Force-based Formulation Research Articles

Push ‘o ver: numerical simulation of the Castel di Lama pushover test through a force-based equivalent frame model

The experimental pushover test performed on the Castel di Lama building is studied by means of a finite element procedure based on the equivalent frame model with the aim to test the ability of this widespread model to reproduce the experimental results and the collapse mechanisms. Pier and spandrel macroelements are modelled as Timoshenko beams with plastic hinges to take into account shear and bending failures. Assuming the nodes as infinitely rigid and resistant, it is possible to model them by introducing properly sized offsets at the ends of pier and spandrel macroelements. A force-based formulation is adopted for the macroelement, taking advantage of its higher performances in terms of accuracy and efficiency with respect to the classical displacement-based method, and the capability of naturally avoiding shear-locking problems.The response of the building walls, either unreinforced or reinforced, is analysed in terms of resisting forces, story displacements and damage patterns, and compared with the experimental results.

Coupled Rigid-Block Analysis: Stability-Aware Design of Complex Discrete-Element Assemblies

The rigid-block equilibrium (RBE) method uses a penalty formulation to measure structural infeasibility or to guide the design of stable discrete-element assemblies from unstable geometry. However, RBE is a purely force-based formulation, and it incorrectly describes stability when complex interface geometries are involved. To overcome this issue, this paper introduces the coupled rigid-block analysis (CRA) method, a more robust approach building upon RBE’s strengths. The CRA method combines equilibrium and kinematics in a penalty formulation in a nonlinear programming problem. An extensive benchmark campaign is used to show how CRA enables accurate modelling of complex three-dimensional discrete-element assemblies formed by rigid blocks. In addition, an interactive stability-aware design process to guide user design towards structurally-sound assemblies is proposed. Finally, the potential of our method for real-world problems are demonstrated by designing complex and scaffolding-free physical models. • CRA method: 3D analysis and design of complex rigid-block assemblies with friction. • Nonlinear constrained optimisation coupling static equilibrium with kinematics. • Penalty formulation to measure structural infeasibility and identify unstable regions. • Stability-aware design process towards more stable design solutions. • Design, fabrication and construction of complex, scaffolding-free assemblies.

Optimization of the structural coupling between RC frames, CLT shear walls and asymmetric friction connections

This paper focuses on the optimum design of the e-CLT technology. The e-CLT technology consists in adding cross laminated timber (CLT) walls to an existing reinforced concrete (RC) infilled frame via asymmetric friction connection (AFC). The authors carried out quasi-static and nonlinear dynamic analyses. The RC frame is modeled in OpenSees by fiber-section-based elements with force-based formulation. The contribution of the infill is simulated using a degrading data-driven Bouc–Wen model with a slip-lock element while the AFC is modelled with a modified Coulomb model. Different types of infill, aspect ratio, scaling, and member size are considered. The benefits of using e-CLT technology are discussed and the ranges of optimum performance of the AFC are estimated. A comparison of the performance of traditional infills with the e-CLT system is presented. The authors provide optimum intervals of the ratio between slip force and in-plane stiffness of the CLT panel, following energy and displacement-based criteria. The seismic displacement demand under various seismic scenario is investigated. Correlations between the RC characteristics and the optimum design ratios bestow possible criteria for the design of the AFC.

Forced-based Shear-flexure-interaction Frame Element for Nonlinear Analysis of Non-ductile Reinforced Concrete Columns

An efficient frame model with inclusion of shear-flexure interaction is proposed here for nonlinear analyses of columns commonly present in reinforced concrete (RC) frame buildings constructed prior to the introduction of modern seismic codes in the Seventies. These columns are usually characterized as flexure-shear critical RC columns with light and non-seismically detailed transverse reinforcement. The proposed frame model is developed within the framework of force-based finite element formulation and follows the Timoshenko beam kinematics hypothesis. In this type of finite element formulation, the internal force fields are related to the element force degrees of freedom through equilibrated force shape functions and there is no need for displacement shape functions, thus eliminating the problem of displacement-field inconsistency and resulting in the locking-free Timoshenko frame element. The fiber-section model is employed to describe axial and flexural responses of the RC section. The modified Mergos-Kappos interaction procedure and the UCSD shear-strength model form the core of the shear-flexure interaction procedure adopted in the present work. Capability, accuracy, and efficiency of the proposed frame element are validated and assessed through correlation studies between experimental and numerical responses of two flexure-shear critical columns under cyclic loadings. Distinct response characteristics inherent to the flexure-shear critical column can be captured well by the proposed frame model. The computational efficiency of the force-based formulation is demonstrated by comparing local and global responses simulated by the proposed force-based frame model with those simulated by the displacement-based frame model.

Micromechanical and multiscale computational modeling for stability analysis of masonry elements

This paper presents two micromechanical and a multiscale finite element models for the analysis of masonry walls under out-of-plane instability effects. A two-dimensional modeling of the wall is considered in all approaches, assuming a cylindrical bending. The micromechanical analyses are performed considering elastic beams to model the bricks and either nonlinear beams or interfaces to model the mortar layers. The beam finite elements rely on the force-based formulation and account for large displacements by making use of the corotational approach. This latter is properly formulated and extended to interface elements to include nonlinear geometry effects. The multiscale model is defined by applying a two-scale beam-to-beam homogenization procedure, developed for masonry elements with periodic brick arrangements. Hence, a Unit Cell made of a single linear elastic brick and a single nonlinear mortar layer is introduced at the microscopic level, which is linked to the macroscale level through a semi-analytic homogenization technique. In all models, a damage formulation with friction plasticity governs the mortar constitutive relationship. Computational details on model implementation and solution procedures are given for all approaches. Correlation studies are performed to assess the proposed numerical micromechanical and multiscale procedures. In particular, the behavior of an unreinforced wall tested under a compressive eccentric load is reproduced and advantages and disadvantages of the proposed approaches are discussed.

A regularized force-based Timoshenko fiber element including flexure-shear interaction for cyclic analysis of RC structures

Reinforced concrete (RC) columns are the most critical structural components in buildings/bridges. For those with small span-to-depth ratios and/or insufficient transverse reinforcing details, it will suffer from shear type or flexure-shear type failure modes, which are challenge to be represented in numerical simulation. For instance, conventional fiber element employs the Euler–Bernoulli beam theory which neglects the shear deformation, thus it will over-estimate the structural responses of flexure-shear and shear-critical columns. This paper presents a new fiber element to include the flexure-shear interaction for cyclic analysis of RC structures. The element adopts a force-based formulation and extends the original fiber element from the Euler-Bernoulli theory basis to the Timoshenko theory basis by introducing shear deformations at the section level. Then the multi-axial softened damage-plasticity model is used for concrete fibers and the uniaxial modified Menegotto–Pinto model is used for reinforcement fibers. The concrete-steel coupling effect that is typical for reinforced concrete subjected to shear, i.e., tension-stiffening and compression softening, are also considered through modifying the material constitutive laws. To overcome the difficulties in force-based element state determination and enhance the computational efficiency, the non-iterative state determination strategy is adopted for the element implementation. Meanwhile, to avoid the localization issue (i.e., integration point-dependency) arising from strain-softening behavior, an interpolatory quadrature is used for the numerical integration of the element. Finally, the element is validated through numerical simulations of a series of column tests under cyclic loadings. The results indicate that the element can retain the performance in modeling flexure-critical columns as conventional Euler–Bernoulli fiber element, while demonstrating significant superiors in modeling flexure-shear and shear-critical columns.

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

SummaryThis paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.

A multiscale force-based curved beam element for masonry arches

This paper presents a Timoshenko beam finite element for nonlinear analysis of planar masonry arches. Considering small displacement and strain assumption, the element governing equations are defined according to a force-based formulation that adopts three different parametrizations of the axis planar curve, permitting the exact description of the element geometry for arbitrarily curved arches. Specific quadrature techniques are illustrated to perform numerical integration over the curved axis. A two-scale arch-to-beam homogenization procedure reproduces the nonlinear response of periodic masonry materials, where an equivalent Timoshenko straight beam describes the behavior of the reference Unit Cell made of a single linear elastic brick and a nonlinear mortar layer. Formation of the hinges characterizing the collapse mechanism of the arch is detected taking advantage of the quadrature rule along the axis and a fracture energy based regularization technique is employed to avoid damage localization.The proposed curved beam model is implemented in a standard FE analysis code and is used to perform several numerical applications. After validating the proposed formulation through benchmarking tests under linear elastic material response, the numerical simulation of six experimental tests is shown, concerning masonry arches characterized by different shapes and undergoing in-plane bending. The numerical results are validated through experimental outcomes and other FE models.

Explicit Fiber Beam-Column Elements for Impact Analysis of Structures

The solution of impact problems requires advanced computational techniques to overcome the difficulties associated with large short-duration loads. In this case, the explicit time integration method is typically used, since it provides a stable solution for problems such as the analysis of structures subjected to shock and impact loads. However, most explicit-based finite elements were developed for continuum models such as membrane and solid elements, which renders the problem computationally expensive. On the other hand, the development of fiber-based beam finite elements allows for the simulation of the global structural behavior with very few degrees of freedoms, while accounting for the detailed material nonlinearity along the element length. However, explicit-based fiber beam elements have not been properly formulated, in particular for the case of the emerging force-based beam element. In this paper, two developed fiber plane beam elements that consider an explicit time integration scheme for the solution of the dynamic equation of motion are presented. The first element uses a displacement-based formulation, while the second element uses a force-based formulation. For the latter case, a new algorithm that eliminates the need for iterations at the element level is proposed. The developed elements require the use of a lumped mass matrix and a small time increment to ensure numerical stability. No iterations or convergence checks are required, which renders the problem numerically efficient. The developed explicit fiber beam-column models, particularly the force-based element, represents a simple yet powerful tool for simulating the nonlinear complex effect of impact loads on structures accurately while using very few finite elements. The traditional implicit method of analysis typically fails to provide numerical stable behavior for such short time duration problems. Two correlation studies are presented to highlight the efficiency of the developed elements in modelling impact problems where the strain rate effect is considered in the material models. These examples confirm the accuracy and efficiency of the presented elements.

Enriched Force-Based Frame Element with Evolutionary Plastic Hinge

AbstractThis paper presents a novel enriched force-based formulation for beam-column elements. The enrichment was developed on the basis of the concept of evolutionary plastic hinges and incorporat...

Nonlinear line-element modeling of flexural reinforced concrete walls

The research presented here developed a model for simulating the nonlinear cyclic response of flexure-controlled concrete walls, which meets the dual objectives of accuracy and computational efficiency. The proposed model represents a significant advancement in that it provides accurate simulation of the dominate failure mechanism exhibited by flexural walls in the laboratory and field: compression-controlled failure characterized by simultaneous crushing of concrete and buckling of longitudinal reinforcement. The first steps in the model development effort comprised assembly of an experimental database and review of current modeling approaches for walls (e.g., lumped plasticity, distributed plasticity, and continuum elements). Model evaluation indicated that the most viable option to achieve accuracy and efficiency was the use of beam–column line elements with fiber-type cross-section models at the integration points. Initially, both displacement-based and force-based element formulations were evaluated; however, the displacement-based formulation resulted in an inaccurate representation of the axial force distribution along the length of the element. Therefore, only the force-based formulation was chosen for further study. The basic model included standard 1D constitutive models for confined concrete, plain concrete and reinforcing steel. Comparing simulated and measured response data showed that the concrete and steel material models must be regularized using a mesh-dependent characteristic length and a material-dependent post-yield energy to enable accurate, mesh-objective simulation of strength loss due to compression failure. The post-yield energy values were determined using relevant experimental data, an important but missing component of prior research on material regularization. The results of this study show that use of the regularized constitutive models significantly improved the accuracy of response predictions.

Force-based higher-order beam element with flexural–shear–torsional interaction in 3D frames. Part I: Theory

An innovative higher-order beam theory, capable of accurately taking into account flexural–shear–torsional interaction, is originally combined with a force-based formulation to derive the corresponding finite element. The selected set of higher-order deformation modes leads to an explicit and direct interaction between three-dimensional shear and normal stresses. Namely, cross-sectional displacement and strain fields are composed of independent and orthogonal modes, which results in unambiguously defined generalised cross-sectional stress-resultants and in a minimisation of the coupling of equilibrium equations. On the basis of work-equivalency to three-dimensional continuum theory, dual one-dimensional higher-order equilibrium and compatibility equations are derived. The former, which govern an advanced form of beam equilibrium, are strictly satisfied via stress fields arising from the solution of the corresponding systems of coupled differential equations. The formulation, which is numerically validated in a companion paper for both linear and nonlinear material response, inherently avoids shear-locking and accurately accounts for span loads. Finally, the superiority of force-based approaches over displacement-based ones, well established for inelastic behaviour, is also demonstrated for the linear elastic case.

Force-based higher-order beam element with flexural-shear-torsional interaction in 3D frames. Part II: Applications

The specific features of the proposed force-based formulation derived in the companion paper, which is applied for the first time to higher-order beam theories, are herein thoroughly validated. Introductory numerical examples illustrate the influence of mesh refinement, boundary conditions, and slenderness ratios for isotropic linear elastic response. Specific higher-order effects—unique to the developed element—are then suitably interpreted, as well as the formulation appropriateness to consider distributed loads and to model three-dimensional behaviour, which is verified with solid finite element analyses. Extensive comparisons against existing proposals, namely other refined higher-order beam theories, emphasize the performance of the proposed approach. Finally, the nonlinear response of the element with a multiaxial J2 linear plasticity material model is analysed, highlighting its advantages in relation to a classical force-based Euler–Bernoulli beam using a one-dimensional plastic material model with kinematic hardening.

Correlation between beam on Winkler-Pasternak foundation and beam on elastic substrate medium with inclusion of microstructure and surface effects

A novel beam-elastic substrate element with inclusion of microstructure and surface energy effects is proposed in this paper. The modified couple stress theory is employed to account for the microstructure-dependent effect of the beam bulk material while Gurtin-Murdoch surface theory is used to capture the surface energy-dependent size effect. Interaction mechanism between the beam and the surrounding substrate medium is represented by the Winkler foundation model. The governing differential equilibrium and compatibility equations of the beam-elastic substrate system are consistently derived based on virtual displacement and virtual force principles, respectively. Both essential and natural boundary conditions of the system are also obtained. Two modified Tonti’s diagrams are presented to provide the big picture of both displacement-based and force-based formulations of the system. Due to similarity between the current problem and the one related to the beam on Winkler-Pasternak foundation, the so-called “natural” beam-Winkler-Pasternak foundation element coined by the authors is employed to perform two numerical simulations to study the characteristics and behaviors of a beam-substrate system with inclusion of microstructure and surface effects.

Simple and efficient finite element modeling of reinforced concrete columns confined with fiber-reinforced polymers

This paper presents a frame finite element (FE) that is able to accurately estimate the load-carrying capacity and ductility of reinforced concrete (RC) circular columns confined with externally-bonded fiber-reinforced polymers (FRP). This frame FE can model collapse mechanisms due to concrete crushing, reinforcement steel yielding, and FRP rupture. The adopted FE considers distributed plasticity with fiber discretization of the cross-sections in the context of a force-based formulation, and uses advanced nonlinear material constitutive models for reinforcing steel and unconfined, steel-confined, and FRP-confined concrete.The adopted frame FE is validated through a comparison between numerical simulations and experimental results available in the literature of the load-carrying capacity of FRP-confined RC columns subjected to axial load only, and both axial and lateral load. The adopted FE is suitable for efficient and accurate modeling and analysis of FRP-confined RC columns, and thus it represents a step toward enabling analysis of real-world large-scale structures containing FRP-confined RC columns, for which more accurate three-dimensional models could be computationally prohibitive.

Force-based FE for large displacement inelastic analysis of two-layer Timoshenko beams with interlayer slips

This paper presents a novel finite element model for the fully material and geometrical nonlinear analysis of shear-deformable two-layer composite planar beam/column members with interlayer slips. We adopt the co-rotational approach where the motion of the element is decomposed into two parts: a rigid body motion which defines a local co-ordinate system and a small deformational motion of the element relative to this local co-ordinate system. The main advantage of this approach is that the transformation matrices relating local and global quantities are independent from the choice of the geometrical linear local element. The effect of transverse shear deformation of the layers is taken into account by assuming that each layer behaves as a Timoshenko beam element. The layers are assumed to be continuously connected and partial interaction is considered by adopting a continuous relationship between the interface shear flow and the corresponding slip. In order to avoid curvature and the shear locking phenomena, the local linear element is derived from the force-based formulation. The present model provides an efficient tool for the elastoplastic buckling analysis of two-layer shear deformable beam/column with arbitrary support and loading conditions. Finally, two numerical applications are presented in order to assess the performance of the proposed formulation.

Discussion of “Simulating Maximum and Residual Displacements of RC Structures: I. Accuracy”

(1) Tazarv and Saiidi state that a unique “best modeling approach” is not proposed in the paper. This is correct; proposing a “best modeling approach” is not intended in the papers. The papers are aimed at investigation of the capabilities and limitations of some common beam-column element modeling strategies. Consequently, development of an ultimate modeling strategy is beyond the scope of the investigation. The authors reviewed the paper once more but could not find any statements suggesting that an approach for accurate estimation of residual displacements is provided. Tazarv and Saiidi ask why the relationship between the small cycle hysteresis response and the bias in the simulated residual displacements is presented based on the results obtained using displacement-based elements and not using force-based elements. The importance of small cycle response is presented based on the measured and simulated response histories of column “A1” tested by Hachem et al. (2003) in Figures 7 and 8. For the test unit “A1,” the maximum and residual displacements obtained from the simulations made using displacement-based and force-based element formulations are marked using “•”symbol in Figures 4 and 5, respectively. In these figures, it can be seen that the maximum and residual displacement obtained using displacement-based and force-based formulations for test unit A1 are practically the same. Therefore, it can be deduced that a very similar plot would be obtained if force-based elements were used instead of displacement-based elements. In brief, the difference would be practically irrelevant for the considered test unit. (2) Tazarv and Saiidi argue that three references which were available at the time of submission of the paper were missed. They provide references to two reports (i.e., Jeong et al. 2008, Lee 2007) and a journal paper (Lee and Billington 2010). The article by Lee and Billington appeared on May 2010 issue of the relevant journal. The submission date of our paper was 7 February 2010 (which is indicated

Development of a Controller Platform for Force-Based Real-Time Hybrid Simulation

Force-based real-time hybrid simulation is a seismic experimental method that combines physical testing using shake tables and dynamic actuators with numerical analysis. The unique aspect of this force-based formulation is that various hybrid simulation techniques, such as dynamic, pseudo-dynamic, and quasi-dynamic testing, can be similarly executed. To implement such a method, the hardware components and the corresponding software were designed and integrated into a modular controller platform. This article focuses on the implementation issues of such formulation. A pilot scale setup was assembled to conduct proof of concept experiments of the controller platform and is presented herein.

Ultimate Load Analysis of Reinforced Concrete Beam with Finite Element

A efficient 3D beam element based on the distributed nonlinearity theory is proposed for nonlinear analysis of reinforcement concrete structures. The sections consist of reinforcing steel and concrete in this formulation and the section stiffness matrices are calculated through the integration of stress-strain relations of concrete and the accumulation of reinforcing steel effect. The force-based formulation is adopted in the evaluation of the element stiffness matrix and the element state determination. The improved Kent-Park model is adopt for the stress-strain relation of concrete, and uniaxial stress–strain relationships proposed by Mander is introduced for reinforcing steel. Finally, the ultimate load of a cantilever reinforced concrete beam subjected to a lateral load was analyzed with the proposed formulation to illustrate its accuracy and computational efficiency.

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.