Displacement-based Formulation Research Articles

Modeling Challenges and Limitation Principles of Reissner’s Mixed Approaches to Laminates

Displacement-based formulations for composite structures are directly linked to the functional called the Principle of Virtual Displacements and present the minimum number of independent modeled fields, which are the displacements. However, these models usually provide an inadequate approximation of the transverse stress fields and suffer from pathological numerical problems. These issues have been addressed with the introduction of mixed variational formulations where independent stress fields are included in the definition of the corresponding functionals. Particularly effective and widely used are Reissner’s mixed variational theorem and Hellinger–Reissner Principle. Both Reissner’s mixed variational theorem and Hellinger–Reissner principle allow the a priori enforcing of the interlaminar equilibrium because of the axiomatic modeling of the transverse stresses. However, these functionals are different and present their own challenges. Under certain circumstances, Reissner’s Mixed Variational Theorem was shown to produce thickness oscillations of the displacements and transverse stresses. It was not clear if Hellinger-Reissner Principle had similar properties. It is shown that both functionals may present numerical oscillations. Moreover, the reconstructed displacement and stress fields are actually identical between Reissner’s Mixed Variational Theorem and Hellinger-Reissner Principle if the orders of the in-plane stresses of the Hellinger–Reissner Prinicple case are sufficiently large. In other words, a Hellinger–Reissner Principle/Reissner’s Mixed Variational Theorem Limitation Principle is shown to exist in addition to the well-studied limitation principle between Hellinger–Reissner principle and principle of virtual displacements.

Time history nonlinear analysis for 2D modelization of an existing building using flexibility and displacement-based formulation

The object of the research is the response of a structure in time between formulations, namely force-based (flexibility-based) and displacement-based. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility based) and displacement based. This research aimed to compare between two formulations namely displacement based and force-based (flexibility-based), in order to show the accuracy and the reliability of the second one. In the case of nonlinear time history response of a residential two-storey reinforced concrete building, which is representative of typical residential building construction in Italy in the 1970’s and 1980’s with the aim of taking advantages of flexibility-based formulation if the comparison proves conclusive. In the course of the research, the former approach is a classical two-node, displacement-based, Euler-Bernoulli frame element with the disadvantage of discretizing the structure thus increasing the number of degrees of freedom and equation, in order to achieve the required precision. The latter is a two-node, force-based, Euler Bernoulli frame element. The main advantage of the second element is that it is «exact» within the relevant frame element theory. This implies that one element per frame member (beam or column) is used in preparing the frame mesh, thus leading to a reduction of the global number of degrees of freedom. Therefore, a reduction in number of equations to solve, resulting in a saving of time with increased precision. An existing two-story reinforced concrete structure subjected to three earthquakes separately is presented in order to carry out a comparative study in terms of time history structure response between the two formulations cited above namely force based (flexibility-based) and displacement-based. As a result of the research, it is shown that with flexibility-based approach no discretization error occurs and all governing equations are satisfied exactly, making a possibility of simpler and faster numerical modeling. In the proposed approach will be of great use in the field of numerical modeling due to its ease and accuracy. This elegant formulation is promising for future work thanks to its advantages.

Analysis of Stress Distribution in a Curved Functionally Graded Porous Beam Using the Unified Shear Deformation Theory

Using unified shear deformation theory (USDT) and a modified power law, the current study examines bending properties of two-dimensional functionally graded curved porous beam. In order to improve accuracy, this method incorporates equilibrium equations, potential energy, and the idea of a neutral surface. The analysis uses a boundary conditions, namely simply supported . A functionally graded beam composed of metal and ceramic with both even and unequal porosity is modeled. The formulation takes into account the symmetrical material gradation, which guarantees alignment between the geometrical and physical neutral surfaces. A displacement-based formulation and energy concepts are used, which leads to a more thorough and accurate beam analysis. This approach effectively regulates the constant changing of material characteristics in FGMs, takes into consideration higher-order shear deformation effects, and does away with the requirement for shear correction factors. As a result, it improves structural behavior predictions, which makes USDT very useful for advanced material applications. The equilibrium equations for the beams are derived using the Hamilton technique and solved with the Kuhn-Tucker conditions.

Kinetic- and Strain-Energy Approaches in the Thermal Analysis of Constrained Mechanical Systems: A Comparative Study

Abstract Despite the unconstrained thermal expansion is assumed stress-free, the conventional FE approach requires formulating elastic forces, and this in turn leads to elastic stresses. A displacement-based formulation, on the other hand, can be used to address this limitation by converting the thermal energy to kinetic energy instead of strain energy. The fundamental differences between the strain- and kinetic-energy approaches are discussed. It is shown that the unconstrained thermal expansion predicted using the kinetic-energy approach is independent of the continuum constitutive model, and consequently, such a formulation can be used for both solids and fluids. The displacement (kinetic) and strain (stress) formulations are discussed to shed light on the mechanism of thermal expansion at the macroscopic level. The thermal-expansion displacement formulation (TEDF) and position-gradient multiplicative decomposition into thermal and mechanical parts are used to compute the thermal stresses due to boundary and motion constraints (BMC). TEDF implementation issues are discussed and constant matrices evaluated at a preprocessing stage after applying sweeping matrix technique to eliminate rigid-body thermal-displacement translational modes are identified. Furthermore, the softening effect due to the constitutive-model dependence on the temperature is investigated at high temperatures. Numerical results are presented to show fundamental differences between the TEDF approach that converts heat energy to kinetic energy and conventional FE approach that converts heat energy to strain energy that produces elastic stresses.

Rate-dependency analysis of mode I delamination by means of different data reduction strategies for the GDCB test method

The loading-rate mechanical response of the mode I delamination in composites and adhesively bonded joints was investigated. The tests were carried out using a recently developed Guided Double Cantilever Beam test method. Three different data reduction methods were proposed and assessed: a displacement-based formulation, a near-crack-tip displacement formulation, and a numerical assessment using the Virtual Crack-Closure Technique method. The methods account for the dynamic effects which may be present. While small differences between the three different methods can be seen, no rate-dependency higher than the uncertainty for the materials and the loading rates considered have been evidenced.

An implicit 3D nodal integration based PFEM (N-PFEM) of natural temporal stability for dynamic analysis of granular flow and landslide problems

The particle finite element method (PFEM) is a robust approach for modelling large deformation problems with free surface evolution. The classical PFEM, however, requires variable mapping from old to new quadrature points when adopting history-dependent material models in granular flow and landslide problems. Although the nodal integration technique circumvents this issue, it makes the PFEM temporal instable in dynamic analysis when using a displacement-based formulation. In this study, we developed a new version of a three-dimensional (3D) Nodal integration based PFEM (N-PFEM) using a mixed variational principle with the final problem resolved in mathematical programming. The proposed N-PFEM not only inherits the benefit from the nodal integration scheme that no variable mapping is required for handling history-dependent models but also is naturally temporal stable requiring no ad-hoc stabilization technique. We simulated a series of benchmark problems to demonstrate its nature of temporal stability as well as other admirable features such as the volumetric-locking free property and capability for tackling extreme configuration changes. Additionally, its application to a 3D landslide with a sensitive clay layer is shown to highlight its robustness.

Rectangular filled composite members made from high strength materials: Behavior and design of columns and beams

The AISC Specification limits the use of steel and concrete materials in filled composite members to concrete strength (fc′) less than or equal to 69 MPa and steel yield strength (Fy) less than or equal to 517 MPa. This paper presents the results of comprehensive analytical investigations conducted using benchmarked numerical models to evaluate the behavior of rectangular filled composite members made from one or both materials exceeding the AISC Specification strength limits. The numerical models are developed and analyzed using the displacement-based formulation of the fiber-based finite element in OpenSees. Prior research has developed effective (phenomenological) stress–strain curves for steel and concrete materials that implicitly account for the effects of steel yielding, local buckling, concrete cracking, crushing and confinement, and the interaction between the two materials in the composite cross-section. This paper first benchmarks the modeling approach and then uses the benchmarked model to conduct analytical parametric studies covering a wide range of parameters, including: member slenderness, section slenderness, section aspect ratio, steel yield stress and concrete compressive strength. The results from the parametric studies are used to evaluate behavior and develop new design equations based on nominal stress distributions over the composite section. The nominal steel stress in compression (Fcr) is expressed as a continuous function of the section slenderness and yield strength, thus eliminating the current AISC Specification’s need for section classification (compact, noncompact and slender). The new design equations are recommended for relaxing the material strength limitations in the next version of the AISC Specification.

Response of shear critical reinforced concrete frames and walls under monotonic loading

This paper focuses on a novel finite element formulation which can predict the bending moment-shear force-axial force interaction of reinforced concrete frames and walls, and validate it against 170 experiments available in literature. This distributed plasticity element is established on force-based finite element method, where the relationship between element nodal forces and section forces are exactly known. Hence, element discretization is nonessential when modelling frames using this formulation, reducing the number of degrees of freedom in the numerical model compared to displacement-based formulations. The computations are carried out at four hierarchical levels, namely structure, element, section and fibre. There are two nested iterative procedures at the structure level and the section level. In the existing formulation, these iterative procedures are computationally demanding due to use of initial stiffness matrices. Furthermore, it uses Modified Compression Field Theory at the fibre level, which has inherent drawbacks compared to its more evolved version, the Disturbed stress Field Model. The current study refines the iterative procedures at structure and section levels to fully operate with tangent stiffness matrices to improve the speed of convergence. In addition, the Modified Compression Field Theory is replaced with the Disturbed stress Field Model at the fibre level to compute fibre resisting force for a given fibre deformation, accounting for both averaged behaviour and local crack slip. The novel element is validated by comparing the predicted results with experimental results of 170 tests found in the literature. It is shown that the novel element predicts the load carrying capacity well with an average experimental-to-predicted load carrying capacity ratio of 0.99 and a coefficient of variation of 12.8%. Furthermore, the element can be used to discuss the different failure mechanisms of reinforced concrete frame elements.

Leveraging spectral analysis to elucidate membrane locking and unlocking in isogeometric finite element formulations of the curved Euler–Bernoulli beam

In this paper, we take a fresh look at using spectral analysis for assessing locking phenomena in finite element formulations. We propose to “measure” locking by comparing the difference between eigenvalue and mode error curves computed on coarse meshes with “asymptotic” error curves computed on “overkill” meshes, both plotted with respect to the normalized mode number. To demonstrate the intimate relation between membrane locking and spectral accuracy, we focus on the example of a circular ring discretized with isogeometric curved Euler–Bernoulli beam elements. We show that the transverse-displacement-dominating modes are locking-prone, while the circumferential-displacement-dominating modes are naturally locking-free. We use eigenvalue and mode errors to assess five isogeometric finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration and three locking-free formulations based on the B-bar, discrete strain gap and Hellinger–Reissner methods. Our study shows that spectral analysis uncovers locking-related effects across the spectrum of eigenvalues and eigenmodes, rigorously characterizing membrane locking in the displacement-based formulation and unlocking in the locking-free formulations.

First-order VEM for Reissner\u2013Mindlin plates

In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

Numerical analysis of steel–concrete composite beams with partial interaction: A plastic-hinge approach

A two-dimensional displacement-based formulation with a plastic-hinge approach for the numerical analysis of composite beams with partial shear connection is presented here. The co-rotational approach is applied in the numerical model to allow large displacements and rotations. The axial and transverse displacement functions are defined to avoid locking problems. The simulation of the materials and shear connection nonlinear behaviors are approached via the strain compatibility method (SCM), where the constitutive relations are explicitly used. The slip in the steel section–concrete slab interface is considered by the axial force decomposition in the cross-section level by the degree of composite action, without introducing degrees of freedom in the finite element. The numerical proposal of the present work is tested by simulating steel–concrete composite beams and comparing the obtained results with the experimental and numerical data already known. This formulation is verified as numerically stable and without locking phenomena, and good convergence with literature results was obtained. However, more refined finite element (FE) meshes are needed.

Reissner’s Mixed Variational Theorem and triangular finite element discretizations: an energetic interpretation

Reissner’s Mixed Variational Theorem (RMVT) presents both displacements and transverse stresses as primary variables. This property allows the a-priori fulfillment of both interlaminar compatibility and equilibrium with potentially excellent numerical performance. However, the triangular finite element based on RMVT has never been assessed from an energetic perspective. This aspect is investigated in the present contribution for the first time: the functional reconstitution technique is extended to RMVT-based triangular elements (retaining transverse normal stress) and it is demonstrated that the elements exhibit significantly lower errors than the corresponding displacement-based formulation. Moreover, the percentage errors on the approximation of the semi-complementary energy is shown to be invariant with the plate thickness-to-width ratio.

Enriched beam finite element models with torsion and shear warping for the analysis of thin-walled structures

This paper presents three beam Finite Element (FE) formulations developed for the analysis of thin-walled structures. These account for out-of-plane cross-section warping by removing the classical rigid body cross-section hypothesis and capture the interaction of axial/bending stress components with shear and torsion.The beam FE models rely on different kinematic assumptions to describe out-of-plane cross-section deformations. Indeed, warping displacement field is interpolated in the element volume according to different approaches, with increasing level of accuracy and detail. First two models adopt a coarse warping description, where warping displacement field is defined as the linear combination of assumed warping profiles and unknown kinematic parameters. In the first model, these are considered as equal to the generalized cross-section torsional curvature and shear strains and a classical displacement-based formulation is adopted to derive the element governing equations. In the second model, warping parameters are assumed as independent kinematic quantities and a mixed approach is considered to derive the FE formulation. Third model, also relying on a mixed formulation, independently interpolates warping by introducing additional degrees of freedom on the cross-section plane, thus, resulting in a richer description of the out-of-plane deformations. This latter is also adopted to propose a numerical procedure for the warping profile evaluation of thin-walled beams subjected to torsional and shear forces, for general cross-section geometry.The efficiency and accuracy of the proposed FE formulations are validated by simulating the response of thin-walled structures under torsion and coupled torsion/shear actions and the influence of the kinematic assumptions characterizing each formulation is discussed.

Displacement-based formulation of Koiter's method: Application to multi-modal post-buckling finite element analysis of plates

Koiter's asymptotic method enables the calculation and deep understanding of the initial post-buckling behaviour of thin-walled structures. For the single-mode asymptotic analysis, Budiansky (1974) presented a clear and general formulation for Koiter's method, based on the expansion of the total potential energy function. The formulation from Budiansky is herein revisited and expanded for the multi-modal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. Given the admittedly difficult implementation of Koiter's method, especially for multi-modal analysis and during the evaluation of the third– and fourth–order tensors involved in Koiter's analysis; the presented study proposes a formulation and notation with close correspondence with the implemented algorithms. The implementation is based on state-of-the-art collaborative tools: Python, NumPy and Cython. The kinematic relations are specialized using von Kármán shell kinematics, and the displacement field variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) finite element, modified to reach third-order interpolation also for the in-plane displacements, using only 4 nodes per element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order fields. The formulation and implementation are verified by comparing results for isotropic and composite plates against established literature. Finally, results for multi-modal displacement fields with up to 5 modes and corresponding post-buckling factors are reported for future reference.

Shear-deformable hybrid finite-element formulation for lateral-torsional buckling analysis of composite thin-walled members

A shear deformable hybrid finite element formulation is developed for the lateral-torsional buckling analysis of fibre-reinforced composite thin-walled members with open cross-section. The method is developed by using the Hellinger–Reissner functional. Comparison to the displacement-based formulations the current hybrid formulation has the advantage of incorporating the shear deformation effects easily by using the strain energy of the shear stress field without modifying the basic kinematic assumptions of the thin-walled beam theory. Numerical results are validated through comparisons with results based on other formulations presented in the literature. Examples illustrate the effects of shear deformations and stacking sequence of the composite layers in predicting bucking loads.

A modelling framework for coupled hydrogen diffusion and mechanical behaviour of engineering components

In this paper, we propose a finite element formulation for solving coupled mechanical/diffusion problems. In particular, we study hydrogen diffusion in metals and its impact on their mechanical behaviour (i.e. hydrogen embrittlement). The formulation can be used to model hydrogen diffusion through a material and its accumulation within different microstructural features of the material (dislocations, precipitates, interfaces, etc.). Further, the effect of hydrogen on the plastic response and cohesive strength of different interfaces can be incorporated. The formulation adopts a standard Galerkin method in the discretisation of both the diffusion and mechanical equilibrium equations. Thus, a displacement-based finite element formulation with chemical potential as an additional degree of freedom, rather than the concentration, is employed. Consequently, the diffusion equation can be expressed fundamentally in terms of the gradient in chemical potential, which reduces the continuity requirements on the shape functions to zero degree, {mathcal {C}}_{0}, i.e. linear functions, compared to the {mathcal {C}}_{1} continuity condition required when concentration is adopted. Additionally, a consistent interface element formulation can be achieved due to the continuity of the chemical potential across the interface—concentration can be discontinuous at an interface which can lead to numerical problems. As a result, the coding of the FE equations is more straightforward. The details of the physical problem, the finite element formulation and constitutive models are initially discussed. Numerical results for various example problems are then presented, in which the efficiency and accuracy of the proposed formulation are explored and a comparison with the concentration-based formulations is presented.

An Efficient Unsplit Perfectly Matched Layer for Finite-Element Time-Domain Modeling of Elastodynamics in Cylindrical Coordinates

The truncation of an unbounded medium into a finite domain and the elimination of reflections from artificial medium boundaries is important in numerical simulations of elastodynamic equations. The perfectly matched layer (PML) technique as a novel absorbing boundary has demonstrated very high efficiency for elastic wave equation models. Based on the stretched coordinate concept, an efficient and unsplit-field perfectly matched layer equation for elastic waves is formulated in a cylindrical coordinate system. By introducing integrated complex variables for radial direction and auxiliary functions, the PML formulation is extended in cylindrical coordinates based on the second-order elastic wave equation with displacements as basic unknowns. Finite-element time-domain modeling of the second-order unsplit PML for the elastodynamic equation is presented, which is a standard displacement-based formulation for the computational domain including a PML region. The formulas for the special cases with 2-D axisymmetric coordinates and polar coordinates are also given. The efficiency of the proposed formulations is illustrated by a numerical example using the finite-element method.

A semi-analytical approach for the analysis of variable-stiffness panels with curvilinear stiffeners

A semi-analytical method is developed for the analysis of composite stiffened panels, where stiffness variability is achieved through a combination of fiber steering and curvilinear stringers. The approach relies upon a 2D plate model for the skin and a 1D beam description for the stiffeners, according to first-order shear deformation theory. The approach is developed in the framework of a displacement-based formulation, where Legendre polynomials are used for approximating the displacement field, and the Ritz method is used for the solution strategy. The resulting tool is characterized by its excellent computational efficiency, and allows problems in free-vibrations, buckling and thermal buckling to be solved with reduced effort. The quality of the results is assessed by an extensive set of comparison against results from the literature, demonstrating the potential of the tool as a means for performing preliminary parametric studies for these variable-stiffness, variable-geometry structures.

An enhanced finite element model for reinforced concrete members under torsion with consistent material parameters

This paper deals with the development of a non-linear finite element model for reinforced concrete members under torsion. Using multi-fiber approach and displacement-based formulation, an enhanced multi-fiber 3D beam is proposed for predicting the behavior of reinforced concrete elements under torsion. The sectional analysis under elastic torsion is considered following Saint-Venant torsional theory for beam in order to take into account the effect of warping phenomenon. Elastic behavior under large displacements is also investigated in this model using a second-order approximation of the Green-Lagrange strains. In the inelastic domain, after cracking, following the space truss theory, the whole element is assumed to act as a tube, meaning the applied torsional moment is resisted only by the shear flow in the wall of the tube. Then, the effective wall thickness of the member after cracking is determined by an empirical formulation developed by the authors. Moreover, the section is discretized into different regions following its material response. In each region, depending on its characteristics, an appropriate constitutive material model is applied. For the concrete, the proposed behavior models are based on the Modified of Compression Field Theory and its extension. In order to correctly predict the torsional response, the authors proposed some modifications in the tensile behavior of concrete, based on experimental tests in torsion. In the elastic domain, the model is validated by comparing to the analytical solution and some results from other researches. In the inelastic field, a good agreement is obtained between a series of experimental tests and numerical results.

F-Bar Aided Edge-Based Smoothed Finite Element Method with 4-Node Tetrahedral Elements for Static Large Deformation Elastoplastic Problems

A new type of smoothed finite element method (S-FEM), F-barES-FEM-T4, is demonstrated in static large deformation elastoplastic cases. F-barES-FEM-T4 combines the edge-based S-FEM (ES-FEM) and the node-based S-FEM (NS-FEM) for 4-node tetrahedral (T4) elements with the aid of the F-bar method in order to resolve the major issues of Selective ES/NS-FEM-T4. As well as most of the other S-FEMs, F-barES-FEM-T4 inherits pure displacement-based formulation and thus has no increase in DOF. Moreover, the cyclic smoothing procedure introduced in F-barES-FEM-T4 is effective to adjust the smoothing level so that pressure checkerboarding (oscillation) is suppressed reasonably. Some examples of static large deformation analyses for elastoplastic materials proof the excellent performance of F-barES-FEM-T4 in contrast to the conventional hybrid T4 element formulation.