Penyelesaian Soal UAS Mekanika Rekayasa IV - 7 Juli 2011 - Yoppy Soleman
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Transcript of Penyelesaian Soal UAS Mekanika Rekayasa IV - 7 Juli 2011 - Yoppy Soleman
Soal Belum Diganti!!!
Derajat Ketidaktentuan Kinematik (degree of kinematic indeterminacy) = 2 Translasi (Join D) dan 2 Translasi (Join C) (Komponen Aksial) Q11 = Q12 = Q21 = Q22 = 140 kN (arah positif) 0 kN 0 kN -60 kN (arah negatif) cos = sin = 0.8575 0.5145
Matriks Vektor Deformasi [A] 0.86 0 -0.86 0 0 D1 0.51 1 0.51 0 0 D2 0 0 0 1 -1 D3 0 1 0 0 0 D4 1 2 3 4 5
Matriks Vektor Deformasi Transpos [A]T 0.86 0.51 0 0 0 1 0 1 -0.86 0.51 0 0 0 0 1 0 0 0 -1 0
Matriks Kekakuan Lokal Elemen [S] 0.34 0 0 0.67 0 0 0 0 0 0.51 0 0 0 0 0 0.2 0 0 0 0 0 0.2 F1 F2 F3 F4 F5
EA
0 0 0
1
2
3
4
5
Matriks Kekakuan Struktur (Global Stiffness Matrix) [K] = [A]T [S] [A]0.86 0 -0.86 0 0 0.51 1 0.51 0 0 0 0 0 1 -1 0 1 0 0 0
x
0.34 0 0 0 0
0 0.67 0 0 0
0 0 0.51 0 0
0 0 0 0.2 0
0 0 0 0 0.2
x
0.86 0.51 0 0
0 1 0 1
-0.86 0.51 0 0
0 0 1 0
0 0 -1 0
Submatrix [K] = [A]T [S]0.29 0.18 0 0 0 0.67 0 0.67 -0.44 0.26 0 0 0 0 0.2 0 0 0 -0.2 0
[K] = [A]T [S] [A]0.63 -0.08 0 0 -0.08 0.89 0 0.67 0 0 0.4 0 0 0.67 0 0.67
EA
[K]-11.65 0.55 0 -0.55 0.55 4.59 0 -4.59 0 0 2.5 0 -0.55 -4.59 0 6.09
EA
Matriks Deformasi [D] = [K]-1 [Q]264.34 352.45 1/EA 0 -442.45 = 1.65 0.55 0 -0.55 0.55 4.59 0 -4.59 0 0 2.5 0 -0.55 -4.59 0 6.09 x 140.00 0.00 0.00 -60.00
Matriks Gaya Internal [H] = [S] [A] [D]139.94 -60 -23.32 0 0 0.34 0 0 0 0 0 0.67 0 0 0 0 0 0.51 0 0 0 0 0 0.2 0 0 0 0 0 0.2 x 0.86 0 -0.86 0 0 0.51 1 0.51 0 0 0 0 0 1 -1 0 1 0 0 0 x 264.34 352.45 0 -442.45
=
Submatrix [H] = [S] [A]0.29 0 -0.44 0 0 0.18 0.67 0.26 0 0 0 0 0 0.2 -0.2 0 0.67 0 0 0
Nilai yang dimasukkan disini adalah reaksi momen internal, sehingga berlawanan arah dengan momen internal atau -[H] atau balikkan tanda momen internal [H]
Gaya Aksial Elemen = - [H] F1 = -139.943 F2 = 60.000 F3 = 23.324 F4 = 0.000 F5 = 0.000
Plot Skema Momen-momen Ujung Elemen Frame -139.94 60 23.32 0 0 0.000 0.000 0.000 0.000 0.000kN kN kN kN kN[COMPRESSION] [TENSION] [TENSION] [NEUTRAL] [NEUTRAL]
=
+
= Perhitungan Manual - Output Aplikasi Program Komputer = 0 %
Soal Belum Diganti!!!
Koefisien Kekakuan Ujung Elemen =
Momen Ujung Elemen Satu Satuan Rotasi
Momen Jepit Ujung
M AB =
1 wL 2 12
A= L1
B
M BA =
1 wL 2 12
M BC =
1 wL 2 12
B= L2
C
M CB =
1 wL 12
2
M CD =
Pab 2 L2
C
D
M DC =
Pba 2 L2
= L3 Derajat Ketidaktentuan Kinematik = Rotasi Join B, C dan D (3 DK) FEMAB = 96.000 kNm FEMBA = -96.000 kNm = 34.667 kNm FEMBC = 130.667 kNmPage 4 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
FEMCB FEMCD FEMDC
= = =
-130.667 kNm 85.000 kNm -85.000 kNm
= =
-45.667 kNm -85.000 kNm
Matriks Vektor Deformasi [A] 0 1 1 0 0 0 D1 0 0 0 1 1 0 D2 0 0 0 0 0 1 D3 1 2 3 4 5 6
Matriks Vektor Deformasi Transpos [A]T 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1
Matriks Kekakuan Lokal Elemen [S] 0.67 0.33 0 0 0 0 1 0.33 0.67 0 0 0 0 2 0 0 0.57 0.29 0 0 3 0 0 0.29 0.57 0 0 4 0 0 0 0 1.5 0.75 5 0 0 0 0 0.75 1.5 6 M1 M2 M3 M4 M5 M6
EI
Matriks Kekakuan Struktur (Global Stiffness Matrix) [K] = [A]T [S] [A]0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0.67 0.33 0 0 0 0 0.33 0.67 0 0 0 0 0 0 0.57 0.29 0 0 0 0 0.29 0.57 0 0 0 0 0 0 1.5 0.75 0 0 0 0 0.75 1.5 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1
x
x
Submatrix [K] = [A]T [S]0.33 0 0 0.67 0 0 0.57 0.29 0 0.29 0.57 0 0 1.5 0.75 0 0.75 1.5
[K] = [A]T [S] [A]1.24 0.29 0 0.29 2.07 0.75 0 0.75 1.5
EI
Page 5 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
Menentukan Matriks Kekakuan Invers [K]-1 yang Metoda Adjoint Metoda ADJOINT 1.24 0.29 0.29 2.07 0 0.75 Rumus : 0 0.75 1.5
[ A]dimana:
1
=
1 1+ adj A = A [ det A
]
[ A] [ A]
1
=a m triksinve rs = jo ad A intT
= d r m ete in an+
Adjo int A =[C ijA ]Cij = ( 1) i + j M ijij
transpos matriks kofaktor =
kofaktor Minor M ij =
= baris ke i kolom ke j= det er min an ( faktor penentu)
[ ] = matriksAlgoritma: 1. Hitung determinan matriks dengan rumus kofaktor Cijminor minor minor
= =
-1.8 3.03
2.07 0.75
0.75 1.5
+
0.4
0.29 0.75
0 1.5
+
0
0.29 2.07
0 0.75
2. Hitung kofaktor elemen-elemen matriks (transpos) 1.24 0.29 0 2.07 = C11 = 0.29 2.07 0.75 0.75 0 0.75 1.5
0.75 1.5
=
2.54
Page 6 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
C12 =
1.24 0.29 0 1.24 0.29 0 1.24 0.29 0 1.24 0.29 0 1.24 0.29 0 1.24 0.29 0 1.24 0.29 0 1.24 0.29 0
0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75 0.29 2.07 0.75
0 0.75 1.5 0 0.75 1.5 0 0.75 1.5 0 0.75 1.5 0 0.75 1.5 0 0.75 1.5 0 0.75 1.5 0 0.75 1.5
=
0.29 0
0.75 1.5
=
-0.43
C13 =
=
0.29 0
2.07 0.75
=
0.21
C21 =
=
0.29 0.75
0 1.5
=
-0.43
C22 =
=
1.24 0
0 1.5
=
1.86
C23 =
=
1.24 0
0.29 0.75
=
-0.93
C31 =
=
0.29 2.07
0 0.75
=
0.21
C32 =
=
1.24 0.29
0 0.75
=
-0.93
C33 =
=
1.24 0.29
0.29 2.07
=
2.48
3. Susun adjoint matriks (adjoint = transpos matriks kofaktor) C11 C12 C13 T C21 C22 C23 [A]+ = = C31 C32 C33 4. Tentukan invers matriks 2.54 0.33 [A]-1 =
C11 C12 C13
C21 C22 C23
C31 C32 C33
=
2.54 -0.43 0.21
-0.43 1.86 -0.93
0.21 -0.93 2.48
-0.43
0.21 =
0.84
-0.14
0.07
Page 7 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
[A]-1 =
0.33
-0.43 0.21
1.86 -0.93
-0.93 2.48
=
-0.14 0.07
0.61 -0.31
-0.31 0.82
Catatan: Cara termudah untuk mengerjakan operasi-operasi matriks (perkalian dan invers) adalah dengan menggunakan spreadsheet Microsoft Office Excel, atau program Mathcad atau program Matlab, atau menggunakan kalkulator program matriks Casio PB-1000, dan sejenisnya.
[K]-10.84 -0.14 0.07 -0.142 0.6133 -0.307 0.07 -0.31 0.82
1/EI
Matriks Deformasi [D] = [K]-1 [Q]29.58 -6.85 1/EI -53.24 = 0.84 -0.14 0.07 -0.14 0.61 -0.31 0.07 -0.31 0.82 x 34.667 -45.667 -85.000
Matriks Gaya Internal [H] = [S] [A] [D]9.86 19.72 14.95 4.54 -50.2 -85 0.67 0.33 0 0 0 0 0.33 0.67 0 0 0 0 0 0 0.57 0.29 0 0 0 0 0.29 0.57 0 0 0 0 0 0 1.5 0.75 0 0 0 0 0.75 1.5 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 x 29.58 -6.85 -53.24
=
x
Submatrix [H] = [S] [A]0.33 0.67 0.57 0.29 0 0 0 0 0.29 0.57 1.5 0.75 0 0 0 0 0.75 1.5
Page 8 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
Nilai yang dimasukkan disini adalah reaksi momen internal, sehingga berlawanan arah dengan momen internal atau -[H] atau balikkan tanda momen internal [H]
Momen Ujung Elemen = FEMij - [H] = -[H] + FEMij MAB MBA MBC MCB MCD MDC = = = = = = 86.140 -115.720 115.720 -135.205 135.205 0.000 -9.86 -19.72 -14.95 -4.54 50.2 85 96.000 kNm -96.000 kNm 130.667 kNm -130.667 kNm 85.000 kNm -85.000 kNm
=
+
Plot Skema Momen-momen Ujung Elemen Beam
Page 9 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
MOMEN JEPIT UJUNG (FIX-END MOMENT)
M BC =
1 wL 2 12
B
C
M CB =
1 wL 12
2
M CE =
1 wL 2 12
C
E
1 M EC = wL 12
2
M CD =
Pab 2 L2
C
D
M DC =
Pba 2 L2
Page 7 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
F
B
1 M BF = wL 2
2
MB = 0
B
A
MA = 0
Derajat Ketidaktentuan Kinematik (degree of kinematic indeterminacy) = 2 Rotasi (Join B dan C) (Hanya pengaruh lentur) FEMEC FEMBF FEMBC FEMCB FEMCE FEMCD FEMDC FEMAB FEMAB = = = = = = = = = -105.000 kNm -40.000 kNm 142.917 kNm -142.917 kNm 105.000 kNm 16.875 kNm -50.625 kNm 0.000 kNm 0.000 kNm = = = = = = -105.000 kNm 102.917 kNm -21.042 kNm -50.625 kNm 0.000 kNm 0.000 kNm
Koefisien Kekakuan Ujung Elemen
=
Momen Ujung Elemen 1 Unit Rotasi
Page 8 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
Matriks Vektor Deformasi [A] 1 0 0 1 1 0 0 0 0 0 D1 0 0 1 1 0 1 0 D2 2 3 4 5 6 7 8
Matriks Vektor Deformasi Transpos [A]T 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0
Matriks Kekakuan Lokal Elemen [S] 1 0.5 0 0 0 0 0 0 1 0.5 1 0 0 0 0 0 0 2 0 0 1.71 0.86 0 0 0 0 3 0 0 0.86 1.71 0 0 0 0 4 0 0 0 0 1 0.5 0 0 5 0 0 0 0 0.5 1 0 0 6 0 0 0 0 0 0 1.33 0.67 7 0 0 0 0 0 0 0.67 1.33 8 M1 M2 M3 M4 M5 M6 M7 M8
EI
Matriks Kekakuan Struktur (Global Stiffness Matrix) [K] = [A]T [S] [A]0 1 1 0 0 0 0 1 1 0.5 0 0 0.5 1 0 0 0 0 1.71 0.86 0 0 0.86 1.71 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1
x
x
Page 9 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
x0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0.5 0 0 0.5 1 0 0 0 0 1.33 0.67 0 0 0.67 1.33
x
Submatrix [K] = [A]T [S]0.5 0 1 0 1.71 0.86 0.86 1.71 0 1 0 0.5 0 1.33 0 0.67
[K] = [A]T [S] [A]2.71 0.86 0.86 4.05
EI
Matriks Kekakuan Invers [K]-1 yang berukuran ordo 2 dapat ditentukan lagi dengan mencari determinannnya saja. Metoda DETERMINAN 2.71 0.86 0.86 4.05Rumus :
[ A]
1
=
1 adjo det A
Aint =
1+ A
[ ]
= (2.5143*2.5143 - 0.8571*0.8571) = 10.25 [A]-1 = 0.1 2.71 -0.86 -0.86 4.05 = 0.265 -0.084 -0.084 0.395
[K]-10.395 -0.084 -0.084 0.265
1/EI
Matriks Deformasi [D] = [K]-1 [Q]42.39 1/EI -14.18 = 0.39 -0.08 -0.08 0.26 x 102.917 -21.042Page 10 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
Matriks Gaya Internal [H] = [S] [A] [D]21.2 42.39 60.52 12.04 -14.18 -7.09 -18.9 -9.45 1 0.5 0 0 0 0 0 0 0.5 1 0 0 0 0 0 0 0 0 1.71 0.86 0 0 0 0 0 0 0.86 1.71 0 0 0 0 0 0 0 0 1 0.5 0 0 0 0 0 0 0.5 1 0 0 0 0 0 0 0 0 1.33 0.67 0 0 0 0 0 0 0.67 1.33 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 x 42.39 -14.18
=
x
Submatrix [H] = [S] [A]0.5 1 1.71 0.86 0 0 0 0 0 0 0.86 1.71 1 0.5 1.33 0.67
Nilai yang dimasukkan disini adalah reaksi momen internal, sehingga berlawanan arah dengan momen internal atau -[H] atau balikkan tanda momen internal [H]
Momen Ujung Elemen = FEMij - [H] = -[H] + FEMij MAB = -21.197 -21.2 0.000kNm
Page 11 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
MBA MBC MCB MCD MDC MCE MEC MBF
= = = = = = = =
-42.393 82.393 -154.952 31.051 -43.537 123.901-95.549 -40.000
=
-42.39 -60.52 -12.04 14.18 7.09 18.9 9.450
+
0.000 142.917 -142.917 16.875 -50.625 105.000
kNm kNm kNm kNm kNm kNm
-105.000 kNm -40.000 kNm
Plot Skema Momen-momen Ujung Elemen Frame
Page 12 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
METODA CROSSJOIN ELEMEN Kekakuan Relatif, EI DISTRIBUTION FACT FIXED-END MOMENT Balancing CARRY-OVER M. FEM Adjusting Bal Siklus 1 Com Bal Siklus 2 Com Bal Siklus 3 Com Bal Siklus 4 Com Bal Siklus 5 Com Bal Siklus 6 Com Bal Siklus 7 Com Bal Siklus 8 Com Bal Siklus 9 Com 34.516 -2.294 -0.085 -0.043 -0.254 -0.127 -0.005 -0.002 -0.014 -0.007 0.000 0.000 -0.001 0.000 0.000 0.000 0.000 0.000 -34.516 -4.588 46.413 -7.310 0.222 -0.136 0.660 -0.405 0.012 -0.008 0.037 -0.022 0.001 0.000 0.002 -0.001 0.000 0.000 0.000 0.000 0.000 -46.413 0.443 -3.655 1.319 -0.068 0.025 -0.203 0.073 -0.004 0.001 -0.011 0.004 0.000 0.000 -0.001 0.000 0.000 0.000 0.000 22.59 45.185 0.785 2.335 0.044 0.129 0.002 0.007 0.000 0.000 0.000 0.392 -0.392 1.168 -1.168 0.022 -0.022 0.065 -0.065 0.001 -0.001 0.004 -0.004 0.000 0.000 0.000 0.000 0.000
AAB0.85
BBA0.85
CBC1.36
DCD2.40
CB1.36
DC2.40
034.52
0.386-34.52
0.61446.41
0.361-46.41
0.63922.59
1.000-45.19 45.19
FINAL MOMENTSSelisih Momen Ujung -
32.04
-39.46
39.46
-48.49
48.49-
0.000
0.0000
0.0000
Page 13 - Stiffness Matrix Analysis (c) Y. Soleman - 2008
Page 14 - Stiffness Matrix Analysis (c) Y. Soleman - 2008