3PC-Article 5: Free Vibration Analysis of Simply Supported Laminated Composite Plates
Naveen Rastogi, PhD
Composites and FEA Expert Leader, Manager, Researcher, Teacher, Mentor, Learner
Overview:
The fifth article talks about the free vibration analyses of simply-supported laminated composite anisotropic plates. The analysis methodology discussed here can be used to obtain the natural frequencies of vibrations of simply-supported laminated composite plates with or without the inclusion of the effects of the applied in-plane edge compression /tension and/or edge shear loads. The edge loads can also include the effects of uniform temperature and/or moisture expansion or contraction. Understanding the natural modes of vibrations is the first step and an important aspect of understanding the more complex dynamic behavior of the composite structures under forced excitation.
Simply-supported boundary conditions are most widely used in the practical analysis of plates and shells.?Closed-form solutions are also useful in validating more complex Finite Element (FEM) analyses.
Sign convention for the applied compressive bi-axial loads and positive shear load is shown below:?
Following combinations of in-plane applied loads (in-plane edge loads can also include the effects of uniform temperature and/or moisture expansion or contraction) can be analyzed using 3pcsolver004 to obtain natural frequencies of vibration and associated modes for simply-supported anisotropic composite laminates:?
Theoretical Background:
Given the laminate stack-up, lamina/ply material properties, temperature and/or moisture differentials and in-plane edge compression /tension and/or edge shear loads as described and shown above, 3pcsolver004 solver calculates natural frequencies of vibrations ?and associated mode shapes or Eigen vectors for simply-supported anisotropic composite laminates. Details of the theoretical approach along with verification examples are provided in the training module 3pcmodule006.
Applications:
The analysis performed by the ?3pcsolver004 ?solver is applicable to a wide variety of ?ply/lamina materials and laminates or composite plates built-up (or fabricated) from a LAMINA that?
(i) has any kind of continuous FIBER such as boron, carbon, graphite, glass, Kevlar, Aramid, polyester, natural fibers, etc.,?
(ii) is in any type of broad form such as unidirectional, bi-directional 2D textile weaves like plain weave, twill and harness, biaxial and triaxial braids, chopped random continuous fibers, non-crimp, nonwoven fabrics, etc.,?
(iii) is impregnated with any RESIN/MATRIX, thermoset or thermoplastic systems such as epoxy, polyester, vinyl ester, polyurethane, phenolic, cyanate ester, bis-maleimide, polyimides, benzoxazine, Acrylic, ABS, Polylactic acid PLA, Polybenzimidazole PBI, Polyether sulfone PES, Polyoxymethylene POM, ?Polyether ether ketone PEEK, Polyetherimide PEI, Polyphenylene oxide PPO, Polyphenylene sulfide PPS, Polystyrene PS, Polypropylene PP, Polyvinyl chloride PVC, Teflon PTFE, etc., and?
(iv) is cured using any MANUFACTURING PROCESS such as Autoclave, Resin Transfer Molding like VARTM, SQRTM, RIM, SRIM, Filament Winding, Pultrusion, Compression Molding, Wet-lay up, etc.?
The analysis performed by 3pcsolver004 ?solver is equally applicable to Hybrid Laminates or plates manufactured from a single or multiple types of lamina materials and/or ply broad forms or fiber types or single or multiple materials systems or their combinations. ?
Examples:
Perform free vibration analysis of a composite laminate having manufactured from a unidirectional carbon/epoxy tape material with following lamina properties, plate dimensions,laminate lay-up and applied edge loads:
o Analysis Options :
For our example, the following inputs are defined:
o Materials: One material with ?E1=1.8e7 psi, E2 = 1.6e6 psi ; G12=G13=8.7e5 psi, G23=6.4e5 psi; ν21=0.3 rho=0.000149
o Plies: ?Two plies of above material oriented at 45 and -45 with t_ply=0.00525 in
o Laminates: ?One laminate with stacking sequences [+/-45]2s
o Panels: One Panel with Laminates 1 and L = 15 in and W = 10 in
o Loads: ?Five Load cases as defined above
o Analysis Options: No of terms for solution M = N = 12 and Number of grid points in x and y = 21 for good quality plots
- Outputs:?
3pcsolver004 solver provides the following standard outputs for each load case:
o Panel geometry
o Analysis options selected
o Material Properties and Laminate Information?
o Laminate [A], [B], [D] stiffness matrices ?
o Effective laminate in-plane and flexural engineering constants?
o Applied loads and First Five (or lowest five) natural frequencies of vibration
o Grid Points coordinates x and y, transverse displacements w for first five natural frequencies of vibration
For our example, for each load case, the following outputs are obtained and written in an output.txt file:
LOADS ID ?PANEL ID ?
1 ????????1 ????????
o Panel Geometry
LENGTH: 15.00 ????????
WIDTH : 10.00 ????
o Analysis Options
m = 12 ???????????
n = 12 ???????????
OUTPUT OPTIONS
NUMBER OF POINTS IN X DIR: 21 ???????????
NUMBER OF POINTS IN Y DIR: 21?
o Material Properties and Laminate Information ????????????????????????????
ID ???????????E1 ???????????E2 ???????????G12 ??????????G23 ??????????G13 ??????????v12 ??????????rho ??????????
aiaa-2009 ????1.80e+07 ?????1.60e+06 ?????8.70e+05 ?????6.40e+05 ?????8.70e+05 ?????0.3000 ???????0.00014900 ??
??????LAMINATE GEOMETRY ?????
STACKING SEQUENCE (PLY ANG): [+45.0 ????????, -45.0 ????????, +45.0 ????????, -45.0 ????????, -45.0 ????????, +45.0 ????????, -45.0 ????????, +45.0 ????????]
STACKING SEQUENCE (PLY MAT): [aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????, aiaa-2009 ????]
TOTAL THICKNESS: 0.0420 ???
领英推荐
TOTAL MASS: 0.0009 ??????
OFFSET: 0.0000 ?????
o Laminate [A], [B], [D] stiffness matrices ?
???????????LAMINATE PROPERTIES ???????????
?????????????????A MATRIX ????????????????
+254160.97 ???+181080.97 ???+0.00 ????????
+181080.97 ???+254160.97 ???+0.00 ????????
+0.00 ????????+0.00 ????????+197298.39 ???
???????A MATRIX - TRANSVERSE SHEAR ???????
+31710.00 ????+0.00 ????????
+0.00 ????????+31710.00 ????
?????????????????B MATRIX ????????????????
+0.00 ????????+0.00 ????????-0.00 ????????
+0.00 ????????+0.00 ????????-0.00 ????????
+0.00 ????????+0.00 ????????+0.00 ????????
?????????????????D MATRIX ????????????????
+37.36 ???????+26.62 ???????+9.57 ????????
+26.62 ???????+37.36 ???????+9.57 ????????
+9.57 ????????+9.57 ????????+29.00 ????
Note that in the first example, the bending-twisting coupling terms ?D16 and D26 are non-zero?
o Effective laminate in-plane and flexural engineering constants ???????????????????????????????????????
Ex ???????????Ey ???????????Gxy ??????????vxy ??????????vyx ??????????Efx ??????????Efy ??????????Gfxy ?????????vfxy ?????????vfyx ?????????
+2.98e+06 ????+2.98e+06 ????+4.70e+06 ????+0.7125 ??????+0.7125 ??????+2.93e+06 ????+2.93e+06 ????+4.23e+06 ????+0.6859 ??????+0.6859 ??
o APPLIED LOADS ?and First Five ?NATURAL FREQUENCY ??????????????????
NUMBER ???????NXX ??????????NYY ??????????NXY ??????????HZ ???????????
1 ????????????0.0000 ???????0.0000 ???????0.0000 ???????67.7402 ??????
2 ????????????0.0000 ???????0.0000 ???????0.0000 ???????130.5990 ?????
3 ????????????0.0000 ???????0.0000 ???????0.0000 ???????187.7086 ?????
4 ????????????0.0000 ???????0.0000 ???????0.0000 ???????219.3856 ?????
5 ????????????0.0000 ???????0.0000 ???????0.0000 ???????271.6349 ???????
o GRID POINTS and TRANSVERSE DISPLACEMENTS ??????????????????????????????????????????
X ????????????Y ????????????W1 ???????????W2 ???????????W3 ???????????W4 ???????????W5 ???????????
0.0000 ???????0.5000 ???????+0.0000e+00 ??+0.0000e+00 ??+0.0000e+00 ??+0.0000e+00 ??+0.0000e+00 ??
0.7500 ???????0.5000 ???????+3.3900e-02 ??+5.5100e-02 ??-7.4100e-02 ??+6.0000e-02 ??+1.2940e-01 ??
1.5000 ???????0.5000 ???????+6.2800e-02 ??+9.7300e-02 ??-1.3760e-01 ??+9.3100e-02 ??+2.2860e-01 ??
2.2500 ???????0.5000 ???????+8.6400e-02 ??+1.2190e-01 ??-1.8910e-01 ??+9.0700e-02 ??+2.8450e-01 ???>>>>>>>Truncated <<<<<<<<<<<<<<<<???????????????????????????????????????????????????
A similar output is obtained for the other load cases and is not repeated here for brevity. The input and output files for such examples can be downloaded from the Resources tab of our website.?
- Visualization:?
Plots of the laminated plates' transverse displacements provide greater insight into the modal behavior of the laminated plates and effects of anisotropy ?Any third-party software such as MS-Excel, TechPlot, MATLAB, etc. can be used to import the ASCII text data and plot ?these quantities. For the examples above, a MATLAB live-script is used to create the following plots, a select few are shown below: ??
???????????????Case I: Natural Frequencies And Mode Shapes [±45]_2s?
??????????????????Case II: Natural Frequencies And Mode Shapes [±45]_2s ?with Edge Tension
???????Case III: Natural Frequencies And Mode Shapes [±45]_2s? with Edge Compression
?????Case IV: Natural Frequencies And Mode Shapes [±45]_2s ??with Positive Edge Shear
Case V: Natural Frequencies And Mode Shapes [±45]_2s with Negative Edge Shear
?Summary of ?Natural Frequencies for the Five Cases for [±45]_2s Laminate
Takeaways:
3pcmodule006 presents comprehensive treatment of analyses of laminated composite plates by discussing various linear and nonlinear, classical, higher-order transverse shear deformable and classical laminated plate theories. More than 700 slides provide key expressions for internal and external virtual work, the strain-displacement relations, constitutive laws, equilibrium equations and associated boundary conditions are presented in concise manner along with many numerical examples ?illustrating the effects of anisotropy on the bending, buckling and free vibrations of rectangular and non-rectangular ?laminated composite ?plates. Full module can be previewed by visiting?
Senior Principal Engineer at Collins Aerospace
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