Porous Beams Research Articles

A finite element approach for forced dynamical responses of porous FG nanocomposite beams resting on viscoelastic foundations

A finite element approach for forced dynamical responses of porous FG nanocomposite beams resting on viscoelastic foundations

Free Vibration of Axially Functionally Graded Porous Beams with Varying Cross-Section

This study aims to determine the natural frequencies of axially functionally graded porous material (FGPM) beams with non-uniform cross-sections under a variety of boundary conditions. Two types of pore distribution were used: even and uneven. Initially, an analytical method was used to determine the natural frequencies of FGPM beams with different cross-sections. The Fredholm integral approach was used to obtain the characteristic equations. Furthermore, the collected results are confirmed and compared to the existing literature. The artificial neural network (ANN) technique is then used to predict the fluctuations in the natural frequency of a clamped–simply supported axially FGPM beam with a non-uniform cross-section. The prediction of natural frequencies by ANN is based on a large dataset of over 1[Formula: see text]100 data points acquired from the analytical solution obtained in this study and data available in the literature. This research conducts a parametric analysis to evaluate the influence of numerous aspects, including as beam characteristics, material properties, geometric details, gradient parameters, and porosity distribution, on functionally graded (FG) beam vibration behavior. Finally, both ANN and computational analysis demonstrate that porosity distributions and non-uniform cross-sectional area have a substantial effect on the natural frequencies of axially FG beams.

Critical lateral‐torsional buckling loads of porous orthotropic rectangular beams containing warping effects

AbstractDue to their elemental porosity, porous materials display individual mechanical properties, rendering them essential components in various engineering applications. So, this study investigates the lateral‐torsional buckling (LTB) sensitivity of porous orthotropic rectangular beams subjected to a concentrated load, including the warping effects. Porosity‐dependent material properties vary along the beam's height via four different porosity distribution patterns. The constitutive equations are obtained using the principle of virtual work based on the classical beam theory containing the warping effect, and the governing equations are solved by performing the Ritz method to obtain the critical LTB load. The porous orthotropic beams and the warping effects add complexity to the analysis. So, the influence of some parameters, such as porosity coefficients, porosity distribution patterns, orthotropy, height‐to‐width ratio, slenderness ratio, and warping coefficients, on the LTB characteristics of porous orthotropic rectangular beams are investigated in detail. The novelty of this study lies in its comprehensive LTB analysis of orthotropic rectangular beams under specific effects, such as porosity and warping factors. The findings offer valuable insights into the LTB characteristics of orthotropic porous beams, contributing to an improved comprehension of their structural behavior and facilitating the design of engineering structures. This study contributes to the composite materials and structural stability analysis, offering potential applications in many engineering domains.

Vibration analysis of functionally graded saturated porous beams using radiative energy transfer method

An energy flow model based on the radiative energy transfer method (RETM) is established for the high-frequency response prediction of finite functionally graded saturated porous beams. The Timoshenko beam theory and the Biot thoery are applied to derive the motion governing equations of the beam. The response of the beam loaded by a high-frequency point force is described by vibrational energy density and intensity. The energy response of the beam is contributed by the rays emitted from the actual source set at load point and then reflected through the fictitious sources at the boundaries. Fictitious sources are determined by the energy balance equations at the boundaries. Numerical examples show that the energy distributions of the saturated porous beams are well predicted by the proposed approach.

A novel hybrid function-based Ritz method for static and free vibration of axially loaded bi-directional functionally graded porous beams

This paper introduces a novel hybrid function, synthesizing trigonometric and Hermitian polynomials, to analyze the structural responses of bi-directional functionally graded porous (2D-FGP) beams. The displacement field of the beam is based on the higher-order shear deformation theory. A power-law relation governs the material variation along the x- and z-directions, with the porosity manifesting in three distinct distributions. The governing equations are deduced through the application of Lagrange’s equation. The proposed hybrid function is developed to delineate the displacement field. The deflections, stresses, buckling loads, and natural frequencies of 2D-FGP beams are computed for diverse gradation exponents in the x- and z-directions, boundary conditions, porosity type, porosity coefficient, and slenderness ratio. A comparative assessment with finite element method results indicates that the findings agree with available data. It substantiates the accuracy and efficiency of the proposed methodology.

The novel Vogel's approximation method integrated with a random forest algorithm in the vibration analysis of a two-directional functionally graded taper porous beam: Assessment

A functionally graded material is a class of composite materials characterized by gradual variations in composition and microstructure, which further induces the respective changes in the material properties. This study focuses on evaluating the vibration behavior of two directional functionally graded taper porous beams (FGTPB). This approach adopts a rectangular cross-section in order to deal with the challenges related to fluctuating material characteristics and geometric tapering in both thickness and width dimensions. The research employs a novel approach that merges Vogel's approximation technique with the Random Forest algorithm, an approach that has not been used in analyzing structural vibrations, establish boundary conditions and solve equations of motion. Comparative results of the suggested beam theory with the existing literature on FGTPB materials such as alumina and SUS304 at various taper, porosity, gradient and width ratios verified it. The material gradation and porosity developed a uniform pattern in the first three modes of fundamental frequencies. Higher gradient indices increased the rigidity and natural frequencies of the beams whereas the porosity index decreased the rigidity, resulting in lower natural frequencies. By combining Vogel's approximation method with machine learning techniques, the study improved vibration behavior analysis in FGTPB. The disciplines of materials and structural engineering are significantly impacted by this.

Impact of porosity distribution and grading parameters on free vibration of functionally graded beams

This study focuses on investigating the free vibration of porous functionally graded (FG) beams based on neutral surface position and the impact of porosity distribution on the properties and performance of materials. To achieve this, the study assumes that the beam’s material characteristics vary continuously along its thickness direction. The volume fraction of constituents is specified using the modified rule of the mixture, which includes porosity volume fraction with changed porosity patterns across the cross-section. The study uses the parabolic shear deformation theory without shear correction factors to derive the equations of motion by applying Hamilton’s principle. The study employs Navier’s method to obtain analytical results for the free vibration of porous FG beams. To verify the proposed formulation, the study compares the results with relevant findings from the literature. The study also examines the relationship between frequency of vibration and grading parameter in homogeneous porosity distributions, as well as the frequency of vibration in beam structures with different porosity patterns. Additionally, numerical examples are provided to investigate the impact of parameters like power-law index, span to depth ratio, porosity distribution pattern, and porosity volume fraction on the natural frequencies of FG beams.

Analysis of Free Vibration Properties of Smart Functionally Graded Porous Beam

Analysis of Free Vibration Properties of Smart Functionally Graded Porous Beam

Dynamics of axially moving spinning microbeams composed of tri-directional graded porous materials with axisymmetric cross-sections and piezoelectric layers in complex fields

The current article appraises the vibration and stability of tri-directional functionally graded porous microscale beams with rectangular cross-sections integrated with piezoelectric layers under spinning and axial movements in complex environments. The microbeam is surrounded by a three-parameter Winkler-Pasternak-Hetenyi medium, and its material characteristics are graded in thickness, width, and longitudinal spatial directions by considering non-uniform and uniform porosity models. Dynamic equations, vibration frequencies, and stability criteria of the system are determined with the aid of the Galerkin approach and Laplace transform. The Campbell diagram and stability maps are drawn. Frequency and stability analyses, as well as comparison and parametric analyses, are conducted. The impacts of piezoelectric voltage, magneto-hygro-thermal fields, axial and tangential distributed follower forces, substrate characteristics, scale parameter, aspect ratio, porosity factor, and material gradation on flutter and divergence instability boundaries are assessed in detail. It is deduced that instability regions are condensed, and the instability threshold is enhanced by fine-adjusting the porosity and material gradient. It is discovered that destructive environmental effects can be alleviated by regulating the piezoelectric voltage. In addition, compared with the case of a square cross-section, the divergence/flutter instability region of the microbeam with a rectangular cross-section is smaller/larger. The outcomes of the present research can be helpful in the design of next-generation bi-gyroscopic systems.

Investigating the bending and buckling behaviors of composite porous beams reinforced with carbon nanotubes and graphene platelets using a TRPIM path following mesh-free approach

The aim of the present work consists in investigating the nonlinear behavior of porous beams reinforced with graphene platelets (GPL) and supported carbon nanotubes (CNT), termed functionally graded graphene platelets reinforced composite beam (FG-GPLRC) and functionally graded nanotube carbon reinforced composite beam (FG-CNTRC), respectively. Notably, the distribution of GPL/CNT is explored in both uniform and non-uniform patterns across the beam's thickness. What sets this research apart is its utilization of a refined beam model as enhanced FSDT incorporating nonlinear shear terms which is a crucial advancement in accurately capturing the post-buckling response in certain boundary conditions, a feature lacking in the existing FSDT literature. Innovatively, the post-buckling load-deflection relationship is derived through the solution of governing equations incorporating cubic nonlinearity. This is achieved by employing Galerkin's method alongside a non-iterative high-order continuation technique based on the asymptotic numerical method coupled with the Tchebychev-radial point interpolation method (TRPIM), using a path-following where the solutions are obtained branch-by-branch by eliminating the need for iterative processes. In essence, this research underscores the pivotal role of porosity and GPL/CNT reinforcement in shaping the post-buckling configuration of both perfect and imperfect nanocomposite beams, thereby advancing our understanding of structural behavior in porous nanocomposite materials. The findings of this study illuminate the significant influence of parameters such as porosity coefficient, porosity distribution, GPL/CNT distribution, and GPL-weight/CNT-volume fraction on the nonlinear buckling behavior of porous beams.

Analysis of Elastic Buckling and Static Bending Properties of Smart Functionally Graded Porous Beam

Background: A Smart Functionally Graded (SFG) porous beam is a Functionally Graded (FG) Porous beam consisting of a piezo-electric layer integrated on the top layer. Aim: This research work addresses the lack of information by examining the bending as well as elastic buckling performance of SFG beams having two distinct Porosity Distributions (PDs). The main purpose of this research work is to study and analyze the bending deflections as well as CBLs of SFG beams with two different PDs, considering various boundary conditions, voltage levels (20V and 100V), and changes in slenderness ratio. Objective: The objectives of this work are as follows: to analyze the impact of variations in voltage levels and slenderness ratios on the critical buckling and bending properties of the SFG beam and to showcase the effect of variation in the slenderness ratio on the dimensionless normal stress through the thickness of the Hinge-Hinge beam. Method: The research work analyzes the elastic buckling as well as static bending of Smart Functionally Graded (SFG) porous beams, considering the equations derived from the Timoshenko beam theory. To simulate the results and analyze the various effects, the ANSYS software has been utilized in this paper. Results: This research work examines how the slenderness ratio impacts the maximum deflection, CBL, along with stress distribution. Experimental data demonstrates that as the slenderness ratio increases, CBL reduces, and maximum deflections in SFG porous beams increase. Also, it has been observed that normal stress distribution shifts from linear to non-linear and changes significantly. Further, the PDs significantly affect the static bending as well as the buckling performance of the beam. The symmetric distribution pattern provides superior buckling capability and enhanced bending resistance compared to the unsymmetric distribution pattern. Additionally, it has been found that as the voltage across the SFG increases, the buckling load increases and the deflection of the beam decreases. Conclusion: This research work has analyzed the effects of slenderness ratio and voltage level on the Critical Buckling Load (CBL) and bending properties of SFG porous beams, considering four different boundary conditions and a fixed set of parameters. The key findings of this paper are that as the slenderness ratio increases, the CBL decreases, and distribution shifts from linear to nonlinear region. Changes are significant, whereas maximum deflection increases. A significant effect is observed in the performance of static bending and buckling of SFG beams. It has been investigated that with an increase in voltage across the SFG beam, the buckling load increases, whereas the maximum deflection of the beam decreases.

Wave propagation in bidirectional functionally graded tapered beams incorporating porosity effect

This study addresses wave propagation in beams made of bidirectional functionally graded materials (BDFG) and having nonuniform cross section by using a quasi-3D analytical solution. For the first time, this aspect will be studied. The mechanical characteristics of the beams are supposed to be variable in both axial and transvers direction according to a specific law depending on the porosity. The mathematical formulation used is based on a displacement field containing indeterminate terms and requiring a few variables to be determined. The thickness and width of the beams are assumed to be linearly variable in the longitudinal direction. The equations governing the simply supported beams are obtained by applying the principle of virtual work and are then analytically solved to obtain the phase velocities and wave frequencies. In addition, the validation results reveal excellent concordance of the proposed theory with those given in the literature. Then, a detailed parametric study is carried out to investigate the influence of the several geometrical and material parameters on the wave propagation in BDFG tapered beams. It is observed that these parameters have significant impact on the wave propagation in BDFG tapered porous beams. These results can be used as benchmark solutions for future studies.

Vibration Analysis of Porous Cu-Si Microcantilever Beams in Fluids Based on Modified Couple Stress Theory.

The vibrations in functionally graded porous Cu-Si microcantilever beams are investigated based on physical neutral plane theory, modified coupled stress theory, and scale distribution theory (MCST&SDT). Porous microcantilever beams define four pore distributions. Considering the physical neutral plane theory, the material properties of the beams are computed through four different power-law distributions. The material properties of microcantilever beams are corrected by scale effects based on modified coupled stress theory. Considering the fluid driving force, the amplitude-frequency response spectra and resonant frequencies of the porous microcantilever beam in three different fluids are obtained based on the Euler-Bernoulli beam theory. The quality factors of porous microcantilever beams in three different fluids are derived by estimating the equation. The computational analysis shows that the presence of pores in microcantilever beams leads to a decrease in Young's modulus. Different pore distributions affect the material properties to different degrees. The gain effect of the scale effect is weakened, but the one-dimensional temperature field and amplitude-frequency response spectra show an increasing trend. The quality factor is decreased by porosity, and the degree of influence of porosity increases as the beam thickness increases. The gradient factor n has a greater effect on the resonant frequency. The effect of porosity on the resonant frequency is negatively correlated when the gradient factor is small (n<1) but positively correlated when the gradient factor is large (n>1).

Examination of Beam Theories for Buckling and Free Vibration of Functionally Graded Porous Beams.

This paper examines the accuracy and effectiveness of various beam theories in predicting the critical buckling loads and fundamental frequencies of functionally graded porous (FGP) beams whose material properties change continuously across the thickness. The beam theories considered are classical beam theory (CBT), first-order shear deformation beam theory (FSDBT), third-order shear deformation beam theory (TSDBT), and the broken-line hypothesis-based shear deformation beam theory (BSDBT). Governing equations for those beam theories are formulated by using the Hamilton's principle and are then solved by means of the generalised differential quadrature method. Finite element simulation solutions are provided as reference results to assess the predictions of those beam theories. Comprehensive numerical results are presented to evaluate the influences of the porosity distribution and coefficient, slenderness ratio, and boundary condition on the difference between theoretical predictions and simulation results. It is found that the differences significantly increase as the porosity coefficient rises, and this effect becomes more noticeable for the rigid beam with a smaller slenderness ratio. Nonetheless, the results produced by the BSDBT are always the closest to simulation ones. The findings in this paper will contribute to the establishment of more refined theories for the mechanical analysis of FGP structures.

Modal characteristics of functionally graded porous Timoshenko beams with variable cross-sections

The study focuses on the free vibration analysis of beams composed of functionally graded porous materials and characterized by a variable cross-section along their length. A broad range of beams is examined encompassing various tapered configurations, porosity profiles, and porosity content. The equations of motion are derived using Hamilton’s principle within the framework of Timoshenko beam theory. These equations are solved semi-analytically using the differential transform method, which has been adapted to incorporate various boundary conditions such as clamped–clamped, clamped–free, clamped–pinned, and pinned–pinned constraints within a general formulation of the beam eigenvalue problem. To validate the proposed solution technique, computed natural frequencies are compared with existing literature results for tapered inhomogeneous beams and uniform porous beams. Notably, new results are obtained for tapered porous beams. In this regard, a comprehensive parametric study explores the influence of various factors on the natural frequencies and mode shapes of functionally graded porous beams with variable cross-sections. These factors include the type of porosity profiles, a range of porosity parameters, cross-section taper ratios, and specific boundary conditions. The findings deepen our understanding of the modal characteristics of functionally graded porous beams, providing valuable guidance for engineering design and structural optimization in relevant applications. Additionally, they may serve as benchmarks for other researchers.

Vibration and buckling analysis of porous metal foam thin-walled beams with closed section

This paper investigates the vibration and buckling analysis of porous metal foam thin-walled box beams. These beams exhibit a unique structural configuration with symmetrical and asymmetrical porosity distributions along their wall thickness, thereby altering the effective mechanical properties. The first-order shear deformable beam theory is employed and the governing equations are derived using the Hamilton’s principle. Numerical results are presented for porous metal foam thin-walled box beams under simply-supported, clamped–clamped and clamped-free boundary conditions. The effects of various porosity parameters, length-to-side and side-to-wall-thickness ratios on the beams’ performance are also examined. A comprehensive comparison between the porous metal foam thin-walled box beams and their counterparts in the form of equivalent homogeneous are presented.

On the Buckling Response of Functionally Graded Carbon Nanotube-reinforced Composite Imperfect Beams

The current inquiry aims to scrutinize the porosity's effect on the buckling response of carbon nanotube reinforced composite (CNTRC) imperfect beams. The developed theories account for higher-order variation of transverse shear strain through the depth of the beam and satisfy the stress-free boundary conditions on the top and bottom surfaces of the beam. Single-walled carbon nanotubes (SWCNTs) are distributed and aligned in a polymeric matrix with various reinforcement patterns to create CNTRC porous beams. The material properties of the functionally graded beam determined using the mixture rule are assumed to vary according to the power law distribution of the volume fraction of the constituents. The mathematical models presented in this study are validated through numerical comparison with existing results. The new buckling results of carbon nanotube-reinforced porous beams are analyzed considering the influence of several parameters, including volume fraction, aspect ratios, degree of porosity, and types of reinforcement. The results stipulate that the above parameters play a significant role in the critical buckling load variation. It is proclaimed that the critical buckling load dwindled as porosity increased and that the X-CNT reinforced beam has a high resistance against buckling compared to other reinforcement types.

On the vibrational behavior of variable thickness FG porous beams with graphene-reinforced nanocomposite facesheets

On the vibrational behavior of variable thickness FG porous beams with graphene-reinforced nanocomposite facesheets

Peridynamic differential operator for stress analysis of imperfect functionally graded porous sandwich beams based on refined zigzag theory

This study focuses on the stress analysis of imperfect functionally graded porous (FGP) sandwich beams using the Peridynamic Differential Operator (PDDO) and Refined Zigzag Theory (RZT). Functionally graded materials (FGMs) can be found in diverse engineering applications since they offer smooth transitions in the mechanical properties of distinct materials, unlike traditional composite materials. Micro-voids and porosities may appear inside the FGMs due to technical challenges during the manufacturing process of such materials. Therefore, understanding the stress variations of the FG sandwich beams with porosities/micro-voids, which are called imperfect FGMs, under loads is of vital importance. In order to achieve the porous configuration in the FG structures, the modified rule of mixtures is applied to calculate the effective material properties of FGMs. The RZT is very suitable for stress analysis of laminated composites, especially for thick and moderately thick beams. It contains only four kinematic variables and eliminates the use of the shear correction factors. In this study, the equilibrium equations of the RZT are solved by means of the PDDO for the stress analysis of imperfect FG sandwich beams having uniform and non-uniform porosity distributions. The PDDO transforms the differential equations into their nonlocal form, namely integral form, providing highly accurate approximations of the derivatives. A comprehensive PDDO analysis is performed for the investigation of the influence of the material variation, and even and uneven porosity distributions on the stress variations of the imperfect FG sandwich beams. It is noted that when the FG core exhibits soft material distributions, the effect of porosity is evident. The even type porosity distribution pattern is more influential on the axial displacement and stress variations in comparison with those of the uneven type porosity distribution pattern.

An enhanced quasi-3D HSDT for free vibration analysis of porous FG-CNT beams on a new concept of orthotropic VE-foundations

The original primary objective of this study is to conduct a comprehensive investigation into the free vibration behavior of beams with porosity, reinforced by carbon nanotubes (CNTs). These beams are supported by an arbitrary orthotropic variable elastic foundation (AOVEF). The focus lies on understanding how the presence of porosity and CNT reinforcement, coupled with the complex support provided by the AOVEF, influences the vibration characteristics of the beams. In addition, we intend to examine the impact of a number of micromechanical models on the vibration properties of these beams. Four carbon nanotube variables are introduced to symbolize several CNT variations. CNTs’ mechanical properties are examined through a variety of micromechanical models. Shear deformation effects are considered into account in the framework of higher-order shear deformation (HSDT) beam theory. The equations of motion are constructed using Hamilton’s concept and the equation system for the FG-CNT beam with supported ends is solved via the Navier methodology. A new approach in elastic foundation modeling is the Arbitrarily Orthotropic Variable Elastic (AOP-VE) foundation. It builds on the Winkler layer concept but introduces the idea of a variable response along the length of the beam foundation. Unlike the Winkler-Pasternak model, AOP-VE allows for control over the directional properties of the Pasternak foundation. The study looks into multiple aspects, involving various distribution kinds of CNTs, the impact of Winkler and Pasternak factors, mode values, side-to-length ratio, porosity, and angle variation. The results indicate that these parameters have a major effect on the inherent vibration features of FG-CNT beams. The study results in various novel findings that improve the understanding of the subject topic and set a standard for future studies.