STRESSES WITI-IIN A RECTANGULAR, HAT
SANDWICI-I PANEL SUI3JECTED.TO A
UNIFORMLY DISTRIEUTED NORMAL LOAD
AND EDGEWISE, DIRECT,
AND WEAR LOADS
January 1953
This Report Is One of a Series
Issued in Cooperation with the
ARMY-NAVY-CIVIL COMMITTEE
on
AIRCRAFT DESIGN CRITERIA
Under the Supervision of the
AERONAUTICAL !WARD
No. 1838
UNITED STATES _DEPARTMENT OF AGRICULTURE
.FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
STRESSES WITHIN A RECTANGULAR, FLAT SAN:1;441CH PANEL
SUBJECTED TO A UNIFORMLY DISTRIBUTED NORMAL LOAD
–
AND EDGEWISE, DIRECT?
AND SBEAR LOADS.
_
By
CHARLES B. NORRIS, Engineer
and
WILLIAM J. ROM, Engineer
Forest Products Laboratory Forest Service
U. S. Department of Agriculture
4110,040110,
Approximate formulas are presented for the deflections, bending moments,
shear loads, and reactions of rectangular sandwich panels subjected to a
uniformly distributed normal load, two edgewise compressive loads, and
an edgewise shear load, The deflection formula was confirmed by tests
on 24 sandwich panels having aluminum facings and end-grain balsa-wood
cores.
Introduction
In many structures involving flat sandwich panels, particularly in aircraft, the panels are subjected to combined edgewise loads and a uniformly
distributed normal load. The presence of the edgewise loads increases the
deflections of the panel from those due to the normal load acting alone,
and, thus, increases the stresses acting within the panel. In order to
design these panels so that the stresses will not exceed prescribed limits,
the effect of the edgewise loads upon them must be determined.
These stresses are functions of their values when the normal load is
applied alone, of the critical value of the edgewise loads, and of the
value of the edgewise loads. Information is available for the determination of these stresses when the normal load is applied alone, and for the
report is one of a series prepared and distributed by the Forest
Products Laboratory under U.S. Navy, Bureau of Aeronautics Order No,
NAer 01336 and U.S. Air Force No, AFL-18(600)-70. Results here reported
are preliminary and may be revised as additional data become available.
2
-bbintainad at Madison, Wis.,, in cooperation with the University of
his
Wisconsin.
Report No. 1838
Agriculture-Madison
computation of the critical edgewise loads, This report presents approximate formulas for the determination of these stresses for edgewise loads
less than their critical value.
Discussion of Mathema tical alis
The deflections, bending moments, and shear loads of a rectangular sandwich panel subjected to a uniformly distributed load over its surface are
increased when edgewise loads are added, Any system of edgewise loads
(uniformly distributed along the edges) may be represented by the three
loads illustrated in figure 1. These loads M I Pp, and Pep) are
expressed in pounds per inch length of panel edge.
It is the purpose of this analysis to express the effect of the edgewise
loads on the deflections, bending moments, and shear loads of the uniform..
ly loaded panel in terms of the critical loads due to Pa, P 13 and Fitp„
acting separately. It is assumed, therefore, that the reader is familiar
with all of the reports given in the references.
If the deflections of the panel are small compared with the thickness of
the panel and the edgewise shear load is omitted, the mathematical
analysis given in Forest Products Laboratory Report No. 1834 (1)–
The abscissas of the curves of figures 4 to 7 of that report substantially
vary directly with the sum of the edgewise loads, and the ordinates become
zero when the abscissas become unity. It follows that the abscissas may
be replaced by the expression 17--- where f is a generalized stress in the
cr
facings of the sandwich and fcr is the critical value of that stress.
If it is assumed that the curve in figure 4 of Report No. 1834 is a
straight line (Which it almost is),, it is given by:
Q
A (1
nb-
er
where. A is the value of the intercept on the
b
axis
lib is the deflection due to bending
Q is Trowittianal to the uniformly distributed load as
defined in: Report No. 1834
3
—Underlined
numbers in parentheses refer to Literature Cited at the end of
this report,
Report No. 1838
-2-
O
Q
_
r
1
b -Alf
fcr
The same sort of approximation applied to figure 5 of Report No. 1834
yields
2
=B
or
=
(1 -
cr
qa2
I
B
1 -
cr
where B is the value of the intercept on the sf axis and the other
symbols are
as
defined in that report.
Using equation (105) of Report No. 1854, the deflection due to shear in
the core is
W =
2QS
2 -
s T (1
4-
ff
cr
cr )B
1 -
where the various symbols not already defined are described in Report
No. 1834.
The total deflection is:
2QS
W = Wb Ws .
A
T 2 (1
+ 0'
1
)B
(1)
1 -
f cr
This equation applies strictly to square isotropic sandwich panels, but it
may be generalized approximately by noting that when f = 0 it must yield
the deflection (W0 ) due to the transverse load alone. Thus
w
(2)
W 1cr
Report No. 1838
-3-
which should be approximately true for all types of rectangular panels.
Of course, this equation is useful only if the correct value of Wo can
be computed in advance. It yields infinite values for W when f = for,
which is consistent with the linear theory used in Report No. 1834. Of
course, if the deflections are large (ever} as large as half the thickness
of the panel) this equation will yield values of W that are too great
because of the restraint imposed by the "skin stresses" that develop.
It is possible to make similar approximations to the other values involved.
Thus, the bending moment at the center of the panel is approximately
M
=
(3)
1 for
The reaction at the center of the side is approximately
1R o 1
1R1
(4)
ff
-or
The shear load on the core at an edge of the panel is approximately
=
1
Qo
(5)
fcr
where the values Wo, Mop 1R011 and Qo are those for the uniformly distributed load acting alone and may be obtained from the appropriate
report (2).
may be approximated by the expression:
The generalized stress ratio f
for
f
f Fa F3 f
T3cr
f
F 0, e l^
faP
f
2
raper
where the numerators are the stresses in the facings of the sandwich
panel and the denominators are the associated critical stresses when
each stress is acting alone. Values of fp p er and fFecr may be computed
by means of Forest Products Laboratory Report No. 1583-B (3), or
approximately by means of Report No. 1583 (3), and fFegicr by mat= of
Report No. 1560 (4). In making these computations, it is assumed that
the uniformly distributed load is of sufficient magnitude to force the panel
to buckle in a single half wave, thus the value of the integer n in Report
1583..B is unity. This expression is taken from equation (1) of Report 1833
Report No. 1838
(2). It was chosen instead of the similar expression in equation (2) of
that report because it agrees in form (if the third term is neglected)
with the abscissas of the curves in figures 4 to 7 of Report No. 1854.
The two expressions are identical if the panel buckles into a single half
wave, which, it is assumed, it is forced to do by the uniformly distributed
normal load.
For experimental verification of equation (2) of the present report, this
equation may be put in the form:
2
fF
+
fF 3cr fFecr
Fa
il7cTr
cr
If the experiment is so made that the ratios of the three stresses to
each other remain constant
fF R = rP
fray
= raj
fFe.
and equation (2) becomes
2
r
1 -
=
a
f
F Jar
-
1
f
Facr
IFe
rap
2
f
(6)
gaper
Measurements of W at various values of fFaare obtained, Values of the
left.-hand member of equation (6) are, computed from these measurements
and plotted against the associated value of f Fa . A similar curve is
plotted by means of the computations indicated by the right-hand member
of equation (6). If the two curves coincide, equation (6) is verified.
Ae'previously stated, equation (6) applies only when both W and Wa are
small. If they are large, verification maybe obtained only in the
neighborhood of the origin of the curves. For comparison in this range,
it is convenient to use the slopes of the curves.
Report No. 1838
.5-
2
rwo )
r
cr rFacr
d(1 df
Tr)
f Zfra
.
F
rft-2-
Fe
rte.
era
er
1
(7)
fFuer
fjCP
It is evident that the accuracy of the term of equation (6) involving
ra p cannot be verified by use of the slopes of the curves at the origin.
Description of lEperimental Technique
The sandwich panels were tested in a manner similar to that described
in Report No. 1833, except that arrangements were made to apply a
uniformly distributed normal load by means of air pressure as well as
the edgewise loads,
The panels 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 cutaway in the form
of a square or rectangle, as shown in figure 2, so that the sandwich panels could buckle over their central areas. These cut-away
areas were orientated at different -.elf's, so that the exposed portions
of the panels were subjected to different values of edgewise normal and
shear loads when the total assembly was subjected to edgewise compression. The opening on one side of the sandwich panel was made air tight
so that a uniformly distributed normal load could be applied to the
sandwich panel by introducing air pressure into this opening,
Twenty-four sandwich panels were tested, all of which had 24ST aluminum,
facings 0,02 inch thick and cores of end-grain balsa wood, The panels
were divided into three classes, numbered 1, 2, end '3, with thicknesses
approximately 7/32, 11/32 0 and 13/32 inch, respectively. Each of these
classes was divided into two groupe 0 one containing Specimens having
square cut-away portions 20 inches on a side/ designated by the letter
8, the other rectangular cut-away portions 20 by 12 inches, designated
by the letter
The square cut-away portions were orientated at angles
of 0 °, 15°, 30 0 , and 45° from the horizontal (perpendicular to the
direction of the load applied 'try the iesting machine) / and the rectangular
cut-► way portions at0° 30° 0 60% and 90° from the horizontal to the
short side of the reetalagle.
'
Report No. 1838
Each specimen was given a number consisting of five parts, for example,
2 B R N 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 cut-away portion was rectangular; the N indicates that a uniformly distributed normal load was applied in the test; and the 30
indicates that the cut-away portion was orientated 30° from the hori.
zontal. The symbols B and N were not necessary to distinguish the
specimens from each other, but were used to distinguish them from the
specimens associated. with Report No. 1833.
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 portions were cut away, were each about 1/2 inch thick. The
two outer lumber panels were about 1 inch thick. The grain direction
of all four lumber panels was perpendicular to the direction of the
load applied by the testing machine, as shown in figure 2. 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 2. 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. Other rosettes were
also mounted on the panel, usually at one of the corners, for check
purposes. The air-tight opening on one side of the sandwich panel was
fitted with suitable connections for the injection of compressed air.
The ends of the completed test specimens were machined square with
those of the sandwich panels and parallel to 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 3. Channel
irons were bolted snugly to the edges of the 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. 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 cut-away portions,- The resistance wire
strain gages were connected to a recorder, as shown in figure 3. The
air-tight opening was connected to a suitable supply of compressed
air and to a manometer for the determination of the pressures applied.
The specimen was placed in the testing machine and held by a slight
load. Air pressure was introduced, and the resulting deflections were
measured by means of the central dial gage. When a prescribed value
of the , deflection was reached, the air pressure was held constant and
load applied by the testing machine. This load was applied in increments. At the end of each increment, the loading was interrupted
while the resistance-wire gages and 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,
Report No. 1838
-7-
After the specimens were tested, they were cut apart, and coupons were'
cut from the sandwich panels parallel to the edges of the cut-away
portions of the lumber paned Two such coupons were cut from each specimen, one parallel to each edge of the cut,6Way portion.. These coupons
were 2 inches Vide and 6 inches long. They were tested in shear to
determine the modulus of rigidity of the- corentiterial in the two direc
tions according po the method described in Forest Products Laboratory
Report NV, 1556.'1
Discussion of Experimental Technique
The experimental technique has the three draw-backs mentioned in
Report No. 1833. . -(1) The exact cOnditiont'at the edges of the cutaway portions of the specimen; are not known. It is known only that
the panel is supported at these edges, and that the support is somewhere
between simple support and fully clamped. (2) Tensile direct stresses
cannot be applied'brythis MethOd, (3) The ratios of the various applied
edgewise stresses to each other have definite values, depending upon
the orientation of the cutaway portion. Other ratios cannot be obtained
by this method; thus, the'range of experiments that can be performed is
restricted.
In general, the readings obtained from the two sets of strain-gage
rosettes located at the centers of the edges of the cut-away portions
were averaged to obtain the three readings for an average rosette.
That these readings were approximately the average for the whole panel
was demonstrated in Report No. 1833 (2). Two of these strains were
plotted agairst the third by using the recorded loads of the test as
reference points, Thus, the ratios of the strains to each other were
determined. These ratios were substantially constant during about the
first half of the test, The ratio of the shear strain to one of the
direct strains was obtained by use of the usual equation applying to
rosettes:
,2e1 e - ea: p
e a0
where e a and e are the direct strains associated with the orthogonal
axes a and 13 ; el is the direct strain associated with an axis at 45°
to both axes a and 0; ea
is the shear strain associated with axes a and
. The ratios of the stresses in the facings to each other were then
found by means of the usual equations:
-Method for Conducting Mechanical Tests of Sandwich Construction at
Normal Temperatures. Forest Products Laboratory Report No. 1556.
Revised February 1950. (p. 12).
Report No. 1838
.8-
fQ, = x ea +
ea
e +e
f = al
X a X (3
fo p = Geap
where E is the modulus of elasticity of the (aluminum) facings
(10,000,000 pounds per square inch); flit the Poisson's ratio of the 2
facings (0.3); G is the modulus of rigidity of the facings, and X = 1 •• .
A sample curve of this nature is shown in figure 4. The shear stress is
plotted against the sum of the two direct stresses, For square panels
the sum of the direct stresses enters the formulas as a unit. It will
be noted from the figure that the ratio of the shear stress to the sum
of the direct stresses remains constant for about half the test and
then changes back and forth. For purposes of comparison betveen the
equations developed and the results of tests only, the early stages of
the tests were used and this ratio was considered to be constant.
wo
In this figure values of the deflection and values of (1 are also
plotted against the sum of the direct stresses. The slope of the latter.
curve in the neighborhood of the origin was compared with computed values
of this slope. It may be noted that after the straight-line portion of
this curve is passed, it is concave upward. This is due to the "skin
stresses" that develop when the deflections are large. The shear term
in equation (6) tends to cause this curve to be concave downward, but
the "skin stresses" develop rapidly enough to counterbalance this
tendency. Thus, the effect of the shear term cannot be determined by
means of these tests.
Discussion of Results of Tests
Table 1 gives the slope of the experimentally determined curve (col. 10)
and the computed slope for simply supported (col. 11) and clamped
(col. 12) edges. It also gives the dimensions of the sandwich cOnstruetions, the moduli of rigidity of the cores of the sandwich panels as
determined by tests of the coupons, the deflection under normal load
alone, the normal load, - the experimentally determined ratios between
the applied stresses, and the value of the parenthetical expression of
the test term of equation (6) for both simply Supported and- clamped
edges.
Report No. 1838
It may be noted from the table that the value of the slope of the
experimentally determined curve lies between the computed values for
simply supported and clamped edges except for specimen 1 B R N 30.
Figure 5 contains plots of these values for the square and the rectangular panels. The experimental values are plotted as ordinates and the
computed values as abscissas. The 45° line indicates the positions of
the points if perfect agreement were obtained between experiment and
computation. Thus, the assumption of clamped edges yields values that
are too great, and the assumption of simply supported edges yields values
that are too small. Because the panel was supported somewhere between
these two conditionsi:At,is assumed that the computations yield, Siabuo,
stantially, the correct values.
Callum 13 and 14 of table I show that the last term of equation (6) is
exceedingly small and, therefore, that a different type of experiment'
would have 'to be devised to show its effect. The shear stresses to
which the panels were subjected were tot nearly great enough to make the
effect of this teriOeVident. : It is an approximate term that is about
correct for square panels and conservative for rectangular panels as
pointed out in Report No. 1833 (2).•
In table 1, the moduli of -rigidity -given in columns 4 and 5 are those
Obtained from the coupons cut from the panels. The panels were considered to be isotropic, so that averages of the two values for each
panel were used in the computations.'
Equations (3), (4), and (5) of the Discussion of Mathematical Analysis
were not checked experimentally. It was assumed that if equation (2)
checked the experimental values with reasonable accuracy, the other
equations would.aaso be reasonably accurate.
Conclusions
The formulas presented for the deflection, bending moment, reaction,
and shear load in a sandwich panel subjected to a system of edgewise
loads and a uniformly distributed normal load seem to yield reasonable
estimates when the deflections are small compared with the thickness of
the panel and conservative estimates for larger deflections.
The effect of the edgewise shear load was so small in the tests that
its confirmation was hot possible, In the light of other Workl however,
the equations seem to yield reasonable results for square panels and
conservative results for rectangular panels for large (as well as small)
shear loads,_ They yield reasonable values when the shear load and
uniformly distributed, normal load only are applied.
Report No. 1838
-10-
Literature Cited
(1)
Behavior of a Rectangular Sandwich Panel Under a Uniform Lateral
Load and Compressive Edge Loads. Forest Products Laboratory
Report No. 1834.
(2)
Critical Loads of a Rectangular, Flat Sandwich Panel Subjected to
Two Direct Loads Combined with a Shear Load. Forest Products
Laboratory Report No. 1833.
(3)
Effects of Shear Deformation in the Core of a Flat Rectangular
Sandwich Panel. (1) Buckling under Compressive End Load.
(2) Deflection under Uniform Transverse Load. Forest Products
Laboratory Report No. 1583.
Stiffness of Flat Panels of Sandwich Construction
Subjected to Uniformly Distributed Loads Normal to
Their Surfaces Simply Supported Edges. Forest
Products Laboratory Report No. 1583-A.
Compressive Buckling of Sandwich Panels Having Facings
of Unequal Thickness. Forest Products Laboratory
Report No. 1583-B.
Deflection Under Uniform Load of Sandwich Panels Having
Facings of Unequal Thickness. Forest Products Laboratory
Report No. 1583-C.
Deflection Under Uniform Load of Sandwich Panels Having
Facings of Moderate Thickness. Forest Products Laboratory
Report No. 1583-D.
(4)
Shear Stability of Flat Panels of Sandwich Construction. Forest
Products Laboratory Report No. 1560.
Report No. 1838
-11-
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Figure 2. --Test specimen. The outer lumber panels will be
glued to the inner lumber panels, which are glued to the
sandwich panel.
Figure 3. --Testing apparatus with test specimen in place.