NASA-CR-1457_MANUAL FOR STRUCTURAL STABILITY ANALYSIS OF SANDWICH PLATES AND SHELLS

_N A S A

_- i_ B

I

!z

CONTRACTOR

REPORT

CASE Fi LE

NASA CR-1457

MANUAL FOR STRUCTURAL

STABILITY ANALYSIS OF

I PLATES AND SHELLSSANDWICH

I by R. T. Sulli,is, G. IFC Smith, a,id E. E. Spier

i _Prepared b),E7 :

_ GENERAL DYNAMICS CORPORATION

_:San Diego, Calif.

_r Manned Spacecraf Ce,zter

E NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • DECEMBER-1969

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_____ ". _ -........ =..... -- •!

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NASA CR-1457

MANUAL FOR STRUCTURAL STABILITY ANALYSIS

OF SANDWICH PLATES AND SHELLS

By R. T. Sullins, G. W. Smith, and E. E. Spier

Distribution of this report is provided in the interest of

information exchange. Responsibility for the contents

resides in the author or organization that prepared it.

Prepared under Contract No. NAS 9-8244 byGENERAL DYNAMICS CORPORATION

San Diego, Calif.

for Manned Spacecraft Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information

Springfield, Virginia 22151 - Price $3.00

ACE

I/

/ ttONEYCOMB..-'_'III Iil tli I _ //" /// _o_ _q-ll II tIL-_ / // /4

_ / /_D 1Rp_lIBCB_]N);

/CORE i {

DEPTH ! / _ _t l--

RIBBON

DIRECTION

HONEYCOMB SANDWICH CONSTRUCTION

iii

ABSTRACT

The basic objective of this study was to develop and compile a manual which

would include practical and up-to-date methods for analyzing the structural

stabilityof sandwich plates and she]Is for typical loading conditions which

might be encountered in aerospace applications. The methods proposed for

use would include known analytical approaches as modified for correlation

with applicable test data.

The data presented here covers recommended design equations and curves

for a wide range of structural configurations and loading conditions, includ-

ing combined loads. In a number of cases, actual test data points are in-

cluded on the design curves to substantiate the recommendations made. For

those items where little or no test data exists the basic analytical approach

is presented along with the notation that this represented the 'best available"

data and should be used with some caution and judgment until substantiated

by test.

The following subjects are among those covered in the manual:

Local Instability

General Instability of Flat Panels

General Instability of Circular Cylinders

General Instability of Truncated Circular Cones

General Instability of Dome-Shaped Shells

Instability of Sandwich Shell Segments

Effects of Cutouts on the General Instability of Sandwich Shells

Inelastic Behavior of Sandwich Plates and Shells

Section

4

TABLE OF CONTENTS

INTRODUCTION .................

i. 1 GE 2:ERA_" .................

I. 2 FAILURE MODES .............

LOCAL INSTABILITY ...............

2.1 INTRACELLULAR BUCKLING (Face Dimpling)

2. i. 1 Sandwich with Honeycomb Core .........

2.1.2 Sandwich with Corrugated Core .........

2.2 FACE WRINKLING .............

2.2.1 Sandwich with Solid or Foam Core (Antisymmetric

Wrinkling) ................

2.2.2 Sandwich with Honeycomb Core (Symmetric

Wrinkling) ................

2.3 SHEAR CRIMPING .............

2.3.1 Basic Principles .............

2.3.2 Design Equations .............

GENERAL INSTABILITY OF FLAT PANELS .......

3.1 RECTANGULAR PLATES ..........

3.1 1 General ................

3.1 2 ITniaxial Edgewise Compression ........

3.1 3 Edgewise Shear ..............

3.1 4 Edgewise Bending Moment ..........

3.1 5 Other Single Loading Conditions ........

3.1 6 Combined Loading Conditions .........

3.2 CIRCULAR PLATES ............

3.2.1 Available Single Loading Conditions .......

3.2.2 Available Combined Loading Conditions ......

3.3 PLATES WITH CUTOUTS ..........

3.3.1 Framed Cutouts ..............

3.3.2 Unframed Cutouts .............

GENERAL INSTABILITY OF CIRCULAR CYLINDERS ....

4.1 GENERAL ................

4.2 AXIAL COMPRESSION ...........


4.2.2 Design Equations and Curves .........

Page

I-i

i-i

1-4

2-1

2-1

2-1

2-8

2-21

2-21

2-28

2-37

2-37

2 -40

3-1

3-1

3-1

3-5

3-25

3-37

3-46

3-47

3-73

3-73

3-73

3-74

3-74

3-75

4-1

4-1

4-5

4-5

4-17

vii

TABLE OF CONTENTS, Cont'd.

Section Page

4.3

4.3.1

4.3.2

4.4

4.4.1

4.4.2

4.5

4.5.1

4.5.2

4.6

4.6.1

4.6.2

4.7

4.7.1

4,7.2

4.7.3

4.7.4

4.7.5

PURE BENDING ..............

Basic Principles .............

Design Equations and Curves .........

EXTERNAL LATERAL PRESSURE .......



TORSION ................



TRANSVERSE SHEAR ............



COMBINED LOADING CONDITIONS .......

General ................

Axial Compression Plus Bending ........

Axial Compression Plus External Lateral Pressure.

Axial Compression Plus Torsion ........

Other Loading Combinations ..........

GENERAL INSTABILITY OF TRUNCATED CIRCULAR

CONES ....................

5.1 AXIAL COMPRESSION ...........



5.2 PURE BENDING ..............



5.3 EXTERNAL LATERAL PRESSURE .......


5,3.2 Design Equations and Curves .........

5.4 TORSION ................



5.5 TRANSVERSE SHEAR ............



5.6 COMBINED LOADING CONDITIONS .......

5.6.1 General ................

5.6.2 Axial Compression Plus Bending ........

4-25

4-25

4-29

4-32

4-32

4-42

4-47

4-47

4-54

4-62

4-62

4-64

4-65

4-65

4-67

4-71

4-89

4-95

5-1

5-1

5-1

5-4

5-6

5-6

5-7

5-9

5-9

5-12

5-14

5-14

5-16

5-18

5-18

5-20

5-21

5-21

5-23

viii

Section

TABLE OF CONTENTS, Cont'd.

5.6.3 Uniform External Hydrostatic Pressure ......

5.6.4 Axial Compression Plus Torsion ........

5.6.5 Other Loading Combinations ..........

GENERAL INSTABILITY OF DOME-SHAPED SHELLS ....

6.1 GENERAL ................

6.2 EXTERNAL PRESSURE ...........



6.3 OTHER LOADING CONDITIONS ........

INSTABILITY OF SANDWICH SHELL SEGMENTS .....

7.1 CYLINDRICAL CURVED PANELS ........

7. i. 1 Axial Compression .............

7.1.2 Other Loading Conditions ...........

7.2 OTHER PANEL CONFIGURATIONS .......

EFFECTS OF CUTOUTS ON THE GENERAL INSTABILITY

OF SANDWICH SHELLS ..............

INELASTIC BEHAVIOR OF SANDWICH PLATES AND

SHE L LS

9.1

9.1.1

9.1.2

9.2

9.2.1

9.2.2

SINGLE LOADING CONDITIONS ........


Design Equations .............

COMBINED LOADING CONDITIONS .......


Suggested Method .............

Page

5-27

5-34

5-39

6-1

6-1

6-3

6-3

6-12

6-15

7-1

7-1

7-1

7-11

7-11

8-1

9-1

9-1

9-1

9-3

9-i0

9-10

9-13

ix

Figure

1.1-1

1.2-1

1.2-2

1.2-3

2.1-1

2.1-4

2.1-5

2.1-6

2.1-7

2.1-8

2.1-9

2.1-10

2.1-11

2.1-12

2.2-3

LIST OF FIGURES

Typical Sandwich Construction ............

Localized Instability Modes .............

Ultimate Failures Precipitated by Face Wrinkling .....

Non-Localized Instability Modes ...........

Critical Stresses for Intracellular Buckling Under

Unia.xi al Compression ...............

Definition of Dimensions ..............

Chart for Determination of Core Cell Size Such That

Intracellular Buckling Will Not Occur .........

Corrugation Configurations .............

Buckling Modes .................

Local Buckling Coefficient for Single-Truss-Core

Sandwich ...................


Sandwich ...................


Sandwich ..................

Local Buckling Coefficient for Double-Truss-Core

Sandwich ...................


Sandwich ...................


Sandwich ...................

Local Buckling Coefficient for Truss-Core

San&rich ...................

Typical Variation of Q vs. q .............

Comparison of Theory vs Test Results for Face Wrinkling

in Sanchvich Constructions Having Solid or Foam Cores.

Parameters for Determination of Face Wrinkling in

Sandwich Constructions Having Solid or Foam Cores ....

xi

P age

1-2

1-5

1-5

1-6

2-3

2-5

2-6

2-8

2-10

2-14

2-15

2-16

2-17

2-18

2-19

2-20

2-24

2-25

2-27

Figure

2.2-4

2.2-5

2.2-6

2.2-7

2.3-1

2.3-2

3.1-1

3.1-2

3.1-3

3.1-4

3.1-5

3.1-6

3.1-7

LIST OFFIGURES,Cont'd.

T_3_icalDesignCurvesfor FaceWrinklingin SanchvichConstructionsHavingHoneycombCores.........

Comparisonof Theoryvs TestResultsfor FaceWrinklingin SandwichConstructionsllaving HoneycombCores .....

Relationship of K6 to Honeycomb Core Properties (Fc/Ee)

and Facing Waviness Parameter (6/te) ..........

Graphs of Equation (2.2-12) for the Wrinkling Stress of

Facings in San_vieh Constructions ttaving ttoncycomb

Cores ....................

Uniaxial Compression ...............

Pure Shear ..................

Elastic Properties and Dimensional Notations for a

Typical Sandwich Panel ..............

K M for a Sandwich Panel with Ends and Sides SimplySupported, Isotropie Facings, and Orthotropie Core,

(R = 0.40) ...................

K M for Sanchvich Panel with Ends and Sides SimplySupported, Isotropic Facings, and Isotropic Core,

(R = 1.00) ...................

K M for Sandwich Panel with Ends and Sides Simply

Supported, Isotropic Facings, and Orthotropic Core,

(R = 2.50) ...................

K M for Sandwich Panel with Ends Simply Supported andSides Clamped, Isotropic Facings, and Orthotropic Core,

(R = 0.40) ...................

K M for Sandwich Panel with Ends Simply Supported and

Sides Clamped, Isotropic Facings, and Isotropic Core,

(R = 1.00) ...................

K M for Sandwich Panel with Ends Simply Supported andSides Clamped, Isotropic Facings, and Orthotropic Core,

(R = 2.50) ...................

Page

2-30

2-31

2 -34

2-35

2 -40

2-41

3-4

3-10

3-11

3-12

3-13

3-14

3-15

xii

Figure

3.1-8

3.1-9

3.1-10

3.1-11

3.1-12

3.1-13

3.1-14

3.1-15

3.1-16

3.1-17

3.1-18

3.1-19

3.1-20

LIST OF FIGURES, Cont'd.

K M for Sandwich Panel with Ends Clamped and Sides Simply


(R = 0.40) ...................


Supported, Isotropic Facings, and Isotropic Core,

(R = 1.00) ...................



(R = 2.50) ...................

K M for Sandwich Panel with Ends and Sides Clamped,

Isotropic Facings, and Orthotropic Core, (R = 0.40) ....


Isotropic Facings, and Isotropic Core, (R = 1.00) .....


Isotropic Facings, and Orthotropic Core, (R = 2.50) ....

K M for Simply Supported Sandwich Panel Having a

Corrugated Core. Core Corrugation Flutes are

Perpendicular to the Load Direction ..........

K M for Simply Supported Sandwich Panel Having aCorrugated Core. Core Corrugation Flutes are

Parallel to the Load Direction ............

KM o for Sandwich Panel With Isotropic Facings inEdgewise Compression ..............

K M for a Sandwich Panel with All Edges Simply Supported,

and an Isotropic Core, (R = 1.00) ...........


and an Orthotropic Core, (R = 2.50) ..........


and with an Orthotropic Core, (R = 0.40) ........


Isotropic Facings and Corrugated Core. Core CorrugationFlutes are Parallel to Side a ............

Page

3-16

3-17

3-18

3-19

3-20

3-21

3-22

3-23

3-24

3-29

3-30

3-31

3-32

xiii

Figure

3.1-21

3.1-2'2

"5.1-23

3.1-24

3.1-25

3.1-26

3 1-27

3 1-28

3 1-29

3 1-30

3 1-31

3 1-32

,,'). 1-33

3 1-34

3.1-35


K M for a Sandwich Panel with All Edges Simt)ly Supported,

Isotropic Facings and Corrugated Core. Core Corrugation

Flutes are Parallel to Side b ............

K M for a Sanchvich Panel with All Edges Clamped, Isotropic

Facings and Isotropic Core, (It = 1.00) .........

K M for a Sandwich Panel with All Edges Clamped, Isotropic

Facings and Orthotropie Core, (R = 2.50) ........

K M for a Sanchvich Panel with All Edges Clamped, Isotropie

Facings and Orthotropic Core, (R = 0.40) ........

K M for a Simply Supported Sanchvich Panel with an Isotropic

Core, (R - 1.00) ................

K M for a Simply Supporled Sandwich Panel with an Orthotropic

Core, (H = 2.50) ................

K M [or a Simply Supported San(kvich Panel with an Orthotropie

Core, (R = 0.40) ................

K M for a Simply Supported Sanchvieh Panel with Corrugated

Core. Core Corrugation Flutes Parallel In Side a .....

Interacti,m Curve for a Honeycomb Core Sanchvich Panel

Subjected to Bimxial Compression ...........

Interaction Curve for a tloneyeomb Core Sandwich Panel

Subjected to Bending and Compression .........

Interaction Curve for a Itoneyeomb Core Sand_vich Panel

Subjected to Compression and Shear ..........

Interaction Curve for a Honeycomb Core Sandwich Panel

Subjected to Bending and Shear ............

Buckling Coefficients for Corrugated Core Sandwich Panels

in Biaxial Compression (a/b = 1/2) ..........


in Biaxial Compression (a/b = 1.0) ..........


in Biaxial Compression (a/b = 2.0) ..........

Page

3-33

3-34

3-35

3-36

3-42

3-43

3 -44

3-45

3-59

3-60

3-61

3-62

3-63

3-64

3-65

xiv

LISTOF FIGURES,Cont'd.

Figure Page

3.1-36

3.1-37

3.1-38

3.1-39

3.1-40

3.1-41

3.1-42

4.1-1

4.2-2

4.2-3

4.2-4


Under Combined Longitudinal Compresmon and Shear with

Longitudinal Core (a/b = 1/2) ..........


Under Combined Longitudinal Compression and Shear with

Longitudinal Core (a/b = 1.0) ..........



Longitudinal Core (a/b = 2.0) ..........



Transverse Core (a/b = 1/2) ..........


Under Combined Longitudinal Compressloh and Shear with

Transverse Core (a/b = 1.0) ..........


Under Combined Longitudinal Compression and Shear with

Transverse Core (a/b = 2.0) ..........


Under Combined Longitudinal Compresmon, Transverse

Compression, and Shear with Longitudinal Core ......

Equilibrium Paths for Axially Compressed Circular

Cylinders ...................

Typical Equilibrium Paths for Circular Cylinders .....

Schematic Representation of Relationship Between K c and

V for 0 _ 1 ..................C

Semi-Logarithmic Plot of _/c vs R/t' for Isotropic (Non-

Sandwich) Cylinders Under Axial Compression ......

Knock-Down Factor 3Zc for Circular Sandwich Cylinders

Subjected to Axial Compression ...........

Comparison of Proposed Design Criterion Against Test

Data for Weak-Core Circular Sandwich Cylinders Sub-

jected to Axial Compression .............

3-66

3-67

3-68

3-69

3-70

3-71

3-72

4-7

4-9

4-10

4-12

XV

Figure

4.2-5

4.2-8

4.2-9

4.3-1

4.3-2

4.4-1

4.4-2

4.4-3

4.4-4

4.4-5

4.5-1


Comparison of Proposed Design Criterion Against a Test

Result for a Weak-Core Circular Sandwich Cylinder

Subjected to Axial Compression ...........

Stresses Involved in Interpretation of Test Data ......

Buckling Coefficient for Axially Compressed Circular

Sandwich Cylinders ................

Design Knock-Down Factor for Circular Sandwich

Cylinders Subjected to Axial Compression ........

Buckling Coefficient for Short Simply-Supported Sandwich

Cylinders Subjected to Axial Compression (0 = 1) ......

Knock-Down Factor _o for Circular Sandwich Cylinders

Subjected to Pure Bending ...........

Design Knock-Down Factor Mo for Circular Sandwich

Cylinders Subjected to Pure Bending ..........

Circular Sandwich Cylinder Subjected to External

Lateral Pressure ..............

Schematic Representation of Log-Log Plot of Cp Versus

L/R for Circular Sandwich Cylinders Subjected to

External Lateral Pressure .............

Buckling Coefficients Cp for Circular Sandwich Cylinders

Subjected to External Lateral Pressure; Isotropic Facings;

Transverse Shear Properties of Core Isotropie or Ortho-

tropic; Vp = 0 .................



Transverse Shear Properties of Core Isotropic or Ortho-

tropic; Vp = 0.05 ................




tropic; Vp = 0.10 ...............

Circular Sandwich Cylinder Subjected to Torsion ......

Page

4-13

4-15

_-19

4-20

4-24

4-28

4-30

4-32

4-36

4-44

4-45

4-46

4-47

xvi

Figure

4.5-2

4.5-3

4.5-4

4.5-5

4.5-6

4.5-7

4.5-8

4.7-3

4.7-4

4.7-5

4.7-6

4.7-7

4.7-8

4.7-9


Typical Log-Log Plot of the Buckling Coefficient K s for

Circular Sandwich Cylinders Subjected to Torsion .....

Buckling Coefficients for Circular Sandwich Cylinders

Subjected to Torsion ...............











Sample Interaction Curve ..............

Design Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus Bending .......



Circular Sandwich Cylinder Subjected to Axial Compression

Plus External Lateral Pressure ..........

Typical Interaction Curves for Circular Sandwich Cylinders

Subjected to Axial Compression Plus External Lateral

Pressure ...................

Interaction Curves for Circular Sandwich Cylinders Subjected

to Axial Compression Plus External Lateral Pressure ....



Interaction Curve for Circular Sandwich Cylinders Subjected




xvii

Page

4-50

4-56

4-57

4-58

4-59

4-60

4-61

4-66

4-68

4-70

4-71

4-74

4-79

4-80

4-81

4-82

Figure

4.7-10

4.7-11

4.7-12

4.7-13

4.7-14

4.7-15

4.7-16

4.7-17

4.7-18

5.1-1

5.1-2

5.2-1

5.3-1

5.3-2

5.4-1

5.5-1

5.6-1

5.6-2



to Axial Compresmon Plus External Lateral Pressure ....

Interaction Curves for Circular Sandwich Cylinders Subjectedto Axial Compress:on Plus External Lateral Pressure ....


to Axial Con:press:on Plus External Lateral Pressure ....


to Axial Con:press:on Plus External Lateral Pressure ....


to Axial Compresmon Plus External Lateral Pressure ....

Interaction C_rves for Circular Sandwich Cylinders Subjected

to Axial Compresswn Plus External Lateral Pressure ....

Circular Sandwich Cylinder Subjected to Axial Compression

Plus Torsion ..................

Conditional Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus Torsion .......

Conservative Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus Torsion .......

Empirical Knock-Down Factors ...........

Truncated Sandwich Cone Subjected to Axial Compression.

Truncated Sandwich Cone Subjected to Pure Bending .....

Truncated Cone Subjected to Uniform External LateralPressure

• , • . • o . , . . . , . . . o . • .

Truncated Sandwich Cone ..............

Truncated Sandwich Cone Subjected to Torsion ......

Truncated Cone Subjected to Transverse Shear ......

Sample Interaction Curve ..............

Design Interaction Curve for Truncated Sandwich Cones


Page

4-83

4-84

4-85

4-86

4-87

4-88

4-89

4-93

4-94

5-2

5-4

5-7

5-9

5-12

5-17

5-18

5-21

5-26

XVIII

Figure

5.6-3

5.6-6

5.6-7

5.6-8

6.1-1

6.2-1

6.2-2

6.2-3

6.2-4

7.1-3

7.1--4


Truncated Cone Subjected to Uniform External Hydrostatic

Pressure ...................

Truncated Sandwich Cone ..............

Truncated Cone Subjected to Axial Compression Plus

Torsion ...................

Conditional Interaction Curve for Truncated Sandwich

Cones Subjected to Axial Compression Plus Torsion .....

Conservative Interaction Curve for Truncated Sandwich

Cones Subjected to Axial Compression Plus Torsion .....

Truncated Cone Subjected to Axial Compression Plus

Bending Plus Torsion .............

Structural Dome Shapes ..............

Sanchvich Dome Subjected to External Pressure ......

Schematic Representation of Relationship Between Kc and V c.

Knock-Down Factor 7d for Sandwich Domes Subjected toUniform External Pressure .............

Buckling Coefficient for Sandwich Domes Subjected to

External Pressure ...............

Cylindrical Panel and Associated Flat-Plate Configuration

Schematic Logarithmic Plot of Schapitz Criterion for Non-

Sandwich Cylindrical Skin Panels ...........

Schematic Logarithmic Plot of Test Data for Cylindrical

Isotropic (Non-Sandwich) Skin Panels Under Axial

Compression ................

Graphical Representation of Equations (7.1-5) through

(7,1-8) ....................

P age

5-27

5-31

5-34

5-38

5-39

5--40

6-1

6-3

6-7

6-9

6-13

7-2

7-3

7-6

7-10

xix

Table

2-1

3'1

4-1

5-1

6-1

7-1

9-1

9-2

9-3

LISTOFTABLES

Summaryof DesignEquationsfor Local Instability Modesof Failure ...................

Summaryof DesignEquationsfor GeneralInstability ofFlat SandwichPanels ...............

Summaryof DesignEquationsfor Instability of CircularCylinders ...................

Summaryof DesignEquationsfor Instability of TruncatedCircular Cones.................

Summaryof DesignEquationsfor Instability of Dome-ShapedShells..................

Summaryof DesignEquationsfor Instability of Cylindrical,CurvedPanels .................

RecommendedPlasticity ReductionFactorsfor LocalInstability Modes ................

RecommendedPlasticity ReductionFactorsfor the GeneralInstability of Flat SanchvichPlates...........

ReeommendedPlasticity ReductionFactorsfor the GeneralInstability of Circular SandwichCylinders, TruncatedCircular SandwichCones,andAxisymmetricSandwichDomes....................

Page

2-42

3-76

4-98

5-42

6-16

7-12

9-7

9-8

9-9

XX

LIST OF SYMBOLS

aR

a

P

b

bR

bf

bP

CL

CO

CP

D

Dq

d

EC

Ef

ES

E t

Panel length, inches. Major semi-axis of an ellipse, inches.

Axial length of a cylindrical panel, inches.

Length of the fiat panel shown in Figure 7. I-i, inches.

Panel width, inches. Minor semi-axis of an ellipse, inches.

Circumferential width of a cylindrical panel, inches.

Pitch of corrugated core, inches.

Width of the flat panel shown in Figure 7. i-i, inches.

Length parameter defined by Equation (4.7-25), dimensionless.

Parameter defined by Equations (4.2-21) and (6.2-19), dimensionless.

Buckling coefficient for sandwich cylinders subjected to external

lateral pressure, dimensionless.

_(Elt 1) (E2t2)h2Bending stiffness of sandwich wall or panel =inch-lbs. )'(Eltl + E2t2)

Shear stiffness of sandwich wall or panel = h2(G )/t , lb/inch.XZ C

Total thickness of sandwich wall or panel (d = t 1 + t2 + tc), inches.

Young's modulus, psi.

Young's modulus of the core in the direction normal to the facings,

psi.

Young's modulus of facing, psi.

Secant modulus of facing, psi.

Tangent modulus of facing, psi.

xxi

E 1 , E 2

ei

FV

(Fv)cr

FC

Ge

Gca

Gcb

Gf

G..

lj

GS

GXZ

Gyz

h

K

KF

Young's moduli for facings I and 2 respectively, psi.

Strain intensity defined by Equation (9.2-2), in./in.

Transverse shear force, Ibs.

Critical transverse shear force, ibs.

Flatwise sandwich strength (the 10wcr of flatwise core compressive,

flatwise core tensile, and flatwise core-to-facing bond strengths),

psi.

Transverse shear modulus of core, psi.

Core shear modulus associated with the plane perpendicular to the

facings and parallel to side a of panel, psi.


facings and parallel to side b of panel, psi.

Elastic shear modulus of facing, psi.


facings and parallel to the direction of loading, psi.

Secant shear modulus of facing, psi.


facings and parallel to the axis of revolution of a cylinder, psi.


axis of revolution of a cylinder, psi.

Distance between middle surfaces of the two facings of a sandwich

construction, inches.

Buckling coefficient for an isotropic (non-sandwich) flat plate,

dimensionless. Buckling coefficient for fiat rectangular sandwich

pane] under edgewise compression (Kc) , edgewise shear (Ks),or

edgewise bending (Eb). K = KF + KM.

Theoretical flat panel buckling coefficient which is dependent on

facing stiffness and panel aspect ratio, dimensionless.

xxii

KM

KC

K I

c

KP

KS

K5

kX

X

kY

k I

Y

L

Le

M

Mcr

M. S.

Ncr

Theoretical fiat panel buckling coefficient which is dependent on

sandwich bending and shear rigidities, panel aspect ralio, and

applied loading, dimensionless.

Buckling coefficient for sandwich cylinders under axial compression

and sandwich domes under external pressure, dimenionlcss.

Buckling coefficient for short sandwich cylinders under axial com-

pression, dimensionless.

Parameter defined by Equation (4.4-2), dimensionless.

Buckling coefficient for sandwich cylinder subjected to torsion,

dimensionless.


Buckling coefficient, dimensionless.

Buckling coefficient associated with compressive stress acting in

the x direction, dimensionless.

Loading coefficient for applied compressive stress which is acting

in the x direction, dimensionless.

Buckling coefficient associated with compressive stress acting in the

y direction, dimensionless.

Loading coefficient for applied compressive stress which is acting in

the y direction, dimensionless.

Over-all length, inches.

Effective length, inches.

Applied bending moment, in-lbs.

Critical bending moment, in-lbs.

Margin of safety, dimensionless.

Critical compressive running load, lbs/inch.

Number of circumferential full-waves in the buckle pattern,

dimensionless.

xxiii

P

pl

pcr

(Per) Empirical

P

Pcr

(Pc r ) Te st

(P×tCL

Py

(PY)cL

Q

q

R

(Rb)cL

RC

(Re)CL

Re

Axial load, lbs.

Equivalent axial load defined by Equation (5.6-32), lbs.

Critical axial load, lbs.

Empirical lower-bound value for critical axial load when acting

alone, ibs.

External pressure, psi.

Critical value for external pressure, psi.

Experimental value for critical external pressure, psi.

Classical theoretical critical pressure for a cylinder subjected to

external pressure acting only on the end closures, psi.

External pressure acting only on the lateral surface of a cylinder,

psi.

Classical theoretical critical pressure for a cylinder subjected to

external pressure acting only on the lateral surface, psi.

The relative minimum, with respect to {, of expression (2.2-2),

dimensionless.

Quantity defined by Equation (2.2-3), dimensionless.

Degree of core shear modulus orthotropicity = G /G _,dimensionless. Radius to middle surface, inehee, a CD

Stress ratio defined by Equation (4.7-9), dimensionless.


Load, stress, or pressure ratios as defined in appropriate sections

of this handbook, dimensionless.

Stress ratios as defined in appropriate sections of this handbook,

dimensionless.

Effective radius, inches.

xxiv

R.

1

RJ

Rlarge

RMax

RP

CL

Rs

(Rs)cL

Rsmall

Rx

RY

R2

ra

T

Ter

(¥cr)Empirical

t

Stress or load ratio for the particular type of loading associated with

the subscript i, dimensionless.

Stress or load ratio for the particular type of loading associated with

the subscript j, dimensionless.

Radius to middle surface at the large end of a truncated conical shell,

measured perpendicular to the axis of revolution, inches.

Maxium radius of curvature for middle surface of a dome-shaped

shell, inches.

Pressure ratios as defined in appropriate sections of this handbook,

dimensionless.

Pressure ratio defined by Equation (4.7-15), dimensionless.

Load or stress ratios as defined in appropriate sections of this

handbook, dimensionless.


Radius to middle surface at the small end of a truncated conical shell,

measured perpendicular to the axis of revolution, inches.

Stress or load ratio corresponding to the x direction, dimensionless.

stress or load ratio corresponding to the y direction, dimensionless.

Middle-surface radius of curvature inthe plane perpendicular to the

meridian, inches.


Cell size of honeycomb core, inches.

External torque, in-lbs.

Critical external torque, in-lbs.

Empirical lower-bound value for critical torque when acting alone,

in-lbs.

Thickness, inches.

xxv

tR

te

tf

tP

t0

t 1, t 2

U

_7

C

_T

P

_7S

V

XZ

V

yz

W

WC

z

zS

Total thickness of the cylindrical panel shown in Figure 7.1-1, inches.

Thickness of core (measured in the direction narmal to the facings),

inches.

Thickness of a single facing, inches.

Total thickness of the fiat panel shown in Figure 7.1-1, inches.

Thickness of material from which corrugated core is formed, inches.

Thicknesses of the respective facings of a sandwich construction

(there is no preference as to which facing is denoted by the subscript

1 or 2), inches.

h 2

Sandwich transverse shear stiffness, defined as U = T-- Ge _ hGe'

lbs. per inch. e

2lr D

Bending and shear rigidity parameter which is defined as V -dimensionless, b 2 U

Parameter defined in Sections 4.2 and 6.2, dimensionless.





Bending and shear rigidity parameter for fiat sandwich panels with

?T2tc (Eltl)(E2t2) T?

corrugated core which is defined as W

dimensionless. Xb 2 Gcb(Eltl+E2t2 )

Running compression load, lbs/inch.



xxvi

7

"Ye

T d

(Ti)Test

Ex

EY

Exy

_Test

Angle of rotation at appropriate joint in corrugated-core sanchvich

construction (sec Figure 2.1-5), degrees. Vertex half-angle of

conical shell, degrees.

Angle of rotation at appropriate joint in corrugated-core sandwich

construction (see Figure 2.1-5), degrees.

Knock-down factor, dimensionless. Ratio = _y/a x, dimensionless.

Knock-down /actor associated with general instability under pure

bending, dimensionless.

Knock-down factor associated with general instability under axial

compression, dimensionless.

Knock-down factor associated with the general instability of a dome-

shaped shell under external pressure, dimensionless.

Knock-down factor determined from a test specimen subjected to the

loading condition corresponding to the subscript i, dimensionless.

Knock-down factor associated with general instability of a cylinder

under uniform external lateral pressure, dimensionless.

Knock-down factor associated with general instability under pure

torsion, dimensionless.

Amplitude of initial waviness in facing, inches.

Normal strain in the x direction, in/in.

Normal strain in the y direction, in/in.

Shear strain in the xy plane, in/in.

Parameter involving the core elastic moduli, core thickness, and

buckle wavelength, dimensionless.

Plasticity reduction factor, dimensionless.

Plasticity reduction factor corresponding to an experimental critical

stress value, dimensionless.

(1 - PaPb ) = (1 - p_) for isotropic facings, dimensionless. Ratio =

r/_×, dimensionless.

xxvii

ve

(l

_b

((_b)c L

_CL

c

(_!

e

((_c)b

(%)c

(_c)CL

(Ycr

crtest

!(I

crtest

Ratio of transverse shear moduli of core Esee Equation (4.2-1)],dimensionless.

Actual Poisson's ratio of facing, dimensionless.

Elastic Poisson's ratio of facing, dimensionless.

Radius of gyration for shell wall of sandwich and non-sandwich

constructions (p _ h/2 for sandwich constructions whose two

facings are of equal thickness), inches.

Stress, psi.

Peak compressive stress due solely to an applied bending moment,psi.

Classical theoretical value for the critical peak compressive stress

under a bending moment acting alone, psi.

Classical value of critical stress, psi.

Uniform compressive stress due solely to an applied axial load,psi.

Effective compressive stress defined by Equation (4.7-38), psi.

Peak axial compressive stress due solely to an applied bendingmoment, psi.

Uniform axial compressive stress due solely to an applied axialload, psi.

Classical theoretical value for the critical uniform compressive

stress under an axial load acting alone, psi.

Critical stress, psi.

Experimental critical stress obtained from a particular test speci-men, psi.

Ex_perimental critical stress which would have been attained had

the test specimen remained elastic, psi.

xxviii

ffcr

x

crimp

1

aH

M

{YMax

O'MI N

predicted

o"P

a R

(ywr

(yX

0 -I

X

(ax)b

CL

(zx)C

Y

Critical value for the compressive stress acting in the x direction,

psi.

Compressive stresses in facings 1 and 2, respectively, in the pres-

ence of the critical loading for general instability (there is no prefer-

ence as to which facing is denoted by the subscript 1 or 2), psi.

Uniaxial compressive stress at which shear crimping occurs in sand-

wich constructions, psi.

Stress intensity defined by Equation (9.2-1), psi.

ttoop membrane stress, psi.

Meridional membrane stress, psi.

Maximum possible critical stress corresponding to a particular

material, psi.

Minimum value of stress for the post-buckling equilibrium path, psi.

Predicted value for critical stress, psi.

Critical buckling stress for a flat plate, psi.

Critical buckling stress for a complete cylinder, psi.

Facing wrinkling stress, psi.

Stress acting in the x direction, psi. Uniform axial compressive

stress due to an applied axial load, psi.

Effective compressive stress defined by Equation (4.7-37), psi.

Peak axial compressive stress due solely to an applied bending

moment, psi.

Classical theoretical value for critical uniform axial compressive

stress when acting alone, psi.

Uniform axial compressive stress due solely to an applied axial

load, psi.

Stress acting in the y direction, psi.

xxix

T

IT

(T)C L

Tcr

IT

cr

Tcrimp

T T

T V

Shear stress, psi.

Effective shear stress defined by Equation (4.7-39), psi.

Classical theoretical value for critical uniform shear stress when

acting alone, psi.

Critical shear stress, psi.

Critical shear stress for an equivalent cylinder subjected to an

applied torque, psi.

Pure shear stress, acting coplanar with the facings, at which shear

crimping occurs in sandwich constructions, psi.

Uniform shear stress due solely to an applied torque, psi.

Peak shear stress due solely to an applied transverse shear force,

psi.

Angular dimension of corrugated core (see Figure 2.1-4), degrees.

Quantity defined by Equation (4.2-10), dimensionless.

Angle of rotation at appropriate joint in corrugated-core sandwich

construction (see Figure 2.1-5), degrees. Parameter defined by

Equation (4.4-3), dimensionless.

XXX

CONVERSION OF U.S. CUSTOMARY UNITS TO THE

INTERNATIONAL SYSTEM OF UNITS 1

(Reference: MIL-HDBK-23)

Quantity

U.S. Customary ConversionUnit Factor 2 SI Unit

Density

Length

Stress

Pressure

IElastieityModuli _Rigidity

Tempe r atu re

The rmal conductivity

lbm/in. 27.68 × 103

lbm/ff 3 16.02

ft 0.3048in. 0.02 54

psi 6. 895 x 103

lb/in. 6. 895 x 103

lb/ft 2 47.88

3psi 6. 895 x 10

(° F + 460) 5/9

Btu in./hr ft 2 ° F 0. 1240

kilograms/meter3 (kg/m33)kilograms/meter ;_ (kg/m)

meters (m)

meters (m)

newtons/meter 2 (N/m 2)




degrees Kelvin (°K)

kg cal/hr m °C

Prefixes to indicate multiples of units are as follows:

Prefix Multiple

giga (G) 109

mcga (M) 10 6

kilo (k) 103

-3milli (m) 10

micro _) 10 -6

1The International System of Units [Syste'me International (S1)] was adopted by the

Eleventh General Conference on Weights and Measures, Paris, October 1960, in

Resolution No. 12.

2Multiply value given in U.S. Customary Unit by conversion factor to obtain

equivalent value in SI unit.

xxxi

1INTRODUCTION

].1 GENERAL

This handbook presents practical methods for the structural stabilityanalysis of

sandwich plates and shells. The configurations and loading conditions covered here

are those which are like]yto be encountered in aerospace applications. Basic equa-

tions, design curves, and comparisons of theory against test data are included.

For the purposes of this handbook, a structural sandwich is defined as a layered

construction formed by bonding two thin facings to a comparatively thick core as

depicted in Figure 1.1-1. The facings provide practically all of the over-all bending

and in-plane extensional rigidity to the sandwich. The core serves to position the

faces at locations removed from the neutral axis, provides virtually all of the trans-

verse shear rigidity of the sandwich, and stabilizes the facings against local buckling.

Thus the structural sandwich concept is quite similar to that of a conventional I

beam. The sandwich core plays a role which is analogous to that of the I beam web

while the sandwich facings perform a function very much like that of the I beam

flanges. The primary difference between these two types of construction lies in the

Numbers in brackets [ ] in the text denote references listed at end of each major

section (1; 2; etc.).

1-1

fact that the transverse shear deflections are usually significant to the sandwich

behavior; whereas, for I beams, these deflections are only important for the special

case of relatively short, deep beams.

JFACING"

Figure 1.1-1. Typical Sandwich Construction

The sandwich is an attractive structural design concept since, by the proper choice

of materials and geometry, constructions having high ratios of stiffness-to-weight

can be achieved. Since rigidity is required to prevent structural instability, the

sandwich is particularly well suited to applications where the loading conditions are

conducive to buckling.

The use of sandwich construction in aerospace vehicles is certainly not a recent

innovation. The British de Havilland Mosquito bomber of World War II employed

1-2

structural sandwich throughout the airframe. In this case, the sandwich was in the

form of birch face sheets bonded to a balsa wood core. Many other airplanes, includ-

ing the B-58, B-70, F-Ill, C-5A, etc., have taken advantage of the high strength-to-

weight ratio enjoyed by sandwich construction. Space vehicle applications have

included the Apollo spacecraft, the Spacecraft LM Adapter (SLA) fairings on the

Centaur and other launch vehicles, as well as propellant tank bulkheads.

In view of the ever increasing application of structural sandwich, it has become desir-

able to assemble a handbook which presents latest design and analysis criteria for the

stability of such construction. The practicing designer and stress analyst need this

information in a form suitable for easy, rapid use. This document is meant to fulfill

that need. However, it should be kept in mind that, in many areas, all practical

problems have not yet been fully resolved and one can only employ what might be re-

ferred to as a "best-available" approach. In these cases it is advisable to supplement

numerical computations with suitable testing. Such areas of uncertainty are identified

in this handbook in the sections dealing with the appropriate configurations and loading

conditions.

In the sections to follow a discussion is given of the basic principles behind the design

equations along with conclusions derived from an analysis of available test data. This

is followed by the design equations along with any limitations on their use. Also, to

facilitate their use, a table of these equations and restrictions immediately precedes

the list of references in Sections 2, 3, 4, and 5 since these sections cover a wide

range of loading conditions and considerations.

1-3

1.2 FAILUREMODES

Structuralinstability of a sandwichconstructioncanmanifestitself in a numberof

differentmodes. Thevariouspossibilitiesare asdescribedbelowandasshownin

Figures1.2-1 through1.2-3.

Intracellular Buckling (Face Dimpling) - This is a localized mode of instability

which occurs only when the core is not continuous. As depicted in Figure 1.2-1, in

the regions directly above core cells (such as those of a honeycomb core), the

facings buckle in plate-like fashion with the cell walls acting as edge supports. The

progressive growth of these buckles can eventually precipitate the buckling mode

identified below as face wrinkling.

Face Wrinkling - This is a localized mode of instability which manifests itself in the

form of short wavelengths in the facings, is not confined to individual cells of

cellular-type cores, and involves the transverse (normal to facings) straining of the

core material. As shown in Figure 1.2-1, one must consider the possible occurrence

of wrinkles which may be either symmetrical or antisymmetrical with respect to the

middle surface of the original undeformed sandwich. As shown in Figure 1.2-2,

final failure from wrinkling will usually result either from crushing of the core,

tensile rupture of the core, or tensile rupture of the core-to-facing bond. However,

if proper care is exercised in the selection of the adhesive system, one can reason-

ably assume that the tensile bond strength will exceed both the tensile and com-

pressive strengths of the core proper.

1-4

SYMMETRIC

B - TensileRuptureof Bond

Figure 1.2-2.

A - Intraceilular Buckling (Face Dimpling)

ANTISYMMETRIC

B - Face Wrinkling

C - Shear Crimping

Figure 1°2-1, Localized Instability Modes

A - Core Crushing

C - Tensile Rupture

of Core Proper

Ultimate Failures Precipitated by Face Wrinkling

1-5

.mm_'

..--m_

D"--'4

i

C

4_

I

,ut|

,.,_|

"_Sl

L_

!

o=Z

1-6

Shear Crimping - Shear crimping is often referred to as a local mode of failure but

is actually a special form of general instability for which the buckle wavelength is

very short due to a low transverse shear modulus for the core. This phenomenon

occurs quite suddenly and usually causes the core to fail in shear; however, it may

also cause a shear failure in the core-to-facing bond. Crimping will sometimes

occur in cases where relatively long-wave general instability first develops. In such

instances the crimp appears because of severe local transverse shear stresses at

the ends of buckle patterns. As the crimp develops, the general buckle may dis-

appear and a post-test examination would then lead to an erroneous conclusion as to

the mechanism which initiated failure.

General Instability - For configurations having no supplementary stiffening (such as

rings) except at the boundaries, the general instability mode is depicted in Figure

1.2-3A. The phenomenon involves over-all bending of the composite wall coupled

with transverse (normal to facings) shear deformations. Usually, transverse exten-

sional strains do not play a significant role in this behavior. Whereas intracellular

buckling and wrinkling are localized phenomena, general instability is of a more

gross nature. Except for the special case cited under the identification "Shear

Crimping", the wavelengths associated with general instability are normally con-

siderably larger than those encountered in intracellular buckling and face wrinkling.

For configurations having supplementary stiffening at locations other than the bound-

aries, the term general instability takes on new significance and reference is also

made to an additional mode identified as panel instability. For this case, general

1-7

instability is asdefinedabovebut with theaddedprovisionthat thebucklepattern

involvessimultaneousradial displacementof boththesandwichwall andtheinter-

mediatestiffeners. As shownin Figure 1.2-3B, theappropriatehalf-wavelengthof

thebucklepatternmustthereforeexceedthe spacingbetweenintermediatestiffeners.

Theexampleusedin Figure 1.2-3B is thatof a sandwichcylinder stiffenedby a

seriesof rings whichhaveinsufficientstiffnessto enforcenodalpointsat their re-

spectivelocations.

Panel Instability - This mode of instability applies only to configurations which have

supplementary stiffening at locations other than the boundaries. Figure 1.2-3C

depicts this mode by again using the example of a sandwich cylinder stiffened by a

series of rings. However, in this case the rings have sufficient stiffness to enforce

nodal points at their respective locations. The rings experience no radial deforma-

tion. Therefore, the half-wavelength of the buckle pattern cannot exceed the spacing

between rings. As in the case of general instability, this mode involves over-all

bending of the composite wall coupled with transverse shear deformations. Here

again, transverse extensional strains do not play a significant role in the behavior.

1-8

2LOCAL INSTABILITY

2.1 INTRACELLULAR BUCKLING (Face Dimpling)

2.1.1 Sandwich with Honeycomb Core

2.1.1.1 Basic Principles

From a practical viewpoint, intracellular buckling can be regarded as flat-plate

behavior. Even where curvature is present, as in the cases of cylinders and spheres,

the honeycomb core cell size will normally be sufficiently small to justify such an

assumption. As noted from Reference 2-1, the critical stress for flat plates can be

expressed in the form

whe re

%r

k

Ef

ve

Crcr - 12(1-v0) _s (2.1-1)

= Critical compressive stress, psi.

= Coefficient which depends on the plate geometry, boundary

conditions, and type of loading, dimensionless.

= Plasticity reduction factor, dimensionless.

= Young's modulus (for facing material in the case of intra-

cellular buckling), psi.

= Elastic Poisson's ratio (for facing material in the case of

intracellular buckling), dimensionless.

2-1

tf = Thicknessof plate (Facingthicknessin the caseof intra-cellular buckling), inches.

s - A selectedcharacteristicdimensionof theplate, inches.

It is convenienthereto combineseveralof theconstantsin Equation(2.1-1) to obtain

r/Ef _tf_ _ (2.1-2O_r K (1- Vea )

)\s/

or

%r (l-re)= K (2.1-3)

_TEf

To apply these equations to the case of intracellular buckling, it is only necessary to

define the dimension s and establish a corresponding value for K. In Reference

2-2, Norris took s to be equal to the honeycomb core cell size. By convention,

this is taken equal to the diameter of the largest circle that can be inscribed within

the cell. Based on the analysis of test data, Norris then chose K = 2.0 for the

case of uniaxial compression. This provides a reasonably good fit to the test results

as shown in Figure 2.1-1 which was taken directly from Reference 2-2. It should be

noted that the choice of K = 2.0 does not provide a lower bound to the data. Six of

the test results fall significantly below the values predicted by the recommended

formula. This situation can be tolerated since the dimpling of several cells in a

honeycomb sandwich construction will not lead to catastrophic failure so long as a

sufficiently large number of cells remain unbuckled. As indicated by the scatter in

Figure 2.1-1, one could reasonably expect the majority of unbuckled cells to possess

considerably greater buckling strengths than would be indicated by the proposed

design curve. Under these conditions, some redistribution of stress would occur

2-2

0.1

0.01

0.001

0.00050.01

Theoryfor Simply_Suppor_d SquarePla_

Legend:Dimplingin Elastic Range• TestDatafor SpruceCorewith

SingleCircular Hole• Test Data for Honeycomb Core

Dimpling Beyond Elastic Range

O Test Data for Spruce Core with

Single Circular Hole

[2 Test Data for Honeycomb Core

1.0

Figure 2.1-1. Critical Stresses for Intracellular Buckling

Under Uniaxial Compression

2-3

but thestructurecouldcontinueto supporttheappliedload. In addition, it is pointed

out that thedimpledregionsretain significantpost-bucklingload-carryingcapability

sincetheybehaveessentiallyasflat plates. This doesnotmean,however,that one

canpermit thedimplesto growwithoutbound. Thepointcanbe reachedwherethese

deformationsprecipitatewrinkling andthis cannotbe tolerated.

It is alsoof importanceto noteherethat mostof thetestdatashownin Figure 2.1-1

wereobtainedfrom sandwichplateshavinga solid sprucecore throughwhicha

singlecircular holewasdrilled to representa corecell. It is questionablethat

suchspecimenstruly simulatethecell edgesupportlikely to beencounteredin

practical honeycombconfigurations. Onlythreedatapointswereobtainedfor speci-

mensactuallyhavinghoneycombcoresand,as shownin Figure 2.1-1, thesepoints

lie in the lower regionof thetotal bandof scatter.

In viewof theforegoingdiscussion,it is evidentthat theuseof Equation2.1-3

togetherwith the selectionof K = 2.0 is certainly not a rigorous approach to the

analysis of intracellular buckling. However, until further work is accomplished in

this area, it is recommended that this criterion be employed as a 'best-available",

approximate design tool.

2-4

2.1.1.2 DesignEquationsandCurves

Thefacingstress atwhich intracellularbucklingwill occurunderuniaxialcompres-

sionis givenby thefollowingsemi-empiricalformula:

_Ef (_)__cr = 2.0 (l_Pe_) (2.1-4)

The dimension s is the diameter of the largest circle that can be inscribed within

the cell shape. For example, in the cases of hexagonal and square cells, s is

measured as shown below.

@Figure 2.1-2. Definition of Dimension s

Solving Equation (2.1-4) for s gives the result

1

[_cr (1- Ve_)] -gs = tf ,,_ _ "j(2.1-5)

This equation may be used to determine the maximum permissible cell size corre-

sponding to particular facing materials and thicknesses. Figure 2.1-3 presents a

family of plots of Equation (2.1-5) for selected values of tf ranging from tf =

0.001 to tf = 0.100.

For elastic cases, use _? = 1. Whenever the behavior is inelastic, the methods of

Section 9 must be employed.

2-5

2.00

1.00

0.80

0.60

0.40

0.30

0.04

0.03

0.02

1 2

"0Ef

Figure 2.1-'3. Chart for Determination of Core Cell Size Such That

Intracellular Buckling Will Not Occur

2-6

When the facings are subjected to biaxial compression, it is recommended that one

use the interaction formula

whe re

R x + Ry = 1 (2.1-6)

Applied Compressive Loading]in Subscript Direction J

Ri = [Critical Compressive Loading (when] (2.1-7)

[acting alone) in Subscript DirectionJ

This straight-line interaction relationship is based on the information provided in

Reference 2-1 for square flat plates. For cases involving shearing stresses which

are coplanar with the facings, it is recommended that the principal stresses first be

computed and that these values then be used in the above interaction equation. When-

ever one of the principal stresses is tensile and the behavior is elastic, the analysis

should be based on the assumption that the compressive principal stress is acting

alone.

2-7

2.1.2 SandwichWith CorrugatedCore

2.1.2.1 BasicPrinciples

This sectiondealswith corrugated-coresandwichconstructionswhosecrosssections

maybe idealizedas shownin Figure2.1-4. For cylinders, the only case treated

here is that where the axis of the corrugations is parallel to the axis of revolution.

For flat plates, however, the corrugations can be oriented in either the longitudinal

or transverse directions.

Single-Truss Doub le-T russ

Figure 2.1-4. Corrugation Configurations

Each of the following loading conditions is considered:

a. Uniaxial compression acting parallel to the axis of the corrugations.

b. Uniaxial compression acting parallel to the facings but normal to the

axis of the corrugations.

c. Biaxial compression resulting from combinations of a and b above.

The design curves presented here are taken directly from Reference 2-3 and are

based entirely on theoretical considerations. No comparisons are made against test

2-8

datato confirm thevalidity of thesesolutions. Until suchsubstantiationis obtained,

therecommendeddesigncurvescanonlybe consideredas a 'best-available"criterion.

It is pointedout, however,thattheredoesnotappearto beanyreasonto suspectthat

testdatawoulddisagreewith thecurves.

AlthoughReference2-3 is devotedsolelyto flat plates, theresults areconsideredto

beapplicableto thecylindrical configurationsshownin Figure 2.1-4 sincethedimen-

sions bf will usuallybe smallwith respectto theradius. Undersuchconditions,

curvatureinfluenceswill benegligible.

Thetheoreticaldevelopmentincludesconsiderationof eachof thebucklingmodes

shownin Figure 2.1-5. Bothof thefollowingpossibilitiesare covered:

a. Thefacesheetsare theunstableelementsandare restrainedby thecore.

b. Thecore is theunstableelementandis restrainedby thefacesheets.

Bucklingis assumedto beaccompaniedby rotationof the joints butwith nodeflection

of the joints. Theanglesbetweenthevariouselementsat anyonejoint are takento

remainunchangedduringbuckling. It is alsoassumedthattheover-all sandwich

dimensionsare sufficientlylarge suchthatendeffectsare negligible.

2-9

cl -oL Q ci 01

Clomped

oL o_ cl ol ol

-cl ci -_ Cl -cl cl

rz _1 i1 ol cl

-el - rx rl -_ _1 cl

,gl o. -el. i::i -oI

-_ cl -cl ol -cI,

__ply t

Single-T russ-Core Double-Truss-Core

(a, fl, and q_ denote angles of rotations at the appropriate joints)

Figure 2.1-5. Buckling Modes

2-10


Thetheoreticalstress at whichintracellularbucklingof thefacingsor bucklingof the

corrugatedcorewill occuris givenby thefollowingformula:

kl zru 77E /tf_ _ (2.1-8)

°_cr - 12(1-V0) \bf]

where

%r

ki

= Critical compressive stress, psi.

= Coefficient which depends upon the geometry and loading

conditions, dimensionless.

_7 = Plasticity reduction factor, dimensionless.

E = Young's modulus of facings and core, psi.

re = Elastic Poisson's ratio of facings and core, psi.

tf = Facing thickness, inches.

bf = Pitch of corrugated core (see Figure 2.1-4), inches.

The only case considered here is that where the two facings are of the same thickness

and the entire sandwich construction (facings and core) is made of a single material.

Figures 2.1-6 through 2.1-12 give values for k i for each of the following loading

combinations:

a. kx when k_ = 0

b. kx when k_ = 0.5

c. kx when k_ = 1.0

d. ky when kx = 0

The coefficients k_ and k_ are defined as follows:

2-11

12(1-Ve _ ) (bf._ _

kx - rr _rIE _-f_ (Applied Compressive ¢rx) (2. I-9)

12(1- re2 ) (Applied Compressive O'y) (2.1-10)ky - _ 17E

The subscript x (for k and k') is used to identify cases where the loading is

directed along the axis of the corrugations (x direction). The subscript y (for k

and k') is used to identify cases where the loading is acting in the y direction which

is parallel to the facings but normal to the axis of the corrugations. For combinations

a through c, separate plots are furnished for single-truss-core and double-truss-

core configurations. For combination d, a single family of curves covers both

arrangements since all of the corresponding applied load is transferred through the

facings. The dashed lines in Figures 2.1-6 through 2.1-11 divide the charts into two

regions. Above the dashed lines, the face sheets are the unstable elements and are

restrained by the core. Below the dashed lines, the core is unstable and is restrained

by the face sheets.

To clarify the design charts given in Figures 2.1-6 through 2.1-12, the following

additional definitions are provided:

to = Thickness of material from which the corrugations are formed

(see Figure 2.1-4), inches.

_b = Angle shown in Figure 2.1-4, degrees.

In addition, the sample problem given below should be helpful to the user of this

handbook.

2-12

Given: SampleProblemData for Single-TrussCore Type SandwichPanel

E = 30 x 106p si to = .016" bf = .700"

ve = .30 tf = .020" ¢ = 65 °

Proportional Limit a= 90,000 psi (ry = 16,300 psi (Compression)

Required: Find acr x ; Assuming _? = 1, one obtains

12ay(1-re2) _k_2 = 12 x 16,300 x ,910 _.70012

k_ = 7r2_E \tf / 9.87 x 1 x 30 x I0' \.020/ = 0.736

%=.o16

tf .020= .800

Using linear interpolation between values given on Figures 2.1-7 and 2.1-8 one

obtains kx = 2.68.

Hence, the critical stress in the x direction (parallel to the corrugation axis) is

kx _2_ E /tf_ 2

aCrx - 12(1-Ve2 ) _bf/

and, assuming _ = 1, one obtains

_Crx 2,68 x 9,87 x 1 x 30 x 106f.O20h 2= 12 x . 910 \. 700/ = 59,300 psi (Compression)

The stress intensity ai (See Section 9) can now be computed as follows:

o"i = _/O-x2 + O'y2 - ax(_y + 31"2

= 103_ f (59.3) 2 + (16.3) 2 - (59.3 x 16.3) + 0 = 53,100 psi

Since this value is below the proportional limit, the assumption _ = 1 is valid.

In cases where the qi value exceeds the proportional limit, the methods of Section

9 must be employed.

2-13

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2-17

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2-20

2.2 FACE WRINKLING

2.2.1 Sandwich With Solid or Foam Core (Antisymmetric Wrinkling)


The problem of face wrinkling has been treated by many investigators dating back as

far as 1940. The most important publications on this subject are listed as References

2-4 through 2-14. For the purposes of this handbook, it was decided that the results

in References 2-7 and 2-9 would be the most useful. The latter applies only to sand-

wich configurations which have solid or foam cores. The development there includes

consideration of both the symmetric and antisymmetric modes along with the influences

from initial waviness of the facings. It is pointed out that, when the core is sufficiently

thick, the wrinkle patterns of the two facings will be independent of each other and the

same critical load is obtained for the symmetric and antisymmetric modes. However,

for sandwiches having thinner cores, the core strains introduced by one facing influ-

ence the wave pattern in the other facing. Under these conditions, it was found that

sandwiches having solid or foam cores can be expected to wrinkle antisymmetrically.

The following governing equation was derived to predict this form of wrinkling for

isotropic facings subjected to uniaxial compression:

1

c_ _r]Ef Ec Gc_ _

{rwr : _L (1-Pea)_ (2.2-1)

whe re,

Crwr = Facing wrinkling stress, psi.

17 = Plasticity reduction factor, dimensionless.

Ef = Young's modulus of facing, psi.

2-21

E c

Gc

_e

The quantity Q

= Young's modulus of the core in the direction normal to the facings, psi.


facings and parallel to the direction of the applied load, psi.

- Elastic Poisson's ratio of facings, dimensionless.

is the relative minimum, with respect to _ , of the expression

_ 16q ( cosh _- 1 )30V ÷ sinh 5(2.2-2)

( cosh_- 1 )1 4 6.4 KS_ 11 sinh _4 5

whe re

and

tc I (1-Ve_)13q = _f Gc LrlEf Ec Gc] (2.2-3)

6Ec (2.2-4)K8 - tc Fc

- Parameter involving the core elastic moduli, core thickness, and

buckle wavelength, dimensionless.

t c Thickness of core, inches.

tf : Thickness of facing, inches.

6 : Amplitude of initial waviness in facing, inches.

F c : Flatwise sandwich strength (the lower of flatwise core compressive,

flatwise core tensile, and flatwise core-to-facing bond strengths),

psi.

The initial waviness plays an important role in the wrinkling phenomenon since it

causes transverse facing deflections to develop even when the applied loading is very

small. As the load increases, these deflections grow at steadily increasing rates and

lead to transverse tensile or compressive failure of the core or tensile rupture of the

2-22

core-to-facingbond. Thesefailures occur, of course, at loadvaluesbelowthepre-

dictionsfrom classical theoryin whichit is assumedthat thefacingsare initially

perfect (K6 = 0).

The results from Reference 2-9 can be summarized in the form of Equation (2.2-1)

accompanied by plots of Q vs q with K 8 as a parameter. A family of such curves

is given in Reference 2-9 and they are of the general shape shown in Figure 2.2-1.

The limiting values established by the straight line 0A correspond to the shear

crimping mode of failure (see Section 2.3). All other points on the curves al"e for

antisymmetric wrinkling. In actual practice, curves of this type do not prove to be

very helpful since the K 8 values appropriate to particular structures are rarely

known. Therefore, in order to provide a practical means for the prediction of face

wrinkling in sandwich constructions having solid or foam cores, it has become com-

mon practice to select a single conservative lower-bound Q based on available test

data. This approach is followed here. Elastic test data selected from Reference 2-9

are plotted in Figure 2.2-2 from which the value Q = 0.50 has been selected as a

safe design value. Additional data are given in Reference 2-6 which are not shown

here but lead to the same value for a lower-bound Q. This is in conformance with

the observation made by Plantema in Reference 2-15 that the value Q = 0.50 has

often been recommended for practical design purposes. However, since much of the

existing test data were obtained from specimens that were not very representative of

configurations likely to be encountered in realistic structures, the selection of Q = o. 50

can only be regarded as a "best-available" approach. In view of the uncertainties

2-23

Q

0q

Figure 2.2-1. Typical Variation of Q vs. q

K 6 = 0

K 8 = Constant

K 6 = Constant

involved, it is recommended that for the verification of final designs, wrinkling tests

be performed on specimens which are truly representative of the actual configuration.

The method presented here for the prediction of wrinkling should only be regarded as

an approximate guideline.

2-24

o

.i

0O

O

0

°,-_

O

o

0 ¢.j

Qo ¢.0 5'qt

2-25

2.2.1.2 Design Equations and Culwes

The following equation may be used to compute the approximate uniaxial compressive

stress at which face wrinkling will occur in sandwich constructions having solid or

foam cores :

awr = Q L (1-pe 2 (2.2-5)

In cases where the amplitude of initial waviness is known, one can use the curves of

Figure 2.2-3 to establish Q. Whenever such information is unavailable, it is recom-

mended that the value Q = 0.50 be used to obtain a lower-bound prediction.

For elastic cases, use }7= 1. Whenever the behavior is inelastic, the methods of


When the facings are subjected to biaxial compression, it is recommended that one use

the interaction formula

Rxa + Ry = 1 (2.2-6)whe re

[Applied Compressive Loading in Subscript Direction]

Ri= iCritieal Compressive Loading (when acting alone) in ? (2.2-7)Subscript Direction

and the y direction corresponds to the direction of maximum compression. This inter-

action relationship is based on the information provided in Reference 2-1 for rectangular

flat plates having very large aspect ratios. For cases involving shearing stresses which

are coplanar with the facings, it is recommended that the principal stresses first be

computed and that these values then be used in the above interaction equation. When-

ever one of the principal stresses is tensile and the behavior is elastic, the analysis

should be based on the assumption that the compressive principal stress is acting alone.

2-26

\

O0L"-

I

///

(y

0

_o

2_

d

2-27

2.2.2 Sandwich With Honeycomb Core (Symmetric Wrinkling)


As noted in Section 2.2.1.1, the results of Reference 2-9 apply only to sandwich con-

figurations which have solid or foam cores. However, the basic theory of that report

is capable of extension to constructions having honeycomb cores and this is accomplished

in Reference 2-7. The extension is achieved by incorporating conditions which recog-

nize that the honeycomb core elastic moduli in the plane parallel to the facings are

very small in comparison with the core elastic moduli in the direction normal to the

facings. Full consideration was given to both the symmetric and antisymmetric wrin-

kling modes along with the influences from initial waviness of the facings. However,

in this case it was found that, except for the region controlled by shear crimping (low q),

symmetric wrinkling develops at stress levels which are lower than those at which the

antisymmetric mode will occur. Based on this observation, the development of Refer-

ence 2-7 resulted in the following equation for the prediction of wrinkling for isotropic

facings in sandwich constructions having honeycomb cores and subjected to uniaxial

compression:

where

1

082(Ec ( Ef,awr = . \r/Ef te/

1 + 0.64 K 8 (2.2-8)

8 E c

K8 - tc Fc (2.2-9)

2-28

and

awr = Facingwrinklingstress, psi.

Ec = Young'smodulusof thecore in thedirectionnormal tothefacings,psi.

tf = Thickness of facing, inches.

77 = Plasticity reduction factor, dimensionless.

Ef = YoungTs modulus of facing, psi.

t c = Thickness of core, inches.

8 = Amplitude of initial waviness in facing, inches.

F c = Flatwise sandwich strength (the lower of flatwise core compres-sive, flatwise core tensile, and flatwise core-to-facing bond

strengths), psi.

Equation (2.2-8) can be used to plot a family of design curves of the form shown in

Figure 2.2-4. It should be noted that the curve for K 8 = 0 is an upper-bound classi-

cal value which is based on the assumption that the facings are initially perfect. This

particular curve agrees very closely with the following symmetrical wrinkling equation

recently obtained by Bartelds and Mayers [2-14] :

awr = 0.86 L_TEf tc j (r/Ef) (2.2-10)

Comparison of Equations (2.2-8) and (2.2-10) shows that, when K8 = 0, the former

gives critical stresses which are approximately 5 percent less than those obtained by

Bartelds and Mayers [2-14].

Numbers in brackets [ ] in the text denote references listed at end of each major

section (1; 2; etc.)

2-29

Figure 2.2-4.

K(_ = 0

= Constant

K 6 = Constant

i

_TEf t c ]

Typical Design Curves for Face Wrinkling" in SandwichConstructions Having Honeycomb Cores

In actual practice, curves of the type shown in Figure 2.2-4 do not prove to be very

helpful since the K5 values appropriate to particular structures are rarely known.

Therefore, in order to provide a practical means for the prediction of face wrinkling

in sandwich constructions having honeycomb cores, a lower-bound approach is taken

in this handbook. For this purpose, test data selected from References 2-7 and 2-10

are plotted in Figure 2.2-5. All of the specimens from Reference 2-7 failed within the

elastic range. Several of these failures occurred by means of shear crimping and

these data were discarded. For the remaining tests reported in Reference 2-7, three

data points are plotted in Figure 2.2-5 for each group of nominally identical specimens.

One point is plotted for the maximum test value for the group, one point for the mini-

mum, and one point for the average. The data from Reference 2-10 were selected in

a similar manner with several added restrictions. A number of these specimens

wrinkled under highly inelastic conditions. Since rather crude plasticity reduction

2-30

00o0

o

oo

o

o

°1°l0

0 o

_-' 0

_._

_=

e_o

or/3

•C_ '_

_._

d

2-31

factors (77 Et/Ef) wereusedin thedatareduction, it wasdecidedto plotdataonly

for thosespecimenswhichwrinkledat stress levelswhere (E t/Ef ) /> 0.85. In addi-

tion, many of the test specimens of Reference 2-10 had very poor eore-to-faeingbonds

as measured by flatwise tensile strengths. It was therefor(, decided to plot data only

for those specimens whose flatwise tensile strengths were at least equal to the flatwise

compressive strengths. Adhesive technology has now advanced to .the point where, with

proper care, one can usually select an adhesive system which satisfies such a require-

ment.

Based on the plot of Figure 2.2-5, the relationship

1

/Eetf _

O'w r = 0.33\r/E f te / (_TSf) (2.2-11)

has been selected here to provide safe design values. This implies that a knock-down

factor of approximately 0.4 is applicable to this wrinkling phenomenon. Obviously,

this is not a rigorous approach to the problem and it would be advisable to base the

design equation on a much wider selection of test data of specimens which were truly

representative of contemporary practical designs. Therefore, Equation (2.2-11) can

only be regarded as a '_est-aw_ilable" approach and it is recommended that, for veri-

fication of final designs, wrinkling tests be performed on specimens that actually dup-

licate the selected sandwich configuration. The method presented here should only be

regarded as an approximate guideline.

2-32

2.2.2.2 Design Equations and Curves

The following equation may be used to compute the approximate uniaxial compressive

stress at which face wrinkling will occur in sandwich constructions having honeycomb

whe re

cores:

z/Ec 0.8 tc/ (,El)

°'wr- 1 + 0.64 K(5 (2.2-12)

(]Ec

K6 - tc Fc (2.2-13)

In cases where the amplitude of initial waviness is known, one can either use these

equations or the curves given in Figures 2.2-6 and 2.2-7 to establish the wrinkling

stress. Both of these figures are taken directly from MIL-HDBK-23 [2-16]. When-

ever the initial waviness is unknown, it is recommended that the following equation be

used to obtain a lower-bound prediction:

awr = 0.33 \_--_f tc/ (UEf) (2.2-14)

For elastic cases, use 77 = 1. Whenever the behavior is inelastic, the methods of


When the facings are subjected to biaxial compression, it is recommended that one use

the interaction formula

where

Ri=

3

R x + Ry = 1 (2.2-15)

[Applied Compressive Loading in Subscript Direction]

Critical Compressive Loading (when acting alone) in]Subscript Direction

(2.2-16)

2-33

i

/// /, ///

z.//, /

CD 0 0 0 0 0 0 0 0 O0

4hl/,/lll "_oCD

LO

CO00

0C_f

jjz//

_I_

o

I>

!

&

2-34

0

II

0

?]

11

.....

!

b b _J

¢£:

0

0oi

and the y direction corresponds to the direction of maximum compression. This

interaction relationship is based on the information provided in Reference 2-1 for

rectangular flat plates having very large aspect ratios. For cases involving shearing

stresses which are coplanar with the facings, it is recommended that the principal

stresses first be computed and that these values then be used in the above interaction

equation. Whenever one of the principal stresses is tensile and the behavior is elastic,

the analysis should be based on the assumption that the compressive principal stress

is acting alone.

2-36

2.3 SHEARCRIMPING

2.3.1 BasicPrinciples

To understandthephenomenonof shearcrimping, onemustkeepin mindthatthis

modeof failure is simplya limiting caseof generalinstability. Theequationsfor

predictingshearcrimpingemergefrom generalinstability theorywhentheanalytical

treatmentextendsinto theregionof lowshearmodulifor thecore. For example,the

theoreticalderivationof Reference2-17, as reformulatedin Section4.2. I. 1of this

handbook,yields theresult that, whenthetwofacingsare of the samematerial, shear

crimpingwill occurin axially compressedsandwichcylinderswhenever

Vc _ 2 (2.3-1)where

_oV c =

Crerimp

h 2 xJtz t2

(ro = *)Ef_ l___pe2(tl + re)

h_

°'crimp - (tl + t2) tc Gxz


Ef = Young's modulus of facings, psi.

h = Distance between middle surfaces of facings, inches.

R = Radius to middle surface of cylindrical sandwich, inches.

t_ and t_ = Thicknesses of the facings (There is no preference as to which

facing is denoted by the subscript 1 or 2.), inches.

Pe = Elastic Poisson's ratio of facings, dimensionless.

(2.3-2)

(2.3-3)

(2.3-4)

2-37

t C

Gxz

facings and crienled in the axial direction, psi.

The critical stress c:m t)(_ dt, termined from the eq_mtion

% r' Kc %

and, when the Inequality (2. :{-1) holds true, Kc: can be computed as follows:

1

Hence,

'l'hickn_,ss of core, inches.

,_. _r-< shear modulus associate,I with the plane perpendicular to the

(2.3-5)

(2.3-6)

'{; rimp

"i?, ....... _'_-7- _r° : 'rcri'nP (2.3-7)

Therefore, when the two facings ",re made of the same material, the following equation

can be written for the c_'itk_'t! stres_ for _h¢,ar crimpir_g in a circular sandwich cylinder

under axial compression:

h _

_r_r : (r . _ixz (2.3-8)crimp 7t: ! t>_) t c

I ' 4 .) • .An equlvalen_ rest)It can be obtained from Referen¢'e :_-t _ for sandwich cylinders sub-

je,"led [(_ _mif,__,'m e>:t,",_i !g_!,,ra! pressure.

of t}l(_ .'a{!til(' Illa[eFi:'J, _}l i, ( ;_n ',vl'Jte

x\' h e 9"e

Gyz

Tim! ,_, ,.:i_,,c_, the two facings are made

rrcr :: (r . - (2 3-9)crimp (t t + re)t e O.vz

;'-,_',, sl_( :z_' ul _d_!t s us,;¢)( iated with the tqane perpendicular to the

'l'<i';',rrcvol,!lion, psi.

In addition, the development of Reference 2-19 leads one to the following formula for

circular sandwich cylinders under pure torsion and having both facings made of the

same material :

- h2 JGxz (2.3-10)-rcr = "rcrim p (t 1 + t2) tc GY z

It should be noted that, although Equations (2.3-8) through (2.3-10) were derived for

sandwich cylinders, all of these final expressions are independent of curvature. Thus,

these equations have a general applicability which is not limited to the cylindrical con-

figuration.

2-39

2.3.2 Design Equations

The following equations may be used to compute the facing stresses at which shear

crimping will occur in sandwich constructions having both facings made of the same

mate rial:

ao For uniaxial compression acting coplanar with the facings (see Figure 2.3-1),

use

h _

_crimp = (t_ + t_) t c Gij (2.3-11)

where

Gij = Core shear modulus associated with the plane perpendicular to

the facings and parallel to the direction of loading, psi.

o-, ps .i___

o-, psi _

O', psi

Figure 2.3-1. Unia×ial Compression

2-40

bo For pure shear acting coplanar with the facings (see Figure 2.3-2), use

h_ (2 3-12)Tcrimp - (t_+ t2) tc _GxzGyz

X

/

Figure 2.3-2. Pure Shear

The foregoing equations are valid regardless of the overall dimensions of the structure.

m addition, no knock-down factors are required since shear crimping is insensitive to

initialimperfections. The predictions from these equations will be somewhat conserva-

tive since their derivations neglect bending of the facings about their own middle sur-

faces. Although such bending is of negligible importance to most sandwich buckling

phenomena, in the case of shear crimping thls influence can be considerable.

Further mention of the shear crimping mode of failure is made in the various sections

on general instability included in this handbook.

2-41

ECJ

_D

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2-42

I

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2-43

RE FERENC ES

2-1

2-2

2-3

2-4

2-5

2-6

2-7

2-8

2-9

Gerard, George and Becker, Herbert, "Handbook of Structural Stability,

Part I - Buckling of Flat Plates," NACA Technical Note 3781, July 1957.

Norris, C. B., "Short-Column Compressive Strength of Sandwich Construc-

tions as Affected by Size of Cells of Honeycomb Core Materials," U. S.

Forest Service Research Note, FPL-026, January 1964.

Anderson, M. S., "Local Instability of the Elements of a Truss-Core

Sandwich Plate," NASA Technical Report R-30, 1959.

Gough, C. S., Elam, C. F., and deBruyne, N. A., "The Stabilization of a

Thin Sheet By a Continuous Supporting Medium," Journal of the Royal Aero-

nautical Society, January 1940.

Williams, D., Leggett, D.M.A., and Hopkins, H. G., "Flat Sandwich Panels

Under Compressive End Loads," Royal Aircraft Establishment Report No.

A.D. 3174, June 1941.

Hoff, N. J. and Mautner, S. E., '_rhe Buckling of Sandwich Type Panels,"

Journal of the Aeronautical Sciences, Vol. 12, No. 3, July 1945.

Norris, C. B., Boller, K. H., and Voss, A. W., 'WVrinkling of the Facings

of Sandwich Construction Subjected to Edgewise Compression," FPL Report

No. 1810-A, June 1953.

Yusuff, S., "Theory of Wrinkling in Sandwich Construction," Journal of the

Royal Aeronautical Society, Vol. 59, January 1955.

Norris, C, B., Ericksen, W. S., March, H. W., Smith, C. B., and Boller,

K. H., "Wz2nkling of the Facings of Sandwich Constructions Subjected to

Edgewise Compression," FPL Report No. 1810, March 1956.

2-45

2-10

2-11

2-12

2-13

2-14

2-15

2-16

2-17

2-18

Jenklnson, P. M. and Kuenzi, E. W,, "Wrinkling of the Facings of Aluminum

and Stainless Steel Sandwich Subjected to Edgewise Compression, " FPL

Report No. 2171, December 1959.

Yusuff, S., "Face Wrinkling and Core Strength in Sandwich Construction,"

Journal of the Royal Aeronautical Society, Vol. 64, March 1960.

Harris, B. J. and Crisman, W. C., "Face-Wrinkling Mode of Buckling of

Sandwich Panels," Proceedings of the American Society of Civil Engineers,

Journal of the Engineering Mechanics Division, June 1965.

Benson, A. S. m_d Mayers, J., "General Instability and Face Wrinkling of

Sandwich Plates - Unified Theory and Applications," AIAA Paper No. 66-138

Presented in New York, New York, January 1966.

Bartelds, G. and Mayers, J., "Unified Theory for the Bending and Buckling of

Sandwich Shells - Application to Axially Compressed Circular Cylindrical

Shells," Department of Aeronautics and Astronautics, Stanford University

Report No. SUDAAR No. 287, November 1966.

Plantema, F. J., Sandwich Construction, John Wiley & Sons, Inc., New York,

Copyright 1966.

U. S. Department of Defense, Structural Sandwich Composites, MIL-HDBK-23,

30 December 1968.

Zatm, J. J. and Kuenzi, E. W., "Classical Buckling of Cylinders of Sandwich

Construction in Axial Compression - Orthotropic Cores," U. S. Forest Service

Research Note FPL-018, November 1963.

Kuenzi, E. W., Bohannan, B., and Stevens, G. H., "Buckling Coefficients for

Sandwich Cylinders of Finite Length Under Uniform External Lateral Pressure,

U. S. Forest Service Research Note FPL-0104, September 1965.

2-46

2-19 March, H. W. and Kuenzi, E. W., "Buckling of Sandwich Cylinders in

Torsion," FPL Report No. 1840, January 1958.

2-47

3GENERAL INSTABILITY OF FLAT PANELS

3.1 RECTANGULAR PLATES

3. i. 1 General

As previously noted, one of the potential modes of failure for sandwich panels is that

of general instability. This occurs when the panel becomes elastically unstable under

the application of certain types of in-plane loads. Further, it should be noted, the

loads which are critical for instability may or may not be of such magnitude as to

cause a failure of the basic materials.

The fiat, rectangular sandwich panel represents that configuration for which the vast

majority of fabrication and test data has been accumulated over the past decade. This

is probably due to the fact that this configuration was best adapted to the structural

needs for a number of applications and that it represented the minimum in fabrication

problems and costs as far as this type of construction is concerned. By the same

token, analytical solutions have been developed for a wide range of loading applications

for flat panels, and an appreciable amount of testing for correlation with these solu-

tions has been accomplished.

As a consequence of this past work, it is now possible to employ the analytical solu-

tions for fiat panels, as given in MIL-HDBK-23, [3-1], with a high degree of con-

fidence. This view is supported by recommendations given in References 3-2 through

3-7, inclusive, for basic panel design. Therefore, with this background in mind, the

3-1

buckling coefficients, K, which will be given in this section for the various plate loading

conditions will be those taken from the applicable sections of Reference 3-1, with no

"knock down" factor to be applied to them.

The development of plate buckling coefficients for sandwich construction requires the

consideration of a ntmlber of factors, some are: 1) the degree of orthotropicitsr of the

face plates, 2) the use of the same or of dissimilar materials for the face plates and,

3) the degree of orthotropicity of the core material. The general equations given in

the following sections account for these possibilities; however, the curves showing K

as a function of (a/b), V, the type of loading and edge support conditions will assume

the use of isotropic faceplate materials since this is largely typical of aerospace

vehicle design practices.

In all cases, the final design of the sandwich panel must comply with the following four

basic design principles, Reference 3-1;

a. The sandwich facings shall be at least thick enough to withstand the chosen

design stresses under the application of the ultimate design loads.

b. The core shall be thick enough and have sufficient shear rigidity and

strength so that over-all sandwich buckling, excessive deflection, and

shear failure will not occur under the design loads.

c. The core shall have high enough moduli of elasticity, and the sandwich

shall have great enough flatwise tensile and compressive strength such

that wrinkling of either facing will not occur under the design loads.

d. If the core is a cellular honeycomb or constructed of corrugated material

and dimpling of the facings is not permissible, the cell size or corrugation

spacing shall be small enough so that dimpling of either facing into the

core spaces will not occur under the design loads.

3-2

Otherrequirementsincludetheuseof moduli of elasticity andstressvaluesrepre-

sentativeof thosevalueswhichprevail undertheconditionsof use. Also, wherethe

stressesarebeyondtheproportionallimit, theappropriatereducedmodulusof elas-

ticity shouldbeused.

Thefollowingsectionsonspecifictypesof panelloadsdefinethe appropriateequations

for eachparticular situationanddiscussusefullimits andotherconsiderations,as

applicable. A summarytable, (Table3-1), listing thepanelinstability equationsgiven

in thevariousparts of this section,alongwith adefinitionof terms, equationlimitations

if any, andreferencesfor theappropriatebucklingcurvesimmediatelyprecedesthe list

of referencesto facilitate useof themanualfor specificproblemsolution.

Figure3.1-1 showselasticpropertiesanddimensionsfor thetypical sandwichpanel

underconsiderationin this section.

3-3

--_ b__..t _

ttf W-bf --q \¢

SINGLE -TRuss

_,_ b[ __ _tf

•/ t¢ _-_-'t(TYPICAL) f

DOUBLE -TRUSS

CORRUGATED CORE CONFIGURATIONS

E b, Gcb

-e_ b

---.--]_ y

CORE --,

tl-_

_h-_

Nv

4-t4

C

.-t-o

_--t2

ItONEYCOMB CORE

Figure 3.1-1. Elastic Properties and Dimensional Notations

for a Typical Sandwich Panel

3-4

3.1.2 Uniaxial Edgewise Compression


The buckling coefficient equations and curves given here for uniaxial edgewise com-

pression arc those originally developed by Ericksen and March, _3-8], and are in-

cluded in the MIL-HDBK-23 documents, issued since then. The basic principles and

assumptions employed in the development of these general instability equations are

noted in the references and are not repeated here except where required to limit their

use because of the original restrictions imposed.

The basic equations for calculation of the allowable sandwich panel edgewise com-

pression loads are given in the following section. Curves for panel buckling coeffi-

cients for panels having isotropic facepk-ttes and both orthotropic and isotropic cores

for various panel edge support conditions follow the equations.


As previously noted, the equations presented in this section arc those developed by

Ericksen and March, and presented in MIL-HDBK-23, as well as in other documents.

Supporting data such as pertinent assumptions and definition of terms are also in-

cluded along with the equations.

Sandwich Panels With Honeycomb Cores

One of the basic assumptions used in the design and analysis of sandwich panels is

that the face plates carry the inplane loads applied and that the core provides that

shear support to the face plates required for them to act as a unit in preventing early

3-5

individual buckling. From this, the edgewise compression capability of the panel is

given by the following equations, which are taken from Section 5.3, Reference 3-1:

N = (_2/b_)(K)(D) (3.1-1)cr

where D is the sandwich bending stiffness. Solving this equation for the facing sires-

ses gives the following:

(E_t_)(E_t_) (h_) (_ (3.1-2)

l'c_, = = _7_K (E_t_ _ E't2)2 (b _) k

For equal facings:

w he re

E I_K (h)_ f

F - (3.1-3)e 4 (b) 2 )_

K

E J

= buckling coefficient = K + KF M

work).

(see definitions in following

1

= (,'afacings.

= effective modulus ()f elasticity for orthotropic

X = (1 - _ta_b)

_a'_b = Poisson's ratio as measured parallel to the subscript direction.

f, 1, 2 = subscripts denoting facings.

h,b = see Figure 3.1-1.

Since the buckling coefficient curves to be presented here are being limited to the case

of isotropic face plates, which is representative of the large majority of structural

sanchvich applications, the affected equations given previously arc revised below for

this situation.

3-6

For isotropic facings:

' = E' = E.' = 77iEi; and =Eai bi 1 _ai _bi = _i

where 77i = plasticity correction factor (see Section 9.0).

As noted above the buckling coefficient for the panel under this loading condition is

given by the equation

K = KF+K M

where

KF - 12 E_t a (E/t2) h_ KM0

(3.1-4)

KM = KM for the case where V=0 Esee Figure (3.1-16)1 (3.1-5)O

Values of K F are generally quite small relative to KM, thus a safe first approximation

is to assume it is equal to zero until a final panel check is made. On this basis, K

= K M may be used to develop initial face plate and core thicknesses for the panel.

K M is a theoretical coefficient which is dependent on the sandwich panel bending and

shear rigidities and panel aspect ratio. Other factors which influence the magnitude

of this coefficient include the panel edge support conditions and the orthotropicity of

the core. A discussion of these considerations along with development of the equations

for calculation of this coefficient are given in References 3-1 and 3-8. This manual

does not propose to repeat these equations here; however, the curves shown in Figures

3.1-2 through 3.1-15 give values of K M as a function of edge support condition, panel

aspect ratio, and the bending-shear rigidity parameter, V which is defined as follows

y_D

V - b2U (3.1-6)

3-7

whichfurther canbewritten as:

-ItCV = (3.1-7)

kb; Ge (E_t,+E;t:)

_2t tc Eftf

V - (for equal facings) (:3.1-7a)2). b 2 G

C

where U is sandwich shear stiffness; Gc is the core shear modulus associated with the

axes parallel to direction of loading (also parallel to panel side of length a) and per-

pendicular to the plane of the panel.

An indication of the influence and importance of the core shear modulus may be obtained

from inspection of the above equations for V m_d the curves giving values of K M given

later. Holding all terms constant except G , an increase in its value reduces the valuee

of V to be used with the buckling coefficient curves, this reduced value then calls for

an increased value of KM"

Sandwich Panels With Corrugated Core

The equations and fornmlas previously given are for sandwich panels with honeycomb

cores; however, they may be adapted to cover the case of panels with corrugated cores

by means of the following modifications:

a. For the case where the corrugation flutes are oriented normal to the direc-

tion of the load application, the shear modulus in the direction parallel to

the flutes, Geb, is very high with respect to the shear modulus parallel to

the direction 2f andl°ading'R Ge :}; thus,= 0.the previous curves may be used byletting Gcb = = Gca/Gcb

b. For the case where the corrugation flutes are parallel to the direction of

loading, the corrugations may be assumed to carry load in a direct pro-

poYtion to their area and elastic modulus. The parameter V for this case

is replaced by the parameter W, which is defined as

3-8

e

= i (3.1-8)W ).b2Gcb (E_t_ + E_t,)

Or, for equal facings,

W = _t cE'[t/2xb 2 Gab (3.1-8a)

Values of K M as a function of (b/a), R = (Gea/Gcb), and V, or W, are given for various

edge support conditions in Figures 3.1-2 through 3.1-15, with Figures 3.1-14 and

3.1-15 representing the case of panels having corrugated cores.

Figure 3.1-16 gives values of KMo as a function of panel aspect ratio and edge support

conditions for use in determining values of K F in order that final values for K may be

obtained for specific designs.

The curves and equations just given may be used in developing a panel design in addi-

tion to checking the adequacy of an existing design; however, this is a slow iterative

process. As a consequence, this manual recommends the use of the design-procedures

approach described in Reference 3-1 since it was specifically developed to expedite the

new design process.

3-9

10

2.5 GcEND

]

1FOR V>0.40 tC =--

M Vo .__ _k .... 1 10 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

V

05

0.150.30

Figure 3.1-2. K M for a Sandwich Panel with Ends and Sides Simply Supported,

Isotropic Facings, and Orthotropic Core, (R = 0.40)

3-10

14

12

10

G

20.60

1 -- 1.00FORV > 1.00 _.'=

0 i .Mi

0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a

b

b

a

.20

.40

Figure 3.i-3. KM for Sandwich Panel with Ends and Sides Simply Supported,

Isotropic Facings, and Isotropic Core, (R = 1.00)

3-11

14

12 I 210 V - rr D

b2U

8

KM X

.... .......... [ l-i

I III

1_ _ !

0.4G i° I

END 'r

I_ _1I- b -I

I

1FOR V >2.50 I( = 1

M Vo t i I

0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2

a bb

a

3

0.10

0.200.30O.5O

2.500

Figure 3.1-4. K M for Sandwich Panel with Ends and Sides Simply Supported,

Isotropic Facings, and Orthotropic Core, (R = 2.50)

3-12

16

14

12

10

2?r D

V m

b 2 U

V

0

_0.05

.10

.150.200.250.30

00 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

ba

ab

Figure 3.1-5. K M for Sandwich Panel with Ends Simply Supported and SidesClamped, Isotropic Facings, and Orthotropic Core, (R=0.40)

3-13

16

14"I ....

12

10

S

1\

2?r D

V m

b 2 U

T I

Gc

END _,

1

FOR V >0.75 KM=__0 a V0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2

a

.-_ ba

v

0

0.I0

0.20

0.300.400.600.75

Figure 3.1-6. K M for Sandwich Panel with Ends Simply Supported and Sides

Clamped, Isotropic Facings, and Isotropic Core, (R=I.00)

3-14

16

14

12

10

8

6\

\

2D

V m

b 2 U

00 0.2 0.4 0.6 0.8 1.0

a

b

i T

i iI l

L_o

0.4GC

END

L_ -- -- -] I

1

FOR V > 1.875 .l,lK_" =_77:1J i

0.8 0.6 0.4 0.2

b

a

V

0

0.10

0.20

0.30

0.601.00

-- 1. 875

0

Figure 3.1-7. K M for Sandwich Panel with Ends Simply Supported and SidesClamped, Isotropic Facings, and Orthotropic Core, (R= 2.50)

3-15

16

14

12

8

6

4

0o 0.2

.30O. 40

1FOR V >0.40 K - V

0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

Figure 3.1-8. K M for Sandwich Panel with Ends Clamped and Sides Simply

Supported, Isotropic Facings, and Orthotropic Core, (R=0.40)

3-16

16

14

12

\

V

2D

b2U

GC

END

1

FORV > i.?0 K M=_0 i V0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2

a b

b a

Figure 3.1-9. K M for Sandwich Panel with Ends Clamped and Sides SimplySupported, Isotropic Facings, and Isotropic Core, (R = 1.00)

3-17

KM

16--

14 ......

12

6

4

2

00 0.2

\\

\

2D

b 2 U

Il,

0.4GC

END

t

0.4

a

b

Figure 3.1-10. K M for Sandwich Panel with Ends Clamped and Sides Simply

Supported, Isotropic Facings, and Orthotropic Core, (R= 2.50)

3-18

14

12 \

\,

;_iA

2 0 °

V = rr D__ 1'

b2U

2.5Ge

END

2 ?1

FoR v > 0.30 % - Vo 10 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

Figure 3.1-11. K M for Sandwich Panel with Ends and Sides Clamped, IsotropicFacings, and Orthotroptc Core, (R = 0.40)

3-19

16

14

12

10

8

6

0

0

.... _.....] 1 i i

b2U

I _'_END

I 1FOR V >0.75 K =--

M \7/ i •

0.2 0.4 0.6 0.8 1.0 0.8 0 6 0.4 0.2 0

a b

b a

0.i0

O. 20

0.300.40

0.50

0.75

Figure 3.1-12. K M for Sandwich Panel with Ends and Sides Clamped, Isotropic

Facings, and Isotropic Core, (R = 1.00)

3-20

16

14

12

8

2D

V =

b 2 U

L

0.4Ge

END _'

1FORV > 1.875 K.o -

0 1 J M v0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2

a b

b a

V

0

O. 10

0.200.30

0.601.00

- --_ 1.875

0

Figure 3.1-13. KM for Sandwich Panel with Ends and Sides Clamped, Isotropic

Facings, and Orthotropic Core, (R = 2.50}

3-21

14I

V =0

12

10

=0.1

\

°v0 \

V_0.4

V _t:

2Tr Dt

c

b 2 h 2 Gca

[ l

+IGb=_

I t? _,

lp

I

t

_L

00 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

Figure 3.1-14. K M for Simply Supported Sandwich Panel Having a CorrugatedCore. Core Corrugation Flutes are Perpendicular to theLoad Direction

3-22

14

12

]0 W

2 DtC

b2 h2 Gob

W

0

0.1

_'--0o2

_- o.4

Figure 3.1-15.

0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

K M for Simply Supported Sandwich Panel Having a CorrugatedCore. Core Corrugation Flutes are Parallel to the Load

Direction

3-23

KM

o

\

I

\

\

0.i 0.2a

b

END t

Figure 3.1-16. KMo for Sandwich Panel with Isotropie Facings

in Edgewise Compression

3-24

3.1.3 Edgewise Shear


As noted earlier in Section 3.1.1, sufficient analysis, design, and testing of flat sand-

wich panels has been accom,lished to demonstrate the adequacy of the analytical

approaches presently in use. Thus, the panel buckling coefficient equations and

curves given in the following paragraphs for edgewise shear are those taken from the

MIL-ttDBK-23 documents presently in use. These equations were originally developed

by Kuenzi and Ericksen E3-13] and employ the same general assumptions as those

described in Section 3.1.1. Specific limitations or restrictions on the use of these

equations will be noted where these require consideration.

The basic equations for use in calculation of the allowable sandwich panel edgewise

shear loads are given in the following section along with applicable background data

and assumptions. Design curves and buckling coefficients for panels having isotropic

faceplates and both orthotropic and isotropic cores for both simply supported and

clamped edge conditions follow the equations.


The design equations presented here are taken from Reference 3-1 and 3-13.

ing data and design constraints are also noted and discussed as required.

Support-

The edgewise shear load carrying capability of a sandwich panel is given by the follow-

ing equation:

N = (Tr_/b_) (Ks) (D) (3.1-9)ser

3-25

where

N = critical edgewise shear load, lb per inchscr

D = sandwich bending stiffness

Solving this for the facing st_'esses gives the following equation:

/ I f(E_tI)(E_t_)(he)E z,

Fsz,_ = _K , + is (Elt 1 E_t_) (be) x

(3.1-10)

Or, for equal facings

y2K (h e) 's EfF -

s 4 (b_)k(3.1-10a)

where

E' is the effective modulus of elasticity of facing at stress F s = 7/E

r_ = plasticity correction factor (Section 9.0)

X = 1-D 2

= _ = p_ = Poisson's ratio of facings

h = distance between facing centroids (Figure 3.1-1)

b = panel width (<a) (Figure 3.1-1)

Ks = KF + KM (Note: These terms differ from those of Section 3.1.2)

whe re

i 3 t

(E_t_3 + E_t2 ) (E'_t_+ E_t_)KMo

K F = 12(E_t_)(E,2t_)h2 (3.1-11)


(tf_KMo

K F = 3h2 (3.l-lla)

KMo = value ofK M for V=W=0

3-26

The equation defining the value of KM is quite complex and involved, being dependent

on panel aspect ratio, (a/b), the number of half-waves, (n), for the minimum energy

buckle pattern, and the panel bending and shear rigidityparameter, (V, or W). This

manual proposes to follow general practice in the literature and provide curves only

for the definitionofthis buckling coefficient. Those interested in the basic equation

and its development will find this in Reference 3-13.

Values of K M are given in Figures 3.1-17 through 3.1-24 as a function of the panel

aspect ratio and the parameter V, or W, for various panel edge support conditions.

These figures cover panels with isotropic faceplates and both isotropic and orthotropie

core, including panels using corrugated flutes for cores. Values of the buckling coeffi-

cient, KMo, may also be obtained from the same set of figures.

The equations defining the parameters V and W are the same as those given in the

previous section for edgewise compression; however, they are repeated below to

The equation numbers previously assigned to them are retainedfacilitate their use.

below

(E_tl)(E'_t2)(Y% tc (3.1-7)

V = k(E[t_ +E_t 2) (b 2) Gca

V = _r_tcE'ft/2),b2 Gca (equal facings)(3.1-7a)

For a sandwich panel with a corrugated core in which the corrugation flutes are parallel

to the edge of length a, the parameter V is replaced by the parameter W which is de-

fined as follows:

3-27

W

)_b2 %b {EIlt_ + E_ta) (3.1-8)


W = rr_t E_t/2kb e (3.1-8a)c Gcb

In checking a particular design for the critical buckling stress, Fscr, Figures 3.1-17

through 3.1-21 should be used for those panels having all edges simply supported.

Curves of K M for sandwich panels having all edges clamped are given in Figures

3.1-22 through 3.1-24. These curves may be interpolated in order to obtain the

buckling coefficients for other values of core orthotropicity, (R = Gea/Gcb), and inter-

mediate values of V or W°

It should be noted that if the resulting value of F is above the proportional limitset

value, the value of E' shall be an effective value based on that stress level, and this

effective value shall be used in computing the value of V, Equation (3.1-7) or (3.1-7a)

or W, Equation (3.1-8) or (3.1-8a), as the case may be. This same effective value

for E I shall also be used in Equation (3.1-10), or (3.1-10a) when calculating the criti-

cal panel buckling stress. Thus, several interations will be required to establish the

actual value of F in those cases where it exceeds the proportional limit.scr

The equations and curves just given may be used in the development of panel designs

as well as in checking an existing design; however, as was the case for uniaxial com-

pression, this is a lengthy iterative process. Thus, this manual recommends the use

of the design-procedures approach described in Reference 3-1 for those cases where

the initiation of new designs is required.

3-28

10

KM

27r 1)

2b U /

0l

/

0.05

0.10

---- 0.40

[0

Figure 3.1-17.

0.2 0.4 0.6 0.8 1.0b

a


and an Isotropie Core, (R = 1.00)

3-29

10V

8

0.4 G

4

0.40

Figure 3.1-18.

b

a

K M for a Sandwich Panel with All Edges Simply Supported,and an Orthotropic Core, (R = 2.50)

3-30

J

ff

10

t _t_L

2D

b2 U

/

//

/

_J" J

V

0.05

O.20

0.40

C0

Figure 3.1-19.

0.2 0.4 b 0.6 0.8 1.0

a


and with an Orthotropic Core, (R = 0.40)

3-31

KM 5

00

10

Figure3.1-20.

i

0.4b

a

K M for a Sandwich Panel with All Edges Simply Supported,Isotropic Facings and Corrugated Core. Core

Corrugation Flutes are Parallel to Side a

3-32

KM

10

9-

7 1

I

4

-r_LI '

_b_

V

i

//

J J_J

ii

2_r t D

c

b 2 h 2 Gca -

//

i /1

f

jlI

11

/" //

V

0

0.05

0.20

O.50

0

0 0.2 0.4 0.6 0.8 1.0b

a

Figure 3.1-2 i. K M for a Sandwich Panel with All Edges Simply Supported,Isotropic Facings and Corrugated Core. Core

Corrugation Flutes are Parallel to Side b

3-33

16

14

12

l Gc

V

0

10

0.05

K M

O. i0

O. 20

0

0 0.2 0.4 0.6 0.8 1.0

b

a

Figure 3.1-22. K M for a Sandwich Panel with All Edges Clamped,

Isotropic Facings and Isotropic Core, (R = 1.00)

3-34

16

14 i

12

10

I

0.4Gc

_-b ---4

IfV ----"--

| i r_

I jjJ

277 D

b 2 U/

/

/V

0

5_

8

6

_JI

_..Iv

7J

II

J

J

0.05

0.10

0.20

o I0 0.2 0.4 0.6 0.8 1.0

b

a

Figure 3.1-23. K M for a Sandwich Panel with All Edges Clamped,

Isotropic Facings and Orthotropic Core, (R=2.50)

3-35

16

14

12

I 2.s IIC

l "_----- _

2_r D

V

0

/

KM

10 0.05

0 0.2 0.4 0.8

0.i0

.2O

0.6

b

a

1.0

Figure 3.1-24. K M for a Sandwich Panel with All Edges Clamped,Isotropic Facings and Orthotropic Core, (R= 0.40)

3-36

3.1.4 Edgewise Bending Moment


The application of an edgewise bending moment to a flat, rectangular sandwich panel

produces a loading condition such as that shown in Figure 3.1-25. This represents a

somewhat different situation from the ones previously covered, since the tension loading

on one half of the panel represents a stabilizing effect. The edge compression load

on the other half of the panel varies linearly from zero at the neutral axis to a maxi-

mum value, N, at the panel edge. It is this compression loading which can produce

panel buckling in the same fashion as the uniaxial compression case; however, the

presence of the panel edge support along the line of maximum loading forces consider-

ation of a more complex failure mode.

These failure mode considerations for this type of loading have been covered in the

development of analytical techniques for the evaluation of flat plates (Reference 3-17).

hlso, as has been previously noted, sufficient analytical development and testing has

been accomplished on flat, rectangular sandwich panels to enable the use of the buckling

coefficients given in Reference 3-1 for this loading condition with complete confidence.

The general equations for the behavior of flat, rectangular honeycomb sandwich panels

under this loading condition were developed by Kimel E3-15] while whose applicable to

panels with a corrugated core were developed by Harris and Auelman, E3-14] and E3-16].

The assumptions employed in the development of the basic equation for the panel sta-

bility coefficient for this loading condition are generally the same as those described

3-37

in Section 3.1.1, with one particulal' exception. This exception requires that the

critical design faeeplate stress, Fcr, shall not exct,ed the elastic buckling stress for

the faceplates. This requirement stresses the fact that the analysis is based on a

linear loading variation across the edge of the panel. Once the elastic buckling stress

is exeeede.d this variation is no longer linear, and extrapolation to a buckling stress

beyond the elastic range of facing stresses cannot be done by using an effective elastic

modulus such as the tangent modulus, in the buckling formulas. Since the proper

extrapolation to stresses beyond the elastic range must consider the variation of

effective elastic modulus across the panel width associated with the stress variation,

the equations and buckling coefficier_ts given here are thus strictly applicable only to

buckling at facing stresses within the elastic range.

The basic equations to be used in the calculation of the allowable sandwich panel edge

loading are given in the following section. Design curves and buckling coefficients

for panels having isotropic faceplates and both isolropic and orthotropic cores based

on simply supported edge conditions follow these equations.


The design equations presented here arc those taken from Reference 3-1. Background

assumptions and any applicable design constraints are also covered.

Using a linear stress variation as previously discussed, the value of N at the panel

edge is given by the equation:

N = 6M/b _ (3.1-12)

3-38

where

N _-_

M

b

load per unit width of edge

= edgewise bending moment

= panel width (Figure 3.1-25)

The edgewise bending load capability of a sandwich panel is given by the following

equation, taken from Reference 3-1:

Ner

where

= (_,/be) (%) (D) (3.1-13)

N = critical edgewise loading, lb per incher

D = sandwiehbending stiffness

The critical faceplate stresses are obtained by solution of the previous equation and

are as follows:

Or, for equal facings,

(E1t_)(E2t_) (h_) (El,s)

Fe_,_ = Y2Kb (Elia +E_t_) 2 (be) 1 (3.1-14)

whe re

F = (3.1-14a)c 4 (b _) ),

E = modulus of elasticity of facing

I = (1 - _)

= Poisson's ratio of facings: /_a = _b assumed above

h = distance between facing centroids

length of loaded edge of panel

K F + K M (Note: The values for these buckling terms differ

from those given in Sections 3.1.2 and 3.1.3)

3-39

(E_tl_+ E_t23)(Elt1+ Eat)KF = 12(E1t_)(E_t2)(he) KMo

(3.1-15)


(tf 2) KM o

KF = 2 h 2(3.1-15a)

where

Values of K M

tion of the parameter \T or W, and the panel aspect ratio.

K M = value ofK M for V=W=0o

for panel buckling are given in Figures 3.1-25 through 3.1-28 as a func-

These cover panels having

isotropic faceplates using both isotropic and orthotropic cores, including those using

corrugated flute-type cores.

The equations defining the parameters V and W are the same as those given in the

previous section for edgewise compression; however, they are repeated below to

facilitate their use. The equation numbers previously assigned to them are retained

below; however, values of E _ are replaced by those of E for this case.

(Elt _ ) (E2t_) (z) tcV = (3.1-7)

k (Eit_+ E_t_)(b2 ) Gca

V = 1r_t e Eftf/2k b 2 Gca (equal facings) (3.1-7a)

For a sandwich panel with a corrugated core in which the corrugation flutes are parallel

to the edge of length a, the parameter V is replaced by the parameter W, which is de-

fined as follows :

3-4O

Or, for equalfacings,

C

w = (3.1-8)I b_Gcb (Elh +E2t _)

W = _r_tc Eft/2X b 2 Geb (3.1-8a)

A particular design may be checked by using the graphs given in Figures 3.1-25 through

3.1-28 to determine the appropriate value of the buckling coefficient to use in Equation

(3.1-15), or (3.1-15a) to compute the critical buckling stress, Fer. This approach,

which involves trial and error solutions by iteration, may also be employed to develop

new panel designs; however, this manual recommends that the design-procedures

approach described in Reference 3-1 be considered since it was set up to facilitate

such design caleulations.

3-41

36 ......

32

28 .......

24

16

V--0

I

°°°I_- C _-- _|_T

2?r D

V --

b 2 U

r

/

"_ n--2/ n--3

i

8

4

0o 0.2 0.4

V=0.215

FOR v => 0.21_ KM = 1.ssG/v

0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

Figure 3.1-25. K M for a Simply Supported Sandwich Panel with anIsotropic Core, (R = 1.00)

3-42

4O

32 !V=0

_4 ! ",

,o"X v=o.osl

16 \ \_.\

12

\

n=l

a -------_

G _e c, ,, |

3_

y2 DV -

b 2 U

A

/f

S n=2vxn=3

-'_..L--o.28

V--0

00 0.2 0.4

Figure 3.1-26.

.4

0.6

a

b

I I----

FOR V _-> 0.4 KM

{ I0.8 1.0

-- 1. ss6/v

0.8 0.6 0.4 0.2 0

b

a

K M for a Simply Supported Sandwich Panel with an

Orthotropic Core, (R = 2.50)

3-43

4O

36

32V=0

2 DV --

2b U

28

KM

0.15

FOR V >015 KM= 1.886/V

0

0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

Figure 3.1-27. KM for a Simply Supported Sandwich Panel with anOrthotropic Core, (R = 0.40)

3-44

4O

36

32

28

24---

2O

16

12

Ill/

v cr rl

\

W=O.05

W=0.20

_W=0.50

a-----_

G --_ 'i

y2t De

b2h 2 Gcb

±

"_.. n=21 n=3

............ L .......................

4

C0

Figure 3.1-28.

0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0

a b

b a

K M for a Simply Supported Sandwich Panel with Corrugated Core.

Core Corrugation Flutes Parallel to Side a

3-45

3.1.5 OtherSingleLoadingConditions

A searchof the literature, aswell ascontactswith a numberofpeoplewhohavebeen

activein theanalyticalmethodsfield for this typeof construction,revealednoother

singleloadingconditionswhichmight leadto panelinstabilityproblems. Consequently,

thepreviously'describedloadingconditionsrepresenttheextentof thefiat panelsta-

bility datawhichwill begivenherefor individualloadingcases.

3-46

3.1.6 Combined Loading Conditions


A study of the effects of combined loadings on the buckling of fiat sandwich panels

requires the consideration of a number of factors. Some are:

a. The mode of failure of the panel under each of the applied loads.

b. The interaction between different modes for precipitation of panel

buckling or failure.

c. The influence of variations in the core shear rigidity values on the

interaction equations for panel instability failure under combined

loadings.

Since little specific testing for biaxial instability modes has been accomplished for

fiat sandwich panels, this part of the manual will provide analytically developed

equations for combination of the stress ratios which are conservative for most appli-

cations. Additionally, some discussion of the considerations involved is included

along with appropriate references in case more specific solutions or background is

needed.

The equations given on the following pages cover the interaction relationships between

the stress ratios, (R i = Ni/Nicr) , for each of the separate loadings which produce

failure by overall panel instability under the action of the combined loads. For the

stress ratio relationships which produce panel failure by local instability only, refer

to Section 2. These latter equations and pertinent discussion are not repeated here

3--47

although the specific equation number and report page are listed below for each of the

local instability modes:

a. Intracellular Buckling:

b. Face Wrinkling (Asymmetric):

c. Face Wrinkling {Symmetric):

Equation (2.1-6), Page 2-7.



It should be noted that there are no known data available for potential panel failures

which might occur as a result of interaction between a local instability situation

arising from the loading applied along one edge in conjunction with a general insta-

bility problem arising from the loading applied along the panel edge perpendicular

to the first one. This situation might occur for panels having very high aspect

ratios; however, most of these would also indicate a potential local instability failure

under the action of the combined loads. In all cases, however, as has been previously

noted, tests should be run to substantiate a final design in all cases where there is

some question as to the structural adequacy of the sandwich component.

The effects of plasticity must be accounted for in calculating the stress ratios, R., toi

be used in the interaction equations which are given in later paragraphs. Reference is

herewith made to the discussion and recommendations given in Section 9.2, COMBINED

LOADING CONDITIONS, in this report and in particular to Equation (9.2-1) or (9.2-1a).

Either of these equations define an effective uniaxial stress, cr., for use in determining1

an effective plasticity reduction factor which accounts for the effects of the biaxial

stress field. Once the value of (y. is known, the plasticity reduction factor, _7, may1

be calculated by means of Equation (9.2-3).

3-48


The design equations and curves for combined loading conditions are separated into

those which should be used for sandwich panels having honeycomb cores and those to

be uscd with panels having corrugated cores. Supporting references are given for

each type and loading condition along with any limitations or restrictions on the use

of the interaction equation.

Sandwich Panels with Honeycomb Cores

The interaction relationships between the stress ratios which define the onset of general

instability buckling of honeycomb core panels under combined loadings are complex

functions of a number of factors. Some of these will be covered briefly here. One of

the prerequisites for the development of the interaction equation is the determination

of the number of half-waves in both the x and y directions for minimum energy plate

buckling. Since each of these is a function of not only its relationship with the other

but is also dependent upon the core shear rigidity parameter, V, the panel aspect

ratio, panel edge support considerations, etc., the establishment of general equations

covering all of these influences presents a formidable problem.

In view of the complexity involved in an exact definition of combined load interactions,

the writers of this manual propose the use of the following simplified stress ratio

relationships for panel buckling. These give somewhat conservative results over the

typical range of aerospace application and have been recommended for general use,

[3-I].

3 -49

A°

Bo

Biaxial Compression.

buckling of a panel subjected to biaxial compression:

R +R =1cx cy

where

Ncr

R = N/Ne cr

The following formula is recommended for estimating

N = Loading along panel edge, lbs/inch.

(3.1-16)

= Critical loading along panel edge, lbs/inch. (See Equation

3.1-1,)

x, y = Subscripts denoting direction of loading. (See Figure 3.1-1.)

A plot of Equation (3.1-16) is given in Figure 3.1-29 to facilitiate its use in

making design checks.

As noted in References 3-1 and 3-23, the above equation is correct for square,

isotropic sandwich panels for which V _ 0. It becomes appreciably conservative

for panels of large aspect ratio, (a/b _ 3.0) and for panels bordering on the weak

core regime (V _> 0.3). For panels with aspect ratios of 2.0 or less, and which

have reasonably stiff honeycomb cores, Equation (3.1-16) provides a satisfactory

method for prediction of the onset of panel buckling.

Bending and Compression. Equation (3.1-17) provides a sufficiently reliable

method for the estimation of panel buckling under the action of combined bending

and compression loads.

R + (RBx)3/2 = 1 (3.1-17)CX

3-50

where

Rc = N/Ncr

R B =

N =

Ncr

(See definition of Terms for Equation 3.1-16.)

(N/Ncr)bending

Load per unit width of edge due to edgewise bending, lbs/in.

Critical edgewise loading on panel due to bending moment,

lbs/in. (See Equation 3.1-13.)

Figure (3.1-30) plots the interaction relationship given by Equation (3.1-17) to

enable its ready use.

References 3-19 and 3-23 are recommended, in case more accurate analysis

of this loading combination is desired.

C. Compression and Shear. The following interaction formula furnishes a depend-

able method for the prediction of panel buckling under this particular combination

of loads:

Rc + (Rs)2 = 1 (3.1-18)

whe re

R = N/Ncr (See definition of terms for Equation 3.1-16.)e

R = (Ns/Nsc r)S

N = Shear loading per unit width of panel edge, lbs/in.s

N = Critical edgewise shear loading, lbs/in. (See Equation 3.1-9.)ser

Equation (3.1-18) is plotted in Figure 3.1-31 to enable it to be more easily used

in the solution of specific problems. References 3-21 and 3-23 develop this

interaction relationship in greater depth for those needing this information.

3-51

O_ Bending and Shear. The following interaction equation represents a close

approximation of tile buckling behavior of panels under combined edgewise

bending and shear loads.

w h e r c

RB

(RB)2+ (Rs)2 = 1

= (N/Ner)bending

Rs = (Ns/Nscr)

(3.1-19)

These terms are defined as before for

Equation (3.1-1 7).

As previously defined for Equation (3.1-18).

Again, as for the previous combined loadings, Equation (3.1-19) is plotted in

Figure 3.1-32 to make it more easily and readily usable. Reference 3-19 pro-

vides additional background information on the development of this interaction

equation.

Sandwich Panels with Corrugated Cores

The interaction equations for predicting the onset of general instability failure for

sandwich panels with corrugated cores involve the consideration of a number of com-

plex relationships also, as for the honeycomb core case. The same influences prevail

for fluted corrugations as before, with the additional consideration that the core shear

modulus normal to the direction of flute orientation is negligible in comparison to the

shear modulus measured parallel to the flutes. Also, the ability of the corrugations

to carry axial loading when it is applied along the axis of the flutes, further compli-

cates the problem since the distribution of this loading between the faceplates and the

flutes depends on the geometry and material thicknesses.

3-52

In view of the magnitude of the problem involved in developing specific equations for

the interaction relationships, this manual will take advantage of the extensive studies

in this area performed by Harris and Auelman, [3-14 and 3-16]. The latter reference

presents interaction equations for the prediction of the onset of panel buckling in the

form'of curves relating the buckling coefficients to each other as a function of panel

aspect ratio, the core bending-shear rigidity parameter, W, the _!ation between load

direction and flute orientation, and the ratio of the loading carried by the flutes with

respect to that carried by the faceplates. These interaction curves are repeated here

in Figures 3.1-33 through 3.1-42 for several values of the shear rigidity parameter,

W, and for the following additional relationships: 1) Panel aspect ratio, a/b = 1/2,

1.0, and 2.0, and 2) Amount of axial load carried by the core corrugations is negligible

(D = bending stiffnesscwith respect to that carried by the faceplates, i.e., De/D = 0.

of corrugations, and D = bending stiffness of sandwich panel.)

A discussion of each of the sets of interaction curves follows.

Ao Biaxial Compression. Interaction curves relating the buckling coefficients for

this combined load condition are given in Figures 3.1-33 through 3.1-35.

Buckling coefficients for other panel aspeo* ratios and different values of W

may be obtained by interpolation.

The following example problem is offered to demonstrate how these curves may

be used to predict the onset of panel buckling.

Given: Panel withN =2000 lbs/in, N =400 lbs/in, a=30 in, b-- 60 in,x y

D = 3.0 × 10 _ lbs/in e, use W = 0 for example problem.

3-53

Figure3.1-33 is usedfor this casesince(a/b)= 1/2. The top line of this figure

applies since W = 0. The interaction equation takes the following general form:

R + R <_1 (3.1-20)cx cy

or

N Nx y

+ __

N Nxcr ycr

1 (3.1-21)

where

Rcx'Rcy

Nxcr

Nycr

= Stress ratios for loads in subscript directions, dimensionless.

= Critical panel loading for loading applied in the x direction, =

(rr_/b _) (Kx) (D), lbs per inch.

= Critical panel loading for the y direction, = (rr_/b _) (%) (D),lbs per inch.

K,Kx y direction.

E; (tf)(t e + tf) 2D = Sandwich bending stiffness =

2 (1 _Nfe)

W = _r_ (t c) (EO (tf)/2 (1 -_Zfe) (b 2) Gcb for equal facings.

Buckling coefficients for loading parallel to the subscript

for equal facings.

' = Effective Young's modulus for faces, psi.Ef

Gcb -- Core shear modulus in the direction parallel to the flutes, psi.

t = Thickness of core, inches.C

tf = Thickness of faceplates, inches.

_Zf = Poisson's ratio of faceplates.

a,b = Panel dimensions, inches.

3-54

Substitutingin Equation(3.1-21):

NX

+

(K x) (_) D/b _)

and, letting

then

NY

(%) (_'_D/b2)

1.0 (3.1-22)

r = N /N (3.1-23)y x

Nxb_ [ 1 r](3.1-24)

Since D, b, N , and r will be known for the design in question, and K and Kx x y

may be obtained from the appropriate curve, Equation (3.1-24) can be used in

checking the panel stability on the basis that the panel margin of safety is the

same for each loading direction. Thus,

From which

(M.S.)x = (M.S.) (3.1-25)Y

(_) 1o:(_yc4lO.Nx. \Ny/

or

Nxer/ == r (3.1-26)

the n

(Ky) (y2D/b 2) K Y-----r -

K

(Kx) (_'_D/b _) x

(3.1-27)

3-55

Returningto thedatagivenfor theexampleproblemto demonstratethemethod

for checkingpanelstability:

r = N /N = (400/2000) = 0.20y x

K /K = 0.20, from Equation (3.1-27)y x

Using Figure 3.1-33, erect a line passing through the origin and having a slope

of K /K = 0.20 and extend it until it intersects the line for W = 0. The coor-y x

dinates of this intersection point, as taken from the figure, are: K = 6.0,X

and K = 1.2.Y

Then,

N = Kxcr X

n

(_2D/bZ) = (6.0) (_r_ × 3.0 × 102/602 )

N = (6.0)(822.0) = 4930 lbs/inchxer

N = K (_r_D/b 2) = (1.2)(822.0) = 986 lbs/inchycr y

Solving Equation (3.1-21) for a panel stability check:

2000 400- 0.406 + 0.406 =: 0.812

4930 986

Since the total is less than 1.0, the panel is stable under the applied loads. The

margin of safety for panel buckling is: M.S. = (1.0/0.812) - 1.0 = +0.232.

3-56

B°

C,

D°

Combined Compression Along Core Flutes and Shear. Figures 3.1-36 through

3.1-38 give eurves showing the interaction relationships between the buckling

coefficients for panels loaded in this manner. Curves for other panel aspect

ratios and values of the shear rigidity parameter may be developed by inter-

polation from those given. Panel stability checks for this combined loading

condition are made in the same manner as for the biaxial compression ease.

This is accomplished by handling the calculations for the t2 term in the inter-s

action equation in the same way as was done for the I_ term in the examplecy

given on page 3-53.

Combined Compression Normal to Core Flutes and Shear. Interaction curves

for the buckling coefficients covering this particular combination of loads are

given in Figures 3.1-39 through 3.1-41. These curves may be interpolated to

obtain values for the specific design under study and the stability checks may

be made in a similar fashion to those for the biaxial compression case. The

method to be used in performing design checks on panels loaded in this manner

is the same as that noted in item (B) above.

Combined Biaxial Compression and Shear. Figure 3.1-42 shows the relation-

ships for the compression and shear buckling coefficients for this loading con-

dition. These curves are for a square panel only, however, as may be noted

from the small change in the values of K between the various values of theY

shear rigidity parameter, W, approximate interpolations may be made on the

basis of ratios obtained from the curves of Figures 3.1-36 through 3.1-38.

3-57

Panelstability checksaremadein basicallythe samemannerasfor the example

problemgivenonpage3-53, exceptthatthe stress ratio, R , is handleddiffer-cy

ently. Thebasic interactionequationfor this conditiontakesthe following

generalform:

where

R andRcx cy

R +R +R _1cx cy s

are asdefinedonpage3-54.

R = (N = /(_/b 2) (Ks) (D)]s xy/Nscr ) [Nxy

K = buckling coefficient for shears

Since, as may be seen in Figure 3.1-42, K is a function of W only for this caseY

and is independent of the values of Kx and Ks, the value for Rcy may be calcu-

lated imm :diately and the interaction equation put in the following form:

Rcx + Rs = (1.0- Rcy ) : Or, Rcx +Rs = C

The design check may now be performed in the same way as for the example

problem on page 3-53, if the R term and calculations are handled in the sames

way as the R term and calculations were handled for the example. It is to becy

noted, however, that the term on the right side of the equation, C, has a value

which is less than 1.0 and this value should be used in place of the 1.0 used in

the example. Thus, assumingR = 0.10, then C = 1.0 -0.1 =0.9, and thecy

margin of safety for panel buckling as calculated on page 3-57 for the example

would now become:

M.S. = (0.90/0.812)- 1.0 = +0.109

3-58

Rcx

0.6 0.80 0.2 0.4 Rcy

Figure 3.1-29. Interaction Curve for a Honeycomb Core Sandwich

Panel Subjected to Biaxial Compression

3-59

0.6

ItCX

0.4

0.2

1.0_:::

r!','_T !_ !! !'

o.8i!.!!!...ii::i_+fl .... i+ .

Ittt]t::iit ::

lll.+'.l; _'.| ',t:

.rg_!!i!!:Fi:i.,+.

+_I : .......144 ;4 _++ ] ,+| :::

F_F!i!!!i:_

U4ttt!iiilii!t0 Hf+H_ii_t+xl _

o

: ' +i

_!i+ i I i!:+'i: +- i_

:ii i_::i 1!

i;I H.

!!! _i: ,_

tit tf: !t

iii tr _

]PI tt

0.2

hi !':Itt _:

;I fl_i i!

?} r_t t[

ii "

1+ ?f

":r !!

r! T[

!-F i:

11 Pt

0°4R

Bx

ft+.+++.

i,+!t

.i..

i '

![;:

II

N_ii["

i!i

+4,)

.la I.

:'I!

TTC_

0.

.... I :: t;+ .... _it_ |:i!: if!!_'_.... + + , '++ .... +P+_ i.

'::;::.t ....... T++[ "I :'' +_+ P [ il +r + ;i,-

::::I:::: t_+: : ,: ,_!_ _ _ +i:: :::!_ilii!: • '

: ;:{!:.. .... : ,:--!: ,,_i,],,i : ::"

:fill+: !:i :':i ;:. ::+_lli-'ii_

:::: til + +_ :,-::: !i",,, i:+ " "+ :

:iiii:i: il!i ::-t:l '+li_li:i: :_++' 111 i':,: '+

+ --.+v'":=;!: ;:,- tit, , + +:;

:::-::::,:ii !]ii _ !_!il!i[i ++:_

_!:iii!L :+++: _i_ +.+;++I+ ...... +,+ +P_

+!t+l+++::'.'.i +:': +!+ ;+++t!G:_',',+...... ::--i , ,+_ +_+ N_:

!! (Rcx + RBx 3/2 : 1.0):_,+.lii:

_; | ' I ::"' _' _t_'l][t JJl.

Irt++'P_+l?': !:++ fff+l_+'f ....

I+++++N+:....... fill+: : _bJ::!+`:!ti:i +::"

++:I++++I++++:N+++:+_,,,+++,: : _,++If+1+ ',_tt

ttTT

+ !+I++++....... + +!+_+4++t++++t+!i++1++FiHtN. ...... ;Ut

_t_;t:'+:: itt_ "_]_ It l_tt I i!::t!t:Ir _ +|:tr [[i[ ' '

_t Iii !,_ii!l _++_!d+:]l+_fi..6 o. 8 1. o


Panel Subjected to Bending and Compression

3-60

RC

![![i[I

ii!

Jill

+÷,.

RS

I!+t:: : iii _!<+!:

- ' !1 ...._ [![ ,, ii:

+':t!tlii! !i'. t!'.: ;;:: [b!:

_;t ii :_i ilti !!!: tr_r,i;

:_ i_ tR + R = 1 O)i[ _:i _ c s "I[ :t_l liT! T.... :: ,lii:*+

. _ : ;+ t_; !_2 .: .- , ]q[ _I_ + , ii

_':_it:_l}_l;ii _.XiitiliaS:.-!i';' t;t: tell :::?='¢tii _!" i

::;ilf!+ ::',. :::;+,+u m;::I1iI2:; ::;; +.1!!Ill: '_+

r!t !!! r i}_+ _iii!!iT:!!!+!

+ , I i , , + i ' _ ' 4 =1 " : l r t! f1 t _ ] i , '

i]4!ti!i::;;i:_i;I+t_;liilli)L+ '.1+ 11+!+'.,.J,i ;++_++++i;+++++l++........ !+,+:t;;_p.+!:_i-]-Ll +

,+++i;+11++1+Pl_+_+0.6 0.8

i! !;;

_iiil,,_+.:

tt _i::"TrY

gt +,+l

++_!!1

1I....tti_

!! iTiiI ¸ !_i

i: !i!!' !111

+:i....1tit

4 4;;;

L +:_i

Lilt

t_ t;:i

L !![

;i 1%I:H _ t_

1,0


Panel Subjected to Compression and Shear

3-61


Panel Subjected to Bending and Shear

3-62

20.0

16.0

12.0

KY

8.0

4.0

W=0

W = 0.05

\

0 2.0

Nx

W

W

Y

___ +. b

-tttt _NY

i

KX

= 0.20-

I

0.50-

4.0

KX

\6.0

L...×

2Nb

X

2_r D

N b 2Y

. Z = --y 2

D

I

(a/b = 1/2)

8.0

Figure 3.1-33. Buckling Coefficients for Corrugated Core Sandwich

Panels in Biaxial Compression (a/b = 1/2)

3-63

5.0


Panels in Biaxial Compression (a/b = 1.0)

3-64

2.0

1.6

1.2

KY

0.8

0.4

NX

Y lL a _1I- -I

b

NY_'_ N b 2

D2

K - 2

\ I

w \ "-,g_,. I

"_...... -.., \'_O. 50 --_

(a/b = 2. O) _ _ \_

o0 1.0 2.0 3.0 4.0

KX

KX


Panels in Biaxial Compression (a/b = 2.0)

3-65

10.0

8.0

(a/b = 1/2)

6.o __

K _ \

× \ "_,

\4.0

\2.0

W=0.20 -

NX

\\\

y_"k- 4

'-_" b 4.._ b-"_---4

xy 2Nb

I xKX

KS

2D

2N b

xy

2D

W = 0

..W = O.05

24.000 8.0 16.0 32.0

KS


Panels Under Combined Longitudinal Compression

and Shear with Longitudinal Core (a/b = 1/2)

3-66

KX

4.0

3.0

2.0

1.0

Y a _1

N _ b

x ' "K = x

_ x 2D2

_ 1>w --io.o_\ g \_ w_-o._o\ jA%

w=o. o_l \ ,w:o(a/b = 1.0)

0 2.0 4.0 6.0 8.0 10. O

KS



and Shear with Longitudinal Core (a/b = 1.0)

3-67

5.0

4.0

3.0

KX

2.0

1.0

0

Y,Ia _1

v I

Nx_'_ *-b(a/b = 2. o) : ...... _ "-

_ xy

_-_'_ _ ! Nxb2

_' _ Kx - 2

_'_'\ ff D

__ N b 2

s _r 2DK xy

.....o,,

,0 2.0 4.0 6.0

KS

8.0



and Shear with Longitudinal Core (a/b = 2.0)

3-68

KX

10.0 •

8.0

(a/b = 1/2)

N _

x.... F I t q b_t t_

I

"" ¢-N _" "-32"_" xxy

i N b 2K - x6.0 "_ x 2

_ 2_ N b

xy_, \ Ks

PW = 0.05 i

I _, 'w_o._o- \0 8.0 16. 0 24.0

KS

4.0

2.0

32.0

Figure 3.1-39. Buckling C(_fficients for Corrugated Core Sandwich


and Shear with Transverse Core (a/b = 1/2)

3-69

KX

--N f I I I /o__ ' b

3.0

2.0

1.0

2

KxNbx2W= 0. Ir DN b 2

= 0.05

_-W=O

W

(a/b = 1.0) \

0 2.0 4,0 6.0 8.0 10.0K

S



and Shear with Transverse Core (a/b = 1.0)

3-70

5.0

4.0

3.0

KX

2.0

1.0

(a/b -- 2.0)

\\

0 2.0

Y iia _1

N x -_ l I l I l _- b

""t., , i i i t"-AL..,,,_ _ ,,_ X

\Nxy

I L _x_K -

x 2_ D

S

w=o._\ _

4.0 6.0

KS

N b 2xy2

17 D

05

8.0



and Shear with Transverse Core (a/b = 2.0)

3-71

2.5

2.0

1.5

K X

1.0

0.5

Y 1

._e_q_

Nx ----- = b(a/b : 1. O)

Y

_N_ _ N b 2

Nxy rr D

N N_ __ Ks =- 2

rr D b2N

2_ ' K - YYKy = 1. 7r 2D

\

K =1.7. W=0

-W = 0.05

\i " ()'20 _ __ K=_ 1"9y

_ 0 50

o ,_,o 4o oo =oK s

Figure 3.1-42. Buckling Coefficients for Corrugated Core Sandwich Panels

Under Combined Longitudinal Compression, Transverse

Compression, and Shear with Longitudinal Core

3-72

3.2 CIRCULARPLATES

3.2.1 AvailableSingleLoadingConditions

A searchof theavailableliterature aswell ascontactswith otherswhoare familiar

with sandwichpanelstability referencesandstudiesin progressuncoverednostability

solutionsfor anysingleloadingcondition. This result mighthavebeenanticipated

sincetheflat, circular sandwichplate hasvery fewapplicationsin aerospacevehicle

structures in whichit mustbestableunderthe appliedloads. Consequently,this

manualmakesno recommendationsfor techniquesto beusedin design,andstrongly

suggeststhatall final configurationsbetestedas requiredto demonstratetheir ade-

quacystructurally.

3.2.2 AvailableCombinedLoadingConditions

Nopanelstability solutionswere foundfor anycombinedloadingconditionsapplicable

to flat, circular platesin thecourseof the literature searchnotedin Section3.2.1.

Consequently,this manualmakesno recommendationsfor possibleanalyticalap-

proacheswhichwoulddescribeanystability limits for circular, flat sandwichplates.

3-73

3.3 PLATESWITHCUTOUTS

3.3.1 FramedCutouts

Whileit is highlydesirableto avoidcutoutsin aerospacestructuresbecauseof the

attendantweightproblemsaswell asuncertaintiesaboutloadpile-up andredistri-

bution, theseare apractical necessitybecauseof accessandother requirementsand

everyeffort shouldbemadeto derive reliabledesignapproacheswhichminimizethese

drawbacks.

Mostgeneralizedsolutionsfor plateswith cutoutsemployframing membersandbase

the analysisonthe assumptionof buckledskinpanelswhichcarry only shearloads.

Obviously,the solutionbecomesmuchmorecomplexwhenskinbucklingdoesnot

occur, aswouldbe thecasefor a framedcutoutin a sandwichpanel. Despitethe

increasedcomplexity,however,solutionsfor the loaddistributionaroundthecutout

canbeobtainedfor variousloadapplicationsawayfrom theopening. Knowingthe

loaddistributionadjacentto the cutoutdoesnotnecessarilyprovideananswerto all

questionsregardingtheadequacyof thedesign,however,particularly in thecaseof

sandwichconstruction.

In thecaseof monocoqueor semi-monocoquepanels, the lateral momentsof inertia

of the framingmembersaregenerallysufficientlygreaterthanthoseof theskin such

thattheymaybe consideredto providelateral supportfor thepaneledge. This is not

necessarilythecasefor sandwichpanels,thussettingupthecaseof afree, or nearly

free, edgefor thepanelandfor whichconditionnogeneralstability solutionsor data

were foundin thecourseof this study.

3-74

It maybepossiblefor specificdesignsto beassessed,onthe basisof goodengineering

judgment,to becritical in local instability rather thanfor generalinstability. This

beingthecase, designchecksmaybemadeonthebasisof the equationsgivenin

Section2. This manualmakesno recommendationsfor thosecaseswherethegeneral

instability modeappearsto controlbeyondthe exerciseof goodjudgmentin the devel-

opmentof thedesign,andsufficient testingasneededto insure its integrity.

3.3.2 UnframedCutouts

Unframedcutoutsin sandwichpanelshaveall of the disadvantagesnotedfor framed

cutoutsandrepresenta muchmoreseriousdesignproblemlocally, insofar asthe

free edgeis concerned.Thewriters of this manualencounteredno instancesinwhich

suchadesignapproachwasusedin primary or secondarystructure and, in general,

recommendavoidanceof this practice. This recommendationis basednotonly onthe

lack of anyanalyticalor test databut alsoonpotentialproblemsof faceplate-corebond

separationalongthefree edgedueto damagewhile in use, adhesivedeterioration,

loadcycling, etc.

3-75

h9

Z

,.m0

[I r]

,--+

%-i0

.,4,-..io,,-I+..o

,B,

rop,-+

,+.9o[,¢]

0o,,-i

¢a9

bB

C.l]

Io.9

0

+ o

I o .._

_ _.5

._ "_ ,._

_u o_ +a_ +a oa

+ii!ff

o "_ m+

+ +.+B _,+_ --+ O "+

o+o:_

8+

B , B_0 _ 0

% e_ o o

0

0 0 I

_t_ _ "_

Eo_.S

d6Z©

Z

<

Z

c

0

0

©

m

ryj

---...

i

Boob+

+o

3-76

M

r..,q

©

o<:

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tm

_i ._

o oE,.b

o

fF

p.

¢,

©U H II II II

M m

0 ° 0 _'

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r_

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'1=H

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i-,].

3-77

0

0

c_

0

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tn

o"

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o

L_

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0

b_

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01

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tn

,_ o_

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r_

r_ Ir

o

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.-o_ o_ >

'-_, °_

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3-78

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b_

©

b_

_o

o

_o

d

o"

H

0

0

L_

0

0

0

II

z

3-79

...... lr

_ ,_

c,l

• gl _ o_ .. N rj

N J_

L_

I m ×

M

g°_,1 ,r/?

,-_ ..o i 0 ,

b'_ _ ,

'ii__j £ _:

g_

_r

god

_ _._II

o

go _

0

°

k -T

8

0

I.

Ca

v

i

J

-_ _ •

Z° Z

,_,_= ._,

t

f'--I

o

L_._J

3-80

o _ o_

._ _ o_ _

_ , _.

_ o_ _ _._o _ _ _ _._

°_ _ _ _.-_ o

b._ • I

0

o _ ,, __•

_ _ _ o

o _ _ o_

_ , _'_0

.o _ _ . _o _ _ "_,._ _ _ _._

o _ ._ _ _.

.._ _ _ _ _o

_ _ ._.__ _ ,_ _

o.

-t

II

_4

D-

_,_,4 I

L! iJl/ \,II

3-81

0

L_

r._ o

o o

M

m

e_o

o

,,,4

o

e_

I

r_

e- *') _1

o_ _ _ _

_ z _ ._ _

• _ _ "_

Z _ ._ _ _

_ °_'

_,_ _ _ ,, _ __ • _ M_ 2_..

0

< mr_ m

u o

- I I

>._ir-'J , , '

z

_._ o

_._

_._

_d

_ zm

_' _ z _'

M I!

_ m

_._ __._ _:

8o

<

8_

._..E

o _.

_ _°

_ _._

_ ° _

hi)' i

I I IJ

3-82

0

C

o_

._ o°

_.o

oo

<i

o_

+

c_

II

t_

0 _ "_

o _ ._ "___

° °_o_ ,, o ._

o

Z _ o_ o

_ _ 0 2

_o _

2

o

z

_ o.

_ z _ _

_ _ z _

C_

xj-F__

I I i I I I_ _

3-83

3-1

3-2

3-3

3-4

3-5

3-6

3-7

3-8

3-9

REFERENCES

U. S. Department of Defense, "Structural San_vich Composites, " MIL-

HDBK-23, 30 December 1968.

General Dynamics Fort Wo,__th Division, Structures Manual - Volume 1,

Methods of Analysis, November 1959.

Konishi, D., et a], Honeycomb Sandwich Structures Manual, North

American Aviation Corp. Los Angeles Division, Report NA58-899.

Structures Manual - Section 11.00: Honeycomb Sandwich Structures,

North American Aviation, Inc. Space and Information Systems Division,

May 1964.

Lockheed-Georgia Company, A Division of Lockheed Aircraft Corp.,

"Sandwich Panel Tests for C-5A Airplane," GELAC Report No. ER 8976,

22 February 1967.

General Dynamics Corporation Forth Worth Division, "Design Allowables

and Methods of Analysis, F-111 Airplane," Report No. FZS-12-141,

1 August 1965.

Hexcel Corp., "Design Handbook for Honeycomb Sandwich Structures,"

Report No. TSB 123, October 1967.

Ericksen, W. S. and March, H. W., "Fffects of Shear Deformation in the

Core of a Flat Rectangular Sandwich Panel - Compressive Buckling of

Sandwich Panels Having Dissimilar Facings of Unequal Thickness," Forest

Products Laboratory Report 1583-B, Revised 1958.

Norris, C. B., "Compressive Buckling Curves for Flat Sandwich Panels

with Isotropic Facings and Isotropic or Orthotropic Cores, " Forest

Products Laboratory Report 1854, Revised 1958.

3-85

3-10

3-11

3-12

3-13

3-14

3-15

3-16

3-17

3-18

Norris, C. B., "Compressive Buckling Curves for Flat Sandwich Panels

with Dissimilar Facings, " Forest Products Laboratory Report 1875,

September 1960.

Kuenzi, E. W., Norris, C. B. and Jenkinson, P. M., "Buckling Coeffi-

cients for Simply Supported and Clamped Flat, Rectangular Sandwich Panels

Under Edgewise Compression, " Forest Products Laboratory Research Note

FPL-070, December 1964.

Jenkinson, P. M. and Kuenzi, E. W., "Buckling Coefficients for Flat,

Rectangular Sandwich Panels with Corrugated Cores Under Edgewise

Compression, " Forest Products Laboratory Research Paper FPL 25,

May 1965.

Kuenzi, E. W., and Ericksen, W. S., "Shear Stability of Flat Panels of

Sandwich Construction, " Forest Products Laboratory Report 1560, 1951.

Harris, L. A. and Auelman, R. R., "Stability of Flat, Simply Supported

Corrugated Core Sandwich Plate Under Combined Longitudinal Compression

and Bending, Transverse Compression and Bending, and Shear," North

American Aviation, Inc., Missile Division, Report STR 67, 1959.

Kimel, W. R., "Elastic Buckling of a Simply Supported Rectangular

Sandwich Panel Subjected to Combined Edgewise Bending and Compression, "

Forest Products Laboratory Report 1857A, 1956.

Harris, L. A., and Auelman, R. R., "Stability of Flat, Simply Supported

Corrugated Core Sandwich Rectangular Plates Under Combined Loadings,"

Journal of Aerospace Sciences, Vol. 27, No. 7, p. 525-534, 1960.

Timoshenko, S. P., and Gere, J. M., Theory of Elastic Stability, McGraw-

Hill Book Company, Inc., 1961, p. 373-379.

Gerard, G., and Becker, H., Handbook of Structural Stability, Part I-

Buckling of Flat Plates, National Advisory Committee for Aeronautics

Technical Note 3781, 1957.

3-86

3-19

3-20

3-21

3-22

3-23

3-24

3-25

Kimel, W. R., "Elastic Bucklingof a SimplySupportedRectangularSand-

wich PanelSubjectedto CombinedEdgewiseBending,Compression,and

Shear," ForestProductsLaboratoryReport1859,1956.

Noel, R. G., "Elastic Stabilityof SimplySupportedFlat RectangularPlates

UnderCritical Combinationsof LongitudinalBending,LongitudinalCom-

pression, andLateral Compression,"Journalof the AeronauticalSciences,

Vol. 19, No. 12, p. 829-834, 1952.

Norris, C. B., and Kommers, W. J., "Critical Loads of a Rectangular,

Flat Sandwich Panel Subjected to Two Direct Loads Combined with a Shear

Load," Forest Products Laboratory Report 1833, 1952.

Norris, C. B., and Kommers, W. J., "Stresses within a Rectangular,

Flat Sandwich Panel Subjected to a Uniformly Distributed Normal Load and

Edgewise, Direct, and Shear Loads, " Forest Products Laboratory Report

1838, 1953.

Plantema, F. J., Sandwich Construction, John Wiley and Sons, Inc., New

York, 1966.

U. S. Department of Defense, "Plastics for Flight Vehicles: Part I, Rein-

forced Plastics, " MIL-HDBK-17, Armed Forces Supply Support Center,

1959, Available from U. S. Government Printing Office, Washington, D. C.

U. S. Department of Defense, "Metallic Materials and Elements for Aero-

space Vehicle Structures," MIL-HDBK-5A, February 1966, Available from

U. S. Government Printing Office, Washington, D. C.

3-87

4GENERAL INSTABILITY OF CIRCULAR CYLINDERS

4.1 GENERAL

In the case of axially compressed, thin-walled, isotropic (non-sandwich) cylinders, it

has long been recognized that test results usually fall far below the predictions from

classical small-deflection theory [4-1].

primarily to

a.

and

b.

These discrepancies are usually attributed

the shape of the post-buckling equilibrium path coupled with the presence

of initial imperfections

the fact that classical small-deflection theory does not account for pre-

buckling discontinuity distortions in the neighborhood of the boundaries.

Neglecting the discontinuity distortions, the equilibrium path for an axially com-

pressed perfect cylinder is of the general shape shown by the solid curve in Figure

4.1-1. This path is linear until point A is reached and general instability occurs at

a stress level crCL equal to the result from classical small-deflection theory

[°'eL - R _/3(1-Ve _) for elastic, isotropic (non-sandwich) cylinders However, if

the cylinder is initially imperfect and the discontinuity distortions are considered, the

behavior will be as shown by curve 0B and buckling will occur at the stress O'cr. The

rrtio (Crcr/erCL) will be dependent upon the magnitude of the initial imperfections pres-

ent in the cylinder. However, since this information is not normally available, one

4-1

usuallyfinds it necessaryto resort to either of thefollowingpracticesto obtainpracti-

cal designvalues:

a. Settheallowablecompressivestress equalto thevaluecrMi N shown in

Figu_e 4. i-I.

b. Use the classical small-deflection value o-CL in conjunction with a suitable

knock-down factor )'e which is based on the results from a large array of

test data. The allowable compressive stress is then obtained from

(rcr = )'c_CL (4.1-I)

AcrCL

Axial

Compressive

Stress

O'cr

O'MiN

0

C

End Shortening

PerfectCylinder

ImperfectCylinder

Figure 4.1-1. Equilibrium Paths for Axially

Compressed Circular Cylinders

For isotropic (non-sandwich) cylinders it is common practice to follow the second of

these approaches and, for such cylinders, the test data shows that _'c is a function of

the radius-to-thickness ratio (R/t).

4-2

In thecaseof sandwichcylindershavingrelatively rigid cores, the behavioris similar

to that of the isotropic(non-sandwich)cylinderandonecanexpectimperfectionsand

boundarydisturbancesto precipitategeneralinstability at compressivestressesbelow

thepredictionsfrom classical small-deflectionsandwichtheory, ttowever, in most

practical applications,the sandwichwall will provideaneffectiverelatively thick shell

sothatthediscrepancieswill notbeaslarge asthosenormallyencounteredin thin-

walledisotropic (non-sandwich)cylinders. In addition,asthe coretransverseshear

rigidity decreases,thedifferencesbetweentest resultsandclassicalpredictionswill

diminish. In theextremecasewhereshearcrimpingoccurs, initial imperfectionsdo

notappearto haveanyinfluence.

Oneof themostprominentof theearly designcriteria developedfor axially compressed

circular sandwichcylindersis that of Reference4-2. This solutionemployedlarge-

deflectiontheorytogetherwith approach(a) cited above(Crcr= °'MIN). However, it is

now rather generally agreed that this criterion often provides design values which are

too conservative. In addition, the theoretical development of Reference 4-3 indicates

that CrMi N can be decreased to essentially zero by including a sufficient number of

terms in the large-deflection displacement functions. Therefore, in recent years, it

has become common practice to design sandwich cylinders by method (b) cited above

[4-4 and 4-5]. This approach, which employs small-deflection theory in conjunction

with an empirical knock-down factor, is likewise followed in this handbook.

In the treatment of various types of external loading, it is important to note that the

characteristics of the equilibrium paths are not identical for cases of axial compres-

sion, torsion, or external radial pressure. For purposes of comparison, Figure 4.1-2

4-3

depictsthegeneralshapesof thesepathsfor eachloadingcondition[4-6] assuming

that thecylindersare initially perfectandthatnodiscontinuitydistortionsarepresent.

DeflectionAxial Compression

Figure 4.1-2.

DeflectionTorsion

DeflectionExternalPressure

Typical Equilibrium Pathsfor Circular Cylinders

Basedonthe relative shapesof thesecurves, onewouldexpectthat, undertorsion or

externalpressure, the cylinderswouldbemuchlesssensitiveto initial imperfections

thanin the caseof axialcompression.This hasbeenborneoutbythe availabletest

datafrom thin-walledisotropic (non-sandwich)cylinders.

4-4

4.2 AXIAL COMPRESSION

4.2.1 Basic Principles

4.2.1.1 Theoretical Considerations

The theoretical basis used here is the classical small-deflection solution of Zahn and

Kuenzi [4-7] which includes the follox_ing assumptions:

a. The facings are isotropic but the core may have orthotropic transverse

shear properties.

b. Bending of the facings about their own middle surfaces can be neglected.

c. The core has infinite extensional stiffness in the direction normal to the

facings.

d. The core e×_tensional and shear rigidities are negligible in directions

parallel to the facings.

e. The cylinder is not extremely short (a quantitative limit is specified in

Section 4.2.2).

f. The approximations of Donnell [4-8] can be applied without introducing sig-

nific ant error.

In this handbook, the final equations of Reference 4-7 have been transformed into

equivalent formulations which should be more meaningful to the user. For those cases

where the core shear moduli satisfy the condition

GxzO- _;1

Gyz

the following expression is obtained:

_cr = Kc(ro

(4.2-1)

(4.2-2)

4-5

where

and

When V c _ 2

When V c > 2

where

h 2 _r}-_l t 2

_o = _Ef--_ l_:-_ee (tl + t_)

1K c = 1 - -_- V c

1

Kc = V--c

o-oV c -

°'crimp

(4.2-3)

(4.2-4)

(4.2-5)

(4.2-6)

t

z

E f=

h=

R=

andt_ =

5 2

_rcrim p - (t 1 + t2)t c Gxz

Plasticity reduction factor, dimensionless.

Young' s modulus of facings, psi.

Distance between middle surfaces of facings, inches.

Radius to middle surface of cylindrical sandwich, inches.

Thicknesses of the facings (There is no preference as to which

facing is denoted by the subscript 1 or 2.), inches.

ue = Elastic Poisson' s ratio of facings, dimensionless.

tc = Thickness of core, inches.

Gxz --

Gy z =

Core shear modulus associated with the plane perpendicular to

the facings and oriented in the axial direction, psi.

Core shear modulus associated with the plane perpendicular tothe axis of revolution, psi.

(4.2-7)

The relationship between K c and V c can be plotted as shown in Figure 4.2-1. It is

important to note that the value V c = 2.0 establishes a dividing line between two

different types of behavior. The region where V c _ 2.0 covers the so-called stiff-core

4-6

1.0

K c

II

I2.0

Vc

Figure 4.2-1. Schematic Representation of Relationship

Between K c and V c for 0_< 1

and moderately-stiff-core sandwich constructions When V e is in the neighborhood of

zero, the core shear stiffness is high and the sandwich exhibits maximum sensitivity to

initial imperfections. Hence, for any given radius-to-thickness ratio, the knock-down

factors applicable to such constructions are of maximum severity. As V e increases

from zero to a value of 2.0, the sensitivity to Imperfections beeomes progressively

less. The domain where V c > 2.0 is the so-called weak-core region where shear

crimping occurs. Sandwich constructions which fall within this category are not influ-

enced by the presence of initial imperfections, and a knock-down factor of unity ean be

applied to such structures. It should be possible to develop a continuous transitional

knock-down relationship which recognizes the variable influence of the core rigidity but

this is beyond the scope of the present handbook.

4-7

4.2.1.2 Empirical Knock-DownFactor

Asnotedin Section4.1, the allowablestress intensitiesfor axially compressed,thin-

walled, isotropic(non-sandwich)cylindersareusually computedusingthefollo_ving

equation:

_cr = _c °-cL

The quantity Yc is referred to as the knock-down factor and this value is generally

recognized to be a function of the radius-to-thickness ratio (R/t). Various investi-

gators have proposed different relationships in this regard. The differences arise out

of the chosen statisticalcriteria and/or out of the particular test data selected as the

empirical basis. One of the most widely used of the relationships proposed to date is

the lower-bound criterion of Seide, et al. [4-9] which can be expressed as follows:

}c = 1 - 0.901(1 - e -¢)

where

1 R

This gives a knock-down curve of the general shape depicted in Figure 4.2-2. For the

purposes of this handbook, it is desired that an empirical means of this type also be

provided for the design of sandwieh cylinders. One of the major obstacles to the

achievement of this objeetivo, is the lack of sufficient sandwich test data for a thorough

empirieal determination. Faced with this deficiency, one finds it expedient to employ

the data from isotropie (non-sandwich) cylinders in conjunction with an effective thick-

ness concept and eorrection factors which are based on the few available sandwich test

points. Toward this end, it is usually assumed that, when V c _< 2.0, equal sensitivity

(4.2-8)

(4.2-9)

(4.2-10)

4-8

1.0

Yc

Log Scale

Figure 4.2-2. Semi-Logarithmic Plot of YC vs R/t for Isotropie (Non-

Sandwich) Cylinders Under Axial Compression

to imperfections results from equivalence of the shell-wall radii of gyration p

h(_ _ for sandwich constructions whose two facings are of equal thickness). There-

fore, the approach taken here is to rewrite Equations (4.2-9) and (4.2-10) in terms

of P. The revised formulations give the plot shown as a dashed curve in Figure

4.2-3. Also sho_ in this figure are the appropriate test points obtained from

axially compressed sandwich cylinders [4-2, 4-10, 4-28] which did not fall into

the weak-core category. Eleven such data points are shown. In addition, two

test points are shown for axially compressed conical sandwich constructions

[4-10] which likewise did not lie in the weak-core region. The conical data are

included in Figure 4.2-3 in view of the scarcity of available test results and

also because the cones were analyzed as equivalent cylinders whose radii were

taken equal to the Rz (finite principal radius of curvature) values at the small end

of the specimens. Based on this limit_i amount of sandwich test data, it is recom-

mended that the solid curve of Figul"e 4.2-3 be used for design purposes. This

4-9

1.0

.8

.7

.6

.3

.2

.1

10 103

Figure 4.2-3. Knock-Down Factor Yc for Circular Sandwich

Cylinders Subjected to Axial Compression

4-10

gives Ycvaluesthat are75percentof thoseobtainedfrom thedashedcurvewhichwas

basedonthe empirical formulaof ,':_eide,eta]. [4-9].

In additionto thetest resultsdescribedabove,a considerablenumberof testpoints

are availablefrom cylindrical sandwichconstructionswhichfall into theweak-core

classification. Asnotedin Section4.2.1.1, the methodsrecommendedin this hand-

bookare suchthat, in theweak-coreregion, noempirical reductionwill beapplied

to thetheoreticalresults of Reference4-7. In order to explorethevalidity of this

approach,plotsare furnishedin Figures4.2-4 and4.2-5 whichcomparetheweak-

core test resultsof References4-2 and4-11againstpredictionsfrom therecom-

mendeddesigncriterion. It canbeseenthat all butoneof thetest results exceed

thepredictedstrengths,andthatthe singleexceptionfailedat 86percentof thepre-

dictedvalue. In manyof the caseswhere (O-CrTest/O-Predicted)> 1.0, althoughthe

discrepanciesmeasuredin unitsof psi werenotvery great, thepercentagediffer-

enceswerequitelarge. This behaviorcanbeexplainedbythe fact that thetheoreti-

cal basis [4-7] proposedin this handbookassumesthat bendingof thefacingsabout

their ownmiddlesurfacescanbeneglected. As shownin Reference4-12, this

assumptioncanbevery conservativein theweak-coreregion. However,in the

interest of simplicity, themethodsof this handbookretain this assumptionespecially

sinceit is a conservativepractice andmostpractical sandwichconstructionswill not

bedesignedasweak-corestructures.

In view of the meager compressive test data available from stiff-core and moderately-

stiff-core sandwich cylinders, the method proposed here is not very reliable when

4-11

Vc < 2.0. Therefore, in suchcasesthemethodcanonlybeconsideredasa "best-

available"approach. On the other hand, where the failure is by shear crimping

(V c _>2.0), the method is quite reliable and will, in fact, usually give conservative

predictions.

2O

15

10

_D

9

8

%

• _,_b.

Test Data from Reference 4-2

I I5 10 15 20

Predicted O'cr, ksi

(Neglecting bending stiffness of individual facings)

Figure 4.2-4. Comparison of Proposed Design Criterion Against Test Data for Weak-

Core Circular Sandwich Cylinders Subjected to Axial Compression

4-12

2OO

lOO

100 200

Test Point fromReference 4-11

Predicted Crcr , ksi

(Neglecting bending stiffness

of individual facings)

Figure 4.2-5. Comparison of Proposed Design Criterion Against

a Test Result for a Weak-Core Circular Sandwich

Cylinder Subjected to Axial Compression

4.2.1.2.1 Interpretation of Test Data

As indicated in the preceding paragraphs, appropriate test data must be used in order

to arrive at practical values for the knock-down factor. However, one can be easily

misled in this endeavor when the test data and/or the classical theoretical predictions

lie in the inelastic region. To demonstrate this point as simply as possible, the pre-

sent discussion is limited to the case of axially compressed circular sandwich cylin-

ders for which V c = 0. Then the recommended design value for the critical stress

can be expressed as follows:

h 2 _ t_ (4.2-11)°'cr = Yc _ Ef R I_Z-Y-Ye2 (tl + t2)

4-13

For anyparticular test specimen,therelatedvaluefor theknock-downfactor shouldbe

computedfrom thefollowingexpressionwhichis obtainedby a simpletranspositionof

C(rCrTest_\ "_Test /

= (4.2-12}

(YC)Test [E h 2vf_-lt_ ]fR _ (t_ t_)

Equation (4.2-11):

The plasticity reduction factor DTest is evaluated at the actual experimental buckling

stress. By inspection of the numerator and denominator of Equation (4.2-12), one can

conclude that this formula may be rewritten in the following more meaningful form:

wEXperimental critical stress value]

hich would have been attained had]the material remained elastic ]

(Yc)Test _lassical theoretical critical" (4.2-13)

| stress value assuming the

L behavior to be elastic

The example illustrated in Figure 4.2-6 should help to clarify this concept. In this

figure, the solid line represents the stress-strain curve for the test specimen mate-

rial. Suppose that this particular specimen buckled at a stress equal to O-CrTest . As

indicated in the figure, it is assumed here that this stress lies in the inelastic region

so that NTest will be less than unity. For the purposes of this discussion, further

assume that _?Test - 0.80. If the material had remained elastic, the experimental

critical stress would have been somewhat higher than OcrTest . This greater value

I

will be denoted as CrcrTest . Then it follows that

°'CrTest °-CrTest- 1.25 (rCrTestCrcrTest - _)Test 0.80

(4.2-14)

4-14

¢rCL

(r_rTest

O'Ma x

_rcrTest

Strain

Figure 4.2-6. Stresses Involved in Interpretation of Test Data

Now let it also be assumed that, using elastic material properties, the classical thee-

retical critical stress equals the value o-CL indicated in Figure 4.2-6. The following

formula would then give the proper value for the experimental knock-down factor:

where

( ¥C)Test -

O'CrTest 1.25 ¢rcrTe st

¢rCL oCL

h 2v/t _ tz

(rCL = Ef _ _pe z {tl +t2)

(4.2-15)

(4.2-16)

4-15

Theabovediscussionis givenhere sincesomeof the resultspresentedin the literature

canbequitemisleading. That is, comparisonsareoftenshownbetweentheactualtest

value (rcrTest (withoutregardasto whetherelastic or not)andthe inelastic classical

theoreticalprediction. For thecaseshownin Figure4.2-6, the latter valuecannot

exceed_Max andthis typeof comparisonmight leadoneto believethat theappropriate

knock-downfactor is very closeto unity. However,useof thecorrect approachas

expressedby Equations(4.2-13)and(4.2-15)givesa muchlower Yc value. For any

given geometry, one could always show very close agreement between O'crTest and

O-Ma x simply by choosing a material with a sufficiently low yield strength and having

a flat post-yield stress-strain curve.

4-16

4.2.2 DesignEquationsandCurves

For simply supportedcircular sandwichcylinderssubjectedto axialcompression,the

critical stressesmaybecomputedfrom the relationshipsgivenonpage4-18wherethe

subscripts1and 2 refer to the separatefacings. "I2aere is no preference as to which

facing is denoted by either subscript. These equations were obtained by a simple ex-

tension of the formulas developed in Reference 4-7 which only considered the case

where the behavior is elastic and the moduli of elasticity are identical for both facings.

The extended versions given in this handbook were derived through the use of equivalent-

thickness concepts based on the ratios of the moduli of the two facings. For cases

where the two facings are not made of the same material, these equations are valid

only when the behavior is elastic (7 = 1). Application to inelastic cases (_? _ 1) can

only be made when both facings are made of the same material. For such configura-

tions, El and E_ will, of course, be equal.

The buckling coefficients K c can be obtained from Figure 4.2-7. Curves are given

Gxzthere for both 0 _< i and 0 = 5 where 0 - Since these two plots are not very

Gyz

different from each other, one may use Figure 4.2-7 to obtain rather accurate esti-

mates of K c when 1< 0 <5.

Whenever V c < 2.0, the knock-down factor Yc can be obtained from Figure 4.2-8.

When V c _> 2.0, use Yc = 1.0.

For elastic cases, use _) = 1. Whenever the behavior is inelastic, the methods of


4-17

! I I I Icq L'_

jO

oJ

v-4I I

17A

v¢--..J

I

-.>

OO

N

O

C

+ IJ

O

I IO,1 o,1

¢7 oL)

g-

¢,J

+

°,.._¢7

_>

4-18

JiJ, i¸ il

........ :!:i!

_9

i

_9

0

_9

<

}.

1

4-19

_'C

1.o ..... _/_'!!I..... !_....' e ..............._i:il :::i :i!_ ii_i ::i

.9 i:, x/(EI tl) (E2 t2) h

• 8 'ki_ i_L[ii_4.!i_i T i_

.............'!t

i i,I

T:t:T!_-T, ,, jklt, F3 dii_t.... _:ii11 i!ii

.2 : J'_' [71_i!:7__ti3 : lit, " 3!7!t7['_i

I0 10_ 10 a

Figure 4.2-8• Design Knock-Down Factor for Circular Sandwich

Cylinders Subjected to Axial Compression

4-20

The critical axial load (in units of pounds) can be computed as follows:

Pcr = 27rR [_cr_ tl + _cre t_] (4.2-26)

In the special case where t_ = t2 --- tf and both facings are made of the same material,

Equations (4.2-17) through (4.2-26) can be simplified to the following:

(rcr = Yc Kc ¢ro (4.2-27)

(_Ef) h (4.2-28)

°'crimp5 2

2 tf t cGxz (4.2-29)

o-oV c - (4.2-30)

arcrimp

Pcr = 41rRtf o-cr (4.2-31)

Equations (4.2-17) through (4.2-31) and Figure 4.2-7 are valid only when the length L

of the cylinder is greater than the length of a single axial half-wave in the buckle pat-

Gxz

tern for the corresponding infinite-length cylinder. For the case where _ - Gy z - 1,

one can apply the following test to determine if the cylinder length is sufficiently large:

When V < 2el

Equations (4.2-17) through (4.2-31)

and Figure 4.2-7 are valid only where

When Vcl > 2

Equations (4.2-17) through (4.2-31)

and Figure 4.2-7 are valid for any

value of L.

4-21

For constructionswhere e _ 1, nocorrespondingnumericalcriterion is presently

available. In suchcases,onecanonly usethe abovetest in conjunctionwithengineer-

ingjudgement. It is helpfulto pointout, however,thatmostpractical sandwichcylin-

ders for aerospaceapplicationswill besufficiently longfor Equations(4.2-17)through

(4.2-31) and Figure 4.2-7 to be valid. In addition, it is comforting to note that the

use of these relationships for shorter cylinders results in conservatism.

Cylinders which fail to meet the foregoing length requirement are usually referred to

as short cylinders. The only means available for the analysis of such sandwich cylin-

ders under axial compression is the solution of Stein and Mayers [4-13] which is only

valid

a. when 0 = 1

and

and

and

b. when both ends of the cylinder are simply supported

c. when both facings are made of the same material

d, the thickness of one facing is not more than twice the thickness of the

other facing.

For short sandwich cylinders which satisfy these conditions, one can use the design

curves of Figure 4.2-9 which involves the following parameters:

Z - 2L_Rh 1-_/_e2 (4.2-32)

4-22

where

and

2rr D

r a -

L2 Dq

• _cr (t_ + t_) Le

Kc = ycrr 2 D

(_ h)(E2 ts)h2

D = r](l_Pe_)[(F__tl) + (F__t_i]

(4.2-33)

(4.2-34)

(4.2-35)

h _

Dq = _-c Gxz (4.2-36)

L = Over-all length of cylinder, inches.

During the preparation of this handbook, no solutions were uncovered for axially com-

pressed sandwich cylinders having any degree of rotational restraint at the boundaries.

However, in most practical aerospace applications, the cylinders will be sufficiently

long for such fixity to have negligible effects on the buckling loads.

4-23

!

KC

......]

70 ---- -_

1

4o- - + ....• _ _

30-: _ II '

t

20-=' i

::ii4' :!

1

10_ _, :

7- --'_-, :

4- ..... i

• i:

.5- _ ::

.4- ,!

I

• 9

1

Z

Figure 4.2-9. Buckling Coefficient for Short Simply-Supl_orted Sandwich Cylinders

Subjected to Axial Compression (0 :: 1)

4-24

4.3 PUIIE BEN1)ING



Based on small-deflection theory, investigations were made in References 4-14, 4-15,

and 4-16 of elastic instability in thin-walled, isotropic (non-sandwich) cylinders sub-

jeeted to axial compressive stresses which vary in the circumferential direction. From

the results of these references, it can be concluded that, regardless of the nature of

the circumferential stress distribution, classical instability is reached when the peak

axial compressive stress satisfies the condition

Et

o" _.6 R (4.3-1)

It should be recalled that the value . 6 Et/R is also obtained from the small-deflection

solution for thin-walled, isotropic (non-sandwich) cylinders subjected to uniform axial

compression. In view of this result, one might reasonably expect that small-deflection

sandwich theory would also indicate that only the peak axial compressive stress need

be considered in cases of pure bending or combined bending and axial compression. It

has been shown in References 4-17, 4-18, and 4-19 that this is indeed the case. Ref-

erences 4-17 and 4-18 demonstrate this for weak-core sandwich cylinders while Ref-

erence 4-19 deals with infinitely long cylinders which fall in the stiff-core and moder-

ately-stiff-core categories. Therefore, for the purposes of this handbook, it is assumed

that the theoretical considerations of Section 4.2 (axial compression) apply equally well

to sandwich cylinders which are subjected to pure bending if the analysis considers only

the peak value of the applied compressive stress. The only differences lie in the em-

pirical knock-dow_l factors recommended for the two cases.

4-25

4.3.1.2 EmpiricalKnock-I)ownFactor

In thecaseof pure bending, only a relatively small portion of the cylinder's circumfer-

ence experiences stress levels which initiate the buckling process. Because of the

consequent reduced probability for peak stresses to coincide with the location of an

imperfection, it is to be expected that the knock-down factors for pure bending will not

be as severe as the corresponding factors for axial load. For thin-walled, isotropic

(non-sandwich) cylinders under pure bending, Seide, et al. [4-9] have proposed the

following lower-bound relationships:

Yb = 1 - 0.731 (1 - e -¢) (4.3-2)

where

= _ (4.3-3)

Comparison against Equations (4.2-9) and (4.2-10) shows that this bending criterion

does indeed give Yb values of lesser severity than those which apply to the axially

compressed cylinders. Following the same approach as that taken in Section 4.2, the

habove equations are rewritten in terms of the shell-wall radius of gyration p(_ for

sandwich constructions whose two facings are of equal thickness).

lationsthen give the plot shown as a dashed curve in Figure 4.3-1.

figure are the appropriate test points from stiff-core sandwich cylinders subjected to

pure bending [4-201. Since only three such data points are available, itwas thought to

be helpful to include the axial compression sandwich data points previously shown in

Figure 4.2-3. To fullyunderstand the information given in Figure 4.3-1, itis im-

portant for the reader to be aware of the data reduction techniques used here. For an

The revised formu-

Also shown in this

4-26

explanation of the procedures used in this handbook, reference should be made to the

related discussion in Section 4.2.1.2.1.

Based on the limited amount of available test data, it is recommended here that the

solid curve shown in Figure 4.3-1 be used for the design of sandwich cylinders sub-

jected to pure bending. This gives T b values that are 75 percent of those obtained

from the dashed curve which is based on the empirical formula of Seide, et al. [4-9].

This is consistent with the practice followed in Section 4.2 for the case of axial com-

pression where the design knock-down factor was likewise taken to be 75 percent of

the value obtained from the corresponding curve derived from Reference 4-9.

In view of the meager test data available from sandwich cylinders under pure bending,

the method proposed in Section 4.3.2 is not very reliable when V c < 2.0. Therefore,

in such cases, the method can only be regarded as a "best-available" approach. On

the other hand, when the failure is by shear crimping (V c >_ 2.0), the method is quite

reliable and will, in fact, usually give conservative predictions.

4-27

• !

(_) Data from Ref. 4-20 (Cylindrical; Pure Bend.)

+ Data from Ref. 4-2 (Cylindrical; Axial Compr.) t

• Data from Ref. 4-10 (Cylindrical; Axial Compr.)_

1010 a

Figure 4.3-1. Knock-Down Factor _ for Circular Sandwich

Cylinders Subjected to Pure Bending

4-28

4.3.2 Design Equations and Curves

For simply supported sandwich cylinders subjected to pure bending, one may use the

same design equations and curves as are given in Section 4.2.2 (for axial compression)

except for the following:

a. For the ease of pure bending, use Figure 4.3.2 to obtain the knock-down

factor Yb whenever V e < 2.0 (When Vc >_2.0, use Yb = 1.0).

bo For the case of pure bending, the critical stresses obtained from the

equations and curves of Section 4.2.2 correspond to the circumferential

location which lies on the compressive side of the neutral axis and is

furthest removed from that axis. Hence the computed stresses are the

peak values within the variable circumferential distribution. Therefore,

when the behavior is elastic, the critical bending moment Mcr can be

computed from the following:

where

Mcr

Mcr = rrR _ [¢rcr 1 t_ + _cr2 t_] (4.3-4)

= Critical bending moment, in.-lbs.

R = Radius to middle surface of sandwich cylinder, inches.

(rcr: and O-cra = Critical compressive stresses in facings 1 and 2, respec-

tively, which result in general instability of the cylinder, psi.

tl and ts = Thicknesses of the facings 1 and 2, respectively, inches.

Note: There is no preference as to which facing is denoted by the

subscripts 1 and 2.

4-29

1.0

.9

6

_b5

.2

.I

I0

(s)P

io_

Figure 4.3-2. Design Knock-Down Factor "¢b for Circular Sandwich

Cylinders Subjected to Pure Bending

4-30

To computel_cr whenthebehavioris inelastic, onemust resort to numericalintegra-

tion techniques.

Sincetheprocedurerecommendedheremakesuseof the methodsof Section4.2.2, all

of the limitations of that sectionare equallyapplicableto the presentcase. Thatis,

onlysimply supportedboundariesare consideredandtheprimary solutionis excess-

ively conservativefor the so--calledshort-cylinderconstructions. In addition,only

very limited meansareavailableto facilitate a quantitativeassessmentof whetheror

not aparticular constructionfalls within the short-cylinderclassification. Further-

more, thecomputationof critical stressesfor short-cylinder constructionscanonly

be accomplishedfor rather specialcasesascited in Section4.2.2.

Asnotedin Section4.2.2, duringthepreparationof this handbook,nosolutionswere

uncoveredfor axially compressedsandwichcylindershavinganydegreeof rotational

restraint at theboundaries.However,it wasalsonotedthat, in mostpractical aero-

spaceapplications,thecylinderswill besufficiently longfor suchfixity to havenegli-

gibleeffectsonthecritical stresses. Thesamesituationexists for thecaseof pure

bending.

4-31

4.4 EXTERNAL LATERAL PRESSURE



This section deals with the loading condition depicted in Figure 4.4-1. Note that the

sandwich cylinder is subjected to external pressure only over the cylindrical surface.

No axial loading is applied. In addition, it is specified that the ends are simply sup-

ported. That is, during buckling, both ends of the cylinder experience no radial dis-

placements and no bending moments.

Both rt_

ends

s imply s uppo

Figure 4.4-1.

p, psi

Circular Sandwich Cylinder Subjected to

External Lateral Pressure

p, psi

The theoretical basis used here is the classical small-deflection solution of Kuenzi,

et al. [4-21] which includes the following assumptions:

a. The facings are isotropic.

b. The facings may be of equal or unequal thicknesses.

c. The facings may be of the same or different materials.

d. Poisson's ratio is the same for both facings.

e. Bending of the facings about their own middle surfaces can be neglected.

f. The core has infinite extensional stiffness m the direction normal to the

facings.

g. The core extensional and shear rigidities are negligible in directions


4-32

h.

i.

j.

The transverse shear properties of the core may be either isotropic or

o rthotropic.

2RThe inequality _- >> 1 is satisfied.

Several additional order-of-magnitude assumptions are valid, as noted

below in connection with Equation (4.4-2).

The solution of Kuenzi, et al. [4-21] draws upon the earlier groundwork laid by

Raville in References 4-22, 4-23, and 4-24. Norris and Zahn used these reports to

develop design curves which are published in References 4-25 and 4-26. The work of

Kuenzi, et al. [4-21] constitutes the latest revision to this series of reports and is

the most up-to-date treatment of the subject. However, the format of their results

has been slightly modified in Reference 4-5 in order to reduce the scope of interpola-

tion required in practical applications. The revised format is used here. However,

the need for interpolation has not been entirely eliminated since separate families are

still required for each of the selected values for Vp [see Equation (4.4-4)].

The final theoretical relationships used in this handbook are as follows:

where

Cp

and

Cp [(Eltl) + (E_)]Pcr = R (l-re _)

(4.4-1)

= Minimum value (with respect to n) of Kp , dimensionless.

_2 (n 2_ 1) (3 + n2 L2 _[/n2 L2 7v:'R2 8--_--_/L\_-'_R 2 1)(nL 1+ _-)-2]+ 9 [1+ (n 2

+

(4.4-2)

+ _]Vp]

4-33

L_/2(Ez tl) (E2 t_) h2

[(Eltl) * (E_t_)laW(4.4-3)

whe re

Per

It

e

Et and E2

tx and te

n

L

h

Gyz

Note:

(Eltl) (Ezt2) hVp =

[(El tl) + (Ez t2)] (1;,seg)R_-Gi, z (4.4-4)

= Critical value of external lateral pressure, psi.

= Radius to middle surface of cylindrical sandwich, inches.

- Elastic Poisson's ratio of facings, dimensionless.


= Young's modtdi of facings 1 and 2, respectively, psi.

= Thicknesses of facings 1 and 2, respectively, inches.

= Number of circumferential full-waves in the buckle pattern,dimensionless.

- Over-all length of cylinder, inches.

- Distance between middle surfaces of facings, inches.

- Core shear modulus associated with the plane perpendicularto the axis of revolution, psi.

There is no preference as to which facing is denoted by thesubscript 1 or 2.

For cases where the two facings are not made of the same material, the foregoing

formulas are valid only when the behavior is elastic (_ = 1). Application to inelastic

cases ( E _ 1) can only be made when both facings are made of the same material.

For such configurations, E_ and E_ will, of course, be equal.

4-34

Equation(4.4-2) constitutesanapproximateexpressionfor Kp sinceit embodiesthe

assumptionscitedearlier in this sectionin additionto thefollowing:

a. Terms containingKp and 4 R2 wereneglected.

b. It wasassumedthat (1 + m2_) = 1, where m is a small whole number.

By using Equation (4.4-2), plots can be generated of the form shown in Figure 4.4-2.

The design curves of this handbook are of this type and were taken directly from Ref-

erence 4-5. It is helpful to note here that lower and upper limits exist for the coeffi-

cient Cp and these are identified in Figure 4.4-2. The lower limit is associated with

long-cylinder behavior. Such configurations are unaffected by the end constraints and

the related critical pressures are equal to those for rings which are subjected to

external pressure. For portions of the cylinders that do not lie in the neighborhoods

of the boundaries, the buckle patterns will be the same as are obtained from such rings.

In this connection, it should be noted that application of the Donnell approximations

[4-8] to non-sandwich rings leads to critical pressures which are 33 percent higher

than the predictions from accurate ring formulations. This is due to the fact that the

related number of circumferential full-waves (n = 2} is not sufficiently high to justify

Donnell's [4-8] assumptions. It is important to observe that the theory of Reference

4-21 retains a sufficient number of terms to accurately predict the buckling of long

cylinders. That is, when Gy z -* o0 {Vp -_ 0) and L/R is large, the critical pressure

is equal to the value obtained from that ring theory which is capable of properly de-

scribing the behavior where n = 2. The upper limit to the curve of Figure 4.4-2 is

associated with the shear crimping mode of failure which involves extremely short

4-35

circumferential wavelengths (n--oo). Specialization of Equations (4.4-i) through

(4.4-4) to this case gives the following formula for the critical compressive running

load Ncr measured in units of lbs/inch:

Ncr

where

= h Gy z (4.4-5)

Ncr = Per It (4.4-6)

By using the approximation h _ tc , it can easily be shown that Equation (4.4-5) is

equivalent to the crimping formula presented earlier as Equation (2.3-9).

Cp

• " Crimping)

Note: Vp = Constant

0 a :- Constant

Lower Limit

(n = 2)

Figure 4.4-2. Schematic Representation of Log-Log Plot of Cp

Versus L/R for Circular Sandwich Cylinders

Subjected to External Lateral Pressure

Another important point which should be noted is that the approximate formula for Kp

[Equation (4.4-2)] does not contain the core shear modulus associated with the plane

perpendicular to the facings and oriented in the axial direction (Gxz). This modulus

has very little influence on cylinders longer than approximately one diameter and has

4-36

therefore disappeared through the approximations made in the development of Refer-

ence 4-21. Thus the theory and design curves presented in this section (Section 4.4)

of the handbook can be considered applicable to sandwich cylinders having cores with

either isotropic or orthotropic transverse shear moduli.

4-37

4.4.1.2 Empirical Knock-DownFactor

In Section4.1 it is pointedoutthat, for circular cylinderssubjectedto external

lateral pressure,theshapeol 0mpost-bucklingequilibriumpathis suchthatone

wouldnotexpectstrongsensitivity to thepresenceof initial imlxerfections.This has

indeedbeenshownto be thecasefor isotropic (non-sandwich)cylinderswherethe

availabletestdatashowrathergoodagreementwith thepredictionsfrom classical

small-deflectiontheory. In viewof this, it hasbecomewidespreadpracticeto either

acceptuncorrectedsmall-deflectiontheoreticalresults asdesignvaluesor to applya

uniformknock-downfactor _p of 0.90 regardlessof theradius-to-thicknessratio.

In Reference4-4 thelatter practiceis also recommendedfor sandwichcylindersand

this approachhaslikewisebeenselectedas thecriterion for this handbook.

Theonly availabletest datafor sandwichcylinderssubjectedto externallateral pres-

sltre are thosegivenin References4-27and 4-28. In thefirst of thesedocuments,

Kazimi reports the results from twospecimenswhichwereidenticalexceptlor the use

of normal-expandedcore in onecylinderwhile theother incorporatedover-expanded

core. Thefollowingresultswereobtained:

Comparison of Theoretical Predictions Versus

Test Results of Kazimi _4-27_

@ @ @ @

Core Type Test Pcr

(psi)

Theoretical Pcr

Based on Ref.

4-5 and "/p : 1.0

(psi)

('(P)Test

(Test Pcr)

Normal-Expanded 17 30.5 .56

Over- Expanded ! 27 30.5 .88

4-38

m

(Theo. Pcr)

: ®-®

Kazimi [4-27] attributes the scatter in his test results to the circumstance whereby

the over-expanded condition gives more uniform core properties than are obtained

from normal-expanded honeycomb. The argument put forth on behalf of this viewpoint

rests on the fact that the over-expanded core exhibits less anticlastic (saddle-type)

deformation in forming the core to the shape of the cylinder.

In Reference 4-28 Jenkinson and Kuenzi report the results obtained from five test

cylinders of nominally identical construction. These cylinders all had glass-reinforced

plastic facings. Each facing was composed of three layers of glass fabric with their

individual orientations controlled to provide a laminate having in-plane properties

which were essentially isotropic. The following results were obtained from these

cylinders :

Comparison of Theoretical Predictions Versus

Test Results of Reference 4-28

® ® ® ®

Cylinder

No.Test Pcr

(psi)

60

52.5

52.5

52.5

52.5

Theoretical Per

Based on Ref.

4-5 and 7p = 1.0

(psi)

55.2

45.2

52.6

45.2

47.6

(YP)Test

(Test Pcr)

(Theo. Pcr)

®+®1.09

1.16

1.00

1.16

1.10

For specimens 2 through 5 it was reported that initial buckling occurred at external

lateral pressures which ranged from 50 to 55 psi. Therefore, in the above tabulation

4-39

it wasassumedthateachof thesefour cylindersbuckledat 52.5psi. In general, the

test valuesare somewhathigherthanthetheoreticalpredictions. This is probably

due to

a.

b.

and

C.

the absence of precise data on the material properties

inaccuracies due to interpolation between the theoretical curves

the fact that the facings were relatively thick in comparison with the

sandwich thickness (_ _ .25).

The foregoing test results from References 4-27 and 4-28 seem to provide added

justification for the use of '{1) = 0.90 as a lower-bound knock-down factor, ltowever,

it would certainly be desirable to supplement these data with additional tests on speci-

mens having small tf/h ratios which would be truly representative of configurations

usually found in realistic full-size sandwich cylinders.

An additional point of interest concerning the use of a unifornl value of Fp = 0.90 is

the fact that shear crimping failures will be insensitive to the presence of initial im-

perfections. Hence, in the region where this mode of failure prevails, one could

safely use the value ,(p = 1.0, especially since the theoretical basis used here neglects

the bending stiffnesses of the facings about their own middle surfaces. However, in-

spection of the design cuz_ves of Section 4.4.2 shows that this type of failure will only

occur for extremely low L/R values. This fact, coupled with considerations of

simplicity and the moderate nature of the value "_p - 0.90, led to the selection here

of a uniform knock-down factor.

4-40

In viewof themeagertestdataavailablefrom sandwichcylinderssubjectedto external

lateral pressure, the methodrecommendedherecanpresentlybe regardedas onlya

"best-available"approach.However,thereappearsto be little reasonto doubtthat

f_'ther testingwouldshowtheseproceduresto bequitereliable.

4-41


For simply supportedcircular sandwichcylinderssubjectedto externallateral pres-

sure, thecritical pressuremaybecomputedfrom theequation

7p T, Cp

Pcr - R(l__e _) [(Eltz) _ (Eet2)]

where

"_p = 0.90

and Cp is obtained from 1,'igures 4.4-3 through 4.4-5.

one must compute the following values:

(E _t_ )(Ea te) he,2

uJ : [(Eltl) + (E2ta)] 2 112

(4.4-7)

In order to use these curves,

(4.4-8)

(E 1 tl )(Ea ta ) h

Vp = TI [(Eltt) +(Eata)] (l-re _) He Gyz(4.4-9)

For elastic cases, use _] = 1. Whenever the behavior is inelastic, the methods of


For cases where the two facings are not made of the same material, the foregoing

formuIas are valid only when the behavior is elastic (,Z - 1). Application to inelastic

cases (_ _ 1) can only be made when both facings are made of the same material.

For such configurations, E_ and Ee will, of course, be equal.

Since separate families of design curves (Cp vs L/R) are provided for only three

values of Vp, one will usually find it necessary to use graphical interpolation or

extrapolation to establish Cp for the configuration of interest. Where desired,

4-42

improved accuracy can be obtained by minimizing Equation (4.4-2) with respect to n

in order to obtain Cp.

The results given by the procedures specified here apply to sandwich cylinders having

cores with either isotropic or orthotropic transverse shear moduli.

4-43

0.01

\5

\ ",v

CIo00] ----*,-

NN

0.000] -

C ......p

O. 00001 ---

Per

\.

'

_x

N <

N-<

Yp rtC

'2 [(I lt]) + (E2t,2)]

I_:] - v )

9OP

I 2 defined by Eq. (4.,t-8)

•] Vp defh]ed by Eq. (,i.4-'J)

w..:

\

\

0.000001--.-- ---1 10

2= O. 0001

/o oJo!oli _-llll,-

---:oTioiiI

X_

100

Figure 4.4-3. Buckling Coefficients Cp for Circular Sandwich Cylinders



tropic; Vp : 0

4-44

0.0

0.00]

O. O00l

CP

0.00001

0,000001

Figure 4.4-4.

:,.-

...... _p ,7c

-- Per - P [(Eltl) + (E2t2)]2

..... R(1 -v e )

_2 defined by Eq. (4.4-8)

V defined by,Eq. (4.4-9)P

10

Buckling Coefficients Cp for Circular Sandwich CylindersSubjected to External Lateral Pressure; Isotropic Facings;

Transverse Shear Properties of Core Isotropic or Ortho-tropic; V = 0.05

P

4-45

0.01

0.001

¢' yz 1

0.0001

CP

0.0000]

0.000001 -- 1001 10

Figure 4.4-5. Buckling Coefficients Cp for Circular Sandwich CylindersSubjected to External Lateral Pressure; Isotropic Facings;


tropic;V = 0.i0P

4-46

4.5 TORSION

4.5.1 BasicPrinciples

4.5. I. 1 TheoreticalConsiderations

This sectiondealswith the loadingconditiondepictedin Figure 4.5-1. Notethat the

onlyconsiderationgivento boundaryconditionsis that, duringbuckling, it is assumed

that no radial displacementsoccurat either end. Further conditionsat theseboundar-

iesare completelydisregarded. This approachshouldbesufficiently accuratefor all

simply supportedcylindersexceptthosewhicharevery short.

T, In.-Lbs. Torque

__ T, In.-Lbs. Torque

It is assumed that, during buckling,no radial displacements occur ateither end.

Figure 4.5-1. Circular Sandwich Cylinder Subjected to Torsion

The buckling of isotropic (non-sandwich), circular cylinders subjected to torsion was

treated by Donnell in Reference 4-8 which has become a standard source of information

concerning reasonable approximations which can be employed in practical thin-shell

theory. Using the Donnell approximations, Gerard [4-29] has investigated the buck-

ling of long circular sandwich cylinders subjected to torsion. This solution gives no

consideration whatsoever to the boundary conditions. Such an approach is valid in

4-47

view of the qssumed extremely long configuration. On the other hand, in Reference

4-30, March and Kuenzi develop small-deflection solutions for sandwich cylinders of

both finite and infinite lengths. The boundary conditions taken lot the finite-length

cylinders are as indicated in Figure 4.5-1. For the purposes of this handbook, Refer-

ence 4-30 is considered to provide the most up-to-date treatment of the subject. The

theoretical design curves given in Section 4.5.2 were taken directly from that report

and embody the following assumptions:

a. The facings are isotropic.

b. The facings are of equal thickness. However, the curves are reasonably

accurate for sandwich cylinders having unequal facings, provided that the

thickness of one facing is not more than twice the other.

c. Young's modulus is the same for both facings.

d. Poissonts ratio is the same for both facings.

e. The core has infinite extensional stiffness in the direction normal to the

facings.

f. The core extensional and shearing stiffncsses are negligible in directions


g. The transverse shear properties of the core may be either isotropie or

o rthotropic.

h. The approximations of Donnell [4-81 can be applied.

The design curves include separate families which respectively neglect and include

bending of the facings about their own middle surfaces. However, for both of these

situations, it is assumed that the facings are thin.

4-48

Thetheoreticalbucklingrelationshipusedhereis

drcr = K s ]] ErR (4.5-1)

which is based on the further assumption that both facings are made of the same mate-

rial. The notation used here is as follows:

rcr = Critical value of facing shear stress, psi.

K s : Torsional buckling coefficient, dimensionless.


Ef - Young's modulus of facings, psi.

d = Total thickness of sandwich wall.

t c

t l and te

R

d t c +tz + te (4.5-2)

= Thickness of core, inches.

: Thicknesses of the facings (There is no preference as to which facing

is denoted by the subscript 1 or 2.), inches.

- Radius to middle surface of sandwich cylinder, inches.

The buckling coefficient K s is arrived at by the minimization of a complicated ex-

pression given in Reference 4-30. This formulation is not reproduced here. However,

it should be noted that the indicated minimization leads to K s values which can be

plotted in the general form shown in Figure 4.5-2 where

L _

Zs - dR (4.5-3)

16 t c tlt_ Ef- (4.5-4)

Vs 15 (tl +t2) Rd Gxz

4-49

and

Gxz

Gyz(4.5-5)

L - Over-all length of cylinder, inches.

Gxz = Core shear modulus associated with the plane perpendicular to the

facings and oriented m the axial direction, psi.

Gy z = Core shear modulus associated with the plane perpendicular to the

axis of revolution, psi.

n Number of circumferential full-waves in the buckle pattern,

d imens ionl es s.

K sUpper Limit (Shear Crimping)

t c/d - Constant I

_ Constant '/

: Constant ]

Long Cylinder

(,1 : 2)

Z S

Figure 4.5-2. Typical Log-Log Plot of the Buckling Coefficient K_

for Circular Sandwich Cylinders Subjected to Torsion

The curves given in Section 4.5.2 are of this type. Note that the upper limit for the

buckling coefficient K s correspends to the shear crimping mode of failure which

involves extremely short circumferential wave-lengths (n -. _o). Specialization of

the buckling equations to this case leads to the following result when itis assumed

that tc/d _ I :

4-50

where

h =

5 2

Tcr = rcrimp - (tz + t_) t c v/Gxz GY z (4.5-6)

Distance between middle surfaces of facings, inches.

In connection with sandwich constructions having large values for the parameter Z s

(long cylinders), it is pointed out that the cylinder will buckle into an oval shape (n = 2)

for which the Donnell approximations [4-8] are no longer valid. To illustrate this

point, attention is drawn to the results obtained for isotropic (non-sandwich), circular

cylinders subjected to torsion. By using the Donnell approximations, Gerard [4-31]

obtains the following result for the critical shear stress of such cylinders:

rcr - (l_Ve_)a/_ E (4.5-7)

In Reference 4-32, Timoshenko presents the following result from a more rigorous

solution which does not invoke the Donnell approximations:

: E (.___.) 3/_rcr 3 _]2 (1-Ve2)a/a (4.5-8)

The more exact result gives a critical stress which is only 87 percent of that given by

the Donnekl approach, This is similar to the situation encountered in the case of exter-

nal lateral pressure (see Section 4.4) where the difference is even more pronounced.

Since the torsional design curves of Section 4.5.2 incorporate the Donnell approxima-

tions, they must be used with caution in the case of long cylinders (n = 2).

4-51

4.5.1.2 Empirical Knock-Down Factor

In Section 4.1 it is pointed out that, for circtdar cylinders subjected to torsion, the

shape of tile post-buckling equilibrium path is such that one would not expect the sensi-

tivity to initial imperfections to be as strong as that encountered in tile case of axial

compression. On the other h:md, tile sensitivity iil torsion would be expected to be

somewhat more severe than is exhibited by circular cylinders under external lateral

pressure, hi the case of isotropic (non-sandwich), circtdar cylinders loaded in torsion

I{eference 4-8 indicates that, over an enormous range of sizes, proportions, and

materials, a lower-bound curve to the available test data can be obtained by taking

60 percent of the values obtained from classical small-deflection theory {_ - 0.60).

Average values of the test data can be approximated by using 80 percent of the classi-

cal theoretical predictions (3's = 0.80)

To date, no test data has been ptd)lished for sandwich cylinders which are of the types

consider_t in this hmld!yook and are subjected to torsion. Therefore, no empirical

basis exists lor the determination of reliable knock-down lactors in such cases.

Based on the moderate drop-off of the post-buclding equilibrium path, some sources

[4-51 recommend that no reduction be employed ( "_s :: 1.0). ttowever, Reference 4-4

takes a more cautious approach in recommending the use of _s .80 for the sand-

wich configuration. This selection was made largely on the basis of the isotropic

(non-sandwich) data. Although this value did not furnish a lower-bound to the isotropic

test points, it is reasonable to expect that the usually greater thicknesses of sandwich

cylinders should lead to more moderate reductions than apply to the isotropic (non-

4-52

sandwich) conligurations. In addition, it should be noted that cylinders under torsion

will continue to supt_)rt considerable torque well into the postbuckled region, ttence

the torsional buckling mechanism should not be nearly so catastropic as the general

instability of axially compressed cylinders. With these several factors in mind, the

value "is - O. 80 has been selected for use in this handbook. In view of the lack of

sandwich test data to substantiate this selection, the methods proposed here can only

be regarded as a "best-available" approach.

4-53


For simply supported circular sandwich cylinders subjected to torsion, the critical

shear stress may be computed from the equation

where

d

rcr = Xs Ks _ Ef _ (4.5-9)

"¢s = 0.80 (4.5-10)

d = tc + t_ + t_ (4.5-11)

and K s is obtained from Figures 4.5-3 through 4.5-8.

one must first compute each of the following values:

L 2

Z s -dR

border to use these cu_,es,

(4.5-12)

V = 16 tc t_ t_ TiEr (4.5-13)s 15 (tl + t_) R d Gxz

Gxz

@ - (4.5-14)Gyz

It is required here that both facings be made of the same material.

For elastic cases, use r1 = 1. Whenever the behavior is inelastic, the methods of


The critical torque Tcr , measured in units of in.-lbs, can be computed from the

following for both elastic and inelastic cases:

Tcr = 2rrR 2 (tl + ts) rcr (4.5-15)

4 -54

Culwesfor K s aregiven for values of @ -0.4; 1.0; and 2.5. Estimates of K s

for other values of @ can be obtained by interpolation.

In addition, curves for K s are given for values of t c/d = 1.0 and 0.7. The former

neglect the contribution from bending of the facings about their own middle surfaces.

The latter may be used to obtain numerical estimates of the conservatism introduced

by neglecting these stiffnesses.

As noted in Section 4.5.1.1, the design curves are somewhat inaccurate in the region

where Z s is large (long cylinders). Some caution should be exercised in the appli-

cation of the curves in this region.

Strictly speaking, Figures 4.5-3 through 4.5-8 apply only when the facings are equal.

However, the curves are reasonably accurate for sandwich cylinders having unequal

facings, provided that the thickness of one facing is not more than twice the other.

4-55

4-56

e_

©

r_

.!

m

El

II II

4-57

0

¢)

_J

¢)

0

0

C)

iZ/

II

0

0

b

or,,i

°,,-I

r..)

4-58

4-59

S

0

£

E)

O(3

4-60

oS

.2

0

_D

"0

.2

e_

.2

0

It_

u

©

©

@

0

m

,.,,-.i

%©

@

©

ca

odI

4-61

4.6 TRANSVERSE SI[EAR


In Reference 4-33, Lundquist reports the results from a series of tests on isotropic

(non-sandwich), circular cylinders subjected to combined transverse shear and bend-

ing. The same type of data is published in Reference ,I-34 for elliptical cylinders.

Both sets of data were obtained from cantilevered cylinders of varied lengths. Extrapo-

lation of these results to the condition of zero bending stress permits a determination

of critical stresses for pure transversc shear loading. It has proven useful to com-

pare these stress values against the theoretical results obtained from small-deflection

theory for isotropic (non-sand_vich), circular cylinder,_ loaded in torsion. Gerard and

Becker [4-35j report that, for nominally identical specimens, such comparisons yield

the following ratios where the theoretical predictions are obtained by using Reference

4-36:

Average of "rer Test Values f¢)r tTransverse Shear Loading

[ i a, 1.(; (4.6-1)Small-Deflection Theoretical TerValues for Torsional Loading

Lower-BoundTcr Test Values for]Transverse Shear Loading

Small-Deflection Theoretical Vc r ]

Values for Torsional Loading j

_ 1.25 (4.6-2)

To properly interpret these ratios, it is pointed out that, for torsional loading, the

shear stress Tcr is uniformly distributed around the circumference. On the other

hand, under transverse shear loading, the shear stress is non-uniform and the value

Tcr then corresponds to the peak intensity which occurs at the neutral axis.

4-62

For the lackof abetter approach,it is recommondedthat Equation(4._;-2)beused[or

thedesignandanalysisof circular sandwichcylindersthat aresubjectedto transverse

shearforces. In suchcases,the requiredsmall-deflectiontheoreticalrcr valuesfor

torsional loadingshouldbeobtainedas specifiedin Section4.5 of this handbookwith

theexceptionthat 7s= 1.0 shouldbeusedhere. Notest dataare availableto sub-

stantiatethe reliability of this practice. Until suchdatadobecomeavailable,onecan

onlyregardthis procedureas a "best-available"approach.

4-63


For simply supportedcircular sandwichcylinderssubjectedto a transverseshear

forceandhavingboth facingsmadeofthe samematerial, thecritical shearstress

maybecomputedfrom theequationd

= 1.25Krcr srlEf -_ (4.6-3)

wherethebucklingcoefficientK is obtainedfrom Figures4.5-3 through4.5-8 ands

thenotationis thesameasthatemployedthroughoutSeetion4.5. As notedin Section

4.5.1.1, thesefigures aresomewhatinaccuratein theregionwhereZ is large (longs

cylinders)andoneshouldexercisesomecautionwhendealingwith suchconfigurations.

Strictly speaking,Figures4.5-3 through4.5-8 applyonlywhenthe facingsare of equal

thickness. Itowever,the curvesare reasonablyaccuratefor san_viehcylindershaving

unequalfacings,providedthatthethicknessof onefacingis notmore thantwice the

other.

For elastic eases,user? = 1. Whenever the behavi'n" is inelastic, the methods of


For elastic eylinders the critical transverse shear force (Fv)cr, measured in units of

pounds, can be computed from the following:

(Fv)er = rrR (t 1 + ta) rer (4.6-4)

To compute (Fv)cr when the behavior is inelastic, one must resort to numerical inte-

gration techniques.

4-64

4.7 COMBINEDLOADINGCONDITIONS

4.7.1 General

For structural memberssubjectedto combinedloads, it is customary to represent

critical loading conditions by means of so-called interaction curves. Figure 4.7-1

shows the graphic format usually used for this purpose. The quantity R. is the ratio1

of an applied load or stress to the critical value for that type of loading when acting

alone. The quantity R. is similarly defined for a second type of loading. Curves ofJ

this form give a very clear picture as to the structural integrity of particular con-

figurations. All computed points which fall within the area bounded by the interaction

curve and the coordinate axes correspond to stable structures. All points lying on or

outside of the interaction curve indicate that buckling will occur. Furthermore, as

shown in Figure 4.7-1, a measure of the margin of safety is given by the ratio of

distances from the actual loading point to the curve and to the origin. For example,

assume that a particular structure is subjected to the combined loading condition

corresponding to point B of Figure 4.7-1.

Then, for proportional increases in R. and R., the margin of safety (M.S.) can be1 j

computed from the following:

(Rj)DM.S. - 1 (4.7-I)

(Rj)B

As an alternative procedure, one might choose to compute a minimum margin of safety

which is based on the assumption that loading beyond point B follows the path BM.

Point M is located in such a position that BM is the shortest line that can be drawn

4-65

between point B and the interaction curve. The minimum margin of safety can then be

calculated as follows:

OB _ BM

Minimum M.S. OB 1 (4.7-2)

RJ

1.0

(Rj) D

(Rj)B

_M

iiiiii

0 R 1.0i

Figure 4.7-1. Smnple Interaction Curve

4-66

4.7.2 Axial Compression Plus Bending

i.7.2.1 Basic Principles

In References 4-17, 4-18, and 4-19 it has been shown that, for circular sandwich

cylinders subjected to axial compression plus bending, the classical theoretical

interaction curve may be accurately described by the equation

(Re)c L + (Rb) CL = 1

where

and

Ue

(Rc)cL -(5c)CZ

_b

{Rb)cL- (Sb)CL

(4.7-3)

(Ye

_b

(_c)CL

((_b) C L

(4.7-4)

(.i. 7-_)

= (4. v-6)((_b) CL ((_e) C L

= Uniform compressive stress due solely to applied axial load,

psi.

= Peak compressive stress due solely to applied bending mo-

ment, psi.

= Classical theoretical value for critical uniform compressive


= Classical theoretical value for critical peak compressive

stress under a bending moment acting alone, psi.

References 4-17 and 4-18 develop the foregoing result for weak-core constructions

which fail in the shear crimping mode. On the other hand, Reference 4-19 deals

with infinitely long cylinders which fall in the stiff-core and the moderately-stiff-

core categories. Since Equation (4.7-3) is written in terms of classical theoretical

4-f;7

allowables,it doesnot includeanyconsiderationof thedetrimentalinfluencesfrom

initial imperfections. For thepullmsesof this handbook,theseinfluencesare treated

by'introducingtheknock-dt_vnfactorsye and Tb (see Figures 4.2-8 and 4.3-2, re-

spectively) to obtain

Rc +Rb = 1 (4.7-7)

where

e

R - (4.7-8)c 'Yc ((_c) C L

(_b

% Yb ((_c) CL (4.7-9)

Therefore, the design interaction curve can be drawn as shown in Figure 4.7-2. Since

no test data is available for sandwich cylinders subjected to combined axial load and

bending, the general validity of this curve has not been experimentally verified. Some

degree of empirical correlation is inherent in the approach since the knock-down fae-

tors 3/c and 7b were established, in part, from sandwich test data (see Sections 4.2

and 4.3). ttc_vever, even these data were few in number. Therefore, until further

experimental substantiation is obtained, the recommended interaction relationship

can only be considered a "best-available" method.

Figure 4.7-2.

lo0 -

%

R c 1.0


Subjected to Axial Compression Plus Bending

4-68


For simply suppol_ed, circular, sand_vich cylinders subjected to axial compression

plus bending, the following interaction equation may be employed:

w he re

Re + Rb = 1 (4.7-10)

(yc

R -

c Tc ((7c) C L

(4.7-11)

au -_b (_c)CL

A plot of Equation (4.7-10) is given in Figure 4.7-3.

(4.7-12)

In Equations (4.7-11) and (4.7-12), the knock-down factors Tc and _b are those ob-

tained from Figures 4.2-8 and 4.3-2, respectively.

The quantity (ffc)CL

Section 4.2.2.

is simply the result obtained by using Tc = 1.0 in the method of

Plasticity considerations should be handled as specified in Section 9.2 except that,

in this case, one may use

aoIlVel Et

r7 = [ 1---2-_1 _ff for short cylinders, and

b. _ --

1

t s

tl-u_] Ef

for moderate-length through long cylinders.

Equation (4.7-10) may be applied to sandwich cylinders of any length. However, length

considerations should be included in the computation of ((_c)CL when the structure falls

into the short-cylinder range (see Section 4.2.2).

4-69

0.8

!i!ii0.6 _' j

i!ilRb

0 0. '2 0.4 0.6 0.8R

C

1.0

Figure 4.7-3. I)esigm Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus tk, nding

4-70

4.7.3 Axial Compression Plus External Lateral Pressure

4.7. :5.1 Basic Principles

This section deals with the loading condition depicted in Figure 4.7-4. The sandwich

cylinder is subjected to uniform external pressure over the cylindrical surface. Axial

loading is imposed as indicated by the [orces P. These forces can originate from any

source including external pressures which are uniformly distributed over the end elos-

ures. In addition, it is specified that the ends of the eylinder are simply supported.

This is, during buckling, the ends are constrained such that they experience no radial

or circumferential displacements and they are free of bending moments.

p , psiY

P, lbs__

/ lllttIlIIlt'(Both Ends Simply Supported

lbs

p , psiY I

Figure 4.7--4. Circular Sandwich Cylinder Subjected to Axial

Compression Plus External Lateral Pressure

The theoretical basis used here is the classical small--deflection solution of Maki

[4-37]. The design curves given in this handbook were taken directly from that source

and embody the following assumptions:

ao

b.

e.

The facings are isotropic.

Both facings are of the same thickness.

Both facings have identical material properties.

4-71

d.

e°

f.

g*

ho

io

Poisson's ratio for the facings is equal to 0.33.

Bending of the facings about their own middle surfaces can be neglected.

The core has infinite extensional stiffness in the direction normal to the

facings.

The core extensional and shear rigidities arc negligible in directions


The transverse shear moduli of the core are the same in the circum-

ferential and longitudinal directions (G = G ).xz yz

The mean radius of the cylinder is large in comparison with the sandwichthickness.

The theoretical relationship derived by Maki [4-37_ is in the form of a complicated

sixth order determinant and no significant advantage would be gained by reproducing

that formulation in this handbook. However, it is important to note that a sufficient

number of terms were retained throughout the derivation to obtain accurate results

when the number of circumferential full-waves equals two (n = 2). If the derivation

had been based on the well-known Donnell approximations [4-81, the results would

not be applicable to structures which buckle in this manner.

The interaction curves given in Reference 4-37 are of the two different types depicted

in Figure 4.7-5 where

Ef tf hV - (4.7-13)

xz 2 (1-.33e) RaGXZ

EftfhV = (4.7-14)yz 2 (1- .33_) RaG

yz

P_ Y

(Rp) CL _y) CL(4.7-15)

4-72

and

(Rc)CLcr

x

(C_x)C L(4.7-16)

Ef : Young's modulus of facings, psi.

tf = Thickness of single facing, inches.

h = I)istancc between middle surfaces of facings, inches.

R = Radius to middle surface of cylindrical sandwich, inches.

G = Core shear modulus associated with the plane perpendicularXZ

to the facings and oriented in the axial direction, psi.

G = Core shear modulus associated with the plane perpendicularyz

to the axis of revolution, psi.

p = Applied external lateral pressure, psi.Y

(1-iy)CL = Classical theoretical value for critical external lateral pressurewhen acting alone, psi.

g = Uniform axial compressive stress due to applied axial load, psi.X

(C_x)C L = Classical theoretical value for critical tmiform axial compres-

sive stress when acting alone, psi.

L = Over-all length of cylinder, inches.

Note : The value .33 appearing in Equations (4.7-13) and (4.7-14) is an

assumed representative value for the elastic Poissonts ratio of

the facings.

Since the curves of Reference 4-37 were developed from a classical, small-deflection

approach, they do not include any consideration of the detrimental effects from initial

imperfections. This is evident from the fact that classical theoretical allowables are

used in the ratios (Rp)cL and (Rc)CL. For the purposes of this handbook, the effects

4-73

1.0

(Rc)CL

\XZ - \VZ - O/ _

[-_ : Constant] _ .

1.0

(Rc)CL

: 4> -._I

% ..o

Jill/ = Constant \

1.0 1.0

(tlp)CL (Rp)CL

Figure 4.7-5. Typical Interaction Curves for Circular Sanchvieh Cylinders Subjected

to Axial Compression Plus External Lateral Pressure

from initial imperfections are introduced through the replacement of (%)CL and

(Re)CL by the ratios Rpand R which are defined as follows:

c

PyR = -

P Tp (Py) CL

(4.7-17)

(y

R = x (4.7-18)

e Yc ((_x) C L

The quantities yp and Yc are the knock-down factors discussed in Sections 4.4 and 4.2,

respectively. Values for yc can be obtained from Figure 4.2-8 while yp may be taken

equal to 0.90.

No test data are available for sandwich cylinders which are of the types considered

here and are subjected to axial compression plus external lateral .ressure. Therefore,

the general validity of the design curves recommended here has not been experimentally

verified. Some degree o[ empirical correlation is inherent in the approach since the

4-74

and were established, in part, from sandwich test data (seeknock-down factors Te Tp

Sections 4.2 and 4.4). ttowever, even these data were few in nmnber. Therefore,

until further eN)erimental substantiation is obtained, the recommended interaction

curves can only be considered as "best-available" criteria.

4-75


For simply supported,circular, sandwichcylinderssubjectedto axialcompression

plus externallateral pressure, onemayemploytheinteractioncurvesof Figures

4.7-6 through4.7-15where

Eftf hxz 2(1-. 33z')R_G

xz

(4.7-19)

Eftf h

Vyz 2(1-.332)R _ Gyz

(4.7-20)

PYR =

P Yp (Py)CL

(4.7-21)

R = x (4.7-22)

c Tc (fix) C L

In Equations (4.7-21) and (4.7-22), the knock-down factor Tc is that obtained from

Figure 4.2-8 while Tp may be taken equal to 0.90.

The quantity (15y)CL is simply the result obtained by using 7p = 1.0 in the methods of

Section 4.4.

The quantity ((_x)CL is simply the result obtained by using Tc = 1.0 in the methods of

Section 4.2.

Plasticity considerations should be handled as specified in Section 9.2.

Figures 4.7-6 through 4.7-12 give interaction curves only for cases where Vxz =

V = 0 (G = G - co). Separate families are provided for each of three selectedyz xz yz

values for the parameter -_ = 50; 160 and 500 . Graphical interpolation may be

4-76

used to obtain results for intermediate values of this parameter. Each family includes

separate curves for ten different values of the ratio --L-- -- = 0.1; 0.'2; --- 1. . In

view of the restrietions on V and V , these curves can only be used to describe thexz yz

behavior of stiff-core eonstruetions. For the purposes of praetieal design and analysis,

it is proposed here that Figures 4.7-6 through 4.7-12 be considered applicable only

when

RtC

-- Vh _ xz

± 0.05 (4.7-23)

RtC

V _ 0.05 (4.7-24)h* yz

\vhcre


It is ex_pected that many realistic sandwich configurations will satisfy these requirements.

Fi_ires 4.7-13 through 4.7-15 present a partial picture of the effects which variations

in V (= V ) will have on the interaction relationships. These figures only treatXZ yZ

cases for which --rrR _= 0.1. }tc_vever, the trends displayed furnish some basis for oneI,

to conjecture that the curves given for V = V = 0 would result in conservative pre-xz yz

dictions if they were applied to sandwich configurations which do not satisfy the In-

equalities (4.7-23) and (4.7-24). However, one should be cautioned againstmaking

sweeping application of this observation in view of the limited scope of the information

shown in Figures 4.7-13 through 4.7-15.

4-77

It shouldbekept in mindthatthe interactioncurvesgivenin Figures4.7-6 through

4.7-12 include C L values ranging only from 0.1 through 1.0. Since

_RC - (4.7-25)

L L

it follows that these curves only embrace the range where

L

3.14 _ -_- <- 31.4 (4.7-2(;)

4-78

h= 50

8 _. Vxz = Vyz : 0,0

__C rrI1

c

O.

0.6

O. ,1

0,2

0 0.2 0.4 0.6R

P

0°8 1,0

Figure 4.7-6. Interaction Curves for Circular Sandwich Cylinders

Subjected to Axial Compression Plus External

Lateral Pressure

4-79

1

I150

11

V = V = 0.0xz yz

'gRC

I_ L

0.6

RC

0.4

0.2

00 0.2 0.4

C L 0.7"

C L 0.8-C O. 9_

0.6 0.8 1.0R

P



I,ateral Pressure

4-80

1.0

0.8

0,6

Rc

0.4

0.2

'0

\

RN = 50h

V = Vxz yz

_RC =

L L

= 0.0

0 0.2 0.4 0.6 0.8 1.0R

P

Figure 4.7-8. Interaction Curve for Circular Sandwich Cylinders


Lateral Pressure

4-81

C

IH

-- 160h

V :-: V :: 0.0- xz yz --

=0.]

0 0.2 0.4 0.6 0.8 1.0R

P



Lateral Pressure

4-82

I

R-- 160

h

e

0.8 = V

C LC_ 0_ 8(_ KC L = 0.6 Vxz ?rRYZ) C L _[[0.6 _ I

0.4 CL

0.2

0.0

00 0.2 0.4 0.6

RP

0.8 1.0

Figure 4.7-10. Interaction Curves for Circular Sandwich

Cylinders Subjected to Axial CompressionPlus External Lateral Pressure

4-83

1.0CLI0. 4

_ jt/ICL = 0.

C L = 0.5'__

IR

5O0h

R

0.8

0.6

c

0.4

0.2

V = V = 0.0xz yz -

CL

_RC

L L

C L = 0.3=0.1

00 0.2 0.4 0.6 0.8 1.0

RP



Lateral Pressure

4-84

RC

0.4

0.2

CL=0.

= 0o0

=0.8

00 0.2 0.4 0.6 0.8 1.0

RP



Lateral Pressure

4-85

RC

l°01

RP



Lateral Pressure

4-86

RC

0.2

0o 0.2 0.4 0.6 0.8 1.0

RP

Figure 4.7-14. Interaction Curves for Circular Sandwich CylindersSubjected to Axial Compression Plus External

Lateral Pressure

4-87

1.0

0.8

0.6

RC

0.4

0.2 --

%

h = 500

CL _R= I--_- =0.1

t5

\

kk --

%

0 0.2 0.4 0,6 0.8 1.0

RP



Lateral Pressure

4-88

4.7.4 Axial CompressionPlusTorsion


This sectiondealswith the loadingconditiondepictedin Figure4.7-1(;. Thesanchvich

cylinder is subjectedto endtorqueTplus axial loadingindicatedbytheforces P.

T, in-lbs Torque T, in-lbs Torque

BothEndsSimplySupported

Figure 4.7-16. Circular Sandwich Cylinder Subjected to

Axial Compression Plus Torsion

In Reference 4-18 Wang, et al. treat this type of problem but only consider the case

of weak-core configurations which fail in the shear crimping mode. In addition they

assume that the cylinder is long so that the boundary conditions can be ignored. This

small-deflection analysis makes use of the Donnell approximations [4-8] to arrive at

the following interaction relationship:

(Rc)CL + 2 = 1(Rs) CL

where

C

(Rc)CL - _(_c)CL

(4.7-2 7)

(4.7-28)

4-89

I- (4.7-29)

(Rs)CL- (_)CL

and

(Ic

((_e)CL

= Uniform axial compressive stress due to applied axial load, psi.

= Classical theoretical value for crilical uniform axial eompres-

sire stress when acting alone, psi.

_- = Uniform shear stress due to applied torque, psi.

(T)CL = Classical theoretical value for critical uniform shear stressduc to torque acting alone, psi.

Because Equation (4.7-27) was developed from a classical, small-deflection approach,

it does not include any consideration of the detrimental effects from initial imperfee-

tions. That is evident from the fact that classical theoretical allc_vables are used in

the ratios (Rc)cL and (Rs)CL. For the purposes of this handbook, the effects from

initial imperfections are introduced through the replacement of (Rc)cL and (Rs)CL by

the ratios R and 1R which are defined as follows:c s

(yc

R -

c "/c(ffc)C L

(4.7-30)

Rs

_ 1- (4.7-31)

7s (_)CL

The quantities _c and _s are the knock-down factors discussed in Sections 4.2 and 4.5,

respectively. Values for Yc can be obtained from Figure 4.2-8 while :Ys may be taken

equal to 0.80. Incorporation of the foregoing substitutions into Equation (4.7-27) then

gives the following interaction relationship for weak-core constructions:

R + R 2 = 1 (4.7-32)c s

4-90

In Reference 4-38, Batdorf, et el. deal with the subject loading condition for thin-

walled, isotropic (non-smldwich), circular cylinders. Since, for such constructions,

transverse shear deformations of the shell wall are of negligible importance, one

might conjecture thai this work could be applied to sandwich cylinders which fall into

the stiff-core category. Based on theoretical considerations modified by test results,

Batdorf, et el. c4-38n: arrived at the same interaction expression as that given above

as Equation (4.7-32). In view of this, one might choose to view Equation (4.7-32) as

a comprehensive interaction formula for sandwich cylinders, ttowever, some caution

should be observed in implementing this viewpoint, partially because of the fact that

only the extremes of transverse shear stiffness of the core have been considered. In

addition, although the interaction relationship for the subject loading condition should

probably be dependent upon a length parameter, no investigations were made to estab-

lish the sandwich cylinder lengths over which Equation (4.7-32) is a reasonable repre-

sentation of the actual behavior. Furthermore, no test data are available for sandwich

cylinders which are of the types considered in this handbook and are subjected to axial

compression plus torsion. Therefore the general validity of Equation (4.7-32) has not

been experimentally verified. Some degree of empirical correlation is inherent in the

approach since the knock-down factor _'e was established, in part, from sandwich test

data (see Section 4.2). ttowever, even these data were few in number. Therefore,

until further theoretical and experimental investigations are accomplished, the inter-

action relationship cited here can only be considered as a "best-available" criterion.

4-91


For simply supported, circular, sandwich cylinders subjected to axial compression

plus torsion, one might choose to employ the interaction formula

R +R _ = 1 (4.7-33)c s

which is plotted in Figure 4.7-17 and where

a

R - c (4.7-34)

c _'c (_c) CL

Rs

- _" (4.7-35)

Ys (_) C L

In Equations (4.7-34) and (4.7-35), the knock-down factor _c is that obtained from

Figure 4.2-8 while _s may be taken equal to 0.80.

The quantity ((_c)CL is simply the result obtained by using )'c = 1.0 in the methods of

Section 4.2.

The quantity (_)

Section 4.5.

is simply the result obtained by using _s = 1.0 in the methods ofCL

Plasticity considerations should be handled as specified in Section 9.2.

Attention is drawn to the fact that, in Section 4.7.4.1, several factors are cited which

shed considerable doubt upon the reliability of results obtained from the indiscriminate

use of Equation (4.7-33) and Figure 4.7-17. In view of these uncertainties, one might

often choose to employ the straight-line interaction formula

R + R = 1 (4.7-36)c s

4-92

which is plotted in Figure 4.7-18. This relationship can be used with confidence for

any length of cylinder and for any region of transverse shear rigidity of the core since

experience has shown that the linear interaction formula is never unconservative for

shell stability problems, ttowever, in many cases it will, of course, introduce execs-

sive conservatism.

(}. (!

CO 0.2 0.4 0. (; 0. _ I . 0

Figure 4.7-17. Conditional Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus Torsion

4-93

0.6

Rs

0 0.2 0.4 0.6 0.8 1.0R

c

Figure 4.7-18. Conservative Interaction Curve for Circular Sandwich Cylinders

Subjected to Axial Compression Plus Torsion

4-94

4.7.5 Other Loading Combinations


In Sections 4.7.3 and 4.7.4 the following combined loading conditions are treated:

a. Axial Compression plus External Lateral Pressure.

b. Axial Compression plus Torsion.

The corresponding interaction relationships can be used for certain additional com-

binations by recognizing that

a. the peak axial stress due to an applied bending moment can be converted

inio an equivalent uniform a.xial stress, and

b. the peak shear stress due to a transverse shear force can be converted

into an equivalent tmiform torsional shear stress.

With this in mind, the design equations and curves of Section 4.7.3.2 can be used for

the combination o[ AXIAL COMPRESSION PLUS BENDLNG PLUS EXTERNAL LATERAL

PRESSURE if one simply substitutes the quantity g_ for (_X X

whe re

and

((_X) C

(Crx) b

%

_zb

, = ((:rx)c +()'c_ ((rx)b (4.7-37)% \rb/

= Uniform axial compressive stress due solely to applied axial

load, psi.

= Peak axial compressive stress due solely to applied bending

moment, psi.

= Knock-down factor associated with axial compression and as

given in Figure 4.2-8, dimensionless.

= Knock-down [actor associated with pure bending and as given in

Figure 4. ;/-2, dimensionless.

4-95

This formula is based on the findings reported in Section 4.3.

In addition, the design equations and curves of Section 4.7.4.2 can be used for the

combination of AXIAL COMPRESSION PLUS BENDING PLUS TORSION PLUS TRANS-

VERSE SHEAR FORCE if one simply substitutes the quantities _' and T _ for (_ and 1",C C

respectively, where

(r' = (_c)c + (_c)be

(4.7-38)

, O. 80T = T T + 1.2----55rV = 7"T + 0"641"V (4.7-39)

and

(gc) c = Uniform axial compressive stress due solely to applied axialload, psi.

(gc) b = Peak axial compressive stress due solely to applied bendingmoment, psi.

_'T = Uniform shear stress due solely to applied torque, psi.

T V = Peak shear stress due solely to applied transverse shear force,psi.

7c and :_b = Knock-down factors specified above.

Equation (4.7-38) is based on the findings reported in Section 4.3 while Equation

(4.7-39) stems from a comparison of Equations (4.5-9) and (4.6-3).

Since no sandwich test data are available to substantiate the foregoing procedures, they

can only be regarded as "best-available" criteria.

4-9G

4.7.5.2 DesignEquationsandCmwes

For thecombinationof AXIAL COMPRESSIONPLUSBENDINGPLUSEXTERNAL

LATERALPRESSURE,substitute(_ for(_ andusethe designequationsandcurvesX X

given in Section 4.7.3.2. The quantity (i is defined follows:as

X

Itowever, the quantity (CTx)CL

0" = (ax) c + ((Ix) bX

(4.7-4O)

used in Section 4.7.3.2 remains as defined in thai

sectior_.

For the combination of AXIAL COMPRESSION PLUS BENDING PLUS TORSION PLUS

TRANSVERSE SIIEAR FORCE, substitute cr I for a and T _for "r in the design equationse c

and curves given in Section 4.7.4.2. The quantities a t and _-_ are defined as follows:e

ue = (<:'c)e\ 'bl(4.7-41)

0.807" = T T + 1.2-----_o"rV = TT + 0.64"r V (4.7-42)

However, the quantities ((Tc)CL and (_)CL used in Section 4.7.4.2 remain as defined

in that section.

The foregoing criteria will still apply, of course, where one or more of the specified

applied loads equal zero.

4-97

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4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

RE FERENCES

Donnell, L. H. and Wan, C. C., "Effect of Imperfections on Buckling of

Thin Cylinders and Columns Under Axial Compression, " Journal of Applied

Mechanics, March 1950.

March, H. W. and Kuenzi, E. W., "Buckling of Cylinders of Sandwich

Construction in Axial Compression, " FPL Report No. 1830, Revised

December 1957.

Hoff, N. J., Madsen, W. A., and Maycrs, J., "The Postbuckling Equilibritm_

of Axially Compressed Circular Cylindrical Shells, " Stanford University

Department of Aeronautics and Astronautics, Report SUDAER No. 221,

February 1965.

Anonymous, "Buckling of Thin-Walled Circular Cylinders, " NASA Space

Vehicle Design Criteria, NASA SP-8007, September 1965.

U. S. Department of Defense, Structural Sandwich Composites, MIL-HDBK-23,

30 December 1968.

Thielemann, W. F., "New Developments in the Nonlinear Theories of the

Buckling of Thin Cylindrical Shells, " Proceedings of the Durand Centennial

Conference, Held at Stanford University, 5-8 August 1969, Pergamon Press,

New York, Copyright 1960.

Zahn, J. J. and Kuenzi, E. W., "Classical Buckling of Cylinders of Sandwich

Construction in Axial Compression --- Orthotropic Cores, " U. S. Forest

Service Research Note, FPL-018, November 1963.

Donnell, L. H., "Stability of Thin-Walled Tubes Under Torsion, " NACA

Technical Report No. 479, 1934.

4-109

4-9

4-10

4-11

4-12

4-13

4-14

4-15

4-16

4-17

Scide, P., Weingarten, V. I., and Morgan, E. J., "Final Report on the

Development of Design Criteria for Elastic Stability of Thin Shell Structures, "

STL-TR-60-0000-19425, 31 December 1960.

Baker, E. H., "Ex]aerimenta] Investigation of Sanchvich Cylinders and Cones

Subjected to Axial Compression, " AIAA Journal, Volume 6, No. 9, September

1968.

Anonymous, "Status Summary Report for R&D Project 6002, " Hexce] Products,

Inc., Advanced Stm_clures Group, 25 July 1963.

Plantema, F. J., Sandwich Construction, John Wiley & Sons, Inc., New York,

Copyright 1906.

Stein, M. and Mayers, J., "Compressive Buckling of Simply Supported Curved

Plates and Cylinders of Sandwich Construction," NACA Technical Note 2601,

January 1952.

Bijlaard, P. P. and Gallagher, R. H., "Elastic Instability of a Cylindrical

Shell Under Arbitrary Circumferential Variation of Axial Stress, " Journal

of the Aerospace Sciences, Vol. 27, 1960.

Abir, l). and Nard(_, S. V., "Thermal Buckling of Circular Cylindrical Shells

Under Circumferential Temperature Gradients, " JcJurnal of the Aerospace

Sciences, Vol. 26, 1959.

Seide, P. and Weingarten, V. I., "On the Buckling of Circular Cylindrical

Shells Under Pure Bending, " Transactions of the A.S.M.E., Journal of

Applied Mechanics, March 1961.

Wang, C. T. and Sullivan, D. P., "Buckling of San_vich Cylinders Under

Bending and Combined Bending and Axial Compression, " Journal of the

Aeronautical Sciences, Vol. 19, July 1952.

4-110

4-18

4-19

4-20

4-21

4-22

4-23

4-24

4-25

4-26

4-27

Wang, C. T., Vaccaro, R. J., and DeSanto, I). F., "Buckling of Sandwich

Cylinders Under Combined Compression, Torsion, and Bending Loads, "

Journal of Applied Mechanics, Vol. 22, September 1955.

C<.llatly, R. A. and Gallagher, R. It., "Sandwich Cylinder Instability Under

Nonuniform Axial Stress, " AIAA Journal, February 1964.

Peterson, J. P. and Andersen, J. K., "Structural Behavior and Buckling

Strength of Itoneyeomb Sandwich Cylinders Subjected to Bending, " NASA

TN D-2926, August 1965.

Kucnzi, E. W., Bohannan, B., and Stevens, G. H., "Buckling Coefficients

for Sandwich Cylinders of Finite Length Under Uniform External Lateral

Pressure," U. S. Foresl Service Research Note FPL-0104, September 1965.

Raville, M. E., "Analysis of Long Cylinders of Sandwich Construction Under

Uniform External Lateral Pressure, " FPL Report No. 1844, November 1954.

Ravillc, M. E., "Supplement to Analysis of Long Cylinders of Sandwich

Construction Under Uniform External Lateral Pressure, " FPL Report No.

1844-A, February 1955.

Raville, M. E., "Buckling of Sandwich Cylinders of Finite Length Under

Uniform External Lateral Pressure, " FPL Report No. 1844-B, May 1955.

Norris, C. B. and Zahn, J. J., "Design Curves for the Buckling of Sandwich

Cylinders of Finite Length Under Uniform External Lateral Pressure, " FPL

Report No. 1869, May 1959.

Norris, C. B. and Zahn, J. J., "Design Curves for the Buckling of Sandwich

Cylinders of Finite Length Under Uniform External Lateral Pressure,"

U. S. Forest Service Research Note FPL-07, 1963.

Kazimi, M. I., "Sandwich Cylinders Part II - Uniformity of the Mechanical

Properties of the Core, " Aerospace Engineering, September 1960.

4-iii

4-28

4-29

4-30

4-31

4-32

4-33

4 -34

4-35

,t-36

4-37

4-38

Jenkinson, P. M. and Kuenzi, E. W., "The Buckling Under External Radial

Pressure and Buckling Under Axial Compression of Sanchvich Cylindrical

Shells, " FPL Report PE-227, January 1962.

C_rard, G., "Torsional Instability of a Long San(hvich Cylinder, " Proc. 1st

U. S. Natl. Congr. Appl. Mech., Copyright 1951.

March, H. W. and Kuenzi, E. W., "Buckling of Sandwich Cylinders in

Torsion, " FPL Report No. 1840 Revised, January 1958.

Gerard, G., Introduction to Structural Stability Theory, McGraw-Itill Book

Company, Inc., New York, Copyright 19(;2.

Timoshenko, S. P. and Gere, J. M., Theory of Elastic Stability, McGraw-

Hill Book Company, Inc., New York, Copyright 1961.

Lundquist, E. E., "Strength Tests of Thin-Walled Duralumin Cylinders in

Combined Transverse Shear and Bending, " NACA TN 523, 1935.

Lundquist, E. E. and Burke, W. F., "StrenKth Tests of Thin-Walled

Duralumin Cylinders of Elliptic Section, " NACA TN 527, 1935.

Gerard, G. and Becker, H., "Handbook of Structural Stability, Part II1 -

Buckling of Curved Plates and Shells, " NACA TN 3783, August 1957.

Batdorf, S. B., Stein, M., and Schildcrout, M., "Critical Stress of Thin-

Walled Cylinders in Torsion, " NACA TN 1.244, 194_.

Maki, A. C., "Elastic Stability of Cylindrical Sandwich Shells Under Axial

and Lateral Load, " U. S. Forest Service Research Note FPL-0173, October

1967.

Batdorf, S. B., Stein, M., and Schildcrout, M., "Critical Combinations of

Torsion and Direct Axial Stress for Thin-Walled Cylinders, " NACA Technical

Note 1345, June 1947.

4-112

5GENERAL INSTABILITY OF TRUNCATED CIRCULAR CONES

5.1 AXIAL COMPRESSION


It appears that no significant theoretical solutions have been published for axially com-

pressed sandwich cones. Therefore, for the purposes of this handbook, the equivalent-

cylinder concept of Seide, eta!. [5-11 has been adopted as a practical expediency.

Based on a large array of test data from thin-walled, isotropie (non-sandwich), trun-

cated cones, Seide, et al. concluded that the critical stresses for such cones can be

taken equal to the values for circular cylinders which satisfy the following conditions:

a. The wall thickness of the equivalent cylinder is equal to that of the cone.

In the ease of sandwich constructions, the logical extension of this con-

dition is that the equivalent cylinder have the same facing and core thick-nesses found in the cone.

b. The radius of the equivalent cylinder is equal to the finite principal radiusof curvature at the small end of the cone.

c. The length of the equivalent cylinder is equal to the slant length of thee one.

In Reference 5-2, Baker presents test data from two axially compressed, truncated

sandwich cones having vertex half-angles equal to 15 degrees. These data were used

in conjunction with the foregoing equivalent-cylinder concept to arrive at knock-down

factors _c" The results are shown in Figure 5.1-i, along with data obtained from

5-1

!!

i i

+ t_t

t !

t

4 J ,

-ill

i!:

I I

2_I

0

!

5-2

axially compressedsandwichcylinders. This figure alsoincludesthedesigncurve

recommendedin Section4.2.2 for suchcylinders. It canbeseenthat the datafrom

theconesare in favorableagreementwith theresults obtainedfrom cylinders. This

providesat leasta small degreeof experimentalsubstantiationfor application(>_the

equivalent-cylinderapproachto sand_vichcones. However,in view of thescarcity of

testpoints from conicalspecimens,this methodcanpresentlybeconsideredasonly

a "best-available"criterion.

5-3


For simply supported, truncated, right-circular, sandwich cones subjected to axial

compression, the critical stresses a and a (for facings 1 and 2 respectively)crl c G

may be computed from the equations and curves of Section 4.2.2, provided that the

following substitutions are made:

SoThe values t_, t_, t , and h are measured as shown in Figure 5.1-2.• C

(There is no preference as to which facing is denoted by the subscripts1 or2.)

6

C°

The radius R is replaced by the effective radius R shown in Figure 5.1-2.e

The length L is replaced by the effective length L shown in Figure 5.1-2.e

P, lbs

Rsmall

Both Ends

Simply Supported

P, lbs

N()F}:

Axis ofRevolution

View A

t 1, 12 t , h, le a_ldI_"lll a] r ' t '

[ c art all o_t a,l_rttl ul IllrJlq (i I

it1( hes w]li}t ¸ _ I_ r41ca_url'd irl

Figure 5.1-2. Truncated Sandwich Cone Subjected to Axial Compression

5-4

The applied axial load P and the computed stresses are associated with the directions

indicated in Figure 5.1-2. In addition, since the maximum stresses occur at the small

end of the cone, the critical values are associated with this location. For both elastic

and inelastic cases, one can therefore write

Pcr = 2_Re ((rcr It_ +_cr_t_) cos2_ (5.1-I)

where

RsmallR -

e cos o_(5. i-2)

It is recommended that the approach specified here be applied only to cases where

_ 30 degrees.

Plasticity reduction factors should always be based on the stress at the small end of

the cone (see Section 9).

5-5

5.2 PURE BENDING


It appears that no significant theoretical solutions have been published for sandwich

cones subjected to pure bending. Therefore, for the purposes of this handbook, the

equivalent-cylinder concept of Seide, et al. has been adopted as a practical expediency.

Based on a large array of test data from thin-walled, isotropic (non-sandwich), trun-

cated cones, Seide, et al. concluded that the critical peak stresses for such cones can

be taken equal to the corresponding values for circular cylinders which satisfy the

following conditions:

a° The wall thickness of the equivalent cylinder is equal to that of the cone.

In the case of sandwich constructions, the logical extension of this condi-

tion is that the equivalent cylinder have the same facing and core thick-nesses as are found in the cone.

b. The radius of the equivalent cylinder is equal to the finite principal radiusof curvature at the small end of the cone.

c. The length of the equivalent cylinder is equal to the slant length of the cone.

No test data are available for sandwich cones which are of the types considered in this

handbook and are subjected to pure bending. Therefore, the validity of the method

recommended here has not been experimentally verified and can only be considered as

a "best-available" approach.

5-6

5.2.2 Design Equations and Cu_wes

For simply supported, truncated, right-circular, san_vich cones subjected to pure

bending, the critical peak stresses _ and ff (for facings 1 and 2, respectively)e r 1 c r>

may be computed from the equations and curves of Section 4.:'.2, provided that the

folh_ving substitutions are made:

a. The values t_, t.z, re, and h are measured as shown in Figure 5.2-1.

(There is no preference as to which facing is denoted by the subseripts

1 or 2.)

b. The radius R is replaced by the effective radius R e shown in Figure 5.2-1.

c. The length L is replaced by the effeetive length L e shown in Figure 5.2-1.

Rsmall

LNote:

[1

"r C - Axisa_nraJll \ r<evoIut ;;_n

View A

tl, t2, t , h, R R , and 1,c small' e e

are all measured in units of inches

while Olis measured in degrees.

Figure 5.2-1. Truncated Sandwich Cone Subjected to Pure Bending

5-7

The applied bending moment M and the computed stresses are associated with the

directions indicated in Figure 5.2-1. In addition, since the maximum stresses occur

at the small end of the cone, the critical values are associated with this location.

When the behavior is elastic, one can therefore write

Mcr = _r R 2e ((Ycr 1 tl +Crcr_t2) cos 3(_ (5.2-1)

whe re

RsmallR = (5.2-2)

e cos (_

To compute M when the behavior is inelastic, one must resort to numerical inte-cr

gration techniques.


o_ _ 30 degrees.

Plasticity reduction factors should always be based on the peak compressive stress at

the small end of the cone (see Section 9).

5-8

5.3 EXTERNAL LATERAL PRESSURE


The loading condition considered here is depicted in Figure 5.3-1. As shown, the cone

is subjected to a uniform external lateral pressure. The axial component of this loading

!

io_ R R

-I-R

smal 1

%F

Rlarge

w c, lbs/in

Figure 5.3-1. Truncated Cone Subjected to Uniform External Lateral Pressure

is reacted by a uniform compressive running load at the large end of the cone. This

results in principal membrane stresses which may be computed as follows, when the

core has a relatively high extensional stiffness in the direction normal to the facings:

P Ram (5.3-!)(YH - (t_ + t_)

(TM (t_ + tp) 2c-_s _ R (5.3-2)

5-9

where

and

R (5.3-3)R_ - cos (_

(YH = Hoop membrane stress, psi.

(rM = Meridional membrane stress, psi.

p = Uniform external lateral pressure, psi.

= Finite principal radius of curvature of middle surface, inches.

= Thicknesses of the facings, inches. {There is no preference as

to which facing is denoted by the subscripts 1 or 2.)

R = Radius of middle surface measured perpendicular to the axis of

revolution, inches.

Radius of middle surIace, at small end of cone, measured per-

pendicular to the axis of revolution, inches.

Radius of middle surface, at large end of cone, measured per-

pendicular to the axis of revolution, inches.

= Vertex half-angle of cone, degrees.

Rsmal 1 =

Rlarg e =

Since the radii R and I_ vary with the axial location, the stresses ¢YH

uniform over the conical surface.

occur at the large end of the cone.

and a M are non-

The maximum values for each of these quantities

It appears that no significant theoretical solutions have been published for the stability

of truncated sandwich cones which are subjected to uniform external hydrostatic pres-

sure. Therefore, for the purposes of this handbook, the equivalent-cylinder approach

suggested in Reference 5-11 has been adopted as a practical expediency. Based on

this method, the critical lateral pressure for the truncated cone may be taken equal

5-10

to that for anequivalentcircular sandwichcylinderwhichsatisfies thefollowing

conditions:

a. Thefacingandcorethicknessesof theequivalentcylinderare thesameasthosefoundin thecone.

b. Thelengthof theequivalentcylinder is equalto theslant lengthof thc conc.

c. Theradiusofthe equivalentcylinder is equalto the averagefinite principalradiusof curvatureof thecone. Thatis,

Rsmal 1 + Rlarg eR = (5.3-4)

e 2 cos

The critical lateral pressure for the equivalent cylinder can be obtained by using the

equations and curves of Section 4.4.2.

Since no test data are available from truncated sandwich cones subjected to external

lateral pressure, the reliability of the foregoing approach has not been experimentally

verified and can only be considered as a "best-available" technique.

5-11


For a simply supported, truncated, right-circular, sandwich cone subjected to uni-

form, external, lateral pressure, the critical pressure may be taken equal to the

critical lateral pressure for an equivalent sandwich cylinder which satisfies the

following:

a,

b.

c.

The values t_, t:., tc, and h are measured as shown in Figure 5.3-2.

The length is taken equal to the slant length L .e

The radius is denoted R and is computed from the formulae

Rsmal 1 + Rlarg eR =

e 2 cos_

where Rsmal 1, Rlarg e, and _ are as shown in Figure 5.3-2.

(5.3-5)

,..---BOTH ENDS

SIMPLY SUPPORTED

NOTE:

t k_\VIEW A

tl, t 2, Ic, h, R, R2, Rsmall,, Rlarge'

a_d Lu aR: all measured it] units of

in{'hcs while _ is measured in degrees,

Figure 5.3-2. Truncated Sandwich Cone

5-12

Thecritical lateral pressurefor theequivalentsandwichcylindercanbeobtainedby

usingtheequation,_andcurvesof Section4.4.2.

Plastieily c_)nsiderationsshouldbehandledas specified in Section 9.2. Tile plasticity

roduclion factor 77 should always be based on theprincit)al nlembrane stresses at the

lat-ge end of the cone where

P R1 arge= in. 3-6)

_I! (L _ t:.) cos q

P Re ( R_m_ll /_M - (L t_) \1 -- (5.3-7)

. ' _ Rlarg e /

It is ruconm_(,nded thai the at)l)roach specified here be applied only to cases where

c_ _ 30 degrees.

5-13

5.4 TORSION

5.i. 1 BasicPrinciples

It appearsthat nosignificantlheoreticalsolutionshavebeen1)ublishedfor sandwich

conessubjectedto torsion. Therefore, for thepurl)(,s(,s,)1this handbook,the

equivalent-cylinderconcepl()t'Seide[5-3i hasbeenadopledasapractical expediency.

Basedontheanalysisof his numericalcomputationsfor lhin-walled, isotropie {non-

sand_vich),truncatedcones,Scideconcludedthat thecritical l¢)rquesfor suchshells

canbe takenequallo thevaluesfor circular cylinderswhichsalisfy thefollowing

conditions:

a.

[?.

c,

The wall thickness of the equivalent cylinder is equal to that of the cone.

In the case of sandwich constructions, the logical extension of this condi-

tion is that the equivalent cylinder have the same facing and core thick-

nesses as arc found in the cone.

The leng-th of the equivalent cylinder is equal to the axial length of the cone.

The radius of the equivalent cylinder is coml)uted from the relationshit)

I[( R >]1'lc :: (Rsmall cosc_)1 _. _ 1 _ R---_---_maH/j- 1 1 )I/larg_eRsmall ,,5.4-1)

where

R = Radius of equivalent cylinder, inches.C

Itsmall

= Radius at small end of cone, inches (measured perpendicular

to the axis of revolution).

Rlarg c :: Radius at large end of cone, inch(_s (measured perpendicularto the axis of revolution).

cy :_ Vertex half-angle of cone, degrees.

5-14

In R(_ference 5-1, Seide, et al. present test results from ten isotropic (non-sandwich),

truncated cones which were subjected to torsion. These tests included specim¢,ns

having vertex half-angles (c_ of both 30 and (;0 degrees. The agreement of these,

results with equivalent-cylinder predictions was similar to that obtained from com-

parisons of test data from isotropic (non-sandwich) cylinders against the corresponding

small-deflection theoretical solutions. For conical sandwich constructions it was

therefore decided to use the same knock-down factor (7 s = 0.80) as was selected in

Section 4.5 for sandwich cylinders under torsion.

No test data are available for sandwich cones which are of the t33)es considered in this

handbook and are subjected to torsion. Therefore the method recommended here has

not been ex-perimentally verified and can only be considered as a "best-available"

approach.

5-15

5.,1.2 Design Equations and Curves

For simply supported, truncated, right-circular sanck_'ich cones subjected to torsion,

the critical torque may be computed from the equation

where

Tcr

R =:

o

t 1 and t 2 =

/

T =cr

T = 2vR z (t t + t._) y_ (5.4-2)er e • cr

Critical torque for sandwich cone subjected to torsion, in-lbs.

Radius of equivalent sandwic:h cylinder, inches r see Equation

(5.,,_-3)].

Ttticknesses of the facings, inches. (There is no preference

as to which faeing is denoted by the subscript 1 or 2.)

Critieal shear stress for equivalent sandwich cylinder when

subjected to torsion, psi. (it should be noted that this value is

not equal to the critical sh,-ar stress of the conical sandwich

construction. )

The radius R is coml)uted frome

Re = (Rsmal 1 c_)scs) 1 +

1 1

[2 (1 + Rlarge_]'- [1(I + Rlar_e_] -_Rsmall/] Rsmall/l

(,_.4-:_')

where Rsmal 1, Rlarg e, and c_ are as shown in Figure 5.4-1.

The stress 7 _ may be computed from the equations and cur_'es of Section 4.5.2 pro-or

vided that

a. The values t_, t_, te, and d are measured as shown in Figure 5.4-1.

bg The radius R is replaced by the effective radius R .e

co The length L is taken equal to the axial length of the cone (see Figure

5.4-1).

5-16

(l, Hl-/bs

Both Ends

Simply Supported / 7R

T, itl-lbs

d

i

small

View A

.!

Axis of

Revolution

N _Tt- i1, 12, it, l/, tl, I, P'srl_al[,

aud Rlarg c art al_ illtasclr_ tl

ill II[lllS O[ il_C lies wllilt' t+_ iS

rneastlrt d i_ (t('_rt cs,

Figure 5.4-1. Truncated Sandwich Cone Subjected to Torsion

In a truncated cone which is subjected to torsion, the maximum shear stress will occur

at the small end. Hence, for sandwich constructions of this type, the critical stress

value is associated with that same location. One can therefore write

Tcr

rcr = 2rr R _ (t_+t 2) (5.4-4)small

where

r = Critical shear stress for truncated sandwich cone when subjectedcr

to torsion, psi.


_ 30 degrees.

Plasticity reduction factors should always be based on the stress at the small end of

the cone (see Section 9).

5-17

:).;_ TRANSVERSE SIIEAR


Th(, case considered here is that of a truncated sandwich cone which is subjected only

to transverse shear forces a,_ shown in Fig_,re 5.5-1. No_c that all transverse see-

tions, such as A-A, are subjected to the same magnitude <)f shear load.

J

____ --

--,.-- A

J

_.--A

l.i

Figxlre 5.5-1. Truncated Cone Subjected to Transverse Shear

This, of course, is a pur_qy h3_pothetical loading condition since it does not result in

over-all static equilibrium of the structure. To obtain the necessary balance of forces

and moments, it is required that an external bending moment also be present. Never-

theless, the hypothetical unbalanced loading system does prove to be of interest since

the combined effects of transverse shear and its associated bending are usually analy-

zed by using an interaction equation. Such a relationship invulves both the critical

peak meridional stress under a bending moment acting alone and the critical peak

shear stress corresponding to the subject artificial loading condition.

It appears that no significant theoretical solutions have been published for sandwich

cones subjected to transverse shear. Therefore, for the purposes of this handbook,

5-18

the conceptusedfor sandwichcylinders (see Section 4.6) will also be adopted here as

a practical expediency. As noted in Section 4.6, the results from a series of tests

[5-4 and 5-5] on isotropic (non-sanchvieh), circular and elliptic cylinders led to the

conclusion [5-6] that

Lower-Bonndrcr TestValues for 1Transverse Shear Loading _ 1.25 (5.5-1)

Small-Deflection Theoretical ]

rcr Values for Torsional Loading]

To properly understand this ratio, it should be observed that for torsional loading of a

thin-walled circular cross section the shear stress Tcr is uniformly distributed around

the circumference. On the other hand, under transverse shear loading, the shear

stress is nonuniform and the value _'cr then corresponds to the peak intensity which

occurs at the neutral axis.

For the lack of a better approach, it was reconm_ended in Section 4.6 that Equation

(5.5-1) be used for the design and analysis of sandwich cylinders that are subjected to

transverse shear forces. For the same reason, it is recommended here that Equa-

tion (5.5-1) also be used for truncated sandwich cones. In the latter ease, the re-

quired small-deflection theoretical Tcr values for torsional loading should be obtained

as specified in Section 5.4, with the exception that 7s must now be taken equal to unity.

No sanchvich test data are available to substantiate the reliability of this practice.

Until such data do become available, one can only regard this proeedm_e as a "best-

available" approach.

5-19


For simply supported,right-circular, truncatedsandwichconessubjectedto trans-

verseshearforces, thecritical peakshearstress maybecomtmtedfrom theequation

= 1.25 (T) (5.5-2)rcr cr Torsion

Ys 1.0

where

(_r) = The critical torsional shear stress obtained by substitutingTorsion Ys = 1.0 throughout the meth_gls cited in Section 5.4, psi.

Ys = 1.0

In a truncated cone which is subjected to transverse shear, the maximum shear stress

will occur at the small end. ttence, the critical stress value is associated with that

1ocation.

Plasticity reduction factors should always be based on the stress at the small end of the

cone (see Section 9).

When the behavior is elastic, the critical transverse shear force .._Fv)cr can be corn-

puted from the following:

(F v) = _ (t_ +%)rcr Rsmall cr(5.5-3)

To compute (Fv)cr when the behavior is inelastic, one must resort to numerical inte-

gration techniques.

5-20

5.6 COMBINED LOADING CONDITIONS

5.6.1 General

For structural members subjected to combined loads, it is customary to represent

critical loading conditions by means of so-called interaction curves. Figure 5.6-1

shows the graphic format usually used for this purpose. The quantity R i is the ratio

of an applied load or stress to the critical value for that type of loading when acting

alone. The quantity" Rj is similarly defined for a second type of loading. Curves of

this form give a very clear picture as to the structural integrity of particular con-

figurations. All computed points which fall within the area bounded by the interaction

curve and the coordinate axes correspond to stable structures. All points lying on

or outside of the interaction curve indicate that buckling will occur. Furthermore,

as shown in Figure 5.6-1, a measure of the margin of safety is given by the ratio of

distances from the actual loading point to the curve and to the origin. For example,

assume that a particular structure is subjected to the combined loading condition

corresponding to point B of Figure 5.6-1.

1.0

D

(Rj)D -- -- -- MI�

(Rj) B --__

Rj 1/111

0 ].0

Ri

Figure 5.6-I. Sample Interaction Curve

5-21

Then, for proportional increases in R. and R., the margin of safety (MS) can be com-1 j

puted from the following:

(Rj) DMS - i

(Rj) B (5.6-1)

As an alternative procedure, one might choose to compute a minimum margin of safety

which is based on the assumption that loading beyond point B follows the path BM.

Point M is located in such a position that BM is the shoo-test line that can be drawn

between point B and the interaction curve. The minimum margin of safety can then

be calculated as follows:

Minimum MS - OB + BM 1OB (5.6-2)

5-22

5.6.2 Axial Compression Plus Bending


In Section 4.7.2 this loading condition is treated for the case of circular sandwich

cylinders. For such configurations, it was concluded that one may use the following

interaction relationship:

R +%=1c

(5.6-3)

where

c

R =c "/c ((_c) C L

(5.6-4)

a b

%=Tb ((_c)C L

(5.6-5)

and

(Yc

_b

(_c)C L

= Uniform compressive stress due solely to applied axial load,

psi.

= Pcak compressive stress due solely to applied bending moment,

psi.

= Classical theoretical value for critical uniform compressive


Tc = Knock-down factor given by Figure 4.2-8, dimensionless.

Tb = Knock-down factor given by Figure 4.3-2, dimensionless.

In this handbook it is proposed that for truncated sandwich cones the cases of pure

bending and of axial load acting alone both be treated by means of an equivalent-

cylinder concept (see Sections 5.1 and 5.2). For both types of loading, the radius

of the equivalent cylinder is taken equal to the finite principal radius of curvature at

the small end of the cone. It should be noted that the maximum stresses from both

5-23

bendingandaxial compressionoccurat this samelocation. In viewof theseseveral

considerations,it is assumedherethatEquations(5.6-2,)through(5.6-5) canbe

appliedto truncatedsanchvichconesif

a,

o"c and gb are bcCh computed for the meridional direction and at the smallend of the cone, and

b.

the values for ;¢e' and Yb' and ((Te)r_T are those which apply to the equivalent

sandwich cylinder described in Sect_i_ns 5.1 and 5.2. (It is important to

keel) in mind that 7c must be taken equal to 1.0 when computing the value((_c) C L • )

Since no test data have been published for truncated, sandwich cones subjected to axial

compression plus bending, the recommended approach has not been experimentally

verified and can only be regarded as a "best-available,, method.

5-24


For simply supported,truncated, right-circular sandwichconessubjectedto axial

compressionplusbending,thefollowinginteractionequationmaybeemployed:

Re+ Rb = 1 (5.6-6)

where

cR - (5.6-7)

c 7e (6c)CL

_b

a b 7t) (5c) (,_.(i-s)CL

Equation (5.6-6) may be used for cones of any length. A plot of this equation is given

in Figure 5.6-2.

The quantity gc is the uniform meridional compressive stress, at the small end of the

cone, due to the axial force acting alone.

The quantity Crb is the peak meridional compressive stress, at the small end of the

cone, due to the bending moment acting alone.

The quantities Ye' _b' and (Cre)CL- are those which apply to the equivalent sandwich

cylinder described in Sections 5.1 and 5.2.

In Equations (5.6-7) and (5.6-8), the knock-down factors )'e and_b are those obtained

from Figures 4.2-8 and 4.3-2, respectively.

The quantity (_c)CL is simply the result obtained by using Yc = 1.0 in the method of

Section 4.2.2.

5-25

Plasticity considerationsshouldbehandledasspecificdin Section9.2 except,that in

this case, one may use

(a)[ l

r? -: [ 1_-]--_ 1 EZ for short cones, and

(b) r7 =

l

Ii- o l VEtEs1---_ l Ef

for moderate-length through long cones.

The plasticity reduction factor 77 should always be based on the peak compressive stress

at the small end of the cone.

1.0t

0.8

It b

0.6

I 'iii

\ ........ T

O. 4 ................... 4-.......... ' ' ' _ ' ' _ -

..... r t

!0.2 1 _

i I '

I

0 i .0 0.2 0.4 0.6 0.8 .0

Re

Figure 5.6-2. Design Interaction Curve for Truncated Sandwich Cones

Subjected to Axial Compression Plus Bending

5-2 6

5.6.3 Uniform External Itydrostatic Pressure


The loading condition considered here is depicted in Figure 5.6-3. As shcavn, the cone

is subjected to a uniform external pressure over the lateral surface and both cnd closures.

p, psi

c_ R Rz

_t_

Both Ends

Simply SuppoEtLd

p, psi

-I..q-_

Rlarg c

Figure 5.6-3. Truncated Cone Subjected to Uniform

External Itydrostatic Pressure

This results in principal membrane stresses which may be computed as follows when

the core has a relatively high extensional stiffness in the direction normal to the

facings:

p R_

CrH - (t_ + t_) (5.6-9)

where

pR._ (5.6-10)cr M - 2 (t_ + t_)

RR - (5.6-11)

2 cos

5-27

and

crM

p -

tI andt --

R :

=: Itoop membrane stress, psi.

- Meridional membrane stress, psi.

Uniform external hydrostatic pressure, psi.

Finite principal radius of curvature of middle surface, inches.

Thicknesses of the facings, inches. (There is no preference

as to which facing is denoted 1)3, the subscripts 1 or 2.)

Radius of middle surface measured perpendicular to the axis

of revolution, inches.

(_ = Vertex half-angle of cone, degrees.

Since the radii [/ and R vary with the axial location, the stresses (Ytt

uniform over the conical surface.

occur at the large end of the cone.

and r; are non-M

The maximum values for each of these quantities


of truncated sandwich c(mes which are subjected lo uniform external hydrostatic pres-

sure. Therefore, i'or lh(, purposes of this handbook, _he equivalent-cylinder at)preach

of Seide, et al. [5-11 has been adopted as apractical expediency. Based on alarge

array of test data from thin-walled, isotropic (non-sandwich), cylinders and truncated

cones, Scide, et al. concluded that the critical hydrostatic pressures for such cones

can be taken equal to the values for equivalent circular cylinders which satisfy" the

following conditions:

a. The wall thickness of the equivalent cylinder is equal to that of the cone.

In the case of sanokvich constructions, the logical extension of this condi-

tion is thai the equivalent cylinder have the same facing and core thick-

nesses as are found in the cone.

5-28

b°

C.

The length of the equivalent cylinder is equal to the slant length of the cone.

The radius of the equivalent cylinder is equal to the average finite principal

radius of eurvature of the cone. That is,

Rsmall + Rlarge (5.6-12)R -:

e 2cos

where

Re

Rsmall

= Radius of middle surface for equivalent cylinder, inches.

= Radius of middle surface at small end of cone (measured

perpendicular to the axis of revolution), inches.

= Radius of middle surface at large end of cone (measured

perpendicular to the taxis of revolution), inches.Rlargc

The critical hydrostatic pressure for the equivalent cylinder can be obtained by using

the equations and curves of Section 4.7.3.

The only available eN)erimental results for conical san&vieh shells under uniform

external hydrostatic pressure are the data from two tests eondueted by North American

Rock, veil, Corp. _5-7 and 5-8_ in conjunction with the Navajo missile program. To

assist in the preparation of this handbook, an analysis was made of the result published

in Reference 5-7. The other specimen was not studied since it was stressed too deeply

into the plastic region. The specimen of Reference 5-7 was also inelastic but the

stresses in this instance were low enough to permit reliable computations. Using the

approach of the present section in conjunction with the plasticity reduction criteria of

Sect{on 9, the design critieal pressure was computed to bc 36.4 psi. This is in satis-

faclot3' agreement with the experimental value of 43.6 psi.

5 ')9

Theforegoingsubstantiates,to avery small degree,thereliability of theequivalent-

cylinderconceptrecommendedhere. However,in viewof the lackof a sufficient

numberof test results, this approachcanpresentlybeconsideredasonly a "best-

available"method.

5-30


I,'or a simply supported, truncated, right-circular sandwich cone subjected 1o uniform,

external, hydrostatic pressure, the critical pressure may be taken equal to that for tin

equivalent sandwich cylinder for which

a. The values t_, t:., t c, and h are measured as shown in Figure 5.6-4.

b. The length is taken equal to the slant length L e

Figx_re 5.6-<1.

of the cone as shown in

C0 The radius is denoted R and is computed from the fornmlae

Rsmal 1 + Rlarg eR =

e 2cos (_(5.6-13)

where Rsmal 1, Rlarg e, and a are as shown in Figure 5.6-4.

, Rlargc

L_L\I_-T

P'smail

R R2

VIEW A

BOTH ENDS

SIMPLY SUPPORTED

NOTE: t 1, l, i , h, R, R,. c 2 Rslnall' Rlarge'

and Le are all measured in units of

inches while o/is measured in degrees.

Figure 5.6-4. Truncated Sandwich Cone

The critical hydrostatic pressure for the equivalent sandwich cylinder can be obtained

from the equations and curves of Section 4.7.3 if the ratios R and R are now definedc p

as follows :

5-31

R = P

c ye (l_x)C L (5.6-14)

w he re

P

R =

p Wp (_y)CL(5. (;-15)

p Uniform, external, hydrostatic pressure applied to lateral

surfaces and end closures of the equivalent sandwich cylinder,

psi.

In Equations (5.6-14) and (5.6-15), the knoek-d(r_cn factor _/e is that obtained from

Figure 4.2-8, while v may be taken equal to 0.90.'p

It should be noted that

or

whe re

(C_x)CL

(Px)CL Re

((rx) CL - 2 (t_ * t:_) (5.6-16)

2 (gx)CL(t ! _ t_)

(ISx)CL = R (5.6-17)e

= Classical theoretieal value for critical uniform axial com-

pressive stress when acting alone on the equivalent sandwich

cylinder. This value can be obtained by using Yc = 1.0 in theequations and curves of Section 4.2.

The value (tSy)c L can be obtained by using Tp = 1.0 in the equations and curves of

Section 4.4.

Plasticity considerations should be handled as specified in Section 9.2. The plasticity

reduction factor r7 should always be based on the principal membrane stresses at the

large end of the cone where,

5-32

pR large(YH= (t1 _ t2) (cos _)

(5.G-18)

P Rlarg e

_M = 2 (t 1 + t_) (cos (_)

(5. (;-19)


_ 30 degrees.

5-33

5.6.4 Axial CompressionPlus Torsion


Theloadingconditionconsideredhereis depictedin Figure 5.t;-5. Theaxial loadP

canoriginatefrom anysourceincludingexternalpressureswhicharedistributed

uniformlyover theendclosures.

P. I bs

T, m-lbs torque

], Hl-lbs (orqtle_

f_ / \\

_.__________ ____ _ _ _ ___ _ P, l bs

Bol:h Ends

Simply Supported

Figure 5.6-5. Truncated Cone Subjected to Axial

Compression Plus Torsion


of truncated sanchvich cones under this combination of loads, ltowevcr, MacCalden

and Matthiesen _5-9] have arrived at certain conclusions for non-sandwich shells under

such loading and, for the purposes of this handbook, these results provide the basis for

an expedient engineering approach to the case of conical sandwich constructions. Based

on a large array of test data from Mylar specimens, MacCalden and Matthiesen con-

cluded that the following interaction relationship could be applied to thin-walled, iso-

tropic (non-sanchvich), truncated cones:

R +R _c S

5-34

= 1(5.6-20)

where

and

Rc

Rs

P (5.6-21):2

m

(Pcr)Empirical

T (5.6-22)

(Ter) Empirical

(Pcr) Empirical

(Tcr) Empirical

= Empirical lcnver-bound value for the critical axial load

when acting alone, lbs.

= Empirical lower-bound value for the critical torque when

acting alone, in-lbs.

This result is identical to that given in Reference 5-10 for thin-walled, isotropic (non-

sandwich) cylinders subjected to axial compression plus torsion. One might, therefore,

conjecture that in the case of sandwich constructions the interaction curves for trun-

cated cones under the subject loading condition are of the same shape as those pre-

sented in Section 4.7.4.2 for circular cylinders. The design equations and curves

recommended here arc based on this premise. That is, one might choose to view the

formula,

R + R2 = 1 (5.6-23)c s

as a comprehensive interaction equation for truncated cones of both isotropic (non-

sanchvich) and sanchvich construction. However, it is important to note here that

MacCalden and Matthicsen observed that the presence of even a very small axial load

made the torsionally-loaded conical shell much more sensitive to imperfections than

was the case when no axial load was applied at all. They, therefore, recommended

that whenever R c is non-zero, the same knock-down factor be employed in computing

5-35

m

(Tcr)Empirica 1 as is used in the calculation of (Pcr)Empirieal" It was further speci-

fied that this single knock-down factor should be taken equal to that which applies for

the case of axial compression acting alone. The same practice is adopted here.

Caution should be exercised in implementing the foregoing recommendations, partially

because only the extremes o[ transverse shear rigidity of the core have been consid-

ered (see Section 4.7.4.1). In addition, although the interaction relationship for the

subject loading condition should probably be dependent upon a length parameter, no

investigations were made to establish the sandwich lengths over which Equation (5.6-23)

is a reasonable representation of the actual behavior. Furthermore, no test data have

been obtained for sandwich cones subjected to axial compression plus torsion. There-

fore, the general validity of Equation (5.6-23) has not been experimentally verified

and can only be regarded as a "best-available" approach.

5-36


For simply supported,truncated,right-circular sandwichconessubjectedto axial

compressionplustorsion, onemight chooseto employthe interactionformLda,

R +R e =1C S

(5.6-24)

which is plotted in Figure 5.6-6 and where,

P

c (Per)Empirical

(5.6-25)

T

- 3,c

(5.6-26)

p = Applied axial load, lbs.

T

(Pcr)Empirical

(Tcr)Empirica 1 =

= Applied torque, in-lbs.

Lower-bound value for the critical axial load when

acting alone. This value can be obtained by using the

equations and curves of Section 5.1.2, lbs.

Lower-bound value for the critical torque when acting

alone. Tkis value can be obtained by using the equa-

tions and curves of Section 5.4.2, in-lbs.

_C

The knock-down factor obtained from Figure 4.2-8

(dimensionless). For the purposes of the present

case, the quantity R (see Figure 4.2-8) must be set

equal to the equivalent radius R e which is computed

as follows:

Rsmall

Re = cosc_(5.6-27)

R = Radius at small end of cone, inches (measured per-

small pendicular to the axis of revolution).

= Vertex half-angle of cone, degrees.

5-37

0

Figure 5.6-6.

O. 2 0.4

Conditional Interaction Curve for Truncated Sandwich Cones

Subjected to Axial Compression :Plus Torsion

The factor _ should be introduced into the demoninator of the ratio R s only when

R c is non-zero. For the special case where no axial load is present (R c 0), R s

should be taken equal to T + (Ter)Empirical"

Attention is drawn to the fact that in Section 5.6.4.1, several factors are cited which

shed considerable doubt upon the reliability of results obtained from the indiscriminate

use of Equation (5.6-24) mid Figure 5.6-6. In view of these uncertainties, one might

often choose to employ the straight-line interaction formula,

R +It =1c s (5.6-28)

which is plotted in Figure 5.6-7. This relationship can be used with confidence for

any length of cone and for any region of transverse shear rigidity of the core, since

5-3 8

experiencehasshownthat the linear interactionformulais neveruneonservativefor

shell stability problems, ttowever, in manyeasesit will, of course, introduce

excessiveconservatism.

Plasticity considerationsshouldbehandledas specifiedin Section9.2. Theplasticity

reductionfactor r{ should always be based on the stresses at the small end of the cone.

0o

0°

0.4

li

(Ii 0.2 0.4 0.6 0.8 1.0

R e

Figure 5.6-7. Conservative Interaction Curve for Truncated Sandwich ConesSubjected to Axial Compression Plus Torsion

5.6.5 Other Loading Combinations


In Section 5.6.4, the combined loading condition of axial compression plus torsion is

treated. The interaction relationships presented there can be used for an additional

5-39

loadingcombination1)5, recognizing that at any given axial location on the cone the

peak meridional stress due to an applied bending moment can be converted into an

equivalent uniform meridional stress. With this in mind, the design equations and

era'yes of Section 5.6.4.2 can be used for the combination of axial compression plus

bending plus torsion which is depicted in Figure 5.6-S.

Fig_lre 5, 6-8.Truncated Cone Subjected to Axial CompressionPlus Bending Plus Torsion

To accomplish this it is simply required tlmt the quantity Rc be redefined as follows:

p#

Rc (tScr) ....

r_mplmcal

whcre

(o. 6-29)

and

(5, 6-30)

P Applied axial load, lbs.

M Applied bending moment, in_-lbs.

Yc = Axial comt)ression tmock-down factor from Figure 4.2-8dimensionless.

Note: Vor the purposes of the present ease, the quantity R (see

Figure 4.2-8) must be set equal to the equivalent radius Re whichis comt)uted as follows:

5-40

Rsmall

Re - cosa(5.6-31)

Tb Bending lo]ock-down factor from Figxlre 4.3-2, dimensionless.

Note: For the pro'poses of the present case, the quantity R (see Fignre

4.3-2) must be set equal to the equivalent radius R e which is computed

as follows:

Rsmall (5.6-32)Re - coset

Rsmall

:: Radius at small end of cone (measured perpendicular to the

axis of revolution), inches.

=: Vertex tmlf-angle of cone, degrees.

The foregoing formula for P_ is based on the principles cited in Section 5.2

Since no sandwich test data are available to substantiate the recommendations made

here, they can only be regarded as a "best-available" criterion.


For simply supported, truncated, right-circular sandwich cones subjected to the

loading condition depicted in Figure 5.6-8, one may use the design equations and

curves of Section 5.6.4.2, except that the quantity R c must now be defined as follows:

p'

R c = - (5.6-33)(Pcr)Empirical

where

_,Tb/ a11

(5.6-34)

5-41

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5-54

REFERENCES

5-1 Seide, P., Weingarten, V. I. , and Morgan, E. J., "Final Report on the Devel-

opment of Design Criteria for Elastic Stability of Thin Shell Structures", STL-

TR-60-0000-19425, 31 December 1960.

5-2 Baker, E. If., "Experimental Investigatim of Sandwich Cylinders and Cones

Subjected to Axial Compression", AIAA Journal, Volume 6, No. 9, September

1968.

5-3 Seide, P., "On the Buckling of Truncated Conical Shells in Torsion", Journal

of Applied Mechanics, June 1962.

5-4 Lundquist, E. E., "Strength Tests of Thin-Walled Duralumin Cylinders in Com-

bined Transverse Shear and Bending", NACA TN 523, 1935.

5-5 Lundquist, E. E. and Burke, W. F., "Strength Tests of Thin-Walled Duralumin

Cylinders of Elliptic Section", NACA TN 527, 1935.

5-6 Gerard, G. and Becker, tt., "Handbook of Structural Stability, Part HI - Buckling

of Curved Plates and Shells", NACA TN 3783, August 1957.

5-7 Anonymous, "Test of XSM-64A Booster Fuel Tank Conical Bulkhead No. 3,"

North American Rockwell, Corp. Report No. AL-2622, 15 September 1957.

5-8 Anonymous, "Test of XSM-64.A Booster Fuel Tank Conical Bulkhead No. 4,"

North American Rockwell, Corp. Report No. AL-2623, 16 September 1957.

5-9 MacCalden, P. B. and Matthiesen, R. B., "Combination Torsion and Axial

Compression Tests of Conical Shells", AIAA Journal, February 1967.

5-10 Batdorf, S. B., Stein, M., and Schildcrout, M., "Critical Combinations of

Torsion and Direct Axial Stress for Thin-Walled Cylinders," NACA Technical

Note 1345, June 1947.

5-55

5-11Baker, E. tt., Cappelli, A. P., Kovalevsky, L., Rish, F. L., andVerette,

R. M. , Shell Analysis Mmmal, Prepared by North American Rockwell Corp.

for the National Aeronautics and Space Administration, Manned Spacecraft

Center, [[ouston, Texas, June 1966.

5-56

6GENERAL INSTABILITY OF DOME-SHAPED SHELLS

6.1 GENERAL

This section deals with dome-shaped shells whose contours are surfaces of revolution.

Figure 6. i--i shows the shapes considered here, all of which are truncated at the equa-

tor. Note that the torispherical shape consists of a lower toroidal segment which

blends into a spherical cap. It is expected that the configurations shown here will

cover the large majority of the dome structures likely to be encountered in aerospace

applications. One should observe that for each of

,) ,?

_---x

, -

Supported or

Clamped

(a) ttemispherical

2 2x y

_2 b2 Y

I.

Boundary Simply-

Supported or

Clamped

(b) Ellipsoidal

_...._Spheri ca 1

t-Boundary Simply-

Supported or

Clamped

(c) Torispherical

Figure 6.1-1. Structural Dome Shapes

these domes the maximum radius of curvature RMa x occurs at the apex. As a practi-

cal engineering expediency, analysis of all the illustrated configurations will be based

on this radius.

6-1

In the case of externally pressurized, thin-walled, isotropic (non-sandwich) domes, it

has long been recognized that the test results normally fall far below the predictions

from classical small-deflection theory for the axisymmelric buckling of complete

spheres. The discrepancies are usually attributed to,

a. the shape of the postbuckling equilibrium path coupled with the presence ofinitial imperfections,

b. the fact that large-deflection analyses of asymmetric behavior yield criti-

cal stresses approximately 20 percent lower than the small-deflection axi-symmetric values, and

c. the fact that classical small-deflection theory does not account for pre-

buckling discontinuity distortions in the neighborhood of the boundary.

This is analogous to the situation described earlier in this handbook (see Section 4.1)

for the case of circular cylinders. For the latter, it has become common practice to

base stability analyses and design procedures on the use of classical small-deflection

theory modified by empirical knock-down factors. This approach was selected in

Section 4.1 for sandwich cylinders and is also adopted here for sandwich domes.

6-2

6.2 EXTERNAL PRESSURE



This section deals with the loading condition depicted in Figure 6.2-1. That is, a uni-

form external pressure acts over the entire surface of the sandwich dome. The net

l lbsw --

_' iI1

+

p, psi

l

lbs l

w j --c in

Figure 6.2-1. Sandwich Dome Subjected to External Pressure

vertical component of this loading is reacted by a uniform rtmning load on the boundary.

From Figure 6.1-1, note that the domes can have either simply-supported or clamped

edges. That is, during buckling the boundary is constrained such that no radial dis-

placements occur. In the simply-supported case, the shell wall is free to rotate along

the boundary whereas for clamped edges such rotations are completely suppressed. It

follows, of course, that intermediate restraints to edge rotation are also acceptable.

The theoretical basis used here is the classical, small--deflection solution by Yao [6-1]

as reformulated by Plantema [6-2]. This result embodies the following assumptions:

ao

b.

C.

The facings are isotropie.

Both facings are of the same thickness.

Both facings have identical material properties.

6-3

d. Bending of the facings about their own middle surfaces can be neglected.

e. The core has infinite extensional stiffness in the direction normal to thefacings.

f. The eore extensional and shear stiffnesses are negligible in directionspacallel to the facings.

g. The transverse shear properties of the core are isotropie.

R11. The inequality 7-- >> 1 is satisfied,

C

W hc re

It _: Radius to middle surface of sandwich sphere, inches.


i. Approximations equivalent to those of I)omlell TG-a] can be applied.

Strictly speaking, this solution was derived for complete sandwich spheres which

exhibit small buckles that are axisymmetric with respect to a radius of the sphere.

The development isolated one such buckle as a free body so that shallow-shell theory

could then be employed. Yao presented his results in a form which is not conducive

to a ready physical interpretation of the phenomena involved. Therefore, Plantema

undertook to ex-press the final relationships in a manner which would foster some

insight in tkis regard, tie was able to sh(rvv that, when the core has isotropie trans-

verse shear stiffness, Yao's solution is identical to the equations given earlier in this

handbook for axially compressed circular sandwich cylinders rsee Equations (4.2-27)

through (4.2-30) and Equations (4.2-4) and (4.2-5)]. TEat is, when the knock-down

factor, Y'd' is included,

(_er = 3/d Kc°'o (6.2-1)

6--4

where

and

O" o

_TEf h

R(6.2-2)

When VC

When VC

---2

>-2

1K = !---V

c 4 e

1K -

c VC

(6.2-3)

(6.2-4)

where

VC

0

crimp

(6.2-5)

(Icr

Ef

5 2

(7 - Gcrimp 2 tft c c

= Critical compressive stress for sandwich sphere, psi.


= Young's modulus of facings, psi.

h = Distance between' middle surfaces of facings, inches.

v = Elastic Poisson's ratio of facings, dimensionless.e

R = Radius to middle surface of sandwich sphere, inches.

tf = Thickness of a single facing, inches.


G = Transverse shear modulus of core, psi.C

(6.2-6)

6-5

Theequivalencebetweenanaxially compressedsandwichcylinderandanexternally

pressurizedsandwichspherehasbeenanalyticallydemonstratedonlyfor thecase

wherethetwo facingshaveidenticalmaterialpropertiesandare of thesamethickness.

If oneassumesthatthis equivalencestill holdstrue whenthe facingsare of different

thicknesses,Equations(4.2-2) through(4.2-7) canthenbeusedhereif Gxzis replaced

by Gc sothat, whentheknock-downfactor Ydis included,

o" = yd K o (6.2-7)cr c o

where

h 2_/tl__7 = (6.2-8)o rtEf R

e

and

When VC

22

1K = 1---V (6.2-9)

c 4 c

When VC

'2

1K - (6.2-10)

c VC

whe re

(70

V - (6.2-11)e (7

crimp

t1

and t2

5 2

- G (6.2-12)crimp (t 1 + t2) t c c

= Thicknesses of the facings (There is no preference as to

which facing is denoted by the subscript 1 or 2.), inches.

6-6

TherelationshipbetweenKe andVe canbeplottedas shownin Figure6.2-2. It is

importantto notethat thevalueVc = 2.0 establishes a dividing line between two

different types of behavior. The region where Vc < 2.0 covers the so-called stiff-

core and moderately-stiff-core sandwich constructions. When V is in the neighbor-C

hood of zero, the core transverse shear stiffness is high and the sandwich exhibits

maximum sensitivity to initial imperfections. As V increases frome

1.0

K e

I

L I2.0

Vc

Figure 6.2-2. Schematic Representation of Relationship Between K and Ve --e

zero to a value of 2.0, this sensitivity becomes progressively less. The domain

where V _ 2.0 is the so-called weak-core region where shear crimping occurs.c

Sandwich constructions which fall within this category are not influenced by the pres-

ence of initial imperfections and a knock-down factor of unity can be applied to such

structures. It should be possible to develop a continuous transitional knock-down

relationship which recognizes the variable influence of the core rigidity but this is

beyond the scope of the present handbook.

6-7

6.2.1.2 Empirical Kmock-D(_vnFactor

As notedin Section6.1, for thepurposesof this handbook,theallowablestressesfor

externallypressurizedsandwichdomesareestablishedby applyinganempirical knock-

downfactor (_d)to the results from classical small-deflectiontheory, ttowever,since

theavailabletest datafrom sandwichdomeconstructionsare very scarce, onecannot

yet determineYdvalueswith a highdegreeof reliability. Theonlyusefuldataun-

coveredduringthepreparationof this handbookare thosewhichwere obtainedby

NorthAmericanRockwelli 6-4_in conjunctionwith theSaturnS-II developmentpro-

gram. Theseresultsgivethe yd values shown in Figure 6.2-3 which inehides two data

points from hemispheres and six data points from domes that were approximately

ellipsoidal. Reference (;-4 includes specimens whose membrane stresses at failure

ranged all the way from the elastic to the deeply plastic zones. In three cases it was

felt that these stresses were too high to permit the computation of reliable plasticity

reduction factors. Therefore, these particular data were discarded and they do not

appear in Figure 6.2-/t. Still another experimental point was discarded because of a

faulty edge condition in the test. In addition, as noted in Figure 6.2-3, two specimens

were subjected to a thernml gradient along with the external pressure. For each of

these domes, the inner facing was at roughly +280°F while the outer facing was at

apt)roximately +10 °F. This gradient was completely neglected in the analysis per-

formed to arrive at the related Yd values. Nevertheless, these results are retained

in Figure 6.2-3 since they fall within the scatter band displayed by the other speci-

mens having the same basic contour.

6-8

1°0

0,

0o

_d

0o

0._

i L ..... Lt • ILEGEND

• ttEMISPHERES AT ROOM TEMPERATURE,, APPROXIMATE EI_LIPSOIDS AT ROOM

TEMPE RATUREAPPROXIMATE EI, LIPSOIDS WITH TItERMAI_

GRADIENT

p = SHE]_I,-WAI,I, RADIUS OF GYRATION

--[]

io,

(p _ h/2 FOR SANDWICtt CONSTRUCTIONS i [ ' '

WtIOSE TWO FACINGS ARr._EOF EQUAL THICKNESS) I ....

• ' ' ' t i i i RECOMMENDED DESIGN. '• ' ........ . _ IVAI,UE FOR EI,I_IPSOIDS] i

, . .... AND TORISt tlERICAI, +

..... '' _ '•: _.. 'liiDOMES '

. • : ...... i ...... RECOMMENDED DESIGN

' _ ' '. i _ ' _ '.. '. i ' _ _"i VALUE FOR HEMISPttERES• [ , r ' • " " •

, I I i

0 100 200 300 400 aO0

(->)

Figure 6.2-3. Knock-Down Factor _/d for Sandwich Domes

Subjected to Uniform External Pressure

6-9

To fully understandtheinformationgivenin Figure6.2-3, it is importantfor the

readerto beawareofthe datareductiontechniquesemployedhere. For anexplanation

of theseprocedures,relbrencemaybemadeto thediscussionin Section4.2.1.2.1.

Althoughthat sectionis concernedwith sandwichcylinders, the samebasicapproach

wasusedin analyzingthedomes.

BasedonFigure6.2-3, it is recommendedthat, exceptwhereshearcrimpingoccurs,

the followingvaluesma3beusedfor Td:

)'d = 0.20 for hemispheres

"Yd = 0.35 for ellipsoids and torispherical domes

(6.2-13)

(6.2-14)

Insufficient data are available to discern any dependence of the knock-down factor on

the ratio RMax/P. ttowever it is quite possible that even a large array of data would

lead to the same conclusion. This would be consistent with the practice usually accepted

for isotropic (non-san&rich) domes.

It is thought that there is physical justification for the hse of a 7d value for hemispheres

which is lower than that for ellipsoids and torispherical domes. This justification lies

in the fact that, for the latter two configurations, the maximum membrane stresses

occur at the apex which is well-removed from the boundary disturbances. On the other

hand, the membrane s_resses in a hemisphere are uniform over the entire surface.

Discontinuity distortions at the boundaries are ignored in classical small-deflection

stability theory but, in reality, these deformations can act somewhat like initial im-

perfections and precipitate buckling. This fact, coupled with the uniform membrane

6-10

stress in thehemisphere,canleadto earlier failure thanwouldbeencounteredfor

shapeswherethepeakmembranestressesdonotextendinto theboundaryregions.

Sincethe recommendedvaluesfor Tdarebasedonmeagertest results, themethod

proposedhereis notvery reliable andcanonlybe regardedas a "best-available"

technique. It shouldonlybeusedasa roughguidelineandfinal designsmustbe

substantiatedby test.

6-11


For sandwichdomesof thetypesshownin Figure6.1-1 andsubjectedto uniform

externalpressure, the critical apex stresses may be computed from the relationships

given in the equations on page (;-14 where the subscripts 1 and 2 refer to the separate

facings. There is no preference as to which facing is denoted by the subscript 1 or 2.

The equations on page 6-14 were obtained by a simple extension of the formulas pre-

sented in Section 6.2.1.1. The extension was accomplished in order to cover some

situations where the two facings arc not made of the same material. This was achieved

through the use of equivalent-thiekness concepts based on the ratios of the moduli for

the respective facings. For cases where the two facings are not made of the same

material, the resulting equations are wtlid only when the behavior is elastie (r_= 1).

Application to inelastic cases (7)/1) can only be made when both facings are made of

the same material. For such configxlrations, E_ and E_, will, of course, be equal.

The buckling coefficients K can be obtained from I"ig_rc (;.2-4.C

The knock-down factor Yd may be chosen as follows:

When V 2.0C

Use ?/d :: 0.20 for hemisl)heres ,

USe_d _ 0.35 forellips_dds andtorispherical domes.

When V _ 2.0e

Use )Jd = 1.0 for hemispheres,

ellipsoids, and torl-

spherical domes.

The quantity RMa x is the maximum principal radius of curvature for the dome and is

measured in units of inches. For all of the shapes shown in Figure 6.1-1, this value

occurs at the apex.

6-12

The formulations given here are based on the assumption that the transverse shear

stiffness of the core is isotropic. However, in most practical sandwich constructions,

this stiffness will vary with direction. In order to apply the given criteria to such

structures, one must select a single effective G c value. Whenever the shear crimping

modc is critical (V c : 2.0), G c must be taken equal to the minimum value for the core.

In all other cases one must rely on engineering judgment in making an appropriate

selection.

The plasticity reduction factor should always be based on the stress at the apex of the

dome. For elastic cases, use r?= 1. Whenever the behavior is inelastic, the methods

of Section 9 must be employed.

Facing 1

Apex a = (i {6.2-15)er 1 Yd Ke ! o i

o" = r?E 1 C (6 2-17)01 0 "

hC _

o RMax

2 _ (Eztl) (Estp)

Facing 2

Apexcr_ =: Yd Kcr_o_

< .:

cr :r/E C% o

(6.2-19)

(6.2-16)

(6.2-18)

crimp_

5 2

= G

Its+ E(_(.)t_] t c c

(7_ O 1

(7crimp_

(6.2-20)

(6.2-22) V =V

C 1 C_

h _

ta+t t

(7o

(7crimp_

GC

(6.2-21)

(6.2-23)

6-14

Thecritical pressurePer (in unitsof psi)maybecomputedasfollows:

2 t2] (6.2-24)Pcr - RMax ECrcrltl +acr2

In the specialcasewheret1= t2_tf andbothfacingsare madeofthe samematerial,

Equations(6.2-15)through(6.2-24)canbe simplifiedto thefollowing:

(6.2-25)= _d Kc(YApex _cr o

(7O

(fiEf) h

ffe_ RMax

(6.2-26)

Crcrimp

5 2- G (6.2-27)

2 tft c c

(7

o (6.2-28)V - ---

e (7c rimp

4 (6.2-29)Pcr - R ((_crtf)

Max

6.3 OTHER LOADING CONDITIONS

No information is available concerning the general instability of dome-shaped sandwich

shells under loading conditions other than that of uniform external pressure which is

covered in Section 6.2.

6-15

3m

o

d_

i

b-,

o o

c,i

'_@ AI

p d-'

c,_

Ph

0 "_ ,_b P'-

0 cq

;,.I, '-'

<: H ,¢

g

c_

" '_ "b

_'-goH

r

_J C,

r_

L_

-__:o

._ _._

g._

lI II

l /I,

H

b

o_D

o

o

U +

M

-,,11--

--0 _q

b

6-15

L;

2,

'-c

b

II

:I

h

rd''

p-

o

el

0

o

II

b _

ob

0

o

o

o

li

,-,o

d_

_._ _ ,_ It

o

o @

6-17

6-1

6-2

6-3

6-4

REFERENCES

Yao, J. C., "Bucklingof SandwichSphereUnderNormalPressure", Journal

of theAerospaceSciences,March1962.

Plantema,F. J., SandwichConstruction, John Wiley & Sons, Inc., New York,

Copyright 1966.

Donnell, L. H., "Stability of Thin-Walled Tubes Under Torsion", NACA

Technical Report No. 479, 1934.

Gonzalez, H. M. and Patton, R. J., "Development and Fabrication of 55-Inch

Diameter Scale Model S-II Common Bulkheads", North American Rockwell,

Corp. Report No. SDL 468, May 1964.

(_-19

7INSTA BILITY OF SANDWI CH S HE LL SE GME N TS

7.1 CYLINDRICAL CURVED PANELS

7.1.1 Axial Compression


It will be helpful here to first consider the case of axially compressed, isotropic

(non-sandwich) skin panels for which all four boundaries are simply supported. In

such cases, the Schapitz criterion _7-1] furnishes a practical means for the corn-

putation of critical stresses. This criterion accounts for the effects of skin-panel

geometry as the transition is made from wide panels, which behave essentially as

flfll cylinders, to narrow panels which approach the behavior of fiat plates. In par-

ticular, Schapitz proposed that one use the following relationships which have been

verified by the rederivation of Reference 7-2:

When

(7.1-1)

then

(l = Uer p

2

c_R

4_P

(7.1-2)

when

then

(YR

Ucr

>2_P

= crR

(7.1-3)

(7.1-4)

7-1

where,

(YP

(rfl

= Critical stress for buckling of a simply supported flat plate of the con-figuration shown in Figure 7.1-1, psi.

Critical stress (in units of psi) for buckling of a simply supported complete

cylinder of radius R, length aR, and thickness t R (see Figure 7.1-1). The

quantities R, aiR, and t R are all measured in units of inches. An empiricalknock-down factor should be incorporated here to account for the detri-

mental effects from initial imperfections.

(Y, psi / a, psi

ap

g, psi

R

R

a =a _aR p

b =b ::bR p

t =t tR p

tP

Figure 7.1-1. Cylindrical Panel and Associated Flat-Plate Configuration

7-2

b. For sandwichpanelswhichfall in themoderately-stiffor weak-corecate-gories, gcr shouldbetakenequalto thehigherof thetwovaluesut)and(_R"

In thecourseof preparingthis handbook,noanalysiswasmadeof test datafrom sand-

wich panels. Therefore,the reliability of this approachhasnotbeenestablished,and,

until ex_perimentalsubstantiationis obtained,onecanonlyregardthe methodasa "best-

available"teebxlique.

In view of the lackof sandwichdatacomparisons,it is informativeto notethata large

collectionof test results from isotropic (non-sandwich)specimensis evaluatedin

Reference7-3 andit is shownthere thatthe Sehapitzcriterion is a reliable approach

for suchpanels. Thetest configurationsembraceda widerangeof , -_ , and

ratios. Narrow, wide, and intermediate panels were included. The K values fell

between those for the ease where all four boundaries are simply supported and the case

where all four boundaries are fully clamped. The results are summarized in the qual-

itative presentation of Figure 7.1-3. This figure shows the general characteristics

and relative positioning for each of the following when displayed in a nondimensional

logarithmic format :

a. The theoretical buckling relationship for fiat plates.

b. The classical, small-deflection, theoretical buckling relationship for

complete cylinders.

c. A lower-bound buckling relationship for complete cylinders. This is

obtained by multiplying the values from b; above, by the empirical

knock-down factor of Reference 7-4.

d. The design curve based on the Sehapitz criterion.

7-5

\

\

\, * _ O"

er t

• .\\

% \

'\De sign Cum'e Based

on Schapitz Criterion

Figure 7.1-3. Schematic Logarithmic Plot of Test Data for Cylindrical Isotropic

(Non-Sanchvich) Skin Panels Under Axial Compression

7-6

Althoughderivedspecificallyfor theeaseof simplesupport,this criterion hasbeen

successfullyemployed[7-3_wheretheboundariesprovidevariousdegreesof rotational

restraint alongwith the conditionof no radial displacement. Thiswasaccomplishedby

simply adjustingtheval_,efor ap to correspondwith the appropriateedgerestraints.

For the case under immediate discussion (non-sandwich skin panels), the Schapitz

criterion can be graphically represented as shown in Figure 7.1-2. A series of design

curves of this type are given in Reference 7-3. The transition curve defined by Equation

(7.1-2) becomes tangent to the full-cylinder curve when gR = 2gp. For (R/t) values

greater than that of the tangency point, the skin panel behaves as a complete cylinder.

For all other (R/t) values, the transitional relationship applies. Note that the transition

curve asymptotically approaches the line for Crp. The quantity K denoted in Figure 7.1-2

!

Figure 7.1-2.

(b) = C°nstat_

:

Schematic Logarithmic Plot of Schapitz Criterion

for Non-Sandwich Cylindrical Skin Panels

7-3

is theconventionallist-plate bucldingcoefficientwhichis oh,pendent upon the aspect

ratio (a/b), 1)oundary c(mditions, and t33)e of loading. Fr,.m_ this fig-ere, it can 1)e

seen that, if the critical stress weco taken equal to the higl_er of the two values crP

and,7 R, one would only be neglecting the tnulsitional strenglh associated with the

cross-hatched region. When (_-_R / =: 1, neglect of this ccmtribution would restllt in a

\ P/

design value which is sl) pc, rccnt of the Schapitz prediction. For all ()/her values of the

ratio {_f{ /, the differences would be less sig_,ificanl. Indeed, for most ranges oi(;tl t\Crp/ \ P,

the conservatism introduced hy neglecting the cross-hatched area would be quite small.

Since lhe Schapitz criterion is: dependent solely on lhe values (71) and cr R, the speculation

is nmde here that on(, might extend its application to cylindrical sandwich panels merely

by computing crp and cJt{ from the sandwich design equations and curves which are pro-

vided in Sections :; al?d .I. tfowever, in making such an extension, one inust recognize

that the behavior of a sandwich panel is dependent upon the (:ore stiffness. For stiff-

core constructions (see So.ellen 4.2), it should be possible to makt. direct application

of Equations (7.1-1) lhr_ugh {7.1-.t). On the other hand, in the weak-core region, the

sandwich panel will lail by shear crimping, and curvature will not contribute to the

buckling strength, in such cases, Equations (7.1-1) through (7.1-4) would yield uneon-

servative predictions. The situation for sandwich constructions having moderately-

stiff cores would, of course, fall somewhere between the foregoing limiting eases.

Consequently it is recommended here that,

a. For stiff-core sandwich panels, Equations (7.1-1) through (7.1-4) can be

applied.

7-4

Also sh(_vn in Figure 7.1-3 are the approximate locations of the test data fr()m the

non-sandwich cylindrical panels of References 7-5 through 7-8. During the course

of the study reported in Reference 7-3, quantitative plots were made for each of these

specimens and the corresponding test points were accurately located on the appropriate

graph. Based on these many different plots, the test points were inserted in Figure

7.1-3 in approximation to their actual positions relative to the several basic curves

and regions of behavior. This figure shows that all but four of the test points which

fall below the design curve lie within the region where the panel beltaves essentially

as a flat plate. Except for those four points, all of the test data for the regions of

transitional and full-cylinder behavior fall between the following t_vo bounds:

a. The recommended design curve.

b. The values which would have been predicted if _R did not incorporate

an empirical knock-down factor.

It is concluded that Figure 7.1-3 verifies the reliability of the Schapitz criterion

for the case of isotropic (non-san&rich) skin panels, even where the boundary con-

ditions include some rotational restraint in addition to the requirement of no radial

displacement. This conclusion is based partly on the fact that the character of flat-

plate buckling is quite different from that exhibited by wide cylindrical panels and

complete cylinders. The fiat plate can continue to support steadily increasing in-

plane loading well into the postbuckling region. This is in contrast to the sudden

drop-off in load usually observed for wide panels and full cylinders. Consequently

the Schapitz criterion utilizes full theoretical predictions as the limiting case of a

7-7

flat plate is approached. One might, therefore, expecl that within this region test data

will display some small degFee of scatter on both sides of th(, design curve, ttowever,

bucause of tile physical behavior cited above, this generally will not lead to any seFious

st ructur,,tl de fieieneies.

7-8


For cylindrical sandwichpanelssubjectedto axial compression,thecritical stress

maybecomputedfrom the following:

Stiff-Core Constructions

When

then

when

then

(3" =cr p

U = Ucr R

(_R

+ 4--_-- ' and

P

(7. i-5)

(7.1-6)

(7.1-7)

(7.1-8)

(y

Weak-Core and Moderately-Stiff-

Core Constructions

The higher of the two]

cr = |values _ and gR ![ P J

(7.1-9)

whe re,

uP

(;R

= Critical axial compressive stress (in units of psi) for the buckling of a

flat sandwich plate which has the same boundary conditions as the cylindri-

cal panel and, except for curvature, is of the same geometry as the cylin-

drical panel (see Figure 7.1-1). No knock-down factor is required in com-

puting this value.

= Critical axial compressive stress (in units of psi) for the buckling of a

complete sandwich cylinder which, except for the circumferential dimen-

sion, is identical to the curved panel. An appropriate empirical knock-

down factor should be incorporated here to account for the detrimental

effects from initial imperfections.

As a rule-of-thumb, one may assume that stiff-core constructions are those which

satisfy the inequality

V <0.25C

where V c is computed as specified in Section 4.2.

7-9

(7. i-i0)

The quantity _p should be computed by using the design equations and curves given in

Section 3.

The quantity (_R should be computed by using the design equations and curves given in

Section 4.

A graphical representation of Equations (7.1-5) through (7.1-8) is provided in Figure

7.1-4.

The method given here applies only where all four boundaries are completely restrained

against radial displacement. Therefore, no free edges are permitted. Any or all of

the four boundaries may include rotational restraint of any degree ranging all the way

from a hinged condition to fully clamped.

4.0

Figure 7.1-4.

+ 4_-

-T.H}--p_

3.0 4.0

(ffR/(_p)

Graphical Representation of Equations (7.1-5) through (7.1-8)

7-10

7.1.2 OtherLoadingConditions

7.].2.1 BasicPrinciples

In thepreparationof this handbook,almostnoconsiderationwasgivento thebuckling

of cylindrical sandwichpanelssubjectedto loadingsotherthanaxial compression.

Therefore,nofirm recommendationscanbemadehere concerningdesignequations

andcurves, llowever, the suggestionis offeredthat, for suchcases,onemight con-

sider anextensionof the conceptspresentedin Section7.1.1. Inparticular, for all

regionsof core stiffness, it mightbepossibleto applytheequation

[The higherof thetwo]Crcr = [values Crp and (_R J

if one simply computes the values _p and o R for the loading condition of interest.

(7.1-11)

In conformance with the restrictions of Section 7.1.1, the foregoing suggestion applies

only when all four boundaries of the panel are completely restrained against radial dis-

placement. Therefore, no free edges are permitted. Any or all of the four boundaries

may include rotational restraint of any degree ranging all the way from a hinged con-

dition to fully clamped.


No rcconm]endations are made here.

7.2 OTHER PANEL CONFIGURATIONS

No information is available concerning the instability of sandwich shell segments of

shapes other than the cylindrical configurations considered in Section 7.1.

7-11

_L _

_ °

_o

c_

q_

_Q

m_

r!

b

7-13

7-1

7-2

7-3

7-4

7-5

7-6

7-7

7-8

REFERENCES

Schapitz, E., Festigkeitslehre f_r den Leichtbau, 2 Aufl., VDI-Verlag GmbtI

D_isseldorf, Copyright 1963.

Spier, E. E. and Smith, G. W., "The Sehapitz Criterion for the Elaslic Buckling

of Isotropie Simply Supported Cylindrical Skin Panels Subjected to Axial Com-

ln'ession", Contract NAS8-11181, General Dynamic Convair division Memo AS-

D-1029, 31 January 1967.

Smith, G. W., Spier, E. E., and Fossum, L. S., "The Stability of Eccentri-

cally Stiffened Circular Cylinders, Volume II - Buckling of Curved Isotropic

Skin Panels; Axial Compression", Contract NAS8-11181, General Dynamics

Convair division Report No. GDC-DDG-67-006, 20 June 1967.

Seide, P., Weingarten, V. I., and Morgan, E. J., "Final Report on the Devel-

opment of Design Criteria for Elastic Stability of Thin Shell Structures", STL-

TR-60-0000-19425, 31 December 1960.

Petcrson, J. P., Whitley, R. O., and Deaton, J. W., "Structural Behavior

and Compressive Strength of Circular Cylinders with Longitudinal Stiffening",

TN-D-1251, May 1962.

Cox, It. L. and Clenshaw, W. J., "Compression Tests on Curved Piates of

Thin Sheet Duralumin", British A.R.C. Technical Report R&M No. 1894,

November 1941.

Crate, It. and Levin, L. R., "Data on Buckling Strength of Curved Sheet in

Compression", NACA WR L-557, 1943.

Jackson, K. B. and Hall, A. H., "Curved Plates in Compression", National

Research Council of Canada Aeronautical Report AR-1, 1947.

7-15

8EFFECTS OF CUTOUTS ON TttE GENERAL INSTABILITY OF

SAND_\rICtt SttELLS

In many practical aerospace shell structures, it is required that cutouts be incorporated

for purposes of access, lightening, venting, etc. However, no theoretical solutions or

experimental data have been published [or the general instability of sandwich shells

having such penetrations. Even in the case of isotropic (non-sandwich) shell struc-

rares, this problem tins received little attention. Some theoretical solutions have been

accomplished concerning the stress distributions around cutouts in isotropic shells

but the authors of this handbook are aware of only one paper (8-1) dealing with the

general instability problem, and this paper is not sufficiently comprehensive to pro

vide a practical design criterion.

An obvious need exists for further theoretical and experimental work to be accomplished

in this area, and, in view of this situation, no related design recommendations can be

made at the present tirne.

8-1

REFERENCE

Snyder, R. E. , "An Experimental Investigation of the Effect of Circular Cutouts

on the Buckling Strength of Circular Cylindrical Shells Loaded in Axial Com-

pression," Thesis submitted to the Graduate Faculty of the Virginia Polyteclmic

Institute in candidacy for the degTee of blaster of Science in Engineering Mechan-

ics, May 1965.8-i

9INELASTIC BEHAVIOR OF SANDWICH PLATES AND SItE_LLS

9.1 SINGLE LOADING CONDITIONS


For structural members stressed beyond the proportional limit of the material, it is

customary to compute critical loads or stresses through the use of so-called plasticity

reduction factors. In this handbook, such factors are denoted by the symbol r7 • In

many cases, appropriate formulas for _7 are established by theoretical derivations

based on plasticity theory but, when this approach proves impractical, one must some-

times resort to empirical expressions. Section 9.1.2 gives the formulations for

which are recommended in this handbook for various sandwich configurations, types

of loading, and modes of instability. These equations are based on the information

provided in References 9-1 through 9-5 for isotropic (non-sandwich) plates and shells.

Application of these reduction factors involves the trial-and-error procedure outlined

a.

b.

c.

below:

First, assume 7) = 1 and compute the critical stress for the appropriate

configuration, loading condition, and mode of failure.

If the critical stress computed in a, above, is less than the proportional

limit of the facing material, no further computations are required, ttow-

ever, if the computed critical stress exceeds the proportional limit, one

must continue as specified below.

Assume a new value for the critical stress which is in excess of the pro-

portional limit but less than the value computed in a, above.

9-1

dp

_°

f.

Based on the stress level assumed in c, above, and the stress-strain curve

for the facing material, compute a value for the appropriate plasticity re-

duction factor. The formulas of Tables 9. i-1 through 9.1-3 can be used

1ol- this purpose.

Using lh(. r_ value computed in d above, recalculate the critical stress.

If the critical stress calculated in e, above, is in reasonable agreement

with the value assumed in e, no further computations are required. How-

ever, if such agreement is not aehieved, one must then repeat the eompu-

ration cycle starting with e. This iterative procedure must be continued

until acceptable agr¢_ement is attained between the assumed and the ealeu-

lated critical stlvsses.

A numerical example of the fc,regoing procedure is provided in Section 9.1.2.

9-2

9.1.2 DesignEquations

Recommendedformulasfor plasticity reductionfactors aregivenin Tables9.1-1

through9.1-3 where

Ef = CompressiveYoung'smodulusof facings,psi.

E = Compressivesecantmodulusof facings,psi.s

E = Compressivetangentmodulusof facings,psi.t

Gf = Elastic shearmodulusof facings,psi.

G = Secantshearmodulusof facings,psi.s

v = Elastic Poisson'sratio of facings,dimensionless.e

v - Actual Poisson's ratio of facings, dimensionless.

Values for v can be obtained by using

v = 0.50- (0_.

or

50 - re) (9.1-1)

= o. o -\ El/(9.1-2)

assumed to be of sufficient length to fall outside the short-cylinder range.

assumed that

al

b.

e.

both facings are of the same thickness,

both facings are made of the same material, and

It is further

the transverse shear properties of the core are isotropic so that

0= (Gxz/Gyz) = 1.

9-3

The technique for applying the plasticity reduction factors is demonstrated below by

means of a numerical example for an axially compressed sandwich cylinder which is

For such cylinders, Section 4.2.2 specifies that the critical stress for general in-

stabilit}, may be computed [rom

(Tcr = Yc Kc(Yo (9.1-3)

where

cro

(71E f) h

(9.1-4)

_/c is obtained from Figure 4.2-8. K e is obtained from Figure 4.2-7 where

Gro

V - (9.1-5)c (Ycrimp

and

5 2

_crimp - 2tft c Gxz (9.1-6)

For the purposes of the present sample problem, assume that

Ef = i0 ×106 psi

v = 0.30e

R = :_2.0"

h = .320"

tf = .020"

t = .300"C

G = 20,000 psiXZ

h

f) - 2 .160"

R 32.0"- - 200

p .160"

Facing Proportional Limit = 25,000 psi

9-4

By using these values and assuming that _ = 1, it is found that

7c = 0.49

cr = 104,9000

= 170,800crimp

V = 104,900/170,800 = .614C

K = 0.85C

The refore,

gcr 7cKcCro = 49 x.85x104,900 43,600

Note that the computed critical stress (43,600 psi) is higher than the proportional limit

(25,000 psi) of the facings. Hence the use of _ = 1 cannot be valid and one must now

proceed on a trial-and-error basis. That is, one must select an assumed critical

stress value which exceeds the proportional limit. For the purposes of this sample

problem, suppose that the value (_cr = 30,000 is selected. By using the stress-strain

curve for the facing material, the corresponding plasticity reduction factor can then

be computed from the following formula which is taken from Table 9.1-3:

1

1--=71 Ef(9.1-7)

Suppose that this gives the result that

= 0. 900

so that one now obtains

7c = 0.49 (remains unchanged)

= .900 ×104,900 = 94,400O

9-5

g = 170,800 (remains unchanged)erimt)

V = 9,t, !t00/170,800 = .553C

K = 0.8Ge

There fore,

(rcr =YcKc cro = .49 ×.86 ×94,400 = :_9,800

Note that the eon_puted critical stress (39,800 psi) does not agree very closely with

the assumed value 1:_0,000 psi). Therefore, another iteration will be performed by

selecting a new assumed critical stress, say 35,000 psi. Suppose that by using

Equation (9.1-7) the corresponding plasticity reduction factor is found to be

r? = 0. 790

so that one now obtains

:/c = 0.,i9 (remains unchanged)

(Yo = "790 ×104,900 = 82,900

cr = 170,800 (remains unchanged)crimp

V = 82,900/170,800 = .486C

K = 0.87C

"rile re fore,

cr = ,/cKcffo = .49 ×.87 ×82,900 = 35,400

Note that the computed critical stress (35,400 psi) is now in reasonable agreement

with the assumed value (35,000 psi). Therefore, no further iterations are required

and the design value for the critical stress is 35,000 psi.

9-6

0

.,..4

0

oU9

0

0

0

0

0

r_

'tJ

0

00

0

I

_b

0

_o

0

,-_0

=o._"_ _ 0

_2J

r_

_ o

O_..-i

¢) _ 0

0

•r_3

c_

o _r..) _

L,-_ _ j

M

_--

[ i

+

,--_I cq+

,-_ I c"qI |

r_

I I

I J

II

r 1

_'_ I_ _

+

+

I

I lli

b_

0

0

t:_

0

o

0

0

0

I.-..4

_L

0

0

r--¢

b_.a

Nr..)

r_

[-,©2;

9-7

*3

@

©

r_

0

,....,

I

c_E_

r

_I_

L J

© pI

Ir

c'o I '._-t._

_1'_

+

_1_1L J

_1_f ]

iI

1

L""* J

][

q3

IJ

l i[

ir

9-8

Table 9-3.

I

Recommended plasticity Reduction Factors for the General Instability

of Circular Sandwich Cylinders, Truncated Circular Sandwich Cones,

and Axisymmctric Sandwich Domes

PlasiicJly Heducliol_ i.'zLclol's

Mo(h:ralc l,englh

Short (:ylinders Through l,ong ('ylinders

_tnd Cones ail(I (:(rues

............... I -u(:' _1 - Lit_ E t

] ,:: L,-:V ] , : L,_-77] Ef

External Laieral I)rcssure _ rl- ll_--_J Ef _4 + ,I Es)

T ors ion

Transverse Shear

External Pressure

ile, mispherical, Ellipsoidal, and

Torispherical Domes (All trun-

cated at the equator)

*This formula for r l is not valid when the cylinder or cone is so short that it

behaves essentially as a long, flat plate. However, it is unlikely that such

configuralions will be encountered in aerospace applications. Furthermore,

it is informative to note that, for such constructions, the given formula for

rl is conservative.

**This formula for rl is not valid when the cylinder or cone is so short that it

behaves essentially as a long, flat plale, ttowever, it is unlikely that such

configurations will he encountered in aerospace applications. Furthermore,

it is informative to note that, for such constructions, the given formul.a is

approximately 13-1)ercent unconscrvative.

9-9

9.2 COMBINED LOADING CONDITIONS


As noted in llefe_'cnce 9-(;, only limited ilfformation is available on the inelastic sta-

l)ility of shell structures subjected to comt)ined loading conditions. A similar situation

exists for flat-plate constructions. Very little theoretical work has been done in these

fields due to the complexity of the problem and, in general, related phtsticity reduc-

tion criteria have not been esklblished, t[owever, in many practical engineering

applications, one is co_ffronted with this type of problem and it becomes necessary to

determine at least a rough estimate of the critical loading conditions. Toward this

end, one should note a fundamental hypothesis of plasticity theory which specifies that,

for a given material and when the stress intensity is increasing (loading condition), the

stress intensity cri is a tmiquely defined, single-valued ftmction of the strain intensity,

e i. When cri is decreasing (unloading condition), the relationship between cr i and e i

is linear as in a purely elastic case. Based on the oetahedral shear law for plane

stress conditions, the stress and strain intensities gi and c i can be defined as follows

[ 9-]] :

,. _<, a:-, _ +3r 2 (9.2-1)1 3' xy

ei ,/,_ ¢2 _ ¢2 + ¢ ¢ +'¢:_ /4 (9.2-2)x y x y xy

It sJ_o'_ld be noted that Eq'_tation (9.2-1) is sometimes written in ti,c following form to

facilitate its use:

[

cri (Crx)y I - 7 + )Z + 3),::

9-10

(9.2-1a)

where

cr - Normal stress in the x direction, psi.x

Normal stress in the y dh'ection, psi.Y

_- : Shear stress in the xy plane, psi.

- Normal strain in the x direction, in/in.

= Normal strain in the y direction, in/in.

= Shear strain in the xy plane, in/in.

y x

x

x

£Y

Exy

From the foregoing discussion it can be concluded that, for the case of increasing

cy (loading condition), the relationship between (yi and e i is identical to the conven-i

tional stress-strain curve obtained from a uniaxial loading test. It should therefore

be evident that although each individual stress component may be less than the propor-

tional limit of the material, the combination of these stresses can give a ffi value which

lies above the proportional limit so that the behavior is actually inelastic. It is im-

portant to keep this phenomenon in mind when deciding whether or not plasticity effects

must be considered.

Lacking a rigorous approach to the subject stability problem, it is conjectured here

that the foregoing generalization of the stress-strain relationship might be used in

conjunction with the plasticity reduction factor

E t

9-11

(9.2-3)

to obtain conservative predictions of inelastic instabilit%_ under combined loadings. The

quantities E t and Ef are as h)llows:

E t Tangent modulus of facing material obtained from the curve of

(_i vs e i at a prescribed value of gi' psi.

Ef Young's modulus of facing material, psi.

The above formula for _ was selected in view of its conservative nature. Since the

overall procedure suggested here is based purely on an engineering estimation, it is

thought that tttis conservatism is well justified.

The details of the suggested approach are outlined in Section 9.2.2. It is important

to keep in mind that this method does not give a rigorous solution, and its reliability

has not been evaluated by comparisons against test data. Therefore, this can only be

regarded as a "best-available" technique and one should be cautious in its application.

9-12

9.2.2 Suggested Method

The method suggested here for analysis of the inelastic stability of sandwich plates

and shells first requires that the conventional stress-strain curve for the facing ma-

terial have the stress coordinates relabeled as cYi and the strain coordinates relabeled

as e. By completely ignoring all plasticity considerations (_ = 1), one should then1

proceed to establish a first-estimate for the critical combined stress condition. This

can be achieved by using the appropriate interaction relationships provided in earlier

sections of tiffs handbook. In performing tiffs computation, the assumption should be

made that for the critical combined stress condition the individual stress components

are in the same ratios to each other as exist for the actual applied loading condition.

That is, during loading, proportionality between the several individual stress compo-

nents is maintained. The stresses from the elastic first-estimate computation must

then be inserted into the equation

cYi = 4a; + (y2 _ cy a + 3-r 2 (9.2-4)y xy

to determine the associated stress intensity value. If this value does not exceed the

proportio_ml limit of the cYi versus e i curve, the first-estimate stress values are in

fact the critical combination. However, if the related a i value exceeds the propor-

tional limit of the cYi versus e. curve, the first-estimate results are not valid and one

must then resort to the following trial-and-error procedure which is similar to that

outlined in Section 9.1:

a. Assume a new value for cri which is in excess of the proportional limit

for the (Yi versus e i curve.

9-13

b °

co

do

For the cri value assumed in a, above, compute the plasticity reduction

factor

(9.2-5)

who re

E t Tangent modulus of the cri versus e i curve, psi.

Ef Elastic modulus of the gi versus e i curve, psi.

Using the 7?, value from b, above, recalculate the critical stress intensity

_.1 This is accomplished by simply multiplying the first-estimate a i valueby _.

If the new value for cri computed in c, above, is in reasonable agreement

with the c_i value assumed in a, above, the related plasticity reduction

factor rl is valid. Then the critical combination of stresses is obtained by

multiplying each of the first-estimate stress components by this rl value.

If the vMue of cyi computed in c, above, is not in reasonable agreement

with the cri value assumed in a, the related plasticity reduction factor is

not valid. One must then repeat the computation cycle starting with a.

This itcratlve procedure must be continued until acceptable agreement is

attained between the assumed and the computed or. values.1

9 -14

REFERENCES

9-1 Gerard, G., "Plastic Stability Theory of Thin Shells," Journal of the Aeronautical

Sciences, April 1957.

9-2 Gerard, G. and Becker, tt., "Handbook of Structural Stability, Part I - Buckling

of Flat Plates," NACA Technical Note 3781, July 1957.

9-3 Gerard, G. and Becket, H., "Handbook of Structural Stability, Part III - Buckl-

ing of Curved PIates and Shells," NACA Technical Note 3783, August 1957.

9-4 Gerard, G., "Compressive and Torsional Buelding of Thin-Wall Cylinders in

Yield Region," NACA Technical Note 3726, August 1956.

9-5 Gerard, G., Introduction to Strueturai Stability Theory, McGraw-Hill Book

Company, Inc., 1962.

9-6 Baker, E. H., Cappelli, A. P., Kovalevsky, L., Rish, F. L., and Verette,

R. M., "Shell Analysis Manual," Prepared by North Ameriean Roe_vell, Corp.

for the National Aeronautics and Space Administration, Manned Spaceeraft

Center, Itouston, Texas, June 1966.

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NASA-CR-1457_MANUAL FOR STRUCTURAL STABILITY ANALYSIS OF SANDWICH PLATES AND SHELLS

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