Geometrically nonlinear static analysis of multi-component structures through variable-kinematics finite elements

Current codes and guidelines provide different methods to perform nonlinear static analysis of structures that require some non-intuitive assumptions in their application. In the present paper an energy-based method for nonlinear static analysis that allows to overcome these assumptions is proposed. In the method the capacity and the demand are both expressed in terms of energy. An energy capacity curve is computed considering that at each step the work of the lateral forces is equal to the structure internal work. The demand is represented in terms of an energy quantity, defined pseudo-energy, that is computed from both the maximum response displacement and the maximum response force. Constant ductility pseudo-energy spectra are introduced as energy demand design spectra, alternative to the input energy demand spectra. The definition of the performance point does not require iterative procedures for equating the internal dissipated energy to the demand energy. For the direct evaluation of the performance point two different operative procedures are proposed. The proposed method is evaluated comparing the earthquake-induced deformations of single degree-of-freedom systems resulting from the application of the presented nonlinear static analysis procedures with those obtained from the time-history analysis and from the application of the EC8 nonlinear static analysis procedure. The method is also applied in the case of a RC plane frame representing the inner frame of a six story building. The results obtained with the proposed method are in good agreement with those computed using nonlinear dynamic analyses, moreover they are characterized by a better accuracy with respect to the results obtained with the method provided by EC8.

A material-based higher-order shear beam model for accurate analyses of FG beams with arbitrary material distribution

This work investigates the dynamic response of various rotating cylindrical structures using low- and high-fidelity one-dimensional (1D) and two-dimensional (2D) finite element models. The adopted mathematical formalism is based on the Carrera Unified Formulation (CUF). CUF offers a procedure to obtain higher-order beam and shell models hierarchically and automatically. These theories are formulated by expanding the unknown variables over the beam cross-section or along the shell thickness. Various beam and shell models can be implemented depending on the choice of the polynomial employed in the expansion. Both Taylor-like and Lagrange polynomials are considered for developing different kinematic models. The linearized equations of motion include the Coriolis and initial stress contributions. Various thick and thin cylinders and disk structures with different boundary conditions are considered. The beam and shell results are compared with analytical and three-dimensional (3D) finite element (FE) solutions from the literature.

A refined beam model for the linear static aeroelastic analysis of generally oriented lifting systems is described in this paper. It is aimed at beam-like structures such as classical and unconventional wing configurations. The structural formulation of refined beam finite elements is embedded in the framework of Carrera Unified Formulation. Increasing accuracy in predicting effects of warping, in-plane deformation is obtained by considering as a free parameter the order of the displacement field expansion over the cross-section. Linear steady aerodynamic loads are described via the Vortex Lattice Method and the transfer to their energetically equivalent structural loads is performed by the Principle of Virtual Displacements. Thanks to the accuracy of refined elements, the coupling of structural and aerodynamic fields is performed via the Infinite Plate Spline method. The procedure involves a set of pseudo-structural points placed on the reference surface of the wing system. Different beam elements as well as different higher-order models are considered for the analysis of various cross-section geometries and loading cases. The structural results are validated with benchmarks retrieved from the classical models and NASTRAN. Aeroelastic results show well agreement with NASTRAN solution for a number of wing configurations. The proposed higher-order model proves its increasing accuracy in predicting aeroelastic responses with respect to analyses based on classical beam theories.

In this study, a new empirical method is presented to obtain Dynamic Increase Factor (DIF) in nonlinear static analysis of structures against sudden removal of a gravity load-bearing element. In this method, DIF is defined as a function of minimum ratio of difference between maximum moment capacity (Mu) and moment demand (Md) to plastic moment capacity (Mp) under unamplified gravity loads of elements. This function determines the residual strength of a damaged building before amplified gravity loads. For each column removal location, a nonlinear dynamic analysis and a step-by-step nonlinear static analysis are carried out and the modified empirical DIF formulas are derived, which correspond to the ratio min of beams in the bays immediately adjacent to the removed column, and at all floors above it. Therefore, the new DIF can be used with nonlinear static analysis instead of nonlinear dynamic analysis to assess the progressive collapse potential of a moment frame structure. The proposed DIF formulas can estimate the real residual strength of a structure based on critical member.

Exact solutions for free vibration analysis of laminated, box and sandwich beams by refined layer-wise theory

This article presents a new higher-order beam model. The present beam model is governed by differential equations that are similar to those present in some existing higher-order beam models; however, the present beam model makes use of a novel method of calculating the transverse shear stiffness, which facilitates the calculation of a shear-warping stiffness without the need for an assumed warping displacement field, and without introducing any additional kinematic variables. The present beam model also facilitates the recovery of the distributions of longitudinal normal stresses and transverse shear stresses. The authors postulate that the bending and shear terms in first-order shear deformation theory represent the first two terms in an infinite series that would constitute an ideal one-dimensional beam model, and it is suggested that the present beam model constitutes the first four terms in this hypothetical infinite series. The present beam model is solved for several example beams, and the results are compared with those of existing classical and higher-order beam models, as well as computational results from finite element analyses. It is shown that the present beam model is able to accurately represent deformed shapes and stress distributions pertaining to beams that exhibit non-trivial shear compliance as well as non-trivial shear-warping stiffness. In the case of laminated composite beams comprising a large number of laminae, the present beam model offers a level of analytical fidelity that is comparable to that of existing zigzag beam models; however, unlike zigzag beam models, the present beam model is equally well suited for the analyses of beams comprising any number of laminae.

Thin-walled beams subjected to load factors and non-structural masses

This paper explores the dynamic behavior of metamaterial-like structures by investigating the evolution of their band gap under the influence of geometrical nonlinearities in the large displacement/rotations field. The study employs a unified framework based on the Carrera Unified Formulation (CUF) and a total Lagrangian approach to develop higher-order onedimensional beam theories that account for geometric nonlinearities. The axis discretization is achieved through a finite element approximation. The equations of motion are solved around nonlinear static equilibrium states, which are determined using a Newton–Raphson algorithm combined with a path-following method of arc-length type. The CUF approach introduces two key innovations that are highly suitable for the evolution of the band gap: 1) Thin-walled structures can be effectively represented using a single one-dimensional beam model, overcoming the common limitations of standard finite elements. This is crucial as three-dimensional solid elements would result in significant computational costs, and twodimensional elements pose limitations for this type of investigation. Finally, employing onedimensional finite elements usually requires a combination of elements, leading to additional mathematical complexities in their connections and lacking geometric precision. 2) CUF enables the use of the full Green-Lagrange strain tensor without the need for assumptions, as is the case with von Karm´ an nonlinearities. ´ The paper specifically compares results obtained with linear and nonlinear stiffness matrices, highlighting the differences. Numerical investigations are conducted on thin-walled structures composed of repeatable cells, assessing mode changes under traction and compression loading. The findings emphasize that the band gap is an inherent property of the equilibrium state, underscoring the necessity of a proper nonlinear analysis for accurately evaluating frequency transitions

This paper proposes an application of already established higher-order models and, namely, investigates the free vibration analysis of civil engineering structures subjected to nonstructural masses. The refined one-dimensional theories adopted are based on the Carrera unified formulation (CUF). The stiffness and mass matrices are obtained by means of the principle of virtual displacements in conjunction with the finite element method (FEM). According to CUF, in fact, their formulation is performed in terms of fundamental nuclei, which depend neither on the adopted class of beam theory nor on the FEM approximation along the axis. In order to account for nonstructural localised masses, the fundamental nucleus of the mass matrix has been opportunely modified. In this work, Lagrange polynomials have been employed in the framework of CUF to develop pure translational displacement-based refined beam models. The models obtained using this approach are referred, in the literature, to as component wise (CW), which allows to model each structural component as a 1D element. The free vibration analysis has been carried out for different cases of nonstructural mass distribution in typical civil buildings. The results obtained are compared with classical Solid/Shell FEM solutions from the commercial code MSC Nastran. The capabilities of the CW models are demonstrated, since this formulation is able to replicate the 3D solid results with enhanced performances in terms of computational efforts. Moreover, the importance of taking into account the correct distribution of the nonstructural masses is shown to be of fundamental importance in vibration analysis of civil buildings.

This paper presents a zeroth-order (non-gradient based) methodology for topology optimization of continuum structures. The solid and void elements in the design domain are represented by binary design variables. As in the SIMP method, design variables are assumed to represent the non-dimensional density of the individual finite elements and their contribution to structural stiffness in the model. The optimization procedure begins by specifying a desired volume fraction over a discretized domain and associated boundary conditions. Based on a randomized distribution of solid and void elements, a finite element analysis is performed to identify the strain energy of individual elements. While keeping the volume fraction fixed, the solid elements with low strain energy are converted into void elements whereas the void elements with high strain energy are converted into solid elements. The strain energy threshold for element exchange is specified at the beginning and can be adjusted in the course of optimization. Through subsequent repetition of analysis and exchange steps, the optimal topology corresponding to minimum compliance emerges with no gray elements to filter out. The general characteristics of the Element Exchange Method are demonstrated through solutions of several benchmark problems.

Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams

Spent fuel pool and storage racks are important nuclear security structures and components. In order to prevent it from structural failure, which includes the loss of the structural integrity of the spent fuel pool and stability of the spent fuel storage racks, also includes the possibility of fallen down of storage racks under seismic loading. Besides the necessary static analysis of structures, the influence of seismic loading on the interaction between water and structure should be fully considered, Especially concerned the analysis of the shaking effect of water sloshing on the storage racks, the displacement and the possibility of fallen down of the storage racks. The present paper is concerned with the problem of modeling the fluid-structure interaction (FSI) in filled liquid and filled with spent fuel pool. The study focuses on the sloshing phenomena and on the coupling computational fluid dynamic (CFD) analysis with the finite element stress analysis (FEA) code LS-DYNA. By the results of the response of seismic, such as the displacement of the storage racks, pressure exerted on the plate of racks and the walls of the pool. This paper also evaluates the seismic performance of the structure and the safety margin. Various numerical methods can be used for analysis of liquid storage pools, among these we mention explicit finite element, implicit Lagrangian-Eulerian, hybrid finite element, Smoothed Particle Hydrodynamics volume of fluid. In this article the coupled sloshing dynamics in a rectangle pool were studied using a model developed in LS-DYNA environment. The main solution methodology is based on explicit time integration. In order to demonstrate the FSI results of the FEA models of the spent fuel pool on seismic analysis, a 3D FEA models were developed. The Finite element model composed of the spent fuel pool (steel plate concrete), spent fuel storage racks, cushion block, water and air. Solid element modeling is used in concrete, cushion block, water and air. Steel plate and storage racks employ the shell element. The constitutive model of solid element is linear elastic. And the constitutive model of fluid element is described by the Gruneisen equation. Arbitrary Lagrangian-Eulerian (ALE) formulation is thought of as algorithms that perform automatic rezoning. It realized the advection of water and air in the ALE multi-material group.

The free vibration analysis of thin- and thick-walled layered structures via a refined one-dimensional (1D) approach is addressed in this paper. Carrera unified formulation (CUF) is employed to introduce higher-order 1D models with a variable order of expansion for the displacement unknowns over the cross section. Classical Euler–Bernoulli (EBBM) and Timoshenko (TBM) beam theories are obtained as particular cases. Different kinds of vibrational modes with increasing half-wave numbers are investigated for short and relatively short cylindrical shells with different cross section geometries and laminations. Numerical results of natural frequencies and modal shapes are provided by using the finite element method (FEM), which permits various boundary conditions to be handled with ease. The analyses highlight that the refinement of the displacement field by means of higher-order terms is fundamental especially to capture vibrational modes that require warping and in-plane deformation to be detected. Classical beam models are not able to predict the realistic dynamic behavior of shells. Comparisons with three-dimensional elasticity solutions and solid finite element solutions prove that CUF provides accuracy in the free vibration analysis of even short, nonhomogeneous thin- and thick-walled shell structures, despite its 1D approach. The results clearly show that bending, radial, axial, and also shell lobe-type modes can be accurately evaluated by variable kinematic 1D CUF models with a remarkably lower computational effort compared to solid FE models.

In intelligent coal mining systems, the primary structural component often comprises a sandwich plate, valued for its high strength, wear resistance, and lightweight properties. This component is vital for maintaining equipment performance in demanding subterranean environments. This research proposes a robust mathematical framework based on the Carrera Unified Formulation (CUF) and the principle of virtual displacements (PVD) to model and optimize the electromechanical behavior of piezoelectric shunt damping systems for subsonic flutter suppression and energy harvesting. Utilizing CUF’s high-order modeling precision, the framework captures the complex aeroelastic and electroelastic couplings under subsonic flow conditions with high fidelity. A multimodal shunt circuit design is developed and optimized to maximize both vibration energy dissipation and electrical energy harvesting, thereby enhancing system stability and extending operational lifespan. The effects of circuit tuning, structural parameters, and external aerodynamic excitations on flutter mitigation and power output are systematically investigated. Comparative numerical results show that CUF-based models offer greater predictive accuracy for multimodal dynamics and superior optimization capabilities relative to traditional finite element methods. This study advances the integration of structural mechanics, electromechanical interactions, and energy harvesting technologies, promoting the development of self-powered, vibration-controlled, and durable mining equipment. The approach supports next-generation energy-efficient mining solutions through rigorous modeling of variable expansions along thickness directions via equivalent single-layer (ESL) and layer-wise (LW) descriptions.