nonlinear analysis and structural optimization of aero ... · nonlinear analysis and structural...

Nonlinear Analysis and Structural Optimization of

Aero-engine Casings Bolted Flange Connections

by

Behzad Fazel

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright by Behzad Fazel 2016

ii

Nonlinear Analysis and Structural Optimization of Aero-engine

Casings Bolted Flange Connections

Behzad Fazel

Master of Applied Science

Department of Mechanical and Industrial Engineering

University of Toronto

2016

Abstract

Bolted flange connections are widely employed as a secure means to fastening structural

components in aero-engine casings. With the preload acting on the bolts, the structural response

behaviour of the bolted flange becomes unpredictable due to the presence of discontinuity and

contact nonlinearity at the joining interfaces of the casings. In this dissertation, a nonlinear

elasto-plastic finite element quasi-static analysis of the aero-engine casings assembly is

established considering the collective effects of the contact and elasto-plastic nonlinearities in

order to investigate the effects of bolt pretension and frictional contact on the deformation

response of the aero-engine casings bolted flange structure subjected to tensile, transverse and

torsional loadings. The contact pressure distributions at the flange interface for different loading

scenarios are identified and investigated. A metaheuristic surrogate-based multi-objective

optimization architecture framework is proposed to efficiently optimize strength and weight

performances considering bolted flange dimensions, bolt preload and frictional contact.

iii

Acknowledgments

Firstly, I would like to express my gratitude to my supervisor Professor Kamran Behdinan for

extending his knowledge, guidance and encouragement to me during the course of my master’s

study. Due to his excellent supervision, insightful comments, and gracious patience, I have been

able to expand my circle of knowledge throughout my graduate study.

I would also like to thank Pratt and Whitney Canada for providing sponsorship of my research.

In addition, I am grateful to NSERC, and University of Toronto for providing

financial assistance to support my research. Their contributions are highly appreciated. I would

like to give special thanks to Nima Bahrani whose support has been indispensable.

Lastly, I would like to thank my parents, and my sister for their continuous support and

encouragement.

iv

Table of Contents

Acknowledgments........................................................................................................................... iii

Table of Contents ............................................................................................................................ iv

List of Figures ................................................................................................................................. vi

List of Tables .................................................................................................................................. ix

List of Appendices ........................................................................................................................... x

Introduction ..............................................................................................................1 Chapter 1

1.1 Background and Motivation ................................................................................................1

1.2 Thesis Outline ......................................................................................................................3

Investigation of Nonlinear Structural Behaviour of Aero-engine Casings Chapter 2Bolted Flange Connections .........................................................................................................4

2.1 Overview ..............................................................................................................................4

2.2 Literature Review.................................................................................................................5

2.2.1 Nonlinear Structural Systems ..................................................................................5

2.2.2 Linearization Techniques in Nonlinear Structural Analysis ..................................12

2.2.3 Contact Formulation for Nonlinear Analysis.........................................................17

2.2.4 Frictional Contact Formulation ..............................................................................23

2.3 Nonlinear Finite Element Model Simulation .....................................................................26

2.3.1 Aero-engine Casings Model Geometry and Material Characteristics ...................27

2.3.2 Aero-engine Casings FE Modeling and Analysis Approach .................................28

2.4 Nonlinear Simulation Results and Discussion ...................................................................31

2.4.1 Nonlinear Response Subjected to Applied External Tensile Loading ...................31

2.4.2 Nonlinear Response Subjected to Applied External Transverse Loading .............38

2.4.3 Nonlinear Response Subjected to Applied External Torsional Loading ...............44

Structural Design Optimization of the Bolted Flange............................................50 Chapter 3

3.1 Overview ............................................................................................................................50

v

3.2 Literature Review...............................................................................................................51

3.2.1 Structural Design Optimization .............................................................................51

3.2.2 Multi-Objective Design Optimization ...................................................................56

3.2.3 Design of Experiments...........................................................................................58

3.2.4 Surrogate Model-Based Optimization ...................................................................61

3.3 Bolted Flange Connection Structural Optimization...........................................................65

3.3.1 CAD-Based Geometric Parametrization ................................................................66

3.3.2 Bolted Flange Geometry Model for Optimization .................................................67

3.3.3 Design of Experiments (DOE) ...............................................................................68

3.3.4 Surrogate Model Construction Methodology ........................................................69

3.3.5 Multi-Objective Structural Optimization of Bolted Flange ...................................69

3.4 Multi-Objective Structural Optimization Results ..............................................................72

3.4.1 Surrogate Model Response Surfaces .....................................................................72

3.4.2 Optimum Bolted Flange Design ............................................................................74

Conclusions and Future Work ...............................................................................76 Chapter 4

4.1 Concluding Remarks..........................................................................................................76

4.2 Future Work .......................................................................................................................78

Appendix A: Force Convergence Plots..........................................................................................84

Appendix B: Multi-Objective Genetic Algorithm (MOGA) Pseudocode .....................................86

vi

List of Figures

Figure 1 - Nonlinearities in structural systems ................................................................................6

Figure 2 - Nonlinear contact condition boundary ............................................................................7

Figure 3 - Geometric nonlinearity condition ...................................................................................8

Figure 4 - Nonlinear elasto-plastic behaviour spring model..........................................................10

Figure 5 - Kinematic hardening and Isotropic hardening model [14] ...........................................11

Figure 6 - Newton-Raphson approach ...........................................................................................14

Figure 7 - Modified Newton-Raphson approach ...........................................................................15

Figure 8 - Incremental Secant approach ........................................................................................17

Figure 9 - Contact between structure bodies..................................................................................18

Figure 10 - Lagrangian function ....................................................................................................21

Figure 11 - Regularized Coulomb friction model ..........................................................................24

Figure 12 - Nonlinear contact condition discretization..................................................................25

Figure 13 - Meshed geometry model of aero-engine casings and bolted flange connection.........27

Figure 14 - Finite element contact and target elements pair ..........................................................29

Figure 15 - External tensile loading boundary conditions and connection close-up section view 32

Figure 16 - Flange deformation response subjected to tensile loading for different bolt preloads32

Figure 17 - Bolt force response subjected to tensile loading .........................................................34

Figure 18 – (a) Flange interface gap distribution (b) Contact pressure distribution at ultimate

loading state ...................................................................................................................................35

vii

Figure 19 – Deformation response with varying bolt/nut-to-flange frictional contact coefficients

........................................................................................................................................................36

Figure 20 - Deformation response with varying flange-to-flange frictional contact coefficients .37

Figure 21 - External transverse loading boundary conditions .......................................................38

Figure 22 - Deformation response on tension side under transverse loading for different bolt

preloads ..........................................................................................................................................39

Figure 23 - Deformation response on compression side under transverse loading for different bolt

preloads ..........................................................................................................................................40

Figure 24 - The bolt load variation by location under transverse loading at ultimate loading state

........................................................................................................................................................41

Figure 25 - a) Flange interface contact pressure distribution contour b) Contact pressure variation

........................................................................................................................................................42

Figure 26 - Deformation response with varying bolt/nut-to-flange frictional contact coefficients

........................................................................................................................................................43


Figure 28 - External torsional loading boundary conditions .........................................................45

Figure 29 - Deformation response under torsional loading for different bolt preloads .................46

Figure 30 – Contact pressure distribution contour.........................................................................47


........................................................................................................................................................48


Figure 33 - Central Composite Design experimental design sampling space ...............................59

Figure 34 - Box-Behnken algorithm design of experiments..........................................................60

viii

Figure 35 - Traditional optimization vs Surrogate-based optimization .........................................61

Figure 36 - Single neuron architecture...........................................................................................64

Figure 37 - Multi- layer Artificial Neural Network architecture ....................................................65

Figure 38 - Bolted flange connection segment baseline design.....................................................67

Figure 39 - Multi-objective design optimization framework of the bolted flange connection ......71

Figure 40 - Kriging Surrogate model response surfaces................................................................72

Figure 41 - Local sensitivity plot for design variables ..................................................................74

Figure 42 - Bolted flange optimum design with specifications converged in Ansys ....................75

Figure 43 - External tensile loading case convergence plot...........................................................84

Figure 44 - External transverse loading case convergence plot.....................................................84

Figure 45 – External torsional loading case convergence plot ......................................................85

ix

List of Tables

Table 1 - Casings Mechanical Properties used in Nonlinear FE Model ........................................27

Table 2 - Bolted flange design parameters.....................................................................................68

Table 3 - Design variable bounds ..................................................................................................70

Table 4 - Multi-Objective Genetic Algorithm (MOGA) parameter specifications .......................70

Table 5 - Optimized bolted flange design specifications ...............................................................74

Table 6 - Optimum design comparison with baseline design ........................................................75

x

List of Appendices

Appendix A: Force Convergence Plots ........................................................................................74

Appendix B: MOGA Pseudocode.................................................................................................77

1

Chapter 1

Introduction

1.1 Background and Motivation

Bolted flange connections are broadly used as a secure fastening means to join structural

components in various mechanical and aerospace engineering applications. The mechanical

characteristics of such connections are rather complex particularly when external loadings are

present in the system. The characteristics and types of the external loadings acting on the aero-

engine casings throughout operation have a significant influence on the overall stress and

deformation behavior of the bolted flange connection. The deformation stiffness of the bolted

flange is known to have a large impact on the noise, vibration performance of the aero-engine

structure due to the inherent discontinuity present within the overall structure [1]. In essence, the

connection introduces nonlinearity in the structural response due to mechanical contact and

friction at the joining interfaces of both bolts and flanges [2] which in turn, gives rise to

uncertainties in structural behavior for the assembled aero-engine casings [3].

Bolts are tightened by applying torque to the bolt shank or nut. As the torque is applied to the

bolt shank, it is elongated which introduces a preload in the structure. In such cases preloaded

bolts are employed to ensure that slippage between the aero-engine casings is avoided. With the

preload acting on the bolts, the structural response behavior of the bolted flange becomes rather

unpredictable due to the presence of discontinuity and contact nonlinearity at the joining

interfaces of the casings [4] . Therefore, it is important to characterize the nonlinear phenomena

in aero-engine casing flange connections.

In essence, contact and friction stresses occur between the contacting interfaces of the bolts and

flanges. Hence, the contact area and the actual stiffness of the contact zone are unidentified prior

to nonlinear analysis. Even though the contact and frictional stress areas are much smaller

compared to the size of the model, the presence of varying stiffness due to the inherent

2

discontinuity and contact region of bolts and flanges causes nonlinear stress and deformation

behavior in the aero-engine casing assembly. Contact deformations between interfaces in bolted

flanges play a significant role in characterization of the structural response of the whole aero-

engine casings structure. Thus far, it has been known that contact deformations cause high

structural nonlinearities in the response of the system when such discontinuous connection

interfaces are present [5]. Since the aero-engine casings bolted flange structure may be subjected

to tensile, transverse and torsional loadings throughout operation the structural resistance of the

casings bolted flange structure in different loading conditions is imperative for the structure’s

vibration, strength and fatigue reliability [6] as the stiffness of the bolted flange essentially varies

depending upon the loading state [7]. It is important to note that the stress and strain in the bolts

may exceed elastic limit and undergo partial yielding during initial fastening and increase further

as the external loadings are exerted while the flanges almost entirely remain within the elastic

region [8], [9]. This gives rise to partial material nonlinearity due to strain-hardening in the bolts.

The traditional approach of the bolted flange structure design is based on elementary design

criterion and empirical analytical equations [4] since only basic structural integrity is considered.

Such methods address the structural strength and fastening capability to a certain extent [10];

however, many other important factors are not given ample attention to in the traditional design

approach, such as weight, stress, fatigue, etc. [11] The traditional approach is also ineffective in

the sense that it does not consider strength-to-weight ratio and cost. In addition, single objective

optimization can be utilized to enhance the performance of a single response but it fails to

explore and optimize other performance parameters. Although it determines the optimal design,

the design is only optimized with respect to a single objective meaning that other performances

are overlooked [12]. Hence, a framework to optimize multiple performances of the bolted flange

structure simultaneously is much demanded to efficiently perform the optimization process. In

order to consider multiple important structural parameters in the bolted flange, a multi-objective

framework is greatly needed to concurrently optimize multiple objectives and accomplish

optimum design by determining design parameters through a robust multi-objective optimization

architecture. So far, the weight and strength are known to be the two most important performance

criteria in the design of bolted flange structures in aero-engine casings.

3

1.2 Thesis Outline

This dissertation consists of four chapters on nonlinear investigation and structural optimization

of aero-engine casings bolted flange connections. The topic and contents of the chapters are

briefly outlined as follows:

Chapter 2 investigates and highlights the effect of bolt preloading, frictional contact of interfaces

along with the contact pressure and frictional stress conditions of the bolted flange interface

when subjected to tensile, transverse and torsional external loadings. In this study, a nonlinear

finite element simulation is used for investigating the nonlinear behaviour of aero-engine casings

bolted flange connection under external tensile, transverse and torsional loadings. The stiffness

variation trends throughout the loading states of the bolted flange connection are identified.

Chapter 3 presents an efficient surrogate-based multi-objective optimization framework proposed

for the bolted flange in aero-engine casings.

Finally, chapter 4 of this dissertation summarizes the findings and concluding remarks of the

nonlinear investigation and structural optimization of bolted flange connections in aero-engine

casings and recommends future work insights and directions.

4

Chapter 2

Investigation of Nonlinear Structural Behaviour of

Aero-engine Casings Bolted Flange Connections

2.1 Overview

Bolted flange connections are widely employed to join structural components in aero-engine

casings. With the preload acting on the bolts, the structural response behavior of the bolted

flange becomes unpredictable due to the presence of discontinuity and contact nonlinearity at the

joining interfaces of the casings. A nonlinear elasto-plastic finite element quasi-static analysis of

the aero-engine casings assembly is established considering the collective effects of the contact

and elasto-plastic nonlinearities in order to investigate the effects of bolt pretension and frictional

contact on the response of the aero-engine casings bolted flange structure subjected to tensile,

transverse and torsional loadings. An isotropic Coulomb frictional contact model is established to

impose nonlinear contact boundary conditions on the bolted flange in conjunction with a

multilinear isotropic hardening approach to simulate nonlinear plastic deformation. The influence

of bolt preload and frictional contact coefficient parameter variations on the nonlinear response

of the bolted flange is discussed. Contact pressure distributions are presented to identify local

stress concentrations at the bolted flange joining interface.

In this study, a nonlinear finite element simulation is used for investigating the nonlinear

behavior of aero-engine casings bolted flange connection under external tensile, transverse and

torsional loadings. The stiffness variation trends throughout the loading states of the bolted

flange connection are identified. This chapter of the dissertation investigates and highlights the

effect of bolt preloading, frictional contact of interfaces along with the contact pressure and

frictional stress conditions of the bolted flange interface when subjected to tensile, transverse and

torsional external loadings.

5

2.2 Literature Review

This section thoroughly presents the literature survey on the study performed on chapter 2 of this

thesis for characterization and classification of nonlinear structural behaviour in nonlinear

mechanical systems. In addition, structural nonlinearity sources in mechanical structures are

presented.

2.2.1 Nonlinear Structural Systems

In essence, nonlinear structural systems are defined as mechanical structures that exhibit a

nonlinear load-deformation or stress-strain relationship response. The structural nonlinearity

sources in nonlinear structures are mainly characterized as contact (changing-status) boundary

condition, geometric (large deformation) nonlinearity and material nonlinearity. A nonlinear

analysis is particularly essential in cases where large deflection results in abrupt changes in

structure’s geometry, permanent deformation of structure after removal of the external loadings

or changes in contact status boundary conditions that in turn, cause variations in the stiffness of a

mechanical structure.

Nonlinearity due to contact status occurs if a nonlinear relation between the external loadings

and the boundary deformation is present within the structural system. As for contact nonlinearity,

a nonlinear analysis is imperative since the contact interaction status and the contact pressures

acting on the interfaces are essentially unknown [13]. Geometric nonlinearities that stem from

large displacements in structures are of great importance as differentiation between the initial and

ultimate states of deformation is required to characterize its structural behaviour. The structure

can also undergo material nonlinearities along with time-dependence or time-independence in

nonlinear behaviour. The nonlinear equations governing the nonlinear behaviour of a structure

can be expressed in an incremental form as follows:

(u) ( ) (u)K d F u K u F (2.1)

where, ∆u and ∆F denote the unknown incremental deformation vector and the known

incremental loading vector, respectively. Different approaches to solution of such structural

systems involve numerical iterative linearization in order to approximate an exact solution. The

6

size and direction of each linearization step can involve numerous iterations depending upon the

nonlinearity type and complexity. The schematic architecture of nonlinearities in structural

problems is presented in Figure 1.

Figure 1 - Nonlinearities in structural systems

As mentioned earlier, several phenomena in mechanical structures exhibit nonlinear behaviour

and linear structural systems are effectively only an approximation of nonlinear systems

subjected to special circumstances. A nonlinear analysis is essential when one or more of

nonlinearities are present within the structural system as a linear analysis would not characterize

the actual structural response. It is important to note that nonlinear structural analysis is relatively

more computationally expensive in compare to a linear structural analysis due to its iterative

approach and convergence.

2.2.1.1 Contact Boundary Condition Nonlinearities

When two structural component bodies collide, contact takes place at their surface interfaces

such that they would not join in a continuous manner. Contact boundary conditions are

imperative to consider in the design of structural systems that involve joining multiple

components in order to compose an assembled larger mechanical system. Contact is effectively

the means to joining components which often involve fasteners, bolts, welds, adhesives, etc. It is

notable that the goal of a contact analysis is to determine the boundaries or regions of contact,

contact stresses acting on the interface and relative motion between interfaces.

7

Contact nonlinearities can be illustrated in two aspects. Primarily, if two separate components are

in contact, the contact load versus deformation plot would resemble a cliff since there is no

contact load acting when two components are away from each other and increases vertically after

the components have contact. In a contact nonlinearity condition, the functional relation is rather

unknown since one-to-one relationship between contact load and deformation does not exist. An

analogous phenomenon occurs for the tangential direction subjected to friction when two

components are sticking together until the tangential load reaches a certain amount, after which

sliding happens without further increase of the tangential load. The rapid change in contact load

along with slippage at interface renders the structural response to exhibit strong nonlinearity

[14]. Nonlinear contact analysis consists of determining contact boundaries and regions between

the interfaces as well as the resulting contact stress distributions at the interface. As depicted in

Figure 2, as the contact load at the interface influences the deformation of nearby regions, this

process requires repetition in order to obtain the contact status of different locations at the

contact interface. Therefore, contact nonlinearity is analyzed in an algorithmic fashion.

Figure 2 - Nonlinear contact condition boundary

For a flexible structural system, equilibrium state can be described as determining a displacement

to minimize potential energy. Thus, contact can essentially be viewed as a constraint in an

optimization problem where the potential energy is minimized while contact condition constraint

is satisfied. This implies that the bodies cannot overlap and penetrate one another. The

constrained optimization problem can then be transformed into an unconstrained optimization

8

problem through employing penalty method or Lagrangian multiplier methods which will be

discussed in detail later in this chapter.

2.2.1.2 Geometric Nonlinearities

Geometric nonlinearities occur when the relationship between kinematic parameters such as

deformation and strains are nonlinear. These nonlinearities generally occur when deformation is

remarkably large. An example of geometric nonlinearity is when an external moment is acting at

the end of a cantilever beam. Since the moment causes a very large rotation in the structural

system, a linear analysis fails to predict the deformation accurately as shown in Figure 3. Under

the large deformation caused by the external moment the beam would see a different geometry

which implies a variation in stiffness of the system.

Figure 3 - Geometric nonlinearity condition

The equation of a large deflection beam can be expressed by the following equation:

2

22 2

2 2

10

2

10

2

d du dwEA q

dx dx dx

d d w d dw du dwEI EA f

dx dx dx dx dx dx

(2.2)

In general, the deformation in such structural systems significantly vary the location and

distribution of loads implying that in order to characterize the structural response, equilibrium

9

condition must be expressed in an integral form with respect to the deformed geometry which is

unknown priori. The general integration form is presented by the following equation:

( , ) ( ) : ( ) da

u u u u (2.3)

Where, is stress, is the strain, u is deformation and is the unknown deformation domain

of integration. Geometric nonlinearities arise from the presence of large strain but finite

displacements and/or rotations, and loss of structural stability.

2.2.1.3 Elasto-plastic Material Nonlinearities

Material nonlinearity occurs when the relationship between stress and strain is nonlinear. The

nonlinear relationship is often termed as the constitutive relation. In linear structural systems, this

relationship is given by the following equation:

E (2.4)

where [E] is the modulus of elasticity matrix. As [E] is a constant term, the relationship between

the stress and strain is linearly proportional to one another. The aforementioned relationship is

representative of the behaviour of elastic materials under an infinitesimal deformation. In other

words, a material is termed nonlinear when the stress and strain relationship cannot be expressed

by a constant matrix. In such case, the modulus of elasticity matrix is dependent upon on the

material deformation state.

Plastic yielding behaviour of materials is associated with material nonlinearities. A common

behaviour of materials is elastic deformation up to a certain point called yield point where any

deformation beyond that limit exhibits irreversible plastic deformation. Figure 4 represents the

plastic behaviour with a spring and a friction instrument. The friction device would only slip

after the stress reaches the yield strength value, σy. The system can be described as follows:

when the stress is below the yield strength, the displacement increases linearly whereas when the

stress exceeds yield, the displacement would increase without raising the stress further meaning

10

that the device would not support any stress above the yield strength point. The displacement is

decreased with the same modulus of elasticity slop after removal of the load.

Figure 4 - Nonlinear elasto-plastic behaviour spring model

Thus, plastic deformation remains in the system after removing the applied loading. This implies

that the stress and strain relationship is dependent upon deformation history. In general, the stress

and strain relationship is represented by stress rate versus strain rate. Materials such as steels or

aluminum alloys, exhibit plastic deformation as the loading exerts stresses beyond the yield point

limit. Since such materials initially undergo elastic deformation and subsequently undergo plastic

deformation beyond a certain threshold [14]. The nonlinear behaviour of such materials is called

elasto-plastic nonlinear behaviour. Since the relationship is expressed by rate, stress state can

only be determined through integration of the stress rate over the previous loading history.

Hence, stress state determination is effectively path dependent. As mentioned earlier, elasto-

plasticity occurs when a material experiences both elastic and plastic deformation. It is also

important to note that in an elasto-plastic analysis elastic and plastic strains are treated separately

from the total strain. Stress is derived after elastic strain is obtained meaning that plastic strain

does not affect the stress.

11

Figure 5 - Kinematic hardening and Isotropic hardening model [14]

Modeling the behaviour of a structural system depends on the objective of the structural analysis.

For instance, if the goal of analysis is to characterize the behaviour upon fracture, it is required to

model all the stress and strain responses but when the goal is to simulate the structural system’s

response subjected to loading without fracture, the material behaviour can be simplified by

considering only the elastic and strain-hardening terms. Figure 5 also represents the elasto-plastic

stress and strain behaviour from a uniaxial tensile test. When an external tensile loading is

applied, the structural behaviour initially exhibits elastic deformation up to reaching the yield

stress, σy. The modulus of elasticity is the slope of the line which is denoted by E. When the

applied external loading is removed, the stress and strain relationship pursues the same curve.

Subsequent to yielding, the plastic portion initiates and the stress increases with the slope of Et

which is called the tangent modulus. Two approaches of modeling elasto-plastic material

behaviour known as the kinematic and the isotropic hardening are schematically indicated in

Figure 5. The kinematic hardening model considers a constant elastic range where the center of

the elastic portion moves parallel to the strain-hardening line as indicated by the dashed line

through the origin. In the isotropic hardening model, the yield stress magnitude for the unloading

is equivalent to the previous yield stress value [15]. Therefore, the elastic range varies and

increases in this model in compare to that of the kinematic hardening model.

12

As for nonlinear behaviour of bolted flange, the accurate prediction of the mechanical response

of bolted flange connections in aero-engines has attracted major interest in the recent years due

to the nonlinear structural behavior observed in the aero-engine casings structural response [16],

[17]. Previous research work has been performed in such connections with analytical and

numerical linear models in which the flanges are joined through a rigid connection between the

interfaces. The linear approach is simple to implement which is why such models are widely

adopted in the industry [18]. However, such models result in a much stiffer connection and fail to

accurately predict the bolted flange structural response as the rigid connection tends to

considerably differ from the realistic nonlinear deformation response of the system [2]. Chavan

[19] performed a linear finite element simulation for a preloaded bolted joint connection and

obtained the linear model force-deformation response subjected to external tensile loading. A

more sophisticated approach proposed by Luan et al. [20] to model the bolted flange considering

the flexibility of the bolts by joining the contact interfaces with nonlinear springs resulting in a

less stiff model than the linear model discussed earlier. Experimental work has been performed

by Van-long et al. [21] to study the behavior of bolted flange joints in structures under

monotonic tensile loading. In addition, Wang et al. [22] suggested that the load-deformation

behavior of a bolted joint connection can be utilized to predict the bolted structural connection’s

mechanical characteristics such as strength and stiffness.

2.2.2 Linearization Techniques in Nonlinear Structural Analysis

2.2.2.1 Newton-Raphson Method

This method is widely utilized in numerical methods to determine nonlinear equation roots. In

general, the majority of numerical analysis techniques for solving nonlinear problems consider a

primary estimate, u0, and compute the increment, Δu, such that the new incremented estimate u0

+Δu, approaches the solution of the nonlinear equation. The nonlinear problem equations are

estimated locally by linearized equations to obtain the increment. The procedure is iteratively

performed so that the nonlinear equations are fully satisfied. Consider an estimate solution is

obtained and denoted by ui. Then, the solution at the subsequent iteration is estimated through a

first-order Taylor’s series given by the following:

13

1( ) ( ) ( )i i i i i

TP u P u K u u f (2.5)

Where iii

T uPuK )()( is known as the tangent stiffness matrix and Δui represents the

increment. The objective here is to compute the increment and replace the solution through

iterations. Subsequently, the terms are rearranged and the linearized equation is identified by the

following:

( )i i

TK u f P u (2.6)

This equation is analogous to the linear systems however, it differs from them in the sense that

the coefficient matrix varies and the equations are solved through iterative incrementing process

[14]. In addition, it is notable that the force is expressed as the subtraction of applied external

load and internal force which is called the residual term of the linearized system of equations.

Subsequent to determination of the increment, a newer estimate solution is found that is given by

the following:

1i i iu u u (2.7)

However, the obtained solution would not be precisely satisfied due to presence of the residual or

unbalance loads term as presented in the following relation:

1 1( )i iR f P u (2.8)

A number of criteria can be utilized to achieve solution convergence. For instance, one criterion

is associated with the increment of solution. A solution is considered to be converged in the case

where the solution increment is less than that of the initial. The convergence criterion is then as

follows:

1 2

1

0 2

1

( )

1 ( )

n i

jj

n

jj

uconv

u

(2.9)

This convergence criterion is reasonable where convergence is occurring at a stagnant rate. As

loads are essentially differentiated from deformations in structural systems, it is more convenient

14

to have deformation convergence than force convergence. Various finite element packages

compute force and deformation convergences to detect if convergence has taken place.

Moreover, the maximum value could also be utilized in place of the presented sum squares

convergence criterion. A schematic representation of the Newton-Raphson linearization method

for nonlinear analysis is indicated in Figure 6.

Figure 6 - Newton-Raphson approach

In brief, the Newton-Raphson approach is systematically performed by the following steps [23]:

1. Designate tolerance value, first estimate and maximum number of iterations.

2. Determine Newton-Raphson residual force.

3. Compute convergence or termination criterion term to proceed further.

4. If iteration number exceeds the specified maximum iteration number, abort process.

5. Obtain the tangent stiffness matrix.

6. If the tangent stiffness matrix determinant is not invertible, abort process.

7. Compute increment term.

8. Update force or deformation solution with the computed increment.

9. Revisit the second for subsequent iterations.

2.2.2.2 Modified Newton-Raphson Method

The Newton-Raphson approach necessitates the formation of the tangent stiffness matrix and the

linearized system must be solved for the solution increment which is often computationally

15

expensive. From a finite element standpoint, constructing and solving the matrix equation have a

high computational cost associated with them. The goal of the modified Newton-Raphson

approach is to reduce the computational cost of linearizing and solving such nonlinear equations.

In the modified Newton-Raphson approach the initial Jacobian matrix is iteratively utilized for

each iteration in substitution for constructing a new Jacobian matrix which circumvents the

necessity to recalculate the Jacobian matrix for iterations [23]. It is beneficial as the

computational cost for solving the nonlinear system is effectively decreased. As with the

modified Newton-Raphson a LU factorization is required to solve the system. Although the LU

factorization process is itself time consuming the substitutions of LU factorization are relatively

convenient to perform.

Through the modified Newton-Raphson approach the decomposed matrix is essentially

preserved and the substitutions are employed with different residual values. Figure 7 indicates a

schematic plot representation of the modified Newton-Raphson approach. As it can be noted

from the plot the method often needs more iterations before convergence is achieved in compare

to the Newton-Raphson method but it is computationally more efficient since each iteration is

evaluated with less time than the Newton-Raphson approach [14]. Furthermore, the modified

Newton-Raphson approach offers reasonable stability with lower chance of divergence.

Figure 7 - Modified Newton-Raphson approach

16

2.2.2.3 Incremental Secant Method

In the incremental secant approach the formation and solving the Jacobian matrix are omitted in

order to decrease computational cost. The goal of the tangent stiffness matrix is to accelerate

convergence, while the residual keeps track of the accuracy [13]. Iterations continue up until the

residual decreases meaning that the equations are satisfied throughout the tolerance bound. The

notion of this approach is to estimate the Jacobian matrix with higher computational efficiency.

This can be accomplished through continuous updating of the matrix utilizing the secant

direction between successive solutions. The key notion of the incremental secant approach can

be exemplified by a single variable. The Jacobian matrix is represented by differentiated

nonlinear term, P, with regards to u. The secant matrix is then computed with a finite difference

approach to find an approximation to the Jacobian matrix as it is presented in the following

equation:

1

1

( ) ( )i ii

s i i

P u P uK

u u

(2.10)

It is important to note that as the denominator of the finite difference equation approaches zero

the Jacobian matrix the Jacobian matrix of the Newton-Raphson method. In addition, for the

initial iteration, the secant approach utilizes the identical Jacobian matrix of the Newton-Raphson

approach. Thereafter, the secant stiffness is employed in the next iterations as opposed to the

tangent stiffness. Figure 8 indicates a schematic plot representation of the incremental secant

approach. The solution increment is presented by the following equation:

))(()()( 1

1i

ii

iii uPf

uPuP

uuu

17

Figure 8 - Incremental Secant approach

The convergence rate for the incremental secant approach is 1.618 that is also known as the

golden ratio. Such strategies in solving nonlinear problems that attempt to evaluate an

approximation to the Jacobian matrix are known as quasi-Newton approaches [23]. Iterations of

such methods are compute faster as they do not require the computation of Jacobian matrices but

they are notorious for slow convergence rate than regular Newton-Raphson method which has

quadratic convergence rate meaning that it offers a considerably faster convergence.

2.2.3 Contact Formulation for Nonlinear Analysis

Contact boundary condition constraint methods and formulations are thoroughly discussed in

this section. A mathematical model of contact boundary condition needs to be defined in order

for simulation of contact conditions with finite element analysis. From a finite element model

standpoint, two main contact formulations are the penalty method and the Lagrange multiplier

method. Other contact formulation methods are generally variants of these methods that are

employed for contact analysis.

A contact problem can be generally characterized as an optimization problem such that the total

internal energy due to elongation of body and the external energy due to external loading on the

structure are to be minimized [24]. A constraint is imposed on the total energy equation of the

structure through designating contact surfaces so that the energy within the system is infinity if

penetration is to ensue. This essentially declines the smoothness of the energy equation. The

18

general form of energy constrained minimization can be expressed by the following:

Minimize ( ), :

( ), :

( ) 0, :

n

n m

n m

f x f

Subjected to h x h

g x h

(2.11)

Where f and h represent the soft constraints and contact constraints, respectively and is the

Euclidean space. The general formulation is elaborated with reference to contact between two

bodies as indicated schematically in Figure 9. In any contact problem, the bodies are considered

to be adequately supported in order to avoid rigid body motion. In this case, the contact is

considered to be between a flexible and a rigid body. In the context of contact mechanics, the

body that has higher flexibility is assumed to be the slave component and the stiffer and more

rigid is assumed to be the master component.

Figure 9 - Contact between structure bodies

Contact can be classified under normal impenetrability and tangential slippage contributions. In

essence, the normal impenetrability condition precludes the slave component from penetration to

the master component, whereas the tangential signifies the frictional contact behaviour at contact

interface. A portion of the slave surface is represented with contact boundary, Γc and a portion of

the master surface is represented with contact boundary, ξ. In this scenario, the contact condition

is characterized with regards to a point x located on the slave boundary and point xc located on

the master boundary. It is notable that the contact point on the master boundary is unknown.

Moreover, in three-dimensional cases two points are necessary to designate the master boundary.

19

One of the objectives in solving contact problems is to determine contact coordinates and contact

loads present at the contact interfaces which involves contact pressure computation. In contact

problems, often times an external known force or deformation is applied to the structural system

and the contact boundary is then unknown which is to be solved. In other words, the contact

statuses of interfaces are unknown prior to the analysis. Determination of the contact coordinates

on the master component corresponding to the slave component is the initial step for analyzing

contact in order to detect contact status of the coordinates in contact which is also known as the

orthogonal projection. For a typical nonlinear contact case, the following equation that is known

as the contact consistency condition is to be solved in order to calculate the contact coordinate:

( ) ( ( )) ( ) 0T

c c c t cx x e (2.12)

where et=t/||t|| denotes the normalized tangential term. After the contact coordinate is determined,

it is required to detect contact condition, which is carried out through evaluation of the length

between the contact coordinates. The normal contact contribution term here is described by

means of the normal gap function that evaluates the normal distance between contact coordinates

which is given by the following equation:

( ( )) ( ) 0,T

n c c t c cg x x e x (2.13)

where en(ξc) denotes the norm of the normal vector on the master boundary of the contact

coordinate. As for the frictional contact contribution, when the contact coordinate travels along

the master surface, a friction load in the lateral direction to the master surface opposes the

relative tangential motion. The tangential slippage term evaluates the relative motion of contact

coordinate on the master surface which is described by the following:

0 0( )t c cg t (2.14)

where t0 and 0

c represent the tangential vector and natural point at the solved increments,

respectively.

20

2.2.3.1 Penalty Method

As mentioned earlier, contact is characterized as a constrained optimization. The penalty method

transforms the constrained optimization problem into an unconstrained optimization problem for

convenience. The penalty method approach penalizes the potential energy proportional to the

constraint violation in a way that the minimum value of the penalized energy virtually satisfies

the contact condition. Note that energy penalization takes place as soon as penetration takes

place. The contact penalization equation for the penetration area is given by the following:

2 21 1

2 2c c

n n t tP g d g d

(2.15)

where ωn and ωt denote the penalty parameters for normal and tangential terms, respectively and

g represents the gap function. The penalty term in this equation results in a penalty method where

the solution arises from the infeasible area. This implies that the normal penetration constraint

would be violated; however the degree of violation declines when the penalty term grows.

Therefore, the constrained contact optimization problem is transformed into an unconstrained

one through augmenting a penalty function to the energy which yields the following:

( ) min (w) min (w) P(w)w w Z

u

(2.16)

Finally, it should be noted that this method is simpler to implement than other methods for

solving contact problems however, it is known to be associated with ill-conditioning when

implemented in some cases.

2.2.3.2 Lagrange Multiplier Method

The Lagrange multiplier method adds to the energy through a product operation of the contact

constraint and Lagrange multiplier that is associated with force in order to set the contact

21

constraint, in a way that the energy minimum satisfies the constraint along with characterizing

the contact load. The Lagrange function is described by the following (refer to Figure 10):

Figure 10 - Lagrangian function

( , ) (x) g(x), (x, ) n mL u f (2.17)

where is the Euclidean space. The optimization problem is obtained by the following:

*

0

( ) ( ) 0( ) maxL(x, )

( ) 0

f x if g xL x

g x

(2.18)

L is at a maximum if λ is zero. If g is negative then the Lagrange function would be increased.

Therefore, the min-max optimization problem becomes the following:

*minimize ( , ) minimize ( )x X x X

L x f x

(2.19)

In order for the solution to be unique the optimal point is needed to be the saddle point of the

Lagrange function. For this minimization problem the λ is the unknown that is to be computed.

The equations are presented in the following system:

( ) ( ) 0

( ) 0 0

xx x

x

f x g x x

g x

(2.20)

22

The derivation for the Lagrange multiplier contact formulation is presented. Then, c is

presented by the following equation:

( ) ( )c c

c n c n cg x d g x d

(2.21)

This method is advantageous in many contact applications as it is independent of defining

contact stiffness and provides remarkable accuracy [24]. Some major drawbacks of this method

include use of excess degrees of freedom for contact constraints as well as zero diagonal that

makes it unsuitable for iterative solving [25]. This method is also known to have over constraint

occurrence in some cases.

2.2.3.3 Augmented Lagrange Method

The Augmented Lagrange method combines the penalty method and Lagrange method in order

to take advantage of simplicity of penalty method as well as the great accuracy of Lagrange

method. This method was first discussed by Hestenes [26] and Powell [27] . The Augmented

Lagrange method requires an initial penalty solution to be carried out. The total contact loads

are exerted as external loads and the penalty method solves the contact condition. The process is

performed until the gap diminishes to a marginal amount. This method is quite beneficial as it

can be utilized for complex multibody contact problems that also allows for desirable negligible

penetration. It often requires a smaller number of iterations to linearize and converge in compare

to the Lagrange method. For normal contact condition, the method is presented by the following

equation:

21( )

2

i i i

c n n

i I

g g

(2.22)

where denotes the normal contact stiffness. It can be observed from the equation that when

is zero the formulation represents the penalty method. Note that the force and stiffness matrices

are adjusted similar to the penalty method with exception of adjustment of force as the Lagrange

method requires. As the solution gets closer to convergence, the penalty term reduces to a

negligible amount. The equation is generalized for finite element formulation by the following:

23

c n n ng g (2.23)

It should be noted that one of the important differences between the penalty method and

Augmented Lagrange method is that the latter allows for determination of contact pressure at

interface meaning that the extra term is essentially insensitive to the value of contact stiffness.

2.2.4 Frictional Contact Formulation

In the presence of friction, the solution is dependent upon the load history acting on the structural

system. The load sequence plays a key role in the sense that frictional load is computed using

both the present and past location of contact coordinate. Therefore, the frictional contact

behaviour is analyzed with respect to the increments of loads.

The Coulomb friction model is widely used to model friction in nonlinear contact finite element

analysis. Since the relation between friction force and slip is discontinuous, it is an arduous

process to perform in terms of iterative solving with the Newton-Raphson method that considers

the solution to be continuous. Alternatively, the Coulomb friction model can be regularized so

that the vertical segment of the Coulomb friction model has a slope associated with it [28] as

indicated in Figure 11. Thus, the tangential slippage is then formulated as follows:

0 T

t c tg t vu e (2.24)

where v denotes the curvature scale (v is 1 for a straight linear contact boundary). The tangential

slippage is then determined with respect to displacement variation as presented in the following:

(u,u) u

c

T

T t t tb vg e d

(2.25)

24

Figure 11 - Regularized Coulomb friction model

where t represents the penalty parameter. As indicated in the plot, the frictional traction load

t tg increases with the tangential slippage. The frictional load varies within a range that is

determined by the normal load multiplication with frictional contact coefficient in the Coulomb

model. As for micro slip case traction load increases with the tangential slip [14]. The stick

condition happens in cases that the following inequality relation is satisfied:

t t t tg g (2.26)

where μ denotes friction coefficient. As for a slippage condition, the contact variational

expression is adjusted. Consequently, a piecewise function is described to represent the frictional

contact conditions as presented in the following equation:

u

(u, u)sgn( ) u

c

c

T

t t t t t t t

TT

n t t t

vg e d if g g

bg vg e d otherwise

(2.27)

25

2.2.4.1 Contact Formulation of Flexible and Rigid Structures

A simpler form of contact problems arises when a flexible structure has contact with a rigid

structure. As mentioned, the flexible structure is usually the apparent choice as the slave

structure while the rigid structure is selected as the master structure in the concept of master-

slave contact formulation. The logic behind this selection scheme is that the flexible structure

incapable of penetrating the rigid structure due to the choice of master and slave [14]. It is also

often practice to describe the master structure boundary as opposed to the whole master structure.

Figure 12 represents the contact discretization of slave and master structures where Гc denotes a

section of the flexible slave boundary that can penetrate the master structure.

Figure 12 - Nonlinear contact condition discretization

In the context of a discretized system, the contact boundary is shown as slave nodes that

penetrate the master structure. The figure indicates one slave node discretized model. Many

methods of imposing contact constraints are available in finite element formulations. In this case,

the contact constraint is described through a contact pair which consists of a master boundary

and a slave node. Furthermore, a linear master boundary that is formed by two nodes is assumed

in the discretization. Thus, a contact pair is described by the master and slave nodes as

1, 2,{ , , }T

s m mX x x x . The master nodes are arranged so that the master structure is positioned on

one side of the boundary between master nodes. The master structure portion region ξ is

described in such a way that it is zero and one at the first and second master nodes, respectively.

The purposes of defining discretized contact pairs include determining contact status and contact

pressure distributions [29]. Determination of contact status is often referred to as contact search

or contact detection in the context of finite element formulation. In a structural system, several

26

slave nodes may be contacting several master boundaries. Thus, contact pairs quantities can be

quite considerable which in turn requires a high computational cost to determine contact status in

a structural system. When the contact pairs are recognized, the contact loads can be computed.

The most general case of contact formulation is between two flexible structures. In this scenario,

both master and slave structures are considered flexible and discretizing the contact system

would be complex since the contact loads are essentially acting on both structures. Therefore, the

system is discretized in such a way that the slave nodes are considered as the boundary nodes of

the slave elements whereas the master boundaries are defined as the edges of the master elements

[14]. As for the master boundary, the contact load is exerted at ξ coordinate of the boundary.

Hence, contact load is transmitted to the master nodes varies based on its position from the

contact coordinate. The contact variational formulations are rearranged to incorporate the

influence of master boundary that are presented by the following equations:

( , ) ( )c

T

N n n n s cb u u g e u u d

(2.28)

0

,( , ) ( ( ) )c

nT T

T t t t s c n c

g tb u u g ve u u e u d

c

(2.29)

The tangential term has a normal term associated with it implying that the contact coordinate is

allowed to travel in normal or tangent direction as a result of the motion of master boundary.

2.3 Nonlinear Finite Element Model Simulation

A nonlinear elasto-plastic finite element model of the aero-engine casings assembly is

established considering the combined effects of the main sources of nonlinearity i.e. contact

(changing-status condition) nonlinearity and material (elasto-plastic) nonlinearity. The aero-

engine casings bolted flange connection structural behavior is investigated subjected to the

tensile, transverse and torsional quasi-static loadings. It is notable that quasi-static analysis is

employed to characterize the structural response of the aero-engine casings in operations such as

thrust, maneuver and landing conditions [30]. In addition, the effect of varying bolt preload and

contact regions friction coefficients at bolt and flange interfaces are studied with the nonlinear

finite element model simulation.

27

2.3.1 Aero-engine Casings Model Geometry and Material Characteristics

The aero-engine compressor and combustor casings meshed geometry model of the bolted flange

connection are shown in Figure 1. The compressor and combustor casings are attached by a 16-

bolted flange connection consisting of two ring flanges and 16 bolts. It is important to note that

some geometric details on the casing components have been neglected since they only have local

effects on behavior of the casing structures. Material properties of casings, flanges and bolts in

the analysis are presented in Table 1. The aero-engine casings material is considered to be a steel

alloy.

Figure 13 - Meshed geometry model of aero-engine casings and bolted flange connection

In order to simulate the elastic-plastic nonlinearity for finite element analysis, the material stress-

strain relation is characterized by a multilinear isotropic constitutive model. The multilinear

isotropic hardening approach along with von Mises yielding criterion is considered to model

plastic deformations.

Table 1 - Casings Mechanical Properties used in Nonlinear FE Model

Aero-engine Compressor and Combustor Casing Material Properties

Young’s Modulus E [MPa] Poisson’s ratio Yield Strength σy [MPa] Ultimate Strength σUTS [MPa]

200000 0.3 250 460

28

2.3.2 Aero-engine Casings FE Modeling and Analysis Approach

In order to simulate the aero-engine casings bolted flange connection behavior subjected to

external loadings in a realistic setting the FE analysis is defined in two quasi-static loading

stages. In the first loading stage the 16 bolts are simultaneously tightened up to a prescribed

preloading amount to the bolt shanks in order to join the two casings at flange interface.

Therefore, the bolt shanks at the preload stage are elongated to carry the prescribed preloading.

In the second loading stage, as the bolts were previously fully tightened up to the prescribed

preload, the external loadings are subsequently applied to the aero-engine casings meaning that

the effect of bolt preloading is essentially carried over to the second external loading stage.

Therefore, the external loading is essentially added incrementally meaning that the stiffness

matrix is updated and adjusted before each increment is applied. In order to solve the nonlinear

model a Newton-Raphson iterative approach implemented in ANSYS is employed where the

iteration process continues until convergence is achieved.

Boundary conditions were imposed on the aero-engine casings model to avoid rigid body

motion. The aero-engine casings boundary conditions are considered to be clamped at combustor

casings end and external loadings were applied to the compressor casings end while bolt preload

is present in the flange connection as shown in Figure 2. Contact boundary conditions were

applied at connection interfaces of bolts, nuts and flanges of the casings. Contact boundary

conditions were also considered to be tangential isotropic contact type with frictional contact for

interfaces. The aero-engine casings FE model was meshed with 52274 8-noded hexahedral

elements and the hexahedral mesh density was refined at contact interfaces of bolts, nuts and

flanges in order to obtain a finer mesh at contact boundary regions. The flange-to-flange contact

is defined as frictional contact interfaces at the ring surfaces and inner spigot edge surfaces using

contact elements. The bolt-to-flange and nut-to-flange contacts were considered to be frictional

with isotropic Coulomb friction model. In addition, the nut-to-bolt contact area was considered to

be bonded and merged between the bolt shanks and nuts. The Augmented Lagrange algorithm

contact formulation was employed in the simulation due to its robustness and efficiency for

frictional contact [31].

29

2.3.2.1 Contact Elements

Contact surfaces were modeled using contact-pair of CONTA174 and TARGE170 elements in

ANSYS. CONTA174 and TARGE170 are 8-noded quadratic contact pairing elements as

indicated schematically in Figure 14. The contact and target elements have identical mechanical

properties as the regular hexahedral elements of the aero-engine casings structure. The target

structure is designated as the body that is softer or has a larger contact area.

Figure 14 - Finite element contact and target elements pair

The bolt/nut-to-flange interfaces were defined by associating the bolt/nut surfaces with contact

elements whereas the flange that has a larger contact area was set to be the target surface. It is

noteworthy that the contact and target elements were refined to a comparable size and density as

it is more desirable for the contact and target elements to be of comparable sizes in order to avoid

further alteration of contact or target elements to match one another. Friction was then imposed

on contact pairing elements in order to simulate frictional contact. Frictional contact enables the

contact surfaces to slide and/or separate under the Coulomb friction model.

2.3.2.2 Pretension Element

The PRETS179 pretension elements were used on the bolt shanks in order to model bolt

preloads. The flanges produce tension in the bolts after bolts are tightened and preloads are

applied. External loadings acting on the bolts can cause elongation or compression in the bolts,

30

however, deformation is noted by the surface or volume elements. As mentioned earlier, the

pretension is then locked to simulate the bolt fastening assembly prior to applying external

loadings and PRETS179 elements are utilized to mimic the assembly procedure in practice.

2.3.2.3 Contact Pressure Distribution at Flange Interface

The flange contact pressure is analyzed through the super element finite element formulation is

presented by the following:

1

Flange Flange Contact Flange Flange ContactS U P U S P

(2.30)

where BoltS represents the stiffness matrix for the nodes at the located flange interface. BoltU

and ContactP denote the deformation and contact pressure load, respectively. It is notable that for

the bolts the same formulation is performed similar to the formulation above. The addition of the

contact loads results in the total contact pressure load given by the following:

Total ContactP P (2.31)

The preload deformation can be formulated by the common deformation represented with a

constant deformation as expressed by the following:

0 1U (2.32)

where 0U represents the common deformations. is a positive deformation and 1 denotes a

vector that is 1 for all entries of the vector. Therefore, the contact state subsequent to preload is

derived as follows:

1

1 11Contact Flange BoltP U U

(2.33)

31

2.4 Nonlinear Simulation Results and Discussion

In order to study the structural response of the bolted flange connections in aero-engine casings

the tensile, transverse and torsional loading scenarios were examined through a nonlinear finite

element model (refer to Appendix A for force convergence plots). The influence of bolt

pretension and frictional contact coefficients on the structural response of the bolted flange

connection as well as the contact pressure and frictional stress were investigated under external

loading cases for the 16-bolted flange connection. The nonlinear simulation results are

thoroughly discussed to demonstrate the influence of the aforementioned parameters on the

significance of nonlinearity in the bolted flange connection structural response.

2.4.1 Nonlinear Response Subjected to Applied External Tensile Loading

The influence of bolt preloading and frictional contact coefficients are studied on the nonlinear

deformation response behavior of the 16-bolted flange connection subjected to an applied

external tensile loading as shown in Figure 15. Figure 15 also indicates the flange connection

close-up section view between the two casings where the deformation response is obtained. The

nonlinear deformation response at bolted flange connection in the tensile loading scenario is then

presented.

2.4.1.1 Effect of Bolt Pretension

The influence of bolt preloading was studied over a range of prescribed bolt preload values. The

contact frictional coefficients of the bolted flange connection interfaces were considered to be

0.15 at all contact interfaces. The deformation response at the bolted connection flange ring

versus the applied external tensile loading is plotted for a range of prescribed bolt preloading

values as shown in Figure 16.

32

Figure 15 - External tensile loading boundary conditions and connection close-up section view

Figure 16 - Flange deformation response subjected to tensile loading for different bolt preloads

As the plot indicates, the curves are classified into two loading regions i.e. Pretension Loading

Region and External Loading Region. The flange bolts are tightened in the first loading region

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 20000 40000 60000 80000 100000

Casin

gs F

lan

ge D

efo

rm

ati

on

(m

m)

Applied External Tensile Loading (N)

Bolt Preload = 2000 N Bolt Preload = 3000 N Bolt Preload = 4000 N

0% Bolt Preload 100%

33

up to their maximum prescribed value upon reaching the second region where the external

loading is applied.

In the pretension loading region, as preload is being applied to the bolts, the flange essentially

undergoes a pre-deformation amount prior to exerting the tensile loading as shown in Figure 3.

On the other hand, in the applied external tensile loading region as the tensile loading approaches

the total bolts force working load, the flanges start to deform and separate as indicated in the

second region, and the bolts would effectively bear the applied tensile loading. Higher bolt

preloading yielded moderately lower flange deformation as indicated by the deformation

response plot. As it can be realized from the plot, the nonlinear tensile stiffness of the bolted

flange structure can effectively be determined by obtaining the ratio of the tensile loading change

to the flange deformation change. Thus, the bolted flange structure stiffness in the pretension

loading region is observed to be larger than that of the external loading region and a nonlinear

hardening effect is seen to occur in the external loading region due to presence of contact

nonlinearity as the effective contact area is increased as also suggested in [5] . As the flanges

start to deflect more from the applied tensile loading, the tensile stiffness of the bolted flange

structure is controlled by the bolts and it is essentially reduced as the tensile loading increases. It

can also be understood from the plot that the tensile stiffness of the bolted flange connection in

both regions does not vary significantly as the bolts preload amount is varied and therefore, the

influence of bolt preloading on bending stiffness is found to be marginal overall.

As for the applied external tensile loading region, all bolts would have identical bolt force

working load variation due to the symmetry of geometry, bolt positions and boundary conditions.

Figure 17 indicates the bolt force working load variation of a single bolt with respect to the

applied external tensile loading.

34

Figure 17 - Bolt force response subjected to tensile loading

As it can be observed from the plot as the external tensile loading is gradually increased and the

flanges start to deform and separate more, the external applied tensile loading would add to the

bolts working load. At lower external tensile loading levels, the bolt forces are initially almost

unchanging because the external loading is being borne by the casings. As the external loading

increases, flange deformation increases and the bolts withstand the external tensile loading.

Hence, the bolt force working loads exhibit a considerable increase. This effect applies to

different bolt preloading cases; however, lower bolt preloads are observed to undergo a slightly

steeper increase than higher preloads. It is important to remark that the total 16 bolt forces at the

ultimate state of external tensile loading do not reach the equilibrium maximum (converge to a

lower bolt force working load) due to partial plastic yielding induced in the bolts [32] as shown

in the plot. The response trend also compares well with experimental and numerical simulations

presented in [8] and [21]. In addition, it is worth noting that as the tensile external loading is

0

1000

2000

3000

4000

5000

6000

7000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Bolt

Force W

ork

ing

Load

(N)


Bolt Preload = 2000 N Bolt Preload = 3000 N

Bolt Preload = 4000 N Bolt Preload = 5000 N

35

increased the bolt shanks experience a loading that is farther away from their cross-section areas

which results in tensile deflection coupled with small bending deflection in the bolts.

2.4.1.2 Bolted Flange Contact Pressure Distribution

As for the tensile loading case, the contact pressure and frictional stress for the flange-to-flange

interface are distributed uniformly due to tension symmetric nature. The contact pressure contour

plot is presented in Figure 18-a. The maximum frictional stress and contact pressure for flange-

to-flange interface were found to increase as the bolt preload increases.

The flange-to-flange contact interface gap and contact pressure contour plots for 3000 N bolt

preload are indicated in Figures 18-a. The contour plot is shown for the maximum external

loading state where gaps are located. It is observed that the maximum gap value for the flange-

to-flange interface decreases as preload is increased.

Figure 18 – (a) Flange interface gap distribution (b) Contact pressure distribution at ultimate

loading state

It is observed from contour plot that the maximum contact pressure and frictional stress at the

ultimate loading state are found to occur close to the bolt and on the outer ring of the flange-to-

flange contact interface as shown in Figure 18-b.

a) b)

36

2.4.1.3 Effect of Frictional Contact

In order to study the influence of frictional contact on the response of the bolted flange structure

the simulation was performed through varying the frictional contact coefficients of bolt/nut-to-

flange, and flange-to-flange contact interfaces.

The nonlinear simulation was performed for the bolt/nut-to-flange interfaces with friction

coefficients of 0.1, 0.15 and 0.3 while the flange-to-flange friction coefficient and bolt preloads

were kept at 0.2 and 3000 N, respectively. The deformation response versus applied external

tensile loading was then plotted to compare the curves as shown in Figure 19. It can be observed

that the nonlinear flange deformation declines only marginally as the frictional contact

coefficient and therefore, it is understood that the effect of bolt/nut-to-flange contact friction

coefficient is insignificant for the external tensile loading scenario.

Figure 19 – Deformation response with varying bolt/nut-to-flange frictional contact coefficients

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 20000 40000 60000 80000 100000

Casin

gs F

lan

ge D

efo

rm

ati

on

(m

m)


Bolt/nut-to-flange Friction Coefficient = 0.1 Bolt/nut-to-flange Friction Coefficient = 0.15

Bolt/nut-to-flange Friction Coefficient = 0.3


37

The nonlinear simulation was then carried out for the flange-to-flange interface with friction

coefficients of 0.05, 0.2 and 0.3 while the bolt/nut-to-flange friction coefficients and bolt

preloads were set to be 0.15 and 3000 N, respectively. The flange deformation versus applied

external tensile loading is indicated in Figure 20. It is observed that the nonlinear flange

deformation behavior exhibits a substantial decrease as the frictional contact coefficient increases

and thus, it is found that the effect of flange-to-flange contact interface friction coefficient is

quite remarkable in compare to the bolt/nut-to-flange contact interfaces.

Figure 20 - Deformation response with varying flange-to-flange frictional contact coefficients

As the nonlinear simulation plot indicates, it is concluded that the effect of flange-to-flange

frictional contact interface on the nonlinear flange deformation response is significant while the

bolt/nut-to-flange frictional contact interface has a negligible effect when subjected to external

tensile loading.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0 20000 40000 60000 80000 100000

Casin

g s

Fla

ng

e D

efo

rm

ati

on

(m

m)


Contact Friction Coefficient at Flange-to-Flange Interface = 0.05




38

2.4.2 Nonlinear Response Subjected to Applied External Transverse Loading

The effect of bolt preloading and frictional contact coefficients are studied on the nonlinear


external transverse bending loading as shown schematically in Figure 8.

Figure 21 - External transverse loading boundary conditions

The effects of bolt pretension and frictional contacts are investigated on the nonlinear

deformation response of the aero-engine casings bolted flange connection subjected to applied

external transverse bending loading. An equivalent transverse force was applied to simulate the

nonlinear deformation response at bolted flange connection for this loading scenario.


The influence of bolt preloading was studied over a range of bolt preload values. The frictional

contact coefficients of the bolted flange connection interfaces were considered to be 0.15 for all

contact interfaces. The deformation response at the bolted connection flange ring versus the

applied external transverse bending loading is plotted for different prescribed bolt preload values

as indicated in Figures 21 and 22 for tension side and compression side of the bolted flange

connection, respectively.

FTransverse

39

Figure 22 - Deformation response on tension side under transverse loading for different bolt

preloads

As shown in the response plots, for both the tension and compression sides of the bolted flange

the lower bolt pretension levels cause the flange to undergo higher deformations however, the

preloading effect on the deformation response is found to be insignificant as the plots indicate.

Furthermore, the nonlinear transverse bending stiffness of the bolted flange structure can

effectively be determined by the ratio of the tensile loading change to the flange deformation

change.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20000 40000 60000 80000 100000

Casin

g F

lan

ge D

efo

rm

ati

on

at

Casin

gsT

en

sio

n S

ide

(mm

)

Applied External Transverse Loading (N)

Bolt Preload = 2000 N Bolt Preload = 3000 N Bolt Preload = 4000 N Bolt Preload = 5000 N


40

Figure 23 - Deformation response on compression side under transverse loading for different bolt

preloads

It is observed that the transverse bending stiffness only increases negligibly with higher bolt

preloads. It can be concluded that bolt preloading has a mostly insignificant influence on the

transverse bending stiffness of the bolted flange structure.

Figure 24 shows the bolt locations on the rear compressor casing on the flange and the applied

transverse bending loading direction. The bolt tension variation by location under transverse

loading (at ultimate loading state) when the bolt preload is 4000 N and contact interfaces

frictional coefficients are considered to be 0.15.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20000 40000 60000 80000 100000

Casin

g F

lan

ge D

efo

rm

ati

on

at

Casin

gs

Com

pressio

n S

ide (

mm

)


Bolt Preload = 2000 N Bolt Preload = 3000 N Bolt Preload = 4000 N Bolt Preload = 5000 N


41

Figure 24 - The bolt load variation by location under transverse loading at ultimate loading state

It can be seen that the maximum bolt force working load develops on bolt 9 being the farthest

bolt located on the tensile opening side of the flange. In addition, symmetric bolt pairs such as

bolts 6 and 12 develop identical bolt forces. It is also notable that the bolt forces are virtually

constant below a transverse loading threshold value. However, as the transverse loading exceeds

this threshold, the bolt forces of bolts 8, 9 and 10 start to increase rapidly while the bolt tensions

of other bolts decrease slightly. Bolt 1 is seen to have the lowest bolt force as it is the farthest

bolt from the neutral plane at the end of the compression side. As it can be observed from the

nonlinear analysis of bolt forces the neutral plane is predicted to be farther up due to nonlinear

considerations while a simple linear model [18] would have its neutral plane located in the center

passing through bolts 5 and 13. It is also evident that the bolt force working load change trends

over the external loadings also vary by bolt preload amount in the sense that lower bolt preload

cases undergo a much steeper increase in response to the external loading than those of lower

bolt preload cases analogous to previous loading case i.e. lower transverse load is needed to

prompt the bolt force working load increase since the flange separation occurs more readily

when bolt preloads are lower.

42


As for the transverse bending loading case, the contact pressure and frictional stress for the

flange-to-flange interface are not distributed uniformly over different bolts due to the transverse

loading nature meaning that the bolts in the tension side of the contact interface undergo higher

stress levels than those farther away towards the compression side of the flange. The maximum

frictional stress and contact pressure for flange-to-flange interface were found to increase overall

as the bolt preload increases. The maximum contact pressure contour plot with 4000 N bolt

preloads at ultimate transverse loading state is shown in Figures 25-a and 25-b.

Figure 25 - a) Flange interface contact pressure distribution contour b) Contact pressure variation

Previous to applying the transverse loading to the structure, the frictional stress and contact

pressure contour plot is uniform and symmetric for all bolts due to preload. It is seen from the

flange contour plot that the maximum contact pressure and frictional stress at flange-to-flange

interface concentrate around the bolt hole located at farthest bolt on the tension side. As a final

point, the transverse load is found to have a drastic effect on flange-to-flange frictional stress and

contact pressure distributions at the interface which was also found to be analogous to the bolt

forces variations by location effect as described earlier.

a) b)

43


The effect of frictional contact coefficients were studied for the external transverse bending

loading model. The nonlinear simulation was carried out with varying the bolt/nut-to-flange

while the flange-to-flange frictional contact coefficient was considered to be constant at 0.15 for

all cases. As it can be observed the deformation response in both locations only increases

marginally as the frictional contact coefficient is reduced. Thus, it is concluded that the effect of

bolt/nut-to-flange frictional contact coefficients of the bolted flange is found to be negligible

implying that the lower friction coefficient produces only slightly higher deformation response in

the bolted flange transverse loading scenario.

The effect of frictional contact coefficients were also investigated for the external transverse

bending loading model. The nonlinear simulation was performed for flange-to-flange frictional

contact coefficients considering while the bolt/nut-to-flange friction coefficients were set to be

0.15. The deformation response plots are indicated in Figures 26 and 27.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20000 40000 60000 80000 100000

Casi

ng F

lan

ge D

efo

rmati

on

at F

lan

ge T

en

sion

Sid

e (

mm

)


Contact Friction Coefficient at Bolt/nut-to-Flange Interfaces = 0.05




44

In the case of flange-to-flange contact, the frictional contact coefficients have a more noticeable

impact on the deformation response more than that of the bolt/nut-to-flange contact coefficients.

Therefore, it can be concluded that the deformation response is rather more sensitive to the

flange-to-flange frictional contact coefficient variation than that the bolt/nut-to-flange interface.


2.4.3 Nonlinear Response Subjected to Applied External Torsional Loading

The effect of bolt preloading and frictional contact coefficients are studied on the nonlinear


external torsional loading. A torsional moment load is applied through the centerline at the top of

the compressor casing as shown in Figure 28. The nonlinear deformation response at bolted

flange connection in the tensile loading scenario is then presented and discussed.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 20000 40000 60000 80000 100000

Casi

ng F

lan

ge

Defo

rmati

on

at

Fla

nge

Te

nsi

on

Sid

e

(mm

)






45

Figure 28 - External torsional loading boundary conditions


The effect of bolt preloading on the torsional deformation response of the bolted flange

connection is studied. As with the other cases, the frictional contact coefficients are considered to

be 0.15 for all contact interfaces. Due to the torsional loading the angular deformation is

measured to determine the deformation response. The angular deformation of the bolted flange

connection is defined as the relative angular displacement of the casing flange with respect to the

centerline of the aero-engine casings. The applied external torsional loading induces shear in the

bolts. Angular deformation response is plotted for different bolt preloading conditions in Figure

29.

46

Figure 29 - Deformation response under torsional loading for different bolt preloads

As for the torsional loading case, the nonlinear simulation plots suggest that the effect of bolt

preloading is noticeable in the torsional loading case. Similar to the previous loading cases, the

torsional stiffness of the bolted flange can also be characterized as the ratio of the applied

external torsional moment change to relative angular deformation change. As defined, it can be

understood from the plot that the torsional stiffness decreases moderately as the external

torsional loading is increased. Initially, all contact interfaces namely, flange-to-flange and

bolt/nut-to-flange interfaces exhibit a microslip condition without any occurrence of macroslip .

The phenomena arise due to the presence of friction at the contact interfaces. Unlike the

macroslip condition, the microslip condition occurs when the areas away from the bolt undergo

slippage and the area close to the bolt hole does not undergo any slippage. On the other hand,

macroslip condition is generally known to occur for loading that induces slippage between the

interfaces [33]. As the applied torsional loading achieves a threshold, slippage takes place at

flange-to-flange contact interface and the frictional contact forces at the flange-to-flange

interface remain fairly steady as loading is being increased. On the other hand, the bolt/nut-to-

flange interfaces are sticking and the frictional loads increase with the applied torsional loading.

0.0003

0.0005

0.0007

0.0009

0.0011

0.0013

0.0015

0.0017

0.0019

0.0021

0 5000000 10000000 15000000 20000000Casin

g F

lan

ge A

ng

ula

r D

efo

rm

ati

on

(R

adia

n)

Applied External Torsional Loading (N.mm)

Bolt Preload = 3000 N Bolt Preload = 4000 N Bolt Preload = 5000 N

47

Thus, the applied torsional loading needs to surpass the frictional loads of bolt/nut-to-flange

interfaces which results in the reduction of torsional stiffness in compare to the early loading

state. Moreover, the torsional stiffness of the bolted flange connection before slipping between

interfaces occurs increases as bolt preload is increased.


The maximum frictional stress, contact pressure for flange-to-flange interface were found to

increase as the bolt preload increases while the gap is essentially reduced. The maximum contact

pressure contour plot is shown in Figure 30 at the torsional loading ultimate state where the

torsional moment is applied in the clockwise direction.

Figure 30 – Contact pressure distribution contour

It is observed from the flange contour plot that the maximum contact pressure and frictional

stress at flange-to-flange interface concentrates in the vicinity of bolt holes in the region shown

as discussed in the previous section.


The nonlinear simulation was then carried out for the flange-to-flange interface with frictional

contact coefficients of 0.1, 0.15 and 0.3 while the bolt/nut-to-flange friction coefficients and bolt

preloads were set to be 0.15 and 3000 N, respectively. The bolted flange angular deformation

response versus the applied external torsional loading with varying frictional contact coefficients

48

of bolt/nut-to-flange and flange-to-flange interfaces are plotted in Figures 31 and 32,

respectively. It can be observed that the bolted flange angular deformation behavior exhibits a

noticeable decrease as the frictional contact coefficient is increased particularly when the

torsional loading approaches higher values. Thus, it is found that the effect of bolt/nut-to-flange

contact interface frictional contact coefficient becomes more evident at higher torsional loading

states showing that higher frictional contact coefficient yields moderately lower angular

deformation of flange.


Similarly, the nonlinear simulation was performed for the flange-to-flange interface with friction

coefficients of 0.05, 0.2 and 0.25 while the bolt/nut-to-flange friction coefficients and bolt

preloads were set to be 0.15 and 3000 N, respectively. When the applied external torsional

loading overcomes the frictional contact loads at the flange-to-flange and the bolt/nut-to-flange

contact interfaces, the frictional stress at all contact interfaces increase with bolt preload and

frictional contact coefficients of interfaces. Initially, the external torsional loading would have to

0.0003

0.0005

0.0007

0.0009

0.0011

0.0013

0.0015

0.0017

0.0019

0.0021

0 5000000 10000000 15000000 20000000

Casin

g F

lan

ge A

ng

ula

r D

efo

rm

ati

on

(R

adia

n)


Contact Friction Coefficient of Bolt/nut-to-Flange Interfaces = 0.1



49

surpass the flange-to-flange frictional contact loads. In essence, when the frictional contact loads

of an interface decrease proportional to frictional contact coefficients, the minimum torsional

loading needed to initiate slipping between the interfaces is effectively reduced.


To sum up the effect of frictional contact results, the frictional contact coefficient at flange-to-

flange interface was found to have a considerable impact on the angular deformation and

torsional stiffness of the bolted flange connection whereas the bolt/nut-to-flange frictional

contact effect was observed to be minor overall.

0.0003

0.0008

0.0013

0.0018

0.0023

0 5000000 10000000 15000000 20000000

Casin

g F

lan

ge A

ng

ula

r D

efo

rm

ati

on

(R

adia

n)


Contact Friction Coefficient of Flange-to-Flange Interfaces = 0.05



50

Chapter 3

Structural Design Optimization of the Bolted Flange

3.1 Overview

The current traditional method of the bolted flange structure design is mostly based on basic

design criterion and empirical equations meaning only structural integrity is considered. These

methods address the structural strength and reliability of the structure to some extent; however,

many other important factors are not directly considered in the traditional design approach, such

as weight, stress, fatigue, etc. The traditional approach can also be inefficient in the sense that it

can cause use of excessive materials at the expense of its weight and cost. Bolt shanks and holes

undergo large stress concentrations during operation where loads are acting on the structure.

Therefore, optimization can be quite beneficial to enhance the performance of casings under

different loading conditions.

The multi-objective optimization of bolted flange structure through structural optimization is

quite beneficial to optimize multiple performances of the bolted flange simultaneously and

achieve optimum structural reliability and weight performances. Strength-to-weight ratio is also

known to be imperative in bolted flange performance. Single objective optimization does not

lend attention to multiple structural performances. In order to accommodate and consider

multiple performances, it is advantageous to propose a framework in multi-objective structural

optimization of bolted flange structures. In order to lend attention to strength, mass, fatigue life,

wear etc. of bolted flange, a multi-objective framework can be employed to strike a balance

between performances and achieve optimum design. As for a bolted flange structure, the mass

and strength are found to be the most important performance parameters. Thus, both strength and

weight parameters are considered for the multi-objective design optimization of the bolted flange

structure in aero-engine casings.

51

3.2 Literature Review

This section presents the literature survey on the study performed on chapter 3 of this thesis for

the structural design optimization of the bolted flange connections. Structural optimization

algorithms and techniques are thoroughly discussed. Topics of design of experiments (DOE),

surrogate model and multi-objective optimization are illustrated in detail.

3.2.1 Structural Design Optimization

Structural design optimization is an imperative practice where design objectives and constraints

criteria are considered in order to obtain the global or local optimal design solution. In general, a

structural design optimization problem can be stated as the following:

1

2minimizes (X)

( ) 0 j 1,2,...,mSubjected to

( ) 0 j 1,2,..., p

n

j

j

x

xDetermine X which f

x

g X

h X

(3.1)

where X is termed the design variables vector set, f(X) is the objective function, g and h

represent inequality and equality constraint functions, respectively.

There are three main classifications of structural optimization problems, namely shape

optimization, size optimization and topology optimization. In a size optimization problem,

geometric parameters that govern the structural system’s geometry are selected for design

variables. Such geometric parameters are often described by mathematical or CAD-based

models. Structural design parameters such as length, width, height diameter, thickness, etc. are

some examples of this category of structural optimization problems. In a size optimization, the

main topology of the structure is preserved meaning that the topology basis is established before

the optimization process. In a shape optimization problem, the shape is optimized through

parameters that govern the shape of the structure meaning that the design shape is altered through

structural shape boundaries which enhances the design space offering a large pool of optimum

52

design candidates. The shape boundaries are often characterized with curves that are

parameterized prior to the optimization process. In this category of optimization problems, only

shape is varied which implies that imposing a new topological feature in the system is not

permitted. In other words, shape optimization consists of selecting the integration domain for the

shape parameters through an optimal approach. As for the topology optimization problem, the

topological characteristics of the structure can be altered to obtain optimum design. Limited

research is available in the structural optimization of bolted flange connections the system. Wang

et al. [34] analyzed the stress optimization of flange joint subjected to temperature fields. The

optimization of fatigue resistance behaviour in the flange was also studied which is described in

[11]. Pedersen [12] proposed a shape optimization problem to improve stress states in bolts and

nuts considering bolt threads local effects.

3.2.1.1 Metaheuristic Structural Optimization

Metaheuristic optimization is widely utilized in structural optimization of various mechanical

and aerospace systems. Metaheuristic optimization algorithms were developed in the recent years

in an effort to address the shortcomings and limitations of the traditional deterministic methods

[35]. Deterministic methods are unable to optimize discrete and discontinuous systems as they

are based on prior computation of gradients. In general, metaheuristic optimization techniques

are often based on behavioural characteristics of natural systems such as biological, molecular

behaviour of genes, insect swarms and flock of birds. As these methods are also termed gradient-

free, knowledge of function values and derivatives are not required prior to optimization process

meaning that these methods are particularly beneficial in cases of discrete feasible space and

non-differentiable objectives or constraints. This section presents the main metaheuristic

methods of optimization in the context of structural design optimization.

3.2.1.1.1 Genetic Algorithm

Genetic algorithm (GA) was devised based on the theory of evolution and the survival-of-the

fittest. It is a stochastic search method that uses natural evolution patter of genes to optimize the

design. Genetic algorithms seek within the design space through utilizing operators that rely on

genetic and natural selection schemes. The three main operators in a genetic algorithm are

53

crossover, mutation and selection. Genetic algorithms initiate search from a population of

encoded solutions rather than a unique point in the global design space. The initial population of

individuals is generated in a random manner. The use of genetic operators results in determining

global optimum solutions based upon the candidates in the current population. The new

generation replace the older population and the evolution process is iteratively performed until

convergence criteria are satisfied.

Selection Operator

The selection process outlines the natural selection or survival of the fittest principle and selects

viable individuals out of the current population to generate the next population according to the

assigned fitness. Design variables are evaluated based upon on the cumulative probability.

Different approaches are employed for selection of individuals. In a roulette-wheel approach a

set of numbers between zero and one is randomly selected to pick the first set of design variables

that would have a cumulative probability greater than the randomized number. In contrast, in a

tournament approach, the fittest individual in a randomized subset of population is chosen

iteratively. Subsequent to selection, crossover and mutation recombine and modify parts of the

individuals to generate new solutions.

Crossover Operator

Crossover process exchanges parts of solutions from two or more parents and recombines these

parts to generate children with a prescribed probability. It essentially generates a new population

by allocating parents properties analogous to biological phenomenon. Single-point, two-point

and cut and splice methods are some approaches that are often implemented for crossover

operations.

Mutation Operator

The generic mutation often alters some properties of individuals to form a new altered set of

solutions. In contrast, mutation operates on a single individual while crossover operates on

54

multiple individuals. In addition, it diversifies the new population by introducing randomness in

the population.

In brief, the systematic implementation of a GA is illustrated by the following steps [36]:

Step 1: Initialize genetic algorithm parameters that are required for the algorithm. These

parameters include population size that indicates the number of individuals, number of

generations necessary for the termination criteria, crossover probability, mutation probability,

number of design variables and the respective bounds for the design variables.

Step 2: Generate random population equal to the specified population size. Each population

member contains the value of all the design variables. This value of design variable is randomly

generated in between the design variable bounds.

Step 3: Obtain the values of the objective function for all the population members. The value of

the objective function indicates the fitness of the individuals. If the problem is a constrained

optimization problem then a specific approach as static penalty, dynamic penalty and adaptive

penalty is used to transform the constrained optimization problem into an unconstrained one.

Step 4: This step is for the selectin procedure to form a mating pool that consists of the

population consisting of best individuals. The commonly used selection schemes are roulette-

wheel, tournament and stochastic selection. The individual having better fitness value will have

more number of copies in the mating pool which in turn increases the chances of mating for fitter

individuals. This step justifies the notion for survival of the fittest.

Step 5: This step is for the crossover where two parents are selected randomly from the mating

pool in order to generate two new off springs. The individuals from the population can proceed

to the crossover step based upon the crossover probability. If the crossover probability is higher,

more individuals are granted the chance to proceed to the crossover procedure.

Step 6: After crossover, mutation step is performed on the individuals of population depending

upon the mutation probability. Note that the mutation probability is generally maintained to be

low so that it does not cause instability within the optimization process.

55

Step 7: Best results obtained are not modified using crossover and mutation operators but can be

replaced if better solutions are obtained during iterations.

Step 8: The steps after step 3 are repeated iteratively until the convergence criterion is satisfied.

3.2.1.1.2 Particle Swarm Optimization

The particle swarm optimization (PSO) concept was originally devised by Kennedy et al. [37]

based on the mutual behaviour of swarm intelligence system as a simulation of a simplified

social system such bird flocks or bee swarm. In that sense, each particle keeps track of its

coordinates in the design space that are associated with the optimum solution. The coordinates of

particles are updated by the following equation model:

1 1

i i i

k k kx x v t (3.2)

where 1

i

kx and

1

i

kv denote the particle position at iteration and its corresponding velocity vector,

respectively. The particle velocity vector is determined by the following:

1 1 1 2 2

inertia termcognitive term social term

( ) ( )i i g ii i k k k kk k

p x p xv wv c r c r

t t

(3.3)

where w is the inertia weight. r1 and r2 are two random numbers between 0 and 1. c1, c2, pi and pg

are the cognitive term, the social term, the best position of particle iteration, and the global best

position in the swarm up to iteration, respectively. Note that the inertia weight has a significant

role in the convergence of the PSO algorithm [38]. Perez et al. [39] proposed a matrix form of

the particle velocity equation by the following:

1 1 2 2 1 1 2 2

1

1 1 2 2 1 1 2 2

1

1

i i i

k k k

i i g

k k k

c r c r w t c r c rx x p

c r c r c r c rwv v p

t t t

(3.4)

Conditions were proposed to ensure the stability and convergence of the PSO algorithm as

follows:

56

1 2

1 2

12

c c

c cw

(3.5)

The inertia can be updated at each iteration or remain constant through the optimization process.

Updating inertia has shown faster convergence towards the optimum solution [38]. It can be

updated by the following relation:

1k kw w (3.6)

where is a constant real number between 0 and 1.

3.2.2 Multi-Objective Design Optimization

In practice, many structural design optimization problems are associated with consideration of

multiple structural performance criteria simultaneously. Multi-objective design optimization is

advantageous in the sense that multiple objectives are optimized concurrently. This implies that

for problems with conflicting criteria, there may be multiple solutions with different

compromises of objectives that govern the performance of the structural system. It is also notable

that design solutions are not arranged in strict hierarchy. A multi-objective design optimization

problem can be stated by the following:

1

2

( )

( )( )

( )

( ) 0 1,2,...,m

( ) 0 1,2,..., p

n

j

k

f x

f xMinimize f x

f x

g x jSubjected to

h x k

(3.7)

57

where n, m and p denote the number of objectives, inequality constraints and equality

constraints, respectively. f(x) is the set of objective functions that is also termed as the value

function. The space where the objective vector lies in is known as the objective space. The

feasible design space is described by the following:

0; 1,2,..., 0; 1,2,...,j kD x g j m and h k p (3.8)

As mentioned earlier, multi-objective optimization offers multiple solutions in the design space

with regards to the objectives of the problem unlike a single objective optimization.

Pareto Optimality

A Pareto optimal set represents a complete set of solutions for the multi-objective design

optimization. Pareto optimal solution makes compromises. They are solutions for which any

enhancement in an objective causes deterioration of at least one other objective. A point x* in the

feasible design space defined above D is called Pareto optimal if and only if there exists no other

point x in the set that decreases at least an objective function without an increase in another.

Pareto Dominance

Dominated design points are described as the design points in the design space for which there

exists at least one feasible design point that is more optimal than them for all objectives. Non-

dominated points are design points which are the most optimal points and no other feasible

design points are as optimal as them. This implies that optimal points are essentially the non-

dominated design points [40].

Metaheuristic optimization algorithms such as GA or PSO are modified to deliver reliable

methods for dealing with multi-objective optimization problems [41]. Various techniques of

solving multi-objective optimization problems enable assignment of importance criteria to

different objectives. A popular approach is to designate a utility function in order to set

constraints and achieve a Pareto optimal set [42]. Some popular techniques of solving multi-

58

objective design optimization problems include Scalarization, ε-constraints, weighted min-max

method, Lexicographic and weighted global criterion techniques [40].

3.2.3 Design of Experiments

Design of experiments (DOE) is a statistical and systematic approach to perform a series of

experiments with bounded design parameters that is greatly beneficial for minimizing the

optimization runs in structural systems. In the context of optimization, an experiment is

described as a series of evaluations where the design parameters are varied based on a certain

sampling algorithm [43] in order to realize the alterations in the response more efficiently.

Therefore, the goal of experiments is to effectively perform design optimization. In this section, a

number of DOE techniques that are suited for structural design optimization purposes are

introduced.

3.2.3.1 Central Composite Design

The central composite design (CCD) is a design of experiments algorithm that is particularly

advantageous for establishing a quadratic model for the response without conducting a three-

level factorial. A central composite design is a two-level full factorial algorithm where the

central point and the star points are augmented [44]. The star points can be illustrated as the

sampling points in which all the design variables except one are considered at a mean. The

remaining design variable value is set in form of distance from the center point between design

bounds. The distance between the center point and every sample can be normalized to 1 which

allows the distance of the star points from the center point to be selected through different

approaches [45] such as central composite scaled (CCS), central composite circumscribed

(CCC), central composite faced (CCF), and central composite inscribed (CCI). A schematic

representation of these approaches in CCD is presented in Figure 33.

59

Figure 33 - Central Composite Design experimental design sampling space

For n design variables 2n star points and a center point are augmented to the full factorial, which

results in a sampling size of 2n+2n+1 for CCD. Acquiring more sample points than these is

required for bilinear interpolations which can be employed to approximate the design response

curvature.

3.2.3.2 Box-Behnken Design

Box-Behnken is a design of experiments algorithm that does not use a fractional factorial or

embedded factorial design. It is based on incomplete three-level factorial. Two-level factorial is

established in conjunction with incomplete block design to form a Box-Behnken design [46].

The DOE algorithm was initially devised in an effort to limit the sample size as the design

parameters number increases. The size of the sample is maintained at a certain value that is

adequate for approximation of the coefficients in a second degree least squares polynomial. A

schematic representation of Box-Behnken is presented in Figure 34.

60

Figure 34 - Box-Behnken algorithm design of experiments

In Box-Behnken DOE algorithm, a block of samples that is associated with a two-level factorial

design is recurred over different sets of design parameters. The design parameters that are

excluded from the factorial design stay at their average level in the block. The type, size and the

block numbers that are computed is dependent upon the design parameter number and it is

selected in order for the rotatability criterion to be satisfied by the design [47]. A DOE is

considered to be rotatable if and only if the variance of the projected response at any point is

proportional to the distance from the center point.

3.2.3.3 Space-Filling Design

In general, space-filling algorithms utilize different techniques to uniformly distribute points

within the design space. Optimal Space-Filling Design (OSF) is classified as a space-filling DOE

algorithm that produces optimal space filling DOE with regards to a certain set of criteria. This

algorithm is based on the classic Latin Hypercube design where the design space is segmented

into an orthogonal grid with a number of elements with similar length per design parameter;

however, it achieves a more uniform space distribution of points than the classic Latin hypercube

algorithm [48]. The Optimal Space-Filling (OSF) design is capable of distributing the design

parameters uniformly within the design space and aims to achieve the maximum vision into the

design with the least points. This renders the OSF quite favourable especially when it is utilized

with sophisticated surrogate modeling techniques such as non-parametric regression, kriging or

artificial neural networks [46] which will be illustrated in the next section.

61

3.2.4 Surrogate Model-Based Optimization

The purpose of surrogate model-based optimization is to address the shortcomings of a direct

design optimization in terms of computational cost and numerical noise [49]. In many

engineering applications, direct optimization is often associated with very high computational

costs. Therefore, surrogate model-based optimization was initially introduced in an effort to

drastically reduce the optimization process and make the optimization computationally efficient

while offering reasonable reliability and accuracy [50]. After conducting design of experiments

on the design and sampling response data, a surrogate model needs to be established in order to

interpolate and fit the design of experiments data. In addition, due to likelihood of local optimum

points, utilizing analysis codes demands higher computational cost and the calculations get more

convoluted [51]. In brief, the goal of surrogate models or meta-models is to substitute the high

fidelity and computationally expensive model for a computationally efficient optimization model

that yet offers reasonable reliability and accuracy (see figure 35). In this part of the dissertation,

main surrogate modeling methodologies that are suited for structural optimization are discussed.

Figure 35 - Traditional optimization vs Surrogate-based optimization

3.2.4.1 Response Surface Model

Response Surface Model (RSM) is also called polynomial interpolation which emerges naturally

and is imperative in establishing reliable and efficient gradient-free algorithms [43]. In the

62

response surface surrogate model, the points are fitted through a polynomial term and a relative

error term as presented by the following:

ˆ( ) ( )s x s x (3.9)

where represents the error with a specific variance and zero mean value. Note that the

response surface model can be of any order. First-order and second-order response surfaces can

be expressed as:

1 2 0

1

ˆ( ) s ...r

T

r i i

i

s x x x x x

(3.10)

1 2 0

1 1

ˆ( ) s ...r r r

T

r i i ij i j

i i i j

s x x x x x x x

(3.11)

where 0 denotes the intercept, r denotes the parameter numbers and ij represents the

interaction coefficients.

3.2.4.2 Kriging

Kriging model is employed in a wide range of engineering optimization problems as it offers

reasonable compactness and computational efficiency in terms of evaluation [52]. This method

was initially developed by Krige [53]. It is classified as a nonparametric method of constructing

surrogate models which employs Gaussian random process to interpolate response data. Kriging

method fundamental response formulation is presented by the following:

ˆ( ) ( ) ( )Ts x g x Z x (3.12)

where g represents the set of known functions, denotes the unknown model parameters set

and Z is the Gaussian stochastic process by normal distribution with a specific variance and zero

mean value. Kriging model suggests that a constant term can be determined to approximate the

relationship between the inputs and responses. Therefore, a random number can be substituted

with the polynomial as:

ˆ( ) ( )xs x Z x (3.13)

63

Additionally, as soon as the kriging surrogate model is established, the stochastic process Z

reports the approximation error of the model which is beneficial for further ameliorating the

model.

3.2.4.3 Radial Basis Functions

Radial Basis Function (RBF) interpolation is a non-parametric regression method which exploits

linear combinations of multiple radial symmetric functions [48], [54]. The fundamental form of

this surrogate model is presented by the following:

1

ˆ( ) ( )r

i

i

i

s x x x

(3.14)

where represents the radially symmetric functions set and is the model parameters vector.

Note that is a square matrix of size r. The radially symmetric function can be chosen in various

forms as follows:

Thin plate spline form 2( ) ln( )i ix x x x (3.15)

Gaussian form 2

2

( )exp( )

2

ix x

(3.16)

Multi-quadratic form 2 2( )ix x (3.17)

Inverse multi-quadratic form 1

2 2 2(( ) )ix x

(3.18)

It is worth noting that the norm of the radially symmetric function is often Euclidean distance,

however, other distance functions can also be applicable.

3.2.4.4 Artificial Neural Network

Another series of surrogate modeling techniques is performed by the artificial intelligence

approach. In the recent years, the artificial neural networks (ANN) have become an attractive

means to addressing engineering optimization problems [35], [55]. A neural network consists of

64

large parallel network of interrelated neurons where a set of inputs is fed into each neuron by

other neurons. Then, an output is determined which is spread to responses. Hence, artificial

neural networks are characterized by means of single neurons, the network interrelations and

weights between the neuron links. ANN consists of an input layer, hidden intermediate layers

that convert the results from input layers to output layers [55]. The aforementioned surrogate

model techniques are based on mathematical models but ANN is inspired from the notion and

structure aspect of biological neural networks. A single neuron is indicated in Figure 36.

Figure 36 - Single neuron architecture

A single neuron is inputted with parameters and a bias value. A weight is associated with each

input. The weighted sum of the inputs governs the neuron state which is presented by the

following:

1

1 2

1

nTT

i i n

i

a w x W X where X x x x

(3.19)

A schematic conversion from the multiple-dimensional inputs to the single-dimensional

response, that the neuron directs to its neighbouring neurons is required that is carried out by a

transfer function. The response of a neuron is associated with its state which in this case is

represented by f(a). The main notion of ANN is that the neural network weight and bias

parameters can be modified in order for the network to exhibit favourable behaviour. The

response of a neuron is usually characterized by a transfer sigmoid function as follows:

1

( )1 a

f ae

(3.20)

65

ANN Architecture

Numerous neurons can be interconnected within a layer and a network can possess multiple

layers. Layers have a weight matrix, a bias matrix and a response associated with them. A three

multiple layer network architecture of ANN is indicated in Figure 37.

Figure 37 - Multi-layer Artificial Neural Network architecture

where the i prefix denotes input layer links, h prefix denote hidden intermediate layer links and o

represents the output response layer links, b and w denote the bias and weight, respectively. In

the recent years, multilayer ANN networks have been adopted due to reliable and accurate

surrogate modeling [56] that can be exploited effectively and trained for structural optimization.

3.3 Bolted Flange Connection Structural Optimization

This section of this dissertation presents the multi-objective structural optimization methodology

and procedures of the bolted flange structure in aero-engine casings. As mentioned, the strength-

to-weight ratio of the bolted flange connection is the most important characteristic in terms of

performance, structural reliability and cost. Thus, weight and strength of the bolted flange are the

objectives of the structural optimization.

66

The multi-objective optimization problem is formulated by the following:

1

1

1

( ,..., )minimize

( ,..., )

with respect to ( ,..., )

Mass n

Von mises n

n

M f x x

g x x

x x

(3.21)

3.3.1 CAD-Based Geometric Parametrization

In order to define geometric design parameters, modern CAD systems can be employed to

parametrize major dimensions, global variables and geometric features of the bolted flange

structure in order to establish a fully parametric model. Since modern CAD systems monitor the

complex geometry many properties of the structure such as mass, volume, center of mass and

moment of inertia can be extracted and parametrized for the optimization process. It is

imperative to use care when geometric parameters are being defined with CAD-based

parametrization as many dimensions and features may be associated with other dimensions and

features. Global variables can effectively be utilized to impose geometric constraints on

dimension such that they do not conflict with other dimensions to create an invalid geometry. For

instance, in the bolted flange structure the bolt hole and diameter must be parametrically

synchronized so that they vary relative to one another. Therefore, CAD-based parametrization is

a powerful tool for optimization of systems that undergo size and dimension variations as

opposed to operations such as addition or removal of features.

Feature-based solid modeling (FBSM) methods can be utilized effectively to form geometrical

features such as boss or cut extrusions, holes, chamfers, fillets, etc. Such topological features are

considered dimension-driven which can be controlled through adjusting dimensional features

[57]. Therefore, a fully parametric bolted flange model along with dimension dependencies and

constraints can be described for major geometrical features which in turn facilitate making

adjustments to dimensions and seek candidate optimal solutions to the optimization problem.

67

Through this approach dimensions are selected as design variable parameters. Throughout the

optimization process, even though making dimension adjustments and reconstructing the

geometry of the bolted flange is not time-consuming for each run, re-meshing and re-imposing

boundary conditions on the new geometry can be a relatively time-consuming process. The

SolidWorks package was employed to perform CAD-based parametrization of studied geometric

parameters through defining global variables and dimensional constraints and dependencies.

3.3.2 Bolted Flange Geometry Model for Optimization

As mentioned, optimization of complex structural systems is often computationally expensive to

implement. It is common optimization practice to simplify the model to a uniform fraction or

segment of the whole model in order to increase computational efficiency in such a way that

design variables and response can also be encompassed. The bolted flange geometry is presented

in form of a fractional slice of the model as shown in Figure 38. Note that the bolted flange is

modeled identical to chapter 2 in terms of material and boundary conditions. The model was

parametrized with SolidWorks and exported to ANSYS for simulation and optimization

processes.

Figure 38 - Bolted flange connection segment baseline design

68

3.3.3 Design of Experiments (DOE)

As mentioned earlier, the design of experiments begins with defining design parameters. The

input parameters and response parameters were described. The bolted flange connection input

design parameters are indicated in Table 2. The input design parameters consist of bolt preload,

bolt shank length, flange thickness and frictional contact coefficients of interfaces. It is notable

that the simulations were performed with identical approach to the simulations in chapter 2 of

this dissertation. The response parameters were set to be equivalent maximum von-Mises stress

and masses for the multi-objective optimization process. The primary step towards multi-

objective optimization is to establish the design of experiments. The optimal space-filling design

(OSF) and custom sampling schemes were selected to be the DOE approach due to its accuracy

and reliability in constructing design of experiments models. A total of 60 design points in the

design space were imposed which were used to obtain output responses for constructing a

surrogate model-based optimization approach.

Table 2 - Bolted flange design parameters

Note that the bolt shank length parameter is subsequently lumped into flange thickness parameter

as they vary relative to each other. Bolt head and nut also vary relative to the bolt diameter

dimension.

Design Parameters Representation

Bolt/Hole Diameter (mm) D

Bolt Shank Length (mm) L

Flange Thickness (mm) t

Bolt Preload Level (N) p

Frictional Contact Coefficient of Bolt/nut-to-Flange /Bolt nut

Frictional Contact Coefficient of Flange-to-Flange Flange

69

3.3.4 Surrogate Model Construction Methodology

The surrogate model was constructed based on DOE design points simulation results. A Kriging

surrogate model was constructed to interpolate the DOE points and achieve response surface

approximations. The variable Kernel approach was selected to construct the Kriging surrogate

model of the bolted flange connection structure. The Kriging surrogate model approximation was

employed due to its proven reliability and suitability in structural optimization problems [52] as

well as significantly lower computational cost for convergence. The response surfaces of the

Kriging surrogate model of the bolted flange are presented in the results section of this

dissertation.

3.3.5 Multi-Objective Structural Optimization of Bolted Flange

The main purpose of multi-objective structural design optimization is to enhance the

performance of structural systems from different aspects in a simultaneous manner. Multi-

objective design optimization is most beneficial when trade-offs are to be made with possible

opposing design objectives. If a single objective optimization is performed, reducing the weight

may result in a considerable increase in stresses acting on the system or vice versa.

The structural optimization was performed through a multi-objective genetic algorithm. MOGA

is a variation of the Non-dominated Sorted Genetic Algorithm II (NSGA-II) which is based on

controlled elitism notion [58]. In MOGA, the Pareto ranking is implemented in Matlab by means

of non-dominated sorting approach that is known to be significantly faster in compare to

traditional Pareto ranking approaches [59] (refer to pseudocode in Appendix B). In addition,

constraint handling utilizes non-dominance approach for the objectives which guarantees that the

feasible candidates are prioritized over the infeasible candidates. The design bounds are indicated

in Table 3.

70

Table 3 - Design variable bounds

The operator characteristics that were utilized for the multi-objective genetic algorithm are

tabulated in Table 4. It is also notable that objectives are ranked and weighted with equal

importance for the bolted flange optimization.

Table 4 - Multi-Objective Genetic Algorithm (MOGA) parameter specifications

Multi-Objective Genetic Algorithm Value

Number of Design Variables 5

Population Size 50

Generations 100

Crossover Rate 0.8

Mutation Rate 0.01

The multi-objective design optimization architecture framework of the bolted flange connection

is presented in Figure 39.

Design Variables Variable

Representation

Lower

Limit

Upper

Limit

Baseline

Value

Bolt/Hole Diameter (mm) D 3 9 4

Flange Thickness (mm) t 3 9 7

Bolt Preload (N) P 500 5000 1000

Frictional Coefficient of Bolt/nut-to-Flange /Bolt nut 0.05 0.4 0.2

Frictional Coefficient of Flange-to-Flange Flange 0.05 0.4 0.15

71

Figure 39 - Multi-objective design optimization framework of the bolted flange connection

72

3.4 Multi-Objective Structural Optimization Results

The multi-objective design optimization of the bolted flange connection is thoroughly discussed

in this section.

3.4.1 Surrogate Model Response Surfaces

The response surface contours of the surrogate model are indicated in Figure 40 over different

design variables and objectives.

Figure 40 - Kriging Surrogate model response surfaces

73

Figure 40 - Kriging Surrogate model response surfaces

It is notable that the response surfaces of the Kriging surrogate model serve as a mathematical

model of design space for the multi-objective optimization.

The local sensitivity plot for the design variables with respect to the response is extracted from

the surrogate model as presented in the following plot in Figure 41.

74

Figure 41 - Local sensitivity plot for design variables

3.4.2 Optimum Bolted Flange Design

The optimum design variables combination and resulting response of objectives are presented in

Table 5. The multi-objective optimization problem solution convergence was achieved with 685

evaluations.

Table 5 - Optimized bolted flange design specifications

Design Variables Variable

Representation

Baseline

Design

Optimum

Design

Bolt/Hole Diameter (mm) D 4 7.175

Flange Thickness (mm) t 7 3.018

Bolt Preload (N) P 1000 2052.60

Frictional Coefficient of Bolt/nut-to-Flange /Bolt nut 0.2 0.2217

Frictional Coefficient of Flange-to-Flange Flange 0.15 0.2281

Bolted Flange Mass (kg) M 1.76 1.57

Bolted Flange Max von-Mises Stress (MPa) Von mises 269.98 231.90

75

Figure 42 - Bolted flange optimum design with specifications converged in Ansys

Table 6 - Optimum design comparison with baseline design

As it can be observed from the tabulated results om Table 6, the multi-objective optimization of

the bolted flange resulted in 14.10% weight reduction of the bolted flange while also improving

its strength as the equivalent von-Mises stress in the bolted flange was reduced by 10.79%. The

proposed multi-objective optimization methodology was implemented in conjuction with a

Kriging surrogate model to approximate the optimum design through the multi-objective genetic

algorithm. Therefore, the success of the proposed multi-objective optimization framework of

bolted flange connections in aero-engine casings suggests that the developed multi-objective

optimization architecture can be employed effectively to remarkably enhance the strength-to-

weight ratio of bolted flange design.

MOGA Optimization Objectives Baseline

Design

Optimum

Design

% Improved

from Baseline

Bolted Flange Mass (kg) 1.76 1.57 14.10 %

Bolted Flange Max von-Mises Stress (MPa) 269.98 231.90 10.79 %

76

Chapter 4

Conclusions and Future Work

4.1 Concluding Remarks

The nonlinear structural response of aero-engine casings bolted flange structure subjected to

external quasi-static tensile, transverse and torsional loadings considering an elastic-plastic finite

element model along with contact nonlinearities of contact interfaces was investigated. The

influence of bolt pretension and frictional contact coefficients of bolt/nut-to-flange and flange-to-

flange on the response of the structure were evaluated and discussed. The contact pressure and

frictional stress response contours were also found over different external loading scenarios. The

findings of the nonlinear investigation chapter of the dissertation are summarized as follows:

In the external tensile loading case, the influence of bolt pretension in the nonlinear deformation

response of casings bolted flange structure was found to be moderate meaning the higher bolt

pretension decreases the flange separation. The influence of frictional contact coefficient of

flange-to-flange interface was reasonably considerable in compare to the bolt/nut-to-flange

interfaces that was rather insignificant. It is also notable that at lower tensile loading levels, the

bolt forces are initially nearly unchanging due to the external loading being endured by the

casings; however, as the external loading increases further, the bolts start to bear the external

tensile loading and the bolt force response exhibits an appreciable increase.

As for the transverse loading case, the effect of bolt preloading on the response was insignificant

meaning that higher bolt preloading only marginally decreases the bolted flange deformation.

The effect of frictional contact coefficients of bolt/nut-to-flange interface was found to be

insignificant while the flange-to-flange interface had a slight influence on the response. In

addition, as the external transverse loading is applied to the casing, the bolt forces working load

response undergoes a steeper increase when bolt preloads are lower meaning that a lower

transverse load is needed to trigger the bolt force working load increase as the flange separation

occurs more readily when bolt preloads are lower.

77

In the torsional loading scenario, the effect of bolt preloading on the response was found to be

appreciable overall. The torsional stiffness of the bolted flange connection before slippage

between interfaces ensues, increases as bolt preload is increased. Furthermore, the effect of

frictional contact coefficient of bolt/nut-to-flange interface on the response was found to be

moderate whereas the same effect for flange-to-flange interface was found to be rather

remarkable.

The contact pressure and frictional stress distributions of flange interface were also identified for

different loading cases. It is important to note that contact pressure at the flange-to-flange

interface is mostly concentrated close to the bolt hole but in different regions around the bolt

depending upon the external loading type. However, in the case of transverse loading the contact

pressure distribution at flange-to-flange interface varies significantly in terms of bolt location

implying that the bolts farther away from the neutral plane in the tension side exhibited higher

contact pressures with the maximum occurring at the farthest bolt.

In the second part of the research work of this dissertation, a surrogate model-based multi-

objective metaheuristic genetic algorithm optimization architecture methodology for the bolted

flange structural optimization was developed in order to enhance the weight and stress

performance of the bolted flange in aero-engine casings. The Kriging surrogate model was

employed to obtain response surfaces of the bolted flange design as well as drastically increasing

the computational efficiency of the multi-objective optimization process. The MOGA algorithm

was utilized as the metaheuristic optimizer for the bolted flange design optimization. The

optimization results indicated that the weight and stress performance objectives of the bolted

flange connection were improved by 14.10% and 10.79% from the baseline design, respectively.

78

4.2 Future Work

In this study, the nonlinear structural behaviour was investigated in order to characterize the

behaviour of bolted flange in aero-engine casings subjected to external loadings. One of the

approaches to take is the extension of the numerical investigation to the whole aero-engine

casings model to characterize the influence on the full structural system. In this context,

experimental analysis is much demanded to simulate and study the effect of parameters in

different loading settings. The proposed methodology of bolted flange multi-objective

optimization can also serve as an architecture for comparative study of different surrogate

modeling techniques and metaheuristic algorithms to evaluate the accuracy, reliability and

performance of different techniques. There is much work to be done in investigating the number

of bolts in the optimization of the full bolted flange at the expense of higher computational cost.

An extension of the multi-objective optimization framework of the bolted flange proposed in this

dissertation would be incorporating stiffness and fatigue life performance in the bolted flange

multi-objective optimization framework.

79

References

[1] V. Ruffini, C. W. Schwingshackl, and J. S. Green, “LDV Measurement of Local Nonlinear Contact Conditions of Flange Joint,” in Topics in Nonlinear Dynamics, Volume 1 SE - 14, vol. 35, G. Kerschen, D. Adams, and A. Carrella, Eds. Springer New York,

2013, pp. 159–168.

[2] C. W. Schwingshackl and E. P. Petrov, “Modeling of flange joints for the nonlinear

dynamic analysis of gas turbine engine casings,” J. Eng. Gas Turbines Power, vol. 134, no. 12, p. 122504, 2012.

[3] R. A. Ibrahim and C. L. Pettit, “Uncertainties and dynamic problems of bolted joints and

other fasteners,” J. Sound Vib., vol. 279, no. 3, pp. 857–936, 2005.

[4] J. H. Bickford, Introduction to the Design and Behavior of Bolted Joints, Fourth Edition:

Non-Gasketed Joints. CRC Press, 2007.

[5] E. Rivin, Stiffness and Damping in Mechanical Design. CRC Press, 1999.

[6] P. Honarmandi, J. W. Zu, and K. Behdinan, “Elasto-Plastic Fatigue Life Improvement of

Bolted Joints and Introducing FBI Method #,” Mech. Based Des. Struct. Mach., vol. 33, no. 3–4, pp. 311–330, Jul. 2005.

[7] I. R. Grosse and L. D. Mitchell, “Nonlinear Axial Stiffness Characteristics of Axisymmetric Bolted Joints,” J. Mech. Des., vol. 112, no. 3, p. 442, Sep. 1990.

[8] N. F. Knight Jr, D. R. Phillips, and I. S. Raju, “Simulating the structural response of a

preloaded bolted joint,” in Proceedings of the 49th AIAA SDM Conference, Shaumburg, Illinois, 2008.

[9] J. Monaghan and B. Duff, “The effects of external loading on a yield tightened joint,” Int. J. Mach. Tools Manuf., vol. 27, no. 4, pp. 443–455, Jan. 1987.

[10] K. H. Brown, C. Morrow, S. Durbin, and A. Baca, “Guideline for Bolted Joint Design and

Analysis: Version 1.0,” Sandia Report, SAND2008-0371, no. January, 2008.

[11] I. Integrated Systems Research, “Optimizing Bolted Joint Geometry for Fatigue

Resistance,” in Tech Brief 080501F, 2008.

[12] N. L. Pedersen, “Optimization of Bolt Stress,” in 10th World Congress on Structural and Multidisciplinary Optimization, 2013.

[13] J. N. Reddy, An Introduction to Nonlinear Finite Element Analysis: with applications to heat transfer, fluid mechanics, and solid mechanics. OUP Oxford, 2014.

[14] N.-H. Kim, Introduction to Nonlinear Finite Element Analysis. Boston, MA: Springer US,

80

2015.

[15] D. Bigoni, Nonlinear Solid Mechanics. Cambridge University Press, 2012.

[16] C. W. Schwingshackl, D. Di Maio, I. Sever, and J. S. Green, “Modeling and Validation of

the Nonlinear Dynamic Behavior of Bolted Flange Joints,” J. Eng. Gas Turbines Power, vol. 135, no. 12, p. 122504, Sep. 2013.

[17] D. Joshi, P. Mahadevan, A. Marathe, and A. Chatterjee, “Unimportance of geometric nonlinearity in analysis of flanged joints with metal-to-metal contact,” Int. J. Press. Vessel. Pip., vol. 84, no. 7, pp. 405–411, Jul. 2007.

[18] Z. Tafheem and K. M. Amanat, “Investigation on bolt tension of flanged pipe joint subjected to bending,” in 4th Annual Paper Meet and 1st Civil Engineering Congress,

2011.

[19] S. G. Chavan, “Stress Analysis of Flanged Joint Using Finite Element Method,” vol. 3, no. 8, pp. 1187–1191, 2014.

[20] Y. Luan, Z.-Q. Guan, G.-D. Cheng, and S. Liu, “A simplified nonlinear dynamic model for the analysis of pipe structures with bolted flange joints,” J. Sound Vib., vol. 331, no. 2,

pp. 325–344, Jan. 2012.

[21] H. Van-Long, J. Jean-Pierre, and D. Jean-François, “Behaviour of bolted flange joints in tubular structures under monotonic, repeated and fatigue loadings I: Experimental tests,”

J. Constr. Steel Res., vol. 85, pp. 1–11, Jun. 2013.

[22] B. Wu, T. Wang, C. Xu, and B. Xu, “Finite Element Analysis of Behavior of Bolted

Flange Connections under Bending Loading,” in Applied Mechanics and Materials, 2010, vol. 26, pp. 1168–1171.

[23] P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive

Algorithms. Springer Science & Business Media, 2011.

[24] E. Madenci and I. Guven, The Finite Element Method and Applications in Engineering

Using ANSYS®. Springer, 2015.

[25] A. Ibrahimbegovic, Nonlinear solid mechanics: theoretical formulations and finite element solution methods, vol. 160. Springer Science & Business Media, 2009.

[26] M. Hestenes, “Multiplier and gradient methods,” J. Optim. Theory Appl., vol. 4, no. 5, pp. 303–320, 1969.

[27] R. Fletcher and M. J. D. Powell, “A rapidly convergent descent method for minimization,” Comput. J., vol. 6, no. 2, pp. 163–168, 1963.

81

[28] A. Konyukhov and R. Izi, Introduction to Computational Contact Mechanics: A Geometrical Approach. Wiley, 2015.

[29] T. A. Laursen, Computational Contact and Impact Mechanics: Fundamentals of Modeling

Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer Science & Business Media, 2013.

[30] A. Schönrock, E. Dascotte, K. Dufour, B. M. W. R. Aeroengines, D.- Dahlewitz, D. Design, S. Dds, and B.- Leuven, “Validation of an Aeroengine Carcass Finite Element Model by Means of Computational Model Updating based on Static Stiffness Testing,” in

7th European Propulsion Forum, 1999.

[31] A. R. Khoei, Extended Finite Element Method: Theory and Applications. John Wiley &

Sons, 2015.

[32] X. Yang and S. A. Nassar, “Elasto-Plastic Clamp Load Analysis of Bolted Joint for Bolt Strain Hardening Material Under Separating Loading,” 2007.

[33] M. Groper, “Microslip and macroslip in bolted joints,” Exp. Mech., vol. 25, no. 2, pp. 171–174, Jun. 1985.

[34] Q. Wang, X. J. Zhou, Z. L. Liu, and H. J. Bi, “Performance Analysis and Optimization of Flange Bolt Joint under the Influence of Inhomogeneous Temperature Field,” in Applied Mechanics and Materials, 2012, vol. 121, pp. 1656–1662.

[35] S. S. Rao, Engineering Optimization: Theory and Practice, Fourth. John Wiley & Sons, 2009.

[36] R. V. Rao and V. J. Savsani, Mechanical Design Optimization Using Advanced Optimization Techniques. Springer Science & Business Media, 2012.

[37] R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in

Proceedings of the sixth international symposium on micro machine and human science, 1995, vol. 1, pp. 39–43.

[38] Y. Feng, Y.-M. Yao, and A.-X. Wang, “Comparing with chaotic inertia weights in particle swarm optimization,” in Machine Learning and Cybernetics, 2007 International Conference on, 2007, vol. 1, pp. 329–333.

[39] R. E. Perez and K. Behdinan, “Particle swarm approach for structural design optimization,” Comput. Struct., vol. 85, no. 19–20, pp. 1579–1588, Oct. 2007.

[40] J. S. Arora, Introduction to Optimum Design. Elsevier, 2012.

[41] C. C. Coello, G. B. Lamont, and D. A. van Veldhuizen, Evolutionary Algorithms for Solving Multi-Objective Problems. Springer Science & Business Media, 2007.

82

[42] A. Konak, D. W. Coit, and A. E. Smith, “Multi-objective optimization using genetic algorithms: A tutorial,” Reliab. Eng. Syst. Saf., vol. 91, no. 9, pp. 992–1007, Sep. 2006.

[43] S. Koziel, D. E. Ciaurri, and L. Leifsson, “Surrogate-based methods,” in Computational

Optimization, Methods and Algorithms, Springer, 2011, pp. 33–59.

[44] T. W. Simpson, J. D. Poplinski, P. N. Koch, and J. K. Allen, “Metamodels for computer-

based engineering design: survey and recommendations,” Eng. Comput., vol. 17, no. 2, pp. 129–150, 2001.

[45] M. Cavazzuti, Optimization Methods: From Theory to Design Scientific and

Technological Aspects in Mechanics. Springer Science & Business Media, 2012.

[46] G. G. Wang and S. Shan, “Review of metamodeling techniques in support of engineering

design optimization,” J. Mech. Des., vol. 129, no. 4, pp. 370–380, 2007.

[47] S. L. C. Ferreira, R. E. Bruns, H. S. Ferreira, G. D. Matos, J. M. David, G. C. Brandao, E. G. P. da Silva, L. A. Portugal, P. S. Dos Reis, and A. S. Souza, “Box-Behnken design: An

alternative for the optimization of analytical methods,” Anal. Chim. Acta, vol. 597, no. 2, pp. 179–186, 2007.

[48] A. Forrester, D. A. Sobester, and A. Keane, Engineering Design via Surrogate Modelling: A Practical Guide. John Wiley & Sons, 2008.

[49] N. V Queipo, R. T. Haftka, W. Shyy, T. Goel, R. Vaidyanathan, and P. K. Tucker,

“Surrogate-based analysis and optimization,” Prog. Aerosp. Sci., vol. 41, no. 1, pp. 1–28, 2005.

[50] Z.-H. Han and K.-S. Zhang, Surrogate-Based Optimization. INTECH Open Access Publisher, 2012.

[51] Y. Mack, T. Goel, W. Shyy, and R. Haftka, “Surrogate Model-Based Optimization

Framework: A Case Study in Aerospace Design,” in Evolutionary Computation in Dynamic and Uncertain Environments SE - 14, vol. 51, S. Yang, Y.-S. Ong, and Y. Jin,

Eds. Springer Berlin Heidelberg, 2007, pp. 323–342.

[52] S. Sakata, F. Ashida, and M. Zako, “Structural optimization using Kriging approximation,” Comput. Methods Appl. Mech. Eng., vol. 192, no. 7–8, pp. 923–939, Feb.

2003.

[53] D. Krige, “A Statistical Approach to Some Basic Mine Valuation Problems on the

Witwatersrand,” J. Chem. Metall. Min. Soc. South Africa, vol. 52, no. 6, pp. 119–139, Dec. 1951.

[54] L. Bajer and M. Holena, “RBF-based surrogate model for evolutionary optimization.,” in

ITAT, 2012, pp. 3–8.

83

[55] P. Hajela and L. Berke, “Neural networks in structural analysis and design: An overview,” Comput. Syst. Eng., vol. 3, no. 1–4, pp. 525–538, Jan. 1992.

[56] Z. Waszczyszyn, Neural networks in the analysis and design of structures. Springer, 1999.

[57] K.-H. Chang, Design Theory and Methods using CAD/CAE: The Computer Aided Engineering Design Series. Academic Press, 2014.

[58] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” Evol. Comput. IEEE Trans., vol. 6, no. 2, pp. 182–197, 2002.

[59] “Multiobjective Genetic Algorithms,” in Network Models and Optimization SE - 1, Springer London, 2008, pp. 1–47.

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Appendix A: Force Convergence Plots

Figure 43 - External tensile loading case convergence plot

Figure 44 - External transverse loading case convergence plot

85

Figure 45 – External torsional loading case convergence plot

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Appendix B: Multi-Objective Genetic Algorithm (MOGA) Pseudocode

Multi-Objective Genetic Algorithm (MOGA)

Initialize population

Assess objective values

Allocate rank depending upon Pareto dominance

Calculate niche count

Allot scaled fitness

For j = 1 to N

Selection through stochastic sampling

Crossover

Mutation

Determine objective values

Allocate rank depending upon Pareto dominance

Calculate niche count

Allot scaled fitness

Designate shared fitness

End

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