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Stability of laminated plates using finite strip method based on a higher-order plate theory
Campsite Structures 34 (1996) 65-16 Copyright 6 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0263-8223f96/$15.00 0263-8223(95)00132-8
Stability of laminated plates using finite strip method based on a higher-order plate theory W. J. Wang repayment of Civil E~~nee~n~ rational Lien-Ho College, ~iaol~ Taiwan
Y. R Tseng Institute of Civil Engineering, Tamkang University, Tamsui, Taiwan &
K. J. Lin Institute of Applied Mathematics, National Chung-Hsing University Taichung, Taiwan A higher-order shear deformable plate finite strip element is developed and employed to investigate the critical buckling load of composite laminated plates. The warping of cross-section and transverse shear deformation can be accurately predicted by the higher-order plate theory. Meanwhile, fewer degrees of freedom in the finite strip method than those in the finite element method are required. The good convergence characteristics of the finite strip approach and the relative efficiency of particular economization schemes are then demonstrated through several numerical results.
1 INTRODUCTION With the characteristics of high specific modulus, high specific strength, and the capability of being tailored, laminated composite materials have been extensively used especially in the design of aircraft structures. Hence, an accurate calculation of the critical buckling load is required in assessing the strength of composite structures. The classical plate theory (CPT) ignores the transverse shear stress, thus usually underprediets deflection and overpredicts natural frequencies and buckling loads except for very thin plates. Mindlin plate theories introduce a correction factor into the shear stress resultants to account for a uniform shear stress through the thickness of the plate. So they do not adequately model the behaviour of highly orthotropic composite structures. There are exact solutions’ for composite material structures to prove the above points. Several higher-order theories2-6 have been developed for the analysis of composite plates. The transverse shear deformation is considered
without the correction factor and the warping of cross-section is also taken into account. A critical evaluation of new plate theories by Bert2 indicated that the theory of Lo et aL3 produced an accurate prediction of the non-linear bending stress dist~bution. Kant and Pandya* and Kant et al.5 developed various Co higher-order plate elements. A refined C1 higher-order plate theory was further proposed by Reddy7 to consider the traction-free condition on the top and bottom surfaces of the plate. All of them can predict the behaviour of laminated plates more accurately and do not need the shear deformation correction factor. Abundent literature can be found for the buckling analysis of composite plates. Jones* evaluated the buckling of unsymmetrically laminated cross-ply rectangular thin plates. Jones et aZ.’ extended the above work to unsymmetrically laminated Noor” angle-ply plates. calculated the buckling loads of thick laminated cross-ply composite plates by using different shear correction factors for different stacking sequences. Putcha and Reddy,l’ Phan and Reddy,12 and Owen and Lit3 introduced several
66
W J. Wang, l! l? Tseng, K. J. Lin
higher-order plate elements for the stability analysis of laminated plates. The finite strip method has been proved to be a more efficient computational tool than the finite element method for prismatic structures. Cheung14 firstly developed the finite strip method of CPT to solve for the natural frequencies of plates. Mawenya and DaviesI applied the finite strip method to the Mindlin plate theory. This work has been extended by Benson and Hinton16 to include vibration and stability applications. Hinton” considered the buckling load of initially stressed plates using the finite strip method based on the Mindlin plate theory. Dawe and Roufaeil18 studied the buckling load of the Mindlin plate including the shear deformation effect. The finite strip method has been incorporated with the higher-order plate theory for the static analysis of laminated plates by Tseng and Wang.” Excellent results were reported. The method of analysis is further extended to determine the buckling load of the composite laminates in this study. The highlight is due to the facts that the higher-order plate theory can accurately predict the behaviour of the composite plates, and the finite strip method not only reduces the computational cost and storage space but also predicts accurate results for prismatic structures. Some comparative benchmark problems are given to demonstrate the adequacy and accuracy of the present study.
to midplane about the x and y axes, respectively, and u”, u*, w*, O$, Oz, 0, are the corresponding higher-order terms in the Taylor series expansion and are defined at the midplane. The positive directions are shown in Fig. 1. The above displacement assumption is written in matrix form as {A>-[L]W
(2)
where {A}=[uvwlT is the 3-D displacement field, {d}=[u, Q w. 0, Q,. Q, U* c* w* O*, fj”,]’ is the midplane displacement field, and [L] is a linear operator matrix. The partial derivative of 3-D displacement field can be presented in the following form as {A.,>=@]
{4/J
(31
where G,,) =[Q),~ uo,Y Q,, Q,~ We., II,,, Q,Y OX., f&,XG,., &,$ u 5 u T, G CT, w 5 WTYSZ,X HZ,, G, Gxl is the partial derivative of the generalized midplane displacement, and [L] is the proper differential operator matrix. Making the usual assumptions, the constitutive relationships for the laminate are co>=@]
{El
(4)
where {g.) = [ox aY o, r-T z,, zYZ]is the 3-D stress, and {r:}=[r;, t;, r;, yg ;:, ~~~1“is the 3-D strain, and [e] is the material property matrix. In the plate analysis, the generalized midplane strain can be defined by {E}=[{FO},{k},{&*},{k*},{~},{~},{~*}17 (5)
2 HIGHER-ORDER PLATE THEORY The displacement field of higher-order plate theory assumed by Lo et al.” is adopted. The plate, of thickness h, is referred to an x, y, z system of rectangular coordinates, where the top and bottom of the plate are the planes z= -+h/2. Specifically, it is assumed that ~(4y,z)=4l(-5Y)
Z.W
+zf&y)
+z2u * (x,y) +z”q?(x,y) ~(~,Y,z)=~o(~,Y)
+zf&,y)
+z2u* (x,y) +z’fqx,y) ~(~,y,z)=%(4Y)
+-a(&Y)
(1)
+z2~*(-%Y)
where u, U, w are the displacements along x, y, and z coordinates, z is the distance from the midplane, u,, v,, W, are the associated midplane displacements, OX,Q,,are the rotations of normal
Fig. 1.
System coordinate
and displacement
field.
67
Stabilityof laminated plates using finite strip method
with the definitions
of
The total potential
{E0)= [&l,Xuo,yuo,y + ro,x 61 {t~*>==[u~x
{~I=[&,,
{k*l=p;,y
14
I=
~y,x+&,,
o;,,
[wo,x
+
0,
I-I=U-V
+
61
(8)
where U is the strain energy, and V is the potential energy due to external load. In the above equation, the strain energy is
2W”lT
~~,,+of,y]’
wo,y
plates
is
T
V~yU~y+U~.JT
oy,x
energy of laminated
T
{~>=[2u”+0,,,2v”+0,,,lT -@*)=[&I,,
+30;
wo,,+30:]T
The midplane strains are also related midplane displacements as follows {C}=[& {d}
to the (6)
=
where [B] is the proper differential operator matrix. The well known constitutive relation of laminated plate can be expressed in the following form as {5>=[D> {El
(7)
where [5] is the midplane stress resultant, and [D], constructed by the proper integral of @] with z, is the material properties matrix of the plate.
3 TOTAL POTENTIAL
TOTAL POTENTIAL
{e}T{o> dA
U=i
(10)
sR
where R is the midplane area of the laminated plates. Substitute the material property into the above equation to obtain {E}T[D] (i) d/4.
U=$
(11)
sR
Since the plate is assumed to be loaded by the stresses o:, ci, z&, the potential energy due to external loading is expressed as
ENERGY
Previous literature considered the critical buckling load, a presumed midplane resultant force, but we consider the buckling load, a presumed in-plane stress. In this paper, we assume that the laminated plates are subjected to uniform in-plane stresses a:, Q;, z$ through the thickness. The superscript ‘0’ represents the unknown prescribed value.
4 INCREMENTAL
where u is the volume of the laminated plate. Integrating eqn (9) through the thickness, the stress {Q} and the strain (1) will become the midplane resultant force (5) and the midplane strain {E}, then
(;+g)] dv (12) where oz, oj, r$ are the unknown cal buckling stresses.
in-plane criti-
ENERGY
According to Washizu,‘” the structure will reach equilibrium at the onset of buckling. When an infinitesimal perturbation is imposed along the bifurcated path, the equilibrium will change into another one. The incremental total potential energy is derived from the difference of these two states. The superscript ‘0’ stands for the onset of buckling and the “’ is after the onset of buckling. The displacements u, 21,w can be separated into two parts, as follows U=UO+U’
(13)
W J. Wang, Y R Tseng, K. J. Lin
68
where u’, v”, w” are the displacements at the initiation of buckling and u’, v’, w’ are the incremental displacements after the onset of the buckling by imposing the infinitesimal perturbation. Similarly, the stresses and strains also can be divided into two parts f&=f3,O+fS: &,=&,o+&: fS,=0,“+0;
Ey=&yo+&;
0,=fJ,o+C7:
&,=&,O+&;
T,=T$+Z&
YlT=Y&+Y&
(14)
z,=z:+z:, J&=y:+y:, zyz= z;z +z;z Yyz= r;z +y;z Therefore,
the total potential
energy II can be separated
into two parts as
l-I=l-I’+AII
(IS)
where II0 is the total potential energy at the onset of the buckling, same as eqn (8), and AII is the additional total potential energy due to external load by imposing an infinitesimal perturbation after the onset of buckling. The potential energy is calculated for the in-plane stresses, not the midplane forces. Therefore, the corresponding membrane strains are chosen to be in-plane Green-Lagrangian strains, not the Von Karman strains, to present the potential energy AII due to external loading as AlI=+
” {.s’}T{a’} dvs
{a;jT{&>
dv
(16)
sV
where {E’} is the incremental strain, (0’) is the incremental stress, {c$}=[c~ oy”z&l’ is the in-plane stress vector at the beginning of buckling, and the corresponding membrane strain {EL} is
(17)
(g(g+(g(z)+(g(g \ But, the relations of incremental &.;=-
au’ 8X
alI’
c’=y
c’=-
z
ay ad a2
strain and incremental
displacement
are still of the form
-r -au’+au’
ax
ay
yq-
alit
aw’
“=-+az I
_
(18)
ax
qv’ aw’
‘Yz-a2
I
ay
It is also assumed that the material small deformation, i.e.
property
after the onset of buckling
is the same as that for
69
Stabilityof laminated plates using finite strip method
(19)
(0’) = [Q] {E’l.
Integrating [e] of the above equation potential energy in eqn (16) becomes + 1 ([B] W))T[D] R
For simplification becomes
through
the thickness,
[B] id’) d/f. purpose,
total
(20)
u ‘, u’, w’ can be replaced
by U, u, W, and the second term of eqn (16)
{cT;}={E,} dv= ” {A.,>‘[$,] {A,,) du s sv where {A,,} is the partial derivative of the 3-D displacement matrix made of critical buckling stresses such as
Substituting
the first term of incremental
eqn (3) into eqn (21) and integrating
(21) field defined in eqn (3), and [O,] is the
it through the z direction, we obtain
(22) where the stress resultant
matrix is I
I
,
x,
s,
0
0
0
0
0
0
0
0
x,
s3
0
0
0
0
x4
s‘j
0
0
Y,
0
0
0
0
x, s*
s2
s1
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
x,
s,
0
0
0
0
x,
s*
0
0
0
0
x,
s3
0
0
0
0
x4
s4
Y,
0
0
0
0
s*
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
s1
0
0
0
0
x,
s1
0
0
0
0
x,
sz
0
0
0
0
x,
s3
0
0
0
0
0
0
0
0
s1
Y,
0
0
0
0
s*
Y,
0
0
0
0
s3
&
0
0
0
0
x,
s2
0
0
0
0
x,
s3
0
0
0
0
x,
s4
0
0
0
0
x,
s5
0
0
s*
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
0
0
x,
s2
0
0
0
0
x,
s3
0
0
0
0
x4
s4
0
0
x5
s5
0
0
s2
Y,
0
0
0
0
s3
Y,
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
x,
s2
0
0
0
0
x,
sj
0
0
0
0
x4
s4
0
0
0
0
0
0
0
0
s2
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
x,
s3
0
0
0
0
x5
s5
0
0
S6
0
0
0
0
0
0
x4
s4
0
0
x,
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
sg
Ye
0
0
0
0
x,
s3
0
0
0
0
x4
s4
0
0
0
0
x5
s5
0
0
0
0
X6
S6
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
S6
Y,
0
0
0
0
x,
s3
s4
0
0
0
0
x5
s5
0
0
0
0
0
0
0
0
s5
Y5
0
0
0
0
0
0
0
0
x4
0
s4
Y4
0
0
s3
Y,
0
0
0
s4
0
0
0
0
x5
s5
0
0
0
0
x,
s,5
0
0
0
0
x,
s,
0
0
Y4
0
0
0
0
s5
Ys
0
0
0
0
s,
Ye
0
0
0
0
s,
Y,
0
0
0
0
x4
s4
0
0
0
0
x5
s5
0
0
0
0
x,
se
0
0
0
0
x7
s7
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
S6
Y,
0
0
0
0
s7
Y,
0
0
x4 s4
I
70
W J. Wang, 2: P Tseng, K. J. Lin
with (X,,X,,X,,X,,X,,X,,X,)=
;
s:
k=l
s
,:*+’ (1, z’, .z2,z3, z4, z5, 2”) dz *
nl
(Yl, Yz, Y,, Y4, Y5, Y,, YT)= 1 c;
hk+’ (1,~‘,z2,z3,z4,zs,zh)
(S,
>
s2, &, S4, Ss, &, &> = ;
dz
c h,
k=l
7;
k=l
i
hh+’ (1, z’, z2, z3, z4, z5, z”)
dz
his
and hk is the vectorial distance from the panel midplane. In the other way, {d,,} can be replaced by midplane displacement &J=[B,]
{d} as
(24)
{d)
where [&,I is the proper differential operator matrix. Using eqn (24), eqn (22) can be transformed to 1 -2-
{d,,)T[~P] {d,,) dA=+ sR
([B,] MT[a,][B,]
(25)
{d) d/I.
s R
Equations
(4) (20) and (25) are combined
s
MI=+ ([i] {d))‘[D] R
-$
[I?]{d) d.A
to obtain the incremental
([&I
displacement
For the interpolations of the semi-analytical finite strip method, the simple polynomials are adopted in the transverse direction and the eigenfunctions are used in the longitudinal direction, with the latter satisfying a priori the end boundary conditions of the strip. Since we use a C” continuous displacement model, there are m harmonic terms and nst nodal points, the midplane displacement for a typical strip is interpolated by nodal displacements as m nst
w= 1 c 1=1
[Nf]{qj}
[B&J Id) d/l.
(26)
R
5 FINITE STRIP ANALYSIS
{dl=[W
energy as
s w-jqo,]
again that the incremental
It is mentioned above formulation.
total potential
(27)
i=l
where m denotes the total number of harmonic terms, nst is the nodal number per strip, [N] is the matrix of shape functions, {q} is the nodal displacement vector, [Nf] is the matrix of shape functions for node i and for the Ith harmonic term, and {qF}=[uGi, uzi, wZ~, Ozi, O_Gz,0%, u 7: vf: w T,’r3;i;f,0 >I ’ is the nodal displacement for node i and for the eth harmonic term. The
{d’} has been replaced
by {d} in the
complete expression of the interpolation can be found in Tseng & Wang. I”) Then, the derivatives of the midplane displacement field are ,?I n.s*
@J= c 1 [B,][Nf]{q:}==[B,]
{q}
(28)
where [Bg] is the relationship matrix between nodal displacement and derivatives of midplane displacement. For laminated plates of y1 strips, substituting eqn (28) and (27) into (26) we obtain the incremental total potential energy represented by nodal displacement field
s.
R + {4)1‘[&]7~[~,,][&]
dA {q)
(29)
Stability of laminated plates using finite strip method where Ri is the midplane area of the ith strip. Equation (29) can be rearranged to get
(30) where
Fl=S
WPI R,
71
6 NUMERICAL RESULTS AND DISCUSSIONS The critical buckling loads of composite plates have been analyzed by using the present higherorder plate finite strip element. The accuracy and applicability of this formulation are demonstrated via problems for which the exact solutions and numerical results are available. Both cross-ply and angle-ply laminates are studied. Two types of orthotropic material properties in principal axes are adopted.
PI ~ 6.1 Material
is the strip elastic stiffness matrix
I
E11/E22=3,10,20,30, G,2=G13=0.6E22
or 40 E33=E22 GZ3=05EZ2
v,2=v23=v,3=0*25 is the strip geometric
stiffness matrix.
The perturbation technique is used to obtain the buckling equation for the finite strip method. The value of incremental potential energy is the extreme relative to infinitesimal perturbation at the initial state of critical buckling loading. Taking the variation of eqn (30) with {q}, with s(AH)=O, the strip equation for the stability of laminated plates is
@l-kl)w=O Assembling of stability is
(31)
all the strips, the global equation
([K] -[Kg]> {r)=O
(32)
where [K] is the structure elastic stiffness matrix, [K,] is the structure geometric stiffness, and Yis the nodal displacement vector. When the laminated plates are loaded by critical buckling in-plane stress 1, then we de6(r) is fine [Kg]= L[K,], and small perturbation any value. We will get det([K] -n[&])=O
(33)
The buckling stress 1, can be evaluated from the generalized eigenvalue problem as shown above. The subspace method is used to find the eigenvalues. It is noticed that the ils are midplane resultant forces N:, Nj, N& in most literature. However, the &s are the in-plane stresses gz,‘,cry0,a& in this paper. The difference between these two is the thickness of plate.
6.2 Material
II
Eli/&=40
E33=E22
G12=G,3=05E22
Gz3=0.2Ez2
v~~=v~~=v~~=O.~~
For the purpose of comparison, non-dimensional critical buckling load N,, is defined as Nc,=;1+2/(E22dz)2 where b is the span in y direction, and h is the thickness. A three-node C” isoparametric strip element in the x direction has been developed, and the continuous eigenfunctions are in the y direction. The selected Gaussian integration technique is used for the element stiffness. For simply supported cross-ply plates, only one term of eigenfunction is required. As for angle-ply plates, three terms of eigenfunctions are used due to the fast convergence for terms, unless stated otherwise. 6.3 Cross-ply composite
plates
The following six cases of square cross-ply laminates of the same thickness per ply are studied. (1) Two-layer, antisymmetric (O/90”), Material I. (2) Four-layer, antisymmetric (O/90/0/90), Material I. antisymmetric (O/900/. . .), (3) Ten-layer,
72
W J. Wang, Y l? Tseng,K. J. Lin Material I. (4) Three-layer, symmetric (O/90/0”), Material I. (5) Nine-layer, symmetric (O/90’/. . .), Material I. (6) Four-layer, symmetric (O/90/90/0”), Material I.
Both symmetric and antisymmetric stacking sequences are included. All the cases are of span-to-thickness ratio b/h= 10. Various b/h ratios are further investigated for (O/90”) and (O/90/90/O”) laminates. The present results are compared with the 3-D elasticity solution by Noor,” closed-form solutions and the other finite element solutions. The predicted critical buckling loads for twolayer [O/90”] laminate of b/h=5 are listed in Table 1. In comparing with the elasticity solution, the present strip method yields quite accurate results. The superiority over the higher-order mixed plate element by Putcha and Reddy ’ ’ is more evident as the material anisotropy ratio E11/E22 increases. It is noted that the CPT results are in considerable error, reaching 20% for E11/E22=40. It is also noticed that the number of eigenfunction terms does not affect the results. Therefore, only one term is used in the subsequent studies of cross-ply laminates. Table 2 tabulates the results for antisymmetric cross-ply [O/90/0/900] laminate, while Table 3 is for six-layer cross-ply [O/90”/. . .] laminate. From the summarized results, excellent accuracy is obtained in both examples. The present results are better than those obtained by higher-
order plate element.12 As the in-plane orthotropy ratio J!?~JJ!Z~~increases in Table 2, the deviation of the other higher-order plate elements is more pronounced. Compared to the elasticity solution,7 the errors of the present predictions do not exceed 2.1% even for the case of highly orthotropic thick plate. Table 4 tabulates the results for antisymmetric cross-ply [O/90/0”] laminate, and Table 5 is for nine-layer cross-ply [O/90”/. . .] laminate. As the in-plane orthotropy ratio E11/E22 increases in Table 4, the deviation of the other higherorder plate elements is also pronounced. Compared to the elasticity solution7 the errors of the present predictions do not exceed 1.03%. Excellent accuracy has been observed in both examples. The effect of span-to-thickness ratio on the dimensionless critical buckling load is displayed in Table 6 with, El L/E22 specified to be 40. The present results are in good agreement with the solution by Phan and Reddy.12 Meanwhile, it is worth pointing out that the CPT over-predicts the critical buckling load for the thick plate. The critical buckling load for symmetric cross-ply [O/90/90/0”] laminated plates subjected to uniaxial or biaxial loadings is listed in Table 7. The critical buckling load for the biaxially loaded case is approximately half that for the uniaxially loaded case. The convergence studies of the number of strips are depicted in Fig. 2, for simply supported cross-ply (O/90”) and (O/90/0/90”) square plates of b/h=20. Fast convergence is observed. Figure 3 shows plots of critical buckling load vs in-plane orthotropy E, ,/E,, for plates with vari-
Table 1. Non-dimensional critical buckling load N,, for simply supported two-layer cross-ply (O/90’) square laminated plates of b/h=lO, subjected to uniaxial loading. NC~=12-bZ/(Ezz-h)2, Material I Sources E,,I&,
Noorlon Putcha Putcha CPT Present Present Present
& Reddy ’ ’ ’ & Reddy”’ (1 term) (2 term) (3 term)
3
10
20
30
40
4.6948 4.7749 4.7718 5.0338 4.6960 4.6960 4.6960
6.1181 6.2721 6.2465 6.7033 6.1300 6.1300 6.1300
7.81968 8.1151 8.0423 8.8158 7.8727 7.8727 7.8727
9.3746 9.8695 9.7347 10.8910 9.4952 9.4952 9.4952
10.8167 115630 11.3530 12.9570 11.0262 11.0262 11.0262
ti Elasticity solution. ‘Higher-order mixed plate element. c First-order mixed plate element. CPT: classical plate theory.
Stability of laminated plates using finite strip method
73
‘Ibble 2. Non-dimensional critical buckling load N, for simply supported four-layer cross-ply (O/~/O/~O) square laminated plates of b/h=lO, subjected to uniaxlal loading. N,=2*bz/(E,,-li)2, Material I Sources
Noor” Putcha & Reddy” Putcha & Reddyr’ CPT Present (1 term)
&l&2 3
10
20
30
40
5.1738 5.2523 5-2543 5.5738 5.1788
9.0164 9.2315 9.2552 10.2950 9.0607
13.7429 14.2540 14.3320 16.9880 13.8944
17.7829 186670 18.8150 23.6750 18.0771
21.2796 225790 22.8060 30.3590 21.7342
Table 3. Non-dimensional critical buckling load N,, for simply supported ten-layer cross-ply (O/90/. . .) square laminated plates of b/lr=lO, subjected to uniaxial loading. Ncr=l-b2/(E22-h)Z, Materiul I Sources
Noor” Putcha & Reddy” Putcha & Reddy” CPT Present (1 term)
EI r&2 3
10
20
30
40
5.3159 5.3882 5.3884 5.7250 5.3189
9.9134 10.0560 10.0600 11*3000 9.9281
15.6685 15.9140 15.9270 19.2770 15.7117
20.6347 20.9860 21.0080 27.2540 20.7149
24.9636 25.4220 25.4500 35.2320 25.0855
Table 4. Non-dimensional critical buckling load N,, for simply supported three-layer cross-ply (O/90/O”)square laminated plates of b/h=lO, h,=h3=h/4, h,=h/2 subjected to uniaxial loading. N,,=12*b2f(E2z~h)Z,Material I Sources
Noor” Putcha & Reddy” Putcha & Reddyr’ Putcha & Red 9” Phan & Reddy’ Owen & Li13” CPT Present (1 term)
3
10
20
30
40
5.3044 5.3950 5.3933 5-3991 5.1143 5.4026 5.7538 5.3254
9.7621 9.9427 9.9406 99652 9.7740 9.9590 11.4920 9.8270
15.0191 15.3001 15-2980 15-3510 15.2980 153201 19.7120 15-1394
19.3040 19.6752 19.6740 19.7560 19.9570 19.6872 27.9360 19.4810
22.8807 23.3398 23.3400 23.4530 23.3400 23.3330 361600 23.1170
“Higher-order plate element.
Table 5. Non-dimensional critical buckling load N,, for simply supported nine-layer cross-ply (O/90/. . .) square laminated plates of b/h=lO, h,=h/S other =h/lO subjected to uniuxial loading. N,,=A*b2t(Ea*h)2, Material I Sources
Noor” Putcha & Reddy” Putcha & Reddy” Putcha & Reddy’” Owen & Lir3 CPT Present (1 term)
&A% 3
10
20
30
40
5.3352 5.4147 5.4313 5-4126 5.4187 5.7538 5.3446
10.0417 10.1991 lo-1970 10.1890 10.1990 11.4920 10.0694
15.9153 16.1745 16.1720 16.1460 16.1560 19.7120 15.9650
20.9614 21.3165 21.3150 21.2650 21.2697 27.9360 21.0332
25.3436 25.7908 25.7~ 25.7150 25.7093 361600 25.4390
ous numbers of layers. As the EIJ& ratios increase, the buckling loads increase. It is then confirmed that the laminated plates are stiffer for higher material anisotropy.
Figure 4 shows plots of critical buckling load vs number of layers for plates with various inplane orthotropies E&!!&, ratios. As the number of layers increases, the buckling load
74
W J. Wang, Y 19 Tseng, K. J. Lin
Table 6. Effect of span-to-thickness b/h ratio non-dimensional critical buckling load N,, for simply supported two-layer cross-ply (O/90°) square laminated plates of b/h=lO, subjected to uniaxial loading. N,,=A~Z?/(&*h) , Material I Sources
blh
CPT Phan & Reddy12* Phan & Reddy”’ Present (1 term)
5
10
20
2.5
50
100
12.628 8.628 8.142 6.172
12.628 11.305 11.099 10.026
12.628 12.268 12.208 12.408
12.628 12.395 12.356 12.600
12.628 12.569 12.559 12.866
12.628 12.614 12.611 12.935
u Higher-order plate theory. ‘First-order plate theory.
Table 7. Non-dimensional critical buckling load NCr for simply supported four-layer cross-ply (O/90/0/90”) square laminated plates of b/h=lO, subjected to uniaxial or biaxial loadings. Nc,=l-b2/(E22-h)2, Material I Sources
E,,lEz
a,O#OCT;=0 fJ*“=lJ; u,o=o a,o#O
3
10
20
30
40
5.179 2.589 5.179
9.061 4.530 9.061
13.894 6.947 13.894
18.077 9.038 18.077
21.734 10.867 21.734
-
5.0
4.5 4.0 3.5
1.0 0.5 0.0 -0.5 1 0
I
I
I
I
I
I
I
I
5
IO
15
4
6
8
10
12
Number
Mash number
Fig. 2.
Mesh convergence study on blh=20, cross-ply [O/90”] and [O/90/0/90”] laminated plates.
increases. The more layers the laminate has, the stiffer it is. When the number of layers is over six, the critical buckling loads of laminated plates will converge to the same value. 6.4 Angle-ply composite plates The following two cases of square angle-ply laminates of the same thickness per ply are studied. (7) Four-layer, antisymmetric Material II.
(Ql- O/Q/- Q),
Fig. 3.
of layers
Effect of material anisotropy on critical buckling load with various ply-numbers.
antisymmetric (8) Six-layer, Material II.
(O/-N..),
The antisymmetric laminates are of span-toratio b/h= 10 and are simply thickness supported. The effects of various lamination orientations are also studied. Since there is no existing elasticity solution, the present results are compared with the analytical solutions by Jones et ~1.~ There is much difference between the present solution and Ref. 9. It is thought to be due to the fact that the shear deformation effect is larger in the angle-ply laminates.
75
Stabilityof laminatedplates using finite stri> method
In the study of four-layer angle-ply (e/-e/e/-e) 1aminates, the normalized critical buckling loading vs orientation angle 6’ is tabulated in Table 8. The non-dimensional buckling loads of six-layer angle-ply laminate versus 0 are tabulated in Table 9. Table 10 shows a detailed study for the four-layer convergence simple supported (45/ - 45”/45/ - 45”) thick square plate using up to 16 strips (in the full plate model) and up to six harmonic terms for each of the normalized critical buckling load. It is indicated that the present results converge stably for both the adopted strip number and eigenfunction terms.
25r $ 2
20
-
E? ._
0
5
IO
15
20
25
30
35
40
I
I
45
50
El 1%
Fig. 4.
Effect of ply-numbers on critical buckling with various in-plane orthotropy ratios.
load
Finally, the effect of stack angles for various ply numbers on the critical buckling load is depicted in Fig. 5. It shows that the critical buckling load of angle-ply laminated plates is not symmetric to 45”.
7 CONCLUSIONS The higher-order finite strip element method established in this study has been used to determine the critical buckling loads of laminated plates. The adopted Co higher-order plate theory is proved to be suitable for the stability analysis of transverse shear deformable lamidisplacement nated plates. Since non-linear field is used, the warping of cross-section can be considered to account for the parabolic variation of transverse shear strain. There is no need for the shear deformation correction factor. It is also mentioned that the Co approach is attractive due to the simplicity and implementation in programming. In the numerical example, the buckling loads of cross-ply laminated plates are very close to those of the analytical solution. On the other hand, the benefit of the finite strip method is obvious in that fewer degrees of freedom are required. However, similar or even better results than the finite element solutions are obtained. The comparative benchmark problems are given to demonstrate the adequacy and accuracy of the present study. It is then concluded that the present formulation
Table 8. Non-dimensional critical buckling load NC, for simply supported four-layer cross-ply (0/ - O/8/ - 0) square laminated plates of b//2=20, subjected to biaxial loadings. Nc,=~~b2/(E,,*h)2, Material II Sources
Jones et a1.9” Present
e 0"
15”
30
45”
10.871 9.334
17.660 13.942
24.912 19.810
28.044 21.675
a Classical plate theory.
Table 9. Non-dimensional critical buckling load N,, for simply supported six-lager cross-ply (e/-8/0/. . .) square laminated plates of b/h=20, subjected to uniaxial loadings. N,,=A-b l(E22*h)Z, Material II Sources
Jones et aL9 Present
0 0
15”
30”
45”
35.831 30.312
41.313 33.967
55.265 41.778
62.455 45.382
76
U? J. Wang, Y I? Tseng, K. .I. Lin
Table 10. Convergence
study of simply supported four-layer angle-ply (45/-45/45/-445’) square laminated b/h=20, subjected to uniaxial loadings. Ncr=l*b2/(E2,-h)2, Material II
Sources
Number
Jones et aL9 Present (1 term) Present (2 term) Present (3 term) Present (4 term) Present (5 term) Present (6 term)
4
8
10
16
56.088 54.446 49.216 44.754 44.752 44.748 44.748
56.088 53.330 48.607 44.265 44.258 44.255 44.254
56.088 52.315 47.993 43.326 43.323 43.321 43.320
56088 52.313 47.973 43.270 43.265 43.264 43.264
56.088 53.312 47.968 43.268 43.264 43.264 43.264
0
g
IO50
0
I
I
I
I
I
I
I
I
1
IO
20
30
40
50
60
70
80
90
Ply-angle
Fig. 5.
Effect
of strips
2
.E ‘5z
plates of
of fibre orientation load.
on critical buckling
offers an efficient and concise mthod for the stability analysis of both thick and thin prismatic laminated plates.
REFERENCES 1. Kulkarni, S. V. & Pagano, N. J., Dynamic characteristics of composite laminates. J. Sound Vib., 23 (1972) 127-43. 2. Bert, C. W., A critical evaluation of new plate theories applied to laminated composites. Composite Structures, 2 (1984) 329-47. 3. Lo, K. H., Christensen, R. M. & Wu, E. M., A higherorder theory of a plate deformation: part II laminated plates. J. Appl. Mech., 44 (1977) 669-76. 4. Kant, T. & Pandya, B. P., A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates. Composite Structures, 9 (1988) 215-46. 5. Kant, T., Ravichandran, B. N. & Pandya, B. P., Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order
theory. Composite Structures, 9 (1988) 319-42. 6. Whitney, J. M. & Sun, C. T., A higher order theory for extensional motion of laminated composites. J. Sound Vib., 30 (1973) 85-97. 7. Reddy, J. N., Energy and Variational Methods in Applied Mechanics, Wiley-Interscience, New York, 1984. 8. Jones, R. M., Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates. AZAA J., 11 (1973) 1626-32. 9. Jones, R. M., Morgan, H. S. & Whitney, J. M., Buckling and vibration of unsymmetrically laminated angle-ply rectangular plates. J. Appl. Mech., 40 (1973) 669-76. 10. Noor, A. K., Stability of multilayered composite plates. Fibre. Sci. T’echnof., 8 (1975) 81-9. 11. Putcha, N. S. & Reddy, J. N., Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory. J. Sound Vib., 66 (1979) 565-76. 12. Phan, N. D. & Reddy, J. N., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher order shear deformation theory. J. Sound Vib., 98 (1985) 157-70. 13. Owen, D. R. J. & Li, Z. H., A refined analysis of laminated plates by finite element displacement methods - II: vibration and stability. Comput. Structures, 26 (1987) 915-23. 14. Cheung, Y. K., The finite strip method in the analysis of elastic plates with two opposite simply supported ends. Proc. Inst. Civil Engrs, 40 (1968) 1-7. 15. Mawenya, A. S. & Davies, J. D., Finite strip analysis of plate bending including transverse shear effects. Build. Sci., 9 (1974) 175-80. 16. Benson, P. R. & Hinton, E., A thick finite strip solution for static, free vibration and stability of problems. Int. J. Num. Meth. Eng., 10 (1976) 665-78. E., Buckling of initially stressed Mindlin 17. Hinton, plates using a finite strip method. Comput. Structures, 8 (1978) 99-105. 18. Dawe, D. J. & Roufaeil, 0. L., Buckling of rectangular Mindlin plates. Comput. Structures, 15 (1982) 461-71. 19. Tseng, Y. P. & Wang, W. J., A refined finite strip method using higher-order plate theory. ht. J. Solids Structures, 30 (1993) 441-52. K., Variational and Methods in Elasticity and 20. Wash& Plasticity, Pergamon Press, Oxford, 1982.
Stability of laminated plates using finite strip method based on a higher-order plate theory W. J. Wang repayment of Civil E~~nee~n~ rational Lien-Ho College, ~iaol~ Taiwan
Y. R Tseng Institute of Civil Engineering, Tamkang University, Tamsui, Taiwan &
K. J. Lin Institute of Applied Mathematics, National Chung-Hsing University Taichung, Taiwan A higher-order shear deformable plate finite strip element is developed and employed to investigate the critical buckling load of composite laminated plates. The warping of cross-section and transverse shear deformation can be accurately predicted by the higher-order plate theory. Meanwhile, fewer degrees of freedom in the finite strip method than those in the finite element method are required. The good convergence characteristics of the finite strip approach and the relative efficiency of particular economization schemes are then demonstrated through several numerical results.
1 INTRODUCTION With the characteristics of high specific modulus, high specific strength, and the capability of being tailored, laminated composite materials have been extensively used especially in the design of aircraft structures. Hence, an accurate calculation of the critical buckling load is required in assessing the strength of composite structures. The classical plate theory (CPT) ignores the transverse shear stress, thus usually underprediets deflection and overpredicts natural frequencies and buckling loads except for very thin plates. Mindlin plate theories introduce a correction factor into the shear stress resultants to account for a uniform shear stress through the thickness of the plate. So they do not adequately model the behaviour of highly orthotropic composite structures. There are exact solutions’ for composite material structures to prove the above points. Several higher-order theories2-6 have been developed for the analysis of composite plates. The transverse shear deformation is considered
without the correction factor and the warping of cross-section is also taken into account. A critical evaluation of new plate theories by Bert2 indicated that the theory of Lo et aL3 produced an accurate prediction of the non-linear bending stress dist~bution. Kant and Pandya* and Kant et al.5 developed various Co higher-order plate elements. A refined C1 higher-order plate theory was further proposed by Reddy7 to consider the traction-free condition on the top and bottom surfaces of the plate. All of them can predict the behaviour of laminated plates more accurately and do not need the shear deformation correction factor. Abundent literature can be found for the buckling analysis of composite plates. Jones* evaluated the buckling of unsymmetrically laminated cross-ply rectangular thin plates. Jones et aZ.’ extended the above work to unsymmetrically laminated Noor” angle-ply plates. calculated the buckling loads of thick laminated cross-ply composite plates by using different shear correction factors for different stacking sequences. Putcha and Reddy,l’ Phan and Reddy,12 and Owen and Lit3 introduced several
66
W J. Wang, l! l? Tseng, K. J. Lin
higher-order plate elements for the stability analysis of laminated plates. The finite strip method has been proved to be a more efficient computational tool than the finite element method for prismatic structures. Cheung14 firstly developed the finite strip method of CPT to solve for the natural frequencies of plates. Mawenya and DaviesI applied the finite strip method to the Mindlin plate theory. This work has been extended by Benson and Hinton16 to include vibration and stability applications. Hinton” considered the buckling load of initially stressed plates using the finite strip method based on the Mindlin plate theory. Dawe and Roufaeil18 studied the buckling load of the Mindlin plate including the shear deformation effect. The finite strip method has been incorporated with the higher-order plate theory for the static analysis of laminated plates by Tseng and Wang.” Excellent results were reported. The method of analysis is further extended to determine the buckling load of the composite laminates in this study. The highlight is due to the facts that the higher-order plate theory can accurately predict the behaviour of the composite plates, and the finite strip method not only reduces the computational cost and storage space but also predicts accurate results for prismatic structures. Some comparative benchmark problems are given to demonstrate the adequacy and accuracy of the present study.
to midplane about the x and y axes, respectively, and u”, u*, w*, O$, Oz, 0, are the corresponding higher-order terms in the Taylor series expansion and are defined at the midplane. The positive directions are shown in Fig. 1. The above displacement assumption is written in matrix form as {A>-[L]W
(2)
where {A}=[uvwlT is the 3-D displacement field, {d}=[u, Q w. 0, Q,. Q, U* c* w* O*, fj”,]’ is the midplane displacement field, and [L] is a linear operator matrix. The partial derivative of 3-D displacement field can be presented in the following form as {A.,>=@]
{4/J
(31
where G,,) =[Q),~ uo,Y Q,, Q,~ We., II,,, Q,Y OX., f&,XG,., &,$ u 5 u T, G CT, w 5 WTYSZ,X HZ,, G, Gxl is the partial derivative of the generalized midplane displacement, and [L] is the proper differential operator matrix. Making the usual assumptions, the constitutive relationships for the laminate are co>=@]
{El
(4)
where {g.) = [ox aY o, r-T z,, zYZ]is the 3-D stress, and {r:}=[r;, t;, r;, yg ;:, ~~~1“is the 3-D strain, and [e] is the material property matrix. In the plate analysis, the generalized midplane strain can be defined by {E}=[{FO},{k},{&*},{k*},{~},{~},{~*}17 (5)
2 HIGHER-ORDER PLATE THEORY The displacement field of higher-order plate theory assumed by Lo et al.” is adopted. The plate, of thickness h, is referred to an x, y, z system of rectangular coordinates, where the top and bottom of the plate are the planes z= -+h/2. Specifically, it is assumed that ~(4y,z)=4l(-5Y)
Z.W
+zf&y)
+z2u * (x,y) +z”q?(x,y) ~(~,Y,z)=~o(~,Y)
+zf&,y)
+z2u* (x,y) +z’fqx,y) ~(~,y,z)=%(4Y)
+-a(&Y)
(1)
+z2~*(-%Y)
where u, U, w are the displacements along x, y, and z coordinates, z is the distance from the midplane, u,, v,, W, are the associated midplane displacements, OX,Q,,are the rotations of normal
Fig. 1.
System coordinate
and displacement
field.
67
Stabilityof laminated plates using finite strip method
with the definitions
of
The total potential
{E0)= [&l,Xuo,yuo,y + ro,x 61 {t~*>==[u~x
{~I=[&,,
{k*l=p;,y
14
I=
~y,x+&,,
o;,,
[wo,x
+
0,
I-I=U-V
+
61
(8)
where U is the strain energy, and V is the potential energy due to external load. In the above equation, the strain energy is
2W”lT
~~,,+of,y]’
wo,y
plates
is
T
V~yU~y+U~.JT
oy,x
energy of laminated
T
{~>=[2u”+0,,,2v”+0,,,lT -@*)=[&I,,
+30;
wo,,+30:]T
The midplane strains are also related midplane displacements as follows {C}=[& {d}
to the (6)
=
where [B] is the proper differential operator matrix. The well known constitutive relation of laminated plate can be expressed in the following form as {5>=[D> {El
(7)
where [5] is the midplane stress resultant, and [D], constructed by the proper integral of @] with z, is the material properties matrix of the plate.
3 TOTAL POTENTIAL
TOTAL POTENTIAL
{e}T{o> dA
U=i
(10)
sR
where R is the midplane area of the laminated plates. Substitute the material property into the above equation to obtain {E}T[D] (i) d/4.
U=$
(11)
sR
Since the plate is assumed to be loaded by the stresses o:, ci, z&, the potential energy due to external loading is expressed as
ENERGY
Previous literature considered the critical buckling load, a presumed midplane resultant force, but we consider the buckling load, a presumed in-plane stress. In this paper, we assume that the laminated plates are subjected to uniform in-plane stresses a:, Q;, z$ through the thickness. The superscript ‘0’ represents the unknown prescribed value.
4 INCREMENTAL
where u is the volume of the laminated plate. Integrating eqn (9) through the thickness, the stress {Q} and the strain (1) will become the midplane resultant force (5) and the midplane strain {E}, then
(;+g)] dv (12) where oz, oj, r$ are the unknown cal buckling stresses.
in-plane criti-
ENERGY
According to Washizu,‘” the structure will reach equilibrium at the onset of buckling. When an infinitesimal perturbation is imposed along the bifurcated path, the equilibrium will change into another one. The incremental total potential energy is derived from the difference of these two states. The superscript ‘0’ stands for the onset of buckling and the “’ is after the onset of buckling. The displacements u, 21,w can be separated into two parts, as follows U=UO+U’
(13)
W J. Wang, Y R Tseng, K. J. Lin
68
where u’, v”, w” are the displacements at the initiation of buckling and u’, v’, w’ are the incremental displacements after the onset of the buckling by imposing the infinitesimal perturbation. Similarly, the stresses and strains also can be divided into two parts f&=f3,O+fS: &,=&,o+&: fS,=0,“+0;
Ey=&yo+&;
0,=fJ,o+C7:
&,=&,O+&;
T,=T$+Z&
YlT=Y&+Y&
(14)
z,=z:+z:, J&=y:+y:, zyz= z;z +z;z Yyz= r;z +y;z Therefore,
the total potential
energy II can be separated
into two parts as
l-I=l-I’+AII
(IS)
where II0 is the total potential energy at the onset of the buckling, same as eqn (8), and AII is the additional total potential energy due to external load by imposing an infinitesimal perturbation after the onset of buckling. The potential energy is calculated for the in-plane stresses, not the midplane forces. Therefore, the corresponding membrane strains are chosen to be in-plane Green-Lagrangian strains, not the Von Karman strains, to present the potential energy AII due to external loading as AlI=+
” {.s’}T{a’} dvs
{a;jT{&>
dv
(16)
sV
where {E’} is the incremental strain, (0’) is the incremental stress, {c$}=[c~ oy”z&l’ is the in-plane stress vector at the beginning of buckling, and the corresponding membrane strain {EL} is
(17)
(g(g+(g(z)+(g(g \ But, the relations of incremental &.;=-
au’ 8X
alI’
c’=y
c’=-
z
ay ad a2
strain and incremental
displacement
are still of the form
-r -au’+au’
ax
ay
yq-
alit
aw’
“=-+az I
_
(18)
ax
qv’ aw’
‘Yz-a2
I
ay
It is also assumed that the material small deformation, i.e.
property
after the onset of buckling
is the same as that for
69
Stabilityof laminated plates using finite strip method
(19)
(0’) = [Q] {E’l.
Integrating [e] of the above equation potential energy in eqn (16) becomes + 1 ([B] W))T[D] R
For simplification becomes
through
the thickness,
[B] id’) d/f. purpose,
total
(20)
u ‘, u’, w’ can be replaced
by U, u, W, and the second term of eqn (16)
{cT;}={E,} dv= ” {A.,>‘[$,] {A,,) du s sv where {A,,} is the partial derivative of the 3-D displacement matrix made of critical buckling stresses such as
Substituting
the first term of incremental
eqn (3) into eqn (21) and integrating
(21) field defined in eqn (3), and [O,] is the
it through the z direction, we obtain
(22) where the stress resultant
matrix is I
I
,
x,
s,
0
0
0
0
0
0
0
0
x,
s3
0
0
0
0
x4
s‘j
0
0
Y,
0
0
0
0
x, s*
s2
s1
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
x,
s,
0
0
0
0
x,
s*
0
0
0
0
x,
s3
0
0
0
0
x4
s4
Y,
0
0
0
0
s*
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
s1
0
0
0
0
x,
s1
0
0
0
0
x,
sz
0
0
0
0
x,
s3
0
0
0
0
0
0
0
0
s1
Y,
0
0
0
0
s*
Y,
0
0
0
0
s3
&
0
0
0
0
x,
s2
0
0
0
0
x,
s3
0
0
0
0
x,
s4
0
0
0
0
x,
s5
0
0
s*
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
0
0
x,
s2
0
0
0
0
x,
s3
0
0
0
0
x4
s4
0
0
x5
s5
0
0
s2
Y,
0
0
0
0
s3
Y,
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
x,
s2
0
0
0
0
x,
sj
0
0
0
0
x4
s4
0
0
0
0
0
0
0
0
s2
Y2
0
0
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
x,
s3
0
0
0
0
x5
s5
0
0
S6
0
0
0
0
0
0
x4
s4
0
0
x,
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
sg
Ye
0
0
0
0
x,
s3
0
0
0
0
x4
s4
0
0
0
0
x5
s5
0
0
0
0
X6
S6
0
0
s3
Y3
0
0
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
S6
Y,
0
0
0
0
x,
s3
s4
0
0
0
0
x5
s5
0
0
0
0
0
0
0
0
s5
Y5
0
0
0
0
0
0
0
0
x4
0
s4
Y4
0
0
s3
Y,
0
0
0
s4
0
0
0
0
x5
s5
0
0
0
0
x,
s,5
0
0
0
0
x,
s,
0
0
Y4
0
0
0
0
s5
Ys
0
0
0
0
s,
Ye
0
0
0
0
s,
Y,
0
0
0
0
x4
s4
0
0
0
0
x5
s5
0
0
0
0
x,
se
0
0
0
0
x7
s7
0
0
s4
Y4
0
0
0
0
s5
Y5
0
0
0
0
S6
Y,
0
0
0
0
s7
Y,
0
0
x4 s4
I
70
W J. Wang, 2: P Tseng, K. J. Lin
with (X,,X,,X,,X,,X,,X,,X,)=
;
s:
k=l
s
,:*+’ (1, z’, .z2,z3, z4, z5, 2”) dz *
nl
(Yl, Yz, Y,, Y4, Y5, Y,, YT)= 1 c;
hk+’ (1,~‘,z2,z3,z4,zs,zh)
(S,
>
s2, &, S4, Ss, &, &> = ;
dz
c h,
k=l
7;
k=l
i
hh+’ (1, z’, z2, z3, z4, z5, z”)
dz
his
and hk is the vectorial distance from the panel midplane. In the other way, {d,,} can be replaced by midplane displacement &J=[B,]
{d} as
(24)
{d)
where [&,I is the proper differential operator matrix. Using eqn (24), eqn (22) can be transformed to 1 -2-
{d,,)T[~P] {d,,) dA=+ sR
([B,] MT[a,][B,]
(25)
{d) d/I.
s R
Equations
(4) (20) and (25) are combined
s
MI=+ ([i] {d))‘[D] R
-$
[I?]{d) d.A
to obtain the incremental
([&I
displacement
For the interpolations of the semi-analytical finite strip method, the simple polynomials are adopted in the transverse direction and the eigenfunctions are used in the longitudinal direction, with the latter satisfying a priori the end boundary conditions of the strip. Since we use a C” continuous displacement model, there are m harmonic terms and nst nodal points, the midplane displacement for a typical strip is interpolated by nodal displacements as m nst
w= 1 c 1=1
[Nf]{qj}
[B&J Id) d/l.
(26)
R
5 FINITE STRIP ANALYSIS
{dl=[W
energy as
s w-jqo,]
again that the incremental
It is mentioned above formulation.
total potential
(27)
i=l
where m denotes the total number of harmonic terms, nst is the nodal number per strip, [N] is the matrix of shape functions, {q} is the nodal displacement vector, [Nf] is the matrix of shape functions for node i and for the Ith harmonic term, and {qF}=[uGi, uzi, wZ~, Ozi, O_Gz,0%, u 7: vf: w T,’r3;i;f,0 >I ’ is the nodal displacement for node i and for the eth harmonic term. The
{d’} has been replaced
by {d} in the
complete expression of the interpolation can be found in Tseng & Wang. I”) Then, the derivatives of the midplane displacement field are ,?I n.s*
@J= c 1 [B,][Nf]{q:}==[B,]
{q}
(28)
where [Bg] is the relationship matrix between nodal displacement and derivatives of midplane displacement. For laminated plates of y1 strips, substituting eqn (28) and (27) into (26) we obtain the incremental total potential energy represented by nodal displacement field
s.
R + {4)1‘[&]7~[~,,][&]
dA {q)
(29)
Stability of laminated plates using finite strip method where Ri is the midplane area of the ith strip. Equation (29) can be rearranged to get
(30) where
Fl=S
WPI R,
71
6 NUMERICAL RESULTS AND DISCUSSIONS The critical buckling loads of composite plates have been analyzed by using the present higherorder plate finite strip element. The accuracy and applicability of this formulation are demonstrated via problems for which the exact solutions and numerical results are available. Both cross-ply and angle-ply laminates are studied. Two types of orthotropic material properties in principal axes are adopted.
PI ~ 6.1 Material
is the strip elastic stiffness matrix
I
E11/E22=3,10,20,30, G,2=G13=0.6E22
or 40 E33=E22 GZ3=05EZ2
v,2=v23=v,3=0*25 is the strip geometric
stiffness matrix.
The perturbation technique is used to obtain the buckling equation for the finite strip method. The value of incremental potential energy is the extreme relative to infinitesimal perturbation at the initial state of critical buckling loading. Taking the variation of eqn (30) with {q}, with s(AH)=O, the strip equation for the stability of laminated plates is
@l-kl)w=O Assembling of stability is
(31)
all the strips, the global equation
([K] -[Kg]> {r)=O
(32)
where [K] is the structure elastic stiffness matrix, [K,] is the structure geometric stiffness, and Yis the nodal displacement vector. When the laminated plates are loaded by critical buckling in-plane stress 1, then we de6(r) is fine [Kg]= L[K,], and small perturbation any value. We will get det([K] -n[&])=O
(33)
The buckling stress 1, can be evaluated from the generalized eigenvalue problem as shown above. The subspace method is used to find the eigenvalues. It is noticed that the ils are midplane resultant forces N:, Nj, N& in most literature. However, the &s are the in-plane stresses gz,‘,cry0,a& in this paper. The difference between these two is the thickness of plate.
6.2 Material
II
Eli/&=40
E33=E22
G12=G,3=05E22
Gz3=0.2Ez2
v~~=v~~=v~~=O.~~
For the purpose of comparison, non-dimensional critical buckling load N,, is defined as Nc,=;1+2/(E22dz)2 where b is the span in y direction, and h is the thickness. A three-node C” isoparametric strip element in the x direction has been developed, and the continuous eigenfunctions are in the y direction. The selected Gaussian integration technique is used for the element stiffness. For simply supported cross-ply plates, only one term of eigenfunction is required. As for angle-ply plates, three terms of eigenfunctions are used due to the fast convergence for terms, unless stated otherwise. 6.3 Cross-ply composite
plates
The following six cases of square cross-ply laminates of the same thickness per ply are studied. (1) Two-layer, antisymmetric (O/90”), Material I. (2) Four-layer, antisymmetric (O/90/0/90), Material I. antisymmetric (O/900/. . .), (3) Ten-layer,
72
W J. Wang, Y l? Tseng,K. J. Lin Material I. (4) Three-layer, symmetric (O/90/0”), Material I. (5) Nine-layer, symmetric (O/90’/. . .), Material I. (6) Four-layer, symmetric (O/90/90/0”), Material I.
Both symmetric and antisymmetric stacking sequences are included. All the cases are of span-to-thickness ratio b/h= 10. Various b/h ratios are further investigated for (O/90”) and (O/90/90/O”) laminates. The present results are compared with the 3-D elasticity solution by Noor,” closed-form solutions and the other finite element solutions. The predicted critical buckling loads for twolayer [O/90”] laminate of b/h=5 are listed in Table 1. In comparing with the elasticity solution, the present strip method yields quite accurate results. The superiority over the higher-order mixed plate element by Putcha and Reddy ’ ’ is more evident as the material anisotropy ratio E11/E22 increases. It is noted that the CPT results are in considerable error, reaching 20% for E11/E22=40. It is also noticed that the number of eigenfunction terms does not affect the results. Therefore, only one term is used in the subsequent studies of cross-ply laminates. Table 2 tabulates the results for antisymmetric cross-ply [O/90/0/900] laminate, while Table 3 is for six-layer cross-ply [O/90”/. . .] laminate. From the summarized results, excellent accuracy is obtained in both examples. The present results are better than those obtained by higher-
order plate element.12 As the in-plane orthotropy ratio J!?~JJ!Z~~increases in Table 2, the deviation of the other higher-order plate elements is more pronounced. Compared to the elasticity solution,7 the errors of the present predictions do not exceed 2.1% even for the case of highly orthotropic thick plate. Table 4 tabulates the results for antisymmetric cross-ply [O/90/0”] laminate, and Table 5 is for nine-layer cross-ply [O/90”/. . .] laminate. As the in-plane orthotropy ratio E11/E22 increases in Table 4, the deviation of the other higherorder plate elements is also pronounced. Compared to the elasticity solution7 the errors of the present predictions do not exceed 1.03%. Excellent accuracy has been observed in both examples. The effect of span-to-thickness ratio on the dimensionless critical buckling load is displayed in Table 6 with, El L/E22 specified to be 40. The present results are in good agreement with the solution by Phan and Reddy.12 Meanwhile, it is worth pointing out that the CPT over-predicts the critical buckling load for the thick plate. The critical buckling load for symmetric cross-ply [O/90/90/0”] laminated plates subjected to uniaxial or biaxial loadings is listed in Table 7. The critical buckling load for the biaxially loaded case is approximately half that for the uniaxially loaded case. The convergence studies of the number of strips are depicted in Fig. 2, for simply supported cross-ply (O/90”) and (O/90/0/90”) square plates of b/h=20. Fast convergence is observed. Figure 3 shows plots of critical buckling load vs in-plane orthotropy E, ,/E,, for plates with vari-
Table 1. Non-dimensional critical buckling load N,, for simply supported two-layer cross-ply (O/90’) square laminated plates of b/h=lO, subjected to uniaxial loading. NC~=12-bZ/(Ezz-h)2, Material I Sources E,,I&,
Noorlon Putcha Putcha CPT Present Present Present
& Reddy ’ ’ ’ & Reddy”’ (1 term) (2 term) (3 term)
3
10
20
30
40
4.6948 4.7749 4.7718 5.0338 4.6960 4.6960 4.6960
6.1181 6.2721 6.2465 6.7033 6.1300 6.1300 6.1300
7.81968 8.1151 8.0423 8.8158 7.8727 7.8727 7.8727
9.3746 9.8695 9.7347 10.8910 9.4952 9.4952 9.4952
10.8167 115630 11.3530 12.9570 11.0262 11.0262 11.0262
ti Elasticity solution. ‘Higher-order mixed plate element. c First-order mixed plate element. CPT: classical plate theory.
Stability of laminated plates using finite strip method
73
‘Ibble 2. Non-dimensional critical buckling load N, for simply supported four-layer cross-ply (O/~/O/~O) square laminated plates of b/h=lO, subjected to uniaxlal loading. N,=2*bz/(E,,-li)2, Material I Sources
Noor” Putcha & Reddy” Putcha & Reddyr’ CPT Present (1 term)
&l&2 3
10
20
30
40
5.1738 5.2523 5-2543 5.5738 5.1788
9.0164 9.2315 9.2552 10.2950 9.0607
13.7429 14.2540 14.3320 16.9880 13.8944
17.7829 186670 18.8150 23.6750 18.0771
21.2796 225790 22.8060 30.3590 21.7342
Table 3. Non-dimensional critical buckling load N,, for simply supported ten-layer cross-ply (O/90/. . .) square laminated plates of b/lr=lO, subjected to uniaxial loading. Ncr=l-b2/(E22-h)Z, Materiul I Sources
Noor” Putcha & Reddy” Putcha & Reddy” CPT Present (1 term)
EI r&2 3
10
20
30
40
5.3159 5.3882 5.3884 5.7250 5.3189
9.9134 10.0560 10.0600 11*3000 9.9281
15.6685 15.9140 15.9270 19.2770 15.7117
20.6347 20.9860 21.0080 27.2540 20.7149
24.9636 25.4220 25.4500 35.2320 25.0855
Table 4. Non-dimensional critical buckling load N,, for simply supported three-layer cross-ply (O/90/O”)square laminated plates of b/h=lO, h,=h3=h/4, h,=h/2 subjected to uniaxial loading. N,,=12*b2f(E2z~h)Z,Material I Sources
Noor” Putcha & Reddy” Putcha & Reddyr’ Putcha & Red 9” Phan & Reddy’ Owen & Li13” CPT Present (1 term)
3
10
20
30
40
5.3044 5.3950 5.3933 5-3991 5.1143 5.4026 5.7538 5.3254
9.7621 9.9427 9.9406 99652 9.7740 9.9590 11.4920 9.8270
15.0191 15.3001 15-2980 15-3510 15.2980 153201 19.7120 15-1394
19.3040 19.6752 19.6740 19.7560 19.9570 19.6872 27.9360 19.4810
22.8807 23.3398 23.3400 23.4530 23.3400 23.3330 361600 23.1170
“Higher-order plate element.
Table 5. Non-dimensional critical buckling load N,, for simply supported nine-layer cross-ply (O/90/. . .) square laminated plates of b/h=lO, h,=h/S other =h/lO subjected to uniuxial loading. N,,=A*b2t(Ea*h)2, Material I Sources
Noor” Putcha & Reddy” Putcha & Reddy” Putcha & Reddy’” Owen & Lir3 CPT Present (1 term)
&A% 3
10
20
30
40
5.3352 5.4147 5.4313 5-4126 5.4187 5.7538 5.3446
10.0417 10.1991 lo-1970 10.1890 10.1990 11.4920 10.0694
15.9153 16.1745 16.1720 16.1460 16.1560 19.7120 15.9650
20.9614 21.3165 21.3150 21.2650 21.2697 27.9360 21.0332
25.3436 25.7908 25.7~ 25.7150 25.7093 361600 25.4390
ous numbers of layers. As the EIJ& ratios increase, the buckling loads increase. It is then confirmed that the laminated plates are stiffer for higher material anisotropy.
Figure 4 shows plots of critical buckling load vs number of layers for plates with various inplane orthotropies E&!!&, ratios. As the number of layers increases, the buckling load
74
W J. Wang, Y 19 Tseng, K. J. Lin
Table 6. Effect of span-to-thickness b/h ratio non-dimensional critical buckling load N,, for simply supported two-layer cross-ply (O/90°) square laminated plates of b/h=lO, subjected to uniaxial loading. N,,=A~Z?/(&*h) , Material I Sources
blh
CPT Phan & Reddy12* Phan & Reddy”’ Present (1 term)
5
10
20
2.5
50
100
12.628 8.628 8.142 6.172
12.628 11.305 11.099 10.026
12.628 12.268 12.208 12.408
12.628 12.395 12.356 12.600
12.628 12.569 12.559 12.866
12.628 12.614 12.611 12.935
u Higher-order plate theory. ‘First-order plate theory.
Table 7. Non-dimensional critical buckling load NCr for simply supported four-layer cross-ply (O/90/0/90”) square laminated plates of b/h=lO, subjected to uniaxial or biaxial loadings. Nc,=l-b2/(E22-h)2, Material I Sources
E,,lEz
a,O#OCT;=0 fJ*“=lJ; u,o=o a,o#O
3
10
20
30
40
5.179 2.589 5.179
9.061 4.530 9.061
13.894 6.947 13.894
18.077 9.038 18.077
21.734 10.867 21.734
-
5.0
4.5 4.0 3.5
1.0 0.5 0.0 -0.5 1 0
I
I
I
I
I
I
I
I
5
IO
15
4
6
8
10
12
Number
Mash number
Fig. 2.
Mesh convergence study on blh=20, cross-ply [O/90”] and [O/90/0/90”] laminated plates.
increases. The more layers the laminate has, the stiffer it is. When the number of layers is over six, the critical buckling loads of laminated plates will converge to the same value. 6.4 Angle-ply composite plates The following two cases of square angle-ply laminates of the same thickness per ply are studied. (7) Four-layer, antisymmetric Material II.
(Ql- O/Q/- Q),
Fig. 3.
of layers
Effect of material anisotropy on critical buckling load with various ply-numbers.
antisymmetric (8) Six-layer, Material II.
(O/-N..),
The antisymmetric laminates are of span-toratio b/h= 10 and are simply thickness supported. The effects of various lamination orientations are also studied. Since there is no existing elasticity solution, the present results are compared with the analytical solutions by Jones et ~1.~ There is much difference between the present solution and Ref. 9. It is thought to be due to the fact that the shear deformation effect is larger in the angle-ply laminates.
75
Stabilityof laminatedplates using finite stri> method
In the study of four-layer angle-ply (e/-e/e/-e) 1aminates, the normalized critical buckling loading vs orientation angle 6’ is tabulated in Table 8. The non-dimensional buckling loads of six-layer angle-ply laminate versus 0 are tabulated in Table 9. Table 10 shows a detailed study for the four-layer convergence simple supported (45/ - 45”/45/ - 45”) thick square plate using up to 16 strips (in the full plate model) and up to six harmonic terms for each of the normalized critical buckling load. It is indicated that the present results converge stably for both the adopted strip number and eigenfunction terms.
25r $ 2
20
-
E? ._
0
5
IO
15
20
25
30
35
40
I
I
45
50
El 1%
Fig. 4.
Effect of ply-numbers on critical buckling with various in-plane orthotropy ratios.
load
Finally, the effect of stack angles for various ply numbers on the critical buckling load is depicted in Fig. 5. It shows that the critical buckling load of angle-ply laminated plates is not symmetric to 45”.
7 CONCLUSIONS The higher-order finite strip element method established in this study has been used to determine the critical buckling loads of laminated plates. The adopted Co higher-order plate theory is proved to be suitable for the stability analysis of transverse shear deformable lamidisplacement nated plates. Since non-linear field is used, the warping of cross-section can be considered to account for the parabolic variation of transverse shear strain. There is no need for the shear deformation correction factor. It is also mentioned that the Co approach is attractive due to the simplicity and implementation in programming. In the numerical example, the buckling loads of cross-ply laminated plates are very close to those of the analytical solution. On the other hand, the benefit of the finite strip method is obvious in that fewer degrees of freedom are required. However, similar or even better results than the finite element solutions are obtained. The comparative benchmark problems are given to demonstrate the adequacy and accuracy of the present study. It is then concluded that the present formulation
Table 8. Non-dimensional critical buckling load NC, for simply supported four-layer cross-ply (0/ - O/8/ - 0) square laminated plates of b//2=20, subjected to biaxial loadings. Nc,=~~b2/(E,,*h)2, Material II Sources
Jones et a1.9” Present
e 0"
15”
30
45”
10.871 9.334
17.660 13.942
24.912 19.810
28.044 21.675
a Classical plate theory.
Table 9. Non-dimensional critical buckling load N,, for simply supported six-lager cross-ply (e/-8/0/. . .) square laminated plates of b/h=20, subjected to uniaxial loadings. N,,=A-b l(E22*h)Z, Material II Sources
Jones et aL9 Present
0 0
15”
30”
45”
35.831 30.312
41.313 33.967
55.265 41.778
62.455 45.382
76
U? J. Wang, Y I? Tseng, K. .I. Lin
Table 10. Convergence
study of simply supported four-layer angle-ply (45/-45/45/-445’) square laminated b/h=20, subjected to uniaxial loadings. Ncr=l*b2/(E2,-h)2, Material II
Sources
Number
Jones et aL9 Present (1 term) Present (2 term) Present (3 term) Present (4 term) Present (5 term) Present (6 term)
4
8
10
16
56.088 54.446 49.216 44.754 44.752 44.748 44.748
56.088 53.330 48.607 44.265 44.258 44.255 44.254
56.088 52.315 47.993 43.326 43.323 43.321 43.320
56088 52.313 47.973 43.270 43.265 43.264 43.264
56.088 53.312 47.968 43.268 43.264 43.264 43.264
0
g
IO50
0
I
I
I
I
I
I
I
I
1
IO
20
30
40
50
60
70
80
90
Ply-angle
Fig. 5.
Effect
of strips
2
.E ‘5z
plates of
of fibre orientation load.
on critical buckling
offers an efficient and concise mthod for the stability analysis of both thick and thin prismatic laminated plates.
REFERENCES 1. Kulkarni, S. V. & Pagano, N. J., Dynamic characteristics of composite laminates. J. Sound Vib., 23 (1972) 127-43. 2. Bert, C. W., A critical evaluation of new plate theories applied to laminated composites. Composite Structures, 2 (1984) 329-47. 3. Lo, K. H., Christensen, R. M. & Wu, E. M., A higherorder theory of a plate deformation: part II laminated plates. J. Appl. Mech., 44 (1977) 669-76. 4. Kant, T. & Pandya, B. P., A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates. Composite Structures, 9 (1988) 215-46. 5. Kant, T., Ravichandran, B. N. & Pandya, B. P., Finite element transient dynamic analysis of isotropic and fibre reinforced composite plates using a higher-order
theory. Composite Structures, 9 (1988) 319-42. 6. Whitney, J. M. & Sun, C. T., A higher order theory for extensional motion of laminated composites. J. Sound Vib., 30 (1973) 85-97. 7. Reddy, J. N., Energy and Variational Methods in Applied Mechanics, Wiley-Interscience, New York, 1984. 8. Jones, R. M., Buckling and vibration of unsymmetrically laminated cross-ply rectangular plates. AZAA J., 11 (1973) 1626-32. 9. Jones, R. M., Morgan, H. S. & Whitney, J. M., Buckling and vibration of unsymmetrically laminated angle-ply rectangular plates. J. Appl. Mech., 40 (1973) 669-76. 10. Noor, A. K., Stability of multilayered composite plates. Fibre. Sci. T’echnof., 8 (1975) 81-9. 11. Putcha, N. S. & Reddy, J. N., Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory. J. Sound Vib., 66 (1979) 565-76. 12. Phan, N. D. & Reddy, J. N., Stability and vibration of isotropic, orthotropic and laminated plates according to a higher order shear deformation theory. J. Sound Vib., 98 (1985) 157-70. 13. Owen, D. R. J. & Li, Z. H., A refined analysis of laminated plates by finite element displacement methods - II: vibration and stability. Comput. Structures, 26 (1987) 915-23. 14. Cheung, Y. K., The finite strip method in the analysis of elastic plates with two opposite simply supported ends. Proc. Inst. Civil Engrs, 40 (1968) 1-7. 15. Mawenya, A. S. & Davies, J. D., Finite strip analysis of plate bending including transverse shear effects. Build. Sci., 9 (1974) 175-80. 16. Benson, P. R. & Hinton, E., A thick finite strip solution for static, free vibration and stability of problems. Int. J. Num. Meth. Eng., 10 (1976) 665-78. E., Buckling of initially stressed Mindlin 17. Hinton, plates using a finite strip method. Comput. Structures, 8 (1978) 99-105. 18. Dawe, D. J. & Roufaeil, 0. L., Buckling of rectangular Mindlin plates. Comput. Structures, 15 (1982) 461-71. 19. Tseng, Y. P. & Wang, W. J., A refined finite strip method using higher-order plate theory. ht. J. Solids Structures, 30 (1993) 441-52. K., Variational and Methods in Elasticity and 20. Wash& Plasticity, Pergamon Press, Oxford, 1982.