fatigue reliability reassessment procedures: state-of-the-art paper
By William G. Byers,1 Fellow, ASCE, Mark J. Marley,2 Member, ASCE,
Jamshid Mohammadi,3 Member, ASCE, Richard J. Nielsen,4 Member, ASCE,
and Shahram Sarkani,5 Member, ASCE
abstract: The need for reassessment of the fatigue life of existing structures is increasing as the
world's infrastructure ages. A fatigue life reassessment typically begins with an assessment of the
current condition of the structure. The condition assessment techniques range from visual
inspection to X-ray inspection or detection of acoustic emissions. The fatigue reliability of the
structure can be estimated from probabilistic fatigue life or fracture mechanics models. The data
obtained from the condition assessment can be combined with these models to estimate the
remaining service life of a structure using Bayes' theorem. Simulation techniques are often used
to facilitate these calculations. If the remaining service life is inadequate, it may be desirable to
repair the structure; however, repairs must be performed carefully to provide the desired benefit.
On the other hand, economic factors may dictate a course of action other than repair, such as
replacing the structure or changing the operation of the structure.
INTRODUCTION
In recent years, the American engineering community has paid a great deal of attention to the
deterioration of the nation's infrastructure and the necessity of developing means for adapting to
its consequences. The first problem encountered is to determine the actual condition of existing,
structures so those that are deficient can be identified and recommended for further examination.
The obvious remedy for such structures is to replace them with new ones incorporating the latest
methods for avoiding the modes of failure responsible for the inadequacy of the existing
structures. However, this is often extremely expensive, especially when the problem of losing the
current structure during construction of the new one is considered. Hence, a preferable solution
may be to repair and/or rehabilitate the older structure. Obviously, economic considerations must
be meshed with technical ones when confronting such situations.
One of the more difficult phenomena to assess is the fatigue effect on structures subjected to
repeated or cyclic load patterns. The precise mechanisms most responsible for fatigue damage
have proven elusive, and the most accepted deterministic methods for predicting such
characteristics as fatigue life remain very approximate, at best. For example, the relationship
between stress range and fatigue life for a given material, which is assumed to be a fixed
property, in fact exhibits considerable variation (Fuchs and Stephens 1980; Dowling 1994). Also,
damage accumulation under irregular amplitude stress ranges has been shown through
experiments to be difficult to predict (Sarkani and Lutes 1988; Sarkani 1990). To make matters
worse, the debilitating effects of common processes such
'Dir., Structures Construction, A. T. & S. F. Railway, 4515 Kansas Ave., Kansas City, KS 66106.
Prin. Engr., Offshore Design, A. S., Billingstadletta 18, N-1361 Bil-lin|stad, Norway.
Prof., Dept. of Civ. and Arch. Engrg., Illinois Inst. of Techno!., Chicago, IL 60616.
"Assoc. Prof., Dept. of Civ. Engrg., Univ. of Idaho, Moscow, ID 83844.1022.
Prof., Dept. of Civ. and Envir. Engrg., The George Washington Univ., washington, DC 20052.
Note. Associate Editor: Bilal M. Ayyub. Discussion open until August 1 '"97. Separate
discussions should be submitted for the individual pa-in this symposium. to extend the closing
date one month, a written 1 must be filed with the ASCE Manager of Journals. The manu-for this
paper was submitted for review and-possible publication January 23, 1995. This paper is part of
the Journal of Structural ' V°l 123- Na 3- March> 1997' ©ASCE, ISSN 0733-9445/ -0271 -0276/S4.00
+ $.50 per page. Paper No. 9982.
as corrosion are virtually impossible to gauge accurately in a deterministic sense. Nevertheless,
fatigue is a problem that must be taken into account when evaluating a structure's current
condition and expected remaining life.
The probabilistic concepts incorporated in fatigue reliability analysis have shown promise for
giving reasonable estimates of the fatigue damage present in and the expected remaining life of
structures (Committee 1982). The first step is to assess the current condition of the structure
through physical inspection and examination of available load and stress data, with particular
attention being paid to the damage that is most likely to result from -fatigue. From this
information, probabilistic methods can be used to obtain estimates of the adequacy of the
existing structure, the need for increased inspection in the future to prevent failure, and the
approximate remaining fatigue life based on projections of the future loads. Fatigue reliability
analysis can also be used in the design stage, to ensure that fatigue has been adequately taken
into account in the conception of a new structure.
Once analysis has provided at least some insight into the present state of a structure, a decision
must be made on a course of action on the basis of the results obtained. One possible course of
action is to repair the structure in the hope of extending its service life. Repairing a structure is
often more complicated and fraught with more pitfalls than designing a new structure. Therefore,
the engineering implications of repairs are discussed briefly.
Other courses of action may include replacing the structure or simply leaving it in operation
without modification. Choosing between these alternatives requires evaluation of the economic
costs and benefits of all the various options, with their additional uncertainties.
The purpose of this paper is to summarize the currently accepted procedures for applying fatigue
reliability analysis. A companion paper (Byers et al. 1997) relates the application of these ideas
to types of structures commonly susceptible to fatigue and fracture problems: railroad bridges,
highway bridges, and offshore structures. Extensive reference will be made to the existing
literature for further information.
CONDITION ASSESSMENT
The fatigue reliability reassessment of a structure typically begins with an assessment of its
current condition. The current condition is a function of the structure's service history. As
described in the companion paper (Byers et al. 1997), the service history can be determined from
a variety of sources depending on the type of structure. Operating records may be available for
offshore structures or railroads; highway bridges may have to rely on traffic counts or weigh-inmotion studies.
The nature of fatigue processes and uncertainties associated , with the prediction of future loads
and the estimation of load histories also require field inspection as a necessary tool for damage
detection and prevention. Inspection may only involve the visual examination of structural
components or may be quite complicated, involving the use of a variety of nondestructive tests
(NDT). When visual inspections without NDT techniques are used, the effectiveness of the
inspection program primarily depends on the inspector's experience and the type of damage
observed in generic classes of structures inspected. In cases where NDT techniques are used, the
effectiveness of the inspection process, to a great extent, depends on the reliability of the selected
technique in damage detection. This reliability is often presented in the form of "probability of
detection" curves or "defect detection" probability values. Inspection results and NDT data are
often used along with structural analyses to determine fatigue damage and fatigue growth in
structural components.
After a repair or rehabilitation has been performed, a rein-spection program is necessary for the
purpose of evaluating the effectiveness of the repair, rehabilitation, and/or crack control method
used. The reinspection closely follows a routine inspection program. However, certain issues
mainly related to the changes in the structure's geometry and perhaps load population are
important and need to be considered in reinspection. At any stage the results of an inspection can
be used as a means to update fatigue reliability and life prediction parameters through an
updating process such as the Bayesian approach.
Several NDT techniques have been implemented in conjunction with the inspection of fatiguecritical structures. The following is a brief description of these methods. The sensitivity of each
of these methods depends on the specific equipment and operators. Therefore, their ability to
detect cracks is described qualitatively instead of providing specific probabil-ity-of-detection
curves for each. In fact, only a few of these methods have been successfully used in field
applications.
Radiographic Inspection
This method is mainly used for the inspection of weldments and detection of weld flaws by
determining porosity, slag, and a lack of fusion penetration. The procedure involves radiating a
weld with X- or gamma-rays and exposing a film placed on the opposite site of the weld. The
amount of radiation reaching the film depends on the amount absorbed by the weld. The method
is generally slow and expensive (Lai 1977).
Electric Inspection Method
This method is mainly used to detect active corrosion and cracks. The rate of corrosion and the
depth of cracks can be detected. The application is primarily in cases where good surface contact
and probe spacing can be achieved (Blitz et al. 1969; Vary 1973).
Dynamic Testing Method
This method is based on evaluation of the dynamic response of the structure under an externally
applied impact or dynamic load. The frequency response of the structure (signature) is examined.
Any change in the response is used as a means to detect damage. The method has been used in
off-shore structures. Recently, it was also applied to bridges [see, for example, Rehm et al.
(1987), Davis (1987), Mazurek and DeWolf (1990), Rubin (1980), Salane and Baldwin (1990)].
In
sured D = 1. This formulation has the advantage of simplicity, but the damage measure D is not
related to a direct physical quantity such as crack length, and it ignores sequence effects.
Theoretically stresses of all ranges can occur in the Palm-gren-Miner damage rule [(2)], and the
random stress range 5 can be described by the probability-density function (PDF) f(s). If пт
cycles occur in time T, then the fraction of those cycles having stress range 5 is nTf(s) ds and the
increment of damage caused by this stress range is then
(3)
Therefore, in the limit the total damage from (2) can be converted to an integral
Reassessment Using Bayesian Updating
The fatigue reassessment process combines information from the inspection process and the
fracture mechanics model to update the probability of failure at the end of the service life. The
updating procedure is based on Bayes' theorem, and requires a probability distribution for the
crack size prior to the inspection/i(a), which can be derived from the original design data and the
random fatigue crack growth model described earlier [(6)]. Also required is the probability of
detection curve for the inspection procedure to be used, P{d\a]. The probability of detection can
be assumed to be conditional on the crack size a. The Bayesian posterior provides the probability
distribution" for the crack size a at the time of rein-spection given that no crack is detected
during the inspection
(4)
Further evaluation of the integral in (4) is possible upon selection of a specific distribution for
PDF f(s) as illustrated in the companion paper (Byers et al. 1997).
Madsen (1984) assumes the stress ranges are Weibull distributed when evaluating (4), and treats
the damage at failure D, as a random variable. Using a limit state that compares the damage
measure from (4) to the random damage at failure Df, Madsen then calculates P{DT a: Df) the
probability of failure in time Т using first-order reliability methods.
With this formulation, it is possible to use Bayesian updating to determine the probability of
failure at the end of an extended service life T2, given that it has not failed at the time of the
reassessment T, (Madsen 19841
(5)
This formulation assumes that the probability of failure increases monotonically over time.
Fracture Mechanics Overview
Linear-elastic fracture mechanics relate the growth of a crack of size a to the number of fatigue
cycles N. The most common relationship is the Paris fatigue crack growth law
(6)
(9)
where P{d\a] = 1 — P{d\a] = probability of not detecting a crack given crack size a.
The probability of failure at the end of an extended service life is equivalent to the probability
that the crack size at this time a-i exceeds the critical crack size ac. If a crack was detected during
the inspection and was determined to be of size a,, the probability of failure is then
(10)
where/((a2|ai) = probability distribution for crack size at the end of the extended service life
given the crack size at inspection was a\. The distribution f^(a2 \ qi) is determined from the
random fatigue crack growth model [(6)].
If no crack was detected during the inspection, the probability of failure is
(H)
where/i(a2|<2) = probability distribution for crack size at the end of the extended service life
given that no crack was found during the inspection. This distribution can also be found from the
random fatigue crack growth model, and the updated probability distribution for the crack size
after inspection is from (9)
where C2 and m2 = material parameters. The range of the stress intensity factor ДАТ is
(7)
where Y(a) = a function of the crack geometry (Broek 1986). Failure is assumed to occur when
the crack size reaches some critical crack size ac. Although most laboratory testing is typically
performed with constant amplitude stress ranges, (6) is often applied to variable stress range
models that ignore sequence effects. It is often advantageous to separate the variables in (6) and
integrate to find the number of cycles, N
(8)
There are a variety of sources of uncertainty in (6) or (8) (Harris 1995). A simple probabilistic
model for the fatigue crack growth model treats the material parameter C2 as a random variable
(Madsen 1983). More sophisticated models treat (6) as a stochastic differential equation and
allow C2 to vary during the crack growth process (Ortiz 1985). Other random models treat the
crack growth as a Markov process (Lin and Yang 1983) or a first-passage problem (Ditlevsen
1986).
PROFILE GUSSET
REMOVE GUSSET
with a proper understanding of material behavior and engineering principles, as improper repairs
can actually weaken the fatigue resistance of the structure.
Repair Strategies
A variety of approaches can be undertaken to repair a fatigue-damaged structure. These
approaches involve strengthening the structure, reducing or accommodating displacements,
removing crack initiators, and others.
Strengthening the member or connection: The engineer must provide an adequate continuous
load path from load to support. Weak links and stress concentrations must be avoided. Stresses
should be transferred to members that are materially and geometrically oriented to adequately
resist the load. For example, lateral bracing elements should cause bending about the strong axis
of the supporting member rather than out-of-plane bending.
In some cases, additional weld material can be provided to strengthen a connection; however, it
can be difficult to achieve satisfactory results, especially in offshore structures. Weld repairs
close to the splash zone have been completed using air welding, sometimes with the aid of
cofferdams (Sharp 1993). Wet welding is performed with stick electrodes in direct contact with
seawater. Repair by wet welding has been used extensively in the Gulf of Mexico but, to date,
only rarely in the North Sea. These welds are subject to rapid cooling and hydrogen uptake,
which has often led to low standard welds, however progress is being made on improving weld
quality. Habitat welding involves welding at hyperbaric pressure from within an underwater dry
habitat. The technique yields good quality welds but preparation is costly.
Reducing or accommodating displacements: In situations where the cracks are caused by
unanticipated secondary displacements, it may be advantageous to reduce the stiffness of the
resisting member to allow it to displace without becoming overstressed. A common example
would be holes that are drilled through the tip of the crack to control crack growth, as shown in
Fig. 1 (Sweeney 1978).
SECTION A-A
FIG. 1. Drilling Holes to Control Fatigue Crack Growth
In some cases crack growth may be slowed by proper maintenance and operation of the structure.
In bridges, periodic
FIG. 2. Reducing Displacements by Removing or Profiling Gusset Plates
maintenance and lubrication of pins and rocker bearings will ensure their proper operation,
reducing secondary displacements elsewhere. In offshore structures, wave loading can be
reduced through removal of appurtenances (boat bumpers and landings), reduction of the number
of conductors, and removal of marine growth (Haagensen 1994).
In some cases it may be possible to reconfigure the connection to reduce the displacements
causing the crack. The possibilities are limited by the structural configuration, but subject to this
constraint, it may be possible to: (1) remove or relocate secondary bracing or gusset plates; or (2)
profile gusset plates to distribute stresses along the connection instead of concentrating them at
the corner of the gusset (see Fig. 2).
Removing crack initiators or propagators: Crack initiators/ propagators include microcracks and
inclusions due to manufacture and fabrication. Removing these may be particularly successful in
welded connections that are both sensitive and prone to imperfections. The imperfections can be
removed by the following methods:
1. Grinding the surface of welds in the connection—the removal of part-through-thickness
fatigue cracks by grinding is often less expensive than clamping or welding. The grinding needs
to be carefully controlled: the site is fatigue critical and the local stress-concentration factor can
be very sensitive to the shape of the final groove. It is important for the groove to be deep
enough for complete removal of the crack, otherwise the crack will rapidly reinitiate. This
method was the least successful in the testing series performed by Fisher et al. (1978).
2. Peening the surface—this will remove or close only very small cracks. As discussed later,
peening may have additional benefits beyond the removal of crack initiators.
3. Gas-arc remelting of the welds in the connection.
Creating compressive residual stresses: Compressive residual stresses in the surface of the weld
can significantly lengthen the initiation and propagation phases of the crack growth. This is
helpful only when the connection has a compressive mean stress and a smaller stress range.
Otherwise, residual compressive stresses created by the repair are overshadowed by the tensile
stresses caused by the load. Compressive residual stresses are usually created by methods 2 and 3
of the previous item. The benefits of gas-arc remelting combine to make this the most effective
weld repair in tests reported by Fisher et al. (1978).
Improper Repairs
Fatigue problems can be caused or aggravated by improper repair procedures. Several cases of
fatigue crack propagation related to welded repairs are described by Byers (1988). The
mechanisms involved out-of-plane bending, high tensile residual stresses, and weld defects.
Some of these mechanisms could be activated without welding.
As indicated, additional stress paths are often created to increase the redundancy of the structural
configuration. Improved redundancy can appreciably improve reliability (Byers 1976; Sweeney
1979; Thoft-Christensen and Morutsu 1986). However, redundancy can be lost if the elements
are welded together since cracks can propagate across the weld.
Other conditions that should be avoided in the design and execution of repairs include: (1)
sudden changes in member stiffness; (2) large residual stresses due to weld cooling or forced
distortion; (3) cracks or crack-like defects in welds; (4) damage caused by excessive drifting in
fitting up bolted connections [e.g., Wyly and Scott (1956)], and (5) out-of-plane bending or shear
caused by deformation resulting from secondary restraint or misfit.
ECONOMIC AND OTHER CONSIDERATIONS
There are a variety of economic factors that may influence the course of action resulting from a
fatigue reliability analysis. Given the complexities of an economic analysis, the factors discussed
in the following paragraphs are quite general. More detailed information can only be provided on
a case-by-case basis. The general outline for such an economic analysis is given by Bea and
Smith (1987). Important decision factors include the following:
Frequency and Extent of Inspections
Any repair scheme suggested by a fatigue reliability reassessment will have an associated
expected fatigue life. Inspection schedules are then based on the expected life of the repair.
Unfortunately, inspections are often very expensive, both on bridges and in offshore structures.
In addition, it is often very difficult to inspect all surfaces of a structure and to locate-, all cracks
on the accessible surfaces. For these reasons it may not be possible to justify the recurring
inspection costs associated with a given repair scheme. In some cases, these factors have lead to
a decision not to repair a structure (Bea et al. 1985).
Chance of Success of Repair
This is an important factor in determining the expected cost or benefit of a rehabilitation effort.
However, only a limited amount of research has been performed on repaired members and
structures. At this point, there is a great deal of uncertainty in reliability analyses of repaired
structures.
Ancillary Costs and Risks of Repairs
Due to the risks of operations during repairs, it is often necessary to take the structure or facility
out of operation while repairs are effected. Even with such precautions the structure and
personnel are exposed to higher risks during repair. There are examples of structures damaged by
crane maneuvers or people being injured while structures are being repaired (Bea et al. 1985). As
mentioned, structural rehabilitation may also increase the uncertainty about the structural
capacity, which in turn would increase the risks of operation over the structure's lifetime.
Strategies Other Than Repair
Economically, it may be preferable to do something besides pair the structure. Other strategies
include the following:
Do Nothing
Depending on a variety of structural and economic factors, it may be advisable not to attempt
any repairs. For example, in offshore structures underwater repair is extremely difficult and
costly. Therefore, a detailed assessment of the criticality of fatigue cracks identified during
inspection is performed before repair is undertaken. If the damage does not represent an
immediate risk to the structure's integrity, the best course of action may be simply to monitor the
crack (e.g., by annual nondestructive evaluation) to determine the rate of growth, if any.
In some cases, various "political" pressures push to undertake some sort of repair. A do-nothing
approach could therefore be difficult to accept. The decision to do nothing will decidedly affect
future operation plans. The acceptance criteria for existing structures may be less stringent than
for new structures; however, a decision not to repair or replace a structure may necessitate a
reduction in its anticipated service life or a reduction in the exposure. Even when these
limitations are included in the economic analysis, the difficulties and risks of undertaking a
repair may lead to a decision to leave the structure as is (Bea and Smith 1987).
Replace the Structure
The high costs of repairing some structures in place combined with the economic advantages of
retaining a fully functioning structure may require the complete replacement of a fatiguedamaged structure. The risk of injury to personnel •'referred to in the treatment of repairs also
exists during replacement.
Reduce Exposure
Changes in operation may significantly reduce exposure to the hazards of a fatigue-damaged
structure. The structure may be used less frequently, fewer personnel may be required to function
on the structure, additional safety measures may be implemented, and inventory and machinery
may be removed from the structure (Bea et al. 1985). In the case of bridges, a future stress range
may be reduced by restricting the weight or speed of vehicles using the bridge. An example of
changes in operation influenced by a fatigue reliability analysis is given in the section on railroad
bridges in the companion paper (Byers et al. 1997).
SUMMARY
The techniques of fatigue and fracture reliability analysis are important tools for evaluating the
condition of existing structures and possibly extending their design lifetimes through
reassessment and rehabilitation.
The reassessment begins with an evaluation of the current condition of the structure and its
service history. Visual inspections along with nondestructive testing can be performed to locate
existing cracks, if any, and determine their size and possibly their growth rate.
Once the condition of the structure has been determined, the remaining service life is evaluated
using either fatigue life methods or probabilistic fracture mechanics. With both methods, it is
possible to incorporate the information from the condition assessment using Bayes' theorem to
update the probability of failure at the end of an extended service life. Since the fracture
mechanics model is based on a physical measure of damage, i.e., crack size, more information
can be incorporated into the updated estimate of the reliability.
If the updated fatigue life or fatigue reliability is not adequate, it may be advisable to repair the
structure. Repairs must be carefully planned and executed in order to improve the fatigue life of
the structure and avoid causing additional problems. Repair strategies include strengthening
connections, changing the connection to avoid or accommodate displacements, removing crack
initiators or propagators, and creating residual stress in a member. Benefits of repairs must be
weighed against those of complete replacement of the structure or leaving the structure as is.
Economic considerations must be weighed in the decision to rehabilitate a structure.
A fatigue reliability approach is very useful in the engineering analysis of the structure, the
evaluation of rehabilitation schemes, and the future operation of the rehabilitated structure.
ACKNOWLEDGMENTS
This paper was conceived and completed by members of the ASCE Subcommittee on Fatigue
and Fracture Reliability. The writers, listed in alphabetical order, would like to acknowledge the
contribution of the previous members of the committee, especially B. F. Spencer and K. Ortiz.
APPENDIX I. REFERENCES
Barsom, J. M., and Rolfe, S. T. (1987). Fracture and fatigue control of structures. Prentice-Hall,
Inc., Englewood Cliffs, N.J.
Bea, R. G., and Smith, С. E. (1987). "AIM (assessment, inspection, maintenance) and reliability
of offshore platforms." Proc., Marine Struct. Reliability Symp., Soc. of Naval Archil, and Marine
Engrs., Jersey City, N.J., 57-74.
Bea, R. G., Litton, R. W., and Vaish, A. K. (1985). "Requalification of existing platforms—OTC
4858." Proc., 17th Annu. Offshore Technol. Con/, Offshore Technol. Conf., Houston, Тех., 163170.
Blitz, J. W. G., and Rogers, D. G. (1969). Electric, magnetic, and visual methods of testing
materials. Butterworth's, London, England.
Broek, D. (1986). Elementary engineering fracture mechanics, 4th Ed., Martinus Nijhoff
Publishers, Dordrecht, The Netherlands.
Byers, W. G. (1976). "Rating and reliability of railway bridges." Methods of structural analysis,
W. E. Saul and A. H. Peyrot, eds., ASCE, New York, N.Y., 2285-2300.
Byers, W. G. (1988). "Bridge repairs—unexpected effects on reliability." Probabilistic methods
in civil engineering, P. D. Spanos, ed., ASCE, New York, N.Y., 305-308.
Byers, W. G., Marley, M. J., Mohammadi, J., Nielsen, R. J., and Sarkani, S. (1977). "Fatigue
reliability reassessment applications: state-of-the-art paper." J. Struct. Engrg., ASCE, 123(3),
277-285.
Collacott, R. A. (1985). Structural integrity monitoring. Chapman and Hall, New York, N.Y.
Committee of Fatigue and Fracture Reliability of the Committee on Structural Safety and
Reliability of the Structural Division. (1982). "Fatigue reliability: introduction." J. Struct. Div.,
ASCE, 108(1), 3-23.
Davis, A. G. (1987). "Application of dynamic test methods to the evaluation of bridge
structures." Bridge evaluation repair and rehabilitation, A. S. Nowak and E. Absi, eds., Univ. of
Michigan, Ann Arbor, Mich.
Department of Energy. (1982). "New fatigue design guidance for steel welded joints in offshore
structures." Recommendations, Her Majesty's Stationary Ofc. (HMSO), London, England.
Ditlevsen, O. (1986). "Random fatigue crack growth—a first-passage problem." Engrg. Fracture
Mech., 23, 467-477.
Dowling, N. E. (1994). Mechanical behavior of materials. Prentice-Hall, Inc., Englewood Cliffs,
N.J.
Fisher, J. W., Pense, A. W., Slockblower, R. E., and Hausammann, H. (1978). "Retrofitting
fatigue damaged bridges." Transp. Res. Rec. 664, Transp. Res. Board, Washington, D.C., 102109.
Fuchs, H. O., and Stephens, R. I. (1980). Metal fatigue in engineering. J. Wiley & Sons, Inc.,
New York, N.Y.
Green, А. Т., Dungan, H. L., and Tetelman, A. S. (1979). "Non-destructive inspection of aircraft
structures and materials via acoustic emission." Int. Adv. in Non-Destructive Testing, 6, 125-177.
Gurney, T. R. (1979). Fatigue of welded structures, 2nd Ed., Cambridge University Press, New
York, N.Y.
Haagensen, P. J. (1994). "Methods for fatigue strength improvement and repair of welded
offshore structures." Proc., 13th Int. Conf. on Offshore Mech. and Arctic Engrg., Am. Soc. of
Mech. Engrs. (ASME), New York, N.Y.
Harris, D. O. (1995). "Probabilistic fracture mechanics." Probabilistic structural mechanics
handbook, C. Sundararajan, ed.. Chapman and Hall, New York, N.Y.
Lai, D. M. (1977). "Non-destructive inspection of steel (phase 10)." Rep., California Dept. of
Transp., Sacramento, Calif.
Lin, Y. K., and Yang, J. N. (1983). "On statistical moments of fatigue
crack propagation." Engrg. Fracture Mech., 18(2), 243-256. Madsen, H. O. (1983).
"Probabilistic and deterministic models for predicting damage accumulation due to time varying
loading." DIALOG 5-82, Danish Engrg. Acad., Lyngby, Denmark. Madsen, H. O. (1984).
"Bayesian fatigue life prediction." Probabilistic methods in the mechanics of solids and
structures, S. Eggwertz and N. C. Lind, eds., Springer-Verlag KG, Berlin, Germany. Madsen, H.
O., Krenk, S., and Lind, N. C. (1986). Methods of structural
safety. Prentice-Hall, Inc., Englewood Cliffs, N.J. Mazurek, D. F., and DeWolf, J. T. (1990).
Experimental study on bridge
monitoring technique." J. Struct. Engrg., 116(9), 2532-2549. Melchers, R. E. (1986). Structural
reliability. John Wiley & Sons, Inc.,
New York, N.Y. Oritz, K. (1985). "On the stochastic modelling of fatigue crack growth,"
PhD dissertation, Stanford Univ., Stanford, Calif. Raju, S. K., Moses, F, and Schilling, C. G.
(1990). "Reliability calibration of fatigue evaluation and design procedures." J. Struct. Engrg.,
ASCE, 116(5), 1356-1369.
Rehm, G., Luz, E., and Bidmon, W. (1987). "Non-destructive test for
early damage detection." Bridge evaluation repair and rehabilitation,
A. S. Nowak and E. Absi, eds. Univ. of Michigan, Ann Arbor, Mich.
Rubin, S. (1980). "Ambient vibration survey of offshore platform." J.
Engrg. Mech. Div., ASCE, 106(6), 425-442.
Salane, H. J., and Baldwin, J. W. (1990). "Identification of modal properties of bridges." J.
Struct. Engrg., ASCE, 116(7), 2008-2021. Sarkani, S. (1990). "Influence of high frequency
components on fatigue
of welded joints." Int. J. of Fatigue, 12(2), 115-120. Sarkani, S., and Lutes, L. D. (1988).
"Fatigue experiments for welded joints under pseudo-narrowband loads." /. Struct. Engrg.,
ASCE, 114(8), 1901-1916.
Sharp, J. V. (1993). "Strengthening and structural repair of ageing North Sea platforms: a
review." Proc., 12th Int. Conf. on Offshore Mech. and Arctic Engrg., Am. Soc. of Mech. Engrgs.
(ASME), New York, N.Y. Sweeney, R. A. P. (1978). "Some examples of detection and repair of
fatigue damage in railway bridge members." Transp. Res. Rec. 676, Transp. Res. Board,
Washington, D.C., 8-14.
Sweeney, R. A. P. (1979). "Importance of bridge redundancy in bridge fracture control." Transp.
Res. Rec. 711, Transp. Res. Board, Washington, D.C., 23-30. Thoft-Christensen, P., and
Morutsu, Y. (1986). Application of structural
systems reliability theory. Spring-Verlag, New York, N.Y. Torng, T. Y, and Wirsching, P. H.
(1991). "Fatigue and fracture reliability and maintainability process." J. Struct. Engrg., ASCE,
117(12), 3804-3822.
Vary, A. (1973). Non-destructive evaluation technique guide. Nat. Aeronautics and Space
Admin., Lewis Res. Ctr., NASA Sp 3079, Cleveland, Ohio.
Wyly, L. Т., and Scott, M. B. (1956). "An investigation of fatigue failures in structural members
of ore bridges under service loadings." Proc., Am. Railway Engrg. Assn., Vol. 57, Am. Railway
Engrg. Assn., Chicago, 111., 175-297.
APPENDIX II. NOTATION
The following symbols are used in this paper:
a = crack size; C\ = fatigue life constant; C2 = fracture mechanics constant; D = damage; d =
detection of crack; /( ) = probability density function; mi = fatigue life exponent; тг = fracture
mechanics exponent; N = fatigue life to failure; n = number of fatigue cycles; P( ) = probability;
5, s = stress range; Y( ) = crack geometry function; and ДАТ = stress intensity factor.
Subscripts
с = critical; / = failure;
i = stress range identifier; and T = time.
fatigue reliability reassessment applications: state-of-the-art paper
By William G. Byers,1 Fellow, ASCE, Mark J. Marley,2 Member, ASCE,
Jamshid Mohammadi,3 Member, ASCE, Richard J. Nielsen,4 Member, ASCE,
and Shahram Sarkani,5 Member, ASCE
abstract: The problem of assessing the damage to and remaining life of structures
subjected to fatigue is explored, with particular emphasis on railroad bridges, highway
bridges, and offshore structures. In railroad structures, the fatigue reliability estimates are
typically calculated based on fatigue life predictions. The resulting predictions are used for
budgeting purposes rather than for scheduling repairs. Examples are given of economic
decision based on fatigue life calculations. Fatigue reliability estimates for highway bridges
are also typically based on fatigue life calculations. Target reliability values have been
suggested and are used in both the design of new bridges and the evaluation of existing
bridges. Given the economics of offshore structures, it is feasible to develop more elaborate
fracture mechanics models. These models have the advantage of explicitly allowing updated
estimates of fatigue reliability based on inspection results. The economics of inspection of
offshore structures also dictates that inspection scheduling be based on fatigue realiability
estimates.
INTRODUCTION
As the world's industrial and transportation structures have aged, the need for
rehabilitation, repair, and replacement of these structures has increased. As a result of this
need, a variety of tools for reassessing the fatigue reliability of structures has emerged
(Byers et al. 1997). Obviously fatigue reliability reassessment is most commonly performed
for structures subjected to fatigue loadings, for example, railroad bridges, highway
bridges, and offshore structures.
However, the application of fatigue reliability reassessment procedures can vary
considerably depending on the type of structure and its loading. For lower-risk, lessvaluable structures, simpler methods are more appropriate; for higher risk or
economically valuable structures, more detailed procedures are justified. Therefore, this
paper examines the currently accepted procedures for applying fatigue reliability analysis
to the aforementioned types of structures—railroad bridges, highway bridges, and offshore
structures. In all cases, extensive reference will be made to the existing literature for
further information.
RAILROAD BRIDGES
Although railroad bridge collapse due to apparent fatigue has been reported (Gasparini
and Fields 1993), most fatigue failures in railroad bridge members do not result in failure
of the bridge because of designed or unintended redundancy (Sweeny 1979). However,
fatigue cracking may significantly reduce capacity and safety. It has major economic
importance due to inspection and repair costs and reduced service life.
'Dir., Structures Construction, A. T. & S. F, Railway, 4515 Kansas Ave., Kansas City, KS
66106.
JPrin.
Engr., Offshore Design, A.S., Billingstadletta 18, N-1361 Bil-""istad, Norway.
Prof., Dept. of Civ. and Arch. Engrg., Illinois Inst. of Technol., Chi-10, IL 60616.
4Assoc.
Prof., Dept. of Civ. Engrg., Univ. of Idaho, Moscow, ID 83844-1022.
; Prof., Dept. of Civ. and Envir. Engrg., The George Washington Univ., Washington, DC
20052.
Note. Associate Editor: Bilal M. Ayyub. Discussion open until August
'• '997. Separate discussions should be submitted for the individual paP^s in this symposium. To extend the closing date one month, a written
fcquest must be filed with the ASCE Manager of Journals. The manu| *cnpt for mjs paper was submitted for review and possible publication
t n January 23, 1995. This paper is part of the Journal of Structural
'' SJ?1'11""11*. V°l 123- No- 3- March- 1997' ©ASCE- ISSN 0733-9445/ "0003-0277-0285/S4.00 + $.50
per page. Paper No. 12948.
Fatigue crack development depends on the local stress range and number of stress cycles.
Some railroad bridges with large replacement costs have been the subjects of extensive
fatigue investigations, including acoustic emission monitoring of critical members. A more
typical management system involves annual visual inspection and a response, when a crack
is found, that depends on the size and location of the crack. In secondary members and at
details of a type having a history of slowly growing or self-limiting cracks, cracks less than
100 mm in length are typically addressed by drilling of the crack tips and continued
regular observation. At other locations, including fracture-critical members, appropriate
repairs are made upon discovery of a crack.
Fatigue Crack Development
FIG. 1. Stringer-to-Floor Beam Connections: (a) Interior Floor Beam; (b) Exterior Floor Beam
The local stress range is determined by the member live load stress range due to traffic and
other operating loads (Jahren and Rooker 1992) and local stress concentration. Most
fatigue cracks are initiated at sharp notches, severely corroded areas, corrosion pits, and
crack-like defects including undercut welds or locations where member stiffness changes
significantly, deformation produces out-of-plane loading, or locally high tensile residual
stresses exist (Albrecht 1982b; AREA 1940; Byers 1979, 1988; Sweeny 1979). With the
exception of accidental damage and some cases of corrosion, these conditions are usually
produced by improper detailing, modifications during fabrication and erection, or repairs
(AREA 1940; Byers 1979, 1988; Shi et al. 1982; Wyly and Scott 1956). In addition to the
loading covered by design assumptions, significant stresses can be produced by
deformation of connected
(12)
Similar to (8), the updated reliabilities in (10) and (11) are often formulated in terms of the
number of cycles to reach the critical crack size (Byers et al. 1997).
Simulation
Given the complexity of the fatigue life model, (4) and (5), and fracture mechanics formulation,
(6)-(12), it may be very difficult to determine the reliability analytically. In many situations, it
may be necessary to determine the reliability through simulation (Ortiz 1985). The efficiency of
the simulation process can be greatly enhanced by importance sampling and other strategies
(Torng and Wirsching 1991).
FATIGUE STRENGTH IMPROVEMENT AND REPAIR
If the remaining fatigue life of the structure is inadequate, as determined by the fatigue
reassessment, it may be advisable to repair a structure or otherwise improve its fatigue strength
rather than replace it. However, the repairs must be undertaken
FIG. 2. Stringer Web Cracks
members. Elongation of through-truss floor systems caused by lower chord deformation
has caused fatigue failures at stringer-to-floor-beam connections (Fig. 1). Out-of-plane
loading from flange rotation caused by deck timber deflection above welded bearing
stiffeners that stop a small distance below the top flange has produced web cracks in steel
stringers (Fig. 2) (Byers 1979). These are typical of loading conditions that are likely to be
overlooked by designers.
The number of stress cycles that can be applied before a fatigue crack reaches a critical size
depends on the stress range per cycle. Since the axle loads on railroad bridges due to empty
and loaded cars can differ by a factor of 4 or more, the number of cycles at various load
levels must be considered. The number of cycles at a stress range will depend on the length
of the influence line for the member and the axle spacing as well as on the axle loads and
the volume of traffic (Brustle and Prucz 1992; Grundy and Chitty 1990).
Fatigue Effect Prediction
Although prediction of future fatigue behavior, including remaining life, would have a
number of practical applications and has been successful in a few cases (Dvorak and
Zimmer 1982; Marek 1982), there are several factors that severely limit the accuracy and
use of such predictions. The large coefficients of variation, up to 40%, of the stress rangefatigue life (S-N) data that provide the basis for such predictions (Albrecht 1983; Albrecht
et al. 1982) reduce confidence in fatigue life estimates. Both past and future loading for
most bridges is uncertain. Determination of appropriate fatigue category for members
subject to corrosion (Albrecht 1982b) and some other members can be difficult. These
considerations limit the usefulness of fatigue effect prediction to the identification of
bridges requiring increased frequency of inspection for fatigue cracks and general
predictions of the probable mortality distribution for a population of similar bridges. The
predicted mortality distribution can be used for budgeting purposes but should not be used
for scheduling replacement or repairs for individual bridges.
One North American railroad with a considerable number of large, relatively old bridges
designed for light live loads uses a fracture mechanics approach to the fatigue analysis of
critical members in conjunction with acoustic emissions testing for crack growth. However,
the more generally used approach for railroad bridges is inspection and repair without a
formal analysis as the cost and time required for analysis of an existing crack are
significant compared to the cost and time usually required for repairs that can, if
appropriate, be completed within a few days after discovery of a crack by visual inspection.
The first step in predicting fatigue behavior of a member is selection of the appropriate
fatigue category and associated S-N curve for the critical detail of the member. Fatigue
categories are identified in several publications (Standard 1989;
278/JOURNAL OF STRUCTURAL ENGINEERING/MARCH 1997
AREA 1993; Fisher 1977). The critical detail may not be at • the location where the
member stress is a maximum, and the 'I stress-live load relationship for various details
must be determined. An additional consideration is comparison of predicted axle spacings
with the influence line for stress at the critical detail.
The second step in predicting future fatigue behavior is estimation of fatigue damage from
prior loading. Estimation of loading history involves both the volume of past traffic, which
can usually be obtained from railroad records, and determination of axle loads and
spacings likely to be associated with this traffic (Brustle and Prucz 1992). A method of load
history development for lines carrying a single predominate traffic type has been suggested
by Ali et al. (1992). They use production records of individual mines for estimating past
traffic for a coal-hauling railroad. With this type of traffic, it is probable that trains will
consist primarily of loaded cars in one direction and empty cars in the other direction, not
in a random mixture of loads and empties. Once the loading history has been developed,
the related damage is obtained from some cumulative damage theory. Although the
Palmgren-Miner damage rule may err in neglecting loading sequence, it is considered
appropriate for assessing fatigue damage to bridges since the uncertainty due to loading
sequence is typically less than the uncertainty due to load magnitude and resistance (Moses
et al. 1987).
The final step is a prediction of future behavior based on estimates of fatigue
characteristics and the portion of fatigue capacity exhausted by prior loading. This
requires selection of a constant amplitude stress range that can be considered equivalent to
the estimated future ensemble of variable amplitude fatigue cycles. Near-term future
loadings are reasonably predictable, but loadings in the more distant future are highly
speculative. Long-range extrapolation from present trends is likely to yield unreasonable
results and completely disregard new car design concepts, such as the articulated cars
developed in the last 25 years. An estimate of the number of future load cycles required to
produce a significant fatigue crack is obtained from estimates of the constant amplitude
stress range equivalent to future load cycles, the fraction of the fatigue life remaining after
the prior loading history, and the appropriate S-N curve for the critical detail. For most
purposes, it is then necessary to estimate the time required to, accumulate this number of
cycles.
A relatively simple application of bridge behavior is estimation of the increase in the cost of
repairing bridge fatigue damage that would result from increasing the loading from freight
cars carrying bulk commodities. Increasing the load carried per car is operating
advantages but increases maintenance costs. Estimation of the increased cost can influence
operating and rate-making decisions. The percentage increase in the cost of repairing
fatigue damage to bridges associated with increasing load per car can be developed
according to the following rationale.
The Palmgren-Miner damage rule described in the companion paper (Byers et al. 1997)
allows the separation of fatigue effects due to the commodity of interest from the effects of
other traffic. If Nt and «i correspond to cycles of interest at one load level, and N2 and n2
correspond to a second load level
(1)
As described in the companion paper, the S-N curve can be described by NS"1 = C,.
Therefore
(2)
For equal fatigue damage, п^пг = (S^S,)"1'. A generally accepted value for ml is 3 (Moses et al.
1987). However there is some evidence that m^ may equal 4 or more for riveted connections
(Sweeny 1990).
At present, the usual bulk commodity car on North American railroads has a capacity of 91 Mg
(100 t) and a loaded gross weight of 1,170 kN (263 kip). If such cars are overloaded by 10% of
their capacity to a gross weight of 1,259 kN, the fatigue damage per loaded car is increased by
25% for mi = 3 and by 34% for m, = 4. The corresponding increases for a given total volume of
lading would be 14 and 22%.
HIGHWAY BRIDGES Introductory Remarks
Although fatigue analysis of bridges is well established, the use of probabilistic methods has
only been considered in recent years. For highway bridges, the techniques of fatigue and fracture
reliability have been applied mainly to: (1) condition assessment and estimation of remaining
useful life of bridges; and (2) development of probability-based design stress ranges for fatiguecritical bridge components.
Studies on fatigue life prediction for highway bridges have utilized various approaches,
including both deterministic and probabilistic methods [e.g., Tung (1970), Moses and Garson
(1973), Cicci and Csagoly (1982), Been and Havens (1974), Woodward and Fisher (1974), Ang
and Munse (1975), Hirt (1982), Albrecht (1982a), Moses et al. (1987)]. The probabilistic method
has been considered by several authors including Wirsching and Yao (1970), Yao (1974, 1980),
Ang and Munse (1975), Moses et al. (1987), Raju et al. (1990), and Munse (1990). The
application of these methods to both condition assessment and design procedures is discussed as
follows.
Condition Assessment
A condition assessment often begins with a physical inspection of the bridge. For purposes of
analysis, it is also useful to gather truck load and component stress data. Accurate truck load data
can be acquired through weigh-in-motion (WIM) systems. An extensive amount of WIM data is
available showing distribution of load by type of truck, number of axles, and gross weight of the
truck [e.g., Cudney (1968), Sny-der et al. (1985), Mohammadi and Shah (1992)].
The truck load data must be related to component stresses at locations believed to be vulnerable
to fatigue damage. Examples of such sections include: steel girders with cover plates, girder
joints and splices, section discontinuities, and heavily loaded truss members. The relationship
between truck loads and member stresses can be obtained by analysis or by direct stress
measurements using strain gauges and advanced data acquisition systems allowing on-site data
reduction and processing. The duration of field data collection for highway bridges is often
limited to a few days. For most bridges this is adequate to obtain a reasonable estimate of stress
or load ranges experienced by a bridge. Such field data would probably not detect the high stress
levels that occur when overloaded trucks are allowed to use the bridge by special permit. The
field data described can be represented with theoretical stress range distribution models. The use
of the beta probability density function has been suggested by Walker (1978, 1980), Ang and
Munse (1975), and Mohammadi et al. (1991).
In deterministic models, the stress distribution data as compiled in field observations are directly
used to determine the fatigue life expended and the remaining life of the structural c°mponent. In
such cases, the number of stress cycles in a 8Pecific range is used along with the S-N diagram to
determine how much fatigue life has been expended for that stress
range and for the period the data was collected. Fatigue life expended for all ranges are then
added. The result represents the damage accumulated over the period the data was collected. This
information can then be used to estimate the fatigue life expended since the bridge started its
service life. In a study, Hahin et al. (1993) used this technique for a number of highway bridges
in Illinois.
Among probabilistic methods, fatigue life estimation using crack initiation is the method of
choice for highway bridges. Methods based on crack growth and fracture mechanics have also
been used, but to a more limited extent [see, e.g., Yamada and Hirt (1982)]. As described in the
companion paper (Byers et al. 1997), the probabilistic method of fatigue analysis consists of: (1)
development of stress-range distributions from field data or simulation; (2) use of the PalmgrenMiner damage rule for fatigue damage analysis along with an appropriate S-N relationship for the
critical structural details; and (3) use of a probability function to describe the reliability of a
critical component and its corresponding fatigue life.
Fatigue Damage Analysis
The Weibull distribution has been used to estimate the reliability of fatigue-critical structural
components for a desired fatigue life (Ang and Munse 1975). The reliability L(ri) of a bridge
component for a desired life of n cycles of random stress range S using the Weibull distribution
is
(3)
in which a = fi1'08; and Г() = gamma function. The parameter П is the coefficient of variation in
fatigue life. It describes the uncertainty in л due to: (1) scatter in the S-N data: (2) variability in
the stress range data; (3) variability in the constants Ct and mi; and (4) uncertainty in the damage
rule. This uncertainty is reported in Ang and Munse (1975) for several fatigue-critical structural
details along with the constants Ct and mi. The parameter П does not include the uncertainty in
the applied stress ranges. The applied stress ranges are defined via a probability density function
that includes the dispersion in the applied loads.
In (3), L(n) is the probability that there will be no fatigue failure after n stress cycles; И is the
mean number of cycles to failure and is derived as follows. Using the continuous form of the
Palmgren-Miner damage law (Byers et al. 1997), the expected damage occurs when the number
of cycles in the time span under consideration is n
(4)
in which SmK = maximum stress range occurring in the component. A fatigue crack initiates when
E[D] - 1. Thus using (4) with E[D] = 1, one obtains
(5)
The formulation presented in (3) and (5) is primarily based on the two-parameter Weibull
distribution model for fatigue life. Wirsching (1995) presents an overview of other available
reliability methods. In particular, Wirsching notes that the Weibull distribution does a poor job of
modeling the fatigue life of welded joints.
Component Reliability
In a practical application, Mohammadi et al. (1991) used (3)_(5) for 15 bridges in Illinois. The
results obtained for the
reliability of a given bridge were used to estimate its remaining useful life for a predetermined
reliability. Appropriate reliability levels were suggested by Moses et al. (1987) and are 99.9 and
97.7% for nonredundant and redundant bridges, respectively. The data on average daily traffic on
a bridge and percentage growth expected in the future permitted Moham-madi et al. (1991) to
compute the remaining fatigue life in terms of years. Information of this sort is especially useful
to bridge engineers so they can make 'necessary decisions on bridges that may require repair or
retrofit.
Moses et al. (1987) proposed a fatigue reliability model to predict the probability that the fatigue
life of steel bridge components will be less than the predicted design life. The procedure follows
the aforementioned steps. However, instead of formulating damage in terms of the stress range
probability density function, damage is described in terms of several random variables such as
bending moment, truck weight, average daily truck traffic, etc. The reliability is then formulated
as a function of the safety margin (the difference between the actual fatigue life and the
calculated life) and is the probability that this margin is greater than zero.
In the evaluation procedure developed by Moses et al. (1987) the effective stress range S, is
computed as follows:
(6)
in which / = fraction of stress range cycles within an interval i; 5,, — midwidth of stress interval
i; and the exponent ml = 3 [see (2)]. Fatigue damage caused by a given number of cycles of the
effective stress range is the same as damage caused by an equal number of different stress ranges
defined by a stress history on the component. Recommendations for the computation of fatigue
truck weight, lateral distribution of the load, and impact are provided in the study. Furthermore,
the study recommends using a reliability factor Rs when calculating the remaining safe life of a
bridge component. This factor is based on the safety index (3. Values of R, are 1.1 and 2 for
redundant and nonredundant systems, respectively. These correspond to reliability values of 97.7
and 99.9 (P = 2 and 3, respectively). The remaining safe life in years Yf is then estimated from
(7)
in which Ta = estimated lifetime average daily truck volume; С = number of cycles per truck
passage; A = age of the bridge in years; К = a constant depending on the structural detail; and/= a
factor to account for the difference between the mean and allowable S—N curve. The factor / = 1
when computing the remaining safe life, and / = 2 when computing the remaining mean life. The
product R,Sr is the factored stress range. Different values for /?, are suggested depending on
whether the structure is considered to be redundant or nonredundant. In any case, /?, = 1 if the
computation is for mean life.
The study also suggested that a more accurate estimate of remaining safe life can be obtained by
dividing the total fatigue life into two periods in which the truck volume and fatigue truck weight
remain constant. These are: (1) a past period from the time of opening of the bridge to the
present; and (2) a future period from the present to the end of fatigue life. The fatigue damage D,
that actually occurs during any calculation period is related to the damage that would have
occurred under present traffic conditions. This means that
(8)
in which Y, = length of calculation period in years; N, = actual
number of cycles for the period; N = number of cycles for the period based on present traffic
conditions; and VV, = fatigue truck weight for the period. Note that D, is measured in years and,
as such, fatigue failure occurs when D, = Y where Y = total fatigue life.
Design Methods in Reassessment and Rehabilitation
Design methods are typically formulated to guide the design of new construction. However,
these design procedures are also useful in the reassessment and rehabilitation of existing bridges.
Therefore, the role of fatigue reliability in design procedures will be examined briefly.
Highway bridges in the U.S. are designed according to the American Association of State
Highway and Transportation Officials (AASHTO) standards (Standard 1992). Welded steel
bridge girders must also meet certain requirements of the American Welding Society (AWS)
Bridge Welding Code (1988). The current AASHTO code design loading consists of two types,
namely the HS-20 and HS-15 standard loadings. In terms of fatigue, the code requires that
structural steel girders and their connections be designed for repeated load application over the
lifetime of the bridge. The standard loading for the repeated load application is the "fatigue
truck," recommended for the design of fatigue-critical bridge components. The gross weight of
the fatigue truck is 54,000 Ib. However, the use of actual data specific to the design site can also
be used as an alternative.
For inspection purposes, AASHTO (Manual 1983) prescribes a procedure for rating the live load
capacity of a bridge. Two levels of rating are described: "operative" and "inventory" ratings.
Fatigue effects are considered in the rating using the permissible fatigue strength of a critical
component. The rating provides an analytical measure for evaluating various components for
their ability to safely carry the live load.
Structural members and details that are considered to be fatigue-critical are categorized and
specified by the code. Fatigue strength of these and other critical details under various load
cycles can be found in Munse (1990). Although such information can be used to determine the
mean fatigue life of a critical structural component, it is stated that the results can be applied only
under very limited conditions. These limitations exist because the stress range is assumed to be
constant in obtaining fatigue life load cycles. One may avoid this shortcoming by describing the
stress range as a random variable, as described earlier for bridge evaluation methodologies.
However, design methods can also benefit from this concept by incorporating an accepted
reliability level to compute a permissible stress range based on a desired fatigue life.
Fatigue reliability approaches applied to design in highway bridges [e.g., Albrecht and Duerling
(1979), Yao (1974, 1980), Moses et al. (1987), Raju et al. (1990)] concern development of a
permissible stress range as described. In several other studies, the fatigue strengths of bridge
components are obtained and compared with various design categories of AASHTO [e.g., Fisher
et al. (1990)] and the AASHTO design specifications are evaluated using statistical approaches
[e.g., Tung and Kusmez (1975)].
The study by Moses et al. (1987) presents a comprehensive discussion on the fatigue evaluation
of steel bridges and a design proceudre based on an accepted reliability level (99.9% for
nonredundant and 97.7% for redundant bridges, as described earlier), truck weight, average daily
truck traffic (ADTT), ADTT growth rate, number of traffic lanes, and the expected design life of
the bridge. The reliability method used for the bridge design procedure is identical to that
described earlier for bridge evaluation. The procedure can be used to compute a nominal stress
range for a desired fatigue life or to estimate the remaining useful life of a bridge. The design
stress range computed via this procedure is multiplied by a reliability factor to provide an
acceptable probability that the actual fatigue life of the member will exceed the desired fatigue
life. If the factored stress range is less than the permissible stress range, there is a high
probability that the actual fatigue life will exceed the desired fatigue life. Otherwise, adjustments
to the bridge design are needed to increase the actual fatigue life at a lower stress range.
The study by Moses et al. (1987) indicates that bridges are mainly subject to a very large number
of relatively small stress cycles. This type of loading condition is referred to as "the region of
concern" and is shown along with the AASHTO fatigue design curves. Several example
problems are then presented to demonstrate the applicability of the proposed methods both for
bridge evaluation and design.
The study by Raju et al. (1990) is in part from Moses et al. (1987). In addition to the description
of the evaluation and design procedure mentioned, this study also calibrated the proposed
methods for redundant and nonredundant bridge components using reliability indices computed
with existing fatigue specifications. The evaluation and design methods presented by Moses et
al. (1987) and Raju et al. (1990) have been incorporated in recent AASHTO guide specifications.
The study by Fisher et al. (1990) deals with the fatigue of nonwelded bridge members. The study
examined data from more than 1,200 previous fatigue tests on full-scale riveted bridge members.
The study finds that the type of riveted detail does not significantly affect the fatigue resistance.
Compared with AASHTO, the study reports that category D is a reasonable lower bound for the
initial fatigue crack development, and that the fatigue strength of a riveted built-up member
effectively exceeds the category D resistance curve. Recommendations are also made for rating
riveted highway bridges for fatigue damage.
OFFSHORE STRUCTURES
There are approximately 7,000 platforms located on the world's continental shelves. A majority
of these are jacket structures: a steel space frame extending from the seabed to just above the sea
surface. Cylindrical tubular members are used because this cross section minimizes
hydrodynamic loads, provides a template for the foundation piles, has the same large buckling
capacity in all directions, and presents a minimum surface area subjected to corrosion. The
connections between the tubulars are welded joints; fatigue damage in jackets occurs at the
tubular joints because of high local stresses in the "hot spots" and the presence of weld defects
(see Fig. 3). The following discussion will focus on tubular joints in jackets; however, similar
principles apply in fatigue reliability assessments of other details or for other platform types.
Collapse or capsizing of mobile offshore units has occurred due to fatigue cracking, in some
cases with tragic consequences (Moan 1981; "Uninspected" 1981). However, most jacket
structures are redundant and collapse due to fatigue is unlikely; nonetheless fatigue damage may
significantly reduce capacity and safety. The cost impact may be very large; underwater
inspection and repair are expensive, and severe damage may necessitate production shutdown
until repairs are effected, or even platform abandonment.
Offshore structures are subjected to fatigue primarily due to the action of waves, on the order of
several million load cycles annually. Hence fatigue loading is a function of the environment, and
is particularly severe, e.g., in the North Sea. Typically, platforms are designed for a service life
of 20-50 years. Major existing structures in the Gulf of Mexico, critical to the Pipeline
infrastructure, date from the 1950s. The development °f the North Sea lagged development in the
Gulf of Mexico; "evertheless many significant structures in the North Sea are
FIG. 3. Tubular Joint In Offshore Structure
approaching 25 years of age. Approximately one-third of these structures are being called upon
for extended service or reuse (Bea and Craig 1993).
In-Service Experience
Historically, fatigue has been the second most common failure mode of an offshore structure's
tubular connections. With the static strength problem having been solved some 20 years ago, and
with aging structures in hostile environments, fatigue is likely to become the predominant failure
mode (Marshall 1992a). Tebbett (1988) presents a summary of damages to U.K. North Sea steel
platforms requiring repair; fatigue was the leading cause of failure, with a total of 31 cases
reported in the period 1966-86. These were major through-thickness cracks in primary structure
(i.e., components essential to the global integrity); many smaller cracks or those in secondary
structure were not included. A common fatigue problem experienced by many North Sea jackets
installed in the 1960s and 1970s was cracking at the first horizontal framing level below the
waterline due to neglect or underestimation of the vertical wave loading (Winkworth and Fisher
1992).
Analysis and Condition Assessment
The objective of the reassessment is twofold: to establish the current condition of the platform
and, on this basis, determine if there is sufficient fatigue reliability for continued operation. The
objective may be met by a combination of analysis and inspection. The analyses should proceed
in progressively more detailed levels (Marshall 1992b):
1. Screening: based on nominal design information, without analysis. Classification and
screening to select those structures to be analyzed in more detail.
2. Design level analysis: with conventional measures of acceptability. The method depends
largely on the severity of the fatigue loading: for moderate loading, a simplified procedure in
which stress concentration factors are determined by parametric equations, calculation of the hot
spot stress for the design wave, and comparison with an allowable stress for the S-N curve to
which the detail is assigned (API 1993; Luyties 1993). For harsher environments, the analysis
calculates damage rates for a sea state, with integration over all sea states and directions to
determine total fatigue damage [see, e.g., Almar-Naess (1985)].
3. Refined analysis: increase accuracy and reduce conser vatism typical in design level analysis.
May include: improved estimates of stress concentration factors by detailed finite-element
analysis; more accurate stress analysis, for example accounting for joint flexibility; and more
precise integration of long-term damage accounting for change of load path (hence stress
concentration factor) with wave height and direction. In some cases refined analysis may include
fracture mechanics or probabilistic assessments.
In any case, the first step in a fatigue reassessment of an offshore structure is information
gathering, and represents a major part of the effort. Information sources include design and
construction documentation (for original structure and modifications), inspection reports, and
maintenance and repair records. Each stage of analysis requires successively more detailed
information.
Offshore platforms are inspected on a regular schedule to identify structural damage or
degradation. API RP2A (1993) classifies underwater inspection as the following.
Level II: general visual inspection by divers or remotely operated vehicles. This may identify
severe fatigue damage, e.g., separated members.
Level III: close visual inspection of preselected areas. Requires cleaning of marine growth, and
may identify major (e.g., through-thickness) fatigue cracks.
Level IV: nondestructive testing. The most common method is magnetic particle inspection
(MPI), used to locate and determine the length of surface cracks. Detection with MPI of cracks
as short as 5 mm is reported (Winkworth and Fisher
1992); laboratory trials indicate a 90% probability of detection of defects with surface length of
50-100 mm (Barnouin et al.
1993).
Inspection reports are a critical input to the analysis, and further inspection may be required for
confidence in the condition assessment. But costs are high; typical in-service inspection
expenditures for a jacket in 100-m water depth in the North Sea is on the order of $500,000 to
$1,000,000 per year, with approximately half the costs associated with inspection for fatigue
cracks (Lotsberg and Marley 1992). Day rates for underwater inspection range from $5,000 to
$20,000 or more in the Gulf of Mexico and to as high as $50,000 in the North Sea (Hennegan et
al. 1993). Some operators opine that too much emphasis is put on finding fatigue cracks, noting
that many "cracks" found by nondestructive testing are, after more detailed and expensive
examination, found to either not be cracks at all, or to be unimportant (Dunn 1983).
Probabilistic Methods
As noted by Wirsching (1988), probabilistic methods are particularly appropriate for application
to marine structures because of the uncertainties in the ocean environment and the historical use
of statistical descriptions of that environment [compare Skjong (1995)]. For jackets, the most
common use of probabilistic methods is for inspection planning. This has recently achieved
widespread application to North Sea jackets [see, e.g., Pedersen et al. (1992) for a description of
its use on the nine jackets in the Tyra field]. Generally, reliability-based inspection planning is
based on a fracture mechanics approach to the calculation of fatigue crack growth; an overview
is given by Kirkemo (1988). An advantage of a fracture mechanics as opposed to an S—N curve
based approach is that the former admits the possibility for Bayesian updating for the fatigue
reliability based on inspection findings (Madsen 1987). These methods are outlined as follows.
Fatigue failure is defined through the limit state function g(Z), which is negative or zero at
failure. Z is the vector of basic variables describing loads, material properties, geometry
variables, statistical estimates, and model uncertainties. The safety margin is defined as M =
g(Z), and the probability of failure is
(9)
where ./j(z) = joint probability density function of Z. For fatigue of offshore structures, the major
uncertainties are related to:
1. Estimation of environmental parameters
2. Calculation of hydrodynamic loads
3. Calculation of structural response
4. Calculation of local stresses (stress concentration factors) and stress intensity factors
5. Analysis of crack growth
The Paris-Erdogan law is commonly adopted for estimating fatigue crack growth
(10)
where a = crack size; N = number of cycles; and C2 and тг = material parameters. The range of
the stress intensity factor АЛГ is a function of the stress range S and the geometry function Y(a).
Separating the variables and integrating gives
(П)
where a0 = initial crack size; and ac = crack size at failure. S"*2 is replaced by £[5™*] for
variable amplitude stress ranges. For offshore structures, it may often be assumed that the longterm distribution of stress ranges is described by a Weibull distribution; then (11) may be written
(12)
where A, and Bs = parameters of f,(s); v0 = zero upcrossing frequency; and Т = time. The safety
margin is
(13)
When the threshold intensity factor ДАГ,А and an uncertainty factor for the geometry function yy are included in the calculation, the safety margin is expressed as
where G(d) = a factor accounting for the threshold (Wirsching 1988): where Г( ) is the
incomplete gamma function.
The probability of failure in the time interval (0, 7)) is calculated as PF = P(Mi ^ 0). This is the
accumulated PF; however, target reliabilities are more rationally established on the basis of an
annual PF. Let Pf(f) denote the annualized failure rate at time t. It may be calculated from
the parametric sensitivity factor is
(15)
(16)
where (3 = reliability index; and Ф( ) = standard normal distribution function.
Reliability Updating through Inspection
New information about crack size following an in-service inspection will give additional
information about the real in-service behavior; this information may be used to update the
calculated reliability of the structure. An inspection may result in either no detection, or the
detection and measurement of a crack
(17)
In the first case, no crack was found in the inspection at time T,, implying that any existing crack
was smaller than the smallest detectable crack size ал, a random variable depending on the
inspection quality. An event margin analogous to the safety margin may be defined for the event
of not finding a crack, as follows:
(18)
A similar event margin Hj applies for the case of detection and measurement of a crack. In this
case the upper limit of integration in (18) is the measured crack size, at, generally a random
quantity due to uncertainties in the measurement.
The indirect information from the inspection is accounted for, in the reliability assessment, by
considering the conditional reliability. For one inspection with no crack detection the updated
probability of failure is [compare Byers et al. (1997)]
(19)
The conditional reliability in (19) can be calculated as the ratio between the reliability for a
parallel system and the reliability for a component. In case of crack detection and measurement
the conditional reliability is P"F = P[M ^ 0|Я, = О].
Updating fatigue reliability on the basis of the results of inspections for cracks in structural
weldments is an example of updating based on relational information. Often, updating based on
direct information can be significantly more effective (Lotsberg and Marley 1992). For example,
by direct stress measurement, the (typically large) uncertainties in the environment parameters,
hydrodynamic loading, and structural response may be greatly reduced. This implies that
updating and inspection planning should be sequential for each structure depending on the
experience with that structure and the general information available.
(16)
where (3 = reliability index; and Ф( ) = standard normal distribution function.
Reliability Updating through Inspection
New information about crack size following an in-service inspection will give additional
information about the real in-service behavior; this information may be used to update the
calculated reliability of the structure. An inspection may result in either no detection, or the
detection and measurement of a crack
(17)
In the first case, no crack was found in the inspection at time T,, implying that any existing crack
was smaller than the smallest detectable crack size ал, a random variable depending on the
inspection quality. An event margin analogous to the safety margin may be defined for the event
of not finding a crack, as follows:
(18)
A similar event margin Hj applies for the case of detection and measurement of a crack. In this
case the upper limit of integration in (18) is the measured crack size, at, generally a random
quantity due to uncertainties in the measurement.
The indirect information from the inspection is accounted for, in the reliability assessment, by
considering the conditional reliability. For one inspection with no crack detection the updated
probability of failure is [compare Byers et al. (1997)]
(19)
The conditional reliability in (19) can be calculated as the ratio between the reliability for a
parallel system and the reliability for a component. In case of crack detection and measurement
the conditional reliability is P"F = P[M ^ 0|Я, = О].
Updating fatigue reliability on the basis of the results of inspections for cracks in structural
weldments is an example of updating based on relational information. Often, updating based on
direct information can be significantly more effective (Lotsberg and Marley 1992). For example,
by direct stress measurement, the (typically large) uncertainties in the environment parameters,
hydrodynamic loading, and structural response may be greatly reduced. This implies that
updating and inspection planning should be sequential for each structure depending on the
experience with that structure and the general information available.
limit states is through a time-variant reliability model (Marley and Moan 1993). Here, the
structure's ultimate capacity is a decreasing function of time due to fatigue, and failure is de-.
fined as the first upcrossing of this threshold by the load process.
SUMMARY
The techniques of fatigue and fracture reliability analysis are important tools for evaluating the
condition of existing structures and possibly extending their design lifetimes through
reassessment and rehabilitation.
In railroad bridges, fatigue crack development is influenced by the stress range in the critical
member, the number of cycles, and secondary effects such as displacement, corrosion, and
accidental damage. Fatigue effect prediction is based on the S—N curve for the critical detail,
past load history, and the stress ranges for those loads, future load prediction, and the prediction
of remaining fatigue life. In current application, the variability of the fatigue life predictions
impede the application of fatigue reliability analysis for repair and maintenance actions on a
specific bridge, but the techniques are useful in making economic decisions regarding operations.
In highway bridges, the methods of fatigue reliability analysis have been used both to assess the
condition of existing bridges and to improve design procedures for new bridges. A condition
assessment begins by: (1) quantifying the loads experienced by the bridge members, and
determining the stress range probability distribution; (2) using the Palmgren-Miner damage rule
for fatigue damage analysis along with an appropriate S—N relationship for the critical structural
details; and (3) using a probability function to describe the reliability of a critical component and
its corresponding fatigue life.
For offshore structures, the costs of inspections and repair are substantially higher than those for
land-based structures. For this reason, the more detailed fracture mechanics models are often
used both to reassess the service life of a structure and to plan repairs, operation, and future
inspections. Since the fracture mechanics model is based on a physical measure of damage it can
be better correlated to inspection results allowing more accurate updating of fatigue reliability
estimates.
ACKNOWLEDGMENTS
This paper was conceived and completed by members of the ASCE Subcommittee on Fatigue
and Fracture Reliability. The writers, listed in alphabetical order, would like to acknowledge the
contribution of previous members of the committee who contributed to this undertaking,
especially B. F. Spencer and K. Ortiz.
APPENDIX I. REFERENCES
Albrecht, P. (1982a). "Fatigue reliability analysis of highway bridges." Rep., Dept. of Civ.
Engrg., Univ. of Maryland, College Park, Md.
Albrecht, P. (1982b). "Predicting the fatigue life of unpainted steel structures." Proc., IABSE
Colloquium Lausanne 1982, Fatigue of Steel and Concrete Structures, Int. Assn. for Bridge and
Struct. Engrg. (IABSE), Zurich, Switzerland, 337-344.
Albrecht, P. (1983). "S-N fatigue reliability analysis of highway bridges." Probabilistic fracture
mechanics and fatigue methods: applications for structural design and maintenance, ASTM STP
798, J. M. Bloom and J. C. Ekvall, eds., ASTM, West Conshohocken, Pa., 184-204.
Albrecht, P. (1986). "Review of fatigue design methods for highway bridges." Rep., Dept. of
Civ. Engrg., Univ. of Maryland, College Park, Md.
Albrecht, P., and Duerling, K. (1979). Probabilistic fatigue design of bridges for truck loading.
Dept. of Civ. Engrg., Univ. of Maryland, College Park, Md.
Albrecht, P., and Yamada, K. (1979). "Simulation of service fatigue loads for short-span
highway bridges." STP671, ASTM, West Conshohocken, Pa.
Albrecht, P., Sahli, A., and Vannoy, D. W. (1982). "Fatigue strength of
retrofitted cover plate ends." Rep. No. FHWA/MD-82/02, Nat. Tech. Information Service,
Springfield, Va.
Al AM Buyukozturk. O., and Martland. C. D. (1992). "A computer
based approach to fatigue life prediction of steel railway bridges." Res.
Rep R92-26, Massachusetts Inst. of Technol. (MIT), Cambridge, Mass.
Almar-Nzss, A. (ed.) (1985). Fatigue handbook. TAPIR, Norwegian Inst.
of Technoi., Trondheim, Norway.
American Petroleum Institute (API). (1993). Recommended practice for planning, designing and
constructing fixed offshore platforms—working stress design, API RP 2A-WSD, 20th Ed.,
Washington, D.C. American Railway Engineering Association (AREA). (1940). "Progress in
design of steel bridge details." Rep. of Comm. 15 Iron and Steel Struct.. Proc., Vol. 41,
Washington, D.C., 412-444. American Railway Engineering Association (AREA). (1993). "Steel
structures." Manual for railway engineering, Washington, D.C. Ang, A. H.-S., and Munse, W.
H. (1975). "Practical reliability basis for structural fatigue." Preprint No. 2494, Proc., Nat.
Struct. Engrg. Сол/, ASCE, New York, N.Y.
Barnouin, В., et al. (1993). "Underwater inspection reliability trials for offshore structures."
Proc., 12th Int. Conf. on Offshore Mech. and Arctic Engrg., Am. Soc. of Mech. Engrs. (ASME),
New York, N.Y. Bea, R. G., and Craig, M. J. K. (1993). "Developments in the assessment and
requalification of offshore platforms." Proc., Offshore Technol. Conf. ОГС 7138. Bridge welding
code ANSI/AASHTO/AWS Dl.5-88. (1988). Am. Welding
Soc. (AWS), Miami, Fla. Brustle, K. E., and Prucz, Z. (1992). "Fatigue of steel bridges." Proc.,
Am. Railway Engrg. Assn. 93(Bull. 736), 177-197. Byers, W. G. (1979). "Structural details and
bridge performance." J.
Struct. Oiv., ASCE, 105(7), 1393-1404.
Byers, W. G. (1988). "Bridge repairs—unexpected effects on reliability." Probabilistic methods
in civil engineering, P. D. Spanos, ed., ASCE, New York, N.Y, 305-308.
Byers, W. G., Marley, M. J., Mohammadi, J., Nielsen, R. J., and Sarkani, S. (1997). "Fatigue
reliability reassessment procedures: a state-of-the-art paper." J. Struct. Engrg., ASCE, 123(3),
271-276. Cicci, F., and Csagoly, P. F. (1982). "Assessment of the fatigue life of a steel girder
bridge." Transp. Res. Rec. 507, Transp. Res. Board, Washington, D.C. Cudney, G. R. (1968).
"Stress histories of highway bridges." J. Struct.
Div., ASCE, 94(12), 2725-2737.
Deen, R. C., and Havens, J. H. (1974). "Fatigue analysis from strain gage data and probability
analysis." Res. Rep. 411, Kentucky Dept. of Transp., Lexington, Ky.
Dunn, P. (1983). "Offshore platform inspection." The Role of Design-Inspection-Redundancy in
Marine Struct. Reliability Proc., Int. Symp. of Nat. Acad. Press, Washington, D.C.
Dvorak, I. J., and Zimmer, D. C. (1982). "Fatigue examination of existing
steel highway bridges." Proc., IABSE Colloquium Lausanne 1982, Int.
Assn. for Bridge and Struct. Engrg. (IABSE), Zurich, Switzerland,
503-510.
Fisher, J. W. (1977). Bridge fatigue guide—design and details. Am. Inst.
of Steel Constr., New York, N.Y. Fisher, J. W. (1984). Fatigue and fracture in steel bridges.
John Wiley
& Sons, Inc., New York, N.Y.
Fisher, J. W., Yen, В. Т., and Wang, D. (1990). "Fatigue strength of riveted bridge members." J.
Struct. Engrg., ASCE, 116(11), 2986-2981.
Gasparini, D. A., and Fields, M. (1993). "Collapse of Ashtabula Bridge on December 29, 1876."
J. Perf. of Constr. Fac., ASCE, 7(2), 109-125.
Grundy, P., and Chitty, G. B. (1990). "Remaining life of a suite of railway bridges." Proc.,
IABSE Workshop Lausanne 1990—Remaining Life of Steel Struct., Int. Assn. for Bridge and
Struct. Engrg. (IABSE), Zurich, Switzerland, 45-56.
Hahin, C., South, J. M., Mohammadi, J., and Polepeddi, R. K. (1993). "Accurate and rapid
determination of fatigue damage in steel bridges." J. Struct. Engrg., ASCE, 119(1), 150-168.
Hennegan, N.. Abadie, W., and Winkworth, W. (1993). "Inspections, surveys, and data
management." Proc., Int. Workshop on Reassessment and Requalification of Offshore
Production Struct., W. A. Dunlap and C. W. Ibbs, eds.. Minerals Mgmt. Service, U.S. Dept. of
Interior, Washington, D.C.
Hirt, M. (1982). "Remaining fatigue life of bridges." Proc., lABSESymp. on Maintenance, Repair
and Rehabilitation of Bridges, Int. Assn. for Bridge and Struct. Engrg. (IABSE), Zurich,
Switzerland. Jahren, С. Т., and Rooker, J. A. (1992). "Fatigue life study for a railroad bridge."
Transp. Res. Rec. 1371, Transp. Res. Board, Washington, D.C., 65-74.
Kirkemo, F. (1988). "Applications of probabilistic fracture mechanics to?'
offshore structures." Appl. Mech. Rev., 41(2).
Lotsberg, I., and Marley, M. J. (1992). "Inservice inspection planning f0r
steel offshore structures using reliability methods." Proc., 6th Int
Conf. on Behavior of Offshore Struct., BOSS'92.
Luyties, W. H. (1993). "An update on allowable fatigue stresses in API
RP2A." Proc., Offshore Tech. Conf., OTC 7154, Offshore Technol.
Conf., Houston, Тех.
Madsen, H. O. (1987). "Model updating in reliability theory." Proc., ICASP5, 5th Int. Conf. on
Applications of Statistics and Probability in Soil and Struct. Engrg., Inst. for Risk Res., Univ. of
Waterloo, Ont., ; Canada. Manual for maintenance inspection of bridges. (1983). Am. Assn. of
State Hwy. and Transp. Officials (AASHTO), Washington, D.C. Marek, P. (1982). "Prediction
of fatigue life in a steel bridge." Proc., IABSE Colloquium Lausanne 1982, Fatigue of Steel and
Concrete Struct., Int. Assn. for Bridge and Struct. Engrg. (IABSE), Zurich, Swit-zerland, 511515.
Marley, M. J., and Moan, T. (1993). "Approximate time variant analysis for fatigue." Proc.,
ICOSSAR '93, 6th Int. Conf. on Struct. Safety and Reliability, A. A. Balkema, Rotterdam, The
Netherlands. Marshall, P. W. (1992a). Design of welded tubular connections. Elsevier
Science Publishing Co., Inc., New York, N.Y.
Marshall, P. W. (1992b). "Screening old offshore platforms: previous approaches and further
thoughts." Proc., Civ. Engr. in the Oceans V, ASCE, New York, N.Y.
Moan, T. (1981). "Analysis of the fatigue failure of the Alexander Kiel-land." Proc., ASME
Winter Annu. Meeting, Am. Soc. of Mech. Engrs. (ASME), New York, N.Y.
Moan, T. (1993). "Reliability and risk analysis for design and operations planning of offshore
structures." Proc., ICOSSAR '93, 6th Int. Conf. on Struct. Safety and Reliability, A. A. Balkema,
Rotterdam, The Netherlands. Mohammadi, J., and Shah, N. (1992). "Statistical evaluation of
truck
overloads." J. Transp. Engrg., ASCE, 118(5), 651-655. Mohammadi, J., Guralnick, S. A., and
Polepeddi, R. (1991). "Effect of increased truck weight upon Illinois highway bridges." Rep. No.
FHWA/IL/RC-013, Dept. of Civ. Engrg., Illinois Inst. of Technol., Chicago, 111.
Moses, F, and Garson, R. C. (1973). "Probability theory for highway bridge fatigue stresses."
Rep. Prepared for Ohio Dept. of Transp., Dept. of Civ. Engrg., Case Western Reserve Univ.,
Cleveland, Ohio. Moses, F., and Verma, D. (1987). "Load capacity evaluation of existing
bridges." Nat. Cooperative Hwy. Res. Program, Rep. No. 301, Transp. Res. Board, Nat. Res.
Council, Washington, D.C. Moses, F, Schilling, C. G., and Raju, K. S. (1987). "Fatigue
evaluation procedures for steel bridges." Nat. Cooperative Hwy. Res. Program, Rep. No. 299,
Transp. Res. Board, Nat. Res. Council, Washington, D.C. Munse, W. H. (1990). "Fatigue, brittle
fracture and lamellar tearing." Structural engineering handbook, 3rd Ed., E. H. Gaylord Jr. and
C. N. Gaylor, eds., McGraw-Hill Book Co., Inc., 4-1-4-24. Pedersen, C., Nielsen, J. A., Riber, J.
P., Madsen, H. O., and Krenk, S. (1992). "Reliability based inspection planning for the Tyra
field." Proc., llth Int. Conf. on Offshore Mech. and Arctic Engrg., Am. Soc. of Mech. Engrs.
(ASME), New York, N.Y.
Raju, S. K., Moses, K, and Schilling, C. G. (1990). "Reliability calibration of fatigue evaluation
and design procedures." J. Struct. Engrg., ASCE, 116(5), 1356-1369.
Shi, Y., Yang, Y, and Chen, Z. (1982). "Fatigue failures of steel railway bridges in China."
Proc., IABSE Colloquium Lausanne 1982, Fatigue of Steel and Concrete Struct., Int. Assn. for
Bridge and Struct. Engrg. (IABSE), Zurich, Switzerland, 517-524.
Skjong, R. (1995). "Applications in offshore structures." Probabilistic structural mechanics
handbook, C. Sundararajan. ed.. Chapman & Hall, Ltd., New York. N.Y.
Snyder, R. E., Linkins, G. E., and Moses, F. (1985). "Loading spectrum
experienced by bridge structures in the United States." Rep. No.
FHWA/RD-85/102, Bridge Weighing Systems, Inc., Warrensville, Ohio.
Standard specifications for highway bridges, 14th Ed. (1989). Am. Assn.
of State Hwy. and Transp. Officials (AASHTO), Washington, D.C. Standard specifications for
highway bridges, 15th Ed. (1992). Am. Assn.
of State Hwy. and Transp. Officials (AASHTO), Washington, D.C. Sweeney, R. A. P. (1979).
"Importance of bridge redundancy in bridge fracture control." Transp. Res. Rec. 711, Transp.
Res. Board, Washington, D.C., 23-30.
Sweeney, R. A. P. (1990). "Update on fatigue issues at Canadian national railways." Proc.,
IABSE Workshop Lausanne 1990—Remaining Fatigue Life of Steel Struct., Int. Assn. for Bridge
and Struct. Engrg. (IABSE), Zurich, Switzerland, 111-116.
Tarricone, P. (1990). "Bridges under surveillance." Civ. Engrg., ASCE, 60(5), 48-51.
Tebbett, I. E. (1988). "Damage and repair trends in fixed steel offshore structures." Proc., Conf.
on Weld Failures, Welding Inst., London, England.
Transportation Research Board (TRB). (1989). "Providing access for large trucks." Spec. Rep.
223, Nat. Res. Council, Washington, D.C.
Tung, C. C. (1970). "Fatigue life of mu Hi lane highway bridges." J. Engrg. Mech. Div., ASCE,
95(6), 1417-1428.
Tung, C. C., and Kusmez, К. M. (1975). "Statistical evaluation of AASHTO fatigue
specifications." Rep. No. FHWA-RD78-S0732, School of Engrg., North Carolina State Univ.,
Raleigh, N.C.
"Uninspected self-elevating mobile drilling unit Ranger I: Collapse and sinking in the Gulf of
Mexico on 10 May 1979 with loss of life." (1981). United States Coast Guard Marine Casualty
Rep. No. USCC 16732/93621, U.S. Coast Guard Ofc. of Investigation and Anal., Washington,
D.C.
Walker, W. H. (1978). "An interim report on studies of stress histories in highway bridges,
Volumes I and II." Struct. Res. Ser, No. 448, Rep. No. UIUC-ENG-78-2009, Univ. of Illinois,
Urbana, 111.
Walker, W. H. (1980). "Stress history studies and the fatigue life expectancy of highway
bridges." Struct. Res. Ser. No. 476, Rep. No. U1UC-Eng-80-2005, Univ. of Illinois, Urbana, 111.
Winkworth, W. J., and Fisher, P. J. (1992). "Inspection and repair of fixed platforms in the North
Sea." Proc., Offshore Technol. Conf., OTC 6937, Offshore Technol Conf., Houston, Тех.
Wirsching, P. H. (1988). "Probability-based fatigue design for marine structures." Marine
Struct., 1(1), 23-45.
Wirsching, P. H. (1995). "Probabilistic fatigue analysis." Probabilistic structural mechanics
handbook, C. Sundararajan, ed., Chapman & Hall, Ltd., New York, N.Y.
Wirsching, P. H., and Yao, J. T. P. (1970). "Statistical methods in structural fatigue." J. Struct.
Div., ASCE, 96(6).
Wirsching, P. H., Ortiz, K., and Chen, Y. K. (1987). "Fracture mechanics fatigue model in a
reliability format." Proc., 6th Int. Conf. on Offshore Mech. and Arctic Engrg., Am. Soc. of
Mech. Engrs. (ASME), New York, N.Y.
Woodward, H. Т., and Fisher, J. W. (1974). "Predictions of fatigue failures in steel bridges."
Fritz Engrg. Lab., Rep. No. 386-12, Lehigh Univ., Bethlehem, Pa.
Wyly, L. Т., and Scott, M. B. (1956). "An investigation of fatigue failures in structural members
of ore bridges under service loadings." Proc., Am. Railway Engrg. Assn., Vol. 57, Am. Railway
Engrg. Assn., Chicago, 111., 175-297.
Yamada, K., and Hirt, M. A. (1982). "Fatigue life estimation using frac-turer mechanics." Proc.,
IABSE Colloquium Lausanne 1982, Fatigue of Steel and Concrete Struct., Int. Assn. for Bridge
and Struct. Engrg. (IABSE), Zurich, Switzerland.
Yang, Y. N. (1993). "Application of reliability methods to fatigue, quality assurance and
maintenance." Proc., ICOSSAR '93, 6th Int. Conf. on Struct. Safety and Reliability, A. A.
Balkema, Rotterdam, The Netherlands.
Yao, J. T. P. (1974). "Fatigue reliability and design." J. Struct. Div., ASCE, 100(9), 1827-1836.
Yao, J. T. P. (1980). "Reliability considerations for fatigue analysis and design of structures."
Rep. No. NSF/RA-800206, School of Civ. Engrg., Purdue Univ., West Lafayette, Ind.
Yen, В. Т., Huang, Т., Lai, L. Y, and Fisher, J. W. (1990). Manual for inspecting bridge for
fatigue damage conditions. Commonwealth of Pennsylvania, Dept. of Transp., Ofc. of Res. and
Spec. Studies, Hams-burg, Pa.
APPENDIX II. NOTATION
The following symbols are used in this paper:
• A = age;
A, = parameter for stress range density function; a = crack size;
B, = parameter for stress range density function;
С = number of cycles per truck passage;
Ci = fatigue life constant;
C2 = fracture mechanics constant;
D = damage;
D, = damage life (years); E[ ] = expected value;
/ = mean stress factor; /( ) = probability density function; G(a) = crack threshold factor; g( ) =
limit state function;
Я = event margin;
К = structural detail factor;
m, = fatigue life exponent;
m2 = fracture mechanics exponent;
L = reliability;
M = margin of safety;
N = fatigue life to failure;
n = number of fatigue cycles;
n = mean number of cycles to failure;
P = probability
R, = reliability factor; 5, s = stress range; T, t = time;
Ta = lifetime average daily truck volume;
W = fatigue truck weight;
Yf = remaining safe life (years); Y( ) = crack geometry function;
Z = vector of random variables;
P = reliability index; Г( ) = gamma function; Г( ) = incomplete gamma function;
yr = crack geometry function; ДАТ = stress intensity factor;
v0 = zero upcrossing rate; Ф( ) = standard normal distribution function; and
П = coefficient of variation of fatigue life.
Subscripts
d = detectable;
F, f = failure;
r = effective range; and
th = threshold.
Superscripts
и = updated.
practice, the method can be costly and is limited in its capability to detect major damage that
alters the structure's dynamic behavior.
Sonic and Ultrasonic Methods
These techniques are primarily used in flaw detection in weldments. They are based on beaming
waves into a component and receiving the reflected waves. Flaws are detected by studying the
reflected waves. These methods require surface preparation and are generally very slow (Lai
1977; Collacott 1985).
Acoustic Emission Method
Using this method, piezoelectric sensors are arranged in an array around the area to be inspected.
These sensors detect stress waves (acoustic emissions) resulting from the energy release during
the cracking process. The source of the energy is strain release in the material itself when the
crack is active (Lai 1977; Collacott 1985).
Dye Penetrant
In this method, a dye penetrant is applied to the surface to be tested and allowed to penetrate into
the cracks. The penetrant is then removed from the surface and a developer suspension (chalk) is
applied. The dye combines with the developer to produce a colored line on the surface,
indicating the presence of a crack. Estimation of the severity and depth of the crack is based on
the interpretation and judgment of the inspector (Lai 1977; Collacott 1985).
REMAINING SERVICE LIFE
Once the current condition of the structure has been assessed, its remaining service life can be
estimated using either fatigue life methods or a fracture mechanics approach (Melchers 1986;
Madsen et al. 1986). A general outline of these methods is presented in the following section.
Specific applications are presented in the companion paper (Byers et al. 1997).
Fatigue Life Estimation
Fatigue life predictions are based on the familiar S-N curves that plot the fatigue life, or number
of cycles to failure, N as a function of the stress range, S. Plotted on log-log paper, the S-N
curves are often approximately linear or bilinear. The linear case can be represented by the
following:
(1)
where C\ = a constant for a given material and fatigue category (Fuchs and Stephens 1980;
Gurney 1979). The exponent m, is, likewise, a material parameter typically ranging from 2 to 4.
A practical probabilistic formulation treats the log of the number of cycles to failure In N as a
normally distributed random variable whose mean varies linearly with S and whose standard
deviation is constant (Department 1982). More sophisticated procedures that treat each of the
material parameters as random variables and allow Bayesian updating of the distributions are
presented by Madsen (1984).
For variable amplitude stress cycles, the Palmgren-Miner damage law is often used
(2)
where n, = number of stress cycles of stress range i; and Л7,- = number of stress cycles to failure
in the structural component if the stress range were s/; N/ is obtained from the component's S-N
curve. Failure is assumed to occur when the damage mea-