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J. Construct. Steel Research 34 (1995) 75—lOl
© 1995 Elsevier Science Limited
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Printed in Malta. All rights reserved
0143-974x/95/$9 so
3. .
ELsEv1ER
A Unified Approach for the Design of Steel Structures
under Low and/or High Cycle Fatigue
Giulio Ballio
Structural Engineering Department, Politecnico di Milano, Piazza Leonardo da Vinci 32
20133 Milan, Italy
&
Carlo A. Castiglioni
Department of Structural & Geotechnical Engineering, ‘La Sapienza’ University, Roma,
Italy
(Received for publication 6 December 1993)
ABS TRACT
In this paper, a method is presented trying to unify both design and damage
assessment methods for high and low cycle fatigue. In particular it is shown that, by
interpreting the stress range A0 as the ideal stress range associated to the real strain
range As in an ideal perfectly elastic material, high and low cycle fatigue test data
can be interpreted by the same Wohler (S—N) lines usually given in recommendations for (high cycle) fatigue design of steel structures. Furthermore, local buckling
can be regarded as a notch eflect, an eflect which is intrinsic to the various shapes,
because it is strictly correlated to their geometrical properties (in particular of the
slenderness ratios b/t and h/tw of the flanges and the web). It is also shown that, in
the case of variable amplitude loading histories reprocessed by the rainflow cycle
counting method, a linear damage cumulation rule together with the previously
defined S—N curves (a procedure usually adopted in high cycle fatigue) can lead to
a reliable collapse criterium for low cycle fatigue also.
1 INTRODUCTION
Since the 19th century, when the use of metals in engineering applications
began to increase, it has been recognised that metal components and
structures subjected to repeated load cycles may fail in service, even
75
76
G. Ballio, C. A. Castiglioni
though they would usually be capable of bearing considerably larger loads
if the loading were of a ‘static’ nature. This type of failure, consisting of
the formation of cracks under the action of varying loads, is known as
‘fatigue’. Eurocode-31 defines fatigue as ‘damage in a structural part,
through gradual crack propagation caused by repeated stress fluctuations’.
To produce a complete list of the types of structures that may sufler from
fatigue failure is nearly impossible; it is, however, interesting to realise that
it has been estimated that nearly 90% of the failures that occur in
engineering components can be attributed to fatigue.2
Examples of structures that may collapse due to fatigue cracking may
conveniently be classified under some of the typical forms of loading which
may cause fatigue failure:
— fluctuating live loads: bridges, cranes and crane runway girders,
engine frames, locomotive underframes, chassis frames and axles of
motorcycles, cars or trucks, ships, aeroplanes, presses, earth-moving
equipment, farm machinery;
— pressure fluctuations: pressure vessels, pipework, containers, aircraft
fuselage;
— temperature fluctuations: process equipment involved with hot or cold
materials, liquids or gases;
— vibrations: rotating machinery, conveyors;
— environmental and atmospheric conditions: chimneys, oflshore platforms.
Depending on a number of factors, these load excursions may be introduced either under stress or strain controlled conditions. Depending on
the number of cycles sustainable to failure, and on their amplitude, we can
distinguish failure for high or low cycle fatigue.
Failure by high cycle fatigue is characterised by a large number of
Withstandable cycles with a nominal stress range Ao in the elastic range
(i.e. under stress controlled conditions, with Ao<2fy, fy being the yield
stress of the material). This is a well-known effect, and has been studied
since 19th century in mechanical engineering applications. In fact, one of
the earliest investigations of stress controlled cyclic loading effects on
fatigue life was carried out by Wohler3 who studied railroad wheel axles,
which were suflering from a number of failures. Civil engineers became
involved with this problem during the 1930s, when the first welded
connections appeared as a sure promise for the future. Since then, a
number of studies of structural details have been carried out in many
countries,“'“ and although only a limited number of typologies of
connections and of structural details can at present be considered
Design of steel structures under low or high cycle fatigue
77
thoroughly investigated, it can be stated that the general aspects of the
problem, and in particular the basic methodologies for assessment and
design, can be considered well established.
Low cycle fatigue is characterised by a small number of cycles to failure,
with large plastic deformations (i.e. under strain controlled conditions,
with strain range Aa>2sy=2fy/E, E being the Young’s modulus of the
material). In general, low cycle fatigue problems in civil engineering
structures arise either under seismic loading or in pressure vessels or under
severe thermal cycling. As extensively discussed and demonstrated by
theoretical and experimental research works (e.g. Refs 12-16), cycles with
large amplitudes in the plastic range are usually connected with local
buckling in structural members. At present, knowledge of low cycle fatigue
behaviour of civil engineering connections is not yet as well established as
the behaviour of high cycle fatigue. In particular, there is no generally
recognised design or damage assessment method for low cycle fatigue, and
a clear definition of a collapse criterion is lacking.
Local buckling can be regarded as inducing a sort of notch effect in the
component. Localised plastic strains can be generated by loading a
component that contains a notch. Regardless of the external mode of
loading (cyclic stress or strain controlled), the plasticity near the notch
root experiences a strain controlled condition dictated by the much larger
surrounding mass of essentially elastic material. A reasonable assumption
can be made that the same number of cycles is needed to develop a crack
at the notch root of an engineering component and in an unnotched
specimen, if the two cracked regions experience the same cyclic stressstrain history. Furthermore, it is important to recognise that fatigue
damage will occur only when cyclic plastic strains are generated in the
crack region.
Starting from these considerations, this paper describes a procedure that
tries to unify design and damage assessment methods for structural details
under high and low cycle fatigue. After discussing the proposed approach,
its experimental validation based on constant and variable amplitude
cyclic test results will be presented. It will be shown that, by transforming
the strain range in an equivalent stress range (Aa*=AaE) computed by
considering the material as indefinitely linear elastic, the experimental test
data obtained under cycles with a constant amplitude in the plastic range
can be interpolated by the same stress range—number of cycles to failure
(S—N) lines usually given in recommendations for the (fatigue) design of
steel structures (e.g. Ref. 1). Furthermore, a linear damage accumulation
model (Miner’s rule), together with the rainflow cycle counting method,
can be adopted for the damage assessment under variable amplitude
loading.
78
G. Ballio, C. A. Castiglioni
2 THE PROPOSED APPROACH
The proposed approach to unify the design and damage assessment
procedures for steel structures under low and/or high cycle fatigue is based
on the two following assumptions:
(1) To know, for a given structural detail (cycled under strain controlled
conditions), the relationship between the number of cycles to failure
Nf and the cycle amplitude in terms of generalised displacement
components As (i.e. of displacements Av or of rotation A6 or of
deformation As). These relationships have the same meaning in high
and in low cycle fatigue with the following difference:
~— in high cycle fatigue the component is subjected to cycles in the
elastic range, with a cycle amplitude As<2sy where Sy is the
value of the displacement component corresponding at first yield
in the material;
— in low cycle fatigue the component is subjected to cycles in the
plastic range, i.e. with an amplitude AS>2sy-
(2) Damage accumulation in a structural detail is a linear function of
the number of cycles sustained by the component itself. This means
that Miner’s rule, usually applied in high cycle fatigue damage
assessment, can also be applied in low cycle fatigue.
An immediate consequence of the second assumption is the definition of
a unified failure criterium for both high and low cycle fatigue: a structural
component fails when Miner’s damage index reaches unity. A consequence
of the first assumption is that it is possible to interpret low cycle fatigue
with the same laws commonly accepted for high cycle fatigue.
In fact, in high cycle fatigue (under stress controlled conditions):
—— a structural component is subjected to load cycles having a constant
amplitude AF;
— the maximum value of the load excursion AF must be lower than
the value Fy associated with the attainment of the yield stress in the
material. FY may be theoretically computed or experimentally evaluated;
— the nominal stress level induced by the external load F may be
computed either theoretically or with conventional methods, leading
to a relationship of the type o=o(F);
— the stress range Aa=a(AF) is finally correlated to the number of
cycles to failure Nf, independently from the yield strength of the
material.
Design of steel structures under low or high cycle fatigue
79
In order to generalise this approach, under the assumption of indefinitely
linear elastic material, it can be written:
do
Ao=AFfi
(1)
with:
do _
_ o(Fy)
dF - cost - FY
It follows that:
Ao = E a(Fy)
(2)
FY
For example, for a simply supported beam of span L, with a section
modulus W and loaded by a concentrated transversal force F applied at
midspan, being o=FL/4W, eqn (2) becomes
AF FyL
AO'—?yZW
In low cycle fatigue (under strain controlled conditions):
~ A structural component is subjected to displacement cycles having
a constant amplitude As.
— The maximum value of the excursion As of the generalised displacement component s is greater than the value Sy associated with the
attainment of the yield stress in the material. Sy may be theoretically
computed or experimentally evaluated.
— If the material can be regarded as an elastic perfectly plastic one (as
in the case of steel), and the hypothesis of concentrated plastic hinge
can be considered realistic (as shown in Refs 17 and 18 for steel
members under seismic loading), it can conventionally be assumed
that strains s are proportional to the generalised displacement
component s, and it can be stated that:
§f=A§
8y Sy
(3)
This equation defines the nominal strain range As in a particular
way, and deserves a detailed discussion. For a simply supported
80
G. Ballio, C. A. Castiglioni
beam with a three-point loading as given in Fig. 1, the ‘curvature moment’ M,= —EJv”=(EI/z)s, and the ‘equilibrium
moment’ M,,=FL/4 are equal as long as e is in the elastic range
(Fig. 1(a)). When s exceeds sy (the yield strain), M, is greater than M,,
in the yielding Zone due to local reduction of the elastic stiffness (Fig.
1(b)). The deflection s could then be calculated from
,_ fifid
_ EJF X
taking account of the nonlinear distribution of M,. The deflection
can, however, also be understood as
_ K3‘M@d
S_ EJ F X
where M;" is assumed to have the same linear distribution as M,
(Fig. 1(b)). In this case, the local reduction of stillness is taken into
account by an equivalent uniform reduction of stiflness along the
total beam length. The peak values s and a* difler accordingly.
The definition of the local strain range As according to eqn (3) is that
of 3*.
—
For an ideally linear elastic material, the relationship between
strains s and the load causing the displacement s can be written as:
Es=o(F)
(4)
E231‘! PLASTIC ZONE
____ M;
____ M,,
F
l
' _ ' _ _ _ _ _ ' "
I
7}‘ ~\-. \§M U‘
'1
4 .4‘ '7/C
_ ~\\,// K
L
(a)
M g
l
X‘ 'l='1\._;._§|l |:_ >_,€:_;*;2- L’ K
"'~-;,:\"_;(-;»~“
M Z:
ME
(bl
Fig. 1. Schematic for ‘curvature moment’ and ‘equilibrium moment’ for (a) an elastic and
(b) an elastic—plastic beam, and identification of the ‘equivalent moment’ M,*, leading to
the definition of As according to eqn (3).
Design of steel structures under low or high cycle fatigue
81
From eqn (4), it follows that:
E8y = o(Fy)
(5)
— The value EAe, which can be interpreted as the stress range
associated in a ideally linear elastic material to the strain range As,
can be finally correlated to the number of cycles to failure Nf.
Because of eqn (3), it can be noticed that:
A
A
EAs=E -S
8,
=_’
o(Fy)
Sy
sy
(6)
Equation (6) is similar to eqn (2) that is valid for high cycle fatigue. It
differs from eqn (2) having considered cyclic displacements instead of cyclic
forces. For example, for the same simply supported beam previously
considered, where v is the deflection caused by the applied load F, eqn (6)
leads to: EAa=(Av/vy)(FyL/4W).
3 EXPERIMENTAL VALIDATION
In order to experimentally validate the proposed approach, tests were
performed at the Structural Engineering Department of Politecnico di
Milano, on full-scale cantilever members 1-6m long, of the commercial
shapes HE220A, HE220B and IPE30O (Table 1), using equipment”
capable of applying horizontal cyclic actions in a quasi-static way.
For evaluating the behaviour of the specimens in the plastic hinge zone,
in addition to the top displacement and applied force, a number of
displacement transducers were connected to their web and flanges (Fig. 2).
The instruments on the specimen flanges were applied on different levels,
at various distances from the base, both at the centre, near the web to
TABLE 1
Geometrical Properties of the Specimen Shapes
Shape
HE220A
HE220B
IPE300
Area
(cmz)
64-3
91-0
53-8
Flange
Web
b
t
b/t
h
1,,
h/rw
220
220
150
11-O
16-O
10-7
20-0
13-8
14-0
188-0
188-0
278-6
7-O
9-5
7-1
26-8
19-8
39-2
82
G. Ballio, C. A. Castiglioni
IDENTIFICATION OF
MEASUREMENT POINTS
___-.__L_-.I_ __
\/(ill
iii
Fizz)
E
9
INSTRUMENT LEVELS (mm)
Q
-
12
~
7
i
l
soo
5
6
zoo
l _
_
,1: is 14 l Tl
I
’
I
=
O
I
_ zoo
1:0
100
E
M
all
SECTION A-A
.
H
’ till 1 rm/r
,3"
7
-5
S
9
"
.2
a
11
:2)
10
-=
Fig. 2. Specimen set-up.
flange connection, to obtain information on plastic rotations and curvature at the plastic hinge, and at the edges, in order to obtain information
regarding torsion and local buckling of the flange plates. Connection of
these instruments was carefully realised in order to avoid measurements of
undesirable displacement components.
Design of steel structures under low or high cycle fatigue
83
The instruments connected to the specimen web were placed at the same
level as those on the flanges, allowing the assessment of web buckling and
its correlation with that of the flange plates.
3.1 Constant amplitude tests
3.1.1 Description of the test results
To date, 34 tests were performed (11 on HEA shapes, 12 on HEB and 11
on IPE) imposing to the specimens displacement cycles with a constant
amplitude Av. Most of these tests were performed (Tables 2-4) with a zero
mean value (vm) of the top displacement, although some replicates were
carried out for HE220A (Table 2) and IPE300 (Table 4) by keeping the
cycle amplitude Av constant and varying the mean value vm.
In addition to the usual hysteresis loops in terms of force applied on the
top versus top displacements, for each specimen the experimental measurements, digitally recorded, were processed following the procedures recommended in Ref. 20 in order to obtain information regarding the resistance
ratio (F/Fy), rigidity ratio, cumulative energy ratio and buckle size which
were plotted versus the number of applied cycles.
In order to highlight the different behaviour of the three shapes under
cyclic loading, some typical results are presented here and discussed with
particular reference to three specimens, HE220A no.3, HE220B no.9
and IPE300 no.5, which were tested under cycle amplitudes resulting
TABLE 2
Summary of HE220A Constant Amplitude Tests
Cyclic tests on HEA
Test
HEA2
HEA3
HEA4
HEA5
HEA6
HEA7
HEA8
HEA12
HEAI3
HEA14
HEA1 5
v,,,(mm)
0000
—l00
— 50
O
A v(mm)
vy(mm)
Fy(kN )
Nf
N;,,
Failure
240
200
280
160
120
340
80
80
200
200
60
20
19
18
18
19
19
19
20
21-4
20-7
22
140
138
135
150
150
143
145
143
143
148
147
16
19
9
41
76
6
165
150
19
24
405
16
19
9
43-5
82
7-5
189
172
21
28-5
436
fimmfifiwmmmmm
84
G. Ballio, C. A. Castiglioni
TABLE 3
Summary of HE220B Constant Amplitude Tests
Cyclic tests on HEB
Test
HEB2
HEB3
HEB4
HEB5
HEB6
HEB7
HEB8
HEB9
HEB 10
HEB11
HEB 14
HEB15
v,.,,(mm)
Av(mm)
0000
240
160
280
200
220
180
180
200
260
300
120
80
vy(mm)
\1
Pit-—*P—*P)—\P *)iI—* I—*l I—*
\O _0\O O 0\IO0O0\lO¢\lI
Fy(kN )
185
185
192
190
190
190
187
190
190
190
187
189
Nf
Nfu
Failure
15
40
7
20
18-5
El’ Oi
;O\
Ln»I-*I\JUJl\.)>—~I\J\O§J10\UJO \]>-*\l@\lO —l>
38-5
28
13
6
65
141
fiigc/itJr/2 ¢/at/13$:/1
Failure
TABLE 4
Summary of IPE300 Constant Amplitude Tests
Cyclic tests on IPE
Test
IPE1
IPE2
IPE3
IPE4
IPE5
IPE6
IPE7
IPE8
IPE12
IPE13
IPE14
vm(mm)
\:00 0 0 0
140
0
Av(mm)
200
160
120
80
140
100
60
240
140
140
40
vy(mm)
Fy(kN )
Nf
Nfu
iytywwmw -l>~U1
148
148
147
142
146
142
147
148
146
146
147
4-5
9
23-5
60
12
32
132
3
12
12
216
9
11
25
64
14
33
136
6
13-5
14-5
)—l *>—lP—-*)—*\—*
13
13
12-8
13-3
12
to|\.> DJ
fit/Jwmgm m m
practically in the same ductility ratio Av/vy. However, before any comment, it is important to remember (Table 1) that HE220A profile has a b/t
ratio of the flanges (20-0) larger than that of HE220B (13-8) and IPE300
(14-0), while its width to thickness ratio for the web (h/tw=26-8) is
intermediate between those of HE220B (h/tw= 19-8) and of IPE300
(h/tw = 39-2).
Design of steel structures under low or high cycle fatigue
85
Figures 3, 4, 5 and 6 respectively show, plotted against the number of
cycles withstood by the specimen:
— the resistance ratio (F/Fy), of the load carrying capacity (F) of the
specimen at each positive reversal normalised on the yield strength
(Fy) conventionally determined following the ECCS recommended
procedure;2°
-Q--»+=HEA 3 AV/Vy=11.11
A-=-Q-H-HEB 9
AV/V§=11.11
1 ' 6 es-5-s-§>_1ri1_§_tA_v/visit-29)
_________________ - I
I
I
I
I
I
5.
~I
II
II
I_______._
/
POSIFyAT
FTIVEREVERSALS
1
OE"'0iv
E"F‘
PP
.0l\Jcaon0
IaIn
'0-L.4;-"
_ _ _ __
IIII
I
II
III
I 'I
I I_ I I_
I
I
' III I
II
I
' Il
I
III
_ ‘II
III
IIII
.
III
II_ II
III
_
I
lI
5. I
III
_ I
III
_
'7I' II
-II
I','-5
IIII
III
r
N4__ _U\_
1I_III
_II
_ .L_ L_ _
TA-1‘
T o:———vI
I——_
I———I
4—_
,_i‘A~_ _lAlA_4 l'_______|
I _
I1\)lII
TTTTT
ITTTTTTT
QTFT TI T TI T T T
U'__lT_'T__TTIT‘____?
TT
JI
IIL
>-'I
I
I.
III
°‘_.
CYCLE
‘I
II
I0-,I_T _ TU\_'
_
Fig. 3. Comparison of typical strength degradation trends for HE220A, HE220B and
IPE300 shapes.
Kw;-MIHEA 3 (Av/v,,=11.11
Aei/MAHEB 9
AV/VY=11.11
1 ' 4 %>_Ii>:E_§>_IA_v/yrsii-291
__________ __
I
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ie
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I I I I
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I’I
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I
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——I4<—I~—~——-— I»-4
OOOO
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.:|14. I_. .I .I L.I .I .I .I‘_I. _=.I . .L.I -_. I. ;t
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D- I___,_ _,_I_ _ P 5'II
sI ID- I III
D_ P'II -A-I -I — :I-III
I- III_ __ _ _ ¢_ _ _
“""r“‘“‘I'*’I
0Ur-r0 ID. I’'I III
I_- D_ 'III
5' TITO‘ ' l '1I5
CYCLE
210
25
30
Fig. 4. Comparison of typical trends of rigidity ratio for HE220A, HE220B and IPE300
shapes.
86
G. Ballio, C. A. Castiglioni
:»@»H+= HEA 3 EAV/VY: ,_.
Wm HEB 9 AV/VY=1 >->—*
>I—~>1 ' 1 <>s—>_@9s>_I1’1~1_§_tA_v/yisii-2
oP‘__
~e-1-»-‘,___
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RATIUMULATOIVEENERGY
C
0.6
F3 w
III
IIII
I I
II
:IVI:
:III
I:/;II II
:II
IIII
I.II:IIIII:II I//I
II
III
o
III
II
I'
_ _ _- _L _ I_ _ '
-4 I4»-_4—I -4——I»—-
U.
10
5IIII
I
II
IIII
II
““F“‘
I“" I" 'I" I'_I" ‘
15
CYCLE
20
25
30
Fig. 5. Comparison of typical trends of absorbed energy ratio for HE220A, HE220B and
IPE300 shapes.
0.0 -1-
BUCKLING AT NEGATIVE REVERSALS
I—— ----—- —--
7
I
_ 30.0 _I _ _ _ _ _ _ _ _ _ 5Q?-$;__,‘,Q
._ _ I I I
——m
(rnrn)
ZE
SI
I I I I
"I
-60.0 »~——~
P’
L
I as
R
CKE
B -90.0 - - - - - I
7
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ra__
I I I I I
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I I I I
I
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9999(>IFLANGE l\=100 mm RIGHT
GGGSOIFLANGE h=200 mm LEFT
>H+-I+-H<lFLAl\'GE lI=200 mm RIGHT
-7- -
ir*~—~-
I__’.__
_;I_____.
'
I I I I
I
SIDE
SIDE
§IDE
I I I
__I__'
I
IPE5
HEA3
I-IE‘B9
I I
_a__
—1200;I--I-I---Ii-IIII-I--III--II--N
0
5
10
15
20
25
r.-=_AL‘-0
CYCLE
I I I I I
=-LU\
Fig. 6. Comparison of typical trends of buckle size for HE220A, HE220B and IPE300
shapes.
the rigidity ratio (k/ky) of the stiffness (k), conventionally determined
following Ref. 20 as the tangent modulus corresponding to the change
of the sign of the applied load, normalised on the elastic initial one (ky);
the cumulative energy ratio (E/Ey)2° of the absorbed cumulative
energy E (defined as the sum of the areas of all the cycles withstood
by the specimen) normalised on the energy (Ey) absorbable, under
the same cycles, by an ideal specimen made of an elastic perfectly
plastic material;
Design of steel structures under low or high cycle fatigue
87
— the buckle size of the flange edges (i.e. the maximum deflection of the
flange due to buckling).
Figure 7 plots, for all the specimens tested under constant amplitude
loading, the cumulative energy ratio versus the number of imposed cycles
1.20
2l"1.10 ~ ,—~~. \ ‘ HEB220
<1Di
X
51.00 { _
I‘
: """"" ~~_I:~~.
Emmi
“‘*-
._
Emmi
0.“/oi
CUMULA
_
--- new rmvns
0.80 '
0,50
. . .. . . . . . , I . . . .. . .. ,. .... . .. . , .... .. . .. .
100
O
1.207
§
:
Q
200
300
N'(VI>+ — VP-)/Vy
400
HEA220
51.10 J
51.605
I“
:
20.00 -_
E000
\
_ . _ _ _ _ _ -_
--c~--
9 ~zo
CUMULA
0.60 -'
0.50
______
--- mu FAILURE
O
. . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . ,
100
200
300
400
N'(V1>+ — VP-)/Vy
1.20 I
g;1.10¢
IPE300
9.
M
;I
Z
.
5 1.00 -f
III
Z 0.90
-I \
.00-]
ECUMULATIVE
O _O~lO
*~_
_
----- __
1%-'''_= 1
--- mm I-*IIn.u1u:
0.80 -
0,50“. . . . . . . . . . , . . . . . . ..... . . . . . . . . , . . . . .
0
100
200
300
N‘(VP+ — V1»-)/Vy
400
Fig. 7. Cumulative energy ratio versus total cumulative excursion in the plastic range
withstood by the specimen, for all beams tested under constant amplitude cycles.
88
G. Ballio, C. A. Castiglioni
times the cyclic excursion in the plastic range (computed as Av—2vy)
divided by vy. The term on the abscissa can be regarded as the total
cumulative excursion in the plastic range withstood by the specimen.
By examining these figures, the following considerations can be drawn,
having a general validity, for the three shapes examined in this study:
(1) Strain hardening effects are much greater in HEB than in HEA
specimens, while IPE ones show intermediate behaviour;
(2) Deterioration elfects, causing a reduction in load carrying capacity
(Fig. 3), stiffness (Fig. 4) and hysteresis loops area (Fig. 5) begin
earlier in HE220A specimens (having larger b/t ratio for the flanges)
than in both IPE300 and HE220B ones; IPE300 beams, however,
characterised by larger h/tw slenderness ratios for the web, although
initially showing an intermediate behaviour between HE220A and
HE220B, experience a much faster degradation.
(3) Local buckling (Fig. 6) starts a few cycles earlier in IPE beams than
in HEA ones, whilst HEB specimens can sustain a larger number of
cycles without buckling. In particular, as the first three cycles were
imposed to the specimens in the elastic range, from Fig. 6 it is
evident that IPE flanges buckle already at the first excursion in the
plastic range, while the HEB shape can sustain nearly seven cycles in
the plastic range before buckling of the flanges occurs. Considering
that the flange b/t ratio of both IPE and HEB is similar, and smaller
than that of HEA shapes, it follows that a major role in governing
local buckling effects is played by the web slenderness ratio h/tw.
Furthermore, buckling develops completely within a few cycles, after
which the size of the buckles stabilises. The number of cycles after
which this stabilisation of the buckles size occurs seems to be
directly dependent on the b/t ratio.
(4) For all three types of profile, once local buckling takes place and
buckle size stabilisation occurs, the hysteresis loops also stabilise,
and the rate of reduction in load carrying capacity decreases with an
increasing number of cycles imposed on the specimen, until a
final stage is reached when the deterioration rate suddenly increases
again and the specimen collapses after a few cycles. In particular,
both IPE and HEA beams clearly show evidence of this type of
behaviour, with a sudden increase of slope in the diagrams, associated with higher deterioration rates, while HEB specimens show a
smoother transition from the (longer) phase of cycle stabilisation to
that leading to collapse.
(5) These differences in behaviour at the final stage, between HEB
specimens and HEA and IPE ones are associated with a difference in
Design of steel structures under low or high cycle fatigue
89
their failure modes. HEA and IPE beams, in fact, generally collapse
by steady crack propagation due to low-cycle fatigue effects; the
HEB specimens, on the contrary, show evidence of some kind of
brittle fracture of both the flange and the web, either at the
specimen-to-base welds or at the plastic hinge where, due to large
localised distortions, surface cracks usually develop a few cycles after
local buckling of the flange plates.
The major consequence of the last two considerations, is the identification
of a possible criterium for defining the number of cycles to failure N,. In
fact, this can be assumed either as the one corresponding to complete
separation of one flange (generally this is the case for HEB shapes) or as
the one corresponding to the sudden (final) increase in the deterioration
trend after cycle stabilisation (generally applicable to HEA and IPE
shapes).
Tables 2-4 summarise all the constant amplitude tests that were
performed during the present study. Together with the test identification,
the mean value vm of the imposed displacement cycles and their amplitude
Av, the tables also report the yield displacement 0, and the yield strength
Fy, conventionally evaluated following the recommended procedures,”
the number of cycles N,“ corresponding at complete separation of one
flange, the conventional number of cycles at failure N,, assumed in the
following calculations and identified as previously stated, and an indication of the failure mode of the specimen: S means that the specimen failed
in the base material, W means that failure was reached at the specimento-base-plate weldments.
3.1.2 Reprocessing the test data
Once the number of cycles to failure N, has been determined, the test data
can be reprocessed with the aim of plotting on a log—log scale N, versus
the ideally elastic stress range given in eqn (6), where the generalised
displacement component is now substituted by the displacement v. For a
cantilever beam, this eqn (6) becomes:
A FL
EA.~;=7"-VYI7
Y
(7)
The domain log (EAs) versus log N is the usual domain for the Wohler
(S—N) curves3 adopted by various International Codes and Standards for
(high cycle) fatigue design of steel structures."21‘2“ In fact, the strain range
As (having the same physical meaning in both high and low cycle fatigue)
has been correlated to the number of cycles to failure N, and is then
90
G. Ballio, C. A. Castiglioni
multiplied by the Young’s modulus E in order to deal with the same
parameters commonly used by designers dealing with high cycle fatigue.
From another point of view, the term EAs=A0'* can be regarded as an
equivalent stress range, associated with the imposed strain range As, in the
case of an ideal indefinitely linear elastic material.
For high cycle fatigue design, the most common structural details have
been grouped into a number of categories (the same category for different
details having a similar fatigue strength), and to each category has been
associated an S~N curve.21‘24 Earlier codes based on such an approach2"24 adopted S—N curves obtained by fitting experimental test
data, resulting in sets of S—N curves having different slopes. The possibility
of unifying the slopes of such curves has recently been recognised, which in
the most recent recommendations1*22’23 is defined by an equation of the
type:
NA0 3 = cost
(8)
In order to verify the first assumption introduced in the previous point 2 of
this paper (equivalence of EAs—N, curves for high and low cycle fatigue),
we try to interpret the experimental test data of the low cycle fatigue tests
performed during the present study, reprocessed according to eqn (7), by
means of the S—N curves proposed by Eurocode-3,1 whose validity is
extrapolated in the low cycle fatigue range (i.e. for the number of cycles N
ranging from 10 to 500, and corresponding stress ranges A0* =EAs > 2fy)~
In high cycle fatigue, different fatigue strength categories implicitly
account for different notch effects, i.e. for different local stress concentrations due to geometry of the detail and/or defects caused by fabrication
procedures. It is supposed that the same consideration holds also in the
case of low cycle fatigue: local buckling can in fact be regarded as a notch
effect, because it induces local stress concentrations in the buckled area (at
plastic hinge location). As already discussed, and in good agreement with
previous results by Yamada,13'16 the different geometries of the crosssections make the specimens more or less vulnerable by loca‘l buckling
effects. This means that each shape, as a function of its geometrical
properties, can be considered as belonging to a definite fatigue strength
category, because it is intrinsically affected by a more or less pronounced
notch effect.
If this assumption is true, it must be expected that the three different
shapes considered in this study belong to three different fatigue strength
categories: HE220B to a higher one, IPE300 to a lower one and HE220A
to an intermediate one. Furthermore, as already discussed, the tested
specimens evidenced two different failure modes: by cracking in the base
Design of steel structures under low or high cycle fatigue
91
material at the plastic hinge locations, or by cracking at the welding of the
reinforcement plates to the specimen (Fig. 8). It must then be expected that
different fatigue strength curves apply to the different failure modes.
Figures 9-11 respectively show the test data for HE220B, HE220A and
IPE300 specimens that failed by cracking in the base material at plastic
hinge locations, fitted by the fatigue strength lines of Eurocode-3,1 which
were extrapolated for low number of cycles to failure by means of eqn (8).
In the same figures, the EC-3 line for base material (category 160) is also
presented, together with some test data obtained by Fisher for rolled
beams.“'5'“ Of course, although both sets of data can be fitted by the
same S—N lines, the experimental data obtained in this research are
grouped together, with the number of cycles to failure ranging between 10
and 5 >< 102, while test data from Refs 4, 5 and 11 were obtained for much
longer fatigue lives (N, >105) and plotted far away from the previous ones.
From Figs 9-11 it can be noticed that, as expected, test data for
HE220B can be fitted by EC-3 line for category 80, and those for IPE300
by that for category 63, while those for HE220A specimens can be
fitted by the line for category 71, intermediate between the two previous
ones. Furthermore by examining Fig. 12, which refers to test data for
specimens failed at weldings, it can be noticed that HE220A and HE220B
Fig. 8. Examples of failure: (A) and (B) at plastic hinge locations, and (C) at welding of the
base reinforcing plate to the specimen.
92
G. Ballio, C. A. Castiglioni
Fig. 8—C0ntd.
specimens can be fitted by category 63 line, while IPE300 specimens are
fitted by line 56, showing a lower fatigue strength. This is probably caused
by the formation of the plastic hinge nearer to the base (i.e. nearer to the
weldment) in IPE specimens than in HE ones. This fact seems again to
confirm the hypothesis that local buckling can be considered as a notch
effect reducing the fatigue strength of the profile, a notch effect that can be
regarded as a function of the geometrical properties of the cross section
(i.e. of the slenderness ratios b/t and h/tw).
However, independently of the category of fatigue resistence pertinent
to each shape, it is important to notice that the slope of the line fitting (in a
log—log plot) the low cycle fatigue test data, reprocessed according to eqn
(7), is nearly —3. This is in good agreement with the results of research on
Design of steel structures under low or high cycle fatigue
93
Fig. 8.—C0ntd.
high cycle fatigue. It follows that both high and low cycle fatigue test data
can be fitted by S—N lines having the same slope -3, the fatigue category
depending on the notch effect (or the stress concentration factor) associated with the various structural details.
3.2 Variable amplitude tests
The second assumption, previously introduced in Section 2, deals with the
possibility of adopting Miner’s rule (based on a linear damage accumulation model) in order to define an acceptable failure criterium in low cycle
fatigue. To assess the validity of this statement, some random displacement
G. Ballio, C. A. Castiglioni
94
4
V
Fisher tests on rolled beams [5]
HEB failed at plastic hinge
-%
3_I
I
»-\
%
\./
E*As
g
Lo
Qlfl
G3
24
EC3-I 60
EC3-100
EC3-90
EC3-80
1
T
I
0
I
1
I
2
3
I
I
4
I-09 (N)
_
I
5
I
s
I
1
s
Fig. 9. HE220B test data fitted by EC-3 S—N curves (specimens failed at plastic hinge)
4
HEIB
“E
€='WI=
FIsher
'
tests on rolled beams [5]
A
HEA faIled at plastic hInge
.
. .
HBIIKI
HEAO5
HEA06
3
liififi
cm» L
LogE*As
2
EC3-I60
EC3-I00
EC3-80
EC3-71
1
_I
0
I
1
I
2
I
3
I
4
I-09 (N)
I
5
I
e
I
7
I
s
Fig. 10. HE220A test data fitted by EC-3 S—N curves (specimens failed at plastic hinge)
Design of steel structures under low or high cycle fatigue
4__.
IPE08
IPE II
I
$
Fisher tests on rolled beams [5] I
I
E
IPE tailed at plastic hinge
IPEOZ
IPEDS
IPE03
IPEOS
IPEO4
IPE07
3_.
A
6%
emf
\/
E*As
g
Lo
2_I
EC3-160
EC3-I 00
EC3-B0
EC3-71
EC3-63
I
_|
I
0
1
I
2
I
I
s
I
4
Log (N)
I
5
I
6
7
I
a
Fig. 11. IPE test data fitted by EC-3 S—N curves (specimens failed at plastic hinge)
A
4
Albrecht tests on non load-carrying fillets welds [28]
Booth tests on cruciformjolnts [27]
0
If
-
Fisher tests on welded cover plates [4]
0
PE!
A
. . -\ .,.
lPE|0P7E14 _
3 —‘
Gurney tests 0n load-carrying fillet welds [29]
I
.
14
\
I
O
u
HEA failed at welds
HEBfai|ed atwelds
|PEfailed atwelds
/\
7
{I
K.‘
Log
E*Ae
\1».
___
.
PI
I
Q ‘
'6; 0
\\\\,,
1
L.
-.1
\\' I“:-
0
EC3-160
‘n.
EC3-100
O7\| ;,;_.
mmmm (7000 wumw 00cn
ui o
1
‘I
0
I
1
I
2
I
3
I
-
4
Log (N)
I
5
I
6
I
7
I
.
8
Fig. 12. Test data for specimens failed at weldings, fitted by EC-3 S—N curves.
96
G. Ballio, C. A. Castiglioni
histories were numerically obtained by means of the dynamic numerical
simulation code,“ adopting artificial accelerograms which were obtained
on the basis of Eurocode-8 recommended spectra.“ This model, set up in
order to simulate the dynamic behaviour of columns under compression
and bending, consists of a rigid bar connected to the ground by a ‘cell’ where
all the deformability of the member is concentrated and a structural lumped
mass applied on the top. The behaviour of the ‘cell’ follows a constitutive
law for the material and accounts for damage, according to the rules and the
model proposed in Ref. 12. Considering a damping factor v=3% and
members with a period T-0-5 s, the oscillograms were numerically
obtained under increasing values of the peak ground acceleration, and
Miner’s damage index was computed using the rainflow method for cycle
counting. When a time history giving a Miner’s damage index greater than 1
was obtained (i.e. indicating failure) as output of the numerical simulation,
such a displacement history was imposed in a quasi-static way to the
specimens under testing. In the earlier tests (IPEl5, HEA1 and HEBI) a
rather different technique was adopted, and more than one accelerogram
obtained under a peak ground acceleration of O-35 g was superimposed on
the specimens, and some cycles had to be added at the end of the test in order
to obtain complete failure of the member. In addition, one cyclic loading
history was considered (IPEl1), with groups of three cycles with a mean
value um: 140 mm and amplitude Av: 80 mm following each other,
generating a loading history with a global mean value v,,,=O (Fig. 13).
A total of ll random tests was performed (four on HEA shapes, four on
IPE and three on HEB), and the corresponding displacement histories are
presented in Figs 13-15.
The experimental results were reprocessed by means of the rainflow
cycle counting method and, based on the transformation given by eqn (7)
and on the EC-3 fatigue strength lines previously identified for the various
profiles, Miner’s damage index associated with the collapse of each
specimen was computed. The obtained results are summarised in Table 5.
In the table, for each test, the failure mode (S=base material at plastic
hinge, W=cracks in welding) and the damage index corresponding to the
EC-3 lines are given.
By examining Table 5, the following conclusions can be drawn:
(1) Miner’s rule gives damage index values with scatters similar to those
commonly found in random high cycle fatigue.
(2) For HEA specimens, all of which collapsed by cracking in the base
material at plastic hinge locations, Miner’s rule correctly allows
prediction of failure in association with EC-3 line for fatigue
strength category 71.
v
r
Design of steel structures under low or high cycle fatigue
20 flO
30.00 -
HEA22O n 1
I0 00
2000
E 000 ._
§ -I000
10.00
V(cm) 000
-20 00
-I0 00
97
HEAZZO n 9
O
-:10 0 U
I) 00
-..-_._._._.---—.-I--1.__._,i.__._..._...__..__,
to on
5,, no
120 go
|@,;_,m
M 0|,
REVERSALS
-2000
0 00
2000I 5 00
I llEA22O n 10
2°“
.50 00
..,. ..,...._
,.
100.00
150.00
200 00
REVERSALS
. .
2:0 ilfl
100 00
051220 n 11
I000
IUIIU
E
;
000
0
50
U
~l000
UIGQ
_
V(
cm) -20.00
=>
woo
-:I0u0
-100%
»---—-I---—.—.--7--_,.,,-_._.-...<._.__...._.
00.00
100.00
150.00
200.00
2:10.00
REVERSALS
--I0.00000 -+-T--I-I-r,-I-W-.-.-..~_,,._-,-.~.-.-.I»._~v-Y.-,....____
50.00
100.00
150.00
200.00
200.00
II0000
REVERSALS
Fig. 13. Random displacement histories for HEA specimens.
20.0
E
E
° HEB220 0 I
IIOIJIJ
. "I,|I,Il'f’|IqIWW,-IlII"'\IlI'III“
j
0
V(cm)
1
-1000i
-20 00
-20 00
I
-10.00D00....._,._»....,...--1-H-»m,.._.._,,,__,,_..,,.,,_,,.,,.,.
mo“
401,0
50-00
‘mm mono 120110 How
REVERSALS
I500
I000
500
Ti
;
“EB22° " 12
000
-500
-I000
-I500
IIIMII
-20.00 I
I100
-LI0 00 ..
U 00
2000
. . . . . - -W . . . . . . . ._.
40.00 5000 00.00
. ..,_.
II:I|I10 innit-0|iuBliI<II100
REVERSALS
llEB22O 0 13
.
_
50 00
. ...<-<...
I00 00
l50.00
.
200.00
REVERSALS
.
250.00
000 00
Fig. 14. Random displacement histories for HEB specimens.
G. Ballio, C. A. Castiglioni
98
10.00
IPE300
20.00
II 9
A
,, nu
E _5 0°
,'|‘
0 10
/\ 10.00
I
5 5.00
Pg
E;
-10.00
W
"-9"
I.'I.rIrI
III I
-5.00 -
_I_-0 no _-.--..-..-.-.----------------------------.
ll 00 20.00 I0 00 00 00 00.00 100.00 120.00 140.00 100.00
REVERSALS
III {I0
IPE300
15.00 -
5 00
IPE300
ll
I0 IIII
n
e I0 - 00
0 "0
15.00] IPE300
ll
lllllllll
I
- I-— ---~I----w----v----—ww1—n-—
2'-I 00
30.00
40-'10
50-00
6° 00
REVERSALS
I0 0°
n
15
10 00
E
:
ii
5.00
-5.00
I0 00
I5 00
20 O0
_,_»,0,I ...__.,_.-.----._..-.._____,_......_-....-r-,-1---..._..-_.--,
0 IIII
£0.00
. 0 I00 nn 12000 I-10.00
“W ii§i'nIi§IlI.s
n
'10
00
'00
I20
REVERSALS
1:0
II'1iI"h 210
Fig. 15. Random displacement histories for IPE specimens.
TABLE 5
Summary of Damage Indexes Corresponding to Specimen Collapse, Computed for
Random Variable Amplitude Tests
Test
160
100
90
80
71
63
56
Failure
HEA1
HEA9
HEA10
HEA11
O-O97
0-152
0-107
0-130
0-395
0-622
O-437
0-534
0 542
0 854
O 599
0 733
0-772
1-202
0 853
1 040
1-104
1-740
1-220
1-490
1-580
2-489
1-746
2-135
2-250
3-544
2-486
3-O39
CDC/)C/JU)
HEBI
0-253
1-040
1 420
2 030
2-900
4-152
5-911
S
HEB12
HEB13
0- 128
0-093
0-522
0-381
0-717
O-523
1-020
O-744
1-460
1-060
2-090
1-524
2-975
2-169
W
W
IPE9
IPE10
IPEll
IPE15
0-082
O-O63
0-082
0-085
0-336
0-280
O-340
0-348
0-461
O-3 85
0-460
0-478
0-657
O-547
0-660
0-680
0-939
0-783
0-940
O-973
1-340
1-120
I-346
I-394
1-914
1-596
1-916
1-984
UJUJUJC/J
Design of steel structures under low or high cycle fatigue
99
(3) For HEB no.1 specimen, collapsed by cracking in the base material
at plastic hinge locations, and the two HEB specimens, collapsed by
cracking in the base welding, Miner’s rule correctly allows prediction of failure respectively in association with EC-3 lines for fatigue
strength categories 80 and 63. In particular, it can be noticed that
the adopted fatigue strength lines lead to damage assessments
largely on the safe side; increasing the fatigue strength of HE220B
specimens by one category (line 90 for failure in base material, line
71 for failure at weldings) results in damage index values still on the
safe side, but nearer to unity.
(4) For IPE specimens, all collapsed by cracking in the base material at
plastic hinge locations, Miner’s rule correctly allows prediction of
failure in association with EC-3 line for fatigue strength category 63.
4 CONCLUSIONS
The obtained results show the validity of the two assumptions on which
the proposed approach is based:
(1) The same S—N curves are valid in high and low cycle fatigue, if an
equivalent stress range Aa*=EAa is considered, associated with an
ideal indefinitely elastic behaviour of the material; in particular, the
slope of these S—N curves in a (log—log) plot is —3.
(2) Miner’s rule can be adopted, together with the previously defined
S—N curves and with a cycle counting method (e.g. rainflow) to define
a unified collapse criterium, valid for both high and low cycle fatigue.
The application of these results and of the proposed method for damage
assessment, to steel structures under seismic loading, may lead to an
overcoming of seismic design methods based on the behaviour factor, as
shown by the same authors in Ref. 30.
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buildings, ENV 1993-l-l:1992 E, CEN, April 1992.
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Press, Cambridge, 1979.
3. Wéhler, A., Zeitschrift fiir Bauwesen, 10, 1860.
G. Ballio, C. A. Castiglioni
Fisher, J. W., Frank, K., Hirt, M. A. & McNamee, B. M., Effect of weldments
on the fatigue strength of steel beams. National Cooperative Highway
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Fisher, J. W., Albrecht, P., Yen, B. T., Klingermann, D. J. & McNamee, B. M.,
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Gurney, T. R., Fatigue tests under variable amplitude loading. The Welding
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Gurney, T. R., Fatigue tests on fillet welded joints to assess the validity of
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Miki, C., Tajima, J., Asahi, K. & Takenouchi, H., Fatigue of large-size
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143-56.
Smith, I. F. C., Bremen, U. & Hirt, M. A., Fatigue thresholds and improvement of welded connections. ICOM 125, Swiss Federal Institute, Lausanne,
CH, March 1984.
Shilling, C. G., Klippstein, K. H., Barsom, J. M. & Blake, G. T., Fatigue of
welded steel bridge members under variable amplitude loading. NCHRP
report 188, Washington DC, 1978.
Background Documentation EC-3, Ch. 9, Doc. 9.2.
Castiglioni, C. A. & DiPalma, N., Steel members under cyclic loads: numerical
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288-312.
Yamada, M., Low cycle fatigue fracture limits of various kinds of structural
members subjected to alternately repeated plastic bending under axial compression as an evaluation basis or design criteria for aseismic capacity. Proc.
IV World Conference on Earthquake Engineering, Santiago, Chile, Vol. 1,
B-2, Jan. 69, pp. 137-51.
Yamada, M., Sakae, K., Tadokoro, T. & Shirakawa, K., Elasto-plastic
bending deformation of wide-flange steel beam-columns subjected to axial
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Yamada, M. & Shirakawa, K., Elasto-plastic bending deformation of wideflange steel beam-columns subjected to axial load. Part II: Alternately
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Ballio, G. & Castiglioni, C. A., Seismic behavior of steel sections. Journal of
Constructional Steel Research 29 (1994) 21-54.
Castiglioni, C. A. & Losa, P. L., Local buckling and structural damage in steel
members under cyclic loading. Proc. of the X World Conference on Earthquake Engineering, Madrid, July 1992, pp. 2891 6.
Ballio, G. & Zandonini, R., An experimental equipment to test steel structural
members and subassemblages subject to cyclic loads. Ingegneria sismica, 2
(May—June 1985) 25-44.
ECCS, T.C.1, T.W.G. 1.3, Recommended testing procedure for assessing the
behavior of structural steel elements under cyclic loads. Publication no. 45,
1986.
Design of steel structures under low or high cycle fatigue
101
BS 5400, Steel concrete and composite bridges, Part 10: Code of practice for
fatigue. British Standards Institute, London, 1980.
ECCS, Recommendations for the fatigue design of structures. Committee TC6
‘Fatigue’, lst edn, 1985.
CNR 10011/85, Costruzioni in Acciaio, Istruzioni per il Calcolo, l’esecuzione,
il collaudo e la manutenzione, Boll. 118, June 1986.
Keating, P. B. & Fisher, J. W., Evaluation of fatigue tests and design criteria
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