Shell Finite Element Research Articles

Effective Modulus Method (EMM) concept applied to thin‐walled steel columns.

AbstractThe design of thin‐walled steel members, subjected to compressive stresses, requires complex analysis to account the interaction between local, distortional and global instabilities. There are simplified methodologies that allow simulating the effects of local and distortional post‐buckling behavior in usual structural members. The Effective Width Method (EWM) performs this simulation by changing the geometry of the cross‐section, being the pioneer methodology for the analysis of thin‐walled members. The Direct Strength Method (DSM) allows the determination of resistant forces through simple‐to‐use formulas, eliminating the calculation of the geometric properties of the section as required by the EWM. This paper presents the Effective Modulus Method (EMM) concept, which simulates the effects of local and distortional post‐buckling, to establish simplified design methodologies, desirably leading to more rigorous results than those provided by the available methodologies. In this approach, local and distortional post‐buckling effects are simulated by changing the mechanical characteristics (namely the modulus of elasticity) of the cross‐section of the members. Specifically, the article presents (i) the proposal of the new design methodology and (ii) the comparison with experimental and numerical results obtained by finite element shell models.

Fourier analysis of membrane locking and unlocking

Despite four decades of research membrane locking remains an important issue hindering the development of effective beam and shell finite elements. In this article, we utilize Fourier analysis of the complete spectrum of natural vibrations and propose a criterion to identify and evaluate the severity of membrane locking. To demonstrate our approach, we utilize primal and mixed Galerkin formulations applied to a circular Euler–Bernoulli ring discretized using uniform, periodic B-splines. By analytically computing the discrete Fourier operators, we obtain an exact representation of the normalized error across the entire spectrum of eigenvalues. Our investigation addresses key questions related to membrane locking, including mode susceptibility, the influence of polynomial order, and the impact of shell/beam thickness and radius of curvature. Furthermore, we compare the effectiveness of mixed and primal Galerkin methods in mitigating locking. By providing insights into the parameters affecting locking and introducing a criterion to evaluate its severity, this research contributes to the development of improved numerical methods for thin beams and shells.

Virtual Dynamics Model for 5-axis Machining of Thin-Walled Blades

Abstract The five-axis ball-end milling dynamics of thin-walled blades is presented. The cutting forces are predicted from the ball end mill–blade geometry engagement maps along the tool path. The Frequency Response Function (FRF) of the thin-walled blade is predicted using Finite Element shell elements, and it is updated along the toolpath as the metal is removed. The predicted cutting forces are applied on both the workpiece and tool FRFs to predict the forced vibrations and chatter stability at each tool location. A simplified method to update the cutter–workpiece engagement (CWE) is used to obtain the three-dimensional stability lobe diagram at each desired point on the blade. The integrated model is used to simulate the 5-axis machining of thin-walled blades in the digital environment. The proposed digital model is experimentally validated by machining a series of thin-walled rectangular plates and a twisted fan blade.

Rotordynamic analyses with variable-kinematic beam and shell finite elements

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.

Differentiable Stripe Patterns for Inverse Design of Structured Surfaces

Stripe patterns are ubiquitous in nature and everyday life. While the synthesis of these patterns has been thoroughly studied in the literature, their potential to control the mechanics of structured materials remains largely unexplored. In this work, we introduce Differentiable Stripe Patterns---a computational approach for automated design of physical surfaces structured with stripe-shaped bi-material distributions. Our method builds on the work by Knöppel and colleagues [2015] for generating globally-continuous and equally-spaced stripe patterns. To unlock the full potential of this design space, we propose a gradient-based optimization tool to automatically compute stripe patterns that best approximate macromechanical performance goals. Specifically, we propose a computational model that combines solid shell finite elements with XFEM for accurate and fully-differentiable modeling of elastic bi-material surfaces. To resolve non-uniqueness problems in the original method, we furthermore propose a robust formulation that yields unique and differentiable stripe patterns. We combine these components with equilibrium state derivatives into an end-to-end differentiable pipeline that enables inverse design of mechanical stripe patterns. We demonstrate our method on a diverse set of examples that illustrate the potential of stripe patterns as a design space for structured materials. Our simulation results are experimentally validated on physical prototypes.

A contact model based on the coefficient of restitution for simulations of bio-prosthetic heart valves.

A new general contact model is proposed for preventing inter-leaflet penetration of bio-prosthetic heart valves (BHV) at the end of the systole, which has the advantage of applying kinematic constraints directly and creating smooth free edges. At the end of each time step, the impenetrability constraints and momentum exchange between the impacting bodies are applied separately based on the coefficient of restitution. The contact method is implemented in a rotation-free, large deformation, and thin shell finite-element (FE) framework based on loop's subdivision surfaces. A nonlinear, anisotropic material model for a BHV is employed which uses Fung-elastic constitutive laws for in-plane and bending responses, respectively. The contact model is verified and validated against several benchmark problems. For a BHV-specific validation, the computed strains on different regions of a BHV under constant pressure are compared with experimentally measured data. Finally, dynamic simulations of BHV under physiological pressure waveform are performed for symmetrical and asymmetrical fiber orientations incorporating the new contact model and compared with the penalty contact method. The proposed contact model provides the coaptation area of a functioning BHV during the closing phase for both of the fiber orientations. Our results show that fiber orientation affects the dynamic of leaflets during the opening and closing phases. A swirling motion for the BHV with asymmetrical fiber orientation is observed, similar to experimental data. To include the fluid effects, fluid-structure interaction (FSI) simulation of the BHV is performed and compared to the dynamic results.

Cross-sectional optimization of perforated pallet rack columns against distortional and global buckling

This paper presents a cross-sectional optimization procedure to increase the load carrying capacity of a perforated rack column under a compression load. The optimization method is based on a Genetic Algorithm (GA) combined with the Generalized Beam Theory (GBT), which allows to take perforations as well as sectional deformations into account. The results indicate that the optimal cross-section geometry depends on the dominant buckling mode. Therefore, a combined optimization criterion to consider distortional and global buckling is defined. Finally, nonlinear shell finite elements analyses are done in order to assess the structural behaviour of optimized cross-sections.

Vibration-based numerical identification of effective orthotropic elastic properties for finite element model of cross-laminated timber panel

The influence of non-gluing of narrow surfaces of boards on all effective orthotropic elastic moduli for shell or solid finite element models of cross-laminated timber (CLT) panels is investigated. This lack of gluing leads to discontinuous layers and severe meshing difficulties. The objective of this study is to create an equivalent homogeneous layer and equivalent material properties in this case. Two finite element (FE) models were built for comparison. The first model, using solid elements, is the reference without gluing the narrow surfaces. The second considers homogeneous layers and equivalent material properties. Thanks to an identification process, all the elastic moduli of the model with homogeneous layers are adjusted to minimize the difference in vibration frequencies between the two models. A method is proposed to significantly speed-up the calculation of first derivatives, which are necessary for the identification process. The process was applied for 36 configurations of CLT panels with different geometrical properties (number of layers, width and thickness of boards) within the practical range. Metamodels are proposed to quickly calculate the effective moduli based on the vibration properties of CLT panels. The Verification and Validation methodology was applied to evaluate the performance of the two simplified FE models using the effective moduli.

Nonlinear earthquake responses of unanchored cylindrical liquid storage tanks on flexible soil

A sophisticated and versatile finite-element model of the nonlinear behavior of unanchored cylindrical liquid storage tanks on flexible soil is proposed and applied for nonlinear earthquake response analysis of the system. The tank is modeled using shell finite elements with material and geometric nonlinearities. The hydrodynamic pressure is represented by acoustic finite elements. The effects of the base uplift are considered by nonlinear springs with negligible and large stiffness coefficients in tension and compression, respectively. The flexible soil is modeled using mechanical springs and dashpots. The effects of base uplift and soil flexibility are examined for typical unanchored tall and broad tanks on flexible soil with various shear-wave velocities. It is observed that the structural responses and hydrodynamic pressure, including the convective pressure of an unanchored tall tank on rigid/flexible soil, are significantly influenced by the base uplift. Although the base uplift in a broad tank is not as large as that in a tall tank, the structural responses and hydrodynamic pressure of an unanchored broad tank can be different from those of an anchored system. However, the convective pressure of the broad tank is not significantly affected by the base uplift. The earthquake responses of the system are principally influenced by the horizontal component of the earthquake ground motion rather than the vertical component.

Optimisation of inverse Finite Element Method for shape sensing of thin-walled cylinders by using Generalised Beam Theory

The inverse Finite Element Method (iFEM) can reconstruct the deformation of structures based on measured strains. This procedure has important applications in Structural Health Monitoring. However, its accuracy depends on the number of the in-situ strain gauges and for shell structures with complex deformation, this number can easily become too high for practical use. Generalised Beam Theory (GBT) can describe the complex deformation of any thin-walled beam member as a linear combination of pre-determined cross-section deformation modes. Instead of shell finite elements, GBT uses bar finite elements. Using these advantages, this paper proposes an iFEM/GBT formulation for the shape sensing of thin-walled cylinders. Great reduction of strain gauges is in this way achieved in comparison with the use of the classic inverse-shell finite elements.

Comparison of various thin-walled composite beam models for thermally induced vibrations of spacecraft boom

A sufficiently accurate, yet computationally efficient prediction of thermomechanical behavior is essential for the structural design of spacecraft. This paper presents a novel modeling approach for thermally induced vibrations of thin-walled composite tubes in spacecraft, in which the generalized beam theory is employed, and meanwhile, the non-conventional shear and extension modes are introduced into it. Furthermore, a cross-sectional temperature model in terms of harmonic functions is developed to provide more accurate temperature field, which is performed by taking into account the strong couplings between average and perturbation temperatures and between perturbation temperatures. Combining the beam model and the temperature model, a thermal–structural finite element formulation for composite boom in space environment is established. By comparison to the finite shell element, the accuracy and characteristic feature of these new models are illustrated. The results show that the additional deformation modes may play a significant role in thermal–structural analysis of thin-walled composite beams.

High-fidelity numerical simulation and experimental validation of a 1600-mm-diameter axial loaded grid stiffened cylindrical shell

Stiffened cylindrical shells are commonly used in the launch vehicle, which are prone to buckling under axial compression load and extremely sensitive to imperfections. Establishing a high-fidelity finite element model is a prerequisite for correctly analyzing buckling load and imperfection sensitivity of stiffened cylindrical shells. This paper proposes a homogenization-based model updating approach for the grid stiffened cylindrical shell with many chamfers, which can establish a high-fidelity shell finite element model that maintains calculation efficiency and accuracy advantages. It uses the equivalent elastic constants of the unit cell as the link to modify the size parameters of the finite element model for the grid stiffened cylindrical shell. Moreover, the axially compressed buckling test of a 1.6-m-diameter orthogrid stiffened cylindrical shell is performed to verify the effectiveness of the proposed method. Through comparison, it could be found that the updated finite element model has a higher analysis accuracy. Specifically, the analysis error of the initial finite element model is −16.85% compared with the test result, while the error of the updated model is only −0.67%. The updated model can be used with full confidence for simulating mechanical performance such as the buckling process and the imperfection sensitivity analysis. Besides, it also indicates the necessity of considering the structural characteristics such as the chamfer in the optimization design process of the grid stiffened cylindrical shell structure.

Direct strength prediction of cold-formed steel fixed-ended singly-symmetric columns buckling in flexural–torsional modes

The work reported in this paper is a sequel of previous numerical investigations carried out by the authors on the post-buckling behaviour, strength and Direct Strength Method (DSM) design of cold-formed steel (CFS) fixed-ended plain and lipped channel columns (both singly-symmetric with respect to the major-axis) buckling in flexural–torsional modes and experiencing global-global interaction – a coupling phenomenon involving major-axis flexural–torsional (FMT) and minor-axis flexural (Fm) buckling that was first unveiled in those studies. In the specific context of plain and lipped channel columns, a DSM-based design approach for columns buckling in FMT modes and failing in either pure FMT or FMT-Fm interactive modes was developed and shown to provide accurate, mostly safe and reliable predictions of the experimental (only a few) and numerical failure loads available. The success of this research effort encouraged the authors to extend the scope of the investigation to fixed-ended columns with other singly-symmetric cross-section shapes, namely return-lip, web-stiffened and web/flange-stiffened lipped channels, hat-sections and rack-sections (deemed representative of any arbitrary singly-symmetric cross-section). The results attained during this investigation are presented and discussed in detail – they concern the post-buckling behaviour, strength and DSM-based design of the columns analysed, including fairly extensive failure load sets (Ansys non-linear shell finite element models are used to obtain them). The main output of this work is a DSM-based design approach for arbitrary CFS fixed-ended singly-symmetric columns exhibiting pure FMT or FMT-Fm interactive failures.

Cross-section resistance and design of stainless steel CHS and EHS at elevated temperatures

The structural behaviour and design of stainless steel circular hollow sections (CHS) and elliptical hollow sections (EHS) at elevated temperatures are investigated in this paper. Shell finite element models of stainless steel CHS and EHS are created and validated against experimental results from the literature, which are subsequently used to generate benchmark structural performance data. Parametric studies are performed on a large number of cold-formed and hot-rolled austenitic, duplex and ferritic stainless steel CHS and EHS subjected to (i) pure axial compression, (ii) pure bending, (iii) combined axial compression and bending and (iv) combined bending and shear at elevated temperatures; the studied cases comprise 24,495 stainless steel CHS and EHS in fire and cover a wide range of cross-section slendernesses and elevated temperature levels. Calibrated against the benchmark structural performance data obtained from the numerical parametric studies, new design proposals for predicting the cross-section resistances of stainless steel CHS and EHS in fire are put forward. The accuracy, safety and reliability of the new design proposals are assessed. It is shown that in comparison to the design provisions of the European structural steel fire design standard EN 1993-1-2, the proposed design methods provide more accurate and safe-sided cross-section resistance predictions for stainless steel CHS and EHS at elevated temperatures.

A layer-wise plasticity-based approach for the analysis of reinforced concrete shell structures using a mixed finite element

The analysis of reinforced concrete shell structures accounting for material nonlinearity is addressed. The structural response is numerically evaluated using a mixed shell finite element and a plasticity-based material behaviour. The finite element is a quadrilateral with four nodes and is based on self-equilibrated assumed stresses. The kinematic fields are interpolated only along the element boundary by polynomials up to the third order. The generalised shell stresses are evaluated through layer-wise integration of the Cauchy pointwise stresses. This allows appropriated three-dimensional elastoplastic constitutive equation to be employed and to include multiple reinforcing layers. The integration of the constitutive laws is performed using a dual decomposition method which preserves the assumed stress interpolation. Results, including civil engineering applications, show that the proposed approach is reliable and accurate in evaluating the static nonlinear equilibrium path. Additionally, the mixed finite element shows a quadratic rate of convergence in the collapse load and low error even for coarse meshes.

NUMERICAL SIMULATION OF OSCILLATIONS OF A PLATE IN RESTING FLUID

This paper presents a numerical solution of the problem of oscillations of a flexible plate placed in a viscous incompressible fluid done with ANSYS. In constructing a finite element model of the plate, both volumetric and shell finite elements, which are more commonly used in engineering applications, are considered. By way of example it is shown that it is possible to transition from a volumetric mesh to a shell mesh without loss of accuracy in solving this problem and similar problems. The considered coupled approach to solving the problem for the shell plate mesh has an important practical application in solving real-world problems of aeroelasticity, because in engineering practice it is much more convenient to represent structures and constructions in the form of beam and shell models. The solution of this problem is of particular importance for the verification of techniques of numerical modelling of coupled aeroelasticity problems.

Application of Finite Element Method to Create a Digital Elevation Model

The generation of a topographical surface or digital elevation model for a given set of points in space is a known problem in civil engineering and topography. In this article, we propose a simple and efficient way to obtain the terrain surface by using a structural shell finite element model, giving advice on how to implement it. The proposed methodology does not need a large number of points to define the terrain, so it is especially suitable to be used with data provided by manual topographical tools. Several examples are developed to demonstrate the easiness and accuracy of the methodology. The digital terrain model of a real landscape is modeled by using different numbers of points (from 49 to 400) using a regular mesh or a randomly generated cloud of points. The results are compared, showing how the proposed methodology creates a sufficiently accurate model, even with a low number of points (compared with the thousands of points handled in a LiDAR representation). A real case application is also shown. As an appendix, the sample code to generate the examples is provided.

Experimental investigation of doubly-symmetric built-up I–girders subjected to high moment gradient

Recent analytical studies have shown that the AISC 360-16 Specification overpredicts the flexural resistance of certain built-up steel I-girders. The largest overpredictions were observed in I‑girders subjected to a high moment gradient having unstiffened webs with a large web slenderness ratio. The objectives of the current research are to provide experimental validation of these strength overpredictions, validate the accuracy of full-nonlinear shell finite element analysis (FEA) solutions, and further investigate the behavior of these susceptible built-up I-girders. To accomplish the objectives, six largescale experimental tests targeted web slenderness ratios ranging from 77 to 192 and a moment gradient factor, Cb, of 1.74 (single curvature (SC) bending) and 2.31 (reverse curvature (RC) bending), calculated using common design equations. Additionally, full nonlinear shell FEA simulations were conducted to evaluate the correlation with the experiments. The tests showed flexural strengths smaller than the predicted resistances from AISC 360, with the predicted resistances ranging from 4% to 16% unconservative for SC members and 9% to 32% unconservative for RC members. Furthermore, the FEA simulation strengths were similar to the strengths measured in the experiments. The results indicate that there are two main phenomena leading to the overpredictions by AISC 360: web distortion effects exacerbated by web shear and the direct scaling of the uniform LTB strength curve by Cb up to the plateau resistance, overestimating the strength gains from moment gradient due to insufficient accounting for the onset of yielding.

Cardiac hypertrophy simulations using parametric and echocardiography-based left ventricle model with shell finite elements

In our paper, we simulated cardiac hypertrophy with the use of shell elements in parametric and echocardiography-based left ventricle (LV) models. The hypertrophy has an impact on the change in the wall thickness, displacement field and the overall functioning of the heart. We computed both eccentric and concentric hypertrophy effects and tracked changes in the ventricle shape and wall thickness. Thickening of the wall was developed under the influence of concentric hypertrophy, while the eccentric hypertrophy produces wall thinning. To model passive stresses we used the recently developed material modal based on the Holzapfel experiments. Also, our specific shell composite finite element models for heart mechanics are much smaller and simpler to use with respect to conventional 3D models. Furthermore, the presented modeling approach of the echocardiography-based LV can serve as the basis for practical applications since it relies on the true patient-specific geometry and experimental constitutive relationships. Our model gives an insight into hypertrophy development in realistic heart geometries, and it has the potential to test medical hypotheses regarding hypertrophy evolution in a healthy and heart with a disease, under the influence of different conditions and parameters.

Mixed Eulerian–Lagrangian modeling of sheet metal roll forming

We propose a nonlinear shell finite element model to simulate sheet metal roll forming, a continuous forming process to produce endless metal profiles. A mixed Eulerian–Lagrangian kinematic description is employed to overcome the drawbacks of the common Lagrangian parametrization. The finite element mesh is detached from the particle motion in axial direction and, thus, facilitates a two-step solution procedure to capture the continuous forming process: First, an equilibrium is sought with the account for contact and plastic flow. Secondly, the material transport is taken into account, which amounts to the integration of an advection problem for the plastic variables. The continuum plasticity model with through-the-thickness integration for the stress resultants guarantees a precise resolution of the forming process in each cross section of the Kirchhoff–Love shell. A series of simulations is carried out to ascertain the convergence of the numerical scheme, to highlight the impact of characteristic parameters and to establish a correspondence to a reference computation with the commercial software Abaqus in a simplified static setting. A physical experiment is devised on an actual roll forming mill to assess the quality of the current computational model.