Narrow Results

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.

The design of some steel shells, like energy absorbers or bumpers, requires the knowledge of their behavior in large deformations. In this paper, the method of sequential limit analysis is presented and applied to axisymmetric shells in order to study their post-collapse response. Although the material behavior is assumed as rigid-plastic, results compare favorably with those produced by elastic-plastic incremental analyses and the procedure appears to be more efficient and numerically stable. Large displacement effects, both of stable and unstable nature, are implicitly accounted for by mesh updating.

This paper develops a novel approach to incorporate the generalized Hoek–Brown criterion in the kinematic approach. Two analytical expressions for the equivalent friction angle and the equivalent cohesion in the context of the Hoek–Brown criterion are derived as a function of the minimum principal stress. As an illustrative example, the stability of a rock slope is investigated. The spatial discretization technique is employed to generate a slope failure mechanism considering the spatial variation of the equivalent friction angle corresponding to the minimum principal stress. In order to simplify this problem, the minimum principal stress is taken as an additional parameter in the optimization process of the kinematical approach. For the numerical modeling, three kinds of stress distributions are considered in the optimization process. The proposed approach is validated by comparison with available numerical results with good agreement. It is also shown that with the uniform distribution of minimum principal stress, the proposed approach gives the same solutions as those by the classical tangential line technique, and with more complicated distributions, the proposed approach yields a better solution than the classical tangential line technique, especially for large values of geological strength index (GSI).

Undrained seismic stability of dual unsupported circular tunnels was investigated in this work using a self-developed code for adaptive finite element limit analysis. The so-called pseudo-static method was used to simulate the seismic effects during an earthquake. Accurate upper and lower bounds of seismic stability factor $N_s$ were obtained by using an adaptive remeshing technique incorporated in the code. Comprehensive parametric studies of the problem variables, including the horizontal seismic coefficient $k_h$, the relative spacing ratio $S$/$D$, the relative depth ratio $H$/$D$ and the strength ratio $\gamma D$/$c_u$ were performed to provide dimensionless design tables for practical uses. In addition, visualized results from AFELA were summarized to reveal how the failure mechanism of dual tunnels would evolve with varying problems variables. Numerical results showed that both the stability and failure mechanism of dual tunnels can be much affected by seismic effects which should be carefully considered for a reasonable design in earthquake zones.

This paper presents a novel numerical procedure for limit analysis of plane problems using edge-based smoothed finite element method (ES-FEM) in combination with second-order cone programming. In the ES-FEM, the discrete weak form is obtained based on the strain smoothing technique over smoothing domains associated with the edges of the elements. Using constant smoothing functions, the incompressibility condition only needs to be enforced at one point in each smoothing domain, and only one Gaussian point is required, ensuring that the size of the resulting optimization problem is kept to a minimum. The discretization problem is transformed into the form of a second-order cone programming problem which can be solved using highly efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plane stress and strain problems.

This paper presents a simple technique for reducing the computational effort while solving any geotechnical stability problem by using the upper bound finite element limit analysis and linear optimization. In the proposed method, the problem domain is discretized into a number of different regions in which a particular order (number of sides) of the polygon is chosen to linearize the Mohr–Coulomb yield criterion. A greater order of the polygon needs to be selected only in that region wherein the rate of the plastic strains becomes higher. The computational effort required to solve the problem with this implementation reduces considerably. By using the proposed method, the bearing capacity has been computed for smooth and rough strip footings and the results are found to be quite satisfactory.

Strip shallow foundations on random heterogeneous soil responding in undrained conditions are analyzed using three-dimensional upper limit analysis and Latin Hypercube sampling. The results obtained considering the three-dimensional variability of soil are compared with results using plane models, showing significant differences in terms of both mean and standard deviation of bearing capacity. An averaged two-dimensional model fitted to a small set of three-dimensional samples is shown to yield accurate predictions of the bearing capacity distribution.

This paper presents a lower bound limit analysis approach for solving an axisymmetric stability problem by using the Drucker–Prager (D–P) yield cone in conjunction with finite elements and nonlinear optimization. In principal stress space, the tip of the yield cone has been smoothened by applying the hyperbolic approximation. The nonlinear optimization has been performed by employing an interior point method based on the logarithmic barrier function. A new proposal has also been given to simulate the D–P yield cone with the Mohr–Coulomb hexagonal yield pyramid. For the sake of illustration, bearing capacity factors N_c, N_q and N_γ have been computed, as a function of ϕ, both for smooth and rough circular foundations. The results obtained from the analysis compare quite well with the solutions reported from literature.

This paper presents a numerical kinematic formulation for computation of collapse load of Mindlin–Reissner plates, that uses a stabilized mesh-free method in combination with second-order cone programming (SOCP). The kinematic formulation is discretized using a moving least squares approximation combined with a stabilized conforming nodal integration scheme, ensuring that shear locking problem at thin plate limit can be removed. The stabilized mesh-free based kinematic formulation is formulated as a conic problem so that it can be solved by an efficient primal-dual interior-point algorithm. To speed up computational progress, an adaptive refinement scheme using dissipation-based error indicator is also performed. The performance of the proposed numerical procedure is illustrated by examining several plates with arbitrary geometries and various boundary conditions.

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretized mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.

Two new computational algorithms for the Limit Analysis (LA) of large-scale 3D truss-frame structures recently proposed by the authors are reconsidered and adapted for a comparison prediction of the elastoplastic response of a strategic beautiful historic infrastructure, namely the Paderno d’Adda bridge (or San Michele bridge), a riveted wrought iron railway viaduct that was built in northern Italy in 1889. The first LA algorithm traces a fully exact evolutive piece-wise linear elastoplastic response of the structure, up to plastic collapse, by reconstructing the true sequence of activation of made-available plastic joints (as a generalization of plastic hinges), in the true spirit of LA. The second LA algorithm develops an independent kinematic iterative approach apt to directly determine the plastic collapse state, in terms of collapse load multiplier and plastic mechanism, based on the upper-bound theorem of LA. Specifically, the marvelous doubly built-in parabolic arch of the bridge is analyzed, under a static loading configuration at try-out stage, and its elastoplastic response is investigated, in terms of evolutive load-displacement curve, collapse load multiplier and plastic collapse mechanism. The two LA algorithms are found to much effectively run and perform, despite the rather large size of the computational model, with a number of dofs in the order of four thousands, by achieving good corresponding matches in terms of the estimate of the load-bearing capacity and of the collapse characteristics of the arch substructure, showing this to constitute a well-set structural element. Moreover, the direct kinematic method displays a rather dramatic performance, in truly precipitating from above onto the collapse load multiplier and rapidly adjusting to the collapse mode, in very few iterations, by a considerable saving of computational time, with respect to the complete evolutive elastoplastic analysis. This shall open up the way for further adoption of such advanced LA tools, with LA regaining a new momentum within the modern optimization analysis of structural design and form-finding problems.

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 $C^0$-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.

This study proposes a pseudo-lower bound method for direct limit analysis of two-dimensional structures and safety evaluation based on isogeometric analysis integrated through Bézier extraction. The key idea in this approach is that the stress field is separated into two parts: fictitious elastic and residual, and then the equilibrium conditions are recast by the weak form. Being different from the displacement approach which employs the kinematic formulation, the approximations based on the stress field satisfy automatically volumetric locking phenomena. Dealing with optimization problems, a second-order cone programming, providing significant advantages of the conic representation for yield criteria, is employed. The examination of various numerical benchmark problems shows an efficient and reliable method for the proposed approach.

New plastic solutions for the bearing capacity factors of conical footings on rock masses obeying the Hoek–Brown yield criterion are presented in the paper by using the lower and upper bound finite element limit analysis. A conical footing has a cone apex angle and is penetrated into a rock mass. The considered dimensionless parameters include the cone apex angle, the yield parameter, and the geological strength index of a rock mass, where the effects of these dimensionless parameters on the bearing capacity factor are compressively investigated. The collapse mechanisms of conical footings on rock masses are examined and discussed in the paper.

The primary aim of this paper is to determine penetration and uplift resistances of two interfering pipelines buried in clay with a linear increase in strength. The advanced finite element limit analysis of upper and lower bound theorems is used to perform new limit analysis solutions for both penetration and uplift resistances of two interfering pipelines. The strength profiles of cohesive soils are the cases of normally consolidated clays in deep water by setting the shear strength at the ground surface to be zero and linearly increased with the depth. The twin pipelines have the same geometries and are simultaneously failed at the same magnitude of the failure uplift or bearing loads. There are three considered input parameters including the spacing between the pipes, the embedded depth of the pipes, and the unit weight of soils. All input parameters have significant influences on the penetration and uplift resistances of two interfering pipelines. Failure mechanisms of the problems are also investigated, and stability charts of the penetration and uplift resistances of two interfering pipelines are produced for practical uses in offshore geotechnical engineering.

This paper concerns the limit analysis of structures in the presence of constitutive instabilities. As a simple and representative illustration, a model problem involving nonassociative frictional rigid blocks is used. The formulation takes the form of a special constrained optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC). In view of the complexity of solving an MPEC directly, we propose its reformulation and solution as a nonlinear programming (NLP) problem via a penalty scheme.