FLAT
No. 1833
June 1952
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Wood Engineering Research
Forest Products Laboratory
Madison, Wisconsin 53705
INFORMATION REVIEWED
AND REAFFIRMED
1958
_ r
rilliiiimutmll.,
FOREST PRODUCTS LABORATORY
MADISON S. WISCONSIN
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE n Cooperation with the Ijnivereity of Wisconsin

CRITICAL LOADS OF A RECTANGULAR, FLAT SANDWICH PANEL
SUBJECTED TO TWO DIRECT LOADS COMBINED WITH A SHEAR LOAD
By
CHARLES B. NORRIS, Engineer and
WILLIAM J. MIXERS, Engineer
Forest Products Laboratory,? Forest Service
U. S. Department of Agriculture
•••••••••
Summary
An approximate formula is presented for the critical loads of a rectangular, flat sandwich panel subjected to two direct loads applied perpendicularly to its edges combined with a shear load applied parallelly to its edges. The formula was confirmed by tests of L5 sandwich panels having aluminum facings and cores of end-grain balsa wood, cellular hard rubber, or cellular cellulose acetate.
Introduction
In many structures involving flat sandwich panels, particularly in aircraft, the panels are subjected to combined edgewise loads. In order to design these panels so that they will not fail elastically by buckling when subjected to such loads, the values of the critical loads must be determined.
Information is available for the determination of the critical loads when sandwich panels are subjected to edgewise compression or edgewise shear. This report presents an approximate formula for the determination of the critical loads when a sandwich panel is subjected to two direct loads applied perpendicularly to its edges combined with a shear load applied parallelly to its edges. Any combination of edgewise loads may be represented by these three loads.
-This progress report is one of a series prepared and distributed by the
Forest Products Laboratory under U. S. Navy, Bureau of Aeronautics Order
No. NAer 01319 and U. S. Air Force No. USAF-18(600)-70. Results here reported are preliminary and may be revised as additional data become available.
Maintained at Madison, Wis., in cooperation with the University of Wisconsin.
Report No. 1833 Agriculture-Madison

Nomenclature
The nomenclature of the ANC-23 bulletin, Sandwich Construction for Aircraft,
Part is used in this report. The general expressions and the special expressions used in the calculations are given in the following list. The special expressions were selected because they assume that the facings are of equal and moderate thicknesses, and they neglect the stresses in the core except the transverse shear stress. The calculations apply to isotropic facings, so that the modulus of elasticity of the facings is the same in all directions, and the usual formula relating the modulus of rigidity and
Poisson's ratio to the modulus of elasticity applies.
t thickness of the sandwich construction t
F12 tF2 thicknesses of the facings of the sandwich construction tc thickness of the core of the sandwich construction
Ep a ,
E F/3 moduli of elasticity of the facings in the directions of the natural axes a. and /3 respectively
G raz,
Ga3z moduli of rigidity of the core associated with the a and z axes and with the i6 and z axes, respectively
F
1 -
Poisson's ratios of the facings a. modulus of rigidity of the facings associated with the a and f axes
11Mm
IM•11,
Hp = E F
(t, tc)
Da
EFa tF(t + tc)2
-
Eva
F tF(t + tc)2
Do .
7 .
4 G FaiotF (t + tc)2
(t + tc)2
4tc
G Caz
• Forest Products Laboratory. Sandwich Construction for Aircraft, Part II.
ANC-23 Bulletin. Munitions Board, Aircraft Committee. 1951.
Report No. 1833 -2-

Upz
_ (t + tc)
2
G
N ab = G Fai e ( t tic)
B 1 = al 1, for isotropic facings
EFp
B = 2B + B
2 3 1
118 a
• 1, for isotropic facings
Dap
B3 = -;
3 VERLEFA
; 0,35, for isotropic facings and P .
= 0,3
V a.
77 2 1\/T,Tp .
7r2tFtcEF
2 _ a u a z 2a2 H F0caz
7T 2
\ i D a
JD
V - a
2 u o bz
;
"
2 tit c \ i
' EFxEF4i
2 a2G
F CO
-
71-
V a2 H
n
DD r:c ;
P
2
_, 12 'f r v
6\ Fa2 (t ' 'C I V 'Fo7F/3 f
F a , f F p direct stresses in the facings in the directions of the natural axes a and
a, respectively f
Fap shear stress in the facings associated with the natural axes a and /3 rat8 f
Fa,5
Ffi
Report No. 1833 -3-

Discussion of Mathematical Analysis
When a sandwich panel is subjected to a combination of edgewise loads, and the loads are kept proportional to one another and increased, the panel will break into a buckle pattern at some combined value of these loads, provided it does not previously fail. The values of the loads at which the buckles form are termed the critical values. The stresses produced in the facings at the critical values of the loads are termed the critical stresses. A great number of different wave patterns are mathematically possible for the buckling of the panel. The particular wave pattern that yields the least values for the loads, or stresses, determines the critical loads or stresses.
Any combination of edgewise loads may be expressed in terms of two direct loads applied perpendicularly to two edges of the panel and a shear load, as shown in figure 1. It is assumed that the natural axes a and
2 of the orthotropic panel are parallel to the edges of the panel with lengths a and b, respectively.
The direct loads pa and P
A
and the shear load P a p are the loads per inch of
•••••• n •
11111 n •• panel edge. The stresses in the facings, f Fa , !E
a l and
!ta2, maybe found by dividing the loads by the sum of the thicknesses of the two facings if the stresses in the core are neglected.
The exact form of the buckle pattern created when these loads reach their critical values is very complicated, so that an approximate formula must be used, such as the familiar interaction formula. This formula is: fF p fFa f
Fper f
Facr fFap
— 2
= 1 f
FaPer
(1) where f
Fa
, f
Fp
, and f
Fap are the three applied stresses, and fm, z CLC f
F or' and fFa3crare the three corresponding critical stresses due to each of the stresses acting alone. This formula, however, yields values of the critical stresses that are too small, because it does not take into account the interference between the three buckle patterns that it assumes are formed.
A closer approximation of the critical stresses may be attained if the interference between the buckle patterns formed by the two direct loads is taken into account by applying equation (A9) of appendix A. The approximate formula
Obtained in this way is:
F
-~,
2 fp. p
+ cll.
+ f Fa.3
= 1 (2)
fFpor
1FaTicr
This equation expresses the parabolic surface referred to in appendix B. It is mathematically identical to equation (1) if the panel is square, and if the
Report No.
1833
-ti-

number of half waves formed at the critical load is unity in both directions for both formulas; otherwise, it yields greater values for the critical stresses. It yields
.
values that are too small if the second term in the lefthand member is about equal to or larger than the first term. This seldom occurs because the squared term is always less than unity, and the value of f
Fa p er is usually much greater than the value of fucr.
For more convenient use, equation (2) may be written in the form:
2
1 ± r a,
9
4 f e f 173 ra.i3
fFa,8cr
1 where K = K
F
.+ Km
(3) tFl3
+ tF2
3
KF
X
F
12 D\/-773 -
C
3
B 1 C 1 I-
2B2C2 B1
2B
2 C 2 + + A [.a +
V
K M -
1 + (B i C i + B 3 C 2 t
+ (BI + B,C
,
2 )V +
V a lfs A
94
(4)
A = C 1
C
3
- B
2
2
C
2
2
+ B
3
C
2
(B
1
C
1
+ 2B 2 C
2
Bi
The equations for C
1
, C
2
, C
3
, and C
I. that should be used when the edges of the panel are simply supported or clamped are given in appendix A. The values of m and n in these equations are integers and represent the number of half waves into which the panel buckles. The combination of these values should be chosen so that the left-hand member of equation (3) is a maximum. Usually, the value of K
F
is so small, compared with the value of Km, that it may be neg-
• approaches infinity, the value of the left-hand member of equation (3) approaches V6; as the value of m approaches infinity, the value of this member approaches raVa.
Since the value of the left-hand member of equation (3) is known, the equation may be solved for the critical value of !I
x if the value of f
Fao cr is found.
The value of this critical shear stress is found from equation 4.221 (B) of
Sandwich Construction for Aircraft, Part 11.2 (If the sandwich panel is isotropic, formula 3.221 (A) may be used.) This equation is:
Report No. 1833 .5-

fr
(n23.3)1/4
11GF a Nip + v = )
(5) where
C3
KM(v
0) = B 1 C l + 2B 2 0 2 + 1:33.
and KM is obtained from equation (4) by using different values of C 1 , C2 , C3, and 0 4 than those given in appendix A. For panels having simply supported edges, these values are:
= c 4
_ 3 r b
2
-.6 for n = 1
'' a`
1
C = CL =
1 b
2
4 n2 1
7 a for other values of n
C 2
3
4 n
6 n
2
2
+1
+1a
2
—6 b` for all values of n
The values for panels having clamped edges are: b2
C1 = 4C4 = 4 7 . for n = 1 c1 = =
16 a2
7 for other values of n
3 (Z1` .
+ 1) b
n 4 + 6n2 + 1 a2
- 3
E ' 3 — n2 +1 b2 for all values of n
The integral value of n that yields the least critical stress is the one to be used. This value of n will probably differ from the value found by equation
(3), because of the approximation involved in that equation.
The proper values of j to use are obtained from figure 2 for panels with simply supported edges and from figure
3 for panels with clamped edges. The parameters for use in these figures are:
Report No. 1833

Thus, the critical value of f is found from equation (3), and the critical values of f Fa and f Fag are found from the known values of r a and :le
The computed critical values for the stresses in the facings are too great if
the proportional-limit stress of the facing material is exceeded. If the
-facings of the sandwich panel are orthotropic, it may be assumed that the
...stress-strain curve of each stress is independent of the other two stresses.
5 The values of the stresses are assumed, and the calculation of the critical stresses is made by using the tangent moduli at the assumed stresses. If the
:critical stress found does not agree with the assumed value, different values of the stresses are assumed, and the calculationls repeated. This cut-andtry method is repeated until the values of the critical stresses found agree with the values of the stresses assumed. The critical stresses found in'this way will again be too great, but they will be less than those found by assuming that the proportional limit has not been exceeded.
If the material of the facings is isotropic, a suitable theory of plasticity may be employed. An estimate of the proportional limit may be obtained from the principal. stresses involved by using Tresca's assumption of yield at a certain shear stress or von Mises' assumption of yield when the energ y of change of shape reaches a certain value. The two principal stresses in the facings are given by: f Fl
1
7 ( f
Fez Fg
1
4 f )
Fp
2
+ 4 fP2/3
2 fF2 = (f F + f FiE3 ) -
V
(fF a f Fi d
2 lifFa/32
If•these two principal stresses are positive (compressive), Tresca f s assumption implies that the proportional limit is reached when the value of principal stress given by equation (6) equals the proportional limit value of the material when subjected to a single compressive stress. In this same range, von Mises' assumption leads to the expression:
Vr ts.
2 fFafFp fFp
2
3f F ap
2
(8)
The proportional limit is reached when the value of this expression reaches the proportional limit value of the material. when subjected to a single campressive stress.
Report No. 1833

Description of Experimental Technique
The sandwich panels tested were glued between panels of redwood lumber, with two lumber panels on each side of the sandwich panels. Portions of the lumber panels adjacent to the sandwich panels were cut away in the form of a square or rectangle, as shown in figure L1, so that the sandwich panels could buckle over their central areas. These cut-away areas were orientated at different angles, so that the exposed portions of the panels were subjected to different values of edgewise normal and shear loads.
Forty-eight sandwich panels were tested: all of which had 24ST aluminum facings
0.02 inch thick. The panels were divided into three classes, numbered 1, 2, and
3, with thicknesses of approximately 7/32, 11/32, and 13/32 inch, respectively. Each of these classes was divided into two groups, those with cores of end-grain balsa wood and those with cores of either cellular hard rubber or cellular cellulose acetate. These groups were designated by letters; B for balsa wood, H for cellular hard rubber, and C for cellular cellulose acetate.
All of the completed specimens were also designated by the letter S, for square cutouts, or R, for rectangular cutouts. The square cutouts were 20 inches on a side, and the rectangular cutouts were 20 by 12 inches. The square cutouts were orientated at angles of 0°, 15°, 30°, and 45° from the horizontal, and the rectangular cutouts were orientated at 0°,
30°, 60°, and
90° from the horizontal to the short side of the rectangle.
Thus, each specimen was given a number consisting of four parts, for example,
2 B R 30. The 2 indicates that the panel was about 11/32 inch thick; the B indicates that the panel had a balsa wood core; the R indicates that the cutout was rectangular; and the 30 indicates that the cutout was orientated 30° from the horizontal.
Both the lumber panels and the sandwich panels were about
36 inches square.
The two lumber panels adjacent to the sandwich panel, in which the cutouts were made, were each about 1/2 inch thick. The two outer panels were about 1 inch thick. The grain direction of all four lumber panels was horizontal, as shown in figure 4. These four panels were glued together and to the sandwich panel.
Before the outer lumber panels were glued on, resistance-wire strain gages were mounted on the sandwich panel, as shown in figure
► .
These gages consisted of four rosettes, two on each side, mounted directly opposite those on the other side, at the centers of two edges of the cutout; and two unidirectional gages, one on each side, mounted vertically at the center of the sandwich panel. The rosettes were used to determine the strains applied at the edges of the cutout, and the unidirectional gages to assist in the determination of the critical load. In some of the first specimens tested, a number of gages were placed along the edges of the cutouts to determine the distribution of the applied strains.
Report No. 1833 -8-

The ends of the completed test specimens were machined square with those of the sandwich panels and parallel with each other, by trimming the sides and ends with a circular saw and finishing the ends with a shaper.
The specimens were tested in the machine shown in figure
5. Channel irons were bolted snugly to the edges of specimens, with spacer blocks at their outer edges, to keep the specimens from buckling as a whole. Three dial gages were mounted on the specimen, with their stems resting on the sandwich panel, to assist in determining the critical load. One of these dial gages was placed at the center of the panel, and the other two were placed at the quarter points of the cutouts. The resistance wire strain gages were connected to a recorder, as shown in figure 5.
The load was applied to the specimen in increments. At the end of each increment, the loading was interrupted while the resistance-wire gages and the dial gages were read. These readings were plotted against the load for analysis.
For most of the specimens, the loading was continued until failure occurred.
After the specimens were tested, they were cut apart, and narrow coupons cut from the sandwich panels parallel to the edges of the cutout in the lumber panels. Four such coupons were cut from each specimen, one parallel to each edge of the cutout. These coupons were about 1 inch wide and of various lengths. They were tested in bending to determine the shear moduli of the core material in the two directions, accor4ng to the method described in
Forest Products Laboratory Report No. 1556.il
Discussion of Experimental Technique
The specimen, composed of a sandwich panel glued between lumber panels, was crushed by the testing machine. The strains produced were approximately vertical with a horizontal Poisson's ratio effect. In some specimens, a number of resistance-wire strain-gage rosettes were mounted along the edges of the cutout. The strains read from these gages indicated that the strains of these edges were about uniformly distributed. Figure 6 is a sketch of a square cutout orientated at 45°. The values of the strains of various positions are indicated graphically and numerically. The average values from the two faces of the sandwich panel are given. It will be seen from these values that the strains were about uniformly distributed along the edges. Similar data were obtained from a number of the specimens.
The testing method used overcomes the usual difficulty of such tests ) in which the mechanism employed to apply the shear load opposes the direct loads and prevents them from acting upon the panel. It, however, has three drawbacks:
(1) The exact conditions at the edges of the cutout are not known. It is known only that the panel is supported at these edges, and that the support is
Sandwich Construction at Normal Temperatures. Forest Products Laboratory
Report No, 1556. Revised
1950. (p. 7.)
Report No. 1833 9 -

somewhere between simple support and fully clamped. (2) Tensile direct stresses cannot be applied by this method. (3) The ratios of the various applied stresses to each other have definite values, depending upon the orientation of the cutout. Other ratios cannot be obtained by this method; thus, the range of experiments that can be performed is restricted.
The critical load was found by picking a point on the load-deflection curve
(fig. 7) just above the knee of the curve or a similar point on the load-strain curve obtained from the central resistance-wire strain gage. When the strains at the critical load, as determined' from the load-strain curve, were greater than the proportional limit value, this curve could not be trusted for determining the critical load, and only the load-deflection curve was used.
The critical load could not be obtained with sufficient accuracy for three specimens, and, therefore, these specimens were omitted from the tables of data.
After the critical load on the specimen was determined, the strains at the various points of the panel at the critical load were obtained from the loadstrain curves for these points. These curves were substantially straight lines, as shown in figure 7, except at strains near the critical load or above the proportional limit value. These straight lines were extrapolated to the critical load, and the strain obtained in this way was used as the strain at the critical load. This process would not have been necessary for strains less than the proportional limit value if the panels had originally been perfectly flat. Because the panels were not perfectly flat, their deflections increased with increasing values of loads below the critical value, and, therefore, the load-strain curves deviated from a straight line at load values below the critical value. The extrapolation method was reasonably accurate, because the strains in the wood panels of the specimen were always less than the proportional limit value of the wood, and, thus, the load shifted only slightly from the sandwich panel to the wood at values near the critical load.
The stresses in the facings of the sandwich panels at the critical load were obtained from the usual equations: fa. =
;1/4-
EP
+
7 es
e
+- • e f
3 l a/3 = Gea/3 where fa and fp are the direct stresses associated with the directions a and fi; fup is the shear stress associated with these directions, and ea , eQ , and are the strains corresponding to these stresses. E is the modulus of elasticity of the (aluminum) facings (10,000,000 pounds per square foot),
/.1. is the Poisson's ratio of the facings (0.3), G is the modulus of rigidity of The facings, and X = 1 -42.
Report No. 1833 -10-

Discussion of Results of Tests
Tables 1 and 2 give the experimental critical stresses and the computed critical stresses from formulas
(3) and (1). They also give the dimensions of the sandwich constructions; the ratios of the experimental critical stress to the proportional limit stress, computed by means of both formulas (6) and (8); and the number of half waves into which the panels will buckle, according to formula (3).
It will be noted from the tables that, for the panels in which the critical stress did not exceed the proportional limit stress, the experimental values usually lie between the two values, one for simply supported edges and the other for clamped edges, computed by means of formula (3). Figure 8 is a plot of the experimental values against the computed values, both divided by the proportional limit stress of 28,000 pounds per square inch, for the square panels. The method of constant shear stress at the proportional limit (Tresca) was'used. The value of 28,000 pounds per square inch, the second proportional limit, was taken from National Advisory Committee for Aeronautics Technical
Note No. 1512.5 Similar plots are given in figure 9 for the rectangular panels with cores of balsa wood and in figure 10 for the rectangular panels with cores of cellular hard rubber. In these plots, the straight lines indicate that the theoretical values for the conditions of perfectly clamped edges or perfectly simply supported edges lie, in general, between the two computed values if the change in the modulus of elasticity at stresses greater than the proportional limit stress is taken into account.
In tables 1 and 2, the moduli of rigidity given in columns
5 and 6 are those determined from the coupons cut from the sandwich panels and tested in bending.
The critical stresses given in columns 7, 8, and 9 were determined from the test data by the methods described in the Discussion of the. Experimental
Technique. The critical stresses listed in columns 10 and 11 are those computed by formula (3) by using a modulus of elasticity of 10,000,000 pounds per square inch and a Poisson's ratio of 0.3. The computed stress f is given and should be compared with the experimental values in column 9. The number of half waves into which the sandwich panel will buckle, according to formula
(3), is given in columns 16 to 19, inclusive.
Columns 12 and 13 of the tables contain values for the critical stresses computed by means of formula (1). It will be noted that this formula very often leads to the same values as formula (3) for square panels. This occurs when both formula (1) and formula (3) predict a single half wave in each direction; otherwise, formula (1) leads to lesser values than formula (3). Columns 14 and 15 give ratios of fto the proportional limit stress. The ratios given in column 14 were found from equation (6), written in the form:
J. A. Stress-strain and Elongation Graphs for Alclad Aluminum Alloy
24ST Sheet. National Advisory Committee for Aeronautics Technical Note
No. 1512 (figs. 1, 15). 1948.
Report No. 1833 -11-

Stress at critical load _ fF
Proportional limit stress
1
4
.
I li )
4- 2
2 \
I(1 ri
)2
+
4w2
The ratios given in column 15 were found from expression (8), written in the form:
Stress at critical load _
Proportional limit stress 28,000 rct 2 + 1 + 3 r ai6
2
In both of these equations, the values of le r a , and r eifi were obtained by using the experimental values given in columns 7, 8, and 9.
The ordinates of
figures 8, 9, and 10 are the ratios listed in column 14. The abscissas for panels with simply supported edges , are the values in column 14 divided by the values in column 9 and multiplied by the values in column 10.
Similarly, the abscissas for panels with clamped edges are the values in column 14 divided by the values in column 9 and multiplied by the values in column 11.
Conclusions
The experimental results for the stresses at the critical load of a flat sandwi.cal panel agree reasonably well with the stresses computed by formula (3).
Formula (1), however, yields some results that are considerably in error. The experimental results are limited to a narrow range by the experimental method.
This range excludes tensile stresses and limits the ratios of the various stresses to those given by the experimental results. Formula (3) is likely to be conservative if the shear stresses exceed this range.
Report No. 1833 -12-

Appendix A
Buckling of a Panel Under Two Edgewise Compressive Loads
Let Px and P y represent the loads per inch of edge on the edges of the panel parallel to the x and z axes, respectively. Then, if the deflection of the panel is w, the work done by P x during buckling is: a ib z
2 dydx
2 0 0 and that done by P is: r
=
Y 2
Pv
(a (b 2 d
— 7) dydx
Y
The total work done by the two loads is:
(Al)
(A2)
W = Wx + Wy
(A3)
Then
W = a fb a y
2 dydx
_y
2 a. jb
(111)
P x x
2 f a yb
2 ifro w
Y
) dydx dydx
(Alt)
Using the notation of Forest Products Laboratory Report No. 1583-B,6 this equation can be written:
2 0 a
2
Dw) dydx [P y + PxCit] o '6 Y
(A5)
When equation (A5) of this report is compared with equation (A17) of report,
No. 1583-B 1 2 it is seen that the formulas for a single compressive load are transformed to those for two compressive loads by the relation:
§.Ericksen, W. S. and March, H. W. Effect of Shear Deformation in the Core of a
Flat Rectangular Sandwich Panel.. Compressive Buckling of Sandwich Panels
Having Facings of Unequal Thickness. Forest Products Laboratory Report No.
1583-B. 1950.
Report No. 1833 -1 3 -

P
37.
P
X
1.4
=
Corresponding to equation (A32) of report No. 1583-Br load coefficients can be defined as:
(A6)
P x
?r
2 1 \EEy a2k
(A7) and
Py
Py n
2
I rE7=y (A8) a2N
Then, according to equation (A6) of this report and equations (A26) and (A32) of report No. 1583-B, ' py + pxC4 = p f
+ pm (A9)
The expressions in the right-hand member of equation (A9) are defined in terms of the quantities (A36) and (A37) of report No. 1583-R4 by equations (A4O) and
(A41) of that report. The quantities (A36) of report No. 1583-B4 must be evaluated for appropriate forms of the deflection function we If the edges of the panel are simply supported, the form mwx w = C sin --- sin a rsa is assumed for the deflection. This function leads to the formulas mi.1132, n2a2
Cn m2.
If .the edges of the panel are all clamped, the form m2b2
= n'a2
(A10)
-1 w - c cos (m - 1)
.
Trx y - cos (m +
7rx
1)-57 cos (n - 1) b- cos (n. + is taken. This function leads to the formulas:
Report No. 1833 14-

Appendix B
Buckling of a Panel Under Two Ed•ewise Compressive Loads and a Shear Load
The buckling of a rectangular sandwich panel under a shear load was discussed in Forest Products Laboratory Report No. 1560.1 With a few alterations, the formulas of that report can be applied to the buckling of the panel under two edgewise compressive loads in addition to a shear load. In the discussion of these alterations, the notation of report No. 1560,7 which differs from that of the main body of this report, will be used. Reference to equation (n) of report No. 15601 will be made by the symbol (1560-n).
The buckling of a panel with all edges simply supported, as treated in
Appendix Al of report No. 1560rt will be considered. Let Nx and represent the edgewise compressive loads per inch of edge on the edges of the panel parallel to the x and x axes, respectively, and N the edgewise shear load per inch of edge. If the deflection of the panel is w, the work done by the applied loads is:
2 a ib
UL -
Nx
— 1
0 0 wy Nv dxdy + T x a b
(.4 dxdy + Nxy a t o 0 0
"aw dxd
This expression replaces the expression (15604.5) for the present loading.
Then, if expressions (1560-Al and 1560.42) are substituted for w, the total potential energy of the panel takes the form:
(B].)
W
> TranCmn
2 NIcabw
2
8a
2 n = 1 n = 1
2p. 2 m `tin
N„ab-n
2
8b2 m = 1 n = 1
+
2,
Lbcy
8 m = 1 n=1 r=1 s-1
(B2) which replaces . (1560-A7). This energy expression leads to the following system of equations, which correspond to (1560415): iKuenzi, E. W. and Ericksen, 17. S. Shear Stability of Flat Panels of Sandwich
Construction. Forest Products Laboratory Report No. 1560. 1951.
Report No. 1833 -16-

aL
C +xy tan b r =
C rs rs
- 0 (B3) mj n = 1, 2,...
where
Nia2X n2 i \of Exz.v.
1:1,Ya
2
X
=
TT
2 Fx7
(BL)
(BY) and the remaining symbols are defined in report No. 1560,1 The terms L m, Lx,
•• n ••• and Lv are buckling load coefficients.
the individual facings about their own middle surfaces. This effect is believed to be negligible in the present problem if the effects of the transverse shear deformations in the core are small. If the expression V f) neglected, and the facings and core materials, are assumed to be isotropic,
(B3) takes the form mn
+ aL
(m2 + n2 a2 )
2 b2 '
11
2 a
2
)
1 +S(m2
,rs
Crsnmn
, 2 kLxm + n2a2, b2
— o (B6) m, n = 1, 2,....
By definition, the symbol H given by (1560-A6) vanishes, unless both (m + r) and (n + r) are odd numbers. The system of equations (B6) therefore separates into two parts, one in which the values of + n) are odd and the other in
Report No. 1833 .17-

which they are even. In order that one of these systems of equations will have a nonzero solution for Cam , its determinant must vanish. For a given combination of Lx and L lesser value of that causes one or the other of the determinants to vanish is the critical value. A critical combination of Lx and taken alone is determined for a rectangular panel by the condition:
Lxm
2 n
2 a2 - b2
(71
2 n212 b4
1 4. s o
22 b2
For a square panel, a critical combination of compressive edge loads is determined by (B7), with m = n = 1; that is, by
(B7)
L x +L
4
1+2S
(B 8) provided 0 < S < 0.3. If the value of S is in this range, the system (B6) leads to an interaction surface plotted to the rectangular coordinates L x, L,, and Lxy. The intersection of this surface with the plane Lxy = 0 is given by
(B8). The intersection of this surface with the plane L x = 0 can be obtained from National Advisory Committee for Aeronautics Technical Note No. 12231 for the case S =0. The close approximation of the latter curve by the formula
L
9.35
2
Lx
4
(B9) suggests that the interaction surface for a square sandwich panel with combined loads can be approximated by a parabolic surface. The surface
9,35
Lxycr - 1 7.356 1
..
(1 + 25)
4
(Lx "1-Ly)
(B10) reduces to (B9) when S = 1 7 = 0, and intersects the plane Lxy = 0 in the line given by (B8). According to the expression (1560-14), its intercepts on the axis Lx = Ly =
0,
S 0 are approximately correct. It therefore appears that,
EBatdorf, S. B. and Stein, Manuel. Critical Combinations of Shear and Direct
Stress for Simply Supported Rectangular Flat Plates. National Advisory
Committee for Aeronautics Technical Note No. 1223. 1948. (Computed values of L (k ) for various values of Lx(kx) are given in table 1 of this report.)
-a -1
• n
• n
• •••••••
Report No. 1833

if the interaction surface is parabolic, (B10) is the logical choice for determining the critical value of la.
In order to obtain further information on the reliability of (B10), computations were made with the determinants of (B3) to obtain the intersection of the interaction surface with the planes Ly = 0 and Lx = Ly. These computations, which were carried out for S = 0 and S = 0.3, indicate that the system in which the values of (m + n) are even numbers yield the critical values of
La.
In each case, the initial (9 x 9) determinant was used, and it was found that the greatest discrepancy between L xycr as a root of such a determinant and as determined by (B10) was of the order of
5 percent. These spot checks indicate that formula (B10) is sufficiently accurate for practical purposes.
The computations recorded in National Advisory Committee for Aeronautics
Technical Note No. 1223 8 indicate that a rectangular panel cannot be handled as simply as a square panel, because the interaction curve between
Lxycr and
Lx may be determined in part of the range of Lx by the root of the determinant for which (m + n) is even and in another part by the root of the determinant in which (m + n) is odd. The use of a parabolic interaction surface for rectangular panels, however, yields conservative results.
Report No. 1833 -19-

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Figure 4.--Test specimen for determining the critical stresses of a sandwich panel. The outer lumber panels will be glued to the inner lumber panels, which are glued to the sandwich panel.

Figure 5.--Testing apparatus with a test specimen in place for determining the critical stresses of a sandwich panel.