1 0.60 -1.00 0.00 0.00 0.00 0.00 0.00 -0.60 0.00
2 0.80 0.00 0.00 0.00 0.00 0.00 1.00 0.80 0.00
3 0.00 1.00 -0.60 0.00 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.80 0.00 0.00 0.00 0.00 0.00 1.00
5 0.00 0.00 1.00 -1.00 0.00 0.00 0.00 0.00 0.00
6 0.00 0.00 0.00 0.00 0.00 0.00 -1.00 0.00 0.00
7 0.00 0.00 0.00 0.00 1.00 -1.00 0.00 0.60 0.00
8 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.80 1.00
9 0.00 0.00 0.60 0.00 0.00 1.00 0.00 0.00 0.00
1.67 0.00 1.67 1.25 0.00 0.00 0.00 -1.25 0.001.00 -0.75 2.00 1.50 0.00 -0.75 0.00 -1.50 0.001.67 -1.25 1.67 2.50 0.00 -1.25 0.00 -2.50 0.00
1.67 -1.25 1.67 2.50 -1.00 -1.25 0.00 -2.50 0.00
0.00 0.00 0.00 -0.75 0.00 0.00 1.00 0.75 1.00
-1.00 0.75 -1.00 -1.50 0.00 0.75 0.00 1.50 1.00
0.00 0.00 0.00 0.00 0.00 -1.00 0.00 0.00 0.00
-1.67 1.25 -1.67 -1.25 0.00 1.25 0.00 1.25 0.00
-1.33 1.00 -1.33 -1.00 0.00 1.00 0.00 2.00 0.00
PEMBUKTIAN INVERS
1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00
matriks identitas
[K]= X9
1.000 0.000 -1.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.375 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-1.000 0.000 2.420 -0.080 -1.000 0.000 0.000 0.000 -0.240
0.000 0.000 0.000 1.015 0.000 0.000 0.000 -0.375 0.240
0.000 0.000 -1.000 0.000 1.180 -0.240 0.000 0.000 0.000
0.000 0.000 0.000 0.000 -0.240 0.695 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 2.000 0.000 -1.000
0.000 0.000 0.000 -0.375 0.000 0.000 0.000 0.375 0.000P9 0.000 0.000 -0.180 0.240 0.000 0.000 -1.000 0.000 2.180
P10 0.000 0.000 0.000 0.000 -0.180 0.240 0.000 0.000 -1.000
1 2 3 4 5 6 7 8= 1 3.578 0.000 2.578 -0.097 2.459 0.422 0.559 -0.097
2 0.000 2.667 0.000 0.000 0.000 0.000 0.000 0.0003 2.578 0.000 2.578 -0.097 2.459 0.422 0.559 -0.0974 -0.376 0.000 -0.376 1.786 -0.404 0.100 -0.361 1.7865 2.450 0.000 2.450 -0.138 3.266 0.655 0.592 -0.1386 0.455 0.000 0.455 0.144 0.688 1.838 -0.117 0.1447 0.502 0.000 0.502 -0.298 0.539 -0.133 0.981 -0.2988 -0.376 0.000 -0.376 1.786 -0.404 0.100 -0.361 4.4539 1.003 0.000 1.003 -0.596 1.078 -0.266 0.963 -0.596
10 1.132 0.000 1.132 -0.556 1.272 -0.500 0.930 -0.556BUKTI INVERS
1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.000.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
x1 x2 x3 x4 x5 x6 x7 x8
P1
P2
P3
P4
P5
P6
P7
P8
[K]-1
TUGAS MATRIKS
0 1.00 [B] =
0 0.00 0 0 0 0 0
9 0.00 0 0 0 0 0
0 0.00 0 0 0 0 0
0 0.00 -1 0 1 0 0
0 0.00 0 0 -1 0 1
-18 0.00 0 1 0 0 0
0 0.00 0 0 3/5 4/5 0
0 0.00 0 0 0 1 0
0 0 - 3/5 4/5 0
15 1.67 0 0 0 0 0 18 1.00 e11 0 0 0 0 - 3/515 1.67
15 1.67
-18 0.00 [S]= 1 0 0 0 0
-9 -1.00 0 1 0 0 0
0 0.00 0 0 1 0 0
-15 -1.67 0 0 0 1 0
-12 -1.33 0 0 0 0 1
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
F11 0 0 0 0 0
[C]= 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
-1.000 0.000 1.000 0.000 0.000
0.000 0.000 -1.000 0.000 1.000
0.000 0.375 0.000 0.000 0.000
0.000 0.000 0.300 0.400 0.000
0.000 0.000 0.000 0.375 0.000
0.000 0.000 -0.300 0.400 0.000
0.000 0.000 0.000 0.000 0.000F11 0.000 0.000 0.000 0.000 -0.300
[A]= 0 0 0 -1 0
0 0 0 0 0
x1 x2 x3 x4 x5
e1
e2
e3
e4
e5
e6
e7
e8
e9
e10
e1 e2 e3 e4 e5
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
e1 e2 e3 e4 e5
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F1 F2 F3 F4 F5
P1
P2
0 0 0 1 -1
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
1 -1 0 0 0
0 0 0 0 0P9 0 1 -1 0 0
X10 P10 0 0 1 0 0
0.000
0.000
0.000
0.000
-0.180
0.240
0.000
0.000 {P}= 10 P1-1.000 0 P21.180 0 P3
0 P49 10 0 P5
1.119 1.237 0 P60.000 0.000 0 P71.119 1.237 -15 P8-0.722 -0.694 0 P91.184 1.368 0 P10-0.233 -0.4660.963 0.925-0.722 -0.694 {x}= 37.2371.925 1.850 0.0001.859 2.719 27.237
-30.5560.00 26.5660.00 2.3870.00 9.4910.00 -70.5560.00 18.9820.00 19.6530.000.000.001.00
P3
P4
P5
P6
P7
P8
TUGAS MATRIKS
x9 x10
0 1 0 0 0
0 -1 0 1 0
0 0 0 -1 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 -1 0 0
0 0 0 3/5 0
1 0 0 0 0 4/5 0 0 0 3/5
e110 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
3/8 0 0 0 0 0
0 1/2 0 0 0 0
0 0 3/8 0 0 0
0 0 0 1/2 0 0
0 0 0 0 3/8 0 0 0 0 0 0 0.5
e9 e10
0.000 1.000 0.000 0.000 0.000
0.000 -1.000 0.000 1.000 0.000
0.000 0.000 0.000 -1.000 1.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000
0.000 0.000 -0.375 0.000 0.000
0.000 0.000 0.000 0.300 0.000
0.375 0.000 0.000 0.000 0.0000.400 0.000 0.000 0.000 0.300
F11
0 0 0 0 0 0
1 0 0 0 0 0
x6 x7 x8
e6 e7 e8 e9 e10
e6 e7 e8
F6 F7 F8 F9 F10
0 0.6 0 -0.8 0 0
0 0.8 1 0.8 0 0
0 0 0 0 0 -0.6
0 0 0 0 1 0.8
0 0 0 0 0 0
0 0 -1 0 0 00 0 0 0.6 0 0
0 0 0 0 0 0.6
{F}= 9.4919.4910.671
-10.000-0.6710.000
-4.05115.000
-14.6990.895
Diketahui :E= 2100 t/cm2 Elemen 1 Elemen 2
Elemen 1 Elemen 2 AE/L 280A (cm2) 100 40 12EI/(L)^3 0.933333333I (cm4) 5000 1000 6EI/(L)^2 140L(m) 500 300 4EI/L 28000cos 0.8 0 2EI/L 14000sin 0.6 -1
Elemen 1420 0 0 -420 0 0
0 1.008 252 0 -1.008 2520 252 84000 0 -252 42000
-420 0 0 420 0 00 -1.008 -252 0 1.008 -2520 252 42000 0 -252 84000
[T]= 0.8 0.6 0 0 0 0-0.6 0.8 0 0 0 0
0 0 1 0 0 00 0 0 0.8 0.6 00 0 0 -0.6 0.8 00 0 0 0 0 1
[TT] = 0.8 -0.6 0 0 0 00.6 0.8 0 0 0 0
0 0 1 0 0 00 0 0 0.8 -0.6 00 0 0 0.6 0.8 00 0 0 0 0 1
[TT] [ki]= 336 -0.6048 -151.2 -336 0.6048 -151.2252 0.8064 201.6 -252 -0.8064 201.6
0 252 84000 0 -252 42000-336 0.6048 151.2 336 -0.6048 151.2-252 -0.8064 -201.6 252 0.8064 -201.6
0 252 42000 0 -252 84000
[kg]= 269.2 201.1 -151.2 -269.2 -201.1 -151.2201.1 151.8 201.6 -201.1 -151.8 201.6
-151.2 201.6 84000.0 151.2 -201.6 42000.0-269.2 -201.1 151.2 269.2 201.1 151.2-201.1 -151.8 -201.6 201.1 151.8 -201.6-151.2 201.6 42000.0 151.2 -201.6 84000.0
269.2 201.1 -151.2 -269.2 -201.1 -151.2201.1 151.8 201.6 -201.1 -151.8 201.6
-151.2 201.6 84000.0 151.2 -201.6 42000.0
a. Matriks kekakuan elemen (member stiffness matrix)
[kl]=
-269.2 -201.1 151.2 269.2 201.1 151.2-201.1 -151.8 -201.6 201.1 151.8 -201.6-151.2 201.6 42000.0 151.2 -201.6 84000.0
Elemen 2280 0 0 -280
0 0.9333 140 00.0000 140.0000 28000.0000 0.0000
-280.0000 0.0000 0.0000 280.00000.0000 -0.9333 -140.0000 0.00000.0000 140.0000 14000.0000 0.0000
[T]= 0 -1 0 01 0 0 00 0 1 00 0 0 00 0 0 10 0 0 0
[kg]= 0.93 0.00 140.00 -0.930.00 280.00 0.00 0.00
140.00 0.00 28000.00 -140.00-0.93 0.00 -140.00 0.930.00 -280.00 0.00 0.00
140.00 0.00 14000.00 -140.00
b. Overall stiffness matrix269.2 201.1 -151.2 -269.2201.1 151.8 201.6 -201.1
-151.2 201.6 84000.0 151.2-269.2 -201.1 151.2 269.2-201.1 -151.8 -201.6 201.1-151.2 201.6 42000.0 151.2
0.93333333330
140
[kl]=
-0.93333333330
140
1 2 3 4F1 1 269.2 201.1 -151.2 -269.2G1 2 201.1 151.8 201.6 -201.1M1 3 -151.2 201.6 84000.0 151.2-4 4 -269.2 -201.1 151.2 270.10 5 -201.1 -151.8 -201.6 201.1
500 6 -151.2 201.6 42000.0 291.2F3 7 0.0 0.0 0.0 -0.9G3 8 0.0 0.0 0.0 0.0M3 9 0.0 0.0 0.0 140.0
Rearrangement-4 = 270.096213 201.11616 291.2 -269.162880 201.11616 431.84512 -201.6 -201.11616
500 291.2 -201.6 112000 -151.2F1 -269.16288 -201.11616 -151.2 269.16288G1 -201.11616 -151.84512 201.6 201.11616M1 151.2 -201.6 42000 -151.2F3 -0.9333 0.0 -140.0 -0.9333G3 0.0 -280.0 0.0 140.0M3 140.0 0.0 14000.0 140.0
Perpindahan1 2 3
-4 = 1 270.096213 201.11616 291.20 2 201.11616 431.84512 -201.6
500 3 291.2 -201.6 112000Invers
U2 = 0.00571074 -0.00266875 -1.965168E-05V2 -0.0026687 0.003564751 1.335529E-05
-1.97E-05 1.34E-05 9.00E-06
F1 = -269.2 -201.1 -151.2G1 -201.1 -151.8 201.6M1 151.2 -201.6 42000.0F3 -0.9 0.0 -140.0G3 0.0 -280.0 0.0M3 140.0 0.0 14000.0
e. Member Forces (Gaya-gaya elemen)
[T] 0.8 0.6 0 0 0
Ɵ2
Elemen 1
-0.6 0.8 0 0 00 0 1 0 00 0 0 0.8 0.60 0 0 -0.6 0.80 0 0 0 0
F1 6.604G1 1.121M1 183.941F2 -6.604G2 -1.121M2 376.321
Elemen 2
[T]0 -1 0 0 01 0 0 0 00 0 1 0 00 0 0 0 -10 0 0 1 00 0 0 0 0
F2 -4.859G2 0.611M2 123.679F3 4.859G3 -0.611M3 59.553
0 0-1 140
-140.0000 14000.00000.0000 0.00000.9333 -140.0000
-140.0000 28000.0000
0 00 00 0
-1 00 00 1
0.00 140.00-280.00 0.00
0.00 14000.000.00 -140.00
280.00 0.000.00 28000.00
-201.1 -151.2-151.8 201.6-201.6 42000.0201.1 151.2151.8 -201.6
-201.6 84000.0
0 140 -0.933333 0 140280 0 0 -280 0
0 28000 -140 0 14000
0 -140 0.9333333 0 -140-280 0 0 280 0
0 14000 -140 0 28000
5 6 7 8 9-201.1 -151.2 0.0 0.0 0.0 0-151.8 201.6 0.0 0.0 0.0 0-201.6 42000.0 0.0 0.0 0.0 0201.1 291.2 -0.9 0.0 140.0 U2431.8 -201.6 0.0 -280.0 0.0 V2
-201.6 112000.0 -140.0 0.0 14000.00.0 -140.0 0.9 0.0 -140.0 0
-280.0 0.0 0.0 280.0 0.0 00.0 14000.0 -140.0 0.0 28000.0 0
-201.11616 151.2 -0.933333 0 140 U2-151.84512 -201.6 0 -280 0 V2
201.6 42000 -140 0 14000201.11616 -151.2 0.0 0.0 0.0 0151.84512 201.6 0.0 0.0 0.0 0
201.6 84000 0.0 0.0 0.0 0-280.0 -140.0 0.9333 280.0 -140.0 0-280.0 14000.0 -140.0 280.0 28000.0 0
0.0 14000.0 -140.0 0.0 28000.0 0
U2V2
-4 = -0.0326690 0.0173526
500 0.0045805
-0.0326688 = 4.6110.01735263 4.8590.00458046 183.941
-0.611-4.85959.553
[Kg]0 269.16288 201.11616 -151.2 -269.1629 -201.1162 -151.2
Ɵ2
Ɵ2
Ɵ2
0 201.11616 151.84512 201.6 -201.1162 -151.8451 201.60 -151.2 201.6 84000 151.2 -201.6 420000 -269.1629 -201.1162 151.2 269.1629 201.1162 151.20 -201.1162 -151.8451 -201.6 201.1162 151.8451 -201.61 -151.2 201.6 42000 151.2 -201.6 84000
[Kg]0 0.9333333 0 140 -0.933333 0 1400 0 280 0 0 -280 00 140 0 28000 -140 0 140000 -0.933333 0 -140 0.933333 0 -1400 0 -280 0 0 280 01 140 0 14000 -140 0 28000
Diketahui :30000400002000030000
E 200000 MPaPx 200 kNPy 600 kNPz -800 kN
Penyelesaian :
Koordinat titik nodal (satuan dalam mm)Nodal X Y Z
1 2000 4000 80002 8000 0 03 8000 6000 04 0 0 05 0 6000 0
Panjang batang dan cosinus sudut arahBatang Nodal
L Li j
1 2 1 10770 -0.557086 0.3713907 0.742781 1160000002 3 1 10198 -0.588348 -0.196116 0.784465 1040000003 4 1 9165 0.2182179 0.4364358 0.872872 840000004 5 1 8485 0.2357023 -0.235702 0.942809 72000000
Elemen 1AE/L 557086.015
557086 0 0 -557086 0 0 f20 0 0 0 0 0 g20 0 0 0 0 0 h2
-557086 0 0 557086 0 0 f10 0 0 0 0 0 g10 0 0 0 0 0 h1
0.31034483 -0.20689655 -0.413793103 -0.310345 0.2068966 0.4137930.137931034 0.275862069 0.2068966 -0.137931 -0.275862
0.551724138 0.4137931 -0.275862 -0.5517240.3103448 -0.206897 -0.413793
0.137931 0.2758620.551724
172888.763 -115259.175 -230518.3508 -172888.8 115259.18 230518.4
SISTEM RANGKA BATANG 3-DIMENSI (SPACE TRUSS)
A1 mm2
A2 mm2
A3 mm2
A4 mm2
λx µx vx
[kl(1)] =
[kg(1)] =
76839.45028 153678.9006 115259.18 -76839.45 -153679307357.8011 230518.35 -153678.9 -307358
172888.76 -115259.2 -23051876839.45 153678.9
307357.8
1 0.310345 λx µx -0.206897 λx vx -0.413793 0.1379312 0.346154 0.115385 -0.461538 0.0384623 0.047619 0.095238 0.190476 0.1904764 0.055556 -0.055556 0.222222 0.055556
Elemen 1AE/L 784464.5
784464.5 0 0 -784465 0 00 0 0 0 0 00 0 0 0 0 0
-784465 0 0 784464.5 0 00 0 0 0 0 00 0 0 0 0 0
0.346154 0.115385 -0.461538 -0.346154 -0.115385 -0.4615380.038462 -0.153846 0.115385 0.038462 -0.153846
0.615385 0.461538 0.153846 -0.6153850.346154 0.115385 -0.461538
0.038462 -0.1538460.888889
271545.4 90515.14 -362061 -271545 -90515.14 -36206130171.71 -120687 90515.14 30171.71 -120687
SPACE TRUSS)
λx2 µx2 vX µx
[kl(2)] =
[kg(2)] =