ON THE HOLED PLATE UNDER TENSION LOAD.
No. 28 Vol.1 Thn. XIV November 2007
ISSN: 0854-8471
ON THE HOLED PLATE UNDER TENSION LOAD
Sabril Haris HG
Laboratorium Material dan Struktur, Jurusan Teknik Sipil, Universitas Andalas
ABSTRACT
Two types of holed plate subjected tension load are examined by using Abaqus numerical software
for non linear calculation. The absence of the element around the hole becomes the interesting
issues such as the effective area and the stress concentration. In this paper, variations on diameter
of hole and stress-strain curve based on Voce’s law are investigated regarding the plate response
in the elastic and plastic region. The maximum load from the numerical work is compared by the
analytical result and a good agreement is obtained.
1.
INTRODUCTION
The holed plate subjected tension load will have
experience the phenomena called by stress
concentration due to the absence of the elements
around the hole. This increasing stress makes those
element will attain yield stress first meanwhile the
others are still in the elastic region. Since the
yielding process occurred, it will affect the response
of the whole plate because yielding elements now
have the less stiffness regarding in the plastic state.
The subject of this paper is to observe the stress
concentration on the holed plate by using the
nonlinear software Abaqus. Two types of holed plate
are subjected static monotonic load until the ultimate
condition reached. A detailed analysis of this
simulation is presented in view of the yielding in the
elements around the hole and the propagation
process to the others.
Several aspects are discussed, such as response
in the elastic region that comprises determination of
first yield-load and significant yield-load, and
response in the plastic region that covers ultimate
conditions of plate. Furthermore, the effect of
diameter of hole and parameter of stress-strain curve
material model are also analyzed.
2.
DESCRIPTION OF MODEL
2.1 Dimension of Plate
There are two types of plate that will be used in
this simulation. The first is plate with one hole in the
center of plate and the other is plate with two-half
hole in the out-fiber side.
Dimension of plates are fixed for all of the
models. They are the length of plate L0 = 300 mm,
the width of plate h = 100 mm, and plate thickness t
= 10 mm. Then, the full cross section area A0 can be
calculated as 1000 mm2.
Diameter of hole (d) are varied for d = 0, 10, 20,
and 30 mm. So, for d = 0 mm, plate type-1 and type2 are identical.
For both types, in the cross section 1-1, area of
plate is defined as the net area (Anet) which is
calculated as Anet = t (h - d). The models are shown
in Fig. 1.
TeknikA
1
d
h = 100 mm
1
L0 = 300 mm
a. Plate Type - 1
b. Plate Type - 2
Figure 1 Types of Plate
2.2 Material
Material used in this paper is the mild steel grade
41, which is widely used in many steel structures
with yield stress σy = 250 MPa. In order to define
the input to Abaqus, Voce’s law [1] is used to
construct stress-strain curve for the material
characteristic. This formula is shown in Eq. (1).
σ = σ 0 + Q (1 - e-C.ε )
(1)
Base on some experimental data of uniaxial
tensile test [2,3], basic parameter in Eq. (1)
determined as follows:
σ0 = 250 MPa
Q = 250
C = 15
21
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
To get the other strain-stress cuurves, Q will be
varied for some valuue, i.e. 75 andd 160 with C is
nt. Plastic strain data are set from 0.00 to the
t
constan
maximu
um value 0.188 with uniform
m increment 0.01
and thenn stress valuess are generated by Voce’s law
w.
Pieccewise-linear relationships
r
fo three types of
for
stress-strain are obtain
ned as well as shown in Fig. 22.
3.
RESPONSE OF PLATE IN
N THE ELAS
STIC
REGION
F two types of
For
o plate, variattion on the diaameter
of holes
h
(d) and pparameter Q in
i the stress strains
stresss relationshipps are obserrved below. First
analy
ysis is observaation in the elastic region annd the
secoond one is in thhe plastic regionn.
E
Areaa
3.1 Effective
60
00
True Stress (MPa)
IS
SSN: 0854-88471
50
00
Q = 2 50
40
00
Q = 16 0
Q =75
30
00
L
Loading
the plate
p
for smalll deformation gives
elasttic response denoted by linear relatioonship
betw
ween load and ddisplacement.
E
Elastic
responnse of the strructures meetts the
form
mula:
σ=E.ε
20
00
10
00
0
0
0.05
0.1
0.15
0.2
Plastic Strain
Figure 2 Stress-strain Curve
C
M while 0.33 is
Elasstic modulus E is 200000 MPa
used foor Poisson’s rattio (μ). For thee yield surfacee, it
followss isotropic hard
dening for worrk hardening and
a
von Mises criterion fo
or the effectivee stress.
2.3 Modeling
By using Abaqus software, plaate is modeled as
w
is regularrly meshed as shown
s
below. To
solid which
handle stress concen
ntration at the area around the
t
hole, finner meshing ellements are cho
osen.
(2)
F uni-axial teension σ can bee defined as the load
For
(F) divided
d
by the cross section area.
a
Meanwhiile ε is
defin
ned as the eelongation or displacement (ΔL)
divid
ded by the lenggth of the platte. Further thenn, it is
easieer to use initiial dimension than instantaaneous
one for the conditioon in this elasttic region.
R
Replacing
σ byy F/A0 and ε byy ΔL/L0, we geet:
⎛ E A0 ⎞
(3)
F = k ΔL ; where k = ⎜
⎟
⎝ L0 ⎠
E (3) shows the linear relaationship betw
Eq.
ween F
and ΔL
Δ by the stifffness k = E A0 / L0.
D
Differences
of cross section area will take place
due to the hole, booth for two typ
pes of plate. Beecause
of thhe hole, thereffore it is necesssary then to define
d
the effective area (Aeff) insteadd of the initial area
(A0) that fulfills Eqq.(3). The stiffn
fness then becoomes k
= E Aeff / L0
F
From
the loadd-displacementt curve plotteed for
eachh model, the stiffness k is easy to obtaain by
lineaarization in thhe elastic part. Some data at
a the
beginning curve are
a taken to construct
c
the linear
equaation of load-ddisplacement. Then, Aeff will
w be
achieved by using the formula. Aeff = k L0 / E.
E The
resullts shown in Taable 1.
Tab
ble 1 Effective Area
Figure 3 Element Messhing
Plate Type
d (mm)
k
Aeff (mm
m2)
No Hole
0
10
20
30
10
20
30
666.6667
661.5578
645.7708
618.6663
661.2260
645.5594
621.0038
1000.000
992.377
968.566
927.999
991.899
968.399
931.566
Type 1
2.4 Loa
ading Conditioon
Thee tension loadin
ng conducted in
i this simulatiion
is subjeected the unifoorm load in th
he one side whhile
the otheer side becom
mes the fixed su
upport. To avooid
unexpeccted additionaal stress regaarding Poissonn’s
effect, the
t vertical and
d lateral fixatio
ons in the suppport
are releeased unless thee nodes in the middle
m
one.
Stattic monotonic loading are suubjected until the
t
ultimatee condition reaached.
Type 2
D
Difference
valuue of Q in thee material propperties
doess not influencee the result sincce the responses are
still in the elastic reegion
I is interestinng then to fin
It
nd the relatioonship
betw
ween Aeff and diameter of hole (d). Forr non
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
dimensiional purpose, Aeff and d arre replaced by its
normaliized value Aefff/Ano and d/h respectively. Ano
is the crross sectional area
a for no hole plate.
IS
SSN: 0854-88471
T
Table
2 givess us the infoormation that stress
conccentrations occcurs due to holle on the platee since
we need
n
the smalllest load to ach
hieve the first yield.
The greater diameeter we have, the higher raatio of
stresss concentratioon we get. It can
c be also seeen the
loadds for plate typee 2 are greater than for plate type
t
1
withh the same diam
meter of hole.
T stress conccentration for both
The
b
types of plate is
show
wn in Fig. 5.
a. Plate Type-1
Figgure 4 Normaalized Effectivee Area vs d/h
From
m the curve in
n Gig.4, we geet the relationshhip
betweenn normalized Aeff and diametter of hole as:
A eff
= 1 - 0.77
A0
⎛d⎞
⎜ ⎟
⎝h⎠
2
((4)
t term 0.77 (d/h)2 in Eq.44 is
We can say that the
duction area duue to hole in the
t
the facttor for the red
plate thhat is limited fo
or this study.
3.2 Firsst Yield
It iss widely know
wn that stress concentration w
will
be takeen place on thee element arouund the hole. By
observing the stress values whicch get from the
t
Abaquss simulation, we
w can see the increasing of the
t
stress due
d to that phen
nomenon.
If we
w assumed the
t
stresses on
n the plates are
a
uniform
m, the load thatt generates firsst yielding (Fyiield)
can be calculated
c
as th
he formula in Eq.
E (5).
Fyield = Aeff . σyield
(5)
Thee yield-load vaalues calculatedd by formula and
a
their nuumerical resultss are shown in Table 2.
b. Plate typee-2
Figure 5 Stress Concentration
3.3 Significant
S
Yieeld on The Looad-Displacem
ment
Curve
T
The
load thatt generates fiirst yielding ((Fyield)
resullted from num
merical result is not meanningful
sincee after that poiint the responsse of the plate is
i still
nearrly linear. The eeffect of local yielding at thee some
of element aroundd the hole, is not significanntly to
channge the stiffnesss of the structu
ure.
S
Significant
yiield point comes
c
after more
additional load. To
T calculate the value off this
signiificant-yield load, two straight liness are
sketcched. First linee represents thee data on elastiic part
befoore the significaant point as weell as the seconnd one
repreesents some daata just after th
he significant point.
For instance, for pplate type 1 with
w 20 mm in holediam
meter and Q = 250, the sign
nificant-yield looad is
210.945 kN while the first yield is 125.075 kN
N. It is
show
wn in Fig. 6.
Table 2 First Yield Lo
oad
Plate Type
d
No Hole
H
0
10
20
30
10
20
30
Typ
pe 1
Typ
pe 2
kN)
Fyield (k
formula
250.000
248.092
242.141
231.999
247.973
242.098
232.889
numerical
n
250.000
153.125
125.075
106.525
155.750
128.400
115.125
raatio
1
1..62
1..94
2..18
1..59
1..89
2..02
Figurre 6 Significannt Yield
No. 28 Vol.1 Thn. XIV November 2007
Significant-yield loads for different value of Q
are almost same because the effect of hardening
which occur in some of element around the hole is
still in the beginning of hardening (in the small
strain). As shown in Fig. (2), in the beginning of
material hardening, the difference between varied Q
is small. It can be seen in the Fig. (7.c) and (7.d).
4.
RESPONSE OF PLATE IN THE PLASTIC
REGION
those vary on of diameter of hole as well as variation
on Q respectively are shown in Fig. (7).
Plate type 1 with one hole in the center of plate,
in general, has less significant-yield load than plate
type 2 with two-half hole in the out fiber side. It is
agree with the load at the first yielding (Table 2).
As shown in Fig. (7.a) and (7.b), for both type,
the greater diameter of hole will make the stiffness
of the load-displacement curve decrease faster. From
the curves, the smooth rounding path is created after
significant-yield point until reach the stage with
nearly linear as the new stiffness of the plate. The
greatest diameter is always positioned on the lowest
curve.
350
350
300
300
250
Load (kN)
Load (kN)
4.1 Yield Propagation
To observe the response of plate in the plastic
region, it is beneficial to spot the load-displacement
curve on the part after significant yield point. The
load displacement curves for two types of plate
ISSN: 0854-8471
d = 0 mm (no hole)
200
d = 10 mm
250
d = 0 mm (no hole)
200
d = 10 mm
d = 20 mm
150
d = 20 mm
150
d = 30 mm
d = 30 mm
100
100
0
2
4
6
8
0
2
displacement (mm)
6
8
b. Plate Type 2, variation on d, Q = 250
300
300
2 50
250
Load (kN)
Load (kN)
a. Plate Type 1, variation on d, Q = 250
200
Q = 250
Q = 160
150
4
displacement (mm)
200
Q = 250
Q = 160
150
Q = 75
Q = 75
10 0
100
0
0 .5
1
1.5
2
displacement (mm)
0
0.5
1
1.5
2
displacement (mm)
c. Plate Type 1, variation on Q, d = 20 mm
d. Plate Type 2, variation on Q, d = 20 mm
Figure 7 Load-displacement Curve for Two Types of Plate
Yield propagation can be analyzed from the
shape of the plate. For plate type 1, on the holedcross section, area of plate is divided into two parts,
the upper and the lower. The distance from the
corner to the out-fiber is (h-d)/2 for both parts. For
plate type 2, on the holed-cross section, there is only
one area of plate from upper-hole side to lower-hole
side and the distance is (h-d). So, propagation model
for plate type 2 produce the greater value of load
TeknikA
than plate type 1. Yield propagation for both type of
plate follows the grey-black contour as shown in
Fig. (8).
a. Plate Type 1
24
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
IS
SSN: 0854-88471
Tablle 3 Maximum
m Load
b. Plate
P
Type 1
wo Types of Plaate
Figuree 8 Yield Propaagation for Tw
As mentioned beefore, variationn on Q has less
effect to
t the load-dissplacement currve of plate affter
the sign
nificant-yield reached.
r
The differences
d
occcur
after that point that iss proportional with
w the valuee of
Q.
T
There
are som
me variation between anallytical
resullts and the nuumerical ones. For Q = 75 - plate
type 1 and Q = 2250 - plate typpe 2, the resullts are
veryy close each othher with differences less thann 1 %.
Meaanwhile, for thee others, the reesults are quitee good
withh differences less than 10 %.
4.2 Maximum Load
Thee maximum loaad in the plate defined when dF
= 0. It can
c be attained
d when diffuse necking occur on
the crosss section with hole (section 1-1 in Fig. 1.).
Diffusee necking for th
hese cases lead
ds to:
dσ
=σ
dε
((6)
Applyin
ng Eq. (6) to Eq.
E (1) will get:
εu =
⎡ Q (1+C ) ⎤
1
ln ⎢
⎥
C ⎣ σ 0 +Q ⎦
σ u = ( σ 0 +Q
Q)
C
1+C
((7)
((8)
a
maximum
m load (Fu) thhen
Tensile strength su and
definedd as:
σu
su =
1 + εu
Fu = s u . A net
5. CONCLUSION
N
N
Numerical
worrks have been done for the holed
platees which are ssubjected the tension load. Some
data in the beginning of the elastic region are used
u
to
ne the stiffnesss of the structtures. It is obttained
defin
that responses in thhe elastic regioon are determinned by
the effective
e
area in the cross seection. The efffect of
paraameter Q in thhe Voce’s law
w is immateriial. A
simp
ple relationshipp between thee full cross section
and the effective one is propossed. The first yield
occuur around the hhole is not signiificant to change the
stiffn
fness of the strructures until the
t significantt-yield
loadd attained. Maxximum load forr the holed plaate can
be determined
d
frrom the num
merical work which
w
show
ws a good agreement with thee analytical ressult.
FERENCES
REF
1.
2.
((9)
(110)
with
Agaain, Anet is the area in the crross section w
hole, ass mentioned previously.
Theese analytical results
r
will be compared to tthe
numericcal ones whicch are obtaineed from Abaqqus
simulattion. Maximum
m load from numerical
n
resuults
are defi
fined when thee critical point, element arouund
the holee, reaches the true uniform strain
s
(εu) as well
w
as (σu).
In Table 3 it iss shown that the results for
diameteer of hole d = 20
2 mm or Anet = 800 mm2.
3.
T.Beylytschkoo, W.K.Liu and Moran, Nonnlinear
finite elemennts for contin
nua and strucctures,
Wiley, 2000.
R.Tornqvist, Design of Crashworthy Ship
Structures, PhhD Thesis, Technical Universsity of
Denmark, 20003.
Civil Enginneering Mateerials Laborratory,
University off Mexico, Ten
nsile Test of Steel,
2007.
ISSN: 0854-8471
ON THE HOLED PLATE UNDER TENSION LOAD
Sabril Haris HG
Laboratorium Material dan Struktur, Jurusan Teknik Sipil, Universitas Andalas
ABSTRACT
Two types of holed plate subjected tension load are examined by using Abaqus numerical software
for non linear calculation. The absence of the element around the hole becomes the interesting
issues such as the effective area and the stress concentration. In this paper, variations on diameter
of hole and stress-strain curve based on Voce’s law are investigated regarding the plate response
in the elastic and plastic region. The maximum load from the numerical work is compared by the
analytical result and a good agreement is obtained.
1.
INTRODUCTION
The holed plate subjected tension load will have
experience the phenomena called by stress
concentration due to the absence of the elements
around the hole. This increasing stress makes those
element will attain yield stress first meanwhile the
others are still in the elastic region. Since the
yielding process occurred, it will affect the response
of the whole plate because yielding elements now
have the less stiffness regarding in the plastic state.
The subject of this paper is to observe the stress
concentration on the holed plate by using the
nonlinear software Abaqus. Two types of holed plate
are subjected static monotonic load until the ultimate
condition reached. A detailed analysis of this
simulation is presented in view of the yielding in the
elements around the hole and the propagation
process to the others.
Several aspects are discussed, such as response
in the elastic region that comprises determination of
first yield-load and significant yield-load, and
response in the plastic region that covers ultimate
conditions of plate. Furthermore, the effect of
diameter of hole and parameter of stress-strain curve
material model are also analyzed.
2.
DESCRIPTION OF MODEL
2.1 Dimension of Plate
There are two types of plate that will be used in
this simulation. The first is plate with one hole in the
center of plate and the other is plate with two-half
hole in the out-fiber side.
Dimension of plates are fixed for all of the
models. They are the length of plate L0 = 300 mm,
the width of plate h = 100 mm, and plate thickness t
= 10 mm. Then, the full cross section area A0 can be
calculated as 1000 mm2.
Diameter of hole (d) are varied for d = 0, 10, 20,
and 30 mm. So, for d = 0 mm, plate type-1 and type2 are identical.
For both types, in the cross section 1-1, area of
plate is defined as the net area (Anet) which is
calculated as Anet = t (h - d). The models are shown
in Fig. 1.
TeknikA
1
d
h = 100 mm
1
L0 = 300 mm
a. Plate Type - 1
b. Plate Type - 2
Figure 1 Types of Plate
2.2 Material
Material used in this paper is the mild steel grade
41, which is widely used in many steel structures
with yield stress σy = 250 MPa. In order to define
the input to Abaqus, Voce’s law [1] is used to
construct stress-strain curve for the material
characteristic. This formula is shown in Eq. (1).
σ = σ 0 + Q (1 - e-C.ε )
(1)
Base on some experimental data of uniaxial
tensile test [2,3], basic parameter in Eq. (1)
determined as follows:
σ0 = 250 MPa
Q = 250
C = 15
21
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
To get the other strain-stress cuurves, Q will be
varied for some valuue, i.e. 75 andd 160 with C is
nt. Plastic strain data are set from 0.00 to the
t
constan
maximu
um value 0.188 with uniform
m increment 0.01
and thenn stress valuess are generated by Voce’s law
w.
Pieccewise-linear relationships
r
fo three types of
for
stress-strain are obtain
ned as well as shown in Fig. 22.
3.
RESPONSE OF PLATE IN
N THE ELAS
STIC
REGION
F two types of
For
o plate, variattion on the diaameter
of holes
h
(d) and pparameter Q in
i the stress strains
stresss relationshipps are obserrved below. First
analy
ysis is observaation in the elastic region annd the
secoond one is in thhe plastic regionn.
E
Areaa
3.1 Effective
60
00
True Stress (MPa)
IS
SSN: 0854-88471
50
00
Q = 2 50
40
00
Q = 16 0
Q =75
30
00
L
Loading
the plate
p
for smalll deformation gives
elasttic response denoted by linear relatioonship
betw
ween load and ddisplacement.
E
Elastic
responnse of the strructures meetts the
form
mula:
σ=E.ε
20
00
10
00
0
0
0.05
0.1
0.15
0.2
Plastic Strain
Figure 2 Stress-strain Curve
C
M while 0.33 is
Elasstic modulus E is 200000 MPa
used foor Poisson’s rattio (μ). For thee yield surfacee, it
followss isotropic hard
dening for worrk hardening and
a
von Mises criterion fo
or the effectivee stress.
2.3 Modeling
By using Abaqus software, plaate is modeled as
w
is regularrly meshed as shown
s
below. To
solid which
handle stress concen
ntration at the area around the
t
hole, finner meshing ellements are cho
osen.
(2)
F uni-axial teension σ can bee defined as the load
For
(F) divided
d
by the cross section area.
a
Meanwhiile ε is
defin
ned as the eelongation or displacement (ΔL)
divid
ded by the lenggth of the platte. Further thenn, it is
easieer to use initiial dimension than instantaaneous
one for the conditioon in this elasttic region.
R
Replacing
σ byy F/A0 and ε byy ΔL/L0, we geet:
⎛ E A0 ⎞
(3)
F = k ΔL ; where k = ⎜
⎟
⎝ L0 ⎠
E (3) shows the linear relaationship betw
Eq.
ween F
and ΔL
Δ by the stifffness k = E A0 / L0.
D
Differences
of cross section area will take place
due to the hole, booth for two typ
pes of plate. Beecause
of thhe hole, thereffore it is necesssary then to define
d
the effective area (Aeff) insteadd of the initial area
(A0) that fulfills Eqq.(3). The stiffn
fness then becoomes k
= E Aeff / L0
F
From
the loadd-displacementt curve plotteed for
eachh model, the stiffness k is easy to obtaain by
lineaarization in thhe elastic part. Some data at
a the
beginning curve are
a taken to construct
c
the linear
equaation of load-ddisplacement. Then, Aeff will
w be
achieved by using the formula. Aeff = k L0 / E.
E The
resullts shown in Taable 1.
Tab
ble 1 Effective Area
Figure 3 Element Messhing
Plate Type
d (mm)
k
Aeff (mm
m2)
No Hole
0
10
20
30
10
20
30
666.6667
661.5578
645.7708
618.6663
661.2260
645.5594
621.0038
1000.000
992.377
968.566
927.999
991.899
968.399
931.566
Type 1
2.4 Loa
ading Conditioon
Thee tension loadin
ng conducted in
i this simulatiion
is subjeected the unifoorm load in th
he one side whhile
the otheer side becom
mes the fixed su
upport. To avooid
unexpeccted additionaal stress regaarding Poissonn’s
effect, the
t vertical and
d lateral fixatio
ons in the suppport
are releeased unless thee nodes in the middle
m
one.
Stattic monotonic loading are suubjected until the
t
ultimatee condition reaached.
Type 2
D
Difference
valuue of Q in thee material propperties
doess not influencee the result sincce the responses are
still in the elastic reegion
I is interestinng then to fin
It
nd the relatioonship
betw
ween Aeff and diameter of hole (d). Forr non
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
dimensiional purpose, Aeff and d arre replaced by its
normaliized value Aefff/Ano and d/h respectively. Ano
is the crross sectional area
a for no hole plate.
IS
SSN: 0854-88471
T
Table
2 givess us the infoormation that stress
conccentrations occcurs due to holle on the platee since
we need
n
the smalllest load to ach
hieve the first yield.
The greater diameeter we have, the higher raatio of
stresss concentratioon we get. It can
c be also seeen the
loadds for plate typee 2 are greater than for plate type
t
1
withh the same diam
meter of hole.
T stress conccentration for both
The
b
types of plate is
show
wn in Fig. 5.
a. Plate Type-1
Figgure 4 Normaalized Effectivee Area vs d/h
From
m the curve in
n Gig.4, we geet the relationshhip
betweenn normalized Aeff and diametter of hole as:
A eff
= 1 - 0.77
A0
⎛d⎞
⎜ ⎟
⎝h⎠
2
((4)
t term 0.77 (d/h)2 in Eq.44 is
We can say that the
duction area duue to hole in the
t
the facttor for the red
plate thhat is limited fo
or this study.
3.2 Firsst Yield
It iss widely know
wn that stress concentration w
will
be takeen place on thee element arouund the hole. By
observing the stress values whicch get from the
t
Abaquss simulation, we
w can see the increasing of the
t
stress due
d to that phen
nomenon.
If we
w assumed the
t
stresses on
n the plates are
a
uniform
m, the load thatt generates firsst yielding (Fyiield)
can be calculated
c
as th
he formula in Eq.
E (5).
Fyield = Aeff . σyield
(5)
Thee yield-load vaalues calculatedd by formula and
a
their nuumerical resultss are shown in Table 2.
b. Plate typee-2
Figure 5 Stress Concentration
3.3 Significant
S
Yieeld on The Looad-Displacem
ment
Curve
T
The
load thatt generates fiirst yielding ((Fyield)
resullted from num
merical result is not meanningful
sincee after that poiint the responsse of the plate is
i still
nearrly linear. The eeffect of local yielding at thee some
of element aroundd the hole, is not significanntly to
channge the stiffnesss of the structu
ure.
S
Significant
yiield point comes
c
after more
additional load. To
T calculate the value off this
signiificant-yield load, two straight liness are
sketcched. First linee represents thee data on elastiic part
befoore the significaant point as weell as the seconnd one
repreesents some daata just after th
he significant point.
For instance, for pplate type 1 with
w 20 mm in holediam
meter and Q = 250, the sign
nificant-yield looad is
210.945 kN while the first yield is 125.075 kN
N. It is
show
wn in Fig. 6.
Table 2 First Yield Lo
oad
Plate Type
d
No Hole
H
0
10
20
30
10
20
30
Typ
pe 1
Typ
pe 2
kN)
Fyield (k
formula
250.000
248.092
242.141
231.999
247.973
242.098
232.889
numerical
n
250.000
153.125
125.075
106.525
155.750
128.400
115.125
raatio
1
1..62
1..94
2..18
1..59
1..89
2..02
Figurre 6 Significannt Yield
No. 28 Vol.1 Thn. XIV November 2007
Significant-yield loads for different value of Q
are almost same because the effect of hardening
which occur in some of element around the hole is
still in the beginning of hardening (in the small
strain). As shown in Fig. (2), in the beginning of
material hardening, the difference between varied Q
is small. It can be seen in the Fig. (7.c) and (7.d).
4.
RESPONSE OF PLATE IN THE PLASTIC
REGION
those vary on of diameter of hole as well as variation
on Q respectively are shown in Fig. (7).
Plate type 1 with one hole in the center of plate,
in general, has less significant-yield load than plate
type 2 with two-half hole in the out fiber side. It is
agree with the load at the first yielding (Table 2).
As shown in Fig. (7.a) and (7.b), for both type,
the greater diameter of hole will make the stiffness
of the load-displacement curve decrease faster. From
the curves, the smooth rounding path is created after
significant-yield point until reach the stage with
nearly linear as the new stiffness of the plate. The
greatest diameter is always positioned on the lowest
curve.
350
350
300
300
250
Load (kN)
Load (kN)
4.1 Yield Propagation
To observe the response of plate in the plastic
region, it is beneficial to spot the load-displacement
curve on the part after significant yield point. The
load displacement curves for two types of plate
ISSN: 0854-8471
d = 0 mm (no hole)
200
d = 10 mm
250
d = 0 mm (no hole)
200
d = 10 mm
d = 20 mm
150
d = 20 mm
150
d = 30 mm
d = 30 mm
100
100
0
2
4
6
8
0
2
displacement (mm)
6
8
b. Plate Type 2, variation on d, Q = 250
300
300
2 50
250
Load (kN)
Load (kN)
a. Plate Type 1, variation on d, Q = 250
200
Q = 250
Q = 160
150
4
displacement (mm)
200
Q = 250
Q = 160
150
Q = 75
Q = 75
10 0
100
0
0 .5
1
1.5
2
displacement (mm)
0
0.5
1
1.5
2
displacement (mm)
c. Plate Type 1, variation on Q, d = 20 mm
d. Plate Type 2, variation on Q, d = 20 mm
Figure 7 Load-displacement Curve for Two Types of Plate
Yield propagation can be analyzed from the
shape of the plate. For plate type 1, on the holedcross section, area of plate is divided into two parts,
the upper and the lower. The distance from the
corner to the out-fiber is (h-d)/2 for both parts. For
plate type 2, on the holed-cross section, there is only
one area of plate from upper-hole side to lower-hole
side and the distance is (h-d). So, propagation model
for plate type 2 produce the greater value of load
TeknikA
than plate type 1. Yield propagation for both type of
plate follows the grey-black contour as shown in
Fig. (8).
a. Plate Type 1
24
No. 28
8 Vol.1 Thn
n. XIV Noveember 20077
IS
SSN: 0854-88471
Tablle 3 Maximum
m Load
b. Plate
P
Type 1
wo Types of Plaate
Figuree 8 Yield Propaagation for Tw
As mentioned beefore, variationn on Q has less
effect to
t the load-dissplacement currve of plate affter
the sign
nificant-yield reached.
r
The differences
d
occcur
after that point that iss proportional with
w the valuee of
Q.
T
There
are som
me variation between anallytical
resullts and the nuumerical ones. For Q = 75 - plate
type 1 and Q = 2250 - plate typpe 2, the resullts are
veryy close each othher with differences less thann 1 %.
Meaanwhile, for thee others, the reesults are quitee good
withh differences less than 10 %.
4.2 Maximum Load
Thee maximum loaad in the plate defined when dF
= 0. It can
c be attained
d when diffuse necking occur on
the crosss section with hole (section 1-1 in Fig. 1.).
Diffusee necking for th
hese cases lead
ds to:
dσ
=σ
dε
((6)
Applyin
ng Eq. (6) to Eq.
E (1) will get:
εu =
⎡ Q (1+C ) ⎤
1
ln ⎢
⎥
C ⎣ σ 0 +Q ⎦
σ u = ( σ 0 +Q
Q)
C
1+C
((7)
((8)
a
maximum
m load (Fu) thhen
Tensile strength su and
definedd as:
σu
su =
1 + εu
Fu = s u . A net
5. CONCLUSION
N
N
Numerical
worrks have been done for the holed
platees which are ssubjected the tension load. Some
data in the beginning of the elastic region are used
u
to
ne the stiffnesss of the structtures. It is obttained
defin
that responses in thhe elastic regioon are determinned by
the effective
e
area in the cross seection. The efffect of
paraameter Q in thhe Voce’s law
w is immateriial. A
simp
ple relationshipp between thee full cross section
and the effective one is propossed. The first yield
occuur around the hhole is not signiificant to change the
stiffn
fness of the strructures until the
t significantt-yield
loadd attained. Maxximum load forr the holed plaate can
be determined
d
frrom the num
merical work which
w
show
ws a good agreement with thee analytical ressult.
FERENCES
REF
1.
2.
((9)
(110)
with
Agaain, Anet is the area in the crross section w
hole, ass mentioned previously.
Theese analytical results
r
will be compared to tthe
numericcal ones whicch are obtaineed from Abaqqus
simulattion. Maximum
m load from numerical
n
resuults
are defi
fined when thee critical point, element arouund
the holee, reaches the true uniform strain
s
(εu) as well
w
as (σu).
In Table 3 it iss shown that the results for
diameteer of hole d = 20
2 mm or Anet = 800 mm2.
3.
T.Beylytschkoo, W.K.Liu and Moran, Nonnlinear
finite elemennts for contin
nua and strucctures,
Wiley, 2000.
R.Tornqvist, Design of Crashworthy Ship
Structures, PhhD Thesis, Technical Universsity of
Denmark, 20003.
Civil Enginneering Mateerials Laborratory,
University off Mexico, Ten
nsile Test of Steel,
2007.