!CULTURE ROOM

TORSION OF SANDWICH PANELS OF

TRAPEZOIDAL, TPIANGUIAP, AND

RECTANGULAR CROSS SECTIONS

June 1960

No. 1874

This Report Is One of a SeriesIssued in Cooperation with theANC-23 PANEL ON COMPOSITE CONSTRUCTIONFOR FLIGHT VEHICLESof the Departments of theAIR FORCE, NAVY, AND COMMERCE

UNITED STATES DEPARTMENT OF AGRICULTUREFOREST PRODUCTS LABORATORY

FOREST SERVICEMADISON 5, WISCONSIN

ID Cooperation the University of Wisconsin

A

TORSION OF SANDWICH PANELS OF TRAPEZOIDAL,

TRIANGULAR, AND RECTANGULAR CROSS SECTIONS!

By

SHUN CHENG, Engineer

Forest Products Laboratory, 2 Forest ServiceU. S. Department of Agriculture

Introduction

A theoretical analysis is made of the torsion of a sandwich panel with atrapezoidal cross section. The facings are of uniform and equal thick-ness. The core is tapered so that its cross section taken perpendicularto the x axis is a trapezoid as shown in figure 1. Its cross section takenperpendicular to the y axis is rectangular. Such sandwich panels areemployed in aircraft for control and stabilization of flight.

McComb (2)3 considered the torsion of shells having reinforcing cores)3of rectangular, triangular, or diamond shape cross section. The shellswere like facings of a sandwich panel except that they were wrappedaround and completely enclosed the cores. Both the shells and the coreswere taken to be isotropic. McComb used the Saint Venant theory oftorsion, which assumes that the distribution of the shear stress is thesame on all cross sections perpendicular to the axis of twist.

-This progress report is one of a series (ANC-23, Item 57-4) preparedand distributed by the Forest Products Laboratory under U. S. Navy,Bureau of Aeronautics Order No. NAer 01967 and U. S. Air ForceContract No. DO 33(616)58-1. Results reported here are preliminaryand may be revised as additional data become available.

?Maintained at Madison, Wis., in cooperation with the University ofWisconsin.

3—Underlined numbers in parentheses refer to Literature Cited at the end

of the text.

Report No. 1874 -1-

Seide (4) considered the torsion of sandwich panels having isotropicfacings of equal thickness, orthotropic cores, and rectangular crosssections. He also used the Saint Venant theory.

The author (1) also considered the torsion of sandwich panels havingisotropic facings of equal thickness, orthotropic cores, and rectangularcross sections. The torque was applied by forces concentrated at thecorners of the panel and a rigorous solution obtained. Numerical re-sults of this solution were compared with those of a similar SaintVenant solution.

The present report is concerned with the torsion of sandwich panelswith trapezoidal cross sections. The facings are taken to be isotropicmembranes. The core is taken to be a honeycomb structure with axesof the cells placed substantially perpendicular to the facings. Becauseof this orientation, the stresses in the core associated with strains inthe plane of the panel are so small that they may be neglected.

The derivation of a system of suitable differential stress strain relationsis carried out by means of the variational theorem of complementaryenergy in conjunction with Lagrangian multipliers (3) (6). These equa-tions apply to any type of loading. Saint Venant torsion is assumed intheir solution for sandwich panels of trapezoidal, triangular, andrectangular cross sections.

The formula for the torsional stiffness of a sandwich panel of rectangu-lar cross section so obtained is similar to Bredt' s formula for thintubular sections (5) and agrees with the infinite series solution givenin the previous work (1).

Notation

x, y, z rectangular coordinates for core (fig. 1).

x l , y i , z 1rectangular coordinates for facings (fig. 1).

b width of sandwich

h half thickness of the smaller side of core (fig. 1).

t thickness of facings.

Report No. 1874 -2-

D

a slope of facing (fig. 1) .

E Young' s modulus of elasticity of the facings.

v Poisson' s ratio of the facings.

+ v ), shear modulus of the facings.

2(1

Gxz Gyz shear moduli of the core.

ox, Ty, T stresses in facings.

Txz Tyz stresses in core.

p load per unit area.

U strain energy.

v1 , v2 , v 3 , ..v6 generalized boundary displacements.

w , (3 , 'I Lagrangian multipliers.

C1, C2 , C3, ..C6 constants.

A1 , B1 , Dl constants.

10 , Ko , K1 modified Bessel functions of first and second kind oforder zero and one.

xz

Gt cos a tan2a

T applied torque

angle of twist.

length of sandwich

2 t, 22Eth (1+-2T.I ) 4Gth (1+2n.

(1- v2 ) v )

Report No. 1874 -3-

Mathematical Analysis

The trapezoidal sandwich panel is shown in figure 1. Let the xy plane bethe middle plane of the core and xiyi plane be the plane of the facing

plate with zi axis along the normal to this plane, as shown in figure 1.

The equation of the xiyi plane is given by

z = h + y tan a

(1)

For points in the plane z i = 0

= x, = cos a '

(2)

Since xj. = x, only x will be used hereafter.

It is assumed that:4—

(1) The core stiffnesses associated with plane stress componentsare negligibly small..

(2) The facings are treated as isotropic solid membranes.

Under these assumptions the strain energy of the sandwich plate is givenby

U

+1-Iff •r 2 •r 2( • xz •yzGXz Gyz )

dx dy dz

where the subscripts f and c refer to the integrations throughout the facelayers and the core.

As Tx, try, and Txy of the core are assumed to be negligibly small, it

follows from the differential equations of equilibrium of the core that

4—These assumptions have been used in many previous analyses and are

known to represent usual sandwich construction having honeycombcores.

(3)

Report No. 1874 -4-

aT XZ

8z0T

= o, Yz - 0az

(4)

the transverse shear stresses do not vary across the thickness of thecore:

Equation 3 becomes, with (2) and (4),

1 bU =if

0 0

t [1

2 1 21COS a

(6.3C2 + Cr - 2v0-- o- ) +—x Y1 G

zTxz2 + 1

G yz Tyz ).1 dx dy

xz G

From summation of moments and forces of a differential element of thesandwich plate, as shown in figure 2, the following equations of equilibri-um are found

ao-x aT _ 0A

(GZ t COS a) xz t (2Z t COS a) - (6)COS a 0Y

t (2z cos a + t) acrYii - 2ZTyz t (2z + t ) DT = 0cos a T3Z

(7)8y

aTxz aTv aT so-+2z + 2T tan a + z2z • + 2t tan a — + 2t sin a Y1yz

8x ay 3x ay

p = 0

(8)

Equations (6), (7), and (8) are three equations for five unknowns--o- cr , T, Txz , and T . To obtain further equations, use is madexof the stress-strain relations. This is done here through the use ofthe variational theorem of complementary energy (3) (6), the Eulerequations of which are equivalent to the required stress-strain relations.The expression to be varied takes the following form:

(5)

Report No. 1874 -5-

ay- y sin a - (z cos a + av = 0 (11)

ay ay

v Tx)

sin aE cos a

I =fit [1 (o- 2 + o- 2 v o- ) + —1 T2 ] T 2 +

cos a E x Y1 x Y1 G Gxz xz

1 2% , aT aT vzT yz ) w k2z xz + 2 -ryz tan a + 2z +

Gyz Ox ay

81:rxacry

2t •1 sin a +2t aT tan a +ayex

[ t (2z + t cos a ) ax+ - COS a

- 2Z T xz t (2Z + t COS cr) + y [t (2z cos a + aryl8y

- 2z-r+ t (Zz t ) aT)3 dx dyb

(v i Txzyz cos a ax

I

v 2 T + v 3 crx)x=o dy - (v4 Tyz + v 5 T + v6o-yi

) y =o dx

x=.9 y=bo

where w, and y are Lagrangian multipliers and are functions of xand y and w is subsequently found to be the deflection in the z direction.Values v1 , v 2, ... v 6 are the generalized boundary displacements of theproblem.

After carrying out the variation 61 = 0, integrating by parts and trans-forming the appropriate surface integrals to line integrals by means ofGreen' s theorem, the variational equations (Euler' a equations) are:

2 (crx l- (2z + t cos a) = 08

(9)

(1 0)

1 p tan a (z + t cos a 813T tan a awG cos a 8x 2 ay

(z + ) 8= 02 cos a

(12)

Report No. 1874 -6-

(13)

(14)

(17)

1 aWT —xz - 13 = 0

Gxz ax

z T

a(WZ)W tan a - zy = 0T y

ayz ay

These equations must be satisfied by values of (Tx, Cry T Txz T yz1

w, /3, and y, which render the complementary energy a minimum.

The boundary conditions (three at each edge of the plate) arising fromthe independent vanishing of each term of the two line integrals given in(9) are at x = 0 and x = 1

v 1 = 2wz

v2 = t (2w tan a + 2z N + t y)Cos a

2t zp + t2gv3 - cos a

at y = 0 and y = b

v4 = 2wz

v5 = 2tzi3 + t2 f3 cos a

v6 = 2w sin a + 2tzy cos a + t2y

Equations (10) to (14), together with equations of equilibrium (6), (7), and(8), represent a complete system of equations for the eight functions CrX

Cr y1 T Txz Tyz w, fi, and

Solutions

1. Trapezoidal Cross Sections

For a sufficiently long plate under torsion, the stresses and strains maybe considered not to vary along the longitudinal axis and therefore areindependent of x. Then crx = 0 and equations of equilibrium (6), (7), and(8) become

Report No. 1874 -7-

T XZ

dc-Y1

t t cos a dT)

dcr

(21)

(22)ZzT yz

dy t (2z cos a + t)

dTT yz

irmtana+z—+t

dy' 1 = 0sin a (23)

dy

The solution of equations (22) and (23) for Tyz and aryl , and the use of theboundary conditions that both stresses are zero when y = 0 and y = b leadsto

1 =T yz= 0 =

(24)

Equations (10), (11), and (14) thus become

813 =0, 0, or f3 = f (y) (25)8x

8w .sm a + y sin a + (z cos a + = 0ay 8y

w tan a - a(wz) ay

- zy = 0 (27)

Equation (27) may be reduced to

Y = Ow

Substituting y = - aw into equation (26) we obtainBy

(27a)

aY = oBy

and hence

or y = F(x) only

w = - yF(x) + F 1 (x) (27b)

because of equation (27a).

(26)

Report No. 1874 -8-

From equations (13) and (25) and the assumption that the stresses areindependent of x, it follows that

W = xF2(Y) F3(Y)

(27c)

In view of the above results of (27a), (27b), and (27c) we obtain

w = - (Cix + C 5) y + C 4x + C6 (28)

= Cix + C 5(29)

where C1 , C 4, C 5 , and C 6 are constants of integration to be determined.

From equations (12), (13), (21), (28), and (29), it follows that

4Gd2T (2z tan a - t sin a) dT xzzT

cry2 z (2z + t cos a) dy tG cos a (2z + t cos a)2

- 2Gxy Ciz(4z + t cos a + )COS a

t (2z + t cos a)2

This equation, with equation (2), can be written as

d 2 T(1- t

+ 2z

dz 2(1+ 2z

cos a) GXZ TdT

a) tG cos a tan 2a (1 + -- cos a) 2cos ) dz

-2C 1 Gxz z + t cos a + t )z 4z cos a

t tang a (1 + t cos a)22z

Since facings are treated as membranes, is relatively small as2z

compared with 1. With this consideration, the above equation may bereduced to

z d2 T dT G Txz -2C1Gxz z

dz 2 dz tG cos a tan 2 a t tan 2 a

(30)

(31)

Report No. 1874 -9-

The complete solution for T with three arbitrary constants C 1 , C2, andC3 is

T = ZCIG cos a (z + 11.•-) + C 21,0 (2 ) + C 3K0 (2 j) (32)

The last two arbitrary constants C2 and C3, which are determined fromthe boundary conditions that T = o at y = o and y = b, can be expressedin terms of Ci as

+ K0 (2 ITITI)

+ b tan a +4-1 Ko Ifr(h+ b tan a))C2 -2 G cos aC I

Io (2 K0 (2 Fir))

10 (2 t/r(h+ b tan a) K0 (2 Lir (h b tan cx))

Io (2 I xr17. )

Io (2 Vr(h+ b tan a))

(h +4.)

(h 4- b tan a 4)C3

= 2 G cos a (33)C l

I0 (2 %5F). ) K0 (2 irr17)

I0 Vr(h + b tan a)) Ko Lir(h + b tan a))

It can be shown that the resultant of the forces distributed over the endsof the plate is zero, and these forces represent a couple, the magnitudeof which is

T = 4t T zdy + t 2 (1 + cost T dy

b b

cos a (34)

If the displacement v1 and v 2 are set to be zero at x = 0, then w and yfrom equations (28) and (29) become

w = - Cixy + C 4x (35)

clx ( 3 6)

Report No. 1874 -10-

The torsional stiffness T may be obtained from

bT8 =(Vi T xz + V2 T ) dy

f (37)x =1

0

Substituting for vi and v2 from equations (15), (16), (35), (36), and for- --.-Txz from equation (21), then integrating by parts and making use of equation(34), we obtain

= 0

( 3 8)

When the value of T from equation (32) is substituted into equation (34) andthe integration carried out, we obtain

zT 2tG cos a + cos 2 z2 z0 tan a

[4 ( z33 + 2

zr +cos a (2 + 7)]

r 2 tan a{. C1

4t C2(rz + 1) I1 (2 rz ) - (rz) Io (2 V1Z- )]

_ C3 [ (rz + 1) Ki (2 rz ) + (rz) Ko (2 FZ 11}C1

t 2 FT (1 + cos t a) [ C (2 )r tan a cos a CT

h- b tan aC3

- K 1 (2 t RT )]Cl

h

where h and h + b tan a are lower and upper limits of the integration of

equation (34) with respect to z .

( 3 9 )

Report No. 1874 -11-

For rz > 25 equation (39) can be expressed approximately as

t2tG cos a 3 2z z t(1 + cos 2 a) (Z 2 Z )1[4( + )0 ta.n a 3 2r cos a 2 r

5

+ 4t(rz)4- C2 2 FT 19 465

[0.2821 — e (1 - +r 2 tan a Cl 512 rz

1611:7

1785 31L5- 105 _)

8192 rz rz 8192 (rz)2a-

C 3 -2 Irz 19 465 1785- 0.8862-- e (1 + + +

Cl 16 ii. z 512 rz 8192 rz , IFT

1315105 t2(rz)7r (1+ cos 2 a) + )] 4'

8192 (rz) 28192 (rz) 2 ./72- r tan a cos a

C 2 2 11-7-z-3 15 105 [0.2821 — e (1 - )Ci 16, 1,/ -"-512rz 8192 rz . rz

C 3 -2 ,//- 0.8862 — e (1 +• 3 15

Cl 161/ 512 rz

+ b tan a105

)1

(40)8192 rz

Report No. 1874 -12-

2. Triangular Cross Sections

For h = 0 and z = y tan a, equation (30) may be written as

d2 T + (y2 t2 cos 2 a) dTy (y t cos a)2 _ Gxz Y21- 2 tan a dy 24 tan2 a dy tG sin a

t cos a + t- 2 Gxz CI y2 (y +

4 tan a 4 sin a)

too

The above differential equation can be solved for T by letting T = Anyn.n = o

For a very small t, as was assumed, the above equation may be reducedto

d 2 T dT GxzT Y + — -

dy2 dy tG sin a

- 2G1 Gxz y

t(41)

Because equation (41) can be obtained from (31) by placing h equal to zero,equations (32) to (40) are applicable to triangular cross sections by settingh = 0.

3. Rectangular Cross Sections

By setting a = 0, equation (30) is reduced to

d2T 4h Gxz T 4Ci h Gxz (42)

dy2 tG (2h + t) 2t (2h + t)

where Ci = according to equation (38).

It can be verified by direct substitution that the general solution ofequation (42) is given by

T = AI sinh y 1 y + B1 cosh Ni y + Dl(43)

where yi 4h- Gxz, D i = G (2h + t)

Gt (2h + t)2

(44)

Report No. 1874 -13-

The first two arbitrary constants Al, and Bi, which are determined fromthe boundary conditions that T = 0 at y = 0 and y = b, can be expressed interms of D 1 as

(cosh yl b - 1)A l - h yi b

D1, B 1 = - D1sin

thus

T = D [ (co

.sh y b 1) sinh y y - cosh y y + 1]

Y 1 J

All equations of equilibrium are satisfied identically except equation (21),which becomes

T t (211 + t) d-r (47)

2h dy

Introducing (46) into (47), results in

tGxz [ (cosh y b - 1)T xz = cosh y 1 y - sinh y] D 1(48)

hG sinh b

The resultant torque due to stresses is equal to the applied torque T

b b

T = f t (2h + t) -rdy - f 2h-rxydy (49)

Substituting T from (46) and -r xz from (48) into equation (49) and then

integrating, gives

D1 =

T (50)

2t (2h + t) [b - 2 (cosh y i b - 1)

] Y 1 sinh y l b

(45)

(46)

Report No. 1874 -14-

The torsional stiffness is obtained from equations (50) and (44)

T = 2t (2h + t) 2 Gb [1 - 0 yib sinhyib

It is evident that equation (51), similar to equation (142) of ForestProducts Laboratory Report No. 1871 (1), is independent of plate length

, Young' s modulus Ec , and shear modulus Gyz of the core.

The results of numerical computations based on equation (51) are givenin table 1. The results based on the infinite series solution (1) are alsolisted in table 1 for comparison with the results obtained by the presentmethod. The numerical computations were for sandwich plate of variouswidths and the following properties:

t = 0.0125 inch

h = 0.25 inch

Gxz = 25,000 pounds per square inch

G = 4 x 10 6 pounds per square inch

The results show a maximum difference in torsional stiffness betweenthe two methods of 3 percent. Their differences can be attributed tosmall errors or lack of using sufficient terms in the infinite series asgiven by equation (142) of Forest Products Laboratory Report No. 1871.It may be concluded that the present result, which is in a form similarto R. Bredt' s formula for thin tubular sections (5), is in excellentagreement with the infinite series solution.

2 (cosh yi b - 1)(51)

Report No. 1874 -15-

Literature Cited

(1) Cheng, S.1959. Torsion of Rectangular Sandwich Plates. Forest Products

Laboratory Report No. 1871.

(2) McComb, H. G., Jr.1956. Torsional Stiffness of Thin-Walled Shells Having Rein-

forcing Cores and Rectangular, Triangular, or DiamondCross Section: U. S. National Advisory Committee forAeronautics, Tech. Note 3749.

(3) Reissner, E.1947. On Bending of Elastic Plates, Vol. V, No. 1, April,

Quarterly of Applied Mathematics.

(4) Seide, P.1956. On the Torsion of Rectangular Sandwich Plates. Journal

of Applied Mechanics, Vol. 23, No. 2.

(5) Timoshenko, S. and Goodier, J. N.1951. Theory of Elasticity. New York, McGraw-Hill.

(6) Wang, C. T.1953. Applied Elasticity. New York, McGraw-Hill.

Report No. 1874 -16- 1.-21

Table 1.--Torsional rigidity of rectangular sandwichplate as computed by two methods

0 7F

0

In. . Lb.-in.2 : Lb.-in.2

1 : 3,500 3,360

2 : 19,000 : 19,100

3 41,920 41,900

4 67,280 : 66,800

5 : 93,200 : 92,300

6 : 119,500 : 118,000

8 172,000 169,000

10 : 224,600 220,000

20 487,200 : 476,000

-Obtained from table 2 of Forest Products Laboratory Report No. 1871.

b method infinite solutionlby present : by series

Report No. 1874

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