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Analysis of tapered sandwich structures with anisotropic composite faces
Compums & Structures Vol. 44. No. 3. pp. 597408, 1992 Printed in Greal Britain.
co45-7949/92 $5.00 + 0.00 0 I992 Pergdmon Press Ltd
ANALYSIS OF TAPERED SANDWICH STRUCTURES WITH ANISOTROPIC COMPOSITE FACES J. S. JEON and C. S. IIoN@ Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea (Received 16 May
1991)
Abstract-A new formulation is presented for the analysis of tapered sandwich structures with anisotropic composite faces. Three local coordinate systems are introduced to describe the independent displacements of each component. The expression for the total potential energy is derived and the Rayleigl-Ritz method is applied to obtain an approximate solution. The analysis takes into account the correlation between the core shear strain and the face normal deflection existing only for the tapered geometry. The present
formulation can be applied to arbitrary boundary conditions. Numerical examples are calculated for various core shear modulus ratios, taper ratios, and slenderness ratios. The correlation between the core shear strain and the face normal deflection is found to he very impor~nt in predicting the deflections and stresses of tapered sandwich structures.
INTRODUCTION Sandwich structures have been widely used in bending dominant structures because of their representative advantage of specific bending stiffness. When they are used in aerospace applications such as wings or control surfaces, the sandwich structures are used as forms of full-depth honey-comb constructions with/without stiffeners according to the load intensity. Such sandwich structures, for aerodynamic reasons, take airfoil shapes and the thickness varies along the chord line from the leading edge to the trailing edge [l]. For the analysis of uniform thickness cases, numerous approximating solutions have been given by many researchers such as Kim and Hong [2] and Monforton and Ibrahim [3]. They have considered the material anisotropy, face bending stiffness and finite bonding stiffness. In contrast to the uniform thickness sandwich structures, tapered sandwich structures have received considerably little attention in spite of their wide application. Until now, tapered sandwich structures have been investigated by few researchers. Akasaka and Asano [4] have analyzed a sandwich pane1 with a sinusoidally varying thickness at all edges simply supported by a frame. They showed the importance of the shell effect on the curvature by the flattening effect. Thereafter, Paydar and Libove have employed the finite difference method to calculate the deflections and core shear stresses for a square plate [5,6] and an annular plate [7]. The bending problem for a j’ Correspondence should be addressed to Professor C. S. Hong.
cantilevered sandwich beam has been analyzed by Libove and Lu 181. By comparing their theory and classical theory using the constitutive equations of locally constant thickness sandwich plates, they emphasized ‘the contribution of the face sheet membrane forces (by virtue of their slope) to the transverse shear’. The previous works for the tapered cases are restrictive in their formulation. The displacements for the upper and lower face are assumed as antisymmetric defo~ations (i.e. u, = -u2) without examining the validity. It imposes artificial constraints on the face displacements. The face bending stiffnesses are not included. It underestimates the structural bending stiffness; in addition the correlation between the core shear strain and face normal deflection is not expressed clearly in the explicit form and the core shear strain has been treated as a variable to be determined numerically. For practical designs and applications of sandwich structures, all the deflections and stress distributions must be known first. Without the knowledge of overall behavior, structural performance is not used to its best advantage and a failure may occur. In the present work, an analytic method has been developed to understand the behavior of tapered sandwich structures with anisotropic composite faces. The face bending stiffness as well as inplane stiffnesses are included and the core is considered as orthotropic. The analysis takes into account the correlation between the core shear strain and the face normal deflection existing only for the tapered geometry. The present formulation can be applied to arbitrary boundary conditions. Some numerical examples are presented for sandwich beams to demonstrate the validity and versatility of the method.
J. S. JEONand C. S. HONG
598 TOTAL
POTENTIAL
ENERGY
An overall view of the sandwich structure under consideration is shown in Fig. I. The assumptions made are as follows: The face thickness is constant and moderately thick to apply the classical laminate theory (CLT). The core thickness varies linearly in the x-direction. The core is orthotropic and in the antiplane stress state (T,, # 0, tYz# 0, all others = 0) f9]. The interfaces between the faces and core are perfectly bonded and the bonding layer thickness has no effect. Each component is under the small deformation. Strain energy of core
The transverse shear stresses in the core are constant through the thickness because the core is inextensible in the thickness direction and deformable only in the transverse shear. When the core is rotated about its reference axes by 4,,, the strain energy in the core can be written as
Fig. I. Geometry of tapered sandwich structure with anisotropic composite faces.
local coordinate systems for each component are introduced as shown in Fig. 2. The normal displacements and curvatures of the upper and lower faces are the same as the previous assumptions. Onfy the directions of the unit vectors of upper and lower faces are different due to the face slope about the structural midplane. This allows that the expression for the normal displacement of each face can be written as one variable. Therefore the six variables for this structure are reduced to five 0
II,(S,Y,n)=u~"(s,Y)-n~
where & = G,, cos* 4, + G,3 sin’ 4, Qd5= (Gu - GD) cos 6, sin +( & = G,, cos2 (p, + Gz, sin’ 4,. Here superscript 0 represents the values at the core midplane, subscript c for the core and Gi), Gz3 are the core transverse shear moduli about its material principal axes. The thickness of the core at any point decreases linearly in the x-direction by the face slope factor of tan p as h(x) = h, - x tan p.
(2)
Substituting eqn (2) into (1) and integrating the core thickness gives
-
x[H;.]}
{;“} dx dy, x.2c
over
(3)
where [Gt] = 2ho[Qij]~, [I$:] = 2 tan @[&],,
i, j = 4,5.
Strain energy of faces
The Rayleigh-Ritz method can be applied to a relatively simple structure such as a uniform beam or plate. But the present structure is complicated by the thickness variation and face slope about the midplane of the structure. To overcome this difficulty three
Fig. 2. Local coordinate systems.
Analysis of tapered sandwich structures where superscript 0 represents the values at the face midplane. By the small deformation condition, the face strain-displacement relations for the upper face are given by
Potential energy by external loads Under the normal distributed load, q. and vertical concentrated load, Q0 in the upper face, potential energy by external loads can be expressed as y,-
Combining
599
JJ
qo(~~~MX~,~)ds dy -~~~~o~Y~~~“~~~~Yo~. (10)
eqns (4) and {S) gives
%S= c:f + 55,
eyy= $ + “‘cyr,
ysv= yf + msy,
Here w” is the structural midplane deflection in the z-direction.
(6) COMPATIBILITY
where
CONDITIONS
Using two different local coordinate systems for the upper face and the core, a point at the interface can be described as
K,=
aho ~$9
a%0 -dyz,
Kvv=
‘csv=
aw as ay
(
uO+h(x) g
-2 n.
Because the faces are analyzed by CLT, the strain energy of the upper face can be written as
-y(l,
>
= usucos /3 - ui sin /I,
tP+ h(x) dw” - yiZ = By,, ( ay > w” = a,, sin /I + ii: cos fl.
(11)
Here u”, a’, w” represent the structural midplane displacements in the x, y, and z-directions, respectively. Similarly, using two different local coordinate systems for the lower face and the core, a point at the interface can be described as
where [&], is the transformed reduced stiffness matrix of the upper face. Expanding eqn (7) gives
Ljly=f
(
g
uo- h(x) dw” ( ay
{~“iIIAijlu{~oI~ h dY
ss
u”-h(x)
) -y;,
-&
=u,,cosfi
>
+ujfsin/3
= $/
w”= -u,,sin~+u~cosfl.
+
ss
+k
~~‘~~~~ijl”~~~ k dY I~)‘[~~jlu{~}h dYv
JT
Combining eqns (11) and (12) gives the explicit form for the core transverse shear strains in terms of the upper and lower face displacements as
(8) _awo -[(u, YXZ-dX -2/z(x) o
where [A,j],, [B,],,, [O,], are the extensional, coupling, and bending stiffness matrices of upper face, respectively, as defined in [lo]. Similarly, the expression for the lower face can be written as
w”+,,uf,=i
ss +
i
- u,,)cos j? - 24 sin j?]
u,,)sin fl + uz cos fl.
(13)
{~“>f[~~~l~{~o}~~ dY
Is
+;
(12)
~~“W,,ltG4 h dy
JJ
f#U4,1,~~~ ds dr-
The key difference from the uniform thickness case is expressed in the first and the third in eqn (13). The transverse shear strain yXZin the core is determined not only from the face inplane displacements but also from the face normal deflection as well by the face slope angle @.In addition the structural deflection w” is related to the face inplane displacements.
@I
J. S. JEON and C. S.
600
For the high taper ratio, the dependence of the core shear strain on the face normal deflection is severely influenced because, in the general bending problem, the value of face normal deflection is much larger (about several hundred times) than those of the inplane displacements. If the face slope angle a is set to 0, they are reduced to the same expression for the uniform thickness sandwich plate in [9]. Figure 3 shows the difference of the mechanism invoking the transverse shear strain y,; in the core. It can be easily shown that it will produce unreasonable results if the classical sandwich plate theory using the constitutive equations of locally constant-thickness (which do not consider the effect of face normal deflection and only consider changing the local bending stiffness) is used in analyzing the tapered sandwich structure.
HONG
By using arbitrary shape functions, several advantages can be retained. It is relatively simple to manage in algebraic calculations and any boundary condition can be dealt with. In addition it forms mathematically complete sets of functions and by taking the sufficient number of expansion terms it converges to an exact solution 1111. For further processes, new functions are defined from the already defined shape functions in eqn (14) as
DISPLACEMENTS ASSUMPTIONS
To apply arbitrary boundary conditions, the displacements of each face are expressed in general form by unknown coefficients, C~,-C~H and the proper shape functions, 4;,,-4;, satisfying the given boundary condition as
Substituting eqns (14) and (15) into the core shear strain in eqn (13) gives r R:’
-T
R-;
Cl” cl” C” Cl”
where
0
n
=> fj=>
;zf
(a) For uniform thickness
(b) For tapered thickness
Fig. 3. Comparison of core shear deformation mechanisms.
I
(16)
601
Analysis of tapered sandwich structures Rearranging equation
Let us rewrite eqn (16) for simplicity as {YOj,=
W{CI.
(17)
For the faces, the strains and curvatures are
{~o}“=[P’]~
1E)I
;”
{cO},= [PI]’
)
the above produces
!?I”
{
{K} = [P21T{CLn~9
In”
I )
(18)
where
kl{Cl = w,
(22)
where [g] is the symmetric equivalent stiffness matrix having 5 x 5 submatrices. Detailed expressions for each submatrix are given in Appendix II. By solving this matrix equation, the unknown coefficients are obtained and all the related quantities can be determined. VERIFICATION
Substituting the above expressions to the strain energy and the potential energy by external loads gives
a linear matrix
OF THE ANALYSIS
In order to show the applicability and to check the validity of the method, some numerical calculations are carried out for several examples. Nine odd expansion terms in the displacement assumptions are taken for sufficient accuracy. The IMSL subroutine DMLIN was used for numerical integration and LEQTZF for the equation solver. Two representative boundary conditions are selected: simply supported and cantilevered sandwich beams. Simply supported uniform sandwich beam
To show the validity of applying arbitrary boundary conditions, simply supported uniform sandwich beams are analyzed at first. The boundary conditions are N, = uI) = MS = 0, at s = 0, c,
(23)
where N, and M, are the force and moment resultant of each face. The selected shape functions satisfying the given boundary conditions are given below
+t{CimY[K81{Cin) ~!,,,,=cos~,
v = -(C)T(F).
SOLUTION
The total potential energy of a system is defined by the summation of the strain energy of the structure itself and the potential energy by external loads as n = UC+ c;, + u, + V.
(20)
For a system to be in a static equilibrium state, the principle of minimum potential energy must be satisfied. It requires the partial differentiation of the total potential energy about each unknown coefficient as
an -=o, ac;"
4i,,=sinE.
(19)
Detailed expressions from [K’] to [K8], {C} and equivalent force vector {F} are given in Appendix I. APPROXIMATE
&,$,=O,
i=l,...,
5.
(21)
c
(24)
The structure is under uniform distributed load. The stresses are compared with those of previous finite element results [12]. The tensile stress at the upper face and the core shear stress are shown in Fig. 4(a) and (b), respectively, under the uniform lateral load. They coincide very satisfactorily. Cantilevered sandwich beams
The comparisons based upon the structural deflections are performed for the tapered sandwich beams with the previous experimental results in [5]. Experimental geometry, loading, and boundary conditions are shown in Fig. 5(a). Because the thickness of the present sandwich structure varies linearly in one direction, the experiment is modeled as in Fig. 5(b). The boundary conditions are
+.@L0, N,=M,=
0
as
ats=O
V,=O,
at s =c,
(25)
602
J. S. JEANand C. S. HONG
0.16
i
(a) Experimental geometry and load 1
0.1
0.2
0.3
0.4
0.5 C_
x/a
51.8
,!
(a) Transverse shear stress in core
-20 -40 -60 -80
(b) Modeled geometry and load
-100
Fig. 5. Modeling of experiment for comparison of ‘quasistiffness’.
-120 -140 -160
’ 0.0
I
I
I
I
0.1
I
0.2
0.3
0.4
0.5
x/o (b) Tensile stress in upper face Fig. 4. Comparison for simply supported sandwich beam with FEM results (HI? = 1.0).
where V, is the resultant shear force of each face. The selected shape functions satisfying the given boundary conditions are given below
Here the V, =0 condition for the faces is not satisfied but the present Rayleigh-Ritz method can
be applied if kinematically admissible functions are selected. From the measurements and calculations at two points apart, its ‘quasi-stiffness’ between the two points is calculated. Its values are listed and compared with previous experimental results in Table 1. The present calculations show more stiff results than those from the experiments with an error of 0.2-4.0%. There may be several reasons: First, the modeling itself from the experiment can produce error because its boundary condition is not exactly the same. Secondly, the core material property used in the experiment is very soft (polystyrene) and the present assumption (tZ2= 0) can produce more stiff results. Thirdly, experimental errors from the apparatus setting and readout from the dial gages can be introduced. Considering the above factors, the present formulation can be said to show good agreement.
Table 1. Comparisons of ‘quasi-stiffness’ between experimental and theoretical results p/G& - 6,) W4 HR
FDM [3] (error, %)
Experimental [3]
Present (error, %)
1.00 1.11 1.43 2.00
1772 (0.00) 1848 (3.88) 2034 (2.42) 2289 (0.3 1)
1772 1779 1986 2282
1773.8 (0.10) 1851.3 (4.06) 2047.5 (3.10) 2332.5 (2.21)
603
Analysis of tapered sandwich structures RESULTS AND DISCUSSIONS
Cantilevered sandwich beams are analyzed for uniform and tapered cases. The used shape functions are the same as in eqn (26). All the stacking sequences are aligned to 0” for best performance, The material properties of graphite/epoxy (T300/5208) used for the faces are E, = 181.0 GPa, E2 = 10.7 GPa, G,, = 7.07 GPa, viz = 0.28. The core is a phenolic honey-comb with material properties of G,, = 110.32 MPa, Gz3= 44.128 MPa [ 131. The dimensions are r,, = tl = 1 mm and To = 25 mm. The remaining dimensions are determined from the geometric parameters. The structure is under uniform normal distributed load. To represent the results, the following normalized quantities are used
0.0
0.2
0.4
0.6
0.8
1.0
(a) Deflections at structural midplane
G,, G& G:,
CR_-2
SR=$
_
4w”E, t, T;
W=qoa4(1 -v$)
627) where To and T, are structure thicknesses at x = 0 and x = a, G$, G,*, are the reference core transverse shear moduli and G,,, Gz3 are varied core shear moduli. Eflect
of core
0.0
0.2
0.4
0.6
0.8
1.0
(b) Transverse shear stresses in core
1.2,
modulusratio
By varying the core modulus ratio, CR, uniform cantilevered sandwich beams have been analyzed (HR = 1.O, SR = 40). The structural deflections, core shear stresses, and tensile stresses in upper face are shown in Fig. 6(a)-(c), respectively. The results show that the core shear modulus gives an influence not on the stresses but only on the deflections for the uniform case. Note that the core shear stresses shown in Fig. 6(a) are zero by its boundary condition at the root and the free end. But near the root, high core shear stresses are developed and the values decrease to zero at the free end. It means that the debonding by the core shear stresses may occur near the root region. The tensile stresses in the lower face are the same as those for the upper face only in the change of the sign; they are not given here.
‘%!O
’ 012 ’ 014
’ 016 ’ ii??
1.0
x/a (c) Tensile stresses in upper face. Fig. 6. Effect of core modulus ratio, CR, for cantilevered sandwich beam (HR = 1.0, SR = 40).
J. S. JEONand C. S. HONG
604
1.2,
0.80 11
L
1.0
0.8 0.6 -
7,z0.4 0.2 0.0 I 0.8
1.0
-3l.0
x/a
I
0.2
I
I
I,
0.4
,
0.6
(
,
0.8
1
x/a (b) Transverse shear stresses in core
(a) Deflections at structural midplane 0.7 ,
1
0.8 ,
0.6 0.0
0.5 CR=0.5 0.4 -
a,,-0.8 0.3 -
-1.6
-.
‘v!
0.0 0.0
I
I
0.2
I
I
I
0.4
I
0.6
I
I
0.8
I
1.0
s/c (c) Tensile stresses in upper face
-2.4' ' ' ' ' ' ' ' ' ' 0.0 0.2 0.4 0.6 0.8
0
s/c (d) Tensile stresses in lower face
Fig. 7. Effect of core modulus ratio, CR, for cantilevered sandwich beam (HR = 2.0,SR = 40).
Tapered cantilevered sandwich beams with a taper ratio of 2 (HR = 2.0)have been analyzed (SR = 40). Structural deflections, core shear stresses, and tensile stresses in both faces are shown in Fig. 7(a)-(d), respectively. The results show that for the tapered case the core shear modulus gives a significant influence not only on the deflections but also on the stresses. The tensile stresses at different faces are not the same in absolute magnitude and have a very different distribution, especially the local positive value in the lower face (Fig. 7d). This is due to the change of local curvature along the span length. Compared with the deflections for the uniform thickness cases (Fig. 6a), those for the tapered case (Fig. 7a) are very small contrary to common sense. The reasons may be described as follows: For the uniform case, the sandwich structure resists lateral load by face bending and core shear stiffness. But for the tapered case, two contradictory effects exist. Firstly, deflection increases by the reduction of
the core volume resisting load by shear stiffness. Secondly, by the face slope about the structural midplane, the vertical component of the face extensional stiffness resists shear load along with the face bending and the core shear stiffness. It induces an increased structural stiffness and in turn the structure deflection decreases. Out of the two effects above, the latter is dominant and the deflection decreases. The structural deflection is greatly influenced by the core shear modulus as shown in Fig. 7(a) because of the correlation in the first of eqn (13). The core shear stresses, shown in Fig. 7(b), are concentrated near the thinner free end. This is the effect of the face normal deflection on the core shear strain. The tensile stresses distribution in the upper face shown in Fig. 7(c) are rearranged. The tensile stresses in the lower face are shown in Fig. 7(d). Contrary to the uniform thickness case, the face stresses in the lower face change their signs and magnitudes along the span direction. That is, high negative stresses are
Analysis of tapered sandwich structures
(a) For uniform thickness (HR = 1.0, SR = 40, CR = 1.0)
605
occur up to the free end. This is due to the curvature change of the structure in its sign along the span length. The essential difference is shown in Fig. 8 by exaggerating the deformed shapes. Figure 8(a) shows the shape for the uniform thickness and Fig. 8(b) for the tapered case. Here we must give attention to the points that the structural curvatures are different for each case. EJect of taper ratio
The taper ratio, HR, is increased from 1.0 to 1.5 (CR = 1.0, SR = 40). The deflection shown in Fig. 9(a) increases up to HR = 1.1 and then decreases (b) For tapered thickness (HR = 2.0, SR = 40, CR = 1.O)
Fig. 8. Comparison of exaggerated deformed shapes for cantilevered sandwich beam (CR = 1.0, SR = 40).
induced at the fixed end and quickly reduced to zero. After those transition points, they change their signs to positive and relatively small positive stresses
,
2.0
1.6
1
as taper ratio increases. This is due to the change of the dominant factor for the structural stiffness (from the stiffness reduction by the core volume change to the increase of the face slope participation). The core shear stresses in Fig. 9(b) are redistributed as the taper ratio increases. The location of the maximum shear stress is moved from the region near the fixed 1.6,
I
I
_HR=l.O 1.2
0.8
w
0.6
0.8
0.4 0.2
0.4
0.0 0.0
0.0
0.2
0.4
0.6
0.8
1.0
-.
?I .O
0.2
X/Q
0.4
0.6
0.6
1.0
da (b) Transverse shear stresses in core
(a) Deflections at structural midplane 1.6,
I
1.2
Gs0.8 -1.6
0.6 0.4
-2.4 0.2 nn _._
0.0
0.2
0.4
0.6
0.6
a/c (c) Tensile stresses in upper face
1.0
-3-a% s/c (d) Tensile stresses in lower face
Fig. 9. Effect of taper ratio, HR, for cantilevered sandwich beam (CR = 1.0, SR = 40).
1.”n
J. S. JIDN and C. S. HONG
606 4.0 ,
I
3.5 3.0 2.5
w
2.0
0.6
1.5
0.4
7.0 0.2
0.5 0.0
0.01 0.0
0.2
0.4
0.6
0.8
1.0
1
0.0
I
I
0.2
X/Q
t
I
0.4
,
0.6
,
I
0.8
,’ ’ 3
X/Q
(b) Transverse shear stresses in core
(a) Deflections at structural midplane
0.0) 0.0
0.2
0.4
0.6
0.8
1.0
x/a (c) Tensile stresses in upper face
Fig. 10. Effect of slenderness ratio, SA, for cantilevered sandwich beam (CR = 1.0, HR = 1.0).
end to the thinner free end after undergoing a process
of smoothing aIong the span. The trends of redistributing tensile stresses in both faces are shown in Fig. 9(c) and fd). For the lower face especially, the transition point moves from the free end to the fixed end as the taper ratio increases. Effect of slenderness ratio By varying the slenderness ratio, SR, a uniform cantilevered sandwich beam has been analyzed (CR = 1.0, HR = 1.0). The structural deflections, core shear stresses and tensile stresses in the upper face are shown in Fig. ~~a)-~c), respectively. The results show that the slenderness ratio gives an influence not on the stresses but only on the deflation for the uniform case. Note that the deflection is larger for short sandwich beams (SR = 10) than for long beams (SR = 40). This is due to the core shear defo~a~ion. That is, for the short sandwich beam, the deflection by the core shear deformation plays a
greater part in the total structural deflection than that for the longer beam. The core shear and tensile stresses in the upper face are almost same for each case. The tensile stresses in the lower face are same as those for the upper face only in the change of the signs; they are not given here. By varying the slenderness ratio, SR, tapered cantilevered sandwich beams with a taper ratio of 2 (HR = 2.0) have been analyzed (CR = 1.0). The structural deflections, core shear and tensile stresses in both faces are shown in Fig. II(a)-(d), respectively. The results show that for the tapered case the slenderness ratio has a sibilant influence not only on the deflections but also on the stresses contrary to the uniform case. The maximum core shear stresses are shown in Fig. 12 by increasing the taper ratio from 1 to 12 (aimost zero in core thickness at the thinner end) (SR = 40). As the taper ratio increases, the maximum value increases linearly up to 8 and reaches to infinity.
Analysis of tapered sandwich structures
0.0 0.0
0.2
0.4
0.6
0.8
1.0
^ b.0
607
0.2
0.4
0.6
0.8
1.0
x/a (b) Transverse shear stresses in core
(a) Deflections at structural midplane 1.0
7.2r
0.5 1.0
0.0 0.8
-.5
i& 0.6
- -1.0 cr,s -1.5 -2.0 -2.5 -3.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
(c) Tensile stresses in upper face
(d) Tensile stresses in lower face
Fig. 11. Effect of slenderness ratio, SR, for cantilevered sandwich beam (CR = 1,O, HR = 2.0).
CONCLUSIONS
12
* I ,2 V
8
6 4 2 0
1 2 3
4
5
6
7
8
9 10 11 12
HR Fig. 12. E&et of taper ratio, HR, on maxirnmn transverse shear stress in core for cantilevered sandwich beam
(SR = 40).
An analytical method is developed for tapered sandwich structures with anisotropic composite faces. Local coordinate systems are introduced for each component of the sandwich structure. The inplane displa~ments for each face are taken as inde~ndent variables. For the tapered case, it is shown that the core shear strain is directly related to the face normal deflection. The present fo~uiation can be applied to arbitrary boundary conditions. Numerical results for the two boundary conditions and several geometric parameters have confirmed the validity of the method. The conventional assumption of antisymmetric deformation must be re-examined. The overall behavior of a tapered sandwich structure must be analyzed by considering the correlation between the core shear strain and the face normal devotion.
J. S. JFDN and C. S.
608
HONG
REFERENCES
(All)
1.
M. J. Salkind, Applications of composite materials. ASTM STP 524 (1973). 2. C. G. Kim and C. S. Hong, Buckling of unbalanced anisotropic sandwich plates with finite bonding stiffness. AIAA Jnl 26, 982-988 (1988). 3. G. R. Monforton and I. M. Ibrahim,
Analysis of sandwich plates with unbalanced cross-ply faces. Inf. J. Mech. Sci. 17, 227-238 (1975). 4. T. Akasaka and K. Asano, Stress and deformation of the sandwich panel having curved faceplates under pressure loading. Recent advances in composites in the United States and Japan. ASTM STP 864, pp. 263-277 (1985). 5. N. Paydar and C. Libove, Stress analysis of sandwich plates with unidirectional thickness variation. ASME, J. Appl. Mech. 53, 609-613 (1986). 6. N. Paydar and C. Libove, Bending of variable thickness. ASME, J. 419-424 (1988). 7. N. Paydar, Stress analysis of annular linearly varying thickness. Inf. J.
8. 9.
IO. 11. 12.
13.
14.
6412) APPENDIX II
The components of symmetric equivalent stiffness matrix defined in eqn (22) are shown below
of sandwich plates Appl.
Mech.
g,, = cos P]K!, -
K:,l+ [K:,l
(A13)
g,, = ~0sPK12
-
%I + K:,l
(A14)
g,, = ~0s BK:,
-
K:,l
(A15)
55,
+ [Kfl
sandwich plates of Solids Struct. 24,
313-320 (1988). C. Libove and C. H. Lu, Beam-like bending of variablethickness sandwich plates, AIAA Jnl27, XX-507 (1989). FI. G. Allen, Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969). R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). B. Baharlou and A. W. Leissa, Vibration and buckling of generally laminated composite plates with arbitrary edge conditions. Inf. J. Mech. Sci. 29, 545-555 (1987). P. J. Hoh and J. P. H. Webber, Finite elements for honeycomb sandwich plates and shells, part 2: Numerical results and testing. Aeronaut. Jnl84, 157-167 (1980). G. Lubin, Handbook of Composites. Van Nostrand Reinhold, New York (1984). F. J. Plantema, Sandwich Construction. The Bending and Buckling of Sandwich Beams, Plates, and Shells. John Wiley, New York (1966).
g,4 = ~0s AK14 - K:41
(A16)
g,, = ~0s PV$
- K:,l
(A17)
g22 = ~0s PK:,
- K&l + K:21
(A18)
g2, = ~0s BK:,
- K&l + K:I
(A19)
g24 = ~0s BK:4 - K:,l
(A20)
g2, = ~0s BW:,
- K&l
(A21)
g,, = cos fl[K:, - +,I + [KS]+ [Kg]
(A22)
g,, = ~0s IW:, -
Kl’
(~23)
VW
(A24)
g,, = ~0s BK:,
K:41+
- K:,l+
gu = cos /l[K$ - K&l + [KY,] APPENDIX I
WI =
IJ
[K*] =
PW?,IIRlrds dy s cosB[R][H:,][R]rds
(AlI dy
Is [K’] =
[P’][A,j],[P’]rds
dy
[P’][Bij],[P2]rds
dy
WI = WI =
Jf ss J‘s
(A2)
WI=
(A4)
jj
[KS1=
(A26)
g,, = ~0s BK:, - K:,l + K&l
(A27)
[#,I
Wf21
WI,1
K141
WI
K:21
V&l
K:41
K:,l
WI21
K:,l
K:41
K:, 1
#,I
[K:J
Khl
K:,l
lG21
W:,l
W$l
[KT,l
K:21
K:,l
K:41
K:,l
E21
[f&l
Ki41
K:,l
K:21
K:,l
K:41
K,2,1
K21
K&l
[f&l
W:,l
K:21
#,I
K:41
6428)
(A3)
ss [K4] =
%I + Kf21
g,, = ~9s B[K:, -
The matrices from [K’] to [I?] and the vectors {C} and {F} defined in eqn (19) are shown below
(~25)
K*l
[P21P,,1.[f’*1’ ds dr
WI
=
(A29)
[~‘l[A,,l,[P’lTd~ dy
cw
(A30)
P-“lPt,l,[P21~d~ dy
(.47)
(A31)
WI =
P21[D,,l,[P21Td~ dy 648) Is {CIr= t{c,!m,‘.{‘%I’, {Gnl’~ {Cm,)‘,{C:nYl (A9) {F}r= ]{FI”Jr, {OjT>{I%“}? @jr,
{0)7
(A101
(~32) (A33)
co45-7949/92 $5.00 + 0.00 0 I992 Pergdmon Press Ltd
ANALYSIS OF TAPERED SANDWICH STRUCTURES WITH ANISOTROPIC COMPOSITE FACES J. S. JEON and C. S. IIoN@ Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, P.O. Box 150, Cheongryang, Seoul, Korea (Received 16 May
1991)
Abstract-A new formulation is presented for the analysis of tapered sandwich structures with anisotropic composite faces. Three local coordinate systems are introduced to describe the independent displacements of each component. The expression for the total potential energy is derived and the Rayleigl-Ritz method is applied to obtain an approximate solution. The analysis takes into account the correlation between the core shear strain and the face normal deflection existing only for the tapered geometry. The present
formulation can be applied to arbitrary boundary conditions. Numerical examples are calculated for various core shear modulus ratios, taper ratios, and slenderness ratios. The correlation between the core shear strain and the face normal deflection is found to he very impor~nt in predicting the deflections and stresses of tapered sandwich structures.
INTRODUCTION Sandwich structures have been widely used in bending dominant structures because of their representative advantage of specific bending stiffness. When they are used in aerospace applications such as wings or control surfaces, the sandwich structures are used as forms of full-depth honey-comb constructions with/without stiffeners according to the load intensity. Such sandwich structures, for aerodynamic reasons, take airfoil shapes and the thickness varies along the chord line from the leading edge to the trailing edge [l]. For the analysis of uniform thickness cases, numerous approximating solutions have been given by many researchers such as Kim and Hong [2] and Monforton and Ibrahim [3]. They have considered the material anisotropy, face bending stiffness and finite bonding stiffness. In contrast to the uniform thickness sandwich structures, tapered sandwich structures have received considerably little attention in spite of their wide application. Until now, tapered sandwich structures have been investigated by few researchers. Akasaka and Asano [4] have analyzed a sandwich pane1 with a sinusoidally varying thickness at all edges simply supported by a frame. They showed the importance of the shell effect on the curvature by the flattening effect. Thereafter, Paydar and Libove have employed the finite difference method to calculate the deflections and core shear stresses for a square plate [5,6] and an annular plate [7]. The bending problem for a j’ Correspondence should be addressed to Professor C. S. Hong.
cantilevered sandwich beam has been analyzed by Libove and Lu 181. By comparing their theory and classical theory using the constitutive equations of locally constant thickness sandwich plates, they emphasized ‘the contribution of the face sheet membrane forces (by virtue of their slope) to the transverse shear’. The previous works for the tapered cases are restrictive in their formulation. The displacements for the upper and lower face are assumed as antisymmetric defo~ations (i.e. u, = -u2) without examining the validity. It imposes artificial constraints on the face displacements. The face bending stiffnesses are not included. It underestimates the structural bending stiffness; in addition the correlation between the core shear strain and face normal deflection is not expressed clearly in the explicit form and the core shear strain has been treated as a variable to be determined numerically. For practical designs and applications of sandwich structures, all the deflections and stress distributions must be known first. Without the knowledge of overall behavior, structural performance is not used to its best advantage and a failure may occur. In the present work, an analytic method has been developed to understand the behavior of tapered sandwich structures with anisotropic composite faces. The face bending stiffness as well as inplane stiffnesses are included and the core is considered as orthotropic. The analysis takes into account the correlation between the core shear strain and the face normal deflection existing only for the tapered geometry. The present formulation can be applied to arbitrary boundary conditions. Some numerical examples are presented for sandwich beams to demonstrate the validity and versatility of the method.
J. S. JEONand C. S. HONG
598 TOTAL
POTENTIAL
ENERGY
An overall view of the sandwich structure under consideration is shown in Fig. I. The assumptions made are as follows: The face thickness is constant and moderately thick to apply the classical laminate theory (CLT). The core thickness varies linearly in the x-direction. The core is orthotropic and in the antiplane stress state (T,, # 0, tYz# 0, all others = 0) f9]. The interfaces between the faces and core are perfectly bonded and the bonding layer thickness has no effect. Each component is under the small deformation. Strain energy of core
The transverse shear stresses in the core are constant through the thickness because the core is inextensible in the thickness direction and deformable only in the transverse shear. When the core is rotated about its reference axes by 4,,, the strain energy in the core can be written as
Fig. I. Geometry of tapered sandwich structure with anisotropic composite faces.
local coordinate systems for each component are introduced as shown in Fig. 2. The normal displacements and curvatures of the upper and lower faces are the same as the previous assumptions. Onfy the directions of the unit vectors of upper and lower faces are different due to the face slope about the structural midplane. This allows that the expression for the normal displacement of each face can be written as one variable. Therefore the six variables for this structure are reduced to five 0
II,(S,Y,n)=u~"(s,Y)-n~
where & = G,, cos* 4, + G,3 sin’ 4, Qd5= (Gu - GD) cos 6, sin +( & = G,, cos2 (p, + Gz, sin’ 4,. Here superscript 0 represents the values at the core midplane, subscript c for the core and Gi), Gz3 are the core transverse shear moduli about its material principal axes. The thickness of the core at any point decreases linearly in the x-direction by the face slope factor of tan p as h(x) = h, - x tan p.
(2)
Substituting eqn (2) into (1) and integrating the core thickness gives
-
x[H;.]}
{;“} dx dy, x.2c
over
(3)
where [Gt] = 2ho[Qij]~, [I$:] = 2 tan @[&],,
i, j = 4,5.
Strain energy of faces
The Rayleigh-Ritz method can be applied to a relatively simple structure such as a uniform beam or plate. But the present structure is complicated by the thickness variation and face slope about the midplane of the structure. To overcome this difficulty three
Fig. 2. Local coordinate systems.
Analysis of tapered sandwich structures where superscript 0 represents the values at the face midplane. By the small deformation condition, the face strain-displacement relations for the upper face are given by
Potential energy by external loads Under the normal distributed load, q. and vertical concentrated load, Q0 in the upper face, potential energy by external loads can be expressed as y,-
Combining
599
JJ
qo(~~~MX~,~)ds dy -~~~~o~Y~~~“~~~~Yo~. (10)
eqns (4) and {S) gives
%S= c:f + 55,
eyy= $ + “‘cyr,
ysv= yf + msy,
Here w” is the structural midplane deflection in the z-direction.
(6) COMPATIBILITY
where
CONDITIONS
Using two different local coordinate systems for the upper face and the core, a point at the interface can be described as
K,=
aho ~$9
a%0 -dyz,
Kvv=
‘csv=
aw as ay
(
uO+h(x) g
-2 n.
Because the faces are analyzed by CLT, the strain energy of the upper face can be written as
-y(l,
>
= usucos /3 - ui sin /I,
tP+ h(x) dw” - yiZ = By,, ( ay > w” = a,, sin /I + ii: cos fl.
(11)
Here u”, a’, w” represent the structural midplane displacements in the x, y, and z-directions, respectively. Similarly, using two different local coordinate systems for the lower face and the core, a point at the interface can be described as
where [&], is the transformed reduced stiffness matrix of the upper face. Expanding eqn (7) gives
Ljly=f
(
g
uo- h(x) dw” ( ay
{~“iIIAijlu{~oI~ h dY
ss
u”-h(x)
) -y;,
-&
=u,,cosfi
>
+ujfsin/3
= $/
w”= -u,,sin~+u~cosfl.
+
ss
+k
~~‘~~~~ijl”~~~ k dY I~)‘[~~jlu{~}h dYv
JT
Combining eqns (11) and (12) gives the explicit form for the core transverse shear strains in terms of the upper and lower face displacements as
(8) _awo -[(u, YXZ-dX -2/z(x) o
where [A,j],, [B,],,, [O,], are the extensional, coupling, and bending stiffness matrices of upper face, respectively, as defined in [lo]. Similarly, the expression for the lower face can be written as
w”+,,uf,=i
ss +
i
- u,,)cos j? - 24 sin j?]
u,,)sin fl + uz cos fl.
(13)
{~“>f[~~~l~{~o}~~ dY
Is
+;
(12)
~~“W,,ltG4 h dy
JJ
f#U4,1,~~~ ds dr-
The key difference from the uniform thickness case is expressed in the first and the third in eqn (13). The transverse shear strain yXZin the core is determined not only from the face inplane displacements but also from the face normal deflection as well by the face slope angle @.In addition the structural deflection w” is related to the face inplane displacements.
@I
J. S. JEON and C. S.
600
For the high taper ratio, the dependence of the core shear strain on the face normal deflection is severely influenced because, in the general bending problem, the value of face normal deflection is much larger (about several hundred times) than those of the inplane displacements. If the face slope angle a is set to 0, they are reduced to the same expression for the uniform thickness sandwich plate in [9]. Figure 3 shows the difference of the mechanism invoking the transverse shear strain y,; in the core. It can be easily shown that it will produce unreasonable results if the classical sandwich plate theory using the constitutive equations of locally constant-thickness (which do not consider the effect of face normal deflection and only consider changing the local bending stiffness) is used in analyzing the tapered sandwich structure.
HONG
By using arbitrary shape functions, several advantages can be retained. It is relatively simple to manage in algebraic calculations and any boundary condition can be dealt with. In addition it forms mathematically complete sets of functions and by taking the sufficient number of expansion terms it converges to an exact solution 1111. For further processes, new functions are defined from the already defined shape functions in eqn (14) as
DISPLACEMENTS ASSUMPTIONS
To apply arbitrary boundary conditions, the displacements of each face are expressed in general form by unknown coefficients, C~,-C~H and the proper shape functions, 4;,,-4;, satisfying the given boundary condition as
Substituting eqns (14) and (15) into the core shear strain in eqn (13) gives r R:’
-T
R-;
Cl” cl” C” Cl”
where
0
n
=> fj=>
;zf
(a) For uniform thickness
(b) For tapered thickness
Fig. 3. Comparison of core shear deformation mechanisms.
I
(16)
601
Analysis of tapered sandwich structures Rearranging equation
Let us rewrite eqn (16) for simplicity as {YOj,=
W{CI.
(17)
For the faces, the strains and curvatures are
{~o}“=[P’]~
1E)I
;”
{cO},= [PI]’
)
the above produces
!?I”
{
{K} = [P21T{CLn~9
In”
I )
(18)
where
kl{Cl = w,
(22)
where [g] is the symmetric equivalent stiffness matrix having 5 x 5 submatrices. Detailed expressions for each submatrix are given in Appendix II. By solving this matrix equation, the unknown coefficients are obtained and all the related quantities can be determined. VERIFICATION
Substituting the above expressions to the strain energy and the potential energy by external loads gives
a linear matrix
OF THE ANALYSIS
In order to show the applicability and to check the validity of the method, some numerical calculations are carried out for several examples. Nine odd expansion terms in the displacement assumptions are taken for sufficient accuracy. The IMSL subroutine DMLIN was used for numerical integration and LEQTZF for the equation solver. Two representative boundary conditions are selected: simply supported and cantilevered sandwich beams. Simply supported uniform sandwich beam
To show the validity of applying arbitrary boundary conditions, simply supported uniform sandwich beams are analyzed at first. The boundary conditions are N, = uI) = MS = 0, at s = 0, c,
(23)
where N, and M, are the force and moment resultant of each face. The selected shape functions satisfying the given boundary conditions are given below
+t{CimY[K81{Cin) ~!,,,,=cos~,
v = -(C)T(F).
SOLUTION
The total potential energy of a system is defined by the summation of the strain energy of the structure itself and the potential energy by external loads as n = UC+ c;, + u, + V.
(20)
For a system to be in a static equilibrium state, the principle of minimum potential energy must be satisfied. It requires the partial differentiation of the total potential energy about each unknown coefficient as
an -=o, ac;"
4i,,=sinE.
(19)
Detailed expressions from [K’] to [K8], {C} and equivalent force vector {F} are given in Appendix I. APPROXIMATE
&,$,=O,
i=l,...,
5.
(21)
c
(24)
The structure is under uniform distributed load. The stresses are compared with those of previous finite element results [12]. The tensile stress at the upper face and the core shear stress are shown in Fig. 4(a) and (b), respectively, under the uniform lateral load. They coincide very satisfactorily. Cantilevered sandwich beams
The comparisons based upon the structural deflections are performed for the tapered sandwich beams with the previous experimental results in [5]. Experimental geometry, loading, and boundary conditions are shown in Fig. 5(a). Because the thickness of the present sandwich structure varies linearly in one direction, the experiment is modeled as in Fig. 5(b). The boundary conditions are
+.@L0, N,=M,=
0
as
ats=O
V,=O,
at s =c,
(25)
602
J. S. JEANand C. S. HONG
0.16
i
(a) Experimental geometry and load 1
0.1
0.2
0.3
0.4
0.5 C_
x/a
51.8
,!
(a) Transverse shear stress in core
-20 -40 -60 -80
(b) Modeled geometry and load
-100
Fig. 5. Modeling of experiment for comparison of ‘quasistiffness’.
-120 -140 -160
’ 0.0
I
I
I
I
0.1
I
0.2
0.3
0.4
0.5
x/o (b) Tensile stress in upper face Fig. 4. Comparison for simply supported sandwich beam with FEM results (HI? = 1.0).
where V, is the resultant shear force of each face. The selected shape functions satisfying the given boundary conditions are given below
Here the V, =0 condition for the faces is not satisfied but the present Rayleigh-Ritz method can
be applied if kinematically admissible functions are selected. From the measurements and calculations at two points apart, its ‘quasi-stiffness’ between the two points is calculated. Its values are listed and compared with previous experimental results in Table 1. The present calculations show more stiff results than those from the experiments with an error of 0.2-4.0%. There may be several reasons: First, the modeling itself from the experiment can produce error because its boundary condition is not exactly the same. Secondly, the core material property used in the experiment is very soft (polystyrene) and the present assumption (tZ2= 0) can produce more stiff results. Thirdly, experimental errors from the apparatus setting and readout from the dial gages can be introduced. Considering the above factors, the present formulation can be said to show good agreement.
Table 1. Comparisons of ‘quasi-stiffness’ between experimental and theoretical results p/G& - 6,) W4 HR
FDM [3] (error, %)
Experimental [3]
Present (error, %)
1.00 1.11 1.43 2.00
1772 (0.00) 1848 (3.88) 2034 (2.42) 2289 (0.3 1)
1772 1779 1986 2282
1773.8 (0.10) 1851.3 (4.06) 2047.5 (3.10) 2332.5 (2.21)
603
Analysis of tapered sandwich structures RESULTS AND DISCUSSIONS
Cantilevered sandwich beams are analyzed for uniform and tapered cases. The used shape functions are the same as in eqn (26). All the stacking sequences are aligned to 0” for best performance, The material properties of graphite/epoxy (T300/5208) used for the faces are E, = 181.0 GPa, E2 = 10.7 GPa, G,, = 7.07 GPa, viz = 0.28. The core is a phenolic honey-comb with material properties of G,, = 110.32 MPa, Gz3= 44.128 MPa [ 131. The dimensions are r,, = tl = 1 mm and To = 25 mm. The remaining dimensions are determined from the geometric parameters. The structure is under uniform normal distributed load. To represent the results, the following normalized quantities are used
0.0
0.2
0.4
0.6
0.8
1.0
(a) Deflections at structural midplane
G,, G& G:,
CR_-2
SR=$
_
4w”E, t, T;
W=qoa4(1 -v$)
627) where To and T, are structure thicknesses at x = 0 and x = a, G$, G,*, are the reference core transverse shear moduli and G,,, Gz3 are varied core shear moduli. Eflect
of core
0.0
0.2
0.4
0.6
0.8
1.0
(b) Transverse shear stresses in core
1.2,
modulusratio
By varying the core modulus ratio, CR, uniform cantilevered sandwich beams have been analyzed (HR = 1.O, SR = 40). The structural deflections, core shear stresses, and tensile stresses in upper face are shown in Fig. 6(a)-(c), respectively. The results show that the core shear modulus gives an influence not on the stresses but only on the deflections for the uniform case. Note that the core shear stresses shown in Fig. 6(a) are zero by its boundary condition at the root and the free end. But near the root, high core shear stresses are developed and the values decrease to zero at the free end. It means that the debonding by the core shear stresses may occur near the root region. The tensile stresses in the lower face are the same as those for the upper face only in the change of the sign; they are not given here.
‘%!O
’ 012 ’ 014
’ 016 ’ ii??
1.0
x/a (c) Tensile stresses in upper face. Fig. 6. Effect of core modulus ratio, CR, for cantilevered sandwich beam (HR = 1.0, SR = 40).
J. S. JEONand C. S. HONG
604
1.2,
0.80 11
L
1.0
0.8 0.6 -
7,z0.4 0.2 0.0 I 0.8
1.0
-3l.0
x/a
I
0.2
I
I
I,
0.4
,
0.6
(
,
0.8
1
x/a (b) Transverse shear stresses in core
(a) Deflections at structural midplane 0.7 ,
1
0.8 ,
0.6 0.0
0.5 CR=0.5 0.4 -
a,,-0.8 0.3 -
-1.6
-.
‘v!
0.0 0.0
I
I
0.2
I
I
I
0.4
I
0.6
I
I
0.8
I
1.0
s/c (c) Tensile stresses in upper face
-2.4' ' ' ' ' ' ' ' ' ' 0.0 0.2 0.4 0.6 0.8
0
s/c (d) Tensile stresses in lower face
Fig. 7. Effect of core modulus ratio, CR, for cantilevered sandwich beam (HR = 2.0,SR = 40).
Tapered cantilevered sandwich beams with a taper ratio of 2 (HR = 2.0)have been analyzed (SR = 40). Structural deflections, core shear stresses, and tensile stresses in both faces are shown in Fig. 7(a)-(d), respectively. The results show that for the tapered case the core shear modulus gives a significant influence not only on the deflections but also on the stresses. The tensile stresses at different faces are not the same in absolute magnitude and have a very different distribution, especially the local positive value in the lower face (Fig. 7d). This is due to the change of local curvature along the span length. Compared with the deflections for the uniform thickness cases (Fig. 6a), those for the tapered case (Fig. 7a) are very small contrary to common sense. The reasons may be described as follows: For the uniform case, the sandwich structure resists lateral load by face bending and core shear stiffness. But for the tapered case, two contradictory effects exist. Firstly, deflection increases by the reduction of
the core volume resisting load by shear stiffness. Secondly, by the face slope about the structural midplane, the vertical component of the face extensional stiffness resists shear load along with the face bending and the core shear stiffness. It induces an increased structural stiffness and in turn the structure deflection decreases. Out of the two effects above, the latter is dominant and the deflection decreases. The structural deflection is greatly influenced by the core shear modulus as shown in Fig. 7(a) because of the correlation in the first of eqn (13). The core shear stresses, shown in Fig. 7(b), are concentrated near the thinner free end. This is the effect of the face normal deflection on the core shear strain. The tensile stresses distribution in the upper face shown in Fig. 7(c) are rearranged. The tensile stresses in the lower face are shown in Fig. 7(d). Contrary to the uniform thickness case, the face stresses in the lower face change their signs and magnitudes along the span direction. That is, high negative stresses are
Analysis of tapered sandwich structures
(a) For uniform thickness (HR = 1.0, SR = 40, CR = 1.0)
605
occur up to the free end. This is due to the curvature change of the structure in its sign along the span length. The essential difference is shown in Fig. 8 by exaggerating the deformed shapes. Figure 8(a) shows the shape for the uniform thickness and Fig. 8(b) for the tapered case. Here we must give attention to the points that the structural curvatures are different for each case. EJect of taper ratio
The taper ratio, HR, is increased from 1.0 to 1.5 (CR = 1.0, SR = 40). The deflection shown in Fig. 9(a) increases up to HR = 1.1 and then decreases (b) For tapered thickness (HR = 2.0, SR = 40, CR = 1.O)
Fig. 8. Comparison of exaggerated deformed shapes for cantilevered sandwich beam (CR = 1.0, SR = 40).
induced at the fixed end and quickly reduced to zero. After those transition points, they change their signs to positive and relatively small positive stresses
,
2.0
1.6
1
as taper ratio increases. This is due to the change of the dominant factor for the structural stiffness (from the stiffness reduction by the core volume change to the increase of the face slope participation). The core shear stresses in Fig. 9(b) are redistributed as the taper ratio increases. The location of the maximum shear stress is moved from the region near the fixed 1.6,
I
I
_HR=l.O 1.2
0.8
w
0.6
0.8
0.4 0.2
0.4
0.0 0.0
0.0
0.2
0.4
0.6
0.8
1.0
-.
?I .O
0.2
X/Q
0.4
0.6
0.6
1.0
da (b) Transverse shear stresses in core
(a) Deflections at structural midplane 1.6,
I
1.2
Gs0.8 -1.6
0.6 0.4
-2.4 0.2 nn _._
0.0
0.2
0.4
0.6
0.6
a/c (c) Tensile stresses in upper face
1.0
-3-a% s/c (d) Tensile stresses in lower face
Fig. 9. Effect of taper ratio, HR, for cantilevered sandwich beam (CR = 1.0, SR = 40).
1.”n
J. S. JIDN and C. S. HONG
606 4.0 ,
I
3.5 3.0 2.5
w
2.0
0.6
1.5
0.4
7.0 0.2
0.5 0.0
0.01 0.0
0.2
0.4
0.6
0.8
1.0
1
0.0
I
I
0.2
X/Q
t
I
0.4
,
0.6
,
I
0.8
,’ ’ 3
X/Q
(b) Transverse shear stresses in core
(a) Deflections at structural midplane
0.0) 0.0
0.2
0.4
0.6
0.8
1.0
x/a (c) Tensile stresses in upper face
Fig. 10. Effect of slenderness ratio, SA, for cantilevered sandwich beam (CR = 1.0, HR = 1.0).
end to the thinner free end after undergoing a process
of smoothing aIong the span. The trends of redistributing tensile stresses in both faces are shown in Fig. 9(c) and fd). For the lower face especially, the transition point moves from the free end to the fixed end as the taper ratio increases. Effect of slenderness ratio By varying the slenderness ratio, SR, a uniform cantilevered sandwich beam has been analyzed (CR = 1.0, HR = 1.0). The structural deflections, core shear stresses and tensile stresses in the upper face are shown in Fig. ~~a)-~c), respectively. The results show that the slenderness ratio gives an influence not on the stresses but only on the deflation for the uniform case. Note that the deflection is larger for short sandwich beams (SR = 10) than for long beams (SR = 40). This is due to the core shear defo~a~ion. That is, for the short sandwich beam, the deflection by the core shear deformation plays a
greater part in the total structural deflection than that for the longer beam. The core shear and tensile stresses in the upper face are almost same for each case. The tensile stresses in the lower face are same as those for the upper face only in the change of the signs; they are not given here. By varying the slenderness ratio, SR, tapered cantilevered sandwich beams with a taper ratio of 2 (HR = 2.0) have been analyzed (CR = 1.0). The structural deflections, core shear and tensile stresses in both faces are shown in Fig. II(a)-(d), respectively. The results show that for the tapered case the slenderness ratio has a sibilant influence not only on the deflections but also on the stresses contrary to the uniform case. The maximum core shear stresses are shown in Fig. 12 by increasing the taper ratio from 1 to 12 (aimost zero in core thickness at the thinner end) (SR = 40). As the taper ratio increases, the maximum value increases linearly up to 8 and reaches to infinity.
Analysis of tapered sandwich structures
0.0 0.0
0.2
0.4
0.6
0.8
1.0
^ b.0
607
0.2
0.4
0.6
0.8
1.0
x/a (b) Transverse shear stresses in core
(a) Deflections at structural midplane 1.0
7.2r
0.5 1.0
0.0 0.8
-.5
i& 0.6
- -1.0 cr,s -1.5 -2.0 -2.5 -3.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
(c) Tensile stresses in upper face
(d) Tensile stresses in lower face
Fig. 11. Effect of slenderness ratio, SR, for cantilevered sandwich beam (CR = 1,O, HR = 2.0).
CONCLUSIONS
12
* I ,2 V
8
6 4 2 0
1 2 3
4
5
6
7
8
9 10 11 12
HR Fig. 12. E&et of taper ratio, HR, on maxirnmn transverse shear stress in core for cantilevered sandwich beam
(SR = 40).
An analytical method is developed for tapered sandwich structures with anisotropic composite faces. Local coordinate systems are introduced for each component of the sandwich structure. The inplane displa~ments for each face are taken as inde~ndent variables. For the tapered case, it is shown that the core shear strain is directly related to the face normal deflection. The present fo~uiation can be applied to arbitrary boundary conditions. Numerical results for the two boundary conditions and several geometric parameters have confirmed the validity of the method. The conventional assumption of antisymmetric deformation must be re-examined. The overall behavior of a tapered sandwich structure must be analyzed by considering the correlation between the core shear strain and the face normal devotion.
J. S. JFDN and C. S.
608
HONG
REFERENCES
(All)
1.
M. J. Salkind, Applications of composite materials. ASTM STP 524 (1973). 2. C. G. Kim and C. S. Hong, Buckling of unbalanced anisotropic sandwich plates with finite bonding stiffness. AIAA Jnl 26, 982-988 (1988). 3. G. R. Monforton and I. M. Ibrahim,
Analysis of sandwich plates with unbalanced cross-ply faces. Inf. J. Mech. Sci. 17, 227-238 (1975). 4. T. Akasaka and K. Asano, Stress and deformation of the sandwich panel having curved faceplates under pressure loading. Recent advances in composites in the United States and Japan. ASTM STP 864, pp. 263-277 (1985). 5. N. Paydar and C. Libove, Stress analysis of sandwich plates with unidirectional thickness variation. ASME, J. Appl. Mech. 53, 609-613 (1986). 6. N. Paydar and C. Libove, Bending of variable thickness. ASME, J. 419-424 (1988). 7. N. Paydar, Stress analysis of annular linearly varying thickness. Inf. J.
8. 9.
IO. 11. 12.
13.
14.
6412) APPENDIX II
The components of symmetric equivalent stiffness matrix defined in eqn (22) are shown below
of sandwich plates Appl.
Mech.
g,, = cos P]K!, -
K:,l+ [K:,l
(A13)
g,, = ~0sPK12
-
%I + K:,l
(A14)
g,, = ~0s BK:,
-
K:,l
(A15)
55,
+ [Kfl
sandwich plates of Solids Struct. 24,
313-320 (1988). C. Libove and C. H. Lu, Beam-like bending of variablethickness sandwich plates, AIAA Jnl27, XX-507 (1989). FI. G. Allen, Analysis and Design of Structural Sandwich Panels. Pergamon Press, Oxford (1969). R. M. Jones, Mechanics of Composite Materials. McGraw-Hill, New York (1975). B. Baharlou and A. W. Leissa, Vibration and buckling of generally laminated composite plates with arbitrary edge conditions. Inf. J. Mech. Sci. 29, 545-555 (1987). P. J. Hoh and J. P. H. Webber, Finite elements for honeycomb sandwich plates and shells, part 2: Numerical results and testing. Aeronaut. Jnl84, 157-167 (1980). G. Lubin, Handbook of Composites. Van Nostrand Reinhold, New York (1984). F. J. Plantema, Sandwich Construction. The Bending and Buckling of Sandwich Beams, Plates, and Shells. John Wiley, New York (1966).
g,4 = ~0s AK14 - K:41
(A16)
g,, = ~0s PV$
- K:,l
(A17)
g22 = ~0s PK:,
- K&l + K:21
(A18)
g2, = ~0s BK:,
- K&l + K:I
(A19)
g24 = ~0s BK:4 - K:,l
(A20)
g2, = ~0s BW:,
- K&l
(A21)
g,, = cos fl[K:, - +,I + [KS]+ [Kg]
(A22)
g,, = ~0s IW:, -
Kl’
(~23)
VW
(A24)
g,, = ~0s BK:,
K:41+
- K:,l+
gu = cos /l[K$ - K&l + [KY,] APPENDIX I
WI =
IJ
[K*] =
PW?,IIRlrds dy s cosB[R][H:,][R]rds
(AlI dy
Is [K’] =
[P’][A,j],[P’]rds
dy
[P’][Bij],[P2]rds
dy
WI = WI =
Jf ss J‘s
(A2)
WI=
(A4)
jj
[KS1=
(A26)
g,, = ~0s BK:, - K:,l + K&l
(A27)
[#,I
Wf21
WI,1
K141
WI
K:21
V&l
K:41
K:,l
WI21
K:,l
K:41
K:, 1
#,I
[K:J
Khl
K:,l
lG21
W:,l
W$l
[KT,l
K:21
K:,l
K:41
K:,l
E21
[f&l
Ki41
K:,l
K:21
K:,l
K:41
K,2,1
K21
K&l
[f&l
W:,l
K:21
#,I
K:41
6428)
(A3)
ss [K4] =
%I + Kf21
g,, = ~9s B[K:, -
The matrices from [K’] to [I?] and the vectors {C} and {F} defined in eqn (19) are shown below
(~25)
K*l
[P21P,,1.[f’*1’ ds dr
WI
=
(A29)
[~‘l[A,,l,[P’lTd~ dy
cw
(A30)
P-“lPt,l,[P21~d~ dy
(.47)
(A31)
WI =
P21[D,,l,[P21Td~ dy 648) Is {CIr= t{c,!m,‘.{‘%I’, {Gnl’~ {Cm,)‘,{C:nYl (A9) {F}r= ]{FI”Jr, {OjT>{I%“}? @jr,
{0)7
(A101
(~32) (A33)