UNITED STATES DEPARTMENT OF AGRICULTURE • FOREST SERVICE • FOREST PRODUCTS LABORATORY • MADISON 5, WIS 

DESIGN CURVES FOR THE BUCKLING 

OF SANDWICH CYLINDERS OF FINITE 

LENGTH UNDER UNIFORM EXTERNAL 

LATERAL PRESSURE 

May 1963 

This Report is One of a Series 

Issued in Cooperation with the 

FOR FLIGHT VEHICLES 

of the Departments of the 

ANC-23 PANEL ON COMPOSITE CONSTRUCTION 

AlR FORCE, NAVY, AND COMMERCE 
DESIGN CURVES FOR THE BUCKLING OF SANDWICH CYLINDERS OF 

FINITE LENGTH UNDER UNIFORM EXTERNAL LATERAL PRESSURE 1 , 2 

BY 

CHARLES B. NORRIS, Engineer 

JOHN J. ZAHN, Engineer 

3 Forest Products Laboratory, Forest Service 

U. S. Department of Agriculture 

Abstract 

This report contains curves and formulas for the calculation of the critical 

external pressure of finite length, circular cylindrical shells with sandwich 

walls. 

vidual stiffness is not taken into account. 

or orthotropic having natural axes in the axial, tangential, and radial directions of the cylinder. 

having isotropic cores and orthotropic cores having certain relative elastic 

properties. If the cores are very rigid, the method yields results that are 

substantially those of von Mises. 

The facings of the sandwich are equal and isotropic and their indiThe sandwich core is isotropic 

Curves are given for cylinders with sandwich walls 

1 

This Note is a revision of a report by the same title, issued May 1959 as 

Forest Products Laboratory Report No. 1869. This revision was made 

by E. W. Kuenzi and B. Bohannan. 

distributed b 

of Naval We pons Order No. 19-61-8041 WEPS and U. S. Air Force 

Contract No. DO 33(616)61-06. Results here reported are preliminary 

and may be revised as additional data become available. 

Wisconsin. 

2 This progress report is one of a series (ANC-23, Item 57-3) prepared and 

the Forest Products Laboratory under U. S. Navy, Bureau 

3 Maintained at Madison, Wis., in cooperation with the University of 

FPL-07 
Introduction 

This report presents design curves for the critical external radial pressure 

of circular cylindrical shells with sandwich walls, calculated according to 4 the formulas developed in Forest Products Laboratory Report 1844 -B ( 9 ) 

The sandwich cylinder walls have isotropic facings and orthotropic or isotropic cores. The natural axes of the orthotropic cores are axial, tangential, and radial. 

von Mises ( 7 , 10 , 14 ) when the core is very rigid. 

These formulas reduce substantially to those developed by 

A great deal of investigative work has been done on isotropic cylindrical 

shells subjected to external pressure since report 1844-B giving theoretical 

analysis of sandwich cylinders was written. It has been found that experiment sometimes yields critical loads that are less than those predicted by 

von Mises' theory ( 13 ). 

experimental cylinders contained imperfections that lowered the critical 

load ( 1 , 2 , 3 , 8 , 12 ). Second, lower energy levels are associated with postbuckling configurations of the cylinder than with those just at buckling, the 

former being reached without the necessity of passing through (snap-through) 

the latter or the energy necessary for passing through the latter being supplied 

by vibration or shocks ( 4 , 5 , 6 , 8 ). 

This has been attributed to two causes. First, the 

The curves published in this report do not consider snap-through buckling or 

cylinders with imperfections. Sandwich cylinders, however, are much more 

perfect than their solid counterparts because they are thicker and the effect 

of an imperfection is in proportion to the ratio of its amplitude to the thickness of the cylindrical shell. 

individual facings. 

cylinders are short, and it is for short cylinders that "snap-through" is 

likely to occur ( 6 ). 

Also, the curves neglect the stiffnesses of the 

These stiffnesses add to the critical loads when the 

Development of Formula from Which 

Design Curves Were Calculated 

The critical external pressure is found by solving equation (51) on page 23 of 

Report No. 1844-B ( 9 ) for the buckling coefficient. 

equation (51) can be simplified if the transverse modulus of elasticity of the 

core (E c ) is assumed to be infinite. For most core materials except possibly for low density foams E c is sufficiently large so that this assumption 

yields values of the critical pressure that are only very slightly too great. 

4 Underlined numbers in parentheses refer to the references. 

The determinant of 

FPL-07 -2- 
Before E c is allowed to approach infinity, the first and fourth columns of the 

and the third column is divided by E c determinant are multiplied by . 

Then when E c approaches infinity, the expressions in rows 3, 4, 5, and 6 in 

column 3 approach zero. 

The expressions in each row, excepting the first, are replaced by new expressions, as indicated by the following formulas in which R represents the 

expression in the row designated by its subscript and in some column. The 

primed values are the new ones to be substituted for the old. 

These substitutions cause 

and 6 and those in column 

the 6-by-6 determinant is 

ant. This determinant is 

the expressions in column 3 and rows 2, 3, 4, 5, 

6 and rows 3, 4, 5, and 6 to become zero, and 

readily reduced (by minors) to a 4-by-4 determinsimplified slightly by replacing the second row of 

expressions by the second row minus the first row. 

After the determinant was written in the above form, a change in parameters 

was made using the following nomenclature: 

d -- thickness of the sandwich 

E -- modulus of elasticity of facing 

G Rz -- modulus of rigidity of core in the radial and axial directions 

G Rθ -- modulus of rigidity of core in the radial and tangential directions 

h -- distance between facing centroids 

qr (1 - μ 2 ) 

2Et k -- 

FPL-07 -3- 
L -- length of cylinder 

n -- number of half waves in the circumference of the cylinder 

q -- the external critical pressure on the cylinder 

r -- mean radius of the sandwich cylinder 

G RZ R 

G Rθ 

t -- thickness of the facings 

Et 

4r (1 - μ 2 ) G Rθ

v -- 

EL -- Poisson' s ratio 

The radii a and b were eliminated by the following equations obtained from 

the geomerry of the cylinder. 

and the following substitutions were made: 

The resulting expressions in the 4-by-4 determinant are: 

Row 1, column 1 

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Row 2, column 1 

Row 3, column 1 

Row 4, column 1 

Row 1, column 2 

Row 2, column 2 

Row 3, column 2 

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Row 4, column 2 

Row 1, column 3 

Row 2, column 3 

Row 3, column 3 

Row 4, column 3 

Row 1, column 4 

FPL-07 -6- 
Row 2, column 4 

Row 3, column 4 

Row 4, column 4 

By setting this determinant equal to zero and solving for k , the critical 

pressure can be found by: 

This represents a theoretical solution for the critical pressure with the 

assumption that the sandwich core modulus of elasticity is infinite. 

The upper and lower limits for k can be determined as given in Forest 

Products Laboratory Reports 1844-A and 1844-B. These limits are for 

cylinders with membrane facings and a core with a modulus of elasticity 

that is infinite. The lower limit is approached as the length of cylinders 

becomes infinite and is given by equation (72) of Report 1844-A as 

FPL-07 -7 - 
The upper limit for k reached as n becomes infinite; it is 

which corresponds to the critical pressure as given on page 28 of Report 

1844-B. 

From this critical pressure the critical hoop compression per unit length 

of cylinder is: 

which is the usual limit imposed on the edge compression of sandwich constructions with membrane facings by the shear instability of the core. 

The above determinant can be further simplified by a few approximations 

without any significant loss of accuracy. Examination of the parameters 

and shows that is large in comparison to 

can be assumed as equal to with only a minor loss of accuracy. 

Making this assumption and substituting β the determinant reduces 

to the following expressions: 

Therefore, any terms of 

Row 1, column 1 

Row 2, column 1 

Row 3, column 1 

FPL-07 -8- 
Row 4, column 1 

Row 1, column 2 

Row 2, column 2 

Row 3, column 2 

Row 4, column 2 

Row 1, column 3 

Row 2, column 3 

Row 3, column 3 

FPL-07 -9- 
Row 4, column 3 

Row 1, column 4 

Row 2, column 4 

Row 3, column 4 

Row 4, column 4 

2 To obtain this value of k , terms containing k2 and k α were neglected. It was 

also assumed that terms ( 1 ± mα )= 1 where m is a small whole number. 

Further simplifications can be made by neglecting the terms of + 

and so that the final approximate solution for k is: 

FPL-07 -10- 

By setting this determinant equal to zero, the approximate solution for k is: 
The limits of this approximate solution are the same as for the exact solution 

if the terms are assumed equal to 

Description of Design Curves 

Using the approximate equation, values for k were determined for various 

L 

r values of V , n , , and α2 . These values of k are plotted (figs. 1 - 5) for 

values of 

to 0, 0.5,-1.0, 2.0, and 4.0. For each of these values of V , curves are 

given for seven values of α2 ranging from 10 -6 to 10 -4 . These curves apply 

to sandwich cylinders with equal isotropic facings and isotropic cores or 

orthotropic cores with their natural axes parallel to the axial, tangential, and 

radial directions of the cylinders. 

ranging from 1 to 100. Curves are given for values of V equal r 

The critical pressure is given by 

From a solution of the exact determinant, it was found that the modulus of 

rigidity of the core associated with the radial and axial directions G Rz has 

very little influence on the critical pressure. It does not enter the formulas 

for the limits of critical pressure. The ratio R = 

G Rz does affect the critical G Rθ

pressure for small values but for reasonable values of R the critical pressure 

is affected only slightly. 

solution for k . 

r 

The ratio, R , does not appear in the approximate 

The accuracy of the approximate solution was checked by making a comparison 

of the approximate and exact solutions for values of V equal to 0, 0.5, and 1.0, 

r t c equal to 50, and equal to 1. This comparison is shown on the top curve of 

FPL-07 -11- 

L 

L 

d d 
figures 1, 2, and 3 where the small circles represent the exact solution. This 

shows that, except for small values, the error encountered with the approximate solution is less than 1 percent. the error 

L 

r 

r For larger values of V and 

L 

r 

is less than is shown here. For the small values of , it is doubtful that 

either solution gives the correct results. 

FPL-07 -12- 1.0-21 

d 
References 

(1) Bodner, S. R., and Berks, W. 

1952. The Effect of Imperfections on the Stress in a Circular Cylindrical 

Shell under Hydrostatic Pressure. 

illus. Polytechnic Institute of Brooklyn. 

PIBAL Report No. 210, 

(2) Donnell, L. H. 

1956. Effect of Imperfections on Buckling of Thin Cylinders under External Pressure. Jour. Appl. Mech., 23: (No. 4). 

(3) Galletly, G. D., and Bart, R. 

1956. Effects of Boundary Conditions and Initial Out-of-Roundness on 

the Strength of Thin-Walled Cylinders Subject to External Hydrostatic Pressure. Jour. Appl. Mech., 23: (No. 3). 

(4) Kempner, Joseph, Pandolai, K. A. V., Patel, S. A., and Crouzet-Pascal, J. 

1957. Post-Buckling Behavior of Circular Cylindrical Shells Under Hydrostatic Pressure. Jour. Aero. Sci., 24: (No. 4). 

(5) Kirstein, A. F., and Wenk, E. 

1956. Observations of Snap-Through Action in Thin Cylindrical Shells 

under External Pressure. David Taylor Model Basin Report 

1062. U. S. Navy Dept. 

(6) Langhaar, H. L., and Boresi, A. P. 

1955. Snap-Through and Post-Buckling Behavior of Cylindrical Shells 

under the Action of External Pressure. T. & A.M. Report No. 

80. Department of Theoretical and Applied Mechanics, Univer - 

sity of Illinois. 

(7) Mises, R. von. 

1929. The Critical External Pressure of Cylindrical Tubes under Uniform 

Radial and Axial Load. Stodola-Festschrift, Zurich. 

(8) Nash, William A. 

1955. Effect of Large Deflections and Initial Imperfections on the Buckling of Cylindrical Shells Subject to Hydrostatic Pressure. 

Aero. Sci., 22: (No. 4). 

Jour. 

(9) Raville, Milton E. 

1954. Analysis of Long Cylinders of Sandwich Construction under Uniform 

External Lateral Pressure. U.S. Forest Products Laboratory 

Reports No. 1844, 23 pp., illus. ; and supplementary Reports NO. 

1844-A (1955) and 1844-B (1955). 

FPL-07 -13- 
(10) Timoshenko, S. 

1936. Theory of Elastic Stability. McGraw-Hill, New York. 

(11) U. S. Forest Products Laboratory 

1955. Sandwich Construction for Aircraft. Part II, Second Edition 1955. 

ANC-23 Bulletin, 115 pp., illus. Superintendent of Documents, 

U. S. Government Printing Office, Washington 25, D. C. 

(12) Volmir, A. S. 

1957. On the Influence of Initial Imperfections on the Stability of 

Cylindrical Shells under External Pressure. 

Nauk SSSR (N. S.) 113: (2)291-293 (translated by M. D. 

Friedman, Inc. , Needham, Mass. , V-124, 5 pp.) 

Dokladi Akad. 

(13) Wenk, E., Slankard, R. C. , and Nash, W. A. 

1954. Experimental Analysis of the Buckling of Cylindrical Shells Subjected to External Hydrostatic Pressure. 

Stress Anal., XII (No. 1). 

Proc. Soc. Exp. 

(14) Windenburg, D. F. , and Trilling, C. 

Pressure. 

56 (No. 11). 

1934. Collapse by Instability of Thin Cylindrical Shells under External 

Trans. American Society of Mechanical Engineers, 

FPL-07 -14- 
Figure 1.--Values of k for V = 0. 

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Figure 2. --Values of k for V = 0 5. 

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Figure 3.--Values of k for V = 1.0. 

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FPL-07 

Figure 4. --Values of k for V = 2.0. 
Figure 5. --Values of k for V = 4.0. 

FPL-07 
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