## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

Modern structural analysis necessitates numerical formulations with advanced nonlinear attributes. To that end, numerous finite elements have been proposed, spanning from classical to hybrid standpoints. In addition to their individual features, all formulations originally stem from an underlying variational principle, which can be deemed as a unified energy metric of the system. The corresponding equations of structural equilibrium define a stationary point of the assumed principle. Following this logic in this work, the total potential energy is directly treated as an objective function, subject to some kinematic compatibility constraints, within the conceptions of nonlinear programming. The only approximated internal field is curvature, whereas displacements occur solely as nodal entities and Lagrange multipliers serve compatibility. Thereby, a new nonlinear programming hybrid element formulation is derived, which uses exact kinematic fields, can incorporate nonlinear assumptions of any extent, and is amenable to various applicable nonlinear programming algorithms. The suggested nonlinear program is presented in detail herein, together with its consistent second-order iterative solution procedure. The results obtained in benchmark nonlinear structural problems are validated and compared with OpenSees flexibility-based elements, showcasing notable performance in terms of accuracy, mesh density discretization, computational speed, and robustness.

To read the full-text of this research,

you can request a copy directly from the authors.

... The present work is an extension of the geometrically exact hybrid formulation presented in [16] in order to account for the effect of shear deformation at the section level. As opposed to deriving the system equations from the Galerkin form after appropriate discretization, in the aforementioned work the problem is originally recast in a nonlinear programming framework, where the total potential energy functional (TPE) is augmented via Lagrange multipliers that enforce satisfaction of the exact kinematic conditions. ...

... In accordance with [16] we then interpolate the curvature field with Lagrange polynomials in order to obtain the rotations φ i at the quadrature points: ...

... An extension to the geometrically exact hybrid element derived in [16] is presented herein that accounts for shear deformations. The system of equilibrium equations is originally derived within a nonlinear programming framework, where the total potential energy functional is discretized and then augmented by the exact kinematic constraints of the physical problem, also in discretized form, and solved by determining the stationary point of the Lagrangian. ...

In the present work, a hybrid beam element based on exact kinematics is developed, accounting for arbitrarily large displacements and rotations, as well as shear deformable cross sections. At selected quadrature points, fiber discretization of the cross sections facilitates efficient computation of the stress resultants for any uniaxial material law. The numerical approximation is carried out through the lens of nonlinear programming, where the enengy functional of the system is treated as the objective function and the exact strain-displacement relations form the set of kinematic constraints. The only interpolated field is curvature, whereas the centerline axial and shear strains, along with the displacement measures at the element edges, are determined by enforcing compatibility through the use of any preferable constrained optimization algorithm. The solution satisfying the necessary optimality conditions is determined by the stationary point of the Lagrangian. A set of numerical examples demonstrates the accuracy and performance of the proposed element against analytical or approximate solutions available in the literature.

... The present work deviates from all previous formulations and suggests a novel hybrid beam element approach based on nonlinear programming (NLP) principles, 34 consistently also accounting for the effect of shear deformation at the section level and considerably extending the work in Reference 35. As opposed to deriving the system equations from the Galerkin form after appropriate discretization, we here recast the problem in a NLP framework by utilizing the underlying variational structure. ...

... Given that for highly nonlinear problems we typically have n ∈ N( [5,10]) for satisfactory accuracy, this results in an element flexibility matrix of dimension dim(B e ) ≤ 13, thus accelerating the analysis considerably. Further discussion on the complexity of this approach can be found in Andriotis et al. 34 Having written the system of equations in the form of Equation (43), implementation of arc-length type schemes is now straightforward. In the present work, we adopt the algorithm proposed by Crisfield 46 whereby the additional equation supplemented to the system is: ...

This work presents a hybrid shear‐flexible beam‐element, capable of capturing arbitrarily large inelastic displacements and rotations of planar frame structures with just one element per member. Following Reissner’s geometrically‐exact theory, the finite element problem is herein formulated within nonlinear programming principles, where the total potential energy is treated as the objective function and the exact strain‐displacement relations are imposed as kinematic constraints. The approximation of integral expressions is conducted by an appropriate quadrature, and by introducing Lagrange multipliers, the Lagrangian of the minimization program is formed and solutions are sought based on the satisfaction of necessary optimality conditions. In addition to displacement degrees of freedom at the two element edge nodes, strain measures of the centroid act as unknown variables at the quadrature points, while only the curvature field is interpolated, to enforce compatibility throughout the element. Inelastic calculations are carried out by numerical integration of the material stress‐strain law at the cross‐section level. The locking‐free behavior of the element is presented and discussed, and its overall performance is demonstrated on a set of well‐known numerical examples. Results are compared with analytical solutions, where available, and outcomes based on flexibility‐based beam elements and quadrilateral elements, verifying the efficiency of the formulation.

... Although possible in some cases regarding second-order derivatives, e.g. [67], in the majority of cases higher-order derivatives are not provided by computational models, such as finite element models. In addition, the computational cost still increases importantly and extra model calls per leapfrog step are usually restrictive for computationally expensive models. ...

... Similar to material inelasticity and degradation effects, geometric nonlinearity is also an important aspect of nonlinear structural analysis, which has been long investigated by many researchers and mostly described by total or updated Lagrangian formulations, often employing corotational schemes (e.g., Crisfield 1991;Neuenhofer and Filippou 1998;Felippa and Haugen 2005;Bathe 2006;Belytschko et al. 2013). Geometric nonlinearities in beam elements can also be incorporated by considering geometrically exact kinematics, as described in the seminal work of Reissner (1972Reissner ( , 1973 and further adopted/modified for finite-element formulations by Simo (1985), Cardona and Geradin (1988), Sivaselvan and Reinhorn (2002), Romero (2008), Andriotis et al. (2018), and Lyritsakis et al. (2021), among others. Some works considering the combined effects of geometric nonlinearities and damage include those of Bratina et al. (2004), Valipour and Foster (2010), and Salehi and Sideris (2018). ...

... Since the model satisfies the plasticity postulates, a consistent evolution of the capacity surface with degradations is now obtained. Eq. (33) is treated as a consistent constitutive law for the hysteretic finite-element formulation herein; however, it can be readily adopted for other modeling approaches, e.g., in fiber elements (Spacone et al. 1996;Scott et al. 2008;Andriotis et al. 2018) to represent stress-strain relationships and model cross section behaviors, among many other applications. Because P h ðxÞ ¼ H D z ðxÞ , the hysteretic deformation vector is obtained as: ...

... Although possible in some cases regarding second-order derivatives, e.g. [67], in the majority of cases higher-order derivatives are not provided by computational models, such as finite element models. In addition, the computational cost still increases importantly and extra model calls per leapfrog step are usually restrictive for computationally expensive models. ...

Preprint also available: https://arxiv.org/abs/2007.00180
------------------------------------------------------------------------------------------------------------------------------------------
Accurate and efficient estimation of rare events probabilities is of significant importance, since often the occurrences of such events have widespread impacts. The focus in this work is on precisely quantifying these probabilities, often encountered in reliability analysis of complex engineering systems, based on an introduced framework termed Approximate Sampling Target with Post-processing Adjustment (ASTPA), which herein is integrated with and supported by gradient-based Hamiltonian Markov Chain Monte Carlo (HMCMC) methods. The basic idea is to construct a relevant target distribution by weighting the high-dimensional random variable space through a one-dimensional output likelihood model, using the limit-state function. To sample from this target distribution, we exploit HMCMC algorithms, a family of MCMC methods that adopts physical system dynamics, rather than solely using a proposal probability distribution, to generate distant sequential samples, and we develop a new Quasi-Newton mass preconditioned HMCMC scheme (QNp-HMCMC), which is particularly efficient and suitable for high-dimensional spaces. To eventually compute the rare event probability, an original post-sampling step is devised using an inverse importance sampling procedure based on the already obtained samples. The statistical properties of the estimator are analyzed as well, and the performance of the proposed methodology is examined in detail and compared against Subset Simulation in a series of challenging low- and high-dimensional problems.

... This is driven by the fact that off-policy algorithms are more sample efficient than their on-policy counterparts, as previously underlined. This attribute can be critical in large engineering systems control, as samples are often drawn from computationally expensive nonlinear and/or dynamic structural models through demanding numerical simulations, e.g. in [68,69]. In addition, as in most standard DRL approaches, experience replay is utilized here for more efficient training. ...

Decision-making for engineering systems can be efficiently formulated as a Markov Decision Process (MDP) or a Partially Observable MDP (POMDP). Typical MDP and POMDP solution procedures utilize offline knowledge about the environment and provide detailed policies for relatively small systems with tractable state and action spaces. However, in large multi-component systems the sizes of these spaces easily explode, as system states and actions scale exponentially with the number of components, whereas environment dynamics are difficult to be described in explicit forms for the entire system and may only be accessible through numerical simulators. In this work, to address these issues, an integrated Deep Reinforcement Learning (DRL) framework is introduced. The Deep Centralized Multi-agent Actor Critic (DCMAC) is developed, an off-policy actor-critic DRL approach, providing efficient life-cycle policies for large multi-component systems operating in high-dimensional spaces. Apart from deep function approximations that parametrize large state spaces, DCMAC also adopts a factorized representation of the system actions, being able to designate individualized component- and subsystem-level decisions, while maintaining a centralized value function for the entire system. DCMAC compares well against Deep Q-Network (DQN) solutions and exact policies, where applicable, and outperforms optimized baselines that are based on time-based, condition-based and periodic policies.

Until recently, linear analysis has been considered sufficient for the static analysis of structural frames. Nonlinear effects, if included, have tended to be considered at the element level rather than at the complete structure level. However, recent changes in codes of practice have been introduced that require a more complete nonlinear analysis to be performed. While these requirements should lead to a more accurate analysis, there has been little guidance given to the type and implementation of such an analysis. Moreover, different implementations have been adopted by various commercial software. In this paper, we discuss the use of mixed finite elements for the large deflection analysis of two-dimensional frames including shear deformation. In particular, we develop a family of elements that can be combined with different nonlinear models and discuss the effects of various assumptions and approximations that are commonly used to simplify the analysis. Examples are given to illustrate the various issues discussed.

Decision-making for engineering systems management can be efficiently formulated using Markov Decision Processes (MDPs) or Partially Observable MDPs (POMDPs). Typical MDP/POMDP solution procedures utilize offline knowledge about the environment and provide detailed policies for relatively small systems with tractable state and action spaces. However, in large multi-component systems the dimensions of these spaces easily explode, as system states and actions scale exponentially with the number of components, whereas environment dynamics are difficult to be described explicitly for the entire system and may, often, only be accessible through computationally expensive numerical simulators. In this work, to address these issues, an integrated Deep Reinforcement Learning (DRL) framework is introduced. The Deep Centralized Multi-agent Actor Critic (DCMAC) is developed, an off-policy actor-critic DRL algorithm that directly probes the state/belief space of the underlying MDP/POMDP, providing efficient life-cycle policies for large multi-component systems operating in high-dimensional spaces. Apart from deep network approximators parametrizing complex functions with vast state spaces, DCMAC also adopts a factorized representation of the system actions, thus being able to designate individualized component- and subsystem-level decisions, while maintaining a centralized value function for the entire system. DCMAC compares well against Deep Q-Network and exact solutions, where applicable, and outperforms optimized baseline policies that incorporate time-based, condition-based, and periodic inspection and maintenance considerations.

This paper deals with the progressive collapse analysis of a tall steel frame following the removal of a corner column according to the alternate load path approach. Several analysis techniques are considered (eigenvalue, material nonlinearities, material and geometric nonlinearities), as well as 2D and 3D modelling of the structural system. It is determined that the collapse mechanism is a loss-of-stability-induced one that can be identified by combining a 3D structural model with an analysis involving both material and geometric nonlinearities. The progressive collapse analysis reveals that after the initial removal of a corner column, its two adjacent columns fail from elastic flexural-torsional buckling at a load lower than the design load. The failure of these two columns is immediately followed by the failure of the next two adjacent columns from elastic flexural–torsional buckling. After the failure of these five columns, the entire structure collapses without the occurrence of any significant plastification. The main contribution is the identification of buckling-induced collapse mechanisms in steel frames involving sequential buckling of multiple columns. This is a type of failure mechanism that has not received appropriate attention because it practically never occurs in properly designed structures without the accidental loss of a column.

This dissertation presents a force-based formulation for inelastic large displacement analysis of planar and spatial frames, and its consistent numerical implementation in a general-purpose finite element program. The main idea of the method is to use force interpolation functions that strictly satisfy equilibrium in the deformed configuration of the element. The appropriate reference frame for establishing these force interpolation functions is a basic coordinate system without rigid body modes. In this system, the element tangent stiffness is non-singular and can be obtained by inversion of the flexibility matrix. The formulation is derived from a geometrically nonlinear form of the Hellinger-Reissner potential, with a nonlinear strain-displacement relation that corresponds to a degenerated form of Green-Lagrange strains. Although the adopted kinematics is based on the assumption of moderately large deformations along the element, rigid body displacements and rotations can be arbitrarily large. This is accomplished with the use of the corotational formulation, in which rigid body modes are separated from element deformations by attaching a reference coordinate system (the basic system) to the element as it deforms. The transformations of displacements and forces between the basic and the global systems are determined with no simplifications regarding the magnitude of the rigid body motion. The non-vectorial nature of rotations in space is handled consistently, through the representation in terms of rotation matrices, rotational vectors and unit quaternions. A new algorithm for the determination of the element resisting forces and tangent stiffness matrix for given trial displacements is proposed. The iterative and non-iterative forms of the algorithm are presented, generalizing earlier procedures for this class of force-based elements. Several planar and spatial problems are studied in order to validate the proposed element. With the present formulation, only one element per structural member is necessary for the analysis of problems with large rigid body rotations and moderate deformations. Furthermore, finite strain problems can also be solved with the proposed formulation, provided that the structural member is subdivided into smaller elements.

In recent years nonlinear dynamic analysis of three-dimensional structural models is used more and more in the assessment of existing structures in zones of high seismic risk and in the development of appropriate retrofit strategies. In this framework beam finite-element models of various degrees of sophistication are used in the description of the hysteretic behavior of structural components under a predominantly uniaxial state of strain and stress. These models are commonly derived with the displacement method of analysis, but recent studies have highlighted the benefits of frame models that are based on force interpolation functions (flexibility approach). These benefits derive from the fact that models with force interpolation functions that reproduce the variation of internal element forces in a strict sense yield the exact solution of the governing equations in the absence of geometric nonlinearity. While the numerical implementation of force-based models at first appears cumbersome, simple examples of nonlinear analysis in this paper offer conclusive proof of the numerical and computational superiority of these models on account of the smaller number of model degrees of freedom for the same degree of accuracy in the global and local response. A numerical implementation that bypasses the iterative nature of the element state determination in recent force-based elements is also introduced, thus further expanding the benefits of flexibility-based nonlinear frame models.

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton's principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.

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.

This paper addresses the development of a hybrid-mixed finite element formulation for the quasi-static geometrically exact
analysis of three-dimensional framed structures with linear elastic behavior. The formulation is based on a modified principle
of stationary total complementary energy, involving, as independent variables, the generalized vectors of stress-resultants
and displacements and, in addition, a set of Lagrange multipliers defined on the element boundaries. The finite element discretization
scheme adopted within the framework of the proposed formulation leads to numerical solutions that strongly satisfy the equilibrium
differential equations in the elements, as well as the equilibrium boundary conditions. This formulation consists, therefore,
in a true equilibrium formulation for large displacements and rotations in space. Furthermore, this formulation is objective,
as it ensures invariance of the strain measures under superposed rigid body rotations, and is not affected by the so-called
shear-locking phenomenon. Also, the proposed formulation produces numerical solutions which are independent of the path of
deformation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their
solutions compared with those obtained using the standard two-node displacement/ rotation-based formulation.
KeywordsThree-dimensional framed structures–One-dimensional beam model–Geometrically exact analysis–Complementary energy principle–Hybrid-mixed finite elements

In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation.
A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements.
The advantages of the approach are illustrated with a few numerical examples.

This survey paper describes recent developments in the area of parametrized variational principles (PVPs) and selected applications to finite-element computational mechanics. A PVP is a variational principle containing free parameters that have no effect on the Euler-Lagrange equations. The theory of single-field PVPs, based on gauge functions (also known as null Lagrangians) is a subset of the Inverse Problem of Variational Calculus that has limited value. On the other hand, multifield PVPs are more interesting from theoretical and practical standpoints. Following a tutorial introduction, the paper describes the recent construction of multifield PVPs in several areas of elasticity and electromagnetics. It then discusses three applications to finite-element computational mechanics: the derivation of high-performance finite elements, the development of element-level error indicators, and the construction of finite element templates. The paper concludes with an overview of open research areas.

Fragility functions indicate the probability of a system exceeding certain damage states given some appropriate measures that characterize recorded or simulated data series. Presented in two main parts, this paper develops fragility functions in their utmost generality, accounting for both (1) multivariate intensity measures with multiple damage states and (2) longitudinal damage state dependencies in time. Without adopting the limiting assumption of common variance to avoid improper function crossings, the first part presents what is here compactly termed as extended fragility functions. As shown, these are best supported by the softmax function for any arbitrary distribution of the exponential family to which the intensity measures of different states may belong, including the typically used normal distribution in the logarithmic scale of intensity measures. In the second part, generalized fragility functions are introduced for cases where multiple system state transitions need to be captured. To that end, dependent Markov and hidden Markov models are employed because they are able to portray longitudinal data dependencies and reveal intrinsic deterioration trends for multiple sequential events. Numerical results are presented, together with underlying implementation details, statistical properties, and practical suggestions.

"Linear and Nonlinear Programming" is considered a classic textbook in Optimization. While it is a classic, it also reflects modern theoretical insights. These insights provide structure to what might otherwise be simply a collection of techniques and results, and this is valuable both as a means for learning existing material and for developing new results. One major insight of this type is the connection between the purely analytical character of an optimization problem, expressed perhaps by properties of the necessary conditions, and the behavior of algorithms used to solve a problem. This was a major theme of the first and second editions. Now the third edition has been completely updated with recent Optimization Methods. Yinyu Ye has written chapters and chapter material on a number of these areas including Interior Point Methods.
This book is designed for either self-study by professionals or classroom work at the undergraduate or graduate level for technical students. Like the field of optimization itself, which involves many classical disciplines, the book should be useful to system analysts, operations researchers, numerical analysts, management scientists, and other specialists.

In this work, a new smooth model for uniaxial concrete behavior that combines plasticity and damage considerations, together with unsymmetrical hysteresis for tension compression and nonlinear unloading, is presented. Softening and stiffness degradation phenomena are handled through a scalar damage-driving variable, which is a function of total strain. Smoothening of the incremental damage behavior is achieved, following similar steps as for Bouc-Wen modeling of classical plasticity, thus exploiting their common mathematical structure. The uniaxial model for concrete, together with the standard steel model exhibiting kinematic hardening, are employed to derive a fiber beam-column element that is used to assemble the numerical model of frame structures. Following the displacement-based approach, the solution of the entire system is established using a standard Newton-Raphson numerical scheme, which incorporates the evolution equations of all fibers elevated at the section, element, and structural level in the inner loop. Numerical results that compare well with existing experimental data are presented, demonstrating the accuracy and efficacy of the proposed formulation.

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist René de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature. Key features: • Combines the two previous volumes into one heavily revised text with obsolete material removed, an improved layout and updated references and notations • Extensive new material on more recent developments in computational mechanics • Easily readable, engineering oriented, with no more details in the main text than necessary to understand the concepts. • Pseudo-code throughout makes the link between theory and algorithms, and the actual implementation. • Accompanied by a website (www.wiley.com/go/deborst) with a Python code, based on the pseudo-code within the book and suitable for solving small-size problems. Non-linear Finite Element Analysis of Solids and Structures, 2nd Edition is an essential reference for practising engineers and researchers that can also be used as a text for undergraduate and graduate students within computational mechanics.

Estimation of fragility functions using dynamic structural analysis is an important step in a number of seismic assessment procedures. This paper discusses the applicability of statistical inference concepts for fragility function estimation, describes appropriate fitting approaches for use with various structural analysis strategies, and studies how to fit fragility functions while minimizing the required number of structural analyses. Illustrative results show that multiple stripe analysis produces more efficient fragility estimates than incremental dynamic analysis for a given number of structural analyses, provided that some knowledge of the building's capacity is available prior to analysis so that relevant portions of the fragility curve can be approximately identified. This finding has other benefits, given that the multiple stripe analysis approach allows for different ground motions to be used for analyses at varying intensity levels, to represent the differing characteristics of low-intensity and high-intensity shaking. The proposed assessment approach also provides a framework for evaluating alternate analysis procedures that may arise in the future.

Boundary-value problems in solid mechanics are often addressed, from both theoretical and numerical points of view, by resorting to displacement/rotation-based variational formulations. For conservative problems, such formulations may be constructed on the basis of the Principle of Stationary Total Potential Energy. Small deformation problems have a unique solution and, as a consequence, their corresponding total potential energies are globally convex. In this case, under the so-called Legendre transform, the total potential energy can be transformed into a globally concave total complementary energy only expressed in terms of stress variables. However, large deformation problems have, in general, for the same boundary conditions, multiple solutions. As a result, their associated total potential energies are globally non-convex. Notwithstanding, the Principle of Stationary Total Potential Energy can still be regarded as a minimum principle, only involving displacement/rotation fields. The existence of a maximum complementary energy principle defined in a truly dual form has been subject of discussion since the first contribution made by Hellinger in 1914. This paper provides a survey of the complementary energy principles and also accounts for the evolution of the complementary-energy based finite element models for geometrically non-linear solid/structural models proposed in the literature over the last 60 years, giving special emphasis to the complementary-energy based methods developed within the framework of the geometrically exact Reissner-Simo beam theory for the analysis of structural frames.

This paper is devoted to the modeling of planar slender beams undergoing large displacements and finite rotations. Transverse shear deformation of beams that is trivial for most slender beams is neglected in the present model, though within the framework of the geometrically exact beam theory proposed by Reissner. A weak form quadrature element formulation is proposed which is characterized by highly efficient numerical integration and differentiation, thus minimizing the number of elements as well as the total degrees-of-freedom. Several typical examples are presented to demonstrate the effectiveness of the beam model and the weak form quadrature element formulation.

This paper presents several beam-column finite element formulations for full nonlinear distributed plasticity analysis of planar frame structures. The fundamental steps within the derivation of displacement-based, flexibility-based, and mixed elements are summarized, These formulations are presented using a total Lagrangian corotational approach. In this context, the element displacements are separated into rigid-body and deformational (or natural) degrees of freedom. The element rigid-body motion is handled separately within the mapping from the corotational to global element frames. This paper focuses on the similarities and differences in the element formulations associated with the element natural degrees of freedom within the corotational frame. The paper focuses specifically on two-dimensional elements based on Euler-Bernoulli kinematics; however, the concepts are also applicable to general beam-column elements for three-dimensional analysis. The equations for the consistent tangent stiffness matrices are presented, and corresponding consistent element state determination algorithms are explained. Numerical examples are provided to compare the performance of the above elements.

Finite-element models of reinforced-concrete beam-columnsoften fail to behave in a stable manner near the point of maximumresistance. Simple numerical examples presented indicate that theconventional displacement formulation when used at the section ormember level is unable to establish solutions associated with softeningbehavior. Thus, such analysis models are often limited in their applicabilityor necessitate the use of unrealistic material laws. A mixedfinite-element scheme is proposed herein that exhibits stable numericalbehavior even when critical regions become ill-conditioned. It is demonstratedthat enforcing equilibrium within the member during statedetermination provides the necessary constraint to obtain a stablesolution. Examples are presented to demonstrate the reliability andrealism of this approach for members subjected to generalized excitationseven where severe deformation softening occurs.

This paper addresses the development of a hybrid-mixed finite-element formulation for the geometrically exact quasi-static analysis of elastic planar framed structures, modeled using the two-dimensional Reissner beam theory. The proposed formulation relies on a modified principle of complementary energy, which involves, as independent variables, the generalized vectors of stress resultants and displacements and, in addition, a set of Lagrange multipliers used to enforce the stress continuity between elements. The adopted finite-element discretization produces numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the static boundary conditions. It consists, therefore, in a true equilibrium formulation for arbitrarily large displacements and rotations. Furthermore, as it does not suffer from shear locking or any other artificial stiffening phenomena, it may be regarded as an alternative to the standard displacement-based formulation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions are compared with those obtained using the standard two-node displacement-based formulation. Numerical analyses of convergence of the proposed finite-element formulation are also included.

In this paper a beam element that accounts for inelastic axial-flexure–shear coupling is presented. The mathematical model is derived from a three-field variational form. The finite element approximation for the beam uses shape functions for section forces that satisfy equilibrium and discontinuous section deformations along the beam. No approximation for the beam displacement field is necessary in the formulation. The coupling of the section forces is achieved through the numerical integration of an inelastic multi-axial material model over the cross-section. The proposed element is free from shear-locking. Examples confirm the accuracy and numerical robustness of the proposed element and showcase the interaction between axial force, shear, and bending moment.

This paper describes a co-rotational formulation for three-dimensional beams in which both the internal force vector and tangent stiffness matrix are consistently derived from the adopted ‘strain measures’. The latter relate to standard beam theory but are embedded in a continuously rotating frame. A set of numerical examples show that the element provides an excellent numerical performance.

Bernoulli–Euler beam theory has long been the standard for the analysis of reticulated structures. The need to accurately compute the non-linear (material and geometric) response of structures has renewed interest in the application of mixed variational approaches to this venerable beam theory. Recent contributions in the literature on mixed methods and the so-called (but quite related) non-linear flexibility methods have left open the question of what is the best approach to the analysis of beams. In this paper we present a consistent computational approach to one-, two-, and three-field variational formulations of non-linear Bernoulli–Euler beam theory, including the effects of non-linear geometry and inelasticity. We examine the question of superiority of methods through a set of benchmark problems with features typical of those encountered in the structural analysis of frames. We conclude that there is no clear winner among the various approaches, even though each has predictable computational strengths. Copyright © 2003 John Wiley & Sons, Ltd.

This paper presents a unified theoretical framework for the corotational (CR) formulation of finite elements in geometrically nonlinear structural analysis. The key assumptions behind CR are: (i) strains from a corotated configuration are small while (ii) the magnitude of rotations from a base configuration is not restricted. Following a historical outline the basic steps of the element independent CR formulation are presented. The element internal force and consistent tangent stiffness matrix are derived by taking variations of the internal energy with respect to nodal freedoms. It is shown that this framework permits the derivation of a set of CR variants through selective simplifications. This set includes some previously used by other investigators. The different variants are compared with respect to a set of desirable qualities, including self-equilibrium in the deformed configuration, tangent stiffness consistency, invariance, symmetrizability, and element independence. We discuss the main benefits of the CR formulation as well as its modeling limitations.

A geometric and material non-linear analysis procedure for framed structures is presented, using a solution algorithm of minimizing the residual displacements. This new non-linear solution technique is believed to be the optimum in the Newton–Raphson scheme since it follows the shortest path to achieve convergence. The concept of the effective tangent stiffness matrix is introduced and is found to be efficient, simple and logical in handling the non-linear analysis of frames with braced members and in separating multiple bifurcation points.

The paper presents a spatial Timoshenko beam element with a total Lagrangian formulation. The element is based on curvature interpolation that is independent of the rigid-body motion of the beam element and simplifies the formulation. The section response is derived from plane section kinematics. A two-node beam element with constant curvature is relatively simple to formulate and exhibits excellent numerical convergence. The formulation is extended to N-node elements with polynomial curvature interpolation. Models with moderate discretization yield results of sufficient accuracy with a small number of iterations at each load step. Generalized second-order stress resultants are identified and the section response takes into account non-linear material behaviour. Green–Lagrange strains are expressed in terms of section curvature and shear distortion, whose first and second variations are functions of node displacements and rotations. A symmetric tangent stiffness matrix is derived by consistent linearization and an iterative acceleration method is used to improve numerical convergence for hyperelastic materials. The comparison of analytical results with numerical simulations in the literature demonstrates the consistency, accuracy and superior numerical performance of the proposed element. Copyright © 2001 John Wiley & Sons, Ltd.

An updated Lagrangian and a total Lagrangian formulation of a three-dimensional beam element are presented for large displacement and large rotation analysis. It is shown that the two formulations yield identical element stiffness matrices and nodal point force vectors, and that the updated Lagragian formulation is computationally more effective. This formulation has been implemented and the resulted of some sample analyses are given.

The paper formulates a one-dimensional large-strain beam theory for plane deformations of plane beams, with rigorous consistency of dynamics and kinematics via application of the principle of virtual work. This formulation is complemented by considerations on how to obtain constitutive equations, and applied to the problem of buckling of circular rings, including the effects of axial normal strain and transverse shearing strain.Das Ziel dieser Arbeit ist eine eindimensionale Theorie mit endlichen Dehnungen und Schubformnderungen, fr ebene Verformungen von ursprnglich ebenen Balken. Das wesentliche der Theorie ist die genaue Vertrglichkeit der dynamischen und kinematischen Gleichungen, insoweit das Prinzip der virtuellen Arbeiten in Frage kommt. Die vorstehenden Entwicklungen sind vervollstndigt durch Betrachtungen ber das Problem der Aufstellung von Spannungs-Formnderungsbeziehungen und durch eine Anwendung auf das Knickproblem des Kreisringes einschliesslich des Einflusses von Axialdehnung und Schubverformung.

A finite element formulation of finite deformation static analysis of plane elastic-plastic frames subjected to static loads is presented, in which the only function to be interpolated is the rotation of the centroid axis of the beam. One of the advantages of such a formulation is that the problem of the field-consistency does not arise. Exact non-linear kinematic relationships of the finite-strain beam theory are used, which assume the Bernoulli hypothesis of plane cross-sections. Finite displacements and rotations as well as finite extensional and bending strains are accounted for. The effects of shear strains and non-conservative loads are at present neglected, yet they can simply be incorporated in the formulation. Because the potential energy of internal forces does not exist with elastic-plastic material, the principle of virtual work is introduced as the basis of the finite element formulation. A generalized principle of virtual work is proposed in which the displacements, rotation, extensional and bending strains, and the Lagrangian multipliers are independent variables. By exploiting the special structure of the equations of the problem, the displacements, the strains and the multipliers are eliminated from the generalized principle of virtual work. A novel principle is obtained in which the rotation becomes the only function to be approximated in its finite element implementation. It is shown that (N−1)-point numerical integration must be employed in conjunction with N-node interpolation polynomials for the rotation, and the Lobatto rule is recommended. Regarding the integration over the cross-section, it is demonstrated by numerical examples that, due to discontinuous integrands, no integration order defined as ‘computationally efficient yet accurate enough’ could be suggested. The theoretical findings and a nice performance of the derived finite elements are illustrated by numerical examples.

Locking phenomena in C0 curved finite elements are studied for displacement, hybrid-stress and mixed formulations. It is shown that for a curved beam element, shear and membrane locking are interrelated and either shear or membrane underintegration can alleviate it. However, reduced shear integration tends to diminish the membrane-flexural coupling which characterizes curved elements. Locking can also be expected in certain types of mixed formulations, the hybrid-stress formulations avoids locking for beams (but not for shells). Methods for avoiding locking are explored and alternatives evaluated.

The geometrically nonlinear analysis of elastic inplane oriented bodies, e.g. beams, frames and arches, is presented in a total Lagrangian co-ordinate system. By adopting a continuum approach, employing a paralinear isoparametric element, the formulation is applicable to structures consisting of straight or curved members. Displacements and rotations are unrestricted in magnitude. The nonlinear equilibrium equations are solved using the Newton-Raphson method for which a number of examples are given. The derivations are extended to include axisymmetric structures.

A new simulation approach, called ‘subset simulation’, is proposed to compute small failure probabilities encountered in reliability analysis of engineering systems. The basic idea is to express the failure probability as a product of larger conditional failure probabilities by introducing intermediate failure events. With a proper choice of the conditional events, the conditional failure probabilities can be made sufficiently large so that they can be estimated by means of simulation with a small number of samples. The original problem of calculating a small failure probability, which is computationally demanding, is reduced to calculating a sequence of conditional probabilities, which can be readily and efficiently estimated by means of simulation. The conditional probabilities cannot be estimated efficiently by a standard Monte Carlo procedure, however, and so a Markov chain Monte Carlo simulation (MCS) technique based on the Metropolis algorithm is presented for their estimation. The proposed method is robust to the number of uncertain parameters and efficient in computing small probabilities. The efficiency of the method is demonstrated by calculating the first-excursion probabilities for a linear oscillator subjected to white noise excitation and for a five-story nonlinear hysteretic shear building under uncertain seismic excitation.

This paper proposes a novel simplified framework for progressive collapse assessment of multi-storey buildings, considering sudden column loss as a design scenario. The proposed framework offers a practical means for assessing structural robustness at various levels of structural idealisation, and importantly it takes the debate on the factors influencing robustness away from the generalities towards the quantifiable. A major feature of the new approach is its ability to accommodate simplified as well as detailed models of the nonlinear structural response, with the additional benefit of allowing incremental assessment over successive levels of structural idealisation. Three main stages are utilised in the proposed assessment framework, including the determination of the nonlinear static response, dynamic assessment using a novel simplified approach, and ductility assessment. The conceptual clarity of the proposed framework sheds considerable light on the adequacy of commonly advocated measures and indicators of structural robustness, culminating in the proposal of a single rational measure of robustness that is applicable to building structures subject to sudden column loss. The companion paper details the application of the new approach to progressive collapse assessment of real steel-framed composite multi-storey buildings, making in the process important conclusions on the inherent robustness of such structures and the adequacy of current design provisions.

Recent studies show that beam finite elements that enforce equilibrium rather than compatibility along the element are better suited for the description of the nonlinear behavior of frame elements. This is particularly true for elements that exhibit loss of strength and stiffness under monotonic and cyclic loads. Existing formulations, however, fail to define a consistent way of implementing the force method in the context of imposed kinematic, rather than static, boundary conditions, as is the case for element models in a standard finite element program. This paper derives the general formulation of a beam finite element from a mixed approach which points the way to the consistent numerical implementation of the element state determination in the context of a standard finite element program. The element state determination centers on a new iterative solution algorithm that is based on residual deformations rather than residual forces at the section and element level. Equilibrium is enforced in a strict sense along the element, while the section constitutive relation is satisfied within a specified tolerance when the algorithm converges. The proposed algorithm is general and can be used with any section constitutive relation.

Riks [1] has recently proposed a new solution procedure for overcoming limit points. To this end, he adds, to the standard equilibrium equations, a constraint equation fixing the length of the incremental load step in load/deflection space. The applied load level becomes an additional variable.The present paper describes a means of modifying Rik's approach so that it is suitable for use with the finite element method. The procedure is applied in conjunction with the modified Newton-Raphson method in both its original and accelerated forms. The resulting techniques not only allow limit points to be passed, but also, improve the convergence characteristics of the unconstrained iterative procedures. Illustrative examples include the large deflection analysis of shallow elastic shells and the collapse analysis of a stiffened steel diaphragm from a box-girder bridge.

A large displacement theory for in-plane elastic plastic frames is formulated. The derivation is based on incremental variational principle using TLA formulation. A beam element with local-global DOF is used in the analysis. The method of solution of the FEM equations is related to computation in the load-configuration space RN+1.Results of two numerical examples are compared with classical solutions.

COMPREHENSIVE COVERAGE OF NONLINEAR PROGRAMMING THEORY AND ALGORITHMS, THOROUGHLY REVISED AND EXPANDED Nonlinear Programming: Theory and Algorithms-now in an extensively updated Third Edition-addresses the problem of optimizing an objective function in the presence of equality and inequality constraints. Many realistic problems cannot be adequately represented as a linear program owing to the nature of the nonlinearity of the objective function and/or the nonlinearity of any constraints. The Third Edition begins with a general introduction to nonlinear programming with illustrative examples and guidelines for model construction. Concentration on the three major parts of nonlinear programming is provided: Convex analysis with discussion of topological properties of convex sets, separation and support of convex sets, polyhedral sets, extreme points and extreme directions of polyhedral sets, and linear programming Optimality conditions and duality with coverage of the nature, interpretation, and value of the classical Fritz John (FJ) and the Karush-Kuhn-Tucker (KKT) optimality conditions; the interrelationships between various proposed constraint qualifications; and Lagrangian duality and saddle point optimality conditions Algorithms and their convergence, with a presentation of algorithms for solving both unconstrained and constrained nonlinear programming problems Important features of the Third Edition include: New topics such as second interior point methods, nonconvex optimization, nondifferentiable optimization, and more Updated discussion and new applications in each chapter Detailed numerical examples and graphical illustrations Essential coverage of modeling and formulating nonlinear programs Simple numerical problems Advanced theoretical exercises The book is a solid reference for professionals as well as a useful text for students in the fields of operations research, management science, industrial engineering, applied mathematics, and also in engineering disciplines that deal with analytical optimization techniques. The logical and self-contained format uniquely covers nonlinear programming techniques with a great depth of information and an abundance of valuable examples and illustrations that showcase the most current advances in nonlinear problems.

Matrix methods of structural analysis: A precis of recent developments

- J H Argyris
- S Kesley
- H Kamel

Argyris, J. H., S. Kesley, and H. Kamel. 1964. Matrix methods of structural
analysis: A precis of recent developments. Oxford, UK: Pergamon
Press.