Higher-Order Finite Element Vibration Analysis of Circular Raft on Winkler Foundation

Application of higher order finite element to the analysis of thin solid circular plates on elastic foundation using the modified Vlasov model in polar coordinate system is attempted. A Matlab code has been written for present formulation that can handle any type of loading and boundary conditions. Validation of the code is carried out after the convergence study. The results are compared to those obtained by other researchers. It is concluded that the present formulation behaves extremely well for thin solid circular plate resting on elastic foundations with good convergence rate and accuracy. The present plate bending element resting on elastic foundation using a higher order finite element is simple and requires less computational efforts, resources and time.

Skew-plate is unavoidable in aerospace, civil, and mechanical engineering. It is challenging to ascertain how significant the dynamic response on a skew plate on an elastic foundation is. This study uses a higher-order finite element method and the modified Vlasov model to investigate the vibration of a four-nodded skewed plate on an elastic foundation. A Matlab algorithm has been written to tackle the boundary conditions within the present formulation. The code undergoes validation through convergence studies. The findings demonstrate a significant level of concordance with previous research. Additionally, a parametric analysis provided tabular and graphical representations of the first ten natural frequency characteristics. Based on the study, the present method is straightforward and excellent for skew plates on Vlasov foundations. It has a reasonable convergence rate, is precise, and requires little computation time and effort. Numerical results of free vibration analysis of skewed plates resting on elastic foundations for various support conditions will demonstrate the effectiveness of the present elements to provide multiple results and serve as a handy reference for future practitioners and design engineers in this field.

Transverse vibration of thin solid and annular circular plate with attached discrete masses

This paper is devoted to the behavior of an axially compressed two-part beamr esting on elastic foundation. The beam was composed of two parts: rigid homogeneous part and sandwich three-layered part. The analytical model of the beam on the elastic foundation has been prepared. Winkler model has been assumed for the foundation description. Moreover, original function of deflection has been introduced and employed by the authors. The critical loads as a function of geometric and mechanical properties of the two-part beam and the parameters of Winkler's foundation were calculated. In addition, numerical analysis has been performed. Sample analytical and numerical calculations have been carried out, demonstrating good compatibility between the results attained with both models. The numerical analysis included the critical loads calculations for the beam with and without the elastic foundation. In both cases, the difference between analytical and numerical values of the critical loads did not exceed 8%.

Static analysis of four-nodded thin rectangular plate element based on Kirchhoff theory resting on Pasternak foundation. All the deformation stiffness matrix of plate and subsoil are evaluated using finite element method. A MATLAB code is developed for present formulation then convergence study is carried out, then validation done and then the static analysis of thin plates resting on Pasternak type foundations. The results, thus obtained, are compared, with the available results obtained by other researchers. Parametric study is done and the maximum deflection, bending moment are presented in tabular and graphical forms. It is concluded that the effect of the soil coefficient on the static analysis of the plates on elastic foundation is generally larger than that of the aspect ratio. It behaves extremely well for thin plate, results are very close to exact solution and convergence rate is high.KeywordsPasternak foundationFinite elementShear parameterSub-grade reaction

Response of an infinite beam on a bilinear elastic foundation: Bridging the gap between the Winkler and tensionless foundation models

The Impact of the Cut-Out Shape on the Dynamic Behavior of Composite Thin Circular Plates

Free vibration analyses of beams on elastic foundation are very common in civil and mechanical engineering. Such types of structures are exposed to dynamic loads is a complex soil-structure interaction problem. In the present study for safe and economical design of such structures, a workable approach for free vibration analysis of beams on Winkler foundation using first-order continuity (C1) two degree of freedom (DOF) per node three nodded beam based on Euler-Bernoulli beam theory (EBBT) is attempted. A Matlab code is developed for the present formulation. The results, thus obtained, are compared with similar studies done by other researchers as well as with exact solution where applicable, which show very good conformity and a maximum difference 0.24% with exact solution for mode eight. It is concluded that the present formulation has rapid convergence regardless of boundary conditions, depth to length ratio of beam and modulus of sub-grade reaction. It performs extremely well for thin beams in terms of ease and consistency and gives a very accurate result with only few elements and within few seconds.

Vibration characteristics of porous FGM plate with variable thickness resting on Pasternak's foundation

The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation

Non-linear vibration and postbuckling of isotrophic thin circular plates on elastic foundations

Applying the ultrasonic machining in gear honing can improve honing speed, reduce cutting force, and avoid blocking. There are two problems leading to the decrease of calculation accuracy in the traditional nonresonant theory of the ultrasonic gear honing. One is that one‐dimensional longitudinal vibration theory and two‐dimensional theory cannot reflect the vibration characteristics of ultrasonic horn and gear comprehensively. And, the other one is that the difference of the analysis dimension between the two theories leads to mismatch of the coupling condition dimension between ultrasonic horn and gear. A free vibration analysis through Chebyshev–Ritz method based on three‐dimensional elasticity theory was presented to analyze the eigenfrequencies of the horn‐gear system in ultrasonic gear honing. In the method, the model of the horn‐gear system was divided into four parts: a solid circular plate, an annular plate, a solid cylinder, and a cone with hole. The eigenvalue equations were derived by using displacement coupling condition between each part under completely free boundary condition. It was found that the eigenfrequencies were highly convergent through convergence study. The hammering method for a modal experiment was used to test the horn‐gear systems’ eigenfrequencies. And, the finite element method was also applied to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, it showed that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of the horn‐gear system in ultrasonic gear honing.

Purpose – The purpose of this paper is to acquire strict upper and lower bounds on quantities of slender beams on Winkler foundation in finite element analysis. Design/methodology/approach – It leans on the dual analysis wherein the constitutive relation error (CRE) is used to perform goal-oriented error estimation. Due to the coupling of the displacement field and the stress field in the equilibrium equations of the beam, the prolongation condition for the stress field which is the key ingredient of CRE estimation is not directly applicable. To circumvent this difficulty, an approximate problem and the solution thereof are introduced, enabling the CRE estimation to proceed. It is shown that the strict bounding property for CRE estimation is preserved and strict bounds of quantities of the beam are obtainable thereafter. Findings – Numerical examples are presented to validate the strict upper and lower bounds for quantities of beams on elastic foundation by dual analysis. Research limitations/implications – This paper deals with one-dimensional (1D) beams on Winkler foundation. Nevertheless, the present work can be naturally extended to analysis of shells and 2D and 3D reaction-diffusion problems for future research. Originality/value – CRE estimation is extended to analysis of beams on elastic foundation by a decoupling strategy; strict upper bounds of global energy norm error for beams on elastic foundation are obtained; strict bounds of quantities for beams on elastic foundation are also obtained; unified representation and corresponding dual analysis of various quantities of the beam are presented; rigorous derivation of admissible stresses for beams is given.

This research, deals with the linear elastic behavior of curved thin beams resting on Winkler foundation with both compressional and tangential resistances. Thin beam theory is extended to include the effect of curvature and externally distributed moments under static conditions. The computer program (CBFFD) coded in fortran_77 is developed to analyze curved thin beams on Winkler foundation by Fourier series and finite difference methods. The results from these methods are plotted with other solutions to compare and check the accuracy of the used methods. INTRODUCION The object of this research is to analyze curved thin beam using finite difference and Fourier series methods. The beam is resting on elastic foundation with Winkler frictional and compresional resistances, and loaded generally (both transverse distributed load and distributed moment). The linear elastic behavior of curved thin beams on elastic foundations is considered. The governing differential equation of curved thin beams (in terms of w only) is developed and converted into finite differences. A computer program in (Fortran language) is developed. This program assembles the finite difference equations to obtain a system of simultaneous algebraic equations and then the solution is obtained by using Gauss elimination method. The deflections and rotations for each node are obtained. The shear and moment are obtained by simple substitutions of the deflections into the finite difference equations of moment and shear. Also, this program used Fourier series method to solve the governing differential equation for simply supported beam and obtain the deflection, moment and shear. The obtained solutions compared with available results to check the accuracy of the used methods. Curved beams are one-dimensional structural elements that can sustain transverse loads by the development of bending, twisting and shearing resistances in the transverse sections of the beam. It's extensively used in engineering and other fields since such beams have many practical applications. The curved beam elements on elastic foundation would be helpful for the analysis of ring foundation of structures such as antennas, water towers structures, transmission towers and various other possible structures and superstructures. These are review of early studies on curved beam.

In this paper, the static analysis of functionally graded (FG) circular plates resting on linear elastic foundation with various edge conditions is carried out by using a semi-analytical approach. The governing differential equations are derived based on the three dimensional theory of elasticity and assuming that the mechanical properties of the material vary exponentially along the thickness direction and Poisson’s ratio remains constant. The solution is obtained by employing the state space method (SSM) to express exactly the plate behavior along the graded direction and the one dimensional differential quadrature method (DQM) to approximate the radial variations of the parameters. The effects of different parameters (e.g., material property gradient index, elastic foundation coefficients, the surfaces conditions (hard or soft surface of the plate on foundation), plate geometric parameters and edges condition) on the deformation and stress distributions of the FG circular plates are investigated.