modul ajar 10 - konstruksi rangka batang
TRANSCRIPT
MODUL AJAR 10
KONSTRUKSI RANGKA BATANG
Tujuan Pembelajaran Umum :Mahasiswa mampu menyelesaikan analisa struktur dengan cara Analisa Struktur Metode Matriks (ASMM)
Tujuan Pembelajaran KhususMahasiswa mampu menyelesaikan struktur Konstruksi Rangka Batang (KRB) 2 Dimensi dengan ASMM
Contoh Soal 1 :Sebuah Konstruksi Rangka Batang Atap seperti pada gambar di bawah ini,hitunglah gaya-gaya dalam yang bekerja dengan cara ASMM
Diketahui :
2 4 m
H = 3 m
1
1.5 2 8 9 2 A = 1
P = 2 ton
5
1.5 3 4 6 7
1 2
2 2
Degree of Freedom
L =
E = kg/m2
m2
1 3
6
54
2
12
Keterangan:
ISIKAN angka 0 atau 1: 1 angka 1 bila melihat jawaban
angka 0 bila jawaban dihilangkan
sementara
1. Menentukan koefisien transformasi matriks kekakuan elemen
ELEMEN V (m) H (m) L (m) C S C.S C.S/L1 0.00 2.00 2.000 1.000 0.000 1.000 0.000 0.000 0.500 0.000 0.0002 0.00 2.00 2.000 1.000 0.000 1.000 0.000 0.000 0.500 0.000 0.0003 1.50 1.00 1.803 0.555 0.832 0.308 0.692 0.462 0.171 0.384 0.2564 -1.50 1.00 1.803 0.555 -0.832 0.308 0.692 -0.462 0.171 0.384 -0.2565 -3.00 0.00 3.000 0.000 -1.000 0.000 1.000 0.000 0.000 0.333 0.0006 1.50 1.00 1.803 0.555 0.832 0.308 0.692 0.462 0.171 0.384 0.2567 -1.50 1.00 1.803 0.555 -0.832 0.308 0.692 -0.462 0.171 0.384 -0.2568 1.50 1.00 1.803 0.555 0.832 0.308 0.692 0.462 0.171 0.384 0.2569 -1.50 1.00 1.803 0.555 -0.832 0.308 0.692 -0.462 0.171 0.384 -0.256
2. Menyusun matriks kekakuan elemen
C2 SS C2/L S2/L
1
2
11
7
8
9
4
3
10
5
6
ELEMEN 1 = ELEMEN 20.500 0.000 -0.500 0.000
= = A.E 0.000 0.000 0.000 0.000-0.500 0.000 0.500 0.0000.000 0.000 0.000 0.000
ELEMEN 3 = ELEMEN 6 = ELEMEN 80.171 0.256 -0.171 -0.256
= = AE 0.256 0.000 -0.256 0.000-0.171 -0.256 0.171 0.256-0.256 0.000 0.256 0.000
ELEMEN 4 = ELEMEN 7 =ELEMEN 90.171 -0.256 -0.171 0.256
= A.E -0.256 0.384 0.256 -0.384-0.171 0.256 0.171 -0.2560.256 -0.384 -0.256 0.384
ELEMEN 50.000 0.000 0.000 0.000
= A.E 0.000 0.333 0.000 -0.3330.000 0.000 0.000 0.0000.000 -0.333 0.000 0.333
3. Menyusun matriks kekakuan global struktur
0.671 0.256 -0.500 0.000 0.000 0.000 -0.171 -0.256 0.000 0.000 0.000 0.0000.256 0.000 0.000 0.000 0.000 0.000 -0.256 0.000 0.000 0.000 0.000 0.000-0.500 0.000 1.341 0.000 -0.500 0.000 -0.171 0.256 -0.171 -0.256 0.000 0.0000.000 0.000 0.000 0.717 0.000 0.000 0.256 -0.384 -0.256 0.000 0.000 -0.333
A.E 0.000 0.000 -0.500 0.000 0.671 -0.256 0.000 0.000 -0.171 0.256 0.000 0.0000.000 0.000 0.000 0.000 -0.256 0.384 0.000 0.000 0.256 -0.384 0.000 0.000-0.171 -0.256 -0.171 0.256 0.000 0.000 0.512 0.512 0.000 0.000 -0.171 -0.256-0.256 0.000 0.256 -0.384 0.000 0.000 0.512 0.384 0.000 0.000 -0.256 0.0000.000 0.000 -0.171 -0.256 -0.171 0.256 0.000 0.000 0.512 -0.256 -0.171 0.2560.000 0.000 -0.256 0.000 0.256 -0.384 0.000 0.000 -0.256 0.768 0.256 -0.3840.000 0.000 0.000 0.000 0.000 0.000 -0.171 -0.256 -0.171 0.256 0.341 0.0000.000 0.000 0.000 -0.333 0.000 0.000 -0.256 0.000 0.256 -0.384 0.000 0.717
[k](1) [k](2)
[k](3) [k](6) [k](8) =
[k](7) = [k](4) = [k](9)
[k](5)
[K]12X12 =
6.71E-01 2.56E-01 -5.00E-01 0.00E+00 0.00E+00 0.00E+00 -1.71E-01 -2.56E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+002.56E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 -2.56E-01 -0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00-5.00E-01 0.00E+00 1.34E+00 0.00E+00 -5.00E-01 0.00E+00 -1.71E-01 2.56E-01 -1.71E-01 -2.56E-01 -0.00E+00 0.00E+000.00E+00 0.00E+00 0.00E+00 7.17E-01 0.00E+00 0.00E+00 2.56E-01 -3.84E-01 -2.56E-01 -0.00E+00 0.00E+00 -3.33E-01
0.00E+00 0.00E+00 -5.00E-01 0.00E+00 6.71E-01 -2.56E-01 0.00E+00 0.00E+00 -1.71E-01 2.56E-01 0.00E+00 0.00E+000.00E+00 0.00E+00 0.00E+00 0.00E+00 -2.56E-01 3.84E-01 0.00E+00 0.00E+00 2.56E-01 -3.84E-01 0.00E+00 0.00E+00-1.71E-01 -2.56E-01 -1.71E-01 2.56E-01 0.00E+00 0.00E+00 5.12E-01 5.12E-01 0.00E+00 0.00E+00 -1.71E-01 -2.56E-01-2.56E-01 -0.00E+00 2.56E-01 -3.84E-01 0.00E+00 0.00E+00 5.12E-01 3.84E-01 0.00E+00 0.00E+00 -2.56E-01 -0.00E+000.00E+00 0.00E+00 -1.71E-01 -2.56E-01 -1.71E-01 2.56E-01 0.00E+00 0.00E+00 5.12E-01 -2.56E-01 -1.71E-01 2.56E-010.00E+00 0.00E+00 -2.56E-01 -0.00E+00 2.56E-01 -3.84E-01 0.00E+00 0.00E+00 -2.56E-01 7.68E-01 2.56E-01 -3.84E-010.00E+00 0.00E+00 -0.00E+00 0.00E+00 0.00E+00 0.00E+00 -1.71E-01 -2.56E-01 -1.71E-01 2.56E-01 3.41E-01 0.00E+000.00E+00 0.00E+00 0.00E+00 -3.33E-01 0.00E+00 0.00E+00 -2.56E-01 -0.00E+00 2.56E-01 -3.84E-01 0.00E+00 7.17E-01
4. Menghitung Reduksi Matriks Kekakuan Global berdasarkan jumlah Degree of Freedom
1.34E+00 0.00E+00 -5.00E-01 -1.71E-01 2.56E-01 -1.71E-01 -2.56E-01 -0.00E+00 0.00E+000.00E+00 7.17E-01 0.00E+00 2.56E-01 -3.84E-01 -2.56E-01 -0.00E+00 0.00E+00 -3.33E-01-5.00E-01 0.00E+00 6.71E-01 0.00E+00 0.00E+00 -1.71E-01 2.56E-01 0.00E+00 0.00E+00-1.71E-01 2.56E-01 0.00E+00 5.12E-01 5.12E-01 0.00E+00 0.00E+00 -1.71E-01 -2.56E-01
2.56E-01 -3.84E-01 0.00E+00 5.12E-01 3.84E-01 0.00E+00 0.00E+00 -2.56E-01 -0.00E+00-1.71E-01 -2.56E-01 -1.71E-01 0.00E+00 0.00E+00 5.12E-01 -2.56E-01 -1.71E-01 2.56E-01-2.56E-01 -0.00E+00 2.56E-01 0.00E+00 0.00E+00 -2.56E-01 7.68E-01 2.56E-01 -3.84E-01-0.00E+00 0.00E+00 0.00E+00 -1.71E-01 -2.56E-01 -1.71E-01 2.56E-01 3.41E-01 0.00E+000.00E+00 -3.33E-01 0.00E+00 -2.56E-01 -0.00E+00 2.56E-01 -3.84E-01 0.00E+00 7.17E-01
5. Menghitung invers Matriks Kekakuan Global Tereduksi
1.63E+00 7.15E-01 1.73E+00 1.44E-01 3.12E-01 2.12E+00 6.64E-02 1.32E+00 -3.36E-017.15E-01 1.58E+00 -4.97E-03 8.98E-01 -9.98E-01 5.62E-01 1.78E+00 -1.36E+00 1.81E+001.73E+00 -4.97E-03 4.25E+00 7.69E-01 7.59E-03 3.08E+00 -1.81E+00 3.29E+00 -1.80E+001.44E-01 8.98E-01 7.69E-01 9.16E-01 1.51E+00 1.41E+00 -7.93E-01 2.89E+00 -1.84E-01
3.12E-01 -9.98E-01 7.59E-03 1.51E+00 -8.07E-01 -5.86E-01 1.99E-01 -2.94E-01 3.90E-012.12E+00 5.62E-01 3.08E+00 1.41E+00 -5.86E-01 5.61E+00 -1.95E-01 3.22E+00 -1.34E+006.64E-02 1.78E+00 -1.81E+00 -7.93E-01 1.99E-01 -1.95E-01 4.74E+00 -3.90E+00 3.15E+00
[K]12X12 =
[K]9X9 =
[K]-19X9 =
1.32E+00 -1.36E+00 3.29E+00 2.89E+00 -2.94E-01 3.22E+00 -3.90E+00 8.69E+00 -2.83E+00-3.36E-01 1.81E+00 -1.80E+00 -1.84E-01 3.90E-01 -1.34E+00 3.15E+00 -2.83E+00 4.34E+00
6. Menghitung Matriks Beban dan Matriks Perpindahan
Matriks Beban { R } Matriks {D} Global
0 -8.42E+01 0.00E+000 -5.19E+03 0.00E+000 7.19E+03 -8.42E+010 -1.06E+03 -5.19E+03
-2000 4.36E+02 7.19E+030 4.25E+03 0.00E+00
-2000 -1.62E+04 -1.06E+030 1.40E+04 4.36E+02
-2000 -1.58E+04 4.25E+03-1.62E+041.40E+04-1.58E+04
7. Menghitung Matriks Gaya Dalam
Matriks Gaya Dalam
0.11
3000.00
0.00
0.00
0.00
3000.00
= 0.00
-2000.00
0.00
-2000.00
{D} = [K]-1 {R}
{R}9X1 = {D}9X1 =
[D]12X1 =
F1H
F1V
F2H
F2V
F3H
F3V
[F]12X1 = F4H
F4V
F5H
F5V
0.00
-2000.00
F6H
F6V
KONSTRUKSI RANGKA BATANG
Tujuan Pembelajaran Khusus
Mahasiswa mampu menyelesaikan struktur Konstruksi Rangka Batang (KRB) 2 Dimensi dengan ASMM
Contoh Soal 2 :
Sebuah Konstruksi Rangka Batang Jembatan Baja seperti pada gambar di bawah ini,
hitunglah gaya-gaya dalam yang bekerja dengan cara ASMM
Diketahui :
4 m
H = 4 m
3 1
A = 1
P = 5 ton
4 4 5 6 7
5 5 5
1 2
4 4
Degree of Freedom
L =
E = kg/m2
m2
1 3
54
2
7
8
9
10
Keterangan:
ISIKAN angka 0 atau 1: 1 angka 1 bila melihat jawaban
angka 0 bila jawaban dihilangkan
sementara
1. Menentukan koefisien transformasi matriks kekakuan elemen
ELEMEN V (m) H (m) L (m) C S C.S C.S/L1 0.00 4.00 4.000 1.000 0.000 1.000 0.000 0.000 0.250 0.000 0.0002 0.00 4.00 4.000 1.000 0.000 1.000 0.000 0.000 0.250 0.000 0.0003 0.00 4.00 4.000 1.000 0.000 1.000 0.000 0.000 0.250 0.000 0.0004 4.00 2.00 4.472 0.447 0.894 0.200 0.800 0.400 0.045 0.179 0.0895 -4.00 2.00 4.472 0.447 -0.894 0.200 0.800 -0.400 0.045 0.179 -0.0896 4.00 2.00 4.472 0.447 0.894 0.200 0.800 0.400 0.045 0.179 0.0897 -4.00 2.00 4.472 0.447 -0.894 0.200 0.800 -0.400 0.045 0.179 -0.089
2. Menyusun matriks kekakuan elemen
ELEMEN 1 = ELEMEN 20.250 0.000 -0.250 0.000
= A E 0.000 0.000 0.000 0.000-0.250 0.000 0.250 0.000
C2 SS C2/L S2/L
[k](1) = [k](2) = [k](3)
1
2
3
4
5
6
0.000 0.000 0.000 0.000
ELEMEN 4 = ELEMEN 6 0.045 0.089 -0.045 -0.089
= = A E 0.089 0.179 -0.089 -0.179-0.045 -0.089 0.045 0.089-0.089 -0.179 0.089 0.179
ELEMEN 5 = ELEMEN 70.045 -0.089 -0.045 0.089
= = A E -0.089 0.179 0.089 -0.179-0.045 0.089 0.045 -0.0890.089 -0.179 -0.089 0.179
3. Menyusun matriks kekakuan global struktur
0.295 0.089 -0.250 0.000 0.000 0.000 -0.045 -0.089 0.000 0.0000.089 0.179 0.000 0.000 0.000 0.000 -0.089 -0.179 0.000 0.000-0.250 0.000 0.589 0.000 -0.250 0.000 -0.045 0.089 -0.045 -0.0890.000 0.000 0.000 0.358 0.000 0.000 0.089 -0.179 -0.089 -0.179
A.E 0.000 0.000 -0.250 0.000 0.295 -0.089 0.000 0.000 -0.045 0.0890.000 0.000 0.000 0.000 -0.089 0.179 0.000 0.000 0.089 -0.179-0.045 -0.089 -0.045 0.089 0.000 0.000 0.339 0.179 0.000 0.000-0.089 -0.179 0.089 -0.179 0.000 0.000 0.179 0.358 0.000 0.0000.000 0.000 -0.045 -0.089 -0.045 0.089 0.000 0.000 0.339 0.0000.000 0.000 -0.089 -0.179 0.089 -0.179 0.000 0.000 0.000 0.358
2.95E-01 8.94E-02 -2.50E-01 0.00E+00 0.00E+00 0.00E+00 -4.47E-02 -8.94E-02 0.00E+00 0.00E+008.94E-02 1.79E-01 0.00E+00 0.00E+00 0.00E+00 0.00E+00 -8.94E-02 -1.79E-01 0.00E+00 0.00E+00-2.50E-01 0.00E+00 5.89E-01 0.00E+00 -2.50E-01 0.00E+00 -4.47E-02 8.94E-02 -4.47E-02 -8.94E-020.00E+00 0.00E+00 0.00E+00 3.58E-01 0.00E+00 0.00E+00 8.94E-02 -1.79E-01 -8.94E-02 -1.79E-01
0.00E+00 0.00E+00 -2.50E-01 0.00E+00 2.95E-01 -8.94E-02 0.00E+00 0.00E+00 -4.47E-02 8.94E-020.00E+00 0.00E+00 0.00E+00 0.00E+00 -8.94E-02 1.79E-01 0.00E+00 0.00E+00 8.94E-02 -1.79E-01-4.47E-02 -8.94E-02 -4.47E-02 8.94E-02 0.00E+00 0.00E+00 3.39E-01 1.79E-01 0.00E+00 0.00E+00-8.94E-02 -1.79E-01 8.94E-02 -1.79E-01 0.00E+00 0.00E+00 1.79E-01 3.58E-01 0.00E+00 0.00E+000.00E+00 0.00E+00 -4.47E-02 -8.94E-02 -4.47E-02 8.94E-02 0.00E+00 0.00E+00 3.39E-01 0.00E+00
[k](4) [k](6)
[k](5) [k](7)
[K]10X10 =
[K]10X10 =
0.00E+00 0.00E+00 -8.94E-02 -1.79E-01 8.94E-02 -1.79E-01 0.00E+00 0.00E+00 0.00E+00 3.58E-01
4. Menghitung Reduksi Matriks Kekakuan Global berdasarkan jumlah Degree of Freedom
5.89E-01 0.00E+00 -2.50E-01 -4.47E-02 8.94E-02 -4.47E-02 -8.94E-020.00E+00 3.58E-01 0.00E+00 8.94E-02 -1.79E-01 -8.94E-02 -1.79E-01-2.50E-01 0.00E+00 2.95E-01 0.00E+00 0.00E+00 -4.47E-02 8.94E-02
-4.47E-02 8.94E-02 0.00E+00 3.39E-01 1.79E-01 0.00E+00 0.00E+008.94E-02 -1.79E-01 0.00E+00 1.79E-01 3.58E-01 0.00E+00 0.00E+00-4.47E-02 -8.94E-02 -4.47E-02 0.00E+00 0.00E+00 3.39E-01 0.00E+00-8.94E-02 -1.79E-01 8.94E-02 0.00E+00 0.00E+00 0.00E+00 3.58E-01
5. Menghitung invers Matriks Kekakuan Global Tereduksi
5.17E+01 -1.80E+02 6.76E+01 1.47E+02 -1.77E+02 -3.18E+01 -9.41E+01-1.80E+02 6.72E+02 -2.41E+02 -5.45E+02 6.53E+02 1.21E+02 3.51E+026.76E+01 -2.41E+02 9.28E+01 1.97E+02 -2.36E+02 -4.24E+01 -1.27E+02
1.47E+02 -5.45E+02 1.97E+02 4.47E+02 -5.33E+02 -9.83E+01 -2.85E+02-1.77E+02 6.53E+02 -2.36E+02 -5.33E+02 6.40E+02 1.18E+02 3.41E+02-3.18E+01 1.21E+02 -4.24E+01 -9.83E+01 1.18E+02 2.52E+01 6.34E+01-9.41E+01 3.51E+02 -1.27E+02 -2.85E+02 3.41E+02 6.34E+01 1.86E+02
6. Menghitung Matriks Beban dan Matriks Perpindahan
Matriks Beban { R } Matriks {D} Global
0 9.01E+05 0.00E+00
-5000 -3.36E+06 0.00E+00
0 1.20E+06 9.01E+05
0 2.73E+06 -3.36E+06
0 -3.27E+06 1.20E+06
0 -6.07E+05 0.00E+00
0 -1.76E+06 2.73E+06
[K]7X7 =
[K]-17X7 =
{D} = [K]-1 {R}
{R}7X1 = {D}7X1 =
[D]10X1 =
-3.27E+06
-6.07E+05
-1.76E+06
7. Menghitung Matriks Gaya Dalam
Matriks Gaya Dalam
0.55
2500.00
0.00
-5000.00
0.00
2500.00
= 0.00
0.00
0.00
0.00
F1H
F1V
F2H
F2V
F3H
F3V
[F]12X1 = F4H
F4V
F5H
F5V
1 2 3 5 6 7 8 9
1 11(1)+11(4) 12(1)+12(4) 13(1) 13(4) 14(4)2 21(1)+21(4) 22(1)+22(4) 23(1) 23(4) 24(4)
3 31(1) 32(1) 33(1)+11(2)+11(5)+11(6) 13(2) 14(2) 13(5) 14(5) 13(6)4 41(1) 42(1) 43(1)+21(2)+21(5)+21(6) 23(2) 24(2) 23(5) 24(5) 23(6)5 31(2) 33(2)+11(7) 34(2)+12(7) 13(7)6 41(2) 43(2)+21(7) 44(2)+22(7) 23(7)
7 31(4) 32(4) 31(5) 33(4)+11(3)+33(5) 34(4)+12(3)+34(5) 13(3)8 41(4) 42(4) 41(5) 43(4)+21(3)+43(5) 44(4)+22(3)+44(5) 23(3)9 31(6) 31(7) 32(7) 31(3) 32(3) 33(3)+33(6)+33(7)
10 41(6) 41(7) 42(7) 41(3) 42(3) 43(3)+43(6)+43(7)
10
14(6)24(6)14(7)24(7)
14(3)24(3)34(3)+34(6)+34(7)44(3)+44(6)+44(7)