Elsevier

Composite Structures

Volume 243, 1 July 2020, 112248
Composite Structures

Comment on the Navier’s solution in “A sinusoidal beam theory for functionally graded sandwich curved beams” (Composite Structures 226 (2019) 111246)

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Abstract

In “A sinusoidal beam theory for functionally graded sandwich curved beams” (Composite Structures 226 (2019) 111246), the Navier’s solution was obtained for a so-called simply-supported curved beam at two ends. However, this case is mathematically indeterminate and hence not worked out. Based on the Timoshenko theory, the governing differential equations are formulated for curved beam problems and then analytically solved. Excellent agreement with the elasticity theory validates the solution for a free-clamped curved beam. Careful study on the simply-supported curved beam shows that the Navier’s solution in the literature is the one when the inappropriate constraint is assumed, which causes a significant error (up to 42%) for a relative large-curvature beam. In this paper, the Navier’s solution is therefore assessed.

Introduction

Curved beams are basic and important structures, which are widely used in design of civil and mechanical engineering such as bridge structures, automobiles and ships [1]. Nowadays, the laminated composite [2], [3], functionally graded [4], [5] and sandwich materials [6], [7] are introduced in the curved beam structures. Different curved beam theories are established such as classical theory [8], Timoshenko theory [3], [5], [9], [10], [11] and various higher-order theories [12], [13], [14], [15], [16] at macro- and micro-scales. Because the mathematically analytic approach becomes more difficult or even impossible for these complex beam problems, the Navier’s solution with series expansions is preferred for researchers to effectively handle them [7], [8], [12], [14], [17]. Particularly, the simply-supported case is regarded as the more suitable boundary conditions for the analysis of bending, buckling and free vibration.

This paper is primarily concerned with the Navier’s solution in [7] which was widely adopted by researchers [7], [8], [12], [14], [17] for a curved beam. For completeness, the governing differential equations are first precisely introduced based on the Timoshenko theory, and the analytic solution is obtained by means of the Mathematica package. The free-clamped case is then solved and validated by the corresponding elasticity solution. For the simply-supported case, the problem with boundary conditions in [7] is mathematically indeterminate. Given an inappropriate constraint with which the problem is determinate and the Navier’s solution in [7] is therefore valid, the solution is obtained for the moveable simply-supported case and then argued.

Section snippets

The Timoshenko theory of curved beam and its analytic solution

As shown in Fig. 1, a circular curved beam with curvature radius r is considered. For simplicity, the rectangular cross section with unit width and thickness h is assumed.

For this curved beam, the generalized strains, namely the axial strain ε0, bending strain κ and shear strain Γ, are [9], [10]ε0=u0(x)+w(x)/rκ=ϕ(x)Γ=ϕ(x)+w(x)-u0(x)/rwhere u0(x), w(x) and ϕ(x) are the three generalized displacements, namely axial displacement, deflection and rotation of cross-section. The prime denotes

Results and comments

In this section, free-clamped and simply-supported curved beams are respectively solved.

Conclusions

In this paper, to check the Navier’s solution in [7], a circular curved beam problem was reformulated by using the Timoshenko theory, and the analytic solution was derived by means of Mathematica. The solution for the simply-supported case indicated that the axial displacement was inappropriately prescribed so that the Navier’s solution could be obtained in [7] for this indeterminate problem, which caused a significant error for a larger-curvature beam.

The Navier’s solution is based on boundary

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundations of China (Grant Nos. 11672221).

Data availability statement

The raw/processed data required to reproduce these findings cannot be shared at this time due to time limitations.

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  • A simplified theory of FG curved beams

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    Citation Excerpt :

    Based on the iso-geometric approach, bending, buckling and vibration of curved beams were evaluated by Luu et al. (2015a), Huynh et al. (2017), Cazzani et al. (2016) and Hosseini et al. (2018). By using the analytical solution to the curved TBT, Pei and Li (2020) studied bending response of the homogenous curved beam and discussed the nature of the Navier's solution under the simply-supported condition. However, the shear correction factor (SCF) must be required for the TBT, which is still controversial for inhomogeneous beams such as FG, laminated composite or sandwiched ones (Hajianmaleki and Qatu, 2013; Sayyad and Ghugal, 2017; Pei et al., 2019).

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