tugas excel analisis struktur

24
1 0.60 -1.00 0.00 0.00 0.00 0.00 0.00 -0.60 2 0.80 0.00 0.00 0.00 0.00 0.00 1.00 0.80 3 0.00 1.00 -0.60 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 5 0.00 0.00 1.00 -1.00 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 7 0.00 0.00 0.00 0.00 1.00 -1.00 0.00 0.60 8 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.80 9 0.00 0.00 0.60 0.00 0.00 1.00 0.00 0.00 1.67 0.00 1.67 1.25 0.00 0.00 0.00 -1.25 1.00 -0.75 2.00 1.50 0.00 -0.75 0.00 -1.50 1.67 -1.25 1.67 2.50 0.00 -1.25 0.00 -2.50 1.67 -1.25 1.67 2.50 -1.00 -1.25 0.00 -2.50 0.00 0.00 0.00 -0.75 0.00 0.00 1.00 0.75 -1.00 0.75 -1.00 -1.50 0.00 0.75 0.00 1.50 0.00 0.00 0.00 0.00 0.00 -1.00 0.00 0.00 -1.67 1.25 -1.67 -1.25 0.00 1.25 0.00 1.25 -1.33 1.00 -1.33 -1.00 0.00 1.00 0.00 2.00 PEMBUKTIAN INVERS 1.00 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 1.00 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 1.00 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 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 matriks identitas

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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

[U]0

00

-0.0326690.017353

0.00458

[U]-0.0326690.017353

0.00458000

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)] =

482747.4 362060.6 120686.9 -482747271545.4 90515.14 -362061

30171.71 -120687697301.8

0.275862 0.551724-0.153846 0.6153850.380952 0.761905

-0.222222 0.888889

vx2