Finite element thermal stress analysis was performed on a rectangular titanium honecomb-core sandwich panel which is subjected to thermal load with a temperature gradient across its depth. The distributions of normal stresses in the face sheets and the face-sheet/sandwich-core interfacial shear stresses are presented. The thermal buckling of the heated face sheet was analyzed by assuming the face sheet to be resting on an elastic foundation representing the sandwich core. Thermal buckling curves and thermal buckling load surface are presented for setting the limit for temperature gradient across the panel depth

Jackson, Raymond H.

Ko, William L.

English

NASA Technical Reports Server

NASA Technical Memorandum 101732
Thermal Behavior of a Titanium
Honeycomb-Core Sandwich Panel
William L. Ko and Raymond H. Jackson
Ames Research Center, Dryden Flight Research Facility, Edwards, California
1991
National Aeronautics and
Space Administration
Ames Research Center
Dryden Flight Research Facility
Edwards, California 93523-0273
https://ntrs.nasa.gov/search.jsp?R=19910008120 2020-03-19T19:52:15+00:00Z

ABSTRACT
F! _ite element thermal stress analysis was performed on a rectangular titanium honeycomb-core sandwich panel
which is subjected to thermal load with a temperature gradient across its depth. The distributions of normal stresses
in the face sheets and the face-sheet/sandwich-core interfacial shear stresses are presented. The thermal buckling of
the heated face sheet was analyzed by assuming the face sheet to be resting on an elastic foundation representing the
sandwich core. Thermal buckling curves and thermal buckling load surface are presented for setting the limit for
temperature gradient across the panel depth.
INTRODUCTION
There are a variety of candidate hot-structural panel concepts for application to hypersonic aircraft such as
the national aero-space plane and the high-speed civil transport. One of the promising candidates is a titanium
honeycomb-core sandwich panel (both face sheets and the sandwich core are made of titanium material). During
high-speed flight, the hot side of the panel will be heated to higher temperatures than the cooler side, resulting in
the buildup of a severe temperature gradient across the panel depth. Under this type of thermal loading, several
phenomena could happen. The typical ones are (1) debonding between the heated face sheet and the sandwich core
if the interfacial shear stresses exceed the shear bonding strength, and (2) thermal buckling of the heated face sheet
if the temperature gradient across the sandwich panel is too severe.
This report presents thermal stress analysis of the titanium sandwich panel, and examines whether these
phenomena will take place under the proposed service heating conditions and laboratory test conditions of the
sandwich panel.
NOMENCLATURE
£I
O,m,n.
b
D
E,
G,
G_, Gvz, Gz_
hc
N_
( N_)_
1%
7"i,
SPAR
G
A U1
length of sandwich panel, in.
Fourier coefficient of assumed deflection function for w( z, y)
width of sandwich panel, in.
face sheet flexural rigidity, 12( 1 - v,_ )' in-lb
modulus of elasticity of face sheet material, lb/in 2
moduli of elasticity of sandwich core, lb/in 2
shear modulus of face sheet material, lb/in 2
shear moduli of sandwich core, lb/in 2
sandwich-core depth, in.
number of buckle half waves in z-direction
panel compressive load in z-direction, lb/in.
panel compressive buckling load in z-direction, lb/in.
panel compressive load in y-direction, lb/in.
number of buckle half waves in y-direction
structural performance and resizing
thickness of face sheets, in.
strain energy of plate bending, in-lb
au2
AW
'lD
,r., y
OLc
OLs
3
%
%
_xzl,V_z, Yxz
Tcxz
strain energy of elastic foundation, in-lb
work done by external forces, in-lb
face sheet out-of-plane deflection, in.
rectangular Cartesian coordinates, in.
coefficient of thermal expansion of sandwich core, in./in.-°F
coefficient of thermal expansion of face sheet, in./in.-°F
foundation modulus, _-_, lbfm.-in 2
weight density of sandwich core, lb/in 3
weight density of face sheet material, lb/in 3
Poisson ratio of face sheet material
Poisson ratios of sandwich core
normal stress in face sheet in the z direction, lbfm 2
interfacial shear stress, lbfln 2
DESCRIPTION OF PROBLEM
Figure 1 shows the rectangular honeycomb-core sandwich panel under consideration. The panel is subjected to
heating with a temperature gradient across its depth. Under such thermal loading, debonding between the heated
face sheet and the sandwich core or thermal buckling of the heated face sheet could occur if the thermal gradient
across the sandwich depth is too steep.
The problem is to perform thermal stress analysis and thermal buckling analysis of the sandwich panel and
to examine whether these two phenomena might take place under the proposed service heating condition of the
sandwich panel.
THERMAL STRESS ANALYSIS
In the thermal stress analysis, the sandwich material will be considered linearly elastic (neglecting thermal creep
effect), and the edges of the sandwich panel will be damped.
The thermal stress analysis will yield compressive stress distribution in the heated face sheet and will yield the
interracial shear stress distribution built up at the interface between the heated face sheet and the sandwich core. The
face sheet peak compressive stress will then be compared with the thermal buckling stress of the heated face sheet,
and the peak interfacial shear stress with the interfacial shear bonding strength.
The structural performance and resizing (SPAR) finite element computer program (ref. 1) was used in the thermal
stress analysis. Figure 2 shows a SPAR finite element model set up for a thin strip of the sandwich panel along the
y = b/2 line. Each of the face sheets was modeled with only one layer of elements in thickness direction, and the
sandwich core with six layers of elements in thickness direction. Because of steep stress gradients near the panel
edges, the element densities were increased in two steps near the panel edge for obtaining smooth stress distributions
in this region.
THERMAL BUCKLING ANALYSIS
When the upper face sheet is heated more than the lower face sheet, the upper face sheet will be under
compression (upper part of fig. 3). If the sandwich core is considered as a spring, the situation could be approx-
2
imatedbythebucklingof a single face sheet on elastic foundation, as shown in the lower pan of figure 3.
In the thermal buckling analysis of the heated face sheet, the energy method will be used. Let A U1, A U2, A W
be, respectively, the strain energy of plate bending, strain energy of elastic foundation, and the work done by the
external forces. At thermal buckling the following energy balance must hold:
aUx + AU2 = AW (1)
These three energy terms may be expressed in terms of plate deflection w(x, y) as follows:
Strain energy of bending A U1"
where D is the face-sheet flexural rigidity given by
2 02 w l dxdy
O Oy;j j (2)
where Es, v,, ts are, respectively, face sheet modulus of elasticity, face sheet Poisson's ratio, and face sheet thickness.
Strain energy of elastic foundation (A 0"2):
 fo°f0AU2 = _ w2dxdy (4)
where 3 is the foundation modulus, given by
Ez
3 = h--/ (5)
where Ez and he are, respectively, modulus of elasticity and depth of the sandwich core.
Work due m external forces (A W):
A W = _ N= k Ox ] + Nv k Oz ] j dxdy (6)
where Nz and Nv are, respectively, the compressive panel loads in the x- and y-direction in the face sheet.
For simply supported edges, the face sheet deflection surface may be represented by the following double Fourier
series (ref. 2):
oo OO
m trx nTry
w( x, y) = _ _ am,, sin -- sin -- (7)
a b
m=l n=l
where am,, is the deflection coefficient, a and b are the length and the width of the sandwich panel, respectively, and
m and n are the number of buckle half waves in the x- and y-directions, respectively.
Substituting equation (7) into equations (2), (4), and (6) yields
DEE := a,,,'* + (8)
AU2: 322 -,, (9)
ra=l n=l
AW "_- "n Em 2 2 Ny _--_ n2 a2 (10)
n= l m= l r_=l J
D = E_t] (3)
12( 1 - v 2)
In the light of equations (8) through (10), the buckling equation (eq. 1) takes on the form
oo0o (m 2 n__2,_2 0ooo
m= ln=l b2 )
m=ln=l (11)
Nz = ,it2 r co co 02 _r co co
Lm=ln=l
This expression becomes a minimum if all the coefficients 0,m,,, except one, are taken to be zero (ref. 2).
Thus, the minimum buckling load is given by
_4 D --_- + _-- +/3
( N=)cv = (12)
N ,e': 7 +
For thermal loading with constrained edges, N= = N u, and equation (12) becomes
For a fixed 0, and b, the value of ( N=)= will become lowest at particular combinations of m and r_
To lind the functional relationship between m and n which gives lowest value of ( Nz)cr, we differentiate equa-
tion (13) with respect to m or r_ and set the derivatives to zero:
O( N_)cr _ O( Nz)cr _ 0 (14)
Om On
from which we obtain
or
m 2 n2 1
= p- (15)
= 1 (16)
which istheequationofellipseinthera-nspaceshown infigure4.
Substitutingequation(15)intoequation(13)yieldsthelowestbucklingloadof (N,,)_.:
(N=)cr = 2_D-'3 (17)
Equation (17) is independent of panel aspect ratio, and is subjected to the condition in equation (15) or (16) which
relate ra to n.
For a square panel 0, = b, and equation (15) becomes
0,2
m 2 + n 2 = _-,V_ (18)
which describes a circular arc in the m-n space for which m 2 1 and n 2 1.
NUMERICAL EXAMPLES
The titanium honeycomb-core sandwich panel, which has been designed and fabricated for testing in the NASA
Ames Research Center's Dryden Thermostructures Research Facility, has the following geometry and material prop-
erties. Geometry:
a = b = 24 in.
he = 0.69 in.
ts = 0.06 in.
Material properties of titanium face sheets:
ITEMS
Es, lb/in 2
Ga, lbfm 2
us
c_s, in./in.-°F
%, lb/in 3
300°F
15.1 x 106
5.7634 x 106
0.31
5.1 x 10 -6
0.16
900°F
11.8 × 106
4.5038 x 106
0.31
5.6 x 10 -6
0.16
Material properties of titanium honeycomb core:
ITEMS 600°F
Ez, lb/in 2 2.7778 x 104
E u, lb/in 2 2.7778 x 104
Ez, lb/in 2 2.7778 x 105
Gx_, lb/in 2 0.00613
G_z, lb/m 2 1.81 x 105
G,x, lb/in 2 0.81967 x 105
vzv 0.658 X 10 -2
_,_, 0.643 x 10 -6
v_z 0.643 x 10 -6
t_o in./in.-°F 5.35 X 10 .6
%, lb/in 3 3.674 x 10-3
The sandwich panel will be subjected to the proposed service thermal loading condition of 300°F on the lower
face (Te = 300 °F), and 900°F on the upper face (Tu = 900 °F). Thus, it will be under the temperature gradient of
(T,_ - Te)/hc = 600°F/he. The four edges of the sandwich panel will be restrained with resisting moments to keep
the panel fiat during thermal loading. This edge condition will induce the highest compressive stresses in the upper
face sheet (away from the edges).
In the finite element computations, dissimilar material properties will be used for the lower and the upper face
sheets. However, for the honeycomb core, the averaged material properties at 600°F will be used.
RESULTS
Figure 5 shows the distributions of the magnitude of the normal stress crfx in the face sheets and the interfacial
shear stress r,.,z in the I/= b/z plane, calculated from the finite element analysis. The normal stress crf_ is maximum
at the panel center with the value (cr/z),,,,_ = 23,193 lb/in, and remains almost constant up to 6 in. from the panel
center,andthenmonotonically decreases to zero at the panel edge. On the other hand, r=j is zero in the panel center
region up to 6 in. from the panel center, and increases rapidly to its peak value of (%xz)maz = 706 lbfm 2 near the
panel edge. It then decreases sharply to zero at the panel edge. For this low value of interfacial shear stress, it is
unlikely that the interracial shear debonding will take place under the proposed service thermal loading.
Next, we will examine whether the compressive stress generated in the upper face sheet will cause it to buckle.
The compressive panel load in the upper face sheet is given by
Nz = Nu = (cry,),_zt, = 1392 lb/in.
From equation (17), the lowest buckling load for the present case is
(Nz)cr = 2x/-D--_= 19,452 lb/in.
for all combinations of ra and r_
Thus, the induced compressive panel load in the upper face sheet is approximately 7 percent of the predicted
buckling load, and the possibility of face sheet buckling is quite remote. Figure 6 shows ( Nz)_r (eq. (13)) plotted as
a function of ra for n = 1 and for n = m. For n = 1 the lowest buckling load occurred at ra = 49, and for n = ra it
occurred at rn = 35. The level of compressive panel load induced in the upper face sheet is shown for comparison
with the predicted buckling loads. Figure 7 shows the buckling load surface plotted in the rn-n space. The lowest
compressive buckling load points form a circular arc at the height of( Nz)c_ = 2 x/_ measured from the m-nplane
(horizontal plane).
CONCLUSIONS
Thermal stress analysis was performed on a titanium honeycomb-core sandwich panel subjected to heating with a
temperature gradient across its depth. The magnitude of the interfacial shear stress induced under the proposed ther-
mal loading was relatively low and would not cause interfacial debonding between the face sheets and the sandwich
COre.
Also, thermal buckling of the heated face sheet was analyzed by assuming the face sheet to be resting on an
elastic foundation which is the sandwich core. The calculated compressive thermal stress induced in the heated face
sheet was approximately 7 percent of the compressive buckling stress predicted from the buckling analysis. Thus,
the heated face sheet is unlikely to buckle under the proposed thermal loading.
Ames Research Center
Dryden Flight Research Facility
National Aeronautics and Space Administration
Edwards, California, September 27, 1990
REFERENCES
1. Whetstone, W.D., SPAR Structural Analysis System Reference Manual, System Level 13A, Vol. 1, Program
Execution, NASA CR-158970-1, 1978.
2. Timoshenko, S.P. and J.M. Gere, Theory of Elastic Stability, McGraw-Hill Book Company, Inc.,
New York, 1961.
6
b 8
Y
i
Figure 1. Rectangular sandwich panel subjected to thermal loading.
T
b
Y
I
I
I
! .0,oo
modeled
I
Tu = 900°F
T1 -- 3000F
Figure 2. Finite element model for a thin strip of sandwich panel along _t= _- plane.
9O0655
7
Heat
llllllllllllllllllllllllllll _,.
II
Nx "_ Nx
.._t_ I
_"JIfIIfffIIIIIfIfIIIfIIIIfIIIt-_°_ = -c-|%
_11111111111111111111
Figure 3. Representation of thermal buckling of heated face sheet of sandwich panel
by compressive buckling of thin plate on an elastic foundation.
F (Nx)©r = 2V-_
_e
_ m
900657
Figure 4. ( Nx)er = Minimum curve for regtangular sandwich panel.
30 x 103
26
24
18 _- 12023,193 fx 1000
Gfx, 16 800
214 I-- Gfx --_ 900°F --/Ib/in 12 _ I
,o_..- _OOF /\ _1,oo'_"n'
i_ 'oo
0
0 1 2 3 4 5 6 7 8 9 10 11 12
X- _, in.a
9O0658
Figure 5. Face sheet normal stress and interfacial shear stress induced
in titanium sandwich panel under thermal load.
9
107
10 e
(Nx)cr 10s
Nx
n=l
N x
1.9452 x 104
104
1.392
10 3
1
fl=m
I
I
J
1
J
•I T. = 900°F
1 /-IIIIIIIIIIIIIIIIIIIII
._ i ,,=
I / 1"/ = 300°F"1
,j__ 35-_
•1 I I I I I
10 20 30
m
I i
40 50
Figure 6. Titanium honeycomb-core sandwich panel face sheet
buckling loads under thermal loading.
I
6O
I I
7O
9OO65'9
10
(Nx)cr
Buckling load
surface
Curve
m
49
II
II
Figure 7. Thermal buckling load surface in m-n space for titanium
honeycomb-core sandwich panel face sheet, a = b.
n
9o066o
11
___ _"_,.__.. Report Documentation Page
1. Report No. 2. Governn'mnt Accession No. 3. Recipient's Catalog No.
NASA TM-101732
4. Title and Subtitle
Thermal Behavior of a Titanium Honeycomb-Core
Sandwich Panel
7. Author(s)
William L. Ko and Raymond H. Jackson
9. Performing Organization Name and Address
NASA Ames Research Center
Dryden Flight Research Facility
P.O. Box 273, Edwards, California 93523-0273
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC 20546-3191
5. Report Date
January 1991
6. Performing Organization Code
8. Performing Organization Report No.
H-1661
10. Work Unit No.
RTOP 532-09-01
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
Finite element thermal stress analysis was performed on a rectangular titanium honeycomb-core
sandwich panel which is subjected to thermal load with a temperature gradient across its depth. The
distributions of normal stresses in the face sheets and the face-sheet/sandwich-core interfacial shear
stresses are presented. The thermal buckling of the heated face sheet was analyzed by assuming the
face sheet to be resting on an elastic foundation representing the sandwich core. Thermal buckling
curves and thermal buckling load surface are presented for setting the limit for temperature gradient
across the panel depth.
17. Key Words (Suggested by Author(s))
Sandwich panel; thermal stresses; thermal
buckling; buckling load curves and surface.
18. Distribution Statement
Unclassified-Unlimited
Subject category 39
19. Security Classif. (of this report)
Unclassified
20. Security Classif. (of this page)
Unclassified
21.No. of Pages
14
22. Price
A02
NASA FORM 1626 OCT M For sale by Ihe National Technical Information Service, Springfield, Virginia 22161-2171


Thermal behavior of a titanium honeycomb-core sandwich panel

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