Monte Carlo Simulation on Ultimate Bending Capacityof Hybrid Composite Girder

XU YIFEI(Master’s Program in Engineering Mechanics and Energy)

Advised by MATSUSHIMA Takashi

Submitted to the Graduate School of
Systems and Information Engineeringin Partial Fulfillment of the Requirements
for the Degree of Master of Scienceat theUniversity of Tsukuba

March 2021

ABSTRACTIn the past, most bridges in Japan were designed using the ASD method (Allowable
Stress Design Method). On the other hand, other countries, like America and China, takethe LSD method (Limit State Design Method) as the main design method of bridges andother structures. Moreover, After the Japanese economic bubble burst, the Japanese
economy came to the great depression, and the Japanese construction industry also faced
great challenges, which required to save material using and low construction cost. Since
the ASD method requires that all sections meet the maximum stress requirements, it will
cost more material.
Due to the external environment of the construction industry and the domestic
economic situation, in order to solve above problems, a new structure called Hybrid
composite girder (HCGs) is developed to design bridge structures. Hybrid composite
girder is a kind of composite girder (NCGs) which contains a hybrid steel girder and aconcrete slab. The hybrid steel girder is composed of a lower flange made by highstrength steel while the upper flange and web are made of normal steel. The objective of
this study is to find a method to value the structural reliability and assess the structural
applicability by comparing different cases. This study does numerical experiments basedon Monte Carlo Method contributing the statistic models. Furthermore, Fiber Method andFEM (Finite Element Method) are adopted to value the structural reliability.
Firstly, in the Chapter 2, the basic contend of computing method is introduced,
including the calculation procedure of Fiber Method, the basic ideas of Finite Element
Method and the application of the Monte Carlo Method. The usage method of Fiber
Method and Finite Element Method are imported in this section, and this section also
discusses the relation variable numbers and convergency about Monte Carlo Method.
The numerical models for simulation are showed in the Chapter 3. Three different
section dimensions (slender, standard and compact) used in Fiber Method calculation andthe two kinds of element type (solid element and shell element) used in Finite Element
Method are introduced.
Chapter 4 is the most important section in this thesis. In Chapter 4, due to the statistical
calculation, the random variables of material and section dimensions are generated toincrease the reliability of the results. The influence of section dimensions on structuralreliability is discussed in this part, and the factors, which influence the structuralreliability, are tried to be found.

iChapter 5 shows the results of simulation based on Fiber method and Finite Element
method. The result of bending ultimate calculation is outlined in Chapter 5.2 and thebuckling ultimate calculation is outlined in Chapter 5.3. Based on the statistical results
and reliability calculation, the reliability of bending capacity about each model could becompared quantitatively. It could be concluded that buckling is an important factor
affecting structural stability. Buckling analysis should be considered for structures whichcontain thin shells part. And the compact model shows to have a better performance torestrain buckling, and compact NCGs shows to have a higher reliability among thosesituations.
Conclusions and future work are outlined in Chapter 6.

ii

FIGURE INDEX
Figure 1.1 Conceptual model of HCGs and NCGs ............................................... 2
Figure 1.2 Rectangular stress distribution during full plastic bending moment ...... 3
Figure 2.1 Conceptual picture of calculating bending capacity ............................. 6
Figure 2.2 The computing process of Fiber Method6) ........................................... 7
Figure 2.3 Relation between numbers of random variations and average, standard
deviation of 𝑓𝑐 ............................................................................................ 9
Figure 2.4 Normal Distribution .......................................................................... 9
Figure 2.5 Log-normal Distribution .................................................................... 9
Figure 2.6 Weibull Distribution ........................................................................... 9
Figure 3.1 Three models of different section dimensions on composite girder ...... 10
Figure 3.2 Element SOILD65 geometry (ANSYS, 2012)12) .................................. 11
Figure 3.3 Element SHELL181 geometry (ANSYS, 2012) 12)................................ 12
Figure 3.4 Conceptual model (One-Girder model) ............................................. 12
Figure 3.5 Conceptual model (Composite beam)13) ............................................ 13
Figure 3.6 Simulation result of displacement about ANSYS about Composite beam
(10 times scale) .......................................................................................... 14
Figure 4.1 The strain-stress curve of concrete .................................................... 16
Figure 4.2 The strain-stress curve of steel.......................................................... 17
Figure 4.3 Conceptual picture of reliability index .............................................. 19
Figure 5.1 Average and Standard Deviation of NCGs and HCGs (Standard model)................................................................................................................. 23
Figure 5.2 Average of NCGs and HCGs (Slender model and Compact model) ..... 23
Figure 5.3 Standard Deviation of NCGs and HCGs (Slender and Compact model)................................................................................................................. 24
Figure 5.4 Kurtosis and Skewness of NCGs and HCGs (Standard model) ............ 24
Figure 5.5 Reliability index and Failure probability (NCGs and HCGs) .............. 26
Figure 5.6 Reliability index and Failure probability (Slender, Standard and Compact)................................................................................................................. 26
Figure 5.7 Histograms of destruction numbers (Slender model) .......................... 27
Figure 5.8 Histograms of destruction numbers (Standard model) ........................ 28iii

Figure 5.9 Histograms of destruction numbers (Compact model) ........................ 28
Figure 5.10 Supplement to One-Girder model (Two-Girder model) ..................... 29
Figure 5.11 Overall conceptual model (Two-Girder model) ................................ 30
Figure 5.12 Total displacement of Standard HCGs model (10 times scale) .......... 31
Figure 5.13 Total mechanical strain of Compact HCGs model (10 times scale) ... 31
Figure 5.14 Graphical illustration of Riks method17) .......................................... 32
Figure 5.15 Relation of displacement and reaction force (Standard HCGs model)32
Figure 5.16 Average and Standard Deviation of NCGs and HCGs (Weibull
Distribution) .............................................................................................. 34
Figure 5.17 Reliability index and Failure probability of FEM (NCGs and HCGs) 35
Figure 5.18 Reliability index and Failure probability (𝛼 = 1.05,1.15) ................ 36
Figure 5.19 Histograms of destruction numbers about FEM (NCGs and HCGs) .. 36iv

TABLE INDEX
Table 3.1 Model properties of Composite beam.................................................. 13
Table 4.1 Parameters of section members (slender (standard compact)).............. 15
Table 4.2 Parameters of concrete ...................................................................... 18
Table 4.3 Parameters of steel ............................................................................ 18
Table 5.1 Histograms of bending capacity (Normal Distribution) ....................... 20
Table 5.2 Histograms of bending capacity (Log-normal Distribution) ................. 21
Table 5.3 Histograms of bending capacity (Weibull Distribution)........................ 21
Table 5.4 Statistics result of Normal Distribution ............................................... 22
Table 5.5 Statistics result of Log-normal Distribution ........................................ 22
Table 5.6 Statistics result of Weibull Distribution ............................................... 22
Table 5.7 Histograms of bending capacity for FEM (Weibull Distribution) .......... 33
Table 5.8 Statistics result of FEM simulation (Weibull Distribution) ................... 33v

CONTENTS
ABSTRACT ............................................................................................................. i
FIGURE INDEX .................................................................................................... iii
TABLE INDEX ....................................................................................................... v
Chapter 1. INTRODUCTION ................................................................................. 1
1.1 Background............................................................................................................ 1
1.2 Previous researches on HCGs .............................................................................. 2
1.3 Objective ................................................................................................................ 3
1.4 Overview ................................................................................................................ 4
Chapter 2. COMPUTING METHOD ..................................................................... 5
2.1 Introduction ........................................................................................................... 5
2.2 Bending Capacity .................................................................................................. 5
2.2.1 Fiber Method ........................................................................................................ 5
2.2.2 Finite Element Method......................................................................................... 6
2.3 Monte Carlo Method ............................................................................................ 8
2.3.1 Probability Density Function ............................................................................... 8
2.3.2 Discussion about Convergency of Simulation Result .......................................... 9
Chapter 3. NUMERICAL MODEL....................................................................... 10
3.1 Fiber Method ....................................................................................................... 10
3.2 Finite Element Method ....................................................................................... 10
3.2.1 Defined Element Type.........................................................................................11
3.2.2 Modeling FEM Model ....................................................................................... 12
3.2.3 Verification of ANSYS Program ........................................................................ 13
Chapter 4. NUMERICAL EXPERIMENT ........................................................... 15
4.1 Introduction ......................................................................................................... 15
4.2 Variables on Computing Model ......................................................................... 15
4.2.1 Variables on Section Members ........................................................................... 15
4.2.2 Variables on Material Properties ........................................................................ 16vi

4.3 Reliability Index and Failure Probability ......................................................... 18
Chapter 5. RESULTS ABOUT SIMULATION ..................................................... 20
5.1 Introduction ......................................................................................................... 20
5.2 Fiber Method ....................................................................................................... 20
5.2.1 Statistical Calculation ........................................................................................ 20
5.2.2 Statistical Results and Discussions .................................................................... 23
5.2.3 Results and Discussions about Structural Reliability ........................................ 25
5.3 Finite Element Method ....................................................................................... 29
5.3.1 From One-Girder Model to Two-Girder Model ................................................. 29
5.3.2 Statistical Calculation and Results ..................................................................... 30
5.3.3 Results and Discussions about Structural Reliability ........................................ 34
Chapter 6. CONLUSION AND FUTUER WORK ................................................ 38
6.1 Conclusion............................................................................................................ 38
6.2 Future Work ........................................................................................................ 40
REFERENCES .......................................................................................................... 41
ACKNOWLEDGE .................................................................................................... 43

vii

Chapter 1. INTRODUCTION
1.1 Background
Civil engineering, as the basis of living and acting of humanbeings, its safety is an eter
nal topic. As an empirical discipline, civil engineering has a long history and is widelyused, but there is a lack of systematic theoretical system for itself. Since modern times,
numerous scientific research achievements have provided theoretical basis for civilengineering discipline. Due to the development and application of new materials, the span,height and strength of constructions have been greatly improved. Therefore, higher
requirements are put forward for structural safety, rationality and reliability in theconstruction industry.
In the past, ASD method (Allowable Stress Design Method) was used for the structure
design in the elastic design of steel bridges in Japan based on the yield point. ASD method,which is a kind of stress-based design method, stipulated that the maximum stress on thesection of structural member in service stage shall not exceed the allowable stress ofmaterial. This method does not consider the nonlinear properties of materials, and ignores
the difference between the actual bearing capacity of the structure and the results
calculated by elastic method. The values of load and allowable stress of materials are
determined by experience and lack of scientific basis. On the other hand, overseas
countries take LSD method (Limit State Design Method) as the mainstream design. LSD
method is a kind of performance-based design method, which divided the limit state ofstructure into bearing capacity limit state and serviceability limit state. The former
requires that the minimum bearing capacity of the structure should not be less than thesection internal force caused by the possible maximum external load. The latter limited
the deformation and crack formation or cracking degree of structure. It could be referredthat LSD method is showed to be a more reasonable design method. It is known that "theRoad Bridge Specifications1)" is widely used to design bridge structures in Japan. In 2017,
it imported LSD method of the background of widely using LSD method in international
bridge design. The LSD method considers one or more than one safety factors, like thevariation of load, material properties and working conditions.
On the other hand, after the Japanese economic bubble burst, the Japanese economy
came to the great depression, and the Japanese construction industry also faced greatchallenges, which required to save material using and low construction cost, there are lotsof attempts for the new material exploitation and reducing costs. According to different
countries, however, the allowable values of steel performance and manufacturing error
1are also different, there is no reference from other situations. Hence, it’s necessary to do
the research independently based on existing design, fabrication standard and material
characteristics to find a new design method.
Therefore, due to the external technical conditions of innovation environment and theplight of the internal construction industry, in order to solve above problems, a new
structure called Hybrid composite girder (HCGs) 2), 3) is developed to design bridge
structures. Hybrid composite girder is a kind of composite girder (NCGs) which contains
a hybrid steel girder and a concrete slab. The hybrid steel girder is composed of a lower
flange made by high-performance steel while the upper flange and web are made ofnormal steel. Since the lower flange is regarded as the most important member for
restraining tension stress, and high-performance steel has a higher yield stress, HCGs
could be considered as a rational structure. And the two models are widely used in thebridge structures design. Both of two structures are showed in Figure 1.1.

concreteconcrete

steel

steel

high-performance
steel

steel

Figure 1.1 Conceptual model of HCGs and NCGs
For these situations, the applicability and rationality of HCGs and NCGs need to be
fully and scientifically discussed.

1.2 Previous researches on HCGs
The research on Hybrid composite girder has been doing for decades and it comes out
primary result. An existing study showed that HCGs are feasible and practical structures.4).
In that study, excellent bending resistance performance is shown at the end of the hybrid
composite girders. In the experiment, local buckling, lateral buckling, and shear buckling2

of web in shear span are showed to do not occur. And the failure is confirmed as the crush
of upper edge about concrete slab. The full plastic bending moment, defined as 𝑀𝑝 ,(Figure 1.2) is regarded as the simple design basis to value the bending capacity.

Square Stress Distribution
0.85𝑓𝑐Neutral Axis

𝜎𝑦

𝜎𝑦
𝜎𝑌
Figure 1.2 Rectangular stress distribution during full plastic bending moment
All of the previous research are about to do the simulation calculate the bending
capacity to value the structural reliability.

1.3 Objective
It is said that hybrid composite girders could reduce 10% use of steel whereas normalcomposite girders could reduce 20% use of steel5). Thus, it’s difficult to conclude that the
HCGs have superiority than the NCGs in all factors. Both of hybrid composite girders
and normal composite girders are suggested to be reasonable structures. It’s difficult topredict the load carrying behavior of structures with nonlinear regions of materials.
Additionally, there is uncertainty in the material and the members.
In order to take advantage of LSD method, the extent of the tolerance to the load
carrying behavior of the structure should be clarified theoretically. The method of
obtaining the probability distribution of the bending capacity is necessary. For thecomposite girder, there are some examples6-8) of the bending capacity distribution basedon the variation of the material parameters.
In the reference 6), there is a new design method for calculating structural bending
3

capacity with the probability variations of material parameters by using Monte Carlosimulation. In this study, three kinds of typical probability density function, like Normal
Distribution, Log-normal Distribution and Weibull Distribution, are selected to set
variations. Using statistic calculation to evaluate the bending capacity. Thus, there are stillseveral problems need to be solved and the objectives of this research are to find a method
to calculate the reliability of composite girder and search the relation between the section
dimension and structural reliability.

1.4 Overview
Chapter 1 is a general introduction of this research. Relevant previous studies andobjective of this thesis are introduced after the background.
In Chapter 2, the basic contend of computing method is introduced, including the
calculation procedure of Fiber Method, the basic ideas of Finite Element Method. Bothtwo simulation method are used combined with Monte Carlo Method.
In Chapter 3 introduces the numerical models for simulation. Three different section
dimensions (slender, standard and compact) used in Fiber Method calculation and the twokinds of element type (solid element and shell element) used for Finite Element Methodmodeling.
In Chapter 4, due to the statistical calculation, the random variables of material andsection dimensions are generated to make the results more reliable. The influence ofsection dimensions on structural reliability is discussed in this part, and the factors, which
influence the structural reliability, are tried to be found.
The simulation results and discussions are showed in Chapter 5. Statistical calculations
are adopted to do the simulation, and the structural reliability could be quantitative
analysis by calculating the reliability index and failure probability.
Conclusions of above work and future work are presented in Chapter 8.

4Chapter 2. COMPUTING METHOD
2.1 Introduction
To quantitatively analyze the structural reliability, numerical experiment should bedone to calculate the bending capacity of each model. This chapter provides computing
methods to calculate structural bending capacity. During the analysis of the linear elastic
stage, the Fiber Method is adopted to calculate the bending capacity. Furthermore, theFinite Element Method (FEM) is a numerical method for modeling continuum with finite
element and is accepted to do the simulation in the nonlinear and elastoplastic stage.
The structure of this chapter is showed as following. In section.2, it will state theapplication of both two simulation method for the numerical experiment. Section.3
introduce the Monte Carlo Method and the combination between the Monte Carlo Method
and both two computing method.

2.2 Bending Capacity
2.2.1 Fiber Method
The Fiber Method is adopted to calculate the bending capacity of hybrid composite
girders (HCGs) and normal composite girders (NCGs).
The Fiber Method is a method to calculate the axial force and the bending moment by
cutting the total length into small parts. First, the crushing of the upper edge of concreteslab is defined as the ultimate limit state. For the fixed ultimate strain, the position, where
the axial force (𝑑𝑦 × 𝑆 × 𝜎𝑦 ) equals zero, is regarded as the neutral axis of the structure(Figure 2.1). Then calculate the bending moment, which is defined as ultimate bending
moment 𝑀𝑢 .
However, the plane holding assumption remains true, and the steel girders are
regarded as not occurring local buckling or lateral buckling and other brittle failure
phenomenon. And the bending capacity could be calculated as 𝑀𝑢 /𝑀𝑝 . By
dimensionless the capacity, it can be used to quantitatively analyze the structuralreliability, so that the difference of section dimensions of structures can be neglected. The
larger 𝑀𝑢 /𝑀𝑝 is, the more the strength of the material could be fully utilized. The
computing process6) of the Fiber Method is showed in Figure 2.2.

5

Concrete Crush Ultimate Limit State

Design Index: 𝑀𝑢 Full-Plastic : 𝑀𝑝
Ultimate
Bending Moment Bending Moment

Figure 2.1 Conceptual picture of calculating bending capacity
The rectangular stress distribution during full plastic bending moment 𝑀𝑝 which isshowed in Figure 1.2, does not necessarily represent the bending moment, but because itis easy to determine by calculating, it is used as an approximation formula for computing
ultimate bending capacity. In this thesis, the mean value of each material parameter iscalculated in the calculation of full plastic bending moment. For the composite girders,
𝑀𝑝 is calculated to normalize the 𝑀𝑢 distribution.
2.2.2 Finite Element Method
FEM is an efficient and commonly used calculation method. It discretizes a continuum
into a set of finite elements to solve continuum mechanics problems. In 1956, when theplane element method was successfully applied to the plane stress analysis of plane frame
by Turner, Clough et al.
In recent years, the application of finite element method in plastic processing has beendeveloped rapidly. The emergence of elastic finite element, rigid plastic finite element,
elastic-plastic finite element, viscoplastic finite element and other related theories andmethods provide powerful tools for analyzing various levels of problems in plastic
processing, including macro quantity, distribution quantity and micro quantity. However,for complex structures and some nonlinear problems, there are still many studies on thefinite element method, which have not yet been standardized and standardized.
6

Figure 2.2 The computing process of Fiber Method6)
For the composite girders, in order to reduce the cost, inevitable, the usage of steel
should be cut. This means that the girder will be construct thinner, thus, the structure ismore likely to occur buckling. Since the Fiber Method only considers the structuralchanges in the linear stage, the nonlinear stage of structural instability has not been
discussed. Large deformation and buckling analysis of structures need to be discussed by
FEM.
Buckling mainly refers to the phenomenon of structural instability under specific loads.
It always accompanies with the plastic deformation, and the structure will suddenly jump
to another random balance state. Nonlinear buckling analysis and linear (or eigenvalue)
buckling analysis are two techniques available in the finite element program forpredicting the buckling mode shape of a structure.9) And it is said that the linear buckling
analysis (by linearized eigenvalue) is generally overestimated. On the contrary, thenonlinear buckling analysis is showed to be more stable and more accurate10). Because of7the complexity and uncertainty of constitutive equation, the calculation of FEM part istaken by ANSYS program.

2.3 Monte Carlo Method
Monte Carlo method is a very important numerical calculation method guided byprobability and statistics theory. It refers to the use of random numbers (or more common
pseudo-random numbers) to solve many computational problems. Theoretically, Monte
Carlo method needs a lot of experiments. The more experiments, the more accurate theresults are.
2.3.1 Probability Density Function
Using Monte Carlo Method could make the results more reliable, modeling probability
density functions. When increasing the number of random variations, the results tend to
be more accurate. In order to see more possibilities of the results, three typical probability
density functions are used. (Normal Distribution, Log-normal Distribution and Weibull
Distribution). And the equations11) of these functions are outlined in Eq. (2.1), (2.2) and(2.3).
𝑓(𝑥) =

1 𝑥−𝜇 2
exp [− (
) ]
2
𝜎
√2𝜋𝜎

(2.1)

1 𝑙𝑛 𝑥 − 𝜆 2
exp [− (
) ]
2
𝜁
√2𝜋𝜁𝑥

(2.2)

1

𝑓(𝑥) =
𝑓(𝑥) =

1

1
1
𝑚𝑥 𝑚−1 exp [− 𝑚 𝑥 𝑚 ]
𝑚
𝛾
𝛾

(2.3)

The parameters 𝜆 , 𝜁, of the Log-normal Distribution and 𝑚 and 𝜂 of Weibull
Distribution are obtained by Eq. (2.4), (2.5), (2.6) and (2.7), respectively.
1
𝜆 = 𝑙𝑛 𝜇 − 𝜁 2
2

(2.4)

𝜎2
)
𝜇2

(2.5)

1
+ 1)
𝑚

(2.6)

2
1
+ 1) − 𝛤 2 ( + 1)
𝑚
𝑚

(2.7)

𝜁 = √𝑙𝑛 (1 +𝜇 = 𝜂𝛤 (𝜎 = 𝜂√𝛤 (8

2.3.2 Discussion about Convergency of Simulation Result
It’s required to set random variations of Monte Carlo simulation. Therefore, it isnecessary to discuss the convergence relationship between random variations and value.
Take the concrete compression strength as an example to set random variations(average:30, standard deviation:1.2), Observe the convergence of the numerical value anddetermine the number of random variations, which are showed in Figure 2.3. (averageand standard deviation)

Figure 2.3 Relation between numbers of random variations and average, standard
deviation of 𝒇𝒄
Due to the results showed above, it could be concluded that the average and standard
deviation could get convergent when the numbers of random variations reaches about
10,000. It shows that with the increase of random numbers, its value is closer to the design
value. And the histograms of the three functions about the example ( 𝑓𝑐 : cconcrete
compression strength) are outlined in Figure 2.4, 2.5 and 2.6.

Figure 2.4 Normal
Distribution

Figure 2.5 Log-normal
Distribution
9

Figure 2.6 Weibull
Distribution

Chapter 3. NUMERICAL MODEL
In this chapter, we will introduce the numerical model of the simulation, including
simulation model of Fiber Method (slender, standard and compact) and the process formodeling FEM model.

3.1 Fiber Method
Based on the previous researches, in order to make the results of numerical simulation
more reliable, more cases need to be considered to discuss the influence of various
parameters on structural reliability. In order to find a more appropriate model for
calculating, the influence of the section dimensions on structural reliability is consideredin the section. Three kinds of section dimensions of both structures (NCGs and HCGs), a
slender one, a standard one and a compact one (showed in Figure 3.1) are selected tomodeling simulation models.

𝑡𝑓𝑡 =15

𝑡𝑓𝑡 =12

slender

𝑡𝑤 =12standard

𝑡𝑓𝑏 =24

𝑡𝑤 =15𝑡𝑓𝑏 =40

𝑡𝑓𝑡 =24compact

𝑡𝑤 =24

𝑡𝑓𝑏 =64

Figure 3.1 Three models of different section dimensions on composite girder
Based the three section dimensions showed above for modeling, using Fiber Method
to do the statistical calculation. Compare the results to get the structure reliability and amore reasonable structure.

3.2 Finite Element Method
From the reference 10), we could know that linear buckling analysis generally yields
unconservative results. Hence, it’s necessary to do the nonlinear buckling analysis. A
10

nonlinear buckling analysis is based of linear buckling analysis. With the increase of theload, the structure occurs the nonlinear deformation and get its ultimate state.
3.2.1 Defined Element TypeFor FEM modeling, there are two kinds of element types adopted for define structure
members. The concrete slab and composite girder are modeled separately.
For the concrete slab, solid element is accepted for modeling. In ANSYS, element
SOILD65 is a common element type used to modeling concrete. This element is used for3D modeling of solids with or without rebar. The solid is capable of cracking in tension
and crushing in compression. Figure 3.2 shows the geometry of SOLID65.

Figure 3.2 Element SOILD65 geometry (ANSYS, 2012)12)On the other hand, for the steel members, shell element is considered to modeling thesteel girder. In ANSYS, element SHELL181 is a common element type used to modelingthin members. Because SHELL181 is first order shear deformation shell and has finite
strain capability. Hence, it is considered as the model for bending and shearing calculation.
Furthermore, SHELL181 can have all applicable nonlinear materials, with both
membrane stresses, enhanced strain effects, and transverse shear stresses, accounted for.
And SHELL181 has a more advanced warping correction algorithm, making the element
stiffness less sensitive to warping, so it is regarded as an appropriate element type forsteel material. Figure 3.3 shows the geometry of SHELL181.

11

Figure 3.3 Element SHELL181 geometry (ANSYS, 2012) 12)
3.2.2 Modeling FEM ModelAt first, we modeling a one-span girder as FEM model. The total length of the structure
is set as 30 meters. Using the solid elements to modeling the concrete slab, and the shellelements are applied to the composite girder modeling. The total structure is set up withmore than 20,000 elements, set constraints on the two edge of bottom flange to avoid
lateral displacement (One end constrained omnidirectional displacement and The otherend constrains longitudinal and lateral displacements), then set the vertical distributedload at one third and two third of the total length, which is easy for structure deformation.
The concrete slab and the composite girder are connected in the fixed way. And theconceptual model is showed in the Figure 3.4.

Figure 3.4 Conceptual model (One-Girder model)
123.2.3 Verification of ANSYS Program
In order to ensure the accuracy of ANSYS calculation, it is necessary to do theverification about ANSYS program. Because there is no definite theoretical solution in
the nonlinear stage, we concentrate on the linear stage of verification.
Verification: Bending of a composite beam

A beam of length 𝑙 and width 𝑤 , made up of two layers of different materials, issubjected to a uniform rise in temperature from 𝑇𝑟𝑒𝑓 to 𝑇𝑜 and a bending moment 𝑀𝑦
at the free-end. Determine the free-end displacement 𝛿 (in the Z-direction) and the Xdirection stresses at the top and bottom surfaces of the layered beam. 𝐸𝑖 and 𝛼𝑖correspond to the Young's modulus and thermal coefficient of expansion for layer 𝑖 ,respectively. And the conceptual model is showed in Figure 3.5, the model properties are
showed in Table 3.1.

Figure 3.5 Conceptual model (Composite beam)13)
Table 3.1 Model properties of Composite beam
Material Properties

Geometric Properties

13

LoadingMaterialⅠ(red):
𝐸1 = 1.2 × 106 𝑝𝑠𝑖−4

𝑙 = 8𝑖𝑛𝑜𝛼1 = 1.8 × 10 𝑖𝑛/𝑖𝑛/ 𝐹

𝑤 = 0.5𝑖𝑛MaterialⅡ(grey):ℎ1 (𝑟𝑒𝑑) = 0.2𝑖𝑛

𝐸2 = 0.4 × 106 𝑝𝑠𝑖ℎ2 (𝑔𝑟𝑒𝑦) = 0.1𝑖𝑛

𝑇𝑜 = 100 𝑜𝐹
𝑇𝑟𝑒𝑓 = 0 𝑜𝐹
𝑀𝑦 = 10𝑖𝑛-𝑙𝑏

𝛼2 = 0.6 × 10−4 𝑖𝑛/𝑖𝑛/ 𝑜𝐹
It refers to the reference 11, using SOLID185 to modeling the composite beam, thedisplacement of 𝑍 direction is calculated as 0.832𝑚𝑚.13)
Then, the simulation result by ANSYS Program is showed in Figure 3.6. And thecomparison and error analysis is stated as following.

Figure 3.6 Simulation result of displacement about ANSYS about Composite beam (10times scale)
It could be observed that the simulation result of maximum displacement is about
0.83181𝑚𝑚 . Thus, the error between the theoretical solution and numerical solution
0.0228%.
Therefore, it can be approximately considered that the results obtained by ANSYS
program are reliable due to the verification result.
14

Chapter 4. NUMERICAL EXPERIMENT
4.1 Introduction
In order to quantitatively calculate the structural reliability and make the results morereliable. Numerical experiment is adopted to do the simulation. To increase the degree ofdata dispersion, we set variations of section members and material properties by theiraverage and standard deviation.
The structure of this chapter is showed as following. Section.2 gives the values (design
value, average and standard deviation) of each parameter. Introduce the relation between
stress and strain of materials (concrete and steel). Section.3 introduces the definition ofreliability index and failure probability and gives the equation for calculating thereliability index.

4.2 Variables on Computing Model
4.2.1 Variables on Section Members
For the computing model, there are three kinds of section dimensions (slender model,
standard model and compact model) and the parameters of section members about eachmodel should be determined.
When determining the section members, it is assumed that the central vertical axis
always exists on the steel web to make the concrete bear the compressive force in theultimate state. In the numerical experiment, there are three kinds of structural section
dimensions, and the parameters of section members14) are outlined in Table 4.1. The data
are slender (standard, compact) in turn.

Table 4.1 Parameters of section members (slender (standard compact))
Variation

AverageStandard Variation

𝜇(mm)

𝜎(mm)

Slab Width：𝑤𝑐

1500

6

0.0040Slab Thickness：𝑡𝑐

160

60.0375Upper Flange Width：𝑤𝑓𝑡

300

4.38

0.014612 (15, 24)

0.175 (0.219, 0.350)

0.014612 (15, 24)

0.175

0.0146

Parameters

Upper Flange Thickness：𝑡𝑓𝑡
Web Thickness：𝑡𝑤

15

Coefficient𝐶𝑉Web Height：𝑑𝑤

1500

21.9

0.0146

Bottom Flange Width：𝑤𝑓𝑏

300

4.38

0.014624 (40, 64)

0.350 (0.584, 0.934)

0.0146

Bottom Flange Thickness：
𝑡𝑓𝑏

Using the parameters outlined above to modeling three different models for evaluating
the structural reliability.
4.2.2 Variables on Material Properties
To do the simulation, the equations of strain-stress about materials15) should be
acknowledged. The strain-stress curve of concrete is showed in the Figure 4.1 and theequations are outlined in Eq. (4.1).

𝜎𝑐0.85𝑓𝑐𝜀𝑐
0.002

0.0035Figure 4.1 The strain-stress curve of concrete𝜀𝑐𝜀𝑐) (2 −) ，𝜀𝑐 ≤ 0.002
0.002
0.002

𝜎𝑐 = 0.85𝑓𝑐 (𝜎𝑐 = 0.85𝑓𝑐 ，0.002 ≤ 𝜀𝑐 ≤ 0.0035
16

(4.1)

where 𝜀𝑐 is the strain of concrete, 𝜎𝑐 is the stress of concrete and 𝑓𝑐 is compression
strength of concrete.
Additionally, the strain-stress curve of normal steel and high-performance steel isshowed in the Figure 4.2 and the equations are outlined in Eq. (4.2).

𝜎𝑠
𝜎y

𝐸st

𝐸

𝜀𝑠
𝜀y

𝜀st

Figure 4.2 The strain-stress curve of steel

𝜎𝑠 = 𝐸𝜀𝑠，𝜀𝑠 ≤ 𝜀𝑦𝜎𝑠 = 𝜎𝑦 ，𝜀𝑦 ≤ 𝜀𝑠 ≤ 𝜀𝑠𝑡𝜎𝑠 1 𝐸𝑠𝑡
𝜀𝑠 𝜀𝑠𝑡=[1 − 𝑒𝑥𝑝 {−𝜉 ( − )}] + 1，𝜀𝑠 ≥ 𝜀𝑠𝑡𝜎𝑦 𝜉 𝐸
𝜀𝑦 𝜀𝑦

(4.2)

where 𝜀𝑠 is the strain of steel, 𝜎𝑠 is the stress of steel, 𝐸 is the young’s modulus ofsteel, 𝜀𝑦 is the yield strain of steel, 𝜎𝑦 is the yield stress of steel, 𝜀𝑠𝑡 is the hardening
strain of steel, 𝐸𝑠𝑡 is the hardening coefficient of steel and 𝜉 is the hardening curvatureof steel.
The statistic parameters of each material are acknowledged from the reference 6). The
parameters of concrete are outlined in Table 4.2, and the parameters of steel are outlinedin Table 4.3.
17

Table 4.2 Parameters of concrete
Parameters
Compression Strength 𝑓𝑐（N/mm2）
Crushing Strain 𝜀𝑐

Average 𝜇Standard Variation 𝜎

Variation Coefficient 𝐶𝑉

30

1.2

0.04

0.0035
Table 4.3 Parameters of steel
steel

AverageStandard𝜇Variation 𝜎

2000002000

293.75high-performance steel
Variation

AverageStandard

Variation𝜇

Variation 𝜎

Coefficient CV

0.01

200000

2000

0.01

23.5

0.08

549

36

0.06560.01850.00490.265

0.00820.0041

0.5

4156

1342

0.323

2000

1170

0.585

0.049

0.027

0.55

0.02

0.025

1.25

Coefficient𝐶𝑉

Young's
Modulus 𝐸
2（N/mm ）
Yield Strength𝜎𝑦 （N/mm2）
hardening strain𝜀𝑠𝑡Hardening
Coefficient2

𝐸𝑠𝑡 （N/mm ）Hardening
Curvature2

𝜉（N/mm ）4.3 Reliability Index and Failure Probability
When it comes to the structural reliability, it always refers to the reliability index 𝛽.
The reliability index is divided by the standard deviation to the average value for thedesigned load. And the conceptual picture is showed in Figure 4.3 while the equation of
the reliability index is outlined in Eq. (4.3).

18

Destruction

Figure 4.3 Conceptual picture of reliability index𝛽=𝜇𝑧 − 𝛼 𝜇𝑧′
=
𝜎𝑧
𝜎𝑧

(4.3)

Where, 𝜇𝑧 and 𝜎𝑧 are the average and standard deviation of bending capacity, 𝛼 is thefactor of safety. It is said that 𝛼 ranges from 0.59-1.0016).
In this part, the probability distribution of bending capacity is calculated by assuming
the normal distribution of all the strength distribution. This means that the probability
density functions of material and member parameters are not considered.
Failure probability is the probability of failure in the sample size, which is assumed tooccur below a certain bending capacity. Reliability index and failure probability are thecriteria for evaluating structural reliability. The structure with high reliability index andlow failure probability is considered as high reliability. And a low reliability and a highfailure probability indicates a low reliability.

19

Chapter 5. RESULTS ABOUT SIMULATION
5.1 Introduction
In this chapter, it provides histograms and other results of the hybrid composite girders
and normal composite girders. Statistical methods were used to evaluate the reliability ofthe structures. After statistical calculation, we could get the histograms of structural
bending capacity. With those results, structural reliability index and failure probability
could be calculated.
The structure of this chapter is showed as following. Section.2 gives the results of twomodels (HCGS and NCGs) in three section dimensions (slender, standard and compact)
and then discuss and analyze the results showed above. In section.3, the problem andsolution of FEM are showed to modeling the FEM model and then the results anddiscussions of FEM are outlined following.

5.2 Fiber Method
5.2.1 Statistical Calculation
In the calculation part of Fiber Method, 10000 random variations were generated forstatistical calculation. And the histograms about bending capacity of each case are
generated for comparison, the histograms of Normal Distribution are outlined in Table
5.1. Those of Log-normal Distribution are outlined in Table 5.2. And those of Weibull
Distribution are outlined in Table 5.3. All of the histograms are in the same horizontal
axis and vertical axis for directly observation. It could be observed that the histograms of
HCGs are narrower than those of NCGs, which indicate less discreteness of data.
Table 5.1 Histograms of bending capacity (Normal Distribution)
𝑀𝑢 /𝑀𝑝SlenderStandard

NCGs

20

Compact

HCGs

Table 5.2 Histograms of bending capacity (Log-normal Distribution)
𝑀𝑢 /𝑀𝑝SlenderStandard

Compact

NCGs

HCGs

Table 5.3 Histograms of bending capacity (Weibull Distribution)
𝑀𝑢 /𝑀𝑝SlenderStandard

NCGs

HCGs

21

Compact

After statistical calculation, the average, standard deviation, kurtosis and skewness are
obtained for analysis and discussion to evaluate the reliability of each case. The statistics
result of Normal distribution is showed in the Table 5.4, the statistics result of Log-normal
distribution is showed in the Table 5.5 and the statistics result of Weibull distribution isshowed in the Table 5.6.
Table 5.4 Statistics result of Normal Distribution
SlenderStandard

Compact

NCG

HCG

NCG

HCG

NCG

HCG

Average

1.0021

0.9921

0.99250.98530.98300.9345

Standard
deviation

0.0617

0.03600.05920.03340.05410.0309

Kurtosis-0.0573

0.0162-0.0356-0.0155-0.0235-0.0433

Skewness-0.0588-0.0644-0.1152-0.0711-0.0950-0.0605

Table 5.5 Statistics result of Log-normal Distribution
SlenderStandard

Compact

NCG

HCG

NCG

HCG

NCG

HCG

Average

1.00200.99180.99250.98490.98300.9345

Standard
deviation

0.06130.03600.05920.03320.05410.0310

Kurtosis0.02870.0149-0.0362-0.0196-0.0282-0.0409

Skewness

0.17080.05030.0106-0.01190.12110.0194

Table 5.6 Statistics result of Weibull Distribution
SlenderStandard

Compact

NCG

HCG

NCG

HCG

NCG

HCG

Average

1.00350.9927

0.99390.98620.98420.9347

Standard
deviation

0.06220.03690.06020.03490.05520.0314

Kurtosis0.94930.6136

1.21170.62551.38750.9705Skewness-0.7785-0.6468-0.8731-0.6314-0.8914-0.707422

5.2.2 Statistical Results and Discussions

NCGs
NCGs

HCGs

HCGs

Figure 5.1 Average and Standard Deviation of NCGs and HCGs (Standard model)
Take the standard model as an example, the upper lines are NCGs, and the lower lines
are HCGs. From the Figure 5.1 (left), it could be concluded that with the same section
dimensions, the average of bending capacity about NCGs is a little larger than that of
HCGs, which means that the NCGs models have a higher bending capacity than those of
HCGs models. And from the Figure 5.1 (right), it could be concluded that with the same
section dimensions, the standard deviation of bending capacity about NCGs is larger thanthat of HCGs, which means that the HCGs models have fewer disperse samples than thoseof NCGs models from the calculation by Fiber method.

Compact NCGs

Slender NCGs

Slender HCGs

Compact HCGs

Figure 5.2 Average of NCGs and HCGs (Slender model and Compact model)
The comparison charts about average of bending capacity are showed in the Figure5.2. From Figure 5.2, it could be concluded that with the same material composing, the23average of bending capacity about slender models is a little larger than that of compact
models, the larger the section dimensions are, the lower the average of bending capacity
is, which means that a thick steel girder could not be made fully use of its bending capacity.

Compact NCGs
Slender NCGs
Slender HCGs

Compact HCGs

Figure 5.3 Standard Deviation of NCGs and HCGs (Slender and Compact model)
The comparison charts about standard deviations of bending capacity are showed in
the Fig 5.3. From Fig 5.3, it could be concluded that with the same material composing,
the standard deviation of bending capacity about slender models is a little larger than thatof compact models, the larger the section dimensions are, the smaller the standard
deviation of bending capacity is, which means that a thick steel girder could provide ahigher structural stability and guarantee a higher data concentration.

NCGs
HCGs
HCGs

NCGs

Figure 5.4 Kurtosis and Skewness of NCGs and HCGs (Standard model)
24

Furthermore, kurtosis and skewness are showed in the Figure 5.4. Kurtosis is thecharacteristic number of the peak value of the probability density distribution curve at the
average value. Intuitively, kurtosis reflects the sharpness of the peak. Because the kurtosisof a normal distribution is equal to 3, 3 is usually subtracted from the kurtosis calculation.
High kurtosis means that the increase of variance is caused by the extreme values of lower
frequency greater than or less than the average value. Hence, due to the Figure 5.4 (left),it could be concluded that Weibull Distribution has more extreme values than other two
kinds of functions, which indicates more abnormal samples.
Skewness represents the degree of asymmetry of probability distribution density curvewith respect to the average value. Intuitively, it is the relative length of the tail of thedensity function curve. When the distribution is symmetrical, the skewness is equal to 0.
When the skewness is greater than 0, the heavy tail is on the right, the distribution is rightskewed. When the skewness is less than 0, the heavy tail is on the left, the distribution isskewed to the left. Hence, due to the Figure 5.4 (right), it could be concluded that Weibull
Distribution is skewed to the left, which indicates there are more values are less than the
average.
This shows that the Weibull Distribution contains more dangerous samples, the designusing Weibull Distribution will make the design safer.
5.2.3 Results and Discussions about Structural Reliability
After the statistical calculation, we could get the reliability index and failure probabilityof each model. Refer to the Chapter 4, according to the reference 16), in order to makethe sample larger, there fixed 𝛼 = 0.9, then the reliability index and failure probabilityof each case could be calculated.
These data are compared in the form of scatter plotted on the same graphs which are
outlined in Figure 5.5 and Figure 5.6. Figure 5.5 is about the comparison of same section
dimensions and Figure 5.6 is about the comparison of same material properties.
From the Figure 5.5, take the compact section dimension as an example, it could beobserved that with same section dimensions, HCGs have lower reliability than the NCGs.
Because concrete will arrive its ultimate state faster with a larger load, such structure
cannot make full use of the bending capacity of steel. On the other hand, the slender modeland standard model of HCGs have higher reliability than such structures of NCGs. Hence,
if the structure section dimensions are selected as compact one, use NCGs are safer. On
the contrary, if the structure section dimensions are selected as slender or standard one,use HCGs are showed to be safer.
25

Compact
HCGs

Risk

Weibull Dist.

Compact
Normal Dist.
Log-normal Dist.
NCGs
Safe
Risk

Safe

Figure 5.5 Reliability index and Failure probability (NCGs and HCGs)
Besides, the failure probability of Weibull Distribution shows to be highest among thethree functions, which means that Weibull Distribution has more failure samples andusing Weibull Distribution for designing tends to be safest.

Risk

Compact
NCGs

Standard
NCGs

Slender
NCGs

Safe
Risk

Safe

Figure 5.6 Reliability index and Failure probability (Slender, Standard and Compact)
26

From the Figure 5.6, it could be observed that with same material properties, theslender model has a higher reliability index and lower failure probability than the standard
one with the same probability distribution, also the standard model has a higher reliability
index and lower failure than the compact one with the same probability distribution.
Hence, it could be concluded that the slender model has the highest reliability among thethree section dimensions with same probability distribution.
And the histograms of destruction numbers are outlined in the Figure 5.7, 5.8 and 5.9.
Using the same horizontal axis (𝛼: factor of safety) and vertical axis to compare eachmodel directly.

NCGsHCGs NCGsHCGsNCGsHCGs

Figure 5.7 Histograms of destruction numbers (Slender model)
From the Figure 5.7 (slender model), when 𝛼 is equal to 0.9, for the HCGs, there arenearly no destructions which is much fewer than that of NCGs. And when 𝛼 is equal to
1.0, the failure number of HCGs is larger than that of NCGs. This means that when thedesign load is small (𝛼 = 0.9), the HCGs have higher reliability with slender section
dimension. And when the design load is large (𝛼 = 1.0) , the NCGs have higher
reliability with slender section dimension.

27

NCGsHCGs NCGsHCGsNCGsHCGs

Figure 5.8 Histograms of destruction numbers (Standard model)
From the Figure 5.8 (standard model), when 𝛼 is equal to 0.9, the failure number of
HCGs is also smaller than that of NCGs. And when 𝛼 is equal to 1.0, the failure numberof HCGs is larger than that of NCGs. This means that when the design load is small (𝛼 =0.9), the HCGs have higher reliability with standard section dimension. And when thedesign load is large (𝛼 = 1.0), the NCGs have higher reliability with slender section
dimension.

NCGsHCGsNCGsHCGsNCGsHCGs

Figure 5.9 Histograms of destruction numbers (Compact model)
28

From the Figure 5.9 (compact model), when 𝛼 is equal to 0.9 and 1.0, the failure
number of HCGs is larger than that of NCGs. This means that for the compact model,using NCGs is showed to have fewer failure samples and could be safer.
It could be concluded that with the same section dimension, when 𝛼 = 0.9, whichindicated a smaller design load, it’s better to design the composite girder as slender andstandard for HCGs, compact for NCGs. And when 𝛼 = 1.0, which indicated a larger
design load, it’s better to design all three kinds of the section dimensions as NCGs.

5.3 Finite Element Method
5.3.1 From One-Girder Model to Two-Girder Model
During the calculation of Fiber Method. one-girder model is adopted as the simulation
model. However, in practical engineering applications, one-girder bridges are usually not
used. Thus, in order to meet the actual needs of the engineering practice, a two-girder
model is accepted for the FEM modeling, the part of steel girder is showed as Figure
5.10. Set stiffeners at the web of I-beam every 5 meters to restrain the deformation ofbuckling, and set cross girder connecting the two span of girders to restrain the lateralbuckling under the large load.

Figure 5.10 Supplement to One-Girder model (Two-Girder model)
And the overall conceptual model is showed in the Figure 5.11. The distributed loads
are set at one third and two third of the top of the concrete slab.

29

Figure 5.11 Overall conceptual model (Two-Girder model)
5.3.2 Statistical Calculation and Results
Refer to the chapter 3.2.1, we modeling the concrete slab with SOLID 65 and the steel
girder with SHELL 181.
Based on the Von Mises yield criterion and associate with its’ liquidity criteria. Using
Multilinear isotropic strengthen criteria to generate material properties. Displacement
control is adopted as the load for the calculation and plenty of loading increments are
given for iterative calculation. The ultimate load is obtained eventually. The nonlinear
calculation follows the criterion of Riks method, the calculation is terminated when thereaction force reaches the peak and the last state is regarded as the limit state of thestructure. The characteristic buckling state is used as the initial structural imperfection.
After calculation, the ultimate bending moment 𝑀𝑢 should be recorded for statistic
calculation. Therefore, it is necessary to determine the limit state of the structure. The
ultimate state determined by Fiber method is the destruction of the upper edge of theconcrete slab. Therefore, it is necessary to determine whether the upper edge of theconcrete slab in the ultimate state obtained by the Riks method is damaged.
The total displacement of standard HCG model due to the simulation result is showed
in Figure 5.12.
It could be observed that the structure occurs large deformation in the last step. Since
the steel girder in the lower part of the compact model has greater strength among thesemodels, the upper part of the concrete slab is more likely to be damaged under the same30

bearing capacity. Therefore, the compact HCGs model is adopted to value the strain of
concrete slab. And the Von Mises total mechanical strain of compact HCGs model isshowed in Figure 5.13.

Figure 5.12 Total displacement of Standard HCGs model (10 times scale)Figure 5.13 Total mechanical strain of Compact HCGs model (10 times scale)31

According to the Figure 5.13, due to the Von Mises total mechanical strain
chromatogram, it could be observed that the maximum total mechanical strain of thecompact HCGs model is posited at the bottom flange of the steel girder. And the totalmechanical strain of the upper edge of concrete slab could be observed to be less than the
crushing strain 𝜀𝑐 =0.035.
Thus, buckling limit state is considered as structural limit state. It could be determinedthat the structural ultimate state is the limit state calculated by the Riks method. The
ultimate bending moment should be calculated by the result of last set about the FEM
simulation. The graphical illustration of Riks method is showed in the Figure 5.14. And
the verification of the relation of displacement and reaction force (standard HCGs model)
is showed in the Figure 5.15.

Peak

Figure 5.14 Graphical illustration of Riks
method17)

Figure 5.15 Relation of displacement andreaction force (Standard HCGs model)

From the Fig 5.15, It could be observed that the curve of the displacement and reaction
force reach its peak value and its shape is similar to the first half of the Figure 5.14. Thus,
it could be concluded that the calculation process of ANSYS satisfies the calculation
criteria of standard Riks method.
And the statistic calculation is adopted for data processing. Since the Weibull
distribution is showed to have a higher design reliability, it is supposed to use Weibull
distribution to generate random variables. And due to the complexity and time
consumption of FEM calculation, 100 random variables of material properties are set by
Weibull distribution. Then the histograms about bending capacity of each case are
generated for discussion, which are showed in Table 5.7.

32

Table 5.7 Histograms of bending capacity for FEM (Weibull Distribution)
𝑀𝑢 /𝑀𝑝SlenderStandard

Compact

NCGs

HCGs

Using the same 𝑥 and 𝑦 axis for comparing each case. After statistical calculation,
the average and standard deviation are obtained for analysis and discussion to evaluate
the reliability of each case.
Since there are cross girder and several stiffeners in the two-girder model than the twospan conceptual model, the bending capacity 𝑀𝑢 /𝑀𝑝 tends to be slightly greater than 1.
The statistics result of FEM simulation is showed in Table 5.8
Table 5.8 Statistics result of FEM simulation (Weibull Distribution)
𝑀𝑢 /𝑀𝑝

NCGs

HCGs

sectionslenderstandard

compact

slenderstandard

compact

average

1.14219

1.1437

1.18028

1.11388

1.12670

1.14051

standard
deviation

0.064128

0.063464

0.062536

0.042917

0.038534

0.038894

33

NCGs

NCGs

HCGs

HCGs

Figure 5.16 Average and Standard Deviation of NCGs and HCGs (Weibull
Distribution)
The comparison charts about average and standard deviation are showed in the Figure
5.16. From Figure 5.16 (left), it could be observed that, with the same section dimensions,the average of bending capacity about NCGs shows to be larger than that of HCGs, whichmeans that the NCGs models have a higher bending capacity than those of HCGs models
in buckling calculations. Furthermore, with the same material composing, the larger thesection dimensions are, the higher the average of bending capacity is, which means thata thick steel girder could give full play to the bending capacity of steel and compressive
capacity of concrete.
And from Figure 5.16 (right), it could be observed that, with the same section
dimensions, the standard deviation of bending capacity about HCGs showed to be lower
than that of NCGs, which means that the HCGs models have fewer disperse samples thanthose of NCGs models in buckling calculations. Besides, with the same material
composing, the larger the section dimensions are, the lower the standard deviation ofbending capacity is, which means that a thick steel girder could provide a higher structural
stability and guarantee a higher data concentration.
5.3.3 Results and Discussions about Structural Reliability
Then, after the statistical calculation, we could get the reliability index and failure
probability of each model. Refer to the Chapter 4, according to the reference 16), in orderto make the sample larger, there fixed 𝛼 = 1.1, then the reliability index and failure
probability of each case could be calculated. These data are compared in the form of
scatter plotted on the same graphs which are outlined in Figure 5.17, where the circles
34

are about NCGs models and the diamonds are about HCGs models, and the blue, red andblack are slender, standard and compact section in turn.

Figure 5.17 Reliability index and Failure probability of FEM (NCGs and HCGs)
From the Figure 5.17, when 𝛼 = 1.1, it could be observed that the distribution of thescatter diagram about structural reliability is regular. From the Chapter 4 we could knowthat a large reliability index and a small failure probability tend to have a higher structuralreliability.
When 𝛼 = 1.1, it could be observed that, with the same material composing, the large
the section dimensions are, the higher the reliability index is and the smaller failure
probability is, the compact models have a higher structural reliability among the threedifferent section dimensions.
Furthermore, with the same section dimensions, the NCGs models have a higher
reliability than that of HCGs models in slender and compact section dimensions, while
the HCGs models have a higher reliability than that of NCGs models in standard section
dimensions. It might be caused by the abnormal samples.
Thus, it could be concluded that the compact models are more reliable in the buckling
calculation, and the compact NCGs shows to have a higher structural reliability than thatof compact HCGs. And the slender models are overvalued for the structural reliability,
they are more likely to occur buckling and lost their bearing capacity.
35

Figure 5.18 Reliability index and Failure probability (𝜶 = 𝟏. 𝟎𝟓, 𝟏. 𝟏𝟓)
And the scatter diagrams of structural reliability about 𝛼 = 1. 𝟎𝟓 and 1.1𝟓 areshowed in Figure 5.18. From the Figure 5.18, it could be observed that with the increaseof design load, buckling resistance performance of NCGs will be more prominent.
Furthermore, the histograms of destruction numbers of FEM are outlined in the Figure
5.19, Using the same horizontal axis (𝛼: factor of safety) and vertical axis to compare
each model directly. This time, to make the samples larger, we fixes 𝛼=1.05, 1.10 and
1.15 to set histograms.

Figure 5.19 Histograms of destruction numbers about FEM (NCGs and HCGs)
The comparison charts about numbers of destruction are showed in the Figure 5.19,
where the left is about NCGs and the right is about HCGs.
From Figure 5.19, it could be observed that with the same section dimensions, thedestruction numbers of NCGs models are fewer than those of HCGs, which means that36

NCGs shows a better performance in buckling analysis while HCGs are more likely toreach the buckling ultimate state.
Furthermore, it could be observed that with the same material composing, the yellow
bar chart is the shortest in a set of data, which means that the compact models occur the
fewest destruction with same situation. The compact models are more reliable in thebuckling calculation, and the compact NCGs shows to have fewest destruction among
these cases.
Thus, for FEM part, involving buckling analysis, compact section could be summarized
to have a higher reliability and a better performance to restrain buckling.
To synthesize the results of Fiber method and FEM, the results of Fiber method are notcompletely credible when the structure has a thin shell part, and the buckling calculation
should be considered in the simulation. The compact NCGs could be concluded to havea highest structural reliability among these situations and have a better performance toresist buckling.

37

Chapter 6. CONLUSION AND FUTUER WORK
6.1 Conclusion
The purpose of this study is to clear the stochastic capacity distribution of the ultimate
limit state of HCGs and to evaluate them by comparing with those of NCGs. To find an
appropriate structure with less material usage and make better use of the bending capacityof steel. Using Monte Carlo Method with Fiber method and FEM to do the simulationand calculate the bending capacity. Using member and material parameters and threekinds of probability functions (Normal Distribution, Log-normal Distribution andWeibull Distribution) to make the results more reliable.
The conclusions could be summarized as follows:
Chapter 1
The background and objective of this research is introduced. The previous researchon HCGs and the significance of the research is introduced.
Chapter 2Chapter 2 introduce the main simulation method used in the research. The idea andcalculation process of Fiber method and FEM are introduced.



Three typical probability functions (Normal Distribution, Log-normal Distribution
and Weibull Distribution) are adopted as distribution for Monte Carlo method
combined the Fiber method and FEM to improve the reliability of the simulation
results.
Chapter 3

The numerical model is introduced in Chapter 3, using MATLAB code to build onegirder model for Fiber method calculation and ANSYS program to build two-girdermode (close to engineering practice) for FEM calculation.

The element type using for FEM calculation is introduced and a verification (bendingof a composite beam) is outlined to verify the accuracy of ANSYS program.
Chapter 4In Chapter 4, the calculation process of Monte Carlo method is introduced. The data
about variables of materials and section members are outlined for following
calculation.



The reliability index and failure probability, which are quantities for evaluating
structural reliability, are introduced for reliability calculation.
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Chapter 5The simulation results are outlined in Chapter 5. Statistical calculation is adopted tovalue the bending capacity of each situation.

For Fiber method, it could be concluded that with the same section dimensions, theaverage and standard deviation of bending capacity about NCGs is a little larger thanthat of HCGs, which means that the NCGs models have a higher bending capacity
than those of HCGs models and HCGs models have fewer disperse samples thanthose of NCGs models.



Furthermore, with the same material composing, the average and standard deviationof bending capacity about slender models is a little larger than that of compact models,the larger the section dimensions are, the lower the average and standard deviationof bending capacity are, which means that a thick steel girder could not be made fullyuse of its bending capacity and provide a higher structural stability and guarantee ahigher data concentration.



Weibull Distribution contains more dangerous samples due to its high kurtosis andleft skewed distribution, which means that using Weibull Distribution considers more
unfavorable values which could make the design safer.



For the reliability calculation, for slender model, when the design load is small (𝛼 =0.9), the HCGs have higher reliability with slender model while when the design
load is large (𝛼 = 1.0), the NCGs have higher reliability. For standard model, withthe small design load, the HCGs have higher reliability with standard model whilewith the large design load, the NCGs have higher reliability. And for compact modelusing NCGs is showed to have fewer failure samples and could be safer. It could beconcluded that with the same section dimension, when 𝛼 = 0.9, which indicated asmaller design load, it’s better to design the composite girder as slender and standard
for HCGs, compact for NCGs while when 𝛼 = 1.0, which indicated a larger designload, it’s better to design all three kinds of the section dimensions as NCGs.



For FEM, using Weibull Distribution (most unfavorable design) to set variables forcalculation. Buckling limit state is considered as structural limit state for all situations
since the upper edge of concrete slab does not reach 𝜀𝑐 in buckling state.

From the statistical result, it could be concluded that, with the same section
dimensions, the average and standard deviation of bending capacity about NCGs
shows to be larger than that of HCGs, which means that the NCGs models have ahigher bending capacity than those of HCGs models and the HCGs models havefewer disperse samples than those of NCGs models in buckling calculations.
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Furthermore, with the same material composing, the larger the section dimensions
are, the higher the average of bending capacity is and the lower the standard deviationof bending capacity is, which means that a thick steel girder could give full play tothe bending capacity of steel and compressive capacity of concrete and provide ahigher structural stability and guarantee a higher data concentration.


For the reliability calculation, it could be concluded that, with the same material
composing, the large the section dimensions are, the higher the reliability index isand the smaller failure probability is, the compact models have a higher structuralreliability among the three different section dimensions. And, with the same section
dimensions, the NCGs models have a higher reliability than that of HCGs models in
slender and compact section dimensions while a lower reliability in standard section
dimensions. And, with the increase of design load, buckling resistance performanceof NCGs will be more prominent. Involving buckling analysis, compact section could
be summarized to have a higher reliability and a better performance to restrain
bucklingTo synthesize the results of Fiber method and FEM, the results of Fiber method are
not completely credible when the structure has a thin shell part, and the buckling
calculation should be considered in the simulation. The compact NCGs could beconcluded to have a highest structural reliability among these situations and have abetter performance to resist buckling.

6.2 Future Work

It is possible to improve the parameters used in numerical simulation. Due to thedifferent statistical data of references, materials and steel manufacturers, it is possible
to further rationalize the results by using more realistic values.



Due to the complexity and time-consuming of the FEM buckling analysis, only 100
variables are generated for simulation in this research. In the future work, we should
try to simplify the calculation process and use more variables to make the results
more reliable.



For the structure with thin shell part, Fiber method is not enough for calculation. In
order to make the result more reliable, buckling analysis of FEM should be added
into calculation. To make fully use of performance of each material, a more
appropriate section dimension could be found for the design of composite girder inthe future work, which should satisfy that the structure achieves the failure of upperedge of concrete slab and buckling ultimate state simultaneously.
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ACKNOWLEDGE
I would like to thank my advisor Prof. MATSUSHIMA for his patience andprofessional guidance and help on my master course and research. It was his kindness
that made me feel the warmth of my home in a foreign country and made me adapt to the
new life and environment quickly. His conscientiousness and hard work in learning has
always been my model, which has stimulated my interest in learning and enhanced mydetermination to overcome difficulties and challenges.
I am also thankful to my co-advisor and mentor Prof. YAMAMOTO Humor and careful
guidance and help on my courses and research. His approachable people helped me
through the most difficult time of studying in Japan. His meticulousness makes me fullof awe and preciseness for academic. Thanks to Prof. YAMAMOTO for his love for mystudy and life. He is a light on my research road and always guides me when I am
confused and depressed. I also thank him for his care and concern for my life, and fortaking me to eat delicious Japanese food many times. Let me know a lot about Japaneseculture and customs.
I am also thankful to my co-advisor Prof. SHOJI. He gave me very useful andsuggestive advises for my research and took a lot of attention for my daily life.
I would like to express my thanks to everybody in Frontier Laboratory. Tanks to Guo
Siliang san and Jiang Xiaoyu san for helping me adapt to the research atmosphere faster,and provide a lot of feasible suggestions and guidance for my research. Thanks to themfor telling me some skills on MATLAB coding.
Thanks to Jiang Hao ran san for his accompany from from just entering the laboratory
to graduation. We've been through a lot, stay up late to write reports, discuss assignments,
travel together, eat delicious food together, watch movies together, and help you catch up
with the girls you like. Thanks for his accompany in my master period.
Thanks to Murakami san, Inoue san, Ota san, Tachibana san and Yu Huaizhi san for
their kindness and care on my life, and we spent a lot of joyful time together.
I would like to thank to my parents for the financial support and mental courage.
Thank you for everyone’s help. I had an unforgettable time studying abroad.

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