Analysis of crack trajectories – Global Homework Experts

Analysis of crack trajectories using layout optimization techniques

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The purpose of this research was to assess a new approach to predicting the probable propagation of fractures in structural elements. This becomes more important in a broad range of problems when structures approach their Ultimate Limit State. Steel supports and joints may develop cracks over time. structural designers might use a trustworthy (in terms of computational labor) approach for forecasting tier propagation in concrete pillars or walls. Also, the directions of the primary stresses in a solid subjected to the same loads and supported in the same manner coincide with the locations of the tension and compression bars. In addition, fractures should start out perpendicular to the direction of the highest tension loads. By doing a Layout Optimisation (LO) study, we may determine the starting direction of the crack and then place a “hole” in the design space facing in that direction. Next, we’ll re-run the study with this “hole” in place and see how the maximum tension shifts as a result. To test if the LO method reliably forecasts the crack’s growth direction, we may conduct the analysis many times. If this works, we may have discovered a promising new way for predicting cracks.



















Abstract 2

Contents. 3

Chapter 1: Introduction. 4

Chapter 2: Literature Review.. 6

Chapter 3: Methodology. 18

Chapter 4: Results and Discussion. 31

Chapter 5: Conclusions and Recommendations. 34

References. 38














Recently, there has been an unbelievable amount of progress in the approaches of structural optimization. When one is trying to optimize a structure, one of the goals that they should have is to decrease the mass of the structure while maintaining its stiffness and satisfying other design criteria within a particular range of possibilities. Utilizing optimization methods may result in improvements to the object’s size, form, and topology (Yang, 2010). While size and form optimization are used to simply maximize the cross-sectional features and the design domain boundaries, respectively, topology optimization is used to determine the number of cavities in continuous structures, as well as their size and location within those structures. Since concrete has a relatively low tensile strength owing to weathering, creep, and aging effects, cracking is to be anticipated in components subjected to service loads. The stiffness, energy absorption, capacity, and ductility of reinforced concrete structures are all negatively impacted by cracking. Loss of a building’s strength and rigidity may cause severe cracks and even collapse. Numerous research and investigations have been conducted to ascertain the impact of crack location and depth on the static and dynamic behavior of concrete structural element.

The growing need for low-cost, light-weight, high-performance structures has led to the categorization of structural optimization techniques into the three broad categories of size, shape, and topology. By reducing or maximizing an objective function while adhering to design restrictions, topology optimization determines the best possible material arrangement within a specified design domain (Réthoré, 2008). Topology optimization has been widely used in several engineering domains, particularly the automotive and aerospace sectors, since it offers the greatest practical and prospective design space compared to other structural optimization approaches. Several topology optimizations models, for instance, have been presented during the last several decades. Shear pivot stiffness in pantographic metamaterials may be predicted with startling accuracy by using topology optimization techniques.

Although some of its fundamental principles were developed about a century ago, layout or topology optimization is one of the most recent and rapidly increasing topics of structural design because it concerns itself with the selection of the ideal configuration for structural systems. This means that it is one of the most important aspects of modern structural design. In spite of the fact that it presents the most challenging design problem from an analytical and computational standpoint, it offers the highest reward. It is essential to make use of the conservative calculations that are included into the standard design standards in order to eliminate the possibility of failure mechanisms. However, more research is required to investigate the mechanical behavior of the structure if non-standard concrete types that are not specified in the design guidelines are used, or if non-static stress or temperature stress is applied (Fagerholt, 2013). Within the framework of the circular economy, a new generation of environmentally friendly concretes that make use of by-products of industrial processes is now under development. Quite a few studies have been carried out in order to assess their performance and possible uses in the building industry. On the other hand, there is a scarcity of research that investigates the mechanical behavior of these organisms at the structural or macro level. During the process of Ultimate Limit State Design, the resistance to shear failure, which is also known as shear capacity, is an essential property that must be verified in SRC beams. In regions where the risk of earthquakes is high, structural monitoring may benefit from the use of non-destructive methods like acoustic emissions. When the shear strength of a material is surpassed, a critical diagonal fracture will occur. As the load continues to increase, the diagonal fracture will continue to expand until it is no longer adequate to induce failure. The load at which a diagonal crack emerges for the very first time is referred to as the diagonal crack load. The beginning of diagonal cracking must be located before one can conduct an analysis of the behavior of structural concrete elements when subjected to shear pressure. On the one hand, the occurrence of the diagonal fracture is a significant factor in determining the overall shear strength of the material (Børvik, 2013). However, the reserve shear strength factor is something that can only be determined with the assistance of the diagonal shear crack load. The reserve shear strength factor assesses the resistance to diagonal shear cracking in comparison to the ultimate shear force. When determining the reserve shear strength factor, it is very necessary to take into account the impact that concrete mixes have on shear resistance, such as the interlocking of the aggregate. The buffer zone that exists in each material between the point at which the first diagonal fracture begins and the point at which shear failure occurs is an important consideration. The reserve shear strength parameter was estimated in subsequent studies while examining the behavior of SRC beams with stirrups in the case of concrete with recycled materials and self-compacting concrete, even though the initial work was performed on high-strength concrete deep beams without stirrups (Jung, 2014). In other words, the reserve shear strength parameter was estimated while examining the behavior of SRC beams with stirrups in the case of concrete with recycled materials and self-compacting concrete (SCC).


The study of the design, construction, operation, and maintenance of a wide variety of buildings and systems, such as houses, bridges, and roads, is referred to as “civil engineering,” and the phrase “civil engineering” refers to the area of study that encompasses this topic. The architecture, engineering, and construction (AEC) sector is frequently criticized for its high labor intensity, poor efficiency, and significant environmental repercussions (Bontempi, 2001). This is despite the fact that this sector is extremely important to the economy. The construction industry is responsible for around nine percent of the world’s gross domestic product (GDP). According to the findings of yet another study, in the year 2017, the construction industry was responsible for approximately 20% of China’s total energy consumption, approximately 23% of the country’s total power consumption, and approximately 30% of the country’s total CO2 emissions (Bolander, 2000). Each of these factors had a significant impact on the environment.

As a direct consequence of this, there has been a surge in the number of initiatives that aim to improve the track record of the civil engineering business in terms of its effects on society, the economy, and the environment. Since the introduction and development of computer systems for structural design and analysis in the 20th century, optimization strategies based on mathematical programming techniques have been developed and utilized in the field of civil engineering in the recent decades (Uta, 2009).

The procedure of getting the very best outcome that can be accomplished is referred to as “optimization,” and the word “optimization” is used to characterize this process. When it comes to projects involving civil engineering, optimization may be carried out throughout any step of the process, beginning with planning and continuing through design, construction, operation, and maintenance. When it comes to making anything as efficient as possible, one of the most common approaches is called structural optimization (Bolourian, 2020). In the context of this study, the term “structural optimization” refers to an optimization strategy that pays no attention to the inherent characteristics of the chosen materials. Instead, the focus is placed on identifying the optimal configuration of structures or structural components in order to achieve a predetermined set of objectives within a specific set of constraints. The structures made by civil engineers rely heavily on the materials they utilize, since such materials are crucial to the functioning of the finished product. The building industry and civil engineering infrastructure both make substantial use of composite site materials that are based on concrete. These materials include plain concrete, reinforced concrete, pre-stressed concrete, and many more (Kaddah, 2019). However, due to the computational challenges involved in taking material distribution into consideration, structural optimization in civil engineering is often only researched for buildings comprised of a single material type. This is because of the nature of the field. The process of optimizing a structure may be segmented into four separate subfields, which are as follows:

Size optimization: also known as sizing optimization, which treats the cross‐sectional areas of structures or structural members as the design variables;

Shape optimization: also known as configuration optimization, which treats the nodal coordinates of structures as the design variables;

Topology optimization: focuses on how nodes or joints are connected and supported, aiming to delete unnecessary structural members to achieve the optimal design;

Multi‐objective optimization: simultaneously considers two or more of the above op‐ timization objectives for better optimization results; an optimization involving size, shape, and topology at the same time is also known as layout optimization.

In the beginning stages of structural optimization research within the domain of civil engineering, the primary focus is placed on mathematical theories and programming methods that are founded on fundamental structures as standards. Particularly, topology optimization has seen increased use as computational and construction techniques have advanced, allowing for the optimization of larger and more complicated civil engineering structures. This has led to an increase in the number of companies that offer topology optimization services. The Qatar National Convention Center (QNCC) in Doha, which is one of the largest civil engineering structures created by generative tools based on topology optimization, utilized topology optimization during the design process in order to minimize structural compliance (Zhou, 2016). The topology optimization was based on iterative 3D Extended Evolutionary Structural Optimization (EESO) algorithms. The Shenzhen CITIC Financial Centre, which can be found in Shenzhen, China, is a further illustration of the use of structural optimization to a large-scale civil engineering building project. Thanks to topology optimization-assisted design, the exo-skeleton truss arrangement was redesigned to be more efficient in its use of materials while preserving the structure’s overall stiffness.

One of the key objectives of structural optimization is to bring down the total cost of the construction. When it comes to construction projects, it is almost always desirable to meet, or even exceed, the criteria of structural performance at a lower cost (Robertson, 2007). According to a number of studies, cutting down on the amount of weight a building has may drastically cut down on construction expenses. In recent years, thanks to a growing focus on environmental issues and sustainable development, the substantial quantity of CO2 emissions produced by the civil engineering sector has made it an additional important goal of structural optimization to minimize the environmental consequences of the sector’s activities. In addition, a number of research publications on structural optimization center their attention on improving certain structural performances, such as mechanical behavior, aerodynamic performance, and dynamic seismic performance, in order to adapt structures to a variety of environments.

Many different optimization solutions have been proposed and put into action in an effort to accomplish the aforementioned aims. As a result of its applicability to combinatorial optimization problems, metaheuristic approaches have quickly risen to prominence as one of the most sought-after optimization strategies in the field of civil engineering structural optimization research. This is due to the fact that combinatorial optimization problems can be solved using these approaches. On the other hand, these metaheuristic techniques come with a number of downsides as well, such as their complexity and their inability for high-dimensional scenarios. As a direct consequence of this, there has been a rise in the amount of research conducted with the objective of improving the effectiveness of optimization strategies (Kumar, 2019). This research can take the form of either refining and expanding upon existing metaheuristic approaches or developing entirely new optimization strategies. For example, Mortazavi developed an extra fuzzy decision mechanism that increased the effectiveness of the interactive search algorithm (ISA) in the process of optimizing the structural size and topology. The fuzzy tuned interactive search algorithm, often known as FTISA, is a hybrid strategy that enhances solution accuracy while simultaneously reducing the amount of computation time required. For the purpose of truss size optimization, Degertekin suggested using not one, but two improved versions of the harmony search technique known as the efficient harmony search algorithm and the self-adaptive harmony search algorithm. Experimental data gathered from a variety of contexts demonstrates that the new approaches are superior than the traditional harmony search algorithm in terms of the amount of money spent on computation, the rate at which convergence is achieved, and the results of optimization (Koyama, 2012). In addition, the transformable triangular mesh (TTM) method offers an explicit topology optimization strategy for structural topology optimization. This approach beats other state-of-the-art algorithms in terms of its capacity to produce the best feasible solution.

Researchers in the field of structural optimization have demonstrated its potential and displayed its successes in the aforementioned works, which aim to improve the productivity and sustainability of civil engineering. These researchers have shown that structural optimization has a lot of potential. However, despite the fact that several research and survey reports were published in this sector, not one of them was able to offer a thorough review of the scientific achievements made on structural optimization (Mohan, 2018). As a result, the purpose of this paper is to conduct a comprehensive review of the literature on structural optimization in the field of civil engineering. This review will include an examination of the optimization objectives and their temporal and spatial trends, an examination of the optimization processes, which consist of four main steps, as well as discussions of research limitations and recommendations for future works.

A significant amount of study has been conducted on diagonal cracking in SRC beams that do not have stirrups. The section height, denoted by h, the concrete compression strength, denoted by fc, and the shear span to effective depth ratio, denoted by a/d, are the primary parameters that influence it. The effect that the ratio of shear span to shear depth has on the resistance of materials to diagonal shear cracking has been the subject of discussion in a number of studies. Cracking along the diagonal in high-strength SRC beams and the influence of early-age shrinkage on the beams (Talischi, 2010). The measurement of the beams’ strength when subjected to flexural loading is a rather basic problem; however, the analysis when shear loads are applied is much more complicated. There is a dearth of analytical explanations about the distribution of shear stress in the cross sections of composite materials. A certain amount of simplification of the problem and the experimental correlations is needed, and because of this, it is necessary to include certain assumptions. This is similar to the situation with the design code equations. A number of expressions that may be utilized to predict the beginning of diagonal cracking can be found in both ACI 318 and Eurocode 2, and both of these building codes were written in the United States (Negri, 2017). It is essential to highlight the fact that these codes are used in the design of genuine projects due to the straightforward and all-encompassing nature of their formulas.

The design of industrial facilities, the placement of equipment and machinery, and even the organization of furniture in professional settings are some of the most fundamental aspects that have a direct influence on productivity and, by extension, the success of a firm. Experts in manufacturing and logistics need to put in an incredible amount of effort in order to maintain their competitive edge (Holmberg, 2014). The layout is an essential part of production in this effort to make improvements, and it has a significant bearing on costs and the reduction of travel lengths, both of which contribute to greater operational efficiency. In this context, “betterment” refers to the process of making something better. The use of the material that involves “staying put” is the most effective usage of it. Ineffective layout design and material management are responsible for between twenty and fifty percent of all the costs associated with production. Finding an efficient answer to the problem of department placement is projected to result in a cut of between 10 and 30 percent in the costs of managing and running the business, in addition to an increase in the effectiveness of the production process.

Errors in the layout design can cause supply interruptions, which can then lead to internal and external customer dissatisfaction, production delays, which can then cause confusing and unnecessary queues and stocks, and high costs related to inefficiency in creating synergy between the physical arrangement set.

Manufacturers are rethinking their production methods in order to make room for newly developed manufacturing technologies or improvements to existing products. Because of this, the planning stage must take a significant amount of time before the final design modifications can be executed. A layout that is well-designed is supposed to help businesses save money by optimizing their use of space, streamlining the movement of workers and machines within the building, streamlining the management of day-to-day operations like task assignment and supply management, reducing the amount of downtime that occurs between processes, and streamlining the overall operation (Bird, 2018). All of these things should help streamline the overall operation.

In light of these positive aspects, the purpose of this article is to investigate the procedures and sources that are engaged in layout management in an attempt to get an understanding of the environment in which it operates and the rules that govern it. The purpose of this investigation is to carry out a comprehensive literature review of the procedures that have been effectively applied and investigated by manufacturing companies over the course of the last ten years (Rezaie, 2020). The findings are extremely restricted to instances of uneven size and single row layout; however, it is anticipated that the practical application of these characteristics will become even more widespread in the future, eventually encompassing a wider variety of scenarios. This is despite the fact that the findings only cover a small portion of the possible circumstances.

It is standard procedure to terminate a shear loading test as soon as a diagonal fracture becomes visible. Once the test has been terminated, the stress level at which the fracture first formed may be recorded. During a three-point flexural test, this may be accomplished by inserting linear crack gauges at the midspan in a direction that is perpendicular to the principal stress axes. The fact that these gauge types are so cumbersome, on the other hand, makes it potentially difficult to include more experimental devices. An alternative experimental strategy might be of interest if the goal is to avoid interrupting the shear cracking load prediction test in order to do a visual examination.

The well-known ‘ground structure’ based truss layout optimization method recently saw the addition of a true modeling of distributed self-weight as one of its optimization criteria. The addition of equally stressed catenaries into the ground structure helps to reduce the number of non-conservative errors that are caused by forgetting about the bending effects that occur inside members that are carrying their own weight (Wang, 2019). It is possible that the self-weight of a structure will play a substantial role in the process of carrying the imposed pressures; nevertheless, this may result in solutions that need components to be arranged in intricate, overlapping patterns. A redesigned formulation for layout optimization is offered here in order to tackle this challenge. This formulation allows for favorable unstressed masses like counterweights to be included in the final design. The equation takes into account the costs of abutments and anchorages, in addition to the frictional support system (Gaynor, 2013). We demonstrate that the method that was presented is efficient by using both benchmark examples and the conceptual design of a simplified long-span bridge construction. This design takes into consideration both the ground anchored and self-anchored choices for the building of the bridge.

At the moment, a significant amount of effort has been put into attempting to anticipate the form development of a fatigue fracture with a traditional numerical analytic approach. On the other hand, not a lot of work has been put into the majorization of propagation spots and increment for the sake of simulation accuracy. In this article, a fresh approach to the error computation is given with the purpose of quantifying the simulation findings and the experimental data. On the basis of this information, the impact of the distribution and amount of propagation sites on the outcomes of the simulation of crack propagation has been analyzed and optimized (Fontanesi, 2013). In addition, in order to get a greater crack development increment by using the calculus approach, a form of equivalent stress intensity factor amplitude expression is produced. This expression is based on the premise that the stress intensity factor increases exponentially with fracture depth. The simulation results are compared with already known expressions of the stress factor, which allows the validity to be checked.

Back in the late 1970s, a pioneering effort was made to optimize computer layouts, sometimes known as “topologies.” However, despite enormous advances in the amount of computer power that is accessible and the implementation of more effective optimization algorithms, even in the present day, the old “ground structure” technique can only be used to solve issues that are relatively small in scale. This is due to the need that the latter must, in general, include every possible member bringing together the nodes in an issue. We provide a solution approach that is not only straightforward but also successful, and it is capable of addressing issues involving enormous numbers of prospective members (such as more than one hundred million) (Prasad, 2003). Despite the fact that the approach is based on the linear programming methodology known as “column generation,” it is presented as an iterative “member addition” method since layout optimization-specific heuristics are used in the process. It is necessary for the first iteration of the procedure to employ a ground structure that has a limited amount of connection. After that, members are added in successive iterations as needed until the (provably optimum) solution is identified.

Limit State:

FORM is a powerful software application for engineers who want to design strong and light components and structures. It is particularly suitable for exploiting the design freedoms associated with additive manufacturing (AM). The software automatically identifies minimum-weight truss designs for specified material stress or deflection limits.

Limit State: FORM provides a number of advantages over other tools:

It can rapidly identify high fidelity optimized lattice design solutions. It produces parametric geometry output, rather than a mesh. Engineers can move quickly from defining the design domain, to an optimized form, and back again, in one fluid workflow. It is easy to refine the optimized design. Optimized forms are highly efficient and light weight – in fact they are very close to theoretical mathematical optima (Leonel, 2010).


There is no need to specify a target volume fraction – the software will automatically determine the optimum component geometry for any given set of input parameters. It includes a range of analysis tools, and models can be easily exported in standard formats for analysis in FEA tools.  The software is built on ANSYS SpaceClaim, a powerful direct modelling CAD package, which offers a rich suite of editing tools. Models can be analyzed in ANSYS Mechanical with a one-click process. The tool has been proven in projects in the aerospace, automotive, space and construction engineering sectors (Velasco‐Hogan, 2018). Using the conservative formulas of standard design codes is a safeguard against failure mechanisms. However, if a non-conventional concrete type that is not specified in the design codes is used or non-static or temperature loading is applied, then the mechanical behavior requires further investigation. Currently, a new generation of green concretes based on the use of industrial by-products is under development in the framework of the circular economy. Several studies have been carried out, in order to assess their behavior and applicability to building. However, research on their mechanical behavior at the structural or macro level is scarce. A key property that must be checked during the Ultimate Limit State Design of Steel Reinforced

Concrete (SRC) beams is resistance to shear failure or shear capacity. For instance, when the risk of earthquakes is high, it would be of interest to implement structural monitoring by means of non-destructive methods such as acoustic emissions. The first sign of shear failure is a critical diagonal crack, the width of which increases until the applied load reaches the ultimate shear strength. The diagonal crack load is referred to as the load at which the first diagonal crack occurs (Portioli, 2016). Determination of the onset of diagonal cracking in structural concrete elements is essential to analyze their behavior under shear loading. On the one hand, the diagonal crack plays a decisive role in the ultimate shear strength. On the other hand, the diagonal shear crack load is necessary to determine the reserve shear strength factor. The reserve shear strength factor is defined as the ratio of the ultimate shear load to the diagonal shear cracking resistance (Farahani, 2017). When analyzing the reserve shear strength factor, the effects of concrete mixtures on shear resistance, such as aggregate interlock, must be taken into account. The safety margin of each material between the first diagonal crack and the ultimate shear failure is a key parameter. Although the pioneering study of high-strength concrete deep beams without stirrups, the reserve shear strength parameter was computed later in other studies when analyzing the behavior of SRC beams with stirrups in the case of concrete with recycled aggregates and self-compacting concrete (SCC).

Identification of cracks

All kinds of engineering structures inevitably have cracks, holes and other defects due to the influence of many factors, such as construction quality, applied load and temperature change (Figure 1 shows some common defects in typical concrete structures). As a representative type of defect, the existence and evolution of cracks will lead to the decrease of structural bearing capacity, the degradation of serviceability and durability, and even lead to more serious consequences such as collapse, endangering the safety of life and property. Therefore, it is of great significance to detect and identify cracks in various structures quickly and accurately (Garcin, 2015).

Crack identification in structures is a typical inverse analysis problem to quantify crack geometries such as the location, size and type of cracks. At present, the most frequently adopted crack identification methods include manual detection technology, ultrasonic detection technology and image recognition technology, but these methods have some limitations. For example, manual detection technology is laborious and inefficient; ultrasonic detection technology will lead to inaccurate measurement under high temperature, and the image recognition method is easily affected by the environment (Dung, 2019). In recent years, the rapid development of numerical technologies and intelligent optimization (IO) schemes provides a new way for crack detection and identification. In the collaborative numerical modeling-IO approaches, the numerical methods are developed to perform forward crack simulation, upon which the required data is obtained and then used in the IO algorithm to further identify the crack configuration. In the following, we will give a brief review on the research status of representative numerical methods and IO schemes for crack modeling and identification.


Depending on a number of circumstances, including the local soil and material conditions and availability, as well as other site-specific considerations, a number of different types of foundations may be suitable for use with ground-anchored systems. These factors include: A gravity-type foundation is supported by a massive anchoring block that is positioned in such a way that the amount of frictional force that exists between its base and the ground is sufficient to sustain the horizontal stress. It is possible that the mass, in this configuration, may be made up of locally accessible and inexpensive materials like sand, which would represent a major cost reduction. The use of direct anchoring into the bedrock is one alternative design that, depending on the conditions of the site, may be acceptable.

The use of counterweights has allowed for the creation of innovative geometries for bridges with reduced spans. For example, the ingenious Alamillo Bridge designed by Calatrava does away with the need for back stays by hanging a cable net from an incline pylon. However, many experts in the subject have pointed out that the form that arises as a result of this situation is very wasteful from a structural standpoint. In the current body of optimization literature, there are very few resources that engineers may consult in order to develop efficient gravity-balanced structural forms. This is true whether the engineers choose to make use of distributed self-weight components or explicit lumped masses.

The literature that already exists on optimization seldom takes into consideration scenarios that use support types other than the usual fixed pin or pin/roller supports. frictional support types where the cost grows with the amount of the reaction force are examined, and different solutions based on these types are given. The cost of the frictional support types increases with the amount of the reaction force. These, on the other hand, need prior acquaintance with the fundamental structure of the optimal solution. In addition, self-weight, which may be employed in combination with an anchoring block to bear horizontal forces, was not taken into consideration. This was a significant omission from the analysis.

The costs of unstressed material, such as those found in anchoring or abutment structures, are included into the distributed self-weight method. This allows for the costs of unstressed material to be taken into consideration in conjunction with the impacts of the shifting boundary conditions. Because of this, it will be feasible to study the prospect of cost reductions in the self-anchored and ground-anchored cases, in addition to the more realistic circumstances in which friction will offer the sole form of restriction. It will be feasible to analyze different situations by making use of materials that are not strained, which will make it possible to include counterweights into the structural solution.

Layout optimization


The classical ground structure-based truss layout optimization procedure is shown diagrammatically below. When distributed self-weight is included, each straight-line connection between nodes is replaced by a pair of equal strength (i.e., equally stressed) catenary elements, one to carry compressive forces and the other tensile forces. However, the resulting problem formulation differs from the standard formulation only in the composition of the coefficient matrices such that linear programming can still be used to obtain solutions; thus, for a problem comprising n nodes and m potential elements the formulation can be written as:

Layout optimization stages: a problem specification; b design domain discretized with grid of nodes; c form of ground structure for a problem without self-weight—employing straight truss members connecting each pair of nodes; d resulting optimal solution; e ground structure for a problem with distributed self-weight—employing two equally stressed catenaries connecting each pair of nodes; f resulting optimal solution, comprising tensile members sagging downwards and compressive members arching upwards due to distributed self-weight.

Member with distributed self-weight: a geometry; b end force in the case of a single equally stressed catenary member. Dashed lines correspond to corresponding member without self-weight.

It should be noted that when self-weight is considered, members that would, in the non-self-weight formulation, have overlapped and been superfluous should now be explicitly included in the model. Such members are included in the ground structure shown, which shows curved elements spanning across two or three nodal divisions (e.g., along the top and bottom edges of the domain). It is evident that, although the end nodes of members may lie on the same straight line, the elements themselves are not coincident, and thus more than one element may exist in the optimal solution.

Finding a crack trajectory

A crack’s path has been mapped out over the course of the last three decades thanks to the application of the variational principle, which is a central tenet of the majority of the competing crack theories. Considerations of both energy and force may provide starting criteria for fracture propagation. A. A. Griffith proposed the concept of an energy fracture criterion in the year 1920, and G. R. Irwin produced the force criterion in the year 1957, concurrently establishing that the two criteria are equivalent. It has been shown that the limiting equilibrium state of a cracked continuous elastic body may be identified by making use of the Irwin force criteria for crack extension in conjunction with the corresponding Griffith energy criterion. [Citation needed] This formulation may be commonly recognized, but it is just one of several that may be used to calculate the limiting equilibrium state of a broken body. Other formulations include: The models created by Leonov and Panasyuk in 1959, Dugdale in 1960, Wells in 1961, Novozhilov in 1969, and McClintock in 1970 are among the most well-known examples of this type of model (1958). Finding the limiting load and the correlated crack propagation direction is a variational problem that is typically reduced to finding the extrema of a function that has multiple variables [9]. The crack propagation problem is formulated on a discrete level in this paper, as opposed to the variational approach that is typically taken, which is more common. The numerical computation is then carried out using a cutting-edge technique known as the Cell Approach, which is a revolutionary numerical method for solving field equations (CM). The absence of the requirement to establish a model for the treatment of the area right before the crack’s leading edge is one of the advantages of adopting a discrete formulation rather than a variational one. Other advantages include:

As an example, take the scenario in which the plastic deformation is confined to a certain area. When examining crack issues for an elastic-perfectly plastic body using the energy equilibrium criteria, a solution is frequently supplied immediately in front of the crack edge. This occurs just before the crack edge. The amount of distance that this area is able to cover is on the order of elastic displacements. In addition, if the elastic-plastic issue is changed into a purely elastic one by reducing the size of the plastic zone in front of the edge, the problem will be solved. This is feasible due to the fact that the action of the plastically deforming material is replaced in the linearized formulation by the action of applied forces along the face of an extra cut, which results in a narrower plastic zone. As a consequence of this, it is brought to the attention of the reader that the plastic non-linear effects region in the model that is being considered varies with the external load and represents a plastically deforming material in which the state of stress and strain must be determined from the solution of an elastic-plastic problem. This information is brought to the reader’s attention as a result of the fact that the model is being considered. In contrast, the discrete formulation does not make any assumptions about the size and geometry of the plastic zone. Instead, the computation is performed immediately, without first translating the issue into an equivalent elastic one.

Crack extension criterion

Numerous criteria, including as the maximum normal stress criterion, the maximal strain criterion, the minimal strain energy density fracture criterion, the maximal strain energy release rate criterion, and the damage law criteria, may be used to calculate the limiting load. The current work investigates the fracture extension condition in the Mohr-Coulomb plane. When calculating the limiting load, the condition of tangency between Mohr’s circle and the Leon limit surface is used (Fig. 2).

In the Mohr-Coulomb plane, the Leon criterion may be expressed as follows: where c represents the cohesion, f represents the compressive strength, tb represents the tensile strength, n represents the shear stress, and sn represents the normal stress on the orientation of the external normal n.

Because the CM domain discretization is of such significant relevance to the physical world, it is possible to calculate the Mohr circle for the tip neighborhood. The CM divides the domain into two parts by making use of two different cell complexes. The first cell complex is a simplicial complex, which means that each cell in that complex only contains a single node from the second cell complex. In order to construct the two meshes that are necessary for the two-dimensional domains, the study in question makes use of a Delaunay and Voronoi mesh generator. The first mesh, known as the Delaunay mesh, is produced when the domain is segmented into triangles. The perimeter of the triangle does not include any extra locations or places (Fig. 3). The Voronoi mesh is the outcome that you get when you take the polygons whose centers are the circumcenters of the main mesh and put them together (Fig. 3). Every point that is geographically close to a Voronoi site is in a position where it is physically closer to that site than it is to any other Voronoi site. Every main vertex has a dual polygon that corresponds to it, and this dual polygon is required to follow the conservation rule. We were able to find the Mohr’s circle that defines the tip neighborhood by inserting a hexagonal piece near the apex, which served as a marker (Ferretti in press, Fig. 3). In order for the mesh generator to function, the hexagonal element must first be sliced into two Delaunay triangles with equal sides. Next, a quasi-regular tip Voronoi cell must be constructed from these triangles (the cell filled in gray in Fig. 3). Because of this, we were able to calculate the relationship between the orientations of the Voronoi cells at the vertices of the tip and the stress field at the tip itself. The tension points provide an exact description of the Mohr’s circle in the Mohr-Coulomb plane with each rotation of the hexagonal element around the tip.

As a consequence of this, we are able to identify the direction of propagation by following the path of the line that starts at the tangent and terminates at Mohr’s pole (Fig. 2).



Topology optimization formulation considering material and construction costs

In order to begin the process of topology optimization, the first step is to formally formulate an optimization problem statement for the design issue. In this investigation, we use a method to topology optimization that is predicated on attaining the lowest possible compliance, which is analogous to achieving the highest possible stiffness. As a result of the fact that this formulation reduces the amount of external effort, or, equivalently, the amount of internal strain energy, it may be used in the construction of structures in which forces are directed along the load path that is the most rigid. The topology optimization process is then used to mesh the design domain; in the hybrid truss-continuum technique, the truss ground structure serves as a stand-in for the steel, while the continuum finite element discretization models the concrete (Figure 2). Calculating the volume percentage c of the continuum elements, where c = 1 indicates compression-carrying concrete and c = 0 indicates non-load-carrying concrete, and the cross-sectional area t of the truss components is the job of the optimizer. Therefore, the design of the steel reinforcement (the ties) is represented by truss elements with non-zero cross-sectional areas, and the path that the compression load will take is represented by continuum elements with a value of c equal to one. (the support beams)

Let’s describe the equilibrium condition as Ku F, where K is the global stiffness matrix that is determined by t and c in the design. In addition, let’s express the external work as F uT, where F are the applied nodal loads and u are the nodal displacements. This will give us the equilibrium condition. Therefore, the optimization problem for the lowest possible compliance (the maximum possible stiffness) is as follows:

where the second constraint is the total cost TC, which is composed of the material cost M and the construction cost C; the third set of constraints is the bounds on the design variables for the continuum elements in the domain (denoted as c), with cmin chosen as a small positive number to preserve the positive definiteness of the global stiffness matrix; and the fourth set of constraints is the bounds on the design variables for the truss elements in the domain (denoted as t). We make use of the formula in order to calculate the cost of the materials within the parameters of the total cost restriction.

where denotes the length of the steel bars and indicates the volume of the element in continuous concrete, and e c and e t are the unit costs of the concrete and steel, respectively. Constraints on material volume use such as this one are rather typical in the minimum compliance conundrum. Asadpoure et al. 23 recently introduced the construction cost to represent the manufacturing cost in discrete structures, and the formula that they used may be found as follows:

Any truss element that has a cross-sectional area that is more than zero is regarded as a needed element, and the function H, also known as the Heaviside step function, is the symbol used to symbolize this concept. Since the continuum elements are what represent the concrete domain, it is essential to notice that this function contains just the truss (steel) components. This is because the concrete domain is the one that is being represented. The term e f is shorthand for the amount of money that must be spent to construct element e. This element in truss constructions accounts for the labor expenditures associated with inserting a member (including the time required for the use of a crane) and establishing two connections (one at each end). Although the amount of the element construction cost will ultimately be determined by the local market and the building procedures used, the purpose of this article is to explain how this cost term may be leveraged to effect the buildability of rebar timescales. Because H is discrete, it has to be regularized before optimizers that are dependent on gradients may utilize it. Following is an explanation of the regularization function that should be used for projection approaches in continuum topology optimization. This function was first proposed by Guest et al. 24.

According to Guest et al. (25), the value of the regularization parameter dictates how well the step function is approximated. This work has the value set to 10. If a steel truss component achieves a non-zero cross-sectional area, this equation gives a magnitude of one, which puts the element’s unit cost onto the total cost function. This is because the equation yields a magnitude of one. When e e c s and 0 e f are both satisfied, it is clear that this total cost limitation is equivalent to the fundamental volume constraint that Yang et al. 20 utilized. In order to enable a comparison between the costs of materials and construction and to highlight the influence of the latter on optimizing the ideal position of reinforcing steel, the fixed and equal to unit values of e c and e s are employed. These values may be found in the equations. Therefore, if you change the value of e and f, the ratio of the cost of constructing to the cost of materials will change. We want to emphasize that the values of these components may be determined by utilizing actual market pricing for concrete, steel, and the cost of labor in the area for placing steel bars.

As a result of this, the Heaviside Projection Method, also known as HPM, is applied (Guest et al. 24, Guest 26) in order to get around the numerical instability of checkerboards and the mesh dependence of continuum components. The adjoint method is used to compute sensitivities (see Gaynor et al. 19 for related equations for the hybrid topology optimization), and the gradient-based optimizer, the Method of Moving Asymptotes (MMA) (Svanberg 27), is utilized because it is efficient for structural optimization. Both of these methods can be found in Svanberg. The algorithmic details for this technique is presented in detail in Guest et al. 25, which can be found here. Last but not least, we want to point out that the hybrid truss-continuum method is based on a bilinear, stress-dependent mechanics model in which the truss components (steel) carry only tension and the continuum elements (concrete) carry only compression. This model assumes that the truss components carry only tension and the continuum elements carry only compression. Concrete is assumed to have a Young’s modulus of 24.9 GPa (3600 ksi) in compression and 2.0 GPa (290 ksi) in tension in the numerical examples presented in this work. On the other hand, steel is assumed to have a modulus of 200 GPa (29000 ksi) in tension but no modulus at all in compression. This assumption was made for the sake of simplicity.

The first numerical example is the classic simply supported beam problem with a load in the middle of the beam, as shown in Fig. 3a. This problem is a simple example of a beam issue. Figure 3b exhibits a typical STM, whereas Figure 3c shows a topology-optimized solution that just takes into consideration the cost of the materials and ignores the cost of the building. We have allowed a fine structural topology to closely approach this and highlight the difference when assessing constructability because it is common knowledge that minimal strain energy topologies will mirror the primary stress trajectories. This was done in order to take advantage of the fact that this is the case. Naturally, simpler topologies might be achieved by making adjustments to the fundamental ground structure (see Gaynor et al. 19 for a discussion on this). The solution may be found in Figure 3d, which depicts the situation where the ratio of unit labor expenses to unit material costs is very large. This design not only has a lower bar count, but it is also more straightforward, and as a result, it would have a cheaper overall construction cost. Because these bars have a considerably larger cross-sectional area, the total cost of the materials is much greater than what is depicted in Fig. 3c. The solution shown in Figure 3e was reached as a result of placing a higher value on inclined rebar in contrast to horizontal and vertical rebar. Due to the fact that this was the situation, the algorithm came to the conclusion that it would not be appropriate to make use of inclined rebar, despite the fact that this specific design example would benefit from the latter’s greater structural efficiency. Keep in mind that two inclined bars were included in the final design because the price was not high enough to negate the structural efficiency of the inclined bars and the accompanying material cost savings that resulted from the usage of the inclined bars.

A standard example of another type is a deep beam that has openings, as can be seen in Figure 4a. Figure 4b illustrates a typical STM that has been constructed using only horizontal and vertical steel ties. Figure 4c illustrates an answer that has been topology-optimized while taking into account only the material cost. It has been noted that it is made up of a significant amount of steel rebar, which lessens the utility of the proposed STM. If you take a look at Fig. 4d, you’ll notice that the STM becomes a great deal easier to understand once you take into account the total cost of materials and labor. If the construction cost of these steel rebar is increased, as shown in Figure 4e, the resulting steel rebar will have a shallower angle of inclination. This will produce a different STM. These findings are preliminary, and they should be interpreted as such; however, they do demonstrate the potential of including labor cost in STM optimization. [Citation needed] A significant challenge is the estimation of labor costs, which are significantly impacted by the economies of individual regions. In this work, we simply express these costs as a ratio to the material costs in order to demonstrate the concept and investigate the tradeoffs that exist between the costs of material and constriction.

In this research, we describe an automated technique for constructing optimum strut-and-tie models in reinforced concrete structures using a performance-based evolutionary topology optimization approach. To demonstrate the utility of the suggested optimization approach, we have offered five examples that span many kinds of reinforced concrete components. Strut-and-tie models derived using the current optimization approach have been demonstrated to be consistent with both analytical solutions and experimental findings. Optimal strut-and-tie models in prestressed concrete structures and reinforced concrete shear walls may also be found using this approach. More research is required in both the theoretical and practical realms to make topology optimization a user-friendly, routine design tool for the concrete industry.


When designing any kind of industrial facility, it is crucial to take into account the complicated nature of the plan. This article presents a mathematical strategy for creating flexible industrial designs. The model considers the interdependence between the block and detailed layout problems. This cutting-edge model considers not only the dissimilar inputs and outputs of each part of the system, but also their dissimilar geometries. By determining where, in a two-dimensional (2D) space, all of the plant’s equipment and pipes may be placed to achieve the lowest feasible connection costs, an optimum plant design can be determined. A variety of topological factors are considered, including the position and orientation of the apparatus, its proximity to other components, the non-overlapping limitations of those components, the input/output connection, the shape, and the available space over a 2D continuous region. Here, production and operations are modeled simultaneously within the bounds of safety and operability.

The challenge of maximizing production within a continuous three-dimensional space has been investigated. The process of developing a broad model that incorporates key features of the existing reality. Rectangular and asymmetrical equipment shapes, input and output connections, equipment orientations, distance constraints, space availability, and allocations over many floors are only some of the topological features that have been represented. We analyzed two cases that included multi-tiered allocations. There are two possible strategies for arranging a building’s residential units: scattering them over an unspecified number of floors, each of which may have a different height, and scattering them across a specified number of levels with uniform heights throughout the building. The model also accounts for potential constraints on safety and operability, as well as the presence of production or operational segments. Ultimately, the model is distinguished by its mixed-integer linear programming (MILP) formulation. The optimal plant layout is guaranteed by solving this issue, which takes into consideration both the capital costs of connection and the operating expenditures, the latter of which are principally related with the costs of flow pumping. As such, the model’s primary objective was cost reduction. This ensures that the final plant structure is economical both initially and over time. Their responses to the numerous set-up problems displayed a wide range of design characteristics. With the use of computers, we were able to quickly determine the best course of action for fixing this sort of issue. While this approach may be time-consuming in more complex cases, it is still an alternative worth exploring. As a result, it’s feasible that future research may want to investigate whether or not modifying the problem’s features or using novel solution strategies can help to speed up the computation time.

Future Work

It is recommended that future theoretical work investigate the impact of material property on the best strut-and-tie models, while also limiting the impact of element mesh size and removal ratios. Furthermore, experimental research is required to learn about the ultimate load capacity of reinforced concrete members constructed using optimum strut-and-tie models created by the current topology optimization method. Results from the tests will be compared to industry standards.

This is a model for addressing design and layout concerns as a single, overarching difficulty at the same time. This formulation is very helpful for modeling design challenges that entail analyzing the particulars of a given plant, and it can be altered with very little effort on the user’s part. The proposed model for the layout problem in a 2D continuous area takes into account important topological aspects such as the orientation of the equipment, the availability of space, the diversity of the inputs and outputs, the diversity of the shapes (both rectangular and irregular), and the safety and operational constraints that are translated into distance constraints (minimum and maximum). The model generates a mixed integer formulation that, given a certain economic goal, zeroes down on the most suitable plant layout. In the context of this discussion, “minimization” refers to the reduction of the total cost of the connection structure as well as the decrease of the unit capital cost of the equipment. In conclusion, we are able to make the observation that a formulation that is rather generic is provided, and that the layout outcomes that are generated from this formulation reflect actual circumstances with a high degree of precision. In further investigations, we want to make advantage of the model’s potential by applying it to the planning of industrial multifunctional batch facilities at the same time as the process of design is being initiated.















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