Tugas Anstruk II
26
Soal 2.75 3.5 4.8 m A = 0.98 E = 20500 A1 E1 L 36.096 0.98 20500 3.5 29.809 A2 E2 L 29.809 0.98 20500 2.75 A3 E3 L 0.98 20500 4.8 A4 E4 L Notasi Matriks Kekakuan Elemen 0.98 20500 2.75 A5 E5 L fi AE/L 0 -AE/L 0 0.98 20500 3.5 gi 0 0 0 0 A6 E6 L fj = -AE/L 0 AE/L 0 0.98 20500 5.532 gj 0 0 0 0 A7 E7 L 0.98 20500 5.532 A8 E8 L 0.98 20500 4.8 A9 E9 L 0.98 20500 5.941 p2 = 2.75 t p3 = 4 t p1 = 2.3 t α3 α1 m² t/m² Sudut α1 = Sudut α2 = Sudut α3 = 2 4 3 2 5 1 9 8 7 6 1 3 4 5 6
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Tugas Analaisa Struktur II Fakultas Teknik Universitas Khairun
Transcript of Tugas Anstruk II
Soal
2.75 m
3.5 m
4.8 m
A = 0.98E = 20500
A1 E1 L 36.0960.98 20500 3.5 29.809A2 E2 L 29.809
0.98 20500 2.75A3 E3 L
0.98 20500 4.8A4 E4 L Notasi Matriks Kekakuan Elemen
0.98 20500 2.75A5 E5 L fi AE/L 0 -AE/L 0 Ui
0.98 20500 3.5 gi 0 0 0 0 ViA6 E6 L fj = -AE/L 0 AE/L 0 Uj
0.98 20500 5.532 gj 0 0 0 0 VjA7 E7 L
0.98 20500 5.532A8 E8 L
0.98 20500 4.8A9 E9 L
0.98 20500 5.941
p2 = 2.75 t p3 = 4 t
p1 = 2.3 t α3
α1
m²t/m²
Sudut α1 =Sudut α2 =Sudut α3 =
2
4
3
2
51 9
8
76
1
3 4
5
6
Elemen 1 dari nodal 1 ke 2A1 E1 L
0.98 20500 3.5
Matriks kekakuan lokal elemen 1
5740.000 0 -5740.000 0
= 0 0 0 0 Sudut elemen 1 = 90-5740.000 0 5740.000 0
0 0 0 0
Transformasi koordinat elemen 1
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0 1 0 0
= -1 0 0 00 0 0 10 0 -1 0
Matriks Transpose Transformasi Koordinat Elemen 1
0 -1 0 0
= 1 0 0 00 0 0 -10 0 1 0
Menghitung Matriks Kekakuan Global Struktur Elemen 1
=
0 0 0 0
= 5740.000 0 -5740.000 00 0 0 0
-5740.000 0 5740.000 0
U1 V1 U2 V2
0 0 0 0 U1
= 0 5740.000 0 -5740.000 V10 0 0 0 U20 -5740.000 0 5740.000 V2
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
OVERALL STIFFNES MATRIKS
U1 V1 U2 V2 U3 V3 U4
2545.991 1458.634 0 0 0 0 01458.634 6575.672 0 -5740 0 0 0
0 0 6919.593 1566.448 0 0 -2734.1760 -5740 1566.448 13942.895 0 -7305.455 -1566.4480 0 0 0 6919.593 -1566.448 -4185.417
= 0 0 0 -7305 -1566.448 8202.895 0.0000 0 -2734.176 -1566.448 -4185.4167 0 6919.5930 0 -1566.448 -897.440 0 0 1566.448
-2545.991 -1458.634 -4185.417 0.000 -2734.176 1566.448 0-1458.634 -835.672 0.000 0.000 1566.448 -897.440 0
0 0 0 0 0 0 00 0 0 0 0 0 0
Unknown Displesment and Reaction
U2 V2 U3 V3 U4 V4 U5
F2 = 2.3 6919.593 1566.448 0.0000 0.0000 -2734.176 -1566.448 -4185.417G2 = 0 1566.448 13942.895 0.0000 -7305.455 -1566.448 -897.440 0.0000F3 = 0 0.0000 0.0000 6919.593 -1566.448 -4185.417 0.0000 -2734.176
G3 = -2.75 = 0.0000 -7305.455 -1566.448 8202.895 0.0000 0.0000 1566.448F4 = -4 -2734.176 -1566.448 -4185.417 0.0000 6919.593 1566.448 0.0000G4 = 0 -1566.448 -897.440 0.0000 0.0000 1566.448 8202.895 0.0000F5 = 0 -4185.417 0.0000 -2734.176 1566.448 0.0000 0.0000 9465.584G5 = 0 0.0000 0.0000 1566.448 -897.440 0.0000 -7305.455 -107.814
6919.593 1566.448 0.0000 0.0000 -2734.176 -1566.448 -4185.4171566.448 13942.895 0.0000 -7305.455 -1566.448 -897.440 0.0000
0.0000 0.0000 6919.593 -1566.448 -4185.417 0.0000 -2734.1760.0000 -7305.455 -1566.448 8202.895 0.0000 0.0000 1566.448
-2734.176 -1566.448 -4185.417 0.0000 6919.593 1566.448 0.0000-1566.448 -897.440 0.0000 0.0000 1566.448 8202.895 0.0000-4185.417 0.0000 -2734.176 1566.448 0.0000 0.0000 9465.584
Invers Matriks
U2 0.001 0.000E+00 0.001 1.4422E-05 0.001 -8.539E-05 0.0004V2 0.000E+00 0.0002 9.981E-05 0.0002 9.981E-05 0.000E+00 -1.573E-19U3 0.001 9.981E-05 0.001 0.0002 0.001 -0.0002 0.001V3 = 1.442E-05 0.0002 0.0002 0.0003 0.0001 -4.734E-06 -2.390E-19U4 0.001 9.981E-05 0.001 0.0001 0.001 -0.0002 0.001V4 -8.539E-05 0 -0.0002 -4.734E-06 -0.0002 0.0003 -9.981E-05U5 0.0004 0.000E+00 0.001 0 0.001 -9.981E-05 0.0004V5 -9.981E-05 0.000E+00 -0.0002 0 -0.0002 0.0002 -9.981E-05
F₁G₁F₂G₂F₃
G₃ F₄G₄F₅G₅F₆G₆
Displacement Kontrol :
U2 -0.001063 1 0 0 0 0V2 -0.000878 0 1 0 0 0U3 -0.002618 0 0 1 0 0V3 = -0.001365 0 0 0 1 0U4 -0.002955 0 0 0 0 1V4 0.000772 0 0 0 0 0U5 -0.000994 0 0 0 0 0V5 0.000569 0 0 0 0 0
Displacement Reaksi tumpuanU1 = 0 F1 = 1.7V1 = 0 F6 = 0U2 = -0.001 G1 = 6.016V2 = -0.001 G6 = -3.266U3 = -0.003V3 = -0.001 Kontrol reaksi tumpuanU4 = -0.003 ΣH = 0 ΣV = 0V4 = 0.001 P1-p3-F1+F6 = 0 G1 - G2 - P2 = 0U5 = -0.001 2.3 - 4 + 1.7 + 0 = 0 6.016 - 3.266 - 2.75 = 0V5 = 0.001 0 = 0 …!!! 0 = 0 … !!!U6 = 0V6 = 0
[ K ] . [ T ]¹ = I
Elemen 2 dari nodal 2 ke 3A2 E2 L
0.98 20500 2.75
Matriks kekakuan lokal elemen 2
7305.455 0 -7305.455 0
= 0 0 0 0 Sudut elemen 2 = 90-7305.455 0 7305.455 0
0 0 0 0
Transformasi koordinat elemen 2
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0 1 0 0
= -1 0 0 00 0 0 10 0 -1 0
Matriks Transpose Transformasi Koordinat Elemen 2
0 -1 0 0
= 1 0 0 00 0 0 -10 0 1 0
Menghitung Matriks Kekakuan Global Struktur Elemen 2
=
0 0 0 0
= 7305.455 0 -7305.455 00 0 0 0
-7305.455 0 7305.455 0
U2 V2 U3 V3
0 0 0 0 U2
= 0 7305.455 0 -7305.455 V20 0 0 0 U30 -7305.455 0 7305.455 V3
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
V4 U5 V5 U6 V6
0 -2545.991 -1458.634 0 0 U10 -1458.634 -835.672 0 0 V1
-1566.448 -4185.417 0.000 0 0 U2-897.440 0.000 0.000 0 0 V2
0.000 -2734.176 1566.448 0 0 U30.000 1566.448 -897.440 0 0 V3
1566.448 0 0 0 0 U48202.895 0 -7305 0 0 V4
0 9465.584 -107.814 0 0 U5-7305.455 -107.814 14778.567 0 -5740 V5
0 0 0 0 0 U60 0 -5740 0 5740 V6
V5
0.0000 6919.593 1566.448 0.0000 0.0000 -2734.176 -1566.448 -4185.417 U20.0000 1566.448 13942.895 0.0000 -7305.455 -1566.448 -897.440 0.000 V2
1566.448 0.0000 0.0000 6919.593 -1566.448 -4185.417 0.0000 -2734.176 U3-897.440 0.0000 -7305.455 -1566.448 8202.895 0.000 0.0000 1566.448 V30.0000 -2734.176 -1566.448 -4185.417 0.0000 6919.593 1566.448 0.0000 U4
-7305.455 -1566.448 -897.440 0.0000 0.0000 1566.448 8202.895 0.0000 V4-107.814 -4185.417 0.0000 -2734.176 1566.448 0.0000 0.0000 9465.584 U5
14778.567 0.0000 0.0000 1566.448 -897.440 0.0000 -7305.455 -107.814 V50.0000 6919.593 1566.448 0.0000 0.0000 -2734.176 -1566.448 -4185.4170.0000 1566.448 13942.895 0.0000 -7305.455 -1566.448 -897.440 0.0000
1566.448 0.0000 0.0000 6919.593 -1566.448 -4185.417 0.0000 -2734.176-897.440 0.0000 -7305.455 -1566.448 8202.895 0.0000 0.0000 1566.4480.0000 -2734.176 -1566.448 -4185.417 0.0000 6919.593 1566.448 0.0000
-7305.455 -1566.448 -897.440 0.0000 0.0000 1566.448 8202.895 0.0000-107.814 -4185.417 0.0000 -2734.176 1566.448 0.0000 0.0000 9465.584
-9.981E-05 2.33.022E-21 0
-0.0001996 0-1.194E-20 X -2.75-0.0001996 -4
0.0002 0-9.981E-05 0
0.0002 0
0 0 00 0 00 0 00 0 0 OK…!!! Terkontrollllll……..!!!!0 0 01 0 00 1 00 0 1
Elemen 3 dari nodal 3 ke 4A3 E4 L
0.98 20500 4.8
Matriks kekakuan lokal elemen 3
4185.417 0 -4185.417 0
= 0 0 0 0 Sudut elemen 3 = 0-4185.417 0 4185.41667 0
0 0 0 0
Transformasi koordinat elemen 3
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
1 0 0 0
= 0 1 0 00 0 1 00 0 0 1
Matriks Transpose Transformasi Koordinat Elemen 3
1 0 0 0
= 0 1 0 00 0 1 00 0 0 1
Menghitung Matriks Kekakuan Global Struktur Elemen 3
=
4185.417 0 -4185.417 0
= 0 0 0 0-4185.417 0 4185.417 0
0 0 0 0
U3 V3 U4 V4
4185.41667 0 -4185.417 0 U3
= 0 0 0 0 V3-4185.41667 0 4185.41667 0 U4
0 0 0 0 V4
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Gaya - gaya batang
Elemen 1 dari nodal 1 ke 2f = [T ] . [Kg ] . [ U ]
f1 0 5740.000 0 -5740.000 0 U1g1 = 0 0 0 0 x 0 V1f2 0 -5740.000 0 5740.000 -0.001 U2g2 0 0 0 0 -0.001 V2
f1 5.042g1 = 0f2 -5.042g2 0
Elemen 2 dari nodal 2 ke 3f = [T ] . [Kg ] . [ U ]
f2 0 7305.455 0 -7305.455 -0.001 U2g2 = 0 0 0 0 x -0.001 V2f3 0 -7305.455 0 7305.455 -0.003 U3g3 0 0 0 0 -0.001 V3
f2 3.558g2 = 0f3 -3.558g3 0
Elemen 3 dari nodal 3 ke 4f = [T ] . [Kg ] . [ U ]
f3 4185.417 0 -4185.417 0 -0.003 U3g3 = 0 0 0 0 x -0.001 V3f4 -4185.417 0 4185.417 0 -0.003 U4g4 0 0 0 0 0.001 V4
f3 1.411g3 = 0f4 -1.411g4 0
Elemen 4 dari nodal 5 ke 4A4 E4 L
0.98 20500 2.75
Matriks kekakuan lokal elemen 4
7305.455 0 -7305.455 0
= 0 0 0 0 Sudut elemen 4 = 90-7305.45 0 7305.455 0
0 0 0 0
Transformasi koordinat elemen 4
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0 1 0 0
= -1 0 0 00 0 0 10 0 -1 0
Matriks Transpose Transformasi Koordinat Elemen 4
0 -1 0 0
= 1 0 0 00 0 0 -10 0 1 0
Menghitung Matriks Kekakuan Global Struktur Elemen 4
=
0 0 0 0
= 7305.455 0 -7305.45 00 0 0 0
-7305.45 0 7305.455 0
U5 V5 U4 V4
0 0 0 0 U5
= 0 7305.455 0 -7305.45 V50 0 0 0 U40 -7305.455 0 7305.455 V4
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Elemen 4 dari nodal 5 ke 4f = [T ] . [Kg ] . [ U ]
f5 0 7305.455 0 -7305.455 -0.001 U5g5 = 0 0 0 0 x 0.001 V5f4 0 -7305.455 0 7305.455 -0.003 U4g4 0 0 0 0 0.001 V4
f5 -1.483g5 = 0f4 1.483g4 0
Elemen 5 dari nodal 6 ke 5f = [T ] . [Kg ] . [ U ]
f6 0 5740.000 0 -5740.000 0 U6g6 = 0 0.000 0 0 x 0 V6f5 0 -5740.000 0 5740.000 -0.001 U5g5 0 0.000 0 0 0.001 V5
f6 -3.266g6 = 0f5 3.266g5 0
Elemen 6 dari nodal 3 ke 5f = [T ] . [Kg ] . [ U ]
f3 3151.108 -1805.314 -3151.108 1805.314 -0.003 U3g3 = 0 0 0 0 x -0.001 V3f5 -3151.108 1805.314 3151.108 -1805.314 -0.001 U5g5 0 0 0 0 0.001 V5
f3 -1.626g3 = 0f5 1.626g5 0
Elemen 5 dari nodal 6 ke 5A5 E5 L
0.98 20500 3.5
Matriks kekakuan lokal elemen 5
5740.000 0 -5740.000 0
= 0 0 0 0 Sudut elemen 5 = 90-5740.000 0 5740.000 0
0 0 0 0
Transformasi koordinat elemen 5
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0 1 0 0
= -1 0 0 00 0 0 10 0 -1 0
Matriks Transpose Transformasi Koordinat Elemen 5
0 -1 0 0
= 1 0 0 00 0 0 -10 0 1 0
Menghitung Matriks Kekakuan Global Struktur Elemen 5
=
0 0 0 0
= 5740.000 0 -5740.00 00 0 0 0
-5740.00 0 5740.000 0
U6 V6 U5 V5
0 0 0 0 U6
= 0 5740.000 0 -5740.00 V60 0 0 0 U50 -5740.00 0 5740.000 V5
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Elemen 7 dari nodal 2 ke 4f = [T ] . [Kg ] . [ U ]
f2 3151.108 1805.314 -3151.108 -1805.314 -0.001 U2g2 = 0 0 0 0 x -0.001 V2f4 -3151.108 -1805.314 3151.108 1805.314 -0.003 U4g4 0 0 0 0 0.001 V4
f2 2.984g2 = 0f4 -2.984g4 0
Elemen 8 dari nodal 2 ke 5f = [T ] . [Kg ] . [ U ]
f2 4185.417 0 -4185.417 0 -0.001 U2g2 = 0 0 0 0 x -0.001 V2f5 -4185.417 0 4185.417 0 -0.001 U5g5 0 0 0 0 0.001 V5
f2 -0.289g2 = 0f5 0.289g5 0
Elemen 9 dari nodal 1 ke 5f = [T ] . [Kg ] . [ U ]
f1 2934.227 1681.060 -2934.227 -1681.060 0 U1g1 = 0 0 0 0 x 0 V1f5 -2934.227 -1681.060 2934.227 1681.060 -0.001 U5g5 0 0 0 0 0.001 V5
f1 1.959g1 = 0f5 -1.959g5 0
Elemen 6 dari nodal 3 ke 5A6 E6 L
0.98 20500 5.532
Matriks kekakuan lokal elemen 6
3631.6164 0 -3631.616 0
= 0 0 0 0 Sudut elemen 6 = 330.191-3631.616 0 3631.616 0
0 0 0 0
Transformasi koordinat elemen 6
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0.868 -0.497 0 0
= 0.497 0.868 0 00 0 0.868 -0.4970 0 0.497 0.868
Matriks Transpose Transformasi Koordinat Elemen 6
0.868 0.497 0 0
= -0.497 0.868 0 00 0 0.868 0.4970 0 -0.497 0.868
Menghitung Matriks Kekakuan Global Struktur Elemen 6
=
3151.108 0 -3151.108 0
= -1805.314 0 1805.314 0-3151.108 0 3151.108 01805.314 0 -1805.314 0
U3 V3 U5 V5
2734.176 -1566.448 -2734.176 1566.448 U3
= -1566.448 897.440 1566.448 -897.440 V3-2734.176 1566.448 2734.176 -1566.448 U51566.448 -897.440 -1566.448 897.440 V5
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Free Body Diagram
P2 = 2.75 t
1.411 -1.411 p3 = 4 t
-1.6261.483
-3.558 -2.984
3.558 -1.4831.626
2.984p1 =2.3 t
-0.289 0.289
-5.042 3.266-1.959
5.042 -3.266
1.959F1 = 1.7 F6 = 0
G1 = 6.016 G6 = -3.266
Displacement diagramU4
U3 V4
U5U2
V5 keteranganV2 U1 = 0
V1 = 0U2 = -0.001V2 = -0.001U3 = -0.003V3 = -0.001U4 = -0.003V4 = 0.001U5 = -0.001V5 = 0.001U6 = 0V6 = 0
1
2
34
5
6
1
2
3
4
5
67
8
9
1
2
3
4
5
6
1
2
3
4
5
67
8
9
Elemen 7 dari nodal 2 ke 4A7 E7 L
0.98 20500 5.532
Matriks kekakuan lokal elemen 7
3631.616 0 -3631.616 0
= 0 0 0 0 Sudut elemen 7 = 29.809-3631.616 0 3631.6164 0
0 0 0 0
Transformasi koordinat elemen 7
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0.868 0.497 0 0
= -0.497 0.868 0 00 0 0.868 0.4970 0 -0.497 0.868
Matriks Transpose Transformasi Koordinat Elemen 7
0.868 -0.497 0 0
= 0.497 0.868 0 00 0 0.868 -0.4970 0 0.497 0.868
Menghitung Matriks Kekakuan Global Struktur Elemen 7
=
3151.108 0 -3151.108 0
= 1805.314 0 -1805.314 0-3151.11 0 3151.108 0
-1805.314 0 1805.314 0
U2 V2 U4 V4
2734.176 1566.448 -2734.176 -1566.448 U2
= 1566.448 897.440 -1566.448 -897.440 V2-2734.176 -1566.448 2734.176 1566.448 U4-1566.448 -897.440 1566.448 897.440 V4
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Elemen 8 dari nodal 2 ke 5A8 E8 L
0.98 20500 4.8
Matriks kekakuan lokal elemen 8
4185.417 0 -4185.417 0
= 0 0 0 0 Sudut elemen 8 = 0-4185.417 0 4185.417 0
0 0 0 0
Transformasi koordinat elemen 8
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
1 0 0 0
= 0 1 0 00 0 1 00 0 0 1
Matriks Transpose Transformasi Koordinat Elemen 8
1 0 0 0
= 0 1 0 00 0 1 00 0 0 1
Menghitung Matriks Kekakuan Global Struktur Elemen 8
=
4185.417 0 -4185.417 0
= 0 0 0 0-4185.417 0 4185.417 0
0 0 0 0
U2 V2 U5 V5
4185.417 0.000 -4185.417 0.000 U2
= 0.000 0.000 0.000 0.000 V2-4185.417 0.000 4185.417 0.000 U5
0.000 0.000 0.000 0.000 V5
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
Elemen 9 dari nodal 1 ke 5A9 E9 L
0.98 20500 5.941
Matriks kekakuan lokal elemen 9
3381.663 0 -3381.663 0
= 0 0 0 0 Sudut elemen 8 = 29.809-3381.663 0 3381.663 0
0 0 0 0
Transformasi koordinat elemen 9
T
cos α sin α 0 0
= -sin α cos α 0 0
0 0 cos α sin α0 0 -sin α cos α
T
0.868 0.497 0 0
= -0.497 0.868 0 00 0 0.868 0.4970 0 -0.497 0.868
Matriks Transpose Transformasi Koordinat Elemen 9
0.868 -0.497 0 0
= 0.497 0.868 0 00 0 0.868 -0.4970 0 0.497 0.868
Menghitung Matriks Kekakuan Global Struktur Elemen 9
=
2934.227 0 -2934.227 0
= 1681.060 0 -1681.060 0-2934.227 0 2934.227 0-1681.060 0 1681.060 0
U1 V1 U5 V5
2545.991 1458.634 -2545.991 -1458.634 U1
= 1458.634 835.672 -1458.634 -835.672 V1-2545.991 -1458.634 2545.991 1458.634 U5-1458.634 -835.672 1458.634 835.672 V5
Kel
T ᵀ
Kg T ᵀ. Kel . T
Kg
Kg
