Limit analysis of masonry arches with finite compressive strength and externally bonded reinforcement

The collapse load of masonry arches with limited compressive strength and externally bonded reinforcement, such as FRP, is evaluated by solving the minimization problem obtained by applying the upper bound theorem of limit analysis. The arch is composed of a finite number of blocks. The nonlinearity of the problem (no-tension material, frictional sliding and crushing) is concentrated in the interface between two adjacent blocks. The crushing in the collapse mechanism is schematised by the interpenetration of the blocks with the formation of hinges at internal or boundary points of the interface. The minimization problem is solved with linear optimization, taking advantages of the robust algorithms offered by linear programming (LP). The optimal solution of the linear programming problem approximates the exact solution to any degree of accuracy. The dual of the minimization problem is also formulated and is solved in order to present the statics (thrust curve, locus of feasible internal reactions, etc.) of the reinforced arch as a consequence of the kinematical assumptions used in the primal minimization problem. Numerical examples are presented in order to show the effectiveness of the proposed method. Finally, it is shown that the results provided by the proposed LP are in good agreement with an experiment on a FRP-strengthened arch characterized by crushing failure of the masonry.     

Limit analysis of FRP strengthened masonry arches via nonlinear and linear programming

The collapse load of masonry arches strengthened with FRP materials is determined. The arch is made of quadrangular blocks and the nonlinearity of the problem (no-tension material, frictional sliding and crushing) is concentrated at the interface between the blocks. Two methods are used to solve the problem. In the first method, a nonlinear programming problem (NLP) is formulated and is solved by using the successive quadratic programming algorithm (SQP) and combinatorial analysis. This method finds the optimal solution in the analysed cases. In the second method, a linear programming problem (LP) is formulated and is solved with classical techniques. LP approximates the optimal solution to any desired degree of accuracy. Although the number of variables of LP is much larger than that of NLP, LP process time can result much lower than NLP process time. Numerical examples are provided in order to show the advantages of the two methods and the effectiveness of FRP strengthening for different arch geometries.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Monte Carlo homogenized limit analysis model for randomly assembled blocks in-plane loaded

A simple rigid-plastic homogenization model for the limit analysis of masonry walls in-plane loaded and constituted by the random assemblage of blocks with variable dimensions is proposed. In the model, blocks constituting a masonry wall are supposed infinitely resistant with a Gaussian distribution of height and length, whereas joints are reduced to interfaces with frictional behavior and limited tensile and compressive strength. Block by block, a representative element of volume (REV) is considered, constituted by a central block interconnected with its neighbors by means of rigid-plastic interfaces. The model is characterized by a few material parameters, is numerically inexpensive and very stable. A sub-class of elementary deformation modes is a-priori chosen in the REV, mimicking typical failures due to joints cracking and crushing. Masonry strength domains are obtained equating the power dissipated in the heterogeneous model with the power dissipated by a fictitious homogeneous macroscopic plate. Due to the inexpensiveness of the approach proposed, Monte Carlo simulations can be repeated on the REV in order to have a stochastic estimation of in-plane masonry strength at different orientations of the bed joints with respect to external loads accounting for the geometrical statistical variability of blocks dimensions. Two cases are discussed, the former consisting on full stochastic REV assemblages (obtained considering a random variability of both blocks height an length) and the latter assuming the presence of a horizontal alignment along bed joints, i.e. allowing blocks height variability only row by row. The case of deterministic blocks height (quasi-periodic texture) can be obtained as a subclass of this latter case. Masonry homogenized failure surfaces are finally implemented in an upper bound FE limit analysis code for the analysis at collapse of entire walls in-plane loaded. Two cases of engineering practice, consisting on the prediction of the failure load of a deep beam and a shear wall arranged with random texture are critically discussed. In particular, homogenization results are compared with those provided by a heterogeneous approach. Good agreement is found both on the failure mechanism and on the distribution of the collapse load.     

A Branch-and-price algorithm for placement routing for a multi-head beam-type component placement tool

We develop a branch-and-price procedure for a placement routing problem for a multi-head beam-type component placement tool. The problem is modelled as an integer programming model with a huge number of variables, each of which corresponds to a placement route. Its linear programming relaxation is solved by a column generation method. For the column generation subproblem to determine the columns to be added, we develop a dynamic programming procedure. We also propose an effective branching rule to partition the current solution space to eliminate the current fractional solution. Through experiments using real tool data, we observe that the LP relaxation solution value is noticeably close to an integer optimal solution value and hence the integer program formulation and the column generation approach are effective.     

Mechanics of hollow concrete block masonry prisms under compression: Review and prospects

The aim of this work is to critically assess the mechanical properties of hollow concrete masonry using experimental results from prisms constructed with blocks of two different strengths and four types of mortar. A key conclusion is that mortar is mostly responsible for the non-linear behavior of masonry. Moreover, a strongly non-linear relationship between masonry elasticity modulus and compressive strength is found, which contradicts the simple linear relation proposed by Eurocode 6 [CEN. Eurocode 6: Design of masonry structures – Part 1 – Common rules for reinforced and unreinforced masonry structures. EN-1996-1-1; 2005.]. The porosity of mortar and the state of stress that mortar undergoes in the process of compressive loading can be responsible for changes in the mechanical properties, such as elasticity modulus and Poisson’s ratio. Finally, different types of mortars induce different failure modes in the masonry prisms and there is clear evidence that the failure of hollow concrete masonry starts after onset of mortar crushing. In order to better reproduce the observed experimental behavior, a tentative model for the mortar Poisson’s ratio variation upon loading is also presented.     

Scheduling parallel processors: An integer linear programming based heuristic for minimizing setup time

The problem of scheduling parallel processors in a make-to-stock environment with sequence setup costs is considered. A new algorithm which formulates a series of 0-1 integer sub problems is proposed and contrasted with an earlier formulation (Dearing and Henderson 1982,1984). Parallels between the sub problem formulations and generalized networks are discussed. The efficiency and quality of the solutions provided were tested using previously published data for a loom assignment problem. The heuristic solution was evaluated against the optimal integer linear programming (ILP) solution, and a rounded linear program (LP) approximation to the optimal solution for several sample problems. Results indicate that the heuristic is efficient, provides near optimal solutions to production planning problems and requires significantly less computing capability than previously reported LP, TLP approaches.     

Large‐deformation and friction analysis of non‐linear elastic cable networks by second‐order cone programming

A new formulation is presented for equilibrium shape analysis of cable networks considering geometrical and material non‐linearities. Friction between cables and joint devices is also considered. The second‐order cone programming (SOCP) problem which has the same solution as that of minimization of total potential energy is solved to obtain the equilibrium configuration. The optimality conditions are derived to verify that the solution satisfies equilibrium conditions and friction laws. Since no assumption on stress state is needed in the proposed method, no process of trial and error is involved. No effort is required to develop any analysis software because SOCP can be solved by using the well‐developed software based on the interior‐point method. Copyright © 2002 John Wiley & Sons, Ltd.     

Incorporation of multipoint constraints into the balancing domain decomposition method and its parallel implementation

The domain decomposition method (DDM) is a major solution algorithm that is used to parallelize the finite element method. In the case of implicit structural analysis using the DDM, the substructuring‐based iterative linear solver is a powerful tool when an effective preconditioner such as the balancing domain decomposition (BDD) method is used. In the present study, a method by which to incorporate a set of linear multipoint constraints (MPC) into the BDD method is proposed. In this method, when an MPC is enforced on the internal degrees of freedom (DOFs) in some subdomains, the DOFs are converted into interface DOFs, that is, all of the DOFs constrained by MPCs become interface DOFs. Then, the interface problem with the set of MPCs for the interface DOFs is solved by the conjugate projected gradient method. In order to combine the above procedure with the preconditioner used in the BDD method, the effect of the MPCs for the interface DOFs is also imposed on the coarse grid problem of the BDD method using the penalty method. A parallel implementation of the present method is also described. Some illustrative examples are solved and good convergence and parallel performance of the present method are demonstrated. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Cohesive单元的石拱桥主拱圈二相数值模拟方法

为解决石砌体材料非均质的描述问题,提出一种基于Cohesive单元的石拱桥主拱圈两相数值模拟方法。视石砌体为两相材料(砌块和砌缝),采用实体单元模拟砌块并引入非线性本构描述其破坏行为,在相邻砌块间插入Cohesive单元考虑砌缝砂浆的剪切和拉伸破坏。通过室内试验与数值模拟对比验证方法的有效性及适用性,分析了砌缝抗剪摩擦系数μ、加载位置等敏感参数对拱桥承载力的影响。结果表明：基于Cohesive单元的石砌体两相数值模型,可以有效描述石砌体材料的非均匀性及石拱桥的破坏过程(尤其是砌缝剪切滑移破坏行为),可为石拱桥极限承载力评估提供重要信息,如荷载-位移曲线、破坏模式等。此外,研究结果还发现主拱圈破坏机制由拱的受弯、受剪特性决定,并与砌缝抗剪摩擦系数μ强相关。     

Approximate limit analysis of full scale FRP-reinforced masonry buildings through a 3D homogenized FE package

A 3D homogenized FE limit analysis software for the numerical prediction of collapse loads and failure mechanisms of entire masonry buildings reinforced with FRP strips is presented. In particular, a two steps approach is adopted: in step I, masonry homogenized failure surfaces are obtained through an admissible kinematic FE approach in the representative element of volume (REV), constituted by a brick interconnected with its six neighbors with finite thickness mortar joints. 8-Noded rigid infinitely resistant parallelepiped elements interconnected with interfaces with frictional behavior and limited tensile and compressive strength are utilized to model the REV. A simple linear programming problem in few variables is obtained, suitable to recover numerically masonry failure surfaces when loaded in- and out-of-plane. In step II, homogenized failure surfaces are implemented in the novel FE kinematic limit analysis software for an inexpensive evaluation of collapse loads of entire buildings. Delamination is considered in the model imposing to FRP–masonry interfaces a limited resistance in agreement with Italian code CNR-DT-200. 6-Noded rigid infinitely resistant 3D wedge-shaped elements are used to model homogenized masonry, whereas FRP strips are modeled by means of triangular 3-noded rigid elements.     

基于离散宏单元的砌体结构高效非线性分析方法

砌体结构由力学性能不同的块体和砂浆构成，材料的各向异性使结构非线性行为体现出高度复杂性。砌体结构非线性分析模型主要包括分离式和整体式两种:分离式模型将块体、砂浆及二者粘结界面分开建模，可以精细化揭示砌体非线性行为和破坏形态，但非线性分析计算量大，多用于局部构件的细部分析和模拟；整体式模型将块体和砂浆假定为连续的匀质体，建模过程简单、易行，适用于整体结构的宏观分析。无论是分离式还是整体式，结构非线性计算分析中大规模刚度矩阵的实时更新与分解降低了分析效率。该文提出了一种基于整体式空间离散宏单元模型的砌体结构高效非线性分析方法，该方法采用剪切单元模拟砌体墙的斜截面剪切破坏模式，采用无厚界面单元模拟砌体墙的正截面弯曲破坏模式、正截面剪切滑移破坏模式和平面外剪扭破坏模式，进一步将剪切单元等效斜向弹簧的轴向变形和无厚界面单元上下表面的相对变形分解为线弹性和非线性两部分，并引入塑性自由度描述分离出的非线性部分，可将任意时刻的切线刚度矩阵表示为初始弹性刚度矩阵的低秩摄动形式，引入Woodbury公式进行求解，该文方法避免了大规模整体刚度矩阵的迭代更新，非线性分析的主要计算量仅集中于小规模非线性矩阵的更新与分解，显著提升了计算效率。     

Surrogated Linear Programming

This paper deals with the development of a completely different technique for solving the linear programming (LP) problem. The general LP problem is replaced by an LP problem having a single constraint called the surrogated linear programming (SLP) problem. The SLP problem retains the same objective function, but is constrained by a convex combination of the original constraints. A special equivalency relationship is derived between the general problem and the surrogated problem in route to the final computational algorithm. The technique is not subject to round off error propagation and has promise for savings in computation time.     

2D analysis for geometrically non-linear elastic problems using the BEM

A boundary element method (BEM) approach for the solution of the elastic problem with geometrical non-linearities is proposed. The geometrical non-linearities that are considered are both finite strains and large displacements. Material non-linearities are not considered in this paper, so the constitutive law employed is Hooke's elastic one and the fundamental solution introduced in the integral equations is the usual one for isotropic linear elasticity. In order to deal with the intricate non-linear equations that govern the problem, an incremental–iterative method is proposed. The equations are linearized and a Total Lagrangian Formulation is adopted. The integral equations of the BEM are developed in an incremental form. The iterative process is necessary in order to achieve a good approximation to the governing equations. The problem of a slab under homogeneous deformation is solved and the results obtained agree with the analytical solution. The problem of a hollow cylinder under internal pressure is also solved and its solution compared with that obtained by a standardized finite element method code.     

Linear Programming Based Genetic Algorithm for the Unequal Area Facility Layout Problem

The facility layout problem (FLP) is generally defined as locating a set of departments in a facility with a given dimension. In this paper, a hybrid genetic algorithm (GA)/linear programming (LP) approach is proposed to solve the FLP on the continuous plane with unequal area departments. This version of the FLP is very difficult to solve optimally due to the large number of binary decision variables in mixed integer programming (MIP) models as well as the lack of tight lower bounds. In this paper, a new encoding scheme, called the location/shape representation, is developed to represent layouts in a GA. This encoding scheme represents relative department positions in the facility based on the centroids and orientations of departments. Once relative department positions are set by the GA, actual department locations and shapes are determined by solving an LP problem. Finally, the output of the LP solution is incorporated into the encoding scheme of the GA. Numerical results are provided for test problems with varying sizes and department shape constraints. The proposed approach is able to either improve on or find the previously best known solutions of several test problems.     

Iterative solution methods for beam angle and fluence map optimization in intensity modulated radiation therapy planning

We present computational approaches for optimizing beam angles and fluence maps in Intensity Modulated Radiation Therapy (IMRT) planning. We assume that the number of angles to be used for the treatment is given by the treatment planner. A mixed integer programming (MIP) model and a linear programming (LP) model are used to find an optimal set of beam angles and their corresponding fluence maps. The MIP model is solved using the branch-and-bound method while the LP model is solved using the interior point method. In order to reduce the computational burden for solving the optimization models, we introduce iterative beam angle elimination algorithms in which an insignificant beam angle is eliminated in each iteration. Other techniques are also explored including feasible set reduction for LP and data reduction. Experiments are made to show the computational advantage of the iterative methods for optimizing angles using real patient cases.     

Dynamic optimization of structures with non-linearities

Dynamic optimal design of structural systems with local non-linearities is proposed and the corresponding mathematical programming is established in this paper. A predictor-corrector method is presented for state and sensitivity analysis of the systems which is based on partitioning and interface prediction of linear and non-linear regions. The non-linear programming problem is linearized and solved by a coarse search SLP approach. A model of an automatic weapon is optimized and some interesting results are given to illustrate the effectiveness of the present method.     

Offshore wind farm electrical cable layout optimization

This article explores an automated approach for the efficient placement of substations and the design of an inter-array electrical collection network for an offshore wind farm through the minimization of the cost. To accomplish this, the problem is represented as a number of sub-problems that are solved in series using a combination of heuristic algorithms. The overall problem is first solved by clustering the turbines to generate valid substation positions. From this, a navigational mesh pathfinding algorithm based on Delaunay triangulation is applied to identify valid cable paths, which are then used in a mixed-integer linear programming problem to solve for a constrained capacitated minimum spanning tree considering all realistic constraints. The final tree that is produced represents the solution to the inter-array cable problem. This method is applied to a planned wind farm to illustrate the suitability of the approach and the resulting layout that is generated.