teori lamina bkh
TRANSCRIPT
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KULIAH
MEKANIKA STRUKTUR KOMPOSIT
05. TEORI LAMINA
Dr. Bambang Kismono Hadi
PT Dirgantara Indonsia
!0"#
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MECHANICS OF COMPOSITE
STRUCTURES
Definitions:
• Isotropic $ a matria% &a'ing simi%ar (ro(rtis in a%% dir)tions. *or
+am(%$ a%,min,m- st%- t).
• Orthotropic $ a matria% &a'ing to (%an o/ smmtr &i)& ar
(r(ndi),%ar to a)& ot&r. E+am(%$ )om(osit matria%s in /ibrdir)tion.
• Anisotropic$ a matria% &a'ing no (%an o/ smmtr. E+am(%$
)om(osit matria%s not in /ibr dir)tion.
Isotro(i) Ort&otro(i) Anisotro(i)
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COMPOSITE MATERIALS
In /ibr dir)tion- a )om(osit matria% is an ort&otro(i) matria%.
1
2
1 $ /ibr dir)tion
2$ (r(ndi),%ar
/ibr dir)tion
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COMPOSITE MATERIAL PROPERTIES
E" $ Mod,%,s o/ %asti)it in /ibr dir)tion
E! $ Mod,%,s o/ %asti)it in (r(ndi),%ar /ibr dir)tion
v "! $ Poisson1s ratio in "2! (%an
3"! $ S&ar mod,%,s in "2! (%an
4In isotro(i) matria%s- it &as on% to matria% (ro(rtis$ E and v
E" 6 tan T& sam (&nomna )an bdran /or ot&r matria%(ro(rtis.
7t-)
1σ
1σ
1σ
1ε
α
α
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Eperi!ent"# $eter!in"tion of E2
E! 6 tan8t-)
2σ
2σ
α
α
2σ
2ε
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Eperi!ent"# $eter!in"tion of %12
P
P&2 P&2
Rostt Strain 3ag
b
S(simn dngan tba% t .
S
0
4512 2ε γ =
t b
P
.
)2/(12 =τ
α
12τ
12γ
α tan12
=G
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PLATE UNDER MULTI'A(IAL LOADIN%S
)Isotropic*
Constit+ti,e E-+"tions for Isotropic
1σ 1σ
1ε
2ε
0
.
12
112
1
1
=
−=−=
=
γ
υσ ε υ ε
σ
ε
E
E
1σ
1σ
2σ
2σ
12τ
12τ
−
−
=
12
2
1
12
2
1
100
01
01
τ
σ σ υ
υ
γ
ε ε
G
E E
E E
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Or$
Stiffness M"trices for Isotropic M"teri"#s
9&r$
( ) ( )
( ) ( )
−−
−−
=
12
2
1
22
22
12
2
1
00
011
011
γ
ε
ε
υ υ υ
υ υ
υ
τ
σ
σ
G
E E
E E
( )υ +=
12
E G
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PLATE UNDER MULTI'A(IAL LOADIN%S
)Orthotropic*
Constit+ti,e E-+"tions for Orthotropic
1σ 1σ
1ε
2ε
0
.
12
1
1121122
1
1
1
=
−=−=
=
γ
σ υ ε υ ε
σ
ε
E
E
1σ
1σ
2σ
2σ
12τ
12τ
−
−
=
12
2
1
12
22
21
1
12
1
12
2
1
100
01
01
τ
σ σ υ
υ
γ
ε ε
G
E E
E E
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Or$
Stiffness M"trices for Orthotropic M"teri"#s
9&r$
( ) ( )
( ) ( )
−−
−−
=
12
2
1
12
2112
2
2112
212
2112
121
2112
1
12
2
1
00
0.1.1
0.1..1
γ
ε
ε
υ υ υ υ υ
υ υ υ υ υ
τ
σ
σ
G
E E
E E
12
1
221 .υ υ
E
E =
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COMPLIANCE MATRI( FOR ORTHOTROPIC
9&r$
=
12
2
1
66
2212
1211
12
2
1
00
00
τ
σ σ
γ
ε ε
S
S S S S
12
66
2
22
2
21
1
1212
1
11
1 ; 1
; 1
GS
E S
E E S
E S
==
−=
−== υ υ
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STIFFNESS MATRI( FOR ORTHOTROPIC
9&r$
=
12
2
1
66
2212
1211
12
2
1
00
00
γ
ε ε
τ
σ σ
Q
QQQQ
1266
2112
222
2112
121
2112
21212
2112
111
; 1
11 ;
1
GQ E
Q
E E Q
E Q
=−
=
−=
−=
−=
υ υ
υ υ
υ
υ υ
υ
υ υ
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E(AMPLE
:arbon2(o+ T;00.! Msi ? E! 6
".5# Msi ? v "! 6 0.!@ ? 3"! 6 0.=! Msi
T&r/or- t& )om(%ian) )o//i)ints ar 4in "
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TRANSFORMED STIFFNESS MATRICES
+
"
!
Trans/ormation o/ strss and strains in arbitrar dir)tion$
and
[ ] [ ]
=
=
xy
y
x
xy
y
x
T T
γ
ε
ε
γ
ε
ε
τ
σ
σ
τ
σ
σ
2
12
2
1
1
12
2
1
[ ] [ ] θ θ sin cos ;
22
2
2
22
22
22
2
22
22
22
1 ==
−−
−=
−−
−= nm
nmmnmn
mnmn
mnnm
T
nmmnmn
mnmn
mnnm
T
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*rom t& sti//nss matri+ ,ation$
T&r/or- /ind$
or
No d/in$
and
or
{ } [ ]{ }11 ε σ Q=
{ } [ ] [ ][ ]{ } x x T QT ε σ 21
1
−=
[ ] [ ]
=
−
xy
y
x
xy
y
x
T
Q
QQ
QQ
T
γ
ε
ε
τ
σ
σ
2
66
2212
1211
1
1
00
0
0
{ } [ ]{ } x x Q ε σ =
{ } [ ] [ ][ ]21
1 T QT Q −=
=
xy
y
x
xy
y
x
QQQ
QQQ
QQQ
γ
ε
ε
τ
σ
σ
662616
262212
161211
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T& indi'id,a% trms ar gi'n b%o$ij
Q
)()22(
)2()2(
)2()2(
)()4(
)2(2
)2(2
44
66
22
6612221166
3
662212
3
66121126
3
662212
3
66121116
44
12
22
66221112
4
22
22
6612
4
1122
42222661241111
mnQnmQQQQQ
nmQQQmnQQQQ
mnQQQnmQQQQ
mnQnmQQQQ
mQnmQQnQQ
nQnmQQmQQ
++−−+=
+−+−−=
+−+−−=
++−+=
+++=+++=
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DISPLACEMENT CHARACTERISTICS
Isotropic Orthotropic Off'"is L"!in"
)Anisotropic*
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E(AMPLE )2*
C"r.on'epo/ T0&23 h"s properties "s fo##o4s: E1 5 1672 Msi 8 E2 5 179
Msi 8 v 12 5 72 8 %12 5 732 Msi "n$ fi.er "n;#e 0o to the ;#o."# "is
T&r/or- t& )om(%ian) )o//i)ints ar 4in "
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OFF'A(IS EN%INEERIN% CONSTANTS
+
"
!
X σ X σ
( )
( )
−+
++
=
+
+−+
=
+
+−+
+−
−+−
=
+
+−+
=
12
1222
2
112
22
1
2
14
12
112
224
1
2
14
12
112
224
12
44
12
1
2
122
2
14
12
1
12
224
1
214
2
2
1
2
G
E mn
E
E nm
E G
E
E m
G
E nmn
E E
E
E n
G
E nmm
mnG
E
E
E mn
E
E
nG
E
nmm
E E
xy
y
xy
x
υ
υ
υ
υ
υ
υ
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Pngar,& s,d,t orintasi srat tr&ada( mod,%,s
%astisitas dan ,atan ba&an om(osit.
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ADA PERTAN8AAN C