## predicting the probable propagation – Global Homework Experts

Analysis of crack trajectories using layout optimization techniques

Dissertation

Student’s Name

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Abstract

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.

Contents

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

Chapter 1: Introduction

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 nondestructive 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).

Chapter 2: Literature Review

Optimization

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

Thispartjinccnsistentwithnresearchpurp.ae gndissertation.cndc

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 allencompassing 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,

contetotthispartneedtbemodfiedaccadingttthe.me 0th articce ,

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 welldesigned 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

The literature review in this part also has nothing to do with the theme of the article, which I emphasized many

times before, so these should be irrelevant. You should look for the results of applying Lo method to analyze

the fracture trajectory through the references I sent you. It can be paraphrase by literature review in these

references or summarized by the results of these articles.

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

The literature review in this part also has

nothing to do with the theme of the article,

which I emphasized many times before, so

these should be irrelevant. You should look

for the results of applying Lo method to

analyze the fracture trajectory through the

references I sent you. It can be rewritten by

literature review in these references or

summarized by the results of these articles.

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.

Chapter 3: Methodology

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

When you modify this part, you can refer to the article written by my tutor and other articles. You can analyze the

optimization processing of the structure by using layopt software or limitstate: form, and then use this method to analyze

the crack trajectory and apply similar data processing methods to modify it.

situations by making use of materials that are not strained,

which will make it possible to include counterweights into the

structural solution.

Layout optimization

Formulation

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:

For this part, we refer to some literatures to analyze the crack with layopt or limitstate: form.

After analyzing the structural optimization with these two software, we can analyze the crack

trajectory. This is probably done through tension and compression truss (I guess so).

So, if we can identify that direction quickly and efficiently from a Layout Optimisation (LO)

analysis(*) we can then position a “hole” in the design space, which represents the initial

direction of the crack. Then we can re-run the analysis with this “hole” and see how the

direction of the maximum tension changes. This way, by running the analysis several times, we

can see if the LO approach accurately predicts the direction of the crack as it grows. If it does,

then we may have a very useful new crack prediction method.

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.

This part is just like what you wrote, but it should be more specific. More examples

should be added, that is, the optimal solution can be generated by setting different

specifications and numbers of examples and data analysis. This part needs a large

number of numbers and pictures to describe, and then the fracture trajectory can be

analyzed in combination with pressure and tension.

lō

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 whole part is to use lo to analyze and predict the crack trajectory, that is,

Lo itself is to find the optimal solution of the structure. However, it is necessary

to find and predict the crack trajectory from this method. If feasible, this is a

new method to predict the crack trajectory. Modify it after understanding my

remarks above.

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 wellknown 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).

My understanding is to analyze the following situations by selecting the initial structure with

different number of components.

1. Crack trajectory analysis when the same load is applied to the same position of these different

structures.

2. Analysis of crack trajectory when different loads are applied at the same position of these

different structures.

3. Crack trajectory analysis when the same load is applied at different positions of these different

structures.

4. Crack trajectory analysis when different loads are applied at different positions of these

different structures.

5. Analysis of crack trajectory when different loads are applied at different positions of the same

structure

6. Analysis of crack trajectory when the same load is applied at different positions of the same

structure.

7. Analysis of crack trajectory when different loads are applied at the same position of the same

structure.

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,

After the above modification and

analysis, it can be changed into the

causes of cracks and how to solve

the problems of cracks, such as the

treatment methods when cracks

appear in different structures.

This part can be written but the number of words should be reduced, because the focus of this paper is not here

but the analysis part I mentioned above.

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.

Chapter 4: Results and Discussion

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

Finally, the remaining parts will be summarized, analyzed and

rewritten according to the above modifications

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.

Chapter 5: Conclusions and Recommendations

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 multitiered 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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