1:19 FftE CO?Y
i.
• 11.: 11•
111 .11
,
March 195,5
,:%F.010/1ATIH
AND REAFFkMED
This Report is One of a Series
Issued in Cooperation with the
AIR FORCE-NAVY-CIVIL SUBCOMMITTEE on
AIRCRAFT !DESIGN CRITERIA
Under the Supervision of the
AIRCRAFT COMMITTEE of the
MUNITIONS BOARD
In Cooperation with the University of Wisconsin

EFFECT OF SHEAR SLIENAH
ON
OF SANDWICH COLUMNSI
LOADS
By
K. E. BOLLER, Engineer and
C. B. NORRIS, Engineer
Forest Products Laboratory,? Forest Service
U. S. Department of Agriculture
ININNA. PM. O.
Summary
A formula developed at the U. S. Forest Products Laboratory for determining maximum load on sandwich columns that have initial irregularities is presented in this report. Tests made in conjunction with development of the formula showed that such columns, which deflect laterally at small loads, may fail, because of shear stresses in their cores, at loads less than the critical buckling load.
Experimentally determined maximum loads for the
50 columns tested were found to agree reasonably well with those calculated by means of the formula for the same columns when the secant modulus of rigidity at failure of the cores was used in the formula. The columns tested consisted of 0.032-inch clad aluminum facings on either 3/16- or 1/2-inch cellular cellulose acetate cores. Their length varied from 10-1/4 to 30 inches and their initial deflection from 0.04
to 0.063 inch.
Introduction
Sandwich panels loaded in edgewise compression have been observed to fail from one of the following causes: (1) local instability of the facings (wrinkling);
(2) over-all instability of the panels (critical buckling); (3) direct compressive stresses greater than the strength of either the facing or the core; or
_This progress report is one of a series prepared and distributed by the Forest
Products Laboratory under U. S. Navy Bureau of Aeronautics No. NBA-PO-NAer
00854 and U. S. Air Force No. USAF-PO-(33-038)49-469E.
Results here reported were obtained during 1950.
_ .
-Aaintained at Madison, Wis., in cooperation with the University of Wisconsin.
Rept. No. 1815
-1- Agriculture-gadison

(4) excessive transverse shear stresses. Previous reports 2 ' h have provided design criteria for the first three causes of failure.
Failure because of transverse shear stress may occur if the panel is not originally perfectly flat. Any initial deflection increases the instant load is applied, and the shear load induced by this deflection may cause shear failure in the core at a load less than that required to cause failure by any of the first three causes. The major difficulty encountered is that the amount of deflection is influenced by the modulus of rigidity of the core, which changes as the core approaches failure. This change takes place only at and near the part of the core which fails -- that is, only in the part near a point of maximum shear load.
It is the purpose of this report to show that sandwich panels subjected to edgewise loads may fail due to transverse shear, and to determine the value of modulus of rigidity which, substituted in equations based on the theory of elasticity, will lead to results that agree well with experiment. In the interest of simplicity, the mathematical development and the tests were limited to columns with fixed ends. It is assumed that the information obtained can be applied to panels. Tests were made on 50 columns and predicted maximum loads compared with the test values. Total stress in the facings of these columns was within the elastic range of the facing material._
Mathematical Development
In dealing with sandwich construction the deflections due to shear should be taken into account. The total deflection y is divided into two parts; that due to bending yb, and that due to shear ys. If the sandwich is initially. 'bent, the total deflection also includes the original deflection, z. The total deflection is:
Y Yb Ys z
(1) and the total curvature is d
2 y d
2 y b d
2 n dx2 dx2 yE; dz
2
( 2 )
N orris, C. B. Wrinkling of Facings of Sandwich Constructions Subjected to
Edgewise Compression. Forest Products Laboratory Rept. No. 1810.
hMarch, H. W. Effect of Shear Deformations in the Core of a Flat Rectangular
Sandwich Panel. Forest Products Laboratory Rept. No. 1563.
Rept. No. 1815 -2-

The curvature due to bending is given by the usual formula:
,2
Yb _ M dx
2
(3) where M is bending moment and D ie bending stiffness of the sandwich construction, shear neglected.
The slope of the deflection curve due to shear is equal to the shear strain y at the neutral surface of the sandwich, thus: dy s = y dx and the curvature due to shear is: d
_ dy dx 2 dx
(4)
Substituting the expressions given by equations
(3) and
(1) in equation (2), the total curvature of the sandwich strip is: d
2 y = M dy dx 2
-15 dx dz
2 dx2
(5)
The shear strain at the neutral axis is given by:
•
AG in which S is the shear load, G the modulus of rigidity of the core material, and A a suitable area such that shear load divided by this area yields the shear stress at the neutral axis. The shear load is equal to the rate of change of the bending moments, thus:
S = dx
And, therefore:
Y =
1 dM
AG dx
Substituting this value in equation
(5) d
2
M AG cT;:f 7
_d2 y
= A ci
-AG d
2 z dx'
Rept. No. 1815
-3-
(6)

Equation (6) is general for beams and columns. It will be applied to a curved column composed of a strip of sandwich material. This column is shown in figure 1. The initial shape of the column is assumed to be given by: a2
2
The slope of this curve at the ends of the column are
(7) s =
+4
2
0
(8)
It will be assumed that the ends of the column are fixed so that these slopes remain constant as the load is applied. The shape of the loaded column is shown in figure 1. The load is applied eccentrically because of the bending moments imposed at the ends'of the strut by the fixity. The bending moment is
M=P (y - e)
Substituting the expressions given in equations (7) and (9) in equation (6):
(9)
+ k
y = k
(e
-
7
(10) in which:
1
D(.
2.
- it)
The solution of this equation is: z
y = Ci sin kx + C2 cos kx + e - 8 I)
7 a2 in which the constants of integration C1 and C2 are to be determined.
Using the conditions that: y =o when x
-
7 y =
D z o )
15
7
(cos
1.0(
(12)
Rept. No. 1815
-4-

The value of the eccentricity, e, can be determined by the condition that the slopes of the column at its ends are given by equation (8). Thus: y = 4 z_ cos kx - cos k a
7 sin k a
(1 3) and the central deflection is obtained when x = o. Thus:
7 k tan -4
The critical load is obtained when yo becomes infinite; that is, when ka -
TT
E
- 2
The critical load is, therefore, a
2
1
1
(15)
The first term in the denominator is the reciprocal of the critical load of a column with clamped ends without correction for deflections due to shear.
The second term makes the proper correction for these deflections. Theoretically the value of A changes slightly with the type of load to which the sandwich strip is subjected. If the thickness of the facings is small with respect to that of the core, the value of A is given with sufficient accuracy by the expression:
= -
2
+ c)b (16) in which h is the width of ings are not the thickness of the sandwich, c the thickness of the core, and b the sandwich strip. This expression applies even if the two facequally thick.
The critical load given by equation (15) may not be reached because the core material may fail in shear. The shear load is given by: s = y
= 4
2 sin kx
sin 0
Thus the maximum shear load is obtained when sin kx = 1
Rept. No. 1815
-5-

or
If this value is equal to
7 is obtained at the end of the strut, where: • sin kx = sin k a
7
Thus the maximum load when ka >
TT is:
P max and when ka < w it is:
T sin k a
(17) p = TAa max U--
0
(18) in which
7. is the shear strength of the core material.
A difficulty arises in the use of equations (17) and (18) in that the modulus of rigidity (G) of the core material is contained in the value of k; and, because the test is carried to failure, the value of this modulus changes with the value of x. Comparison of these equations with the results of tests indicates that the use of the secant modulus of rigidity at failure yields reasonable estimates of the maximum load.
The load P max is also contained in the value of k in equations (17) and (18), so that their solution involves a cut-and-try process.
The value of z o
plays an important part in these equations. If the strut is perfectly straight at the start (z o = o) the equations yield very Irge values of the maximum load, and the critical load given by equation (15) is then the maximum load. Thus, if the strut is nearly straight at the start, the value of z o
should be accurately known. In the experimental work subsequently described, the values of z o
were computed by use of known values of the bending moment at the center of the column. These bending moments were obtained from measurements of strain during the test. The formula used is derived as follows:
The bending moment is given by equation (9). An expression for (y - e) can be
Obtained from equation (12) and is: y- e=
4 z a
0 cos kx
7
8 D z a o
7
Rept. No. 1815 -6-

Thus, the bending moment at the center of the strut is: o
0
Dz
1 s o
.
sin k a
2 a u.lo„ for these small values:
M o
-
8 z PD
-2- E
Thus, an accurate value of z o can be obtained from the-slope of the curve relating load to bending moment at low values of the load.
(19)
Experimental Investigation
Test Specimens
The test specimens were divided into three groups. Group 1 included 20 columns that consisted of 0.032-inch clad aluminum facings and 3/16-inch cellular cellulose acetate cores. The density of the cellular cellulose acetate was between 6 and 7 pounds per cubic foot. The columns were 18-1/8 inches long and about 1 inch wide. They were cut from a single flat plate 36 inches square. This group was subdivided into four sets of five columns each.
All of the columns in a set were tested with their ends at the same slope.
The slope was varied from 0 to 0.113 inch per inch. •
Group 2 included 30 columns that consisted of 0.032-inch clad aluminum facings and 1/2-inch cellular cellulose acetate cores of the same quality as those used in group 1. They were cut from a single flat plate 36 inches square.
This group was subdivided into six sets of five columns each. Two sets of columns were cut 10-1/4 inches long, two sets 20-1/4 inches long, and two sets
30 inches long. The slope at which each set of columns was tested varied from
0.0184 inch per inch for the 10-1/4-inch columns to 0.0847 inch per inch for the 30-inch columns.
Group 3 included six shear coupons which consisted of 1/2- by 2- by 6-inch cellular cellulose acetate cores bonded between 1/4-inch steel plates. The cellular cellulose acetate in these specimens was selected at random from the material which was incorporated in the flat plates. These coupons were used to obtain stress-strain curves (fig. 2).
All columns and coupons were conditioned in a room at
75°
F. and 64 percent relative humidity, and were tested immediately thereafter.
Rept. No. 1815
-7-

Experimental Procedure
The apparatus that supported each column is shown in the test set-up in figure
3. Each end of the column was supported by two 3-inch angle irons, which were drawn together by four bolts to clamp the ends of the column (hereafter called clamps). The clamps, in turn, mere bolted to a 5/8- by 4- by 10-1/2inch bearing plate..
In order to develop initial deflections, the apparatus was made to provide various degrees of slope at the bearing ends of the columns. The apparatus was constructed so that the bearing plates could be rotated. They rested against knife edges about which the clamps and plates could be made to rotate by manipulation of the wing nuts and bolts. These bolts connected the plates to the stationary I beams on the test machine. The clamps were oriented over the knife edge so that the center line of the column coincided with the axis of rotation.
Measurements were taken of slope, literal deflection, strain, and head movement.
The slope of the bearing plates was determined from the measurement on dials located 10 inches from the axis. The amount of lateral deflection was measured as the movement of the center of the column in a horizontal direction with respect to the column's ends. This measurement was made by reading the steel scale, over which a wire was drawn tight from the edge of a top clamp to the edge of a bottom clamp. The readings were made through a telescope. Strain measurements were obtained on the facings at midlength of the column by means of electric strain gages. The vertical movement of the upper clamps with respect to the lower clamps was observed as a guide in adjusting the rate of head movement and increments of loading.
Before the columns were subjected to loads, the clamps were alined by securely fastening one face of the top set of clamps in a vertical plane with one face of the bottom set of clamps. In this state the bearing ends had no elope.
Then after a column was inserted between the jaws, the top and bottom bearing plates were brought into contact with the extreme ends of the column. With a small load on the column, the clamps were tightened to the column and to the bearing plates. The bearing plates were then rotated, one clockwise and the other counter clockwise, to the desired slope. Application of this slope to the column, of course, produced a curvature of the column as a whole. When columns were tested with small slopes, relatively small initial loads maintained the desired curvature; but for columns with large slopes, larger initial loads were found to be necessary.
The columns were loaded continuously to failure at a uniform rate of vertical head movement. During this period, data were taken at equal increments of head movement. At the maximum load the core failed in shear locally -- that is, over a short increment of length in the area between the edge of the clamp and the center of the column. These failures were the typical offset type.
The six shear coupons were tested for their shear strength and stress-strain
' curves according to the procedure in Forest Products Laboratory Rept. No.
, Methods of Test for Determining Strength Properties of Core Material for
1555,
Sandwich Construction at Normal Temperatures," revised October 1948.
Rept. No. 1815
-8-

Prediction of Maximum Loads
Formula (17) was used to predict the maximum loads of the test columns. It is repeated as follows with explanations:
P
(h + C)
8 z
OC a sin in which
0
2 a
2D 0 -
(h + c)Gs
P = maximum load in pounds per inch of width h = thickness of sandwich in inches, measured for each specimen c = thickness of the core in inches
T = shear strength of core material in pounds per square inch.
An average value of 90 pounds per square inch was obtained from the six shear coupons a = the unsupported length of the column between clamps in inches. Columns in this investigation were 5.74 inches longer than the reported lengths because each end was clamped by 3-inch angle irons z = observed initial lateral deflection at the center of the column in inches zoc = computed initial lateral deflection at the center of the column in inches. This was computed in two ways; (1) from the slope at the bearing ends and the column's length according to equation (8), thus: zoc = sa /14;
(2) for the set of five columns which were tested with zero slope, z oo was computed from a moment-load relationship, given by equation. (19), thus: c
/16 P'D. In this equation,
Mo/P 1 is the slope of the moment-load curve at the lower values of the applied load, P'; the moments, M o , were computed from the strain measurements taken at the center of the column with electric strain gages; Go , the initial tangent modulus of rigidity of the core, was found to be
5,000 pounds per square inch from tests of the six shear coupons
Rept. No. 1815 -9-

D = stiffness of the column
E f b
12 A.
(h3 - c3)
In this equation, since aluminum was used as the facing material, 10 million pounds per square inch was substituted for Ef/Xf. The width is designated by b
Gs = secant modulus of rigidity of the core material at its maximum strength. The average value from six shear coupons was 2,900 pounds per square inch.
It is, of course, apparent that the value of the maximum load on the column is obtained by trial and error, because sin 0 is a function of the load, P. The solution is obtained by assuming successive values of P in the expression for
9 until an assumed value of P equals the computed value. Although more than one value of P will satisfy the equation, only the smallest one is applicable.
That is, values of
0 should be in the second quadrant for the columns tested; however, a few values were in'the first quadrant. This means, according to the derivation of the formula, that the point of centraflexure is outside the column's length. Therefore, when this condition occurs, the maximum shear stress actually is at the end of the column and the value of 9 does not exceed w.
2
Results and Discussion
Tables 1 and 2 present the loads and related data for each of the 50 test columns. The tables show the dimensions . of the columns, the slopes at their ends, their lateral deflections, and their loads. The initial and final values of the lateral deflections are listed. Computed values of lateral deflection also are listed because they are more accurate than the observed values considering the method of measuring them and the method of applying the slope to the columns.
The maximum load given in the tables is significant because (1) it was the highest load supported by the column and (2) it marked the beginning of the shear failure. If the columns had been tested with dead loads no further observations could have been made. Since they were loaded in a testing machine, however, the maximum load did not cause complete destruction. After the maximum load was attained additional compressive head movement caused the core to fail progressively until the column reached its final load.
Figure 1 presents a graphical comparison of the observed and computed maximum loads. This figure shows that the observed values are in good agreement with the computed values. It shows that four of every five observed values lie between 90 and 110 percent of the computed ones. This also shows that the maximum loads were predicted satisfactorily by the formula, which took into account the shear strength of the core.
Rept. No. 1815 -10-

It is apparent by comparing initial deflections with the associated loads that the smaller loads were obtained from columns having the greater initial deflections, and the larger ones from columns having the lesser initial deflections.
Initial deflection varied from apparently nothing to a distance equal to the thickness of the column. The five columns that had no apparent initial deflection and were tested with zero slope developed appreciable lateral deflection due to local irregularities. These columns have particular significance because they had irregularities that were too small to be detected by ordinary observations but, nevertheless, were large enough to affect the maximum load.
If these five columns had been perfectly flat, their maximum load would have been limited by other causes of failure. One such cause is the elastic instability of the columns. The critical buckling load of each of these five columns, according to equation (15), is 586 pounds per inch of width. Computations based on other causes of failure lead to still greater maximum loads.
The Observed maximum loads, however, were between 128 and 485 pounds per inch of width, which are about 80 percent of the critical buckling load. The critical buckling load of each of the other columns in group 1 is also 586 pounds per inch of width because their critical load is independent of their initial deflection. These columns, however, failed before reaching their critical load at maximum loads dependent upon their initial deflection. The columns in group 2 also failed at maximum loads less than the critical loads, which showed that these maximum loads were also dependent on initial deflection. The formulas developed here, which include shear strength, show how the maximum loads vary with initial deflections.
I will be noted that in tables 1 and 2 the computed values of the initial deflections do not always agree well with the observed values. The computed values are based on equation (7), which may not truly represent the initial shape of the column. They are, in a sense, fictitious; but they are the proper values for substitution in equations (17) or (18) because these equations contain equation (7).
When the data were analyzed, several values for modulus of rigidity of the core material were tried. These moduli were obtained from the stress-strain curve shown in figure 3. Among those tried were the modulus at low stresses, the tangent modulus at one-half the ultimate strain, and the secant modulus at failure. The last of these gave the best correlation between the test data and equations (17) or (18). It seems reasonable to assume that the use of this modulus for any core material will give approximate correlation between computed and test values.
The formulas derived can be used for design purposes if a design value of initial lack of flatness is determined. Its use will be similar to that of the design value of eccentricity used in the design of structural columns. It is possible that the results obtained for columns can be extended to include panels of sandwich construction subjected to edgewise loads.
Rept. No. 1/3155 -11-

Conclusions
The formulas derived and the results of the tests made show that sandwich columns may fail in transverse shear at loads less than those consistent with other types of failure. These shear failures may be caused by initial deflections that are exceedingly small. The results obtained can be used for design purposes if standard initial deflections are established. The secant modulus of rigidity of the core material at failure should be used in the calculations. It is possible that the method developed can be extended to include sandwich panels subjected to edgewise loads.
Rept. No. 1815 -12-

Table 1. --Compari
Li on of ma.Y.-irrza loaus due to she-sr failures in s,:inciwich columns r:.032-inch clad 21:f:,T aluminum facings on 3/16-inch cellular cellulose acetate cores
Dimension, : Slope : Lateral deflections : of column= : of
Length: Thick
-: clamps : Observed : Com- :
: puted :
: ness
Column loads
Observed Com-
: puted a : h :
::
:Initial:Final:Initial: Maximum : Final zo yo zoc Po
P-
:
: Maximum
Pc
In. : In. :In.per in.: In.
: In. : In. :Lb.per in.:Lb.per in.:Lb.per in.
: of width : of width : of width
: :
16.11 :0.276 : 0
18.11 : .275 : 0
:
: 0
: C
: :
:0.33 :0.0570 :
: .35 : .0891 :
•
452
431
•
:
.
397
391 :
:
:
412
394
18.11 : .273 : 0 : 0 : .35 :
.0649 : 436 : 397 : 401
18.11 : .275 : 0
18.11 : .271 : 0
: 0
: 0
: .39 : .0520 :
: .36 : .0571 :
485
428 :
: 433 :
395 :
412
400
•
•
18.11 : .273 : .0188
: .08
: .41 : .081 :
.0188
: .07
: .32 : .0851 :
406
156
: 388
3'6 :
: 392
16.11 : .272 :
18.11 :
.275 :
389
.0188
: .06
: .34 : .0851 : 39 1 1384 : 396
18.11 : .273 : .0188
: .07
: .40 : .0651 : 324 : 310 : 392
18.11 :
.275 : .0188
: .07
• .37 :
.0851 :
399 : 390 : 396
18.11 :
.270 : .0847
: .27
: .67
.3835 :
2314
18.11 : .270 : .0847
: .27
: .56
• .3835
• 210
• 227
206 :
: 244
244
18.11 : .272 : .0847
: .23
• .55
• .3835
207 207 : 246
18.11 : .270 : .0847
: .28
: .57 : .3E:35 : 234 : 232 : 244
18.11 : .272
.0847
: .23
• .55 : .3e35
231 : 228 : 246
18.11 : .270 : .1130
: .38
: .80 : .5116 : 204 :
18.11 : .269 : .1130
: .26
:.55 : .5116 :
20; :
18.11 : .276 : .1130
: .27
: .60 :.511: :
18.11 : .276 : .1130
: .28
: .63 :
18.11 : .276 : .1130
:
:
219
233
:
No data obtained
202 : 190
207 : 189
218
•
232
194
194
1
-Test columns were about 1 inch wide.
Rept. No. 1815

Table 2.--Comparison of maximum loads due to shear failures in sandwich columns having 0.032-inch clad 24ST aluminum facings on 1/2-inch cellular cellulose acetate cores
Dimension of column- :
Slope : Lateral deflections : of : : clamps : Observed : Com- :
Length:Thick-:
: ness s : : puted :
: a : h :
:
Column loads
Observed
:
: Computed
:Initial:Final:Initial: Maximum : Final : Maximum
: z o : y o : z oc
: Po : Pf : Pc
In. : In. :In.per in.: In. : In. : In. :Lb.per in.:Lb.per in.:Lb.per in.
. : . : : : of width : of width : of width
.
10.26 :0.546 :
10.26 : .543 :
10.26 : .543 7
10.26 : .545 :
. .
0.0184 : 0.03 :0.10 :0.0472 : 1,253 : 1,253 : 1,310
.0184 : .04 : .16 : .0472 : 1,264 : 1,252 : 1,302
.0184 : .03 : .13 : .0472: 1,186 : 1,186 : 1,302
.0184 : .03 : .11 : .0472 : 1,277 : 1,277 : 1,307
10.26 : .547 : .0184 : .03 : .11 : .0472 : 1,194 . 1,194 : 1,313
10.26 : .546 :
10.26 : .548 :
10.26 : .552 :
980 : 980 1,181
10.26 : .554 : .0373 : .07 : .16 : .0957 : 1,052 : 1,052 : 1,202
10.26 : .555 :
.0373 : .07 : .15 • .0957 :
.0373 : .07 : .14 : .0957 :
.0373 : .07 : .14 : .0957.
984 :
1,061 :
'977 : 1,186
1,061 : 1;197
.0373 : .07 : .16 : .0957. 1,126 : 1,109 : 1,204
. . :
• • : . .
20.26 : .542 : .0242 : .13 • .43 : .1226 : 825 : 819 : 963
20.26 : .542 : .0242 : .13 : .45 : .1226 : 917 : 917 :
20.26 : .541 : .0242 : .13 : .43 : .1226 : 932 : 929 :
20.26 : .548 : .0242 : .12 : .43 : .1226 : 1,043 : 1,043 :
20.26 :.545 : .0242 : .13 : .50 : .1226 : 1,071 : 1,065 :
963
961
978
971
20.26 : .543 : .0565 : .23 : .55 : .2862 :
20.26 : .538 : .0565 : .18 : .50 ; .2862 :
20.26 : .548 :
20.26 : .542 :
20.26 :.545 ;
.0565 : .17 : .50 : .2862 :
.0565 : .21 :.46 : .2862 ;
.0565 : .23 : .60 : .2862 :
30.01 : .538 : .0338 : .23 :1.15 : .2536 :
30.01 : .539 : .0338 : .21 : .96 : .2536 1
30.01 : .538 : .0338 : .27 :1.02 : .2536 :
30.01 : .539 : .0338 : .26 : .98 : .2536 :
30.01 : .538 : .0338 : .20 :1.19 : .2536 :
748
720 :
735 :
664 :
791 :
634 :
705 :
627
611
665 :
744 :
720 :
735 :
664 :
785 :
603 :
665 :
611 :
585 :
622 :
757
748
766
755
760
674
676
674
676
674
30.01 : .533: .0847 : .47 :1.29 : .6335 :
30.01 : .533 : .0847 : .37 :1.31 : .6335 :
30.01 : .530 : .0847 : .37 :1.29 : .6335 :
30.01 : .535 : .0847 : .44 :1.33 : .6335 :
30.01 : .541 : .0847 : .43 :1.65 : .6335 :
461 :
411 :
432 :
463 :
500 :
446 :
411 :
418 :
459 :
490 :
482
482
478
484
492
1
-Test columns were about 1 inch wide.
Rept. No. 1815

Figure 3.--Apparatus for supporting the bearing ends of sandwich columns and for providing various initial deflections in the columns.
Z M 78892 F