Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method

Doeva, Olga; Masjedi, Pedram Khaneh

doi:10.1007/s00707-021-03043-z

Cited by 7 publications

(3 citation statements)

References 63 publications

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“…In Ref. [23], the static study of composite beams over elastic foundations with varying stiffness has been performed out using the homotopy approach. A Pasternak‐type elastic foundation was used to support FG beams with changing cross‐sections and provided an analytical solution for investigating their vibrations [24].…”

Section: Introductionmentioning

confidence: 99%

Analytical and computational analysis of vibrational behavior of axially loaded beams placed on elastic foundations

Alahmadi,

Nawaz,

Kanwal

et al. 2024

Z Angew Math Mech

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In this study, the vibrational behavior of Rayleigh and shear beams placed over the Hetényi and Pasternak foundations is investigated, with special attention paid to the effects of axial stresses (both tensile and compressive). The uniqueness of the work comes from the thorough consideration of how axial forces, shear deformation, rotational inertia, and foundation factors affect the eigenfrequencies and eigenmodes that control beam vibrations. Eigenfrequencies and eigenmodes are determined through computational analyses with the Galerkin finite element method (GFEM) employing quadratic and cubic polynomials. Comparisons between finite element results and analytical outcomes, both in generalized and specialized contexts, serve to demonstrate the correctness and effectiveness of the approximate solution. The study shows that the Hetényi foundation's influence on eigenfrequency depends on foundation stiffness and relative beam magnitudes, and that shear deformation affects eigenfrequencies more than rotary inertia does. Higher frequencies are largely unaffected by compressive and tensile stresses, but the fundamental frequency is rigorously influenced. Further demonstrating its accuracy, the finite element approach shows great agreement in forecasting eigenmodes and eigenfrequencies. A variety of real‐world situations involving elastically confined beams resting on elastic foundations and being subjected to axial forces are the focus of the study. Axial forces commonly result from external loads, thermal expansion, or structural interactions in real‐world settings. Therefore, it is crucial to understand how they affect beam vibration when designing structures that can endure expected stresses and vibrations while maintaining safety and performance criteria.

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Section: Introductionmentioning

confidence: 99%

Analytical and computational analysis of vibrational behavior of axially loaded beams placed on elastic foundations

Alahmadi,

Nawaz,

Kanwal

et al. 2024

Z Angew Math Mech

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“…Asif et al [23] explored the dispersion of elastic waves in an inhomogeneous multilayered plate over a Winkler elastic foundation with imperfect interfacial conditions. Doeve et al [24] to statistically analyze composite beams on elastic foundations with variable stiffness. Doyle and Pavlovic [25] performed dynamic analysis of beams on partial elastic foundations.…”

Section: Introductionmentioning

confidence: 99%

Effects of shear deformation and rotary inertia on elastically constrained beam resting on pasternak foundation

et al. 2023

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This study investigates the free vibrations of elastically constrained shear and Rayleigh beams placed on the Pasternak foundation. Of particular interest, it is aimed to analyze the influence of shear strain, rotational inertia, elastic stiffness, and shear layer on the naturalfrequencies and eigenmodes of beam vibrations. For this purpose, the eigenfrequencies and eigenmodes are determined using analytical and numerical techniques. A finite element scheme is developed employing quadratic and cubic polynomials for slope and transverse displacement, respectively. The efficiency and accuracy of the finite element method are illustrated by comparing it with the analytical results for generalized and special cases. The underlying model analysis justifies that the natural frequencies of the beam vibration depend only on the geometry of the Rayleigh beam, while these frequencies depend on the physical and geometric properties of the shear beam. However, the natural frequencies of the Euler-Bernoulli depend solely on the geometric conditions of the beam.

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“…The homotopy analysis method (HAM) was used to obtain the solution for the large deformation of a cantilever beam made of axially functionally graded material [44] and a nonlinear beam subjected to a coplanar terminal load consisting of a moment, an axial compressive force, and a transverse force [45]. Closed-form series solutions for the static deflection of anisotropic composite beams resting on elastic foundations were obtained by both the homotopy analysis method (HAM) and iterative HAM (iHAM) [46]. The iterative homotopy analysis method (iHAM) was used to obtain analytical solutions for the arbitrary large deflection of geometrically exact beams subjected to distributed and tip loads based on follower and conservative loading scenarios [47].…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

Yin-shan¹,

Li²,

Huo³

et al. 2022

Applied Sciences

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An improved homotopy analysis method (IHAM) is proposed to solve the nonlinear differential equation, especially for the case when nonlinearity is strong in this paper. As an application, the method was used to derive explicit solutions to the rotation angle of a cantilever beam under point load at the free end. Compared with the traditional homotopy method, the derivation includes two steps. A new nonlinear differential equation is firstly constructed based on the current nonlinear differential equation of the rotation angle and the auxiliary quadratic nonlinear differential equation. In the second step, a high-order non-linear iterative homotopy differential equation is established based on the new non-linear differential equation and the auxiliary linear differential equation. The analytical solution to the rotation angle is then derived by solving this high-order homotopy equation. In addition, the convergence range can be extended significantly by the homotopy–Páde approximation. Compared with the traditional homotopy analysis method, the current improved method not only speeds up the convergence of the solution, but also enlarges the convergence range. For the large deflection problem of beams, the new solution for the rotation angle is more approachable to the engineering designers than the implicit exact solution by the Euler–Bernoulli law. It should have significant practical applications in the design of long bridges or high-rise buildings to minimize the potential error due to the extreme length of the beam-like structures.

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Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method

Cited by 7 publications

References 63 publications

Analytical and computational analysis of vibrational behavior of axially loaded beams placed on elastic foundations

Analytical and computational analysis of vibrational behavior of axially loaded beams placed on elastic foundations

Effects of shear deformation and rotary inertia on elastically constrained beam resting on pasternak foundation

Explicit Solutions to Large Deformation of Cantilever Beams by Improved Homotopy Analysis Method I: Rotation Angle

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