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8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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Data beban
P1 = 50 kNP2 = 100 kN
q1 = 0 kN/m
q2 = 0 kN/m
Properties penampang
Bahan / material :
No. Element E G b (m) h (m)
1 25000000 0.5 0.5
2 25000000 0.4 0.4
3 25000000 0.5 0.5
4 25000000 0.4 0.4
5 25000000 0.5 0.5
25000000 0.4 0.4
! 25000000 0.5 0.5
" 25000000 0.4 0.4
# 25000000 0.4 0.4
10 25000000 0.4 0.4
PENYELESAIAN :
STEP 1 : Discretize and Global Degrees of reedom !D"#
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D" ses$ai s$mb$ global str$%t$r dan tent$%an &eban ata$ 'omen p
No.titik ($oint) %&'
1
1 1 0 ran. *
2 2 0 ran. +
3 ,1 0 -ot.
24 ,2 50 ran. *5 ,3 0 ran. +
,4 0 -ot.
3
! ,5 0 ran. *
" , 0 ran. +
# ,! 0 -ot.
4
10 10 0 ran. *
11 11 0 ran. +
12 ," 0 -ot.
5
13 ,# 0 ran. *
14 ,10 0 ran. +15 ,11 0 -ot.
1 ,12 0 ran. *
1! ,13 0 ran. +
1" ,14 0 -ot.
!
1# 1# 0 ran. *
20 20 0 ran. +
21 ,15 0 -ot.
"
22 22 0 ran. *
23 23 0 ran. +
24 ,1 0 -ot.
#
25 ,1! 50 ran. *
2 ,1" 100 ran. +
2! ,1# 0 -ot.
%i.Global i
Beban /omen
rahBeban/omen
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10
2" ,20 0 ran. *
2# ,21 100 ran. +
30 ,22 0 -ot.
Nomor +oint $nt$% setiap element
Element !i#(oint
Sudut (deg)n6 n6 k
1 1 2 #0
2 2 3 0
3 3 4 2!0
4 3 5 0
5 5 ! 2!0
5 0
! " 2!0
" 3 # #0
# # 10 0
10 10 5 2!0
Displacement titi% s$mb$ global str$%t$r pada tiap element
Element +
1 1 2 3 4
2 4 5 !
3 ! " # 10
4 ! " # 13
5 13 14 15 1#
13 14 15 1
! 1 1! 1" 22
" ! " # 25
# 25 2 2! 2"
10 2" 2# 30 13
,e%ap Elemen dalam 'atri%s -e%a%$an &atang
Batan6 EA / L 4.EI / L 2.EI / L 6.EI / L^2
1 152500 13020".333 5104.1! 4""2".125
2 .! 35555.555 1!!!!.!!!!!!" """"."""""""#
3 152500 13020".333 5104.1! 4""2".125
4 .! 35555.555 1!!!!.!!!!!!" """"."""""""#
5 152500 13020".333 5104.1! 4""2".125 .! 35555.555 1!!!!.!!!!!!" """"."""""""#
! 152500 13020".333 5104.1! 4""2".125
" 1000000 53333.3333 2.! 20000
# .! 35555.555 1!!!!.!!!!!!" """"."""""""#
10 1000000 53333.3333 2.! 20000
'eng)it$ng Ga.a dan 'omen /+$ng !A%ibat beban l$ar di sepan+ang b
ELE'ENT 1
7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka
Ga.a dan 'omen /+$ng pada Element 1 :
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/nit : kNm
;a
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ELE'ENT 7
7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka
Ga.a dan 'omen /+$ng pada Element 7 :
/nit : kNm;a
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10 0 0 0 0
o>al i 1 2 3
Element ke 1 1 2 ,1
Element ke 2 ,2 ,3 ,4Element ke 3 ,5 , ,!
Element ke 4 ,5 , ,!
Element ke 5 ,# ,10 ,11
Element ke ,# ,10 ,11
Element ke ! ,12 ,13 ,14
Element ke " ,5 , ,!
Element ke # ,1! ,1" ,1#
Element ke 10 ,20 ,21 ,22
Element ?1
o>al ai 1 2 3 4
Global ai 1 2 3 4
%&' 0 0 ,1 ,2
Element ?2
o>al ai 1 2 3 4
Global ai 4 5 !
%&' ,2 ,3 ,4 ,5
Element ?3
o>al ai 1 2 3 4Global ai ! " # 10
%&' ,5 , ,! 0
Element ?4
o>al ai 1 2 3 4
Global ai ! " # 13
%&' ,5 , ,! ,#
Element ?5
o>al ai 1 2 3 4Global ai 13 14 15 1#
%&' ,# ,10 ,11 0
Element ?
o>al ai 1 2 3 4
Global ai 13 14 15 1
%&' ,# ,10 ,11 ,12
Element ?!
o>al ai 1 2 3 4
Global ai ! " # 25%&' ,12 ,13 ,14 0
7eeaian S& L"-AL element ,en6an S& GL"&AL dan D" ara) glob
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Element ?"
o>al ai 1 2 3 4
Global ai ! " # 25
%&' ,5 , ,! ,1!
Element ?#
o>al ai 1 2 3 4Global ai 25 2 2! 2"
%&' ,1! ,1" ,1# ,20
Element ?10
o>al ai 1 2 3 4
Global ai 2" 2# 30 13
%&' ,20 ,21 ,22 ,#
Element 1 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
.125!E01! 1 0 0
1 .12E01! 0 0
@-A = 0 0 1 0
0 0 0 .125!42E01!
0 0 0 1
0 0 0 0
.125!E01! 1 0 0
1 .12E01! 0 0
0 0 1 0
0 0 0 .125!42E01!
0 0 0 1
0 0 0 0
Element 2 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 00 0 0 0
'atri%s Transformasi ;,< dan ;,
T
< tiap element
>o α1
in α1
in α1
>o α1
>o α1
in α1
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
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1 0 0 0
0 1 0 0
@-A = 0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
Element 3 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
@-A = 0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
Element 4 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
1 0 0 0
0 1 0 0@-A = 0 0 1 0
0 0 0 1
@-A =
>o α1
in α1
in α1 >o α1
>o α1
in α1
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
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0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 10 0 0 0
0 0 0 0
Element 5 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 00 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
@-A = 0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
Element 6 :
0 0
0 0
- = 0 0 1 00 0 0
0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
@-A = 0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
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0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
Element 7 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0@-A = 0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
0 0 1 0
0 0 0 1."3!!2E01
0 0 0 10 0 0 0
Element 8 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
.125!E01! 1 0 0
1 .12E01! 0 0
@-A = 0 0 1 0
0 0 0 .125!42E01!
0 0 0 1
0 0 0 0
.125!E01! 1 0 0
1 .12E01! 0 0
0 0 1 00 0 0 .125!42E01!
0 0 0 1
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
@-A =
>o α1
in α1
in α1
>o α1
>o α1
in α1
@-A =
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0 0 0 0
Element 9 :
0 0
0 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
@-A = 0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
Element 1 :
0 00 0
- = 0 0 1 0
0 0 0
0 0 0
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0@-A = 0 0 1 0
0 0 0 1."3!!2E01
0 0 0 1
0 0 0 0
1."3"E01 1 0 0
1 1."3"E01 0 0
0 0 1 0
0 0 0 1."3!!2E010 0 0 1
0 0 0 0
>o α1
in α1
in α1
>o α1
>o α1
in α1
@-A =
>o α1 in α1 in α
1>o α
1
>o α1
in α1
@-A =
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===> DALA' S
Batan6 = 1
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
152500 13020".333 5104.1! 4""2".125
D = >o theta = .125!42E01!
D8 = in theta = 1
o>al ai 1 2 3
1 152500 0 0
2 0 24414.025 4""2".125
3 0 4""2".125 13020".333333
4 152500 0 0
5 0 24414.025 4""2".125 0 4""2".125 5104.1!
1 2 ,1
Global ai 1 2 3
1 24414.025 #.421#1"E011 4""2".125
2 #.422E011 152500 2.##10"5E012
3 4""2".125 2.##10"5E012 13020".333333
4 24414.025 #.421#2E011 4""2".125
5 #.422E011 152500 2.##10#E012
4""2".125 2.##10"5E012 5104.1!
===> DALA' S
Batan6 = 2
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
.! 35555.555 1!!!!.!!!!!!" """"."""""""#
D = >o theta = 1
D8 = in theta = 0
o>al ai 1 2 3
1 .! 0 0
2 0 2#2.#2#2# """"."""""""#
3 0 """"."""""""# 35555.555555
4 .! 0 0
5 0 2#2.#2#3 """".""""""#
0 """"."""""""# 1!!!!.!!!!!!"
,2 ,3 ,4Global ai 4 5
4 .! 0 0
0it$ng matri%s %e%a%$an batang 1 ;S'1<
@1A =
@1
A =
0it$ng matri%s %e%a%$an batang 2 ;S'2<
@2A =
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0 """"."""""""# 1!!!!.!!!!!!"
,5 , ,!
Global ai ! " #
! .! 0 0
" 0 2#2.#2#2# """"."""""""#
# 0 """"."""""""# 35555.55555513 .! 0 0
14 0 2#2.#2#3 """".""""""#
15 0 """"."""""""# 1!!!!.!!!!!!"
===> DALA' S
Batan6 = 5
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
152500 13020".333 5104.1! 4""2".125
D = >o theta = 1."3!!2E01
D8 = in theta = 1
o>al ai 1 2 3
1 152500 0 0
2 0 24414.025 4""2".125
3 0 4""2".125 13020".333333
4 152500 0 0
5 0 24414.025 4""2".125 0 4""2".125 5104.1!
,# ,10 ,11
Global ai 13 14 15
13 24414.025 2."25!5E010 4""2".125
14 2."2!E010 152500 ".#!32E012
15 4""2".125 ".#!32E012 13020".333333
1# 24414.025 2."25"E010 4""2".125
20 2."2!E010 152500 ".#!3255E012
21 4""2".125 ".#!32E012 5104.1!
===> DALA' S
Batan6 =
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
.! 35555.555 1!!!!.!!!!!!" """"."""""""#
D = >o theta = 1
D8 = in theta = 0
o>al ai 1 2 3
1 .! 0 0
@4A =
0it$ng matri%s %e%a%$an batang 5 ;S'5<
@5A =
@5
A =
0it$ng matri%s %e%a%$an batang 6 ;S'6
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2 0 2#2.#2#2# """"."""""""#
3 0 """"."""""""# 35555.555555
4 .! 0 0
5 0 2#2.#2#3 """".""""""#
0 """"."""""""# 1!!!!.!!!!!!"
,# ,10 ,11Global ai 13 14 15
13 .! 0 0
14 0 2#2.#2#2# """"."""""""#
15 0 """"."""""""# 35555.555555
1 .! 0 0
1! 0 2#2.#2#3 """".""""""#
1" 0 """"."""""""# 1!!!!.!!!!!!"
===> DALA' S
Batan6 = !
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
152500 13020".333 5104.1! 4""2".125
D = >o theta = 1."3!!2E01
D8 = in theta = 1
o>al ai 1 2 3
1 152500 0 02 0 24414.025 4""2".125
3 0 4""2".125 13020".333333
4 152500 0 0
5 0 24414.025 4""2".125
0 4""2".125 5104.1!
,12 ,13 ,14
Global ai 1 1! 1"
1 24414.025 2."25!5E010 4""2".125
1! 2."2!E010 152500 ".#!32E0121" 4""2".125 ".#!32E012 13020".333333
22 24414.025 2."25"E010 4""2".125
23 2."2!E010 152500 ".#!3255E012
24 4""2".125 ".#!32E012 5104.1!
===> DALA' S
Batan6 = "
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C21000000 53333.3333 2.! 20000
@A =
@
A =
0it$ng matri%s %e%a%$an batang 7 ;S'7<
@!A =
@!
A =
0it$ng matri%s %e%a%$an batang 8 ;S'8
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Batan6 = 10
Elemen ,alam matrik kekakan batan6 :
E / 4.E / 2.E / .E / C2
1000000 53333.3333 2.! 20000
D = >o theta = 1."3!!2E01
D8 = in theta = 1
o>al ai 1 2 3
1 1000000 0 0
2 0 10000 20000
3 0 20000 53333.3333333
4 1000000 0 0
5 0 10000 20000
0 20000 2.!
,20 ,21 ,22Global ai 2" 2# 30
2" 10000 1."1#345E010 20000
2# 1."1#E010 1000000 3.!545E012
30 20000 3.!545E012 53333.3333333
13 10000 1."1#35E010 20000
14 1."1#E010 1000000 3.!5445E012
15 20000 3.!545E012 2.!
IE,,",!?L"rtkan eai mb 6lobal/trktr ,en6an %&' 8an6 ,i,ahlkan
%&' : 1 2 ,1
%i.Global ai : 1 2 3
%itata lan6 manal ==F ,1 ,2 ,3
3 4 5
3 13020".333 4""2".125 2.##10#E012
4 4""2".125 #10"0.!2#1! #.421#1"E011
5 2.##1E012 #.421#1"E011 15542.#2# 5104.1! 4""2".125 """"."""""""#
! 0 .! 0
" 0 0 2#2.#2#3
# 0 0 """"."""""""#
12 0 0 0
13 0 0 0
14 0 0 0
15 0 0 0
1 0 0 0
1! 0 0 0
1" 0 0 0
21 0 0 0
24 0 0 0
@10
A =
@10
A =
'AT,I-S -E-A-/AN ST,/-T/, ;S'S<
@A =
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25 0 0 0
2 0 0 0
2! 0 0 0
2" 0 0 0
2# 0 0 0
30 0 0 0
1 4""2".125 24414.025 #.421#2E0112 2.##1E012 #.421#2E011 152500
10 0 0 0
11 0 0 0
1# 0 0 0
20 0 0 0
22 0 0 0
23 0 0 0
0it$ng 'atri%s &eban (oint* Nodal
Nodal forces:
a A%ibat beban di sepan+ang bentang !%ed$a $+$ng di%e
Element: 1
0
0
= 0
00
0
ranormai b lokal ke 6lobal ==F
.125!42E01!
1
= 0
0
00
Global ai
1
2
= 3
4
5
Element: 2
@A1
@A1
@A1
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0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1
0
= 0
0
0
0
Global ai
4
5
=
!
"
#
Element: 3
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1."3!!2E01
1
= 0
0
0
0
Global ai
!
"
= #10
11
@A2
@A2
@A1
@A1
@A1
@
A1
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12
Element: 4
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1
0
= 0
0
0
0
Global ai
!
"
= #
13
14
15
Element: 5
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1."3!!2E01
1
= 0
0
0
0
Global ai
13
14
@A1
@A1
@A1
@A1
@A1
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= 15
1#
20
21
Element: 6
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1
0
= 0
0
0
0
Global ai
13
14
= 15
1
1!
1"
Element: 7
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1."3!!2E01
1
= 0
00
0
@A1
@A1
@A1
@A1
@A1
@A1
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Global ai
1
1!
= 1"
22
2324
Element: 8
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
.125!42E01!
1
= 0
0
0
0
Global ai
!
"
= #
25
2
2!
Element: 9
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1
0
@A1
@A1
@A1
@A1
@A1
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= 0
0
0
0
Global ai
252
= 2!
2"
2#
30
Element: 1
0
0
= 0
0
0
0
ranormai b lokal ke 6lobal ==F
1."3!!2E01
1
= 0
0
0
0
Global ai
2"
2#
= 30
13
14
15
S$s$n matri%s beban e%iFalen
$oint b Global Element 1
1 1 0
1 2 0
1 3 0
2 4 0
e = 2 5 0 2 0
3 ! 0
@A1
@A1
@A1
@A1
@A1
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3 " 0
3 # 0
4 10 0
4 11 0
4 12 0
5 13 0
5 14 05 15 0
1 0
1! 0
1" 0
! 1# 0
! 20 0
! 21 0
" 22 0
" 23 0
" 24 0
# 25 0
# 2 0
# 2! 0
10 2" 0
10 2# 0
10 30 0
e = 0 1 1
0 2 2
0 3 ,1
0 4 ,20 5 ,3
0 ,4
0 ! ,5
0 " ,
0 # ,!
0 10 10
0 11 11
0 12 ,"
0 13 ,#
0 14 ,10
0 15 ,11
0 1 ,12
0 1! ,13
0 1" ,14
0 1# 1#
0 20 20
0 21 ,15
0 22 22
0 23 23
0 24 ,1
0 25 ,1!0 2 ,1"
0 2! ,1#
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0 2" ,20
0 2# ,21
0 30 ,22
b A%ibat &eban di +oint !%ed$a $+$ng di%e%ang#
$oint b Global1 1 0
1 2 0
1 3 0
2 4 50
= 2 5 0
2 0
3 ! 0
3 " 0
3 # 0
4 10 0
4 11 0
4 12 0
5 13 0
5 14 0
5 15 0
1 0
1! 0
1" 0
! 1# 0
! 20 0
! 21 0" 22 0
" 23 0
" 24 0
# 25 50
# 2 100
# 2! 0
10 2" 0
10 2# 100
10 30 0
c &eban %ombinasi ata$ gab$ngan !Ae C A+#
$oint b Global %&'
1 1 1
1 2 2
1 3 ,1
2 4 ,2
> = 2 5 ,3
2 ,4
3 ! ,5
3 " ,3 # ,!
4 10 10
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4 11 11
4 12 ,"
5 13 ,#
5 14 ,10
5 15 ,11
1 ,12
1! ,13 1" ,14
! 1# 1#
! 20 20
! 21 ,15
" 22 22
" 23 23
" 24 ,1
# 25 ,1!
# 2 ,1"
# 2! ,1#
10 2" ,20
10 2# ,21
10 30 ,22
ata lan6 > : %i9la>ement /
,1 0
,2 50
,3 0
,4 0 > = ,5 0
, 0
,! 0
," 0
,# 0
,10 0
,11 0
,12 0
,13 0
,14 0
,15 0
,1 0
,1! 50
,1" 100
,1# 0
,20 0
,21 100
,22 0
1 0
2 0
10 011 0
1# 0
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20 0
22 0
23 0
0.00001!4 2.!15E005 2.2#41#E00" !."#443#E00!
2.!15E005 #.34"E005 1.4415"E00! 1.5"105E005
2.2#4E00" 1.44E00! .3#3"!!E00! .3#2"0E00"
!."#44E00! 1.5"1E005 .3#2"0E00" 0.0000102!#
2."1E005 #.225E005 1.454151E00! 1.55!"#E005
1."#3E00" 1.021E00" 0.000000001 4.551323E00"
3.244E00 #.#!#E00 4.12432E00" #.#4"111E00!
".42##E00 2.#1E005 3.3#0#1E00" 5.344#E00
2.5#E005 #.151E005 1.444!"2E00! 1.544!4E005
1.1001E011 !."4#E00# 1.5"345E010 5.#0#00E00#
2."#5#E00 #.#!0E00 1.55#4!E00" 1."02E00
2.51E005 #.124E005 1.440501E00! 1.5401E005
4.1"5!E00" 1.441E00! 2.2#2E010 2.4323"1E00"
4.4!5!E00 1.540E005 2.4323"E00" 2.5##"5E00
".524E00 2.#33E005 4.3"20E00" 4.#4#!2!E00
!.!051E00 2.51E005 4.1"5#E00" 4.4!5!40E00
3.40"E005 0.000115"4 2.13353E00! 0.00001"!1
2.#"2E00" 2.522E00" 1.0#!"1"E00# 4.0!2244E00"
5.24!"E00! 1."#E00 0.000000001 3.!2!!!E00!
3.40"E005 0.000115"4 2.133#25E00! 0.00001"!22
1.0#03E00" 2.!5"E00" 2.5"5"!E010 1.11"21E00#!.141"E00! 2.10!E00 1.0!!14E00" 1.522405E00!
-ea>tion:
GAYAGAYA DALA' /(/NG ELE'EN
&atang: 1
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
1 1 0.0000002 2 0.000000
3 ,1 0.00300
4 ,2 0.0104
5 ,3 0.00001"
,4 0.001!30
%eormai trktr ,alam arah B &7 ,i batan6 1H
.12E01! 1 01 .125!42E01! 0
0 0 1
@%A1 =
M 1 = R . DMS 1
∆M 1
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0 0 0
0 0 0
0 0 0
b lokal %&'
0.000000 1 1
0.000000 2 20.00300 3 ,1
0.00001" 4 ,2
0.0104 5 ,3
0.001!30 ,4
Ga.a $+$ng batang
0 1525000 0
0 0
0 152500
0 0
0 0
0 2!.##!"522#
0 21.35#!25521
0 4.232E0140 2!.##!"522"#
0 21.35#!2552
0 ".543"#020"5
&atang: 2
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
4 ,2 0.0104
5 ,3 0.00001"
,4 0.001!30! ,5 0.010423
" , 0.000054
# ,! 0.001443
%eormai trktr ,alam arah B &7 ,i batan6 2H
1 0 0
0 1 0
0 0 10 0 0
0 0 0
∆M 1
AM 1 = AML 1 + SM 1 . M 1
@1A =
@1A =
@%A2 =
M 2 = R . DMS 2
∆M 2
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0 0 0
b lokal %&'
0.0104 1 ,2
0.00001" 2 ,3
0.001!30 3 ,4
0.010423 4 ,50.000054 5 ,
0.001443 ,!
Ga.a $+$ng batang
0 .!
0 0
0 00 .!
0 0
0 0
0 2".3402!44!#
0 2!.##!"522#
0 ".543"#020"
0 2".3402!44"
0 2!.##!"522"#0 "1.44324!15
&atang: 3
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
! ,5 0.010423
" , 0.000054
# ,! 0.001443
10 10 0.000000
11 11 0.00000012 ," 0.0031"!
%eormai trktr ,alam arah B &7 ,i batan6 3H
1."3"E01 1 0
1 1."3!!2E01 0
0 0 1
0 0 0
0 0 00 0 0
∆M 2
AM 2 = AML 2 + SM 2 . M 2
@2A =
@2A =
@%A3 =
M 3 = R . DMS 3
∆M !
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b lokal %&'
0.000054 1 ,5
0.010423 2 ,
0.001443 3 ,!
0.000000 4 10
0.000000 5 11
0.0031"! ,"
Ga.a $+$ng batang
0 152500
0 0
0 0
0 152500
0 00 0
0 "4.05204"20#3
0 2".3"0#5#3#11
0 113.523"3!54
0 "4.05204"20#
0 2".3"0#5#3#1
0 1.!05303E013
&atang: 4
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
! ,5 0.010423
" , 0.000054
# ,! 0.001443
13 ,# 0.0103"
14 ,10 0.0000!4
15 ,11 0.001435
%eormai trktr ,alam arah B &7 ,i batan6 4H
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.010423 1 ,5
∆M !
AM 3 = AML 3 + SM 3 . M 3
@3A =
@3A =
@%A4 =
M 4 = R . DMS 4
∆M 4
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0.000054 2 ,
0.001443 3 ,!
0.0103" 4 ,#
0.0000!4 5 ,10
0.001435 ,11
Ga.a $+$ng batang
0 .!
0 0
0 0
0 .!
0 0
0 0
0 24.#334"!!#4#
0 25.52"5"5421
0 !.5"121!#
0 24.#334"!!#5
0 25.52"5"54214
0 !.514!0034#
&atang: 5
%eormai trktr ,alam arah B G&B ,i batan6entkan no. b 6lobal ,ahl
13 ,# 0.0103"
14 ,10 0.0000!4
15 ,11 0.001435
1# 1# 0.000000
20 20 0.000000
21 ,15 0.0031!!
%eormai trktr ,alam arah B &7 ,i batan6 5H
1."3"E01 1 0
1 1."3!!2E01 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.0000!4 1 ,#0.0103" 2 ,10
0.001435 3 ,11
∆M 4
AM 4 = AML 4 + SM 4 . M 4
@4A =
@4A =
@%A5 =
M 5 = R . DMS 5
∆M "
∆M "
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0.000000 4 1#
0.000000 5 20
0.0031!! ,15
Ga.a $+$ng batang
0 152500
0 0
0 0
0 152500
0 0
0 0
0 115.""#4!22"0 2".34!"4!3#2
0 113.3#13"#5"
0 115.""#4!22!
0 2".34!"4!3#2
0 5."434E014
&atang: 6
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
13 ,# 0.0103"14 ,10 0.0000!4
15 ,11 0.001435
1 ,12 0.010354
1! ,13 0.00001"
1" ,14 0.001!02
%eormai trktr ,alam arah B &7 ,i batan6 H
1 0 00 1 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.0103" 1 ,#
0.0000!4 2 ,10
0.001435 3 ,110.010354 4 ,12
0.00001" 5 ,13
AM 5 = AML 5 + SM 5 . M 5
@5A =
@5A =
@%A =
M 6 = R . DMS 6
∆M 6
∆M 6
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0.001!02 ,14
Ga.a $+$ng batang
0 .!0 0
0 0
0 .!
0 0
0 0
0 21.352204"
0 2".05335!52
0 "1.!#!131"5
0 21.352205
0 2".05335!51
0 ".540""25#
&atang: 7
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
1 ,12 0.010354
1! ,13 0.00001"
1" ,14 0.001!02
22 22 0.000000
23 23 0.000000
24 ,1 0.003031
%eormai trktr ,alam arah B &7 ,i batan6 !H
1."3"E01 1 0
1 1."3!!2E01 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.00001" 1 ,12
0.010354 2 ,13
0.001!02 3 ,14
0.000000 4 22
0.000000 5 230.003031 ,1
AM 6 = AML 6 + SM 6 . M 6
@A =
@A =
@%A! =
M 7 = R . DMS 7
∆M #
∆M #
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Ga.a $+$ng batang
0 152500
0 0
0 0
0 152500
0 0
0 0
0 2".05335!51
0 21.352204"
0 ".540""25#1
0 2".05335!52
0 21.352205
0 1.!05303E013
&atang: 8
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
! ,5 0.010423
" , 0.000054
# ,! 0.001443
25 ,1! 0.01!"#3
2 ,1" 0.000135
2! ,1# 0.001044
%eormai trktr ,alam arah B &7 ,i batan6 "H
.12E01! 1 0
1 .125!42E01! 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.000054 1 ,5
0.010423 2 ,
0.001443 3 ,!
0.000135 4 ,1!
0.01!"#3 5 ,1"
0.001044 ,1#
Ga.a $+$ng batang
AM 7 = AML 7 + SM 7 . M 7
@!A =
@!A =
@%A" =
M 8 = R . DMS 8
∆M $
∆M $
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0 1000000
0 0
0 0
0 1000000
0 0
0 0
0 "1.5"2!!!401"
0 24.#5041#!3"2
0 44.5!221!!#3
0 "1.5"2!!!402
0 24.#5041#!3"
0 55.22545!1!33
&atang: 9
%eormai trktr ,alam arah B G&B ,i batan6
entkan no. b 6lobal ,ahl
25 ,1! 0.01!"#3
2 ,1" 0.000135
2! ,1# 0.001044
2" ,20 0.01!"5
2# ,21 0.0001#3
30 ,22 0.00104!
%eormai trktr ,alam arah B &7 ,i batan6 #H
1 0 0
0 1 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.01!"#3 1 ,1!
0.000135 2 ,1"
0.001044 3 ,1#
0.01!"5 4 ,20
0.0001#3 5 ,21
0.00104! ,22
Ga.a $+$ng batang
AM 8 = AML 8 + SM 8 . M 8
@"A =
@"A =
@%A# =
M 9 = R . DMS 9
∆M %
∆M %
AM 9 = AML 9 + SM 9 . M 9
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0 .!
0 0
0 0
0 .!
0 0
0 0
0 25.04#5"021"
0 1".41!2225#"
0 55.22545!1!3
0 25.04#5"022
0 1".41!2225#"2
0 55.2!!"!"41
&atang: 1
%eormai trktr ,alam arah B G&B ,i batan6entkan no. b 6lobal ,ahl
2" ,20 0.01!"5
2# ,21 0.0001#3
30 ,22 0.00104!
13 ,# 0.0103"
14 ,10 0.0000!4
15 ,11 0.001435
%eormai trktr ,alam arah B &7 ,i batan6 10
1."3"E01 1 0
1 1."3!!2E01 0
0 0 1
0 0 0
0 0 0
0 0 0
b lokal %&'
0.0001#3 10.01!"5 2
0.00104! 3
0.0000!4 4
0.0103" 5
0.001435
Ga.a $+$ng batang
0 1000000
0 0
@#A =
@#A =
@%A10 =
M 10 = R . DMS 10
∆M 1&
∆M 1&
AM 10 = AML 10 + SM 10 . M 10
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0 0
0 1000000
0 0
0 0
0 11".41!2225#"0 25.04#5"021"
0 55.2!!"!"411
0 11".41!222
0 25.04#5"022
0 44.#20442313
Element = 1 2
0asil 'an$al 2!.##" 2".34
21.3 2!.##"4.23E014 ".543#
2!.##" 2".34
21.3 2!.##"
".543# "1.4432
0asil SAP2 2!.##" 2".34
21.3 2!.##"
5."4E014 ".543#
2!.##" 2".34
21.3 2!.##"".543# "1.4432
TA&LE: Element orces rames
rame Station "$tp$tBase
et m et
1 0 %E%
1 4 %E%
2 0 %E%
2 %E%
3 0 %E%
3 4 %E%
4 0 %E%
4 %E%
5 0 %E%
5 4 %E%
0 %E%
%E%
! 0 %E%
! 4 %E%
" 0 %E%" 4 %E%
# 0 %E%
@10
A =
@10
A =
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39/332
(m2) (m) (m4) 7eteran6an
2.500000E01 4 5.20"333E03 1 (5050)
1.00000E01 2.133333E03 2 (4040)
2.500000E01 4 5.20"333E03 1 (5050)
1.00000E01 2.133333E03 2 (4040)
2.500000E01 4 5.20"333E03 1 (5050)
1.00000E01 2.133333E03 2 (4040)
2.500000E01 4 5.20"333E03 1 (5050)
1.00000E01 4 2.133333E03 2 (4040)
1.00000E01 2.133333E03 2 (4040)
1.00000E01 4 2.133333E03 2 (4040)
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Defined DOF
after boundary conditions
da Titi% (oint Str$%t$r dalam Ara) S$mb$ Global*S$mb$ Str$%t$r
TA&LE: (oint Boordinates
(oint BoordS.s BoordT.pe or,et et et m
1 G&B Darteian #
2 G&B Darteian #
3 G&B Darteian 3
4 G&B Darteian 3
5 G&B Darteian 3
G&B Darteian #
! G&B Darteian 3
" G&B Darteian #
# G&B Darteian 310 G&B Darteian 3
TA&LE: BonnectiFit. rame
rame (ointI (oint( IsB$rFed
et et et +e/No
1 1 2 No
2 2 3 No
3 4 3 No
4 3 5 No
5 ! 5 No
5 No
! " No
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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" 3 # No
# # 10 No
10 5 10 No
%
5 ==F Global ai
" # ==F Global ai
11 12 ==F Global ai
14 15 ==F Global ai
20 21 ==F Global ai
1! 1" ==F Global ai
23 24 ==F Global ai
2 2! ==F Global ai
2# 30 ==F Global ai
14 15 ==F Global ai
12.EI / L^! Sudut (deg) D D8
24414.025 #0 .125!4E01! 1
2#2.#2#2# 0 1 0
24414.025 2!0 1."3#!E01 1
2#2.#2#2# 0 1 0
24414.025 2!0 1."3#!E01 12#2.#2#2# 0 1 0
24414.025 2!0 1."3#!E01 1
10000 #0 .12323E01! 1
2#2.#2#2# 0 1 0
10000 2!0 1."3#!E01 1
entang dengan $+$ng di%e%ang*di+epit# !DALA' A,A0 S/'&/ L"-AL# :
rekai n6 terkekan6 eman8a = 0
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
43/332
rekai n6 terkekan6 eman8a = 0
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
44/332
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
45/332
5
2 2!
,1" ,1#
5 2# 30
,21 ,22
5
14 15
,10 ,11
0 0
0 0
0 0
0
0
0 1
0 0
0 0
0 0
1 0
.125!42E01! 0
0 1
0 0
0 0
0 0
1 0
.125!42E01! 0
0 1
0 0
0 0
0 0
0
00 1
in α1
>o α1
in α1
>o α1
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0
0
0 1
0 0
0 0
0 0
1 0
1."3!!2E01 0
0 1
0 0
0 0
0 0
1 0
1."3!!2E01 0
0 1
0 0
0 0
0 0
0
0
0 1
0 0
0 00 0
0 0
in α1
>o α1
in α1
>o α1
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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1 0
0 1
0 0
0 0
0 0
0 01 0
0 1
0 0
0 0
0 0
0
00 1
0 0
0 0
0 0
1 0
1."3!!2E01 0
0 1
0 0
0 0
0 0
1 0
1."3!!2E01 0
0 1
0 0
0 0
0 00
0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
0 0
in α1
>o α1
in α1
>o α1
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0
0
0 1
0 0
0 00 0
1 0
1."3!!2E01 0
0 1
0 0
0 0
0 0
1 0
1."3!!2E01 00 1
0 0
0 0
0 0
0
0
0 1
0 0
0 0
0 0
1 0
.125!42E01! 0
0 1
0 0
0 0
0 01 0
.125!42E01! 0
in α1
>o α1
in α1
>o α1
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
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0 1
0 0
0 0
0 0
0
0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
0 00 0
0 0
0 ,/'/S /'/'
0
0 1
0 0 0
0 0 = 00 0
1 0 0
1."3!!2E01 0 0
0 1
0 0
0 0
0 0
1 01."3!!2E01 0
0 1
in α1
>o α1
in α1
>o α1
u2
1
E/1
E/1
21
11
MS
MS
S
S
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/'&/ L"-AL
12.E / C3
24414.025
4 5
152500 0 0
0 24414.025 4""2".125
0 4""2".125 5104.1!
152500 0 0
0 24414.025 4""2".1250 4""2".125 13020".33333
,2 ,3 ,4
4 5
24414.025 #.421#E011 4""2".125
#.421#2E011 152500 2.##10#E012
4""2".125 2.##11E012 5104.1!
24414.025 #.421#2E011 4""2".125
#.421#1"E011 152500 2.##10#E012
4""2".125 2.##11E012 13020".33333
/'&/ L"-AL
12.E / C3
2#2.#2#2#
4 5
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
,5 , ,!! " #
.! 0 0
=
61
51
41
31
MS
MS
MS
MS
S
S
S
8/18/2019 Finite Element (Direct Stiffness) Kekakuan Langsung
51/332
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
/'&/ L"-AL 3
3 13020".333333
4 4""2".125
5 2.##10#E012
12.E / C3 5104.1!
24414.025 ! 0
" 0
# 0
12 0
13 0
4 5 14 0152500 0 0 15 0
0 24414.025 4""2".125 1 0
0 4""2".125 5104.1! 1! 0
152500 0 0 1" 0
0 24414.025 4""2".125 21 0
0 4""2".125 13020".33333 24 0
25 0
10 11 ," 2 0
10 11 12 2! 0
24414.025 2."2E010 4""2".125 2" 02."25"E010 152500 ".#!32E012 2# 0
4""2".125 ".#!32E012 5104.1! 30 0
24414.025 2."25"E010 4""2".125
2."25!5E010 152500 ".#!32E012
4""2".125 ".#!32E012 13020".33333
/'&/ L"-AL
12.E / C3
2#2.#2#2#
4 5
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!".! 0 0
0 2#2.#2#3 """"."""""#
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0 """"."""""# 35555.55555
,# ,10 ,11
13 14 15
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!".! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
/'&/ L"-AL
12.E / C3
24414.025
4 5
152500 0 0
0 24414.025 4""2".125
0 4""2".125 5104.1!
152500 0 0
0 24414.025 4""2".1250 4""2".125 13020".33333
1# 20 ,15
1# 20 21
24414.025 2."2E010 4""2".125
2."25"E010 152500 ".#!32E012
4""2".125 ".#!32E012 5104.1!
24414.025 2."25"E010 4""2".125
2."25!5E010 152500 ".#!32E012
4""2".125 ".#!32E012 13020".33333
/'&/ L"-AL
12.E / C3
2#2.#2#2#
4 5
.! 0 0
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0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
,12 ,13 ,141 1! 1"
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
/'&/ L"-AL
12.E / C3
24414.025
4 5
152500 0 00 24414.025 4""2".125
0 4""2".125 5104.1!
152500 0 0
0 24414.025 4""2".125
0 4""2".125 13020".33333
22 23 ,1
22 23 24
24414.025 2."2E010 4""2".125
2."25"E010 152500 ".#!32E0124""2".125 ".#!32E012 5104.1!
24414.025 2."25"E010 4""2".125
2."25!5E010 152500 ".#!32E012
4""2".125 ".#!32E012 13020".33333
/'&/ L"-AL
12.E / C310000
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4 5
1000000 0 0
0 10000 20000
0 20000 2.!1000000 0 0
0 10000 20000
0 20000 53333.333333
,1! ,1" ,1#
25 2 2!
10000 .045E011 20000
.044"E011 1000000 1.22515E012
20000 1.2251E012 2.!
10000 .044"E011 20000.044"5E011 1000000 1.22515E012
20000 1.2251E012 53333.333333
/'&/ L"-AL
12.E / C3
2#2.#2#2#
4 5
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
,20 ,21 ,22
2" 2# 30
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""# 1!!!!.!!!!!"
.! 0 0
0 2#2.#2#3 """"."""""#
0 """"."""""# 35555.55555
/'&/ L"-AL
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12.E / C3
10000
4 5
1000000 0 0
0 10000 20000
0 20000 2.!
1000000 0 0
0 10000 20000
0 20000 53333.333333
,# ,10 ,1113 14 15
10000 1."1#3E010 20000
1."1#35E010 1000000 3.!545E012
20000 3.!545E012 2.!
10000 1."1#35E010 20000
1."1#345E010 1000000 3.!545E012
20000 3.!545E012 53333.333333
-/P!@D68@D@49:@(@54'ATB0!E@67@D@48:@(@48#ALSE##CIE,,",!?L""-/P!@
,2 ,3 ,4 ,5 , ,!
4 5 ! " #
,4 ,5 , ,! ," ,#
! " # 12 13
5104.1! 0 0 0 0 0
4""2".125 .! 0 0 0 0
""""."""""""# 0 2#2.#2#3 """".""""""# 0 015!3."""""# 0 """"."""""# 1!!!!.!!!!!" 0 0
0 13!!4!.3# 3.43302E010 2""2".125 4""2".125 .!
"""".""""""# 3.43302E010 25"425.#25# 1.22515E012 ".#!32E012 0
1!!!!.!!!!!!" 2""2".125 1.22515E012 25452.!!!!" 5104.1! 0
0 4""2".125 ".#!32E012 5104.1! 13020".333333 0
0 .! 0 0 0 13!!4!.3#5"3
0 0 2#2.#2#3 """"."""""# 0 4.45#21E010
0 0 """".""""""# 1!!!!.!!!!!" 0 2""2".125
0 0 0 0 0 .!
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 4""2".125
0 0 0 0 0 0
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0 10000 .044"E011 20000 0 0
0 .045E011 1000000 1.2251E012 0 0
0 20000 1.22515E012 2.! 0 0
0 0 0 0 0 10000
0 0 0 0 0 1."1#35E010
0 0 0 0 0 20000
4""2".125 0 0 0 0 02.##10"5E012 0 0 0 0 0
0 24414.025 2."25"E010 4""2".125 4""2".125 0
0 2."2E010 152500 ".#!32E012 ".#!3255E012 0
0 0 0 0 0 24414.025
0 0 0 0 0 2."25"E010
0 0 0 0 0 0
0 0 0 0 0 0
%ang#
o>al ai
1
2
3
45
1 0 0 0 0
.125!42E01! 0 0 0 0
0 1 0 0 0
0 0 .125!4E01! 1 0
0 0 1 .125!4E01! 00 0 0 0 1
0
0
0
0
0
0
o>al ai
AMS 1 ' . AM
L 1
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1
2
3
4
5
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0
0
0
0
0
0
o>al ai
1
2
3
4
5
1 0 0 0 0
1."3!!2E01 0 0 0 0
0 1 0 0 0
0 0 1."3!!2E01 1 0
0 0 1 1."3!!E01 0
0 0 0 0 1
0
0
00
0
AMS 1 ' . AM
L 1
AMS 1 ' . AM
L 1
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0
o>al ai
1
2
3
4
5
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0
0
0
0
0
0
o>al ai
1
2
3
4
5
1 0 0 0 0
1."3!!2E01 0 0 0 0
0 1 0 0 0
0 0 1."3!!2E01 1 0
0 0 1 1."3!!E01 0
0 0 0 0 1
0
0
AMS 1 ' . AM
L 1
AMS 1 ' . AM
L 1
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0
0
0
0
o>al ai
1
2
3
4
5
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0
0
0
0
0
0
o>al ai
1
2
3
4
5
1 0 0 0 0
1."3!!2E01 0 0 0 0
0 1 0 0 0
0 0 1."3!!2E01 1 00 0 1 1."3!!E01 0
0 0 0 0 1
AMS 1 ' . AM
L 1
AMS 1 ' . AM
L 1
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0
0
0
0
00
o>al ai
1
2
3
4
5
1 0 0 0 0
.125!42E01! 0 0 0 0
0 1 0 0 0
0 0 .125!4E01! 1 0
0 0 1 .125!4E01! 0
0 0 0 0 1
0
0
0
0
0
0
o>al ai
1
2
3
4
5
0 0 0 0 0
1 0 0 0 0
AMS 1 ' . AM
L 1
AMS 1 ' . AM
L 1
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0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
00
0
0
0
0
o>al ai
1
2
3
4
5
1 0 0 0 0
1."3!!2E01 0 0 0 0
0 1 0 0 0
0 0 1."3!!2E01 1 0
0 0 1 1."3!!E01 0
0 0 0 0 1
0
0
0
0
0
0
Element 2 Element 3 Element 4
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0
AMS 1 ' . AM
L 1
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0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
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%&'1 1
2 1
,1 1
,2 2
,3 2
,4 2
,5 3
, 3
,! 3
10 4
11 4
," 4
,# 5
,10 5
,11 5
,12
,13
,14
1# !
20 !
,15 !22 "
23 "
,1 "
,1! #
,1" #
,1# #
,20 10
,21 10
,22 10
0 0 0
0 0 0
0 0 0
0 50 50
0 0 = 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
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0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 50 50
0 100 100
0 0 0
0 0 0
0 100 100
0 0 0
eormai trktr
,1 = 0.00300 1
,2 = 0.0104 2
,3 = 0.00001" 2
,4 = 0.001!30 2,5 = 0.010423 3
, = 0.000054 3
,! = 0.001443 3
," = 0.0031"! 4
,# = 0.0103" 5
,10 = 0.0000!4 5
,11 = 0.001435 5
,12 = 0.010354
,13 = 0.00001"
,14 = 0.001!02
,15 = 0.0031!! !
,1 = 0.003031 "
,1! = 0.01!"#3 #
,1" = 0.000135 #
,1# = 0.001044 #
,20 = 0.01!"5 10
,21 = 0.0001#3 10
,22 = 0.00104! 10
F S
FF*1.A
F+
%' =
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.125!42E01! 1 0 0.0104
1 .125!4E01! 0 0.00001"
0 0 1 0.001!30
0 0 152500 0 024414.025 4""2".125 0 24414.025 4""2".125
4""2".125 13020".3333 0 4""2".125 5104.1!
0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 5104.1! 0 4""2".125 13020".333333
b lokal %&' / b 6lobal
2!.##!"5 1 1
21.35#!3 2 2
= 4.233E014 3 ,12!.##!"5 4 ,2
21.35#!3 5 ,3
".543"#0 ,4
H bb :
bb :
0 0 0 0.01045#"5!
0 0 0 1.!#1"3E005
0 0 0 0.001!301 0 0 0.010423
0 1 0 0.000054
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0 0 1 0.001443
0 0 .! 0 0
2#2.#2#2# """"."""""# 0 2#2.#2#3 """"."""""""#
""""."""""""# 35555.5555 0 """"."""""# 1!!!!.!!!!!!" 0 0 .! 0 0
2#2.#2#3 """"."""""# 0 2#2.#2#3 """".""""""#
""""."""""""# 1!!!!.!!!!" 0 """"."""""# 35555.555555
b lokal %&' / b 6lobal
2".3402! 1 ,2
2!.##!"5 2 ,3
= ".543"#0 3 ,4
2".3402! 4 ,5
2!.##!"5 5 ,"1.44324! ,!
H bb :
bb :
0 0 0 0.01042343#!
0 0 0 5.3!#33E005
0 0 0 0.001443
1."3!!2E01 1 0 0.000000
1 1."3!!E01 0 0.0000000 0 1 0.0031"!
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0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 13020".3333 0 4""2".125 5104.1!
0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".1254""2".125 5104.1! 0 4""2".125 13020".333333
b lokal %&' / b 6lobal
"4.05204" 1 ,5
2".3"0#5# 2 ,
= 113.523"3" 3 ,!
"4.05204" 4 10
2".3"0#5# 5 11
1.!0530E013 ,"
H bb :
bb :
0 0 0 0.01042343#!
0 0 0 5.3!#33E005
0 0 0 0.001443
1 0 0 0.0103"
0 1 0 0.0000!4
0 0 1 0.001435
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0 0 .! 0 0
2#2.#2#2# """"."""""# 0 2#2.#2#3 """"."""""""#
""""."""""""# 35555.5555 0 """"."""""# 1!!!!.!!!!!!"
0 0 .! 0 0
2#2.#2#3 """"."""""# 0 2#2.#2#3 """".""""""#
""""."""""""# 1!!!!.!!!!" 0 """"."""""# 35555.555555
b lokal %&' / b 6lobal
24.#334"" 1 ,5
25.52"5"5 2 ,
= !.5"12 3 ,!
24.#334"" 4 ,#
25.52"5"5 5 ,10
!.514!0035 ,11
H bb :
bb :
0 0 0 0.0103"03#4
0 0 0 !.41#3E005
0 0 0 0.001435
1."3!!2E01 1 0 0.000000
1 1."3!!E01 0 0.000000
0 0 1 0.0031!!
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0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 13020".3333 0 4""2".125 5104.1!
0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 5104.1! 0 4""2".125 13020".333333
b lokal %&' / b 6lobal
115.""#4!2 1 ,#2".34!"4! 2 ,10
= 113.3#13#0 3 ,11
115.""#4!2 4 1#
2".34!"4! 5 20
5."43E014 ,15
H bb :
bb :
0 0 0 0.0103"03#40 0 0 !.41#3E005
0 0 0 0.001435
1 0 0 0.010354
0 1 0 0.00001"
0 0 1 0.001!02
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0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 13020".3333 0 4""2".125 5104.1!
0 0 152500 0 0
24414.025 4""2".125 0 24414.025 4""2".125
4""2".125 5104.1! 0 4""2".125 13020".333333
b lokal %&' / b 6lobal
2".0533 1 ,12
21.35221 2 ,13
= ".540""3 3 ,14
2".0533 4 22
21.35221 5 23
1.!0530E013 ,1
H bb :
bb :
0 0 0 0.01042343#!
0 0 0 5.3!#33E005
0 0 0 0.001443
.125!42E01! 1 0 0.01!"#3
1 .125!4E01! 0 0.000135
0 0 1 0.001044
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0 0 1000000 0 0
10000 20000 0 10000 20000
20000 53333.33333 0 20000 2.!
0 0 1000000 0 0
10000 20000 0 10000 20000
20000 2.! 0 20000 53333.3333333
b lokal %&' / b 6lobal
"1.5"2!!! 1 ,5
24.#50420 2 ,
= 44.5!222 3 ,!
"1.5"2!!! 4 ,1!
24.#50420 5 ,1"
55.22545!1! ,1#
H bb :
bb :
0 0 0 0.01!"#32#22
0 0 0 0.0001353!1
0 0 0 0.001044
1 0 0 0.01!"5
0 1 0 0.0001#3
0 0 1 0.00104!
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20000 53333.33333 0 20000 2.!
0 0 1000000 0 0
10000 20000 0 10000 20000
20000 2.! 0 20000 53333.3333333
b lokal %&' / b 6lobal
11".41!223 1 ,2025.04#5"0 2 ,21
= 55.2!!"!" 3 ,22
11".41!223 4 ,#
25.04#5"0 5 ,10
44.#204423 ,11
3 4 5
"4.052 24.#33 115.""#
2".3"1 25.52# 2".34"113.523" !.5" 113.3#14
"4.052 24.#33 115.""#
2".3"1 25.52# 2".34"
1.!0530E013 !.514! 5."434E014
"4.052 24.#33 115.""#
2".3"1 25.52# 2".34"
0.0000 !.5" 0.0000
"4.052 24.#33 115.""#
2".3"1 25.52# 2".34"113.523" !.514! 113.3#14
BaseT.pe P ?2 ?3 T '2
et 7N 7N 7N 7Nm 7Nm
intati> 2!.##" 21.3 0 0 0
intati> 2!.##" 21.3 0 0 0
intati> 2".34 2!.##" 0 0 0
intati> 2".34 2!.##" 0 0 0
intati> "4.052 2".3"1 0 0 0
intati> "4.052 2".3"1 0 0 0
intati> 24.#33 25.52# 0 0 0
intati> 24.#33 25.52# 0 0 0
intati> 115.""# 2".34" 0 0 0
intati> 115.""# 2".34" 0 0 0
intati> 21.35 2".05 0 0 0
intati> 21.35 2".05 0 0 0
intati> 2".05 21.35 0 0 0
intati> 2".05 21.35 0 0 0
intati> "1.5"3 24.#5 0 0 0intati> "1.5"3 24.#5 0 0 0
intati> 25.05 1".41! 0 0 0
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intati> 25.05 1".41! 0 0 0
intati> 11".41! 25.05 0 0 0
intati> 11".41! 25.05 0 0 0
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Y H Special(t Global GlobalY GlobalHm m +e/No m m m
0 0 No # 0 0
0 4 No # 0 4
0 4 No 3 0 4
0 0 No 3 0 0
0 4 No 3 0 4
0 4 No # 0 4
0 0 No 3 0 0
0 0 No # 0 0
0 " No 3 0 "0 " No 3 0 "
Lengt) Bentroid BentroidY BentroidH
m m m m
4 # 0 2
0 4
4 3 0 2
0 0 4
4 3 0 2
0 4
4 # 0 2
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4 5 ! " #
4""2".125 2.##10#E012 5104.1! 0
#10"0.!2#1! #.421#1"E011 4""2".125 .!
#.421#1"E011 15542.#2# """"."""""""#
4""2".125 """"."""""""# 15!3."""""#
.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""""# 1!!!!.!!!!!!"
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
4""2".125 2.##10#E012 5104.1! 0 0 0
#10"0.!2#1! #.421#1"E011 4""2".125 .!
#.421#1"E011 15542.#2# """"."""""""#
4""2".125 """"."""""""# 15!3."""""#.! 0 0
0 2#2.#2#3 """".""""""#
0 """"."""""""# 1!!!!.!!!!!!"
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
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0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
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68@D@58:@(@63'ATB0!E@67@D@57:@(@57#ALSE##
10 11 ," ,# ,10 ,11
10 11 12 13 14 15
,10 ,11 ,12 ,13 ,14 ,15
14 15 1 1! 1" 21
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
2#2.#2#3 """"."""""""# 0 0 0 0
"""".""""""# 1!!!!.!!!!!!" 0 0 0 0
0 0 0 0 0 0
4.45#21E010 2""2".125 .! 0 0 4""2".125
25"425.#25#3 3.!5445E012 0 2#2.#2#3 """"."""""""# ".#!32E012
3.!5445E012 25452.!!!!!" 0 """".""""""# 1!!!!.!!!!!!" 5104.1!
0 0 #10"0.!2#1! 2."25!5E010 4""2".125 0
2#2.#2#3 """".""""""# 2."25!5E010 15542.#2# """".""""""# 0
""""."""""""# 1!!!!.!!!!!!" 4""2".125 """".""""""# 15!3."""""# 0
".#!32E012 5104.1! 0 0 0 13020".333333
0 0 4""2".125 ".#!32E012 5104.1! 0
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0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
1."1#35E010 20000 0 0 0 0
1000000 3.!545E012 0 0 0 0
3.!5445E012 2.! 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
2."25"E010 4""2".125 0 0 0 4""2".125
152500 ".#!3255E012 0 0 0 ".#!3255E012
0 0 24414.025 2."25"E010 4""2".125 0
0 0 2."25"E010 152500 ".#!3255E012 0
0
0
0
0
00
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0
0
0
0
0
0
0
0
0
0
0
0
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0
0
0
0
0
0
0
0
0
00
0
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0
0
0
0
0
0
0
0
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0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
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0.00300 (oint "$tp$tBase BaseT.pe /1
0.0104 et et et m
0.00001" 1 %E% intati> 0
0.001!30 2 %E% intati> 0.01040.010423 3 %E% intati> 0.010423
0.000054 4 %E% intati> 0
0.001443 5 %E% intati> 0.0103"
0.0031"! %E% intati> 0.010354
0.0103" ! %E% intati> 0
0.0000!4 " %E% intati> 0
0.001435 # %E% intati> 0.01!"#3
0.010354 10 %E% intati> 0.01!"5
0.00001"
0.001!02
0.0031!!
0.003031
0.01!"#3
0.000135
0.001044
0.01!"5
0.0001#3
0.00104!
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2."#5"5E00 2.514!E005 4.1"5"#E00" 4.4!5!40E00 ".52445E00 !.!05141E00
#.#!03#E00 #.12412E005 1.44050E00! 1.5401E005 0.00002#332 2.514!E005
1.55#4!E00" 1.440501E00! 2.2#2E010 2.4323"E00" 4.3"20E00" 4.1"5#E00"
1."02E00 1.5401E005 2.4323"1E00" 2.5##"5E00 4.#4#!2!E00 4.4!5!40E00
#.####3E00 #.15125"E005 1.444!"E00! 1.544!4E005 2.#41#3E005 2.5#35E005
3.42340E00" !."4#33E00# 1.5"345E010 5.#0#002E00# 0.00000002 1.1000"E011
0.00000104# #.#!03#E00 1.55#4!2E00" 1."02E00 3.2254"0E00 2."#5"5E00
3.2254"0E00 0.00002#332 4.3"1#5E00" 4.#4#!2!E00 #.41#4#!E00 ".52445E00
1.0033!E005 #.22540#E005 1.45415E00! 1.55!"#E005 2.#"12E005 2."05"E005
".5"550!E00# 1.0214##E00" 0.000000001 4.55132E00" !.53341E00# 1."#200E00"
.25"0E00 #.#!""1E00 4.124322E00" #.#4"111E00! .2#!22"E00! 3.2444#E00
#.#!""1E00 #.34!!2"E005 1.441E00! 1.5"105E005 2.#05#E005 2.!14"!E005
4.124322E00" 1.441E00! .3#3"!!E00! .3#2"04E00" 3.3#0#0E00" 2.2#41"#E00"
#.#4"111E00! 1.5"105E005 .3#2"04E00" 0.0000102!# 5.344#E00 !."#443#E00!
.2#!22"E00! 2.#05#E005 3.3#0#0E00" 5.344#E00 1."4#55#E005 ".42#"#E00
3.2444#E00 2.!14"!E005 2.2#41"#E00" !."#443#E00! ".42#"#E00 0.00001!4
1."!""5E005 0.000115"352 2.133#3E00! 0.00001"!22 3.41"45E005 3.40!!2E005
4.!1#!4E00" 2.!5"42E00" 2.5"5"!E010 1.11"210E00# 1.314442E00" 1.0#033E00"
1.!1##E00 2.10!4E00 1.0!!13!E00" 1.522405E00! .2#04E00" !.141!"2E00!
1.""3!4E005 0.000115"3!! 2.1335E00! 0.00001"!1 3.4111E005 3.40"11E005
2.154"#3E00" 2.521#0E00" 1.0#!"1"E00# 4.0!224E00" .!1533E010 2.#"1"33E00"2.353#"E00! 1."#4E00 0.000000001 3.!2!!!E00! ".3022!E00! 5.24!"3!E00!
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0.000054
0.010423
0.001443
0.000000
0.0000000.0031"!
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0.010423
0.000054
0.001443
0.0103"
0.0000!4
0.001435
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0.0000!4
0.0103"
0.001435
0.000000
0.000000
0.0031!!
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0.00001"
0.010354
0.001!02
0.000000
0.000000
0.003031
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0.000054
0.010423
0.001443
0.000135
0.01!"#3
0.001044
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0.01!"#3
0.000135
0.001044
0.01!"5
0.0001#3
0.00104!
0.0001#3
0.01!"5
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55.2!!# #1
44.#204 101 0
55.2!!# 101 4
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G/IDet
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12 13 14 15 1 1!
0 0 0 0 0 0
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,12 ,13 ,14 1# 20 ,15
1 1! 1" 1# 20 21
,1 ,1! ,1" ,1# ,20 ,21
24 25 2 2! 2" 2#
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 0
0 10000 .044"E011 20000 0 0
0 .044"E011 1000000 1.22515E012 0 0
0 20000 1.22515E012 2.! 0 0
0 0 0 0 0 0
0 0 0 0 10000 1."E010
0 0 0 0 1."1#35E010 1000000
0 0 0 0 20000 3.!E012
4""2".125 0 0 0 0 0
".#!32E012 0 0 0 0 0
5104.1! 0 0 0 0 0
0 0 0 0 0 0
13020".333333 0 0 0 0 0
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0 !.! .044"5E011 20000 .! 0
0 .044"5E011 1002#2.#2# """"."""""# 0 2#2.#3
0 20000 """"."""""""# """"".""""# 0 """".""#
0 .! 0 0 !.! 1."2E010
0 0 2#2.#2#3 """"."""""# 1."1#345E010 1002#3
0 0 """"."""""""# 1!!!!.!!!!" 20000 """".""#
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
4""2".125 0 0 0 0 0
".#!3255E012 0 0 0 0 0
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Element " Element # Element 10
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0 =0 0 0
0 0 0
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0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 00 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
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/2 /3 ,1 ,2 ,3
m m -a,ian -a,ian -a,ian
0 0 0 0.0030 0
0 0.00001" 0 0.001!3 00 0.000054 0 0.001443 0
0 0 0 0.0031"! 0
0 0.0000!4 0 0.001435 0
0 0.00001" 0 0.001!02 0
0 0 0 0.0031!! 0
0 0 0 0.003031 0
0 0.000135 0 0.001044 0
0 0.0001#3 0 0.00104! 0
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3.40"11E005 2.#"1"3E00" 5.24!"3!E00! 3.40!!E005 1.0#0334E00" !.14E00!
0.000115"3!! 2.521"#!E00" 1."#4E00 0.000115"352 2.!5"4E00" 2.1E00
2.13353E00! 1.0#!"1"E00# 0.000000001 2.133#3E00! 2.5"5"!E010 1.1E00"
0.00001"!1 4.0!2244E00" 3.!2!!!E00! 0.00001"!22 1.11"21E00# 1.52E00!
0.00011212" 2.#411E00" 1.#001!E00 0.000112101 2.!"!"1E00" 2.1E00
2.12!03E00! .3!545"E00! 0.0000000#4 2.12#13E00! 1.14##1E00# ".4E00"
1.""3!4E005 2.154"#E00" 2.353#"E00! 1."!"#E005 4.!1#!43E00" 1.!2E00
3.4111E005 .!15335E010 ".3022!E00! 3.41"5E005 1.31444E00" .3E00"
0.000112101 2.!"!"12E00" 2.1211"E00 0.00011212" 2.#412E00" 1.#E00
2.12#13E00! 1.14##1E00# ".4100E00" 2.12!E00! .3!545"E00! #.40E00"
1."!""5E005 4.!1#!4E00" 1.!1##E00 1.""3!E005 2.154"#3E00" 2.4E00!
0.000115"352 2.!5"42E00" 2.10!4E00 0.000115"3!! 2.521#0E00" 1.#E00
2.133#3E00! 2.5"5"!E010 1.0!!13!E00" 2.134E00! 1.0#!"1"E00# #.5"E010
0.00001"!22 1.11"210E00# 1.522405E00! 0.00001"!1 4.0!224E00" 3.!3E00!
3.41"45E005 1.314442E00" .2#04E00" 3.411E005 .!1533E010 ".30E00!
3.40!!2E005 1.0#033E00" !.141!"2E00! 3.40"1E005 2.#"1"33E00" 5.25E00!
0.00024202!! 5.455"0"E00! 1.#003"E005 0.0002412" 5.45"24E00! 1.#E005
5.455"0"E00! 1.3!00E00 2.2#"0E00! 5.45"24E00! 2.41213E00# 2.2E00!
1.#003"E005 2.2#"0E00! 1.51332E005 1.""12E005 2.1#"4E00! 2.53E00!
0.0002412!# 5.45"235E00! 1.""125E005 0.00024202" 5.455"1E00! 1.#E005
5.45"24E00! 2.41213E00# 2.1#"4E00! 5.455"E00! 1.3!00E00 2.30E00!1.""125E005 2.1#"E00! 2.52"223E00! 1.#004E005 2.2#!#"E00! 1.52E005
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1" 21 24 25 2 2! 2" 2# 30
0 0 0 0 0 0 0 0 0
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22 23 ,1 ,1! ,1" ,1# ,20 ,21 ,22
22 23 24 25 2 2! 2" 2# 30
,22 1 2 10 11 1# 20 22 23
30 1 2 10 11 1# 20 22 23
0 4""2".13 2.##E012 0 0 0 0 0 0
0 24414.0 #.4E011 0 0 0 0 0 0
0 #.4E011 152500 0 0 0 0 0 00 4""2".13 2.##E012 0 0 0 0 0 0
0 0 0 24414.0 3E010 0 0 0 0
0 0 0 3E010 152500 0 0 0 0
0 0 0 4""2".13 ".#!E012 0 0 0 0
0 0 0 4""2".13 ".#!E012 0 0 0 0
20000 0 0 0 0 24414.0 2."E010 0 0
3."E012 0 0 0 0 3E010 152500 0 0
2.! 0 0 0 0 4""2".13 ".#!E012 0 0
0 0 0 0 0 0 0 24414.0 3E010
0 0 0 0 0 0 0 3E010 152500
0 0 0 0 0 0 0 4""2".13 ".#!E012
0 0 0 0 0 4""2".13 ".#!E012 0 0
0 0 0 0 0 0 0 4""2".13 ".#!E012
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0 0 0 0 0 0 0 0 0
""""."""# 0 0 0 0 0 0 0 0
1!!!!.!!" 0 0 0 0 0 0 0 0
20000 0 0 0 0 0 0 0 0
"""".""# 0 0 0 0 0 0 0 0
""""".""# 0 0 0 0 0 0 0 0
0 24414.03 #.42E011 0 0 0 0 0 00 #.42E011 152500 0 0 0 0 0 0
0 0 0 24414.0 2."3E010 0 0 0 0
0 0 0 2."3E010 152500 0 0 0 0
0 0 0 0 0 24414.0 2."3E010 0 0
0 0 0 0 0 2."3E010 152500 0 0
0 0 0 0 0 0 0 24414.0 2."3E010
0 0 0 0 0 0 0 2."3E010 152500
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0
0
0
0
00
0
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0
0
0
0
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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0 0.0030 &7
50 0.0104 &7
0 1.!#E005 &7
0 0.001!3 &7
0 0.010423 &7
0 5E005 &7
0 0.001443 &7
0 0.0031"! &7
0 0.0103" &7
0 !E005 &7
0 0.001435 &7
0 0.010354 &7
0 2E005 &7
0 0.001!02 &7
0 0.0031!! &7
0 0.003031 &7
50 0.01!"#3 &7
100 0.000135 &7
0 0.001044 &7
0 0.01!"5 &7
100 0.0001#3 &70 0.00104! &7
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3 ,1
4 ,2
5 ,3 ,4
! ,5
" ,
# ,!
12 ,"
13 ,#
14 ,10
15 ,11
1 ,12
1! ,13
1" ,14
21 ,15
24 ,1
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25 ,1!
2 ,1"
2! ,1#
2" ,20
2# ,21
30 ,22
1 12 2
10 10
11 11
1# 1#
20 20
22 22
23 23
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,20
,21,22
,#
,10
,11
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Data beban
P1 = 20 kN
q1 = 10 kN/m
Properties penampang
Bahan / material :
No. Element E G b (m) h (m) (m2)
1 2!000 0.3 0. 1."00000E01
2 2!000 0.3 0. 1."00000E01
3 2!000 0.3 0. 1."00000E01
PENYELESAIAN :
STEP 1 : Discretize and Global Degrees of reedom !D"#
Defined DOF
after boundary
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D" ses$ai s$mb$ global str$%t$r dan tent$%an &eban ata$ 'omen pada Tit
%&'
11 1 0 ran. +
2 2 0 -ot.
23 3 0 ran. +
4 ,1 0 -ot.
35 5 0 ran. +
,2 0 -ot.
4! ,3 20 ran. +
" ,4 0 -ot.
Nomor +oint $nt$% setiap element Displacement titi% s$mb$ global str$%t$
Element !i#(oint
,t (,e6)(
n6 n6 k +
1 1 2 0 1 2
2 2 3 0 3 4
3 3 4 0 5
,e%ap Elemen dalam 'atri%s -e%a%$an &atang
Batan6 4.E / 2.E / .E / C2 12.E / C3 ,t (,e6)
1 !2.# 3.45 13."!5 3.41!1"!5 0
2 2#1. 145." 21".! 21".! 0
3 2#1. 145." 21".! 21".! 0
'eng)it$ng Ga.a dan 'omen /+$ng !A%ibat beban l$ar di sepan+ang bentan
ELE'ENT 1
7arena ti,ak a,a 6a8a ,ie9anan6 bentan6 reebo,8 maka rekai
Ga.a dan 'omen /+$ng pada Element 1 :
/nit : kNm
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Ga.a dan 'omen /+$ng pada Element 2 :
/nit : kNm
al ai 1 2 3 4
Global ai 1 2 3 4
%&' 0 0 ,3 ,4
===> DALA' S/'&/ L"
Batan6 = 1
Elemen ,alam matrik kekakan batan6 :4.E / 2.E / .E / C2 12.E / C3
!2.# 3.45 13."!5 3.41!1"!5
7eeaian S& L"-AL element ,en6an S& GL"&AL dan D" ara) global
0it$ng matri%s %e%a%$an batang 1 ;S'1
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D = >o theta = 1
D8 = in theta = 0
o>al ai 1 2 3 4
1 3.41!1"!5 13."!5 3.41!1"!5 13."!5
2 13."!5 !2.# 13."!5 3.45
3 3.41!1"!5 13."!5 3.41!1"!5 13."!5
4 13."!5 3.45 13."!5 !2.#
1 2 3 ,1
Global ai 1 2 3 4
1 3.41!1"!5 13."!5 3.41!1"!5 13."!5
2 13."!5 !2.# 13."!5 3.45
3 3.41!1"!5 13."!5 3.41!1"!5 13."!5
4 13."!5 3.45 13."!5 !2.#
===> DALA' S/'&/ L"
Batan6 = 2
Elemen ,alam matrik kekakan batan6 :
4.E / 2.E / .E / C2 12.E / C3
2#1. 145." 21".! 21".!
D = >o theta = 1
D8 = in theta = 0
o>al ai 1 2 3 4
1 21".! 21".! 21".! 21".!
2 21".! 2#1. 21".! 145."
3 21".! 21".! 21".! 21".!
4 21".! 145." 21".! 2#1.
3 ,1 5 ,2
Global ai 3 4 5
3 21".! 21".! 21".! 21".!
4 21".! 2#1. 21".! 145."
5 21".! 21".! 21".! 21".!
21".! 145." 21".! 2#1.
===> DALA' S/'&/ L"
Batan6 = 3
Elemen ,alam matrik kekakan batan6 :
4.E / 2.E / .E / C2 12.E / C3
2#1. 145." 21".! 21".!
D = >o theta = 1
D8 = in theta = 0
@1A =
@1
A =
0it$ng matri%s %e%a%$an batang 2 ;S'2<
@2A =
@2
A =
0it$ng matri%s %e%a%$an batang 3 ;S'3
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o>al ai 1 2 3 4
1 21".! 21".! 21".! 21".!
2 21".! 2#1. 21".! 145."
3 21".! 21".! 21".! 21".!
4 21".! 145." 21".! 2#1.
5 ,2 ,3 ,4
Global ai 5 ! "
5 21".! 21".! 21".! 21".!
21".! 2#1. 21".! 145."
! 21".! 21".! 21".! 21".!
" 21".! 145." 21".! 2#1.
IE,,",!?L""-/P!@D6
rtkan eai mb 6lobal/trktr ,en6an %&' 8an6 ,i,ahlkan
%&' : 1 2 3 ,1
%i.Global ai : 1 2 3 4
%itata lan6 manal ==F ,1 ,2 ,3 ,4
4 ! "
4 34.5 145." 0 0
145." 5"3.2 21".! 145."
! 0 21".! 21".! 21".!
" 0 145." 21".! 2#1.1 13."!5 0 0 0
2 3.45 0 0 0
3 205.03125 21".! 0 0
5 21".! 0 21".! 21".!
0it$ng 'atri%s &eban (oint* Nodal
Nodal forces:
a A%ibat beban di sepan+ang bentang !%ed$a $+$ng di%e%ang#
Element: 1
o>al ai
0 1
= 0 2
0 3
0 4
7arena 9a,a balok mener mb lokal batan6 =
Global ai
1 0
@3A =
@3
A =
'AT,I-S -E-A-/AN ST,/-T/, ;S'S<
@A =
@A1
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= 2 0
3 0
4 0
Element: 2
o>al ai
10 1
= 3.33333333 2
10 3
3.3333333 4
7arena 9a,a balok mener mb lokal batan6 =
Global ai
3 10
= 4 3.3333333333
5 10
3.333333333
Element: 3
o>al ai
0 1
= 0 2
0 3
0 4
7arena 9a,a balok mener mb lokal batan6 =
Global ai
5 0
= 0
! 0
" 0
S$s$n matri%s beban e%iFalen
$oint b Global Element 1 Element 2
1 1 0 0
1 2 0 0
2 3 0 10
2 4 0 3.3333333333
e = 3 5 0 10
3 0 3.333333333
4 ! 0 0
4 " 0 0
e = 0 1 10 2 2
10 3 3
@A1
@A2
@A2
@A3
@A3
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3.333333333 4 ,1
10 5 5
3.333333333 ,2
0 ! ,3
0 " ,4
b A%ibat &eban di +oint !%ed$a $+$ng di%e%ang#
$oint b Global %&'
1 1 0 1
1 2 0 2
2 3 0 3
2 4 0 ,1
= 3 5 0 5
3 0 ,2
4 ! 20 ,3
4 " 0 ,4
c &eban %ombinasi ata$ gab$ngan !Ae C A+#
$oint b Global %&'
1 1 1 0
1 2 2 0
2 3 3 10
2 4 ,1 3.333333333
> = 3 5 5 10
3 ,2 3.3333333333
4 ! ,3 04 " ,4 0
ata lan6 > : %i9la>ement / %eormai
,1 3.3333333
,2 3.33333333
,3 20
,4 0
> = 1 0
2 0
3 10
5 10
-ea>tion:
GAYAGAYA DALA' /(/NG ELE'EN
&atang: 1%eormai trktr ,alam arah B G&B ,i batan6 1H bb :
entkan no. b 6lobal ,ahl
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1 1 0.000000
2 2 0.000000
3 3 0.000000
4 ,1 0.051440
%eormai trktr ,alam arah B &7 ,i batan6 1H bb :
7arena 9a,a balok mener mb lokal batan6 = mb 6lobal t
b lokal %&'
0.000000 1 1
0.000000 2 2
% 1 = 0.000000 3 3
0.051440 4 ,1
Ga.a $+$ng batang
0 3.41!1"!5 13."!5
0 13."!5 !2.#
0 3.41!1"!5 13."!5
0 13."!5 3.45
0 0.!03125
0 1."!5
0 0.!03125 =
0 3.!5
&atang: 2
%eormai trktr ,alam arah B G&B ,i batan6 1H bb :
entkan no. b 6lobal ,ahl
3 3 0.000000
4 ,1 0.051440
5 5 0.000000
,2 0.15143
%eormai trktr ,alam arah B &7 ,i batan6 1H bb :
7arena 9a,a balok mener mb lokal batan6 = mb 6lobal t
b lokal %&'
0.000000 1 3
0.051440 2 4
% 2 = 0.000000 3 5
0.15143 4 ,2
Ga.a $+$ng batang
@%A1 =
A' 1 = A'L 1 C S' 1 D' 1
@1A =
@1A =
@%A2 =
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10 21".! 21".!
3.33333333 21".! 2#1.
10 21".! 21".!
3.3333333 21".! 145."
10 21."!5
3.33333333 !.0"33333
10 21."!5 =
3.3333333 3.!
&atang: 3
%eormai trktr ,alam arah B G&B ,i batan6 1H bb :
entkan no. b 6lobal ,ahl
5 5 0.000000
,2 0.15143
! ,3 0."!24
" ,4 0.425"12
%eormai trktr ,alam arah B &7 ,i batan6 1H bb :
7arena 9a,a balok mener mb lokal batan6 = mb 6lobal t
b lokal %&'0.000000 1 5
0.15143 2
% 3 = 0."!24