Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Tensor-decomposition based matrix computation: A fast method for the isogeometric FSDT analysis of laminated composite plate

Correction of stiffness and mass matrices utilizing simulated measured modal data

This paper is concerned with the coupled thermo-mechanical stress analysis of functionally graded (FG) gas turbine rotor shaft system. Gas turbine shaft may expose in high temperature environments which demands to use functionally graded materials (FGMs). The aim of the present work is to study the stresses developed in the FG turbine shaft due to temperature variations and mechanical loading due to unbalance masses. For the present analysis aluminum oxide (Al2O3) and stainless steel (SUS304) are taken as shaft materials, power law gradation is used for the determination of FG material properties of the turbine shaft. Three nodded Timoshenko beam element with six degree of freedom (DOF) per node is considered for the finite element modelling of FG shaft. First order shear deformation theory (FSDT) is used with rotary inertia, strain and kinetic energy. Solution for governing equation of motion is obtained by the Hamilton principle. Complete MATLAB code has been developed for thermosmechanical stress analysis. Comparative study between steel shaft and FG shaft have been carried out. Normal stress (σxx) on plane perpendicular to axial direction, shear stress (τxr) on plane perpendicular to axial direction in radial direction and shear stress (τxθ) on plane perpendicular to axial direction in circumferential direction are obtained against time and along radius of shaft. Also these stresses are obtained for different parameters like power law indexes and speed of rotation of shaft.

Detailed formulation and coding of exact finite element is carried out to study the static behavior of a layered beam structure. The beam element is modelled based on the first-order shear deformation theory and it is assumed to be composed of three layers whereas the middle layer is made of functionally graded material (FGM), i.e. with variable elastic properties in the thickness direction. The shape of the FGM mechanical properties variation in the thickness direction takes the form of exponential or power-law. The governing equations and boundary conditions are derived by applying the virtual work principle. Variations of displacements along the beam and stresses across the depth due to mechanical loadings are investigated. Comparative examples are carried out to highlight the static behavior difference between FGM layered beams and pure metal-ceramic beams.

<div class="htmlview paragraph">With highly complex structures, it is sometimes desirable to derive a dynamic model of the system from experimental vibration data. This paper presents algorithms for synthesizing the mass and stiffness matrices from experimentally derived modal data in a way which preserves the physical significance of the individual mass and stiffness elements. The synthesizing procedures allow for the incorporation of other mass and stiffness data, whether empirical or based on the analyst's insight. The mass and stiffness matrices are derived for a cantilever beam example and are compared with those obtained using earlier techniques.</div>

On the vibration and stability of FGM cylindrical shells under external pressures with mixed boundary conditions by using FOSDT

This study presents the development and validation of an open-source finite element framework implemented in Python for the analysis of laminated composite plates using First Order Shear Deformation Theory (FSDT). The formulation employs graded quadrilateral meshing with eight-noded serendipity elements and incorporates full laminate constitutive modeling through computation of the A, B, and D stiffness matrices along with transverse shear effects. The framework enables automated geometry generation for plates with geometric discontinuities, global stiffness assembly, displacement solution, plylevel stress recovery, and contour visualization. To assess numerical accuracy, the developed solver is benchmarked against the layered shell formulation (SHELL181) in ANSYS under identical geometry, mesh density, material properties, boundary conditions, and loading. Comparisons are performed for in-plane and transverse displacements as well as normal stress components in laminated plies for a composite plate containing a central circular cutout. The Python-based results demonstrate strong agreement with the commercial solver, with displacement deviations below 1% and stress deviations within 5%. The study highlights the feasibility of using open-source computational tools for small- to medium-scale composite structural analyses, reducing dependency on high-cost licensed software for preliminary design studies and parametric investigations. Furthermore, the developed framework is extended toward the generation of high-fidelity finite element datasets to enable data-driven modeling. Specifically, the solver infrastructure is being utilized to create structured simulation databases for training Graph Neural Network (GNN)-based predictive models aimed at replacing computationally expensive finite element simulations. This integration establishes a pathway toward rapid surrogate modeling of composite structures while preserving physics-informed accuracy

In this study, an enhanced first order shear deformation theory via the strain energy transformation (EFSDT) in Laplace domain is presented for the efficient yet accurate viscoelastic FEM analysis of laminated composite plates. The main objective herein is to systematically derive the improved strain energy expression of first order shear deformation theory (FSDT) based on a conventional Reissner-Mindlin’s plate theory. To this end, an efficient higher order plate theory (EHOPT) is utilized to obtain the in-plane warping functions and synthesized into the strain energy of FSDT to improve their performance. The relationships between the two independent theories are systematically established via the strain energy transformation and the least-square approximation of the mean displacement. And the convolution theorem of Laplace transformation is applied to circumvent the overwhelming complexity of dealing with visco-elastic materials which is expressed as the Bolzmann superposition integral equation. The proposed EFSDT in Laplace domain has the same computational advantage of the conventional FSDT, while allowing us to improve the local distributions of the visco-elastic responses via the recovery procedure. The numerical results obtained herein are compared to those available in literature.

Nonlinear forced response of doubly-curved laminated panels composed of cnt patterned layers within first order shear deformation theory

Mechanical and thermal buckling of functionally graded axisymmetric shells

This paper proposes a method to identify mass and stiffness matrices of structures. Element mass and stiffness matrices have mass and stiffness distributions. Therefore, a ratio between two elements in these matrices has a proper value. Element mass and stiffness matrices are divided into several groups considering this relation. A group is represented by a sparse matrix (called group matrix) with only elements in the group matrix. Mass and stiffness matrices are modified group by group. The objective function to be minimized is the sum of each norm of difference between analytical and identified group matrices. A numerical example is given to show the effectiveness of the method.

We consider damped elastodynamic networks where the damping matrix is assumed to be a non-negative linear combination of the stiffness and mass matrices (also known as Rayleigh or proportional damping). We give here a characterization of the frequency response of such networks. We also answer the synthesis question for such networks, i.e., how to construct a Rayleigh damped elastodynamic network with a given frequency response. Our analysis shows that not all damped elastodynamic networks can be realized when the proportionality constants between the damping matrix and the mass and stiffness matrices are fixed.

A method for the dynamic analysis of flexible legged locomotion systems that accounts for the rotary inertia and shear deformation effects is presented. The motion of the flexible components in the legged vehicle is described using a set of inertia-variant Timoshenko beams that undergo large rotations. A shape function that accounts for the combined effect of rotary inertia and shear is employed to describe the deformation relative to a selected component reference and the rigid-body modes of the shape function are eliminated using a set of reference conditions. Kinetic and strain energies are derived for each Timoshenko beam, thus identifying the beam mass and stiffness matrices which account for the rotary inertia and shear deformation effects. A new set of time-invariant matrices that describe the nonlinear inertia coupling between the reference motion and elastic deformation and account for the rotary inertia and shear is developed and it is shown that the form of these matrices as well as the mass and stiffness matrices are significantly affected by the inclusion of rotary inertia and shear. Numerical experimentations indicate that shear and rotary inertia can have a significant effect on the dynamics of flexible legged locomotion.

The classical continuum theory not only underestimates the stiffness of microscale structures such as microbeams but is also unable to capture the size dependency, a phenomenon observed in these structures. Hence, the non-classical continuum theories such as the strain gradient elasticity have been developed. In this paper, a Timoshenko beam finite element is developed based on the strain gradient theory and employed to evaluate the mechanical behavior of microbeams used in microelectromechanical systems. The new beam element is a comprehensive beam element that recovers the formulations of strain gradient Euler–Bernoulli beam element, modified couple stress (another non-classical theory) Timoshenko and Euler–Bernoulli beam elements, and also classical Timoshenko and Euler–Bernoulli beam elements; note that the shear-locking phenomenon will not happen for the new Timoshenko beam element. The stiffness and mass matrices of the new element are derived in closed forms by following an energy-based approach and using Hamilton’s principle. It is noted that unlike the classical beam elements, the stiffness matrix of the new element has a size-dependent nature that can capture the size-dependent behavior of microbeams. The shape functions of the newly developed beam element are determined by solving the equilibrium equations of strain gradient Timoshenko beams, which brings about a size-dependent characteristic for them. The new beam element is employed to evaluate the static deflection of a microcantilever, and the results are compared to the experimental data as well as the results obtained by using the classical beam element and the couple stress plane element. The new beam element is also implemented to calculate the static deflection, vibration frequency, and pull-in voltage of electrostatically actuated microbeams. The current results are compared to the experimental data as well as the classical FEM outcomes. It is observed that the results of the new element are in excellent agreement with the experimental data while the gap between the experimental and classical FEM results is significant.

In order to present an efficient, practical technique to determine progressive failure mechanism of structures, modelling of member deterioration by using a spring system is outlined. The procedure uses updates of member stiffness and mass matrices as well as the random load vector in incremental forms. In this procedure, the assembly process produces redistributions of the system stiffness and mass matrices, and the load vector. In the calculation of response spectral values, the original forms remain unchanged. Inversion of the stiffness matrix is calculated by using the Neumann expansion solution in which the original stiffness matrix is inverted only once so that a considerable computation time is saved in the whole calculation process. An incremental solution technique is presented for spectral analyses of both static and dynamic sensitive structures. In the case of dynamic analysis, special attention is paid to estimations of modified natural frequencies and mode shapes of deteriorated structures, which may affect response spectral values considerably. The technique, which is presented in the paper, is attractive in practical applications and can be efficiently used in the reliability calculation as well, and also it can be successfully used to determine a progressive failure mechanism of the structure.