Bulletin of Earthquake Science and Engineering

Bulletin of Earthquake Science and Engineering

Analysis of Buckling and Vibrations of Non-Prismatic Beam on Winkler-Pasternak Foundation with Finite Element and Riley-Ritz Methods

Document Type : Research Article

Authors
Amir Hossein Taherkhani 1
Majid Amin Afshar 2
1 M.Sc. Graduate, Earthquake Engineering, Department of Civil Engineering, Imam Khomeini International University, Qazvin, Iran
2 Assistant Professor, Faculty of Engineering and Technology, Department of Civil Engineering, Imam Khomeini International University, Qazvin, Iran
10.48303/bese.2024.2011051.1134
Abstract
In the analysis of structures such as building foundations, railway rails, reservoirs, airport runways and docks, it is necessary to consider the effect of the elastic foundation in modeling. For this reason, different theories such as Winkler, Pasternak, and Reisner have been introduced. On the other hand, a member with a variable cross-section has a higher load-bearing capacity than a prismatic member with a larger cross-section. Nowadays, engineers use members with variable cross-sections in the design of structural members to minimize the consumption of materials, to reduce the weight of the structure, and to increase the critical buckling load capacity. Numerous studies have been conducted to investigate the critical load capacity of compressive members, particularly focusing on the stability of non-prismatic columns on elastic foundations. For the first time, Timoshenko, Wang, Bazzant, Morley, and Dinnik studied the critical buckling load of elastic columns based on closed-form solutions or numerical approximations of the governing differential equation. In these studies, researchers aimed to provide simplified relations for practical use by designers. In the present paper, the vibrations and stability of a non-prismatic beam on a Winkler-Pasternak foundation are examined. The variations in the moment of inertia and cross-sectional area of the beam are incorporated into the equation in three cases, as functions of the initial moment of inertia and initial cross-sectional area. The effects of displacement and vertical pressure of the soil are modeled through elastic springs based on the Winkler model, and shear deformations are taken into account according to the Pasternak model. To solve the governing differential equation, both the finite element method and the Rayleigh-Ritz method are employed. In the finite element method, third-order Hermite interpolation functions are used, while in the Rayleigh-Ritz method, a power series expansion is employed as the shape function. Finally, eigenvalue-solving techniques are applied to determine the response parameters, including dimensionless natural frequency and effective length factor. All calculations, including the computation of material stiffness matrices, geometric stiffness, mass, and Winkler-Pasternak foundation stiffness, were carried out using MATLAB coding.The results indicate that the simultaneous increase in the section slope coefficient, Winkler spring constant, and Pasternak shear constant under various support conditions leads to an increase in the effective length and a decrease in the beam's buckling load capacity. Additionally, the simultaneous increase in the section slope coefficient, Winkler spring constant, and Pasternak shear constant, depending on the type of variations in the moment of inertia along the member, results in either an increase or decrease in the dimensionless natural frequency.Both the finite element method and Rayleigh-Ritz method were used to solve the governing equation. The Rayleigh-Ritz method, compared to the finite element method, is more efficient. With fewer terms, the Rayleigh-Ritz method provides a more accurate calculation of the target parameters in comparison to the finite element method. All parameters involved in this study, including the section slope coefficient (β), Winkler spring constant (k_w), Pasternak shear constant (k_p), effective length factor (K), and natural frequency (ω), are dimensionless. The ninth topic (year 2020) and the tenth topic (year 2022) of Iran's national building regulations do not provide relations for analyzing the stability and vibrations of non-prismatic beams on the Winkler-Pasternak foundation. In this paper, aligned curves are used to present the results. The findings of this study serve as practical research that can be used by engineers and designers for the design of non-prismatic beams on Winkler-Pasternak foundations.
Furthermore, a practical example demonstrating aligned curves to calculate the critical buckling load capacity and natural frequency of the system is provided. The results of previous research are used for verification. There is an acceptable agreement between the present results and previous research.
Keywords
Stability
Natural Frequency
Hamilton's Principle
Finite Elements
Eigenvalue Analysis

Subjects

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  • Receive Date 08 September 2023
  • Revise Date 26 December 2023
  • Accept Date 02 January 2024