Nonlinear Analysis Of Frame Structures Research Articles

A co-rotational formulation for quasi-steady aerodynamic nonlinear analysis of frame structures

The design of structures submitted to aerodynamic loads usually requires the development of specific computational models considering fluid-structure interactions. Models using structural frame elements are developed in several relevant applications such as, the design of advanced aircraft wings, wind turbine blades or power transmission lines. In the case of flexible frame structures submitted to fluid flows, the computation of inertial and aerodynamic forces for large displacements and rotations is a challenging task. In this article, we present a novel formulation for the efficient computation of aerodynamic forces in frame structures, coupling the co-rotational framework with the quasi-steady theory. A numerical procedure is provided considering a tangent matrix for the aerodynamic forces. This formulation is implemented in the open-source library ONSAS, allowing users to reproduce the results or solve other frame nonlinear dynamic problems. The proposed formulation and its implementation are validated through the resolution of four numerical examples. The formulation and the numerical procedure proposed efficiently provide accurate solutions for these challenging problems with large displacements and rotations.

An Efficient Reduced-Order Multi-Scale Simulation Approach for Nonlinear Analysis of Frame Structures with Nonlinear Localization

ABSTRACT This paper presents a reduced-order multi-scale simulation approach for nonlinear analysis of structures with local nonlinearities. Using it, the traditional global structural analysis can be changed to that of a reduced-order system where only nodal displacements of nonlinear elements are solved using the Newton–Raphson method in a single iterative procedure. Meanwhile, two nonlinear force/stiffness correctors are introduced based on the stress decomposition to describe material nonlinearities of frame members. In addition, two simulation ways are used to deal with the nonlinearities of the yielding frame members. Finally, the static and transient structural systems are simulated to verify the approach.

A co-rotational curved beam element for geometrically nonlinear analysis of framed structures

Curved beams are sometimes used in practical framed structures due to good mechanical properties and artistic design. In a framed structure, curved beams may undergo large displacement and experience nonlinear behavior as same as the other straight slender beam-column members. Thus, a geometrically nonlinear curved beam element plays an important role in the analysis of framed structures with curved beams. However, most existing curved beam elements are not accurate enough and still need several or even dozens of elements to accurately describe the behavior of a curved beam with a large subtended angle. To fill this gap, this paper presents a novel geometrically nonlinear curved beam element based on the element-independent co-rotational (EICR) method. The proposed element can simulate a curved beam using only one single element in the analysis and design of most practical framed structures. Moreover, this element is directly derived in a Cartesian coordinate system and can be directly and conveniently used with straight beam-column elements in nonlinear structural analysis. In this manuscript, the derivation of the proposed geometrically nonlinear curved beam element is detailed and several benchmark problems are proposed to verify its accuracy and efficiency.

Rational nonlinear analysis of framed structures and curved beams considering joint equilibrium in deformed state

Based on the continuum mechanics principles, a rigorous formulation is presented for the linearized stiffness equation of three-dimensional beam elements with account taken of the joint moment equilibrium in the deformed configuration C2. By sticking to the Bernoulli–Euler hypothesis of plane sections and elasticity definitions for stress resultants, the bending moments and torque of the element are shown to be quasi- and semi-tangential, respectively, in the updated Lagrangian formulation. Further, by invoking the moment equilibrium conditions for structural nodes at C2, the induced moment matrix that first appears to be antisymmetric on the element level turns out to be symmetric upon assembly of all elements on the structural level. The joint equilibrium conditions at C2, as represented by the induced moment matrix, are central not only to the out-of-plane buckling analysis of angled frames, but also to the simulation of curved beams by the straight-beam elements. Examples on the buckling of angled frames and curved beams are provided to support the theory presented.

Qualified Geometric Stiffness for Linear Buckling and Second-Order Nonlinear Analysis of Framed Structures

There exist various potential energy formulations dealing with the linear buckling and second-order nonlinear analysis of framed structures with different degrees of refinement in the kinematic model. However, the geometric stiffnesses derived often give rise to different structural behaviors, which indeed represents a confusion regarding their qualified usage in bifurcation and post-buckling analysis. This study aims to carry out a comprehensive evaluation of the validity of the geometric stiffness for use at the predictor and corrector phrases of an incremental analysis based on the rigid-body motion test. To remove the unbalanced element forces caused by the nonqualified geometric stiffness, a supplementary correction matrix is developed according to simple kinematic and static analysis in the context of rigid rotations. An updated Lagrangian approach-based force recovery procedure (FRP) is presented for updating the element forces with improved reliability and efficiency, when using relatively large step size. Some benchmark problems which exhibit compound three-dimensional nonlinear behavior of framed structures are solved to clarify the capabilities of rigid-body qualified and nonqualified geometric stiffnesses along with the existing FRPs for different mesh sizes, step sizes and load patterns. It is shown that the proposed procedure can be adopted to predict correct buckling loads and post-buckling equilibrium paths without adding extra computational costs.

Evaluation of the plastic hinge length of steel-concrete composite beams under hogging moment

Lumped plasticity is one of the most commonly used approaches for the nonlinear analysis of framed structures, but its effectiveness depends on the reliability of the plastic rotational capacity. In the case of RC or steel structures, this approach is based on well-known and reliable formulations, but no specific rules have been established for steel-concrete composite structures. This paper reports the results of a detailed analysis of the nonlinear behaviour of the section and entire element of a composite beam with the goal of determining its flexural ductility. In particular, the composite beam was examined under a hogging moment that results in its worst performance due to the concrete experiencing tension and the steel compression. An FE model is used to study the element; it was considered all the features of the materials and the geometry of the section to develop a parametric analysis that enabled to study the effect of different parameters on the rotational capacities of the composite beams. This formulation considers the local phenomenon of profile buckling and the global dimensions of the composite section. Finally, the obtained results allowed to define an equivalent plastic hinge length that provides the available plastic rotation when multiplied by the plastic curvature of the cross-section.

A spectral stochastic formulation for nonlinear framed structures

The present paper proposes a stochastic formulation which enables the effective coupling of spectral stochastic finite elements with geometrically nonlinear analysis of framed structures. This is achieved by projecting the stochastic part of the incremental displacements, formulated in the framework of a Newton–Raphson solution scheme, to a polynomial chaos expansion basis. The nonlinear solution is consequently achieved by introducing these expansion coefficients in the iterative solution algorithm as unknowns to be evaluated. With this approach, an augmented system of nonlinear equations is produced and the values of the coefficients are obtained through convergence of the iterative algorithm. In a similar to the deterministic problems manner, the nonlinear load or displacement control algorithms are extended to their stochastic counterparts and a discussion on the appropriate choice of such algorithms depending on the problem at hand is presented. This work focuses on stochastic beam finite element systems in which uncertainty is considered in the system properties. The efficiency of the proposed methodology is demonstrated in benchmark problems with strong geometrically nonlinear behavior. In order to verify the validity of the proposed approach the results obtained are compared to those of Monte Carlo simulations and fair accuracy can be reported.

Software complexes and new approaches to non-linear analysis of framed structures

There is a great number of non-linear tasks which can be solved with the use of computer-based simulation and finite elements method (FEM). But in order to ensure high speed and accuracy of calculations it is necessary to use new mathematical models and algorithms. The analysis of the known FEM forms and approaches to numerical solution of tasks based on force method is performed in this work. Represented in the document is formulation of the finite-element stress method and algorithmization of the classical force method for calculation of framed structures based on the equation of strain compatibility – contour forces method. Statement of problems which can be more effective from the point of view of elastic-plastic deformations at numerical solution is considered. In particular, the hybrid finite-element method based on the generalized Mohr formula for determination of long bar stiffness coefficient with account of non-linear behavior. The main relationships for creation of finite-element method equation system in the form of force method are represented.

Improved displacement based alternative to force based finite element for nonlinear analysis of framed structures

This paper presents an improved displacement based element (iDBE) for nonlinear finite element analysis of framed structures, which, like the force based elements (FBE), needs neither high order shape functions nor the refinement of the finite element mesh in straight members to attain an accurate solution. To achieve this objective, to the strain fields based on the displacements shape functions of the conventional displacement based element (cDBE) corrective fields are added, which are determined at the element level by the principle of virtual forces. Simple examples involving tapered elements, nonlinear axial-bending interaction and elasto-plastic behavior, are included to illustrate the application of iDBE and compare its performance to cDBE and FBE.

Geometrically exact curved beam element using internal force field defined in deformed configuration

This paper presents a study on the development of high-performance finite elements for geometrically nonlinear analysis of frame structures with curved members. Based on the geometrically exact beam theory, a highly efficient and accurate mixed finite element is developed. A new approach is proposed for constructing the independent internal force field by including major terms satisfying equilibrium conditions in the deformed configuration. An element-level equilibrium iteration procedure is employed for the condensation of element internal degrees of freedom during the nonlinear solution. Numerical results are presented to demonstrate the excellent performance of the element developed, and it is shown that even when each structural member is modelled with just one element, accurate solutions can still be achieved.

A Large Rotation Matrix for Nonlinear Framed Structures, Part 1: Theoretical Derivation

In this paper, a large rotation matrix for geometric nonlinear analysis of frame structures is developed by introducing the Rodrigues formula proposed by Argyris to modify the original rotation scheme of Bathe and Bolourchi. In the deduction, the large rigid body rotation of the beam element is decomposed into the relative translational displacements and the axial rotation respectively. Using Rodrigues formula, the large rotation matrix of rigid body can be calculated and sequentially the nodal transformation matrix of the beam element can be derived. And with this transformation matrix, the equilibrium equations of the beam elements established in the current configuration can be transformed into the original configuration, and then assembled and solved finally. The validity verification of the method would be presented and discussed in the part 2 of this paper.

Mixed Dimensional Hierarchic Partitioned Nonlinear Analysis of Frames with Masonry Infill

This paper presents the mixed-dimensional partitioned nonlinear analysis of framed structures with masonry infill. The analysis method is elaborated with the help of an example frame made of steel sections with brick masonry infill. The structure is partitioned in a hierarchic manner using a domain decomposition technique based on dual partition super-elements. The frame is subjected to a load increasing constantly on its free end. The final deflected shape after the analysis is presented that shows a separation occurring at the location of the mortar between bricks. Contours showing plastic work flow at the interfaces are also presented.

Error estimation for geometrically non‐linear analysis of framed structures

AbstractThis paper is concerned with the estimation of errors that arise in the geometrically non‐linear analysis of framed structures using 1D beam‐column finite elements. A quartic element is used to illustrate specific issues of error estimation relating to geometrically non‐linear analysis, this element exhibiting most of the sources of error that arise with other 1D elements. After a brief description of the quartic element, the sources of error that it manifests in geometrically non‐linear analysis are highlighted. The first source of error is due to equilibrium defaults, including contributions from finite element discretization and from the approximation of equivalent loads, whereas the second source of error is due to compatibility defaults. Next, a stress recovery method, termed the ‘Exact equilibrium recovery’ (EER) method, is proposed for the estimation of equilibrium errors due to finite element discretization. This method, specifically developed for framed structures, is element based and is shown to be more effective than other patch‐based recovery methods. An overall error estimation method is then proposed, providing separate yet combinable measures for equilibrium and compatibility errors. Finally, several examples are presented to demonstrate the effectiveness of the proposed error estimation method in the geometrically non‐linear analysis of framed structures, illustrating also the relative importance of the various sources of error that are manifested in typical problems. Copyright © 2004 John Wiley & Sons, Ltd.

ESA formulation for large displacement analysis of framed structures with elastic–plasticity

In this work, a spatial beam element for geometrically and materially non-linear analysis of framed structures is presented in this work. The equilibrium equations of a straight beam element are formulated using an updated Lagrangian (UL) incremental description. Internal moments are represented as the resultants of stresses calculated by engineering theories: Euler–Bernoulli–Navier theory for bending and Saint-Venant theory for torsion. Although the element developed can undergo large displacements and rotations, strains are assumed to be small. The non-linear cross-sectional displacement field including large rotation effects is introduced in the analysis, resulting in the geometric potential of bending and torsional moments which corresponds to that of semitangential behaviour. In such a way, the joint equilibrium of non-collinear elements is provided. For the force recovering, the external stiffness approach (ESA) is presented as an alternative to the common natural deformation approach (NDA). Material non-linearity is introduced for an elastic–perfectly plastic material through the plastic hinge formation at finite element nodes and for this a new plastic reduction matrix of the element is determined. The interaction of element forces at a hinge and the possibility of elastic unloading are taken into account. The effectiveness of the numerical algorithm discussed is validated through the test problem.

Derivation of the higher-order stiffness matrix of a space frame element

A physical concept, the rigid body rule, is applied for the derivation of the higher-order stiffness matrix of a space frame element. The derivation has a physical meaning that is the higher-order stiffness matrix can be derived by regarding that there is a set of incremental nodal forces existing on the element, then the element undergoes a small rigid body rotation. The incremental forces should keep their magnitude and follow the rigid body motions. Then taking advantage of the established geometric stiffness matrix derived by former researchers, the higher-order stiffness matrix can be analogy derived in explicit forms without any difficulty. It can be used at the forces recovery stage in the geometric nonlinear analysis of frame structures. Meanwhile an effective numerical method, the generalized displacement control method, was adopted to trace the load–deflection curves of structures. Some famous numerical examples were tested by the element which included the proposed higher-order stiffness matrix in tangential stiffness matrix.

Recent developments in geometrically nonlinear and postbuckling analysis of framed structures

Geometric nonlinear analysis of structures is not a simple extension from its counterpart of linear analysis. In this article, some research works conducted primarily in the past two decades on the geometric nonlinear analysis of framed structures that are readily available to the authors, including, in particular, those conducted by the senior author and coworkers, will be briefly reviewed. To highlight the key features of geometric nonlinear analysis, each of the papers cited will be reviewed according to one or more of the following categories: a) analytical or semi-analytical works, b) formulation of incremental nonlinear theory, c) discrete vs connected element and procedure of assembly, d) joint equilibrium conditions in the deformed configuration, e) rigid body test for nonlinear finite elements, f) key phases in incremental-iterative analysis, g) force recovery procedure, h) strategy for incremental-iterative approaches, i) rigid body-qualified geometric stiffness matrix, j) formulation and simulation for curved beam problems, k) special considerations for truss structures, and l) other related considerations. Throughout this article, emphasis will be placed on the theories and procedures leading to solution of the load-deflection response of structures, which may involve multi-looping curves in the postbuckling range. In fact, a nonlinear analysis using incremental-iterative schemes need not be as complicated as we think. If due account can be taken of the rigid body behaviors at each stage, then the whole process of incremental-iterative analysis can be made simpler and more efficient. Even when the postbuckling behavior of structures is of concern, the use of an accurate elastic stiffness matrix plus a rigid-body-qualified geometric stiffness matrix can always yield satisfactory results. There are 122 references cited in this review article.

Geometric Nonlinear Analysis of Frame Structures by Pseudodistortions

In this paper, the previously described pseudodistortion method is extended for the case of geometric nonlinearity. The general theory of stability is followed to account for axial force-bending interaction and P − Δ effects. The resulting second-order plastic hinge analysis may be solved by the imposition of pseudodistortions on members, and the complete behavior of the structure may be modeled by a single elastic analysis of the original frame and the determination of the appropriate pseudodistortions. The advanced theory of the pseudodistortion method is derived and then applied to sample frames. Results are compared to a conventional nonlinear analysis involving reformulation of the global stiffness matrix.

Nonlinear Analysis of Frame Structures by Pseudodistortions

A method is developed to reanalyze frame structures without requiring reformulation of the global stiffness matrix. The method is particularly well suited to those situations where a limited number of members are changed at each step in an iterative optimization or reliability process. It is equally applicable to those incremental loading process analyses where plastic hinges develop in the members. The method involves the imposition of pseudodistortions that have the same effect on displacements and member forces at all joints as the actual structural modification. The approach is an extension of the virtual distortion method that had previously been developed for pin-jointed structures. A single elastic analysis and determination of appropriate pseudodistortions is sufficient to derive the behavior of the modified structure, thus representing a significant efficiency in computational effort. The pseudodistortion method is applied to sample frames to illustrate its application for member replacement and plastic hinge formulation. A companion paper extends the method to include geometric nonlinearity.

Geometrically Nonlinear Flexibility-Based Frame Finite Element

The paper presents a new element for geometrically nonlinear analysis of frame structures. The proposed formulation is flexibility based and uses force interpolation functions for the bending moment variation that depend on the transverse displacements and strictly satisfy equilibrium in the deformed configuration. The derivation of the governing equations and their consistent linearization are substantially more involved than for stiffness-based elements. Nonetheless, the element offers significant advantages over existing stiffness-based approaches, since no discretization error occurs and all governing equations are satisfied exactly. Consequently, fewer elements are needed to yield results of comparable accuracy. This is demonstrated with the analysis of several simple example structures by comparing the results from flexibility and stiffness-based elements.

Modeling of Masonry Infill Panels for Structural Analysis

An analytical macromodel based on an equivalent strut approach integrated with a smooth hysteretic model is proposed for representing masonry infill panels in nonlinear analysis of frame structures. The hysteresis model uses degrading control parameters for stiffness and strength degradation and slip “pinching” that can be implemented to replicate a wide range of hysteretic force-displacement behavior resulting from different design and geometry. The paper presents the development of the hysteretic model and the definitions of the control parameters, which can be determined using any suitable theoretical model for masonry infills. An available theoretical model for simplified engineering evaluation of masonry infilled frames was explored for estimating the control parameters of the proposed macromodel. The macromodel was incorporated in a nonlinear structural analysis program, IDARC2D Version 4.0, for quasi-static cyclic and dynamic analysis of masonry infilled frames. Simulations of experimental force-deformation behavior of prototype infill frame subassemblies are performed to validate the proposed model. A lightly reinforced concrete frame structure is analyzed for strong ground motions to evaluate the influence of masonry infill panels on the dynamic response.