Load Multiplier Research Articles

Comprehensive sensitivity analysis of rotational stability of a super-deep underground spherical structure considering uncertainty

Jiangmen Underground Neutrino Observatory central detector is located 700 m below the ground and also submerged into an ultrapure water pool. The main structure of the Jiangmen Underground Neutrino Observatory central detector is a hybrid spherical shell that is vulnerable to rotation under the buoyancy effect. The influences of the model parameters on the rotational stability of this complex and unique structure are investigated. Since the model parameters are inevitably subjected to many sources of uncertainties (e.g. manufacturing tolerances and geometrical imperfections), the parameter uncertainty is taken into account. In addition, linear and nonlinear rotational stabilities of this super-deep underground spherical structure are also under consideration. Specifically, the critical loading multiplier is used as the evaluation indicator of linear rotational stability and the load proportionality factor- θ curve is considered as the evaluation indicator of nonlinear rotational stability. The sensitivity of linear and nonlinear rotational stabilities to uncertain parameters is systematically studied in terms of univariate and multivariate global sensitivity analyses. The univariate global sensitivity analysis is able to evaluate the effects of uncertain parameters on each evaluation indicator, whereas multivariate global sensitivity analysis enables to assess the global influence of uncertain parameters on all evaluation indicators. A polynomial chaos expansion surrogate model is utilized to replace the time-consuming simulation model for analytical implementation of the univariate and multivariate global sensitivity analyses. The present polynomial chaos expansion-based univariate and multivariate global sensitivity analyses effectively and efficiently reveal the sensitivity of the rotational stability of this super-deep underground spherical structure to uncertain parameters, and provide a practical method for comprehensive sensitivity analysis of similar structures.

Force-Based Seismic Evaluation of Retrofitting Interventions of Historic Masonry Castles by 3D Rigid Block Limit Analysis

This paper deals with the force-based assessment of collapse mechanisms and strengthening interventions of the historic masonry castle “Bussi sul Tirino” (Abruzzi, Italy) using rigid block limit analysis (RBLA). The structure, which is a fortified palace dating back to the 11th century, has experienced severe earthquakes over the centuries and was hit once again in 2009 by the L’Aquila earthquake. Based on the historical analysis and the results of in situ investigations, a spatial rigid block model of an entire structural unit was generated using the in-house software LiABlock_3D. The software is a MATLAB® based tool for three-dimensional RBLA, which provides as outputs collapse failure modes and collapse load multipliers. In addition, a specific routine was developed for the purpose of the study to compute the participating mass ratio and the spectral acceleration that activated the failure mechanisms. The results of the numerical analysis were used to address three different retrofitting interventions, based on the use of connection elements and ties that, according to the minimum intervention principle, progressively enhanced the seismic capacity. Comparisons in terms of seismic safety indices are finally provided in order to give a quantitative measure of the effectiveness of the adopted retrofitting strategies.

Determination of Collapse Load of Engineering Structures using Iterative Node-based Smoothed Finite Element Analysis Method

This paper presents an iterative limit analysis scheme modeled within a three-node node-based smoothed finite element framework incorporating a so-called modified elastic compensation method to determine the ultimate bearing capacity of ductile structures. More explicitly, the approach iteratively reduces those elemental Young’s modulus associated with a high-stress intensity such that the inelastic behavior of the elastic-perfectly plastic material model is taken into account. The approach provides good accuracy and convergence of solution through a series of standard linear elastic analyses that make possible the application in large-scale engineering structures, even the one with complex geometric configurations, and hence eminently accessible to structural engineers. As a result, complex inelastic analyses are eliminated and computational cost is significantly reduced. Several benchmark examples, one of which reported, illustrates the efficiency and robustness of the iterative limit analysis approach in tracing the load multipliers of in-plane ductile structures from the beginning until convergence of collapse load solution or failure state occurs. The yield areas at the ultimate state gradually appear in the graph of elastic modulus distribution.

Seismic capacity of buttressed masonry arches

Masonry buttressed arches have been designed in the past for vertical actions deriving from gravity loads and horizontal actions determined by the thrust of the arch. Horizontal actions consequent to earthquake motions were not usually taken into account in the design of these structures. This, despite of the high vulnerability of such typology, frequently characterizing monumental buildings.This paper addresses the problem of the capacity of buttressed masonry arches with a circular shape and different angles of embrace, under horizontal actions induced by seismic motions.To this aim, in the framework of Limit Analysis for masonry structures, the seismic analysis of a large sample of buttressed arches is provided and discussed.The outcomes of numerical analysis have shown that the activated failure mechanisms change in presence of horizontal actions. The results of the analysed large sample are very consistent in terms of lateral capacity since the horizontal load multiplier increases with the thickness of the buttresses and decreases with the buttress slenderness. The paper provides an unexpected result for thin arches since in this case semi-circular arches become more vulnerable than segmental.

Efficient meta-heuristic mesh adaptation strategies for NURBS upper–bound limit analysis of curved three-dimensional masonry structures

We propose a new adaptive NURBS upper-bound limit analysis approach for the assessment of general three-dimensional curved masonry structures, based on different meta-heuristic mesh adaptation schemes. The method, which can easily be integrated within CAD modeling environments, allows to establish the actual failure mechanism and load bearing capacity of a masonry structure by iteratively adjusting a tentative pattern of yield lines, defined upon a suitable initial mesh of NURBS rigid elements, by means of a suitable meta-heuristic algorithm which searches for the minimum collapse load multiplier, thus enforcing the upper-bound theorem of limit analysis. In particular, we investigate and discuss the efficiency of several meta-heuristic algorithms in delivering the optimal solution: a specifically devised Prey Predator Algorithm (PPA) is compared with the Particle Swarm Optimization (PSO) Algorithm, the Firefly Algorithm (FA) and a suitable Genetic Algorithm (GA). Four masonry vaults have been chosen as case studies. In particular, the modified PPA proves to be the most efficient mesh-adjustment scheme for the proposed adaptive NURBS-based limit analysis procedure.

Parametric Analysis on Local Mechanisms of Masonry Churches in Teramo (Italy)

Local collapse mechanisms related to the out-of-plane response of walls are commonly observed in existing masonry buildings subjected to earthquakes. In such structures, the lack of proper connections among orthogonal walls and between walls and floors does not allow a global box-type behaviour of the building to develop, which would be governed by the in-plane response of walls. In this paper, parametric linear kinematic analyses on the main local mechanisms of masonry churches were performed with the aim to evaluate the corresponding horizontal load multipliers. This study was conducted on 12 masonry churches, located in Teramo (Italy) and affected by the 2016 Central Italy earthquake, whose main out-of-plane collapse mechanisms, namely facade overturning, vertical bending, corner overturning and roof gable wall overturning, have been analysed. For each mechanism, parametric analysis was carried out on varying heights and thicknesses of walls. Firstly, the acceleration values activating the considered mechanisms were calculated in order to conduct checks prescribed by the current Italian standard. Subsequently, on the basis of the obtained results, simple analytical procedures to determine load collapse multiplier for each mechanism were drawn. Finally, ranges of suitable values of both the thickness and height of walls were found in order to always satisfy seismic checks.

Two solution strategies to improve the computational performance of sequentially linear analysis for quasi‐brittle structures

SummarySequentially linear analysis (SLA), an event‐by‐event procedure for finite element (FE) simulation of quasi‐brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λcrit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low‐rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two‐dimensional (2D) and three‐dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury‐based direct solver over its counterparts, especially for large 3D problems.

Design and thermal fluid structure interaction analysis of liquid nitrogen cryostat of cryogenic molecular sieve bed adsorber for hydrogen isotopes removal system

Efficient design of Tritium Extraction System (TES) for the fuel cycle of any fusion reactor is very important to maintain the tritium breeding ratio and hence sustain the fusion reaction. Hydrogen Isotopes Removal System (HIRS) for Indian Tritium Breeder Blanket removes Q2 (Q = H, D or T) and impurities using Cryogenic Molecular Sieve Bed (CMSB) adsorber at 77 K. The CMSB is maintained at liquid nitrogen temperature using a double walled cryostat made up of SS304. The paper describes the design and thermal Fluid Structure Interaction (FSI) analysis of cryostat assembly for CMSB of HIRS. The coupled analysis performed in this work involves solving for the fluid domain and transferring the results to ANSYS Thermal-Static structural set up to determine the stresses and displacement due to combined effects in the system. The mechanical design of the cryostat components is analytically performed using ASME codes. The velocity, pressure drop and time taken to cool the CMSB are determined by solving the fluid and energy equations in ANSYS Fluent analysis system. The solutions are imported into ANSYS Thermal-Static structural analysis system and the thermal-structural stresses and deformations are determined considering the temperature, pressure and acceleration loads. The space between inner and outer vessel is maintained at vacuum, which might lead to buckling. So, the critical buckling load multiplier factor is determined. These results are used in fabricating the complete cryostat system for CMSB of HIRS.

Deterministic and reliability-based design of necessary support pressures for tunnel faces

This paper provides methods for the deterministic and reliability-based design of the support pressures necessary to prevent tunnel face collapse. The deterministic method is developed by extending the use of the unique load multiplier, which is embedded within OptumG2/G3 with the intention of determining the maximum load that can be supported by a system. Both two-dimensional and three-dimensional examples are presented to illustrate the applications. The obtained solutions are validated according to those derived from the existing methods. The reliability-based method is developed by incorporating the Response Surface Method and the advanced first-order second-moment reliability method into the bisection algorithm, which continuously updates the support pressure within previously determined brackets until the difference between the computed reliability index and the user-defined value is less than a specified tolerance. Two-dimensional reliability-based support pressure is compared and validated via Monte Carlo simulations, whereas the three-dimensional solution is compared with the relationship between the support pressure and the resulting reliability index provided in the existing literature. Finally, a parametric study is carried out to investigate the influences of factors on the required support pressure.

Geometrically nonlinear behavior of lattice domes coupled with local Eulerian instability

Abstract Structural analysis is an intricate subject when nonlinearities occur. They make the structural behavior complex and may have important consequences in the design choice as well. Especially for lattice domes, as snap-through phenomena and local Eulerian instabilities generally affect the structural response, linear analysis is not enough. In this paper, a semi-analytical formulation is used in order to study the geometrically nonlinear behavior of lattice domes subject to vertical loads. The formulation is derived from the equilibrium equations written in the deformed configuration, considering large displacements and taking also into account local buckling conditions. The resulted system of equations, being strongly nonlinear, has been solved by means of a numerical procedure, based on a mixed load-displacement control scheme, leading to the evaluation of the complete equilibrium path. The influence of geometrical parameters on the critical load multiplier and shape of the load-displacement curve is also discussed. In particular, a complex equilibrium path for a sixteen-member five-node lattice structure is analyzed, which is characterized by several branches which can generate ‘snapping’ conditions.

A digital tool based on Genetic Algorithms and Limit Analysis for the seismic assessment of historic masonry buildings

New technologies are changing the way engineers work within the construction sector. Newly developed software solutions have provided effective methods to explore the design space at the interface between Structural Engineering and Architecture, allowing more efficient design strategies. These technologies are based on the integration of parametric generation and visualisation of geometries with powerful numerical solvers, employing user-customised routines. While the construction industry is rapidly moving the design of new construction towards a fully digitalised process, the assessment and the analysis of existing structures with such tools are still largely unexplored. In this context, a visual script for the structural assessment of out-of-plane mechanisms in historic masonry structures subject to seismic loading has recently been proposed by the authors. This relies on two successive steps of analysis, which are integrated into a digital work-flow. Datasets describing the geometric configuration of masonry structures are employed to automatically generate a non-linear Finite Element (FE) model and investigate possible collapse modes. A preliminary global analysis is performed using the commercial software ABAQUS CAE. This, in combination with the Control Surface Method (CSM), allows identifying the most likely failure mechanisms which are described by the geometry of the macro-blocks. The parametric modelling of the macro-blocks geometry allows exploring the domain of possible solutions using the upper bound method of limit analysis. A Genetic Algorithms (GA) solver is used to refine the geometry of the macro-blocks and search the minimum of the upper-bound load multipliers, which guarantees equilibrium. The script is implemented in the visual programming environment offered by Rhino3D+Grasshopper. In this paper, a set of parametric analyses considering various input variables such as friction coefficient and opening incidence are performed to verify both the sensitivity and the accuracy of the proposed method.

A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures

This paper presents a quasi-static formulation, based on integrated radial basis functions (iRBF) and conic programming, for direct analysis of 2D and 3D structures. The iRBF shape functions are stabilized using stabilized conforming nodal integration (SCNI) technique. Equilibrium equations, boundary conditions and yield criterion associated with the residual stresses are directly enforced at scattered nodes without any special treatment. The safety load multipliers are determined by solving the conic optimization problems resulting from static shakedown theorem. The computation cost in terms of the number of variables and optimization CPU time is kept to a minimum. Various benchmark problems in two and three dimensions are investigated, showing the efficiency of the presented numerical approach.

Evolutive and Kinematic Limit Analysis Algorithms for Large-Scale 3D Truss-Frame Structures: Comparison Application to Historic Iron Bridge Arch

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.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

Abstract The numerical manifold method (NMM) can be viewed as an inherent continuous-discontinuous numerical method, which is based on two cover systems including mathematical and physical covers. Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants. Therefore, higher-order NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy. In addition, it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics. Thus, some other enriched local approximations are introduced to model the stress singularity at the crack tip. Generally, higher-order NMM, especially first-order NMM wherein local approximations are first-order polynomials, has the linear dependence problems as other partition of unit (PUM) based numerical methods does. To overcome this problem, an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method (FEM), which has no linear dependence issue. Meanwhile, the stresses at the nodes of mathematical mesh (the nodal stresses in FEM) are continuous and the degrees of freedom defined on the physical patches are physically meaningful. Next, the extended NMM is employed to solve multiple crack propagation problems. It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation. Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM. The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions. Thus the effectiveness and correctness of the developed NMM have been validated.

The Bending of Beams in Finite Elasticity

In this paper the analysis for the anticlastic bending under constant curvature of nonlinear solids and beams, presented by Lanzoni, Tarantino (J. Elast. 131:137–170, 2018), is extended and further developed for the class of slender beams. Following a semi-inverse approach, the problem is studied by a three-dimensional kinematic model for the longitudinal inflexion, which is based on the hypothesis that cross sections deform preserving their planarity. A compressible Mooney-Rivlin law is assumed for the stored energy function and from the equilibrium equations, the free parameter of the kinematic model is computed. Thus, taking into account the three-dimensionality of the beam, explicit formulae for the displacement field, the stretches and stresses in every point of the body, following both Lagrangian and Eulerian description, are derived. Subsequently, slender beams under variable curvature were examined, focusing on the local determination of the curvature and bending moment along the deformed beam axis. The governing equations take the form of a coupled system of three equations in integral form, which is solved numerically. The proposed analysis allows to study a very wide class of equilibrium problems for nonlinear beams under different restraint conditions and subject to generic external load systems. By way of example, the Euler beam and a cantilever beam loaded by a dead or live (follower) concentrated force applied at the free end have been considered, showing the shape assumed by the beam as the load multiplier increases.

Analysis of stability against rotation of a spherical shell structure subjected to buoyancy

The main structure of the central detector at the Jiangmen Underground Neutrino Observatory (JUNO) is a spherical shell structure, which has a giant acrylic spherical shell connected to a stainless-steel (SS) reticulated shell with 590 SS rods. The acrylic spherical shell is submerged into water and filled with detection liquid and prone to rotation subjected to considerable buoyancy. The stability against rotation of this super-deep underground spherical shell structure needs to be fully investigated. In this study, an effective and practical method consisting of parametric analysis and optimization procedure is proposed to improve the stability against rotation. Specifically, two indicators, namely, the critical loading multiplier and the rotation angle, are proposed to evaluate the stability against rotation of the acrylic spherical shell in linear and nonlinear stability analyses, respectively. Then, parametric analysis is performed to assess the sensitivity of the stability against rotation to four parameters of interest (i.e., rod outer end constraints, liquid level difference, disc spring stiffness, and rod deviation). Based on the obtained parametric analysis results, the liquid level difference and disc spring stiffness are finally selected as design variables for the subsequent optimization process to further improve the stability against rotation. An efficient scheme combining design variable discretization and exhaustive method is adopted to identify the optimal variable values at which the acrylic spherical shell has good stability against rotation. The results indicate that the proposed method is efficient and effective for optimization of stability against rotation of such super-deep underground spherical shell structure. The proposed method is practical and easy to use for structural designers, and provides an efficient approach to the stability design of such rod-connected spherical shell structures.

Non‐proportional loading in sequentially linear analysis for 3D stress states

SummarySequentially linear analysis (SLA), a non‐incremental‐iterative approach towards finite element simulation of quasi‐brittle materials, is based on sequentially identifying a critical integration point in the model, to reduce its strength and stiffness, and the associated critical load multiplier (λcrit), to scale the linear analysis results. In this article, two novel methods are presented to enable SLA simulations for non‐proportional loading situations in a three‐dimensional fixed smeared crack framework. In the first approach, the cubic function in the load multiplier is analytically solved for real roots using trigonometric solutions or the Cardano method. In the second approach, the load multiplier is expressed as a function of the inclination of a potential damage plane and is deduced using a constrained optimization approach. The first method is preferred over the second for the validation studies due to computational efficiency and accuracy reasons. A three‐point bending beam test, with and without prestress, and an RC slab tested in shear, with and without axial loads, are used as benchmarks. The proposed solution method shows good agreement with the experiments in terms of force‐displacement curves and damage evolution.

Stress Compensation Method for Structural Shakedown Analysis

This paper presents a novel direct method called the stress compensation method (SCM) for structural shakedown analysis. Being different from the popular direct method of mathematical programming, the SCM just carries out some iterative calculations. Making full use of static shakedown theorem, the residual stress field is constructed via solving the modified global equilibrium equations. An effective and robust iteration control technique is adopted to generate a sequence of decreasing load multipliers. The numerical procedure is incorporated into the ABAQUS platform via some user subroutines. The shakedown problems for a cantilever beam, a symmetric continuous beam and a practical shell with nozzles are effectively solved and analyzed. These results are compared to the analytical solutions and those found in literatures. Both the incremental collapse mechanism and the alternating plasticity mechanism are revealed to determine the shakedown boundaries. Numerical examples show that the SCM is of numerical stability, good accuracy, high computational efficiency, and can effectively perform shakedown analysis of large-scale practical engineering structures.

On the analysis of masonry arches

The behaviour of masonry arches structures is studied through the homogenising theory, applying the newest results of symbolic calculation. This procedure, based on the solution suggested in 1982 as an extension of Castigliano's theorem, takes advantage of considering the stony ashlar and mortar structure as a homogenous, non-tensile resistant body: as a consequence, a different reacting configuration will be needed for each load multiplier. Thus, it is urged to fix a modified structure, able to contain the arch funicular in determining the statically compatible solution to support traditional tensile stress methods. This work ends with an analysis on the behaviour of a well-known existing monumental structure, Saint-Martin Bridge upon Lys.

The structural reliability analysis using explicit neural state functions

The present study considers the problems of stability and reliability of spatial truss susceptible to stability loss from the condition of node snapping. In the reliability analysis of structure, uncertain parameters, such us load magnitudes, cross-sectional area, modulus of elasticity are represented by random variables. Random variables are not correlated. The criterion of structural failure is expressed by the condition of non-exceeding the admissible load multiplier. In the performed analyses explicit form of the random variables function were used. To formulate explicit limit state functions the neural networks is used. In the paper only the time independent component reliability analysis problems are considered. The NUMPRESS software, created at the IFTR PAS, was used in the reliability analysis. The Hasofer-Lind index in conjunction with transformation method in the FORM was used as a reliability measure. The primary research method is the FORM method. In order to verify the correctness of the calculation SORM and Monte Carlo methods are used. The values of reliability index for different descriptions of mathematical model of the structure were determined. The sensitivity of reliability index to the random variables is defined.