j2007-estimation of cable safety factors

8/10/2019 J2007-Estimation of Cable Safety Factors

1/22

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 70:11121133

Published online 7 November 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1928

Estimation of cable safety factors of suspension bridges usingartificial neural network-based inverse reliability method

Jin Cheng1, ,, C. S. Cai2 and Ru-Cheng Xiao1

1Department of Bridge Engineering, Tongji University, Shanghai 200092, China2Department of Civil and Environmental Engineering, 3418H CEBA, Louisiana State University,

Baton Rouge, LA 70803, U.S.A.

SUMMARY

The design of the main cables of suspension bridges is based on the verification of the rules defined bystandard specifications, where cable safety factors are introduced to ensure safety. However, the currentbridge design standards have been developed to ensure structural safety by defining a target reliabilityindex. In other words, the structural reliability level is specified as a target to be satisfied by the designer.Thus, calibration of cable safety factors is needed to guarantee the specified reliability of main cables.This study proposes an efficient and accurate algorithm to solve the calibration problem of cable safetyfactors of suspension bridges. Uncertainties of the structure and load parameters are incorporated inthe calculation model. The proposed algorithm integrates the concepts of the inverse reliability method,non-linear finite element method, and artificial neural networks method. The accuracy and efficiency ofthis method with reference to an example long-span suspension bridge are studied and numerical resultshave validated its superiority over the conventional deterministic method or inverse reliability methodwith Gimsings simplified approach. Finally, some important parameters in the proposed method are also

discussed. Copyrightq

2006 John Wiley & Sons, Ltd.

Received 28 February 2006; Revised 27 September 2006; Accepted 3 October 2006

KEY WORDS: inverse reliability method; cable safety factor; suspension bridges; target reliability index;uncertainties; artificial neural networks; finite element method

1. INTRODUCTION

The safety of main cables under different load conditions is one of the major concerns in the

design of suspension bridges. In the design of the main cables of suspension bridges, a cable

Correspondence to: Jin Cheng, Department of Bridge Engineering, Tongji University, Shanghai 200092, China.E-mail: [email protected]

Contract/grant sponsor: National Natural Science Foundation of China; contract/grant number: 50408037

Copyright q 2006 John Wiley & Sons, Ltd.


2/22

ESTIMATION OF CABLE SAFETY FACTORS OF SUSPENSION BRIDGES 1113

safety factor()is introduced to ensure an acceptable safety margin. Traditionally, the cable safety

factor is estimated using a deterministic model where all structural parameters are deterministic. In

reality, however, there are uncertainties in design variables. These uncertainties include geometric

properties such as cross-sectional properties and dimensions, material mechanical properties such

as modulus and strength, and load magnitude and distribution, etc. Thus, the deterministic modelcannot provide complete information regarding the cable safety factor. Therefore, the estimation

of the cable safety factor should be more rationally studied under a probabilistic viewpoint.

Probabilistic analysis provides a tool for incorporating structural modelling uncertainties in the

estimation of the cable safety factor by describing the uncertainties as random variables. Matteo

et al. [1] presented a methodology to estimate the current safety factor for the main suspension

cables. The methodology estimates the cable safety factor using ductile wire and ductile brittle

wire models, following two different approaches within each model: a Monte Carlo simulation

approach and an extreme-value-distribution approach. Haight et al.[2]used a type-I extreme value

distribution to compute cable safety factors for four suspension bridges. Cremona [3] presented

a probabilistic approach for cable residual strength assessment. The approach is applied to the

Tancarville suspension cables by taking into account the tensile test results, inspection data, and

weight-in-motion records. The application of a probabilistic approach in cable safety factors has

contributed to more realistic representations of the actual reliability of the main cables.

The current bridge design standards have been developed to ensure structural safety by defining

a target reliability index [4]. In other words, the structural reliability level is specified as a target

to be satisfied by the designer. Thus, calibration of cable safety factors is needed to guarantee

the specified reliability of the main cables. Although the above-mentioned probabilistic methods

have been very successful, the common drawback is their inability to deal with the calibration

problem of cable safety factors. These findings provide motivation for developing a methodology

for calibrating cable safety factors satisfying a prescribed reliability level.

The inverse reliability method can be pursued for the calibration of cable safety factors satisfying

the prescribed reliability. The basic idea of the method is to determine the unknown parameters

considered in the design such that a prescribed target reliability index is reached. In recent years,many efforts have been made on the development of the method and/or application of the method

in different design problems. Winterstein et al.[5]utilized this method for the estimation of design

loads associated with specified target reliability levels for offshore structures. An extension of

the method was developed by Der Kiureghian et al. [6] for general limit state functions. Li and

Foschi [7] introduced an inverse reliability method for determining the design parameters, and

applied it to problems of earthquake and offshore engineering. Fitzwater et al. [8] applied inverse

reliability methods for extreme loads on pitch- and stall-regulated wind turbines. More recently,

Saranyasoontorn and Manuel[9]extended the inverse reliability method to estimate nominal loads

for the design of wind turbines against ultimate limit states.

Although the inverse reliability method found some applications for different design problems,

its application to calibration of cable safety factors of suspension bridges has only been realized

very recently by Cheng and Xiao [10], in which they proposed an inverse reliability method toestimate the cable safety factors of suspension bridges satisfying a prescribed reliability. However,

their inverse reliability method uses the Gimsings simplified approach [11] for assessing cable

safety factors of suspension bridges by ignoring geometric non-linear effects. Moreover, only

limited design parameters can be considered in their method. Therefore, the purpose of the current

study is to develop an improved inverse reliability method for estimating cable safety factors of

suspension bridges satisfying the prescribed reliability.

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 70:11121133

DOI: 10.1002/nme


3/22

1114 J. CHENG, C. S. CAI AND R.-C. XIAO

Generally, the accuracy of the inverse reliability problem solution can only be as good as the

quality of approximation of the response function. This quality depend on the shape of the response

function and the experimental sampling points. Therefore, the accuracy and efficiency depend on

the structural problem to be solved. Since the focus of this paper is long-span suspension bridges,

complex structural behaviour may not be captured accurately by a low-order polynomial. Artificialneural networks (ANN), as discussed later, can adapt to more complex limit state functions that

may not be represented well by means of a low-order polynomial. Therefore, an ANN model is

used to determine an approximate response surface function in the present study.

In the proposed method, geometric non-linearity is considered by using non-linear finite element

method. An ANN model is applied to approximate the maximum cable tension function. The explicit

formulation of the maximum cable tension in the main span is derived by using the parameters

of the ANN model. Once the explicit maximum cable tension functions are found, the actual limit

state function is explicit in terms of the basic random variables and the unknown cable safety

factor. Thus, the conventional inverse reliability method such as inverse first-order reliability

method (inverse FORM method) is easily applied to solve the calibration problem of cable safety

factors. The accuracy and efficiency of the proposed method are demonstrated through a long-span

suspension bridge. The advantages of the proposed method over the deterministic method and the

conventional inverse reliability method are highlighted. Finally, some important parameters in the

proposed method are also discussed.

2. ESTIMATION OF CABLE SAFETY FACTORS FOR SUSPENSION BRIDGES

For illustrative purposes, two estimation methods of cable safety factors mentioned previously

are briefly reviewed in this section. One is deterministic analysis method and the other is inverse

FORM method with Gimsings simplified approach.

2.1. Estimation based on deterministic analysis method

The estimation of the cable safety factors based on deterministic analysis method is one of the most

frequently used approaches. As described in the previous sections, this method uses the assumption

of complete determinacy of structural parameters. In general, the method may be categorized into

two main types: (1) linear method (Gimsings simplified method [11]) and (2) non-linear finite

element method. They can be correspondingly expressed as

=8f Acc

(0.5(g+ p)+c)l

l2 +16f2(1)

=Acc

Tc(X)(2)

where g and p are the uniform dead and live loads acting on the stiffening girder, respectively,

c is the uniform dead load acting on the main cable, l is the length of the main span, f is the sag

of the main cable, c is the rupture strength of the main cable, Ac is the cross-sectional area of

the main cable, Xis the design variable, and Tc is the maximum cable tension in the main span.

It should be noted that Equation (1) includes limited design parameters only. Other design

parameters such as material properties are not considered. In addition, geometric non-linear effects


DOI: 10.1002/nme


4/22


are not considered in Equation (1) either. However, using Equation (2), it is straightforward

to consider the complicated geometric arrangements, various sources of non-linearity, different

materials, and different load types.

2.2. Estimation based on inverse FORM method with Gimsings simplified approach

The cable safety factors obtained with Equations (1) and (2) do not explicitly account for the

uncertainties occurring in nature. In addition, the above deterministic method cannot be used to

solve the calibration problem of cable safety factors. To overcome these limitations, an estimation

method based on the inverse FORM method with Gimsings simplified method has been proposed

as an alternative. In the following, only a brief description of the method is given. The more

detailed theoretical description and the iteration procedure may be obtained from [10].

For a target reliability indext, the calibration problem of cable safety factors can be described as

finding the cable safety factor , which minimizes t=

and subject to G(, ) = 0 (3)

where G(, ) represents the transformation of the limit state function g(X, ) from the original

space to the space of standard normal variable, is the vector of standard normal variables, and X

represents the vector of basic random variables. Here, the five parameters in Equation (1), namely

g, c, p, Ac, and c are treated as random variables.

Referring to Equation (1), the following limit state equation for the main cable is used:

G= c(0.5(g+ p)+c)l

l2 +16f2

8f Ac(4)

The inverse FORM method involves an iterative algorithm given by the following recursive

formulae:

k+1 = k +k dk (5)

k+1 = k +k dk (6)

dk = tG(

k, k)

G(k, k)k (7)

dk =[G(

k, k),k] G (k, k)+tG(k, k)

G(k, k)

(8)

where k is the vector of the standard normal variables at the kth iteration, k represents the cable

safety factor at the kth iteration, is a vector of gradient operators with respect to , is avector of gradient operators with respect to , [, ]is the inner product of two vectors, and k is the

step size at kth iteration, which is determined through a line search algorithm proposed by Der

Kiureghianet al. [6]. The algorithm proceeds iteratively until convergence is achieved, i.e. when

(k+1 k2 + |k+1 k|2)1/2

(k+12 + |k+1|2)1/2 (9)


DOI: 10.1002/nme


5/22


where is a small control parameter assigned by the user. From the writers previous experience,

a value of = 104 to 103 usually provides a satisfactory estimation.

3. PROPOSED ESTIMATION METHOD OF CABLE SAFETY FACTORS

In practice, the limit state function G(, ) is of the form

G= c Tc(X)

Ac(10)

when the limit state function in Equation (10) is explicitly expressed in terms of the random

variables, Equation (3) can be solved easily using the above-mentioned inverse FORM method.

Unfortunately, the maximum cable tension in the main span, Tc, is usually an implicit function

of the random variables. In other words, the closed-form solutions of the maximum cable tension

in the main span are not available. The solution of Equation (3) is then not straightforward. To

overcome the problem of using the inverse FORM, an ANN-based inverse FORM is developed.In the proposed method, the ANN method is applied to approximate the maximum cable tension

function. As a result, the explicit formulation of the maximum cable tension in the main span is

derived by using the parameters of the ANN model. Once the explicit maximum cable tension

functions are found, the limit state function in Equation (10) is explicit in terms of the basic random

variables and the unknown cable safety factor. Thus, the solution of Equation (3) can conveniently

be obtained by using the inverse FORM. Some of the important elements of the proposed method

are briefly discussed in the following sections.

3.1. Establishment of ANN model

The ANN method is an information processing technique based on the way the biological nervous

systems, such as the human brain, process information [12]. This technique as a whole has thecapability to respond to input stimuli, produce the corresponding response, and adapt to the

changing environment by learning from experience. More detailed description of the ANN method

can be found in [13].

There are a number of ANN paradigms. A multilayer feed-forward backpropagation network,

which is one of the well known and the most widely used ANN paradigms, is used in this study.

Some of the essential features of the proposed ANN are discussed very briefly in the following

sections.

3.1.1. ANN architecture and training algorithm. The proposed ANN structure consists of an input

layer, one hidden layer, and an output layer. Each layer has its corresponding units (neurons or

nodes) and weight connections. The number of neurons or nodes in the input and output layers is

determined by the number of input and output parameters, respectively. However, the selection ofan optimal number of neurons or nodes in the hidden layer is a difficult task. There is no general

rule for selecting the number of neurons or nodes in a hidden layer. It depends on the complexity

of the system being modelled. In this paper, the optimal number of neurons or nodes in the hidden

layer is determined by a trial-and-error process. Figure 1 shows a typical architecture of an ANN

model, in which the left column is the input layer, the right most column is the output layer, and

the middle one is the hidden layer.


DOI: 10.1002/nme


6/22


X1

)(XTc

X2

INPUT LAYER HIDDEN LAYER OUTPUT LAYER

Figure 1. Typical architecture of an ANN model.

Each neuron in the network operates by taking the sum of its weighted inputs and passing the

result through a non-linear activation function (transfer function). In this study, unless stated, a

logistic transfer function f(z) =1/(1 + ez )is used to transfer the values of the input layer nodes

to the hidden layer nodes, whereas the linear transfer function f(z) =z is adopted to transfer the

values from the hidden layer to the output layer.

The training phase of the proposed model is implemented by using a training algorithm. In this

study, a backpropagation training algorithm is used to train all models. To make sure that the model

is well trained and has the capability to generalize, the Bayesian regularization is implemented

in the above backpropagation training algorithm. The Bayesian regularization minimizes a linear

combination of squared errors on the training samples and network weights. It also modifies

the linear combination so that at the end of the training, the resulting network model has good

generalization qualities. A more detailed discussion of the Bayesian regularization can be foundin the literature, e.g. MacKay [14].

The training process is terminated when any of the following conditions is satisfied: the maximum

number of iterations (epochs) is reached; the performance gradient falls below a minimum value;

or the performance is minimized to the target value.

3.1.2. Data preparation and processing. In this study, a set of input and output data is prepared

for developing the ANN model. A sub-set of data is used for training, while the other for testing

the model. For the purpose of simplicity, all input data (design variables) are randomly generated

from the distribution of the variables for a given problem. All output data (cable safety factors)

are obtained by using the above-mentioned non-linear finite element methods.

To improve the training process of the model, all training data need to be scaled before presenting

them to the model, the following scaling equation is used:

Pscaled =P P

P(11)

where P and P scaled are the un-scaled and scaled values of the training data, respectively; P is

the mean value of the training data; and P is the standard deviation of the training data.


DOI: 10.1002/nme


7/22


3.1.3. Evaluation of the ANN performance. Once the ANN model is trained, the relationship

between the maximum cable tension in the main span and the various design variables can be

readily retrieved by using the model. The next step is to validate and evaluate the trained model.

This can be done by using common error metrics such as the mean absolute error (MAE) or

root-mean-squared error (RMSE). The two error functions can be expressed as follows:

MAE =

ni =1

mj =1|Pi j Ti j |

nm(12)

RMSE =

ni =1

mj =1(Pi j Ti j )

2

nm(13)

where n is the number of patterns in the validation data (i.e. the test data); m the number of

components in the output vector; P the output vector from the ANN model; and T the desired

output vector from the deterministic non-linear finite element analysis. It should be noted that the

above-mentioned scaling equation is used in computing the MAE and RMSE.

3.2. Explicit formation of the maximum cable tension function

The explicit formation of the maximum cable tension function is derived by using the parameters

(inputs, weights and scaled values) of the proposed ANN model (see Equation (14)). All necessary

parameters are obtained from the trained ANN model. The weights and bias values in the derivations

of the ANN model-based formulations are given in Tables I and II. Each input is multiplied by a

connection weight.

Suppose that the transpose of the vector of input variables is, XT = (X1,X2)and the transpose

of the vector of output variables is TTc = (Tc(X)). As mentioned earlier, a logistic transfer function

f(z) = 1/(1 + ez )is used to transfer the values of the input layer nodes to the hidden layer nodes,

whereas the linear transfer function g(z) =z is adopted to transfer the values from the hidden layer

Table I. Weight and bias values between the input and hidden layers.

Number of hidden layer nodes (i )

Weights 1 2 3

w1i w1(1, 1) w1(1, 2) w1(1, 3)w2i w1(2, 1) w1(2, 2) w1(2, 3)Bias b1(1, 1) b1(2, 1) b1(3, 1)

Table II. Weight values between the output and hidden layers.


Weights 1 2 3

wi w2(1, 1) w2(1, 2) w2(1, 3)Bias b2


DOI: 10.1002/nme


8/22


to the output layer. For an ANN model with one hidden layer (see Figure 1) with three

hidden nodes, the explicit formation of an approximate limit state function Tc(Xscaled) can be

expressed as

Tc(Xscaled) =K1w2(1, 1)+ K2w2(1, 2)+ K3w2(1, 3)+b2 (14)

where

K1 =1

1+e(w1(1,1)Xscaled1 +w1(2,1)X

scaled2 +b1(1,1))

K2 =1

1+e(w1(1,2)Xscaled1 +w1(2,2)X

scaled2 +b1(2,1))

K3 =1

1+e(w1(1,3)Xscaled1 +w1(2,3)X

scaled2 +b1(3,1))

Note that the maximum cable tension function obtained by using Equation (14) has been scaledand should be un-scaled before it is used.

3.3. Procedure for the proposed method

The procedure of the proposed method is as follows:

(1) Construct a database of deterministic maximum cable tension including the training data

sets and the test data sets.

(2) Determine the architecture of the ANN model and training algorithm.

(3) Train the ANN model with the training data sets.

(4) Evaluate and validate the ANN model with the test data sets.

(5) Extract the explicit formulation of the maximum cable tension function by the use of

well-trained ANN model parameters.

(6) After the explicit maximum cable tension function is determined, the inverse FORM is

applied to solve the calibration problem of cable safety factors defined in Equation (3) with

the limit state function of Equation (10).

In order to implement the method proposed above, a computer program named ECFSP is

developed by combining a deterministic non-linear finite element analysis program NASAB [15]

and the proposed algorithm. A flow chart for the program is given in Figure 2. The capabilities of

the proposed method and those of the other estimation methods mentioned earlier are compared

in Table III.

4. APPLICATION TO AN EXAMPLE SUSPENSION BRIDGE

4.1. Description of the example bridge

The example suspension bridge studied here is the Jiang Yin Bridge, with a 1385 m central span

length, which is one of the longest suspension bridges in China. The bridge span arrangements

are 336+1385+309 m. The elevation view of the bridge is shown in Figure 3.


DOI: 10.1002/nme


9/22


ANN Model

Determine the architecture of

the ANN model and training

algorithm

Train the ANN model with the

training data sets

Evaluate and validate the

ANN model with the test data sets

Extract the explicit formulation of

maximum cable tension function

(Equation 14)

Solve Eq.(3) to obtain cable safety factor by using

inverse FORM method

Input statistics of the random

variables

Construct a database of training

and test data sets using Program

NASAB

Obtain the explicit formulation of

limit state function

(Equation 10)

Figure 2. Flowchart of the proposed method (Program ECFSP).

The deck cross-section is an aerodynamically shaped closed box steel girder with 36 .9 m wide

and 3.0 m high. The distance between the two cables is 32.5 m; and the hanger spacing is 16.0 m.

For more details of the bridge, the reader can refer [16, 17].


DOI: 10.1002/nme


10/22


TableIII.Comparisonofthecharacteristicsofdifferentmethods.

Characteristicsofdifferent

Deterministic

InverseFORM

with

methods

meth

od

Gimsingssimplifi

edmethod

Proposedmethod

Considerationofge

ometric

Ye

s

No

Yes

non-linearityofstru

cture

Considerationofpa

rametric

No

Yes

Yes

uncertainty

Limitstatefunction

Explicittype

Implicit

type

Accuracyofcalcula

tion

Overestim

atethe

Accurate

Accurate

cablesafe

tyfactor

Numbersofrandom

variables

Onlyconsiders

tructural

Considerall

structural

parametersinEq

uation(1)

parametersof

suspension

asrandomvariables

bridgesasrandomvariables

Appliedtypesofsu

spensionbridges

Alltyp

esof

Onlyearth-an

chored

Alltypesofsuspension

suspensionbridges

suspensionbridges

bridges


DOI: 10.1002/nme


11/22


Figure 3. Elevation of Jiang Yin Bridge (unit: m).

Table IV. Random variables and their statistical properties.

Mean Coefficient of variation Distribution type Sources

Ac 0.4825m2 0.05 Lognormal Assumed

g 180.0 kN/m 0.08 Normalp 33.5 kN/m 0.05 Extreme type Ic 1678 MPa 0.05 Normal Imai [18]

4.2. Finite element modelling

A three-dimensional finite element model has been established for the Jiang Yin Bridge [17].

Three-dimensional beam elements were used to model the two bridge towers. The cables and

suspenders were modelled with three-dimensional truss elements accounting for the geometric

non-linearity due to the cable sag. The bridge deck is represented by a single line of beams and

the cross-section properties of the bridge deck are assigned to the beams as equivalent properties.The connections between the bridge components and the supports of the bridge were properly

modelled.

4.3. Uncertainties of the example bridge

As it is well known, the bridge material properties and geometric parameters may fluctuate in

the vicinity of the nominal values, caused during the process of structural element manufacturing

and erection of the bridge. Therefore, system parameters should be treated as random rather than

deterministic variables. Besides, the various loads acting on the bridge are also random variables

due to their nature and/or insufficient measured data. However, with the increase in the number

of random variables, the number of training sample sets required to evaluate the ANN model

increases. For the sake of simplicity, only four design parameters, namely, the uniform dead loadacting on the stiffening girder, g, the uniform live load, p, the cross-sectional area of the main

cable, Ac, and the rupture strength of the main cable, c, are treated as random variables in the

subsequent study. Table IV shows the statistics of these random variables. As the objective of this

study is to propose an efficient method for estimating the cable safety factor of suspension bridges

satisfying prescribed reliability levels, all random parameters in the following analysis are based

on arbitrary but typical values. On the other hand, since the determination of the correlation of the


DOI: 10.1002/nme


12/22


Table V. Range of input and output variables for theexample bridge.

Variables Range of values (kN/m)

g 156.02211.44p 30.8736.76

random parameters is a difficult task without sufficient data, using the independence assumption

can greatly simplify the reliability assessment. Therefore, all random parameters in this paper

are treated as stochastically independent from each other. Note that the mean values of these

parameters are used in the following deterministic analysis.

4.4. Some parameters in the proposed method

To estimate the cable safety factor using the proposed method, the target reliability level needs tobe specified for the limit state considered in this study. Nowaket al. [19] recommended a target

component reliability index of 3.5 and a target system reliability index of 5.5 in the ultimate limit

states for bridge structures. Li et al. [20] suggested that the target reliability index for bridge

structures should lie in an approximate range of 3.25.2. A target reliability index of 3.5 has been

used in the calibration of the OHBD code and the AASHTO code for bridges [21]. Based on these

data, a target reliability index of 3.5 is used in this study unless otherwise stated.

In the following analyses, unless otherwise stated, the mean values shown in Table IV are

selected as the initial values for the random variables, = 3.5, and the proposed ANN model has

two input nodes, three hidden nodes, and one output node. The number of training data sets is

20, and the data sets for testing the model are eight. The ranges of variables in these data sets are

shown in Table V.

4.5. Verification of the proposed method

In the literature, no available benchmark calibration problems of cable safety factors under pre-

scribed reliability for suspension bridges are available for the verification of the study. One way to

verify the proposed method is to make separate verifications for the estimation of the maximum

cable force in the main span by the ANN model and for the accuracy and efficiency of inverse

FORM method used in the proposed method.

This verification study is performed to determine the feasibility of using the ANN model to

estimate the maximum cable force in the main span of the example suspension bridge and the

verification of the accuracy and efficiency of the inverse FORM method is referred to [10]. The

performance of training and test data sets and error parameters of the ANN model are given inFigures 4(a) and (b) and Table VI, respectively. As can be seen, the correlation factor for both sets

is quite high, which proves the high accuracy of the ANN model.

The weights and bias values in the derivations of Tc(scaledg ,p

scaled) formulations are given in

Tables VII and VIII. Substituting these values into Equation (14) one obtains the following:

Tc(scaledg ,p

scaled) = 3.3760K14.4853K2+3.3184K31.1543 (15)


DOI: 10.1002/nme


13/22


260000 270000 280000 290000 300000 310000 320000

260000

270000

280000

290000

300000

310000

320000

Maximumcableforceinthemainspan,Tc

fromnonlinearfiniteelementanalys

is(KN)

Predicted maximum cable force in the main span, Tc(KN)

Training data

260000 270000 280000 290000 300000 310000

260000

270000

280000

290000

300000

310000

Maximumcableforceinthemainspan,Tc

fromnonlinearfinite

elementanalysis(KN)

Predicted maximum cable force in the main span, Tc(KN)

Test data

(a)

(b)

Figure 4. Comparison between the ANN model and non-linear finite element analysis, Tc for the example

bridge using: (a) training data; and (b) test data.

where

K1 =1

1+e(0.5365scaledg +0.1355p

scaled1.5982)


DOI: 10.1002/nme


14/22


Table VI. A summary of the performance of the ANN model for MAE andRMSE, both for the training and test data.

Maximum cable force in the main span, Tc

Training data Test data

MAE 5.8e4 8.03e004RMSE 0.7e3 1.2e3

Table VII. Weight and bias values between input and hidden layers for theexample bridge.


Weights 1 2 3

w1i 0.5365 0.4903 0.3740

w2i 0.1355 0.1223 0.0596Bias 1.5982 1.4408 0.2552

Table VIII. Weight values between output and hidden layersfor the example bridge.


Weights 1 2 3

wi 3.3760 4.4853 3.3184Bias 1.1543

K2 =1

1+e(0.4903scaledg 0.1223p

scaled1.4408)

K3 =1

1+e(0.3740scaledg 0.0596p

scaled0.2552)

Note that the above Tc(scaledg ,p

scaled) has been scaled and should be un-scaled before it is

used. For this purpose, substituting Equation (11) into Equation (15), Tc(scaledg , p

scaled)becomes

Tc(g,p)

Tc(g,p) = 45 656.8155K160 659.0842K2+44 877.4343K3+271 771.1610 (16)

where

K1 =1

1+e(0.03897g0.08773p+11.6747)


DOI: 10.1002/nme


15/22


Table IX. Comparison of the maximum cable force in the main span, Tc (KN) obtained by the derivedformulation, Gimsings simplified method, and deterministic non-linear finite element method.

DeterministicGimsings simplified non-linear finite

Group no. Derived formulation method element method

1 293 953.75 295 121.7 293 9502 292 281.21 293 287.7 292 2803 287 538.84 288 057.1 287 5304 292 681.65 293 707.3 292 6605 292 643.36 293 697.1 292 6606 282 496.44 282 481.9 282 4507 281 228.19 281 061.3 281 1508 279 561.07 279 330.1 279 5709 266 131.06 264 367.9 265 850

10 281 846.25 281 792.4 281 820

K2 =1

1+e(0.03561g+0.07922p7.7356)

K3 =1

1+e(0.02717g+0.03860p+3.9300)

To check the accuracy of the derived Tc(g,p) formulation, we can generate 10 groups of

random samples simultaneously. Each group consists of two random variables(g,p). The function

value ofTc(g,p)obtained by the derived formulation, the Gimsings simplified method, and the

deterministic non-linear finite element analysis are summarized in Table IX. Reasonable agreements

have been observed, indicating that both the derived formulation and the Gimsings simplified

method for estimating the maximum cable force in the main span is adequate from a deterministic

point of view. In addition, it should be noted that the predicted results using the derived formulationshow more favourable agreement with the deterministic non-linear finite element data than the

predicted results using the Gimsings simplified method.

4.6. Sensitivity study of the proposed method

The proposed method is mainly based on the following parameters: (1) the initial value of; (2)

the number of nodes in the hidden layer; (3) the number of training data sets; and (4) the transfer

function. In this section, different values are chosen for these parameters in order to determine

their effects on the estimated cable safety factors of the example bridge.

4.6.1. Sensitivity on the initial value of. Since the initial value of used in the proposed method

is chosen arbitrarily, it is necessary to investigate the effect of the initial value of on the finalresults. For this purpose, three different initial values of are used: 4.5, 3.5 and 1.5. The variations

of the estimated values with the iteration number are shown in Figure 5 for different initial values

of. It can be seen that the iteration number may slightly increase for some cases. However, the

convergence is achieved and the accuracy of the results is the same regardless of the initial value

of. The results indicate that while the initial value of could have a major effect on the rate of

convergence of the proposed algorithm, the accuracy of the proposed algorithm is not influenced


DOI: 10.1002/nme


16/22


1 2 3 4 5

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Iterative number

Initial value of =4.5



Figure 5. Variations of the cable safety factor for the example bridge versus the number ofiterations for different initial values of.

Table X. Estimated value of for different numbers of nodesin the hidden layer.

Numbers of nodes in the hidden layer

Estimated value 3 6 9

2.11 2.11 2.12

by the initial value of . Therefore, the proposed algorithm can be used to solve the calibration

problem of cable safety factors considered in this paper.

4.6.2. Sensitivity on the number of nodes in the hidden layer. In this study, three different numbers

of nodes in the hidden layer ranging from three to nine are investigated, and the results are listed in

Table X. From this table, it can be seen that changing the numbers of nodes in the hidden layer has

little effect on the accuracy of the predicted values of. Using more than three nodes in the hidden

layer results in little improvement in accuracy, and possibly results in a loss of generalization.

4.6.3. Sensitivity on the number of training data sets. Keeping other parameters unchanged andonly varying the number of training data sets, the predicted values of are compared in Table XI.

It is clear that changing the numbers of training data sets has a significant effect on the accuracy

of the estimated values of . The best accuracy is associated with at least 10 training data sets.

With less than 10 training data sets, the ANN model in the proposed method seems unable to be

trained adequately and leads to a poor estimated value of. By reducing the number of training

data sets from 10 to 5, the error increases from 0.9% to 9.5%.


DOI: 10.1002/nme


17/22


Table XI. Estimated value of for different number of training data sets.

Number of training data sets

Estimated value 5 10 15 20

2.31 2.13 2.12 2.11

Table XII. Transfer functions used in this paper.

Name Equations

Logistic g(z) = 1/(1+ez )

Tangent g(z) = 2/(1+e2z )1

Table XIII. Value of obtained by applying different

transfer functions.

Transfer function

Logistic 2.11Tangent 2.11

Table XIV. Comparisons of different methods for cable safety factor, .

t= 3.5 t= 4.0 t= 4.5

Inverse FORM Inverse FORM Inverse FORMwith Gimsings with Gimsings with Gimsings

Different Deterministic simplified Proposed simplified Proposed simplified Proposedmethod method method method method method method method

2.84 2.10 2.11 2.00 2.02 1.92 1.94

4.6.4. Sensitivity on transfer functions. The fourth important parameter of the proposed method

is the transfer function which is used to transfer the values of the input layer nodes to the hidden

layer nodes. In this study, two types of transfer functions tabulated in Table XII are tested for

the given example bridge. The results are listed in Table XIII. As shown, the results associated

with the logistic transfer function are almost the same as those associated with the tangent transfer

function.

4.7. Estimation of cable safety factors using different methods

This section examines three methods of estimating cable safety factors of suspension bridges

and gives the comparisons of the accuracy and reliability of the three methods. Three different

values of the target reliability index are used: 3.5, 4.0, and 4.5, and the results are presented in

Table XIV. From this table, it can be seen that: (1) deterministic method considerably overestimates


DOI: 10.1002/nme


18/22


0.48 0.49 0.50 0.51 0.52 0.53

2.10

2.15

2.20

2.25

2.30

2.35

mean value of the cable cross-section, Ac(m2)(a)

178 180 182 184 186 188 190 192 194 196 198 200

1.96

1.98

2.00

2.02

2.04

2.06

2.08

2.10

2.12

Mean value of the dead load, (KN/m)(b)

33.0 33.5 34.0 34.5 35.0 35.5 36.0 36.5 37.0

2.070

2.075

2.080

2.085

2.090

2.095

2.100

Mean value of the live load, p (KN/m)(c) (d)1 66 0 1 68 0 1 70 0 1 72 0 1 74 0 1 76 0 1 78 0 1 80 0 1 82 0 1 84 0 1 86 0

2.10

2.15

2.20

2.25

2.30

2.35

Mean value of rupture strength, c(MPa)

Figure 6. Influence of the mean values of the different random variables on the cable safety factor for the

example bridge: (a) cable cross-sectional area; (b) dead load; (c) live load; and (d) rupture strength.

the cable safety factors of the bridge; (2) as the target reliability index increases, the estimated

cable safety factor decreases; and (3) there exists small difference between the results estimated

by the inverse FORM method with Gimsings simplified approach and the proposed method.

4.8. Effect of mean value of random variables on cable safety factors

Keeping the coefficients of variation of all random variables unchanged and only the mean values

of all random variables are varied, the estimated cable safety factors are compared in Figure 6. As

the mean values of random variables Ac and c increase, the estimated cable safety factors also

increase. However, an opposite effect is observed for the random variables and p. As the mean

values of the random variables and p decrease, the estimated cable safety factors increase.

4.9. Effect of standard deviation of random variables on cable safety factors

Figure 7 shows the estimated cable safety factors having the same coefficients of variation for all

random variables but different standard deviation. As the standard deviation of all random variables

increases, the estimated cable safety factors decrease.


DOI: 10.1002/nme


19/22


20/22


and (3) better in obtaining deterministic estimation results than using the Gimsings simplified

method.

As an application of the proposed algorithm, the cable safety factor of a long-span suspension

bridge is estimated. The effects of various parameters on the estimated cable safety factor of

suspension bridges are investigated, and the following conclusions are drawn:

(1) The validity and the accuracy of the proposed ANN-based inverse FORM method were

verified by making separate verifications for the estimation of the maximum cable force in

the main span by an ANN model and for the accuracy and efficiency of inverse FORM

method used in proposed method. Therefore, the proposed method would be effective in

estimating cable safety factors of suspension bridges satisfying prescribed reliability levels.

(2) Deterministic method considerably overestimates the cable safety factor of suspension

bridges. The actual cable safety factor of suspension bridges should be estimated based

on the proposed method.

(3) Estimation of the cable safety factors of suspension bridges using the proposed method

offers theoretically more accurate results than that estimated by the inverse FORM method

with Gimsings simplified approach. However, from the view of engineering practice, theinverse FORM method with Gimsings simplified approach can be used for estimating cable

safety factors of suspension bridges satisfying prescribed reliability levels, which offers

acceptable results in many cases.

(4) The target reliability index has a significant influence on the cable safety factor of suspension

bridges. As the target reliability index increases, the estimated cable safety factor decreases.

(5) The sensitivity study of the proposed method indicates that the estimated cable safety factor

is sensitive to the numbers of training data sets in the ANN model. The estimated errors

increase as the numbers of training data sets decrease. However, the estimated cable safety

factor is insensitive to the initial value of cable safety factors, the number of nodes in the

hidden layer, and the transfer function.

(6) The cable safety factor of suspension bridges is highly influenced by the statistics of some

random variables such as the uniform dead load acting on the stiffening girder, g, theuniform live load, p, the cross-sectional area of the main cable, Ac, and rupture strength

of the main cable, c. Therefore, the accurate determination of the distribution of such

parameters is very important to obtain reliable estimation results.

Although emphasis in this study was placed on the calibration of cable safety factors of earth-

anchored suspension bridges, the proposed algorithm offers immediate applications to other types of

suspension bridges such as self-anchored suspension bridges as well as other calibration problems.

It should be noted that the proposed algorithm is not intended as a replacement for existing inverse

reliability methods, but as a possible complement of these methods. Further development of the

proposed algorithm is needed for the estimation of cable safety factors of suspension bridges.

APPENDIX A: DERIVATION OF EQUATION (1) [11]

The cable safety factor is obtained as the ratio of the cable strength (Tu ) over the maximum

cable tension in the main span (Tc), namely

=Tu

Tc(A1)


DOI: 10.1002/nme


21/22


For simplicity, the maximum cable tension in the main span is given as [11]

Tc=(0.5(g+ p)+c)l

l2 +16f2

8f(A2)

The cable strength (Tu ) can be expressed as

Tu= Acc (A3)

Substituting (A2) and (A3) into (A1), can be written as

=8f Acc

(0.5(g+ p)+c)l

l2 +16f2

ACKNOWLEDGEMENTS

This work has been supported by the National Natural Science Foundation of China under grant number50408037. This support is much appreciated.

REFERENCES

1. Matteo J, Deodatis G, Billington DP. Safety analysis of suspension-bridge cables: Williamsburg bridge. Journal

of Structural Engineering (ASCE) 1994; 120(11):31973211.

2. Haight RQ, Billington DP, Khazem D. Cable safety factors for suspension bridges. Journal of Bridge Engineering

(ASCE) 1997; 2(4):157167.

3. Cremona C. Probabilistic approach for cable residual strength assessment. Engineering Structures 2003; 25:

377384.

4. Reid SG. Specification of design criteria based on probabilistic measures of design performance. Structural Safety

2002; 24:333345.

5. Winterstein SR, Ude TC, Cornell CA, Bjerager P, Haver S. Environmental contours for extreme response: inverse

FORM with omission factors. Proceedings of the ICOSSAR-93, Innsbruck, 1993.6. Der Kiureghian A, Zhang Y, Li C-C. Inverse reliability problem. Journal of Engineering Mechanics (ASCE)

1994; 120(5):11541159.

7. Li H, Foschi RO. An inverse reliability method and its application. Structural Safety 1998; 20:257270.

8. Fitzwater LM, Cornell CA, Veers PS. Using environmental contours to predict extreme events on wind turbines.

Wind Energy Symposium, AIAA/ASME, 2003; 244258.

9. Saranyasoontorn K, Manuel L. Efficient models for wind turbine extreme loads using inverse reliability. Journal

of Wind Engineering and Industrials Aerodynamics 2004; 92:789804.

10. Cheng J, Xiao R-C. Application of inverse reliability method to estimation of cable safety factors of long span

suspension bridges. Structural Engineering and Mechanics 2006; 23(2):195207.

11. Gimsing N. Cable Supported Bridges: Concept and Design. Wiley: New York, 1997.

12. Adhikary BB, Mutsuyoshi H. Artificial neural networks for the prediction of shear capacity of steel strengthened

RC beams. Construction and Building Material 2004; 18:409417.

13. Flood I, Kartam N. Neural networks in civil engineering I: principles and understandings. Journal of Computing

in Civil Engineering (ASCE) 1994 8(2):131148.14. MacKay DJC. Bayesian interpolation. Neural Computation 1992; 4(3):415447.

15. Cheng J, Xiao R-C, Xiang H-F, Jiang J-J. NASAB: a finite element software for the nonlinear aerostatic stability

analysis of cable-supported bridge. Advances in Engineering Software 2003; 34:287296.

16. Xiang HF et al. Wind-resistant Study on the Jiang Yin suspension bridge. Research Report of Tongji University,

Shanghai, China, 1995 (in Chinese).

17. Cheng J, Jiang J-J, Xiao R-C, Xiang H-F. Nonlinear aerostatic stability analysis of Jiang Yin suspension bridge.

Engineering Structures 2002; 24:773781.


DOI: 10.1002/nme


22/22


18. Imai K. Reliability analysis of geometrically nonlinear structures with application to suspension bridges.

Ph.D. Thesis, Department of Civil, Environmental, and Architectural Engineering, University of Colorado,

Boulder, CO, 1999.

19. Nowak AS, Szerszen MM, Park CH. Target safety levels for bridges. Proceedings of the Seventh International

Conference on Structural Safety and Reliability, Kyoto, Japan, 1997; 18971903.20. Li YH, Bao WG, Guo XW, Cheng XY. Structural Reliability for Highway Bridges and Probability-Based Limit

State Design. Peoples Communication Press: Beijing, 1997 (in Chinese).

21. Casas JR. Reliability-based partial safety factors in cantilever construction of concrete bridges. Journal of

Structural Engineering (ASCE) 1997; 123(3):305312.


DOI: 10.1002/nme

j2007-estimation of cable safety factors

Documents