A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures

A 3D model for the evaluation of the non-linear behavior of masonry double curvature structures is presented. In the model, the heterogeneous assemblage of blocks is substituted with a macroscopically equivalent homogeneous non-linear material. At the meso-scale, a curved running bond representative element of volume (REV) constituted by a central block interconnected with its six neighbors is discretized through of a few six-noded rigid wedge elements and rectangular interfaces. Non linearity is concentrated exclusively on joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. The macroscopic homogenous masonry behavior is then evaluated on the REV imposing separately increasing internal actions (in-plane membrane actions, meridian and parallel bending, torsion and out-of-plane shear). This simplified approach allows to estimate heuristically the macroscopic stress–strain behavior of masonry at the meso-scale. The non-linear behavior so obtained is then implemented at a structural level in a novel FE non-linear code, relying on an assemblage of rigid infinitely resistant six-noded wedge elements and non-linear interfaces, exhibiting deterioration of the mechanical properties.Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. To fully assess numerical results, additional non-linear FE analyses are presented. In particular, a simplified model is proposed, which relies in performing at a structural level a preliminary limit analysis – which allows to identify the failure mechanism – and subsequently in modeling masonry through elastic elements and non-linear interfaces placed only in correspondence or near the failure mechanism provided by limit analysis. Simulations performed through an equivalent macroscopic material with orthotropic behavior and possible softening are also presented, along with existing experimental evidences (where available), in order to have a full insight into the capabilities and limitations of the approach proposed.     

The application of graph theory to the limit analysis for torsion of thin-walled bars with complex cross-sections

Graph theory is employed in this paper as a means to establish the topological model of complex thin-walled cross-sections. On this basis, the upper and lower bound theorems of the plastic limit analysis are applied to the analysis of the plastic limit shear flows on the cross-section of thin-walled bars under St. Venant torsion. Corresponding mathematical programming problems are formulated and their duality is shown. After solving the linear programming problem corresponding to the lower bound theorem, the limit torsional moment of a thin-walled cross-section can be calculated according to the shear stress distribution in the limit state. The formula for calculating the limit torsional moment is given. Furthermore, the limit state of thin-walled cross-sections under St. Venant torsion is also discussed and the concept of the limit tree is introduced. A computer program has been developed by the author. Results calculated by the program for typical complex cross-sections are given.     

Limit analysis by linear programming of 3D masonry structures with associative friction laws and torsion interaction effects

In this paper, a formulation for limit analysis of three-dimensional masonry structures discretized as rigid block assemblages interacting through no-tension and frictional contact interfaces is developed. Linear and piecewise linearized yield functions are used for rocking, sliding and torsion failure. A simple yield condition has been defined to take into account interaction effects of shear force with torsion and bending moment. Associative flow rules are considered for strain rates. On the basis of the developed governing equations, the limit analysis problem has been formulated as a nonlinear mathematical program. An iterative solution procedure based on linear programming is used to solve the limit analysis problem and to take into account nonlinearities due to the influence of bending moments and shear stresses on torsion strength. The results of experimental investigations on out-of-plane masonry walls constrained at one edge and different examples from literature were considered for validation. Comparison with existing formulations is carried out.     

A kinematic FE limit analysis model for thick English bond masonry walls

Two-wythes masonry walls arranged in English bond texture were often used in the past as bearing panels in seismic area. On the other hand, earthquake surveys have demonstrated that masonry strength under horizontal actions is usually insufficient, causing premature collapses of masonry buildings, often ascribed to out-of-plane actions. Furthermore, many codes of practice impose for new brickwork walls a minimal slenderness, which for instance is fixed by the Italian O.P.C.M. 3431 equal to 12 for artificial bricks and 10 for natural blocks masonry.For the above reasons, the analysis at failure of English bond brickwork walls under out-of-plane actions is a topic that deserves consideration, despite the fact that almost the totality of the studies of masonry at failure is devoted to running bond arrangements. Furthermore, it must be noted that an approach based on the analysis of running bond texture – in comparison with English bond pattern – is not suitable for the investigation of the behavior of bearing panels.In this framework, in the present paper, a Reissner–Mindlin kinematic limit analysis approach is presented for the derivation of the macroscopic failure surfaces of two-wythes masonry arranged in English bond texture. In particular, the behavior of a 3D system constituted by infinitely resistant bricks connected by joints reduced to interfaces with frictional behavior and limited tensile/compressive strength is identified with a 2D Reissner–Mindlin plate. In this way, assuming both an associated flow rule for the constituent materials and a finite subclass of possible deformation modes, an upper bound approximation of macroscopic English bond masonry failure surfaces is obtained as a function of macroscopic bending moments, torsion and shear forces.Several examples of technical relevance are treated both at a cell level and at a structural level, addressing the differences in terms of collapse loads and failure surfaces due to different textures and constituent laws for joints. Finally, two meaningful structural examples consisting of a panel in cylindrical flexion and a masonry slab constrained at three edges and out-of-plane loaded are discussed. A detailed comparison in terms of deformed shapes at collapse and failure loads between a 2D FE Reissner–Mindlin limit analysis approach and a full 3D heterogeneous FE model shows the reliability of the results obtained using the kinematic identification approach proposed.     

NONSMOOTH MODEL FOR PLASTIC LIMIT ANALYSIS AND ITS SMOOTHING ALGORITHM

By means of Lagrange duality theory of the convex program, a dual problem of Hill's maximum plastic work principle under Mises' yield condition has been derived and whereby a non-differentiable convex optimization model for the limit analysis is developed. With this model, it is not necessary to linearize the yield condition and its discrete form becomes a minimization problem of the sum of Euclidean norms subject to linear constraints. Aimed at resolving the non-differentiability of Euclidean norms, a smoothing algorithm for the limit analysis of perfect-plastic continuum media is proposed. Its efficiency is demonstrated by computing the limit load factor and the collapse state for some plane stress and plain strain problems.     

Three-dimensional limit analysis of rigid blocks assemblages. Part II: Load-path following solution procedure and validation

A novel solution procedure for the non-associated limit analysis of rigid blocks assemblages is proposed. This proposal produces better solutions than previously proposed procedures and it is also able to provide an insight into the structural behaviour prior to failure. The limit analysis model proposed in Part I of this paper and the solution procedure are validated through illustrative examples in three-dimensional masonry piers and walls. The use of limit analysis for three-dimensional problems incorporating non-associated flow rules and a coupled yield surface is novel in the literature.     

Classical and sequential limit analysis revisited

Classical limit analysis applies to ideal plastic materials, and within a linearized geometrical framework implying small displacements and strains. Sequential limit analysis was proposed as a heuristic extension to materials exhibiting strain hardening, and within a fully general geometrical framework involving large displacements and strains. The purpose of this paper is to study and clearly state the precise conditions permitting such an extension. This is done by comparing the evolution equations of the full elastic–plastic problem, the equations of classical limit analysis, and those of sequential limit analysis. The main conclusion is that, whereas classical limit analysis applies to materials exhibiting elasticity – in the absence of hardening and within a linearized geometrical framework –, sequential limit analysis, to be applicable, strictly prohibits the presence of elasticity – although it tolerates strain hardening and large displacements and strains. For a given mechanical situation, the relevance of sequential limit analysis therefore essentially depends upon the importance of the elastic–plastic coupling in the specific case considered.     

Computational analysis of stress-based forming limit curves

This article, through computational analyses, examines the validity of using the stress-based and extended stress-based forming limit curves to predict the onset of necking during proportional loading of sheet metal. To this end, a model material consisting of a homogeneous zone and a zone that has voids (material inhomogeneity) is proposed and used to simulate necking under plane strain and uni-axial stress load paths. Results of the in-plane loading computations are used to construct a strain-based formability limit curve for the model material. This limit curve is transformed into principal stress space using the procedure due to Stoughton [Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 42, 1–27]. The stress-based limit curve is then transformed into equivalent stress and mean stress space to obtain an Extended Stress-Based Limit Curve (XSFLC). When subjected to three-dimensional loading, the model material is observed to display a variety of responses. From these responses, a criterion for the applicability of the XSFLC to predict the onset of necking in the model material when it is subjected to three-dimensional loading is obtained. In the context of straight tube hydroforming, to provide support for the use of the XSFLC, it is demonstrated that the criterion is satisfied.     

Kinematic FE limit analysis homogenization model for masonry walls reinforced with continuous FRP grids

A homogenization model for periodic masonry structures reinforced with continuous FRP grids is presented. Starting from the observation that a continuous grid preserves the periodicity of the internal masonry layer, rigid-plastic homogenization is applied directly on a multi-layer heterogeneous representative element of volume (REV) constituted by bricks, finite thickness mortar joints and external FRP grids. In particular, reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints and the FRP grid is applied on the external surfaces of the REV. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces.A FE strategy is adopted at a cell level, modeling joints and bricks with six-noded wedge shaped elements and the FRP grid through rigid infinitely resistant truss elements connected node by node with bricks and mortar. A possible jump of velocities is assumed at the interfaces between contiguous wedge and truss elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for FRP bars some formulas available in the literature are adopted to reproduce the delamination of the truss from the support.Two meaningful structural examples are considered to show the capabilities of the procedure proposed, namely a reinforced masonry deep beam (0°/90° continuous reinforcement) and a masonry beam in simple flexion for which experimental data are available. Good agreement is found between present model and alternative numerical approaches.     

Kinematic limit analysis of frictional materials using nonlinear programming

In this paper, a nonlinear numerical technique is developed to calculate the plastic limit loads and failure modes of frictional materials by means of mathematical programming, limit analysis and the conventional displacement-based finite element method. The analysis is based on a general yield function which can take the form of the Mohr–Coulomb or Drucker–Prager criterion. By using an associated flow rule, a general nonlinear yield criterion can be directly introduced into the kinematic theorem of limit analysis without linearization. The plastic dissipation power can then be expressed in terms of kinematically admissible velocity fields and a nonlinear optimization formulation is obtained. The nonlinear formulation only has one constraint and requires considerably less computational effort than a linear programming formulation. The calculation is based entirely on kinematically admissible velocities without calculation of the stress field. The finite element formulation of kinematic limit analysis is developed and solved as a nonlinear mathematical programming problem subject to a single equality constraint. The objective function corresponds to the plastic dissipation power which is then minimized to give an upper bound to the true limit load. An effective, direct iterative algorithm for kinematic limit analysis is proposed in this paper to solve the resulting nonlinear mathematical programming problem. The effectiveness and efficiency of the proposed method have been illustrated through a number of numerical examples.     

A powerful force-based approach for the limit analysis of three-dimensional frames

For the estimation of the strength of a structure, one could avoid detailed elastoplastic analysis and resort, instead, to direct limit analysis methods that are formulated within linear programming. This work describes the application of the force method to the limit analysis of three-dimensional frames. For the limit analysis of a framed structure, the force method, being an equilibrium-based approach, is better suited than the displacement method and results, generally, to faster solutions. Nevertheless, the latter has been used mostly, since it has a better potential for automation. The difficulty for the direct computerization of the force method is to automatically pick up the structure’s redundant forces. Graph theory concepts may be used to accomplish this task, and a numerical procedure was proposed for the optimal plastic design of plane frames. An analogous approach is developed herein for the limit analysis of space frames which is computationally more cumbersome than the limit analysis of plane frames. The proposed procedure results in hypersparse matrices, and in conjunction with the kinematic upper bound linear program which is solved by a sparse solver, the proposed method appears computationally very efficient. It is also proved that it is much more effective than any displacement-based formulation. The robustness and efficiency of the approach are testified by numerical examples for grillages and multi-storey frames that are included.     

联合极限状态下基于最大熵可靠度的易损性分析

考虑工程需求参数(EDP)的前四阶矩,提出基于最大熵可靠度理论的地震易损性分析方法.基于SAP2000建立钢筋混凝土框剪模型,选择最大层间位移角和最大层加速度衡量结构的联合性能极限状态,建立极限状态方程.不对EDP的分布进行人为假定,在不同峰值加速度(PGA)下计算两种EDP的前四阶矩,并作为约束条件,建立极限状态方程...     

Extended bounding theorems of limit analysis

This paper studies the bounding problems of the complete solu-tion of limit analysis for a rigid-perfectly plastic medium,allowing for the discontinuity of plastic flow.A generalizedvariational principle involving conditions of the rigid-plas-tic interface and the discontinuous surface of a velocityfield has been advanced for the mixed-boundary value problem.Based on this principle,a set of variational formulae of li-mit analysis is established.The safety factors obtained bythese formulae lie between the upper and lower bounds obtainedby the classical bounding theorems with the same kinematicallyand statically admissible field.Moreover,extended bounding theorems have been derivedand proved,which hold a broader stress and velocity field thanthe statically and kinematically admissible field.The corol-laries of these theorems indicate the relationship between thevariational solution and the complete solution of limit analy-sis.Applications of these theorems show that a close approxi-mation can be obtained     

A Reissner–Mindlin limit analysis model for out-of-plane loaded running bond masonry walls

Earthquake surveys have demonstrated that the lack of out-of-plane strength is a primary cause of failure in many traditional forms of masonry. Moreover, bearing walls are relatively thick and, as a matter of fact, many codes of practice impose a minimal slenderness for them, as for instance the recent Italian O.P.C.M. 3431 [2005. Ulteriori modifiche ed integrazioni all’OPCM 3274/03 (in Italian) and O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica (in Italian)], in which the upper bound slenderness is fixed respectively equal to 12 for artificial bricks and 10 for natural blocks masonry. In this context, a formulation at failure for regular assemblages of bricks based both on homogenization and Reissner–Mindlin theory seems particularly attractive. In this paper a kinematic limit analysis approach under the hypotheses of the thick plate theory is developed for the derivation of the macroscopic failure surfaces of masonry out-of-plane loaded. The behavior of a 3D system of blocks connected by interfaces is identified with a 2D Reissner–Mindlin plate. Infinitely resistant blocks connected by interfaces (joints) with a Mohr–Coulomb failure criterion with tension cut-off and compressive cap are considered. Finally, an associated flow rule for joints is adopted. In this way, the macroscopic masonry failure surface is obtained as a function of the macroscopic bending moments, torsional moments and shear forces by means of a linear programming problem in which the internal power dissipated is minimized, once that a subclass of possible deformation modes is a priori chosen. Several examples of technical relevance are presented and comparisons with previously developed Kirchhoff–Love static [Milani, G., Lourenço, P.B., Tralli, A., 2006b. A homogenization approach for the limit analysis of out-of-plane loaded masonry walls. J. Struct. Eng. ASCE (in press)] and kinematic [Sab, K., 2003.Yield design of thin periodic plates by a homogenisation technique and an application to masonry walls. C.R. Mech. 331, 641–646] failure surfaces are provided. Finally, two meaningful structural examples are reported, the first concerning a masonry wall under cylindrical flexion, the second consisting of a rectangular plate with a central opening out-of-plane loaded. For both cases, the influence of the shear strength on the collapse load is estimated.     

外包钢-砼组合梁纯扭作用下抗扭刚度的实验及数值模拟

为了研究新型外包钢-砼T形截面组合梁在纯扭作用下的变形性能,设计了5根不同配箍率的的足尺悬臂组合梁。通过对5根悬臂梁的抗扭性能的实验研究,得到了组合梁的扭矩-扭率关系曲线。利用有限元分析软件ANSYS,对组合梁的抗扭性能进行了非线性有限元分析,得到了混凝土与外包钢在极限阶段的应力云图。根据实验以及有限元结果分析了组合梁在整个加载过程中扭转刚度的变化。基于现行砼结构设计规范,提出了组合梁从开裂到极限阶段抗扭刚度的计算公式,可供组合梁受扭设计参考。把有限元模型和公式的计算结果与实验结果进行比较,三者吻合较好。     

A computational model for the limit analysis of three-dimensional masonry structures

This paper extends previous work on the limit analysis of ductile frames and plane masonry arches to the limit analysis of three-dimensional masonry structures. A lower-bound approach is developed which can handle three-dimensional collapse mechanisms involving any combination of sliding, twisting and hingeing at the block interfaces. A computer program for determining the collapse load of such structures is used to study (a) the equilibrium limits of a block with four contact points resting on an inclined plane and (b) the collapse of a semicircular arch of four blocks. The paper also describes experimental and computational work on a radially symmetric model dome of 380 blocks subject to foundation settlement.

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Sommario Il presentre contributo estende al campo delle structture tridimensionali in muratura un precedente lavoro sull'analisi limite di telai duttili ed archi in muratura piani. Si e' sviluppato un approccio statico che analizza meccanismi di collasso tridimensionale ottenuti per combinazione dei meccanismi semplici di scorrimento e rotazione nel piano e fuori dal piano delle superfici di interfaccia tra i blocchi. Si descrivono (a) i limiti di equilibrio di un blocco con 4 punti di contatto su base inclinata, (b) le condizioni di collasso di un arco semicircolare costituito da quattro blocchi, applicando un programma di calcolo redatto per l'analisi e la definizione del carico di collasso di tali strutture. La terza parte dell'articolo presenta il lavoro sperimentale e di calcolo sviluppato su un modello di cupola a simmetria radiale costituita da 380 blocchi soggetta a cedimenti fondali.

     

3D macro and micro-block models for limit analysis of out-of-plane loaded masonry walls with non-associative Coulomb friction

In this paper, a masonry system composed of a façade wall connected with two sidewalls and subjected to out-of-plane loading is investigated within the framework of three-dimensional limit analysis. Two different modeling approaches, namely macro and micro-block models, are adopted. A rigid-perfectly plastic model with dry contact interfaces governed by Coulomb failure criterion is assumed for masonry walls with regular units and staggering (non-standard limit analysis). Three classes of failure modes are investigated, involving rocking, sliding, twisting failure and combinations of them. The macro-block model is based on the assumption that the failure involves a number of cracks which separate the structure into a few macro-blocks and all the possible relative motions among micro-blocks are concentrated along the cracks. Two limiting conditions for the ultimate load factor are kinematically computed by use of minimization routines. The micro-block model is based on a concave contact formulation in which contact points are located at the corners of interfaces, allowing failure modes involving opening and sliding to be simulated. An iterative solution procedure is used to solve the non-associative friction problem, with second order cone programming (SOCP) used to allow the conic yield function to be solved directly. Both models are validated against experimental outcomes from the literature. A parametric analysis is carried out in order to highlight the influence of each geometrical and mechanical parameter on the prevalence of a mechanism over the other. The presence of an unrestrained horizontal floor system with different orientations is also analyzed.     

A mathematical programming algorithm for limit analysis

This paper deals with the limit analyses of perfect rigid-plastic continua. Based on the kinematic theorem of the limit analysis theory, a mathematical programming finite element formula for determining the upper bound load multiplier has been established, and an iteration algorithm proposed accordingly. In this algorithm the plastic and rigid zones are distinguished for every iteration step, and the goal function is modified gradually. The difficulties caused by the nonsmoothness of the goal function are overcome. Some examples solved by this algorithm are presented. The project supported by National Natural Science Foundation of China.     

Dynamic limit analysis formulation for impact simulation of structural members

This paper introduces an extended concept of limit analysis to deal with the dynamic equilibrium condition considering the inertia and strain-rate effect for dynamic behavior of structures. The conventional limit analysis method has been applied to only static collapse analysis of structures without consideration of dynamic effects in the structural behavior. A dynamic formulation for the limit analysis has been derived for incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ-α method. The dynamic material behavior is described by the Johnson–Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. Simulations have been carried out for impact analysis of a Taylor bar and an S-rail and their numerical results are compared with elasto-plastic explicit analysis results by LS-DYNA3D. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial code compared. The crashworthiness of a structure with the rate-dependent constitutive model is also compared to that with the quasi-static constitutive relation in order to investigate the dynamic effect on deformation of structures.     

Limit analysis of multi-layered plates. Part II: Shear effects

In the first part of this work [Dallot, J., Sab, K., 2007. Limit analysis of multi-layered plates. Part I: the homogenized Love-Kirchhoff model. J. Mech. Phys. Solids, in press, doi:10.1016/j.jmps.2007.05.005], the limit analysis of a multi-layered plastic plate submitted to out-of-plane loads was studied. The authors have shown that a homogeneous equivalent Love-Kirchhoff plate can be substituted for the heterogeneous multi-layered plate, as the slenderness (length-to-thickness) ratio goes to infinity. In fact, the out-of-plane shear stresses are shown to become asymptotically negligible when compared to in-plane stresses, as the slenderness ratio goes to infinity. Actually, failure of thick multi-layered structures often occurs by shearing in the core layers and sliding at the interfaces between the layers. Both shearing and sliding are caused by the out-of-plane shear stresses. The purpose of the present paper is to build an enhanced Multi-particular Model for Multi-layered Material (M4) taking into account shear stress effects. In this model, each layer is seen as a Reissner-Mindlin plate interacting with its neighboring layers through interfaces. The proposed model is asymptotically consistent with the homogenized Love-Kirchhoff model described in the first part of the work, as the slenderness ratio goes to infinity. Kinematic and static methods for the determination of the limit load of a thick multi-layered plate which is submitted to out-of-plane distributed forces are described. The special case of multi-layered plates under cylindrical bending conditions is studied. These conditions lead to simplifications which often allow for the analytical resolution of the Love-Kirchhoff and the M4 limit analysis problems. The benefit of the proposed M4 model is demonstrated on an example. A comparison between the heterogeneous 3D model, the Love-Kirchhoff model and the M4 model is performed on a three-layer sandwich plate under cylindrical bending conditions. Finite element calculations are used to solve the 3D problem, while both the Love-Kirchhoff and the M4 problems are analytically solved. It is shown that, when the contrast between the core and the skins strengths is high, the Love-Kirchhoff model fails to capture the plastic collapse modes that cause the ruin of the sandwich plate. These modes are well captured by the M4 model which predicts limit loads that are very consistent with the limit loads predicted by the heterogeneous 3D model (the relative error is found to be smaller than 1%).