NASA Technical Memorandum 4643

Predictions of Thermal Buckling 

Strengths of Hypersonic Aircraft 

Sandwich Panels Using Minimum 

Potential Energy and Finite 

Element Methods

William L. Ko

May 1995
National Aeronautics and 

Space Administration

Office of Management

Scientific and Technical 

Information Program

1995

NASA Technical Memorandum 4643

Predictions of Thermal Buckling 

Strengths of Hypersonic Aircraft 

Sandwich Panels Using Minimum 

Potential Energy and Finite 

Element Methods

William L. Ko

Dryden Flight Research Center

Edwards, California
iii

CONTENTS

PAGE

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

DESCRIPTION OF PROBLEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

RAYLEIGH–RITZ THERMAL BUCKLING ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Panel Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Deformation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Thermal Buckling Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

FINITE ELEMENT THERMAL BUCKLING ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Finite Element Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Eigenvalue Extractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

NUMERICAL EXAMPLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Eigenvalue Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Buckling Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

APPENDIX A—COEFFICIENTS OF CHARACTERISTIC EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

APPENDIX B—BUCKLING EQUATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

TABLES

1. Sizes of three finite element models A, B, and C (c.f., fig. 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Buckling temperatures of sandwich panels calculated using minimum energy

and finite element models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

FIGURES

1. Rectangular honeycomb-core sandwich panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2. Rectangular sandwich panel under thermal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. Four types of edge conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4. Edge distortions of sandwich panel under different edge conditions;

no edge distortions for 4C condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
iv

5. Quarter-panel and half-panel regions for finite element models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6. Three finite element models generated for sandwich panels of different aspect ratios . . . . . . . . . . . . . . . . 19

7. Modeling of sandwich panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8. Simulation of different edge conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

9. Convergence curves of eigenvalue iterations; 4C condition; b/a = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10. Increase of processor time with number of eigenvalue iterations;

ELXSI 6400 computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

11. Buckled shapes of b/a = 1 sandwich panel under different edge conditions;

half panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

12. Buckled shapes of b/a = 2 sandwich panel under different edge conditions;

full panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13. Buckled shapes of b/a = 3 sandwich panel under different edge conditions;

half panel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

14. Buckling temperature curves for titanium sandwich panels under different

edge conditions; a = constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
ABSTRACT

Thermal buckling characteristics of hypersonic aircraft sandwich panels of various aspect ratios were investigated. The panel is fastened at its four edges to the substructures under four different edge conditions and is subjected to uniform temperature loading. Minimum potential energy theory and finite element methods were used to

calculate the panel buckling temperatures. The two methods gave fairly close buckling temperatures. However, the

finite element method gave slightly lower buckling temperatures than those given by the minimum potential energy

theory. The reasons for this slight discrepancy in eigensolutions are discussed in detail. In addition, the effect of

eigenshifting on the eigenvalue convergence rate is discussed.

NOMENCLATURE

Fourier coefficients of trial function for w, in.

extensional stiffnesses of sandwich panel, 

 lb/in

a length of sandwich panel, in.

coefficients of characteristic equations

b width of sandwich panel, in.

Fourier coefficients of trial function for in/in

c shift factor in eigenvalue extractions

Fourier coefficients of trial function for in/in

bending stiffnesses of sandwich panel, 

in–lb

transverse shear stiffnesses in xz-, yz-planes, lb/in

flexural stiffness parameter, in–lb

relative displacements of actual face sheets in x-, y-, and z-directions, in.

relative displacements of finite element face sheets in x-, y-, and z-directions, in.

effective Young’s moduli of honeycomb core, lb/in2

Young’s moduli of face sheets, lb/in2

effective Young’s modulus of finite element sandwich core in z-direction, lb/in2

effective shear moduli of honeycomb core, lb/in2

Amn Akl ,

Aij A11

2t

sEx

1 νxyνyx – ------------------------- A12 , 2t

sνyxEx

1 νxyνyx – = = -------------------------

A21

2t

sνxyEy

1 νxy – νyx

----------------------- A22 , 2t

sEy

1 νxyνyx – ------------------------- A66 , 2t

sGxy == = ,

amnkl

ij

Bmn Bkl , γ xz,

Cmn Ckl , γ yz,

Dij D11

2ExIs

1 νxyνyx – ------------------------- D12 , 2νyxExIs

1 νxyνyx – = = -------------------------

D21

2νxyEyIs

1 νxyνyx – ------------------------- D22 , 2EyIs1 νxyνyx – ------------------------- D66 , 2GxyIs === ,

DQx DQy , DQx hcGcxz DQy , hcGcyz = = ,

D* D11D22,

d1 d2 d3 , ,

d'

1 d'2 d'3 , ,

Ecx Ecy Ecz , ,

Ex Ey ,

E'

cz

Gcxy Gcxz Gcyz , ,
2

effective transverse shear moduli of finite element sandwich core, lb/in2

shear modulus of face sheets, lb/in2

h depth of sandwich panel = distance between middle planes of two face sheets, in.

depth of honeycomb core, in.

i index, 1, 2, 3, ...

moment of inertia, per unit width, of a face sheet taken with respect to horizontal centroidal 

axis of the sandwich panel, in4/in

j index, 1, 2, 3, ...

JLOC joint location (used in figures and tables)

K system stiffness matrix

k index, 1, 2, 3, ...

system initial stress stiffness matrix corresponding to a particular applied force condition

compressive buckling load factors in x- and y-directions, 

shear buckling load factor, 

l index, 1, 2, 3, ...

m number of buckle half waves in x-direction

bending moment intensities, (in–lb)/in

n number of buckle half waves in y-direction

thermal forces, lb/in

SPAR structural performance and resizing finite element computer program

thickness of sandwich face sheets, in.

displacement components in x-, y-, and z-directions, respectively, in.

X displacement vector

rectangular Cartesian coordinates

global x- and y-coordinates for finite element model

coefficients of thermal expansion, in/in-°F

coefficients of thermal shear distortion, in/in-°F

transverse shear strain in xz- and yz-planes, in/in

temperature rise, °F

assumed buckling temperature, °F

G'

cxz G'cyz ,

Gxy

hc hc h ts = – ,

Is

Is

1

4

--t

sh

2 1

12-----t

s

3 = + ,

Kg

kx, ky kx

Nx

T

a

2

π

2

D* ------------- ky , Ny

T

a

2

π

2

D* = = -------------

kxy kxy

Nxy

T

a

2

π

2

D* = ---------------

Mx My ,

Nx

T NyT NxyT , ,

t

s

u, v, w

x, y, z

xG yG ,

αx αy ,

αxy

γ xz γ yz ,

∆T

∆Ta
3

critical buckling temperature, °F

numerical coefficient of in 

numerical factor in buckling equation, which changes with the edge condition

eigenvalue of i-th buckling mode

Poisson ratios of face sheets, also used for those of sandwich panel

Poisson ratios of honeycomb core

numerical coefficient of in 

specific weight of titanium honeycomb core, lb/in3

specific weight of titanium material, lb/in3

INTRODUCTION

Hypersonic aircraft structural panels are subjected not only to aerodynamic loading (mechanical loading), but

also to aerodynamic heating (thermal loading). These structural panels are usually called hot structural panels because they operate at elevated temperatures. For certain cases, the thermal load could be the primary load, and therefore, it could be a key factor in the design of the hot structures. When a monolithic hot structure is subjected to

uniform temperature field and is allowed to expand freely, no thermal stresses can be generated in the structural panel. When the temperature field is nonuniform, thermal stresses can build up in the panel even if it can expand freely.

In actual applications, the structural panels are attached to relatively cooler substructures (i.e., spars and ribs, both

of which act as heat sinks); the panels are, therefore, constrained from free expansion. These constraints will cause

thermal stresses to build up in the panels. The heating over an individual panel surface is usually relatively uniform;

however, the panel surface temperatures are seldom uniform over the entire panel surface because the panel edges

are attached to the heat sinks (i.e., relatively cooler substructures). The temperature rise over the panel surface will

then have a plateau in a large central region and will taper down near the cooler edges. That is, the temperature rise

profile over the panel surface will look like a truncated dome shape. High-intensity thermal loading could induce

(1) thermal buckling, (2) material degradation, (3) thermal creep, (4) thermal yielding, (5) thermal cracking after

cooling down, etc. Excess thermal deformation caused by thermal buckling could disturb the airflow field, creating

localized hot spots that could degrade the panel's structural performance.

When the structural panels are applied as the hypersonic aircraft wing skins, the aerodynamic loading during

hypersonic flight will cause the wing upper panels to be under combined spanwise compression (resulting from wing

bending) and shear (resulting from wing torsion). On the other hand, the wing lower skin panels will be subjected

to combined spanwise tension and shear loading. Under thermal loading, both the wing upper and lower skin panels

will be under mainly biaxial compression with certain localized shear. The thermal loading will increase the mechanical compressive stresses in the wing upper panels, and tend to reduce the mechanical compressive stresses in the

wing lower panels. Thus, the thermomechanical buckling characteristics of the hot structural panels are a critical

concern in the hypersonic aircraft wing structural panels.

Thermal buckling problems of single plates (continuous or laminated composites) were investigated by several

authors in recent years (refs. 1–6), and thermomechanical buckling characteristics of the hot structural sandwich

panels were analyzed extensively by Ko and Jackson (refs. 7–12). Using the minimum potential energy method, Ko

and Jackson developed thermomechanical buckling equations for orthotropic rectangular sandwich panels subjected

to combined mechanical compressive and shear loading, or under thermal loading (refs. 11–12).

∆Tcr

ζ Ny

T

amnkl

11

η

λi

νxy νyz ,

νcxy νcyz νcxz , ,

ξ Nx

T

amnkl

11

ρHc

ρTi
4

This report investigates the thermal buckling characteristics of uniformly heated rectangular titanium sandwich

panels of different aspect ratios supported under four different edge conditions. The thermal buckling loads (or buckling temperatures) will be calculated using Ko–Jackson thermal buckling equations (refs. 11–12) and the finite element method for the purpose of validating the Ko-Jackson theory. The thermal buckling solutions obtained from the

two methods will be compared and their discrepancies discussed.

DESCRIPTION OF PROBLEM

Figure 1 shows the geometry of the hypersonic aircraft rectangular sandwich panel. The sandwich panel has

length a, width b, and depth h. It is fabricated with titanium face sheets of the same thicknesses joined together

through a titanium honeycomb core of depth using the enhanced diffusion bonding process (ref. 13). Figure 2

shows the combined thermal forces acting on the middle plane of the sandwich panel.

For the thermal buckling analysis, the panel will be subjected to uniform temperature field under four different

edge conditions shown in figure 3. The edges in the x- and y-directions are defined, respectively, as sides and ends.

The minimum potential energy thermal buckling theory developed by Ko and Jackson (refs. 11–12) and the finite

element method will be used to calculate panel buckling temperatures, and the eigensolutions based on the two

methods will be compared.

RAYLEIGH–RITZ THERMAL BUCKLING ANALYSIS

In the thermal buckling analysis of sandwich panels conducted by Ko and Jackson (refs. 11–12), the extensional

and bending stiffnesses are provided by the two face sheets, and the panel transverse shear stiffness is provided by

the sandwich core only.

Panel Boundary Conditions

 The four sets of boundary conditions used in the Ko and Jackson thermal buckling analysis (refs. 11–12) are

given below.

Case 1: Four edges simply supported (4S condition)

(1)

(2)

Case 2: Four edges clamped (4C condition)

(3)

(4)

Case 3: Two sides clamped, two ends simply supported (2C2S condition)

(5)

(6)

t

s,

hc

x 0, a: uvwMx γ yz = == = = = 0

y 0, b: uvwMy γ xz = == = = = 0

x 0, a: uvw

∂w

∂x

------ γ xz γ yz = == = = = = 0

y 0, b: uvw

∂w

∂y

------ γ xz γ yz = == = = = = 0

x 0, a: uvwMx γ yz = == = = = 0

y 0, b: uvw

∂w

∂y

------ γ xz γ yz = == = = = = 0
5

Case 4: Two sides simply supported, two ends clamped (2S2C condition)

(7)

(8)

For anisotropic sandwich panels, cases 3 and 4 will give different thermal buckling solutions.

Deformation Functions

For satisfying the different sets of boundary conditions (1) through (8), the associated deformation functions

 chosen by Ko and Jackson (refs. 11–12) in the thermal buckling analysis of the sandwich panels have

the following forms:

Case 1: 4S condition

(9)

(10)

(11)

Case 2: 4C condition

(12)

(13)

(14)

x 0, a: uvw

∂w

∂x

------ γ xz γ yz = == = = = = 0

y 0, b: uvwMy γ xz = == = = = 0

w, γ xz, γ yz { }

wxy ( ) , Amn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= ∑

γ xz( ) x y, Bmn

mπx

a

---------- nπy

b cos sin---------

n = 1

∞

∑ m = 1

∞

= ∑

γ yz( ) x y, Cmn

mπx

a

---------- nπy

b sin cos---------

n = 1

∞

∑ m = 1

∞

= ∑

wxy ( ) , πx

a

----- πy

b

----- Amn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= sin sin ∑

γ xz( ) x y, πx

a

----- πy

b

----- Bmn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= cos sin ∑

πx

a

----- πy

b

----- mBmn

mπx

a

---------- nπy

b cos sin---------

n = 1

∞

∑ m = 1

∞

+ sin sin ∑

γ yz( ) x y, πx

a

----- πy

b

----- Cmn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= sin cos ∑

πx

a

----- πy

b

----- nCmn

mπx

a

---------- nπy

b sin cos---------

n = 1

∞

∑ m = 1

∞

+ sin sin ∑
6

Case 3: 2C2S condition

(15)

(16)

(17)

Case 4: 2S2C condition

(18)

(19)

(20)

The choice of these four sets of deformation functions, each of which satisfies the associated boundary conditions (1) through (8), is for the mathematical amenability of the eigenvalue solutions. As shown in figure 4, the zero

transverse shear distortions (i.e., or ) at the panel edges cannot be enforced simultaneously in the

actual panel deformations, except for the 4C condition.

Thermal Buckling Equations

The thermal buckling equations developed by Ko and Jackson (refs. 11–12) for uniformly heated and constrained rectangular orthotropic sandwich panels using the Rayleigh–Ritz method, written in terms of temperature

rise for each set of integral values {m, n} (or mode shape), have the following form

(21)

wxy ( ) , πy

b

----- Amn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= sin ∑

γ xz( ) x y, πy

b

----- Bmn

mπx

a

---------- nπy

b cos sin---------

n = 1

∞

∑ m = 1

∞

= sin ∑

γ yz( ) x y, πy

b

----- Cmn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= cos ∑

πy

b

----- nCmnsinmπx

a

---------- nπy

b cos---------

n = 1

∞

∑ m = 1

∞

+ sin ∑

wxy ( ) , πx

a

----- Amn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= sin ∑

γ xz( ) x y, πx

a

----- Bmn

mπx

a

---------- nπy

b sin sin---------

n = 1

∞

∑ m = 1

∞

= cos ∑

πx

a

----- mBmn

mπx

a

cos---------- nπy

b sin---------

n = 1

∞

∑ m = 1

∞

+ sin ∑

γ yz( ) x y, πx

a

sin----- Cmn

mπx

a

---------- nπy

b sin cos---------

n = 1

∞

∑ m = 1

∞

= ∑

γ xz = 0 γ yz = 0

∆T

Mmnkl

∆T --------------- Pmnkl δmnkl + + Akl

l

∑

k

∑ = 0
7

The bending-stiffness parameter and the extensional stiffness parameter in equation (21) are

defined as

(22)

(23)

The characteristic coefficients (i, j = 1, 2, 3) appearing in equation (22) are defined in appendix A, and

in equation (22) is the first part of containing no thermal loading terms (i.e., terms containing

). The parameters in equation (23) are, respectively, the numerical coefficients of the load factors

 contained in The values of change with the indicial and edge conditions (ref. 12).

The numerical parameter appearing in equations (22) and (23), and the special delta function 

appearing in equation (21), are defined for different edge conditions as (ref. 12)

Case 1: 4S condition

(24)

Case 2: 4C condition

(25)

Case 3: 2C2S condition

(26)

Case 4: 2S2C condition

(27)

Mmnkl Pmnkl

Mmnkl

ab

ηA66αxy

--------------------- amnkl

–11 amnkl

12

amnkl

23

amnkl

31

amnkl

21

amnkl

33 ( ) – amnkl13

amnkl

21

amnkl

32

amnkl

22

amnkl

31

+ ( ) –

amnkl

22

amnkl

33

amnkl

23

amnkl

32 –

≡ + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- 





classical

thin plate

theory term transverse shear effect terms

Pmnkl

ab

ηA66αxy

--------------------- ξ( ) m k, A11αx A12αy ( )ζ + ( ) n l , A21αx A22αy ≡ [ ] + ( ) +

amnkl

ij

amnkl

–11

amnkl

11 ,

kx, ky { } ξζ { } ,

kx, ky { } amnkl

11 . { } ξ ζ,

η, δmnkl,

η 32 =

δmnkl

mnkl

m

2 k2 ( ) – n2 l2 ( ) –

= == --------------------------------------------; odd m k ± , n l ± odd

η ( ) 16 3

2 = -------------

δmnkl

mnkl m2 k

2 [ ] + – 2 n2 l2 [ ] + – 2

m

2 k2 ( ) – n2 l2 ( ) – ( ) m k +2 [ ] – 4 ( ) m k – 2 [ ] – 4 ( ) n l +2 [ ] – 4 ( ) n l – 2 [ ] – 4

= ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ; odd, m k ± = n l ± = odd

η 83 =

δmnkl

mnkl 2 n

2 l2 [ ] – ( ) +

m

2 k2 ( ) – n2 l2 ( ) – ( ) n l +

2 [ ] – 4 ( ) n l – 2 [ ] – 4

= = -------------------------------------------------------------------------------------------------------------; odd, m k ± n l ± = odd

η 83 =

δmnkl

mnkl 2 m

2 k2 [ ] – ( ) +

m

2 k2 ( ) – n2 l2 ( ) – ( ) m k +

2 [ ] – 4 ( ) m k – 2 [ ] – 4

= = ------------------------------------------------------------------------------------------------------------------; odd, m k ± n l ± = odd
8

In the thermal buckling equation (21), both and terms contain temperature dependent material

properties. Thus, in the eigenvalue solution process using equation (21), one has to assume a buckling temperature

 and use the material properties corresponding to as inputs to calculate the buckling temperature 

This material property iteration process must continue until approaches Thus, in the thermal buckling,

the eigenvalue solution process is a multi-step process. However, in the mechanical buckling, only one step eigenvalue solution process is required.

When the coefficient of thermal shear distortion is zero (i.e., ), the buckling equation (21) takes on

the form

(28)

where and are defined as

(29)

(30)

When equation (28) is reduced to the isotropic case with no transverse shear effects, the buckling temperature

 will be independent of the material’s modulus of elasticity (ref. 14).

The characteristic equation (21) forms a system of infinite number of simultaneous homogeneous equations,

each of which is associated with each indicial combination of {m, n}. Those simultaneous equations, generated from

equation (21), have the following characteristics. The first two terms of equation (21) are

nonzero only for the indicial conditions {m = k or m – k = 2} and {n = l or n – l = 2} based on the indicial constraints

for (appendix A). Thus, if (m ± n) is even, then (k ± l) is also even, and if (m ± n) is odd, then (k ± l) is also

odd. The special delta function in the third term of equation (21) is nonzero only when (m ± k) is odd and

(n ± l) is odd. It follows that (m ± k) ± (n ± l) = (m ± n) ± (k ± l) = even. This implies that if (m ± n) is even, then

(k ± l) is also even, and if (m ± n) is odd, so also is (k ± l). Because of these indicial characteristics, there is no coupling between the even case (i.e., symmetric buckling) and the odd case (i.e., antisymmetric buckling). Thus, the

simultaneous equations generated from equation (21) may be divided into two groups that are independent of each

other—one group for which (m ± n) is even, and the other group for which (m ± n) is odd (refs. 11–12).

For the deflection coefficients to have nontrivial solutions for given aspect ratio b/a, the determinant of coefficients of unknown of the simultaneous homogeneous equations written out from equation (21) must vanish.

The largest eigenvalue thus obtained will give the lowest buckling temperature The determinants of

the coefficients of the simultaneous equations written out from equation (21) up to order 12 are given in appendix B

for the cases m ± n = even (symmetric buckling) and m ± n = odd (antisymmetric buckling) for different edge conditions. The determinants of order 12 were found to give sufficiently accurate eigensolutions and, therefore, the determinants were truncated at order 12 in the present eigenvalue extractions. In appendix B one notices that for the

4S edge condition only, terms form the diagonal terms of the determinants, and the nonzero

off-diagonal terms consist only of numerical values given by However, for the rest of edge conditions,

 not only appear in the diagonal terms, but also in the off-diagonal terms (mixed with the

numerical terms associated with ).

Mmnkl Pmnkl

∆Ta ∆Ta ∆Tcr .

∆Ta ∆Tcr .

αxy = 0

Mmnkl

∆T --------------- + Pmnkl

l

∑

k

∑ Akl = 0

Mmnkl Pmnkl

Mmnkl amnkl

11

 + 

amnkl

12

amnkl

23

amnkl

31

amnkl

21

amnkl

33 ( ) – amnkl13

amnkl

21

amnkl

32

amnkl

22

amnkl

31 – ( ) +

amnkl

22

amnkl

33

amnkl

23

amnkl

32 –

≡ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------

Pmnkl ξ( ) m k, A11αx A12αy ( )ζ + ( ) n l , A21αx A22αy ≡ [ ] + ( ) +

∆Tcr

Mmnkl ∆T Pmnkl ( ) ⁄ +

amnkl

ij

δmnkl

Akl

Akl

1/∆T ∆Tcr .

Mmnkl ∆T Pmnkl ( ) ⁄ +

δmnkl.

Mmnkl ∆T Pmnkl ( ) ⁄ +

δmnkl
9

FINITE ELEMENT THERMAL BUCKLING ANALYSIS

The structural performance and resizing (SPAR) finite element computer program (ref. 15) was used in the finite

element thermal buckling analysis of the sandwich panels.

Finite Element Modeling

To gather thermal buckling data of sandwich panels having a wide range of aspect ratios b/a, three basic finite

element models of different b/a were set up so that each model would cover certain limited range of b/a. Changing

b/a of each model was accomplished by simply modifying length b and keeping length a constant. Sometimes more

elements had to be added to an overstretched model to maintain proper element aspect ratios. An overstretched

model without additional elements could result in local buckling of slender element cells rather than global buckling

of the panels (i.e., local buckling temperature is less than the global buckling temperature). For low b/a (<1.8) and

high b/a (>2.9) aspect ratio panels (figs. 5(a) and 5(c)) for which the lowest buckling mode were symmetric, only

the quarter panels were modeled. The SPAR constraint commands SYMMETRY PLANE = 1 and SYMMETRY

PLANE = 2 (ref. 15) were then used to generate the full panels. For moderate b/a (1.8 ≤ b/a ≤ 2.9) aspect ratio panels (fig. 5(b)) for which the lowest buckling mode could be either symmetric or antisymmetric, half panels were

modeled. The SPAR constraints command SYMMETRY PLANE = 1 was then used to generate the full panels. For

purely antisymmetric buckling, one can model only a quarter panel and use the constraint commands SYMMETRY

PLANE = 1 and ANTISYMMETRY PLANE = 2 to generate the whole panel. However, such quarter-panel models

were not used in gathering the buckling data because they consistently gave somewhat higher buckling temperatures

than those given by the half-panel models. Figure 6 shows the three basic finite element models set up for the sandwich panels. Both models A and C are the quarter-panel models, but model B is a half-panel model. From these basic models, several modified models (not model shown) were also set up for handling certain aspect ratios and edge

conditions.

Each face sheet of the sandwich panel was modeled with one layer of E43 elements (quadrilateral combined

membrane and bending elements) and the sandwich core with one layer of S81 elements (hexahedron (or brick) elements), which connect the upper and the lower face-sheet elements E43. Because the joint locations of those face

sheet elements E43 are located in the middle planes of the respective face sheets, the finite element core depth will

then be h instead of the actual depth (fig. 7). Thus, to simulate the actual relative displacements (or maintain same

stiffness) between the two face sheets in the sandwich thickness direction and the x- and y-directions (i.e., 

and and fig. 7), the thickness elastic modulus and the transverse shear moduli and

 of the S81 elements had to be increased slightly according to the following relationships:

(31)

(32)

(33)

One can also model the sandwich core with one layer of S81 elements having the exact depth and then connect the gaps between E43 joints and S81 joints with rigid elements. However, this alternative modeling method requires twice as many total joint locations, and therefore, it was not used.

hc

d3 d'

3 =

d1 d'

1 = d2 d'2 = , E'cz, G'cxz

G'

cyz

E'

cz Ecz

h

hc

= ----

G'

cxz Gcxz

h

hc

= ----

G'

cyz Gcyz

h

hc

= ----

hc,
10

For simply supported edges, free rotation and free transverse shear deformation must be allowed (fig. 8(a)). To

simulate this type of edge, pin-ended rigid rods were attached to the panel edge for connecting the two face sheets

(sandwich core carries no extensional and bending stiffnesses), and then the midpoints of the rigid rods were pinjointed to fix points lying in the sandwich middle plane (fig. 8(a)). Each pin-ended rigid rod was modeled with two

identical E22 elements (beam element for which the intrinsic stiffness matrix is given). To simulate the rigidity of

the rod, extensional and transverse shear stiffnesses of the E22 elements were made very large. The pin-joint condition at the face sheet edges was simulated by assigning zero values to the rotational spring constants in the stiffness

matrix for the E22 elements. The pin-joint condition at the middle-plane fixed points was simulated by relaxing the

three rotational constraints. Two methods were used to connect the ends of the E22 elements to the middle plane

fixed points. In the first method (center drawing of fig. 8(a)), the first joint of each E22 element was connected to

the associated joint of E43 element and its end point to the panel middle-plane fixed point. In the second method

(right-hand drawing of fig. 8(a)), the ends of the two identical E22 elements, whose first joints were connected to

the upper and the lower face sheets, were connected together to the middle-plane fixed point through E25 element

(zero length element used to elastically connect geometrically coincidental joints). The stiffnesses of the E25 were

made so large that the two E22 elements, connected together by the E25 element, will behave like one rigid rod.

These two types of simply supported edge simulations were found to give identical thermal buckling solutions. In

most of the buckling data gathering, the first edge simulation was used because it required less joint locations.

For the clamped edge (fig. 8(b)), the edges of the two face sheets were built into fixed vertical walls to generate

the desired constraints of zero slope, zero in-plane displacements, and zero transverse shear deformations.

Table 1 shows the sizes of the three finite element models set up for the sandwich panels of different b/a.

To make sure that the above finite element models gave accurate eigensolutions, the sandwich cores of model A

was modified to two layers of S81 elements to investigate the eigensolution convergencies. It turned out that both

the basic and the modified models gave practically identical eigensolutions. Because the modified model required

about three times longer computational time, it was not used in the actual buckling data gatherings.

Eigenvalue Extractions

The eigenvalue equation for buckling problems is of the form

(34)

where

Table 1. Sizes of three finite element models A, B, and C (c.f., fig. 6).

Feature Model A Model B Model C

JLOC 2178 4850 4050

E43 2048 4608 3840

S81 1024 2304 1920

= system initial stress stiffness matrix (or differential stiffness matrix), corresponding to particular 

applied force condition (e.g., thermal loading), and in general a function of

K = system stiffness matrix

X = displacement vector

= eigenvalues for various buckling modes

λKgX KX + 0 =

Kg

X

λi
11

The eigenvalues (i = 1,2,3,...) are the load factors by which the static load (mechanical or thermal) must be

multiplied to produce buckling loads corresponding to various buckling modes. Namely, if the applied temperature

load is then the buckling temperature for the i-th buckling mode is obtained from

(35)

If it is desired to find eigenvalues in the neighborhood of c, then the following shifted eigenvalue equation may be

used.

(36)

In the eigenvalue extractions, the SPAR program uses an iterative process consisting of a Stodola matrix iteration

procedure, followed by a Rayleigh–Ritz procedure, and then followed by a second Stodola procedure. This process

results in successively refined approximations of i eigenvectors associated with the i eigenvalues of equation (34)

closest to zero. Reference 15 describes the details of this process.

NUMERICAL EXAMPLES

The hot structural sandwich panels analyzed were fabricated with titanium face sheets and titanium sandwich

core having the following geometrical and material properties.

Geometry:

Material properties:

a = 24 in. 

b = varying

h = 0.75 in.

=

= 0.06 in.

Face sheets

70 °F 900 °F*

0.31 0.31

0 0

0.16 0.16

*Mach 15 flight temperature.

λi

∆T, ∆Tcr

∆Tcr λi = ∆T

( ) λ – c KgX Kc Kg + 0 ( ) + X =

hc h ts – 0.69 in. =

t

s

Ex Ey lb/in2 = , 16 106 × 13.1 106 ×

Gxy lb/in2 , 6.2 106 × 5.0 106 ×

νxy νyx =

αx αy = , in/in-°F 4.85 10–6 × 5.35 10–6 ×

αxy, in/in-°F

ρTi lb/in3 ,
12

The main objective of the present report is to study the general trend of thermal buckling characteristics of sandwich panels under different edge conditions, and to validate Ko-Jackson theory (ref. 12). Because of the lack of material property data at high temperatures, the material iteration process was not performed, and the face sheet

properties at 900 °F and sandwich core properties at 600 °F were used in the buckling temperature calculations.

RESULTS

Eigenvalue Iterations

In finite element eigenvalue extractions, the maximum number of iterations was set to be 100. For most cases

(with or without eigenshifting), however, the eigenvalues converged well below 100 iterations based on the convergence criterion (| λi – λi–1 | / λi) < 10–4. Figure 9 compares the convergence curves of eigenvalue iterations with and

without shifting for the square panel (b/a = 1) under the 4C condition. With shifting, the number of iterations could

be reduced from 14 to 9 iterations. For certain cases, the number of eigenvalue iterations with shifting turned out to

be very close or even identical to that without shifting. For certain problems, such as the thermocryogenic buckling

of cryogenic tanks (ref. 16), eigenshifting could drastically reduce the number of eigenvalue iterations (i.e., reduction in computer time). For the present sandwich panel buckling problem, however, the reduction in the number of

eigenvalue iterations through the eigenshifting turned out to be relatively small or negligible.

In most of the thermal buckling data gathering, the eigenshifting method was used. The shifting factors used

were near the values of the buckling temperatures predicted from the energy theory. In figure 10 the ELXSI 6400

computer processor times are plotted as functions of the number of eigenvalue iterations for the four edge conditions.

The processor time per iteration (i.e., slope of the data fitting line) for the 4S, 4C, 2C2S, and 2S2C conditions are,

respectively, 2.45, 1.88, 2.48, and 2.06 minutes.

Honeycomb core (properties at 600 °F)

=

=

=

=

=

=

=

=

=

=

=

=

Ecx 2.7778 104 lb/in2 ×

Ecy 2.7778 104 lb/in2 ×

Ecz 2.7778 105 lb/in2 ×

Gcxy 0.00613 lb/in2

Gcyz 0.81967 105 lb/in2 ×

Gcxz 1.81 105 lb/in2 ×

νcxy 0.658 10–2 ×

νcyz 0.643 10–6 ×

νcxz 0.643 10–6 ×

αx αy = 5.37 10–6 × in/in-°F

αxy 0 in/in-°F

ρHc 3.674 10–3 × lb/in3
13

Buckling Temperatures

Figures 11, 12, and 13, respectively, show the lowest-mode buckling shapes of sandwich panels of aspect ratios

b/a = 1, 2, and 3 under the four different edge conditions. Notice that at the simply supported edges of the 4S case

and at the clamped edges of the 2C2S and 2S2C cases, the transverse shear deformations cannot be zero (fig. 4). At

higher b/a (figs. 12 and 13), the 4C and 2S2C cases required finer element models for obtaining global buckling. The

square panel (b/a = 1, fig. 11), under all the four different edge conditions, buckled symmetrically with {m = 1, n =

1} buckling mode. For the b/a = 2 rectangular panel (fig. 12), the 4S, 4C and 2C2S conditions still induced symmetrical buckling mode of {m = 1, n = 1}. However, under the 2S2C condition, the rectangular panel buckled antisymmetrically under {m = 1, n = 2} buckling mode. For the slender panel of b/a = 3 (fig. 13), both 4S and 2C2S

conditions continued to induce symmetrical buckling mode of {m = 1, n = 1}. However, under the 4C and 2S2C

conditions, the multiple symmetrical buckling mode of {m = 1, n = 3} turned out to be the lowest buckling mode. In

figure 14 the thermal buckling temperatures calculated using the minimum energy method (solid curves) and the finite element method (circular symbols) are plotted as functions of panel aspect ratio b/a for the four cases of edge

conditions. Notice that for the high values of b/a, the thermal buckling solutions for the 2C2S and 2S2C cases approach those of the 4S and 4C cases, respectively, because the constraint effects of the shorter edges of the slender

panels diminish. The buckling solutions obtained from the two methods compare fairly well. The average solution

difference between the two methods are 3.87%, 1.62%, 2.04% and 2.71% respectively for the 4S, 4C, 2C2S and

2S2C cases. The finite element method tends to give slightly lower buckling temperatures than those given by the

minimum-energy method. The reason could be the following: (1) the finite element method allows deformations in

the sandwich thickness direction, which the minimum energy theory ignores, (2) the theoretical edge conditions of

zero and (eqs. (1), (2), (5)–(8)) cannot be enforced properly in the finite element edge constraints for the

4S, 2C2S, and 2S2C conditions (fig 4), and (3) the finite element modeling assumptions. For the 4C case only, all

the theoretical edge conditions could be enforced in the finite element edge constraints. For 4S, 4C and 2C2S cases,

the discrepancy of the eignsolutions between the two methods is larger at the low values of b/a, and gradually diminishes at high values of b/a. This solution discrepancy is minimum for the 4C case, and maximum for the 4S case

(because both and cannot be zero at the edges of the finite element modes). For the 2S2C case, the solution

discrepancy is almost unaffected by the change of b/a. Table 2 lists the buckling temperatures calculated from

the minimum energy and the finite element methods. 

Table 2. Buckling temperatures of sandwich panels calculated using minimum energy and finite

element models.

°F

4S 4C 2C2S 2S2C

b/a

Energy

theory

Finite

element

Energy

theory

Finite

element

Energy

theory

Finite

element

Energy

theory

Finite

element

0.5 1297 1207 2541 2498 2424 2366 1512 1464

0.6 1051 970 2169 2126 1995 1938 1334 1286

0.7 885 815 1889 1847 1654 1603 1232 1187

0.8 769 710 1683 1645 1387 1343 1175 1132

0.9 682 637 1535 1499 1181 1143 1144 1105

1.0 622 583 1428 1396 1021 988 1128 1093

1.1 575 547 1352 1322 897 866 1122 1090

1.2 538 513 1297 1271 799 774 1121 1092

1.4 486 465 1232 1209 662 640 1126 1102

1.6 451 436 1199 1179 573 560 1136 1116

γ xz γ yz

γ xz γ yz

∆Tcr

∆Tcr ,
14

CONCLUSIONS

Thermal buckling characteristics of hypersonic aircraft honeycomb-core sandwich panels subjected to uniform

temperature loading were analyzed using minimum energy theory and finite element methods. The thermal buckling

curves were generated for titanium sandwich panels of various aspect ratios. The two methods predicted very close

buckling temperatures, and thus, the Ko-Jackson theory was validated. The finite element method tended to give

slightly lower buckling temperatures than those given by the minimum energy theory. The slight discrepancies in

the eigensolutions between the two methods could be attributed to the following:

1. The minimum energy theory does not consider deformations in the panel thickness direction, whereas the

finite element method does. 

2. The theoretical zero transverse shear deformations at the panel edges cannot be enforced in the finite

element models with simply supported edges and cannot be enforced simultaneously in the finite elements

models with mixed simply supported and clamped edges.

3. Assumptions made in finite element modeling.

The discrepancy of the eigensolutions between the minimum energy theory and the finite element method is the

largest for the simply supported edge condition, because the zero transverse shear deformations at the panel edges

cannot be constrained in the finite element models. This solution discrepancy is larger at low values of b/a and gradually decreases as b/a increases. For the sandwich panels the eigenshifting has small effect on the improvement of

the eigenvalue convergence rate.

The author gratefully acknowledges the contribution of the late Raymond H. Jackson, NASA mathematician, in

setting up computer programs for the eigenvalue extractions.

Dryden Flight Research Center

National Aeronautics and Space Administration

Edwards, California, June 14, 1994

Table 2. Concluded.

°F

4S 4C 2C2S 2S2C

b/a

Energy

theory

Finite

element

Energy

theory

Finite

element

Energy

theory

Finite

element

Energy

theory

Finite

element

1.8 427 416 1183 1165 514 504 1144 1130

2.0 409 403 1175 1160 473 471 1128 1096

2.2 396 391 1172 1158 444 439 1122 1092

2.4 386 382 1171 1156 423 419 1121 1094

2.6 379 376 1164 1155 407 404 1122 1099

2.8 372 370 1156 1151 395 393 1126 1100

3.0 367 366 1150 1145 386 384 1128 1095

4.0 353 348 1140 1121 360 360 1124 1100

∆Tcr ,
15

940387

Figure 1. Rectangular honeycomb-core sandwich panel.

940455

Figure 2. Rectangular sandwich panel under thermal loading.
16

940463 940464

940465 940466

Figure 3. Four types of edge conditions.

(a) Four edges simply supported (4S condition). (b) Four edges clamped (4C condition).

(c) Two sides clamped, two ends simply supported 

(2C2S condition).

(d) Two sides simply supported, two ends clamped 

(2S2C condition).
17

940388

(a) 4S condition.

940389

(b) 4C condition.

940390

(c) 2C2S condition.

940391

(d) 2S2C condition.

Figure 4. Edge distortions of sandwich panel under different edge conditions; no edge distortions for 4C condition.
18

940460

(a) Symmetric buckling (m = 1, n = 1).

940461

(b) Antisymmetric buckling (m = 1, n = 2).

940462

(c) Symmetric buckling (m = 1, n = 3).

Figure 5. Quarter-panel and half-panel regions for finite element models. 
19

940448

(a) Model A—Quarter-panel model for {m = 1, n = 1}.

940449

(b) Model B—Half-panel model for {m = 1, n = 2}.

940450

(c) Model C—Quarter-panel model for {m = 1, n = 3}.

Figure 6. Three finite element models generated for sandwich panels of different aspect ratios.
20

940392

Figure 7. Modeling of sandwich panel.
21

940393

(a) Simply supported edge.

940394

(b) Clamped edge.

Figure 8. Simulation of different edge conditions. 
22

940397

Figure 9. Convergence curves of eigenvalue iterations; 4C condition; b/a = 1.

940398

Figure 10. Increase of processor time with number of eigenvalue iterations; ELXSI 6400 computer.
23

940451

(a) 4S condition {m = 1, n = 1}.

940452

(b) 4C condition {m = 1, n = 1}.

Figure 11. Buckled shapes of b/a = 1 sandwich panel under different edge conditions; half panel.
24

940453

(c) 2C2S condition {m = 1, n = 1}.

940454

(d) 2S2C condition {m = 1, n = 1}.

Figure 11. Concluded.
25

940456

(a) 4S condition {m = 1, n = 1}.

940457

(b) 4C condition {m = 1, n = 1}.

Figure 12. Buckled shapes of b/a = 2 sandwich panel under different edge conditions; full panel.
26

940458

(c) 2C2S condition {m = 1, n = 1}.

940459

(d) 2S2C condition {m = 1, n = 2}.

Figure 12. Concluded.
27

940403

(a) 4S condition {m = 1, n = 1}.

940404

(b) 4C condition {m = 1, n = 3}.

Figure 13. Buckled shapes of b/a = 3 sandwich panel under different edge conditions; half panel. 
28

940405

(c) 2C2S condition {m = 1, n = 1}.

940406

(d) 2S2C condition {m = 1, n = 3}.

Figure 13. Concluded.
29

940474

Figure 14. Buckling temperature curves for titanium sandwich panels under different edge conditions; a = constant.
30

APPENDIX A

COEFFICIENTS OF CHARACTERISTIC EQUATIONS

The characteristic coefficients appearing in equation (22) are defined in the following for different indicial and edge conditions (ref. 12).

Case 1: 4S condition

(1) m = k, n = l

(A-1)

(2) m ≠ k, n ≠ l

(A-2)

amnkl

ij

amnmn

11 D11

mπ

a

-------    4

2 D12 2D66 ( ) +

mπ

a

-------    2 nπ

b

------    2

D22

nπ

b

------    4

= + +

D* π

a

--    2

kx

mπ

a

-------    2

ky

nπ

b

------    2

– +

amnmn

12

amnmn

21 D11

mπ

a

-------    3

D12 2D66 ( ) +

mπ

a

-------     nπ

b

------    2

= = – +

amnmn

13

amnmn

31 D22

nπ

b

------    3

D12 2D66 ( ) +

mπ

a

-------    2 nπ

b

------     = = – +

amnmn

22 D11

mπ

a

-------    2

D66

nπ

b

------    2

DQx = + +

amnmn

23

amnmn

32 D12 D66 ( ) +

mπ

a

-------     nπ

b

------     = =

amnmn

33 D22

nπ

b

------    2

D66

mπ

a

-------    2

DQy = + +

amnkl

ij = 0
31

Case 2: 4C condition

(1) m = n = k = l = 1

(A-3)

(2) m = k = 1, n = l ≠ 1

(A-4)

a1111

11 12D11

π

a

--    4

8 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

12D22

π

b

--    4

= + +

 3D* π

a

--    2

kx

π

a

--    2

ky

π

b

--    2

– +

a1111

12

a1111

21 12D11

π

a

--    3

4 D12 2D66 ( ) +

π

a

--     π

b

--    2

= = – +

a1111

13

a1111

31 12D22

π

b

--    3

4 D12 2D66 ( ) +

π

a

--    2 π

b

--     = = – +

a1111

22 12D11

π

a

--    2

4D66

π

b

--    2

3DQx = + +

a1111

23

a1111

32 4 D12 D66 ( ) +

π

a

--     π

b

--     = =

a1111

33 12D22

π

b

--    2

4D66

π

a

--    2

3DQy = + +

a1n1n

11 8D11

π

a

--    4

4 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

1 n

2 ( ) +

3

2

--D22

π

b

--    4

1 n

2 ( ) +

2

4n

2 = + + [ ] +

D* π

a

--    2

2kx

π

a

--    2 3

2

--k

y

π

b

--    2

1 n

2 – + ( ) +

a1n1n

12

a1n1n

21 8D11

π

a

--    3

2 D12 2D66 ( ) +

π

a

--     π

b

--    2

1 n

2 = = – + ( ) +

a1n1n

13

a1n1n

31 3

2

--D22

π

b

--    3

1 n

2 ( ) +

2

4n

2 [ ] + 2 D12 2D66 ( ) +

π

a

--    2 π

b

--     1 n

2

+ ( ) +

    = = –  

a1n1n

22 8D11

π

a

--    2

2D66

π

b

--    2

1 n

2 ( ) + 2DQx = + +

a1n1n

23

a1n1n

32 2 D12 D66 ( ) +

π

a

--     π

b

--     1 n

2 = = ( ) +

a1n1n

33 3

2

--D22

π

b

--    2

1 n

2 ( ) +

2

4n

2 [ ] + 2D66

π

a

--    2

1 n

2 ( ) +

3

2

--DQy 1 n

2 = + + ( ) +
32

(3) m = k ≠ 1, n = l = 1

(A-5)

(4) m = k ≠ 1, n = l ≠ 1

(A-6)

am1m1

11 3

2

--D11

π

a

--    4

1 m

2 ( ) +

2

4m

2 [ ] + 4 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

1 m

2 ( ) + 8D22

π

b

--    4

= + +

D* π

a

--    2 3

2

--kx

π

a

--    2

1 m

2 ( ) + 2ky

π

b

--    2

– +

am1m1

12

am1m1

21 3

2

--D11

π

a

--    3

1 m

2 ( ) +

2

4m

2 [ ] + 2 D12 2D66 ( ) +

π

a

--     π

b

--    2

1 m

2

+ ( ) +

      = = –

am1m1

13

am1m1

31 8D22

π

b

--    3

2 D12 2D66 ( ) +

π

a

--    2 π

b

--     1 m

2 = = – + ( ) +

am1m1

22 3

2

--D11

π

a

--    2

1 m

2 ( ) +

2

4m

2 [ ] + 2D66

π

b

--    2

1 m

2 ( ) +

3

2

--DQx 1 m

2 = + + ( ) +

am1m1

23

am1m1

32 2 D12 D66 ( ) +

π

a

--     π

b

--     1 m

2 = = ( ) +

am1m1

33 8D22

π

b

--    2

2D66

π

a

--    2

1 m

2 ( ) + 2DQy = + +

amnmn

11 D11

π

a

--    4

1 m

2 ( ) +

2

4m

2 [ ] + 2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

1 m

2 ( ) + 1 n2 = + ( ) +

 + D22

π

b

--    4

1 n

2 ( ) +

2

4n

2 [ ] + D * π

a

--    2

kx

π

a

--    2

1 m

2 ( ) + ky

π

b

--    2

1 n

2 – + ( ) +

amnmn

12

amnmn

21 D11

π

a

--    3

1 m

2 ( ) +

2

4m

2 [ ] + D12 2D66 ( ) +

π

a

--     π

b

--    2

1 m

2 ( ) + 1 n2

+ ( ) +

      = = –

amnmn

13

amnmn

31

D22

π

b

--    3

1 n

2 ( ) +

2

4n

2 [ ] + D12 2D66 ( ) +

π

a

--    2 π

b

--     1 m

2 ( ) + 1 n2

+ ( ) +

      = = –

amnmn

22 D11

π

a

--    2

1 m

2 ( ) +

2

4m

2 [ ] + D66

π

b

--    2

1 m

2 ( ) + 1 n2 ( ) + DQx 1 m2 = + + ( ) +

amnmn

23

amnmn

32 D12 D66 ( ) +

π

a

--     π

b

--     1 m

2 ( ) + 1 n2 = = ( ) +

amnmn

33 D22

π

b

--    2

1 n

2 ( ) +

2

4n

2 [ ] + 2D66

π

a

--    2

1 m

2 ( ) + 1 n2 ( ) + DQy 1 n2 = + + ( ) +
33

(5) m = k = 1, n – l = 2

(A-7)

(6) m = k ≠ 1, n – l = 2

(A-8)

a1n1l

11 4D11

π

a

--    4

2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

( ) 1 + l 2 3

4

--D22

π

b

--    4

( ) 1 + l 4 = – + +

 +D* π

a

--    2

kx

π

a

--    2 3

4

--ky

π

b

--    2

( ) 1 + l 2+

a1n1l

12

a1n1l

21 4D11

π

a

--    3

D12 2D66 ( ) +

π

a

--     π

b

--    2

( ) 1 + l 2 = = +

a1n1l

13

a1n1l

31 3

4

--D22

π

b

--    3

( ) 1 + l 4 D12 2D66 ( ) +

π

a

--    2 π

b

--    ( ) 1 + l 2 = = +

a1n1l

22 4D11

π

a

--    2

D66

π

b

--    2

( ) 1 + l 2 DQx = – + +

a1n1l

23

a1n1l

32 D12 D66 ( ) +

π

a

--     π

b

--    ( ) 1 + l 2 = = –

a1n1l

33 3

4

--D22

π

b

--    2

( ) 1 + l 4 D66

π

a

--    2

( ) 1 + l 2 3

4

--DQy( ) 1 + l 2 = – + +

amnml

11 1

2

-- D11

π

4

--    4

1 m

2 ( ) +

2

4m

2 [ ] + 2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

1 m

2 ( ) + ( ) 1 + l 2

+





 = –

 + D22

π

b

--    4

( ) 1 + l 4





 D*

2

------ π

a

--    2

kx

π

a

--    2

1 m

2 ( ) + ky

π

b

--    2

( ) 1 + l 2+ +

amnml

12

amnml

21 1

2

-- D11

π

a

--    3

1 m

2 ( ) +

2

4m

2 [ ] + D12 2D66 ( ) +

π

a

--     π

b

--    2

1 m

2 ( ) + ( ) 1 + l 2

+

      = =

amnml

13

amnml

31 1

2

-- D22

π

b

--    3

( ) 1 + l 4 D12 2D66 ( ) +

π

a

--    2 π

b

--     + 1 m

2 ( ) + ( ) 1 + l 2

      = =

amnml

22 1

2

-- D11

π

a

--    2

1 m

2 ( ) +

2

4m

2 [ ] + D66

π

b

--    2

1 m

2 ( ) + ( ) 1 + l 2

+ DQx 1 m

2

+ ( ) +

      = –

amnml

23

amnml

32 1

2

-- D12 D66 ( ) + π

a

--     π

b

--     1 m

2 ( ) + ( ) 1 + l 2 = = –

amnml

33 1

2

-- D22

π

b

--    2

( ) 1 + l 4 D66 + π

a

--    2

1 m

2 ( ) + ( ) 1 + l 2 DQy( ) 1 + l 2 = – +
34

(7) m – k = 2, n = l = 1

(A-9)

(8) m – k = 2, n = l ≠ 1

(A-10)

am1k1

11 3

4

--D11

π

a

--    4

( ) 1 + k 4 2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

( ) 1 + k 2 4D22

π

b

--    4

= – + +

 + D* π

a

--    2 3

4

--kx

π

a

--    2

( ) 1 + k 2 ky

π

b

--    2

+

am1k1

12

amlkl

21 3

4

--D11

π

a

--    3

( ) 1 + k 4 D12 2D66 ( ) +

π

a

--     π

b

--    2

( ) 1 + k 2 = = +

am1k1

13

amlkl

31 4D22

π

b

--    3

D12 2D66 ( ) +

π

a

--    2 π

b

--    ( ) 1 + k 2 = = +

am1k1

22 3

4

--D11

π

a

--    2

( ) 1 + k 4 D66

π

b

--    2

( ) 1 + k 2 3

4

--DQx( ) 1 + k 2 = – + +

am1k1

23

amlkl

32 D12 D66 ( ) +

π

a

--     π

b

--    ( ) 1 + k 2 = = –

am1k1

33 4D22

π

b

--    2

D66

π

a

--    2

( ) 1 + k 2 DQy = – + +

amnkn

11 1

2

-- D11

π

a

--    4

( ) 1 + k 4 2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

( ) 1 + k 2 1 n

2

+ ( ) +





 = –

 + D22

π

b

--    4

1 n

2 ( ) +

2

4n

2 [ ] +





 D*

2

------ π

a

--    2

kx

π

a

--    2

( ) 1 + k 2 ky

π

b

--    2

1 n

2

+ + ( ) +

amnkn

12

amnkn

21 1

2

-- D11

π

a

--    3

( ) 1 + k 4 D12 2D66 ( ) +

π

a

--     π

b

--    2

( ) 1 + k 2 1 n

2 = = + ( ) +

amnkn

13

amnkn

31 1

2

-- D22

π

b

--    3

1 n

2 ( ) +

2

4n

2 [ ] + D12 2D66 ( ) +

π

a

--    2 π

b

--    ( ) 1 + k 2 1 n

2 = = + ( ) +

amnkn

22 1

2 – -- D11

π

a

--    2

( ) 1 + k 4 D66

π

b

--    2

( ) 1 + k 2 1 n

2 ( ) + DQx( ) 1 + k 2 = + +

amnkn

23

amnkn

32 1

2

-- D12 D66 ( ) +

π

a

--     π

b

--    ( ) 1 + k 2 1 n

2 = = – ( ) +

amnkn

33 1

2 – -- D22

π

b

--    2

1 n

2 ( ) +

2

4n

2 [ ] + + D66

π

a

--    2

1 + k

2 ( ) 1 n2 ( ) + + DQy 1 n2 ( ) +

      =
35

(9) m – k = 2, n – l = 2

(A-11)

Case 3: 2C2S condition

(1) m = k, n = l = 1

(A-12)

amnkl

11 1

4

-- D11

π

a

--    4

( ) 1 + k 4 2 D12 2D66 ( ) +

π

a

--    2 π

b

--    2

( ) 1 + k 2( ) 1 + l 2 = +

 + D22

π

b

--    4

( ) 1 + l 4 – D*

4

------ π

a

--    2

kx

π

a

--    2

( ) 1 + k 2 ky

π

b

--    2

( ) 1 + l 2+

amnkl

12

amnkl

21 – 1

4

-- D11

π

a

--    3

( ) 1 + k 4 D12 2D66 ( ) +

π

a

--     π

b

--    2

( ) 1 + k 2( ) 1 + l 2 = = +

amnkl

13

amnkl

31 – 1

4

-- D22

π

b

--    3

( ) 1 + l 4 + D12 2D66 ( ) +

π

a

--    2 π

b

--    ( ) 1 + k 2

( ) 1 + l 2 = =

amnkl

22 1

4

-- D11

π

a

--    2

( ) 1 + k 4 D66

π

b

--    2

( ) 1 + k 2( ) 1 + l 2 DQx( ) 1 + k 2 = + +

amnkl

23

amnkl

32 1

4

-- D12 D66 ( ) +

π

a

--     π

b

--    ( ) 1 + k 2

( ) 1 + l 2 = =

amnkl

33 1

4

-- D22

π

b

--    2

( ) 1 + l 4 D66

π

a

--    2

( ) 1 + k 2( ) 1 + l 2 DQy( ) 1 + l 2 = + +

am1m1

11 3D11

mπ

a

-------    4

8 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--    2

16D22

π

b

--    4

= + +

D* π

a

--    2

3kx

mπ

a

-------    2

4ky

π

b

--    2

– +

am1m1

12

am1m1

21 3D11

mπ

a

-------    3

4 D12 2D66 ( ) +

mπ

a

-------     π

b

--    2

= = – +

am1m1

13

am1m1

31 16D22

π

b

--    3

4 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--     = = – +

am1m1

22 3D11

mπ

a

-------    2

4D66

π

b

--    2

3DQx = + +

am1m1

23

am1m1

32 4 D12 D66 ( ) +

mπ

a

-------     π

b

--     = =

am1m1

33 16D22

π

b

--    2

4D66

mπ

a

-------    2

4DQx = + +
36

(2) m = k, n = l ≠ 1

(A-13)

(3) m = k, n – l = 2

(A-14)

amnmn

11 2D11

mπ

a

-------    4

4 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--    2

1 n

2 ( ) + 2D22

π

b

--    4

1 n

2 ( ) +

2

4n

2 = + + [ ] +

 2D* π

a

--    2

kx

mπ

a

-------    2

ky

π

b

--    2

1 n

2 – + ( ) +

amnmn

12

amnmn

21 2D11

mπ

a

-------    3

2 D12 2D66 ( ) +

mπ

a

-------     π

b

--    2

1 n

2 = = – + ( ) +

amnmn

13

amnmn

31 2D22

π

b

--    3

1 n

2 ( ) +

2

4n

2 [ ] + 2 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--     1 n

2

+ ( ) +

      = = –

amnmn

22 2D11

mπ

a

-------    2

2D66

π

b

--    2

1 n

2 ( ) + 2DQx = + +

amnmn

23

amnmn

32 2 D12 D66 ( ) +

mπ

a

-------     π

b

--     1 n

2 = = ( ) +

amnmn

33 2D22

π

b

--    2

1 n

2 ( ) +

2

4n

2 [ ] + 2D66

mπ

a

-------    2

1 n

2 ( ) + 2DQy 1 n2 = + + ( ) +

amnml

11 D11

mπ

a

-------    4

2 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--    2

( ) 1 + l 2 D22

π

b

--    4

( ) 1 + l 4 = – + +

D* π

a

--    2

kx

mπ

a

-------    2

ky

π

b

--    2

( ) 1 + l 2+ +

amnml

12

amnml

21 D11

mπ

a

-------    3

D12 2D66 ( ) +

mπ

a

-------     π

b

--    2

( ) 1 + l 2 = = +

amnml

13

amnml

31 D22

π

b

--    3

( ) 1 + l 4 D12 2D66 ( ) +

mπ

a

-------    2 π

b

--    ( ) 1 + l 2 = = +

amnml

22 D11

mπ

a

-------    2

D66

π

b

--    2

( ) 1 + l 2 DQx = – + +

amnml

23

amnml

32 D12 D66 ( ) +

mπ

a

-------     π

b

--    ( ) 1 + l 2 = = –

amnml

33 D22

π

b

--    2

( ) 1 + l 4 D66

mπ

a

-------    2

( ) 1 + l 2 DQy + +1( ) + l 2 = –
37

Case 4: 2S2C condition

(1) m = k = 1, n = 1

(A-15)

(2) m = k ≠ 1, n = l

(A-16)

a1n1n

11 16D11

π

a

--    4

8 D12 2D66 ( ) +

π

a

--    2 nπ

b

------    2

3D22

nπ

b

------    4

= + +

D* π

a

--    2

4kx

π

a

--    2

3ky

nπ

b

------    2

– +

a1n1n

12

a1n1n

21 16D11

π

a

--    3

4 D12 2D66 ( ) +

π

a

--     nπ

b

------    2

= = – +

a1n1n

13

a1n1n

31 3D22

nπ

b

------    3

4 D12 2D66 ( ) +

π

a

--    2 nπ

b

------     = = – +

a1n1n

22 16D11

π

a

--    2

4D66

nπ

b

------    2

4DQx = + +

a1n1n

23

a1n1n

32 4 D12 D66 ( ) +

π

a

--     nπ

b

------     = =

a1n1n

33 3D22

nπ

b

------    2

4D66

π

a

--    2

3DQx = + +

amnmn

11 2D11

π

a

--    4

1 m

2 ( ) +

2

4m

2 [ ] + 4 D12 2D66 ( ) +

π

a

--    2 nπ

b

------    2

1 m

2 ( ) + 2D22

nπ

b

------    4

= + +

 2D* π

a

--    2

kx

π

a

--    2

1 m

2 ( ) + ky

nπ

b

------    2

– +

amnmn

12

amnmn

21 2D11

π

a

--    3

1 m

2 ( ) +

2

4m

2 [ ] + 2 D12 2D66 ( ) +

π

a

--     nπ

b

------    2

1 m

2

+ ( ) +

      = = –

amnmn

13

amnmn

31 2D22

nπ

b

------    3

2 D12 2D66 ( ) + 1 m

2 ( ) +

π

a

--    2 nπ

b

------     = = – +

amnmn

22 2D11

π

a

--    2

1 m

2 ( ) +

2

4m

2 [ ] + 2D66

nπ

b

------    2

1 m

2 ( ) + 2DQy 1 m2 = + + ( ) +

amnmn

23

amnmn

32 2 D12 D66 ( ) +

π

a

--     nπ

b

------     1 m

2 = = ( ) +

amnmn

33 2D22

nπ

b

------    2

2D66

π

a

--    2

1 m

2 ( ) + 2DQy = + +
38

(3) m – k = 2, n = l

(A-17)

amnkn

11 D11

π

a

--    4

( ) 1 + k 4 2 D12 2D66 ( ) +

π

a

--    2 nπ

b

------    2

( ) 1 + k 2 D22

nπ

b

------    4

= – + +

D* π

a

--    2

kx

π

a

--    2

( ) 1 + k 2 ky

nπ

b

------    2

+ +

amnkn

12

amnkn

21 D11

π

a

--    3

( ) 1 + k 4 D12 2D66 ( ) +

π

a

--     nπ

b

------    2

( ) 1 + k 2 = = +

amnkn

13

amnkn

31 D22

nπ

b

------    3

D12 2D66 ( ) +

π

a

--    2 nπ

b

------    ( ) 1 + k 2 = = +

amnkn

22 D11

π

a

--    2

( ) 1 + k 4 D66

nπ

b

------    2

( ) 1 + k 2 DQx( ) 1 + k 2 = – + +

amnkn

23

amnkn

32 D12 D66 ( ) +

π

a

--     nπ

b

------    ( ) 1 + k 2 = = –

amnkn

33 D22

nπ

b

------    2

D66

π

a

--    2

( ) 1 + k 2 DQy = – + +
39

APPENDIX B

BUCKLING EQUATIONS

The buckling equations (eigenvalue solution equations) written out from equation (21) up to order 12 (i.e., 12 ×

12 matrices) for the cases m ± n = even (symmetric buckling) and m ± n = odd (antisymmetric buckling) for different

edge conditions are given on the following pages (ref. 12).
kling)

0 0 0 0 0 0 0

0 0 0 0 0 0

00 0

0 0 0 0 0

0 00 0

0 0

= 0

00 0

Symmetry

0

0 0

0

 (B-1)

A13 A22 A31 A15 A24 A33 A42 A51 A35 A44 A53

49--

8

45-----

8

45-----

16

225--------

1313

T

-------- P1313 +

45

– --

87--

8

25

– -----

1635-----

M2222

∆T

-------------- P2222 +

45

– --

2063

– -----

3625-----

2063

– -----

47--

47--

M3131

∆T

-------------- P3131 +

8

25

– -----

87---

1635-----

M1515

∆T

-------------- P1515 +

4027

– -----

8

63

– -----

1627

– -----

M2424

∆T

-------------- P2424 +

7235

– -----

8

63

– -----

83--

120147

– --------

M3333

∆T

-------------- P3333 +

7235

– -----

144

49

--------

M4242

∆T

-------------- P4242 +

4027

– -----

120147

– --------

83--

M5151

∆T

-------------- P5151 +

1627

– -----

M3535

∆T

-------------- P3535 +

8021

– -----

M4444

∆T

-------------- P4444 +

8021

– -----

M5353

∆T

-------------- P5353 +
uckling) (4S)

0 0 0 0 0

– 0000 0

– 0000

000 0

000

00 0

= 0

0 0

Symmetry

0 0

0

0

 (B-2)

A21 A14 A23 A32 A41 A16 A25 A34 A43 A52 A61

49

– --

45--

8

45

– -----

2063-----

8

25-----

4

35

– -----

2121

T

-------- P2121 +

8

45-----

45--

4

35

– -----

8

25-----

2063-----

M1414

∆T

-------------- P1414 +

87--

16

225

– --------

4027-----

1635

– -----

8

175

– --------

M2323

∆T

-------------- P2323 +

3625

– -----

49

– ---

7235-----

47

– --

M3232

∆T

-------------- P3232 +

87

– --

47

– --

7235-----

49

– --

M4141

∆T

-------------- P4141 +

8

175

– --------

1635

– -----

4027-----

M1616

∆T

-------------- P1616 +

2011

– -----

8

45

– -----

36

1225

– -----------

M2525

∆T

-------------- P2525 +

83

– --

100441

– --------

M3434

∆T

-------------- P3434 +

144

49

– --------

8

45

– -----

M4343

∆T

-------------- P4343 +

83

– --

M5252

∆T

-------------- P5252 +

2011

– -----

M6161

∆T

-------------- P6161 +
kling)

0 0 0 0

0 0

0 0

0 0

= 0

Symmetry

0

 (B-3)

A13 A22 A31 A15 A24 A33 A42 A51 A35 A44 A53

1113

T

-------- P1113 +

4

225--------

M1131

∆T

-------------- P1131 +

8

1575

– -----------

M1133

∆T

-------------- P1133 +

8

1575

– -----------

16

11025--------------

1313

T

-------- P1313 +

44

1575

– -----------

M1331

∆T

-------------- P1331 +

M1315

∆T

-------------- P1315 +

184

4725-----------

M1333

∆T

-------------- P1333 +

88

11025--------------

M1335

∆T

-------------- P1335 +

368

33075

– --------------

M2222

∆T

-------------- P2222 +

44

1575

– -----------

4

525--------

M2224

∆T

-------------- P2224 +

484

11025--------------

M2242

∆T

-------------- P2242 +

4

525--------

44

3675

– -----------

M2244

∆T

-------------- P2244 +

44

3675

– -----------

M3131

∆T

-------------- P3131 +

88

11025--------------

M3133

∆T

-------------- P3133 +

184

4725-----------

M3151

∆T

-------------- P3151 +

368

33075

– --------------

M3153

∆T

-------------- P3153 +

M1515

∆T

-------------- P1515 +

104

2079

– -----------

M1533

∆T

-------------- P1533 +

8

3675

– -----------

M1535

∆T

-------------- P1535 +

208

14553--------------

M2424

∆T

-------------- P2424 +

2024

33075

– --------------

M2442

∆T

-------------- P2442 +

8

3675

– -----------

104

1323-----------

M2444

∆T

-------------- P2444 +

184

11025--------------

M3333

∆T

-------------- P3333 +

2024

33075

– --------------

M3351

∆T

-------------- P3351 +

M3335

∆T

-------------- P3335 +

8464

99225--------------

M3353

∆T

-------------- P3353 +

M4242

∆T

-------------- P4242 +

104

2079

– -----------

184

11025--------------

M4244

∆T

-------------- P4244 +

104

1323-----------

M5151

∆T

-------------- P5151 +

208

14553--------------

M5153

∆T

-------------- P5153 +

M3535

∆T

-------------- P3535 +

4784

43659

– --------------

M3553

∆T

-------------- P3553 +

M4444

∆T

-------------- P4444 +

4784

43659

– --------------

M5353

∆T

-------------- P5353 +
uckling) (4C)

0 0

0 0

0

0

0

0

= 0

0

Symmetry

0

 (B-4)

A21 A14 A23 A32 A41 A16 A25 A34 A43 A52 A61

4

225

– --------

M1214

∆T

-------------- P1214 +

44

1575-----------

M1232

∆T

-------------- P1232 +

8

1575-----------

4

525

– --------

M1234

∆T

-------------- P1234 +

88

11025

– --------------

4

4725-----------

2121

T

-------- P2121 +

8

1575-----------

M2123

∆T

-------------- P2123 +

44

1575-----------

M2141

∆T

-------------- P2141 +

4

4725-----------

88

11025

– --------------

M2143

∆T

-------------- P2143 +

4

525

– --------

M1414

∆T

-------------- P1414 +

184

4725

– -----------

M1432

∆T

-------------- P1432 +

16

11025

– --------------

M1416

∆T

-------------- P1416 +

104

2079-----------

M1434

∆T

-------------- P1434 +

368

33075--------------

8

33075

– --------------

M2323

∆T

-------------- P2323 +

484

11025

– --------------

M2341

∆T

-------------- P2341 +

172

17325--------------

M2325

∆T

-------------- P2325 +

2024

33075--------------

M2343

∆T

-------------- P2343 +

44

3675-----------

M3232

∆T

-------------- P3232 +

184

4725

– -----------

44

3675-----------

M3234

∆T

-------------- P3234 +

2024

33075--------------

M3252

∆T

-------------- P3252 +

172

17325--------------

M4141

∆T

-------------- P4141 +

8

33075

– --------------

368

33075--------------

M4143

∆T

-------------- P4143 +

104

2079-----------

M4161

∆T

-------------- P4161 +

M1616

∆T

-------------- P1616 +

236

3861

– -----------

M1634

∆T

-------------- P1634 +

344

121275

– -----------------

4

99225

– --------------

M2525

∆T

-------------- P2525 +

104

1323

– -----------

M2543

∆T

-------------- P2543 +

4

1225

– -----------

M3434

∆T

-------------- P3434 +

8464

99225

– --------------

M3452

∆T

-------------- P3452 +

344

121275

– -----------------

M4343

∆T

-------------- P4343 +

104

1323

– -----------

M4361

∆T

-------------- P4361 +

M5252

∆T

-------------- P5252 +

236

3861

– -----------

M6161

∆T

-------------- P6161 +
kling)

0 0 0 0 0 00 0 0 0 0

0 0

0 0 0 0

0 00 0

0 0

= 0

0 0

Symmetry

– 

0

0

 (B-5)

A13 A22 A31 A15 A24 A33 A42 A51 A35 A44 A53

1113

T

-------- P1113 +

4

45-----

8

315

– --------

8

225--------

16

1575

– -----------

1313

T

-------- P1313 +

44

315

– --------

M1315

∆T

-------------- P1315 +

184945--------

88

1575

– -----------

368

4725-----------

M2222

∆T

-------------- P2222 +

4

25

– -----

4

105--------

M2224

∆T

-------------- P2224 +

44

175--------

4

63

–-----

12

175

– --------

44

441--------

M3131

∆T

-------------- P3131 +

8

175--------

M3133

∆T

-------------- P3133 +

8

35-----

16

245

– --------

M1515

∆T

-------------- P1515 +

520

2079

– -----------

8

525--------

208

2079

– -----------

M2424

∆T

-------------- P2424 +

184525

– --------

8

441--------

104231--------

184

1323

– -----------

M3333

∆T

-------------- P3333 +

88

245

– --------

M3335

∆T

-------------- P3335 +

368735--------

M4242

∆T

-------------- P4242 +

8

27-----

24

245--------

M4244

∆T

-------------- P4244 +

88

189--------

M5151

∆T

-------------- P5151 +

16

189--------

M5153

∆T

-------------- P5153 +

M3535

∆T

-------------- P3535 +

10401617

– -----------

M4444

∆T

-------------- P4444 +

368567

– --------

M5353

∆T

-------------- P5353 +
uckling) (2C2S)

0 0 0 0

000 0

0 0 0

0 0 – 0

0 0

0 0

= 0

0 0

Symmetry

0 0

0

0

 (B-6)

A21 A14 A23 A32 A41 A16 A25 A34 A43 A52 A61

4

25

– -----

M1214

∆T

-------------- P1214 +

44

315--------

8

225

– --------

4

105

– --------

88

1575-----------

4

175

– --------

2121

T

-------- P2121 +

8

315--------

M2123

∆T

-------------- P2123 +

4

25-----

4

925--------

8

175

– --------

4

63-----

M1414

∆T

-------------- P1414 +

184945

– --------

16

1575-----------

M1416

∆T

-------------- P1416 +

520

2079-----------

368

4725

– -----------

8

1225-----------

M2323

∆T

-------------- P2323 +

44

175

– --------

172

3465-----------

M2325

∆T

-------------- P2325 +

184525--------

44

441--------

M3232

∆T

-------------- P3232 +

8

35

– -----

12

175--------

M3234

∆T

-------------- P3234 +

88

245--------

4

45

– -----

M4141

∆T

-------------- P4141 +

8

4725-----------

16

245--------

M4143

∆T

-------------- P4143 +

8

27-----

M1616

∆T

-------------- P1616 +

11803861

– -----------

344

17325--------------

4

3675-----------

M2525

∆T

-------------- P2525 +

104231

– --------

4

147--------

M3434

∆T

-------------- P3434 +

368735

– --------

8

315--------

M4343

∆T

-------------- P4343 +

88

189

– --------

M5252

∆T

-------------- P5252 +

4

11

– -----

M6161

∆T

-------------- P6161 +
kling)

0 0 0 0 0 0

0 0 0 0 0

0 0

0 0 0 0

00 0

0

= 0

0 0

Symmetry

0

0 0

0

 (B-7)

A13 A22 A31 A15 A24 A33 A42 A51 A35 A44 A53

4

45-----

M1131

∆T

-------------- P1131 +

8

225--------

8

315

– --------

16

1575

– -----------

1313

T

-------- P1313 +

4

25

– -----

8

35-----

M1333

∆T

-------------- P1333 +

8

175--------

16

245

– --------

M2222

∆T

-------------- P2222 +

44

315

– --------

4

63

– -----

44

175--------

M2242

∆T

-------------- P2242 +

4

105--------

44

441--------

12

175

– --------

M3131

∆T

-------------- P3131 +

88

1575

– -----------

184945--------

M3151

∆T

-------------- P3151 +

368

4725-----------

M1515

∆T

-------------- P1515 +

8

27

– -----

8

441--------

M1535

∆T

-------------- P1535 +

16

189--------

M2424

∆T

-------------- P2424 +

88

245

– --------

8

525--------

88

189--------

M2444

∆T

-------------- P2444 +

24

245--------

M3333

∆T

-------------- P3333 +

184525

– --------

368735--------

M3353

∆T

-------------- P3353 +

M4242

∆T

-------------- P4242 +

520

2079

– -----------

184

1323

– -----------

104231--------

M5151

∆T

-------------- P5151 +

208

2079

– -----------

M3535

∆T

-------------- P3535 +

368567

– --------

M4444

∆T

-------------- P4444 +

10401617

– -----------

M5353

∆T

-------------- P5353 +
uckling) (2S2C)

0 0 0 0

0 0 0 0

0 0 0

00 0

0 0

0 0

= 0

0 0

Symmetry

0 0

0

0

 (B-8)

A21 A14 A23 A32 A41 A16 A25 A34 A43 A52 A61

4

45

– -----

4

25-----

M1232

∆T

-------------- P1232 +

8

315--------

4

63-----

8

175

– --------

4

945--------

2121

T

-------- P2121 +

8

225

– --------

44

315--------

M2141

∆T

-------------- P2141 +

4

175

– --------

88

1575-----------

4

105

– --------

M1414

∆T

-------------- P1414 +

8

35

– -----

16

1575-----------

8

27-----

M1434

∆T

-------------- P1434 +

16

245--------

8

4725-----------

M2323

∆T

-------------- P2323 +

44

175

– --------

4

45

– -----

88

245--------

M2343

∆T

-------------- P2343 +

12

175--------

M3232

∆T

-------------- P3232 +

184945

– --------

44

441

– --------

184525--------

M3252

∆T

-------------- P3252 +

172

3465-----------

M4141

∆T

-------------- P4141 +

8

1225-----------

368

4725

– -----------

520

2079-----------

M4161

∆T

-------------- P4161 +

M1616

∆T

-------------- P1616 +

4

11

– -----

8

315--------

4

3675-----------

M2525

∆T

-------------- P2525 +

88

189

– --------

4

147--------

M3434

∆T

-------------- P3434 +

368735

– --------

344

17325--------------

M4343

∆T

-------------- P4343 +

104231

– --------

M5252

∆T

-------------- P5252 +

11803861

– -----------

M6161

∆T

-------------- P6161 +
48

REFERENCES

1. Gowda, R. M. Siddaveere and K. A. V. Pandalai, Thermal Buckling of Orthotropic Plates, Studies in

Structural Mechanics, Hoff's 65th Anniversary Volume, Indian Institute of Technology, Madras-36,

India, 1970, pp. 9–44.

2. Tauchert, T. R. and N. N. Huang, “Thermal Buckling and Postbuckling Behavior of Antisymmetric

Angle-Ply Laminates,” Proc. Internat'l Symp. Composite Materials and Structures, Beijing, China,

June 1986, pp. 357–362.

3. Tauchert, T. R. and N. N. Huang, “Thermal Buckling of Symmetric Angle-Ply Laminated Plates,”

Composite Structures, I. H. Marshall-editor, Elsevier Applied Science, London, 1987, pp. 1424–1435.

4. Thangaratnam, Kari R., Palaninathan, and J. Ramachandran, “Thermal Buckling of Composite Laminated Plates,” Computers and Structures, Vol. 32, No. 5, 1989, pp. 1117–1124.

5. Huang, N.N. and T.R. Tauchert, “Postbuckling Response of Antisymmetric Angle-Ply Laminates to

Uniform Temperature Loading,” Acta Mechanica, Vol. 72, 1988, pp. 173–183.

6. Tauchert, Theodore R., “Thermal Stresses in Plates—Statical Problems,” in Thermal Stresses I, Vol. 1,

Elsevier Sciences Publishing Co., New York, 1986, pp. 23–141.

7. Ko, William L. and Raymond H. Jackson, Thermal Behavior of a Titanium Honeycomb-Core Sandwich Panel, NASA TM-101732, 1991.

8. Ko, William L. and Raymond H. Jackson, “Combined Compressive and Shear Buckling Analysis of

Hypersonic Aircraft Structural Sandwich Panels,” AIAA Paper No. 92-2487-CP. Presented at the 33rd

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Dallas,

Texas, April 13–15, 1992; also NASA TM-4290, 1991.

9. Ko, William L. and Raymond H. Jackson, Combined-Load Buckling Behavior of Metal-Matrix Composite Sandwich Panels Under Different Thermal Environments, NASA TM-4321, 1991.

10. Ko, William L. and Raymond H. Jackson, “Compressive and Shear Buckling Analysis of Metal Matrix

Composite Sandwich Panels under Different Thermal Environments,” Composite Structures, Vol. 25,

July 1993, pp. 227–239.

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Edge Conditions,” Proc. 1st Pacific International Conference on Aerospace Science and Technology,

Tainan, Taiwan, Dec. 6–9, 1993. 

12. Ko, William L. and Raymond H. Jackson, Mechanical and Thermal Buckling of Rectangular Sandwich Panels Under Different Edge Conditions, NASA TM-4535, 1994.

13. Tenney, D. R., W. B. Lisagor, and S. C. Dixon, “Materials and Structures for Hypersonic Vehicles,”

J. Aircraft, Vol. 26, no. 11, Nov. 1989, pp. 953–970.
49

14. Thornton, Earl A. “Thermal Buckling of Plates and Shells,” Applied Mechanics Review, Vol. 46,

No. 10, October 1993, pp. 485–506.

15. Whetstone, W. D., SPAR Structural Analysis System Reference Manual, System Level 13A, Vol. 1,

Program Execution, NASA CR-158970-1, Dec. 1978.

16. Ko, William L. “Thermocryogenic Buckling and Stress Analyses of 9 Partially Filled Cryogenic Tank

Subjected to Cylindrical Strip Heating,” Proceedings of 2nd Thermal Structures Conference,

Charlottesville, Virginia, Oct. 18–20, 1994. Also, NASA TM-4579, Oct. 1994.
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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Predictions of Thermal Buckling Strengths of Hypersonic Aircraft Sandwich

Panels Using Minimum Potential Energy and Finite Element Methods

WU 505-70-63

William L. Ko

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

H-2009

National Aeronautics and Space Administration

Washington, DC 20546-0001 NASA TM-4643

Thermal buckling characteristics of hypersonic aircraft sandwich panels of various aspect ratios were investigated. The

panel is fastened at its four edges to the substructures under four different edge conditions and is subjected to uniform

temperature loading. Minimum potential energy theory and finite element methods were used to calculate the panel

buckling temperatures. The two methods gave fairly close buckling temperatures. However, the finite element method

gave slightly lower buckling temperatures than those given by the minimum potential energy theory. The reasons for this

slight discrepancy in eigensolutions are discussed in detail. In addition, the effect of eigenshifting on the eigenvalue

convergence rate is discussed.

Different edge conditions, Finite element method, Minimum potential energy theory,

Sandwich panels, Thermal bucklings A04

54

Unclassified Unclassified Unclassified Unlimited

May 1995 Technical Memorandum

Available from the NASA Center for AeroSpace Information, 800 Elkridge Landing Road, 

Linthicum Heights, MD 21090; (301)621-0390

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Subject Category 39