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The effect of thickness coordinate to radius ratio on free vibration of moderately thick and deep doubly curved cross-ply laminated shell

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Abstract

This paper presents the free vibration analysis of moderately thick and deep doubly curved laminated shell with \(C^{0}\) finite element model based on higher-order shear deformation theory. The strain–displacement relationships are developed using the accurate equations of elastic deformation of shell structure. An eight-noded isoparametric shell element with nine degrees of freedom per node is used to formulate the present model. The effect of incorporating the ratio of thickness coordinate to radius of curvature (z / R) in the strain components has been taken into account in the present study. The numerical results in terms of natural frequencies obtained by the present formulations are compared with those available in the published literature to validate the proposed model.

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Authors and Affiliations

  1. Department of Civil Engineering, University Institute of Technology, The University of Burdwan, Burdwan, 713104, India

    Sandipan Nath Thakur

  2. Department of Civil Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    Chaitali Ray

Authors
  1. Sandipan Nath Thakur
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  2. Chaitali Ray
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Correspondence to Chaitali Ray.

Appendices

Appendix 1

Rigidity matrix of laminates using model 1:

(19)
$$\begin{aligned} \left\{ {{\begin{array}{l} {{\varvec{Q}}_{{\varvec{s}}} } \\ {{\varvec{Q}}_{{\varvec{r}}} } \\ {{\varvec{S}}_{{\varvec{s}}} } \\ {{\varvec{S}}_{{\varvec{r}}} } \\ {{\varvec{Q}}_{{\varvec{s}}}^*} \\ {{\varvec{Q}}_{{\varvec{r}}}^*} \\ {{\varvec{S}}_{{\varvec{s}}}^*} \\ {{\varvec{S}}_{{\varvec{r}}}^*} \\ \end{array} }} \right\}= & {} \left[ {{\begin{array}{llllllll} {\overline{{\varvec{A}}_\mathbf{44}}} &{}\quad {\varvec{A}}_\mathbf{45} &{}\quad {\overline{{\varvec{B}}_\mathbf{44}}}&{}\quad {\varvec{B}}_\mathbf{45} &{}\quad {\overline{{\varvec{C}}_\mathbf{44}}}&{}\quad {\varvec{C}}_\mathbf{45} &{}\quad {\overline{{\varvec{D}}_\mathbf{44}}}&{}\quad {\varvec{D}}_\mathbf{45}\\ {\varvec{A}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{A}}_\mathbf{55}}} &{}\quad {\varvec{B}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{B}}_\mathbf{55}}}&{}\quad {\varvec{C}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{C}}_\mathbf{55}}} &{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{D}}_\mathbf{55}}}\\ {\overline{{\varvec{B}}_\mathbf{44}}} &{}\quad {\varvec{B}}_\mathbf{45} &{}\quad {\overline{{\varvec{C}}_\mathbf{44}}}&{}\quad {\varvec{C}}_\mathbf{45} &{}\quad {\overline{{\varvec{D}}_\mathbf{44}}}&{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\overline{{\varvec{E}}_\mathbf{44}}}&{}\quad {\varvec{E}}_\mathbf{45}\\ {\varvec{B}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{B}}_\mathbf{55}}} &{}\quad {\varvec{C}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{C}}_\mathbf{55}}}&{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{D}}_\mathbf{55}}} &{}\quad {\varvec{E}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{E}}_\mathbf{55}}}\\ {\overline{{\varvec{C}}_\mathbf{44}}} &{}\quad {\varvec{C}}_\mathbf{45} &{}\quad {\overline{{\varvec{D}}_\mathbf{44}}}&{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\overline{{\varvec{E}}_\mathbf{44}}}&{}\quad {\varvec{E}}_\mathbf{45} &{}\quad {\overline{{\varvec{F}}_\mathbf{44}}}&{}\quad {\varvec{F}}_\mathbf{45}\\ {\varvec{C}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{C}}_\mathbf{55}}} &{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{D}}_\mathbf{55}}}&{}\quad {\varvec{E}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{E}}_\mathbf{55}}} &{}\quad {\varvec{F}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{F}}_\mathbf{55}}}\\ {\overline{{\varvec{D}}_\mathbf{44}}} &{}\quad {\varvec{D}}_\mathbf{45} &{}\quad {\overline{{\varvec{E}}_\mathbf{44}}}&{}\quad {\varvec{E}}_\mathbf{45} &{}\quad {\overline{{\varvec{F}}_\mathbf{44}}}&{}\quad {\varvec{F}}_\mathbf{45} &{}\quad {\overline{{\varvec{G}}_\mathbf{44}}}&{}\quad {\varvec{G}}_\mathbf{45}\\ {\varvec{D}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{D}}_\mathbf{55}}} &{}\quad {\varvec{E}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{E}}_\mathbf{55}}}&{}\quad {\varvec{F}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{F}}_\mathbf{55}}} &{}\quad {\varvec{G}}_\mathbf{45} &{}\quad {\widetilde{{\varvec{G}}_\mathbf{55}}}\\ \end{array} }} \right] \left\{ {{\begin{array}{l} {{\varvec{\epsilon }}_{{\varvec{szo}}} } \\ {{\varvec{\epsilon }}_{{\varvec{rzo}}} } \\ {{\varvec{\kappa }}_{{\varvec{sz} }}} \\ {{\varvec{\kappa }}_{{\varvec{rz} }}} \\ {{\varvec{\epsilon }}_{{\varvec{szo}}}^*} \\ {{\varvec{\epsilon }}_{{\varvec{rzo}}}^*} \\ {{\varvec{\kappa }}_{{\varvec{sz}}}^*} \\ {{\varvec{\kappa }}_{{\varvec{rz}}}^*} \\ \end{array} }} \right\} \end{aligned}$$
(20)

where the values of above matrix coefficients are given by

$$\begin{aligned} ( {A_{ij}, B_{ij}, C_{ij}, D_{ij}, E_{ij}, F_{ij}, G_{ij}, H_{ij} } )=\mathop \sum \limits _{k=1}^{n} {\bar{Q}}_{ij}^{\left( k \right) } \mathop \int \limits _{z_{k} }^{z_{k+1} } ( {1,z,z^{2},z^{3},z^{4},z^{5},z^{6},z^{7}} )\mathrm{d}z \end{aligned}$$

and

$$\begin{aligned} {\overline{A_{ij}}}= & {} A_{ij} -C_{0} B_{ij},\quad {\widetilde{A_{ij} }} =A_{ij} +C_{0} B_{ij}\\ {\overline{B_{ij}}}= & {} B_{ij} -C_{0} C_{ij},\quad {\widetilde{B_{ij} }} =B_{ij} +C_{0} C_{ij}\\ {\overline{C_{ij}}}= & {} C_{ij} -C_{0} D_{ij},\quad {\widetilde{C_{ij} }} =C_{ij} +C_{0} D_{ij}\\ {\overline{D_{ij}}}= & {} D_{ij} -C_{0} E_{ij},\quad {\widetilde{D_{ij} }} =D_{ij} +C_{0} E_{ij}\\ {\overline{E_{ij}}}= & {} E_{ij} -C_{0} F_{ij},\quad {\widetilde{E_{ij} }} =E_{ij} +C_{0} F_{ij}\\ {\overline{F_{ij}}}= & {} F_{ij} -C_{0} G_{ij},\quad {\widetilde{F_{ij} }} =F_{ij} +C_{0} G_{ij}\\ {\overline{G_{ij}}}= & {} G_{ij} -C_{0} H_{ij},\quad {\widetilde{G_{ij} }} =G_{ij} +C_{0} H_{ij} \end{aligned}$$

where \(i, j= 1, 2, 4, 5, 6\) and \(C_0 =\left( {\frac{1}{R_{s} }-\frac{1}{R_{r} }} \right) \).

Appendix 2

Rigidity matrix of laminates using model 2:

(21)
$$\begin{aligned} \left\{ {{\begin{array}{l} {{\varvec{Q}}_{{\varvec{s}}}} \\ {{\varvec{Q}}_{{\varvec{r}}}} \\ {{\varvec{S}}_{{\varvec{s}}}} \\ {{\varvec{S}}_{{\varvec{r}}} } \\ {{\varvec{Q}}_{{\varvec{s}}}^*} \\ {{\varvec{Q}}_{{\varvec{r}}}^*} \\ {{\varvec{S}}_{{\varvec{s}}}^*} \\ {{\varvec{S}}_{{\varvec{r}}}^*} \\ \end{array} }} \right\}= & {} \left[ {{\begin{array}{llllllll} {\varvec{A}}_\mathbf{44}&{}\quad {\varvec{A}}_\mathbf{45}&{}\quad {\varvec{B}}_\mathbf{44}&{}\quad {\varvec{B}}_\mathbf{45}&{}\quad {\varvec{C}}_\mathbf{44}&{}\quad {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{44}&{}\quad {\varvec{D}}_\mathbf{45}\\ {\varvec{A}}_\mathbf{45}&{}\quad {\varvec{A}}_\mathbf{55}&{}\quad {\varvec{B}}_\mathbf{45}&{}\quad {\varvec{B}}_\mathbf{55}&{}\quad {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{C}}_\mathbf{55}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{55}\\ {\varvec{B}}_\mathbf{44}&{}\quad {\varvec{B}}_\mathbf{45}&{}\quad {\varvec{C}}_\mathbf{44}&{}\quad {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{44}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{44}&{}\quad {\varvec{E}}_\mathbf{45}\\ {\varvec{B}}_\mathbf{45}&{}\quad {\varvec{B}}_\mathbf{55}&{}\quad {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{C}}_\mathbf{55}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{55}&{}\quad {\varvec{E}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{55}\\ {\varvec{C}}_\mathbf{44}&{}\quad {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{44}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{44}&{}\quad {\varvec{E}}_\mathbf{45}&{}\quad {\varvec{F}}_\mathbf{44}&{}\quad {\varvec{F}}_\mathbf{45}\\ {\varvec{C}}_\mathbf{45}&{}\quad {\varvec{C}}_\mathbf{55}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{55}&{}\quad {\varvec{E}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{55}&{}\quad {\varvec{F}}_\mathbf{45}&{}\quad {\varvec{F}}_\mathbf{55}\\ {\varvec{D}}_\mathbf{44}&{}\quad {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{44}&{}\quad {\varvec{E}}_\mathbf{45}&{}\quad {\varvec{F}}_\mathbf{44}&{}\quad {\varvec{F}}_\mathbf{45}&{}\quad {\varvec{G}}_\mathbf{44}&{}\quad {\varvec{G}}_\mathbf{45}\\ {\varvec{D}}_\mathbf{45}&{}\quad {\varvec{D}}_\mathbf{55}&{}\quad {\varvec{E}}_\mathbf{45}&{}\quad {\varvec{E}}_\mathbf{55}&{}\quad {\varvec{F}}_\mathbf{45}&{}\quad {\varvec{F}}_\mathbf{55}&{}\quad {\varvec{G}}_\mathbf{45}&{}\quad {\varvec{G}}_\mathbf{55}\\ \end{array} }} \right] \left\{ {{\begin{array}{l} {{\varvec{\epsilon }}_{{\varvec{szo}}} } \\ {{\varvec{\epsilon }}_{{\varvec{rzo}}} } \\ {{\varvec{\kappa }}_{{\varvec{sz}}}} \\ {{\varvec{\kappa }}_{{\varvec{rz}}}} \\ {{\varvec{\epsilon }}_{{\varvec{szo}}}^*} \\ {{\varvec{\epsilon }}_{{\varvec{rzo}}}^*} \\ {{\varvec{\kappa }}_{{\varvec{sz}}}^*} \\ {{\varvec{\kappa }}_{{\varvec{rz}}}^*} \\ \end{array} }} \right\} \end{aligned}$$
(22)

where the values of above matrix coefficients are given by

$$\begin{aligned} ( {A_{ij}, B_{ij}, C_{ij}, D_{ij}, E_{ij}, F_{ij}, G_{ij} } )=\mathop \sum \limits _{k=1}^{n} {\bar{Q}}_{ij}^{\left( k \right) } \mathop \int \limits _{z_{k} }^{z_{k+1} } ( {1,z,z^{2},z^{3},z^{4},z^{5},z^{6}} )\mathrm{d}z \end{aligned}$$

where \(i, j= 1, 2, 4, 5, 6\).

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Thakur, S.N., Ray, C. The effect of thickness coordinate to radius ratio on free vibration of moderately thick and deep doubly curved cross-ply laminated shell. Arch Appl Mech 86, 1119–1132 (2016). https://doi.org/10.1007/s00419-015-1082-8

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