DEFLECTION (ft). Figure 5.12 - Elasto-Plastic Response for. Composite Steel Girder Bridge. R eserve. S ...... Mirza, S. A., Hatzinikolas, M. and MacGregor, J. G., ".
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•T Dissertation Information Service University Microfilms International A Bell & Howell Information Company 300 N. Zeeb Road, Ann Arbor, Michigan 48106
8621387
Tantaw i, Hasan Mohammad
ULTIMATE STRENGTH OF HIGHWAY GIRDER BRIDGES
The University of Michigan
University Microfilms International
300 N. Zeeb Road, Ann Arbor, Ml 48106
Ph.D.
1986
ULTIMATE STRENGTH OF HIGHWAY GIRDER BRIDGES
by Hasan Mohammad Tantawi
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Civil Engineering) in The University of Michigan 1986
Doctoral Committee: Professor Professor Professor Professor Professor
Andrzej S. Novak, Chairman Subhash C. Goel Niels C. Lind, University of Waterloo-Canada Stephen M. Pollock Wadi S. Rumman
RULES REGARDING THE USE OF MICROFILMED DISSERTATIONS
Microfilmed or bound copies of doctoral dissertations submitted to The University of Michigan and made available through University Micro films International or The University of Michigan are open for inspection, but they are to be used only with due regard for the rights of the author. Extensive copying of the dissertation or publication of material in excess of standard copyright limits, whether or not the dissertation has been copy righted, must have been approved by the author as well as by the Dean of the Graduate SchooL Proper credit must be given to the author if any material from the dissertation is used in subsequent written or published work.
DEDICATED TO MY MOTHER AND THE BLESSED MEMORY OF MY FATHER , TO WHOM I OWE EVERY THING.
ACKNOWLEDGEMENTS
I express cordially my indebtedness, gratitude and appreciation to Professor Andrzej S. Nowak for his inspiration to do this work, his generous help, unlimited kindness and valuable guidance during this work.
I am grateful to Professors Subhash C. Goel, Niels C. Lind, Stephen M. Pollock and Wadi S. Rumman, members of the dissertation committee,
for reviewing the dissertation
and offering helpful suggestions and comments. The valuable help and instructions of professor Niels C. Lind (University of Waterloo- Canada) are deeply appreciated.
I also wish to record my sincere gratitude and deep thanks to the Jordan Army and Mutah University for giving me the scholarship which fulfilled my dream of further studies.
My sincere thanks and gratitude are to my wife and my children for their patience and support during my study.
Finally I would like to thank all my colleagues specially, Vahid Sattary, Medhat Boutros and Adel El-Tayem who have helped me in some way or another during the research.
TABLE OF CONTENTS DEDICATION iii
ACKNOWLEDGEMENTS LIST OF TABLES
vi
LIST OF FIGURES
vi i x
LIST OF APPENDICES
xi
NOTATIONS CHAPTER I.
INTRODUCTION 1.1 1.2
II.
2.4
43
Introduction Dead Load Live Load Dynamic Load Load Combinations Transverse Location of the Truck Traffic
BRIDGE DECK M O D E L ....................................72 4.1 4.2 4.3 4.4
V.
12
Limit State Function and Failure Probability Safety Analysis Concepts Simulation and Approximation Techniques Used in Calculating the Statistical Moments of Independent Random Variables Simulation and Approximation Techniques Used in Calculating the Statistical Moments of dependent Random Variables
BRIDGE LOAD M O D E L S ................ 3.1 3.2 3.3 3.4 3.5 3.6
IV.
Historical Development Thesis Outline
FUNDAMENTALS OF STRUCTURAL RELIABILITY 2.1 2.2 2.3
III.
1
The Grid Model Geometric Properties ofthe Grid Elements Strength of the Grid Elements Strength Reliability of Single Elements
ULTIMATE RESISTANCE MODEL 5.1 5.2
OF BRIDGE SYSTEM . .
General Description Ultimate Strength Analysis
104
5.3 VI.
APPLICATION TO BRIDGE SYSTEM EVALUATION. 6.1 6.2
VII.
Ultimate Strength Reliability ...
Evaluation using the Developed Approach Bridge Capacity Rating
CONCLUSIONS AND R E C O M M E N D A T I O N S ............. 7.1 7.2 7.3
142
. 157
Research Results Summary of the Conclusions Recommendations
A P P E N D I C E S .................................................. 163 B I B L I O G R A P H Y ................................................ 178
V
LIST OF TABLES Table
2.1
Results of Example 1 ..................
2.2
Results of Example 2
3.1
Number of Trucks as Function of Time .
3.2
Parameters of Live l o a d ................................ 52
3.3
Input Data for Eg. 3 . 3 .............................
3.4
50 Year Live Load Models
3.5
Dynamic Load F a c t o r ..........................
65
3.6
Typical Percentage Values of Major Load C o m p o n e n t s ...........................
68
. . .
.
.
35
.......................... 37 ........... 49
55
......................... 64
4.1
Statistical Characteristics of Input Data for Section Analysis .*............................... 100
6.1
Strength of Grid Components...............
6.2
145
Results of Simulation for a Reinforced Concrete B r i d g e .....................................
147
6.3
Reliability Analysis without Correlation ........
153
6.4
Reliability Analysis with Correlation
153
...........
LIST OF FIGURES Figure
1.1
Flowchart of the Research P l a n ..................
2.1
Schematic Illustration of the Fundamental C a s e ............................
10
.
14
2.2
Hasofer and Lind Reliability I n d e x ................ 21
2.3
Continous Beam with Three S p a n s .................... 36
3.1
Distribution Functions of Moments for Various Spans ......................................
48
3.2
Configuration of the Heaviest Commercial V e h i c l e s ............................................. 50
3.3
Upper tails of moment distributions from Truck Survey........................................
51
3.4
Extrapolated live load d i s t r i b u t i o n s ...............53
3.5
Typical Distribution of Load ratios (60 Ft. s p a n . ) ........................................ 58
3.6
Distribution of the arbitrary-point-in-time and 50 year load ratio, LR^, for 60 ft. span . .
61
3.7
Distribution of 50 year live load for 80 ft. s p a n ........................................... 62
3.8
Distribution of 50 year live load for 125 ft. s p a n ......................................... 63
3.9
Typical Curb Distance Histogram for Interstate Highway 1-94.......................... 71
3.10
Distribution of Curb Distance on Normal Probability P a p e r ......................................71
4.1
Typical Bridge System and The Corresponding Grid M o d e l ..............................................73
4.2
Plate Element and the Equivalent Framework Model
4.3
Stress-Strainss Relationships
.....................
. 75 80
4.4
Typical Composite Steel Section ...................
81
4.5
Typical Sections Commonly Used in Prestressed Concrete Bridges....................................... 82
4.6
Typical Strain Diagram in a Composite Section.
4.7
Typical Moment-Curvature Relationship for Composite Steel Concrete Section ..................
84
4.8
Actual S e c t i o n ............................
87
4.9
Idealized Section ..................................
87
4.10
Torsion Versus Angle of Twistrelationship.......... 89
. . 83
4.11 Typical Thin Section under different state of torsional stresses................................. 91 4.12 Idealized Thin Section under different state of torsional stresses................................. 91 4.13
Composite Concrete Steel Section ..................
4.14 Typical Torsion Vs. Angle of Twist Relationship for Composite Steel S e c t i o n ....................
95
. 98
4.15
Typical Moment-Curvature Relationships
4.16
Distribution of the Maximum Load per Girder . . . 103
5.1
Coordinate System .................................
5.2
Nonprismatic Beam E l e m e n t .......................... 110
5.3
........
101
108
Reduced Nonprismatic Element Stiffness Matrix ...................................
114
5.4
Wheel Load D i s t r i b u t i o n ............................ 115
5.5
Clamped Beam Load Versus Deflection Curve . . . .
5.6
Plate E l e m e n t
120
. 122
5.7
Plate Grid M o d e l ..................................... 124
5.8
Effect of Mesh Size on the Failure L o a d ........... 125
5.9
Effect of Mesh Size on the Accuracy of the Results Compared with the Yield Line Theory . . . 125
5.10
Details of the Bridge M o d e l ....................... 126
5.11
Load Versus Deflection
(Three Models)
...........
127
5.12 Elasto-Plastic Response for Composite Steel Girder B r i d g e ................................. 129 5.13 Example of a Series S y s t e m ........................ 131 5.14 Example of a Parallel S y s t e m .......... ...
132
5.15 Reliability Model for Series System in Parallel
. 132
5.16 Reliability Model for Parallel System
. 133
inSeries
5.17 Typical Cross Section of a Composite Steel Girder B r i d g e ................................. 135 6.1
Reinforced Concrete B r i d g e .......................... 143
6.2
Bridge Grid M o d e l ....................... .......... 144
6.3
Transverse Location of the T r u c k ................. 146
6.4
Composite Steel Girder Bridge
6.5
Prestressed Concrete Bridge .......................
A.1
Graphical Rackwirz and Fiessler Algorithim . . .
....................
150 152 .166
LIST OF APPENDICES Appendix A. Second Moment Reliability Index .................
164
A.l Cornel Reliability Index A . 2 Determination of the Design Point Using Rackwitz and Fessler Algorithim B. Generation of Correlated Random Variables.
. . . 170
B.l Transformation to Uncorrelated Space B.2 Generation of Correlated Lognormal Variables B.3 Generation of Jointly Distributed Normal and Lognormal Variables. C. Reliability of Structural S y s t e m .................. C.l Series System C.2 Parallel Systems C.3 Complex Systems
x
173
LIST OF NOTATIONS
BM
: maximum base length of a truck
CDF
: cumulative distribution function
COV
: coefficient of variation
COV(X.,X..)
: covariance of X. and X. i J : covariance matrix
X E
: Young's modulus of elasticity
E[
]
FX ,fX
: expected value of a random variable : probability distribution function and density function of variable X
fxryp
: joint probability density function of random variables X and Y
G
: shear modulus
i
: moment of inertia of a section
lra
LR
B
LR0 M MP
: Truck Moment / HS20 Moment : Truck Moment / Base Length
Moment
: Truck Moment /OHBDC Moment : bending moment : plastic bending moment
MTC
: Ministry of Transportation and Communications, Ontario, Canada
OHBDC
: Ontario Highway Bridge Design Code
PDF
: probability density function
Q
: mean value of the combined load effect
: live load : impact load Qg
:dead load
R
:ultimate resistance of the bridge
R
:mean value of the resistance
VR
:coefficient of variation of resistance
Vq
:
coefficient of variation of load effect
j3
:reliability index
6r
:permanent deflection of the bridge at ultimate load
S>,0
: cumulative distribution function and density function for the standard normal variable
u
:Poison's ratio
pv
v
:linear correlation coefficient between
1f 2 variables
T
and ^
[ ]
:transpose of a matrix
[ I-"*-
:inverse of a matrix
I I
:absolute value
xii
CHAPTER I
INTRODUCTION
1.1
1.1.1
Historical Development
Reliability Theory
Traditionally,
structural design relies on deterministic
analysis. Suitable dimensions, material properties, and loads are assumed, and analysis is then performed to provide a more or less detailed description of the structure. The uncertainty in load and resistance was accounted for by introducing the so called load and resistance factors. However, fluctuations of loads, variability of material properties, and uncertainties regarding analytical models all contribute to a generally small probability that the structure does not perform as intended, however, this small probability has significant impact on other aspects such as life, budget, etc. In response to this problem, methods have been developed to deal with the statistical nature of loads and material properties and, more recently, a general framework for comparing and combining these statistical effects has emerged [66].
The structural reliability is defined as the probability that the structure will perform satisfactorily for a
1
2 specified period of time under a stated set of load conditions.
It has been used to develop economic and
efficient design methods incorporating probability theory together with methods of advanced structural analysis. Structural reliability theory has attracted academic interest since 1960. Lind et al.
[64] defined the
problem of rational design of a code as that of finding a set of best values of the load and resistance factors. Cornell [18] suggested the use of a second moment format to define the reliability index. Lind [61] showed that the Cornell's reliability index requirement could be used to derive a set of safety factors for load and resistance. The year of 1974 saw the publication of the first standards in limit states format based on probabilistic rationale [14].
However,
until recently most of the work was
concentrated on finding the reliability index based on the behavior of individual elements. Moses [80] presented some simplified analysis and approximations for evaluating system reliability. The system reliability is indispensable information for adjusting the element's safety factors to account for its presence in the structural assemblage so that the overall probability of failure remains roughly equal to what is desired of the element.
Gorman [33] utilized rigid plastic hinge theory to evaluate the resistance of a redundant building structure
3 subjected to a proportional loading. The mean and the standard deviation of the system resistance were evaluated in terms of the statistical properties of their components. Several simulation techniques were used or developed. Some of these techniques will be extended for application to a redundant bridge system.
Kam [52] presented an approximate method for numerically integrating the probability of failure for a simple framed structure.
He considered three kinds of limit states in the
reliability analysis of frames. The first of these limit states is first yielding in the structural system;
the
second limit state is the formation of the first plastic hinge in the structure; and the third limit state is the formation of a local mechanism or instability of the structure. The incremental load procedure was utilized to construct the limit state surface. The resistance of the structure was treated as a deterministic quantity while the loads were treated as random variables. The method was demonstrated by analyzing a two-dimensional frame to construct the three limit states of a structure in the load space.
Lin [59] developed an approach to determine the reliability of redundant ductile framed structures. His approach was based on determining the limit state function in load space and evaluating the probability of failure directly for the overall structural system rather than the
4
individual failure modes.
Lin determined first the limit
state function in load space for the structure with mean resistances by performing nonlinear structural analysis along various load paths.
For each load path, a random
variable representing the distance from the origin to the limit state was introduced to account for the randomness in the strength of the structure. The probability of failure conditioned on a particular limit state is calculated by integrating the joint density function of the load over the region beyond that limit state. The total probability of failure was then calculated as the sum of all conditioned failure probabilities weighted by their respective probability of occurrence.
1.1.2
Bridge Deck Analysis
The ultimate strength of the bridge system is the sum of the strength at the elastic stage and the strength reserve beyond the elastic limit up to failure.
The concept of
bridge ultimate strength involves nonlinear behavior under different loading conditions. The method of modeling and analysis of the bridge structure should combine accuracy and simplicity along practical and realistic lines.
Many attempts have been made to simulate bridge system behavior including: finite element methods Wegmuller [112], finite strip method Nowak and Boutros [86], finite deference method Heins and Kuo [44], orthotropic plate theory Heins and Looney [46], simple yield line theory Reddy and Hendry
5
[100], and the grid analogy [9,51, 8]. The grid analogy was chosen as structural analysis tool in this research. Although it is necessarily approximate,
it has the
advantage of almost complete generality of applications.
The method of grid analysis involves the idealization of the bridge deck through its representation as a plane grillage of discrete interconnected beams. The grid method allows for any support conditions. The bridge analysis can be carried out with comparative ease.
The structural gridwork has evoked intense interest since the earliest stages of development. Several methods were developed approaching the solution of the grid problem from various viewpoints. Bares and Massonnet
[9] mentioned
that until circa 1955 most of the solutions to the problem of the bridge grids, with few or man y main girders, were based on the assumption that the torsional resistance of the element can be neglected. The torsional rigidity may, without noticeable error, be ignored in bridge structures with noncomposite steel girders and reinforced concrete. The torsional rigidity was considered by several researchers as mentioned by Bares and Massonnet
[9] but this approach
was so complicated and thus making them of limited use in practice.
The general availability of the digital computer
has revived the grid method. A number of computer programs have been developed using the grid analogy for design purposes based on elastic analysis
[21].
6
Recently, Jaeger and Bakht [51], provided a basic framework for using the grid model to simulate the behavior of the bridge system in the elastic stage. Some features of their work will be used in this study.
The traditional assumption of linear elastic behavior can produce inaccurate or misleading results that do not reflect the true structural response. Nonlinear analysis is more costly to perform than a linear elastic analysis. However, even a relatively simple nonlinear analysis yields a more reliable structural response than linear analysis [93].
Most of the work mentioned earlier deals with the bridge analysis using the grid model in the elastic condition.
In reliability analysis a programmable analytical model to analyze the bridge system under the effect of traffic loading is required. The grid model was found to be suitable for the purpose. Although the model involves some approximations,
it provides an economic ( in terms of
computing time) and acceptable level of accuracy.
In this study the grid model was extended to include the nonlinear response of the bridge system in a practical way.
Very early experimental and theoretical work in the area of nonlinear analysis of a grid system was done by Heyman (1953). Heyman based his analysis on the assumptions of plastic hinges and the effect of torsional rigidity. He
7
proved that the interaction between bending moment and torsion has no importance in predicting the actual capacity of the grid system. Later, Hollinger and Mangelsdorf
[49]
found that an assumption concerning the deterioration of the torsional rigidity with respect to different levels of loading could be used to update the torsional rigidity of a structural element.
The method assuming the formation of plastic hinges has received a great deal of interest in recent years [84, 48, 67].
Gorman [33] developed a method for identifying significant failure modes based on rigid plastic hinges. They applied their method successfully to different cases of buildings subjected to different loading conditions.
They
used approximate methods to calculate the first two statistical moments of the structural resistance. Some of these approximation methods will be generalized in this thesis for application in the reliability analysis of a bridge system.
1.2 Thesis Outline
1.2.1
Definition of the Problem
Highway bridges are currently designed according to codes based on the elastic behavior of the material. There is a considerable amount of strength reserve beyond the elastic stage and before the ultimate capacity of the system
i .
8 is reached. For example, a recent field study done by the Ministry of Transportation and Communications of Ontario, Canada, showed that some of the bridges, posted with a maximum capacity of 8 tons, were loaded up to 300 tons before reaching the load carrying capacity. Accurate evaluation of bridge systems is important for decisions concerning replacement, rehabilitation or posting.
The problem investigated in this study is the determination of an overall reliability measure for the structural system. Safety is measured in terms of a reliability index, 0. The structures considered are bridges modeled as grillages. Three types of bridges are considered composite steel girders, prestressed concrete girders and reinforced concrete beams. The model includes material nonlinearity, ductility of components and deterioration of torsional rigidity with the increase of flexural stress in section.
The basic random variables considered in this study are load magnitude,
load position and component strength.
The
component strengths considered in this study are the ultimate flexural, torsional and shear capacities. These random variables are characterized by their mean and covariance values.
1.2.2
Objectives of the Research
The three main objectives of this study are:
9 1. To develop an approach for evaluation of the bridge load carrying capacity,
2. To develop a practical reliability analysis method for short and medium span bridges, and
3. To establish
practical rating criteria based on the
available bridge loading data, the actual conditions of the bridge and a predefined target reliability level.
The calculation of structural system reliability involves combining the effect of load and resistance. The • analysis includes a complex interaction between the system reliability and element reliability. A large number of variables are required to describe the system.
The
reliability of a structural system is investigated using Monte-Carlo simulation and a discrete distribution methods. The research plan is presented in Fig. 1.1.
1.2.3
Organization of the Thesis
The work described in this thesis addresses three separate topics. The first topic is the bridge loading based on the available models in literature and an a survey conducted by a research team of the University of Michigan (see Sec. 3.6). It is described in Chapter III. topic (Chapters IV and first part of Chapter V)
The second is the
evaluation of the bridge ultimate load carrying capacity. The third topic (the remainder of Chapter V)
is the
10 t
RESEARCH PLAN
DETERMINISTIC MODELS
BRIDGE D E C K
SECTION ANALYSIS
BRIDGE SYSTEM
MODEL
MODEL
MODEL
RELIABILITY MODELS
LIVE
LOAD
CO M P O N E N T
MODEL
STRENGTH
BRIDGE
MODEL
RESISTANCE
MODEL
BRIDGE CAPACITY RATING M O D E L
APPLICATION TO PRACTICAL CASES
ULTIMATE STRENGTH EVALUATION
CONCLUSIONS
BRIDGE CAPACITY RATING
AN D RECOMMENDATIONS
Figure 1.1 - Flowchart of the Research Plan
11 extension of Rosenblueth point estimation method to model the ultimate resistance of a redundant bridge system with various levels of correlation between elements.
The work is organized into seven chapters and an appendix.
In Chapter II the necessary knowledge about
reliability theory and Rosenblueth1s point estimates is presented.
Results of a special survey of the transverse
position of trucks on a bridge is given in Chapter III. Chapter IV includes modelling the bridge deck into a grid system, properties of the grid elements and evaluation of the bridge elements'
strength.
The reliability analysis
of the bridge system is described in chapter V.
The applications of the developed model to actual structures is presented in Chapter VI. A discussion of the results and conclusions can be found in Chapter VII.
Additional Derivations and methods are given in the appendices.
CHAPTER II
FUNDAMENTALS OF STRUCTURAL RELIABILITY
A highway bridge structure may be subjected during its life-time to loads (or actions) which may cause a change of condition or state of the structure. This change may go from the undamaged state to a state of deterioration, damage in varying degrees, to failure or collapse.
Ultimate strength
analysis involves determining the response of a structure subjected to an external live load. Predicting the ultimate strength of a structural system is usually associated with a considerable degree of uncertainty. This is mostly due to variability of material properties and simplifications used in the method of analysis. Predicting the future load condition that might act on a bridge system is also a process that involves a great deal of uncertainty. Hence the risk of unacceptable combinations of load and resistance can only be evaluated by reliability analysis.
2.1. Limit State Function and Failure Probability
Reliability analysis requires the determination of the relevant load and resistance parameters, and the functional relationship among them. If, for a given structure, all the actions (loads) can be lumped into one single random
12
13 variable, Q, and the strength of the structure into another single random variable, R, then the state of the structure can be determined by a variable Z,
Z = R - Q
(2.1)
The failure function can be formulated as
g(R,Q) = R - Q
(2.2)
The limit state is defined as the boundary between safety and failure as determined by:
g(R,Q) = 0
(2.3)
The event of failure corresponds to:
g(R,Q)
R) = / / fn p (q,r)dqdr = Jdqj fn R (q,r)dr 1 Q>R Q 'R 0 0 Q 'R
(2.5)
where f^ R is the joint probability density function of the random variables Q and R as shown in Fig. 2.1. If Q and R are statistically independent, then fQ , R (1-r> *
and Eq.
(2.5) becomes
(2.6)
14
q=r
q
Volume = IJ Figure 2.1 - Schematic Illustration of the Fundamental Case [5]
! fR (r)ar
dq = / fQ (q)FR (q)dq (2.7)
where fg and fR are the density functions of Q and R respectively, and FR is the distribution function of R. Reliability P
d
is the complement of the probability of
failure,
(2.8)
ps=1-p f
In general, R and Q are functions of other random variables.
In this case, the limit state function, Eq. 2.3,
can be expressed in terms of the set of n basic variables as shown in Eq. 2.9.
Therefore,
the probability of failure, Pj,
is given by the
multiple integral
X
I' 2'
where f ^ x ^ X j ,
• • • f
xn )dxld x 2'
dx. n
(2 .10 )
..., xn > is the joint probability density
function of the random vector X, and the integration is performed over the region where g(X) are
statistically independent, Eq. 2.10 may be replaced by
• • • 9
fx u n)dx h
2.2. Safety Analysis Concepts
2.2.1
General Concepts
In practice, Eq. 2.11, presents two problems. First, usually there is insufficient data to define the joint probability density function for the n basic variables. Usually, there is hardly enough information to be confident about the marginal distributions and the covariance matrix. Secondly, even if the joint density function is known, or in the case of Eq. 2.11 the marginal densities, the multi-
16 dimensional integration required may be extremely timeconsuming.
Analytical solutions do not exist for the
majority of practical problems and the analyst must resort to the numerical methods by using level 2 or special level 3 methods
(explained next) depending on the calculations
performed and the approximations made [107], A short review of level 2 and 3 numerical methods is presented further in this chapter.
2.2.2
Level 2 Methods
Level 2 methods provide powerful tools to solve a wide range of practical problems. At this level only the means, and the covariances, C^j, of the random variables are used.
It is often convenient to use the reliability index, j3, instead of probability of failure, P f , to characterize the reliability. The generalized reliability index was defined by Ditlevsen [22] in terms of the probability of failure as:
0 = - $ 1 (Pf )
(2.12)
where $ is the standard normal CDF.
In case of Eq. 2.2, and if both R and Q are independent and normal random variables, becomes:
the reliability index, 0,
17 (R - Q) 0^ R
+ °Q
(2.13)
where R, aR = the mean and the standard deviation of the resistance,
and
Q, Oq = the mean and the standard deviation of the load.
If both R and Q are lognormal random variables, a reliability index can be derived as follows. First,
g = in S = lnR . lnQ
(214)
g is a normally distributed random variable with mean g = InR - InQ
(2.15)
In general if X is a random variable, then (see Benjamin and Cornell
[10])
ln(x) = ln(X) - |ln(l + V^)
(2.16)
and
2
2
aln(X) = ln(1 + VX J where Vx = coefficient of variation of X. Hence
(2.17)
18
g = ln(R) - |ln(l + V*) - ln(Q) + |ln(l + V*)
(1 + v j ) g = In- - Tpln(1 ♦ v ‘ >
»
g = In
(2.19)
P o * 1
(-) Q VR + 1
(2 .20 )
alnQ
(2 .21 )
But
= v/fflnR +
°g
hence
R In (-) Q
/VQ + 1
VR + 1
0
=
_
=
A In [(VR + 1 ) (VQ + 1)]
(2 .2 2 )
In a more practical way, the following simplifying assumptions can be made with a reasonable accuracy (Benjamin and Cornell [10]);
19
ln(x) s* ln(x)
(2.23)
(2.24)
Then Eg. 2.22 gives the useful approximation
R InQ
•"S
* VQ
(2.25)
If R and Q are neither normal nor lognormal, then 0 can be calculated using a special numerical procedures (Madsen Krenk and Lind 1986).
The limit state function g(X), equation (2.9), always linear.
is not
In case of a nonlinear limit state function,
a Taylor expansion with linear terms can be used. Then, the first order approximations of the mean and the variance of the limit state function can be expressed as follows [66]:
G ~ g(X*,X*,
• • • f
(2.26)
(2.27)
and the reliability index is
20
(2.28)
where the partial derivatives, gjjp are evaluated at the •ff ^ linearization point X^, Xj is the mean of the basic variable X. and COV(X.,X.) 1
1
is the covariance of X. and X.
3
1
3
The reliability index 0 evaluated using Eqs. 2.26, 2.27 and 2.28 is approximate and it is called a first-order second-moment reliability index. is the mean-value point,
If the linearization point
then the reliability index is
called a mean-value first-order second-moment reliability index [66]. A better estimate to the reliability index was provided by Hasofer and Lind [41]. First, the the set of basic variables Xj is transformed to a set of normalized variables with zero mean and unit variance. The new set Z = (Z^, Z 2 , ..., Zn ) is obtained by applying the following transformation :
..., n (2.29)
Where X^ and = Y (x^f ^
2
, ••», x. +
, ..., x ^ ),
i.e. the function evaluated with the ith variable at its mean value plus one standard deviation, while the rest of the variables are at their mean values.
27
• • •»
n ( i + v,2yj ) - i i=l i
(2.38)
where
V
The difference between this method and the 2n point estimates is that in this method it is required to work with one variable at a time while keeping the other variables at their mean values
(in the other method this restraint is not
required). For n greater than 2, the 2n+l point estimate requires only a small fraction of the number of function evaluations required for the 2n point estimate.
2.3.4. Three Point Estimates
Gorman [33] derived an expression for a three point estimate of a random variable, exact for the first four moments (mean, m; variance o ; skewness, a 3 ; and the kurtosis, a 4 ). Let Y be a given function, Y = y(XlfX 2 ,... x n )f of n random variables. To obtain the first two moments of Y based on the three point estimate method, the following steps may be followed.
28 1. For each variable determine the three probability mass concentrations
(P+ at x = X + ), (PQ at x = X Q ) and (P_a
= X_) to comply with the first five moments of X:
P + + P 0 +P- - 1
(2.39)
p+x + + P 0 X 0 + P - X - = X
(2.40)
p+ (X+
X)2 +
0 (x 0
X ) 2 + P (X
p+(X+
X)3 ♦ P 0 (x 0
X ) 3 + P (X
p
y \2_
2
X) "
X
(2.41)
*»3* a3x°x
(2.42)
P + (X+ - X)4 + P 0 (X0 - X ) 4 . P.(X. - X > 4= «4x°x
(2.43)
where, X^ = X + a X X
= X - a X
by setting X q = X, solution of these equations yields the parameters of the three point distribution, P_, P + , P q , X + , and X _ , as
1 +
3x M
a4x “ a 3X
(2.44)
29
a4x " a3X
1 -
(2.45)
3x
M a 4x " a 3X
(2.46)
X_ = X (1 - -^(M-a3x) (2.47)
X+ = X (1 + -— (M + a3x) (2.48)
where
M = v/4(a4x -
2
2
a3X + a 3X
(2.49)
2. Y _ , Y q and Y + , the function values evaluated at X_ ,Xg and X + respectively, are now calculated. Next, the mean is approximated by Y s*Yq .
E[(Y - Y ) k ] « P_(Y_-Y)k+P0 (Y0-Y)k +P+ (Y+-Y)k
(2.50)
Results for the mean and variance are not very sensitive to the skewness and kurtosis of X (Rosenblueth 1981). These
30 statistics are often not available and estimates are uncertain.
It may be considered possible to set skewness
equal to zero and disregard Eq. 2.51, giving the simple expression:
p* - £••
p0
■ ! and p- - s
X_ = X - /3 ax ; X Q = X and X + = X + /3 ax
(2.51)
(2>52)
As before, the three parameters Y + , Y q , and Y_ are estimated as the functions of X, Y(X+ ), Y(X q ) and Y(X_) respectively. From those, the mean and the variance are approximated by
Y = P +Y+ + P0Y 0 + P_Y_
»Y = P+ (Y+ " Y)2 +
2.3.5.
P 0 (Y0 " Y)2 + P - (Y- " Y)2
Numerical Examples
The following two numerical examples are presented to demonstrate the procedure and compare the results of the aforementioned methods. Example 1
As a numerical example consider the example of Rosenblueth's
(see Gorman [33])
31
with X^ and X 2 are uncorrelated and lognormally distributed.
The mean and standard deviations of function Y will be evaluated by using the exact method, Simulation,
2n point
estimates and 2n+l point estimates. The following numerical values are assigned to the parameters: = 10;
= 2? Xj =4 and ax =0.5
Solution using exact m e t h o d . The mean value of Y is
uY
= (Xj^2 (X2 )2 (1 + V 2 )(1+V2 )= 1 2 = 100 (16)
(1 + 0.04)(1 + 0.015625) = 1690
The variance is
°Y = (* 1 )4 (i2 )4 (1 + VX 1 )2 (1+VX 2 )2 E where
E = exp I n ((1 + V
(1+ V
- 1
\
Insertion of the parameters gives oy = 839 The coefficient of variation is
Vy = /(l + 0.04)4 (1 + 0.015625)4 -1 = .4947
32
Solution using 2n point estimate. The 2n possible combinations of two random variables are four, arranged in a
ro II
(10 + 2)2 (4
Y1 -
+ o .
binary pattern of the sign of the standard deviations, 2916
(10 - 2)2 (4 + 0,5)2 = 1296
OJ II
Y2 =
(10 + 2)2 (4 - 0.5)2 = 1764
II
(10 - 2)2 (4 - 0.5)2 =
784
If the correlation is zero then the weighting function for each variable equals 0.25.
The mean value of the function Y is calculated using Eq. 2.34; Y = | [ 1 2 2 (4.5)2 + 82 (4.5)2 + 122 (3.5)2 + 8 2 (3.5)2] =
= 2916 + 1296 + 1764 + 784 = 169Q
The variance of Y is given by o2 = E[Y2 ] - (Y)2
= |(29162 + 12962 + 17642 + 7842 ) - 16902 = 621155 The coefficient of variation is ✓621155 Vv - -------- = .4664. * 1690
33
Solution by 2n + 1 point estimate m e t h o d . From Eq. 2.36, Yj can be evaluated as
Y* = (10 + 2)2 (4)2 = 2304? Y" = (10 - 2)2 (4)2 = 1024
Y* = (10)2 (4 + 0.5)2 = 2025; Y ~ = (10)2 (4 - 0.5)2 = 1225
2304 + 1024 2304 - 1024 Y, = ------------ = 1664; Vv = = .3846 1 2 *1 2304 + 1024
2025 + 1225 Y, = ------------ = 1625; Vv *
2
x2
2025 - 1225 =
= .2462 2025 + 1225
Y Q = 102 (42 ) = 1600
from Eq. 2.37, the mean value of the function H is
1664 H = (1600)
( 1600
1625
)
) = 1690 1600
The coefficient of variation of H is computed using Eq. 2.38
VR = (/(-l + (1 + (.3846)2 ) (1 + (.2462)2 )) = .4663
Solution using a three point estimate. A three point estimate is made in a manner similar to a 2n+l point
34 estimate, except that the function evaluated at the mean is used with each variable. By using this method the mean value of Y and its coefficient of variation were calculated using Eqs.
are found to be,
Y Q = 102 (42 ) = 1600
Y*
=16(10 +
2/3)2
= 2900.5
Y~
=16(10 -
2/3)2
= 683.49
Yj
=100(4 +
.5»/3)2
= 2367.82
Y“
=100(4 -
.5j/3)2
= 982.2
Y 1 = |(2900.5) + |(1600)
+ |(683.49) = 1664
Y 2 = |(2367.8) + §(1600)
+ §(982.20) = 1625
Y ■ 1600 (if§ffHii§§>
- 1690
-§(yI-v 2+!2 + i(y2 - y2>2 ' 161245 Vv = Y1
°Y
= .3884 and Vv
Yx
= 2
°Y
= .2471
Y2
Vv = /(I + V 2 )(1 + V 2 ) - 1 = .4703
X
*1
*2
Solution bv using simulation. Monte Carlo technique is used to generate 1000 simulations of the function. The mean and standard deviation of the sample were calculated and the values are:
35
Y = 1703.9,
and Vy - 0.5166.
For the sake of comparing the accuracy of the approximate methods compared to that of the exact method the results of this example are presented in Table 2.1.
Table 2.1 - Results of Example 1
No.
1 2
3
Method
Mean value Coefficient of variation
Exact Monte Carlo 1000 points 2n point estimate
1690.0
0.4947
1703.9
0.5166
1690.0
0.4664
4
2n+l point estimate
1690.0
0.4663
5
three point estimate
1690.0
0.4703
Example 2
The maximum negative bending moment, M fi, (over an intermediate support, B) in a continuous beam shown in Fig. 2.3,
is estimated in terms of the span lengths and load
intensities as follows
+ W 2 (L2.2L3 )l | - W 3L 2l | mb
---------------------------------------4(4(L1+L2 )(L2+L3 )-L2)
36
Figure 2.3 - Continuous Beam with three Spans. Let all the variables in the above equation be considered as lognormals. Then, the means, standard deviations and coefficients of variation of these variables are as follows variable
L1 L2 L3 W1
w2 W3
mean value
standard deviation
coefficient of variation
30.0
ft.
3.60 ft.
12
%
45.0
ft.
4.50 ft.
10
%
25.0
ft.
5.00 ft.
20
%
4.0
k/ft.
.60 k/ft.
15
%
6.0
k/ft.
.72 k/ft.
12
%
3.0
k/ft.
.54 k/ft.
18
%
The statistics of M fi were calculated using four methods: Monte Carlo simulation with 1000 points,
2n point estimate,
2n+l point estimate and three point estimate. The results are presented in Table 2.2.
37 Table 2.2 - Results of Example 2
NO. Method
1
2
Monte Carlo (1000 points) 2n point estimate
Mean value Coefficient of variation
870.33
0.1933
869.76
0.2030
3
2n+l point estimate
869.80
0.2026
4
three point estimate
869.78
0.2037
Examples one and two indicate that Rosenblueth's point estimates give sufficiently accurate results. Therefore, Rosenblueth's 2n+l point estimate is utilized in the bridge system reliability analysis.
2.4 Simulation and Approximation Techniques Used in Calculating the Statistical Moments of dependent Random Variables
The random variables dealt with in the previous sections were assumed to be independent.
In practical applications
basic variables will often be correlated. Correlation sometimes affects the value of the reliability index, appreciably.
0,
In this section the concept of correlation is
investigated.
Let the random variable Y be a linear function of a number of random variables X.,..., X„ such that: l n
38
Y = a 0 + a1X 1 +,
..., + anXn ,
(2.53)
Then the expected value E[Y] is given by
n E[Y] = an + Z a.E[X.] U i=l 1
(2.54)
and the variance Var[Y] by
Var[Y] =
n 5 n n Z af Var[X.] + Z^Za.a.Cov[X.,X.] i-1 1 1 i*j 1 3 1 3
(2.55)
Consider now the case of a set of n mutually correlated random variables X = ( X ^Xj,..., X R ) with expected values E[X^],
i=l,2,..., n, and covariance matrix Cx . If the Cx is
diagonal then there is no correlation between any pair of the variables.
Var(Xx )
Cov(X
n
If general Cx can be written as:
Cov(XlfX 2 ) ... Cov(XlfXn )
.X,) Cov(X_,x,) l n 2
... V a r ( X ) n
A set of correlated random variables (Xlf X 2 ,
(2.56)
XR )
can be transformed to a set of uncorrelated variables (Y^, Y 2' ***' Y n^ that
means of a special transformation X, such
39
Y = ST X
(2.57)
and
ey = XT ex I
(2.58)
where the diagonal elements of Cy are equal to the eigenvalues of Cx , and %. is an orthogonal matrix with column vectors equal to the orthonormal eigenvectors of Cx .
The expected values of the vector Y can be evaluated as shown in Eq. 2.59 [107].
E[Y] =
at
E[X]
(2.59)
Consequently, a set of correlated random variables can be transformed to a new set of uncorrelated random variables by using the above mentioned transformation. The reliability analysis can then be carried out for the new uncorrelated variables.
The generation of statistically independent random variables is a direct extension of the methods used to generate a single random variable using any of the procedures mentioned in the previous section. This section describes how correlated random numbers can be generated in two cases:
40 A. Normally distributed variables. Let Z be a function of n correlated normal random variables, Z = f(X^, X 2 ,..., X n ). If the means and covariance of the random variables {X} are known. Let the mean vector be: {X} = { X x X 2 , ..., Xn } and let the covariance matrix of the X variables to be as given by Eq. 2.56. Then the approximation to find the statistics of Z is carried out in six steps:
1. Obtain the eigenvalues and the eigenvector of the covariance matrix of X variables. The eigenvalues represent the variance vector of an independent set {Y}.
2. Evaluate the mean values of {Y} using Eq. 2.57.
3. Once the mean and variance of {Y} are evaluated, the problem becomes similar to that of a set of independent random variables.
In this case any of the
techniques mentioned in Sec. 2.3 can be applied to obtain a new set of independent random variables Y.
4. The new set of Y is transformed to the original set variables X. This is done using the inverse of Eq. 2.57 such that:
{X} = [A] {Y}
5. Evaluate the value of the function Z.
(2.60)*
41 6. Repeat steps 3 and 4 to either get enough points to evaluate the statistics of function Z (case of Monte Carlo) or the limit of the method (case of Rosenblueth's methods).
B. Lognormally Distributed variables.
In this case, the
random variables can be transformed to normal variables. Eqs. 2.16-2.17 can be used to find the mean and the standard deviation of the new transformed variables. Let Z be a certain function of a set of random variables, such that, Z = f(Zlf Z2 , ..., Zn ), where Z ^ ,
Z 2 , ..., Zn are lognormally distributed. Let X
be another function of X^, X 2 , ..., X n such that X = In Z, X 1 = In Z^, X 2 = In Z2 and so on then, the covariance of any pair of the X variables can be written as:
Cy
Y
12
“
P v
Y
12
^ Y
*1
^Y
*2
(2.61)
2
Cy y 12
= Py y *1*2
2
/ltt(1 + V„ ) In(1 + V„ ) Z1
2
(2.62)
Where pv v is approximately equal =* p_ „ * 1*2 12
In this transformation the lognormal random variables are replaced by normal random variables with a new mean, standard deviation and covariance matrix. The procedure explained above for normal random variables can now be
42 carried out for the new variables. The mean and standard de-'iation of the output function will be in terms of the logarithmic space,
i. e. the output will be in terms of In Z
and a(ln Z ) . To get the mean and standard deviation of Z, the inverse of the transformation mentioned in Egs. 2.16 and 2.17 can be used, namely
Z = e x p [ H T z + |ln(l+v2)]
4
=
^
2
(exp^alnZ^ “
More details on this area are presented in Appendix B.
(2>63)
(2.64)
CHAPTER III BRIDGE LOAD MODELS
3.1
Introduction
The major load components for highway bridges are dead load,
live load with impact, environmental load (wind,
earthquake), special loads (braking forces, collision). The load effects for highway bridges vary with the span length of the bridge.
In case of long span bridges,
the source of
the most critical loads are strong winds, dead load and earthquakes. For medium and short spans (up to 300 ft), the major loads are live load and dead load. The manner in which heavy trucks are positioned on the bridge is also important. This includes the possibility of occurrence of two trucks side-by-side or in one lane at the same time. For short spans
(less than 100 ft),
it is particularly important
how the truck weights are distributed on the structure. case of very short spans (less than 20 ft),
In
it is the axle
weight along with side-by-side occurrence that dominates the live load.
In addition to truck weight and configuration, three important characteristics of bridge load are: The dynamic factor, the distribution of load to individual members and
43
44 the transverse position of truck on the bridge. The magnitude of the dynamic effect depends on the bridge stiffness,
roughness of the wearing surface, vehicle
flexibility, vehicle weight and speed. The load distribution factor depends on several parameters: The stiffness of slab and girders, the stiffness of diaphragms, the degree of composite action and the transverse position of truck in the lane.
The load combination model is based on Turkstra's rule which is discussed in Sec. 3.5.
Statistical models developed further are based on the available truck survey (Csagoly and Knobel 1981), survey of overweight truck files (citations) of the Michigan State Police (University of Michigan, unpublished), survey of transverse position of trucks (Al-Zaid, the author and Nowak) and data available from the weigh-in-motion study [30].
3.2
Dead Load
Dead load, D, is the gravity load of the structural and nonstructural elements permanently connected to the bridge. Because of different degrees of variation and for convenience, Nowak and Zhou [90] classify the components of D as follows: = weight of factory made elements (steel sections, precast concrete members);
45
Dj = weight of cast-in-place concrete members; Dg = weight of the wearing surface (asphalt); = other weights.
The statistical parameters of D are well established for buildings.
It has been found that D is normally distributed
with the mean-to-nominal ratio,
X = 1.05, and the
coefficient of variation, VD = 0.05 [24], In safety analysis it is recommended to use VD = 0.10, rather than 0.05, because of human errors.
The dead load model for bridges is based on the available data from literature [90]. The normal distribution was found to fit the upper tail of all components of the dead load, D.
The mean-to-nominal ratio, XD , is assumed to
vary as follows: XD1 XD2 XD3
= 1.03
and
= 1.05
and
= 1.0
and
VD1 VD2 VD3
= 4% = 8% = 25%
Nowak and Zhou [90] further assumed another factor to take care of the uncertainty associated with evaluating the above factors.
This factor is called
the influence factor,
its mean to nominal ratio, X, is estimated at 1.0 and the coefficient of variation is estimated at two per cent.
46
3.3
Live Load
Live load, L, covers a range of forces produced by vehicles moving on the bridge. The effect of live load depends on wheel force, wheel geometry (configuration), position of the vehicles on the bridge
(transverse and
longitudinal), number of vehicles on the bridge
(multiple
presence), stiffness of the deck (slab) and stiffness of the girders.
Because of this complexity, the effect of each
parameter is discussed separately.
Three
live load models are considered in this
thesis. The first was developed by the MTC Task Force [35], in conjunction with the development of the OHBDC [92]. Nowak and Zhou [90] have revised this model, developing a new approach to predict the extreme values. was developed by Ghosn and Moses
The second model
[30]. The model was based
on a multi-dimensional stochastic process approach that utilizes the bridge measurement data to obtain the maximum lifetime distribution of loads on multilane bridges. Several factors were considered including the occurrence of side-by-side traffic. The third model is a proposed approach based on a survey of the overweight vehicles in the State of Michigan.
3.3.1
Nowak and Zhou's model
[90]
In this model the static component of the load effect is considered. The model is based on Ontario survey data
47 [20]. The concept of Ontario truck formula to determine the equivalent base length, BM , [19] is used in this model to develop the bending moment spectra. Values of BM were calculated for whole trucks and for combinations of axles. For each pair of weight and equivalent length, the maximum value of the midspan moment was calculated. The distributions of the resulting moments are shown in Fig.
3.1
on a normal probability paper. The vertical scale is the inverse of the standard normal distribution function.
The
horizontal scale is the ratio of the moment to the nominal (OHBDC) moment.
The total number of trucks in life time of the structure has been established by the committees working on the OHBDC. The results are given in Table 3.1.
The Ontario surveyed trucks (10000) was considered to represent an about weekly traffic on class A highway.
It was
assumed that the distribution of the bending moment shown in Fig. 3.1 model the arbitrary point in time moment per lane for a class A highway. If the bridge design life time is considered to be 50 year, then the distribution of the maximum 50 year live load, LgQ, can be considered as exponential based on the findings of Grouni Ft
[35],
(x) = 1 - e“x 50
(3.1)
48
50 y e a r
liv e
load
le v e l
40 60
12, 6 , 18
All spans in m eters 24
M om eiit/O H B D C
Moment Ratio
1.0 60 40
30
Figure 3.1 - Distribution Functions of Moments for Various Spans [90]
49 Table 3.1 - Number of Trucks as Function of Time
Class of the Road +
Total Numl >er of Trucks Per day
.
Per 50 Years
A
over 1000
over 20 million
B
over 250
over 5 million
over 50
over 1 million
less than 50
less than 1 million
C1
C2
+ Classification of roads according to the OHBDC (1983).
This corresponds to a straight line on the exponential scale. Therefore, to predict the mean Lgg a line was fitted to the upper tail of the surveyed data. This approximation is accepted assuming that the exponential function fits the upper tail of the load distribution over the region that contains the design point
(see Ditlevsen [23]). However, the
coefficient of variation of Lgg was assumed to be = 11%.
Six heaviest trucks were selected from the Ontario data. The axle configuration and weights are shown in Fig. 3.2.
The ratios of the maximum midspan moments were calculated for different spans. The calculated moment were divided by the OHBDC nominal moments, so that the ratios
50
42.7
81.0
{ 3.89
81.0
86.3
J.1.85 { 3.3
86.3 82.7 46.3
77.4
]l.83 ]l.3 ]l.3 [ 3.05
56.0 M
[ 1.91 [
KN
Vehicle No. (1) L=18.41M and W=639.6KN 41.4
1
92.5 3.51
89.0
j.1.4
102.3
It 3.76
96.5
j l .85 j
100.8
89.4
[ 1.85
3.38
[
M KN
Vehicle No. (2) L=15.75M and W=611.2KN 67.2
64.9
| 1.83
j
131.7 3.43
132.6
j l .83 j
49.8 2.67.1,
100.1
109.0
[ 1.85 [
2.74
M KN
Vehicle No. (3) L-18.41M and H»639.6KN 86.7
44.5 |
3.48
85.4
[l.43 [
72.9 3.96
|
184.6 2.46
| 2.44
96.1 |
M KN
Vehicle No. (4) L-18.41M and W»639.6KN 42.7
92.1
[4.06
90.3
90.3
ll.37 [ 3.00
90.3
101.4
[ 1.8 j 1.58
j
M KN
Vehicle No. (5) L«=18.41M and W=639.6KN 49.9 1
72.9 3.53
72.9
111.2
90.7
[ 1.5s[ 3.05 [ 1.8s| 1.37
90.7 [
Vehicle No. (6) L-11.35M and W=487.9KN
Figure 3.2 - Configuration of the Heaviest Vehicles in the 1975 Survey [90]
M KN
51 were dimensionless.
The ratios were plotted on the
logarithmic scale. The results for several simple spans ranging from 3m to 60m are shown in Fig. 3.3. Horizontal scale is the ratio of the moments, while the vertical scale is the logarithm of the number of trucks exceeding the moment ratio.
The Numbers Refer to Spans in Meters
-0 60,
W o 3 »h—
o
u.
i—
H—
o
40,
o
-1
o -Q
E
S3
E 3 z
60.
3
O
z
E ^
0.5
s O) o
Moment/OHBDC Moment Ratio Figure 3.3 - Upper tails of Moment Distributions from Truck Survey [90] Nowak and Zhou [90] assumed that the 50 year population of trucks is 600 times larger than the number of Ontario truck survey. They determined the 50 year live load, L 5 q , by extrapolating a straight line for larger moment ratios. They determine the mean which corresponds to I n (600). The distributions of moment due to an arbitrary- point-in-time truck Fig. 3.1 are extrapolated to predict the maximum 50 year live load on the normal probability paper in Fig. 3.4.
52
Values of the mean and coefficient of variation for the dimensionless ratio (actual moment due to truck / OHBDC nominal moment) were determined for different spans and for both arbitrary-point-in-time and maximum 50 year live load. The results are summarized in Table 3.2.
Table 3.2 - Parameters of Live Load [90]
Arbitrary-point-in-time 50 year maximum Span mean/nominal coefficient mean/nominal coefficient m ratio of variation ratio of variation
3
0.44
0.20
1.0
0.14
6
0.48
0.21
1.05
0.125
9
0.52
0.23
1.12
0.11
12
0.48
0.27
1.12
0.11
18
0.42
0.35
1.12
0.11
24
0.39
0.40
1.12
0.11
30
0.37
0.45
1.12
0.11
40
0.33
0.50
1.12
0.11
60
0.32
0.50
1.12
0.11
3.3.2
Ghosn and Moses' Model [30 3
Ghosn and Moses [30] used a multi-dimensional stochastic process approach that utilizes bridge measurement data to obtain the maximum lifetime (50 year) distribution of loads on multilane bridges. Several factors were included in Ghosn and Moses' model , including the possibility of side-by-side
53
liv e load
le v e l
Normal Distribution
Function
50 y e a r
12, 6 , 1 8
All spans in m eters
Inverse
24
M om ent/O HB DC
Moment Ratio
1.0 60 40 >
30
Figure 3.4 - Extrapolated Live Load Distributions [90]
54
traffic occurrence. Ghosn and Moses'
explained the median of
the maximum bridge moment in a 50 year period in the following expression:
M “ a m W >95 H g I G r
Where, m
= factor to account for the randomness in the axle configuration of random truck traffic,
a
= the maximum moment effect of the standard simulation truck with a unit gross weight.
W 9g
= truck weight value corresponding to the upper 5% of the gross weight histogram.
H
= factor that relates the moments of the standard truck (W 9g) to the estimated maximum lifetime load on a bridge,
g
= distribution factor.
I
= impact (dynamic) factor.
Gr
= growth factor to account
for the fact
that truck weights have been increasing in time.
Values of the parameter are presented in Table 3.3.
The model presented in Eq. 3.2 is used to generate the load spectrum per girder. The girder load spectrum is used in this study to calculate the reliability index of the
55
Table 3.3 - Input Data for Eq. 3.3 [30] 1 Variab .e
a (1 ’
m
12 >
.95
H (* ’
g (2 ) 1(2’
40
a
3.54
4.94
8.4
13.4
m
1.03
1.08
1.06
.2
.18
.17
V
1..... 150
18.4
24.4
30.9
1.02
1.0
1.0
.99
.12
.10
.08
.07
W 75.25 75.25 75.25
75.25
75.25
75.25
75.25
H
2.63
2.69
2.75
2.78
•2.8
2.86
2.87
9
.3
.3
.3
.3
.3
.3
.3
i
1.2
1.2
1.2
1.2
1.2
1.2
1.2
1.15
1.15
1.15
1.15
1.15
1.15
1.15
5r
1’
1— .. 125
Q ( 2J r
r..... I .... jSpan in ft. 60 80 100
30
vm m
w
r
r
282.1 449.8 806.5 1164.9 1524.0 1973.2 2475.1
(1 ’Deterministic variables [30] ( 2 ’Coefficient of variation of these variables do not affected by the span length. Their values are, .1, .07, .1 and .1, for Vw , VH , V^, Vj and VGr respectively.
individual girder. Statistical parameters (mean and standard deviation) of Eq. 3.2 are presented in Table 3.3 for different spans.
56 3.3.3 Proposed Approach
The 50 year maximum live load model has been developed in this study based on citations of overweight trucks in the state of Michigan.
Overweight trucks were evaluated as part
of a general plan to collect data on highway traffic loading. There were 1600 citations available, as the Michigan State Police keeps them on file for about 6 months only.
The maximum moment and the corresponding critical
position were determined for different spans. Dimensionless factors, denoted as load ratios, LR, were calculated as the ratio of the moment due to actual truck, M
t , to the
specified moment. Three specified moments were considered: moment due to AASHTO Truck HS20,
moment due to
Ontario Truck, M 0HBDC an(* moment corresponding to Ontario equivalent base length formula, M fl.
M act
LR a = — —
MHS20
LRfi =
M act
LRq
B
M act
= — -- ■
M' OHBDC
The moment due to AASHTO HS20 is considered as the basis for live load model in this study. The other two specified moments are used for comparison.
Based on the findings of Ditlevsen [23], the cumulative distribution functions of load ratios LRA , LRfl and LR q , are plotted on normal probability paper for different spans. Ditlevsen has shown that any (smooth) distribution of
57
load or resistance may be replaced by its normal tail approximation for the purpose of evaluating the probability of failure. The normal tail approximation of a distribution is the normal distribution that has the same value and the same density at the design point. The design point is defined as the limit of an iterative process called the Rackwitz-Fiessler algorithm [66]. The design point is normally unique for a pair of smooth distributions. Thus,
if
the distribution of load (or of resistance) have the same normal tail approximation they yield the same probability of failure and hence the same reliability index. Typical distributions of live load ratios are plotted on normal probability paper for 60 ft. span as shown in Fig. 3.5.
These distributions represent the arbitrary-point-in time truck loading for three months. The 50 year live load model considering the live load ratio, LRA , is presented. The following assumptions were made:
1. The current arbitrary point-in-time live load distribution is representative for the 50 years to come.
2. Three months truck loadings are independent random variables.
3. On short and medium span bridges, only one heavy truck is considered to cause the maximum bending moment.
58
/ M-AASHTC
CM
INVERSE NORMAL DISTRIBUTION FUNCTION
/ M-BMAX / M-ONTARIO CM
o
CO
CM
0.00
0.60
1.20
1.80
LOAD RATIO
Figure 3.5 - Typical Distribution of the Load Ratio (60 ft. Span)
2.40
3.00
59 4. The surveyed truck is considered as representative sample of a three month population. The 50 year population of trucks is therefor assumed to be 200 times larger than the surveyed number.
Hence, the 50 year live load distribution, Fgg(x),
is given
by:
F 50*x * " ^F ^x ^ 20°
(3.3)
where,
F(x) = three month distribution of live load.
The 50 year load distribution is then evaluated based on Eq. 3.3. A design point is chosen in the upper tail of the 50 year load distribution.
A straight line is fitted
tangent to the original distribution at this point [23], Hence, the mean and standard deviation are evaluated. Different spans are considered.
To compare the three models, Ghosn and Moses' model Eq. 3.2 was modified to obtain the maximum 50 year live load in terms of Load ratio, LRA , rather than moment. Hence, Eq. 3.2 becomes
ip .
a m W Qc H »
MRS20
(3.4)
60 where,
T = maximum live load moment in the bridge measured in terms of the moment due to AASHTO Truck HS20. Other parameters are the same as in Eq. 3.2.
Simulation was used to generate the cumulative distributions of the three models. The results are plotted on normal probability paper,
in Figs. 3.6 - 3.8. The three
models are compared in Table 3.4.
3.4.
Dynamic Load
Dynamic effect is considered as an integral part of the live load model. The major factors affecting dynamic load on a bridge include surface condition (bumps, potholes), natural frequency of the bridge (span length, stiffness, mass) and dynamics of the vehicle (suspension, shock absorbers). Practically it is impossible to predict the percentage contribution of these three factors. Traditionally, dynamic effect is measured by an frequency of the bridge,
impact factor.
It was calculated as
function of the loaded length of the bridge [2]. OHBDC [92] specifies the dynamic load allowance as a function of the natural frequency of the bridge.
Nowak and Lind [89] considered the mean impact effect,!, as a fraction of the mean largest live load effect, L. They found that this fraction is a function of the first flexural frequency of the bridge. They presented an approximate
4.20
61
NOW AK and ZHOU MODEL
3.40 0.60
0.20
1.00
1.80
2.60
PROPOSED M ODEL
1.40 -
2.20 -
3.00 -
INVERSE
-
NORMAL
DISTRIBUTION
FUNCTION
GHOSN and MOSES MODEL
0.00
1.00
2.00
3.00
4.00
5.00
6.00
LOAD RATIO Figure 3.6 - Distribution of 50 Year Load Ratio for 60 ft. Span (Comparative Results)
7.00
62
NORMAL
DISTRIBUTION
FUNCTION
rvi
* NOW AK and ZHOU MODEL ♦ GHOSN and MOSES MODEL * PROPOSED M ODEL
CO CM
00
CM
■
INVERSE
(O
CM CM*
CO
0.00
1.00
2.00
3.00
4.00
5.00
6.00
LOAD RATIO Figure 3.7 - Distribution of 50 Year Load Ratio for 80 ft. Span (Comparative Results)
7.00
NOW AK and ZHOU MODEL
C\l
FUNCTION
00
PROPOSED M ODEL
o CN
3 =0.00
Strain /1000
Prestressing Steel
........................................... 10.00
20.00
30.00
40.00
S0.00
Strain /!000
Structural Steel
Figure 4.3 - Stress-Strain Relationships
60.00
70.0
Actual Section
Idealized Section
Figure 4.4 - Typical Composite Steel Section
i-1
= the bottom strain obtained in the previous step; = the unbalanced internal force resulting in the present step; and
F i-1
= the unbalanced internal force resulting in the previous step; The procedure is repeated for increasing top strain
level until enough points are obtained to establish the moment-curvature relationship. A typical curve is plotted in Fig. 4.7.
Two assumptions were made in the analysis:
82
m Composite Section
T Section
w m ?
P #
‘"”§ |
w m
if'
te f
m l m
Double T Section
v.-v.v., fe&
.jW
1 L*.«*•••••
>•••••*
w m m m Box Section
Figure 4.5 - Typical Sections commonly used in Prestressed Concrete Bridges
83
Top Strain
X
Bottom Strain Figure 4.6 - Typical Strain Diagram in a Composite Section 1). The effective flange width (bg ) can be used rather than the total flange width to account for the effect of shear lag. The effective flange width was studied by Jaeger and Bakht [51] and Heins [44], Heins defined the effective flange width as follows. For interior girder
2be = b C 61? b + 0.702)
(4 ,
For exterior girder
be = b(0.873 - £)
(4.9)
84
z 2sT-^
i).00
*©
\ ° ;* ■? , ^ / o V ’ O
o °‘ n '
'o '- 0
* < * ' 4
W3CX135
1200.00
2400.00
INCREMENTAL
3600.00
4800.00
9999.99
MOMENT (Kip.In) (XICT1)
7139.99
84
10.00
20.00
30.00
CURVATURE.
40.00
Figure 4.7 - Typical Moment-Curvature Relationship for Composite Steel Section
50.00
85
jr = aspect ratio; b
= specified flange width;
L
= span length;
bg
= half of the effective flange
width in the case of
interior girder, or flange width in the case of an exterior girder. Based on the above expressions,
it was found that the ratio
of the effective flange width to the total width can be taken as equal to 0.74 if the aspect ratio is within the range of five percent to seven per cent.
2). There is a complete composite action between concrete and steel section.
The effect of slip was neglected based on experimental and theoretical work done by Kurata and Shoda [56] who observed that slip is not important in the ultimate loading capacity analysis. Yam and Chapman [115] found that the ultimate capacity because of slip is reduced by six per cent to eleven per cent with respect to the ultimate capacity.
4.3.2
Torsion-Twist Relationship
Two methods are considered. One is developed by Michell and Collins [72] and extended here to be used in the analysis of composite prestressed concrete sections and reinforced concrete T-Beam section. The second one is developed in this study to be applied on composite steel
86 section and denoted as the incremental twist method. The two methods are explained in the following section:
Extended Michell and Collins Diagonal Compression Field Theory. (DCFT).
Mitchell and Collins [72] developed the DCFT to predict the true characteristic of a reinforced concrete section under a pure torsion. The DCFT is extended in this work to predict the behavior of a composite prestressed and reinforced concrete T-beam section. Several assumptions are made:
1. The elastic theory is applicable to the whole section in the pre-cracking stage.
2. Torsional shear is resisted by diagonal compression in concrete and the tensile stresses in the transverse and longitudinal steel.
3. It is assumed that the behavior of the actual section, shown in Fig. 4.8 can be modeled as shown in Fig. 4.9.
4. The over hanging parts of the slab bL and bR Fig. 4.10 can not carry any more torsion after the elastic stage.
Hence, only the web part will take
any increment in the torsional loading.
87
Figure 4.8 - Actual Section
12.5
"
i
r
Figure 4.9 - Idealized Section
88 5. The stirrups are closed and the longitudinal steel is uniformly distributed.
The procedure of the DCFT is programmed and tested with some other theories on a simple L shaped reinforced concrete section Fig. 4.10. The procedure is summarized as:
1. Compute the elastic torsional capacity of the whole section using the elastic theory,
2. Compute the angle of twist corresponding to this torsional capacity, and
3. Using the DCFT, compute the torsion vs twist relationship for the web part of the section under increasing torsional loading.
The Incremental Twist A p p roach.
This method is suggested in this work to predict the behavior of a composite steel section under an incremental torsional stresses. The method is approximate and it is modified based on the findings of Heyman [47]. Heyman found that,
if the stress-strain relationship in shear is elasto-
plastic, then the maximum shear stress that can occur has a value of
tq
, where
tq
is the shear yield stress in the
structural steel. Further, Heyman assumed that the phenomenon of torsional stresses can be represented by imagining that planes of constant slope (equal to the yield stress) are constructed around the edge of the membrane. As
1000-
80 0-
G
•H
4i
I
Torsion
20
6 00 -
4 00-
200-
rad./in 14
10 Twist
Figure 4.10 - Torsion Vs. Angle of Twist Relationship
10
90 the torsion is increased, then the pressure is i n c r e a s e d * ->■ above that corresponding to the elastic limit, the membrane will come into contact with the roof slopes. Further increase in pressure will result in further areas becoming plastic.
In fully plastic condition, the shear stresses
become parallel to the edges. The Phenomenon is illustrated in Fig. 4.11. The findings of Heyman is further idealised in this research as shown in Fig. 4.12. This idealization forms the basis of the suggested method where .
The concrete slab failure in torsion is assumed to be ductile (i.e., the concrete will retain its yielding torsion). Hence, any additional torsion beyond the elastic torsional capacity of the section will be resisted by the steel section only. There fore the theory of thin walled section can be applied. A twist angle is applied incrementally on the section. The response of the section is calculated in terms of torsion at each incremental step.
The procedure is described in the following steps:
1). Calculate the initial angle of twist (0j) which is enough to cause the concrete flange to be yielded in torsion. $ =
Tntax G X K
where, Tm a V = 5 i/f~ lUoX c as defined by Hsu (1984);
(4.10)
91
k 3
Jl
k
»
E =
T
Elastic State
Initial Plastification
k
_L
k
a
T Partial Plastification
Complete Plastification
Figure 4.11 - Typical Thin Section under Different State of Torsional Stresses [47]
k a
T Elastic State
k
Initial Plastification
H ii. k a
1
Partial Plastification
T Complete Plastification
Figure 4.12 - Idealized Thin Section Under Different State of Torsional Stresses (Used in this Study)
92 f
= Maximum compressive strength of concrete
c
G
= Shear Modulus in elastic stage
X
= Thickness of concrete slab:
8
1
K n2 cosh n2ffxY
(4.11)
where, Y « total width of the flange;
K
Y 1 if — > 5 X
and
n = number of the series terms
2). Compute the total torsional response of the section due the initial angle of twist 9 j
N
1
3
T e = 2 (I G i 9 I X i Y i C i } e i=l 3 1 1 1 1 1
(4.12
where, N
= number of parts forming the section. 1,1.
Xj = width of i
part and Yj is its length and
its shear modulus; 9 j = initial angle of twist computed from Eq.
4.10
T g = maximum elastic torsional capacity of the section.
192 X.
xY. uaau
(4.13)
93 3). Calculate the maximum torsional stress in steel for a given r using Eq. 4.10.
4). Check the penetration of the yield in the structural steel by comparing the calculated rmax with the maximum allowable torsional stresses in steel. the calculated
max
If
is greater than allowable then s
calculate the width of the elastic portion X 0 of each part of the section from Eq. 4.10.
5). Calculate torsional capacity of the steel parts using the sand-hill analogy. For a thin walled rectangular section, the torsion capacity of the yielded part Tp is
T . * 2 rmax (3Y - X) (4.14)
Where, T
= Maximum torsional capacity of a rectangular section if the section is completely yielded the rest of the parameters are as before.
Tmax = yield stress
/3
structural steel in shear
94 = flexural yield stress.
The capacity of the rest of the steel is calculated using the elastic theory.
If the steel parts of the cross section is partially yielded,
then Eq. 4.15 is applied to calculate the torsional
capacity.
1
max
(3Y1 - X 1 ) +
6
t
e (4.15)
where, T
is the torsion obtained from Eq. 4.11 is the torsion obtained from Eq. 4.10, replacing X and Y
by
and Y^.
The procedure mentioned above is repeated until all steel is yielded. Example 4.1 is presented to clarify the procedure.
Example 4.1
The procedure will be demonstrated on a composite steel section as shown in Fig. 4.12. The given data is: concrete strength,
f
c
= 4000 psi steel yield stress, F
section dimensions are as shown in Fig. 4.14
Solution
y
= 36 ksi
95
7.25 " .74"
I W 3 3 x 118
31.38 "
.74" .74"
Figure 4.13 - Composite concrete steel section The first step is to get the value of r which corresponds to the maximum torsional capacity of the concrete flange, = 5i/4000 = 316.2 psi
t
ei
~ 1584400 x 8 = 2,5x10
5 rad/inch
192 x 7.25 _5 „ Q . 7T5 X 84
„ tanh
2 x 7.25
= .946
192 X . CJ1
- i steel flanges ~ 1
e
5 v
X
r*
concrete
= 1
1
C
Lanu ff x I I »48 _ nr Afl tann 2 x .74 " *96 11.
2)
[Ke ] - [ T l X H T ]
( 5
3)
Where, I f {Fg } and {dg }
element end forces and deformation.
1
indicates local coordinate system.
The element stiffness matrix,
[Ke ], is symmetrical about the
diagonal and its general form is
108
Sll S21
S22
S31
S32
S33
S41
S42
S43
S44
S51
S52
S53
S54
S61
S62
S63
S64
[Kg ] =
(5.4) The definition of the parameters in Eq. 5.4 is presented later in this Chapter. The coordinate system considered in this analysis is presented in Fig. 5.1.
b - Global Coordinates
End (i)
a - Local Coordinates
End (j)
Figure 5.1 - Coordinate System
109 The rotational matrix,
[T], with respect to these
coordinates is given by:
Cos0
0
Sin0
0
0
0
0
1
0
0
0
0
-Sin0
0
Cos 0
0
0
0
0
Cos0
0
Sin0
[T] = 0
0
0
0
0
0
1
0
0
0
0
-Sin0
0
Cos0 (5.5)
where, 0 = the angle between the element X-axis and the global X-axis.
The stiffness matrix for a member with a plastic hinge is formed as for a nonprismatic beam element.
Numerical
integration is used to form the flexibility matrix of such an element. The element is discretized into ten segments. The bending moment and the curvature are calculated at the end of each segment.
The elements of the flexibility matrix,
[F], of a
nonprismatic beam element (Fig. 5.2) are: T Fll =
.
10 M. M, 9 M. M. + 4 2 ^ + 2 2 — i] EI1 i=2,2 E I £ i-1,2 EIj
F21 = ^ [ 4
10 M. M. 9 M. M. 2 -1 + 2 2 -1 ] i=2,2 El. i»l,2 EX.
(5.6)
(5.7)
a - Nonprismatic Beam Element
1 1 5 ^ b - Unit Moment at End i (Ml)
c - Curvature Due to Unit Moment at End i ( H i ) El
lliUII
b - Unit Moment at End j(M2)
^
c - Curvature Due to Unit Moment at End i (H ?) J E l'
Figure 5.2 - Nonprismatic Beam Element
Ill
(5.8)
Where, Fll = the rotation at end 1, due to unit moment at the same end. F21 = the rotation at end 2, due to unit moment at end
1
.
F22 = the rotation at end 2, due to unit moment at same end. FI2 = F21. EIj = flexural rigidity at location i of the beam, L
=length of the beam element,
1 . 0
0 . 0
0.9
0 . 1
0 . 8
0 . 2
0.7
0.3 0.4 0.5
0 . 6
0.5 0.4 0.3
and M . = 1
0 . 6
0.7
0 . 2
0 . 8
0 . 1
0.9
0 . 0
1 . 0
The inverse of the flexibility matrix, element stiffness matrix,
is the beam
[C], with respect to two flexural
coordinates. The elements of the 22,
F22 Cll
[F],
[C], matrix are:
F12 ; Cl 2 =
(Fll F22 - FI2 F21)
(Fll F22 - FI2 F21)
112
Fll C21 = C12;C22 = --------------------- . (Fll F22 - FI2 F21)
The elements of the member stiffness matrix presented earlier in Eq. 5.4 , can be established as follows:
GJ GJ GJ Sll = — ; S41 = ---- ; S44 = — ; L L L
Cll + 2 C12 + C22 Cll + C12 ; S22 = ---------- 5 -------- ; S32 = L L
Cll + 2 C12 + C22 C22 + C12 ------------ 5 -------- ? S62 = ? L L
S52
Cll + C l 2 S33 = Cll; S53
; S63 = C21;
Cll + 2 C12 + C22 C22 + C12 S55 = ---------- 5 ; S65 = ------------L L
S6
6
= C22.
Where, GJ
= elastic torsional rigidity if no torsional plastic hinge is formed,
113 - plastic torsional hinge otherwise
An assemblage of the element stiffness matrix is presented in Fig. 5.3.
5.2.2 Distribution of Wheel Loads
A special consideration is given to load partitioning. When the position of a load does not coincide with a grid node,
it will be distributed linearly in the
longitudinal direction without taking into account the moments that are associated with distribution. The load will be distributed nonlinearly in the transverse direction,
i.e
the moments associated with the apportioning will be as counted for [51]. This idealization is reasonably accurate if the bridge is divided in the longitudinal direction into panels of length not more than 1.5 times the spacing of longitudinal elements [21]. The method of load idealization is explained in Fig. 5.4. The loads and the moments associated with the apportioning are listed below:
P b c d2
P b(S + 2c) (5.10)
(5.11)
P a c d2
P a(S + 2c)
(5.12)
(C11+2C12+C22) ,2
(C11+C12) L
(C11+C12) L
(C11+2C12+C22)
,2
Cll
JG L
(C22+C12) L
(C11+C12) L
Cll
JG L (C11+2C12+C22)
,2 (C22+C12) L
(C11+C12) L C21
(C11+2C12+C22) ,2
(C22+C12) L
Figure 5.3 - Reduced Nonprismatic Element's Stiffness Matrix
(C22+C12) L C22
115
Pb
Pa
c - Load Apportioning in
a - Location of the Load
Longitudinal Direction
g - Load Apportioning the four corners
Figure 5.4 - Wheel Load Distribution
F a d e 2 4
L S
2
= E_b - P '
L
4
1
(5.13)
Where, a, b, c, and d are defined in Fig. 5.4.
The components of the distributed load, M ^ , P^, M 2 , P 2 / M 2 , P2 ,
and P4 , form the external actions on the grid
nodes.
These actions will be used to form the overall load
vector,
{P }.
5.2.3 Analysis Procedure
The elasto-plastic analysis of the composite bridges under study was performed using a step by step load incremental procedure.
Initially the bridge was analyzed
116 elastically. The minimum value of the load intensity that will cause any point on the structure to plastify is then determined.
The load is incremented so that at each step, one or more additional plastic hinges will form. The formation of plastic hinges (torsional or flexural)
is determined by
comparing the element loads with the element resistances. The response of each element in the failure path is considered to be linear within each load increment. The load pattern is considered constant which is the AASHTO Truck configuration (this approach has the advantage of handling various of truck configurations).
For
each load increment, the different member stiffness matrices are updated to account for the newly formed plastic hinges. The procedure is repeated until an unacceptable level of permanent deformation has occurred (one per cent of the span length).
A computer program has been developed to perform the elasto-plastic analysis. The program flowchart consists of the following steps:
1. Form the element stiffness matrix, Fig. 5.3,
2. Form the rotational matrix of each element [T],
3. Transform stiffness matrices using Eq. 5.3,
117 4. Place the element stiffness matrices in the overall stiffness matrix,
5. Form the load vector and solve the system of simultaneous equations to get the structure deformations.
6
. Calculate the member end forces using Eq. 5.2
7. Check the end forces of each element to find the maximum
incremental ratio needed to cause forming of
the first plastic hinge using the following equation:
|AF | r = F limit state “ lFoldlI
(5.14)
Where, AF= response due to incremental load; F q 1(j = cumulative actions from previous steps. F limitstate = e ^t^ler y i el(* moment, ultimate torsion or ultimate shear
8
. Increase all the structural element forces using the following equation : p
AF ^^
_ p new '
old
r
Where, F««,, “ total new cumulative actions, new '
(5.15)
9. Trace the plastic hinges and form the reduced stiffness matrix of the individual elements.
10. Repeat steps five to nine until either a singular matrix or an unacceptable level of permanent deformation is obtained.
5.2.4
Method Verification and Examples
The developed procedure is demonstrated on three types of structures. The structures were analyzed and the results were compared with other well known methods, such as yield line theory, virtual work and the finite element method. These examples serve to illustrate the degree of validity of the grid method.
Example 1 :
A clamped beam is loaded unsymmetrically by a point load (P) as shown in Fig.
5.5. The ultimate load was calculated
using the statical and the virtual displacement method. The results were compared with the grid method where a good agreement has been observed.
Static method:
(1) The first plastic hinge
The first plastic hinge will form where moment takes its maximum value which is at point A. Mv P = — 14.4
If M
= 4820 k.ft then P = 334.7 K. y
(2) The second plastic hinge
After the first plastic hinge has been formed the clamped beam can be modeled as M fi = AP . a . b(L + a)/
a propped cantilever.
21 2
Mg = 16.8 8P M c = AP . a (2 - 3a/L +
a 3 /L 3 )
M c = 17.28 AP
The second plastic hinge will be formed at point C when: 4820 - 3855.74 AP = --------------- = 55.8 kips 17.28
(3)The third plastic hinge
After the first and the second plastic hinges are formed the beam can be modelled as a cantilever beam . Mg = AP x b - 60 DP hence,
120
AP = 4280 - 937.44 - 3213.12 = 11.16 kips The total cumulative load is: 11.16 + 55.8 + 334.7 = 401.66 kip Virtual Displacement Method: The external work, E . W . , is P A. The internal work,
I.W.,
is
2(1.5 My 9 + M p )
But, E.W. = I.W. hence P = 401.67 kip. Grid Method. The structure has been analyzed using grid program. The results at different levels of loading are identical to those obtained by other methods as shown in Fig. 5.5.
450 375 300 2 225 150 60'
j40'
0.1
0.2
Deflection in ft. Figure 5.5 - Clamped Beam, Load Vs. Deflection Curve
121
Example 2 :
A concrete slab is simply supported on four sides with loads and geometrical details as shown in Fig. 5.6
Material and section properties:
1.1 reinforcement ratio
p
= ---
100 I concrete strength Reinforcing steel yield
f
= 4000 psi
strength Fy = 60 ksi
Slab thickness
t
=
Effective depth ofreinforcement
d
= 4.75 inches
The yielding moment per unit length, Mp
6
inches
= 12.5 k.ft.
Yield line theory: w
C
1
= -------------Ly
( C 2
+ C3) 2
Where, Wn
= maximum failure uniform load per unit area (in k s f ),
W
= total collapse load (kips),
M p X = yielding moment in the X direction Mpy = yielding moment in the Y direction Lx
Ly
= Length of element in the X direction
= Length of element in the Y direction
Cj = 24 My
122
Simply Supported Slab
T t = 6" J-
As = .6 2 ”s q ./ft. Typical Cross Section in The Slab
Figure 5.6 _ Plate Element
Hence, Wn = 1.6927 Total load on the slab, W = W„ L L n x y W = 1.6827
(12)
(15) = 304.7 Kips.
Grid method
The plate structure has been divided into sub-elements. Four cases are considered; very coarse mesh, coarse mesh , medium mesh and fine mesh. Fig. 5.7 show a typical case of grid mesh (coarse mesh). The properties of the structural elements were calculated using Eqs. 4.1 - 4.6.
The load versus deflection for different mesh size is presented in Fig. 5.8. The conclusion is, as the grid mesh gets finer the failure load predicted by the grid method gets closer to that of the yield line theory. The effect of the mesh size on the results is presented in Fig. 5.9. McCarth and Traina [1984] observed that the results obtained using yield line theory form a lower bound to the results obtained using fine grid mesh. The same conclusion is obtained in this example where the fine mesh failure load
124
3.75'
3 .7 5 ’
T
© —
3’
t 3’
1
Figure 5.7 - Plate Grid Model was 308.0 kips, whereas the yield line theory gave a value of 304.7 kips. Example 3 ;
In this example a 15 ft. composite steel girder highway bridge model is considered. The bridge consists of five longitudinal girders.
The
girders are 1.5 ft. apart. The concrete deck slab is 1.75 inch thick.
The bridge cross section is shown in Fig. 5.10.
In the analysis the bridge is represented by longitudinal and transverse elements.
Strength properties
of the different elements were calculated using the section analysis methods (Sec. 4.3). The results obtained by using the grid method agree well with the ones reported by
125
a.
QS
Ui oc u z M
1
Fine Mesh
2
Medium Mesh
3
Coarse Mesh
4 Very Coarse Mesh 5 Yield Line Theory
0.25
0.50
1.00
0.75
1.25
1.50
DEFLECTION
Figure 5.8 - Effect of Mesh Size on the Failure Load
*\
%
25
\
\
X
\ \ \
o 13 W
»•1
*2
* .3
» f-« *4
»
►
.5
l/(area of the mesh panel)
Figure 5.9 - Effect of Hesh Size on the Accuracy of the Results Compared with the Yield Line Theory
a -Transverse Cross Section
1.75“? " HQO .188
8.0"
— ► ««—
.135"
.188^-
T
2.281 b
-Details ofCross Section
Figure 5.10 - Details of the Bridge Model
127
i i
o
P in kips
Yield Line Method
o cvi o ©
Load
x Finite Element
Method
o
/
CO
o CO
o Grid Method
o c\i o
d
0.0
0.2
0.4
0.6
0.8
Deflection in Inches
Figure 5.11 - Load Vs. Deflection (Three Models)
128 Wegmuller [112] using finite element approach. The failure load obtained using grid program 27.4 kips, versus 28.5 using the finite element method. The simplified yield line [100] gave a value of 27.0 kips. Fig. 5.11 shows a comparison between the three results.
Example 4 :
In this example a full scale bridge of 60 ft span is analyzed. The bridge consists of five girders spaced at 6.75ft.
Details of a typical section of the bridge is shown
in Fig. 5.12. The bridge grid model and the properties of its components were calculated as in example 3. The bridge is analyzed. The elasto-plastic response of the bridge system is presented in Fig. 5.12.
Observationsi
The following observations can be made from the analysis in the preceding four examples:
1. The developed grid analysis method showed a good agreement with the available experimental data,
2. Comparison between the developed method and other methods (FEM and Yield Line Theory)
reveals that the
method is efficient with respect to computer time and generality of application.
720.00
800.00
129
Reserve Strength
560.00 480.00 240.00
320.00
400.00
Initial Yielding
,# 5 « t * » at 18'
160.00
*S a t s"
PL. to * o .a
80.00
I
Deflection due to Dead Load Permanent Deflection
0.00
TOTAL
LIVE
LOAD
ON
THE BRIDGE
(kips)
640.00
Ultimate Strength
— 0— 1----- 1----- 1----- 1----- 1----- h-’--- 1----- 1----- 1----- 1
0.00
0.20
0.40
0.60
0.80
DEFLECTIO N (ft)
Figure 5.12 - Elasto-Plastic Response for Composite Steel Girder Bridge
1.00
130
3. The strength reserve in the bridge system is considerable and may be as high as 25 per cent (Fig. 5.12).
5.3
Ultimate Strength Reliability
5.3.1 Background
The preceding approach was devoted to a deterministic analysis. The ultimate strength of a bridge system was obtained while all the material properties and load parameters were at their specified values.
In this section
the uncertainty associated with determining the values of these parameters will be considered.
Traditionally,
in highway bridges,
reliability was
checked by checking the reliability of single elements. practice,
In
a structure is a system with many interacting
components. Failure of a structural system may involve one or more failure modes. The structural model
should consider
all the relevant failure modes.
From the reliability point of view, the fundamental models of structural system are series, parallel and combined systems. A series system (weakest link) fails whenever any of its components fails. An example of a series system is a determinant truss as shown in Fig. 5.13.
In a
parallel system the failure of one component does not lead to a total failure. An example of a parallel system is a statically indeterminate frames. Fig. 5.14. Combined system
131 is a combination of parallel and series. An example is an indeterminate truss with three panels, two diagonals in each panel. Figs. 5.15-5.16 show typical examples of combined system. Classification of a structural system depends also on other factors. Some of them are: ductility of the system components, degree of redundancy,
load characteristics and
mode of failure. The term ductility is defined in this section as the ability of the component to hold its load after its ultimate strength is reached. A review of the fundamental system classes is presented in Appendix A.
P 1
2
n
I
(a)
p
(b)
Figure 5.13 - Example of a Series System: a Determinate Truss b) System Reliability Model)
132
(a)
(b)
Figure 5.14 - Example of Parallel System:) a Statically Indeterminate Structure) b System Reliability Model)
1,2
1,3
2.1
K.1
Figure 5.15 - Reliability Model for Series System in Parallel
133
sl.l
—
—
S2,l —
S1.2
—
Si,n —
Sl,3
—
—
—
S2,3
s2,n —
— *f— •
—
SM
—
:
:
sk,2
sk,3
•
—
Sk.n —
Figure 5.16 - Reliability Model for Parallel System in Series.
5.3.2
Proposed Approach for Evaluating System Reliability The approach is based on evaluating the two basic
parameters of the limit state function (Eq. 2.2), the resistance, R, and the load effect, Q. The statistics of the load effect has been investigated in Chapter III. Statistics of the bridge resistance will be determined next. 1). Statistics of System Resistance, R.
The resistance of the structure is a function of several random variables, such that, R = f(Xlf X 2 , ....... Xn )
134 where, X^, X 2 ,..etc, are the strength of individual components
(deck slab, girders load configuration and
transverse load location). The statistics of the components' strengths was calculated as shown in Chapter IV. The AASHTO HS20 configuration will be considered. The statistics of R (mean and standard deviation)
is to be determined. The usual
approximate formulas [10] are obtained from Taylor expansion of the function about the expectation of the random variables. This approach imposes excessive restrictions on the function (existence and continuity of the first few derivatives) and requires the computation of derivatives.
In
case of bridge system analysis with complex interaction of different components, Expansion.
it is almost impossible to use Taylor
This difficulty has been overcome by using
Rosenblueth's point estimates [101]. Rosenblueth developed his method (2n point estimate) for independent random variables.
a function with
This method has been extended
in this study to handle the case of dependent random variables. This modification together with Rosenblueth point estimate have been explained in Chapter II.
Demonstration of the procedure
Let the system shown in Fig. 5.17 represent a bridge deck with five girders. The strength of the components are:
Xj, X2, ... Xg.
135
P
X,
Xj
X,
x4
^
Figure 5.17 - Typical Cross Section in Composite Steel Girder Bridge Let the covariance of the X.'s be £v (Eq. 2.42). The j A VVi subscript, j, is used to refer to the j component. The histogram representing the transverse location of the truck (Fig. 3.9)
is discretized into six intervals. The
probability of the truck to be in each location, P^, calculated from the histogram. refer to the i ^
The subscript,
is
i, is used to
location of the truck.
The general theory of the procedure is explained in Sec. 2.3.3. The application is explained in the following steps: 1. Compute the bridge resistance, Y^, with all Xj's at their mean values.
Decompose the covariance matrix to transfer the correlated random variables to a new set of independent random variables,
Z 2 ,....Zg.
The mean values of the independent variables, m are obtained as, {Z} = [A] .{X} (see Section
{Z},
2.3.3). The variances of the Z^ are equal to the eigenvalues of the matrix A.
Increase Z. with one standard deviation, J
o„,
other
variables are kept at their mean values. Compute the equivalent set of Xj's where, {X} = [A]{Z }
Compute the resistance of the bridge system using the nonlinear analysis program,
Repeat step 3 with Zj
decreased by one standard
deviation.
Repeat step 4 to obtain Y T j .
Calculate the average value of the resistance corresponds to j*"*1 girder such that:
137
(5.17) 9. Calculate the average value of the bridge resistance, R'-j,
for the truck being in location i. Yij
y.
n —
(5.18)
-i + n (i+v?.) j=n
^
(5.19)
where, n = number of components.
10. Repeat steps 4 through 9 for other truck locations.
11. The overall resistance of the bridge is calculated as:
(5.20) where, k
= number of discreated intervals of the
transverse truck locations
2). Statistics of Load Effects.
The different components of load effects have been studied in Chapter III. The mean and standard deviation of load effect has been established for different spans. The
138 distribution of the load effect is considered as Lognormal [30].
3). Reliability Analysis. Once, the two variables of the limit state function have been evaluated,
then any of reliability index,/?, models
(normal model, Eq. 2.13, or the lognormal model, Eq. 2.22) can be used to determine the reliability index, /?.
The obtained reliability index, /?, gives an estimate to the reliability of a bridge system even if the distribution of the load and resistance are neither normal nor lognormal [30].
5.3.3
Proposed Approach for Bridge Capacity Rating
Bridges that have questionable load carrying capacity are checked by bridge capacity rating. Rating is performed by a combination of field inspection and analytical study as guided by AASHTO [2,3]. The application of these guidelines will be illustrated using the reliability based analysis.
There are two levels of bridge capacity rating [3,114]:
1). The inventory rating which is defined as the load that produces a stress in the critical bridge element of 0.55 times the yield stress or the allowable stress used in design.
2). The operating rating which is defined as the maximum load that should be allowed on a bridge under any
139 circumstances and should not exceed 0.75 times the yield stress.
Lathia [57] classified the rating into three levels;
in
addition to the two levels mentioned above the third level which he introduced was, the " Safe Load Capacity", defined as the actual load that can be carried safely by the structure on a long-term basis under actual traffic conditions.
The rating factors defined by AASHTO [3] are:
inv (5.21)
RF
°-75 Mu - “p opr
(5.22)
where RF. mv RFQp r Mu
inventory rating factor = operating rating factor, = ultimate moment capacity of girder = moment due to dead load effect
M L+I
= moment created by vehicle load + impact
The AASHTO operating rating procedure addresses the capacity of an individual girder.
In this approach the load
which will cause the first plastic hinge to be formed in the
140 bridge system, Py, will be considered instead of the ultimate moment capacity of an individual girder. The limit state function can be formed to reflect the requirements that the stress does not exceed the yield stress everywhere in the structure. Using the AASHTO operating rating factor, Eq. 5.22, gives:
where Qr
- Q d ♦ RF Q l i
dd
= dead load effect,
Qli
= effect of live load plus impact load
The values of the mean and standard deviation of Py are determined in a similar way to that mentioned in Sec. 5.3.2.
The mean value of the dead load can be obtained from actual measurement of the bridge under consideration. The coefficient of variation of the dead load is recommended to be = .06 [30].
The truck load model mentioned in Eq. 3.2 is considered to represent the five year period. The mean and standard deviation is obtained by using Monte Carlo simulation. The input data for this model is presented in table 3.3.
The mean value of the limit state function can be written as:
141
g = py - Qd - RF Qli
(5.24)
and its standard deviation is
The reliability index is given by: 9 0 = _ X ___________ >/ln(1 + VX>
(B.6)
173
APPENDIX C
RELIABILITY OF STRUCTURAL SYSTEMS
C.1
Series System
For a series system to survive, modes must survive.
all possible failure
Therefore, the probability of failure
of an n failure mode system is [Garson, 1980]
Pp = l-p(S1 nS2 n ...nSn )
(C.l)
i V*
where p(
denotes the survival event for the.i
mode,
) is the probability of the events in parentheses.
and Eq.
C.l can be expressed as
Pp — 1—p(S^/S2n...Sj,j)p(S2/S2fi...nS^) ... p(S^)
2j
In general,
n-1
Pp = 1-
n
p( S . / S i+1n . ..nSn p(Sn>
(C. 3)
If it is now assumed that the survival events, S^'s, are mutually independent, then Eq. C.3 takes the form
174
n
p_ « l- n p(s.) *
i=l
1
(C.4)
If the survival events are perfectly dependent,
then
the probability of failure of the system is
Pp = 1 - min p ( S i )
i = l,n
The probabilities of failure given by Eqs. C.4 and C.5 are usually called the upper and lower simple bounds for series systems,
respectively [Thoft-Chrestensen,
C.2 In parallel systems,
1982],
Parallel Systems it is important to distinguish
between systems with ductile elements and systems with brittle elements. maintains
An element is said to be ductile if it
its load carrying capacity level after failure,
and brittle if it becomes ineffective after failure.
This
criterion of brittle elements complicated the reliability analysis of such systems.
In some cases
(e.g., structures
with a low degree of statical indeterminacy),
the brittle
failure of one element will usually result in the failure of other elements.
If this is the case, the system behavior is
like a series system.
175
For a perfectly ductile parallel system to fail, modes must fail.
Therefore,
all
the probability of failure of
an n mode system is
PF = P(Fi nF2n * • •nFV
(C. 6)
where F^ denotes the failure event for the ith mode.
Eq.
A . 6 can be expressed as
Pp = p(F1/F2 n ...nFn )p(F2/ F 3n . ..nFfi) ... p(Fn )
(C.7)
In general,
n-1 PF
iJ1 P(Fi/Fi+in-“ nFn) P(V (C. 8)
If it is assumed that the failure events, F^'s, are mutually independent,
Pp = *
then Eq. A . 8 takes the form
n II p ( F . ) i=l 1
(C. 9)
If the failure events are perfectly dependent, then the probability of failure of the system is
Pp = min p ( F i )
i = l,n
(C.10)
176
Eqs. C.9 and C.10 give the lower and upper simple bounds for the parallel system with ductile elements, respectively.
C.3
Complex Systems
Complex systems are combinations of series and parallel systems.
The interaction of failure modes and the
definition of failure of the structural system determine its failure path and the type of combination of simple systems within the structure. A complex system in which a series of one of each of n components
is required for operation,
and where there are K
of such identical series is shown in Fig. 5.15.
If the
components within a series are assumed to be independent, the reliability of the i
+•Y\
series
is given by
i=i t •
tK
n
=
ps i
n
p(s.
j=l
.) ,
l '3
• •
(C.ll)
4-h where p(S. .)“probability of survival of the j component j +•h of the i series. Thus the system reliability is
k P( S
i -
n i=l
[i-P_ ] Si
177
n
k
i -
n
1 -
i=l
n
p(s.
j-l
•) ' 3 .
(C.12)
assuming that the k series are independent. Next consider a system which has n subsystems of k parallel components in series as shown in Fig.
5.14.
The
+* V i
j
parallel subsystem will have a reliability of
1 -
n
[1 - p(S.
.)] ,
j=l,...,n
i=l
(C.13)
^ V*
assuming that the k components of the j
subsystem are
independent. If it is assumed that the j subsystems are independent, then the reliability of the system is
n
ps ~ s
n j-i
Pg sj
n
n
i
-
n i=l
(i -p(s. .)} (C.14)
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