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In this paper, the standard Peridynamic Timoshenko beam model accounting for the shear deformation is chosen to describe the thick beam kinematics. Unfortunately, when applied to very thin beam structures, the standard Peridynamics (PD) encounters the shear locking phenomenon, leading to incorrect solutions.

PD differs from classical continuum mechanics and other nonlocal theories that do not involve spatial derivatives of the displacement field. PD is based on the integral equation instead of differential equations to handle discontinuities and other singularities.

The shear locking can be successfully alleviated using the developed selective integration method. In particular, this technique has been implemented in the standard PD, which allows an accurate result for a wide range of slenderness from very thin to thick (10 < L/t < 10³) structures. It can also accelerate the computational time for particular dynamic problems using fewer neighboring integration particles. Several numerical examples are solved to demonstrate the effectiveness of the proposed method for modeling beam structures.

The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam available in the literature, especially for very thin structures. A new alternative for the alleviation of shear locking in the Peridynamic Timoshenko beam, using selective integration. Hence the developed Peridynamic Timoshenko beam model is effective for thin and thick structures. A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

1. The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam proposed in the literature, especially for very thin structures.

2. The developed Peridynamic Timoshenko beam model based on selective integration is effective for thin and thick structures.

3. A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

The paper highlights the severe shear locking phenomenon in the Peridynamic Timoshenko beam proposed in the literature, especially for very thin structures.

The developed Peridynamic Timoshenko beam model based on selective integration is effective for thin and thick structures.

A new peridynamic formulation for the low-velocity impact beam models is presented and validated.

This study aims to perform dynamic response analysis of damaged rigid-frame bridges under multiple moving loads using analytical based transfer matrix method (TMM). The effects of crack depth, moving load velocity and damping on the dynamic response of the model are discussed. The dynamic amplifications are investigated for various damage scenarios in addition to displacement time-histories.

Timoshenko beam theory (TBT) and Rayleigh-Love bar theory (RLBT) are used for bending and axial vibrations, respectively. The cracks are modeled using rotational and extensional springs. The structure is simplified into an equivalent single degree of freedom (SDOF) system using exact mode shapes to perform forced vibration analysis according to moving load convoy.

The results are compared to experimental data from literature for different damaged beam under moving load scenarios where a good agreement is observed. The proposed approach is also verified using the results from previous studies for free vibration analysis of cracked frames as well as dynamic response of cracked beams subjected to moving load. The importance of using TBT and RLBT instead of Euler–Bernoulli beam theory (EBT) and classical bar theory (CBT) is revealed. The results show that peak dynamic response at mid-span of the beam is more sensitive to crack length when compared to moving load velocity and damping properties.

The combination of TMM and modal superposition is presented for dynamic response analysis of damaged rigid-frame bridges subjected to moving convoy loading. The effectiveness of transfer matrix formulations for the free vibration analysis of this model shows that proposed approach may be extended to free and forced vibration analysis of more complicated structures such as rigid-frame bridges supported by piles and having multiple cracks.

Among different types of engineering structures, plates play a significant role. Their analysis necessitates numerical modeling with finite elements, such as triangular, quadrangular or sector plate elements, owing to the intricate geometrical shapes and applied loads. The scope of this study is the development of a new rectangular finite element for thin plate bending based on the strain approach using Airy's function. It is called a rectangular plate finite element using Airy function (RPFEUAF) and has four nodes. Each node had three degrees of freedom: one transverse displacement (w) and two normal rotations (x, y).

Equilibrium conditions are used to generate the interpolation functions for the fields of strain, displacements and stresses. The evolution of the Airy function solutions yielded the selection of these polynomial bi-harmonic functions. The variational principle and the analytical integration approach are used to evaluate the basic stiffness matrix.

The numerical findings for thin plates quickly approach the Kirchhoff solution. The results obtained are compared to the analytical solution based on Kirchhoff theory.

The efficiency of the strain based approach using Airy's function is confirmed, and the robustness of the presented element RPFEUAF is demonstrated. Because of this, the current element is more reliable, better suited for computations and especially intriguing for modeling this kind of structure.

Hoping to uncover the physical principles of the vibration of the functionally graded material (FGM) microplate, by which the authors can make contributions to the design and manufacturing process in factories like micro-electro-mechanical system (MEMS) and other industries.

The authors design a method by establishing a reasonable mathematical model of the physical microplate composed of a porous FGM.

The authors discover that the porosity, the distributions of porosity, the power law of the FGM and the length-to-thickness ratio all affect the natural frequency of the vibration of the microplate, but in different ways.

Originally proposed a model of the micro FGM plate considering the different distributions of the porosity and scale effect and analyzed the vibration frequency of it.

A unified framework for solving the bending, buckling and vibration problems of rectangular thin plates (RTPs) with four free edges (FFFF), including isotropic RTPs, orthotropic rectangular thin plates (ORTPs) and nano-rectangular plates, is established by using the symplectic superposition method (SSM).

The original fourth-order partial differential equation is first rewritten into Hamiltonian system. The class of boundary value problems of the original equation is decomposed into three subproblems, and each subproblem is given the corresponding symplectic eigenvalues and symplectic eigenvectors by using the separation variable method in Hamiltonian system. The symplectic orthogonality and completeness of symplectic eigen-vectors are proved. Then, the symplectic eigenvector expansion method is applied to solve the each subproblem. Then, the symplectic superposition solution of the boundary value problem of the original fourth-order partial differential equation is given through superposing analytical solutions of three foundation plates.

The bending, vibration and buckling problems of the rectangular nano-plate/isotropic rectangular thin plate/orthotropic rectangular thin plate with FFFF can be solved by the unified symplectic superposition solution respectively.

The symplectic superposition solution obtained is a reference solution to verify the feasibility of other methods. At the same time, it can be used for parameter analysis to deeply understand the mechanical behavior of related RTPs. The advantages of this method are as follows: (1) It provides a systematic framework for solving the boundary value problem of a class of fourth-order partial differential equations. It is expected to solve more complicated boundary value problems of partial differential equations. (2) SSM uses series expansion of symplectic eigenvectors to accurately describe the solution. Moreover, symplectic eigenvectors are orthogonal and directly reflect the orthogonal relationship of vibration modes. (3) The SSM can be carried to bending, buckling and free vibration problems of the same plate with other boundary conditions.

In the current manuscript, the authors examine the Belleville spring with the variable thickness. The thickness is assumed to be variable along the meridional and parallel coordinates of conical coordinate system. The calculation of the Belleville springs includes the cases of the free gliding edges and the edges on cylindric curbs, which constrain the radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.

In the current manuscript, the authors examine the Belleville spring with the variable thickness. The calculation of the Belleville springs investigates the free gliding edges and the edges on cylindric curbs with the constrained radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations.

The developed equations demonstrate that the shift of the inversion point to the inside edge does not influence the bending of the cone. On the contrary, the character of the extensional deformation (circumferential strain) of the middle surface alternates significantly. The extension of the middle surface of free gliding spring occurs outside the inversion. The middle surface of the free gliding spring squeezes inside the inversion point. Contrarily, the complete middle surface of the disk spring on the cylindric curb extends. This behavior influences considerably the function of the spring.

A slotted disk spring consists of two segments: a disk segment and a number of lever arm segments. Currently, the calculation of slotted disk spring is based on the SAE formula (SAE, 1996). This formula is limited to a straight slotted disk spring with freely gliding inner and outer edges.

The equations developed here are based on common assumptions and are simple enough to be applied to the industrial calculations. The developed method is applicable for disk springs with radially constrained edges. The vertical displacements of a disk spring result from an axial load uniformly distributed on inner and outer edges. The method could be directly applied for calculation of slotted disk springs.

The nonlinear governing equations for the of Belleville spring centres were derived. The equations describe the deformation and stresses of thin and moderately thick washers. The variation method is applicable for the disc springs with free gliding and rigidly constrained edges. The developed method is applicable for Belleville spring with radially constrained edges. The vertical displacements of a disc spring result from an axial load uniformly distributed on inner and outer edges.

The purpose is to explore the free vibration behaviour of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The goal of this study is to gain a better knowledge of the dynamic response of nanoscale structures made of functionally graded materials and porous features. The Rayleigh-Ritz approach is used in this study to generate realistic mathematical models that take elastic foundation support into account. This research can contribute to the design and optimization of advanced nanomaterials with potential applications in engineering and technology by providing insights into the influence of material composition, porosity and foundation support on the vibrational properties of nanoplates.

A systematic methodology is proposed to evaluate the free vibration characteristics of elastic foundation-supported porous functionally graded nanoplates using the Rayleigh-Ritz approach. The study began by developing the mathematical model, adding material properties and establishing governing equations using the Rayleigh-Ritz approach. Numerical approaches to solve the problem are used, using finite element methods. The results are compared to current solutions or experimental data to validate the process. The results are also analysed, keeping the influence of factors on vibration characteristics in mind. The findings are summarized and avenues for future research are suggested, ensuring a robust investigation within the constraints.

The Rayleigh-Ritz technique is used to investigate the free vibration properties of elastic foundation-supported porous functionally graded nanoplates. The findings show that differences in material composition, porosity and foundation support have a significant impact on the vibrational behaviour of nanoplates. The Rayleigh-Ritz approach is good at modelling and predicting these properties. Furthermore, the study emphasizes the possibility of customizing nanoplate qualities to optimize certain vibrational responses, providing useful insights for engineering applications. These findings expand understanding of dynamic behaviours in nanoscale structures, making it easier to build innovative materials with specific features for a wide range of industrial applications.

The novel aspect of this research is the incorporation of elastic foundation support, porous structures and functionally graded materials into the setting of nanoplate free vibrations, utilizing the Rayleigh-Ritz technique. Few research have looked into this complex combo. By tackling complicated interactions, the research pushes boundaries, providing a unique insight into the dynamic behaviour of nanoscale objects. This novel approach allows for a better understanding of the interconnected effects of material composition, porosity and foundation support on free vibrations, paving the way for the development of tailored nanomaterials with specific vibrational properties for advanced engineering and technology applications.

This study addresses the reliability of a composite fiber (carbon fibers/epoxy matrix) at microscopic level, with a specific focus on its behavior under compressive stresses. The primary goal is to investigate the factors that influence the reliability of the composite, specifically considering the effects of initial fiber deformation and fiber volume fraction.

The analysis involves a multi-step approach. Initially, micromechanics theory is employed to derive limit state equations that define the stress levels at which the fiber remains within an acceptable range of deformation. To assess the composite's structural reliability, a dedicated code is developed using the Monte Carlo method, incorporating random variables.

Results highlight the significance of initial fiber deformation and volume fraction on the composite's reliability. They indicate that the level of initial deformation of the fibers plays a crucial role in determining the composite reliability. A fiber with 0.5% initial deformation exhibits the ability to endure up to 28% additional stress compared to a fiber with 1% initial deformation. Conversely, a higher fiber volume fraction contributes positively to the composite's reliability. A composite with 60% fiber content and 0.5% initial deformation can support up to 40% additional stress compared to a composite containing 40% fibers with the same deformation.

The study's originality lies in its comprehensive exploration of the factors affecting the reliability of carbon fiber-epoxy matrix composites under compressive stresses. The integration of micromechanics theory and the Monte Carlo method for structural reliability analysis contributes to a thorough understanding of the composite's behavior. The findings shed light on the critical roles played by initial fiber deformation and fiber volume fraction in determining the overall reliability of the composite. Additionally, the study underscores the importance of careful fiber placement during the manufacturing process and emphasizes the role of volume fraction in ensuring the final product's reliability.

The purpose of this paper is to develop an organized structure for damage detection of a cracked cantilever beam using finite element method and experimental method technique.

Due to presence of cracks the dynamic characteristics of structure change. The change in dynamic behavior has been used as one of the criteria of fault diagnosis for structures. Major characteristics of the structure which undergo change due to presence of crack are: natural frequencies, the amplitude responses due to vibration and the mode shapes. Therefore, an attempt has been made to formulate a smart technique for minimizing the amplitude of vibration for crack cantilever beam structures. In the analysis both single and double cracks are taken into account.

The results of the active vibration control experiments proved that piezoelectric sensor/actuator pair is an effective sensor and actuator configuration for active vibration control to reduce the amplitude of vibration for closed-loop system.

It is necessary that structures must safely work during its service life, but damages initiate a breakdown period on the structures which directly affect the industrial growth. It is a recognized fact that dynamic behavior of structures changes due to presence of crack. It has been observed that the presence of cracks in structures or in machine members leads to operational problem as well as premature failure.

The purpose of this study is to numerically model the rigid pavement resting on two-parameter soil and to examine its modal parameters.

This study is carried out using a one-dimensional beam element with three rotational and three translational degrees of freedom based on the finite element method. MATLAB programming is used to perform the free vibration analysis of the rigid pavement.

Cyclic frequency and their corresponding mode shapes were determined. It has been investigated how cyclic frequency changes as a result of variations in the thickness, span length of pavement, shear modulus, modulus of subgrade, different boundary conditions and element discretization. Thickness of the pavement and span length has greater effect on the cyclic frequency. Maximum increase of 29.7% is found on increasing the thickness, whereas the cyclic frequency decreases by 63.49% on increasing span length of pavement.

The pavement's free vibration is the sole subject of the current investigation. This study limits for the preliminary design phase of rigid pavements, where a complete three-dimensional finite element analysis is unnecessary. The current approach can be extended to future research using a different method, such as finite element grilling technique, mesh-free technique on reinforced concrete pavements or jointed concrete pavements.

The finite element approach adopted in this paper involves six degrees of freedom for each node. Furthermore, to the best of the authors’ knowledge, no prior study has done seven separate parametric investigations on the modal analysis of rigid pavement resting on two-parameter soil.