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FATIGUE ANALYSIS OF STEEL RAILWAY GIRDER BRIDGES
Stephen M. Dick, P.E., S.E., Ph.D., Manager of Bridges and Transit, Baker Engineering
Inc., Chicago, Illinois
Steven L. McCabe, Ph.D., P.E., Professor and Chair, Department of Civil, Architectural,
and Environmental Engineering, University of Kansas, Lawrence, Kansas
ABSTRACT Fatigue analysis of steel girder bridges has focused on examination at midspan with the
assumption that damage in that region is representative of the entire span. Recent research has
shown that fatigue effects on typical girders can vary significantly along the entire length from
indications at midspan. The behavior of maximum moment along the span is well-established
theory while the variation in live-load moment (moment range) along a span length has not been
explored in detail. New research examined moment range behavior along the length of a span
and its implications for estimating fatigue life.
The results of the research are applied to the typical unit-train loading for this report.
The application demonstrates that other points along a span can be critical for fatigue life
estimation versus estimating only the midspan. For riveted and welded girders with partial
length cover plates, fatigue susceptibility at other points is important to examine due to section
changes and the potentially fatigue prone connections.
This paper includes discussion on the effects of span length being loaded compared to
the car length providing that loading. This is referred to as the LS/LO ratio. This ratio is
fundamental in understanding behavior of unit-train loadings on girder spans, especially in
determination of critical areas for fatigue behavior.
INTRODUCTION
Fatigue of older steel girder spans is a concern for most railroad bridge engineers. Older designs
of girder spans still provide excellent service, but the continuing increase in allowable railcar
weights creates the potential for accumulating fatigue damage at faster rates than previously
expected. Additionally, traffic in the last decade has been in excess of historical trends.
Given the allowable stresses used to design older girders and their evolution to present
day, overall load capacity is not generally in question. Close examination of many girders is
necessary, however, given their age, physical condition, and the aggregate tonnage they have
seen. Increased axle loadings will place additional importance on investigating girders for
fatigue given the high number of cycles these bridges have experienced since their installation.
Investigating fatigue life at points other than midspan is important for girders. The
midspan has been used as an indicator of fatigue life because overall maximum moment occurs
in that region. Variation in live-load moment (moment range), caused by the changing location
of wheel loads on the bridge as a train traverses, is not always involved in fatigue, given the
necessity of the potential fatigue cycles to exceed the magnitude of the minimum fatigue stress,
or constant amplitude fatigue limit. Historically, the check of moment range has occurred at
midspan without a formal emphasis on investigation at other locations.
Generally, the midspan region of a girder is not susceptible to fatigue damage. This
region usually has adequate fatigue strength since cover plates are continuous through this
region, and a full section is in place. For riveted girders, cover plates on tension flanges are
generally not full length, with significant addition of section occurring between the quarter
points and midspan. Similarly, welded girders with partial length cover plates follow the same
philosophy of adding section as needed, although the number of cover plates is usually limited
to one plate. The terminating welds at the ends of the partial length cover plates are quite
susceptible to fatigue damage.
Fatigue analysis for railroad bridges has four major components to explore when
estimating the potential number of cycles from the loading side:
• Maximum live-load moment
• Magnitude of live-load moment range
• Live-load impact
• Span response
This paper focuses on maximum live-load moment and live-load moment range. The typical
coal unit train is utilized here, on simply supported spans ranging from 50 feet to 100 feet
illustrating how points other than midspan can be analyzed.
GENERAL BACKGROUND
The coal unit train is seen on most railroads in today’s environment, with its characteristics seen
in trains hauling other bulk commodities; primarily agricultural and mineral products. The
configuration of the coal car is the same car used in the fatigue recommendations in the
AREMA manual (1). For this paper, the assumed unit train uses General Motors SD70
locomotives and the unit train coal car. For this analysis, the numerical dimensions of the
locomotive and coal car are modified slightly to facilitate computation. The length dimensions
used in this analysis along with the numerical values are shown in Figure 1.
LO
FIGURE 1. Dimensions Used For Analysis
LO - Overall length of railroad car measured over the pulling face of the coupler.TC - Length between the center pin on the trucks, known as the truck center distance.SI - Inboard Axle Spacing, the distance between the inside axles of the railroad car.SO - Outboard Axle Spacing, the distance between the outside axles of the railroad car.ST - Truck Axle Spacing, the distance between the adjacent axles of a truck.n - number of axlesP - axle load
LO
TC TC
ST ST ST STSO/2 SO/2SI SO SI
SD7070.0011.007.00
31.006.0070.0
COAL53.006.755.80
34.654.00
71.50
LOSOSTSInP
All dimensions are in feet and kips.
Figure 2 shows the variation in live-load moment with time for midspan and quarter
point for a train of three locomotives and five cars traversing a 50-foot span, while Figure 3
shows the same locations for the 100-foot span. Significant differences are apparent in the
moment ranges that occur between the types of equipment, along with the location and span
length.
Given a longer span length, the maximum moment will increase in magnitude. What is
also apparent from the figures is that the relative magnitude of maximum moment between the
different types of equipment will vary. On shorter spans, the maximum moment is more
influenced by the axle spacings while on the longer span the more influenced by the equipment
unit weight. Differences in span length make the moment range vary for the same equipment,
but Figures 2 and 3 show that different locations on the same span have different results in
relation to moment range.
Table 1 provides the magnitudes of maximum moment and moment range at 0.25L,
0.375L, and 0.50L (midspan) for the unit train on spans from 50 feet to 100 feet in 10-foot
increments. The behaviors in the results, especially for moment range, are striking.
For maximum moment, the generally accepted theoretical behavior is displayed. For
both locomotive and coal car, the maximum moment increases with increasing span length and
increases toward the middle of the span. For some of the examples, the maximum moment at
the 0.375L exceeds midspan. This is not totally unexpected. This is an example of where the
absolute maximum moment is located closer to 0.375L than midspan. For all cases, maximum
moment at 0.25L is less than either 0.375L or midspan.
Moment range behavior varies, however. Some span lengths display the same behavior
as maximum moment for both locomotive and coal car. For the locomotive, the range at 0.25L
is less than the other locations except for a span length of 100 feet, where the magnitude at
0.25L
Ben
ding
Mom
ent
CarsLocomotives Midspan
Quarter Point
Time
FIGURE 2. Moment Trace For A Unit Coal Train On A 50-Foot Span
Ben
ding
Mom
ent
CarsLocomotives
Midspan
Quarter Point
Time
FIGURE 3. Moment Trace For A Unit Coal Train On A 100-Foot Span
TABLE 1. Maximum Moment And Moment Range
Maximum Moment
SD70 Locomotive Coal Car
Span Span Position Span Position
Length 0.25L 0.375L 0.50L 0.25L 0.375L 0.50L
50 2135 2585 2625 2025 2574 2678 60 2923 3570 3675 2562 3244 3393 70 3710 4554 4725 3192 3915 4108 80 4498 5539 5775 3991 4700 4823 90 5364 6528 6825 4966 5725 5660
100 6458 7757 8050 6038 7066 6827
Moment Range
SD70 Locomotive Coal Car
Span Span Position Span Position
Length 0.25L 0.375L 0.50L 0.25L 0.375L 0.50L
50 1505 1832 1785 1530 1958 1995 60 1768 2082 2100 1509 1963 1986 70 1907 2218 2050 1424 1591 1629 80 1768 2074 2100 1508 1303 962 90 1584 1750 1785 1768 1256 369
100 1628 1536 1435 1995 1524 106 Units are feet and kip-feet
exceeds the other values. For the coal car, however, this behavior is more pronounced. Table 2
shows the ratio of the span length to the car length or LS/LO. Using LS/LO as a guide to the
phenomenon provides a behavioral pattern. Moment range behavior is similar to maximum
moment behavior up to a value of LS/LO of approximately 1.0. Beyond that, the moment range
behavior shows variability in magnitude, based on span position and span length. Those
relationships are explored further.
MAXIMUM MOMENT
Calculation of maximum moment on a simply supported beam is established theory using
influence lines combined with locating the centroid of the loading group. From that, the
absolute maximum moment can be found for any loading group on any span length. For a
fatigue investigation on any arbitrary span, however, the moment must be found at a number of
locations that are not points of absolute maximum. Also, analysis can get tedious when having
to calculate
maximum moments at more than one location, requiring loads to be repositioned for each point
being analyzed.
An approximation method has been developed (2) that precludes performing separate
calculations at each point. The requisite data is the absolute maximum moment for the load on
the bridge and the distance from midspan to the point of absolute maximum. With knowing
those two pieces of information, an approximation is made using the sine function. The formula
for the sinusoidal approximation is:
=
csxzx XL
zMM
2–sin,
π
TABLE 2. LS/LO Values
Span Locomotive Coal Car
Length 70 feet 53 feet
50 0.714 0.943
60 0.857 1.132 70 1.000 1.321 80 1.143 1.509 90 1.286 1.698
100 1.429 1.887
where:
Mx, z = maximum moment at point z (between 0 and L/2).
Mx = absolute maximum moment for the span
Xc = distance from centerline of span to the point of absolute maximum moment
The method for finding absolute maximum moment given a specified loading pattern takes into
account some variables. For a given loading group and a span length, the possibility exists that
subgroups of the overall loading group may actually produce the absolute maximum. An
example may be a span length that is long enough to have a four-axle car on it with all axles on
the span. There may be another grouping of the same car configuration, however, where only
three axles of that car produces the absolute maximum moment. Included in reference (2) area a
series of equations based on loading subgroups of the basic pattern. The basic loading pattern is
either a car or locomotive, with characteristics shown in Figure 1. The subgroups can be those
groups of axles that compose a single axle, a truck, or a truck and axles from another truck.
For each subgroup a series of formulae can be developed. These include the formula for
absolute maximum moment, location of absolute maximum moment relative to midspan, and the
minimum span length for that particular loading group. For certain groups of axles, it is not
always the case that the maximum number of axles fitting on a span will generate the highest
maximum moment. Figures 4 and 5 show the absolute maximum moment curve versus LS/LO
for the locomotive and coal car. An examination of those figures show that certain portions of
the curve show a linear rate of change while other portions have varying rates of change. This is
the influence of the location of absolute maximum moment and the axle group generating that
moment. The linear portions of those figures show where one loading group controls absolute
Abs
olut
e M
axim
um M
omen
t (k-
ft)
0
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
2000
4000
6000
8000
10000
12000
14000
16000
LS/LO
FIGURE 4. Maximum Moment Versus LS/LO Ratio For SD70 Locomotive
Abs
olut
e M
axim
um M
omen
t (k-
ft)
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
2000
4000
6000
8000
0
LS/LO
FIGURE 5. Maximum Moment Versus LS/LO Ratio For Coal Car
maximum moment for a range of span length while the changing slope in other parts indicates
that the controlling loading group changes.
Table 3 compares the actual and approximated moments for the coal car and locomotive
for the span lengths and span positions, using the magnitude and position of the absolute
maximum moment. For both the locomotive and coal car, the approximation at 0.25L provides
the most error; the highest error being less than ten percent. At the other points of 0.375L and
0.50L, however, the approximation method provides very accurate approximations. Given the
locations of most cover plate cutoffs, the approximation of maximum moment should provide
accurate values. For initial investigations of girders and fatigue, the use of the sine wave
approximation appears warranted, given that more precise calculations can be made if required.
MOMENT RANGE
As seen in Table 1, the behavior of moment range is not similar to maximum moment. Moment
range behavior does not appear to follow any pattern. A more appropriate method is to examine
the behavior of moment range graphically instead of using tables.
Figure 6 shows moment range versus span position for both locomotive and coal car for a
50-foot span length. This graphical representation allows one to examine the entire span for
moment range behavior, showing that the locomotive and the coal car both produce similar
moment range curves, even though the locomotive has substantially more total weight.
Figure 7 shows the same curve for a span length of 100 feet. This figure provides
interesting results. The coal car produces a maximum moment range for the span, but only for a
short distance in the vicinity of the quarter point. The moment range for the coal car at midspan
is close to zero indicating that the bending moment at midspan is close to constant. This
TABLE 3. Comparison Of Actual And Estimated Moments
Coal Car
Span Position
Span 0.25L 0.375L 0.50L Length Actual Approx Percent Actual Approx Percent Actual Approx Percent
(ft) (kip-ft) (kip-ft) Diff. (kip-ft) (kip-ft) Diff. (kip-ft) (kip-ft) Diff.
50 2025 2011 0.7 2574 2568 0.2 2678 2676 0.1 60 2562 2518 1.7 3244 3230 0.4 3393 3391 0.1 70 3192 3026 5.2 3915 3892 0.6 4108 4106 0.0 80 3991 3609 9.6 4700 4609 1.9 4823 4803 0.4 90 4966 4597 7.4 5725 5702 0.4 5660 5650 0.2
100 6030 5811 3.6 7055 7065 -0.1 6820 6752 1.0
Locomotive
Span Position
Span 0.25L 0.375L 0.50L Length Actual Approx Percent Actual Approx Percent Actual Approx Percent
(ft) (kip-ft) (kip-ft) Diff. (kip-ft) (kip-ft) Diff. (kip-ft) (kip-ft) Diff.
50 2135 2077 2.7 2585 2607 -0.9 2625 2638 -0.5 60 2923 2837 2.9 3570 3590 -0.6 3675 3681 -0.2 70 3710 3589 3.3 4554 4567 -0.3 4725 4728 -0.1 80 4498 4341 3.5 5539 5544 -0.1 5775 5777 0.0 90 5364 5096 5.0 6528 6523 0.1 6825 6823 0.0
100 6458 6082 5.8 7757 7732 0.3 8050 7995 0.7
Mom
ent R
ange
(k-f
t)
Span Position (x/L)
FIGURE 6. Moment Range Versus Span Position On 50-Foot Span
0
0 0.1 0.2 0.3 0.4 0.5
500
1000
1500
2000
2500
Coal Car
Locomotive
2500
Coal Car
Locomotive
2000
Mom
ent R
ange
(k-
ft)
1500
1000
500
0
0 0.1 0.2 0.3 0.4 0.5
Span Position (x/L)
FIGURE 7. Moment Range Versus Span Position On 100-Foot Span
correlates with Figure 3 showing high live-load moment variation at the quarter point while
midspan variation is minimal. The moment range curve for the locomotive does not produce the
maximum value of the coal car, but the maximum value of the locomotive is consistent from
0.2L to midspan.
Figure 8 displays moment range versus span position for the locomotive for the span
length range of 50 to 100 feet. The largest overall moment range shown in that figure is for a
span length of 70 feet, equal to an LS/LO ratio equal to 1.0. As shown in Table 2, the range of
the LS/LO ratio for the locomotive on these spans is 0.714 to 1.429. The curves for 60 feet and
80 feet produce similar curves while the 50-foot and 90-foot curves produce similar results for a
portion of the span length. The 100-foot span produces the higher magnitude of moment range
for the portion of the span between the supports and 0.2L, but then maintains a consistent and
mostly equivalent magnitude from that point to midspan.
Figure 9 shows the moment range curves for the coal car on the same span lengths. The
range of LS/LO ratio for the coal car is 0.943 to 1.887. The moment range curves for the coal car
show different behavior than the locomotive curves. For the coal car, the curves display a
behavior where the moment range at midspan steadily decreases as LS/LO increases while the
moment range at 0.25L steadily increases. The question is whether similar values of LS/LO for
dissimilar equipment produce similar behaviors. Figure 10 shows the moment range curves for
the locomotive and coal car at LS/LO = 1.0. The curves show similar behaviors with the
maximum moment range occurring at a point SO/2 away from 0.50L. Although the point of
maximum moment range is at a different location from the point of absolute maximum moment,
the curve has similar characteristics to an absolute maximum moment curve.
2500
Span Position (x/L)
0 0.1 0.2 0.3 0.4 0.5
70 feet
100 feet50 feet
80 feet90 feet
60 feet2000
Mom
ent R
ange
(k-
ft)
1500
1000
500
0
FIGURE 8. Moment Range Versus Span Position For SD70 Locomotive
2500M
omen
t Ran
ge (
k-ft)
Span Position (x/L)
FIGURE 9. Moment Range Versus Span Position For Coal Car
0
50 feet
60 feet
100 feet
90 feet
70 feet
80 feet
2000
1500
1000
500
0 0.1 0.2 0.3 0.4 0.5
2500M
omen
t Ran
ge (
k-ft)
Span Position (x/L)
FIGURE 10. Moment Range Versus Span Position For LS/LO = 1.0
0
0 0.1 0.2 0.3 0.4 0.5
500
1000
1500
2000Locomotive
Coal Car
Figure 11 displays the moment range curves for the locomotive and coal car for an LS/LO ratio of
2.0. Both locomotive and coal car display similar behaviors with no moment range at midspan.
While the midspan moment range is zero, the moment range in the vicinity of the quarter point
is the same magnitude as the moment range near midspan where LS/LO = 1.0. In fact, for an
LS/LO = 2.0, the moment range curve is the same curve between the support and 0.25L as the
curve for LS/LO = 1.0 between the support and 0.50L. For the portion between 0.25L and 0.50L,
the moment range curve is the mirror image of the basic moment range curve for LS/LO = 1.0.
The explanation of this behavior is shown in Figure 12. That figure shows influence
lines for moment for LS/LO = 2.0 and 4.0. For LS/LO = 2.0, the midspan moment is explored.
Given the conditions for a span that is twice as long as a loading pattern that will repetitively
load it, the slopes on each side of midspan are equal and opposite, and will always have the
same amount of load on it. Given those conditions, location of the loads has no effect on
variable moment magnitude because the moment has equal rates of increase and decrease due to
the slopes on either side of midspan. Given no change in moment is created, a constant moment
exists at midspan, hence moment range is zero.
The other part of Figure 12 shows an influence line for a quarter point where LS/LO = 4.0.
The length of the influence line is such that the left portion of the influence line has one car
length on it at all times, while the right side has three car lengths. The slopes of the two sides
are such that the left side has three times the rate of change of the right side. With the load
distribution combined with the slopes on each side, the movement of the loads does not vary the
moment magnitude, creating a constant moment at the quarter point along with midspan.
The behavior of moment range has implications for fatigue analysis. Moment range
exhibits a cyclic behavior based on LS/LO. This implies that an absolute maximum moment
2500
Locomotive
Coal Car
2000
Mom
ent R
ange
(k-
ft)
1500
1000
500
0
0 0.1 0.2 0.3 0.4 0.5
Span Position (x/L)
FIGURE 11. Moment Range Versus Span Position For LS/LO = 2.0
21
21
X
FIGURE 12. Influence Lines For Moment Behavior At Midspan And Quarter Point
Influence Line for LS = 2LO at Midspan
Influence Line for LS = 4LO at Quarter Point
3LOLO
43
41
X
LOLO
range exists. Although current methods are concerned with midspan behavior, it is apparent that
other locations require investigation due to variation in live-load moment.
The absolute maximum moment range is exhibited on spans where the value of LS/LO is
an integer. The magnitude of absolute maximum moment range is:
+=
o
ooIAM L
SSSnPR
44–
4
2
When LS/LO is an integer, maximum moment range at any point along the span can be
calculated as per maximum moment using a sine wave approximation. The curve assumes its
shape based on the peak occurring at a distance SO/2 away from midspan based on LS/LO = 1.0.
For values of LS/LO other than integer, the application of the formula is not as clean. The
approximation of the moment range at any point along the span requires the use of influence
charts providing upper and lower bounds for the shape of the moment range curve. Figure 13 is
an example of the influence chart with a series of lines that delineate the upper and lower bound
of moment range based on span position and decimal ordinate of the LS/LO ratio. The decimal
ordinate is the decimal remainder of the ratio beyond the whole value. The shape of the chart is
determined by the railcar characteristics. This includes the number of axles, the ratio of SO/LO
and the ratio of ST/LO. The chart is created individually for those characteristics.
An example using a chart is shown in Figure 14. The chosen upper bound is based on
the decimal ordinate value of 0.143. The approximation uses line segments on the influence
chart without duplicating the actual moment range curvature. The reason for this is that the
point (or points) of inflection is predefined upon LS/LO and the car dimensions. Also, a function
that develops the curvilinear approximation is not easily defined. Even with the use of line
segments, the approximation is reasonably close to the actual moment range envelope for the
given
Mom
ent R
ange
Rat
io
0.4,0.6
0.3,0.7
0.2,0.8
0.1,0.9
0
0 0.05
0.9
1 0.1, 0.9
0.2, 0.8
0.3, 0.7
0.4, 0.6
0.50.5
0.4, 0.6
0.3, 0.7
0.2, 0.80.1, 0.9
0.7
0.8
0.6
0.5
0.4
0.3
0.2
0.1
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.4
L S/L
OD
ecim
al O
rdin
ate
Span Position (x/L)
FIGURE 13. Typical Moment Range Envelope For LS/LO Spectrum
Mom
ent R
ange
Rat
io
0.8
0.050
0
0.9
0.7
0.6
0.5
0.4
0.1
0.2
0.3
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
1.0
1.1
Actual
Approximate
Actual
Approximate
Approximate =Actual
Span Position (x/L)
FIGURE 14. Approximation Of Moment Range For SD70 Locomotive (80-Foot Span)
conditions. Its use in determining approximate moment ranges at position of cover plate cutoffs
provides for determination of whether the cycling due to live load will create damaging cycles.
SUMMARY
The purpose of the research was the development of a method to approximate maximum
moment and moment range magnitudes at any point along a girder. Given conditions of riveted
and welded girders with partial length cover plates, knowing the envelope for maximum moment
and moment range at the locations of cover plate cutoffs is paramount. The work was limited to
working with unit-train configurations.
The research shows that the maximum moment at any point along a span is estimated
using a sine wave approximation based on finding the absolute maximum moment for the
loadings on the bridge along with the point of absolute maximum moment. Moment range
shows a periodic behavior when examining its relation to LS/LO. The behavior includes an
absolute maximum moment range for any typical railcar configuration when LS/LO is an integer
value. The behaviors of both maximum moment and moment range are such that computer
programs are not necessary to provide the results needed to perform a fatigue analysis.
REFERENCES
1. American Railway Engineering and Maintenance of Way Association (AREMA) (2000),
Manual for Railway Engineering, Chapter 15, Washington, D.C.
2. Dick, S. M. (2002), “Bending Moment Approximation Analysis for Use in Fatigue Life
Evaluation of Steel Railway Girder Bridges”, Ph.D. Dissertation, University of Kansas,
Lawrence, Kansas.
List of Tables
TABLE 1. Maximum Moment and Moment Range
TABLE 2. LS/LO Values
TABLE 3. Comparison Of Actual And Estimated Moments
List of Figures
FIGURE 1. Dimensions Used for Analysis
FIGURE 2. Moment Trace For A Unit Coal Train On A 50-Foot Span
FIGURE 3. Moment Trace For A Unit Coal Train On A 100-Foot Span
FIGURE 4. Maximum Moment Versus LS/LO Ratio For SD70 Locomotive
FIGURE 5. Maximum Moment Versus LS/LO Ratio For Coal Car
FIGURE 6. Moment Range Versus Span Position On 50-Foot Span
FIGURE 7. Moment Range Versus Span Position On 100-Foot Span
FIGURE 8. Moment Range Versus Span Position For SD70 Locomotive
FIGURE 9. Moment Range Versus Span Position For Coal Car
FIGURE 10. Moment Range Versus Span Position For LS/LO = 1.0
FIGURE 11. Moment Range Versus Span Position For LS/LO = 2.0
FIGURE 12. Influence Lines For Moment Behavior At Midspan And Quarter Point
FIGURE 13. Typical Moment Range Influence Chart for LS/LO Spectrum
FIGURE 14. Approximation Of Moment Range For SD70 Locomotive (80-Foot Span)
