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Fatigue hazards in welded plate crane runway girders – Locations, causes and
calculations
Article in Archives of Civil and Mechanical Engineering · January 2018
DOI: 10.1016/j.acme.2017.05.003
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archives of civil and mechanical engineering 18 (2018) 69–82
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Original Research Article
Fatigue hazards in welded plate crane runway
girders – Locations, causes and calculations
Kazimierz Rykaluk a, Krzysztof Marcinczak b,*, Sławomir Rowiński b
a
b
Wrocław University of Environmental and Life Sciences, C.K. Norwida 25, 50-375 Wrocław, Poland
Wrocław University of Technology, Wybrzeże Wyspiańskiego 25, 50-370 Wrocław, Poland
article info
abstract
Article history:
Steel crane runway beams compared with other building structures are exposed to extreme-
Received 14 October 2016
ly complex load-stress conditions. It turns out, that significant from the point of view of the
Accepted 12 May 2017
resistance of the crane runway beams is a cyclic nature of fluctuating loads, which leads to
Available online
formation of numerous cracks and damages. This effect is especially characteristic for webs
Keywords:
by overall bending that causes normal and shear stresses – sx, txz, and by crane wheel
Crane runway beams
eccentric load that produces respectively stresses – sz,x, so,x, to,xz. Stress components
Fatigue mechanism
produced by overall bending are determined as I kind stress, whereas the stress components
Fatigue cracks
from the crane wheel load are introduced as II kind stress. Such a combination of stresses
Fatigue strength
lowers the fatigue strength of the web, which is ignored by many rules specified in
in plate I – cross sections of crane runway beams. The complex state of stresses is generated
standards. Limited fatigue strength is observable, among others, in crane rails splices.
The results of numerical analyses obtained as II kind stresses in the web located directly
beneath the crane rails splices that occur as: orthogonal contact, bevel contact and stepped
bevel contact as well, confirmed the complexity of the issue. Following that, other factors,
not being defined yet, but affecting the stress state of the both crane rail and crane runway
beam are scheduled to be studied, as for instance, the eccentric load induced by crane trolley
in mentioned above elements.
© 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
1.
Factors that influence the lowering of the
fatigue strength of welded steel structures
When compared to other types of structures, crane runway
beams operate under very complex load-stress conditions.
One of the parameters describing the conditions of operation
that affects fatigue loads is load spectrum. Fig. 1 shows,
according to [1,2], schematic scatter bands of the relative
stress variation ranges of stresses Dsi/Dsmax in crane runway
beams, and railway bridges. The latter are widely regarded in
the construction industry as structures of heavy fatigue
exposure. The presented charts show (Fig. 1) that the stress
intensities in crane runway beams, over the variation range
spectrum, are much larger than the intensities in the
spectrum of railway bridges. Detailed spectrum histograms
* Corresponding author.
E-mail address: krzysztof.marcinczak@pwr.edu.pl (K. Marcinczak).
http://dx.doi.org/10.1016/j.acme.2017.05.003
1644-9665/© 2017 Politechnika Wrocławska. Published by Elsevier Sp. z o.o. All rights reserved.
70
archives of civil and mechanical engineering 18 (2018) 69–82
Nomenclature
a
aw
a1
br
bs, ts
cs
ey
ez
fy
Fz
hr
HT
hw ; tw
Jf
Jr
Jy
K
L
leff
spacing of long transverse ribs
design weld thickness
axial spacing of transverse ribs
width of crane rail foot
width and thickness of a single rib
leg of right-angled triangular notch for a weld
eccentricity of vertical wheel load
distance from the top of rail surface to the webtop flange joint
nominal steel yield strength
vertical wheel load
depth of the crane rail
transverse crane wheel load
depth and thickness of the beam's web
second moment of area of an upper crane beam
flange
second moment of area of the rail about its
horizontal centroid axis
second moment of area about y–y axis
drive force
theoretical span length of a beam
length of the uniform distribution of stresses sz,
x
lw;eff
MT
My
N
effective length of the weld
torque
bending moment about y–y axis
design life time of a beam expressed as a number of cycles
number of cycles of a permanent fatigue
N0
strength
i-cycle of fatigue load
ni
number of wheels in one crane runway beam
nk
parameter
pi(i=a,k)
relative number of investigated crane runway
r
beams
yield stress
Re
ultimate static tensile stress
Rm
load due the crane self-weight
Qc
equivalent fatigue load
Qe
load-bearing capacity of the long, transverse rib
Qgr
Qh
Hoist load
range of crane load variability at i-cycle
DQi
characteristic value of the maximum wheel
Qmax
load
static moment of the cross-section portion over
S
the z–z coordinate
thickness of the flange in a cross-section
tf
shear force
Vz
distance of a butt weld from external surface of
z1
the top flange
effective concentration factor
bk
initial deflection (flexure) of the web
Do
w1, w2
dynamic factor due to self-weight of crane and
hoist load, respectively
dynamic fatigue factor
wfat
wfat,1, wfat,2 dynamic fatigue factor for a self-weight of the
crane and to hoist load, respectively
li
xt
equivalent factors of fatigue damages
distance from main maximum shear stresses
t1,2 to axis of the beam's support
relief factor
c
partial factor for equivalent constant amplitude
gFf
stress range DsE, DtE
gMf
partial factor for fatigue strength DsC, DtC
gM0
partial factor of load-bearing capacity of the
cross-section
normal stress of the I kind in the longitudinal
sx
direction – x
equivalent normal stress for 2 million cycles
sE,2
normal stress of the I kind due to bending
sM,x
moment My in the beam cross-section
local normal stress of the II kind in the web,
so,x
directly beneath the concentrated force Fz along
x-axis
normal stress due to torque
sT
sT,x
normal stress due to torque along x–x axis
normal stress due to torque along z–z axis
sT,z
normal stress of the I-kind beneath the force Fz
sz,0
sz,x
vertical normal stress of the II-kind induced by
force Fz at x distance from applying point
Ds, Dsi amplitude normal stresses range under cyclic
load and in i-cycle, respectively
maximum amplitude normal stresses range unDsmax
der cyclic load during the life time
Dsx,E,2, Dsz,0,E,2
Dsx,C
Dsz,0,E,2
DsT,E,2
Dsz,C
DsL, DtL
DsC, DtC
DsE, DtE
t1,2
tT,xz
tV,xz
to,xz
txz
tQ
Dt
Dtxz,E,2
Dt1,2
DtQ
equivalent constant amplitude stress range related to 2 million cycles along x–x and z–z axis,
respectively
reference value of the fatigue strength along x–x
axis
equivalent constant amplitude stress range
along z–z axis related to 2 million cycles
equivalent constant amplitude stress range due
to torque related to 2 million cycles
reference value of the fatigue strength along z–z
axis
constant fatigue strength at NL cycles
reference value fatigue strength at Nc = 2 million
cycles
equivalent constant amplitude stress range related to nmax
main shear stress
shear stress due to the torque at plane xz
shear stress of the I kind under transverse force
Vz
local shear stress of the II kind at xz plane under
the concentrated force Fz
local shear stress of the I kind at xz plane
shear stress in the weld
amplitude shear stress range under cycle load
equivalent amplitude shear stress range at xz
plane related to 2 million cycles
amplitude main shear stress range
amplitude shear stress range in the weld under
cyclic load
archives of civil and mechanical engineering 18 (2018) 69–82
Fig. 1 – Scatter bands of spectrum of relative variation
ranges of stresses Dsi/Dsmax depending on relative number
of cycles ni/N for crane bridges, crane runway beams, and
railway bridges [2].
Fig. 2 – Histograms of loads of relative variability range of
loads DQi/Qmax depending on the relative number of cycles
ni/N [3].
Fig. 3 – Curve of crack intensity in welded plate crane
runway girders [7].
significant variation of the crystalline structure in the
particular areas of thermal influence of the welded contact.
Additionally, internal welding stresses and micro cracks are
left in the heat affected zone. These factors reduce the
dependence of the fatigue strength on the static tensile
strength Rm (that is, on the type of material) which can be
observed during fatigue tests of standard samples, e.g.
according to the standard [3].
The high level of fatigue interactions and the negative
effects of welding cause the welded crane runway beams,
called plate girders, to crack in a relatively short period of time
from the beginning of their operation. Macroscopic fatigue
cracks appear even after two years of operation and the crack
frequency is intensified between the sixth and twelfth year of
operation, as shown in Fig. 3 [7] depicting the relationship
between the relative number of investigated beams with r
cracks and the time of their operation. Such a fast response to
the fatigue load is resulting from stress concentration around
micro cracks [4], and embrittlement of steel under cyclic
loading conditions [8]. The effective concentration factor bk [9]
in crack locations, increases with the increase of the lifetime of
the structure.
2.
for cranes with lifting classes HC1–HC4, according to [3], and
for the main girders of the railway bridge with a span length of
60 meters, according to [4], is shown in the Fig. 2. The lifting
class of a crane takes into account the working day-time, ratio
of loads lifted to the safe load during one-day work, and the
frequency of lifting.
The second, multi-parameter factor causing a decrease in
the fatigue strength of steel constructions is welding [5]. Remelting of the base material in high gradient thermal heat,
mixed with the additional material and rapid cooling, causes
71
Webs in welded plate crane runway girders
On the basis of the results of investigation of failures in the
used crane tracks (see [7,10]) and on the basis of the results of
experimental tests on models, and theoretical analyses (see
[11]), it can be concluded that the most fatigue cracking prone
component of the crane track beams is a web. Increased
sensitivity of the web in relation to the beam flanges should be
attributed to the effects of:
1) Cyclically variable, moving drive wheel loads of the crane
bridge
2) Web breathing in intercostal panels
72
archives of civil and mechanical engineering 18 (2018) 69–82
by the Broudy's formula [12]. Assuming the local coordinate
system in such a way that the z-axis coincides with the wheel
load Fz, the formula is obtained [12]
s z;x ¼ 2:6c
7
Fz X
kpx
p cos
;
leff
tw leff k¼1 k
(3)
where the length of the pressure distribution on the web under
the upper flange of the beam leff is
leff
sffiffiffiffiffiffiffiffiffiffiffiffiffi
3 Jf þ Jr
;
¼ 3:25
tw
(4)
wherein Jf and Jr are respectively the second moments of crosssectional area of the upper flange and the crane rail about their
own horizontal centerlines.
The c modifier as a relief factor includes the impact of
transverse ribs having an axial spacing a1, which is for the web
expressed as:
Jf þ Jr 1
:
c ¼ 0:95 1 þ 23 3
a1 tw
Fig. 4 – Eccentric impacts of the crane wheel on the web in
crane runway beam.
3) The torque of the upper flange of the beam which is caused
by the eccentric crane wheel load in the relation to the
vertical axis of the web, and by eccentric horizontal impacts
of the wheels on the rail head (Fig. 4).
3.
Stresses inside the web, outside the rail
dilatation
General stresses due to the bending of the beam i.e., normal sx
due to actions in sections of bending moments My, and shear
tV,xz due to the shearing forces Vz, are generated in the beam
web. In any of the fibers of the cross-section that have the z
coordinate, stresses are expressed in the formulas of the
mechanics of materials, as formulas of Navier and Żurawski,
respectively.
s M;x ¼
tV;xz
My z
;
Jy
Vz S
¼
;
tw Jy
(5)
The smaller their relative spacing a1 =tw , the bigger relief
effect of the ribs(c < 1.0).
The spacing is dependent from the
parameter pa ¼ Jf þ Jr =t4w . The maximum relative spacing of
ribs a1 =tw for a number of selected values of the pa parameter is
presented in Table 1.
The cosine series coefficients pk (3) have non-zero values
only when k is odd. Thus [12], p1 = 11/16, p3 = 13/64, p5 = 7/96,
p7 = 7/192.
On the basis of formula (3), when c = 1.0, the values of
relative stresses are obtained as s z;x =ð2; 6Fz =ðtw leff ÞÞ in chosen
points of relative axis of abscissas x/leff given in Table 2 and
Fig. 5.
The stresses sz,x non-uniformly distributed within the leff
length generate consequently the normal stresses in the
vertical direction so,x and shear stress to,xz in the web panel,
considered as a disc. At the x distance from the force Fz is [13],
s o;x ¼
t o;xz ¼
x
0:637leff
!2
s z;x ;
(6)
x
s z;x ;
0:637leff
(7)
where sz,x is expressed by the formula (3).
(1)
Table 1 – Maximum relative ribs spacing pa.
(2)
Drive crane wheel load causes local stresses sz,x, so,x and
to,xz, (as a result of non-uniform distribution of stresses of the I
kind) that can be determined on the basis of the theory of
elasticity. The most important in testing the conditions that
determine the resistance of the beam (strength of the cross
section and stability of the web) are the stresses sz,x expressed
pa
250
500
750
1000
1250
a1 =tw
47.8
60.2
68.9
75.9
81.7
Table 2 – Relative values of the stress sz,x.
x/leff
0.0
0.1
0.2
0.3
0.4
0.5
s z;x =ð2:6Fz =ðtw leff ÞÞ
1.0
0.752
0.409
0.246
0.092
0.0
archives of civil and mechanical engineering 18 (2018) 69–82
73
Fig. 5 – Distribution of relative values of the stress sz,x in the steel web in area ABCD.
Maximum values of stresses s0,x and t0,xz are:
max s o;x ¼ 1:25s z;x ;
max t o;xz ¼ 1:325txz ;
(8)
and are located from the action of Fz force at the distance of
0.637leff and 0.358leff, respectively.
The other biggest components of the stress caused by
torque MT are nominal stress sT,x and shear stress tT,xz [13]
s T;x ¼ 0:3 s T;z ;
(13)
t T;xz ¼ 0:25 s T;z ;
(14)
The torque (Fig. 6) is:
MT ¼ Fz ey þ HT ez ;
(9)
where HT is the horizontal transverse load of the crane wheel.
These forces include the loading on the bridge through the
wheel flange or through horizontal guide rollers, the friction
forces by chamfering of the bridge, and the inertial force of the
bridge trolley with suspended weight due to acceleration or
braking. The eccentricities ey and ez in design standards of
substructures of crane tracks shall be respectively 0.25br and
0.75hr. The wear of the rail head height in the last 25 years of
the track operation causes a non-uniform distribution over the
length of the panel between the ribs and uneven distribution of
stress variables sT,x, sT,z, and tT,xz through the web thickness
(Fig. 7).
The most important in the assessment of fatigue stresses
are stresses bending the web out of its plane sT,z. Their
maximum value in the beam cross-section passing through
the Fz force line can be calculated using the formula given in
the standard [14].
s T;z ¼
2MT tw
;
Jf þ Jr
(10)
or in the standard [15]
s T;z ¼
6MT
htanh h;
at2w
(11)
where
#0:5
"
0:75at3w
sinh2 ðphw =aÞ
h¼
;
sinhð2phw =aÞð2phw =aÞ
Jf
(12)
Fig. 6 – Load arrangement in the beam cross-section to
obtain the torque.
74
archives of civil and mechanical engineering 18 (2018) 69–82
Fig. 7 – Graph of non-uniformly distributed stresses over the length of the web panel between the ribs to,xz, so,x, and unevenly
variable through the thickness of the web stresses sT,x, sT,z and tT,xz.
On the basis of the extensive theoretical and experimental
analysis, it was stated that the main shear stress due the
torque is in the web panel at 0.4–0.5 L of a distance from the
support axis. In this web panel the states of stresses are
analyzed in two cross - sections localized at 0.2a 0.35leff of
distance from the adjacent rib (Fig. 8). The considered section
is located from the support of the beam, at distance of xt,
depending on the ratio between the upper flange tf thickness
and the web tw thickness, as described by expression (15).
2
3
tf
tf
tf
xt
¼ 0:6520:189 þ 0:04
0:003
:
L
tw
tw
tw
(15)
In case of tf =tw ¼ 1; 2; 3; 4 and 5, the ratios xt/L = 0.500,
0.410, 0.364, 0.344 i 0.332 are obtained respectively.
4.
Causes of fatigue stresses
During variable recurring stresses at the place of considered
construction detail, which is the joint between the web and the
flange in the relevant beam (Fig. 10), one can distinguish two
life cycles – incubation period, characterized by a microstructural deterioration of intermolecular bonds and ending with
the crack initiation, and the period of propagation of a crack,
resulting in destroying local or global structural components.
Final stages of both periods are presented in fatigue life graphs,
shown respectively as French and Wöhler curves (Fig. 9). With
high stress ranges of stress variation Ds (or Dt), the incubation
period is relatively short and the propagation period –
relatively long. With low stress ranges of stress variation
however, it looks differently – the period of crack incubation is
relatively long and the propagation period is relatively short.
With stresses that remain close to the stable fatigue limit DsL
(or DtL) the period of cracking initiation can be regarded as the
damage period of the component. In short, it is assumed that
French and Wöhler curves have one common point with
coordinates ðNo ; Ds L Þ.
A fatigue crack is initiated in the first crystal grain on the
surface of the element being under the influence of main shear
stress. It is directed diagonally to the direction (at the angle of
458) of main shear stresses. After reaching the next few
archives of civil and mechanical engineering 18 (2018) 69–82
75
Fig. 8 – Location of main shear stresses due to torque within the length of the runway crane beam.
force at the distance of 0.35leff from the application point of
load Fz, evaluated by its smooth, not abrupt, transition from
one side of the load to the other [13].
The web panel pre-bent in the shape of initial deflection D,
allowed by the requirements of execution [17] is a basic
tolerance (tolerance ensuring the fulfillment of the calculation
assumptions of EC 3 pack concerning the both strength and
stability of beams) set between adjacent transverse ribs, where:
D¼
adjacent grains, it changes to the direction of a right angle to
the first main normal stress and begins propagation [16]. The
fatigue crack initiation is caused by shear stresses because in
pffiffiffi
this case, the endurance of the material is 1= 3 lower than by
normal stresses.
In a complex state of stress, shear stresses are decisive. In a
free-supported, single-span crane runway beam, the stress t1,2
will be equal to the stress variation range Dt1,2 which is
described by expression:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
X 2
X 2
sx sz þ 4
t xz ;
(20)
The amplification of this deflection occurs during two-way
compression with stresses sx and sz,x.
Fig. 9 – French and Wöhler fatigue strength curves.
Dt 1;2 ¼ 0:5
h2w
¼ 0:0075 hw 0:9tw
ð16; 000tw Þ
(16)
where
X
s x ¼ s M;x þ s o;x þ s T;x ;
(17)
X
s z ¼ s z;x þ s T;z ;
(18)
X
t xz ¼ 0:3t V;xz þ t o;xz þ t T;xz
(19)
In accordance with the requirements of the standard [15], it
is recommended to design crane runway beams with web
slenderness hw =tw not exceeding 120, which will prevent web
breathing. The relative technological bend ratio D=tw according to the standard [17] should be no larger than the basic
tolerance ensuring the fulfillment of the calculation assumptions concerning the both strength and stability. This should
not initiate fatigue cracking [18], and assuming that the
eccentricity ey does not occur, then there is no torque of the
upper flange. Meanwhile, according to the standard [15]
calculated value of stress due the load Fz should be assumed
for the eccentricity ey (Fig. 6).
One cannot ignore the rippling in the track axis plan,
allowed by the standard [17] up to 10 mm over the distance of
2,0 m. If taking into account the amplification effect of the
three above mentioned factors on the initial bend of the web
panel, we will find actual favourable conditions for web
breathing and thereby acceleration of initiation of cracking
under the upper flange [18,19].
5.
The reduction coefficient 0.3 in the component on the right
side of the expression (19) includes the value of the diagonal
Locations of fatigue stresses
If the most common web fatigue cracks can be limited to one
panel set between the long transverse ribs, then two groups of
fatigue cracks can be distinguished (Fig. 10).
76
archives of civil and mechanical engineering 18 (2018) 69–82
Fig. 10 – Fatigue cracks in crane runway beams.
The first group of cracks are:
- horizontal cracks 1 of the heat affected zone of the joint,
directly under the edge of the longitudinal flange welds,
where technological undercut and diverse microstructure
occur (Fig. 10 according to [20]),
- horizontal cracks 2 in longitudinal flange welds near the
middle of the web panel,
- diagonal cracks 3, when they occur at the vicinity of the ends
of the panel [16],
- horizontal cracks 4at the bottom of the web next to the
beveled termination of the rib (fig. 10 according to [21]),
- horizontal cracks under the bottom end of a long or short
transverse rib.
fatigue category from 36 to 71 MPa due to the vertical
stresses of the compression,
c) Compressing the transverse rib spacing by adding short ribs
in the panel between long ribs (Fig. 12a), lowering thus the
maximum compression stress sz,0 [12] c times,
The second group of cracks are:
- vertical cracks 5 of the lower end of the weld connecting the
vertical rib with the web,
- transverse cracks 6 of the lower flange by the half-round
relief opening in the web.
- horizontal cracks 7 of the top flange next to the gusset plates,
used for attaching the horizontal bracing of the crane
runway girder.
The effects of fatigue stresses of crane runway beams in
terms of unconditional durability [22] will be lower if the
following are used:
a) Rolled sections at the transition of the upper flange into the
web (Fig. 11), which increase the fatigue category of this
construction detail under vertical stresses sz,x from 36 to
71 MPa [23],
b) Full penetration butt welds instead of fillet welds connecting the upper flange with the web, which increases the
Fig. 11 – Rolled sections at the transition of the upper flange
into the web.
77
archives of civil and mechanical engineering 18 (2018) 69–82
weld, localized from the outer surface of the upper flange at
distance z1, where vertical, normal stresses are:
s z1 ¼ s z;0
Fig. 12 – (a) Short ribs situated between long ribs of the
panel, (b) Longitudinal ribs within 0.2 hw from the upper
flange.
Ad a)
After using longitudinally cut rolled I-section, the condition
of fatigue strength should be checked for the longitudinal butt
(21)
If lifting rails of SD type [24] are used, then the most suitable
rolled cross-section is HL, that allows proper mutual fastening
by clamps Lp4 [24].
Ad b)
In the tee joint, which is a combination of a beam flange
with a web, there is a horizontal gap with high stress
concentration at its ends between the double-side welds (fillet
or partial penetration butt weld) (Westergaard issue [25]). In
these local areas with altered microstructure, the material has
a reduced static and fatigue strength.
Ad c)
The maximum spacing of ribs a1 for which the coefficient c
will be bigger than 1.0 (Table 1) may be obtained on the basis of
the transformed expression (5). The load-bearing capacity of
the transverse long rib can be determined on the basis of the
formula [26]
Q gr ¼
d) Longitudinal ribs within 0.2 hw from the upper flange,
without short transverse ribs, that reduce the effects of
torque at the upper beam zone and the effects of the
bending moment acting out of the web plane, produced by
the compressive stresses at its initial deflection D (Fig. 12b).
e) Fillet rib's welds with design thickness (effective) not
smaller than 0.4 tw .
f) Stiffening protecting the lower flange from twisting by
using vertical fins connecting lower ends of ribs with the
flange by longwise welds with higher fatigue category than
the crosswise ones (Fig. 13).
leff
leff
z1 2
1
:
þ 2z1
hw
h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
i
2 bf t2f =tw þ 4bf tw þ 2:8bs ts f y
g M0
;
(22)
Ad d)
The longitudinal rib welded to the transverse ribs stiffens
the initially deflected web (while bending it out of its plane),
which prevents horizontal cracks in the area of thermal
influence of the web (above the weld) or in the weld itself [18].
In order to avoid the fillet weld accumulation in the cruciform
connection, it is reasonable to replace the produced in this way
welded I-section (upper flange + longitudinal rib + a portion of
a web between them) by rolled I-beam [27].
Ad e)
In tee and cruciform fillet weld joints of design thickness
smaller than the above mentioned, the fatigue cracks are
initiated in the root of the weld [28]. Therefore, initiated and
propagating cracks cannot be seen for a long time after
Fig. 13 – Stiffening the lower flange against its twisting by using vertical fins.
78
archives of civil and mechanical engineering 18 (2018) 69–82
Fig. 14 – Numerical model: (a) view in the ABAQUS with a continuous rail [mm], (b) cross-section with dimensions [mm], (c)
meshed model.
technical inspection. Thus, a greater risk of danger arises,
because cracks reaching the weld's face may have dimensions
close or equal to the critical, as a brittle form of cracking.
Ad f)
The cross-sections of crane runway beams are usually
mono-symmetric with a weaker lower flange that is more
vulnerable to twisting than the more rigid upper flange with a
rail. Twisting of the lower flange, that results in web cracking
under long ribs, is caused by horizontal actions of crane wheels
[34]. Even ribs adapted to the flange, or welded using crosswise
welds do not prevent the lower flange from twisting (Fig. 13).
6.
Field joints of the crane track
Because of fatigue cracks of both the upper web zone and
upper plate, a very dangerous area is located always in the field
joint of the crane track [7,27,30]. The first fatigue cracks occur
under such joint and this fact should be attributed to large
values of local compression stresses sz,0 and accompanying
them stresses so,x, to,xz.
In order to analyze the impact of the rail joint on the steel
web stressing, a numerical model of steel beam segment with
the length of 800 mm together with crane rail (SD75 type [24])
was created in the ABAQUS 6.14-2 [31]. The model assumes
material of elastic–plastic characteristics according to Prandtl's model [32]. The strength parameters are assumed as for
steel S355 [33]: fy = 355 MPa, fu = 510 MPa, and material constants E = 210 GPa, W = 0.3.
The load was modeled as a crane wheel moving along the
rail and causing the vertical force of 100 kN. It was assumed
that the force denoted by Fz is centrally applied to the rail head
over the central plane of the web. A static nature of the load
was assumed in the model. Calculation models were constructed out of SOLID elements of type C3D8R (elements with
20 nodes) [31]. Longitudinal welds in the flanges-web junction
in I-section beam were modeled as an equivalent surface of
contact using function Constraints – Type: Tie. The adopted
geometrical assumptions are presented in Fig. 14. The rail to
upper beam flange connection was modeled as not carrying
the force of delamination (sliding interconnection). The elastic
pad under the rail was not modeled. Numerical model does not
take into account the 25% wear of the rail head.
The analysis of convergence of the influence, that the size
of the finite element mesh has on the compression stress
values in the steel at the places of load application (directly
beneath the top plate) was carried out for the model. The
percentage differences presented in Table 3 refer to the result
for the size e = 2 mm. Finally, a mesh of 5 mm of the finite
element mesh was selected for the analyses.
After obtaining a satisfactory convergence of the stresses in
the beam web by using an analytical method in accordance
with the standard [15] for the model with a continuous rail, the
analysis has been extended for a crane wheel travelling
through 3 different crane rail splices in the form of: orthogonal
contact, bevel contact, and stepped bevel contact (Fig. 15). It
has been assumed that the gap between the rails is 2 mm. It is
the largest permissible gap by [34,35].
Maps of local compressive stresses in the web under the
load of passing crane wheel were determined (Table 4).
Extreme values of stresses were observed for the bevel and
stepped bevel contacts, when positioning the wheel centrally
Table 3 – The influence that the ES mesh has on stresses.
Analytical
calculation
[15] [MPa]
43.52
Mesh size
[mm]
sz,0
[MPa]
Difference
in results [%]
2
5
10
15
20
43.26
43.49
43.97
45.04
46.68
0
0.53
1.64
4.11
7.91
archives of civil and mechanical engineering 18 (2018) 69–82
79
Fig. 15 – Splices of the crane rails under consideration.
Table 4 – Distributions of the local compressive stresses.
in the gap between rails (xr-denotes the distance from the axis
of the beam support to the axis of the gap between rails). The
maximum stress in the orthogonal contact occur at the
moment of the whole wheel reaction, influencing one of the
rail ends that are located in the contact. Therefore, as visible in
Fig. 16, the area of reading the stresses is misaligned with other
cases. Maximum stress values for each contact are listed in
Table 3. Fig. 16 shows distributions of local compression
Fig. 16 – Summary of distributions of local compression stresses.
80
archives of civil and mechanical engineering 18 (2018) 69–82
stresses. Stresses were read in first nodes of steel web finite
elements, directly beneath the bottom surface of the top flange
of the crane runway beam.
7.
Analytical methods of assessing fatigue in
the web of crane runway beam
Assessment of fatigue in the web of the beam can be carried
out in a simplified way by calculating the equivalent fatigue
load Qe of constant amplitude at 2 million cycles. The
characteristic value of the maximum wheel load Qmax takes
into account the number of wheels on one side according to
the standard [37], while the equivalent damage factor li, for the
class load spectrum Si, takes into account only those classes
that cause significant fatigue damages S4 S9.
The equivalent impact Qe is determined according to the
standard [37] on the basis of the dynamic fatigue factor wfat, as
’fat ¼ maxð’1 ; ’2 Þ:
Q e ¼ ’fat li Q max ;
(23)
or separating the wheel load Qmax on the portion of the crane
self-weight Qc with determined for it wfat,1, and the lifted load
Qh with a corresponding wfat,2, [38]:
Q e ¼ li ’fat;1 Q c;max þ ’fat;2 Q h;max ;
(24)
In the case of the first constructional detail (tee-connection
of the upper flange with the web) it is recommended in the
standard to calculate the equivalent stress at 2 million cycles
sE,2 from the formula:
s E;2 ¼
(26)
Wherein the stress from the torque MT should be calculated
not only from the eccentric vertical pressure Fz but also from
the transverse horizontal force HT such as the largest from the
location eccentric total weight transported with respect to the
drive force K or from the bridge chamfering or from the trolley
inertia.
sT ¼
2tw
Q ey þ 0:75HT hr ;
Jf þ Jr r
(27)
The condition of fatigue strength in the standard [22] will
have the following form:
Ds E;2
1:0;
Ds C =g Mf
(28)
where DsC should be 150 MPa for all steel grades.
All components of the stress condition have to be taken into
account while testing the static strength.
s x þ s o;x where
’fat;1 ¼ 0:5ð1 þ ’1 Þ:
fy
g M0
;
s z;0 þ s T fy
g M0
;
fy
t xz þ t o;xz þ t T;xz pffiffiffi
;
3g M0
(29)
’fat;2 ¼ 0:5ð1 þ ’2 Þ:
The fatigue assessment should be carried out for the most
sensitive constructional details of two welded structural
components of the beam: tee-connection of the upper flange
with the web, and cruciform connection of the transverse rib
with the web. There are complex states of stresses in both of
them, such as effects of a general bending of the beam by force
Qe – sx, txz (due to cross-sectional bending moment My and
cross-sectional shearing force Vz) and local effects from drive
vertical wheel pressure – sz,x, and the torque – sT, with
additional local effects so,x, to,xz from the non-uniform distribution of stresses sz,x along the x axis, and finally – tT,xz from the
non-uniform distribution of stresses sT along the x axis and in
the direction of the web thickness. Both of mentioned above
cases are neither described in the standard [22] nor in
regulations of the International Institute of Welding [36].
There is a proposal for testing the fatigue strength in
complex state of stresses presented in the manual [29], taking
into account the components of stresses Dsx and Dtxz from
general bending, DsT from twisting, Dsz,0 and Dto,xz = 0.2Dsz,0
from local effects, and also considering the number nk of crane
wheels on one beam, we obtain:
3
3
3 Ds
3 Ds
x;E;2
z;0;E;2 þ Ds T;E;2
þ nk g Ff g Mf
g Ff g Mf
Ds x;C
Ds z;C
5 !
5 Dt
xz;E;2 þ 0:2Ds z;0;E;2
þ nk g Ff g Mf
1:0
Dt C
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s 2x þ 0:36t2xz þ 0:8s z;0 þ s T ;
(25)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f
2
2
y
;
s x þ s o;x þ s 2z;0 s x þ s o;x s z;0 þ 3 t xz þ t o;xz g M0
(30)
Components of the local stresses have to be assumed with
the following values:
s o;x ¼ 0:25s z;0 ;
t o;xz ¼ 0:3s z;0 ;
tT;xz ¼ 0:25s T ;
(31)
In case of the second constructional detail (cruciform
connection of the transverse rib with the web), in which fillet
welds are under shear,
tQ ¼
Fz
;
4aw lw;eff
(32)
where aw and lw;eff mean respectively, the effective (design)
thickness and length of the weld, and the web is only treated
with horizontal stresses sx (when the force Fz acts at the rib
axis, the stresses sz,0 decrease radically at the expense of tQ)
then one can use the condition of the standard [22] for orthotropic bridges by calculating the variation range of the first
main stress [1]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
;
Ds E;2 ¼ 0:5 Ds x þ ðDs x Þ2 þ 4 Dt Q
(33)
where, in free-supported, single-span crane runway beams,
the variation ranges are equal to stresses themselves because
minimum horizontal stresses are zero.
archives of civil and mechanical engineering 18 (2018) 69–82
In the formula (33), the sx stresses should be determined at
the top and at the bottom of the vertical weld near the rib and
the fatigue category in the condition (28) should be taken as
DsC = 56 MPa.
8.
Summary
The article presents the broader problem of fatigue cracks in
the crane runway beam. Special attention is paid to the
complex state of stresses in the steel webs of those structures,
from which the process of initiation and propagation of fatigue
cracks may start.
Analytical formulas were defined to determine those
stresses. The way of determining the fatigue strength of the
crane runway beam by expression 16 in the compression
portion of the web, described by expressions 17, 18, 19 was
verified experimentally [13]. The differences between calculated values and those obtained by test do not exceed 15%. On
this basis the fatigue strength of the crane runway beam was
recommended to be verified in terms of the standard [14] as
more objective in comparison with the assessment given by
[22].
The most common causes of fatigue cracking have been
collected and listed (e.g. rippling in the rail axis plane, web
breathing) and their locations in the structure (including the
field joint of the crane rail, the area of the web just beneath
the flange of a crane runway beam). Extensive numerical
analyses have been conducted in order to analyze the impact
of the shape of the crane rail splice on the strength of the
steel web in greater detail. For each type of contact
(orthogonal, bevel and stepped bevel), an increase in tested
compressive stresses sz,0 in relation to the case where the rail
was continuous, was observed. The smallest increase in
stresses occurs in the stepped bevel contact (approx. 50%),
the largest occurs in the orthogonal contact (approx. 150%)
(see Table 4).
The problem analyzed by the authors requires further
analyses and research in order to determine the detailed
design guidelines which are aimed at improving the functioning of crane runway beams under cyclically variable load.
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