Supplement to 

EFFECTS OF SHEAR DEFORMATION 

IN THE CORE OF A FLAT RECTANGULAR 

SANDWICH PANEL 

DEFLECTION UNDER UNIFORM LOAD OF SANDWICH PANELS 

HAVING FACINGS OF MODERATE THICKNESS 

AIR 

December 1951 

This Report is One of a Series 

Issued in Cooperation with 

FORCE-NAVY-CIVIL SUBCOMMITTEE 

on 

AIRCRAFT DESIGN CRITERIA 

Under the Supervision of the 

AIRCRAFT COMMITTEE 

of the 

MUNITIONS BOARD 

NO. 1583-D 

UNITED STATES DEPARTMENT OF AGRICULTURE 

FOREST SERVICE 

FOREST PRODUCTS LABORATORY 

Madison 5, Wisconsin 

In Cooperation with the University of Wisconsin 
Supplement to 

EFFECTS OF SHEAR DEFORMATION IN THE CORE OF A 

FLAT RECTANGULAR SANDWICH PANEL 

Deflection Under Uniform Load of Sandwich Panels 

1 Having Facings of Moderate Thickness 

By 

WILHELM S. ERICKSEN, Mathematician 

Forest Products Laboratory,2 Forest Service 

U. S. Department of Agriculture 

Introduction 

This report supplements the mathematical analysis of the deflection of a 

simply supported rectangular sandwich panel under uniform transverse load 

given in U. S. Forest Products Laboratory Report No. 1583-C.3 Formulas are 

derived and curves presented for determining the central deflection and 

the maximum stresses and strains in the facings and core of a panel having 

isotropic core and facing materials. These formulas and curves incorporate 

the effects of the transverse shear deformations in the core. They also

take into account the effects of the bending of the facings about their 

1This progress report is one of a aeries prepared and distributed by the 

Forest Product6 Laboratory under U. E. Navy Bureau of Aeronautics Order 

No. NAer 01019, 01077, and 00938, Amendments Nos. 1 and 2, and U. S. Air 

Force No. USAF-(33-038) (51-4066-E and 51-4326-E). Results here reported 

are preliminary and may be revised as additional data becomes available. 

2Maintained at Madison, Wis, , in cooperation with the University of 

Wisconsin. 

3Ericksen, W. S., Deflection Under Uniform Load of Sandwich Panels Having 

Facings of Unequal Thickness. Forest Products Laboratory Report 

No. 1583-C, 1950. 

Report No. 1583-D Agriculture-Madison 
own middle surfaces and therefore extend into the range of moderately thick 

facings. In the analysis the facings are considered as thin plates in the 

sense that transverse shear deformations within them are neglected. 

The analysis in Report No. 1583-C led to a double Fourier expansion for the 

deflection. This expunsion was transformed to the M. Levy form, which is more 

suitable for use in computations, under the assumptions that the facing and 

core materials were isotropic, and the facings could be considered as membranes. In the present report, a similar transformation is applied to the 

double Fourier series derived under the assumption that the facings bend about 

their own middle surfaces. 

Specific formulas for the components of stress and strain in the facings and 

core were not derived in Report No. 1583-C. Such formulas, however, are 

readily derived from other formulas developed in that report and are given 

in the present report in double Fourier form for a sandwich panel having 

orthotropic core and facing materials. These Fourier expansions, reduced to 

the case of isotropic core and facing materials, are then transformed into 

forms more suitable for computations. 

The mathematical analysis in Report No, 1583-C, and consequently that in the 

present report, are based on a number of simplifying assumptions that include 

the following: (a) the effect of the transverse shear deformations in the 

facings is negligible; (b) the effect of the bending of the core is negligible; 

(c) the transverse shear strains in the core are constant over the thickness 

of the core; and (d) the core and facing materials are not stressed beyond their 

proportional limits. These assumptions have been widely used in sandwichplate theory and have been proved applicable to various types of problems by 

comparisons with results obtained in actual tests. The possibility that one 

or more of these assumptions may not be applicable to a particular construction 

and load should be kept in mind, however, in the application of results derived 

under such assumptions. 

In the present analysis, a number of operations are performed on single and 

double series. These operations have been proved valid by considerations not 

included in this report. 

Notation 

dimensions of the panel 

thickness of the core 

Youngf

's modulus of isotropic facings 

components of strain in facings i, i = 1, 2 

components of shear strain in the core 

thicknesses of' facings 

Report No. 1583-D 
uniform load per unit area 

coordinates with axes shown in figure 1 

normal deflection of panel 

coordinates with axes shown in figure 2 

shear moduli of orthotropic core 

shear modulus of isotropic core 

Poisson’s ratio of isotropic facing material 

Results and Discussion 

Formulas for determining the central deflection and the components of stress 

and strain in the facings and core of a uniformly loaded sandwich panel are 

given in this section. These formulas, which are derived in this report, apply 

to a rectangular panel having simply supported edges, isotropic facings, and 

either a core material that is isotropic or one in which 

The following parameters, in addition to those defined in the notation, determine the deflection and components of stress and strain: 

(1) 

(2) 

(3) 

(4) 

Report No. 1583-D -3-
The parameters have the following significance: If is the sum of the moments 

of inertia of the facings about their respective middle surfaces; I is the 

moment of inertia of the spaced facings about the centroidal axis with the 

stiffness of the individual facings 

facings 

neglected; S is the shear-flexibility 

co-efficient of a core, and is a parameter that determines mainly the effect 

of the bending of the about their own middle surfaces. 

The central deflection is given by the formula, 

(5) 

in which parameters A1 and A2 are determined from figures 3 and 4, respectively. 

The strain components in facing i, i = 1 or 2 according as the thickness of 

the facing is f1 or f2 as shown in figure 2, are maximum with respect to x and 

y at the center of the panel. Reference to (3.4), (3.10), and (3.13) indicates 

that these components can be expressed in the forms 

(6) 

(7) 

in which parameters B1, B2, C1, and C2 we determined from figures 5, 6, 7, and 

8 respectively, and 

(8) 

Report No. 1583-D -4-
Figure 2 indicates the origins of the coordinates Z1 and Z2. The corresponding 

components of stress are given in terms of the same quantities by the formulas 

(9) 

and 

(10) 

The maximum shear-strain component in the core occurs at the center of the 

longer side, length b, and is given by the formula 

(11) 

in which parameter D is determined from figure 9. The strain component eyz evaluated at x = a y = 0, is given by the formula 2 , 

(12) 

in which the parameter F is determined from figure 10. 

1. The M. Levy Form of the Expansion for the Deflection Under 

Uniform Transverse Load. 

The double Fourier form of the expansion for the deflection under uniform load 

of a simply supported panel having; isotropic facing and core materials can be 

obtained from formulas (A38), (A41), and (A42) of Report No. 1583-C in the form 

(1.1) 

where 

Report No. 1583-D -5-
(1.2) 

(1.3) 

(1.4) 

(1.5) 

x and y are coordinates with axes as shown in figure 1, and the symbols 

denote summation over all positive odd integers m and n, 

respectively. Formula (1.1) can be mitten 

(1.6) 

where 

(1.7) 

and 

(1.8) 

Report No. 1583-D -6-
The M. Levy form of the expansion for w is obtained from (1.6) by summing with 

respect to n. In this summation, it is convenient to consider w as the sum 

(1.9) 

where 

(1.10) 

and 

Consider series (1.10)' and let 

(1.11) 

(1.12) 

where 

Evidently 

(1.13) 

(1.14) 

With the use of (1.13), series (1.12) can be written 

(1.15) 

Report No. 1583-D -7-
From equation (A49), in Report No. 1583-C 

(1.16) 

where 

(1.17) 

and from (A48) of the same report, 

(1.18) 

From the last expression, it is evident that 

(1.19) 

where 

(1.20) 

Substituting formulas (1.16), (1.18), and (1.19) into formula (1.15) gives 

Report No. 1583-D -8-
(1.21) 

If this expression is substituted into (1.14), the terms that are independent 

of hyperbolic function can be summed. The required summations are: 

(1.22) 

(1.23) 

and 

(1.24) 

Formula (1.24) was obtained from (1.18) by interpreting The substitution of (1.21) into (1.14) and the use of (1.22), (1.23), and (1.24) gives 

Report No. 1583-D -9-
(1.25) 

The formula for w1 for a plate of width a and of infinite length consists of 

the terms in this expression that have been summed with respect to m. 

The summation of the series (1.11) with respect to and partially with respect 

to m can be accomplished similarly. 

Let 

so that 

With the use of (1.13), 

(1.26) 

(1.27) 

(1.28) 

and, therefore, with the use of (1.18) and (1.19), 

(1.29) 

Report No, 1583-D -10-
The substitution of formula (1.29) into (1.27) and the use of (1.23) and 

(1.24), then gives 

(1.30) 

The formula for w2 for a plate of infinite length consists of the terms in 

this expression that have been summed with respect to m. 

The central deflection, w is obtained by evaluating w 

From ( 1.25), (1.30), and (1.8), 

(1.31) 

where 

(1.32) 

(1.33) 

Plots of A1 and A2 as functions of for various values of are given in 

figures 3 and 4, respectively. 

2, The Components of Strain in the Facings and Core of Sandwich Panels 

Having Orthotropic Core and Facing Laterials. 

The stain components in the facings of a sandwich panel having orthotropic 

core and facing materials are obtained from equation (A4) and (A5) of Report 

No. 1583-C in the forms: 

Report No, 1583-D -11-
(2.1) 

(2.2) 

(2.3) 

and 

(2.4) 

In these expressions, the superscripts 1 and 2 refer to the facings 1 and 2, 

respectively, Z1 and Z2 are coordinates with axes and origins as indicated 

in figure 2, wmn is a term of series (A38) of 

ameters Kmn, hmn, qmn, and rmn are determined 

The components of shear strain in the core, 

and 

are obtained from equation (A6) of Report No. 

Combinations of the parameters kmn, hmn, qmn, 

the evaluation of the above strain components 

forms from (A24) of Report No. 1583-C: 

Report No. 1583-C, and the parby (A24) of Report No. 1583-C. 

(2.5) 

(2.6) 

1583-C 

and rmn that are required for 

are determined in the following 

Report No. 1583-D -12-
(2.7) 

(2.8) 

where 

and 

(2.9) 

The expressions for hmnrmn and hmn are obtained by substituting Sx for Sy, m2 for 

for a, and vice versa in (2.7) and (2.9), respectively. Finally, 

kmn(c - qm) and hmn(c - qmn) are obtained from kmnqm and hmnrmn, respectively 

by substituting f1 for f2 and f2 for f1 in the latter expressions. Definitions 

and a that appear in formulas (2.7), (2.8), and 

formulas (A32) of Report No. 1583-C. 

of the symbols 

(2.9) are given in 

The Components of Strain in the Facings of Sandwich panels 

Having Isotropic Core and Facing Materials 

If the core and facing materials are isotropic, 

and 

Report No. 1583-D 

(3.1) 
where is the Poisson's ratio of the facing material and S is given by formula 

(1.5). Equation (2.7) then reduces to 

(3.2) 

The expression for kmn(c - qmn)is obtained from equation (3.2) by substituting 

f1 for f2 and f2 for f1. In the case under consideration, wmn is a term in 

the series (1.6). From formulas (2.1) and (2.3) therefore, the strain components 

i = 1, 2, may be written as 

where 

(3.3) 

(3.4) 

Similarly, from formulas (2.2) and (2.3), 

(3.5) 

where i = 1, 2, and Mi and N'i are defined by (3.4). 

Series (3.3) and (3.5) can be summed with respect to n by methods similar to 

those used in section 1, It is observed from series (1.10), (1.11), and (3.3) 

however, that 

(3.6) 

Report No, 1583-D -14-
and from (1.10), (1.11), and (3.5) that 

(3.7) 

(i) (i) Therefore, the representations of exx and eyy that are summed with respect to 

n and partially with respect to m can be obtained by using formulas (1.25) and 

(1.30) in formulas (3.6) and (3.7). Results obtained in this manner are given 

as follows: 

(3.8 

(3.9) 

Report No, 1583-D -15-
The terms in formula (3.8) that have been summed with respect to m represent 

the strain component of a plateofwidth a and of infinite length. 

The maximum values of the strain components with respect to x and y occur at 

the center of the plate where Formulas for these components 

can be written: 

(3.10) 

where 

(3.11) 

(3.12) 

and 

(3.13) 

where 

(3.14) 

(3.15) 

The parameters B1, B2, C1, and C2 are plotted as functions of for various 

values of in figures 5, 6, 7, and 8 respectively. 

The stresses in the facings can be obtained from the strain components by use 

of the formulas 

(3.16) 

(3.17) 

Report No, 1583-D -16-
If (3.10) and (3.13) are substituted in expressions (3.16) and (3.17), the 

maximum stresses in the facing (with respect to x and y) are obtained in the 

forms 

(3.18) 

(3.19) 

4. The Components of Shear Strain in the Core of a Sandwich Panel 

Having Isotropic Core and Facing Materials 

The relations (3.1), which exist in the isotropic case, reduce formula (2.9) 

to the form 

(4.1) 

With the use of this expression and with wmn a term of series (1.6), series 

(2.5) takes the form 

(4.2) 

This series can be summed with respect to n and partially with respect to m 

by the method used in section 1. It is observed from (1.11), however, that 

(4.3) 

and, therefore, from (1.30), 

(4.4) 

Report No. 1583-D -17-
The maximum shear strain occurs at the center of one of the edges x = o or 

(4.5) 

where 

(4.6) 

This parameter is plot'ted as a function of for various values of in figure 

9. 

Similarly, it is found from (2.6) and (4.1) that 

(4.7) 

and, therefore, from (1.30), 

(4.8) 

The maximum of this strain component occurs at the center of the shorter sides 

of the panel. At y = o, x = 2 

a , 

(4.9) 

where 

(4.10) 

The parameter F is plotted in figure 10. 

Report No. 1583-D -18-
5. The Reaction at the Edges of a Sandwich Panel 

Having Isotropic Core and Facing Materials 

In the present analysis, it is assumed that the effect of the bending of the 

core is negligible and, therefore, that bending and twisting moments are 

carried entirely by the facings. These moments are given by the formulas: 

(5-1) 

(5.2) 

(5.3) 

where indicates integration over the facing of thickness fi, and the 

(i) origin of z may be chosen at any convenient point. The stresses Xx and 

Y(i), i = 1, 2, are obtained from (3.16) and (3.17), respectively, with the y 

use of (3.6) and (3.7). The shear stresses Xy

(i) , i = 1, 2, are given by 

the formula 

(5.4) 

' where Mi and Ni are defined by (3.4). This formula is derived from equation 

(A4) of Report No. 1583-C with the use of (3.2). 

The reactions at the edges of the panel are computed from (5.1), (5.2), and 

(5.3) by the usual formulas. When these evaluations are made, it is found 

that the reactions are independent of the parameters S and and are the same 

as those of an ordinary plate. Numerical values of these reactions may be 

obtained from Timoshenko1 , table 5. 

1 Timoshenko, S. Theory of Plates and Shells, New York, 1940, 

Report No. 1583-D -19-
Figure 1.-Section of panel parallel 

to facings. 

(ZM 85798 F) 

Rept. No. 1583-D 
Z M 85798 F 
Figure 2.--Cross section of loaded sandwich 

panel. 
Z M 88373F 


Figure 4.--The parameter A2 in formula (5), 

Rept. No. 1583-D 
Z M 88375 F 
Figure 5.--The parameter B1 in formula (6), 

Rept. No. 1583-D 
Z M 88376 F 
Figure 6.--The parameter B2 in formula (6), 

Rept. No. 1583-D 
Z M 88377 F 
Figure 7.--The parameter C1 in formula (7). 

Rept. No. 1583-D 

Figure 8.--The parameter C 2 in formula (7), 

Rept. No. 1583-D 

Figure 9.--The parameter D in formula (11), 

Rept. No. 1583-D 

Figure 10.--The parameter F in formula (11), 

Rept. No. 1583-D