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The paper proposes an efficient sequential edge-smoothed finite element (ES-FE) analysis method enhanced with the smoothed recovery stress field to determine the collapse load limit of inelastic structures. The approach applies a modified concept of elastic compensation algorithm developed within the computationally advantageous ES-FE modelling framework and recovery stress field enhancement. It solely performs a series of standard elastic analyses to successively converge the limit load of structures at plastic collapse. Each analysis systematically adjusts the values of elastic moduli associated with some critical elements as determined by the intensity of stress resultants and hence considers inelastic stress distributions. The computing efforts are as modest as would be required to furnish standard linear elastic analysis procedures. What is important is that the proposed modulus smoothing technique enables the fast assembly of modified stiffness formulations, and the recovery stress field ensures the smoothed C0-continuous admissible stress conditions enhancing the satisfaction of yield conformity over an entire element. Various successfully solved numerical examples highlight the accuracy and efficiency of the analysis framework in overcoming the numerical challenges associated with stress singularity and volumetric locking phenomena under incompressibility conditions. They also address the determination of a lower-bound limit for sufficiently fine ES-FE models.

The majority of topology optimization methods for porous infill designs is based on the assumption of deterministic loads. However, in practice, quantities such as positions, weights, and directions of applied loads may change accidentally. Deterministic load-based designs might deliver poor structural performance under loading uncertainties. Such uncertain factors need to be taken into account in topological optimization to seek robust results. This paper presents a novel robust concurrent topology optimization method for the design of uniform/non-uniform porous infills under the accidental change of loads. A combination of moving morphable bars (MMBs) and loading uncertainties is proposed to directly model multiscale structures and seek robust designs. The macro- and microscopic structures can be simultaneously optimized through the minimization of the weighted sum of the expected compliance and standard deviation. The geometries of adaptive geometric components (AGCs) are straightforwardly optimized. The AGCs consist of two classes of geometric components: macroscopic bars describing the overall structure and microscopic bars describing the material microstructures. Automatic mesh-refinement is utilized to enhance computing efficiency. Numerical examples demonstrate that robust porous design can be obtained with only one global volume constraint while the material continuity of neighboring unit cells and the structural porosity can be maintained without additional constraints. The robust designs yield a more robust structural performance along with a smaller standard deviation compared with deterministic porous designs under loading uncertainties.

The paper proposes a novel numerical approach that incorporates the use of a modified elastic compensation method, within a polygon scaled boundary finite element framework, to determine the maximum load capacity of structures at plastic collapse. The distinctive feature of the proposed scheme is its effective computational ability in performing a series of successive elastic analyses by systematically adjusting elastic material properties of structures up to failure. The quadtree structural discretization within a polygon scaled boundary finite element platform enables model construction of sophisticated geometries at modest computing effort and thus the effective analysis of large-scale structures. The approach overcomes the challenges associated with stress singularity and locking phenomena under incompressibility conditions, even in the presence of high-order nonlinear yield loci. The robustness and accuracy of the proposed scheme are validated through a number of benchmarks and available practical engineering applications in 2D and 3D spaces. These illustrate the influences of some key algorithmic parameters, and the satisfaction of a lower-bound limit given by the present analysis method for a sufficiently fine discretization.

The paper proposes a novel automatic adaptive recovery stress edge-smoothed finite element method (ES-FEM) that determines the maximum load capacity of inelastic structures at plastic collapse. This approach performs solely a series of elastic ES-FEM analyses with systematic modulus variations (considering the influences of stress singularity) to converge the collapse load solutions. The smoothed [Formula: see text]-continuous recovery stress field ensures the satisfaction of static admissible stress and yield conformity conditions underpinning lower-bound limit analysis theorems. A modified modulus error function within the newest node bisection algorithm enables automatic mesh refining and coarsening constructions, and fast convergence to the lower-bound collapse limit.

The paper presents a simple yet efficient optimal design method based evolutionary structural optimization (ESO) to determine the minimum member sizes complying with an AISC-LRFD (an acronym for American Institute of Steel Construction – Load and Resistance Factor Design) specification. The influences of nonlinear (second-order approximation) geometry are realistically incorporated using an AISC 2010 direct analysis method. The optimization minimizes the total weight of structures subjected to the challenges associated with the simultaneous presence of nonlinear design formulations and discrete design variables realistically considering the availability of standard steel sections. This leads to a non-convex and/or non-smooth optimization program. The ESO algorithm overcomes the problem by performing iterative elimination procedures of ineffective sections, where the feasible design domain progressively converges to the minimum distribution of steel members satisfying limit state (ultimate strength and serviceability) criteria. The ESO design process is encoded within a visual basis application (VBA) framework having a direct interface with the commercial analysis and design software, called SAP2000. An illustrative example motivated by practical engineering applications has been successfully proceeded and hence highlighted robustness of the proposed approach in obtaining the optimal design of steel structures under applied forces.

In this paper, three different variants of the canonical particle swarm optimization (PSO) algorithm are compared in terms of their optimal solutions to engineering optimization problems. The PSO often face to premature convergence due to its weak capability in local exploitation, which leads to a low optimization precision or even failure. Moreover, the computational burden is also one of the obstacles for the PSO when solving complex problems. To overcome these drawbacks, the original PSO algorithm is integrated with the knowledge sharing and auxiliary mechanisms of comprehensive learning (CL) and Gaussian local search (GLS) strategies to form the so-called GLS-PSO, CLPSO, and GLS-CLPSO algorithms. To demonstrate the accuracy and robustness of the three variants, their performance is tested on a benchmark spatial truss structure with continuous variables. The obtained optimization results are then compared with those by other PSO variants in the literature.

The purpose of this study was to investigate the properties of fresh and hardened geopolymer concrete made with granite fines waste. In order to reduce the consumption of natural aggregates, the granite waste sourced from the stone industry was used as fine aggregates for making geopolymer concrete cured at room temperature. Natural river sand was replaced with the granite waste at 25% and 50% by weight. The effect of moisture conditions of fine aggregates on the properties of the geopolymer concrete were also investigated. The fine aggregates were prepared in three different moisture states: saturated surface dry (SSD), air-dry (AD) and oven-dry (OD). It was found that the slump flow of the geopolymer concrete increased with the increasing proportion of the granite waste in the mixture. Meanwhile, the initial and the final setting times were not influenced by the inclusion of the granite waste except when the fine aggregates were saturated. The presence of angular shaped particles enhanced the early compressive strength of geopolymer concrete with 50% replacement of natural sand by the granite waste. Furthermore, the fire test results showed that the geopolymer concrete with the granite waste had a similar spalling behavior to the one made entirely with the natural sand, but tended to show inferior post-fire residual strengths, particularly when the oven-dried aggregates were used.

The paper proposes the sequential elastic analyses run within the node-based smoothed finite element (NS-FE) framework incorporating an automatic adaptive mesh scheme to converge the lower-bound collapse load limit of plane-strain structures. The approach is familiar to practical designers as it solely involves a series of linear elastic analysis solves. Each of which performs a modified version of elastic compensation method that appropriately considers the modulus variations of some critical members. The nonlinear constitutive formulations describing intrinsic material properties can be directly accommodated. The NS-FE method simply adopts the low-order displacement functions, whilst the mesh adaptive algorithm employs the newest-bisection criteria with the novel modulus variation error indicators. A number of numerical examples and benchmarks involving the plane-strain structures with vertex and curve geometries illustrate the accuracy and efficiency of the proposed method. The approach overcomes the challenges associated with stress singularity and volumetric locking phenomena in an incompressibility condition. The collapse load solution converges to the lower-bound limit at modest computing efforts, where the final NS-FE layout (namely fine meshes over the areas developing high modulus variation rates and at the same time coarse meshes for those undergoing elasticity) depicts the failure lines of structures and hence mechanisms.

A numerical technique, based upon a scaled boundary finite element (SBFE) method, is proposed for the analysis of a two-dimensional, infinite, elastic, layered medium under arbitrary surface loads and surface energy effects. The governing equations of the bulk layers and the material surface/interfaces were formulated from linear elasticity and Gurtin-Murdoch surface elasticity theories, respectively. The SBFE formulation was established in a general context to handle both homogenous and functionally graded bulk materials. Enforcing the continuity of the displacements and tractions along the material interfaces together with the surface/interface equations resulted in a system of non-homogeneous, linear, ordinary differential equations (ODEs) governing the unknown nodal functions of the layer medium. A general solution of the final system of ODEs was constructed via a standard procedure and then used to form a system of linear algebraic equations governing the nodal boundary data. To facilitate the treatment of surface loads over a finite region, the subdomain technique was applied to obtain the governing equations for the whole layered medium. A selected set of results is reported to demonstrate the accuracy, convergence, and performance of the proposed technique.

Semi-rigid connections can often be a more economical solution for a framing system than one with either fully fixed connections or fully pinned ones. In view of the fact that the properties of such ductile and partial-strength connections are not known accurately, this paper presents a method for the obtention of both upper and lower bound responses of semi-rigid frames for possible variations in their moment-rotation properties. The latter are thus assumed to be known within some key upper and lower bound values, namely a constitutive law that is still deterministic but is described in terms of a so-called “interval” model. A mathematical programming approach is used to formulate and solve the problem. In particular, for each load level, a pair of nonstandard optimization problems known as interval mathematical programs with equilibrium constraints (or interval MPECs) are solved to provide the required bounds. A number of examples are provided to highlight the important effects of considering uncertainties in semi-rigid connection properties.

This paper presents a robust safety assessment for engineering structures involving fuzzy uncertainties. Uncertain applied loads and yielding capacities of structural elements are modelled as fuzzy variables with associated membership functions representing possibility distributions. A new computation-orientated methodology, namely the α-level collapse assessment (α-level CA) approach, is developed to provide structural safety profile by constructing membership function of the structural collapse load limit accommodating fuzzy uncertainties. The proposed method firstly utilizes the α-level strategy to transform the fuzzy limit analysis into a series of interval limit analyses. By implementing the concept of robust and optimistic optimizations, a mathematical programming (MP) scheme is proposed to explicitly capture the upper and lower bounds of the collapse load limit at each α-sublevel. Subsequently, the membership function of the collapse load limit is established by using the upper and lower bounds obtained from the series of α-sublevel calculations. The proposed α-level mathematical programming scheme preserves the quality of sharpness of the bounds of collapse load limit at each α-sublevel, which consequently provides a rigorous evaluation on the fuzzy profile of the safety of engineering structures against structural collapse. Numbers of numerical examples, motivated by real-world engineering applications, have been investigated to illustrate the accuracy, efficiency and applicability of the proposed method.

A new hybrid probabilistic and interval computational scheme is proposed to robustly assess the stability of engineering structures involving mixture of random and interval variables. Such hybrid approach possesses noticeable flexibility by directly implementing primitive information on uncertain system parameters, thus the validity of the structural safety assessment against uncertainties can be improved. By implementing the different types of uncertainties, the presented approach is able to separately investigate the effects of random and interval variables acting upon the overall structural stability. A unified interval stochastic sampling (UISS) approach is proposed to calculate the statistical characteristics (i.e., mean and standard deviation) of the lower and upper bounds of the linear bifurcation buckling load of engineering structure involving hybrid uncertain system parameters. Consequently, the stability profile of engineering structure against various uncertainties can be parametrically established, such that the bounds on the maximum structural buckling load at any particular percentile of probability can be effectively determined. Both academic and practically motivated engineering structures have been thoroughly investigated by rigorously establishing the corresponding structural stability profiles, so the applicability and accuracy of the proposed method can be critically justified.

This paper proposes a simple inelastic analysis approach to efficiently map out the complete nonlinear post-collapse (strain-softening) response and the maximum load capacity of axially loaded concrete encased steel composite columns (stub and slender). The scheme simultaneously incorporates the influences of difficult instabilizing phenomena such as concrete confinement, initial geometric imperfection, geometric nonlinearity, buckling of reinforcement bars and local buckling of structural steel, on the overall behavior of the composite columns. The proposed numerical method adopts fiber element discretization and an iterative Muller's algorithm with an additional adaptive technique that robustly yields solution convergence. The accuracy of the proposed analysis scheme is validated through comparisons with various available experimental benchmarks. Finally, a parametric study of various key parameters on the overall behaviors of the composite columns is conducted.

The present work considers the effects of uncertainties in a classical elastoplastic analysis. The uncertainties are related to the applied forces and the plastic material capacities, both of which are taken to lie within deterministic but bounded intervals. The primary aim is to obtain both upper and lower bound responses (e.g. of the displacement history). Our interval elastoplastic problem is formulated as an optimization problem which we refer to as an interval mathematical program with equilibrium constraints or interval-MPEC. We develop an efficient and robust algorithm to process such interval-MPEC accurately with any readily available nonlinear programming solver.

The paper proposes a pair of novel mathematical programming based approaches which directly determine the worst collapse load limit in one case, and the best limit in the other case of rigid perfectly-plastic structures subjected to uncertain-but-bounded applied loads using an Info-Gap model. The methods take advantage of the important properties in which the worst collapse load limit defined for an uncertain static formulation is equivalent to the most favourable solution of the uncertain kinematic limit analysis problem, and vice versa for the best collapse load limit. The formulation for capturing the worst collapse load limit (robust worst case solution) takes the form as a mathematical program with equilibrium constraints that is processed using a penalty algorithm, whilst that for the best collapse load limit is a standard linear programming problem. The efficiency and robustness of the proposed schemes are evidenced from a number of successfully solved examples.

This paper presents the non-deterministic dynamic analysis and reliability assessment of structures with uncertain-but-bounded parameters under stochastic process excitations. Random ground acceleration from earthquake motion is adopted to illustrate the stochastic process force. The exact change ranges of natural frequencies, random vibration displacement and stress responses of structures are investigated under the interval analysis framework. Formulations for structural reliability are developed considering the safe boundary and structural random vibration responses as interval parameters. An improved particle swarm optimization algorithm, namely randomised lower sequence initialized high-order nonlinear particle swarm optimization algorithm, is employed to capture the better bounds of structural dynamic characteristics, random vibration responses and reliability. Three numerical examples are used to demonstrate the presented method for interval random vibration analysis and reliability assessment of structures. The accuracy of the results obtained by the presented method is verified by the randomised Quasi-Monte Carlo simulation method (QMCSM) and direct Monte Carlo simulation method (MCSM).

The paper proposes three novel optimal retrofitting strategies for moment resisting frames using cross braces to satisfy given serviceability limitations. The influences of 2nd-order geometric effects are included. The well-known “ground structure” method is used to generate an initial network of possible brace layouts and automatically determine the best one using an optimization algorithm. Of special note is the use of a constraint to impose a realistic design criterion, in which a brace can be selected only if its area lies within specified bound limits. This in effect eliminates members with non-practical (too small) sizes. The formulation is cast as a challenging instance of a mixed integer nonlinear programming problem, which accommodates simultaneously continuous and binary variables. A two-step optimization algorithm is proposed as an efficient and robust solution approach. The safety and integrity of the retrofitted structures are assessed using step-by-step nonlinear elastic analyses.

In this paper, a finite-element-based computational method is proposed for time-dependent structural stability analysis of a concrete-filled steel tubular (CFST) arch with uncertain parameters. Specifically, the targeted uncertainty includes the mercurial effects of the creep and shrinkage of the concrete core, which inevitably affect the structural performance of the CFST arch. The structural stability of the composite arch is systematically investigated under the influence of uncertain creep and shrinkage in a time-dependent fashion. The proposed computational scheme efficiently establishes the quantitative long-term stability envelope for CFST arches against uncertain viscoelastic effects. In order to demonstrate the effectiveness and efficiency of the proposed time-dependent structural stability analysis for CFST arches, practically motivated numerical examples are thoroughly investigated throughout this work.

This paper proposes an interval analysis scheme to map out the complete bound spectrum of the most maximum and most minimum responses of geometrically nonlinear elastoplastic structures subjected to both interval applied loads and interval inelastic material properties. The proposed heuristic method uses a finite-step holonomic formulation under pseudodisplacement control. Geometric nonlinearity is modeled using a conventional second-order approximation. The analysis thus determines directly the most maximum and most minimum bound solutions by processing a pair of optimization problems, known as interval mathematical programs with equilibrium constraints or interval MPECs. The simultaneous presence of complementarity constraints and interval data is the main cause of difficulties (associated with nonconvex and/or nonsmooth optimization programs) underpinning the interval MPECs considered. The simple solution approach proposed reformulates the interval MPECs into their noninterval nonlinear program...

SUMMARY This paper presents a novel mathematical programming approach for the static stability analysis of structures with uncertainties within the framework of FEM. The considered uncertain parameters are material properties, geometry of element cross section, and loading conditions, all of which are described by an interval model. The proposed method formulates the two cases of interest, namely, worst and best buckling load calculation, into a pair of mathematical programming problems. Two straightforward advantages are exhibited by such formulations. The first advantage is that the proposed formulation can overcome the interference on the sharpness of bounds of the buckling load due to the interval dependence issue. The second benefit is that the information of uncertain parameters causing the extremities of buckling load can always be retrieved as by-products of the uncertain stability analysis. Some numerical examples are presented to illustrate the capability of the proposed method on various structures and the sharpness of the bounds of the buckling load factors. The efficiency and effectiveness of the proposed method are also demonstrated through comparison with the classical Monte Carlo simulation method. Copyright © 2014 John Wiley & Sons, Ltd.

Traditional elastoplastic analysis presumes that all structural data are known exactly. However, these values often cannot be predicted precisely: they are influenced by such factors as manufacturing errors, material defects, and environmental changes. Ignoring these may lead to inaccurate (overly conservative or nonconservative) results. It is thus important that the effects of uncertain data be quantified in a reliable assessment of structural safety. This paper presents the elastoplastic analysis of skeletal frames with uncertaintiesassumed to be interval quantities (i.e. with known upper and lower bounds). Starting from the well-known mixed complementarity program (MCP) statement of the state problem without uncertainties, we extend this to an optimisation problem formulation involving complementarity constraints (that represent the plastic nature of the ductile material). Calculations for both upper and lower bounds of displacements corresponding to monotonically increasing loads are computed. The final results are checked through a comparison with interval limit analyses and Monte Carlo simulations.

This paper presents an efficient and accurate numerical technique for determining the mechanical response of an infinite elastic layer under surface loading and surface stress effects. The governing equation of the bulk is formulated from the classical linear elasticity theory via a SBFE technique whereas that of the material surface is obtained from a complete version of Gurtin-Murdoch surface elasticity theory. By enforcing the continuity at the interface of the material surface and the bulk, it leads to a system of linear non-homogenous ordinary differential equations governing the nodal functions. A general solution of the resulting system of ODEs is constructed via standard procedures and then used together with the boundary conditions to form a system of linear algebraic equations governing nodal degrees of freedom. To investigate the accuracy and convergence of the proposed method, selected scenarios are solved and obtained numerical results are reported and discussed.

This paper proposes a pair of novel mathematical programming approaches to automatically identify the worst load combination corresponding to the extreme maximum and minimum responses of nonlinear structural behavior under multiple load cases. The proposed schemes encompass various assumptions founded on nonlinear solid mechanics, including 2nd-order nonlinear geometry, elastoplastic materials and contact-like conditions. Motivated by practical engineering purposes, the analysis further accounts for dependent pattern loads to obtain accurate bound predictions. Each of the critical bounds can be captured within a single step by processing a challenging nonconvex and nonsmooth optimization problem, known as a 0–1 mathematical program with equilibrium constraints or 0–1 MPEC. A penalty algorithm is proposed to furnish the optimal solution of the 0–1 MPEC. This involves iteratively solving a series of reformulated mixed integer nonlinear programming or MINLP problems to enforce the complementarity condition. Four numerical examples are provided to illustrate application and robustness of the proposed schemes.

The paper presents novel mathematical programming approaches for interval limit analysis that are guaranteed to furnish sharp (extreme) bounds to the collapse load of structures subjected to uncertain but bounded parameters. The formulation is cast as a pair of linear programs with interval coefficients. We discuss when sharp collapse load bounds occur when the interval parameters are at the extreme limits of their respective intervals. A mixed 0–1 programming approach is first proposed to compute the minimum collapse limit, and, more importantly, we then develop a pair of robust and efficient nonlinear programming schemes that capture maximum collapse loads in one case and minimum collapse loads in the other.

This paper presents a mathematical programming based technique for the minimum weight (volume) topology optimization of truss-like structures such that strain softening material properties, that can lead to severe physical instability, can be accommodated. In addition, satisfaction of such serviceability criteria as limited displacements at some specific points is ensured. The problem is formulated in terms of truss member cross-sectional areas. This leads to a challenging nonconvex and nonsmooth optimization problem, known as a mathematical program with equilibrium constraints (MPEC). A two-step optimization algorithm is proposed to overcome the problems typically associated with nondefiniteness of some key matrices and at the same time nondifferentiability of the mathematical system. Each step involves updating the ground structure and solving the MPEC using a penalized nonlinear programming (NLP) approach. Some numerical examples are provided to illustrate application, robustness and efficiency of the proposed scheme. The safety and integrity of the designed topologically optimal structures are validated using appropriate stepwise holonomic elastoplastic analyses.

The evolutive second-order geometry elastoplastic analysis of flexibly connected planar structures is performed to investigate their cyclic behavior under quasistatic loading conditions. The formulation is cast as a mathematical programming problem, involving so-called “complementarity” constraints. A “fictitious force” concept, that preserves static-kinematic duality, is used to describe geometric nonlinearity. A second-order approximation, deemed sufficiently accurate for practical structures, is adopted. A semirigid connection is idealized as a zero-length elastoplastic element attached to either or both ends of a beam element. Inelasticity is captured through the familiar generalized plastic hinge concept, and a piecewise linear approximation of the nonlinear plastic capacity domain is used to represent the yield condition. A number of examples concerning realistic structures and benchmark cases is provided to check the validity and applicability of the proposed method and to study the cyclic behavior of flexibly connected planar frames.

This paper describes a mathematical programing based approach for the direct limit load evaluation of a structural system under simultaneous contact and limited displacement conditions. The contact model we adopt can simulate either a classical unilateral (nonassociative) Coulomb friction situation or a cohesive fracture idealization at the potential discontinuity interface between contacting bodies. The discrete FE model is constructed using locking free mixed finite elements. The main feature and novelty of our proposed approach is to compute in a single step the maximum load capacity of the structure, such that both the imposed displacement limitations and the contact conditions are satisfied. In essence, the formulation is a nontrivial extension of classical limit analysis. The analysis is cast in its most natural form, namely in mixed static-kinematic variables, and leads to, what is known in the mathematical programing literature, as a mathematical program with equilibrium constraints or MPEC. A nonlinear programing (NLP) based algorithm is proposed to solve the MPEC. Two examples are provided to illustrate application of the proposed scheme, and some comments regarding the various advantages of the adopted mathematical programing framework are made.

A mathematical-programming (MP)-based approach is proposed to directly determine the two extreme [(1) minimum, and (2) maximum] bounds of some static response quantity for elastic structures subject to the simultaneous application of interval applied forces and interval elastic material properties. Such a direct determination scheme reformulates the interval finite-element (FE) analysis into a pair of standard nonlinear programming (NLP) problems that can be solved by any available NLP code. Not only does the proposed method advantageously bypass any computationally expensive combinatorial searches to capture the response limits, but it can also directly accommodate any dependency in the interval data. Some examples are presented to illustrate the robustness and efficiency of the method.

The paper proposes a novel approach for the interval limit analysis of rigid-perfectly plastic structures with (nonprobabilistic) uncertain but bounded forces and yield capacities that vary within given continuous ranges. The discrete model is constructed within a polygon-scaled boundary finite element framework, which advantageously provides coarse mesh accuracy even in the presence of stress singularities and complex geometry. The interval analysis proposed is based on a so-called convex model for the direct determination of both maximum and minimum collapse load limits of the structures involved. The formulation for this interval limit analysis takes the form of a pair of optimization problems, known as linear programs with interval coefficients (LPICs). This paper proposes a robust and efficient reformulation of the original LPICs into standard nonlinear programming (NLP) problems with bounded constraints that can be solved using any NLP code. The proposed NLP approach can capture, within a single step, the maximum collapse load limit in one case and the minimum collapse load limit in the other, and thus eliminates the need for any combinatorial search schemes.

This paper presents a mathematical programming–based approach for optimal retrofitting of steel structures with braces, subject to some given system performance criteria. The aim is to ensure the safety of the post-retrofitted structures under applied forces and limited displacement conditions. The proposed scheme uses a simple form of the classical ground structure–type concept to accommodate possible brace locations. Three rehabilitation cases are studied, each of which is formulated as an instance of a nonconvex and nonsmooth optimization problem generally referred to as a mathematical program with equilibrium constraints or MPEC. In spite of the fact that this type of problem is known to be challenging to solve in the mathematical programming literature, a simple, efficient, and robust approach to process is proposed. The system performance of all retrofitted structures is validated using exact nonholonomic evolutive analyses.

This paper extends classical limit analysis to structures for which some supports are subjected to ''nonstandard'' unilateral frictional contact with the ground. A typical and commonly adopted model is nonassociative Coulomb friction. For such cases, the use of the classical bound theorems is not possible. Moreover, simply solving the governing equations as a mixed complementarity problem (MCP) does not guarantee that the best bound has been calculated. We have therefore developed an approach that attempts to compute, in a single step, the critical (least) upper bound solution by formulating and solving an instance of the challenging class of optimization problems, known as a mathematical program with equilibrium constraints (MPEC). Two examples are provided to illustrate application of the proposed scheme, as well as to highlight some key features of such structures.

This paper presents two complementary mathematical programming based approaches for the accurate safety assessment of semirigid elastoplastic frames under quasistatic loads. The inelastic behavior of the flexible connections and material plasticity are accommodated through piecewise linearized nonlinear yield surfaces. As is necessary for this class of structures, geometric nonlinearity is taken into account. Moreover, only a 2nd-order geometric approximation is included as this is sufficiently accurate for practical structures. The work described has a twofold contribution. First, we develop an algorithm that can robustly and efficiently process the complete (path-dependent) nonholonomic response of the structure in a stepwise (path-independent) holonomic fashion. The governing formulation is cast in mixed static-kinematic variables and leads naturally to what is known in the mathematical programming literature as a mixed complementarity problem (MCP). The novelty of the proposed algorithm is that it processes the MCP directly without using some iterative (and often cumbersome) predictor–corrector procedure. Second, in the spirit of simplified analyses, the classical limit analysis approach is extended to compute the limit load multiplier under the simultaneous influence of joint flexibility, material and geometric nonlinearities, and limited ductility. Our formulation is an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints (MPEC). Various nonlinear programming based algorithms are proposed to solve the MPEC. Finally, four numerical examples, concerning practical structures and benchmark cases, are provided to illustrate application of the analyses as well as to validate the accuracy and robustness of the proposed schemes.

This paper presents a mathematical programming based approach for the analysis of elastoplastic softening frames in the presence of geometric nonlinearity. Arbitrarily large deformations, albeit within a small strain regime, can be accommodated, if necessary. For the sake of efficiency, and without undue loss of accuracy, the algorithm processes the nonholonomic (path-dependent) problem in a stepwise holonomic (path-independent) fashion. This analysis capability has been used to investigate what level or order of geometric nonlinearity needs to be adopted to obtain sufficiently accurate results for practical frames. Two examples are provided for illustrative purposes.

A number of recent papers (see, e.g. (Int. J. Mech. Sci. 2007; 49:454–465; Eur. J. Mech. A/Solids 2008; 27:859–881; Eng. Struct. 2008; 30:664–674; Int. J. Mech. Sci. 2009; 51:179–191)) have shown that classical limit analysis can be extended to incorporate such important features as geometric non-linearity, softening and various so-called ductility constraints. The generic formulation takes the form of a challenging (nonconvex and nonsmooth) optimization problem referred to in the mathematical programming literature as a mathematical program with equilibrium constraints (MPEC). Similar to a classical limit analysis, the aim is to compute in a single step a bound (upper bound, in the case of the extended problem) to the maximum load. The solution algorithm so far proposed to solve the MPEC is to convert it into an iterative non-linear programming problem and attempts to process this using a standard non-linear optimizer. Motivated by the fact that no method is guaranteed to solve such MPECs and by the need to avoid the use of an optimization approach, which is unfamiliar to most practising engineers, we propose, in the present paper, a novel numerical scheme to solve the MPEC as a constrained non-linear system of equations. We illustrate the application of this approach using the simple class of elastoplastic softening skeletal structures for which certain ductility conditions are prescribed. Copyright © 2009 John Wiley & Sons, Ltd.

The present paper extends the classical limit analysis of plane frames to account for 2nd-order geometric nonlinearity. Any specified displacement limits can also be included in the proposed analysis. The effect of combined bending moment and axial force is accommodated in the adopted piecewise linearized yield condition, albeit still assumed as perfectly plastic. The main feature of the novel approach proposed is to compute simultaneously, in a single step, the maximum load and corresponding deformations of the structure under limited displacement conditions. The problem is cast as an instance of the challenging class of (nonconvex and nonsmooth) mathematical programs with equilibrium constraints (MPECs). Various nonlinear programming based algorithms are proposed to solve the MPEC. Four numerical examples are provided to illustrate application of the proposed limit analysis approach and to highlight the necessity of considering geometric nonlinearity for a more realistic assessment of structural behavior.