤ β ≤ 0.968

0.862 ≤ (D22/D11)

1/4 ≤ 1.00

0.446 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 0.999

0.958 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 1.09
14

for the nine material systems and p = 1,2, ..., 8. For both families of quasi-isotropic laminates, 

(A22/A11)

1/4 = 1 and μ = 1, regardless of the lamina material system and the number of plies. 

Thus, it follows that the heterogeneity parameter has the range 0.862 ≤ I ≤ 1.00 for both families 

of quasi-isotropic laminates, regardless of the lamina material system and the number of plies.

 Two other laminate families considered herein are the [(±45/02)p]S axially stiff laminates and 

the [(±45/902)p]S circumferentially stiff laminates with integer values for the index p = 1, 2, ..., 8. 

Results obtained in the present study for the [(±45/02)p]S laminates yielded: 

1.04 ≤ β ≤ 2.24

0.531 ≤ μ ≤ 0.987

1.01 ≤ I ≤ 1.72

0.687 ≤ (A22/A11)

1/4 ≤ 0.941

for the nine material systems and p = 1,2, ..., 8. For the [(±45/902)p]S laminates, the same ranges 

for β, μ, and were obtained. In contrast, the ranges 0.582 ≤ I ≤ 0.989, 

1.06 ≤ (A22/A11)

1/4 ≤ 1.45 were obtained for the nine material systems and p = 1, 2, ..., 8.

 Sandwich-like laminates with pairs of 0-deg and 90-deg plies lumped between pairs of angle 

plies were also considered in the present study. In particular, results were obtained in the present 

study for [(+θ/−θ)2/(0/90)4]S laminates that revealed: 

0.083 ≤ β ≤ 1.58

1.22 ≤ μ ≤ 16.89

0.543 ≤ I ≤ 1.50

0.845 ≤ (A22/A11)

1/4 ≤ 1.18

0.875 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 0.956

0.901 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 0.988

1 − νm

2 A11A22 h

4

144D11D22

1

4
15

for the nine material systems and for 0 ≤ θ ≤ 90 deg. Results were also obtained in the present 

study for 24-ply [90p/0q]S laminates that indicate: 

0.065 ≤ β ≤ 0.697

1.71 ≤ μ ≤ 17.85

1.04 ≤ I ≤ 3.42

0.568 ≤ (A22/A11)

1/4 ≤ 1.76

for the nine material systems and for p = 1, 2, ..., 11. Similarly, results for 24-ply [0p/90q]S 

laminates that exhibit the same ranges for β, μ, (A22/A11)

1/4, and were 

obtained. In contrast, the range of the heterogeneity parameter given by 0.292 ≤ I ≤ 0.962 was 

obtained for the nine material systems and for p = 1, 2, ..., 11. 

Generic Buckling Results

 The nondimensional buckling loads obtained by minimization of equation (16) with respect to 

m and n are shown in figures 4-75 as a function of the stiffness-weighted length-to-radius ratio 

αm defined by equation (6). Six curves are shown in each figure that correspond to values of the 

orthotropy parameter μ = - 1, 0, 1, 5, 10 and 20. Figures 4-75 are also composed of three groups 

of related curves. In particular, figures 4-27 correspond to results for the stiffness-weighted 

thinness ratio ρ = 25, figures 28-51 correspond to results for ρ = 100, and figures 52-75 

correspond to results for ρ = 1000. Within each of these three groups, 24 curves are presented for 

combinations of the values of the orthotropy parameter β = 0, 1, 2, and 3 and for values of the 

nonhomogeneity parameter I = 0.25, 0.5, 1, 1.5, 2, and 4. In computing these results, equations 

(6), (8), and (19) were used to obtain 

 The results in figures 4-75 generally show a series of festoon curves in which the amplitudes of 

the festoons attenuate to a negligible magnitude as the stiffness-weighted length-to-radius ratio αm 

increases. Likewise, the nondimensional buckling loads forming a given curve attenuate to the 

constant value associated with the corresponding infinitely long cylinder that buckles into a local 

0.988 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 1.18

.829 ≤ 1 − νm

2 A11A22 h

4

144D11D22

1

4

≤ 1.69

1 − νm

2 A11A22 h

4

144D11D22

1

4

αb = Iαm.
16

shell mode, not an overall column mode. The results in figures 4-75 also generally show that the 

festoons making up the curves become less prominent as β decreases, as μ increases, or as I 

increases for each group of curves corresponding to a given value of ρ. Additionally, the festoons 

become generally less pronounced for a group of curves as the value of ρ increases.

“Master Curves”

 As evidenced by the parameter ranges presented herein, the numerous results appearing in 

figures 4-75 are applicable to a very broad range of symmetrically laminated cylinders. However, 

it is desirable to reduce these results into a compact form suitable for preliminary design that 

captures the essential trends and that provides adequate estimates of the buckling resistance. This 

reduction was performed in the present study by observing that each festoon curve appearing in 

figures 4-75 has a useful, practical lower bound that is given by the buckling load of the 

corresponding infinitely long cylinder. These lower-bound buckling loads were obtained in the 

present study for selected values of β, μ, ρ, and I by minimizing equation (16) with respect to 

m and n for relatively large values of αm. Moreover, the curves in figures 4-75 with the 

pronounced festoons required significantly larger values of αm than the curves that exhibit rapid 

attenuation of the buckling load with increasing values of αm. This process typically involves 

minimization over a very large set of values for m and n. As a result, the determination of a 

buckling load following this process involves a large number of computations.

 The approximate buckling loads for infinitely long cylinders that are given herein by the 

smallest positive value of obtained from equations (27), (34b), and (42) require significantly 

fewer computations than the minimization of equation (16). To demonstrate the accuracy of these 

approximate buckling loads, several attempts were made during the present study to find a concise 

means of comparison applicable to the wide range of values for β, μ, ρ, and I appearing in 

figures 4-75. The comparisons presented in figures 76-79 were found to be adequate. An 

important step in arriving at the form of these figures was found when parametric studies revealed 

that plots of versus μ have the same shape for a fixed value of β and any value of ρ. In 

addition, it was found that identical sets of curves were produced for a given values of I and its 

reciprocal. Moreover, equation (35a) revealed that for an isotropic material. These 

facts led to the use of as the ordinate in figures 76-79.

 Four pairs of red and blue curves are shown in each of figures 76-79. Each pair corresponds to 

a given value of the orthotropy parameter β. The blue curves correspond to buckling loads 

obtained by minimizing equation (16) with respect to m and n for αm = 100. The red curves 

correspond to the smallest positive value obtained from equation (27) for short-wave-length, 

axisymmetric buckle patterns, from equation (34b) for short-wave-length, non-axisymmetric 

buckle patterns, and from equation (42) for long-wave-length, axisymmetric buckle patterns. All 

pcr

pcr

π2pcr

4 3 ρ

= 1

π2pcr

4 3 ρ

pcr
17

of the curves shown in these figures were computed for a value of ρ = 100. In solving equation 

(34b), it was found that equation (34a) exhibited only one positive-valued root.

 Inspection of the results used to obtain figures 76-79 indicated differences in buckling loads for 

each pair of curves that are less than 3%. Similar results were obtained for values of ρ = 1000 

and 25 which exhibited differences in buckling loads for each corresponding pair of curves that 

are less than 3% and 8%, respectively. Overall, the differences in each pair of corresponding 

buckling loads obtained from the two solution methods are insignificant compared to the inherrent 

variability in the material properties. It is important to observe that the results in figures 76-79 are 

applicable to an enormous number of laminate constructions. Moreover, the compact form of 

figures 76-79 make the design space easier to navigate and interpret.

 The results presented in figures 76-79 show that the nondimensional buckling load 

decreases as the orthotropy parameters β or μ decrease. Moreover, the effect of decreasing μ 

on the nondimensional buckling load is reduced as the nonhomogeneity parameter I becomes 

greater or less than a value equal to one. The further I departs from I = 1, the more the 

nondimensional buckling loads are reduced.

Approximate Formulas for Critical Stresses

 The results presented in figures 76-79 are also useful for estimating the critical stress in a 

cylinder incipient to buckling. Specifically, approximate formulas for the critical stress of a 

geometrically perfect cylinder, are obtained by using with equation (12) to get

 (43)

An insightful, alternate form of equation (43) is obtained by using equation (11) to get

 (44a)

where

 (44b)

π2pcr

4 3 ρ

σx

cr

, σx

cr

= Nx

cr

h

σx

cr

= pcrπ2

R2h

D11D22

σx

cr

= cx

Eh

R

cx ≡ 1 − νm

2

3

144 A11A22D11D22

E

4

h

8

1

4

π2pcr

4 3 ρ
18

and the term in parentheses is the ordinate used in figures 76-79. The symbol denotes a 

convenient reference modulus of elasticity and is included in equations (44) to provide some 

freedom in the way one may want to represent the critical stress for a given problem. For 

example, consider an isotropic material with a modulus E and Poisson’s ratio ν. For this 

material, and using = E in equations (44) yields the well-known formula

 (45)

that is often used in the design of isotropic cylinders. For a given class of laminated cylinders, 

picking the reference modulus to be the effective axial modulus of a specific laminate configuration in the class may yield a desired physical interpretation of equation (44a) that is useful in the 

design of these cylinders.

Concluding Remarks

 A set of nondimensional linear-bifurcation buckling equations for balanced, symmetrically 

laminated cylinders subjected to uniform axial compression loads have been presented that are 

based on Donnell’s equations. These equations presume negligible shell-wall anisotropies and 

have been solved exactly for the practical case of simply supported ends. Nondimensional 

quantities have been used to characterize the buckling behavior that consist of a stiffnessweighted length-to-radius parameter, a stiffness-weighted shell-thinness parameter, a shell-wall 

nonhomogeneity parameter, two parameters that quantify membrane and bending orthotropies, 

and a nondimensional buckling load. Results that establish the range of the nondimensional 

parameters appearing in the buckling analysis have also been presented that encompass a wide 

range of laminated-wall constructions. Numerous generic plots of nondimensional buckling load 

versus a stiffness-weighted length-to-radius ratio are presented for various combinations of the 

remaining parameters governing the behavior. These plots span the established ranges of the 

nondimensional parameters and are expected to include many practical cases of interest to 

designers. Additionally, these plots show how the parameter values affect the distribution and size 

of the festoons forming each response curve and how they affect the attenuation of each response 

curve to the corresponding solution for an infinitely long cylinder. 

 To aid in preliminary design studies, approximate formulas for the nondimensional buckling 

load have been derived. These formulas give the attenuated buckling response of an infinitely 

long cylinder in terms of the previously described nondimensional parameters and have been 

validated against the corresponding exact solution. An important finding of the present study is 

that a relatively small number of “master curves” have been identified that give a nondimensional 

measure of the buckling load of an infinitely long cylinder as a function of the orthotropy and wall 

inhomogeneity parameters. In particular, these curves reduce greatly the complexity of the 

design-variable space as compared to representations that use dimensional quantities as design 

variables. Approximate formulas for the critical stresses have also been given that utilize the 

E

π2pcr

4 3 ρ

= 1 E

σx

cr

= 1

3 1 − ν2

Eh

R ≈ 0.6

Eh

R
19

information in these curves. As a result of their inherent simplicity, these “master curves” are 

anticipated to be useful in the ongoing development of buckling-design technology.

References

1. Nemeth, M. P. and Mikulas, M. M., Jr.: Simple Formulas and Results for Buckling-Resistance 

and Stiffness Design of Compression-Loaded Laminated-Composite Cylinders. NASA/TP2009-215778, August 2009.

2. Mikulas, M. M., Jr.; Nemeth, M. P.; Oremont, L.; and Jegley, D. C.: Effect of Boundary 

Conditions on the Axial Compression Buckling of Homogeneous Orthotropic Composite 

Cylinders in the Long Column Range. Proceedings of the AIAA/ASME/ASCE/AHS/ASC 52nd 

Structures, Structural Dynamics, and Materials Conference, 2011, AIAA paper 2011-2126.

3. Anon.: Buckling of Thin-Walled Circular Cylinders. NASA Space Vehicle Design Criteria, 

NASA SP-8007, September 1965, revised August 1968.

4. Nemeth, M. P.: Nondimensional Parameters and Equations for Buckling of Symmetrically 

Laminated Thin Elastic Shallow Shells. NASA TM 104060, 1991.

5. Jones, R. M.: Mechanics of Composite Materials. Second ed., Taylor & Francis, 1999.

6. Jones, R. M.: Buckling of Bars, Plates, and Shells. Bull Ridge Publishing, 2006.

7. Nemeth, M. P.: Nondimensional Parameters and Equations for Nonlinear and Bifurcation 

Analyses of Thin Anistropic Quasi-Shallow Shells. NASA/TP-2010-216726, 2010.
20

Table 1. Lamina Properties

* The symbols L and T denote the longitudinal fiber and transverse matrix directions of a specially orthotropic lamina, respectively.

Lamina 

property*

Material Systems

Boron-Al S-glassepoxy

Kevlar 

49-epoxy

IM7/

5260

AS4/

3502

AS4/

3501-6

BoronepoxyIM7/

PETI-5

P-100/

3502

EL, Msi 33 7.5 11.02 22.1 18.5 20.01 29.58 20.35 53.5

ET, Msi 21 1.7 0.8 1.457 1.64 1.30 2.68 1.16 0.73

νLT 0.23 0.25 0.34 0.258 0.30 0.30 0.23 0.29 0.31

GLT, Msi 7.0 0.80 0.33 0.860 0.87 1.03 0.81 0.61 0.76

R

h

θ

x

Typical lamina fiber

y

L

Figure 1. Cylinder geometry, surface coordinate system, and lamina fiber orientation.
21

n = 1 n = 2 n = 3 n = 4

n = 5 n = 6 n = 7 n = 8

n = 9 n = 10 n = 11 n = 12

n = 13 n = 14 n = 15 n = 16

Figure 2. Circumferential waveforms used in equation (12).
22

L

2πR mcr = 2

Axial half-wave length

Figure 3. Typical buckle pattern of a compression-loaded cylinder.

Buckle

(a) Three-dimensional rendering of a buckling mode

(b) Contour plot of the radial displacement

ncr = 4
23

0 2 4 6 8

0

30

40

50

Figure 4. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 0, I = 0.25, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

30

40

50

Figure 5. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 1, I = 0.25, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0
24

0 2 4 6 8

0

30

40

50

Figure 6. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 2, I = 0.25, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

30

40

50

Figure 7. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 3, I = 0.25, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0
25

0 2 4 6 8

0

30

40

50

Figure 8. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 0, I = 0.5, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

30

40

50

Figure 9. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 1, I = 0.5, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0
26

0 2 4 6 8

0

30

40

50

Figure 10. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 2, I = 0.5, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

30

40

50

Figure 11. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 3, I = 0.5, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

0
27

0 1 2 3 4

0

20

25

30

Figure 12. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 0, I = 1, and ρ = 25.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

10

0 1 2 3 4

0

20

25

30

Figure 13. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 1, I = 1, and ρ = 25.

μ = -1 and 0 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

1

10
28

0 1 2 3 4

0

20

25

30

Figure 14. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 2, I = 1, and ρ = 25.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

10

0 1 2 3 4

0

20

25

30

Figure 15. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 3, I = 1, and ρ = 25.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

10
29

0 1 2 3 4

0

20

25

30

Figure 16. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 0, I = 1.5, and ρ = 25.

μ = -1 and 0 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

10 1

0 1 2 3 4

0

20

25

30

Figure 17. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 1, I = 1.5, and ρ = 25.

μ = -1 and 0 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

10

1
30

0 1 2 3 4

0

20

25

30

Figure 18. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 2, I = 1.5, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

10

0 1 2 3 4

0

20

25

30

Figure 19. Effects of parameters αm and μ on nondimensional buckling loads on

5

15

cylinders with β = 3, I = 1.5, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

10

5
31

0 1 2 3 4

0

15

20

Figure 20. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 0, I = 2, and ρ = 25.

μ = -1 and 0 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1

0 1 2 3 4

0

15

20

Figure 21. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 1, I = 2, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5
32

0 1 2 3 4

0

15

20

Figure 22. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 2, I = 2, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

5

0 1 2 3 4

0

15

20

Figure 23. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 3, I = 2, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

5
33

0 1 2 3 4

0

15

20

Figure 24. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 0, I = 4, and ρ = 25.

μ = -1 and 0 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5 1

0 1 2 3 4

0

15

20

Figure 25. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 1, I = 4, and ρ = 25.

μ = -1, 0, and 1 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5
34

0 1 2 3 4

0

15

20

Figure 26. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 2, I = 4, and ρ = 25.

μ = -1, 0, 1, and 5 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

0 1 2 3 4

0

15

20

Figure 27. Effects of parameters αm and μ on nondimensional buckling loads on

5

10

cylinders with β = 3, I = 4, and ρ = 25.

μ = -1, 0, 1, and 5 

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10
35

0 2 4 6 8

0

60

80

100

Figure 28. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 0, I = 0.25, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

60

80

100

Figure 29. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 1, I = 0.25, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0
36

0 2 4 6 8

0

60

80

100

Figure 30. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 2, I = 0.25, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

60

80

100

Figure 31. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 3, I = 0.25, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

0
37

0 2 4 6 8

0

60

80

100

Figure 32. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 0, I = 0.5, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 2 4 6 8

0

60

80

100

Figure 33. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 1, I = 0.5, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

0
38

0 2 4 6 8

0

60

80

100

Figure 34. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 2, I = 0.5, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

0

0 2 4 6 8

0

60

80

100

Figure 35. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 3, I = 0.5, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

0
39

0 1 2 3 4

0

60

80

100

Figure 36. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 0, I = 1, and ρ = 100.

μ = -1

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10 μ = 20

0

0 1 2 3 4

0

60

80

100

Figure 37. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 1, I = 1, and ρ = 100.

μ = -1 and 0

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20

1
40

0 1 2 3 4

0

60

80

100

Figure 38. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 2, I = 1, and ρ = 100.

μ = -1, 0, and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

0 1 2 3 4

0

60

80

100

Figure 39. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 3, I = 1, and ρ = 100.

μ = -1, 0, and 1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5

10

μ = 20
41

0 1 2 3 4

0

60

80

100

Figure 40. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 0, I = 1.5, and ρ = 100.

μ = -1 and 0

1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5 10

μ = 20

0 1 2 3 4

0

60

80

100

Figure 41. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 1, I = 1.5, and ρ = 100.

μ = -1, 0 and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20
42

0 1 2 3 4

0

60

80

100

Figure 42. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 2, I = 1.5, and ρ = 100.

μ = -1, 0 and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

0 1 2 3 4

0

60

80

100

Figure 43. Effects of parameters αm and μ on nondimensional buckling loads on

20

40

cylinders with β = 3, I = 1.5, and ρ = 100.

μ = -1, 0 and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20
43

0 1 2 3 4

0

30

40

50

Figure 44. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 0, I = 2, and ρ = 100.

μ = -1 and 0

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

1

0 1 2 3 4

0

30

40

50

Figure 45. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 1, I = 2, and ρ = 100.

μ = -1 and 0

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

1
44

0 1 2 3 4

0

30

40

50

Figure 46. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 2, I = 2, and ρ = 100.

μ = -1, 0, and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

0 1 2 3 4

0

30

40

50

Figure 47. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 3, I = 2, and ρ = 100.

μ = -1, 0, and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20
45

0 1 2 3 4

0

30

40

50

Figure 48. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 0, I = 4, and ρ = 100.

μ = -1 and 0

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

1

0 1 2 3 4

0

30

40

50

Figure 49. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 1, I = 4, and ρ = 100.

μ = -1, 0 and 1

5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20
46

0 1 2 3 4

0

30

40

50

Figure 50. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 2, I = 4, and ρ = 100.

μ = -1, 0, 1, and 5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10

μ = 20

0 1 2 3 4

0

30

40

50

Figure 51. Effects of parameters αm and μ on nondimensional buckling loads on

10

20

cylinders with β = 3, I = 4, and ρ = 100.

μ = -1, 0, 1, 5, and 10

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20
47

0 1 2 3 4

0

600

800

1000

Figure 52. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 0, I = .25, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

5 10 μ = 20

1 0

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

0 1 2 3 4

0

600

800

1000

Figure 53. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 1, I = .25, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

1

0
48

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

0 1 2 3 4

0

600

800

1000

Figure 54. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 2, I = .25, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

1

0

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

0 1 2 3 4

0

600

800

1000

Figure 55. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 3, I = .25, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

1

0
49

0 1 2 3 4

0

600

800

1000

Figure 56. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 0, I = 0.5, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5 1

0

0 1 2 3 4

0

600

800

1000

Figure 57. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 1, I = 0.5, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1

0
50

0 1 2 3 4

0

600

800

1000

Figure 58. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 2, I = 0.5, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1

0

0 1 2 3 4

0

600

800

1000

Figure 59. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 3, I = 0.5, and ρ = 1000.

μ = -1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

1

0
51

0 1 2 3 4

0

600

800

1000

Figure 60. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 0, I = 1, and ρ = 1000.

μ = -1 and 0

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1

0 1 2 3 4

0

600

800

1000

Figure 61. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 1, I = 1, and ρ = 1000.

μ = -1, 0, and 1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5
52

0 1 2 3 4

0

600

800

1000

Figure 62. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 2, I = 1, and ρ = 1000.

μ = -1, 0, and 1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

0 1 2 3 4

0

600

800

1000

Figure 63. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 3, I = 1, and ρ = 1000.

μ = -1, 0, and 1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5
53

0 1 2 3 4

0

600

800

1000

Figure 64. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 0, I = 1.5, and ρ = 1000.

μ = -1 and 0 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1

0 1 2 3 4

0

600

800

1000

Figure 65. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 1, I = 1.5, and ρ = 1000.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5
54

0 1 2 3 4

0

600

800

1000

Figure 66. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 2, I = 1.5, and ρ = 1000.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

0 1 2 3 4

0

600

800

1000

Figure 67. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 3, I = 1.5, and ρ = 1000.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5
55

0 1 2 3 4

0

600

800

1000

Figure 68. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 0, I = 2, and ρ = 1000.

μ = -1 and 0 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5 1

0 1 2 3 4

0

600

800

1000

Figure 69. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 1, I = 2, and ρ = 1000.

μ = -1 and 0 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

10 μ = 20

5

1
56

0 1 2 3 4

0

600

800

1000

Figure 70. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 2, I = 2, and ρ = 1000.

μ = -1, 0, and 1 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20 10

5

0 1 2 3 4

0

600

800

1000

Figure 71. Effects of parameters αm and μ on nondimensional buckling loads on

200

400

cylinders with β = 3, I = 2, and ρ = 1000.

μ = -1, 0, 1, and 5 pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10
57

0 1 2 3 4

0

300

400

500

Figure 72. Effects of parameters αm and μ on nondimensional buckling loads on

100

200

cylinders with β = 0, I = 4, and ρ = 1000.

μ = -1 and 0

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5

1

0 1 2 3 4

0

300

400

500

Figure 73. Effects of parameters αm and μ on nondimensional buckling loads on

100

200

cylinders with β = 1, I = 4, and ρ = 1000.

μ = -1, 0, and 1

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

5
58

0 1 2 3 4

0

300

400

500

Figure 74. Effects of parameters αm and μ on nondimensional buckling loads on

100

200

cylinders with β = 2, I = 4, and ρ = 1000.

μ = -1, 0, 1, and 5

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20

10

0 1 2 3 4

0

300

400

500

Figure 75. Effects of parameters αm and μ on nondimensional buckling loads on

100

200

cylinders with β = 3, I = 4, and ρ = 1000.

μ = -1, 0, 1, 5, and 10

pcr = Nx

cr R2

π2 D11D22

Stiffness-weighted length-to-radius ratio, αm = L

R

A22

A11

1

4

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

μ = 20
59

-5 0 5 15 20

0

0.6

0.8

1

Figure 76. Nondimensional buckling loads of infinitely long cylinders with I = 1. 

0.2

0.4

Nondimensional parameter, 

β = 3

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

π2pcr

4 3 ρ

10

β = 0

2

1

Nondimensional

buckling load,

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

pcr = Nx

cr R2

π2

D11D22

Isotropic cylinder

-5 0 5 15 20

0

0.6

0.8

1

Figure 77. Nondimensional buckling loads of infinitely long cylinders with I = 0.5 and I = 2. 

0.2

0.4

Nondimensional parameter, 

β = 3

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

π2pcr

4 3 ρ

10

β = 0

2

1

Nondimensional

buckling load,

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

pcr = Nx

cr R2

π2

D11D22
60

-5 0 5 15 20

0

0.6

0.8

1

Figure 78. Nondimensional buckling loads of infinitely long cylinders with I = 0.67 and I = 1.5. 

0.2

0.4

Nondimensional parameter, 

β = 3

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

π2pcr

4 3 ρ

10

β = 0

2

1

Nondimensional

buckling load,

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

pcr = Nx

cr R2

π2

D11D22

-5 0 5 15 20

0

0.6

0.8

1

Figure 79. Nondimensional buckling loads of infinitely long cylinders with I = 0.25 and I = 4. 

0.2

0.4

Nondimensional parameter, 

β = 3

μ = A11A22 − A12

2

− 2A12A66

2A66 A11A22

π2pcr

4 3 ρ

10

β = 0

2

1

Nondimensional

buckling load,

ρ = R

h 1 − νm

2 A11A22 h

4

144D11D22

1

4

β = D12 + 2D66

D11D22

I = A11D22

D11A22

1

2

pcr = Nx

cr R2

π2

D11D22
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

An Approximate Solution and Master Curves for Buckling of 

Symmetrically Laminated Composite Cylinders 

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Nemeth, Michael P.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20250

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 39

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

Nondimensional buckling equations for symmetrically laminated cylinders with negligible shell-wall anisotropies and subjected to uniform 

axial compression are presented. These equations are solved exactly for the practical case of simply supported ends and nondimensional 

quantities are used to characterize the buckling behavior. Numerous generic plots of nondimensional buckling load versus a 

stiffness-weighted length-to-radius ratio are presented for various combinations of other parameters. These plots show how the parameter 

values affect the distribution and size of the festoons forming each response curve and how they affect the attenuation of each response 

curve to the corresponding solution for an infinitely long cylinder. Approximate formulas for the nondimensional buckling load are also 

derived that give the buckling response of an infinitely long cylinder in terms of the nondimensional parameters. A relatively small number 

of “master curves” are presented that give a nondimensional measure of the buckling load of an infinitely long cylinder as a function of the 

orthotropy and wall inhomogeneity parameters. 

15. SUBJECT TERMS

Buckling; Laminated composites; Shells

18. NUMBER

 OF 

 PAGES

65

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

869021.04.07.01.13

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TM-2013-217999

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and 

Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person 

shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 05 - 2013