U.S. DEPARTMENT OF AGRICULTURE • FOREST SERVICE • FOREST PRODUCTS LABORATORY • MADISON, WIS. 

In Cooperation with the University of Wisconsin 

U. S. FOREST SERVICE 

RESEARCH NOTE 

FPL-0173 

OCTOBER 1967 

ELASTIC STABILITY OF CYLINDRICAL 

SANDWICH SHELLS UNDER AXIAL 

AND LATERAL LOAD 
Summary 

Presents a linear solution for determining the effect of combined 

axial and lateral loads under which a cylindrical sandwich shell 

will buckle. The facings of the sandwich cylinder are treated as 

homogeneous isotropic cylindrical shells and the core as an orthotropic elastic body. The characteristic determinant that represents 

the solution to the problem is solved numerically. Curves are given 

that show how the buckling load changes as the parameters of the 

problem change. 
ELASTIC STABILITY OF CYLINDRICAL SANDWICH SHELLS 

UNDER AXIAL AND LATERAL LOAD1 

By 

A. CARL MAKI, Engineer 

2 Forest Products Laboratory, Forest Service 

U.S. Department of Agriculture 

Introduction 

Sandwich construction is a result of the search for a strong, stiff, and yet light weight 

material. It is usually made by gluing relatively thin sheets of a strong material to the 

faces of relatively thick but light weight, and often weak, material. The outer sheets are 

called facings and the inner layer is called the core. 

Such a layered system presents difficult design problems. What is offered here is a 

straightforward method for dealing with some of these problems. 

The problem to which the method is applied is that of the elastic stability of a sandwich 

cylinder under uniform external lateral load and uniform axial load. 

Notation 

r, θ, z radial, tangential, and longitudinal coordinates, respectively 

a radius to middle surface of outer facing 

b radius to middle surface of inner facing 

r mean radius 

m 

t thickness of each facing 

1

This Research Note is a revision, under the same title, of Forest Products Laboratory 

Report 1852, issued in 1955. It was originally prepared by Everett E. Haft, and 

issued as one of a series by the Forest Products Laboratory in cooperation with the 

U.S. Navy, Bureau of Aeronautics. 

2

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

FPL-0173 
L length of cylinder 

E modulus of elasticity of facings 

μ Poisson’s ratio of facings 

G modulus of rigidity of facings 

Ec 

modulus of elasticity of core in direction normal to facings 

Grθ 

modulus of rigidity of core in rθ plane 

G modulus of rigidity of core in rz plane rz 

q intensity of uniform external lateral loading 

k 

σr 

normal stress in core in radial direction 

τ rθ, τ rz 

transverse shear stresses in core 

u, v, w radial, tangential, and longitudinal displacements, respectively 

n number of waves in circumference of buckled cylinder 

m number of half waves in length of buckled cylinder 

V rθ

V 

rz 

λ mγ

γ

δnθ

FPL-0173 -2-
-- 

δz 

N N , N normal forces and shear force per unit length of facing θ, z θz 

Qθ , Q z 

M , M θ z 

M , M 

zθ θz 

X, Y, Z 

A 

R 

P 

transverse shear forces per unit length of facing 

bending moments per unit length of facing 

twisting moments per unit length of facing 

surface forces per unit area of facing 

the value of the lateral load parameter φ

1 

when the axial load parameter 

φ

2 is equivalent to zero 

the value of the axial load parameter φ

2 when the lateral load parameter φ

1 is equivalent to zero 

natural logarithm 

total axial load 

A, B, C, D, K, E, A' , B' A", B" arbitrary constants 

note any of the above term that appear with a prime (as N ' ) refer to the inner facing. z 

FPL-0173 -3-

β

φ1

φ2

φ1

φ2

1n

 

 

h
Z M 105 662 

Figure I.--Sandrich cylinder. 

M 133 627 

Figure 2.--Differential elements of core and facings before buckling. 

FPL-0173 -4-
Mathematical Analysis 

As previously stated, the core is relatively weak. Because of the high strength of the 

facings the core need carry little tension or compression except in a direction perpendicular to the facings. The facings are able to resist shearing deformation in their plane 

and it is necessary only that the core be able to resist shear in the radial direction in 

planes perpendicular to the facings. In this analysis, the core is considered to be an 

orthotropic elastic body. It is unable to resist deformations other than those just mentioned. 

Interdependence of the core and the facings is gained by equating their displacements 

at the interfaces. To simplify the analysis, the core is assumed to extend to the middle 

surface of each facing. 

Figure 1 shows the cylinder and the coordinates that are used. 

Prebuckling Stresses 

Before buckling occurs, the cylinder is in a state of uniform compression. The axial 

load is carried. by the facings since the core material is assumed to be incapable of 

carrying load in this direction. With facings of like material, the stress is the same in 

both facings. Since the facings have the same thickness, then the loading per unit length 

of facing, N or N z ' , will be the same, This means that for a total load P 

z 

The calculation of stresses due to the lateral pressure is a problem in rotational 

symmetry. Differential elements of the core and of the facings are shown in figure 2. 

Summing forces in the radial direction gives for the core 

for the outer facing 

and for the inner facing 

FPL-0173 -5-
, these equations can be solved for s r , N θ, and N θ

' . 

As P and q increase, N , N ', N , N ' and σ also increase. Eventually a condition my z z θ θ r 

be reached where a slight increase in load caues the cylinder to lose its state of uniform 

compression and buckle as a result of elastic instability. This buckling is assumed to 

cause only a small change in the stress distribution. These small changes will now be 

considered. 

Buckling Stresses 

The core.--A free body diagram of an element of the core is shown in figure 3. 

Neglecting term which are products of more than three difterentials, a summation 

of forces in the r, θ and z direction gives 

(1) 

(2) 

(3) 

Equation (2) my be integrated to give 

(4) 

3 RaviIle, M. E. Analysis of long cylinders of sandwich construction under uniform 

external lateral pressure, Forest Products Laboratory Report 1844, 1954. 

FPL-0173 -6-
Figure 3.--DifferentiaI element of deformed core. Z M 105 664 

Equation (3) may be integrated to give 

(5) 

(6) 

(7) 

(8) 

It is convenient to assume the displacements u, v, and w in the form 

(9) 

(10) 

(11) 

This form will permit a unique determination of f1 (r), f2 (r), and f3 (r); assumes n circumferential waves and m longitudinal half waves upon buckling; results in zero displacements in the radial and circumferential directions at the ends; and imposes no 

moment upon the facings at the ends, 

FPL-0173 -7-
From a consideration of equations (4), (5), (7), (8), (9), (10), and (11) it can be shown that 

(12) 

(13) 

Substituting equation (9) into (6) and then equations (6), (12), and (13) into equation (1) gives 

(14) 

which upon integration shows that 

(15) 

Equations (9), (10), and (12) are substituted into equation (7) to give 

(16) 

from which 

(17) 

Equations (9), (11), and (13) are substituted into equation (8) to give 

(18) 

from which 

(19) 

It is convenient to have the constants of f1(r), f2(r), and f3 (r) in nondinnaensional form. 

Redefining the constants , the following form is obtained. 

(20) 

FPL-0173 -8-
-9-
(21) 

(22) 

The facings.--Since the problem of stability of homogeneous cylindrical shells has 

been solved by other authors, it is only necessary in the stability analysis of cylindrical 

sandwich shells to consider it as a composite of two homogeneous shells bonded together 

by an elastic core subject to the compatibility requirements of equal displacements at 

their interfaces, which, to simplify the analysis, is assumed to extend to the middle 

surface of each facing. 

Free body diagrams of a facing element showing the sense of the forces and moments 

are found in figures 4 and 5. 

Z M 106 302 

Figure 5.--Differentlal element of facing showing moments and twists. 

The equations of equilibrium for the facings can be obtained from a mathematical 

theory of thin shells. It is felt that for a small deflection analysis the theories presented 

by various authors differ principally in second-order effects, as is evidenced by comparing the work of Flugge4 and Timoshenko5 on the buckling of cylindrical shells. 

4

FIugge, W. Stresses in Shells, Springer-Verlag, Berlin, 1962. 

5

Timoshenko, S. Theory of Elastic Stability, McGraw-Hill, 1936. 

FPL-0173 -10-
As a result, the equations of equilibrium as presented by Timoshenko and subject to 

the necessary transformations will be used here. Such theory, when applied to cylindrical 

shells of radius r; yields the following differential equations: 

(23) 

(24) 

(25) 

(26) 

(27) 

(28) 

FPL-0173 -11-
Assuming that resultant forces other than Nθ and Nz are small and neglecting products 

of these forces and the displacements u, v, and w which are also small, and assuming 

further a membrane analysis where the bending moments and transverse shear forces 

in the individual facings are neglected, the equilibrium equations can be 

reduced to: 

(29) 

(30) 

(31) 

Evaluating equations (29), (30), and (31) for each facing or when r = a and r = b and 

satisfying the compatibility requirements at these interfaces provide the six equations 

necessary for the determination of the six arbitrary constants found in expressions 20, 

21, and 22. 

Mathematical Analysis 

N 

z 

and N θ of equations (29), and (31) are 

respectively, for the outer facing and by 

replaced by 

pectively, for the inner facing (primes denote inner facing). This is necessary because 

the forces in the buckled shell are the prebuckling forces plus the forces due to buckling. 

The N z

, N z Nq, and N θ ' are small forces due to buckling which are later to be expressed 

in terms of displacements, (which are assumed to have differed negligibly from the 

prebuckled state), and the product of these small forces and the displacements are also 

neglected. Expressions (29), (30), and (31), as evaluated for the outer facing then become: 

(32) 

(33) 

(34) 

FPL-0173 -12-
As is customary in such stability problems, the stretching of the middle surface is 

taken into account by assuming that the prebuckling stress resultants are very large in 

comparison with the other stress resultants; thus the terms 

and 

should be substituted for the quantities and qa(1 - k), respectively, and the surface 

forces X, Y, and Z should be multiplied by (1 + εθ)(l + εz) in equations (32) to (34). In 

these expressions 

(35) 

Finally, the expressions for the forces, moments, and twists along with the relationships for the surface faces: 

(36) 

are expressed in terms of u, v, and w and substituted in (32), (33), and (34). They lead to 

the following equations for the outer facing: 

(37) 

(38) 

(39) 

where φ and φ are as defined under notations. 1 2 

FPL-0173 -13-
To achieve proper interaction between the core and the facings, the displacements of 

the middle surfaces of the facing are set equal to the displacements of the core at r = a 

and r = b. 

Thus, displacements u, v, and w in equations (37), (38), and (39) are replaced by equations (20), (21), and (22) evaluated at r = a. In this manner, three equations in six arbitrary 

constants (A, B, C, D, E, and F) are written for the outer facing. In a like fashion, three 

equations are written for the inner facing. These equations can then be written in matrix 

form as: 

(40) 

where the a

ij’s are defined in terms of geometric, loading, and material parameters. 

Equations (40) are satisfied if the constants A, B, C, D, F, and L are all equal to zero. 

This represents the uniformly compressed circular form of equilibrium of the cylinder. 

A buckled form of equilibrium is possible only if equations (40) yield nonzero solutions 

for the constants. This requires that the determinant of the coefficients of these constants 

be equal to zero. 

It is possible to find simultaneous minimum values (absolute sense) of φ

1 and φ

2 for 

which these six equations will be satisfied for any values of the arbitrary constants. 

Mathematically this means that for such a combination of loads and material parameters 

the deflections are indeterminate. The shell becomes elastically unstable and the loads 

that bring about this condition are the desired critical loads. 

Numerical Computations 

A literal solution of the sixth order determinant for the eigenvalues (φ

1

, φ2) is not 

feasible. A numerical solution, from which curves may be drawn, is possible if a digital 

computer is used. 

For sandwich construction, particularly for that utilizing honeycomb cores, the equations can be greatly simplified by assuming E = ∞. This substitution is common with c 

this type of construction due to the small relative displacement occurring between the 

FPL-0173 -14-
sandwich facings. Indications are that analyses based on this assumption are sufficiently 

accurate in most ranges, but for relatively short cylinders having weak cores such an 

assumption may result in serious error. 

Poisson's ratio for the facings can be taken as one-third and the value of k closely 

approximated by the fraction one-half for most cylinders where the radius is large 

compared to the sandwich thickness h. 

Making these substitutions, the a aij's in equations (40) are found to be 

FPL-0173 -15-
FPL-0173 -16-
The problem is now reduced to developing a search technique whereby minimum values 

of φ 1 and φ

2 can be determined for given values of V 

rq’ V 

rz , ∆, and γ, such that the 

determinant goes to zero. 

This problem was programed for the digital computer and results are presented in 

figures 6, 7, and 8. The solutions show a linear relationship between φ

1

and φ

2

as was 

found previously for isotropic shells 5,6 wherein the slopes of the lines are dependent 

upon values of number of buckles producing minimum loads. The lowest of these lines 

are shown in figures 6, 7, and 8 for chosen values of γ. These figures were then used as 

a basis for plotting figures 9, 10, 11, 13, 14, 16, and 17, which clarify the effect of the 

interaction of the loads by presenting them in terms of ratios. 

The effect of special cases such as the hydrostatic case can be seen most clearly on 

figures 6, 7, and 8 by the use of a load ratio line where the slope of the line and the 

desired load ratio are equivalent. The hydrostatic case, for example, has a load ratio of 

one-fourth or 

Discussion of Results 

For illustrative purposes, the three values of of 0.0001, 0.00001, and 0.000001 

were chosen to represent the range of delta. For a given delta then, figures are presented 

which show the effect on the critical loads of (1) varying the value of γ while holding the 

values of Vrz and Vrθ a constant, and (2) varying the value of V rz 

and V 

rθ while holding 

the value of a constant. 

The results of this analysis agree favorably with those previously presented 6,7 for 

lateral pressure and axial load acting separately. These values then represent the values 

used for φ 1 and φ 2 in the figures. 

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

Cylinders of Finite Length Under Unifom External Lateral Pressure. U.S. Forest 

Service Research Note FPL-0104. 1965. Forest Prod. Lab. 

7 Zahn, John W.,and Kuenzi, Edward W. Classical Buckling of Cylinders of Sandwich 

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

Note FPL-018. 1963. Forest Prod; Lab. 

FPE-0173 -17- .67-30 

M 133 630 

Figure 7.--Effect of combined loading for D2 = 0.0001. 

M 133 639 

Figure 9.--Effect on critical loads of varying parameter γ. 
M 133 635 

Figure 10.--Effect on critical loads of varying parameter γ.
M 133 629 

Figure 11.--Effect on critical loads of varying parameter γ. 
M 133 640 

Figure 12.--Effect on critical loads of varying parameters Vrθ and Vrz. 
M 133 642 

Figure 13.--Effect on the critical loads of varying parametere γ. 
M 133 634 

Figure 14.--Effect on critical loads of varying parameter γ. 
M 133 641 

Figure 15.--Effect on critical loads of varying parameters Vrθ and Vrz. 
M 133 636 

Figure 16.--Effect on critical loads of varying parameter γ. 
M 133 637 

Figure 17.--Effect on critical loads of varying parameter γ. 
Figure 18.--Effect on critical loads of varying parameters Vrθ and Vrz. 
M 133 628 

Figure 19.--Classical buckling coefficient for sandwich cylinders with isotropic facings 

and orthotropic core. 
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is dedicated to the principle of multiple use management of the Nation's 

forest resources for sustained yields 

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