Limit analysis of masonry structures with free discontinuities

We formulate a novel procedure for the limit analysis of two-dimensional masonry structures subject to arbitrary loading conditions. The proposed approach works in the framework of free discontinuity methods, on examining collapse mechanisms that exhibit free crack opening discontinuities. The load bearing capacity and the collapse mechanism of the structure are obtained through a fully variational approach, by minimizing a kinetic functional that admits the collapse crack pattern as a variable. Numerical examples illustrate the practical application of the proposed procedure to the limit analysis of a variety of masonry walls and arches subject to foundation settlements, vertical and horizontal forces.     

Limit analysis of arch-beam structures by dynamic programming

Summary We study one-dimensional structures like arch-beams in the limit state of plastic collapse, on the ground of a two-dimensional yielding surface (bending moment and normal generalized stress). The proposed method, which is able to give a numerical solution of the problem of finding the limit load, rests on the upper bound theorem of limit analysis and uses dynamic programming. We examine also some questions linked with numerical procedures. A future work devoted to applications will complete the treatment.

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Sommario Si studiano le strutture monodimensionali ad arco nello stato limite di collasso plastico, sulla base del dominio bidimensionale (momento flettente e sforzo normale). Il metodo proposto, atto a fornire una soluzione numerica del problema della ricerca del carico limite, si fonda sul teorema cinematico dell'analisi limite e sull'impiego della programmazione dinamica. Si prendono anche in esame talune questioni connesse con l'algoritmo di calcolo. Un successivo lavoro di carattere applicativo completerà la trattazione.

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This paper has been supported by National (italian) Research Council (C.N.R.).     

Plastic analysis of structures and duality in linear programming

Summary The possibility of formulating the static and kinematic methods of plastic analysis theory as dual linear programming problems is extended to the cases of monodimensional structures under combined stresses with piecewise linear yield curves (or surfaces).     

Limit analysis of anisotropic soils using finite elements and linear programming

The main purpose of this paper is to present a finite element formulation of the bound theorems which allows for the variation of soil strength with direction. To achieve this objective, the conventional isotropic Mohr-Coulomb yield criterion is generalised to include the effect of strength anisotropy. The finite element limit analysis formulation using the modified anisotropic yield criterion is then developed. Several examples are given in the paper to illustrate the capability and effectiveness of the proposed numerical procedure for computing rigorous bounds for anisotropic soils.     

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.     

Limit analysis and design of structures having elements with random distribution of yield stresses,with associated and non-associated flow laws

Summary A generic structure is considered, made up of elements with rigid perfectly plastic standard material and random plasticity constant, however distributed around the average value. The limit load of structures having elements of average plasticity constant is shown to be greater than or equal to the average value of the limit loads of all the feasible structures, thus generalising a theorem [1] formulated for framed structures.The minimum volume of structures having elements of average plasticity constant is then shown to be less than or equal to the average value of the minimum volumes of all the feasible structures. Then the earlier theorems are extended to take in non standard materials.

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Sommario Si considera una generica struttura composta da elementi di materiale rigido-plastico standard a costante di plasticità aleatoria e comunque distribuita attorno al valor medio.Si dimostra che il carico di collasso della struttura dotata di elementi a costante di plasticità media è maggiore o uguale al valor medio dei carichi limite di tutte le strutture possibili, generalizzando un teorema [1] formulato per le travature reticolari. Si dimostra poi che il minimo volume della struttura dotata di elementi a costante di plasticità media è minore o uguale al valor medio dei minimi volumi di tutte le strutture possibili. Si estendono quindi i teoremi precedenti al caso di materiali non standard.

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This work has been sponsored by C.N.R. (the Italian Council of Research).     

Limit analysis of masonry vaults by means of curved shell finite elements and homogenization

The study of masonry vaults should take into account the essentials of the material “masonry” – i.e. heterogeneity, almost no resistance to tension combined with a good compressive strength and a high friction coefficient, as well as the overall importance of the geometry for achieving the equilibrium.In this paper, a new six-noded triangular curved element, specifically developed for the kinematic limit analysis of masonry shells, is presented. Plastic dissipation is allowed only at the interfaces (generalized cylindrical hinges) between adjoining elements for combined membrane actions, bending moment, torsion and out-of-plane shear, as it is required for the analysis of thick (Reissner–Mindlin) shells. An upper bound of the collapse load is so obtained, since, looking at the dual formulation, the admissibility of the stress state is imposed only at the element boundaries. Masonry strength domain at each interface between contiguous triangular elements is evaluated resorting to a suitable upper bound FE homogenization procedure. The model is assessed through several numerical simulations on a number of masonry shells experimentally tested until collapse. In particular, the dependence of the collapse multiplier on the mesh and on the material parameters (sensitivity analysis) is thoroughly discussed.     

Limit analysis of plates with piecewise linear yield surface

Summary Rigid-plastic plates having a piecewise linear yield surface are studied by limit analysis. Extremum principles for the evaluation of specific dissipation power are defined by means of linear programming concepts. The law governing the plastic collapse of plates is formulated on the basis of the well-known kinematic theorem of limit analysis. A general procedure for the approximate determination of the collapse load is proposed. The paper ends with a brief numeric investigation of the uniformly loaded square plate.

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Sommario Si studiano attraverso l'analisi limite le piastre costituite di materiale rigido-plastico aventi superficie di snervamento di tipo poliedrico. I principi di estremo riguardanti la valutazione della potenza specifica di dissipazione vengono definiti attraverso i concetti della programmazione lineare. Attraverso il noto teorema cinematico dell'analisi limite, si formula la legge che regola il collasso plastico delle piastre. Viene proposto un procedimento per la determinazione approssimata del carico di collasso del tutto generale. Si conclude infine con una breve indagine numerica relativa al caso di piastra quadrata con carico uniforme.

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The results presented in the paper form part of a Research supported by the National Research Council (C.N.R.).     

三维偏心结构的Pushover分析

Pushover分析方法近年来引起了地震工程界的广泛兴趣,我国新的抗震设计规范已将其作为大震下结构弹塑性变形分析的一种方法。结构不规则性是影响Pushover分析应用的一个主要因素,而加载模式的选取则是Pushover分析的一个关键问题。本文采用FEMA273建议的3种加载模式对两个典型的三维偏心结构（刚度偏心和质量偏心）进行了Pushover分析,并与非线性动力时程分析的结果进行了比较,给出了三维偏心结构Pushover分析加载模式选取的建议。     

Limit analysis of continuous media with piecewise linear yeld condition

Summary The problem of limit analysis of continua formed by rigid-perfectly plastic material with piecewise yield surface is discussed. With the use of the known concepts of linear programming, the extremum principles concerning the determination of specific dissipation power are defined in unitary form. The basic principles governing the problem of plastic collapse are then expressed through a simple formulation based on the well known kinematical theorem of limit analysis. Finally, an approximate general procedure is suggested for the calculation of the collapse load of continuous media.

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Sommario Si affronta il problema dell'analisi limite dei mezzi continui costituiti di materiale rigido perfettamente plastico aventi superficie di snervamento di tipo poliedrico. Si definiscono in forma unitaria, avvalendosi di noti concetti di programmazione lineare, i principi di estremo riguardanti la valutazione della potenza specifica di dissipazione. Si deducono poi i principi fondamentali che governano il problema del collasso plastico attraverso una semplice formulazione basata sul ben noto teorema cinematico dell'analisi limite. Si suggerisce infine un procedimento approssimato tutt'affatto generale per la valutazione del carico di collasso di un qualsiasi mezzo continuo.

     

Homogenization towards a mechanistic Rigid Body and Spring Model (HRBSM) for the non-linear dynamic analysis of 3D masonry structures

A mechanistic model with rigid elements and interfaces suitable for the non-linear dynamic analysis of full scale 3D masonry buildings is presented. The model relies into two steps: in the first step, a simplified homogenization is performed at the meso-scale to deduce the mechanical properties of a macroscopic material, to be used in structural applications; the second step relies into the implementation of a Rigid Body and Spring Model (RBSM) constituted by rigid elements linked with homogenized interfaces. In the homogenization step, a running bond elementary cell is discretized with 24 three-node plane-stress elastic triangular elements and non-linear interfaces representing mortar joints. It is shown how the mechanical problem in the unit cell is characterized by few displacement variables and how homogenized stress–strain curves can be evaluated by means of a semi-analytical approach. The second step relies on the implementation of the homogenized curves into a RBSM, where an entire masonry structure can be analyzed in the non-linear dynamic range through a discretization with rigid elements and inelastic interfaces. Non-linear structural analyses are conducted on a church façade interconnected with a portion of the perpendicular walls and on a small masonry building, for which experimental and numerical data are available in the literature, in order to show how quite reliable results may be obtained with a limited computational effort.     

Towards a multiscale analysis of periodic masonry brickwork: A FEM algorithm with damage and friction

The aim of this paper is to present an effort towards a multiscale model of the inelastic behaviour of masonry brick panels and the relative solution algorithm. The essential features of the inelastic behaviour, such as the damage development within the bed joints and the frictional dissipation over cracks’ faces, are taken into account. Micromechanical solutions are adopted in order to trace the guidelines for modelling the mentioned phenomena, and FEM analyses of large-scale panels are shown.     

Numerical analysis of the interaction of 3D compressible bubble clusters

Based on the bubble dynamic theory and the compressible two-phase flow solver of the open source software Open FOAM, a numerical simulation study is carried out on the interactions of bubble clusters in a closed volume. The bubble dynamics and interactions of a single bubble, two bubbles, and four bubbles are investigated under the working conditions without and with the presence of a free surface. Through a parametric study, the qualitative patterns of the variations of the bubble collapse period,the volume compressibility, the bubble pressure peak value, and the breakdown, fusion,and separation phenomena with the parameters such as the bubble pressure, the radius size, the bubble spacing, and the distance from the free surface are obtained. The main factors affecting the bubble morphology and the dynamic characteristics are summarized from numerous parameter experiments. It is shown that, in the absence of a free surface,the main factors are the relative size of the bubbles, the pressure of the liquid, and the pressure differences among the bubbles, while in the presence of a free surface, the main factor is the pressure of the liquid between the upper surface of the bubble and the free surface.     

Limit equilibrium analysis and the minimum thickness of circular masonry arches to withstand lateral inertial loading

In this paper, we compute the location of the imminent hinges and the minimum thickness, t, of a circular masonry arch with mid-thickness radius, R, and embracing angle, β, which can just sustain its own weight together with a given level of a horizontal ground acceleration, ε g. Motivated from the recent growing interest in identifying the limit equilibrium states of historic structures in earthquake prone areas, this paper shows that the value of the minimum horizontal acceleration that is needed to convert an arch with slenderness (t/R, β) into a four-hinge mechanism depends on the direction of the rupture at the imminent hinge locations. This result is obtained with a variational formulation and the application of the principle of stationary potential energy, and it is shown that a circular arch becomes a mechanism with vertical ruptures when subjected to a horizontal ground acceleration that is slightly lower than the horizontal acceleration needed to create a mechanism with radial ruptures. The paper explains that the multiplicity on the solution for the minimum uplift acceleration is a direct consequence of the multiple possible ways that a masonry arch with finite thickness may rupture at a given location. The paper further confirms that the results obtained with commercially available distinct element software are in very good agreement with the rigorous solution.     

Topology optimization of 3D structures with design-dependent loads

Topology optimization of continuum structures with design-dependent loads has long been a challenge. In this paper, the topology optimization of 3D structures subjected to design-dependent loads is investigated. A boundary search scheme is proposed for 3D problems, by means of which the load surface can be identified effectively and efficiently, and the difficulties arising in other approaches can be overcome. The load surfaces are made up of the boundaries of finite elements and the loads can be directly applied to corresponding element nodes, which leads to great convenience in the application of this method. Finally, the effectiveness and efficiency of the proposed method is validated by several numerical examples.     

Nonstationary pandom vibpation analysis of linear elastic structures with finite element method

At present, the finite element method is an efficient method for analyzing structural dynamic problems. When the physical quantities such as displacements and stresses are resolved in the spectra and the dynamic matrices are obtained in spectral resolving form, the relative equations cannot be solved by the vibration mode resolving method as usual. For solving such problems, a general method is put forward in this paper. The excitations considered with respect to nonstationary processes are as follows: P(t)={P_i(t)},P_i(t)=a_i(t)P_i(t), a_i(t) is a time function already known. We make Fourier transformation for the discretized equations obtained by finite element method, and by utilizing the behaviour of orthogonal increment of spectral quantities in random process^[1], some formulas of relations about the spectra of excitation and response are derived. The cross power spectral denisty matrices of responses can be found by these formulas, then the structrual safety analysis can be made. When a_i(t)=l (i= 1,2,…n), the. method stated in this paper will be reduced to that which is used in the special case of stationary process.