Beam Element Formulation Research Articles

Refined beam finite element model for thin-walled multi-cell box girders considering distortion and secondary distortional moment deformation effect

In this paper, a novel one-dimensional refined beam finite element model (FEM) accounting for non-uniform distortional warping and secondary distortional moment deformation effect (SDMDE) is formulated for the distortional analysis of thin-walled multi-cell box girders with cantilevered flanges. The extra distortional angle is introduced to incorporate SDMDE, and the validity of which is proved through rigorous theoretical derivation. The in-plane and out-plane deformations at any point around the edge of transverse section built upon the distortional center are unified in the same form and described in detail. The governing differential equations in terms of two generalized displacements, distortional angle χ and primary distortional angle curvature Θ, are derived based on the principle of minimum potential energy (MPE) and solved exactly so as to provide non-polynomial shape functions for the establishment of element stiffness matrix and nodal load vector. A series of multi-cell rectangular hollow section (RHS) girders are analyzed and particular extensions are given to single-cell and multi-cell bridge girders with cantilevered flanges. Comparisons to the results generated from corresponding ones provided by pioneering work or shell FEM analyses not only validate the correctness and reliability of the present beam element formulations but also highlight the significance of the inclusion of SDMDE when the cantilevered flanges are arranged.

A selectively reduced degree basis for efficient mixed nonlinear isogeometric beam formulations with extensible directors

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu–Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results.

Algorithm for the plastic analysis of arbitrary steel cross sections based on finite element formulations

AbstractIn structural engineering verifications, the plastic capacity of steel members may be utilized for cross sections of class 1 and 2. In literature, algorithms for the primary torsional capacity considering different types of numerical procedures, such as finite element methods, have mainly been addressed. However, there is a lack of explicit illustration of algorithms for cross sections considering combined loading, i.e. procedures for bending and torsional analysis. This paper therefore presents an algorithm for the plastic analysis connected to a cross‐sectional finite element approach. The algorithm considers the description of bilinear material laws, accounting for ideal‐plastic behavior as well as effects of isotropic hardening. Cross sections are discretized by isoparametric two‐dimensional nine‐node elements with curvilinear boundaries, which allow describing any geometry with high accuracy. The study proves that the plastic algorithm combined with the isoparametric elements accurately represents the behavior of cross sections. In addition, precise cross‐sectional capacities of hot‐rolled U‐sections gained by the methodology, i.e. plastic torsional moments and plastic bending moments, are presented. In future research, this methodology will be combined with finite element formulations for beams.

Deep Neural Network Direct Stiffness Method: Fundamental Steps into Beam Design

AbstractThis paper presents a novel approach for performing beam element analysis that considers the nonlinear deformation behavior of various RHS and SHS sections made of mild to high‐strength steel: the Deep Neural Network Direct Stiffness method (DNN‐DSM), which uses Deep Neural Networks (DNN), a subset of machine learning algorithms and more general artificial intelligence approaches, to predict the nonlinear stiffness matrix terms in a beam element formulation for implementation in the direct stiffness matrix (DSM). These predictions are made using trained DNN models drawn from an extensive pool of geometric‐material nonlinear simulations with additional imperfections (GMNIA) using shell‐based models. Initial implementations of this method are able to predict the nonlinear load‐displacement and moment‐rotation behavior of various cross sections with high accuracy. This combines the precision of shell analysis with the computational efficiency of beam element analysis. Previous publications have already demonstrated the suitability and advantages of this method, albeit on a small scale for local buckling prediction. This work goes beyond previous studies and focuses on finite beam element design. This includes a review of the modeling approaches of DNN‐DSM using experimental and numerical results from literature.

The transformation matrix in the 7DOFs beam formulation

A non-negligible percentage of steel products for civil and industrial constructions is realised with members characterised by a mono-symmetric cross-section. The non-coincidence between the centroid and the shear centre reflects in a quite complex behaviour that is remarkably influenced by the warping effects. Usually, frame design is based on the output data of commercial Finite Element Analysis Packages (FEAPs) but only few of them offer a refined beam element formulation able to reproduce the response of non-bisymmetric cross-section members (like the 7 degrees of freedom, DOFs, beam element). As a consequence, warping effects are usually neglected in routine design, leading in several cases to assess non-correct internal stresses, global displacements and critical buckling multipliers.This paper is focussed on mono-symmetric cross-section members and deals with the influence of the interaction between axial force, bending moments and bimoment in linear elastic range. In particular, the two strategies that FE developers usually adopt, in the 7 DOFs beam formulation, to pass from the local to the global reference system are herein shortly introduced and discussed. The effects of the associated transformation matrices have been investigated in few examples, also reproduced via closed expressions derived from the mixed torsion theory and FE shell models. Paper outcomes show that the results in terms of generalised displacements, internal forces and, consequently, stress distribution along the member cross-section are significantly different, strictly depending on the adopted transformation matrix. The importance of a correct evaluation of the bimoment distribution is stressed also by the fact that the new edition of the EN1993-1-1, expected in the next months, includes also the bimoment contribution for member checks.

A Two-Dimensional Corotational Beam Formulation Based on the Local Frame of Special Euclidean Group SE(2)

AbstractThe corotational frame method is widely used in the simulation of flexible multibody dynamics. Its core idea is to separate the rigid motion from the flexible deformation so that it can make fully exploit a large number of excellent local finite elements. The essence of the conventional corotational frame method is the projection relationship between the element frame and the global frame. This paper explores another coordinate projection method for two-dimensional (2D) corotational beam element. The projection relationship between the element frame and the local frame in the framework of Lie algebra se(2) has been proposed. Based on the description of SE(2), the formulation of corotational beam element and integration algorithm is presented. The local frame description greatly reduces the nonlinearity of the formula by eliminating the effect of the rigid body motion on the projection matrix, internal force and inertial force. Several examples of large deformation and large rotation are performed, and it is found that the step-size convergence and iterative efficiency of SE(2) description are improved compared with R3 description. Moreover, some examples are used given to verify that the frame invariance brought by SE(2) is valuable for improving computing efficiency. The presented transformation method can easily extend to other 2D elements.

FORM-FINDING PROBLEM USING BEAM ELEMENTS OF FINITE ELEMENT TECHNIQUE ASSUMING NODAL COORDINATES: HYBRID TENSILE STRUCTURES WITH ACTIVE BENDNG BEAM, MEMBRANE, CABLE ELEMENTS

The finite element technology assuming nodal coordinate is one of a useful method for form-finding of tensile structures, which does not require coordinates transformation and is described by a simple formulation. Furthermore, bending-active structures is known as a form-finding problem for hybrid tensile structures. It is possible to realize lightweight structures with self-equilibrium by the prestressing tension of the membrane and cables or the temporary external force during construction. In this paper, we drive the formulation of beam elements with the finite element technique assuming coordinate values, and apply the method to hybrid bending-active structures.

Influence of partial shear connection on the behaviour of U-shaped steel concrete beams with L-shaped shear connectors

This paper presents a study on the influence of partial shear connection on the behaviour of the U-shaped steel concrete beams (USCB) with L-shaped shear connectors. An experimental campaign of three full-scale experimental tests on the USCB with a degree of shear connection ranging from 0.4 to 1 has been conducted under sagging bending moment. The test results showed high ductility of the USCB with both full and partial shear connections. The failure mode of the USCB was governed by the plastic buckling of the upper flanges of the U-shaped steel profile at the location of the shear connectors. The plastic moment resistance of the USCB with partial shear connection predicted by the full plastic analysis approach and by the simplified approach in Eurocode 4 agreed well with the experimental results. Furthermore, the flexural stiffness of the USCB with partial connections predicted by American and Australian standards fit well with the experimental results. A numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was also adapted to determine the behaviour of the USCB. This model was validated against experimental tests as well as against the analytical approaches. It was finally used to carry out a parametrical study in order to investigate the effect of selected parameters such as the degree of shear connection, the concrete strength and the steel strength of U-shaped profile.

Efficient beam element model for analysis of composite beam with partial shear connectivity

A two-node beam element formulation with eight degrees of freedom was developed and implemented into MATLAB to simulate the nonlinear behavior of the composite. The beam element includes the reinforced concrete slab, the concrete reinforcement, the steel beam, and the shear connectors. One dimension-softening model was used to simulate the concrete, the bilinear isotropic plastic model was used to simulate the steel, and a multi-linear spring was used to simulate the partial shear connectivity. Experimental results were used to validate the numerical model, which was able to closely predict the experimental response by a 2% difference. The model was able to simulate several material nonlinearities such as the plasticity of the steel beam and cracking and crushing of concrete slab. A parametric study was performed to investigate the effects of strength of shear connector, concrete slab thickness, steel beam depth, and the concrete compressive strength on the overall response of steel–concrete composite beams. It was found that the variation in shear connector strength significantly affects the response of the composite beams. In addition, the composite beam stiffness and strength increased with the increase of the steel beam depth, significantly.

An isogeometric finite element formulation for frictionless contact of Cosserat rods with unconstrained directors

This paper presents an isogeometric finite element formulation for nonlinear beams with impenetrability constraints, based on the kinematics of Cosserat rods with unconstrained directors. The beam cross-sectional deformation is represented by director vectors of an arbitrary order. For the frictionless lateral beam-to-beam contact, a surface-to-surface contact algorithm combined with an active set strategy and a penalty method is employed. The lateral boundary surface of the beam is parameterized by its axis and cross-sectional boundary curves with NURBS basis functions having at least C^2-continuity, which yields a continuous surface metric and curvature for the closest point projection. Three-dimensional constitutive laws of hyperelastic materials are considered. Several numerical examples verify the accuracy and efficiency of the proposed beam contact formulation in comparison to brick element solutions. The lateral contact pressure distribution of the beam formulation is in excellent agreement with the contact pressure of the brick element formulation while requiring much less degrees-of-freedom.

Self-expandable stent for thrombus removal modeling: Solid or beam finite elements?

The performance of self-expandable stents is being increasingly studied by means of finite-element analysis. As for peripheral stents, transcatheter valves and stent-grafts, there are numerous computational studies for setting up a proper model, this information is missing for stent-retrievers used in the procedure of thrombus removal in cerebral arteries. It is well known that the selection of the appropriate finite-element dimensions (topology) and formulations (typology) is a fundamental step to set up accurate and reliable computational simulations. In this context, a thorough verification analysis is here proposed, aimed at investigating how the different element typologies and topologies - available to model a stent-retriever - affect simulation results. Hexahedral and beam element formulations were analyzed first individually by virtually replicating a crimping test on the device, and then by replicating the thrombectomy procedure aiming at removing a thrombus from a cerebral vessel. In particular, three discretization refinements for each element type and different element formulations including both full and reduced integration were investigated and compared in terms of the resultant radial force of the stent and the stress field generated in the thrombus. The sensitivity analysis on the element formulation performed with the crimping simulations allowed the identification of the optimal setting for each element family. Both setting lead to similar results in terms of stent performance in the virtual thrombectomy and should be used in future studies simulating the mechanical thrombectomy with stent-retrievers. The carried out virtual thrombectomy procedures confirmed that the beam element formulation results were sufficiently accurate to model the radial force and the performance of the stent-retriever during the procedure. For different self-expandable stents, hexahedral formulation could be essential in stress analysis.

In-plane pipe whip: Post-failure dynamic response

Pressurised pipes are ubiquitous in chemical, nuclear and power plants. A sudden failure and release of high-speed fluid cause large inelastic displacements characterised by a whipping-type motion, which can ultimately hinder the surrounding structural and functional systems. In this paper, a beam user element has been developed, implemented and applied to analyse the in-plane flexural dynamic response of pipe whip. The two-dimensional Euler–Bernoulli beam element is based on the corotational kinematic formulation and elastoplastic constitutive models that include metal plasticity and moment–curvature relationships obtained from numerical bending tests, which highlighted the existence of two new dimensionless groups that govern the flexural response of slender pipes and enable the creation of moment–curvature master curves for thick and thin pipes. The corotational beam element formulation is compared against an analytical rigid-perfectly plastic model, numerical simulations employing shell elements and available experimental results, showing very good accuracy in the prediction of the inelastic deformation response of pipe whip and its hazardous area of influence. Furthermore, parametric studies are performed to investigate the effect of load intensity, cross-sectional geometry and concentrated tip mass on the post-failure deformation modes, plastic hinge formation and extension of the hazard zone. The presented results represent valid tools to assess the safety of industrial piping systems undergoing failure, and to optimally design pipe whip restraints.

On a virtual element formulation for trusses and beams

The virtual element method (VEM) was developed not too long ago, starting with the paper [2] related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements; however, it has so far not applied to one-dimensional structures like trusses and beams. Here we study several VEM elements suitable for trusses and beams and show that the virtual element methodology produces elements that are equivalent to well-known finite elements but also elements that are different, especially for higher-order ansatz functions. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns as finite beam elements. Furthermore, the formulation allows to compute nonlinear structural problems undergoing large deflections and rotations.

Joint modelling in advanced thin-walled beam models

A methodology for thin-walled frame analysis using displacement modes of beams at joint interfaces has recently been proposed. This paper introduces an implementation that uses a specific advanced beam element based on semi-analytical beam mode solutions of a generalised beam theory. The end cross-sections of the beam elements consist of wall elements with displacement degrees of freedom that can be joined to conventional finite shell element models of the joints. However, before joining the beam element and joint model at the interface, both beam and joint model are transformed into cross-section displacement mode degrees of freedom. This transformation enables the use of a reduced mode space at the interface as well as in the beam. The quadratic nature of the beam mode equations of the beam element leads to a non-unique number of beam displacement modes. However, this paper introduces an exact mode selection technique that leads to a unique set of modes. Necessary details of the beam element formulation are included in order to make a self-contained description of the mode selection technique. Mode reduction through a length scale dependent choice of the number of exponentially decaying distortional modes is investigated and discussed for three types of joint design in four examples. The three joint design types investigated are based on in-plane membrane action, flexural out-of-plane action and a combination of both membrane and flexural action.

A perturbation-based stochastic nonlinear beam element formulation using the B-spline wavelet on the interval finite element method

A perturbation-based stochastic nonlinear beam element formulation using the B-spline wavelet on the interval finite element method

Vibration analysis of functionally graded beams using a higher-order shear deformable beam model with rational shear stress distribution

This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams. In the element model proposed, the stress equilibrium equation is employed to derive the rational distribution of transverse shear stress. The beam element formulation oriented with vibration analysis is derived on the basis of the variational principle with mixed energy. To visualise the effectiveness and applicability of the proposed mixed beam element model in vibration analysis, numerical examples have been studied. According to the numerical results, in comparison with the classical beam element model and the beam element models based on the conventional first-order or higher-order shear deformation theory, the vibration frequencies obtained using the mixed beam element model are significantly more satisfactory. Moreover, the comparative study demonstrates that, for vibration analysis of FG sandwich beams, the rational distribution of transverse shear stress is of decisive significance to achieve high-precision solutions. With the mixed beam element used, the load–frequency relationship of FG sandwich beams is discussed. As indicated by the results, in addition to the axial force, the bending moment exerts a significant difference upon the vibration frequencies of the FG beams.

Three-dimensional nonlinear mixed 6-DOF beam element for thin-walled members

This paper presents a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees. This formulation is presented using a corotational total Lagrangian approach and implemented in the OpenSees corotational framework. In this context, a basic coordinate system is lined up with the element chord and translates and rotates as the element deforms. The element tangent stiffness matrix and resisting forces in the basic system are derived through linearization of the two-field Hellinger-Reissner variational principle. The displacement shape functions are cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation. The generalized stress resultant shape functions are linear for moments and constant for axial force and torque with the P - δ effect considered, which are developed from equilibrium equations. The fiber section method with uniaxial constitutive laws is adopted to account for material nonlinearity. Since the degrees-of-freedom in the basic system are defined with respect to different reference points, all element responses are transformed to acting about the shear center before conducting the corotational transformation. The mixed element is validated through a number of experimental and numerical examples.

Simplified Nonlinear Simulation of Rectangular Concrete-Filled Steel Tubular Columns

In detailed three-dimensional (3D) finite-element modeling, two-dimensional and/or 3D elements are widely used because of their high accuracy and ease of use despite the high computational cost. In contrast, simplified modeling based on fiber beam element (FBE) formulation is preferred for developing macro models to simulate structural frames due to its simplicity and computational efficiency. As for the FBE simulation of concrete-filled steel tubular (CFST) columns, its accuracy largely depends on the input steel and concrete material models, which should implicitly consider the material nonlinearity and interaction between the steel and concrete components. The authors have previously developed a FBE model for circular CFST columns, and this paper is a continuation of the previous work. In this paper, uniaxial effective stress–strain relationships are developed for the steel and concrete materials in rectangular CFST columns based on rigorous analysis of data generated from 3D finite-element modeling of stub columns. Thus, the material models have implicitly considered the effects of yielding and local buckling of the steel tube and passive confinement to the concrete. Meanwhile, the size effect is also considered in the concrete model. The accuracy of the proposed material models is verified against a database of rectangular CFST stub columns covering a wide range of material and geometric parameters. The developed material models are further used to simulate rectangular CFST slender columns and beam-columns, and a reasonably good agreement is achieved between the experimental and predicted load–deformation curves.

Efficient mortar‐based algorithms for embedding 1D fibers into 3D volumes

AbstractMany composite materials are based on 1D fibers being embedded into 3D solid volumes, e.g. carbon‐fiber reinforced plastics in aerospace engineering or fiber‐reinforced concrete in civil engineering to name only two prominent examples. The present contribution highlights the most important numerical methods and algorithmic building blocks for an efficient analysis of such systems based on cutting‐edge finite element formulations for nonlinear beams and a novel beam‐to‐solid volume coupling approach inspired by classical mortar methods. A particular emphasis is put on the efficient parallel implementation for large‐scale simulations, which includes suitable procedures for domain partitioning and geometry‐based search.

Mixed formulation for geometric and material nonlinearity of shear-critical reinforced concrete columns

This study presents a new beam element formulation following a Hellinger-Reissner variational principle for shear critical reinforced concrete members, and considering large displacement and rotation effects. The corotational formulation is used to describe the large displacement at the element nodal level and degenerated Green-Lagrange strain measures are used at the basic element level. A new displacement shape function has been developed considering section kinematics and compatibility conditions to satisfy stability criteria. New explicit equations for the internal geometric stiffness matrix to consider large deformation effect has been developed. A robust state determination algorithm for the large deformation mixed-based formulation is proposed. The Timoshenko beam theory has been adopted to consider shear deformations in deriving the section kinematics equations. The multi-axial stress state in reinforced concrete fibre has been simulated through the fixed crack smeared softened membrane model. The developed shear fibre beam element is capable of accurately reproducing the experimentally-observed large displacement effect on the post-peak shear strength degradation in the softening region. The accuracy and efficiency of the mixed-based formulation was evaluated by examining the responses at local and global levels for both monotonic and reverse cyclic loading conditions along with the process of reproducing experimentally-observed failure modes.