!CULTURE ROOM
TORSION OF SANDWICH PANELS OF
TRAPEZOIDAL, TPIANGUIAP, AND
RECTANGULAR CROSS SECTIONS
June 1960
No. 1874
This Report Is One of a Series
Issued in Cooperation with the
ANC-23 PANEL ON COMPOSITE CONSTRUCTION
FOR FLIGHT VEHICLES
of the Departments of the
AIR FORCE, NAVY, AND COMMERCE
A
FOREST PRODUCTS LABORATORY
MADISON 5, WISCONSIN
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
I D Cooperation
the University of Wisconsin
TORSION OF SANDWICH PANELS OF TRAPEZOIDAL,
TRIANGULAR, AND RECTANGULAR CROSS SECTIONS!
By
SHUN CHENG, Engineer
Forest Products Laboratory, 2 Forest Service
U. S. Department of Agriculture
Introduction
A theoretical analysis is made of the torsion of a sandwich panel with a
trapezoidal cross section. The facings are of uniform and equal thickness. The core is tapered so that its cross section taken perpendicular
to the x axis is a trapezoid as shown in figure 1. Its cross section taken
perpendicular to the y axis is rectangular. Such sandwich panels are
employed in aircraft for control and stabilization of flight.
McComb (2)3
)3 considered the torsion of shells having reinforcing cores
of rectangular, triangular, or diamond shape cross section. The shells
were like facings of a sandwich panel except that they were wrapped
around and completely enclosed the cores. Both the shells and the cores
were taken to be isotropic. McComb used the Saint Venant theory of
torsion, which assumes that the distribution of the shear stress is the
same on all cross sections perpendicular to the axis of twist.
-This progress report is one of a series (ANC-23, Item 57-4) prepared
and distributed by the Forest Products Laboratory under U. S. Navy,
Bureau of Aeronautics Order No. NAer 01967 and U. S. Air Force
Contract No. DO 33(616)58-1. Results reported here are preliminary
and may be revised as additional data become available.
?Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
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 isotropic
facings of equal thickness, orthotropic cores, and rectangular cross
sections. He also used the Saint Venant theory.
The author (1) also considered the torsion of sandwich panels having
isotropic facings of equal thickness, orthotropic cores, and rectangular
cross sections. The torque was applied by forces concentrated at the
corners of the panel and a rigorous solution obtained. Numerical results of this solution were compared with those of a similar Saint
Venant solution.
The present report is concerned with the torsion of sandwich panels
with trapezoidal cross sections. The facings are taken to be isotropic
membranes. The core is taken to be a honeycomb structure with axes
of the cells placed substantially perpendicular to the facings. Because
of this orientation, the stresses in the core associated with strains in
the plane of the panel are so small that they may be neglected.
The derivation of a system of suitable differential stress strain relations
is carried out by means of the variational theorem of complementary
energy in conjunction with Lagrangian multipliers (3) (6). These equations apply to any type of loading. Saint Venant torsion is assumed in
their solution for sandwich panels of trapezoidal, triangular, and
rectangular cross sections.
The formula for the torsional stiffness of a sandwich panel of rectangular cross section so obtained is similar to Bredt' s formula for thin
tubular sections (5) and agrees with the infinite series solution given
in 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-
a
slope of facing (fig. 1) .
E
Young' s modulus of elasticity of the facings.
v
Poisson' s ratio of the facings.
, shear modulus of the facings.
2(1
+v)
Gxz Gyz
shear moduli of the core.
ox, Ty, T
stresses in facings.
Txz T yz
stresses in core.
p
load per unit area.
U
strain energy.
v1 , v 2 , v 3 , ..v6 generalized boundary displacements.
Lagrangian multipliers.
w , (3 , 'I
C1, C2 , C3, ..C6 constants.
A1 , B 1 , D lconstants.
1 0 , K o ,
K1
modified Bessel functions of first and second kind of
order zero and one.
xz
Gt cos a tan 2 a
T
applied torque
angle of twist.
length of sandwich
2
2Eth (1+-2T.I )
t, 2
4Gth (1+2n.
D
(1- v 2 )
Report No. 1874
v)
-3-
Mathematical Analysis
The trapezoidal sandwich panel is shown in figure 1. Let the xy plane be
the 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 components
are negligibly small..
(2) The facings are treated as isotropic solid membranes.
Under these assumptions the strain energy of the sandwich plate is given
by
U
•r
+1-Iff
( • xz
2
2 •r
•yz
dx dy dz
GXz Gyz )
(3)
where the subscripts f and c refer to the integrations throughout the face
layers and the core.
and Txy of the core are assumed to be negligibly small, it
follows from the differential equations of equilibrium of the core that
As Tx, try,
4
—These assumptions have been used in many previous analyses and are
known to represent usual sandwich construction having honeycomb
cores.
Report No. 1874
-4-
the transverse shear stresses do not vary across the thickness of the
core:
aT XZ
8z
= o,
0T
Yz - 0
az
(4)
Equation 3 becomes, with (2) and (4),
1 b
2 - 2v0-- o- ) +—1 2
t U =if
[1 (6.3C2 + Cr
x Y1 G
1COS a
00
zTxz 2
G xz
Tyz ).1 dx dy
+1
G
yz
(5)
From summation of moments and forces of a differential element of the
sandwich plate, as shown in figure 2, the following equations of equilibrium are found
COS a
t (2z
(GZ
t COS a)
ao-x
cos a + t) acrYii -
xz t (2Z t COS a)
2ZTyz
8y
t (2z + t
cos
aT _
0Y
A
) DT = 0
a
aTxz
(7)
T3Z
so- Y1
aTvz
aT
2t
sin
a
—
+
2z
•
+
2t
tan
a
+ 2T
tan
a
+
yz
ay
3x
ay
8x
p=0
2z
(6)
- 0
+
(8)
Equations (6), (7), and (8) are three equations for five unknowns-ox cr , T, Txz , and T . To obtain further equations, use is made
of the stress-strain relations. This is done here through the use of
the variational theorem of complementary energy (3) (6), the Euler
equations of which are equivalent to the required stress-strain relations.
The expression to be varied takes the following form:
Report No. 1874
-5-
I =fit
cos
2%
1
T yz )
Gyz
) + —1 T 2 ]
v o-
x Y1 G
2
(o- 2 + oY1
E x
[1
a
T 2
Gxz xz
aT vz
aT
xz + 2 -ryz tan a + 2z
w , k2z
ay
Ox
2t aT tan a + 2t
ex
acry•1
sin a +
ay
[ t
+ +
- 2Z T xz t (2Z + t COS cr)
COS a
+
+
(2z + t cos a ) 81:rx
ax
y [t (2z cos a +
aryl
8y
b
-
2z-r+
yz
t (Zz t ) aT )3
cos a ax
dx
(v i
dy
Txz
I
v 2 T + v 3 crx)x=o dy -
x=.9
o
(v4 T yz + v 5 T + v 6 o- yi ) y =o dx
y=b
(9)
and y are Lagrangian multipliers and are functions of x
where w,
and y and w is subsequently found to be the deflection in the z direction.
Values v 1 , v 2 , ... v 6 are the generalized boundary displacements of the
problem.
After carrying out the variation 61 = 0, integrating by parts and transforming the appropriate surface integrals to line integrals by means of
Green' s theorem, the variational equations (Euler' a equations) are:
2 (crx
l-
v Tx)
E cos a
1
G cos a
(z + T
2 cos a
Report No. 1874
(2z + t cos a) 8
sin a
ay
ay
tan a aw
8x
(1 0)
=0
- y sin a - (z cos a +
p tan a (z + t cos a
2
a v = 0
ay
(11)
813
ay
(12)
) 8= 0
-6-
1
Gxz
z
aW
T xz
—
y
W
T
T
ax
- 13 = 0
tan a -
yz
(13)
a(WZ)
(14)
zy = 0
aay
These equations must be satisfied by values of (T x,
Cry T T xz T yz
1
w, /3, and y, which render the complementary energy a minimum.
The boundary conditions (three at each edge of the plate) arising from
the 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
Cos
a
y)
2t
2
v3 - cos a zp + t g
(17)
at y = 0 and y = b
v 4 = 2wz
v5 = 2tzi3 + t 2 f3 cos a
v6 = 2w sin a + 2tzy cos a + t 2 y
Equations (10) to (14), together with equations of equilibrium (6), (7), and
(8), represent a complete system of equations for the eight functions
1 T Txz Tyz w, fi, and
CrX
Cr y
Solutions
1. Trapezoidal Cross Sections
For a sufficiently long plate under torsion, the stresses and strains may
be considered not to vary along the longitudinal axis and therefore are
independent of x. Then crx = 0 and equations of equilibrium (6), (7), and
(8) become
Report No. 1874
-7-
t
T XZ
t cos a) dT
dc-
ZzT yz
Y1
dy
T yz
(21)
(22)
t (2z cos a + t)
dcr
d irm
' 1 sin a
tana+z—+t
dy
dy
=
0
(23)
The solution of equations (22) and (23) for T yz and aryl , and the use of the
boundary conditions that both stresses are zero when y = 0 and y = b leads
to
1 =T yz= 0
=
(24)
Equations (10), (11), and (14) thus become
813 =0,
0, or f3 = f (y)
8x
(25)
8wsm
. a + y sin a + (z cos a +
=0
8y
ay
(26)
w tan a - a(wz) - zy = 0
ay
(27)
Equation (27) may be reduced to
(27a)
Y = Ow
Substituting y = - aw into equation (26) we obtain
By
aY = o
By
or y = F(x) only
and hence
w
= -
(27b)
yF(x) + F 1 (x)
because of equation (27a).
Report No. 1874
-8-
From equations (13) and (25) and the assumption that the stresses are
independent of x, it follows that
(27c)
W = xF 2(Y) F3(Y)
In view of the above results of (27a), (27b), and (27c) we obtain
(28)
w = - (C ix + C 5) y + C 4x + C6
= 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
4G xz zT
a - t sin a) dT dy tG cos a (2z + t cos a)2
cry2 z (2z + t cos a)
d 2 T (2z tan
- 2Gxy C iz(4z + t cos a + )
COS a
t (2z + t cos a)2
(30)
This equation, with equation (2), can be written as
(1- t cos a)
GXZ T
d 2 T + 2z
dT
cosa)) dz
dz(
2 1+
tG cos a tan 2 a (1 + -- cos a) 2
2z
cos a + t
)
-2C 1 Gxz z + tz
4z cos a
t tan g a (1 + t cos a)2
2z
is relatively small as
2z
compared with 1. With this consideration, the above equation may be
reduced to
Since facings are treated as membranes, G
xz T
z d 2 T dT dz 2 dz tG cos a tan 2 a
Report No. 1874
-9-
-2C1Gxz z
t tan 2 a
(31)
The complete solution for
C3 is
T =
T
with three arbitrary constants C 1 , C2, and
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 from
the boundary conditions that T = o at y = o and y = b, can be expressed
in terms of Ci as
C2 -2 G cos a
CI
C3
Cl
+
K0 (2 ITITI)
+ b tan a +4-1
Ko
Io (2
)
Ifr(h+ b tan a))
K0 (2 Fir)
10 (2 t/r(h+ b tan a)
K0
(2 Lir (h
Io (2 I r17
x .)
(h +4.)
Io (2 Vr(h+ b tan a))
(h 4- b tan a 4)
b
tan cx))
(33)
= 2 G cos a
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 ends
of the plate is zero, and these forces represent a couple, the magnitude
of which is
b
b
2
(1
+
cost
(34)
T = 4t
T zdy + t
T dy
a
cos
If the displacement v1 and v 2 are set to be zero at x = 0, then w and y
from equations (28) and (29) become
(35)
w = - Cixy + C 4x
( 3 6)
clx
Report No. 1874
-10-
The torsional stiffness
T
may be obtained from
b
T xz + V2 T ) dy
f
T8 =(Vi
x =1
(37)
0
Substituting for vi and v2 from equations (15), (16), (35), (36), and for
-
--.Tx z 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) and
the integration carried out, we obtain
T
0
2tG cos a
tan a
4t 2
z
3 +zr +
[4 ( z3
C2
+ cos 2
cos a
z2 z
(2 + 7)]
rz ) - (rz) I o (2 V1Z-)]
(rz + 1) I 1 (2
r 2 tan a{. C1
_ C3 [
C1
(rz + 1) K i (2 rz ) + (rz) K o (2 FZ 11}
t 2 FT (1 + cos t a) [ C
r tan a cos a
CT
(2
)
h- b tan a
-
C3
K 1 (2 t RT )]
Cl
(39)
h
where h and h + b tan a are lower and upper limits of the integration of
equation (34) with respect to z .
Report No. 1874
-11-
For rz > 25 equation (39) can be expressed approximately as
t2tG cos a [4( z 3 + z 2
3
2r )
ta.n a
0
t(1 + cos 2
cos a
5
+ 4t(rz)4 C2
r 2 tan a
_
2 FT
[0.2821 — e
(1 - Cl
a)
19
1:7
161
(Z 2
2
+
r
465
512 rz
105
31L5- 8192 (rz)2a-
1785 8192 rz rz
Z )1
)
C 3 -2 Irz
465 1785
19
+
(1 + +
- 0.8862-- e
Cl
512
rz
8192 rz , IFT
16 ii
. z
1
t2(rz)7r
(1+ cos 2 a)
5
31510
+
)] 4'
r tan a cos a
8192 (rz) 28192 (rz) 2 ./72-
C2 2
[0.2821 Ci
— e
1-7-z1
/
C 3 -2 ,/
0.8862
—
e
C l
105 )1
8192 rz
Report No. 1874
(1 -
3 15
16, ,/1 -"-512rz
(1 +•
3 /
161
105
)
8192 rz . rz
15
512 rz
+ b tan a
(40)
-12-
2. Triangular Cross Sections
For h = 0 and z = y tan a, equation (30) may be written as
y
t cos a)2
(y
2 tan a
-
t2 cos 2 a) dT _ Gxz Y21T
dy 24 tan 2 a dy tG sin a
d2
+
2 Gxz CI y 2 (y +
(y2
t
t cos a +
4 tan a
4 sin a)
t
oo
Anyn.
The above differential equation can be solved for T by letting T =
n=o
For a very small t, as was assumed, the above equation may be reduced
to
d 2 T dT GxzT
—Y dy 2 + dy
tG sin a
- 2G1 Gxz y
(41)
t
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 setting
h = 0.
3. Rectangular Cross Sections
By setting a = 0, equation (30) is reduced to
d2 T 4h Gxz T
4C i h Gxz
(42)
dy 2 tG (2h + t) 2t (2h + t)
where Ci = according to equation (38).
It can be verified by direct substitution that the general solution of
equation (42) is given by
T = AI sinh y
where yi - 1
y +
B1 cosh Ni y
4h Gxz, D i = G (2h + t)
Gt (2h + t)2
Report No. 1874
-13-
+
Dl(43)
(44)
The first two arbitrary constants Al, and Bi, which are determined from
the boundary conditions that T = 0 at y = 0 and y = b, can be expressed in
terms of D 1 as
Al -
(cosh y l b - 1)
D1, B 1 = - D1
sin
h yi b
(45)
thus
T=
D[
(co sh y b 1)
sinh y y - cosh y y + 1]
.
Y
1
(46)
J
All equations of equilibrium are satisfied identically except equation (21),
which becomes
T
t (211 + t)
2h
d-r
dy
(47)
Introducing (46) into (47), results in
tGxz [ (cosh y b - 1) cosh y 1 y - sinh y] D 1(48)
sinh b
hG
T xz =
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
T
D1 =
(50)
2 (cosh y i b - 1)
2t (2h + t) [b -
Report No. 1874
Y 1 sinh y l b
-14-
]
The torsional stiffness is obtained from equations (50) and (44)
T 0
2 (cosh y i b - 1)
= 2t (2h + t) 2Gb [1 - (51)
yib sinhyib
It is evident that equation (51), similar to equation (142) of Forest
Products Laboratory Report No. 1871 (1), is independent of plate length
, Young' s modulus E c , and shear modulus G yz of the core.
The results of numerical computations based on equation (51) are given
in table 1. The results based on the infinite series solution (1) are also
listed in table 1 for comparison with the results obtained by the present
method. The numerical computations were for sandwich plate of various
widths 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 between
the two methods of 3 percent. Their differences can be attributed to
small errors or lack of using sufficient terms in the infinite series as
given by equation (142) of Forest Products Laboratory Report No. 1871.
It may be concluded that the present result, which is in a form similar
to R. Bredt' s formula for thin tubular sections (5), is in excellent
agreement with the infinite series solution.
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 Reinforcing Cores and Rectangular, Triangular, or Diamond
Cross Section: U. S. National Advisory Committee for
Aeronautics, 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 sandwich
plate as computed by two methods
b
by present method :
0
0
by infinite series solutionl
7F
In.
.
Lb.-in.2
1
:
3,500
2
:
19,000
:
Lb.-in.2
3,360
:
19,100
3
41,920
41,900
4
67,280
:
66,800
5
:
93,200
:
92,300
6
:
119,500
:
118,000
8
10
:
20
172,000
169,000
224,600
220,000
487,200
:
476,000
-Obtained from table 2 of Forest Products Laboratory Report No. 1871.
Report No. 1874
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