# Frontiers of Structural and Civil Engineering

 Front. Struct. Civ. Eng.    2020, Vol. 14 Issue (3) : 675-689     https://doi.org/10.1007/s11709-020-0625-4
 RESEARCH ARTICLE
Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses
Iman FATTAHI, Hamid Reza MIRDAMADI(), Hamid ABDOLLAHI
Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 8415683111, Iran
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 Abstract Analyzing static and dynamic problems including composite structures has been of high significance in research efforts and industrial applications. In this article, equivalent single layer approach is utilized for dynamic finite element procedures of 3D composite beam as the building block of numerous composite structures. In this model, both displacement and strain fields are decomposed into cross-sectional and longitudinal components, called consistent geometric decomposition theorem. Then, the model is discretized using finite element procedures. Two local coordinate systems and a global one are defined to decouple mechanical degrees of freedom. Furthermore, from the viewpoint of consistent geometric decomposition theorem, the transformation and element mass matrices for those systems are introduced here for the first time. The same decomposition idea can be used for developing element stiffness matrix. Finally, comprehensive validations are conducted for the theory against experimental and numerical results in two case studies and for various conditions. Corresponding Author(s): Hamid Reza MIRDAMADI Just Accepted Date: 22 April 2020   Online First Date: 17 June 2020    Issue Date: 13 July 2020
 Cite this article: Iman FATTAHI,Hamid Reza MIRDAMADI,Hamid ABDOLLAHI. Application of consistent geometric decomposition theorem to dynamic finite element of 3D composite beam based on experimental and numerical analyses[J]. Front. Struct. Civ. Eng., 2020, 14(3): 675-689. URL: http://journal.hep.com.cn/fsce/EN/10.1007/s11709-020-0625-4 http://journal.hep.com.cn/fsce/EN/Y2020/V14/I3/675
 Fig.1  The cross section of the 3-D beam with a piezoelectric layer. Fig.2  The composite cantilevered beam used for numerical validation. Tab.1  Specifications of the composite cantilevered beam used for numerical validation Fig.3  Natural frequencies of the composite cantilevered beam in different modes: (a) natural frequency; (b) error between theoretical model (MATLAB code) and ABAQUS results. Fig.4  The first mode shape of the structure: (a) ABAQUS and (b) theoretical model. Fig.5  The second mode shape of the structure: (a) ABAQUS and (b) theoretical model. Fig.6  The third mode shape of the cantilevered harvester: (a) ABAQUS and (b) theoretical model. Fig.7  The results of ($azrel/ azbase$) over a frequency range for the composite cantilevered beam for (a) c = 0.1 mm, (b) c = 0.2 mm, and (c) c = 0.4 mm. Fig.8  Parametric study and numerical validation for different lengths of the beam: L = 15, 24, and 33 mm. Fig.9  Numerical validation of the beam with two different substructures (aluminum and steel). Fig.10  The composite cantilevered beam prototype used for experimental validations. Tab.2  Specifications of the composite cantilevered beam used for experimental validation Fig.11  The whole test setup of the composite cantilevered beam used for experimental validation. Fig.12  The results of tip displacement for experimental validation for excitation amplitudes of (a) 0.07 mm, (b) 0.1 mm, and (c) 0.13 mm.