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Quasi-3D analysis of free vibration of anisotropic plates
Composite Structures 74 (2006) 449–457 www.elsevier.com/locate/compstruct
Quasi-3D analysis of free vibration of anisotropic plates Luciano Demasi
*
Department of Aeronautics and Astronautics, Guggenheim Building, University of Washington, Seattle, WA 98195-2400, USA Available online 13 June 2005
Abstract The subject of the present work is multilayered finite elements that are able to furnish an accurate description of both strain and stress fields. The formulation of the finite elements is based upon ReissnerÕs mixed variational theorem (RMVT), which allows one to assume two independent fields for displacements and transverse stress variables. The resulting advanced finite elements can describe, a priori, the interlaminar continuous transverse shear and normal stress fields, and the so called C 0z -requirements can be satisfied. This paper is mainly concerned about the vibrations of multilayered plates. The present FEM formulation is validated by comparing the results with both the literature and the commercial code NASTRAN. To conduct the assessment, five challenging benchmarks with different boundary conditions and lamination schemes are considered. For each benchmark, the first four nondimensional frequencies are calculated and a few modes are presented. It can be concluded that the present quasi-3D layer-wise formulation has a very good agreement with respect to the more computationally expensive three-dimensional FEM formulations. Moreover, the displacement and transverse stresses are computed a priori without any operation of post-processing. Therefore, the present results can be used as reference numbers for testing other new FEM models. 2005 Elsevier Ltd. All rights reserved. Keywords: FEM; RMVT; Transverse stresses; Mixed formulation; Layer-wise models; NASTRAN; C 0z requirements
1. Introduction Composite structures combine light weight, high stiffness, high structural efficiency and durability, and, therefore, have been used to build large portion of aerospace as well as automotive and ship vehicles. As far as two-dimensional modeling of multilayered flat structures is concerned, there are a number of requirements that should be considered for an accurate description of their stress and strain fields. First, anisotropic multilayered structures possess higher shear and normal flexibility than traditional isotropic one-layer ones. In fact, classical two-dimensional analyses of plates and shells based on Cauchy–Poisson– Kirchoff [1–3] assumptions (see Classical Lamination *
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Theory (CLT) [4]), are inadequate to predict the global response of thick plates, Reissner–Mindlin [5–7], even though accounting for transverse shear deformations can lead to very inaccurate conclusions as far as local response of thick layered structures are concerned. Second, the intrinsic discontinuity of the mechanical properties in the thickness direction of plate puts further difficulties on the two-dimensional modeling of layered structures. Exact three-dimensional solutions (see [8–11]) have shown that the displacements and the transverse stresses are C0-continuous functions in the thickness z direction (see Fig. 1 for the geometry and notations). In [12–14] these facts were referred to as C 0z requirements. Literature often marks these requirements as zig–zag form for the displacements u = [ux uy uz]T and interlaminar continuity (i.e., equilibrium) for the transverse stresses rn = [rzx rzy rzz]T. Many equivalent single-layer and layer-wise theories have been proposed
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L. Demasi / Composite Structures 74 (2006) 449–457
Fig. 1. Geometry and notations.
and directed to overcome the limitation of CLT and FSDT and include, partially or completely, the above C 0z requirements. For a complete and detailed overview, see [15]. This paper assesses the vibrations of multilayered plates. The present FEM layer-wise mixed formulation is validated by comparing the results with both the literature and the commercial code NASTRAN. As previously discussed, thick composite plates require advanced theories/formulations, in order to capture all C 0z requirements. Therefore, the effectiveness of such quasi-3D layer-wise mixed formulation is demonstrated through five challenging benchmarks involving different boundary conditions and lamination schemes. This paper is organized as follows: • Section 2: C 0z requirements. The C 0z requirements are introduced. • Section 3: ReissnerÕs mixed variational theorem. (RMVT) The RMVT is introduced. • Section 4: Layer-wise displacement and transverse stress assumptions. The displacement and stress field assumptions are detailed and illustrated. • Section 5: RMVT: Equilibrium and compatibility equations. The equilibrium and compatibility equations, in the FEM approximation, are reported. • Section 6: Free vibrations problem. The free vibrations problem is formulated using the present mixed formulation. Applying the static condensation technique, the problem is transformed into the standard form. • Section 7: Results and discussion. The free vibrations problem is solved for several different cases. Comparison with the literature results is presented and five benchmarks are discussed. • Section 8: Conclusion. The principal properties and results are discussed.
2. C 0z requirements The layered construction (see Fig. 1) introduces complicating effects [12,13,16]. Transverse discontinuous
mechanical properties, in fact, cause displacement fields (see Fig. 2(a)) in the thickness direction which can exhibit rapid changes and different slopes in correspondence to each layer interface. This is known as the zig–zag form of displacement field in the thickness direction. Although in-plane stresses (see Fig. 2(b)) can, in general, be discontinuous, equilibrium reasons demand continuous transverse stresses (see Figs. 2(c) and 3). This is often referred to, in the literature, as interlaminar continuity (IC) of transverse shear and normal stresses. Fig. 2 shows, from a qualitative point of view, what could be the scenario of displacement and transverse stresses distributions in a multilayered structure in exact solutions. Figs. 2 and 3 show that both displacements and transverse stresses, due to compatibility and equilibrium reasons, respectively, are C0-continuous functions in the thickness z direction. These conditions are referred to as C 0z requirements. The fulfillment of C 0z requirements is a crucial point in two-dimensional modeling of multilayered structures. Summarizing, the C 0z requirements are: • Compatibility: interlaminar continuity of the displacements. • Zig–zag form of the displacement field (due to the discontinuous mechanical properties in the thickness direction). • Equilibrium: interlaminar continuity of the transverse stresses.
3. ReissnerÕs mixed variational theorem (RMVT) In the previous section, it was shown that the fulfillment of C 0z requirements is very important in the multilayered structure models. Therefore, it is fundamental to have a tool which allows to model the transverse stresses and enforce the continuity through the thickness. Starting from the Hellinger–Reissner functional (see [17]), with a few transformations (partial Legendre transformation) it is not difficult to obtain the ReissnerÕs mixed variational theorem [18,19]. In the dynamic case, the RMVT can be expressed as Z ðdeTpG rpH þ deTnG rnM þ drTnM ðenG enH ÞÞ dV V Z ð1Þ þ q€udu dV ¼ dLe V
The subscripts ‘‘p’’ and ‘‘n’’ mean that the quantities are ‘‘plane’’ or ‘‘out-of-plane’’; the subscript ‘‘G’’ means that the strains e are calculated using the geometric relations (which express the strains as functions of the displacements); the subscript ‘‘H’’ underlines that the quantities are computed via HookeÕs law; and, finally,
L. Demasi / Composite Structures 74 (2006) 449–457 (a)
(b)
451 (c)
Fig. 2. C 0z requirements. Displacement and stress fields in the thickness plate direction. Three-layered plate.
Fig. 3. Details of the stress states at the interface between two consecutive layers.
the subscript ‘‘M’’ means that the stresses are those of the assumed model (the model is chosen such that it satisfies the IC requirement). q is the material density, u are the displacements, €u are the accelerations, dLe is the external virtual work and V denotes the three-dimensional multilayered body volume. The relevant quantities involved in RMVT are explicitly reported: • Stresses and strains rp ¼ ½rxx ryy rxy T ; T
ep ¼ ½exx eyy exy ;
rn ¼ ½rzx rzy rzz T T
en ¼ ½ezx ezy ezz
ð2Þ
e np ep þ C e nn en rn ¼ C
ð3Þ
• Mixed form of Hooke’s law rp ¼ C pp ep þ C pn rn en ¼ C np ep þ C nn rn
e pn ð C e nn Þ1 C pn ¼ C e nn Þ1 C e np C np ¼ ð C
ð5Þ
e nn Þ1 C nn ¼ ð C • Strain–displacement relations ep ¼ Dp u en ¼ Dn u ¼ ðDnX þ Dnz Þu
ð6Þ
where Dp, DnX and Dnz are differential matrices.
4. Layer-wise displacement and transverse stress assumptions
• Hooke’s law e pp ep þ C e pn en rp ¼ C
• Relations between the two forms of Hooke’s law e pp C e pn ðC ~ nn Þ1 C e np C pp ¼ C
ð4Þ
4.1. Displacement assumptions The displacement components ukx , uky and ukz of the k-layer (the total number of layers is indicated by Nl) are postulated, in the z-direction, according to the expansion
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L. Demasi / Composite Structures 74 (2006) 449–457
ukx ¼ F t ukxt þ F b ukxb þ F r ukxr ¼ F s ukxs uky ¼ F t ukyt þ F b ukyb þ F r ukyr ¼ F s ukys ukz
¼
F t ukzt
þ
F b ukzb
þ
F r ukzr
¼
F s ukzs
ð7Þ
s ¼ t; b; r; r ¼ 2; . . . ; N 1; k ¼ 1; . . . ; N l The subscripts t and b denote values related to the top and bottom layer surfaces, respectively. These two terms consist of the linear part of the expansion. The thickness functions Fs(fk) are defined at the k-layer level as Ft ¼
P0 þ P1 ; 2
Fb ¼
P0 P1 ; 2
F r ¼ P r P r2 ; r ¼ 2; 3; . . . ; N
fk ¼
þ1; F t ¼ 1; F b ¼ 0; F r ¼ 0 1; F t ¼ 0; F b ¼ 1; F r ¼ 0
ð9Þ
s ¼ t; b; r;
ð10Þ
In the FEM approximation, the displacements are expressed in terms of their nodal values via shape functions uks ¼ N i qksi
ði ¼ 1; 2; . . . ; N n Þ
ð11Þ
where Nn denotes the number of nodes in the element,1 while qksi is the vector of the nodal displacement qksi ¼ ½qkux si qkuy si qkuz si
T
5. RMVT: Equilibrium and compatibility equations Imposing the definition of virtual variations, ReissnerÕs mixed variational theorem leads to the following equilibrium and compatibility equations:
kssij k kssij k dgkT si : K ru qsj þ K rr g sj ¼ 0
r ¼ 2; . . . ; N 1;
k ¼ 1; . . . ; N l
Notice that the previous assumptions for the displacement and stress fields satisfy a priori the C 0z requirements. The interlaminar continuity of the transverse stresses and displacements is imposed in the assemblage process of the FEM matrices. For completeness, the indicial notation is explained in Fig. 4. In this paper, a particular case of the general theory, previously explained, is here considered. The corresponding theory is called LM4 (L = layer-wise, M = mixed, N = 4).
kssij k kssij k kssij k €qsj ¼ Pksi dqkT si : K uu qsj þ K ur g sj þ M
Eq. (7) can be written in a compact way uk ¼ F s uks
ð15Þ
ð8Þ
in which Pj = Pj(fk) is the Legendre polynomial of j-order defined in the fk domain: 1 6 fk 6 +1. The chosen functions have the following properties:
where h iT g ksi ¼ gkzxsi gkzysi gkzzsi
ð16Þ
where Pksi is the vector of external loads, qksi and g ksi are the nodal displacements and transverse stress variables kssij kssij (layer k), and the matrices K kssij and K kssij uu , K ur , K ru rr (3 · 3 matrices) are the fundamental nuclei. Mkssij is the fundamental nucleus which allows one (by expanding the indexes) to write the consistent mass matrix. More details about the derivation of the matrices can be found in [16] and [20]. The expansion of the layer matrix from the correspondent 3 · 3 fundamental nuclei is shown in Fig. 5. In Fig. 6 the assemblage from layer to multilayered level is shown.
ð12Þ
4.2. Transverse stress assumptions The transverse stresses rkzx , rkzy and rkzz are written as rknM ¼ F s rkns ;
s ¼ t; b; r;
r ¼ 2; . . . ; N 1;
k ¼ 1; . . . ; N l
ð13Þ
As has been done for the displacements, the nodal values are obtained using the relation rkns ¼ N i g ksi
1
ði ¼ 1; 2; . . . ; N n Þ
In this paper, only the case Nn = 9 will be considered.
ð14Þ
Fig. 4. Summary of the introduced approximations and related indicial notations.
L. Demasi / Composite Structures 74 (2006) 449–457
453
at layer level and assembling the obtained matrices (see Figs. 5 and 6), the resulting matrices, at multilayered element level, are K uu ; K ur ; K ru ; K rr ; M
ð18Þ
Eq. (16), at multilayered element level, can be written as (notice that in the free vibrations problem the external loads are zero) dq : K uu q þ K ur g þ M€q ¼ 0 dg : K ru q þ K rr g ¼ 0
ð19Þ
where q and g are the displacements and transverse stresses at multilayered element level. At this stage it is possible to apply the static condensation technique as follows: 1
½K uu K ur ðK rr Þ K ru q þ M€q ¼ 0
ð20Þ
By introducing 1
KI mixed ¼ ½K uu K ur ðK rr Þ K ru
ð21Þ
the governing equation written in terms of only displacement variables is obtained KI q¼0 mixed q þ M€ Fig. 5. Expansion of the layer matrix from the correspondent 3 · 3 fundamental nuclei via s and s indexes.
ð22Þ
At structure level, one has S S S K IS q ¼ 0S mixed q þ M €
ð23Þ
which is the ‘‘classical’’ vibrations problem.
7. Results and discussion 7.1. Data of the treated problems 7.1.1. Materials The materials that are used in this paper are: • MAT 1 EL GLT GLz GTT ¼ 40; ¼ ¼ 0.50; ¼ 0.60; ET ET ET ET tLT ¼ tLz ¼ tTT ¼ 0.25 • MAT 2
Fig. 6. Assemblage from layer to multilayered level for a three-layered plate.
7.1.2. Lamination schemes The lamination schemes that are used in this paper are:
6. Free vibrations problem Expanding the layer matrices K kssij uu ;
kssij kssij K kssij ur ; K ru ; K rr
EL GLT GLz GTT ¼ 25; ¼ ¼ 0.50; ¼ 0.20; ET ET ET ET tLT ¼ tLz ¼ tTT ¼ 0.25
ð17Þ
• LS 1: +0/+90, h1 = h2 = h/2. • LS 2: +0/+45, h1 = h2 = h/2.
454
L. Demasi / Composite Structures 74 (2006) 449–457 Table 3 Benchmark 2. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
Fig. 7. Boundary conditions.
Table 1 qffiffiffiffiffiffiffi 4 Circular frequency parameter x Ea hq2
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.995 8.908 8.841 8.823
12.884 12.816 12.858 12.782
15.505 15.416 15.402 15.329
17.417 17.408 17.708 17.510
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.009 4.953 4.945 4.925
5.993 5.953 5.958 5.937
10.056 10.007 10.073 10.013
10.636 10.583 10.650 10.592
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.517 2.506 2.503 2.500
5.810 5.801 5.828 5.809
8.051 8.003 7.979 7.961
8.469 8.449 8.552 8.518
T
a/h
4
10
20
100
LM4a LM4
7.345 7.351
10.088 10.094
10.859 10.867
11.151 11.160
Comparison with the analytical solution (LM4a). Boundary conditions: SSSS. Material: MAT 1. Lamination scheme: LS1. Mesh: MESH 1.
Table 2 Benchmark 1. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
9.117 9.012 8.916 8.900
14.355 14.264 14.237 14.169
14.355 14.264 14.237 14.169
18.241 18.233 18.300 18.203
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
6.436 6.346 6.267 6.255
6.689 6.633 6.585 6.573
8.507 8.492 8.535 8.516
11.510 11.488 11.489 11.462
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.818 2.804 2.792 2.790
7.542 7.494 7.456 7.443
7.792 7.735 7.689 7.675
11.053 11.032 11.031 11.006
• LS 3: 45/+45, h1 = h2 = h/2. • LS 4: 30/+45, h1 = h2 = h/2. • LS 5: 30/+30, h1 = h2 = h/2.
7.1.3. Mesh In this paper, two different FEM elements are compared. The first FEM element is LM4 (see above) which is a two-dimensional element (Q9). Therefore, a twodimensional mesh is required when such element is used. The second FEM element is a three-dimensional NASTRAN element (EXA with 8 nodes). Therefore, a three-dimensional mesh is required when such element is used.
Table 4 Benchmark 3. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.935 8.834 8.754 8.735
14.302 14.199 14.186 14.109
14.302 14.199 14.186 14.109
18.660 18.636 18.705 18.599
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.874 5.795 5.773 5.749
7.342 7.271 7.255 7.229
11.486 11.459 11.484 11.446
11.848 11.723 11.793 11.684
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.863 2.839 2.833 2.826
6.461 6.434 6.456 6.431
7.829 7.758 7.733 7.707
8.313 8.291 8.488 8.427
• • • •
MESH MESH MESH MESH
1: 2: 3: 4:
4 · 4 (LM4 only). 5 · 5 (LM4 only). 16 · 16 · 4 (NASTRAN only). 24 · 24 · 6 (NASTRAN only).
For example, MESH 4 has 24 EXA elements in the x direction, 24 EXA elements in the y direction and 6 EXA elements in the z direction. In all cases, the plates are square plates with edge a and thickness h. 7.1.4. Boundary conditions The boundary conditions are explained in Fig. 7. 7.1.5. Description of the benchmarks Five different benchmarks are proposed. The details are the following:
L. Demasi / Composite Structures 74 (2006) 449–457 Table 5 Benchmark 4. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.959 8.859 8.779 8.760
13.700 13.606 13.611 13.534
14.893 14.786 14.759 14.684
18.588 18.566 18.650 18.540
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.367 5.300 5.291 5.267
7.008 6.946 6.940 6.914
11.007 10.894 10.982 10.873
11.924 11.893 11.910 11.873
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.840 2.817 2.811 2.805
6.287 6.264 6.285 6.262
7.964 7.897 7.874 7.850
8.392 8.364 8.543 8.486
Table 6 Benchmark 5. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
455
Fig. 8. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.993 8.896 8.819 8.801
13.101 13.020 13.046 12.970
15.459 15.354 15.320 15.247
18.326 18.298 18.701 18.466
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
4.830 4.777 4.782 4.759
6.702 6.647 6.659 6.632
10.066 9.968 10.080 9.970
12.312 12.230 12.325 12.228
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.890 2.868 2.864 2.858
6.227 6.202 6.220 6.198
8.160 8.099 8.079 8.057
9.626 9.563 9.732 9.663
• Benchmark 1 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 1. • Benchmark 2 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 2. • Benchmark 3 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 3. • Benchmark 4 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 4.
Fig. 9. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
• Benchmark 5 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 5.
7.2. Comparison with some results available in the literature The present FEM formulation is compared with the analytical results found by Carrera [21]. As shown in Table 1, the correlation for both thick and thin plates is very good.
456
L. Demasi / Composite Structures 74 (2006) 449–457
Fig. 10. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
Fig. 12. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
Fig. 11. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
7.3. Proposed benchmarks: comparison with NASTRAN The present FEM formulation is compared with NASTRAN. Different mesh and boundary conditions are analyzed. In all cases (see Tables 2–6) the correlation between LM4 and NASTRAN is excellent. It is possible to observe that the frequency parameters change significantly if the boundary conditions are changed. The modes, for the benchmark 5, are plotted2 in Figs. 8– 13. Finally, a comparison with the degree of freedom (DOF) used in both LM4 and NASTRAN is reported in Table 7. As can be seen, LM4 can capture the fre2 In the LM4 case, the modes are plotted at the top surface of the plate.
Fig. 13. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
Table 7 DOF of the used FEM models FEM model
DOF
LM4 (MESH 1) LM4 (MESH 2) NASTRAN (MESH 3) NASTRAN (MESH 4)
2187 3267 8670 26250
quencies and modes with good accuracy using a significantly less number of DOF. Moreover, LM4 is a two-
L. Demasi / Composite Structures 74 (2006) 449–457
dimensional formulation, while NASTRAN (in this paper) is a three-dimensional formulation.
8. Conclusion The present quasi-three-dimensional layer-wise formulation has shown very good correlation with the analytical results available in the literature for both thin and thick plates. Five challenging benchmarks with different lamination schemes and boundary conditions have been proposed. In all cases, the present FEM model has shown very good agreement with the commercial code NASTRAN. Considering the results, it can be concluded that RMVT is a very powerful tool: the C 0z requirements are satisfied a priori and no post processing operations are required in order to calculate the transverse stresses. Therefore, the benchmarks reported here can be used as reference numbers for future FEM formulations.
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Quasi-3D analysis of free vibration of anisotropic plates Luciano Demasi
*
Department of Aeronautics and Astronautics, Guggenheim Building, University of Washington, Seattle, WA 98195-2400, USA Available online 13 June 2005
Abstract The subject of the present work is multilayered finite elements that are able to furnish an accurate description of both strain and stress fields. The formulation of the finite elements is based upon ReissnerÕs mixed variational theorem (RMVT), which allows one to assume two independent fields for displacements and transverse stress variables. The resulting advanced finite elements can describe, a priori, the interlaminar continuous transverse shear and normal stress fields, and the so called C 0z -requirements can be satisfied. This paper is mainly concerned about the vibrations of multilayered plates. The present FEM formulation is validated by comparing the results with both the literature and the commercial code NASTRAN. To conduct the assessment, five challenging benchmarks with different boundary conditions and lamination schemes are considered. For each benchmark, the first four nondimensional frequencies are calculated and a few modes are presented. It can be concluded that the present quasi-3D layer-wise formulation has a very good agreement with respect to the more computationally expensive three-dimensional FEM formulations. Moreover, the displacement and transverse stresses are computed a priori without any operation of post-processing. Therefore, the present results can be used as reference numbers for testing other new FEM models. 2005 Elsevier Ltd. All rights reserved. Keywords: FEM; RMVT; Transverse stresses; Mixed formulation; Layer-wise models; NASTRAN; C 0z requirements
1. Introduction Composite structures combine light weight, high stiffness, high structural efficiency and durability, and, therefore, have been used to build large portion of aerospace as well as automotive and ship vehicles. As far as two-dimensional modeling of multilayered flat structures is concerned, there are a number of requirements that should be considered for an accurate description of their stress and strain fields. First, anisotropic multilayered structures possess higher shear and normal flexibility than traditional isotropic one-layer ones. In fact, classical two-dimensional analyses of plates and shells based on Cauchy–Poisson– Kirchoff [1–3] assumptions (see Classical Lamination *
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Theory (CLT) [4]), are inadequate to predict the global response of thick plates, Reissner–Mindlin [5–7], even though accounting for transverse shear deformations can lead to very inaccurate conclusions as far as local response of thick layered structures are concerned. Second, the intrinsic discontinuity of the mechanical properties in the thickness direction of plate puts further difficulties on the two-dimensional modeling of layered structures. Exact three-dimensional solutions (see [8–11]) have shown that the displacements and the transverse stresses are C0-continuous functions in the thickness z direction (see Fig. 1 for the geometry and notations). In [12–14] these facts were referred to as C 0z requirements. Literature often marks these requirements as zig–zag form for the displacements u = [ux uy uz]T and interlaminar continuity (i.e., equilibrium) for the transverse stresses rn = [rzx rzy rzz]T. Many equivalent single-layer and layer-wise theories have been proposed
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L. Demasi / Composite Structures 74 (2006) 449–457
Fig. 1. Geometry and notations.
and directed to overcome the limitation of CLT and FSDT and include, partially or completely, the above C 0z requirements. For a complete and detailed overview, see [15]. This paper assesses the vibrations of multilayered plates. The present FEM layer-wise mixed formulation is validated by comparing the results with both the literature and the commercial code NASTRAN. As previously discussed, thick composite plates require advanced theories/formulations, in order to capture all C 0z requirements. Therefore, the effectiveness of such quasi-3D layer-wise mixed formulation is demonstrated through five challenging benchmarks involving different boundary conditions and lamination schemes. This paper is organized as follows: • Section 2: C 0z requirements. The C 0z requirements are introduced. • Section 3: ReissnerÕs mixed variational theorem. (RMVT) The RMVT is introduced. • Section 4: Layer-wise displacement and transverse stress assumptions. The displacement and stress field assumptions are detailed and illustrated. • Section 5: RMVT: Equilibrium and compatibility equations. The equilibrium and compatibility equations, in the FEM approximation, are reported. • Section 6: Free vibrations problem. The free vibrations problem is formulated using the present mixed formulation. Applying the static condensation technique, the problem is transformed into the standard form. • Section 7: Results and discussion. The free vibrations problem is solved for several different cases. Comparison with the literature results is presented and five benchmarks are discussed. • Section 8: Conclusion. The principal properties and results are discussed.
2. C 0z requirements The layered construction (see Fig. 1) introduces complicating effects [12,13,16]. Transverse discontinuous
mechanical properties, in fact, cause displacement fields (see Fig. 2(a)) in the thickness direction which can exhibit rapid changes and different slopes in correspondence to each layer interface. This is known as the zig–zag form of displacement field in the thickness direction. Although in-plane stresses (see Fig. 2(b)) can, in general, be discontinuous, equilibrium reasons demand continuous transverse stresses (see Figs. 2(c) and 3). This is often referred to, in the literature, as interlaminar continuity (IC) of transverse shear and normal stresses. Fig. 2 shows, from a qualitative point of view, what could be the scenario of displacement and transverse stresses distributions in a multilayered structure in exact solutions. Figs. 2 and 3 show that both displacements and transverse stresses, due to compatibility and equilibrium reasons, respectively, are C0-continuous functions in the thickness z direction. These conditions are referred to as C 0z requirements. The fulfillment of C 0z requirements is a crucial point in two-dimensional modeling of multilayered structures. Summarizing, the C 0z requirements are: • Compatibility: interlaminar continuity of the displacements. • Zig–zag form of the displacement field (due to the discontinuous mechanical properties in the thickness direction). • Equilibrium: interlaminar continuity of the transverse stresses.
3. ReissnerÕs mixed variational theorem (RMVT) In the previous section, it was shown that the fulfillment of C 0z requirements is very important in the multilayered structure models. Therefore, it is fundamental to have a tool which allows to model the transverse stresses and enforce the continuity through the thickness. Starting from the Hellinger–Reissner functional (see [17]), with a few transformations (partial Legendre transformation) it is not difficult to obtain the ReissnerÕs mixed variational theorem [18,19]. In the dynamic case, the RMVT can be expressed as Z ðdeTpG rpH þ deTnG rnM þ drTnM ðenG enH ÞÞ dV V Z ð1Þ þ q€udu dV ¼ dLe V
The subscripts ‘‘p’’ and ‘‘n’’ mean that the quantities are ‘‘plane’’ or ‘‘out-of-plane’’; the subscript ‘‘G’’ means that the strains e are calculated using the geometric relations (which express the strains as functions of the displacements); the subscript ‘‘H’’ underlines that the quantities are computed via HookeÕs law; and, finally,
L. Demasi / Composite Structures 74 (2006) 449–457 (a)
(b)
451 (c)
Fig. 2. C 0z requirements. Displacement and stress fields in the thickness plate direction. Three-layered plate.
Fig. 3. Details of the stress states at the interface between two consecutive layers.
the subscript ‘‘M’’ means that the stresses are those of the assumed model (the model is chosen such that it satisfies the IC requirement). q is the material density, u are the displacements, €u are the accelerations, dLe is the external virtual work and V denotes the three-dimensional multilayered body volume. The relevant quantities involved in RMVT are explicitly reported: • Stresses and strains rp ¼ ½rxx ryy rxy T ; T
ep ¼ ½exx eyy exy ;
rn ¼ ½rzx rzy rzz T T
en ¼ ½ezx ezy ezz
ð2Þ
e np ep þ C e nn en rn ¼ C
ð3Þ
• Mixed form of Hooke’s law rp ¼ C pp ep þ C pn rn en ¼ C np ep þ C nn rn
e pn ð C e nn Þ1 C pn ¼ C e nn Þ1 C e np C np ¼ ð C
ð5Þ
e nn Þ1 C nn ¼ ð C • Strain–displacement relations ep ¼ Dp u en ¼ Dn u ¼ ðDnX þ Dnz Þu
ð6Þ
where Dp, DnX and Dnz are differential matrices.
4. Layer-wise displacement and transverse stress assumptions
• Hooke’s law e pp ep þ C e pn en rp ¼ C
• Relations between the two forms of Hooke’s law e pp C e pn ðC ~ nn Þ1 C e np C pp ¼ C
ð4Þ
4.1. Displacement assumptions The displacement components ukx , uky and ukz of the k-layer (the total number of layers is indicated by Nl) are postulated, in the z-direction, according to the expansion
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L. Demasi / Composite Structures 74 (2006) 449–457
ukx ¼ F t ukxt þ F b ukxb þ F r ukxr ¼ F s ukxs uky ¼ F t ukyt þ F b ukyb þ F r ukyr ¼ F s ukys ukz
¼
F t ukzt
þ
F b ukzb
þ
F r ukzr
¼
F s ukzs
ð7Þ
s ¼ t; b; r; r ¼ 2; . . . ; N 1; k ¼ 1; . . . ; N l The subscripts t and b denote values related to the top and bottom layer surfaces, respectively. These two terms consist of the linear part of the expansion. The thickness functions Fs(fk) are defined at the k-layer level as Ft ¼
P0 þ P1 ; 2
Fb ¼
P0 P1 ; 2
F r ¼ P r P r2 ; r ¼ 2; 3; . . . ; N
fk ¼
þ1; F t ¼ 1; F b ¼ 0; F r ¼ 0 1; F t ¼ 0; F b ¼ 1; F r ¼ 0
ð9Þ
s ¼ t; b; r;
ð10Þ
In the FEM approximation, the displacements are expressed in terms of their nodal values via shape functions uks ¼ N i qksi
ði ¼ 1; 2; . . . ; N n Þ
ð11Þ
where Nn denotes the number of nodes in the element,1 while qksi is the vector of the nodal displacement qksi ¼ ½qkux si qkuy si qkuz si
T
5. RMVT: Equilibrium and compatibility equations Imposing the definition of virtual variations, ReissnerÕs mixed variational theorem leads to the following equilibrium and compatibility equations:
kssij k kssij k dgkT si : K ru qsj þ K rr g sj ¼ 0
r ¼ 2; . . . ; N 1;
k ¼ 1; . . . ; N l
Notice that the previous assumptions for the displacement and stress fields satisfy a priori the C 0z requirements. The interlaminar continuity of the transverse stresses and displacements is imposed in the assemblage process of the FEM matrices. For completeness, the indicial notation is explained in Fig. 4. In this paper, a particular case of the general theory, previously explained, is here considered. The corresponding theory is called LM4 (L = layer-wise, M = mixed, N = 4).
kssij k kssij k kssij k €qsj ¼ Pksi dqkT si : K uu qsj þ K ur g sj þ M
Eq. (7) can be written in a compact way uk ¼ F s uks
ð15Þ
ð8Þ
in which Pj = Pj(fk) is the Legendre polynomial of j-order defined in the fk domain: 1 6 fk 6 +1. The chosen functions have the following properties:
where h iT g ksi ¼ gkzxsi gkzysi gkzzsi
ð16Þ
where Pksi is the vector of external loads, qksi and g ksi are the nodal displacements and transverse stress variables kssij kssij (layer k), and the matrices K kssij and K kssij uu , K ur , K ru rr (3 · 3 matrices) are the fundamental nuclei. Mkssij is the fundamental nucleus which allows one (by expanding the indexes) to write the consistent mass matrix. More details about the derivation of the matrices can be found in [16] and [20]. The expansion of the layer matrix from the correspondent 3 · 3 fundamental nuclei is shown in Fig. 5. In Fig. 6 the assemblage from layer to multilayered level is shown.
ð12Þ
4.2. Transverse stress assumptions The transverse stresses rkzx , rkzy and rkzz are written as rknM ¼ F s rkns ;
s ¼ t; b; r;
r ¼ 2; . . . ; N 1;
k ¼ 1; . . . ; N l
ð13Þ
As has been done for the displacements, the nodal values are obtained using the relation rkns ¼ N i g ksi
1
ði ¼ 1; 2; . . . ; N n Þ
In this paper, only the case Nn = 9 will be considered.
ð14Þ
Fig. 4. Summary of the introduced approximations and related indicial notations.
L. Demasi / Composite Structures 74 (2006) 449–457
453
at layer level and assembling the obtained matrices (see Figs. 5 and 6), the resulting matrices, at multilayered element level, are K uu ; K ur ; K ru ; K rr ; M
ð18Þ
Eq. (16), at multilayered element level, can be written as (notice that in the free vibrations problem the external loads are zero) dq : K uu q þ K ur g þ M€q ¼ 0 dg : K ru q þ K rr g ¼ 0
ð19Þ
where q and g are the displacements and transverse stresses at multilayered element level. At this stage it is possible to apply the static condensation technique as follows: 1
½K uu K ur ðK rr Þ K ru q þ M€q ¼ 0
ð20Þ
By introducing 1
KI mixed ¼ ½K uu K ur ðK rr Þ K ru
ð21Þ
the governing equation written in terms of only displacement variables is obtained KI q¼0 mixed q þ M€ Fig. 5. Expansion of the layer matrix from the correspondent 3 · 3 fundamental nuclei via s and s indexes.
ð22Þ
At structure level, one has S S S K IS q ¼ 0S mixed q þ M €
ð23Þ
which is the ‘‘classical’’ vibrations problem.
7. Results and discussion 7.1. Data of the treated problems 7.1.1. Materials The materials that are used in this paper are: • MAT 1 EL GLT GLz GTT ¼ 40; ¼ ¼ 0.50; ¼ 0.60; ET ET ET ET tLT ¼ tLz ¼ tTT ¼ 0.25 • MAT 2
Fig. 6. Assemblage from layer to multilayered level for a three-layered plate.
7.1.2. Lamination schemes The lamination schemes that are used in this paper are:
6. Free vibrations problem Expanding the layer matrices K kssij uu ;
kssij kssij K kssij ur ; K ru ; K rr
EL GLT GLz GTT ¼ 25; ¼ ¼ 0.50; ¼ 0.20; ET ET ET ET tLT ¼ tLz ¼ tTT ¼ 0.25
ð17Þ
• LS 1: +0/+90, h1 = h2 = h/2. • LS 2: +0/+45, h1 = h2 = h/2.
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L. Demasi / Composite Structures 74 (2006) 449–457 Table 3 Benchmark 2. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
Fig. 7. Boundary conditions.
Table 1 qffiffiffiffiffiffiffi 4 Circular frequency parameter x Ea hq2
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.995 8.908 8.841 8.823
12.884 12.816 12.858 12.782
15.505 15.416 15.402 15.329
17.417 17.408 17.708 17.510
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.009 4.953 4.945 4.925
5.993 5.953 5.958 5.937
10.056 10.007 10.073 10.013
10.636 10.583 10.650 10.592
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.517 2.506 2.503 2.500
5.810 5.801 5.828 5.809
8.051 8.003 7.979 7.961
8.469 8.449 8.552 8.518
T
a/h
4
10
20
100
LM4a LM4
7.345 7.351
10.088 10.094
10.859 10.867
11.151 11.160
Comparison with the analytical solution (LM4a). Boundary conditions: SSSS. Material: MAT 1. Lamination scheme: LS1. Mesh: MESH 1.
Table 2 Benchmark 1. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
9.117 9.012 8.916 8.900
14.355 14.264 14.237 14.169
14.355 14.264 14.237 14.169
18.241 18.233 18.300 18.203
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
6.436 6.346 6.267 6.255
6.689 6.633 6.585 6.573
8.507 8.492 8.535 8.516
11.510 11.488 11.489 11.462
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.818 2.804 2.792 2.790
7.542 7.494 7.456 7.443
7.792 7.735 7.689 7.675
11.053 11.032 11.031 11.006
• LS 3: 45/+45, h1 = h2 = h/2. • LS 4: 30/+45, h1 = h2 = h/2. • LS 5: 30/+30, h1 = h2 = h/2.
7.1.3. Mesh In this paper, two different FEM elements are compared. The first FEM element is LM4 (see above) which is a two-dimensional element (Q9). Therefore, a twodimensional mesh is required when such element is used. The second FEM element is a three-dimensional NASTRAN element (EXA with 8 nodes). Therefore, a three-dimensional mesh is required when such element is used.
Table 4 Benchmark 3. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.935 8.834 8.754 8.735
14.302 14.199 14.186 14.109
14.302 14.199 14.186 14.109
18.660 18.636 18.705 18.599
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.874 5.795 5.773 5.749
7.342 7.271 7.255 7.229
11.486 11.459 11.484 11.446
11.848 11.723 11.793 11.684
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.863 2.839 2.833 2.826
6.461 6.434 6.456 6.431
7.829 7.758 7.733 7.707
8.313 8.291 8.488 8.427
• • • •
MESH MESH MESH MESH
1: 2: 3: 4:
4 · 4 (LM4 only). 5 · 5 (LM4 only). 16 · 16 · 4 (NASTRAN only). 24 · 24 · 6 (NASTRAN only).
For example, MESH 4 has 24 EXA elements in the x direction, 24 EXA elements in the y direction and 6 EXA elements in the z direction. In all cases, the plates are square plates with edge a and thickness h. 7.1.4. Boundary conditions The boundary conditions are explained in Fig. 7. 7.1.5. Description of the benchmarks Five different benchmarks are proposed. The details are the following:
L. Demasi / Composite Structures 74 (2006) 449–457 Table 5 Benchmark 4. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.959 8.859 8.779 8.760
13.700 13.606 13.611 13.534
14.893 14.786 14.759 14.684
18.588 18.566 18.650 18.540
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
5.367 5.300 5.291 5.267
7.008 6.946 6.940 6.914
11.007 10.894 10.982 10.873
11.924 11.893 11.910 11.873
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.840 2.817 2.811 2.805
6.287 6.264 6.285 6.262
7.964 7.897 7.874 7.850
8.392 8.364 8.543 8.486
Table 6 Benchmark 5. Comparison LM4 vs NASTRAN qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffi ffi 4 4 4 Frequency parameter x1 Ea hq2 x2 Ea hq2 x3 Ea hq2
x4
CCCC case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
T
455
Fig. 8. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
qffiffiffiffiffiffiffi ffi 4 a q ET h2
T
T
8.993 8.896 8.819 8.801
13.101 13.020 13.046 12.970
15.459 15.354 15.320 15.247
18.326 18.298 18.701 18.466
CFCF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
4.830 4.777 4.782 4.759
6.702 6.647 6.659 6.632
10.066 9.968 10.080 9.970
12.312 12.230 12.325 12.228
CCFF case NASTRAN (MESH 3) NASTRAN (MESH 4) LM4 (MESH 1) LM4 (MESH 2)
2.890 2.868 2.864 2.858
6.227 6.202 6.220 6.198
8.160 8.099 8.079 8.057
9.626 9.563 9.732 9.663
• Benchmark 1 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 1. • Benchmark 2 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 2. • Benchmark 3 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 3. • Benchmark 4 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 4.
Fig. 9. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
• Benchmark 5 Square plate (edge a and thickness h). Boundary conditions: CCCC, CFCF, CCFF. Material MAT 2. Lamination scheme LS 5.
7.2. Comparison with some results available in the literature The present FEM formulation is compared with the analytical results found by Carrera [21]. As shown in Table 1, the correlation for both thick and thin plates is very good.
456
L. Demasi / Composite Structures 74 (2006) 449–457
Fig. 10. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
Fig. 12. Benchmark 5: Modes 1 and 2. LM4 (MESH 2) vs NASTRAN (MESH 4).
Fig. 11. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
7.3. Proposed benchmarks: comparison with NASTRAN The present FEM formulation is compared with NASTRAN. Different mesh and boundary conditions are analyzed. In all cases (see Tables 2–6) the correlation between LM4 and NASTRAN is excellent. It is possible to observe that the frequency parameters change significantly if the boundary conditions are changed. The modes, for the benchmark 5, are plotted2 in Figs. 8– 13. Finally, a comparison with the degree of freedom (DOF) used in both LM4 and NASTRAN is reported in Table 7. As can be seen, LM4 can capture the fre2 In the LM4 case, the modes are plotted at the top surface of the plate.
Fig. 13. Benchmark 5: Modes 3 and 4. LM4 (MESH 2) vs NASTRAN (MESH 4).
Table 7 DOF of the used FEM models FEM model
DOF
LM4 (MESH 1) LM4 (MESH 2) NASTRAN (MESH 3) NASTRAN (MESH 4)
2187 3267 8670 26250
quencies and modes with good accuracy using a significantly less number of DOF. Moreover, LM4 is a two-
L. Demasi / Composite Structures 74 (2006) 449–457
dimensional formulation, while NASTRAN (in this paper) is a three-dimensional formulation.
8. Conclusion The present quasi-three-dimensional layer-wise formulation has shown very good correlation with the analytical results available in the literature for both thin and thick plates. Five challenging benchmarks with different lamination schemes and boundary conditions have been proposed. In all cases, the present FEM model has shown very good agreement with the commercial code NASTRAN. Considering the results, it can be concluded that RMVT is a very powerful tool: the C 0z requirements are satisfied a priori and no post processing operations are required in order to calculate the transverse stresses. Therefore, the benchmarks reported here can be used as reference numbers for future FEM formulations.
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