portal 2d buku matlab hal.71.xlsx
TRANSCRIPT
PENYELESAIAN UAS "MEKANIKA REKAYASA V" (reguler)Semester Gasal 2005/2006 - Tgl. 21 Januari 2006
STEP 1 : Discretize and Global Degrees of Freedom (DOF)
Defined DOF
after boundary conditions
PENYELESAIAN :
Properties penampangBahan / material :
E = 2.10E+06 kg/cm2
Batang 1 :
120 cm2E = 2.10E+06 kg/cm2
1.07E+05 cm4
400 cm
0 degrees
A1 =
I1 =
L1 =
theta1 =
Rekap Elemen dalam Matriks Kekakuan Batang
Batang EA / L 4.EI / L 2.EI / L 6.EI / L^2 12.EI / L^3 Sudut (deg)1 6.29E+05 2.24E+09 1.12E+09 8.38E+06 4.19E+04 02 4.45E+05 1.58E+09 7.90E+08 4.19E+06 1.48E+04 3153
Data beban
P1 = 500 kgP2 = 0q = 100 kg/cmM = 0
Gaya dan Momen Ujung pada Element #1:Ha = 0 kg Hb = 0 kgVa = 20000 kg Vb = 20000 kgMa = 1333333 kg.cm Mb = -1333333.333 kg.cm
Gaya dan Momen Ujung pada Element #2:Ha = 0 T Hb = 0 TVa = 0 T Vb = 0 TMa = 0 T.m Mb = 0 T.m
Gaya dan Momen Ujung pada Element #3:Ha = 0 T Hb = 0 TVa = 0 T Vb = 0 TMa = 0 T.m Mb = 0 T.m
Rekap Gaya Ujung Batang (Sumbu Lokal)
No. BtgUjung kiri (i) Ujung kanan (j)
Beban FX(i) Beban FY(i) Beban MZ(i) Beban FX(j) Beban FY(j) Beban MZ(j)1 0 20000 1333333 0 20000 -13333332 0 0 0 0 0 03 0 0 0 0 0 0
Beban atau Momen pada Titik Joint Struktur dalam Arah Sumbu Global/Sumbu StrukturNo. Titik : 1 2
Dis. Sb Global : 1 2 3 4 5 6Beban/Momen : 0 0 0 500 0 0
Tentukan parameter yang dibutuhkan :
a. Sumbu global struktur pada tiap elementb. DOF sesuai sumbu global strukturc. Kesesuaian sb lokal element dengan DOF arah global
d.
a. Displacement titik sumbu global struktur pada tiap element
Matriks Transformasi [R] dan [RT] tiap element
Element i j1 1 2 3 4 52 4 5 6 7 83
b. DOF sesuai sumbu global struktur
Global axis 1 2 3 4 5DOF 1 2 3 d1 d2
c.
Local Axis 1 2 3 4Element ke 1 1 2 3 d1Element ke 2 d1.CxCy+d2CxCyd1.CxCy+d2CxCy d3 d1.CxCy+d2CxCyElement ke 3
Element #1Local axis 1 2 3 4 5
Global axis 1 2 3 4 5DOF 0 0 0 d1 d2
Element #2Local axis 1 2 3 4 5
Global axis 4.Cx 5Cy 5 6 7 8DOF d1Cx 0 d2 0 0
d.
Element 1:
R = 0 0 0
0 0 00 0 1 0 0
0 0 0
0 0 00 0 0 0 0
Element 2:
R = 0 0 0
0 0 00 0 1 0 0
0 0 0
0 0 00 0 0 0 0
Kesesuaian SB LOKAL element dengan SB GLOBAL dan DOF arah global
Matriks Transformasi [R] dan [RT] tiap element
cos 1
sin 1
- sin 1
cos 1
cos 1
sin 1
- sin 1
cos 1
cos 1
sin 1
- sin 1
cos 1
cos 1
sin 1
- sin 1
cos 1
Element 3:
R = 0 0 0
0 0 00 0 1 0 0
0 0 0
0 0 00 0 0 0 0
===> DALAM SUMBU LOKALBatang = 1
Elemen dalam matriks kekakuan batang :EA / L 4.EI / L 2.EI / L 6.EI / L^2 12.EI / L^3
6.29E+05 2.24E+09 1.12E+09 8.38E+06 4.19E+04
Cx = cos theta = 1Cy = sin theta = 0
Local axis 1 2 3 4 51 628800 0 0 -628800 02 0 41921.31 8384262 0 -41921.31
3 0 8384262 2.2E+009 0 -83842624 -628800 0 0 628800 05 0 -41921.31 -8384262 0 41921.316 0 8384262 1.1E+009 0 -8384262
===> DALAM SUMBU LOKALBatang = 2
Elemen dalam matriks kekakuan batang :EA / L 4.EI / L 2.EI / L 6.EI / L^2 12.EI / L^3
4.45E+05 1.58E+09 7.90E+08 4.19E+06 1.48E+04
Cx = cos theta = 0.707107Cy = sin theta = -0.707107
Local axis 1 2 3 4 51 444628.7 0 0 -444628.744 02 0 14821.42 4192131 0 -14821.42
3 0 4192131 1.6E+009 0 -41921314 -444629 0 0 444628.74401 05 0 -14821.42 -4192131 0 14821.426 0 4192131 7.9E+008 0 -4192131
Urutkan sesuai sumbu global/struktur dengan DOF yang didahulukan
cos 1
sin 1
- sin 1
cos 1
cos 1
sin 1
- sin 1
cos 1
Hitung matriks kekakuan batang 1 [SM1]
[SM1] =
Hitung matriks kekakuan batang 2 [SM2]
[SM2] =
MATRIKS KEKAKUAN STRUKTUR [SMS]
DOF 1 2 3 d1 d2Global axis 1 2 3 4 5
d1 d2 d3 d4 d54 5 6 7 9
4 858525.1 -214904 2964284 -229725.0826 29642845 -214904 271646.4 -5419978 214903.66136 29642846 2964284 -5419978 3.8E+009 -2964284.258 7.9E+0087 -229725 214903.7 -2964284 229725.08265 -2964284
9 2964284 2964284 7.9E+008 -2964284.258 1.6E+0091 -628800 0 0 0 02 0 -41921.31 8384262 0 03 0 -8384262 1.1E+009 0 08 214903.7 -229725 -2964284 -214903.6614 -2964284
0 0 0 0 00 0 0 0 00 0 0 0 0
Hitung Matriks Beban Joint/ NodalNodal forces:
a. - Akibat beban di sepanjang bentang (kedua ujung dikekang)
Element: 1Local axis
0 120000 2
= 1333333 30 4
20000 5-1333333 6
Element: 2Local axis
0 10 2
= 0 30 40 50 6
[SMS] =
[AML]1
[AML]2
Susun matriks beban ekivalen
Joint Sb Global Element 1 Element 21 1 0 01 2 20000 01 3 1333333 02 4 0 0
-Ae = 2 5 20000 + 02 6 -1333333 03 7 0 03 8 0 03 9 0 0
b. - Akibat Beban di joint (kedua ujung dikekang)
Joint Sb Global DOF1 1 0 11 2 0 21 3 0 32 4 500 d1
Aj = 2 5 0 d22 6 0 d33 7 0 d43 8 0 83 9 0 d5
c. - Beban kombinasi atau gabungan (Ae + Aj)
Joint Sb Global DOF1 1 1 01 2 2 -200001 3 3 -1333333.3332 4 d1 0
Ac = 2 5 d2 -20000 +2 6 d3 1333333.33333 7 d4 03 8 8 03 9 d5 0
Displacement / Deformasi Struktur
d1 = 0.0008
d2 = -0.4404d3 = -0.0003d4 = 0.4318d5 = 0.0018
Reactions:
DF = SFF-1.AFC
DF =
GAYA-GAYA DALAM UJUNG ELEMEN
Batang: 1
Deformasi struktur dalam arah SUMBU GLOBAL di batang 1, sbb :Tentukan no. Sb global dahulu
1 1 02 2 0
3 3 04 d1 0.0007955 d2 -0.4403946 d3 -0.000312
Gaya ujung batang
0 628800 0 020000 0 41921.31 8384262
1333333 + 0 8384262 2.2E+0090 -628800 0 0
20000 0 -41921.31 -8384262-1333333 0 8384262 1.1E+009
Batang: 2
Deformasi struktur dalam arah SUMBU GLOBAL di batang 2, sbb :Tentukan no. Sb global dahulu
4 d1 0.0007955 d2 -0.440394
6 d3 -0.0003127 d4 0.4318478 8 09 d5 0.00179
Gaya ujung batang
0 444628.7 0 00 0 14821.421289 4192131
0 + 0 4192131 1.6E+0090 -444629 0 00 0 -14821.42129 -41921310 0 4192131 7.9E+008
SELESAI
[DMS]1 =
AM 1 = AML 1 + SM 1 . M 1
[AM1] =
[DMS]2 =
AM 2 = AML 2 + SM 2 . M 2
[AM2] =
Defined DOF
after boundary conditions
45
Batang 2 : TAN 1
120 m2 SIN 0.707107E = 2.10E+06 t/m2 COS 0.707107
1.07E+05 m4
565.6854 cm
315 degrees
A2 =
I2 =
L2 =
theta2 =
versi buku matlab
Cx Cy1 0
0.707107 -0.707107
Beban atau Momen pada Titik Joint Struktur dalam Arah Sumbu Global/Sumbu Struktur3
7 8 90 0 0
6 ==> Global axis9 ==> Global axis
==> Global axis
6 7 8 9d3 d4 8 d5
5 6d2 d3 <=== DOF sesuai sb global
d1.CxCy+d2CxCy 9 <=== DOF sesuai sb global <=== DOF sesuai sb global
66d3
6 Uraian sb global :9 Uraian sb 4 = 1.Cx + 2.Cy0 Uraian sb 5 = 1.Cy + 2.Cx
Uraian sb 7 = 4.Cx + 5.CyUraian sb 8 = 4.Cy + 5.Cx
0 [R] = 1 0 0 0 0 0
0 0 1 0 0 0 00 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 1 01 0 0 0 0 0 1
0 [R] = 0.707107 -0.707107 0 0 0 0
0 0.707107 0.707107 0 0 0 00 0 0 1 0 0 0
0 0 0 0 0.707107 -0.707107 0
0 0 0 0 0.707107 0.707107 01 0 0 0 0 0 1
SB GLOBAL dan DOF arah global
0 [R] = 0 0 0 0 0 0
0 0 0 0 0 0 00 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 01 0 0 0 0 0 1
1 2 3 d16 Global axis 1 2 3 4
0 1 628800 0 0 -6288008384262 2 0 41921.31 8384262 0
1.1E+009 3 0 8384262 2.2E+009 00 4 -628800 0 0 628800
-8384262 5 0 -41921.31 -8384262 02.2E+009 6 0 8384262 1.1E+009 0
d1 d2 d3 d4
6 Global axis 4 5 6 70 4 229725.1 -214904 2964284 -229725
4192131 5 -214904 229725.1 2964284 214903.7
7.9E+008 6 2964284 2964284 1.6E+009 -29642840 7 -229725 214903.7 -2964284 229725.1
-4192131 8 214903.7 -229725 -2964284 -2149041.6E+009 9 2964284 2964284 7.9E+008 -2964284
[SMS1
] =
[SMS2
] =
d3 d4 8 d5 0 0 06 7 8 9 0 0 0
1 2 3 8 <=== Ditata ulang manual1 2 3 8
-628800 0 0 214903.7 4 d10 -41921.31 -8384262 -229725 5 d20 8384262 1.1E+009 -2964284 6 d30 0 0 -214904 7 d4
0 0 0 -2964284 9 d5628800 0 0 0 1 1
0 41921.31 8384262 0 2 20 8384262 2.2E+009 0 3 30 0 0 229725.1 8 80 0 0 0 00 0 0 0 00 0 0 0 0
Transformasi sb lokal ke global ==>
1 0 0 0 0 00 1 0 0 0 0
= 0 0 1 0 0 0 x0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1
Transformasi sb lokal ke global ==>
0.707107 0.707107 0 0 0 0-0.707107 0.707107 0 0 0 0
= 0 0 1 0 0 0 x0 0 0 0.707107 0.707107 00 0 0 -0.707107 0.707107 00 0 0 0 0 1
AMS 1 = RT . AML 1
[AMS]1
AMS 2 = RT . AML 2
[AMS]2
0 0 1 120000 -20000 2 2
1333333 -1333333 3 30 0 4 d1
+ = 20000 Ae = -20000 5 d2-1333333 1333333 6 d3
0 0 7 d40 0 8 80 0 9 d5
Tata ulang Ac :0 0 d1 5000 -20000 d2 -200000 -1333333 d3 1333333
500 500 d4 00 = -20000 Ac = d5 00 1333333 1 00 0 2 -200000 0 3 -13333330 0 8 0
Deformasi struktur dalam arah SUMBU LOKAL di batang 1, sbb :
1 0 0 0 0 00 1 0 0 0 0
0 0 1 0 0 0 x0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1
-628800 0 0 0 0 -5000 -41921.31 8384262 0 20000 15846.19
0 -8384262 1.1E+009 x 0 = 1333333 + 3343619628800 0 0 0.000795 0 500
0 41921.31 -8384262 -0.440394 20000 -15846.190 -8384262 2.2E+009 -0.000312 -1333333 2994857
Deformasi struktur dalam arah SUMBU LOKAL di batang 2, sbb :
0.707107 -0.707107 0 0 0 00.707107 0.707107 0 0 0 0
0 0 1 0 0 0 x0 0 0 0.707107 -0.707107 00 0 0 0.707107 0.707107 00 0 0 0 0 1
-444629 0 0 0.311968 0 2937.1870 -14821.42 4192131 -0.310843 0 -2937.187
0 -4192131 7.9E+008 x -0.000312 = 0 + -1661524444628.7 0 0 0.305362 0 -2937.187
0 14821.42 -4192131 0.305362 0 2937.1870 -4192131 1.6E+009 0.00179 0 -5E-010
M 1 = R . DMS 1
M 1 =
M 2 = R . DMS 2
M 2 =
ATAN 45
ASIN 45ACOS 45
1 0 0 0 0 0
0 1 0 0 0 00 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 00 0 0 0 0 1
0.707107 0.707107 0 0 0 0
-0.707107 0.707107 0 0 0 00 0 1 0 0 0
0 0 0 0.707107 0.707107 0
0 0 0 -0.707107 0.707107 00 0 0 0 0 1
[R]T =
[R]T =
0 0 0 0 0 0
0 0 0 0 0 00 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 0 00 0 0 0 0 1
d2 d35 6
0 0-41921.31 8384262
-8384262 1.12E+0090 0
41921.31 -8384262-8384262 2.24E+009
8 d5
8 9214903.7 2964284.3 ### ### ### ### ### ###-229725 2964284.3 ### ### ### ### ### ###
-2964284 790475802 ### ### ### ### ### ###-214904 -2964284 ### ### ### ### ### ###
229725.1 -2964284 ### ### ### ### ### ###-2964284 1.58E+009 ### ### ### ### ### ###
[R]T =
Cek: OK<=== Ditata ulang manual
### ### ###### ### ###### ### ###
Global axis0 1 0
20000 2 20000
1333333 = 3 13333330 4 0
20000 5 20000-1333333 6 -1333333
Global axis0 4 00 5 0
0 = 6 00 7 00 8 00 9 0
0 500.00 0 ###0 ###
Cek: OK0 0.00
-0.00021 (0.44)0.000589 (0.00)
sb lokal DOF0 0 1 1 - 0 0 2 2 -
0 = 0 3 3 - 0.000795 0.000795 4 d1 - -0.440394 -0.440394 5 d2 - -0.000312 -0.000312 6 d3 -
sb lokal DOF / sb global Cek :-500 1 1 -0.0005 -
35846.19 2 2 0.035846 -
= 4676951.9 3 3 4.676952 - 500 4 d1 0.0005 -
4153.8101 5 d2 0.004154 - 1661524 6 d3 1.661524 -
sb lokal DOF0.000795 0.311968 1d1.CxCy+d2CxCy (0.75)-0.440394 -0.310843 2d1.CxCy+d2CxCy 0.31
-0.000312 = -0.000312 3 d3 - 0.431847 0.305362 4d1.CxCy+d2CxCy 0.31
0 0.305362 5d1.CxCy+d2CxCy 0.74 0.00179 0.00179 6 9 -
sb lokal DOF / sb global Cek :2937.1873 1d1.CxCy+d2CxCy 0.002937 - -2937.187 2d1.CxCy+d2CxCy -0.002937 -
= -1661524 3 d3 -1.661524 - -2937.187 4d1.CxCy+d2CxCy -0.002937 - 2937.1873 5d1.CxCy+d2CxCy 0.002937 - -4.66E-010 6 9 -5E-016 -
1 0 0 0 0 0 628800 00 1 0 0 0 0 0 41921.31
0 0 1 0 0 0 0 83842620 0 0 1 0 0 -628800 00 0 0 0 1 0 0 -41921.310 0 0 0 0 1 0 8384262
0.707107 0.707107 0 0 0 0 444628.7 0
-0.707107 0.707107 0 0 0 0 0 14821.420 0 1 0 0 0 0 41921310 0 0 0.70710678 0.707107 0 -444629 00 0 0 -0.7071068 0.707107 0 0 -14821.420 0 0 0 0 1 0 4192131
[R]T [SM] [R]
[R]T [SM] [R]
OK -0.0005OK 0.0358
OK 4.6770OK 0.0005OK 0.0042OK 1.6615
OK 0.0029OK -0.0029
OK -1.6615OK -0.0029OK 0.0029OK 0.0000
0 -628800 0 0 1 0 0 08384262 0 -41921.31 8384262 0 1 0 0
2.2E+009 0 -8384262 1.1E+009 0 0 1 00 628800 0 0 0 0 0 1
-8384262 0 41921.31 -8384262 0 0 0 01.1E+009 0 -8384262 2.2E+009 0 0 0 0
0 -444629 0 0 1 -1 0 0
4192131 0 -14821.42 4192131 1 1 0 01.6E+009 0 -4192131 7.9E+008 0 0 1 0
0 444628.7 0 0 0 0 0 1-4192131 0 14821.42 -4192131 0 0 0 17.9E+008 0 -4192131 1.6E+009 0 0 0 0
0 0 628800 0 0 -628800 0 00 0 0 41921.31 8384262 0 -41921.31 8384262
0 0 0 8384262 2.2E+009 0 -8384262 1.1E+0090 0 -628800 0 0 628800 0 01 0 0 -41921.31 -8384262 0 41921.31 -83842620 1 0 8384262 1.1E+009 0 -8384262 2.2E+009
0 0 314400 10480.33 2964284 -314400 -10480.33 2964284
0 0 -314400 10480.33 2964284 314400 -10480.33 29642840 0 0 4192131 1.6E+009 0 -4192131 7.9E+008-1 0 -314400 -10480.33 -2964284 314400 10480.33 -29642841 0 314400 -10480.33 -2964284 -314400 10480.33 -29642840 1 0 4192131 7.9E+008 0 -4192131 1.6E+009
1 0 0 0 0 0 6288000 1 0 0 0 0 0
0 0 1 0 0 0 = 00 0 0 1 0 0 -6288000 0 0 0 1 0 00 0 0 0 0 1 0
0.707107 -0.707107 0 0 0 0 229725.1
0.707107 0.707107 0 0 0 0 -2149040 0 1 0 0 0 = 29642840 0 0 0.707107 -0.707107 0 -2297250 0 0 0.707107 0.707107 0 214903.70 0 0 0 0 1 2964284
0 0 -628800 0 0 - - - 41921.31 8384262 0 -41921.31 8384262 - - -
8384262 2.2E+009 0 -8384262 1.1E+009 - - - 0 0 628800 0 0 - - -
-41921.31 -8384262 0 41921.31 -8384262 - - - 8384262 1.1E+009 0 -8384262 2.2E+009 - - -
-214904 2964284 -229725 214903.7 2964284 - - -
229725.1 2964284 214903.7 -229725 2964284 - - - 2964284 1.6E+009 -2964284 -2964284 7.9E+008 - - - 214903.7 -2964284 229725.1 -214904 -2964284 - - - -229725 -2964284 -214904 229725.1 -2964284 - - - 2964284 7.9E+008 -2964284 -2964284 1.6E+009 - - -
- - - - - -
- - - - - - - - - - - -
- - -
- - - - - - - - - - - - - - -
