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The purpose of this paper is to analyze the frequency and response of hexagonal cell honeycomb structures under Hamilton system.

Taking orthotropic sandwich cylinder as the analytical model, the basic equilibrium equations are transformed into Hamilton system, where the canonical transformation, the extended Wittrick‐Williams algorithm and the precise integration method can be applied to calculate the frequency and the responses of the honeycomb sandwich structures.

For the cellular structures, the basic frequency is the most important which can be affected greatly by the wave number. It is also found that the displacement mode shape is dominated by the radial displacement and the axial principal stress is much higher than that of the radial or the circumferential principal stress for the sandwich cylinders.

A new solution procedure is proposed for the cellular structures by constructing the Hamilton matrix in the cylindrical coordinates. The analysis system is thus transformed from Lagrange to Hamilton.

The purpose of this paper is to provide an efficient numerical solution for dynamic properties of sandwich tubes with honeycomb cores and investigate the effects of material distribution and relative density on the dynamic properties of the structure.

By introducing dual variables and applying the variational principle, the canonical equations of Hamiltonian system are constructed. The precise integration algorithm and extended Wittrick-Williams algorithm are adopted to solve the equations and obtain the dispersion relations of sandwich tubes. The effects of the material distribution and the relative density on the non-dimensional frequencies of the sandwich tubes are investigated.

The validity of the procedure and programs is verified by comparing with other works. Dispersion relations of the typical sandwich tubes are obtained. Dramatic differences are observed as the material distribution and relative density of the sandwich structures vary.

The work gains insight into the role of symplectic analysis in the structural dynamic properties and expects to provide new opportunities for the optimal design of sandwich tubes with honeycomb cores in engineering applications.

In this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.

An adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.

The results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.

The proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

This study aims to develop an adaptive finite element method for structural eigenproblems of cracked Euler–Bernoulli beams via the superconvergent patch recovery displacement technique. This research comprises the numerical algorithm and experimental results for free vibration problems (forward eigenproblems) and damage detection problems (inverse eigenproblems).

The weakened properties analogy is used to describe cracks in this model. The adaptive strategy proposed in this paper provides accurate, efficient and reliable eigensolutions of frequency and mode (i.e. eigenpairs as eigenvalue and eigenfunction) for Euler–Bernoulli beams with multiple cracks. Based on the frequency measurement method for damage detection, using the difference between the actual and computed frequencies of cracked beams, the inverse eigenproblems are solved iteratively for identifying the residuals of locations and sizes of the cracks by the Newton–Raphson iteration technique. In the crack detection, the estimated residuals are added to obtain reliable results, which is an iteration process that will be expedited by more accurate frequency solutions based on the proposed method for free vibration problems.

Numerical results are presented for free vibration problems and damage detection problems of representative non-uniform and geometrically stepped Euler–Bernoulli beams with multiple cracks to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

The proposed combination of methodologies described in the paper leads to a very powerful approach for free vibration and damage detection of beams with cracks, introducing the mesh refinement, that can be extended to deal with the damage detection of frame structures.

Domain decomposition of finite element models (FEM) for parallel computing are often performed using graph theory and algebraic graph theory. This paper aims to present a new method for such decomposition, where a combination of algebraic graph theory and differential equations is employed.

In the present method, a combination of graph theory and differential equations is employed. The proposed method transforms the eigenvalue problem involved in decomposing FEM by the algebraic graph method, into a specific initial value problem of an ordinary differential equation.

The transformation of this paper enables many advanced numerical methods for ordinary differential equations to be used in the computation of the eigenproblems.

Combining two different tools, namely algebraic graph theory and differential equations, results in an efficient and accurate method for decomposing the FEM which is a combinatorial optimization problem. Examples are included to illustrate the efficiency of the present method.

This study aimed to solve the engineering problem of free vibration disturbance and local mesh refinement induced by microcrack damage in circularly curved beams. The accurate identification of the crack damage depth, number and location depends on high-precision frequency and vibration mode solutions; therefore, it is critical to obtain these reliable solutions. The high-precision finite element method for the free vibration of cracked beams needs to be developed to grasp and control error information in the conventional solutions and the non-uniform mesh generation near the cracks. Moreover, the influence of multi-crack damage on the natural frequency and vibration mode of a circularly curved beam needs to be detected.

A scheme for cross-sectional damage defects in a circularly curved beam was established to simulate the depth, location and the number of multiple cracks by implementing cross-section reduction induced by microcrack damage. In addition, the h-version finite element mesh adaptive analysis method of the Timoshenko beam was developed. The superconvergent solution of the vibration mode of the cracked curved beam was obtained using the superconvergent patch recovery displacement method to determine the finite element solution. The superconvergent solution of the frequency was obtained by computing the Rayleigh quotient. The superconvergent solution of the eigenfunction was used to estimate the error of the finite element solution in the energy norm. The mesh was then subdivided to generate an improved mesh based on the error. Accordingly, the final optimised meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained. Lastly, the disturbance behaviour of multi-crack damage on the vibration mode of a circularly curved beam was also studied.

Numerical results of the free vibration and damage disturbance of cracked curved beams with cracks were obtained. The influences of crack damage depth, crack damage number and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam were quantitatively analysed. Numerical examples indicate that the vibration mode and frequency of the beam would be disturbed in the region close to the crack damage, and a greater crack depth translates to a larger frequency change. For multi-crack beams, the number and distribution of cracks also affect the vibration mode and natural frequency. The adaptive method can use a relatively dense mesh near the crack to adapt to the change in the vibration mode near the crack, thus verifying the efficacy, accuracy and reliability of the method.

The proposed combination of methodologies provides an extremely robust approach for free vibration of beams with cracks. The non-uniform mesh refinement in the adaptive method can adapt to changes in the vibration mode caused by crack damage. Moreover, the proposed method can adaptively divide a relatively fine mesh at the crack, which is applied to investigating free vibration under various curved beam angles and crack damage distribution conditions. The proposed method can be extended to crack damage detection of 2D plate and shell structures and three-dimensional structures with cracks.

The purpose of this paper is to develop and apply an exact finite strip (F‐a FSM) for the buckling and initial post‐buckling analyses of box section struts.

The Von‐Karman's equilibrium equation is solved exactly to obtain the buckling loads and deflection modes for the struts. The investigation is then extended to an initial post‐buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. Through the solution of the Von‐Karman's compatibility equation, the in‐plane displacement functions are developed in terms of the unknown coefficient. These in‐plane and out‐of‐plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient.

The F‐a FSM is applied to analyze the buckling and initial post‐buckling behavior of some representative box sections for which the results were also obtained through the application of a semi‐energy finite strip method (S‐e FSM). For a given degree of accuracy in the results, however, the F‐a FSM analysis requires less computational effort.

In the present F‐a FSM, only one of the calculated deflection modes is used for the initial post‐buckling study.

A very useful and computationally economical methodology is developed for the initial design of struts which encounter post‐buckling.

The originality of the paper is the fact that by incorporating a rigorous buckling solution into the Von‐Karman's compatibility equation, and solving it, a fairly efficient method for post‐buckling stiffness calculation is achieved.

The purpose of this paper is to identify the crack in beam-like structures before the complete failure or damage occurs to the structure. The beam-like structure plays an important role in modern architecture; hence, the safety of this structure is much dependent on the safety of the beam. Hence, predicting the cracks is much more important for the safety of the overall structure.

In the present work, the regression analysis has been carried out through LASSO and Ridge regression models. Both the statistical models have been well implemented in the detection of crack depth and crack location. A cantilever beam-like structure has been taken for the analysis in which the first three natural frequencies have been considered as the independent variable and crack location and depth is used as the dependent variable. The first three natural frequencies, f₁, f₂ and f₃ are used as an independent variable. The crack location and crack depth are estimated though the regressor models and the accuracy are compared, to verify the correctness of the estimation.

As stated in the purpose of work, the main aim of the present work is to identify the crack parameters using an inverse technique, which will be more effective and will provide the results with less time. The data used for regression analysis are obtained from theoretical analysis and later the theoretical results are also verified through experimental analysis. The regression model developed is tested for its Bias Variance Trade-off (“Bias” – Overfitting, “variance” – generalization). The regression results have been compared with the theoretical results to check the robustness in the subsequent result section.

The idea is an amalgamation of existing and well-established technologies, that is aimed to achieve better performance for the given task. A regressor is trained from the data obtained through numerical simulation. The model is developed taking bias variance trade-off into consideration. This generalized model gives us very much acceptable performance.

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 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.