Solid-shell Element Research Articles

A study on solid-shell finite element formulations applied to nonlinear thermoelastic analysis of thin-walled structures

The solid-shell elements have shown great advantages in three-dimensional (3D) finite element (FE) simulation of thin-walled structures. To overcome the numerical lockings for 3D element with a large span-thickness ratio, the assumed natural strain (ANS) method combined by either the hybrid stress (HS) or enhanced assumed strain (EAS) formulations are commonly used. In this work, two types of finite element formulations of the solid-shell elements are developed for nonlinear thermoelastic analysis of thin-walled structures, which are termed as the “ANS+HS” and “ANS+EAS” formulations. Accordingly, two eight-node solid-shell elements (CSSH8 and CSSE8) are developed based on the “ANS+HS” and “ANS+EAS” formulations, respectively. The Green–Lagrange displacement–strain relation is applied to involve the geometrical nonlinearities, and both the thermal expansion and temperature-dependent material properties are considered. Nonlinear thermoelastic equilibrium equations are constructed in the framework of the two types of finite element formulations using the Hellinger–Reissner and conventional variational principles, respectively. The proposed method can easily realize three different coupling analyses for thermal–mechanical loads by modifying the parameters of nonlinear thermoelastic equilibrium equations. Numerical examples demonstrate that the proposed method using CSSH8 and CSSE8 elements traces the nonlinear thermoelastic response of the isogrid stiffened panel as accurately as the shell, solid-shell, and solid elements of ABAQUS. Furthermore, the path-following capability of the proposed method using the CSSH8 with “ANS+HS” formulation is slightly superior to that using the CSSE8 with “ANS+EAS” formulation, but both better than ABAQUS.

An efficient isogeometric buckling optimization framework for multi-patch laminated shells: Leveraging 3D solid elements, lamination parameters and penalty method

Buckling is a common failure mode of laminated shell structures during service, and isogeometric analysis methods are highly suitable for the buckling analysis due to their accurate geometric modeling capability and high-order continuity. However, the isogeometric buckling analysis for a multi-patch structure and its optimal design still face challenges in computational efficiency. To address this issue, this paper proposes an integrated framework for the buckling optimization of multi-patch laminated shells, which leverages 3D solid elements, lamination parameters, and the penalty coupling method. Solid-shell elements have advantages in shell analysis due to their simplicity, absence of rotational degrees of freedom, and adaptive layering in the thickness direction. Lamination parameters simplify the optimization of fiber orientations in laminated shells by condensing the design variable space and improving the convexity of the optimization problem. The penalty method can handle the coupling in multiple patches at a relatively low cost. Nevertheless, lamination parameters are rarely applied to the case of solid-shell elements. In addition, despite using lamination parameters, the number of design variables for multi-patch structures remains significantly large. To tackle these issues, this study derives the solid-shell formulation in lamination parameters and the sensitivity analysis formulae in the reverse mode. Experiment validation is conducted to assess the accuracy of the proposed method based on lamination parameters with solid elements, the feasibility of the multi-patch optimization model, and the computational efficiency of optimization sensitivity based on the reverse mode compared to the forward counterpart common in the literature.

Establishment and validation of mixed solid degenerate shell elements

Establishment and validation of mixed solid degenerate shell elements

A novel hybrid-stress method using an eight-node solid-shell element for nonlinear thermoelastic analysis of composite laminated thin-walled structures

Composite laminated thin-walled structures, widely used in high-speed aircrafts, undergo a complex thermal–mechanical coupling environment. Geometrical nonlinearities with a thermal effect bring significant challenge to finite element analysis of structures. In this paper, a novel hybrid-stress method based on the solid-shell element is proposed for nonlinear thermoelastic analysis. An eight-node solid-shell element (CSSH8) is developed based on the assumed natural strain method and hybrid-stress formulations to overcome various locking problems and achieve an effective 3D simulation for structures with a large span-thickness ratio. The Green–Lagrange displacement-strain relation is selected to take the geometrical nonlinearities into account. The modified generalized laminate constitutive model is extended to consider both the thermal expansion and temperature-dependent material properties. A temperature variation along the laminate thickness can also be assumed in the constitutive model. Nonlinear thermoelastic equilibrium equations are derived using the Hellinger–Reissner variational principle, in which five different coupling cases for thermal–mechanical loads can be fully involved. Numerical examples demonstrate that the proposed method with CSSH8 element is insensitive to various distorted meshes and numerically robust to pass the buckling point; meanwhile large step sizes can be achieved in the path-following nonlinear thermoelastic analysis.

A hybrid-stress formulation based reduced-order method using a solid-shell element for geometrically nonlinear buckling analysis

A hybrid-stress formulation based reduced-order method using a solid-shell element for geometrically nonlinear buckling analysis

Stress–displacement stabilized finite element analysis of thin structures using Solid-Shell elements, Part II: Finite strain hyperelasticity

This work is the second of a two-part research project focused on modeling solid-shell elements using a stabilized two-field finite element formulation. The first part introduces a stabilization technique based on the Variational Multiscale framework, which is proven to effectively address numerical locking in infinitesimal strain problems. The primary objective of the study was to characterize the inherent numerical locking effects of solid-shell elements in order to comprehensively understand their triggers and how stabilized mixed formulations can overcome them. In this current phase of the work, the concept is extended to finite strain solid dynamics involving hyperelastic materials. The aim of introducing this method is to obtain a robust stabilized mixed formulation that enhances the accuracy of the stress field. This improved formulation holds great potential for accurately approximating shell structures undergoing finite deformations. To this end, three techniques based in the Variational Multiscale stabilization framework are presented. These stabilized formulations allow circumventing the compatibility restriction of interpolating spaces of the unknowns inherent to mixed formulations, thus allowing any combination of them. The accuracy of the stress field is successfully enhanced while maintaining the accuracy of the displacement field. These improvements are also inherited to the solid-shell elements, providing locking-free approximation of thin structures.

Finite Element Modelling and Simulation of Tunnel Gates of Dam Structures in ABAQUS Using Reduced-Integrated 8-Node Hexahedral Solid-Shell Element

Tunnel gates used for water regulation are very important and critical parts of dam safety. This is because they are exposed to the hydrostatic pressure of the total height of the water in the reservoir. In this case study, a nonlinear finite element modelling and simulation of the diversion tunnel gate of the Gokdere Bridge Dam in Adana, Turkey, are performed to investigate the performance and capacity of the structure before collapse. The maximum water level in the reservoir is 85 meters. A 3D finite element modelling of the gate structure was created, considering the details of the construction project. Both the concrete structure and the anchorages between the concrete of the first and second phases of the gate structure and the steel profiles in the gate supports are modelled to obtain composite behavior, bond stresses, and a more accurate load and stress distribution. For the nonlinear finite element modelling in ABAQUS, C3D8R reduced-integrated 8-node hexahedral solid elements with concrete damage and tension stiffening are used. For the simulation, linear and nonlinear capacity analyses of the gate structure are performed, and the stresses, strains, deformations, and crack propagation in concrete and steel are investigated. It is found that nonlinear analysis and finite element modelling of anchors for capacity and load transfer are important in the simulation of gate structures to prevent tunnel collapse. It is suggested that dam monitoring and control systems and the use of multiple gates are recommended when a problem occurs in the operation of a gate in the diversion tunnel gates of a dam structure.

Three-dimensional nonlinear stress analysis of functionally graded shells subjected to displacement-dependent loads

The nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading, developed in the companion paper, is here extended to the stress analysis of functionally graded material (FGM) shells. The term GeX means that the middle surface is described by analytical functions, including spline functions, and, therefore, the coefficients of the first and second fundamental forms can be taken exactly at element nodes. The FGM solid-shell element formulation is based on the choice of N sampling surfaces (SaS) parallel to the middle surface and located at Chebyshev polynomial nodes within the shell as well as on the outer surfaces, to introduce the displacements of these surfaces as basic unknowns. The use of Lagrange polynomials of degree N–1 in the distributions of displacements, strains, stresses, and elastic coefficients through the thickness of the shell leads to an efficient 3N-parameter shell formulation. To develop the FGM four-node solid-shell element based on the hybrid-mixed method, in which strains and stresses are utilized as primary variables, the Hu-Washizu variational principle is applied. Due to the analytical integration used to evaluate the tangent stiffness matrix, the developed FGM solid-shell element subjected to displacement-dependent loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered benchmark problems. It has been established that the difference between the second Piola-Kirchhoff and Cauchy stress tensors in non-conservative problems for FGM shells undergoing moderately large strains can be significant.

Internal-interfacial cracking interaction: Combined phase-field and discontinuous Galerkin/cohesive zone modeling

Interaction simulations between internal and interfacial cracks arising in laminated composites have remained a challenging task in the computational mechanics community. Toward this end, this paper proposes a novel explicit approach that combines a length scale-insensitive phase-field model and a discontinuous Galerkin/cohesive zone model to respectively account for brittle or cohesive fracture of bulk materials and interfacial debonding. The presented approach advocates the incorporation of a discontinuous Galerkin term to overcome the artificial compliance problem inherent in existing phase-field methods that normally adopt intrinsic cohesive zone models for interfacial debonding. The capacity of our approach is further exploited by innovatively integrating the approach with an efficient six-node pentahedral solid-shell element, enabling a natural simulation of through-thickness cracks in thin shell structures with high-aspect-ratio elements. Implementation challenge, arising from the stress decomposition of the solid-shell element, of such integration is well addressed. In light of the considerable computational cost required for a phase-field-based method, the proposed approach is mapped to graphics processing units to improve the computational efficiency. Representative numerical examples have demonstrated the capability of the proposed approach in simulating mode-I and mixed-mode cohesive cracks, as well as brittle fractures in solids and shells. It is also shown that the presented approach is capable of effectively addressing the artificial compliance issue and capturing the competitive behavior between internal and interfacial cracks in a physically consistent manner.

Electromechanical behavior of piezolaminated shell structures with imperfect functionally graded porous materials using an improved solid-shell element

This work introduces a solid-shell element with eight nodes for the bending behavior of structures embedded with piezoelectric layers using the FSDT theory. The mechanical degrees of freedom are coupled, through the constitutive equations, with the piezoelectric one considering the linear distribution of the electric potential along the piezoelectric face's thickness. In order to subdue shear and thickness locking, the Assumed Natural Strain (ANS) method as well as the Enhanced Assumed Strain (EAS) method are adopted. Bonding the external faces of shell structures with piezofaces accentuates the smart aspect of solid shells. The middle part of the structure is made of a Functionally Graded Material (FGM) with distributed pores along the thickness. The formulated finite element model is implemented to investigate the effects of some parameters on the bending response of solid shells such as the impact of various configurations of porous FGM, the power law index of FGM and some geometrical parameters of solid shell structures.

Large deflection response of functionally graded porous structures considering geometrical nonlinearity using an improved FSDT

This study’s objective is to propose a full 3D finite element modeling for examining the geometrically nonlinear (NL) response of functionally graded (FG) porous materials with plates, ring, and cylindrical shapes. An improved first-order shear deformation theory (IFSDT) is utilized in order to develop the 3D solid-shell element. The governing equations are developed in a way that offers a parabolic distribution of the transverse shear strains through the FG porous shell thickness and a zero condition of the transverse shear stresses on the top and bottom surfaces. The given IFSDT solid-shell element has the ability to model any kind of shell structures and incorporates a 3D material law. In this study, two types of porosity distribution functions are considered. The effectiveness of the present IFSDT solid-shell element is demonstrated through linear (L) and NL numerical examples. The effect of the porosity distribution, porosity coefficient, power-law index, geometrical design parameters, and boundary conditions on the geometrically NL response of FG porous shells structures is evaluated.

Improved-FSDT-based solid-shell element for buckling analysis of plate, spherical cap, and cylindrical shell of FG porous materials

This study's objective is to propose a full three-dimensional finite element modeling for examining the buckling behaviors of functionally graded (FG) porous materials with plates, spherical cap, and cylindrical shapes. An improved first-order shear deformation theory (IFSDT) is utilized in order to develop the three-dimensional solid-shell element. The governing equations are developed in a way that offers a parabolic distribution of the transverse shear strains through the FG porous shell thickness and a zero condition of the transverse shear stresses on the top and bottom surfaces. The given IFSDT solid-shell element has the ability to model any kind of shell structures and incorporates a three-dimensional material law. In this study, two types of porosity distribution functions are considered. The performance of the current IFSDT solid-shell element has been demonstrated by examining the buckling behavior of Functionally Graded Porous plates, spherical caps, and cylindrical shells. The effect of the porosity distribution, porosity coefficient, power-law index, geometrical design parameters and boundary conditions on the buckling behaviors of FG porous shells structures is evaluated.

Dynamic analysis of piezolaminated shell structures reinforced with agglomerated carbon nanotubes using an enhanced solid-shell element

Dynamic analysis of piezolaminated shell structures reinforced with agglomerated carbon nanotubes using an enhanced solid-shell element

An assessment of PWR fuel assembly spacer grid strength by FEM models

Since Fukushima accident, integrity of fuel assembly for seismic load became very important. This study assesses the structural integrity of mid spacer grid (SG) of PLUS7 fuel assembly (FA) which is the weakest part of fuel assembly structural components. Spacer grid FEM models were developed by 3D solid element and 3D shell element. From the FEM analysis, it was found that the structural strength for normal direction was much larger than for shear direction. At core outlet, the normal stiffness was estimated to 36kN/mm for 3D solid model and 33kN/mm for 3D shell model, and the shear stiffness was estimated to 49 N/mm for 3D solid model and 75 N/mm for 3D shell model. From this result normal stiffness is a lot bigger than shear stiffness, and spacer grid structure is more vulnerable against shear loads. It is recommended to do shear test or analysis of fuel assembly in the development stage.

An efficient solid shell material point method for large deformation of thin structures

AbstractThe standard material point method (MPM) encounters severe numerical difficulties in simulating shell structures. In order to overcome the shortcomings of locking effects, the discretization size of background grid should be small enough, usually smaller than 1/5 of the shell thickness, which however will lead to prohibitive computational cost. A novel solid shell material point method (SSMPM) is proposed to efficiently model the large deformation of thin structures. The SSMPM describes the material domain of shell structures by shell particles with hexahedral particle domains. The locking treatments of solid shell element are then introduced in SSMPM, which results in the correction of strain field throughout the shell thickness. Namely, the assumed natural strain (ANS) method is adopted to eliminate the shear locking and trapezoidal locking, while the enhanced assumed strain (EAS) method is employed to eliminate the thickness locking. With the precise description of bending modes, a single layer of particles and a coarse background grid could be used in shell structure simulations with the SSMPM, which dramatically increases the computational efficiency. A local multi‐mesh contact method is presented to naturally couple SSMPM and MPM for the contact situations of shells with other objects. Several numerical examples, including beam vibration, pinched cylinder with free edges and full hemispherical shell, are performed to verify and validate the SSMPM, which shows that the SSMPM considerably outperforms the standard MPM in these situations. A fluid–structure interaction problem and the penetration of a thin plate are investigated based on the contact method and the results are in good agreement with those in the literature.

An unsymmetric 8-node hexahedral solid-shell element based on ANS and incompatible concepts for thin shell analysis

In this paper, an incompatible unsymmetric 8-node hexahedral solid-shell element is proposed with different sets of trial and test functions. The trial functions used in the 8-node hexahedral solid element in our previous work are adopted with minor revision. In addition, to eliminate the transverse shear and trapezoidal locking in the calculation of thin shell structures, the test functions are revised by the assumed natural strain method. The present solid-shell element inherits advantages from the original solid element and also alleviates the typical locking phenomena for shell elements. Numerical investigations show that the present element is insensitive to mesh distortion and has rapid convergence. In the tests of thin shell structures with bending, it can also provide locking-free predictions with high precision. Finally, the present element is extended to geometrically nonlinear analysis and its accuracy is demonstrated by a simple numerical example.

High-accurate FE simulation on compressive behavior of steel cruciform column with welding imperfection

Due to its various advantages, welding is commonly used in the construction for steel connection. However, the local temperature difference in the welding causes welding deformation and residual stress. The welding deformation and residual stress can significantly impact the load-bearing performance of members. Therefore, it is important to evaluate those behaviors with high precision. In the previous studies, thermal elastic-plastic analysis was used to evaluate welding deformation and residual stress in simulated welded specimens and the compressive simulation was applied to obtain, for example, the load-bearing performance. Nevertheless, there are limitations in assessing structures made up of thin plates. This is because the simulation of welding is typically modeled using solid elements, while the simulation of loading analysis uses shell elements. As a result, these analysis steps are not directly related, leading to deficiencies in the evaluation process. In this study, the solid element model and shell element model were modeled to simulate the welding deformation and residual stress. The corresponding load-bearing capacity results obtained from the continuous loading analysis were compared with the results from experiment to verify the consistency. According to the results, shell models that perform welding analysis simulation followed by continuous compressive analysis are better able to replicate experimental results in terms of out-of-plane deformation and maximum load-carrying capacity under compressive analysis. This is in contrast to solid models, which fall short in reproducing the same results. Moreover, the shell model is also more efficient than the solid model in terms of time saved during analysis and calculation.

Topology optimisation for robust design of large composite structures

The manufacturing of composite structures can lead to material defects. Some of these defects, due to their small size, can go undetected by current non-destructive testing techniques; especially in very large and geometrically complex components. Therefore, it is of paramount importance to design composite components to resist defects by accounting for them as early as possible. Topology optimisation is a methodology whereby complex features, such as cracks and debonds, can be accounted for in the design. Therefore, this work proposes a new approach for the design of very large composite structures. This new approach is enabled by a solid shell element formulation with boundary tracking capabilities (through the floating node and the level-set methods) and optimisation capabilities (through design sensitivity analysis of an energy release rate objective function). Furthermore, this work proposes a new algorithm and modelling workflow that has the ability to handle large multi-scale models. The proposed methodology shows advantages as the ability to track the topology explicitly, and the ability to increase defect tolerance of the design. This works shows the new methodology is capable of handling complex structures, such as a multi-scale stringer run-out model from an aircraft wing-box, while a kissing bond defect is present.

Numerical Simulation of Train-Induced Structural Vibration

Train-induced structural vibration caused more and more attentions nowadays, and the present paper developed a numerical approach of train-induced structural vibration simulation with ANSYS software. The finite element model is composed by buildings, rail, sleepers, ballast and several layers of soil, and the moving load is represented by wheel-rail coupling forces, and both solid element, shell element and beam element are used in the finite element model. Acceleration time history and vibration levels of the key point on the buildings are given, and effect of different wheel-rail coupling forces and the dynamic response of the building are also numerically investigated. Comparisons of free boundary, symmetric boundary, fixed boundary and transmitting boundary conditions on the dynamic responses are also illustrated. Numerical results show that, both boundary condition and load type could affect the dynamic response of the structures near the train tracks, and validation test should be performed firstly to determine the proper boundary condition and load amplitude and frequency. The proposed numerical method provides a powerful technic tool for numerically predicting the dynamic response of train-induce structural vibration, and could also be employed to evaluate the vibration control results of different approaches.

Isogeometric stiffness matrix formulations for buckling analysis of multi-layered shells based on lamination parameters for solid-shell elements

Multi-layered composite shells are extensively used in many industrial fields since they have superior performance in applications. As load-carrying components, buckling failure is one of the most common failure modes of multi-layered shells. Hence, the layup design of multi-layered shells based on optimal buckling factors is widely concerned, where lamination parameters are commonly used in the representation of stiffness. However, the common formulations of laminate stiffness are based on the Classical Laminate Plate Theory (CLT) or the First-order Shear Deformation Plate Theory (FSDT). This paper aims at developing stiffness formulations based on 3D continuum solid theory. The accuracy and efficiency of the stiffness formulations derived in this study are analysed and compared to those obtained by layer-by-layer integration and summation of stiffness calculations.