Prediction of free vibration responses of orthotropic stiffened flat panels

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ScienceDirect Materials Today: Proceedings 5 (2018) 20170–20176

www.materialstoday.com/proceedings

ICMPC_2018

Prediction of free vibration responses of orthotropic stiffened flat panels Sudhansu S. Patroa*, Ranjan K. Beheraa, Nitin Sharmaa a

School of Mechanical Engineering,Kalinga Institute of Industrial Technology, Deemed to be University, Bhubaneswar, Odisha-751024, India

Abstract In this paper, the free vibration characteristics of stiffened orthotropic plates is investigated. The flat panel geometry is modelled by using ANSYS parametric design language (APDL) code based on First Order Shear Deformation Theory (FSDT) midplane kinematics. The flat panel as well as the stiffeners have been modelled using an eight noded serendipity shell element. Firstly, the validity of the present model is established along with the convergence. Further, several numerical examples have been solved and discussed in detail to bring out the effect of lamination scheme, eccentricity, modular ratio and stiffener spacing on the free vibration responses of stiffened laminated composite flat panels. © 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization. Keywords: stiffened orthotrpic flat panel, FSDT, natrual frequency, modal analysis, APDL

1. Introduction Composite laminates is used worldwide as structural material for various industry due to the combination of good mechanical properties, high stiffness, strength-to-weight ratio, resistance to corrosion and long fatigue life. Composite laminated plate structures with stiffeners are extensively used in many modern engineering constructions such as aerospace structures, ship hulls, civil engineering structures, etc. Plates are reinforced with stiffeners to enhance the load carrying capacity or to meet special purpose stiffness and strength. Also, by adding stiffeners to composite plates, vibration response can be eliminated or reduced without increasing the plate thickness. The main purpose of designing stiffened plate to resist vibration due to dynamic loads. The stiffeners influence vibration behaviour to a great extent. That is why a large number of numerical and analytical model for the analysis

* Corresponding author. Tel.: +91-8984358186 E-mail address: [email protected] 2214-7853© 2018 Elsevier Ltd. All rights reserved. Selection and/or Peer-review under responsibility of Materials Processing and characterization.

Sudhansu S. Patro et al./ Materials Today: Proceedings 5 (2018) 20170–20176

Nomenclature E1, E2, E3 G12, G13, G23

ν12 ρ e

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Young's Modulus Shear Modulus Poisson's ratio Density eccentricity

stiffened laminated plates have been reported in the literature[1-5]. Chao et al [6] has studied three typical cases for the stiffened rectangular laminated plates namely (1) specially orthotropic laminates, (2) anti-symmetric crossply laminates and (3) laminates with multiple isotropic layers and calculated frequencies. He has neglected extension-shear and bending-twisting coupling effects in his investigation and also modeled stiffeners using simple beam theory. Lorenzo et al [7] analyzed free vibration of stiffened plates by a combined analytical and numerical method. He modeled the plate using classical thin plate theory, where the shear deformation effect and rotary inertia are neglected and stiffening beams are modeled as per Euler–Bernoulli beams. Many authors [8-10] have employed First-order Shear Deformation Theory (FSDT) to establish Finite Element Modeling for free vibration analysis of the laminated composite plates. Pandit et al [10] considered shear deformation by employing FSDT for the analysis of free vibration of laminated composites. Guo meiwen et al [11] investigated free vibration on stiffened laminated using the layered (zigzag) finite element method based on the first order shear deformation theory. Both authors [10-11] used nine-node iso-parametric degenerated flat shell element to model layers of the laminated plate. Many works [12-14] have been reported involves using higher-order shear deformation theories (HSDT). In the present work, an attempt is made for an effective and efficient method to model orthotropic stiffened laminated panels to predict natural frequency using a shell element available with commercial finite element package ANSYS. 2. Formulation of equations The laminated plate with stiffener, considered for the present investigation is shown in the figure 1. Plates and stiffeners have orthotropic properties. Three different lamination scheme are used in this investigation. The stiffener eccentricity can be defined by e= (hs-h)/2, where hs is the stiffener depth and h is plate thickness.

Figure 1. Diagram of stiffened composite plate 2.1. Plate and Stiffener vibration The laminated composite flat panels considered here consists of orthotropic layers of equal thickness. The boundary conditions are varied with different cases. In the present investigation, extension-shear and twistingbending coupling effects are neglected in all the cases of investigation. The stiffeners are defined as to consists of

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reinforcing rib and a strip of rib width in the plating. Shear due to extension and twisting due to bending are neglected. , , , , , , , , , , (1) , , , , , where, u, v and w are the displacement vector of any point in the layer at time t along x, y and z direction respectively. Modal analysis is used for this structure and its modal parameters are calculated by solving the eigen value equations: 0

(2)

where [K], [M], ω and {φ} are the stiffness matrix, mass matrix, natural frequency of vibration and corresponding mode shape vector, respectively. 3. Results and Discussions The modal analysis of the laminated composites flat panel with single and double eccentric stiffeners has been performed by using ANSYS 15. The convergence test and validation with previously reported results have been conducted to establish the accuracy and efficacy of the present analysis. The effect of stiffener eccentricity, modular ratio (E1/E2) and spacing between stiffener on modal frequencies for three different boundary conditions (SSSS, CCCC, SCSC) have been investigated. The properties of glass/epoxy composite material have been used for the current analysis.

Figure 2. Convergence of natural frequencies for eccentrically stiffened plate with (a) Single Stiffener, (b) Double Stiffeners 3.1. Convergence and Validation studies of natural frequencies In this section, the natural frequencies are computed for different mesh divisions to check the convergence behavior of present study. The convergence study is made for four different boundary conditions (SSSS, CCCC, SCSC, CFFF). Further, the computed values of frequencies are compared with the available numerical data for the validation of current study. A glass/epoxy composite with single and double eccentric Y-stiffeners, which was previously reported by Chao[6], has been selected for convergence and validation of present method. The material properties of glass/epoxy is presented in Table 1.

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E1

E2=E3

G12=G13

G23

ν12

ρ

9.71GPa

3.25 GPa

0.9025 GPa

0.2356 GPa

0.29

1347 kgm-3

Table 1. Material properties of glass/epoxy composites The geometry of the stiffened plate, lamination scheme (plate and stiffener) and material properties are kept similar to the corresponding reference considered for the study. The values of natural frequencies for first modes are computed for different mesh sizes for different boundary conditions and presented in Figure 2(a) and 2(b). It can be observed that as number of elements in the discretization is increased, the natural frequencies are converging on the values. A mesh (14×14) is employed to compute frequencies for plates with single stiffener and a mesh (21×21) is employed for plates with double stiffeners throughout the present analysis. The comparison of natural frequencies for composite plates with single and double eccentric stiffeners, computed in this study with the corresponding reference are presented in Table 2. It is clearly observed that the present values of natural frequencies are in good agreement with the reference results for all modes of frequencies. Table 2. Validation of natural frequency

Eccentricity 1

2

3

4

5

Modes Chao [6] Present Difference (%) Chao [6] Present Difference (%) Chao [6] Present Difference (%) Chao [6] Present Difference (%) Chao [6] Present Difference (%)

F11

Single Stiffener F12

56.856 55.939 1.639 59.148 58.418 1.250 63.600 62.902 1.109 69.129 69.07 0.085 75.199 76.196 -1.308

198.205 194.27 2.025 205.892 202.79 1.529 215.738 212.97 1.299 225.583 220.84 2.148 237.588 225.76 5.239

F11

Double Stiffeners F21 F12

F22

58.902 56.521 4.214 62.252 60.656 2.632 66.679 67.994 -1.933 75.933 78.018 -2.672 87.868 89.761 -2.108

95.379 94.280 1.166 96.582 95.636 0.989 99.936 98.294 1.671 105.434 102.33 3.034 113.079 107.53 5.160

225.184 222.560 1.179 239.263 230.33 3.878 259.784 240.74 7.910 280.838 276.18 1.686 296.526 278.61 6.430

200.509 196.830 1.869 213.515 212.64 0.412 237.255 233.94 1.417 265.817 250.55 6.093 288.480 258.52 11.589

3.2. Influence of eccentricity on natural frequencies The influence of stiffener eccentricity on natural frequencies of a four layered anti-symmetric cross-ply [00/900]2 laminated composite flat panel with single and double stiffeners is investigated in this example. The plate dimensions are taken as a=0.4 m, b=0.3m, h=0.0034m and ws=0.003m. The height of stiffener (hs) is varied such that eccentricity (e)=1, 2, 3, 4, 5. The boundary condition used in this example is simply supported type (SSSS). The variation of natural frequencies with different eccentricity is shown in the Figure 3 (a) and (b). From the figure, it

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can be observed that the values of frequencies for all modes are increasing with eccentricity. The change in frequencies is greater at higher eccentricity. Also, as the number of stiffeners increases the value of frequencies have increased for all modes of frequency except F32 mode. From Figure 3(a), it can be observed that frequency values for F12, F22, F31 are converging at high value of eccentricity.

Figure 3. Influence of eccentricity on natural frequencies for eccentrically stiffened plate with (a) Single Stiffener, (b) Double Stiffeners 3.3. Influence of modular ratio on natural frequencies The effect of modular ratio on natural frequencies of a four layered symmetric angle-ply [450/-450]s laminated composite flat panel with single and double stiffeners is investigated in this example. The plate dimensions are taken as a=0.4 m, b=0.3m, h=0.0034m, ws=0.003m and hs=.008+h. A clamped (CCCC) type boundary condition is used in this calculation. The natural frequencies are computed for E1/E2= 5, 10, 20, 30, 40.

Figure 4. Influence of modular ratio on natural frequencies for eccentrically stiffened plate with (a) Single Stiffener, (b) Double Stiffeners The variation of natural frequencies with modular ratio is shown in the Figure 4 (a) and (b). It can be clearly observed that the values of frequencies are decreasing with modular ratio for modes of frequency. The variation in

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the values of frequencies is large at lower value of modular ratio. For higher value of modular ratio, it can be observed little or no variation in values of frequencies. 3.4. Influence of spacing between two stiffeners on natural frequencies It is evident from the above section that number of stiffener has influence on the frequencies. In this section, the effect of spacing between stiffeners on frequencies of a three layered cross-ply [900/00/900] laminated composite flat panel with double stiffeners is investigated. The plate dimensions are taken as a=0.4 m, b=0.3m, h=0.0034m, ws=0.003m and hs=.008+h. A SCSC type boundary condition is used in this calculation. The natural frequencies are computed for Spacing = a/6, a/5, a/4, a/3, a/2. The variation of natural frequencies with spacing is shown in the Figure 5.

Figure 5. Influence of spacing on natural frequencies for eccentrically stiffened plate with Double Stiffeners The values of first mode frequency is increasing with the spacing up to a/4, where it attains the maximum value and then decreases with increasing spacing. Similar trends can be observed for F41 and F12 modes. For other modes of frequencies the values are increasing with spacing. 4. Conclusion In this study, vibration characteristics of eccentrically stiffened laminated composite is investigated in ANSYS using APDL code. First, the vibration responses computed are compared and validated with available benchmark results. Then, the values of frequencies are computed for different values of eccentricity, modular ratio and spacing of stiffeners. It is observed that the values of frequencies for all modes are increasing with eccentricity. The change in values of frequencies is greater at higher eccentricity. It is also observed that for all modes of frequencies the values of frequencies are decreasing with modular ratio. The variation in the values of frequencies is large at lower value of modular ratio, where as there is a little or no variation is observed in the values of frequencies at higher value of modular ratio. It is interesting to note that the values of first mode frequency has an increasing trend with the spacing value equal to a/4, then decreases with increasing spacing.

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Sudhansu S. Patro et al./ Materials Today: Proceedings 5 (2018) 20170–20176

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