Lifetime Reliability Based Life-Cycle Cost Effective Optimum...
16
Transcript of Lifetime Reliability Based Life-Cycle Cost Effective Optimum...
Steel Structures 6 (2006) 337-352 www.kssc.or.kr
Lifetime Reliability Based Life-Cycle Cost Effective
Optimum Design of Orthotropic Steel Deck Bridges
Kwang-Min Lee, Hyo-Nam Cho, Hyun-Ho Choi* and Hyoung-Jun An
1Assistant Manager, Daelim Industrial Co. Ltd., 146-12, Susong-Dong, Jongno-Gu, Seoul 100-732, Korea2Professor, Department of Civil & Environmental Engineering; Hanyang University, An-San 425-791, Korea
3Senior Researcher, Highway Design Evaluation Office, Korea Highway Corporation, Seongnam, Kyung-gi, Korea4Director, Civil Structure Dept, Saman Corporation, Byeolyangdong, Gwacheonsi, Gyeonggido, Korea
Abstract
This paper presents a realistic and practical lifetime reliability based LCC-effective optimum design of orthotropic steel deckbridges. To consider the time variant conditions extreme live load model and a corrosion propagation model consideringcorrosion initiation, corrosion rate, and repainting effect are adopted in this study. The methodology proposed in the paper isapplied to the optimum design problem of an actual orthotropic steel deck bridge with 7 continuous spans using various designparameters such as local corrosion environments, number of average daily traffic volume, and type of steel. From the numericalinvestigations, it may be concluded that the local corrosion environments, number of truck traffic significantly influence theLCC-effective optimum design of orthotropic steel deck bridges, they should be considered as crucial parameters for theoptimum LCC-effective design of orthotropic steel deck bridges.
Keywords: Life-Cycle Cost, Indirect Cost Model, Steel Bridge, Optimization, Time-variant Reliability, Corrosion Model
1. Introduction
Traditionally, the main objective of optimal design of
bridges is usually to select member sizes for the optimal
proportioning of the structural members so as to achieve
the minimum initial cost design that meets all the
performance requirements specified in the design code. A
number of researchers have made efforts to develop
optimization algorithms applicable to the initial cost
optimum design of bridges (Farkas, 1996; Cho et al.,
1999; etc.). However, it seems that the LCC-effective
optimum design models for bridges have not been
extensively investigated so far, and thus its established
models and methodology are not available. Theoretically,
the LCC modeling of bridges considering time effect
such as time-variant degrading resistance and stochastic
extreme load effects is extremely difficult, simply because
the time effect incur various failures related with strength,
fatigue, corrosion, local buckling, stability, etc. throughout
the life span of the structure, which, in turn, bring forth
highly complicated cost and uncertainty assessment that
often involves the lack of cost data associated with
various direct and indirect losses, and the absence of
uncertainty data available for the assessment as well.
Recently, author (Cho et al., 2001) performed a time-
invariant reliability-based optimum design of deck and
girder system for minimizing the LCC of orthotropic steel
deck bridges. However, these studies ignored the influence
of stochastic loading history and degrading resistance
with various time-variant rates of deterioration and/or
damages during the life span of a structure. Thus, this
paper presents a realistic and practical LCC methodology
for lifetime reliability based LCC-effective optimum
design of orthotropic steel deck bridges. The LCC functions
considered in the LCC optimization consist of initial cost,
expected life-cycle maintenance cost and expected life-
cycle rehabilitation costs including repair/replacement
costs, loss of contents or fatality and injury losses, road
user costs, and indirect socio-economic losses.
For the assessment of the life-cycle rehabilitation costs,
since the annual probability of failure which depends
upon the prior and updated load and resistance histories
should be accounted for, extreme live load model is
adopted, which is based on the statistics of extreme
values where the probability of encountering a large truck
at the extreme upper tail of the distribution increases as
the number of trucks passing over the bridge increases.
And a modified corrosion propagation model considering
corrosion initiation, corrosion rate, and repainting effect is
proposed in this study. The proposed LCC methodology in
this study is applied to the optimum design problem of an
actual orthotropic steel deck bridge with 7 continuous
*Corresponding authorTel: +82-2-2230-4697; Fax: +82-2-2230-4199E-mail: [email protected]
338 Kwang-Min Lee et al.
spans (80 m + 120 m + 140 m + 160 m + 140 m + 120 m
+ 80 m = 840 m), and sensitivity analyses are performed
with discussions, to investigate the effects of design
parameters such as local corrosion environments, number
of average daily traffic volume, and type of steel on the
LCC-effectiveness.
2. Formulation of LCC-Effective Optimum Design of Orthotropic Steel Deck Bridges
2.1. Design variables
An orthotropic steel deck bridge consists of main
girders and deck system. The design variables for a main
girder are web height (h = ax2 + b), web thickness (tw),
and lower flange thickness (tlf) in this study. And, the
design variables for the deck system are taken as deck
plate thickness (tdp), U-rib type and spacing (ru, lr), and
floor beam dimensions and spacing (hfb, tfb1, tfb2, lf). The
main girder elements have many design variables since
the design section of the girders varies along the entire
span of the bridge. Thus, in this study, the numbers of
design variables for the main girder are reduced by
introducing variable linking technique and parabolic
curve approximation (Cho et al., 2001).
2.2. Life-cycle cost formulation
An orthotropic steel deck bridge design may be based
on a comparison of risks (costs) with benefits. An optimal
design solution, chosen from multiple alternative designs,
can then be found by minimizing LCC. Such a decision
analysis can be formulated in a number of ways by
considering various costs or benefits, which can also be
referred to as a whole life costing, life time cost-benefit
or cost-benefit-risk analysis. LCC may be used to assess
the ‘cost-effectiveness’ of design decisions. If the benefits
of each alternative are the same, then the total expected
LCC up to the life span Tlife of an orthotropic steel deck
bridge can be formulated as follows:
(1)
where = total expected LCC which are
functions of the vector of design variable and life span
Tlife; = initial cost; = expected maintenance
costs for the item j; = expected rehabilitation
(or failure) cost for limit state k; and r = discount rate
As presented in Eq. (1), the total expected LCC of an
orthotropic steel deck bridge involves not only the initial
cost but also the expected maintenance costs and the
expected rehabilitation costs over the life span of the
structure, which requires the assessment of the updated
annual probability of failure for possible limit states
considered in the cost model such as strength, fatigue,
serviceability, buckling, stability, etc. and of associated
costs as well. Since costs may occur at different time, it
is necessary for all costs to be discounted using discount
rate r.
And then the formulation for the LCC-effective optimum
design of a steel bridge can be represented as follows:
Minimize (2-a)
Subject to j = 1, 2, ..., j (2-b)
k = 1, 2, ..., K (2-c)
(2-e)
E CT X Tlife,( )[ ] CI X( )
E CMjX t,( )[ ]
j 1=
J
∑ E CFkX t,( )[ ]
k 1=
K
∑+
1 r+( )t
-------------------------------------------------------------------------
t 1=
Tlife
∑+=
E CT X Tlife,( )[ ]
X
CI X( ) E CMjX t,( )[ ]
E CFkX t,( )[ ]
E CT X Tlife,( )[ ]
Gj X( ) 0≤
pFkX( ) p
Fk
allow≤
XLX X
U≤ ≤
Figure 1. Design variables.
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 339
where = the vector of design variables; = j-th
constraint (i.e., allowable stresses, combined stress limits,
geometric limits, etc.; see Section 2.3); = probability
of failure for limit state k considered in the design (i.e.,
strength, fatigue, serviceability, buckling, stability, etc.);
= allowable probability of failure for the limit state
k; and , = the lower and upper bounds
2.2.1. Initial cost
The initial costs involved in design and construction of
the structural and non-structural components of the
bridges such as planning and design, construction of
foundation, superstructure, substructure, accessories, etc.
can be included in the initial cost. Thus, in general, the
initial cost can be formulated as in the following
equation:
(3)
where = planning and design cost; =
construction cost; and = testing cost before opening
to traffic
The construction cost should include all the labor,
materials, equipment, construction site management and
quality control costs involved in the actual construction of
the bridge. The construction costs can be expressed in
terms of the initial cost, which may be estimated at about
90% of the initial costs, and the design cost and load
testing cost are also typically assumed as a percentage of
construction cost (Ministry of Science and Technology,
2004).
2.2.2. Expected maintenance cost
Maintenance actions may be performed periodically to
maintain bridge performance from structural deteriorations/
degradations due to environmental stressors (e.g., corrosion,
sulfate attack, alkai-silica reaction, and freeze-thaw cycle
attack, etc.). Similar to the design cost and load testing
cost, the expected maintenance costs have also been
estimated as a function of the construction cost (Leeming
and Mouchel, 1993; Piringer, 1993; etc.). However, if the
suggestions are followed, then the more cost would be
required as the construction cost increases, which may
result in a contradiction in the assumption. Meanwhile,
Wen and Kang (1997) assumed that, in their LCC analysis,
the dependence of maintenance cost on the design variables
under consideration would be generally weak, and thus
they did not consider the maintenance cost.
However, since the maintenance costs over a life span
may be high, depending upon the quality of the design
and construction of steel bridge under environmental
stressors, the costs must be carefully regarded as one of
the important costs in the evaluation of the LCC. In this
study, the maintenance costs are classified depending on
whether the costs which are related to design variable
(e.g., painting cost, pavement, etc.) or not (e.g., periodic
inspection, bridge cleaning, etc.). However, the latter
might have a minor or no influence on the optimum
solution in the LCC optimum design of steel bridges.
Accordingly, they can be ignored in the LCC formulation.
Therefore, in this study, only the expected maintenance
costs which are related to design variables are considered
as follows:
(4)
where = maintenance cost for the item j; and
= maintenance probability at time t for the item j
2.2.3. Expected rehabilitation cost
Expected rehabilitation costs may arise as a result of
annual probability of failure or damages of various
critical limit states (strength, buckling, fatigue, and etc.)
that may occur any time during life span of a steel bridge.
Even though the rehabilitation costs do not have to be
considered under normal circumstances, these costs should
still be considered in an economic analysis as insurance
costs (Merchers, 1987). The expected rehabilitations can
be obtained from the direct/indirect rehabilitation costs
and the annual probability of failure for failure limit state
k considered in the design as follows:
(5)
where k = index for a failure limit state; =
rehabilitation cost for failure limit state k; =
updated annual probability of failure at time any t (i.e.,
probability that the failure will occur (annual) during time
interval t1 conditional on updated loads or resistance); and
T = survived years which can be expressed as t − t1
Meanwhile, in Eq. (5), the direct/indirect rehabilitation
costs can be expressed as follows (Lee et al., 2004):
(6)
where = direct rehabilitation cost; =
indirect rehabilitation cost; CH, CU, CE = loss of contents
or fatality and injury losses, road user cost, and indirect
socio-economic losses, respectively; = increased
accident rate during rehabilitation activities; and =
period of rehabilitation activities
The direct rehabilitation cost and the loss of
contents or fatality and injury losses cost CH could be
evaluated in a relatively easy way. The direct rehabilitation
costs could be estimated based on the various
sources available, such as the Construction Software
Research’s (CSR, http://www.csr.co.kr) price information,
opinions of the experts, and also obtained from various
references (OCFM - Office of Construction and Facility
Management, 2002; LISTEC - Korea Infrastructure Safety
X Gj X( )
pFkX( )
pFk
allow
XL
XU
CI X( ) CID X( )= CIT X( ) CIC X( )+ +
CID X( ) CIC X( )
CIT X( )
E CMjX t,( )[ ] CMj
X( ) fMjt( )⋅=
CMjX( )
fMjt( )
E CFkX t,( )[ ] CFk
X( ) pFkX t T,( )⋅=
CFkX( )
pFkX t T,( )
CFkX( )
CFkX( ) CDRk
X( ) CIRkX( )+=
CDRkX( ) CH rrk X( )⋅ CU trk X( )⋅ CE trk X( )⋅+ +[ ]+=
CDRkX( ) CIRk
X( )
rrk X( )
trk X( )
CDRkX( )
CDRkX( )
340 Kwang-Min Lee et al.
and Technology Corporation, 2000). And the loss of contents
or fatality and injury losses cost CH can be evaluated
based on the research results of Korea Transport Institute,
using the human capital approach of the traffic accident
cost data (KOTI, http://traffic.metro.seoul.kr).
2.2.4. Road user cost and socio-economic losses model
for bridge structures
In Eq. (6), the formulation of rehabilitation cost involves
the assessment of road user cost and indirect socio-
economic losses. For an individual structure, like a
building structure, it can be argued that only the owner’s
cost may be relevant to a failure of the structure and thus
it might have a minor influence on public user cost or
socio-economic losses. However, when it comes to
infrastructure such as bridge, tunnel, water delivery, and
underground facilities, etc., the situation becomes
completely different precisely because those infrastructures
are primary public investments that provide vital service
to entire urban areas. Thus, the indirect costs accruing to
the public user of these infrastructural systems should
also be accounted for.
In general, road user costs consist of 5 major cost items
-namely, vehicle operating costs, time delay costs, safety
and accident costs, comfort and convenience costs, and
environmental costs (Berthelot et al., 1996). Among the
items, time delay costs and vehicle operating costs have
been generally considered as major cost items of the road
user cost (Cho et al., 2001). To evaluate the rational road
user costs, the essential factors such as traffic network,
location of bridge, and the information on rehabilitations
(e.g., work zone condition, detour rate, the change of traffic
capacity of traffic network, etc.) must be considered. The
road user cost can be expressed as follows (Lee et al.,
2004):
(7-a)
(7-b)
(7-c)
, (7-d)
where CU =daily road user cost; CTDC =daily time delay
cost; CVOC = vehicle operating cost; i = an index for route
in network; j = an index of types of vehicles which should
be classified into those for business or non-business such
as owner car for business, owner car for non-business,
taxi, bus for business, bus for non-business, small truck,
and large truck etc.; , = number of passengers in
vehicle; , = Average Daily Traffic Volume (ADTV);
= average unit value of time per the user; =
average operator wages for each type of vehicle; =
average unit fuel cost per unit length; ri = detour rate
form original route to i-th route; , = the
additional time delay on the original route and detour
route; lo, ldi = the route length of bridge route (the route
including bridge) and detour route; , = the average
traffic speed on the original route during normal
condition and rehabilitation activity; and , = the
average traffic speed on detour route during normal
condition and rehabilitation activity, respectively.
As expressed in Eq. (7), the information about the
traffic network conditions, such as the average traffic
speed on the original and detour routes during normal
condition and rehabilitation activity, and the detour rates
from original route to detour route, etc., is necessary for
the evaluation of the road user cost. These data are
undoubtedly functions of traffic volume, number of
detour routes, number of detour route lanes, and length of
detour route, etc., which can be obtained from traffic
network analysis. For the purpose, EMME/2 v5.1 (Inro
LTD, 1999) is used in this study.
Indirect socio-economic losses are the result of multiplier
or ripple effect on the economy caused by functional
failure of a structure. The indirect socio-economic losses
analysis have been done with appropriate models of the
regional economics which include (i) input-output (I-O),
(ii) social accounting matrix (SAM) models, (iii) computable
general equilibrium (CGE) model, and (iv) other macro-
econometric models. The I-O model is frequently selected
for various economic impact analyses and is widely
applied throughout the world (Kuribayashi and Tazaki,
1983). The I-O tables which represent site-specific
business and foreign trade are required for the indirect
socio-economic losses analysis using the I-O model.
However, generally, the I-O tables is made only from the
view point of macro economic analysis rather than micro
economic analysis for an area where traffic flowing is
influenced. As an alternative way, the suggestion from the
reference (Seskin, 1990) may be adopted for the
approximate but reasonable assessment of the cost. In the
reference, it is reported that socio-economic losses could
range approximately from 50 to 150% of road user costs.
CU CTDC= CVOC+
CTDC
nP0j
j 1=
J
∑ T0j u1j
⋅ ⋅⎩ ⎭⎨ ⎬⎧ ⎫
1 ri
i 1=
I
∑–⎝ ⎠⎜ ⎟⎛ ⎞
∆td0⋅ ⋅ +
ri nP0jT0j⋅ ⋅ nPij
Tij⋅+( ) u1j
⋅[ ]
j 1=
J
∑⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
I
∑ ∆tdi⋅
=
CVOC
T0j u2j u
3jld0+( )
j 1=
J
∑ 1 ri
i 1=
I
∑–⎝ ⎠⎜ ⎟⎛ ⎞
∆td0⋅ ⋅ +
ri T0j⋅ Tij+( ) u2j⋅[ ]
j 1=
J
∑⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
I
∑ ∆tdi⋅ +
ri T0j ldild0–( )⋅ Tij+( ) u
3j⋅[ ]
j 1=
J
∑⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
I
∑ ∆tdi⋅
=
∆tdildi
vdwi
-------ldi
vdni
-------–= ∆td0l0
v0w
------l0
v0n
------–=
nP0j
nPij
T0j Tij
u1j
u2j
u3j
∆td0 ∆tdi
v0n
v0w
vdnivdwi
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 341
2.3. Design constraints
In this study, for the optimum design of an orthotropic
steel deck bridge, the behavior and side constraints
include various allowable stresses, combined stress, fatigue
stress, deflection limits, geometric limits, etc. based on
the Korean Bridge Design Specification (Korea Road &
Transportation Association, 2000). The design constraints
of girder and deck system are summarized in Tables 1
and 2, respectively.
3. Time Variant Reliability Model and Optimization Algorithm
3.1. Time variant reliability model for orthotropic
steel deck bridges
3.1.1. Basic limit state functions
The evaluation of expected rehabilitation cost utilizing
Eq. (5) requires the reliability analysis for all failure limit
states in the cost model. The strength limit states in terms
of bending and/or shear of a main girder of orthotropic
steel deck bridges can be defined as the ultimate limit
state failures of flange and web. Thus, the strength limits
state for flexure and shear strength of main girder can be
formulated as following Eq (8) and (9), respectively:
(8)
(9)
where , = flexural and shear strength of
girder; , = correction factor for adjusting any
bias and incorporating the uncertainty involved in the
assessment of and ; , = bending
moment and shear due to design dead load for considered
load type l (i.e. weight of steel, asphalt on steel deck, etc.)
of main girder; , = bending moment and shear
due to design live load of main girder; , =
the correction factors for adjusting the bias and uncertainties
in the estimated and ; , = the
correction factors for adjusting the bias and uncertainties
g .( ) MGnNG
MR⋅ MG
Dl
l 1=
L
∑ NGMDl
⋅– MGLL
NGMLL
Ibeam⋅ ⋅–=
g .( ) SGnNG
SR⋅ SG
Dl
l 1=
L
∑ NGSDl
⋅– SGLL
NGSLL
Ibeam⋅ ⋅–=
MGn
SGn
NGMR
NGSR
MGn
SGn
MGDl
SGDl
MGLL
SGLL
NGMDl
NGMLL
MGDl
MGLL
NGSDl
NGSLL
Table 1. Design constraint of main girder (Korea Road & Transportation Association, 2000)
Design constraints Remarks
Bending stressfs = bending stress for structural steel
fsa = allowable bending stress for structural steel
Shear stressτs = shear stress for structural steel
τsa = allowable shear stress for structural steel
Combined stress
fs = bending stress for structural steel
fsa = allowable bending stress for structural steel
τs = shear stress for structural steel
τsa = allowable shear stress for structural steel
Local buckling fbld = allowable local buckling stress
Fatigue stressfmax, fmin = maximum and minimum stress by live load, respectively
fsa = allowable fatigue stress
Deflection∆max = maximum deflection by live load
∆a = allowable defection(L/500)
Specification forminimum thickness
bw, tw = width and thickness of the web, respectively
bf = width of the flange
tcf, ttf = thickness of the tension and compression flange, respectively
Specification formaximum thickness
t = thickness of all structural member
tmax = maximum thickness of all structural member
g1
fs
fsa----- 1.0 0≤–=
g2
τsτsa------ 1.0 0≤–=
g3
fs
fsa-----
⎝ ⎠⎛ ⎞
2 τsτsa------
⎝ ⎠⎛ ⎞
1.2 0≤–+=
g4
fs
fbld------- 1.0 0≤–=
g5
fmax
fmin
–( )ffa
------------------------- 1.0 0≤–=
g6
∆max
∆a
----------- 1.0 0≤–=
g7
bw
220--------- tw 0≤–=
g8
bf
48fn---------- tcf 0≤–=
g9
bf
80n--------- ttf 0≤–=
g10
t
tmax
--------- 1.0 0≤–=
342 Kwang-Min Lee et al.
in the estimated and ; and = uncertainty
factor for girder impact
In Eq. (8) and (9), ultimate flexural and shear strength
of main girder depends on many criteria such as: whether
the section is compact or non-compact, braced or
unbraced, stiffened or unstiffened, composite or non-
composite, etc. Detailed description for estimation of the
ultimate flexural and shear strength can be found in the
references (AASHTO, 1996; AISC, 1994).
The fatigue failure of orthotropic steel deck bridges
may be defined as failure of the connection weld of the
closed ribs shown in Fig. 2. The connections of closed rib
in the floor beam and closed rib in the deck system are
classified as the category C and B, respectively, which
can be specified in the current AASHTO fatigue
specification.
For the evaluation of fatigue reliability index, the simple
formula by the Pedro Albrecht’s S-N fatigue reliability
(Albrecht, 1983) is used as follows:
(10)
where β = reliability index, N,Nd = number of cycles for
equivalent stress range and loading cycle, respectively;
Sτ = standard deviation of the difference between the
resistance and load, which can be expressed as
; SR, SQ = standard deviation of the
number of cycles for resistance and load, respectively;
and m = slop of S-N line (transformation coefficient)
3.1.2. Time effect of extreme live load model
The reliability of a bridge is only valid for a specific
point in time. The maximum value of the live load is
expected to increase over time, and the bridge deteriorates
through aging, increased use, and specific degradation
mechanisms such as fatigue and corrosion, etc. The live
load model used in this study (Nowak, 1993) accounts for
the increased effects of maximum shear and moment as
more trucks pass over the bridge. This model is based on
statistics of extreme values where the probability of
encountering a heavy truck at the extreme upper tail of
the distribution increases as the number of trucks passing
over the bridge increases. The live load model uses the
actual daily truck traffic and the actual bridge span
lengths to predict the maximum moment and shear effects
over time in a bridge. To apply the live load model to
reliability analysis of a bridge, only the mean and standard
deviation of the extreme distribution are needed. For
instance, using central and dispersion of the Type I
extreme distribution, the mean µMn and standard deviation
σMn can be computed as (Ang and Tang, 1984)
(11-a)
(11-b)
SGDl
SGLL
Ibeam
βNlog Nlog–
Sτ
-----------------------------=
SR( )2 m SQ'( )2+
µMn σ un⋅ µ γ σ⋅ αn⁄( )+ +=
σMn π 6⁄( ) σ αn⁄( )⋅=
Table 2. Design constraint of steel deck (Korea Road & Transportation Association, 2000)
Design constraints Remarks
Bending stress ofU-rib
fd = stress of U-rib in the analysis of orthotropic steel deck
fm = stress of U-rib in the analysis of main girder
fall = allowable stress of structural steel
Bending stress of cross beam
ff = floor beams bending stress
fall = allowable stress of floor beam considering local buckling
Fatigue stressfmax, fmin = maximum and minimum stresses, respectively
ffa = allowable fatigue stress
Bending stress constraint of floor-beam
ff = bending stress of floor beam
fbld = allowable stress of floor beam considering local buckling
Shear stress of floor beam
τfsa = shear stress of floor beam
τfs = allowable stress of floor beam considering local buckling
Combined bending stressff, fcal = bending stress and allowable bending stress of floor beam
τfsa, τfs = shear stress and allowable shear stress of floor beam
Specification for mini-mum thickness
tfb, tmdp = minimum thickness of floor beams and deck plate, respectively
tfb, tdp = thickness of floor beams and deck plate, respectively
g11
fd
fall------
fn
fall------ 1.0 0≤–+=
g12
ff
fcal------ 1.0 0≤–=
g13
fmax
fmin
–( )ffa
------------------------- 1.0 0≤–=
g14
ff
fbld------- 1.0 0≤–=
g15
τfsτfsa------- 1.0 0≤–=
g16
ffs
ffsa------⎝ ⎠⎛ ⎞
2 τfsτfsa-------⎝ ⎠⎛ ⎞ 1.2 0≤–+=
g17
tmfb
tmax
--------- 1.0 0≤–=
g18
tmfb
tdp-------- 1.0 0≤–=
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 343
where µ, σ = the initial mean and standard value of live
load effect, respectively; n = cumulative number of truck
crossing a bridge;
; ;
and γ = 0.577216 (the Euler number)
3.1.3. Time-variant resistance model
Except for high performance steel such as anti-
corrosion weathering steel, almost all structural steel are
subject to corrosion to some degree. As long as structural
steel bridges are concerned, corrosion reduces the original
thickness of the webs and flanges of the steel girders. Fig
3 and 4 show typical locations where corrosion can occur
on a steel bridge (Kayser and Nowak, 1989). The type of
corrosion that will most likely occur at mid-span of steel
bridge is section loss on the surface of bottom flange and
on the bottom at around one quarter of the web. Typical
section loss near the end support is characterized by
section loss over the whole web surface and on the
surface of the bottom flange, as shown in Fig 4.
Measurement of remaining thickness of corroded
structural steel is an important problem. If undetected
over a period of time, corrosion will weaken the webs and
flanges of steel girder and possibly lead to dangerous
structural failures. Severity of corrosion, in general,
depends on the type of steel, local corrosion environment
affecting steel condition including temperature and
relative humidity, and exposure conditions (Albrecht and
Naeemi, 1984). A number of researchers have pursued
extensive studies to predict time-variant corrosion propagation
to capture the actual corrosion process (Albrecht and
Naeemi, 1984; Ellingwood et al., 1999; etc.). However,
these studies ignore the influence of periodic repainting
effect on the corrosion process. Thus, in this study, based
on the previous study (Ellingwood et al., 1999), a
modified corrosion propagation model is introduced as
follows:
for
(12-a)
otherwise (12-b)
where pi(t) = corrosion propagation depth in µm at time
t years during i-th repainting period; C = random corrosion
rate parameter; m = random time-order parameter; and TCI,
TREP = random corrosion initiation and periodic repainting
period, respectively
In order to obtain mean and standard deviation of time-
variant resistance, such as ultimate flexural strength, shear
strength, etc., based on Eq. (12), Monte Carlo Simulation
(MCS) with a sample size 50,000 is used for this study.
3.1.4. Annual probability of failure and reliability
assessment
It may be noted that, in Eq. (5), the evaluation of the
annual probabilities of failure for failure limit state k
considered in the cost model are essential to estimate the
expected rehabilitation costs. The probability that a steel
bridge will fail in t subsequent years, given that it has
survived T years of loads, can be expressed as follows
(Stewart, 2001b):
αn
2 n( )ln⋅= un
αn
n( )ln[ ]ln 4π( )ln+
2 αn
⋅-----------------------------------------–=
pit( ) C t i T
REP⋅– T
CI–( )
m⋅=
i( ) TREP
⋅ TCI
+ t≤ i 1+( ) TREP
⋅<
pit( ) p
i 1–i T
REP⋅( )=
Figure 2. Connection weld of closed rib.
Figure 3. Corrosion of steel bridge.
Figure 4. Typical location of corrosion on a steel bridge(Kayser and Nowak, 1989).
344 Kwang-Min Lee et al.
(13)
where , = the cumulative probability
of failure of survice proven bridges anytime during this
time interval, respectively
In Eq. (13), the cumulative life-time probability of
failure can be obtained by combinding the extreme live
load model, the time-variant resistance model and the
limit state models proposed in this study (Mori and
Ellingwood, 1993; etc.). Various numerical reliability methods
available can be applied to the reliability analysis for
strength and serviceability limit state functions. For this
specific study, in the evaluation of element reliability for
the strength and serviceability of a steel bridge, the
advanced first order second moment (AFOSM) method is
used, with the Rackwitz-Fiessler normal tail approximation
for non-Gaussian independent variates (Rackwitz and
Fiessler, 1978) and with direct iteration for correlated varieties
(Ang and Tang, 1984). Meanwhile, as aforementioned,
for the evaluation of fatigue reliability of deck system of
orthotropic steel deck bridges, the simple formula by the
Pedro Albrecht’s S-N fatigue reliability (Albrecht, 1983)
such as Eq. (10) is applied in this study.
Note that the system models have been usually used, in
general, for reliability analysis for the strength failure of
a bridge. However, because the rehabilitation of shear or
bending moment failure of a steel bridge is usually not an
event of structure collapse but that of local limit state
failure, the element level reliability analysis may be more
reasonable than the system level reliability in the case of
LCC-effective optimization of steel bridges
3.2. Optimization algorithm for the LCC-effective
optimum design of orthotropic steel deck bridge
A conceptual flow chart of the LCC-effective optimum
design algorithm of orthotropic steel deck bridges is shown
in Fig. 5. The algorithm essentially consists of structural
analysis module, lifetime reliability analysis module,
optimization module, and LCC evaluation module. As
shown in the figure, in this study, Finite Element Method
(FEM) and Perikan-Esslinger Method are used as the
structural analysis. For solving the LCC optimization
problem, a computer program, called Automated Design
Synthesis (ADS) (Vanderplaats, 1986), is used for the
optimum design of the bridge. The Augmented Lagrange
Multiplier (ALM) method is found to be very effective,
together with Broydon-Fletcher-Goldfarb-Shanno (BFGS)
for the unconstrained minimization, and the golden section
method for one-dimensional search (Cho, 1998).
4. Illustrative Example and Discussions
To demonstrate the effect of the LCC optimization in
the optimal framework and design of orthotropic steel
deck bridges, an orthotropic steel deck bridge having
seven continuous spans (80 m + 120 m + 140 m + 160 m
+ 140 m + 120 m + 80 m) and a total length of 840m is
considered as an illustrative example. The general data
for the bridge is shown in Table 3. Bridge profile and
design group, and typical section of the bridge are also
shown in Fig 6.
pFkX t T,( )
pf X T, t1
+( ) pf X T,( )–
1 pf X T,( )–
-----------------------------------------------=
pf X T, t1
+( ) pf X T,( )
Figure 5. LCC-effective optimum design algorithm oforthotropic steel deck bridges.
Table 3. General data of numerical example
Bridge type Seven-span continuous orthotropic steel deck bridge
Bridge length (m) 80 + 120 + 140 + 160 + 140 + 120 + 80 = 840 No of rib/type 20-22 ea/closed rib
Bridge width (m) 15.5 Design Load DB/DL-24 (HS-20 / 0.75)
No. of lane 4 Type of Steel SM490(fa=1900kgf/cm2)
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 345
In this study, though the example bridge is an actual
bridge constructed in a region (Wan-do, Korea), it is
assumed that the bridge will be constructed as a part of a
typical urban highway which has large ADTV or a typical
rural highway which has relatively moderate ADTV. Fig
7 and 8 show a typical highway network and modeling
for traffic network analysis used for this study. And Table
4 and 5, respectively, represent the expected traffic
volumes of typical urban and rural highways for 20 years
after construction, respectively.
4.1. Data for estimating life-cycle costs
Recently, in Korea, the construction of steel bridges
using high-strength, high-performance steels (e.g., anti-
corrosion weathering steel, TMCP steel, etc) keep
increasing. Thus, in this study, for comparing the LCC-
effectiveness of steel bridge using conventional structural
steel with that using weathering steel, the construction
unit costs are investigated. The estimation of unit costs is
based on the price information of the Research Institute
of Industrial Science and Technology (RIST, 1998). The
Figure 6. Bridge profile and design group, and typical section of example bridge.
Figure 7. Typical highway network near the brige site. Figure 8. Modeling for traffic network analysis.
346 Kwang-Min Lee et al.
design cost and load testing cost are assumed as 7% and
3% of construction cost, respectively (Ministry of Science
and Technology, 2004).
As aforementioned, the rehabilitation costs can be
classified into direct and indirect rehabilitation costs.
Direct rehabilitation costs such as repair and replacement
costs result from the damage of a bridge. Indirect
rehabilitation costs can also be classified into loss of
contents or fatality and injury losses, road user costs, and
indirect socio-economic losses.
Based on the various sources available, such as the
CSR’s price information, the opinions of the field experts
who engaged in the construction, and the references
available (OCFM, 2002; KISTEC, 2000; KIST and
KISTEC, 2001), in order to estimate rehabilitation costs,
the required data related to each failure limit state, such
as countermeasures for rehabilitation, unit direct
rehabilitation cost, expected period for rehabilitation, and
work-zone condition during rehabilitation activity, are
Table 4. Expected future ADTV of typical rural highway
Year 2007 2011 2015 2019 2023 2027
Car
I 15,419 18,487 21,276 23,850 25,936 26,758
II 15,606 19,047 21,914 24,562 26,711 27,557
III 14,424 17,596 20,262 22,716 24,703 25,486
Bus
I 01,793 02,076 02,290 02,456 02,554 02,575
II 01,847 02,139 02,358 02,529 02,630 02,653
III 01,707 01,977 02,180 02,339 02,433 02,452
Truck
I 10,057 11,984 13,531 14,882 15,883 16,238
II 10,360 12,347 13,937 15,325 16,358 16,723
III 09,576 11,406 12,886 14,174 15,129 15,467
Descriptions I : V intersection ~ location of the bridge; II : location of the bridge ~ J; III : J~ IV intersection
Table 5. Expected future ADTV of typical urban highway
Year 2007 2011 2015 2019 2023 2027
Car
I 29,450 42,486 55,760 69,408 81,320 86,162
II 29,762 43,774 57,432 71,480 83,750 88,734
III 27,508 40,430 53,103 66,108 77,455 82,065
Bus
I 03,419 04,771 06,002 07,147 08,008 08,292
II 03,522 04,916 06,180 07,360 08,246 08,543
III 03,225 04,543 05,713 06,807 07,628 07,895
Truck
I 19,179 27,541 35,462 43,310 49,800 52,286
II 19,757 28,376 36,525 44,599 51,289 53,848
III 18,262 26,213 33,772 41,249 47,436 49,804
Descriptions I : V intersection ~ location of the bridge; II : location of the bridge ~ J; III : J~ IV intersection
Table 6. Construction unit costs for orthotropic steel deckbridge
Structuralsteel
Weathering steel
Labor cost ($/ton) 1330.9 1330.9
Material cost ($/ton) 419.9 519.2
Paint or repainting cost ($/m2) 236.0 -
Table 7. Data related to limit states for estimating rehabilitation costs
Limit state Strength of girder Fatigue of steel deck
CountermeasuresRetrofit Bolting repair
Structural steel Weathering steel Structural steel Weathering Steel
Unit direct rehabilitation cost ($)1,750/ton + 32.1/ton
(disposal)1,850/ton + 32.1/ton
(disposal)4,500 per location
Expected periods for rehabilitation 3 Month 3 day
Work-zone condition Partially Traffic Closing (2-lanes) -
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 347
summarized in Table 7. Moreover, major parameters for
the estimation of indirect rehabilitation costs are also
summarized in Table 8.
Table 9 shows the result of the rehabilitation costs for
the actual design of the bridge. In the estimation of
indirect rehabilitation costs, since the I-O Table for local
areas except for the metropolitan area (Seoul) are not
investigated, the results from the reference (Seskin, 1990)
are used in this paper. In these example applications, the
ratios of the socio-economic losses to the road user costs
are assumed to be 0.5 and 1.5, respectively, for moderate
and large ADTV regions. It may be noted that the indirect
cost dominates the rehabilitation cost if partial traffic
closing is to occur, because, as shown in Table 9, it is
about 97~99% of the total rehabilitation cost. Thus, it
should be regarded as one of the most important costs in
the evaluation of the expected life-cycle cost.
In this study, all the uncertainties of the basic random
variables of resistance and load effects described above
are based on the uncertainty data available in the literature
(Nowak, 1993, 1994; Zokaie et al., 1991; Albrecht, 1983;
Albracht and Naeemi, 1984; Maunsell Ltd, 1999). The
resulting mean and coefficient of variation (C.O.V) of
each parameter for resistance and load effects with the
assumed distributions are summarized in Table 10.
Table 8. Major parameters for estimating indirect costs
Parameter Value Unit Reference
Discount rate 4.00 % KIST and KISTEC (2001)
The traffic accident cost 0.12 Million $KOTI
(http://traffic.metro. seoul.kr)Traffic accident rate during repair work activity 2.2 Million vehicle/kilometers
Traffic accident rate during normal condition 1.9 Million vehicle/kilometers
The value of fatality 3.5 Million $Lee and Shim (1997)
The value of injury 0.021 Million $
The hourly driver cost 21.52 $/person KOTI
Table 9. Total rehabilitation cost for countermeasures (unit = Million $)
Countermeasures forrehabilitation
Retrofit Bolting repair
Large ADTV Moderate ADTV Large ADTV Moderate ADTV
Direct rehabilitation cost 0.381 (3.4%) 0.381 (0.7%) 0.0045 (100 %) 0.0045 (100%)
Indirect rehabilitation cost 10.691 (96.6%) 53.454 (99.3%) - -
Total rehabilitation cost 11.072 (100%) 53.835 (100%) 0.0045 (100%) 0.0045 (100%)
Table 10. Uncertainties of random variables
Definition and Units of random variables Type Mean C.O.V Reference
Yield stress of steel in girders (kgf/cm2) Fy N** 3552.0 0.12 Nowak (1993)
Model uncertainty: flexure in steel γmfg N 1.11 0.11 Nowak (1993)
Model uncertainty: shear in steel γmsg N 1.14 0.12 Nowak (1993)
Uncertainty factor: weight of asphalt λasph N 1.00 0.25 Nowak (1993)
Uncertainty factor: impact on girders positive moment part (at span 1/2/3/4/5/6/7)
Ibeam N1.125/1.094/1.083/1.075/1.083/1.094/
1.0830.10
Nowak et al.
(1994)
Uncertainty factor: impact on girders negative moment part (at pier 1/2/3/4/5/6)
Ibeam N1.107/1.088/1.079/1.079/1.088/1.107
0.10Nowak et al.
(1994)
Parameter and uncertainty factor related to fatigue reliability for the connection type B
m,/SR/S'Q LN*** 3.372/1.0/1.00.0/0.147*/
0.0492*Albrecht(1983)
Parameter and uncertainty factor related to fatigue reliability for the connection type C
m,/SR/S'Q LN 3.250/1.0/1.00.0/0.0628*/
0.0492*Albrecht(1983)
Corrosion Initiation TCI LN 15 0.3 Nowak (1999)
Duration of Repainting Action TREP N 20 0.25Maunsell Ltd.
(1999)
Corrosion rate/time-order parameter (rural) C, M N 34/0.65 0.09/0.1 Albracht and Naeemi (1984)Corrosion rate/time-order parameter (urban) C, M N 80/0.593 0.42/0.4
* : Standard deviation, ** N: Normal, *** LN: Log-Normal
348 Kwang-Min Lee et al.
4.2. Optimization results and discussions
4.2.1. Effect of local corrosion environment on LCC-
effective optimum design in moderate ADTV regions
LCC-effective optimum design of an orthotropic steel
deck bridge may be affected by local corrosion environment.
Thus, in this study, to demonstrate the effects of local
corrosion environment on the cost-effectiveness of the
LCC optimum designs of the illustrative bridge various
LCC optimizations are performed considering 5 different
cases A~E (see Table 11). For estimating expected
rehabilitation costs and the mean and C.O.V of the
maximum live load moment/shear, the moderate ADTV
given in Table 4 is only used in this section in order to
focus on the effects of local corrosion environment. In
addition to the 5 cases, since the LCC optimum design
can be alternatively achieved by the initial cost
optimization with a reasonable allowable stress ratio
(= design stress/allowable stress) (Cho, 1998), which may
be called ‘Equivalent LCC optimization’, the total expected
LCCs for the 8 different levels of allowable stress ratio
(65% to 100% with increments of 5%) are estimated
based on the Equivalent LCC optimization procedure
considering 3 different cases I~III (see Table 11).
Table 12 shows the optimal stress ratio and total expected
LCC of various types and spacing of closed rib and floor
beam spacing for Cases I~III, respectively. As shown in
the table, the optimum design point for Cases I, II, and III
are achieved similarly at design stress 80% of the
allowable stress, closed rib type 4, closed rib spacing
62 cm, and floor beam spacing 3.0 m.
The design results of all different 8 cases are summarized
in table 13 for Case A~E and Cases I~III at the optimum
LCC design point (80% of the allowable stress, closed rib
type 4, closed rib spacing 62 cm, and floor beam spacing
3.0 m), respectively, in terms of the optimum values of
the design variables and estimated costs. Based on the
investigation of the design results for each case, the
following observations can be made on the characteristics
of LCC-effective design of orthotropic steel deck bridges.
First, note that for deck thickness, the type of closed
ribs and spacing, and the floor beams spacing of the deck
system, Case B (initial cost optimization) results in 14
mm, type 2, 62 cm and 250 cm, respectively, whereas
Case C~E (LCC optimizations) provide 17 mm, type 4,
Table 11. Considered cases in LCC optimum designs of the illustrative bridge
Case ID Design methodology Type of steel Corrosion environment
Case A Conventional design
Structural SteelUrban corrosion Env.Case B Initial cost optimum design
Case C
LCC optimum designCase D Rural corrosion Env.
Case E Weathering Steel -
Case I
Equivalent LCC optimizationStructural Steel
Urban corrosion Env.
Case II Rural corrosion Env.
Case III Weathering Steel -
Table 12. Total expected LCC and optimal stress ratio (moderate ADTV, unit = $)
Case IDFloor beam
spacing
Closed rib type and spacing
Type2(1)-62cm Type2 -68 cm Type3(2)-62 cm Type3-68cm Type4(3)-62 cm Type4-68cm
Case I
2.00 m 17,051,700.0 17,331,906.20 16,679,071.20 17,059,637.50 16,507,663.80 17,074,588.80
2.50 m 17,204,726.2 17,455,780.00 16,602,446.20 16,951,377.50 16,605,062.50 17,001,558.80
3.00 m 17,010,212.5 17,607,005.00 16,463,618.80 16,802,643.8016,358,268.80
(80%)16,481,670.00
Case II
2.00 m 15,757,333.8 16,320,785.00 15,608,547.50 15,762,695.00 15,947,588.80 15,843,988.80
2.50 m 15,681,620.0 15,887,265.00 15,406,751.20 15,619,622.50 15,504,420.00 15,635,346.20
3.00 m 15,554,103.8 16,325,280.00 15,394,972.50 15,537,582.5015,263,847.50
(80%)15,392,320.00
Case III
2.00 m 12,173,312.5 12,857,621.20 12,034,742.50 12,199,967.50 12,398,200.00 12,278,658.80
2.50 m 12,103,677.5 12,306,816.20 11,807,528.80 12,002,933.80 11,945,785.00 12,062,257.50
3.00 m 11,965,522.5 12,827,601.20 11,717,763.80 11,926,952.5011,655,681.20
(80%)11,828,176.20
* Total expected LCC** Optimal stress ratio(1) Closed Rib Type 2 : A-A’-H-t-R =320-204-260-6-40(2) Closed Rib Type 3 : A-A’-H-t-R =324-216.5-242-8-40(3) Closed Rib Type 4 : A-A’-H-t-R =324-207.7-262-8-40
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 349
62 cm and 300 cm, respectively, as the result. From the
result, in the aspect of LCC optimal framework of the
bridge, it may be advantageous to increase strength by
larger deck thickness and stiffer closed rib rather than to
increase strength by narrower floor beam spacing. These
trends may be attributed to more expensive expected
rehabilitation cost of girder strength then that of fatigue of
steel deck (see table 9). Next, note that, for the height of
web at pier 2 (h2), Cases B result in 7.00 m, whereas
Cases C~E (LCC optimization) provide 8.87~8.91 m. It
may be observed that the height of web by the LCC
optimization provide a significantly higher than that by
initial cost optimization.
Also, it may be noted that, although the local corrosion
environments are different, the optimum values of the
design variables from Cases C, D and E, respectively, are
the same. Thus, to investigate the causes of these results,
the sensitivity analyses for the effect of the ADTV on the
optimum design are performed in the next section, since
the ADTV is one of major parameters in the estimation of
expected rehabilitation cost and the mean and C.O.V of
the maximum live load effects.
In order to examine the relative effects of the various
design cases on the LCC costs, all the costs such as the initial
cost, expected maintenance cost, expected rehabilitation
cost (expected retrofit cost, expected fatigue repair cost),
and total expected LCC of each design case, as shown in
Table 13 and Fig. 9, need to be compared with one
another. It is found in the tables that the initial cost from
Case B (initial cost optimization) is 9.372 Million $,
Table 13. Results of optimum design of the illustrative example bridge
Case
Conven-tional design
(Case A)
Initial cost optimumdesign
(Case B)
LCC optimization and Equivalent LCC optimization
Urban Env. Rural Env. No Corrosion
Case C Case I Case D Case II Case E Case III
Types of closed rib/Spacing (m) 3/0.62 2/0.62 4/0.62 4/0.62 4/0.62 4/0.62 4/0.62 4/0.62
Floor beam spacing (m) 2.0 2.5 3.0 3.0 3.0 3.0 3.0 3.0
Deck thickness (mm) 14 14 17 17 17 17 17 17
Girder
Lower flangeThickness(mm)
Design group1/2/3
14/18/14 10/10/10 10/10/10 10/10/10 10/10/10 10/10/10 10/10/10 10/10/10
Design group4/5/6
20/26/20 10/14/10 10/16/10 10/16/10 10/16/10 10/16/10 10/16/10 10/16/10
Design group7/8/9
16/22/28 10/10/17 10/10/18 10/10/18 10/10/18 10/10/18 10/10/18 10/10/18
Design group10/11/12
24/18/24 10/10/10 13/14/21 13/14/20 13/14/20 13/14/20 13/14/20 13/14/20
Design group13/14/15
30/20/26 11/10/15 28/12/19 28/12/19 28/12/19 28/12/19 28/12/19 28/12/19
Web thickness (mm) 18 14 15 15 15 15 15 15
Height (m)
b1 3.00 3.26 4.00 4.00 4.00 4.00 4.00 4.00
b2 3.50 3.92 4.50 4.50 4.50 4.50 4.50 4.50
h1 6.50 6.01 7.23 7.22 7.21 7.22 7.20 7.21
h2 7.50 7.00 8.91 8.89 8.88 8.89 8.87 8.89
h3 8.50 11.00 11.00 11.00 11.00 11.00 11.00 11.0
FloorBeam
Height (m) 1.00 1.12 1.18 1.18 1.18 1.18 1.18 1.18
Web thickness (mm) 14 10 1.0 10 10 10 10 10
Lower flange Thickness (mm) 14 10 1.0 10 10 10 10 10
Cost (Million $)
Initial cost 11.5820 9.3715 10.8954 10.8473 10.8212 10.8473 11.3990 11.4246
Expectedmaintenance cost
3.8143 5.3374 5.7113 5.6627 5.6477 5.6627 0.0000 0.0000
Expectedretrofit cost
3.8725 39.4555 1.9567 1.9667 0.2606 0.2596 0.2501 0.2484
Expected fatigue repair cost
0.0001 0.0036 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002
Total Expected LCC (Million $) 19.2689 54.1680 18.5637 18.4769 16.7295 16.7696 11.6493 11.6731
350 Kwang-Min Lee et al.
while the initial costs from Cases C~D (LCC optimizations)
are 10.895~10.821 Million $. Therefore, it can be observed
that the initial cost of the initial cost optimization is
decreased by about 16.26~15.47% compared with the
initial costs of the LCC optimizations. Thus it may be
stated that, in general, the initial costs of the LCC
optimizations may be slightly increased, as expected,
compared with those of the initial cost optimizations.
Meanwhile, the total expected LCC from Cases B
(initial cost optimization) is 54.168 Million $, but those
from Cases C~D (LCC optimizations) are 18.563~16.730
Million $. Therefore, it may be definitely stated that, in
terms of the total expected LCC, the LCC optimizations
are much more economical than the initial cost optimizations
(by about 34.27~30.88%) because the failure probabilities
(for strength and fatigue) in the initial cost optimizations
are much higher than those in the LCC optimizations.
Also, note that Table 13 and Fig 9 clearly show the
advantages of weathering steel when used for orthotropic
steel deck bridge design. Though the initial cost of the
weathering steel from Case E is more expensive by about
5.07~4.42% compared with those of Cases C~D, the total
expected LCC from Cases E is much more economical by
about 59.35~43.61%. These trends may be evidently
attributed to more expensive expected maintenance cost
(repainting cost) and expected rehabilitation cost (due to
the disadvantage of the structural resistance degradation
as survival age increase) of conventional structural steel
compared with those of weathering steel.
Also, it has been found that the total expected LCCs
from equivalent optimum LCC design (Cases I~III) are
very close to that of the LCC optimization (Cases C~E)
only with the difference of around 0.47~0.20%. Thus, as
an alternative approach to the LCC optimization, the initial
cost optimization with a reasonable allowable stress ratio,
i.e. the stress limit to be used for conventional design in
practice, can be utilized to implicitly accomplish a near
optimum LCC of an orthotropic steel deck bridges.
4.2.2. Effect of local corrosion environment on LCC-
effective optimum design in large ADTV regions
In the previous section, it was shown that the optimum
values of optimum designs from Cases C~E (LCC
optimizations) are all the same. Thus, in this section, for
the investigation of the causes of these results, the design
conditions of the moderate ADTV are replaced by those
of large ADTV, and then only the economical aspects are
focused on and discussed. For comparing relative effects
of ADTV on the optimum LCC of orthotropic steel deck
bridge, additional 3 different cases are considered and
summarized in Table 14.
Table 15 shows the results of total expected LCC of
initial cost optimizations of Cases IV~VI (cases in which
large ADTV is applied). As expected, for all cases, closed
rib type 4, closed rib spacing 62 cm, and floor beam
spacing 3.0 m are designed as the optimum framework of
the bridge. Note that the optimum design point for Cases
IV, V, and VI are achieved at design stress 75% (initial
cost = 12.354 Million $), 80% (initial cost = 10.847 Million
$), and 80% (initial cost = 11.425 Million $) of the allowable
stress, respectively. It shows different result compared
with those of Case I (urban corrosion environment) where
the moderate ADTV is applied (see Table 12). This result
means the local corrosion environment affects to the
optimum LCC design, as the ratio of expected rehabilitation
cost to total expected LCC becomes large in these case.
Also, comparing Case IV and VI, it should be noted that
the weathering steel is advantageous not only in view of
initial cost but also in view of LCC effectiveness, since
the LCC optimum design for weathering steel is achieved
at the higher allowable stress level than that of conventional
structural steel for urban corrosion environment. From
the results, it may be concluded that the weathering steel,
which requires virtually minimal maintenance, is
economically advantageous in view of the LCC-effective
optimum design of an orthotropic steel deck bridge
especially in large ADTV and urban corrosion environment
region.
5. Conclusion
This paper presents a realistic and practical LCC
methodology for lifetime reliability based LCC-effective
optimum design of orthotropic steel deck bridges. From
the illustrative example and discussions, some important
conclusions would be pointed out as follows:
Figure 9. Comparison of costs (Moderate ADTV).
Table 14. Cases considered in the equivalent LCC optimization procedure
Case ID ADTV Type of steel Corrosion environment
Case IV
LargeConventional structural steel
Urban corrosion env.
Case V Rural corrosion env.
Case VI Anti-corrosion Weathering Steel -
Lifetime Reliability Based Life-Cycle Cost Effective Optimum Design of Orthotropic Steel Deck Bridges 351
(1) The optimum structural framework of the LCC
optimization may be significantly different from that of
conventional design and also that of initial cost
optimization. For orthotropic steel deck bridges, from the
standpoint of LCC-effectiveness, it may be advantageous
to increase girder strength (i.e., increase deck thickness,
strength of closed rib, girder height, etc.) rather than to
increase deck fatigue strength (i.e., decrease floor beams
spacing, etc.). These trends may be attributed to more
expensive expected rehabilitation cost due to girder
strength failure than the cost due to fatigue failure of steel
deck.
(2) Although the initial costs of LCC optimizations are
slightly more costly than that of initial cost optimization,
it may be positively stated that, in terms of the total
expected LCC, the expected costs of the LCC optimizations
are much more economical than those of the initial cost
optimization, mainly because the probability of failure for
the limit states considered in the initial cost optimization
are usually much higher than those of the LCC optimization,
and thus the light-weight design will become prone to
deteriorations and failures of orthotropic steel deck
bridges under environmental stressors.
(3) As an alternative approach to LCC optimization, the
initial cost optimum design with optimal level of design
stress can be utilized to implicitly accomplish the optimum
LCC design of orthotropic steel deck bridges, and the
approximate optimum LCC design would be efficiently
used in practice by design engineers.
(4) Since the local corrosion environments, number of
truck traffic significantly influence the LCC-effective
optimum design of orthotropic steel deck bridges, they
should be considered as crucial parameters for the
optimum LCC-effective design of orthotropic steel deck
bridges.
(5) Also, it should be point out that the LCC-effective
bridge design with high-performance steel, such as anti-
corrosion weathering steel and simplified details may
result in almost maintenance-free bridge during life-span,
but bring forth somewhat higher initial cost. In conclusion,
it is almost evident that, when truly integrated high-
performance steels (corrosion resistance, fatigue-resistance,
high strength, high energy absorption, etc) become available
in near future, reasonable moderate-weight design with
optimal proportioning and minimal details, that may
guarantee virtually maintenance/rehabilitation free bridge
except routine bridge management, will become a new
paradigm for steel bridge design in the coming era, in
view of true LCC-effectiveness.
References
Albrecht, P. and Naeemi, A. H. (1984), “Performance of
Weathering Steel in Bridges,” National Cooperative
Highway Research, Report 272Ang, A. H-S., and Tang, W. H. (1984), “Probability
Concepts in Engineering Planning and Design,” Vol. I
and II, John Wiley
Berthelot, C. F., Sparks, G. A., Blomme, T., Kajner, L., andNickeson, M. (1996), “Mechanistic-probabilistic vehicle
operating cost model,” Journal of Transportation
Engineering, ASCE, 1996; 122(5): 337-341Cho, H. N. (1998), Development of Programs for Computer
Aided Optimum Design of Steel Box Girder Bridges,
Sam-bo engineering, 1st editionCho, H. N., Ang, A. H-S., Lim, J. K., and Lee, K. M. (2001),
“Reliability-Based Optimal Seismic Design and
Upgrading of Continuous PC bridges Based on MinimumExpected Life-Cycle Costs,” Proc. of ICOSSAR01
Table 15. Total expected LCC and optimal stress ratio (Large ADTV, unit = $)
Case IDFloor beam
spacing
Closed rib type and spacing
Type2 (1)-62cm Type2 -68 cm Type3(2)-62 cm Type3-68cm Type4(3)-62 cm Type4-68cm
Case IV
2.00 m 18,468,986.20 19,315,112.50 18,404,566.20 19,104,562.50 18,300,353.80 18,779,156.20
2.50 m 18,373,712.50 19,032,637.50 18,571,705.00 18,311,475.00 18,301,797.50 18,383,177.50
3.00 m 18,636,045.00 19,645,715.00 18,157,865.00 18,515,792.5017,924,015.00
(75%)18,178,520.00
Case V
2.00 m 17,269,688.80 17,444,970.00 17,004,443.80 17,156,721.20 17,310,775.00 17,386,282.50
2.50 m 17,311,331.20 17,640,667.50 16,952,948.80 17,451,200.00 17,012,011.20 17,165,812.50
3.00 m 16,983,882.50 17,592,223.80 17,305,598.80 16,927,701.2016,902,316.20
(80%)16,911,040.00
Case VI
2.00 m 13,723,152.50 13,935,166.20 13,446,561.20 13,600,423.80 13,730,580.00 13,844,742.50
2.50 m 13,821,473.80 14,075,307.50 13,420,423.80 13,816,570.00 13,393,286.20 13,629,245.00
3.00 m 13,417,100.00 14,086,691.20 13,791,303.80 13,373,968.8013,297,750.00
(80%)13,344,162.50
* Total expected LCC** Optimal stress ratio(1) Closed Rib Type 2 : A-A’-H-t-R =320-204-260-6-40(2) Closed Rib Type 3 : A-A’-H-t-R =324-216.5-242-8-40(3) Closed Rib Type 4 : A-A’-H-t-R =324-207.7-262-8-40
352 Kwang-Min Lee et al.
Cho, H. N., Lee, D. H., Chung J. S., and Min, D. H. (1999),
“Multi-objective and multi-level optimization for
orthotropic steel plate deck bridges,” 6th International
Conference on Steel & Space Structures, pp 313-320
Cho, H. N., Min, D. H., and Lee, K. M. (2001), “Optimum
Life-Cycle Cost Design of Orthotropic Steel Deck
Bridges,” International Journal of Korean Society of Steel
Construction, Vol.1 No.2, pp. 141~153
Ellingwood, B. R. and Mori, Y. (1993), “Methodology for
Reliability Based Condition Assessment,”
Ellingwood, E. R. and Naus, D. J. (1999), “Condition
Assessment and Maintenance of Aging Structure in
Critical Facilities A probabilistic Approach,” Case Study
in Optimal Design and Maintenance Planning of Civil
Infrastructure Systems, ASCE, pp 45-56
Farkas, J. (1996), “Optimum design of welding bridges,”
Welding in the World, 38: 295-306.
INRO (1999), “EMME/2 User's Manual, Software Release
9.0,” INRO Consultants Inc., Montreal, Canada
Kayser, J. R. and Nowak, A. S. (1989), “Capacity Loss Due
to Corrosion in Steel-Girder Bridges,” Journal of
Structural Engineering, ASCE, Vol. 115, pp. 992-1006
Korea Infrastructure Safety and Technology Corporation
(2000), “Method for Extension of Service life of Road
Bridges,” Final Report to Korea Road and Transportation
Association
Korea Institute Construction Technology and Korea
Infrastructure Safety and Technology Corporation (2001),
“An Enhanced Strategy for Infra Facilities Safety
Management System based on LCC Concept,” Final
Report to Korea Road and Transportation Association
Korea Roan and Transportation Association (2000), “Korean
Bridge Design Specification”
Kuribayashi, E. and T. Tazaki (1983), “Outline of the
earthquake disaster,” p. 67-90 in: Report on the Disaster
Caused by the Miyagi-ken-oki Earthquake of 1978,
Report No. 159, Public Work Research Institute, Ministry
of Construction, Japan
Lee, K. M., Cho, H. N., and Choi, Y. M. (2004), “Life-Cycle
Cost Effective Optimum Design of Steel Bridges,”
Journal of constructional Steel Research, Vol. 60, No. 11,
1585-1613
Lee, S. B. and Shim, J. I. (1997), “Trends and important
parameters of traffic accident costs,” Report 97-09, Korea
Traffic Development Institute (in Korean)
Leeming, M. B., Mouchel, L. G. (1993), “The application of
life cycle costing to bridges,” Bridge Management,
Thomas Telford, London, 574-583
Maunsell LTD. and Transport Research Laboratory (2000),
“Optimum Maintenance Strategies for different Bridge
Type: Bridge Data,” Final Report, The Highways
Agency, London
Melchers, R. E. (1987), “Structural Reliability, Analysis and
Prediction,” Ellis Horwood Ltd., West Sussex, England
Ministry of Science and Technology (2004), “The standard
engineering business cost,” Final Report
Nowak, A.S. (1993), “Calibration of LRFD Bridge Design
Code,” National Cooperative Highway Research
Program, Final Report 12-33
Office of Construction and Facilities Management (2002),
“Data for Design Guide to Maintenance and Repair of
Infrastructure,” Final Report
Piringer, S. (1993), “Whole-Life Costing of Steel Bridges,”
Bridge Management, Thomas Telford, London, U.K.,
584-593
Rackwitz, R. and Fiessler, B. (1978), “Structural reliability
under combined random load sequences,” Computers and
Structures, Pergamon Press, 9: 489-494
Research institute of Industrial Science and Technology
(1998), “Guidelines for the Use of Uncoated Weathering
Steel Bridge,” Final Report
Seskin, S. N (1990), “Comprehensive framework for
highway economic impact assessment methods and
result,” Transportation Research Record 1274,
Transportation Research Board, Washington, D.C., 24-34
Stewart M.G. (2001b), “Reliability-based assessment of
ageing bridges using risk ranking and life cycle cost
decision analyses,” Reliability Engineering and System
Safety Vol. 77, pp 263-273
Vanderplaats, G. N. (1986), ADS: A FORTRAN Program for
Automated Design Synthesis, Engineering Design
Optimization, Inc, Santa Barbara, California
Wen, Y. K. and Kang, Y. K. (1997), “Optimal seismic design
based on life-cycle cost,” Proc. of the International
Workshop on Optimal Performance of Civil Infrastructure
Systems, ASCE, Portland, Oregon, 194-210.
Zokaiw, T., Imbsen, R. and Osterkamp, T. (1991),
“Distribution of Wheel Loads on Highway Bridges,”
Transportation Research Record 1290, Vol. 1, 3rd Bridge
Engineering Conference, Denver CO, March 10-13,
Transportation Research Board, Washington D.C.
