Free Vibration of Laminated Composite and Hybrid Axisymmetric Shell Structures

In a recent study published in the journal Composite Structure, researchers from Portugal developed a model to study free vibrations in laminated and hybrid axisymmetric composite structures.

Study: Free Vibrations Analysis of Composite and Hybrid Axisymmetric Shells. Image Credit: Gorodenkoff/

Revolutionary Composite Laminates

Revolutionary shells are used in various applications such as aerospace structural systems, pressure vessels, and cooling towers. Revolutionary structures of composite materials are developed using functionally graded material by combining two different materials and they have high structural performance ability, possess low thermal conductivity, and strong mechanical resistance.

The present study discusses the free vibrations analysis of axisymmetric composite laminates and hybrid axisymmetric structures. The researchers also proposed a model to solve the free vibration of composite laminates, and the solutions were discussed using different composite laminates cases.


In the present study, researchers developed the composite model of functionally graded materials (FGM) by combining two distinct isotropic material phases, such as ceramic and metal. The material properties of an FGM structure were assumed to change continuously throughout its thickness, depending on the volume fraction of constituent materials. The virtual layer approach was developed by continuously varying the materials mixture using a certain number of discrete layers (NF) along the thickness direction. Additionally, only 20 virtual layers were used across all applications.

The stress-strain relationship used in this study was based on the elasticity matrix in the local system and the stress and strain vectors. In addition to this, in the case of laminated composite materials, the stress-strain relationship was obtained after converting principal material directions to the laminate’s local system.

The researchers evaluated the local displacement of a point on the middle surface of the meridian plane in axisymmetric shells using three components in the meridional, circumferential, and normal directions, respectively. The displacement field for a generic point was defined by time coordinates, using an equivalent single-layer model and incorporating transverse shear strains through the imposed constraints.

To develop the model, numerical solutions were obtained by expanding variables in the Fourier series and then using conical frustum finite elements in the meridional direction and the circumferential direction. The finite element solution used conical frustum with dual nodal circles and ten degrees of freedom per nodal circle.

The geometry of axisymmetric structures was modeled with a small number of finite elements, and the integration procedures of them used one Gauss point. Additionally, the variation in thickness properties in FGM laminas was modeled with a limited number of virtual layers, resulting in very high computational efficiency.

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Applications and Results

The researchers calculated the natural frequencies of a laminated cylindrical shell for two cases, clamped and simply supported at both ends. The results obtained with the models such as the current first-order shear deformation theory  (FSDT) model, the cone penetration tests (CPT), and the high-order were similar to the researchers’ proposed model.

Researchers also investigated the natural frequencies of a laminated conical shell with both ends clamped. The results in this case by solution obtained from the proposed model were also similar to those obtained from previous literature.

For certain cases, free vibration results were not found in the previous studies, which could be used for comparison with the researchers’ proposed model. Hence, researchers used Abaqus software to validate the results. Hybrid cylinder results showed similarity between the developed model and the Abaqus software results.

Similarly, the fundamental frequency results of the clammed hybrid cone were comparable between the model presented by the researchers’ and those using Abaqus.

Free vibration studies using the hybrid hemispherical cap, hybrid cone-cylinder shell, toroidal shell, and hybrid cooling tower were also analyzed. These results showed an excellent comparison between the model presented in this study and Abaqus results.


The current study aimed to present an efficient finite element model for natural free-vibration analysis of laminated composites and multi-material axisymmetric shell structures.

The researchers developed the model by separating the variables in a Fourier series solution in the circumferential direction and meridional direction. The final model was an axisymmetric model consisting of two nodes and 10 degrees of freedom at each node after applying finite element analysis.

A significant number of applications were presented in the study, and the results obtained by the model presented in the study were compared to solutions provided by alternative models in the previous studies and solutions obtained with the finite element program Abaqus.


Moita, J.S., Araujo, A.L., Correia, V.F., Mota Soares, C.M., Free Vibrations Analysis of Composite and Hybrid Axisymmetric Shells, Composite Structures (2022),

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Chinmay Saraf

Written by

Chinmay Saraf

Chinmay Saraf is a science writer based in Indore, India. His academic background is in mechanical engineering, and he has extensive experience in fused deposition-based additive manufacturing. His research focuses on post-processing methods for fused deposition modeling to improve mechanical and electrical properties of 3D printed parts. He has also worked on composite 3D printing, bioprinting, and food printing technologies. Chinmay holds an M.Tech. in computer-aided design and computer-aided manufacturing and is passionate about 3D printing, new product development, material science, and sustainability. He also has a keen interest in "Frugal Designs" to improve the existing engineering systems.  


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