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FEM analysis of steel members considering damage accumulation effects under cyclic loading
Advances in Steel Structures, Vol. I Chan, Teng and Chung (Eds.) © 2002 Published by Elsevier Science Ltd.
105
FEM ANALYSIS OF STEEL MEMBERS CONSIDERING DAMAGE ACCUMULATION EFFECTS UNDER CYCLIC LOADING Z. Y. Shen
and Z. S. Song
State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China
ABSTRACT Due to the steady growth of micro level deterioration, the properties of structural steel, such as yield stress. Young's modulus and hardening coefficient will deteriorate under cyclic loading, and fatigue failure maybe take place at a low number of cycles. Based on continuum damage theory and experimental results, a cumulative damage mechanics model, as measured by effective plastic strain, is suggested. A general FEM analysis procedure on damage cumulation is presented and a damage crack criterion is proposed. Cantilever members subjected to cyclic loading are numerically simulated with a FEM program which takes damage cumulation into consideration. The analytical results agree very well with the experiments, which demonstrates that the cumulative damage mechanics model presented in the paper is accurate enough for simulating the behavior of steel structures under seismic loading.
KEYWORDS low cycles fatigue, damage cumulation, finite element method
INTRODUCTION The properties of structural steel, such as yield stress. Young's modulus and hardening coefficient will deteriorate under cyclic loading. For understanding the performance of steel structures under cyclic loading, a damage mechanics method has been used to study the damage and its cumulation in steel members. Due to the complication of factors related to damage, different definitions of the damage index D have been proposed. Kachanov(1986) was the first one who pointed out that the damage can
106
be described as the ratio of effective area to real area. Rabotnov (1969) defined the damage index by the ratio of damaged area to real area. Park and Ang (1985), Park, et al (1985) used a linear combination of deformation and energy to express the damage index. Based on the proposal of Park and Ang (1985), Kumar and Usami (1994) suggested an improved expression. As for the rule for the cumulation of damage, linear and nonhnear cumulation equations have been proposed by Miner, Chaboche and Coflfm-Manson (Yu and Feng, 1997). But all of these results are very difficult to use in practical structural analysis. Recently, based on experimental results, Shen and Dong (1997) suggested a uniaxial cumulative damage mechanics model for steel subjected to cyclic loading, which can be appUed in steel structural analysis. In this paper, the model is broadened to fit triaxial stress conditions and a FEM analysis procedure taking the damage cumulation into consideration is presented.
CUMULATIVE DAMAGE MECHANICS MODEL OF STRUCTURAL STEEL UNDER CYCLIC LOADING The model established by Shen and Dong (1997) can be expressed as follows. The damage index D is computed according to the plastic strain and hysteretic energy dissipation of steel pP
N
^„
1=1
P
^„
where A^ is the total number of half cycles which cause plastic strain, p is the weighted value of the /th half cycle, ef is the plastic strain during the /th half cycle, £^ is the largest plastic strain during all half-cycles and e^ is the ultimate plastic strain of the material. The effects of damage on Young's modulus, yield strength and strain hardening coefficient can be expressed as £"> ={\-^,D)E,
(2)
(3)
^P
^^^o+^sZ^
(4)
1 = 1 ^u
where EQ and E^ are the Young's modulus in respect of/)=0 and D respectively, o^,^ is the initial yield stress when Z)=0, af is the yield stress in respect of Z), ko and k are the hardening coefficients in respect ofZ)=0 andZ) and ^ j , i^, ^3 are material parameters. Figure 1 is the hysteretic model of steel allowing for damage cumulation given by Shen and Dong
107
(1997)
o nth half cycle
th half cyile
Figure 1: Hysteretic model of steel considering damage cumulation The hysteretic models of steel shown in Figure 1 and Eqn. 1 to Eqn. 4 are derived from cyclic loading tests in one-dimension. For the condition of triaxial stresses and proportional loading, the above results can also be valid if ef and ej are substituted with equivalent plastic strain pi and /?«, and a is substituted with equivalent stress a .
p. Pi •
ttPu
(5)
(6)
\dP ith-half cycle
dp = I -de^de^
(7)
" F l - ^ 2 ) ' +(^2 - ^ 3 ) ' +(^3 -^1)']
(8)
According to the energy damage theory, the relationship between/?« and e^ is (Yu and Feng, 1997) Pu-<'K
(9)
R^, is called the triaxial stress index, which reflects the effect of triaxial stress ratio.
R^=l(l + 3
v)-h3(l-2v)(^Y (J^„
(10)
108 where, v is the Poisson ratio of the material, a^ is the hydrostatic stress and a^ =cr^^/3 . If cr^= (72 =0-3, from Eqn. 8, then % = 0 and R^=^. In this case, the damage index should not be calculated using Eqn. 5. Based on the plastic damage theory, the release rate of damage strain energy can be expressed as (Yu and Feng, 1997)
-Y = -
^(l + vK^,+3(l-2v>7^
2E{\-D)
(11)
If a^ = a^ = a^ = a^, a =0, then Eqn. 11 becomes
y j l z ^
02)
2E{\-D)
and for uniaxial tension, since a = a and a^ = al3, Eqn. 11 becomes
Assuming that the value of -Y is the same when damage fracture occurs in uniaxial and triaxial conditions, then the hydrostatic stress a^ at fracture is
^mu = >J/ , V
(14)
where, cr„ is the ultimate strength of the material. Let damage index be 1 at fracture and assuming its value is proportional to the hydrostatic stress, then the increment of damage index under triaxial stress conditions can be calculated from
AD =
^ =
^^-^^-^
—
In practical analysis, Eqn. 15 will be employed if R^ is larger than a specified number.
(15)
109
CONSTITUTIVE RELATION OF STRUCTURAL STEEL WITH DAMAGE CUMULATION If the material is of the Von Mises-type, the subsequent yield function can be expressed as F^f-h
(16)
/ — v .
(17)
h-^cj]{p)
(18)
where, Sj, is the deviatoric stress tensor and a/p), i.e. a f is the current yield stress. It is assumed that the total strain tensor can be separated into elastic and plastic components, E ,' and E ,J, respectively, and the differential of the plastic strain tensor is given by the plastic flow rule
(19) By differentiating Eqn. 16, the following expression is obtained
^^ dG^^--G^^dp^O
do^^
'
(20)
3 ' dp
where ^
da
- .
(21)
•^ = E'
(22)
dp
E^ IS the plastic modulus of the material, which may be expressed as
p _
_E^E^_^^JcE^ 1^ j ^ _ 1^ rvj^ rrD r^tD rpD frpD E^ -E''' ~ E"" -kE""
-
F ^ i
j^ ^^ \-k
where E ^^ is the tangent modulus in respect of the damage index D.
^^^)
110
Substituting Eqn. 19 into Eqn. 7, then
dp = {-de ^de ^f'
= dX
i^^^f'
(24)
and Hook's law of material with cumulative damage may be written as dffij = C,„dEl, = C,„ids,i -de'J
df Ci0dSi,~Cij„dX^
(25)
E" v-E" <5,.d"„ +-6J., 1 +v '• (l + v)(l-2v)
(26)
= where C,,,
d ,j is the Kronecker delta. Putting Eqn. 3, 22, 23, 24, 25 into Eqn. 19, the scalar fiinction dX can be solved
§-c,A. dX =-
sl/2
^f-C..„-^ + da, "-da,, 3(1-A)
^-{\-i,D){^-£,Dy,-,,E,
(27)
(ida,^da,^j
substituting Eqn. 27 into Eqn. 25, then (28)
da,, = C,J,cfe „ where
(29)
C
da ^^ OG ^^
=aa,^
^
da,,
CI, is named as plastic tensor
3(1-^)
^T,da,,da,^j
Nl/2
(30)
Ill
NONLINEAR FINITE ELEMENT FORMULATIONS BASED ON CUMULATIVE DAMAGE MECHANICS MODEL Taking damage cumulation, elastoplasticity and large displacements into account, the updated Lagrangian finite element formulation can be written (31) where (32)
\-'D
(33)
.^, =-C^,,+.^,,)
f%
(34)
^-t^k.f'^k,
and Uj is the displacement at node /
FEM ANALYSIS OF STEEL MEMBERS SUBJECTED CYCLIC LOADING To verify the cumulative damage mechanics model and constitutive relations suggested, a nonlinear program was developed in which a 20-nodes solid element was employed and used to provide numerical results which were compared with the results obtained from experiments conducted by Chen (2000). Description of Test Specimens To understand the effects of damage cumulation on steel members, two series of cyclic tests, SI and S2 respectively, were conducted in the State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University. Outline of a specimen is shown in Figure 2 and design dimensions are shown in TABLE 1.
, — ^ .
'^
. . . „ „ „ .
^ti-^^
1-1
^
.
h
__: __\
L
I
i
i3 n
^r
^\ ^ :
id
\^^m_
2-2
Figure 2: Outline of specimen
112
TABLE 1 DESIGN DIMENSIONS OF TEST SPECIMENS
Specimen
SI S2
L (mm) 1100 1100
H (mm) 1300 1300
b (mm)
d (mm)
tf (mm)
h ("^i")
54 42
140 120
6 6
6 6
Cyclic loading was applied to the top of the specimen along the weak direction and the testing procedure was controlled by the displacement measured at that point, which is shown as Figure 3.
(b)
(a)
Figure 3: Testing Procedure Sl-1: SI-2: SI-3: Sl-4: SI-5: S2-1: S2-2: S2-3: S2-4: S2-5:
constant displacement amplitude, 5=50 mm; constant displacement amplitude, 6=60 mm; constant displacement amplitude, 5=40 mm; inconstant displacement amplitude, 5=50 mm(5 inconstant displacement amplitude, 5=60 mm(5 inconstant displacement amplitude, 5=40 mm(5 constant displacement amplitude, 5=50 mm; constant displacement amplitude, 5=60 mm; constant displacement amplitude, 5=65 mm; inconstant displacement amplitude, 5=70 mm(5
cycles)^5=60 mm(5 cycles)^5=50 mm(to failure); cycles)^5=50 mm(5 cycles)-^5=60 mm(to failure); cycles)^5-60 mm--5=40 mm(to failure);
cycles)---5=65 mm(5 cycles)^5=70 mm(to failure);
Finite Element Subdivision To simplify the computation, only that part above the stiffened plate at the bottom of the column was considered in the FEM analysis. The boundary conditions are assumed as fixed. Due to symmetry, the fmite element model was generated for only half of the specimen and necessary constraints are imposed on the plane of symmetry. The material properties of steel and the parameters in the cumulative damage model used in the FEM analysis are listed in TABLE 2 and TABLE 3, respectively. TABLE 2 MATERIAL PROPERTIES OF STEEL
o}o(Mpa)
cr„(Mpa)
EoCMpa)
<(°^)
310
445
196784
24.185
113
TABLES PARAMETERS IN THE CUMULATIVE DAMAGE MODEL
p 0.0081
r
^1
6
^3
kfj
0.227
0.119
0.000073
Eqn. 35
1.44
rj
0.041
k, = : 0 . 0 1 4 - 0 . 1 6 5 k , + l - 1 2 c -2.88k:,
(35)
Criterion of Damage Crack and Treatments Damage in steel was found to along with the increase of plastic strain and the number of cycles. This increase is irreversible. According to the cumulative damage mechanics model presented in this paper, a damage crack occurs at a gauss point if the value of the damage index at this point is greater than 1. Comparisons betyveen FEM and Experimental Results The number of half cycles during which a damage crack was initiated is shown in TABLE 4, and hysteretic curves of FEM analysis and experiment before the crack occurs are shown in Figure 4 to Figure 13 TABLE 4 NUMBER OF HALF CYCLES DURING WHICH DAMAGE CRACK INITIATED
Specimen
Calculated values
Sl-1 Sl-2 Sl-3 Sl-4 Sl-5 S2-1 S2-2 S2-3 S2-4 S2-5
25 20 40 22 21 51 44 30 22 16
Experimental results 24 14 32 12 16 48 16 24 16 8
(a) experimental curve (b) FEM curve Figure 4: Hysteretic curves of specimen SI — 1
114
20
40
60
80
(a) experimental curve (b) FEM curve Figure 5: Hysteretic curves of specimen SI —2
10
5
f0 r -5
.,.
•
' f^ 1 M
\ M*^
• -40
-1S
-20
0 u(mm)
20
(a) experimental curve (b) FEM curve Figure 6; Hysteretic curves of specimen SI —3
u(mm)
u(mm)
(a) experimental curve (b) FEM curve Figure 7: Hysteretic curves of specimen SI —4
.--/lA
If
-
-40
-20
-
0
:
-
J
20
40
u(mm)
(a) experimental curve (b) FEM curve Figure 8: Hysteretic curves of specimen SI —5
40
60
115
-60
-40
-20
0
20
u(mm)
(a) experimental curve (b) FEM curve Figure 9: Hysteretic curves of specimen S2— 1
-80
-60
-40
-20
0
20
40
60
80
(a) experimental curve (b) FEM curve Figure 10: Hysteretic curves of specimen S2—2
(a) experimental curve (b) FEM curve Figure 11: Hysteretic curves of specimen S2—3
;
'I..,
(a) experimental curve (b) FEM curve Figure 12: Hysteretic curves of specimen S2—4
40
116
-80
-60
-40
-20
(a) experimental curve (b) FEM curve Figure 13: Hysteretic curves of specimen S2—5 From TABLE 4 and Figure 4 to Figure 13, it can be seen that the FEM analytical results are in good agreement with the experimental ones. The cumulative damage mechanics model suggested in this paper is accurate enough to evaluate the damage cumulation in the structural steel and to predict initiation of damage cracks under cyclic loading. The seismic behavior of steel structures, therefore, can be simulated with an acceptable error for an earthquake, and repeated earthquakes by using of damage model presented.
CONCLUSION Based on the experiments and energy damage theory, a cumulative damage model which caters for triaxial stress conditions is proposed in this paper. A general procedure for computing damage cumulation in steel members with finite element method is suggested. Cantilever members subjected cyclic loading at their ends were simulated numerically with a FEM program. The analytical results showed good agreement with the experimental results.
ACKNOWLEDGEMENT The project is financially supported by National Science Foundation of China as a key project (59895410).
References Chen, R. Y. (2000). Damage Cumulation Analysis of Tall Steel Mega-Structiires under Seismic Actions. Doctoral Dissertation, Tongji University, (in Chinese) Kachanov, L. M. (1986). Introduction to Continuum Damage Mechanics, Martinus Nijhoflf Publishers, Dordrecht Kumar, S. and Usami, T. (1994). A Note on Evaluation of Damage in Steel Structures under Cyclic Loa.dmg, JSCE Journal of Structural Engineering, 40A, 177-188 Lemaitre, J. and Chaboche, J. L. (1985). Mecanique des Materiaux Solides, Dunod, Paris (Chinese Translation, 1997) Park, Y. J. and Ang, A. H. S. (1985). Mechanistic Seismic Damage Model for Reinforced Concrete,
117
ASCE Journal of Structural Engineering, 111: 4, 722-739. Park, Y. J., Ang, A. H. S. and Wen, Y. K. (1985). Seismic Damage Analysis of Reinforced Concrete BviMmgs, ASCE Journal of Structural Engineering, 111: 4. lAO-lSl. Rabotnov, Y N.(1969) Creep Rupture. Proceeding of the Twelfth Internal Congress of Applied Mechanics, lUTAM, 342-349. Shen, Z. Y and Dong, B. (1997). An Experiment-Based Cumulative Damage Mechanics Model of Steel under Cyclic Loading. Advances in Structural Engineering. 1:1, 39-46 Yu, S. W. and Feng, X. Q. (1997). Damage Mechanics. Tsinghua University, Beijing, China(in Chinese)
105
FEM ANALYSIS OF STEEL MEMBERS CONSIDERING DAMAGE ACCUMULATION EFFECTS UNDER CYCLIC LOADING Z. Y. Shen
and Z. S. Song
State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China
ABSTRACT Due to the steady growth of micro level deterioration, the properties of structural steel, such as yield stress. Young's modulus and hardening coefficient will deteriorate under cyclic loading, and fatigue failure maybe take place at a low number of cycles. Based on continuum damage theory and experimental results, a cumulative damage mechanics model, as measured by effective plastic strain, is suggested. A general FEM analysis procedure on damage cumulation is presented and a damage crack criterion is proposed. Cantilever members subjected to cyclic loading are numerically simulated with a FEM program which takes damage cumulation into consideration. The analytical results agree very well with the experiments, which demonstrates that the cumulative damage mechanics model presented in the paper is accurate enough for simulating the behavior of steel structures under seismic loading.
KEYWORDS low cycles fatigue, damage cumulation, finite element method
INTRODUCTION The properties of structural steel, such as yield stress. Young's modulus and hardening coefficient will deteriorate under cyclic loading. For understanding the performance of steel structures under cyclic loading, a damage mechanics method has been used to study the damage and its cumulation in steel members. Due to the complication of factors related to damage, different definitions of the damage index D have been proposed. Kachanov(1986) was the first one who pointed out that the damage can
106
be described as the ratio of effective area to real area. Rabotnov (1969) defined the damage index by the ratio of damaged area to real area. Park and Ang (1985), Park, et al (1985) used a linear combination of deformation and energy to express the damage index. Based on the proposal of Park and Ang (1985), Kumar and Usami (1994) suggested an improved expression. As for the rule for the cumulation of damage, linear and nonhnear cumulation equations have been proposed by Miner, Chaboche and Coflfm-Manson (Yu and Feng, 1997). But all of these results are very difficult to use in practical structural analysis. Recently, based on experimental results, Shen and Dong (1997) suggested a uniaxial cumulative damage mechanics model for steel subjected to cyclic loading, which can be appUed in steel structural analysis. In this paper, the model is broadened to fit triaxial stress conditions and a FEM analysis procedure taking the damage cumulation into consideration is presented.
CUMULATIVE DAMAGE MECHANICS MODEL OF STRUCTURAL STEEL UNDER CYCLIC LOADING The model established by Shen and Dong (1997) can be expressed as follows. The damage index D is computed according to the plastic strain and hysteretic energy dissipation of steel pP
N
^„
1=1
P
^„
where A^ is the total number of half cycles which cause plastic strain, p is the weighted value of the /th half cycle, ef is the plastic strain during the /th half cycle, £^ is the largest plastic strain during all half-cycles and e^ is the ultimate plastic strain of the material. The effects of damage on Young's modulus, yield strength and strain hardening coefficient can be expressed as £"> ={\-^,D)E,
(2)
(3)
^P
^^^o+^sZ^
(4)
1 = 1 ^u
where EQ and E^ are the Young's modulus in respect of/)=0 and D respectively, o^,^ is the initial yield stress when Z)=0, af is the yield stress in respect of Z), ko and k are the hardening coefficients in respect ofZ)=0 andZ) and ^ j , i^, ^3 are material parameters. Figure 1 is the hysteretic model of steel allowing for damage cumulation given by Shen and Dong
107
(1997)
o nth half cycle
th half cyile
Figure 1: Hysteretic model of steel considering damage cumulation The hysteretic models of steel shown in Figure 1 and Eqn. 1 to Eqn. 4 are derived from cyclic loading tests in one-dimension. For the condition of triaxial stresses and proportional loading, the above results can also be valid if ef and ej are substituted with equivalent plastic strain pi and /?«, and a is substituted with equivalent stress a .
p. Pi •
ttPu
(5)
(6)
\dP ith-half cycle
dp = I -de^de^
(7)
" F l - ^ 2 ) ' +(^2 - ^ 3 ) ' +(^3 -^1)']
(8)
According to the energy damage theory, the relationship between/?« and e^ is (Yu and Feng, 1997) Pu-<'K
(9)
R^, is called the triaxial stress index, which reflects the effect of triaxial stress ratio.
R^=l(l + 3
v)-h3(l-2v)(^Y (J^„
(10)
108 where, v is the Poisson ratio of the material, a^ is the hydrostatic stress and a^ =cr^^/3 . If cr^= (72 =0-3, from Eqn. 8, then % = 0 and R^=^. In this case, the damage index should not be calculated using Eqn. 5. Based on the plastic damage theory, the release rate of damage strain energy can be expressed as (Yu and Feng, 1997)
-Y = -
^(l + vK^,+3(l-2v>7^
2E{\-D)
(11)
If a^ = a^ = a^ = a^, a =0, then Eqn. 11 becomes
y j l z ^
02)
2E{\-D)
and for uniaxial tension, since a = a and a^ = al3, Eqn. 11 becomes
Assuming that the value of -Y is the same when damage fracture occurs in uniaxial and triaxial conditions, then the hydrostatic stress a^ at fracture is
^mu = >J/ , V
(14)
where, cr„ is the ultimate strength of the material. Let damage index be 1 at fracture and assuming its value is proportional to the hydrostatic stress, then the increment of damage index under triaxial stress conditions can be calculated from
AD =
^ =
^^-^^-^
—
In practical analysis, Eqn. 15 will be employed if R^ is larger than a specified number.
(15)
109
CONSTITUTIVE RELATION OF STRUCTURAL STEEL WITH DAMAGE CUMULATION If the material is of the Von Mises-type, the subsequent yield function can be expressed as F^f-h
(16)
/ — v .
(17)
h-^cj]{p)
(18)
where, Sj, is the deviatoric stress tensor and a/p), i.e. a f is the current yield stress. It is assumed that the total strain tensor can be separated into elastic and plastic components, E ,' and E ,J, respectively, and the differential of the plastic strain tensor is given by the plastic flow rule
(19) By differentiating Eqn. 16, the following expression is obtained
^^ dG^^--G^^dp^O
do^^
'
(20)
3 ' dp
where ^
da
- .
(21)
•^ = E'
(22)
dp
E^ IS the plastic modulus of the material, which may be expressed as
p _
_E^E^_^^JcE^ 1^ j ^ _ 1^ rvj^ rrD r^tD rpD frpD E^ -E''' ~ E"" -kE""
-
F ^ i
j^ ^^ \-k
where E ^^ is the tangent modulus in respect of the damage index D.
^^^)
110
Substituting Eqn. 19 into Eqn. 7, then
dp = {-de ^de ^f'
= dX
i^^^f'
(24)
and Hook's law of material with cumulative damage may be written as dffij = C,„dEl, = C,„ids,i -de'J
df Ci0dSi,~Cij„dX^
(25)
E" v-E" <5,.d"„ +-6J., 1 +v '• (l + v)(l-2v)
(26)
= where C,,,
d ,j is the Kronecker delta. Putting Eqn. 3, 22, 23, 24, 25 into Eqn. 19, the scalar fiinction dX can be solved
§-c,A. dX =-
sl/2
^f-C..„-^ + da, "-da,, 3(1-A)
^-{\-i,D){^-£,Dy,-,,E,
(27)
(ida,^da,^j
substituting Eqn. 27 into Eqn. 25, then (28)
da,, = C,J,cfe „ where
(29)
C
da ^^ OG ^^
=aa,^
^
da,,
CI, is named as plastic tensor
3(1-^)
^T,da,,da,^j
Nl/2
(30)
Ill
NONLINEAR FINITE ELEMENT FORMULATIONS BASED ON CUMULATIVE DAMAGE MECHANICS MODEL Taking damage cumulation, elastoplasticity and large displacements into account, the updated Lagrangian finite element formulation can be written (31) where (32)
\-'D
(33)
.^, =-C^,,+.^,,)
f%
(34)
^-t^k.f'^k,
and Uj is the displacement at node /
FEM ANALYSIS OF STEEL MEMBERS SUBJECTED CYCLIC LOADING To verify the cumulative damage mechanics model and constitutive relations suggested, a nonlinear program was developed in which a 20-nodes solid element was employed and used to provide numerical results which were compared with the results obtained from experiments conducted by Chen (2000). Description of Test Specimens To understand the effects of damage cumulation on steel members, two series of cyclic tests, SI and S2 respectively, were conducted in the State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University. Outline of a specimen is shown in Figure 2 and design dimensions are shown in TABLE 1.
, — ^ .
'^
. . . „ „ „ .
^ti-^^
1-1
^
.
h
__: __\
L
I
i
i3 n
^r
^\ ^ :
id
\^^m_
2-2
Figure 2: Outline of specimen
112
TABLE 1 DESIGN DIMENSIONS OF TEST SPECIMENS
Specimen
SI S2
L (mm) 1100 1100
H (mm) 1300 1300
b (mm)
d (mm)
tf (mm)
h ("^i")
54 42
140 120
6 6
6 6
Cyclic loading was applied to the top of the specimen along the weak direction and the testing procedure was controlled by the displacement measured at that point, which is shown as Figure 3.
(b)
(a)
Figure 3: Testing Procedure Sl-1: SI-2: SI-3: Sl-4: SI-5: S2-1: S2-2: S2-3: S2-4: S2-5:
constant displacement amplitude, 5=50 mm; constant displacement amplitude, 6=60 mm; constant displacement amplitude, 5=40 mm; inconstant displacement amplitude, 5=50 mm(5 inconstant displacement amplitude, 5=60 mm(5 inconstant displacement amplitude, 5=40 mm(5 constant displacement amplitude, 5=50 mm; constant displacement amplitude, 5=60 mm; constant displacement amplitude, 5=65 mm; inconstant displacement amplitude, 5=70 mm(5
cycles)^5=60 mm(5 cycles)^5=50 mm(to failure); cycles)^5=50 mm(5 cycles)-^5=60 mm(to failure); cycles)^5-60 mm--5=40 mm(to failure);
cycles)---5=65 mm(5 cycles)^5=70 mm(to failure);
Finite Element Subdivision To simplify the computation, only that part above the stiffened plate at the bottom of the column was considered in the FEM analysis. The boundary conditions are assumed as fixed. Due to symmetry, the fmite element model was generated for only half of the specimen and necessary constraints are imposed on the plane of symmetry. The material properties of steel and the parameters in the cumulative damage model used in the FEM analysis are listed in TABLE 2 and TABLE 3, respectively. TABLE 2 MATERIAL PROPERTIES OF STEEL
o}o(Mpa)
cr„(Mpa)
EoCMpa)
<(°^)
310
445
196784
24.185
113
TABLES PARAMETERS IN THE CUMULATIVE DAMAGE MODEL
p 0.0081
r
^1
6
^3
kfj
0.227
0.119
0.000073
Eqn. 35
1.44
rj
0.041
k, = : 0 . 0 1 4 - 0 . 1 6 5 k , + l - 1 2 c -2.88k:,
(35)
Criterion of Damage Crack and Treatments Damage in steel was found to along with the increase of plastic strain and the number of cycles. This increase is irreversible. According to the cumulative damage mechanics model presented in this paper, a damage crack occurs at a gauss point if the value of the damage index at this point is greater than 1. Comparisons betyveen FEM and Experimental Results The number of half cycles during which a damage crack was initiated is shown in TABLE 4, and hysteretic curves of FEM analysis and experiment before the crack occurs are shown in Figure 4 to Figure 13 TABLE 4 NUMBER OF HALF CYCLES DURING WHICH DAMAGE CRACK INITIATED
Specimen
Calculated values
Sl-1 Sl-2 Sl-3 Sl-4 Sl-5 S2-1 S2-2 S2-3 S2-4 S2-5
25 20 40 22 21 51 44 30 22 16
Experimental results 24 14 32 12 16 48 16 24 16 8
(a) experimental curve (b) FEM curve Figure 4: Hysteretic curves of specimen SI — 1
114
20
40
60
80
(a) experimental curve (b) FEM curve Figure 5: Hysteretic curves of specimen SI —2
10
5
f0 r -5
.,.
•
' f^ 1 M
\ M*^
• -40
-1S
-20
0 u(mm)
20
(a) experimental curve (b) FEM curve Figure 6; Hysteretic curves of specimen SI —3
u(mm)
u(mm)
(a) experimental curve (b) FEM curve Figure 7: Hysteretic curves of specimen SI —4
.--/lA
If
-
-40
-20
-
0
:
-
J
20
40
u(mm)
(a) experimental curve (b) FEM curve Figure 8: Hysteretic curves of specimen SI —5
40
60
115
-60
-40
-20
0
20
u(mm)
(a) experimental curve (b) FEM curve Figure 9: Hysteretic curves of specimen S2— 1
-80
-60
-40
-20
0
20
40
60
80
(a) experimental curve (b) FEM curve Figure 10: Hysteretic curves of specimen S2—2
(a) experimental curve (b) FEM curve Figure 11: Hysteretic curves of specimen S2—3
;
'I..,
(a) experimental curve (b) FEM curve Figure 12: Hysteretic curves of specimen S2—4
40
116
-80
-60
-40
-20
(a) experimental curve (b) FEM curve Figure 13: Hysteretic curves of specimen S2—5 From TABLE 4 and Figure 4 to Figure 13, it can be seen that the FEM analytical results are in good agreement with the experimental ones. The cumulative damage mechanics model suggested in this paper is accurate enough to evaluate the damage cumulation in the structural steel and to predict initiation of damage cracks under cyclic loading. The seismic behavior of steel structures, therefore, can be simulated with an acceptable error for an earthquake, and repeated earthquakes by using of damage model presented.
CONCLUSION Based on the experiments and energy damage theory, a cumulative damage model which caters for triaxial stress conditions is proposed in this paper. A general procedure for computing damage cumulation in steel members with finite element method is suggested. Cantilever members subjected cyclic loading at their ends were simulated numerically with a FEM program. The analytical results showed good agreement with the experimental results.
ACKNOWLEDGEMENT The project is financially supported by National Science Foundation of China as a key project (59895410).
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