Discontinuity Layout Optimization Research Articles

Probabilistic assessment of seismic earth pressure against backfills with a nonlinear failure criterion

A novel approach for the probabilistic assessment of seismic earth pressure against nonlinear backfills is introduced in this study. To implement reliability-based design for the retaining structures under seismic loading condition, nonlinear upper bound analysis is adopted to obtain the seismic earth pressure through optimization procedure. Firstly, rigorous analytical function is formulated to reveal the normal-shear stress information along the failure surface in soil with nonlinear failure criterion. Subsequently, a kinematic equation is put forward to estimate the seismic earth pressures by balancing the external work rate and the internal energy dissipation rate. The solutions are verified by the Discontinuity Layout Optimization numerical modellings based on the derived stress data from the proposed method. To consider the probability analysis of nonlinear backfill properties, the moment method is presented by combining the adaptive dimension decomposition with the direct integral method. According to the estimated first four statistical moments, the approximated probability density function of the performance function is determined contently. Finally, the failure probability of seismic earth pressure is calculated based on the proposed moment method. Two numerical examples demonstrate that the moment method can be adapted to the characteristics of nonlinear backfills and further improve the accuracy of reliability estimation by introducing higher-order moments analytically.

Application of artificial neural networks for predicting the bearing capacity of the tip of a pile embedded in a rock mass

Addressing the problem of the bearing capacity of the tip of a pile in rock needs to consider a full-nonlinear behavior of the rock, as the Hoek and Brown failure criterion, and including the various geometric parameters defining the system geometry. It usually needs the use of complex numerical models. They require a high level of training and expertise and are commonly out of the reach of engineers on the day-to-day calculation procedures. Technical codes usually recommend simpler formulae based on one or two rock parameters, their accuracy relying on particular and local empirical testing. This research develops an artificial neural network (ANN) that solves the previous difficulties. The ANN is based on a group of 1440 calculations using the novel numerical procedure Discontinuity Layout Optimization (DLO). This numerical method can efficiently reproduce the non-linear behavior of the rock (including rock type, uniaxial compressive strength, and geological strength index) for every configuration of the model (foundation width, pile embedment, and height of the overlying soil). Once the ANN is trained and optimized, it can easily predict any result within its range of applicability. Furthermore, it can be reduced to a simple set of recursive equations to be implemented in a spreadsheet. The proposed ANN has only one hidden layer besides the input and output layers, with (6)-(8)-(1) neurons, respectively. It gives highly accurate results with minimum cost and can be a useful tool for engineers and scientists in the foundation field.

Pile tip bearing capacity in rock masses: analytical and numerical analysis comparison

This paper researches the different approaches to calculate the ultimate bearing capacity at the pile tip embedded in a homogenous rock mass characterized by the modified Hoek and Brown failure criterion. The main objective is to analyze the performance of the novel Discontinuity Layout Optimization method (DLO), which directly calculates the limit load, instead of using a load convergence scheme, being first applied here to the complex problem of piles in rock. The DLO results are validated against the available analytical solution and the Finite Difference Method (FDM) results, for wide range of geometric and geotechnical parameters. Perfect plasticity, plane strain conditions, associated flow rule, the modified Hoek and Brown failure criterion, and weightless rock media are considered in all models. The comparison between solutions is expressed by a percentage difference considering the analytical solution or the numerical solution as a reference. Results relative to DLO and FDM differences against the analytical solutions point towards some limitations of the analytical solution and that the numerical model does not reproduce exactly the analytical solution hypotheses. From the analysis, DLO demonstrates to be a very efficient and accurate tool to address the tip-of-pile bearing capacity, presenting considerable advantages over other methods.

Two-dimensional Analysis of the Group Interaction Effects Between End-bearing Piles Embedded in a Rock Mass

This research is a two-dimensional analysis of the interaction effect in a group of piles socketed in a rock mass when studying the end-bearing capacity of the group. This problem was extensively studied for deep foundations in soils, but very little is published about foundations in rock. The study was conducted in 2D to establish the basis and criteria to analyze the problem, as well as understand the main factors and failure mechanisms involved when several piles work together in a foundation. The factors included in the analysis are, among others: the parameters of the rock mass, the pile embedment, and the pile separation. The research highlights shaft resistance as one of the key factors in the interaction. It proves that neglecting shaft resistance changes the interaction influence and the failure mechanism which may not be on the safe side. The research includes an analysis of the failure modes of different pile configurations using the discontinuity layout optimization (DLO) method, by taking advantage of the method’s ability to show the wedge configuration at the failure. The minimum and maximum efficiencies are characterized in terms of the pile separation values, thereby proposing a procedure to obtain the absolute minimum efficiency of the group. A second procedure is also presented for the approximate prediction of the bearing capacity of the pile tip, for a group with a particular number of piles and properties, by considering the contribution of the corner piles and the intermediate piles.

Yield-line mechanisms and design proposal for endplate connections in rectangular hollow section members in tension

Hollow section members are popular elements in steel constructions, especially when they are mainly loaded by normal forces, like simply supported columns or truss members. The individual hollow section members can be combined by bolted endplate connections. If the bolts are arranged at all four sides of the rectangular hollow section, the design rules for the T-stub model from the widely used Eurocode EN1993–1-8 cannot be applied due to the missing failure mechanisms (yield-line mechanisms) of the whole plate. The presented research deals with this problem. A parameter study on different endplate layouts with and without chamfers has been done using a verified numerical model. The numerical model is based on the discontinuity layout optimization (DLO) method and delivers all the relevant yield-line mechanisms of the plate and bolts. Based on these observed failure mechanisms, a simple and Eurocode conform design proposal has been worked out for endplates with and without chamfers. The obtained resistances from the design proposal and the DLO calculations are within a deviation of approximately 10%.

Retracted: Stability analysis of blocky structure system using discontinuity layout optimization

Retracted: Stability analysis of blocky structure system using discontinuity layout optimization

Featured Cover

The cover image is based on the Research Article Virtual displacement based discontinuity layout optimization by Yiming Zhang et al., https://doi.org/10.1002/nme.7084.

Limit analysis of masonry walls using discontinuity layout optimization and homogenization

AbstractA numerical limit analysis model for masonry walls subject to in‐plane loading is posed as a discontinuity layout optimization (DLO) problem, with the masonry conveniently modeled using a smeared continuum (“macromodeling”) approach and a homogenized yield surface. Unlike finite element limit analysis, DLO is formulated entirely in terms of discontinuities and can produce accurate solutions for problems involving singularities naturally, without the need for mesh refinement. In the homogenized model presented, masonry joints are reduced to interfaces, with sliding governed by an associative friction flow rule and blocks are assumed to be infinitely resistant. The model takes account of the interlock ratio of the masonry blocks, their aspect ratio and the cohesion and coefficient of friction of interfaces in both the vertical and horizontal directions. Results from the proposed model are compared with those from the literature, showing that complex failure mechanisms can be identified and that safe estimates of load carrying capacity can be obtained. Finally, to demonstrate the utility of the proposed modeling approach, it is applied to more complex problems involving interactions with other elements, such as voussoir arches and weak underlying soil layers.

Yield‐line Mechanisms of Endplate Connections with Eight Bolts and Circular Bolt Configurations Using DLO

AbstractDesigning bolted endplate connections is a typical problem in practice. Nevertheless, the calculation of such endplates is not that easy. Due to the deformation of the plate, the resultant prying forces influence the resistance of the connection. The so‐called T‐stub model considers prying effects and is included in the EN 1993‐1‐8 for typical endplates with two bolts and rectangular bolt configurations. It was found in the literature that circular bolt configurations can have some benefits compared to rectangular ones, like a better stress distribution in the plate and bolts, resulting in higher resistances. Systematic research on the evolution of the yield‐line mechanisms is still not available, which is the focus of the presented study. We investigated and compared the yield‐line mechanisms and ultimate loads of typical rectangular and circular bolt configurations. A parameter study has been worked out by varying the endplate thickness for each case. The calculations were made with the Discontinuity Layout Optimization (DLO) concept, directly offering the kinematic mechanisms and corresponding ultimate loads. Finally, we gave some simple recommendations for calculating the ultimate loads of circular bolt configurations.

Virtual displacement based discontinuity layout optimization

AbstractDiscontinuity layout optimization (DLO) is a relatively new upper bound limit analysis method. Compared to classic topology optimization methods, aimed at obtaining the optimum design of a structure by considering its self‐weight, building cost, or bearing capacity, DLO optimizes the failure pattern of the structure under specific loading conditions and constraints by minimizing the dissipation energy. In this work, we present a modified DLO algorithm that contains all of the advantages of DLO. It is referred to virtual displacement‐based discontinuity layout optimization (VDLO). VDLO takes the stress state of a loaded structure as a snapshot and correspondingly provides the optimum failure pattern, which greatly extends the application potential of DLO. Numerical examples indicate the effectiveness and flexibility of VDLO. It is regarded as a highly promising supplemental tool for other numerical methods in element‐/node‐based frameworks.

Parametric Analysis of Failure Loads of Masonry Textures by Means of Discontinuity Layout Optimization (DLO).

Several masonry structures of cultural and historical interest are made with a non-periodic masonry material. In the case of periodic textures, several methods are available to estimate the strength of the masonry; however, in the case of non-periodic masonry, few methods are available, and they are frequently difficult to use. In the present paper we propose using discontinuity layout optimization (DLO) to estimate the failure load and mechanism of a masonry wall made with non-periodic texture. We developed a parametric analysis to account for the main features involved in the estimation of failure: in particular we considered three different textures (periodic, quasi-periodic, and chaotic), variable height-to-width ratio of the wall (from 0 to 3) and of the blocks (from 0.25 to 1), different mechanical properties of mortar joints and blocks, and possible presence of a load on the top. The results highlight the importance of the parameters considered in the analysis, both on the values of the failure load and on the failure mechanism. Therefore, it is found that DLO can be an useful and affordable method in order to assess the mechanical strength of masonry wall made with non-periodic textures.

Discontinuity layout optimization using unstructured meshes and material layering in 2D

Discontinuity layout optimization using unstructured meshes and material layering in 2D

The stress function basis of the upper bound theorem of plasticity

The discontinuous slip-line form of upper bound plasticity analysis is considered using an equilibrium of forces approach. It is demonstrated that the underlying basis of the approach can be written in terms of stress functions that provide a continuum stress state interpretation of the upper bound solution. An alternative proof of the upper bound theorem, applicable to both associative and non-associative materials, using stress functions is presented. The broader nature of the equilibrium form and the strict conditions under which it is valid are discussed, including examination of the apparent omission of moment equilibrium and associativity in many equilibrium form solutions. Finally, the relationship of the stress function formulation to the output of the computational limit analysis method discontinuity layout optimisation (DLO) and the potential to use the stress function formulation to derive a form of lower bound solution from an upper bound analysis are demonstrated.

Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms

Segmental arched forms composed of discrete units are among the most common construction systems, ranging from historic masonry vaults to contemporary precast concrete shells. Simple fabrication, transport, and assembly have particularly made these structural systems convenient choices to construct infrastructures such as bridges in challenging environmental conditions. The most important drawback of segmental vaults is basically the poor mechanical behaviour at the joints connecting their constituent segments. The influence of the joint shape and location on structural performances has been widely explored in the literature, including studies on different stereotomy, bond patterns, and interlocking joint shapes. To date, however, a few methods have been developed to design optimal joint layouts, but they are limited to extremely limited geometric parameters and material properties. To remedy this, this paper presents a novel method to design the strongest joint layout in 2D arched structures while allowing joints to take on a range of diverse shapes. To do so, a masonry arched form is represented as a layout of potential joints, and the optimization problems developed based on the two plastic methods of classic limit analysis and discontinuity layout optimization find the joint layout that corresponds to the maximum load-bearing capacity.

Bearing Capacity of Volcanic Pyroclasts Using the Discontinuity Layout Optimization Method

The failure criterion of low-density volcanic rocks differs radically from that of conventional rocks by manifesting collapse under isotropic stress. In this way, the shapes of the failure model do not reveal a continuously increasing growth of deviating stress with the isotropic stress, but they reach a maximum value, after which they decrease until they vanish under the isotropic collapse pressure. As a consequence, engineering applications require the implementation of numerical codes and the resolution of associated numerical difficulties. This article presents the problem of the bearing capacity of a foundation on a low-density volcanic rock using the DLO (discontinuity layout optimization) numerical method. The analysis of results shows the ability of the DLO method to solve the numerical difficulties associated with the complex failure criteria, so that the convergence and stability of the solution can be achieved without generating high computational costs. Additionally, a discussion of the DLO results is also presented, demonstrating forms of failure on the ground following the real collapses in these volcanic materials. In addition, numerical validation was performed with the finite difference method, using FLAC, and with an analytical method using simplified configurations, obtaining good contrast results, with the DLO method performing better. In this way, an adequate and reliable resolution technique is provided to face the problem of bearing capacity in low-density volcanic rocks, overcoming limitations referred to in the technical literature regarding the difficulty of treating highly non-linear and non-monotonic numerical criteria, which allows the introduction of isotropic collapse failure.

The study on yield acceleration and permanent displacement of broken-back gravity quay walls – a case study

In the recently developed performance-based design method, permanent displacement is a key parameter for evaluating seismic performance quay walls, which can be achieved by fully dynamic and simple pseudo-static limit equilibrium (PS-LE) based methods. PS-LE methods have some ambiguities concerning the calculation of yield acceleration (ay ); therefore, the present study shows an alternative approach to determining ay based on the recently proposed limit analysis technique: Discontinuity Layout Optimization (DLO). The seismic performance of quay walls located in the Pars Petrochemical Port is evaluated as a case study. Firstly, according to the wall’s geometry, a modified expression of ay for PS-LE methods is provided. The parametric studies illustrate that the DLO gives a higher acceleration (ay = 0.271 g) than the LE method (ay =0.196 g). Then, a simple dynamic Finite Element (FE) analysis is also conducted to assess the permanent displacement. Comparison of resulting displacements with pseudo-dynamic (PD) methods indicated that accurate acceleration could lead to good agreement between Whitman and Liao as a PD (dmax=21cm) and FE (dmax=23.46 cm) analysis. Eventually, it is demonstrated that besides retaining the features of simple analysis, the DLO technique is more adaptable to complex geometries and offers a more realistic failure mechanism than the PS-LE approach.

Stability analysis and safety factor prediction of excavation supported by inclined piles in clay

The inclined retaining pile is an innovative excavation retaining structure free of internal braces. It can significantly reduce the cost and construction time of excavation. However, the stability of inclined-pile-supported excavations has not been well investigated. Discontinuity layout optimization was used to analyse the failure mode and the influencing factors of the stability of the excavation supported by inclined piles in clay. This shows that overturning failure and overall failure are the two possible failure modes of excavations supported by inclined piles. The failure mode gradually changes from overturning failure to overall failure as the incline angle increases. The stability of the excavation supported by inclined piles is positively correlated with the undrained shear strength and negatively correlated with the weight of the soil, the embedded depth of the inclined pile, the excavation depth ratio and the incline angle, whereas the influence of the excavation width can be neglected. Furthermore, a simple dimensionless formula is proposed based on 11,880 cases covering the most common values of the main influencing factors. It is demonstrated that the formula can predict the safety factor of inclined-pile-supported excavation well.

Generalized method for identifying yield-line patterns in T-stubs using discontinuity layout optimization

The so-called “plate in bending” is an essential component in typical steel connections. The underlying concept is the T-stub model. Typically, the resistance of a T-stub is obtained with the upper bound theorem, which means that a kinematic failure mechanism must be assumed. The most relevant failure patterns and the corresponding resistances of typical T-stub configurations can be found in the standards, like the EN1993–1-8. A general method for identifying the correct yield-line pattern and for calculating the resistance for any T-stub configuration is still not available. To overcome this unsatisfactory situation, such a generalized method has been worked out in the presented research. This was made by extending the “Discontinuity Layout Optimization” (DLO) procedure by including the internal work dissipated by the bolts. The approach was validated on a broad range of relevant applications, such as typical unstiffened and stiffened T-stubs, as well as more complex configurations, which are not ruled by the standard. The obtained results were compared with results from experiments and analytical solutions from the literature. Overall, the modified DLO approach is able to reproduce the failure patterns and the ultimate loads of the investigated bolt configuration correctly.

Application of discontinuity layout optimization method to bearing capacity of shallow foundations on rock masses

AbstractThe discontinuity layout optimization (DLO) method is applied to obtain the bearing capacity of rock masses where it is necessary to consider a non‐linear resistance law. In rock mechanics, it is widely accepted the modified Hoek & Brown failure criterion, developed for homogeneous and isotropic rock masses. The results obtained with the DLO method are compared with those obtained from the analytical solution and using the finite differences method (FDM). To validate the DLO numerical method, the same hypotheses of the analytical solution are adopted: plane strain conditions, associated flow rule, Hoek & Brown material, and weightless rock. The research analyses the results of a numerical and analytical study based on a sensitivity study varying the three parameters that characterize the rock mass (rock type, uniaxial compressive strength, and geological strength index) and shows the need to adopt an adequate linearization of the non‐linear failure criterion in the numerical calculations. Furthermore, numerical results are obtained considering the self‐weight of the rock mass, using both DLO, considering the intermediate secant linearization proposed in this investigation, and FDM, implemented in a widely accepted geotechnical software. After comparing the results, the advantages and limitations of the DLO method are pointed out.

Ultimate bearing capacity of two interfering strip footings on sand overlying clay

Foundations may be constructed at close spacings. Prior investigations have presented insights into the interference effect of two nearby shallow foundations on homogenous soils. However, soils are often deposited in layers, such as a sand layer overlying clay. Hence, the interference effect of two adjacent foundations on their ultimate bearing capacity for sand overlying clay has been a relevant topic, which is complex because it is tied to a coupling effect of the geometric configurations and the soil properties. In this paper, the ultimate bearing capacity of two interfering strip footings on sand overlying clay is determined using an upper-bound limit state plasticity method known as discontinuity layout optimization (DLO). The solution is illustrated as design charts constructed through the use of dimensionless parameters. Parametric studies are performed to explore the influence of geometric and strength parameters on the bearing capacity and the failure mechanism. A simplified model is developed based on the data obtained from DLO. Validations are carried out to demonstrate the accuracy of the developed model using various statistical criteria, including coefficient of determination (R2), arithmetic mean and coefficient of variation.