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.

     

A node-to-surface linear complementarity problem approach for general three-dimensional contact analysis

Meccanica - Contact between two deformable bodies is a complex nonlinear problem especially when rigid-body motion and geometric errors of components are taken into consideration. In this paper, an...     

A fast and general upper-bound limit analysis approach for out-of-plane loaded masonry walls

A new general approach for the limit analysis of out-of-plane loaded masonry walls based on an upper bound formulation is presented. A given masonry wall of generic form presenting openings of arbitrary shape is described through its Non-Uniform Rational B-Spline (NURBS) representation in the three-dimensional Euclidean space. A lattice of nodes is defined in the parameters space together with possible fracture lines. An initial set of rigid elements initially subdividing the original wall geometry is identified accordingly. A homogenized upper bound limit analysis formulation, which takes into account the main characteristics of masonry material such as very low resistance in traction and anisotropic behavior is deduced. Moreover the effect of vertical loads and membrane stresses is considered, assuming internal dissipation allowed exclusively along element edges. A number of technically meaningful examples prove that a good estimate of the collapse load multiplier is obtained, provided that the initial net of yield lines is suitably adjusted by means of a meta-heuristic approach (i.e. a Genetic Algorithm, GA) in order to enforce that element edges accurately represent the actual failure mechanism.     

A force-based failure criterion for spot weld design

This paper suggests a very simple, force-based formula that combines four failure modes into one dimensionless equation to govern spot weld failure under general static loading conditions. The four failure modes are shear, rotation, normal and peel. The normal separation mode and the peel mode are corresponding to mode I (opening mode). The tensile/shear mode is mode II (sliding mode), and the in-plane rotation mode is mode III (tearing mode). Test coupons and test fixtures are designed and tested to establish and verify this equation. To further verify this equation, a long difficult to understand automotive spot weld failure problem was studied. Applying finite element-calculated resultant loads to the proposed formula resulted in analytical values that correlated very well with the long time field observed spot weld failures. This analytical prediction reasonably explained the spot weld failure mechanism and provided good design directions to improve the durability of the auto structure.     

A modified upper bound approach to limit analysis for plane strain deeply cracked specimens

The classical upper bound approach of limit analysis is based on assumption of rigid blocks of deformation that move between lines of tangential displacement discontinuity. This assumption leads to considerable simplification but often at cost of higher estimate of the actual load. Moreover, in many cases, it does not give a correct shape of the plastic field. In order to overcome these limitations a modified upper bound approach is proposed in this article. The proposed approach is basically an energetic approach but unlike the classical upper bound approach it is capable of including presence of statically governed stress field. As an application, of proposed approach, theoretical plane strain solutions are presented for deeply cracked fracture mechanics specimens (single edge cracked specimen in pure bending – SE (PB), single edge cracked specimen in three-point bending – SE (B), and compact tension – C (T) specimens). Plane strain plasticity problem in rigid elastic–plastic mono-material (homogeneous) was solved to evaluate useful parameters like limit load, plastic eta function (η_p) and plastic rotation factor (r_p) and in bi-material (mismatch welds) to evaluate mismatch limit load, for deeply cracked specimens. New kinematically admissible velocity fields are proposed for SE (B) and C (T) specimens. Proposed theoretical solutions were confirmed by classical slip-line field solutions, wherever available, and by detailed elastic–plastic finite element analysis with Von-Mises yield criterion. Good agreement was found between proposed solutions and results obtained from the classical slip-line field theory and finite element analysis.     

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.     

Lower bound limit analysis of three-dimensional elastoplastic structures by boundary element method

IntroductionThelimitanalysisofstructuresisoneofthemostpracticalandusefulbranchesinplasticity .Ithasimportantapplicationbackgroundforproblemssuchasthedeterminationofloadcarryingcapacityandplasticformingofmetal.Thepurposeofthelimitanalysisofstructuresistoprovidereliabletheoreticalbasesforengineeringdesignandsafetyassessment.Asasimplifiedmethodforelastoplasticproblems,limitanalysisneednotrequirethehistoryofloadandcancomputethelimitloadsdirectlyinsteadofelastoplasticincrementalcomputationwhichisus…     

建立基于力的变截面梁单元质量矩阵的新方法

基于位移的有限梁单元中三次Hermite插值函数不能有效地描述变截面梁单元内部位移变化,只能通过加密网格增加单元数解决,会造成计算量增大。基于力的有限梁单元由于使用的力插值函数不受截面形状变化的影响,在处理变截面梁时有很大优势,可以得到精确的位移插值函数,利用较少的单元可以达到很高的精度,解决了基于位移的有限梁单元在处理变截面梁时的不足。本文得到了考虑剪切变形的位移插值函数和考虑转动惯量的一致质量矩阵。利用算例验证了本文理论的正确性和高效性。     

On exactness of the kinematical approach in the structural shakedown and limit analysis

Summary Applying the concepts of limit analysis and shakedown, appropriately defined parameters called generalized stresses and strains are used instead of the actual stresses and strains. It is investigated if the resulting approximations introduce too large errors and if the errors follow only from the simplifying kinematical hypotheses adopted when defining the generalized variables.

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Zur Genauigkeit der kinematischen Methode bei Shakedown und Traglast-Verfahren

Übersicht Bei der Anwendung von Shakedown- oder Traglast-Verfahren werden durch passend gewählte Parameter verallgemeinerte Spannungen und Verzerrungen definiert. Es wird untersucht, ob durch diese Approximation unzulässig große Fehler eingeführt werden.

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This paper was presented at the Second Symposium on Inelastic Solids and Structures held in Bad Honnef in September 1981     

An algorithm for the shakedown analysis of plane frames

Summary A quite straightforward procedure for the shakedown analysis of plane frames is presented. Plastic deformability is assumed as due to the bending moment only and lumped in a finite number of sections of the structure. This procedure is also suitable for the limit analysis problems of frames.

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Sommario Si propone un procedimento speditivo per il calcolo a shakedown di strutture intelaiate piane, sotto l'ipotesi di deformabilità plastica concentrata in sezioni isolate e dipendente dal solo momento flettente. Il procedimento è anche applicabile alla risoluzione di problemi di calcolo a rottura di telai.

     

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.     

Continuum limit for three-dimensional mass-spring networks and discrete Korn's inequality

In a bounded domain Ω⊂R³ we consider a discrete network of a large number of concentrated masses (particles) connected by elastic springs. We provide sufficient conditions on the geometry of the array of particles, under which the network admits a rigorous continuum limit. Our proof is based on the discrete Korn's inequality. Proof of this inequality is the key point of our consideration. In particular, we derive an explicit upper bound on the Korn's constant. For generic non-periodic arrays of particles we describe the continuum limit in terms of the local energy characteristic on the mesoscale (intermediate scale between the interparticle distances (small scale) and the domain sizes (large scale)), which represents local energy in the neighborhood of a point. For a periodic array of particles we compute coefficients in the limiting continuum problems in terms of the elastic constants of the springs.     

A novel beam finite element with singularities for the dynamic analysis of discontinuous frames

In this work, a model of the stepped Timoshenko beam in presence of deflection and rotation discontinuities along the span is presented. The proposed model relies on the adoption of Heaviside’s and Dirac’s delta distributions to model abrupt and concentrated, both flexural and shear, stiffness discontinuities of the beam that lead to exact closed-form solutions of the elastic response in presence of static loads. Based on the latter solutions, a novel beam element for the analysis of frame structures with an arbitrary distribution of singularities is here proposed. In particular, the presented closed-form solutions are exploited to formulate the displacement shape functions of the beam element and the relevant explicit form of the stiffness matrix. The proposed beam element is adopted for a finite element discretization of discontinuous framed structures. In particular, by means of the introduction of a mass matrix consistent with the adopted shape functions, the presented model allows also the dynamic analysis of framed structures in presence of deflection and rotation discontinuities and abrupt variations of the cross-section. The presented formulation can also be easily employed to conduct a dynamic analysis of damaged frame structures in which the distributed and concentrated damage distributions are modelled by means of equivalent discontinuities. As an example, a simple portal frame, under different damage scenarios, has been analysed and the results in terms of frequency and vibration modes have been compared with exact results to show the accuracy of the presented discontinuous beam element.     

The nonlinear numerical analysis method for frames

This paper gives the direct formulas of stiffness matrixes of two kinds of Kirchhoff nonlinear elements under total-Lagrange coordinate. For the first one, it includes not only the quadric terms of increments of strain and displacement but also the influence of rotations. For the second one, it is simplified and its nonlinear is considered by taking into account the influence of axial force on the equilibrium equation in the linear beam theory. The nonlinear equation obtained from both of the above-said elements is solved by mixed Newton-Raphson method, and by comparing the results obtained from two kinds of nonlinear beam some important conclusions that we can know how to use them right are given in our paper.     

Complex loading beyond the elastic limit on three-dimensional trajectories

Determinative equations of the plasticity theory are proposed to describe some complex deformation processes along the three-dimensional trajectories of small torsion and arbitrary curvature. Methods of concretization have been developed for the scalar functions. Assumptions and hypotheses obtained for determinative equations, ways of scalar function concretization, and determinative equations themselves have been experimentally substantiated. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 16–25, December, 1998.     

A new approach for the limit to tree height using a liquid nanolayer model

Liquids in contact with solids are submitted to intermolecular forces inferring density gradients at the walls. The van der Waals forces make liquid heterogeneous, the stress tensor is not any more spherical as in homogeneous bulks and it is possible to obtain stable thin liquid films wetting vertical walls up to altitudes that incompressible fluid models are not forecasting. Application to micro tubes of xylem enables to understand why the ascent of sap is possible for very high trees like sequoias or giant eucalyptus.      

A three-dimensional photoelastic method for analysis of differential-contraction stresses

The property of homogeneous and isotropic contraction accompanying the slow polymerization of a photoelastic epoxy resin is utilized to produce a photoelastic model of the same size and shape, at the elevated cure temperature, as the container in which it was cast. Reducing the temperature of the bonded model-container composite structure through the epoxy material transition-temperature range results in frozen-stress photoelastic patterns which correspond to the forces of mutual elastic restraint of differential thermal contraction. The requirements for model-prototype similarity and the model-calibration method are discussed. Particular experiments with calibration specimens and with more complex structures in two and three dimensions are described. The validity of the technique is further demonstrated by correlation with a three-dimensional numerical solution. The properties of a material that was specially developed for use in this new technique are given.     

A bipotential approach for plastic limit loads of strip footings with non-associated materials

This paper is concerned with a bipotential approach for estimating the plastic collapse loads of a half-space made with a non-associated Mohr–Coulomb material and indented by a rigid punch. In geotechnics, this problem is called the bearing capacity of shallow strip footing for which the analytical solution is derived by Prandtl (1920) [46] and Hill (1950) [35] in the context of associated plasticity. However, when the plastic model is not associated, no analytical methods have yet been developed. Here we explore this issue in a rigorous mathematical framework coupling the bipotential concept and limit analysis. First, the method proposed makes use of the method of characteristics to build a statically and plastically admissible stress field that enables a lower estimate of the plastic limit loads. Next, the extended kinematic theorem of limit analysis to non-standard plasticity is applied to derive an upper quasi-bound of the collapse loads. For this aim, the internal rate of plastic dissipation is obtained thanks to the bipotential functional depending on both a trial stress field and a Prandtl-like collapse mechanism. The analytic estimates are compared to the formulae and numerical results provided in literature.