Download - Mekanika Rekayasa 4 ( Metode Matriks )

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Page 1: Mekanika Rekayasa 4 ( Metode Matriks )

E = 200000 MpaA = 11000 cm2

Batang 1L = 5000 mmEA/L = 440000sin α = 0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 440000 0 -440000 0

= 440000 0 0 0 0 = 0 0 0 0-1 0 1 0 -440000 0 440000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 0.8 0 0 0.6 -0.8 0 0

[T] = -0.8 0.6 0 0 0.8 0.6 0 00 0 0.6 0.8 0 0 0.6 -0.80 0 -0.8 0.6 0 0 0.8 0.6

264000 0 -264000 0= 352000 0 -352000 0

-264000 0 264000 0-352000 0 352000 0

u1 v1 u2 v2158400 211200 -158400 -211200 u1

[Kg] = [T] = 211200 281600 -211200 -281600 v1-158400 -211200 158400 211200 u2-211200 -281600 211200 281600 v2

Batang 2L = 500 cm

[KI,1]

[TT] =

[TT].[K1]

[TT] [K1]

4 T

800 cm

300 cm1

3

2

Page 2: Mekanika Rekayasa 4 ( Metode Matriks )

EA/L = 4400000sin α = 0.6cos α = 0.8

Matriks kekakuan1 0 -1 0 440000 0 -440000 0

= 4400000 0 0 0 0 = 0 0 0 0-1 0 1 0 -440000 0 440000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 -0.8 0 0 0.6 0.8 0 0

[T] = 0.8 0.6 0 0 -0.8 0.6 0 00 0 0.6 -0.8 0 0 0.6 0.80 0 0.8 0.6 0 0 -0.8 0.6

264000 0 -264000 0= -352000 0 352000 0

-264000 0 264000 0352000 0 -352000 0

u2 v2 u3 v3158400 -211200 -158400 211200 u2

[Kg] = [T] = -211200 281600 211200 -281600 v2-158400 211200 158400 -211200 u3211200 -281600 -211200 281600 v3

Batang 3L = 800 cmEA/L = 2750000sin α = 0cos α = 1

Matriks kekakuan1 0 -1 0 2750000 0 -3E+06 0

= 2750000 0 0 0 0 = 0 0 0 0-1 0 1 0 -2750000 0 2750000 00 0 0 0 0 0 0 0

Matriks transformasi1 0 0 0 1 0 0 0

[T] = 0 1 0 0 0 1 0 00 0 1 0 0 0 1 00 0 0 1 0 0 0 1

2750000 0 -2750000 0= 0 0 0 0

-2750000 0 2750000 00 0 0 0

u1 v1 u3 v3

[KI,2]

[TT] =

[TT].[K2]

[TT] [K2]

[KI,3]

[TT] =

[TT].[K3]

Page 3: Mekanika Rekayasa 4 ( Metode Matriks )

2750000 0 -2750000 0 u1[Kg] = [T] = 0 0 0 0 v1

-3E+06 0 2750000 0 u30 0 0 0 v3

Overall Stiffnessu1 v1 u2 v2 u3 v3

F1 2908400 2E+05 -158400 -211200 -2750000 0 u1 0G1 211200 3E+05 -211200 -281600 0 0 v1 00 = -158400 -2E+05 316800 0 -158400 211200 u2 U2-4 -211200 -3E+05 0 563200 211200 -281600 v2 V20 -2750000 0 -158400 211200 2908400 -211200 u3 U3

G3 0 0 211200 -281600 -211200 281600 v3 0

Rearrangementu2 v2 u3 u1 v1 v3

0 316800 0 -158400 -158400 -211200 211200 U2-4 0 6E+05 211200 -211200 -281600 -281600 V20 = -158400 2E+05 2908400 -3E+06 0 -211200 U3

F1 -158400 -2E+05 -3E+06 2908400 211200 0 0G1 -211200 -3E+05 0 211200 281600 0 0G3 211200 -3E+05 -211200 0 0 281600 0

Unknown Displacement and Reaction

0 316800 0 -158400 U2-4 = 0 6E+05 211200 V20 -158400 2E+05 2908400 U3

U2 3.2475E-06 -7E-08 1.82E-07 0V2 = -6.818E-08 2E-06 -1.4E-07 -4U3 1.8182E-07 -1E-07 3.64E-07 0

U2 2.7273E-07V2 = -7.307E-06U3 5.4545E-07

F1 -158400 -2E+05 -3E+06 2.727E-07 F1G1 = -211200 -3E+05 0 -7.307E-06 G1 =G3 211200 -3E+05 -211200 5.455E-07 G3

[TT] [K3]

Page 4: Mekanika Rekayasa 4 ( Metode Matriks )

Gaya BatangBatang 1

f1g1 =f2g2

f1 0.6 0.8 0 0 158400 211200 -2E+05 -2E+05g1 = -0.8 0.6 0 0 211200 281600 -2E+05 -3E+05f2 0 0 0.6 0.8 -158400 -211200 2E+05 2E+05g2 0 0 -0.8 0.6 -211200 -281600 2E+05 3E+05

264000 4E+05 -264000 -352000 00 0 0 0 0

-264000 -4E+05 264000 352000 2.727E-070 0 0 0 -7.307E-06

f1 2.5g1 = 0f2 -2.5g2 0

Batang 2

f2g2 =f3g3

f2 0.6 -0.8 0 0 158400 -211200 -2E+05 2E+05g2 = 0.8 0.6 0 0 -211200 281600 2E+05 -3E+05f3 0 0 0.6 -0.8 -158400 211200 2E+05 -2E+05g3 0 0 0.8 0.6 211200 -281600 -2E+05 3E+05

264000 -4E+05 -264000 352000 2.727E-070 0 0 0 -7.307E-06

-264000 4E+05 264000 -352000 5.455E-070 0 0 0 0

f2 2.5g2 = 0

[T1] [Kg1] [U1]

[T2] [Kg2] [U2]

Page 5: Mekanika Rekayasa 4 ( Metode Matriks )

f3 -2.5g3 0

Batang 3

f1g1 =f3g3

f1 1 0 0 0 2750000 0 -3E+06 0g1 = 0 1 0 0 0 0 0 0f3 0 0 1 0 -2750000 0 3E+06 0g3 0 0 0 1 0 0 0 0

2750000 0 -3E+06 0 00 0 0 0 0

-2750000 0 2750000 0 5.455E-070 0 0 0 0

f1 -1.5g1 = 0f3 1.5g3 0

[T3] [Kg3] [U3]

2 = -3.333 T

2 T 2 T

1 = -3.333 T

3 = 2.667 T

Page 6: Mekanika Rekayasa 4 ( Metode Matriks )

luas 1 6000luas 2 5000

A 11000

1 0 0 0 1 0 0 0[T] 0 1 0 0 0 1 0 0

0 0 1 0 0 0 1 00 0 0 1 0 0 0 1

440000 0 -440000 0[Kg] 0 0 0 0

-440000 0 440000 00 0 0 0

[TT] =

Page 7: Mekanika Rekayasa 4 ( Metode Matriks )

022

Page 8: Mekanika Rekayasa 4 ( Metode Matriks )

00

3E-07 u2-7E-06 v2

3E-07-7E-065E-07

0

Page 9: Mekanika Rekayasa 4 ( Metode Matriks )

00

5E-070

Page 10: Mekanika Rekayasa 4 ( Metode Matriks )

E = 200000 MpaA = 11000 cm2

Batang 1L = 3000 mmEA/L = 733333.3sin α = 0cos α = 1

Matriks kekakuan

1 0 -1 0 733333.3 0= 733333.3 0 0 0 0 0 0

-1 0 1 0 -733333.3 00 0 0 0 0 0

1 0 0 0 1 0[T] = 0 1 0 0 0 1

0 0 1 0 0 00 0 0 1 0 0

733333.3 0 -733333.3 0 733333.3 00 0 0 0 [Kg] 0 0

-733333.3 0 733333.3 0 -733333.3 00 0 0 0 0 0

Batang 2L = 5000EA/L = 440000sin α = 0cos α = 1

matriks kekakuan

1 0 -1 0 440000 0= 440000 0 0 0 0 = 0 0

-1 0 1 0 -440000 0

[KI,1]

[TT] =

[TT].[K1]

[KI,2]

Page 11: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 0 0

1 0 0 0 1 0[T] = 0 1 0 0 0 1

0 0 1 0 0 00 0 0 1 0 0

440000 0 -440000 0 440000 0= 0 0 0 0 [Kg] 0 0

-440000 0 440000 0 -440000 00 0 0 0 0 0

Batang 3L = 3000EA/L = 733333.3sin α = 0cos α = 1

matriks kekakuan

1 0 -1 0 733333.3 0= 733333.3 0 0 0 0 = 0 0

-1 0 1 0 -733333 00 0 0 0 0 0

1 0 0 0 1 0[T] = 0 1 0 0 0 1

0 0 1 0 0 00 0 0 1 0 0

733333.3 0 -733333 0 733333.3 0= 0 0 0 0 [Kg] 0 0

-733333 0 733333.3 0 -733333 0

[TT] =

[TT].[K2]

[KI,3]

[TT] =

[TT].[K3]

Page 12: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 0 0

Batang 4L = 5000EA/L = 440000sin α = 0.8cos α = 0.6

matriks kekakuan

1 0 -1 0 440000 0= 440000 0 0 0 0 = 0 0

-1 0 1 0 -440000 00 0 0 0 0 0

0.6 0.8 0 0 0.6 -0.8[T] = -0.8 0.6 0 0 0.8 0.6

0 0 0.6 0.8 0 00 0 -0.8 0.6 0 0

264000 0 -264000 0 158400 211200= 352000 0 -352000 0 [Kg] 211200 281600

-264000 0 264000 0 -158400 -211200-352000 0 352000 0 -211200 -281600

Batang 5L = 4000EA/L = 550000sin α = 1cos α = 0

matriks kekakuan

1 0 -1 0 550000 0= 550000 0 0 0 0 = 0 0

[KI,4]

[TT] =

[TT].[K4]

[KI,5]

Page 13: Mekanika Rekayasa 4 ( Metode Matriks )

-1 0 1 0 -550000 00 0 0 0 0 0

0 1 0 0 0 -1[T] = -1 0 0 0 1 0

0 0 0 1 0 00 0 -1 0 0 0

0 0 0 0 0 0= 550000 0 -550000 0 [Kg] 0 550000

0 0 0 0 0 0-550000 0 550000 0 0 -550000

Batang 6L = 5385.16EA/L = 408530.1sin α = 0.37cos α = 0.93

matriks kekakuan

1 0 -1 0 408530.1 0= 408530.1 0 0 0 0 = 0 0

-1 0 1 0 -408530 00 0 0 0 0 0

0.93 0.37 0 0 0.93 -0.37[T] = -0.37 0.93 0 0 0.37 0.93

0 0 0.93 0.37 0 00 0 -0.37 0.93 0 0

379933 0 -379933 0 353337.7 140575.2= 151156.1 0 -151156 0 [Kg] 140575.2 55927.77

-379933 0 379933 0 -353338 -140575-151156 0 151156.1 0 -140575 -55927.77

[TT] =

[TT].[K5]

[KI,6]

[TT] =

[TT].[K6]

Page 14: Mekanika Rekayasa 4 ( Metode Matriks )

Batang 7L = 5385.16EA/L = 408530.1sin α = 0.37cos α = 0.93

matriks kekakuan

1 0 -1 0 408530.1 0= 408530.1 0 0 0 0 = 0 0

-1 0 1 0 -408530 00 0 0 0 0 0

0.93 -0.37 0 0 0.93 0.37[T] = 0.37 0.93 0 0 -0.37 0.93

0 0 0.93 -0.37 0 00 0 0.37 0.93 0 0

379933 0 -379933 0 353337.7 -140575= -151156 0 151156.1 0 [Kg] -140575 55927.77

-379933 0 379933 0 -353338 140575.2151156.1 0 -151156 0 140575.2 -55927.77

Batang 8L = 4000EA/L = 550000sin α = 1cos α = 0

matriks kekakuan

1 0 -1 0 550000 0= 550000 0 0 0 0 = 0 0

-1 0 1 0 -550000 00 0 0 0 0 0

[KI,7]

[TT] =

[TT].[K7]

[KI,8]

Page 15: Mekanika Rekayasa 4 ( Metode Matriks )

0 1 0 0 0 -1[T] = -1 0 0 0 1 0

0 0 0 1 0 00 0 -1 0 0 0

0 0 0 0 0 0= 550000 0 -550000 0 [Kg] 0 550000

0 0 0 0 0 0-550000 0 550000 0 0 -550000

[TT] =

[TT].[K8]

Page 16: Mekanika Rekayasa 4 ( Metode Matriks )

-733333.3 00 0

733333.3 00 0

0 00 01 00 1

-733333.3 00 0

733333.3 00 0

-440000 00 0

440000 0

Page 17: Mekanika Rekayasa 4 ( Metode Matriks )

0 0

0 00 01 00 1

-440000 00 0

440000 00 0

-733333 00 0

733333.3 00 0

0 00 01 00 1

-733333 00 0

733333.3 0

Page 18: Mekanika Rekayasa 4 ( Metode Matriks )

0 0

-440000 00 0

440000 00 0

0 00 0

0.6 -0.80.8 0.6

-158400 -211200-211200 -281600158400 211200211200 281600

-550000 00 0

Page 19: Mekanika Rekayasa 4 ( Metode Matriks )

550000 00 0

0 00 00 -11 0

0 00 -5500000 00 550000

-408530 00 0

408530.1 00 0

0 00 0

0.93 -0.370.37 0.93

-353338 -140575-140575 -55927.77

353337.7 140575.2140575.2 55927.77

Page 20: Mekanika Rekayasa 4 ( Metode Matriks )

-408530 00 0

408530.1 00 0

0 00 0

0.93 0.37-0.37 0.93

-353338 140575.2140575.2 -55927.77353337.7 -140575-140575 55927.77

-550000 00 0

550000 00 0

Page 21: Mekanika Rekayasa 4 ( Metode Matriks )

0 00 00 -11 0

0 00 -5500000 00 550000

Page 22: Mekanika Rekayasa 4 ( Metode Matriks )

Unknown Displacement and Reaction

0 1783467 422400 -733333.3 0 0 0 0 -316800 -422400 0 0 U20 422400 1113200 0 0 0 0 -550000 -422400 -563200 0 0 V20 -733333.3 0 1783467 -422400 -733333.3 0 0 -316800 422400 0 0 U30 0 0 -422400 1113200 0 0 0 422400 -563200 0 -550000 V30 0 0 -733333.3 0 891733.33 0 0 0 0 -158400 211200 U4

300 = 0 0 0 0 0 1208533 -211200 -316800 422400 -733333.3 0 U50 0 -550000 0 0 0 -211200 1394800 422400 -563200 0 0 V50 -316800 -422400 -316800 422400 0 -316800 422400 1267200 0 -316800 -422400 U60 -422400 -563200 422400 -563200 0 422400 -563200 0 2252800 -422400 -563200 V60 0 0 0 0 -158400 -733333.3 0 -316800 -422400 1208533 211200 U7

-700 0 0 0 -550000 211200 0 0 -422400 -563200 211200 1394800 V7

U2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0V2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0U3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0V3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0U4 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0U5 = #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 300V5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0U6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0V6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0U7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0V7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! -700

U2 #VALUE!V2 #VALUE!U3 #VALUE!V3 #VALUE!U4 #VALUE!U5 = #VALUE!V5 #VALUE!U6 #VALUE!V6 #VALUE!U7 #VALUE!V7 #VALUE!

Gaya BatangBatang 1

f1g1 =f2g2

f1 1 0 0 0 733333.33 0 -733333.3 0 0g1 = 0 1 0 0 0 0 0 0 0f2 0 0 1 0 -733333.3 0 733333.33 0 #VALUE!g2 0 0 0 1 0 0 0 0 #VALUE!

733333.33 0 -733333.3 0 0

[T1] [Kg1] [U1]

Page 23: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 0-733333.3 0 733333.33 0 #VALUE!

0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f2 #VALUE!g2 #VALUE!

Batang 2f2 1 0 0 0 733333.33 0 -733333.3 0 #VALUE!g2 = 0 1 0 0 0 0 0 0 #VALUE!f3 0 0 1 0 -733333.3 0 733333.33 0 #VALUE!g3 0 0 0 1 0 0 0 0 #VALUE!

733333.33 0 -733333.3 0 #VALUE!0 0 0 0 #VALUE!

-733333.3 0 733333.33 0 #VALUE!0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 3f3 1 0 0 0 733333.33 0 -733333.3 0 #VALUE!g3 = 0 1 0 0 0 0 0 0 #VALUE!f4 0 0 1 0 -733333.3 0 733333.33 0 #VALUE!g4 0 0 0 1 0 0 0 0 #VALUE!

733333.33 0 -733333.3 0 #VALUE!0 0 0 0 #VALUE!

-733333.3 0 733333.33 0 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f4 #VALUE!g4 #VALUE!

Batang 4f1 0.6 0.8 0 0 158400 211200 -158400 -211200 0g1 = -0.8 0.6 0 0 211200 281600 -211200 -281600 0f5 0 0 0.6 0.8 -158400 -211200 158400 211200 #VALUE!g5 0 0 -0.8 0.6 -211200 -281600 211200 281600 #VALUE!

264000 352000 -264000 -352000 0

Page 24: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 0-264000 -352000 264000 352000 #VALUE!

0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f5 #VALUE!g5 #VALUE!

Batang 5f2 0 1 0 0 0 0 0 0 #VALUE!g2 = -1 0 0 0 0 550000 0 -550000 #VALUE!f5 0 0 0 1 0 0 0 0 #VALUE!g5 0 0 -1 0 0 -550000 0 550000 #VALUE!

0 550000 0 -550000 #VALUE!0 0 0 0 #VALUE!0 -550000 0 550000 #VALUE!0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f5 #VALUE!g5 #VALUE!

Batang 6f2 0.6 0.8 0 0 316800 422400 -316800 -422400 #VALUE!g2 = -0.8 0.6 0 0 422400 563200 -422400 -563200 #VALUE!f6 0 0 0.6 0.8 -316800 -422400 316800 422400 #VALUE!g6 0 0 -0.8 0.6 -422400 -563200 422400 563200 #VALUE!

528000 704000 -528000 -704000 #VALUE!0 0 0 0 #VALUE!

-528000 -704000 528000 704000 #VALUE!0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f6 #VALUE!g6 #VALUE!

Batang 7f6 0.6 -0.8 0 0 316800 -422400 -316800 422400 #VALUE!g6 = 0.8 0.6 0 0 -422400 563200 422400 -563200 #VALUE!f3 0 0 0.6 -0.8 -316800 422400 316800 -422400 #VALUE!g3 0 0 0.8 0.6 422400 -563200 -422400 563200 #VALUE!

528000 -704000 -528000 704000 #VALUE!0 0 0 0 #VALUE!

-528000 704000 528000 -704000 #VALUE!0 0 0 0 #VALUE!

Page 25: Mekanika Rekayasa 4 ( Metode Matriks )

f6 #VALUE!g6 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 8f3 0 1 0 0 0 0 0 0 #VALUE!g3 = -1 0 0 0 0 550000 0 -550000 #VALUE!f7 0 0 0 1 0 0 0 0 #VALUE!g7 0 0 -1 0 0 -550000 0 550000 #VALUE!

0 550000 0 -550000 #VALUE!0 0 0 0 #VALUE!0 -550000 0 550000 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f7 #VALUE!g7 #VALUE!

Batang 9f7 0.6 -0.8 0 0 158400 -211200 -158400 211200 #VALUE!g7 = 0.8 0.6 0 0 -211200 281600 211200 -281600 #VALUE!f4 0 0 0.6 -0.8 -158400 211200 158400 -211200 #VALUE!g4 0 0 0.8 0.6 211200 -281600 -211200 281600 0

264000 -352000 -264000 352000 #VALUE!0 0 0 0 #VALUE!

-264000 352000 264000 -352000 #VALUE!0 0 0 0 0

f7 #VALUE!g7 = #VALUE!f4 #VALUE!g4 #VALUE!

Batang 10f5 0.6 -0.8 0 0 316800 -422400 -316800 422400 #VALUE!g5 = 0.8 0.6 0 0 -422400 563200 422400 -563200 #VALUE!f6 0 0 0.6 -0.8 -316800 422400 316800 -422400 #VALUE!g6 0 0 0.8 0.6 422400 -563200 -422400 563200 #VALUE!

528000 -704000 -528000 704000 #VALUE!0 0 0 0 #VALUE!

-528000 704000 528000 -704000 #VALUE!0 0 0 0 #VALUE!

f5 #VALUE!g5 = #VALUE!f6 #VALUE!

Page 26: Mekanika Rekayasa 4 ( Metode Matriks )

g6 #VALUE!

Batang 11f6 0.6 0.8 0 0 316800 422400 -316800 -422400 #VALUE!g6 = -0.8 0.6 0 0 422400 563200 -422400 -563200 #VALUE!f7 0 0 0.6 0.8 -316800 -422400 316800 422400 #VALUE!g7 0 0 -0.8 0.6 -422400 -563200 422400 563200 #VALUE!

528000 704000 -528000 -704000 #VALUE!0 0 0 0 #VALUE!

-528000 -704000 528000 704000 #VALUE!0 0 0 0 #VALUE!

f6 #VALUE!g6 = #VALUE!f7 #VALUE!g7 #VALUE!

Batang 12f5 1 0 0 0 733333.33 0 -733333.3 0 #VALUE!g5 = 0 1 0 0 0 0 0 0 #VALUE!f7 0 0 1 0 -733333.3 0 733333.33 0 #VALUE!g7 0 0 0 1 0 0 0 0 #VALUE!

733333.33 0 -733333.3 0 #VALUE!0 0 0 0 #VALUE!

-733333.3 0 733333.33 0 #VALUE!0 0 0 0 #VALUE!

f5 #VALUE!g5 = #VALUE!f7 #VALUE!g7 #VALUE!

Reaksi Perletakan

#VALUE!#VALUE!#VALUE!#VALUE!

F1 -733333.3 0 0 0 0 -158400 -211200 0 0 0 0 #VALUE! #VALUE!G1 = 0 0 0 0 0 -211200 -281600 0 0 0 0 #VALUE! = #VALUE!G4 0 0 0 0 -211200 0 0 0 0 211200 -281600 #VALUE! #VALUE!

#VALUE!#VALUE!#VALUE!#VALUE!

Page 27: Mekanika Rekayasa 4 ( Metode Matriks )

E = 200000 MpaA = 11000 cm2

Batang 1L = 4000 mmEA/L = 550000sin α = 0 (S)cos α = 1 (C)

CS -(CS) 1 0 -1 0= EA/L CS S² -(CS) (S²) = 550000 0 0 0 0

-(CS) CS -1 0 1 0-(CS) (S²) CS S² 0 0 0 0

u1 v1 u2 v21 0 0 0 550000 0 -550000 0 u1

= 0 1 0 0 [Kg,1] = 0 0 0 0 v10 0 1 0 -550000 0 550000 0 u20 0 0 1 0 0 0 0 v2

Batang 2L = 4000 mmEA/L = 550000sin α = 0 (S)cos α = 1 (C)

CS -(CS) 1 0 -1 0= EA/L CS S² -(CS) (S²) = 550000 0 0 0 0

-(CS) CS -1 0 1 0-(CS) (S²) CS S² 0 0 0 0

u2 v2 u3 v31 0 0 0 550000 0 -550000 0 u2

= 0 1 0 0 [Kg,2] = 0 0 0 0 v20 0 1 0 -550000 0 550000 0 u30 0 0 1 0 0 0 0 v3

Batang 3L = 4000 mmEA/L = 550000sin α = 0 (S)cos α = 1 (C)

CS -(CS) 1 0 -1 0= EA/L CS S² -(CS) (S²) = 550000 0 0 0 0

-(CS) CS -1 0 1 0-(CS) (S²) CS S² 0 0 0 0

u3 v3 u4 v41 0 0 0 550000 0 -550000 0 u3

= 0 1 0 0 [Kg,3] = 0 0 0 0 v30 0 1 0 -550000 0 550000 0 u40 0 0 1 0 0 0 0 v4

Batang 4L = 4000 mmEA/L = 550000sin α = 0 (S)cos α = 1 (C)

CS -(CS) 1 0 -1 0= EA/L CS S² -(CS) (S²) = 550000 0 0 0 0

-(CS) CS -1 0 1 0-(CS) (S²) CS S² 0 0 0 0

u4 v4 u5 v51 0 0 0 550000 0 -550000 0 u4

= 0 1 0 0 [Kg,4] = 0 0 0 0 v40 0 1 0 -550000 0 550000 0 u50 0 0 1 0 0 0 0 v5

Batang 5L = 5000 mmEA/L = 440000sin α = 0.6 (S)

C² -(C²)[Kg,1]

-(C²) C²

[T1]

C² -(C²)[Kg,2]

-(C²) C²

[T2]

C² -(C²)[Kg,3]

-(C²) C²

[T3]

C² -(C²)[Kg,4]

-(C²) C²

[T4]

Page 28: Mekanika Rekayasa 4 ( Metode Matriks )

cos α = 0.8 (C)

CS -(CS) 0.64 0.48 -0.64 -0.48= EA/L CS S² -(CS) (S²) = 440000 0.48 0.36 -0.48 -0.36

-(CS) CS -0.64 -0.48 0.64 0.48-(CS) (S²) CS S² -0.48 -0.36 0.48 0.36

u1 v1 u6 v60.8 0.6 0 0 281600 211200 -281600 -211200 u1

= -0.6 0.8 0 0 [Kg,5] = 211200 158400 -211200 -158400 v10 0 0.8 0.6 -281600 -211200 281600 211200 u60 0 -0.6 0.8 -211200 -158400 211200 158400 v6

Batang 6L = 3000 mmEA/L = 733333.33sin α = 1 (S)cos α = 0 (C)

CS -(CS) 0 0 0 0= EA/L CS S² -(CS) (S²) = 733333.33 0 1 0 -1

-(CS) CS 0 0 0 0-(CS) (S²) CS S² 0 -1 0 1

u2 v2 u6 v60 1 0 0 0 0 0 0 u2

= -1 0 0 0 [Kg,6] = 0 733333.33 0 -733333 v20 0 0 1 0 0 0 0 u60 0 -1 0 0 -733333 0 733333.33 v6

Batang 7L = 3000 mmEA/L = 733333.33sin α = -0.6 (S)cos α = 0.8 (C)

CS -(CS) 0.64 -0.48 -0.64 0.48= EA/L CS S² -(CS) (S²) = 733333.33 -0.48 0.36 0.48 -0.36

-(CS) CS -0.64 0.48 0.64 -0.48-(CS) (S²) CS S² 0.48 -0.36 -0.48 0.36

u6 v6 u3 v30.8 -0.6 0 0 469333.33 -352000 -469333 352000 u6

= 0.6 0.8 0 0 [Kg,7] = -352000 264000 352000 -264000 v60 0 0.8 -0.6 -469333 352000 469333.33 -352000 u30 0 0.6 0.8 352000 -264000 -352000 264000 v3

Batang 8L = 6000 mmEA/L = 366666.67sin α = 1 (S)cos α = 0 (C)

CS -(CS) 0 0 0 0= EA/L CS S² -(CS) (S²) = 366666.67 0 1 0 -1

-(CS) CS 0 0 0 0-(CS) (S²) CS S² 0 -1 0 1

u3 v3 u7 v70 1 0 0 0 0 0 0 u3

= -1 0 0 0 [Kg,8] = 0 366666.67 0 -366667 v30 0 0 1 0 0 0 0 u70 0 -1 0 0 -366667 0 366666.67 v7

Batang 9L = 5000 mmEA/L = 440000sin α = 0.6 (S)cos α = 0.8 (C)

CS -(CS) 0.64 0.48 -0.64 -0.48= EA/L CS S² -(CS) (S²) = 440000 0.48 0.36 -0.48 -0.36

-(CS) CS -0.64 -0.48 0.64 0.48

C² -(C²)[Kg,5]

-(C²) C²

[T5]

C² -(C²)[Kg,6]

-(C²) C²

[T6]

C² -(C²)[Kg,7]

-(C²) C²

[T7]

C² -(C²)[Kg,8]

-(C²) C²

[T8]

C² -(C²)[Kg,9]

-(C²) C²

Page 29: Mekanika Rekayasa 4 ( Metode Matriks )

-(CS) (S²) CS S² -0.48 -0.36 0.48 0.36

u3 v3 u8 v80.8 0.6 0 0 281600 211200 -281600 -211200 u3

= -0.6 0.8 0 0 [Kg,9] = 211200 158400 -211200 -158400 v30 0 0.8 0.6 -281600 -211200 281600 211200 u80 0 -0.6 0.8 -211200 -158400 211200 158400 v8

Batang 10L = 3000 mmEA/L = 733333.33sin α = 1 (S)cos α = 0 (C)

CS -(CS) 0 0 0 0= EA/L CS S² -(CS) (S²) = 733333.33 0 1 0 -1

-(CS) CS 0 0 0 0-(CS) (S²) CS S² 0 -1 0 1

u4 v4 u8 v80 1 0 0 0 0 0 0 u4

= -1 0 0 0 [Kg,10] = 0 733333.33 0 -733333 v40 0 0 1 0 0 0 0 u80 0 -1 0 0 -733333 0 733333.33 v8

Batang 11L = 5000 mmEA/L = 440000sin α = -0.6 (S)cos α = 0.8 (C)

CS -(CS) 0.64 -0.48 -0.64 0.48= EA/L CS S² -(CS) (S²) = 440000 -0.48 0.36 0.48 -0.36

-(CS) CS -0.64 0.48 0.64 -0.48-(CS) (S²) CS S² 0.48 -0.36 -0.48 0.36

u8 v8 u5 v50.8 -0.6 0 0 281600 -211200 -281600 211200 u8

= 0.6 0.8 0 0 [Kg,11] = -211200 158400 211200 -158400 v80 0 0.8 -0.6 -281600 211200 281600 -211200 u50 0 0.6 0.8 211200 -158400 -211200 158400 v5

Batang 12L = 5000 mmEA/L = 440000sin α = 0.6 (S)cos α = 0.8 (C)

CS -(CS) 0.64 0.48 -0.64 -0.48= EA/L CS S² -(CS) (S²) = 440000 0.48 0.36 -0.48 -0.36

-(CS) CS -0.64 -0.48 0.64 0.48-(CS) (S²) CS S² -0.48 -0.36 0.48 0.36

u6 v6 u7 v70.8 0.6 0 0 281600 211200 -281600 -211200 u6

= -0.6 0.8 0 0 [Kg,12] = 211200 158400 -211200 -158400 v60 0 0.8 0.6 -281600 -211200 281600 211200 u70 0 -0.6 0.8 -211200 -158400 211200 158400 v7

Batang 13L = 5000 mmEA/L = 440000sin α = -0.6 (S)cos α = 0.8 (C)

CS -(CS) 0.64 -0.48 -0.64 0.48= EA/L CS S² -(CS) (S²) = 440000 -0.48 0.36 0.48 -0.36

-(CS) CS -0.64 0.48 0.64 -0.48-(CS) (S²) CS S² 0.48 -0.36 -0.48 0.36

u7 v7 u8 v80.8 -0.6 0 0 281600 -211200 -281600 211200 u7

= 0.6 0.8 0 0 [Kg,13] = -211200 158400 211200 -158400 v70 0 0.8 -0.6 -281600 211200 281600 -211200 u80 0 0.6 0.8 211200 -158400 -211200 158400 v8

[T9]

C² -(C²)[Kg,10]

-(C²) C²

[T10]

C² -(C²)[Kg,11]

-(C²) C²

[T11]

C² -(C²)[Kg,12]

-(C²) C²

[T12]

C² -(C²)[Kg,13]

-(C²) C²

[T13]

Page 30: Mekanika Rekayasa 4 ( Metode Matriks )

Overall Stiffnessu1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6

1 F1 831600 211200 -550000 0 -2816002 G1 211200 158400 0 0 -2112003 0 -550000 0 1100000 0 -550000 0 04 0 0 0 0 733333.33 0 0 05 0 -550000 0 1850933 -140800 -550000 0 -4693336 0 0 0 -140800 789066.67 0 0 3520007 0 -550000 0 1100000 0 -550000 08 0 = 0 0 0 733333.33 0 09 0 -550000 0 831600 -211200

10 G5 0 0 -211200 15840011 0 -281600 -211200 0 0 -469333 352000 103253312 0 -211200 -158400 0 -733333 352000 -264000 7040013 500 0 0 -28160014 0 0 -366667 -21120015 0 -281600 -211200 0 0 -281600 21120016 -1000 -211200 -158400 0 -733333 211200 -158400

1 2 3 4 5 6 7 8 9 10 11

Rearrangement

3 0 1100000 0 -550000 0 0 0 0 0 0 0 04 0 0 733333.33 0 0 0 0 0 0 -733333 0 05 0 -550000 0 1850933 -140800 -550000 0 0 -469333 352000 0 06 0 0 0 -140800 789066.67 0 0 0 352000 -264000 0 -3666677 0 0 0 -550000 0 1100000 0 -550000 0 0 0 08 0 0 0 0 0 0 733333.33 0 0 0 0 09 0 0 0 0 0 -550000 0 831600 0 0 0 0

11 0 = 0 0 -469333 352000 0 0 0 1032533 70400 -281600 -21120012 0 0 -733333 352000 -264000 0 0 0 70400 1314133 -211200 -15840013 500 0 0 0 0 0 0 0 -281600 -211200 563200 014 0 0 0 0 -366667 0 0 0 -211200 -158400 0 683466.6715 0 0 0 -281600 -211200 0 0 -281600 0 0 -281600 21120016 -1000 0 0 -211200 -158400 0 -733333 211200 0 0 211200 -1584001 F1 -550000 0 0 0 0 0 0 -281600 -211200 0 02 G1 0 0 0 0 0 0 0 -211200 -158400 0 0

10 G5 0 0 0 0 0 0 -211200 0 0 0 03 4 5 6 7 8 9 11 12 13 14

Unknown Displacement and Reaction

0 1100000 0 -550000 0 0 0 0 0 0 0 00 0 733333.33 0 0 0 0 0 0 -733333 0 00 -550000 0 1850933 -140800 -550000 0 0 -469333 352000 0 00 0 0 -140800 789066.67 0 0 0 352000 -264000 0 -3666670 0 0 -550000 0 1100000 0 -550000 0 0 0 00 0 0 0 0 0 733333.33 0 0 0 0 00 = 0 0 0 0 -550000 0 831600 0 0 0 00 0 0 -469333 352000 0 0 0 1032533 70400 -281600 -2112000 0 -733333 352000 -264000 0 0 0 70400 1314133 -211200 -158400

500 0 0 0 0 0 0 0 -281600 -211200 563200 00 0 0 0 -366667 0 0 0 -211200 -158400 0 683466.670 0 0 -281600 -211200 0 0 -281600 0 0 -281600 211200

-1000 0 0 -211200 -158400 0 -733333 211200 0 0 211200 -158400

U2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!U3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!U4 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V4 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

Page 31: Mekanika Rekayasa 4 ( Metode Matriks )

U5 = #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!U6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!U7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!U8 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!V8 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE!

U2 #VALUE!V2 #VALUE!U3 #VALUE!V3 #VALUE!U4 #VALUE!V4 = #VALUE!U5 #VALUE!U6 #VALUE!V6 #VALUE!U7 #VALUE!V7 #VALUE!U8 #VALUE!V8 #VALUE!

Gaya BatangBatang 1

f1g1 =f2g2

f1 1 0 0 0 550000 0 -550000 0 0g1 = 0 1 0 0 0 0 0 0 0f2 0 0 1 0 -550000 0 550000 0 #VALUE!g2 0 0 0 1 0 0 0 0 #VALUE!

550000 0 -550000 0 00 0 0 0 0

-550000 0 550000 0 #VALUE!0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f2 #VALUE!g2 #VALUE!

Batang 2

f2 1 0 0 0 550000 0 -550000 0 #VALUE!g2 = 0 1 0 0 0 0 0 0 #VALUE!f3 0 0 1 0 -550000 0 550000 0 #VALUE!g3 0 0 0 1 0 0 0 0 #VALUE!

550000 0 -550000 0 #VALUE!0 0 0 0 #VALUE!

-550000 0 550000 0 #VALUE!0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 3

f3 1 0 0 0 550000 0 -550000 0 #VALUE!g3 = 0 1 0 0 0 0 0 0 #VALUE!

[T1] [Kg1] [U1]

Page 32: Mekanika Rekayasa 4 ( Metode Matriks )

f4 0 0 1 0 -550000 0 550000 0 #VALUE!g4 0 0 0 1 0 0 0 0 #VALUE!

550000 0 -550000 0 #VALUE!0 0 0 0 #VALUE!

-550000 0 550000 0 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f4 #VALUE!g4 #VALUE!

Batang 4

f4 1 0 0 0 550000 0 -550000 0 #VALUE!g4 = 0 1 0 0 0 0 0 0 #VALUE!f5 0 0 1 0 -550000 0 550000 0 #VALUE!g5 0 0 0 1 0 0 0 0 0

550000 0 -550000 0 #VALUE!0 0 0 0 #VALUE!

-550000 0 550000 0 #VALUE!0 0 0 0 0

f4 #VALUE!g4 = #VALUE!f5 #VALUE!g5 #VALUE!

Batang 5

f1 0.8 0.6 0 0 281600 211200 -281600 -211200 0g1 = -0.6 0.8 0 0 211200 158400 -211200 -158400 0f6 0 0 0.8 0.6 -281600 -211200 281600 211200 #VALUE!g6 0 0 -0.6 0.8 -211200 -158400 211200 158400 #VALUE!

352000 264000 -352000 -264000 00 0 0 0 0

-352000 -264000 352000 264000 #VALUE!0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f6 #VALUE!g6 #VALUE!

Batang 6

f2 0 1 0 0 0 0 0 0 #VALUE!g2 = -1 0 0 0 0 733333.33 0 -733333 #VALUE!f6 0 0 0 1 0 0 0 0 #VALUE!g6 0 0 -1 0 0 -733333 0 733333.33 #VALUE!

0 733333.33 0 -733333 #VALUE!0 0 0 0 #VALUE!0 -733333 0 733333.33 #VALUE!0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f6 #VALUE!g6 #VALUE!

Batang 7

f6 0.8 -0.6 0 0 469333.33 -352000 -469333 352000 #VALUE!g6 = 0.6 0.8 0 0 -352000 264000 352000 -264000 #VALUE!f3 0 0 0.8 -0.6 -469333 352000 469333.33 -352000 #VALUE!g3 0 0 0.6 0.8 352000 -264000 -352000 264000 #VALUE!

586666.67 -440000 -586667 440000 #VALUE!0 0 0 0 #VALUE!

-586667 440000 586666.67 -440000 #VALUE!0 0 0 0 #VALUE!

Page 33: Mekanika Rekayasa 4 ( Metode Matriks )

f6 #VALUE!g6 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 8

f3 0 1 0 0 0 0 0 0 #VALUE!g3 = -1 0 0 0 0 366666.67 0 -366667 #VALUE!f7 0 0 0 1 0 0 0 0 #VALUE!g7 0 0 -1 0 0 -366667 0 366666.67 #VALUE!

0 366666.67 0 -366667 #VALUE!0 0 0 0 #VALUE!0 -366667 0 366666.67 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f7 #VALUE!g7 #VALUE!

Batang 9

f3 0.8 0.6 0 0 281600 211200 -281600 -211200 #VALUE!g3 = -0.6 0.8 0 0 211200 158400 -211200 -158400 #VALUE!f8 0 0 0.8 0.6 -281600 -211200 281600 211200 #VALUE!g8 0 0 -0.6 0.8 -211200 -158400 211200 158400 #VALUE!

352000 264000 -352000 -264000 #VALUE!0 0 0 0 #VALUE!

-352000 -264000 352000 264000 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f8 #VALUE!g8 #VALUE!

Batang 10

f4 0 1 0 0 0 0 0 0 #VALUE!g4 = -1 0 0 0 0 733333.33 0 -733333 #VALUE!f8 0 0 0 1 0 0 0 0 #VALUE!g8 0 0 -1 0 0 -733333 0 733333.33 #VALUE!

0 733333.33 0 -733333 #VALUE!0 0 0 0 #VALUE!0 -733333 0 733333.33 #VALUE!0 0 0 0 #VALUE!

f4 #VALUE!g4 = #VALUE!f8 #VALUE!g8 #VALUE!

Batang 11

f8 0.8 -0.6 0 0 281600 -211200 -281600 211200 #VALUE!g8 = 0.6 0.8 0 0 -211200 158400 211200 -158400 #VALUE!f5 0 0 0.8 -0.6 -281600 211200 281600 -211200 #VALUE!g5 0 0 0.6 0.8 211200 -158400 -211200 158400 0

352000 -264000 -352000 264000 #VALUE!0 0 0 0 #VALUE!

-352000 264000 352000 -264000 #VALUE!0 0 0 0 0

f8 #VALUE!g8 = #VALUE!f5 #VALUE!g5 #VALUE!

Page 34: Mekanika Rekayasa 4 ( Metode Matriks )

Batang 12

f6 0.8 0.6 0 0 281600 211200 -281600 -211200 #VALUE!g6 = -0.6 0.8 0 0 211200 158400 -211200 -158400 #VALUE!f7 0 0 0.8 0.6 -281600 -211200 281600 211200 #VALUE!g7 0 0 -0.6 0.8 -211200 -158400 211200 158400 #VALUE!

352000 264000 -352000 -264000 #VALUE!0 0 0 0 #VALUE!

-352000 -264000 352000 264000 #VALUE!0 0 0 0 #VALUE!

f6 #VALUE!g6 = #VALUE!f7 #VALUE!g7 #VALUE!

Batang 13

f7 0.8 -0.6 0 0 281600 -211200 -281600 211200 #VALUE!g7 = 0.6 0.8 0 0 -211200 158400 211200 -158400 #VALUE!f8 0 0 0.8 -0.6 -281600 211200 281600 -211200 #VALUE!g8 0 0 0.6 0.8 211200 -158400 -211200 158400 #VALUE!

352000 -264000 -352000 264000 #VALUE!0 0 0 0 #VALUE!

-352000 264000 352000 -264000 #VALUE!0 0 0 0 #VALUE!

f7 #VALUE!g7 = #VALUE!f8 #VALUE!g8 #VALUE!

Reaksi Perletakan

F1 -550000 0 0 0 0 0 0 -281600 -211200 0 0G1 = 0 0 0 0 0 0 0 -211200 -158400 0 0G5 0 0 0 0 0 0 -211200 0 0 0 0

Page 35: Mekanika Rekayasa 4 ( Metode Matriks )

Hasil Komparasi Matriks dan SAP2000

Batang Matriks SAP20001 #VALUE! 583.332 #VALUE! 583.333 #VALUE! 12504 #VALUE! 12505 #VALUE! -104.176 #VALUE! 07 #VALUE! 4.547E-138 #VALUE! 5009 #VALUE! -833.33

10 #VALUE! 011 #VALUE! -1562.512 #VALUE! -104.1713 #VALUE! -729.17

Page 36: Mekanika Rekayasa 4 ( Metode Matriks )

v6 u7 v7 u8 v8-211200 u1 0-158400 v1 0

0 u2 U2-733333 v2 V2352000 0 0 -281600 -211200 u3 U3-264000 0 -366667 -211200 -158400 v3 V3

0 0 u4 U40 -733333 v4 V4

-281600 211200 u5 U5211200 -158400 v5 0

70400 -281600 -211200 u6 U61314133 -211200 -158400 v6 V6-211200 563200 0 -281600 211200 u7 U7-158400 0 683466.67 211200 -158400 v7 V7

-281600 211200 844800 -211200 u8 U8211200 -158400 -211200 1208533 v8 V8

12 13 14 15 16

0 0 -550000 0 0 U20 0 0 0 0 V2

-281600 -211200 0 0 0 U3-211200 -158400 0 0 0 V3

0 0 0 0 0 U40 -733333 0 0 0 V4

-281600 211200 0 0 -211200 U50 0 -281600 -211200 0 U60 0 -211200 -158400 0 V6

-281600 211200 0 0 0 U7211200 -158400 0 0 0 V7844800 -211200 0 0 211200 U8-211200 1208533 0 0 -158400 V8

0 0 831600 211200 0 00 0 211200 158400 0 0

211200 -158400 0 0 158400 015 16 1 2 10

0 0 U20 0 V2

-281600 -211200 U3-211200 -158400 V3

0 0 U40 -733333 V4

-281600 211200 U50 0 U60 0 V6

-281600 211200 U7211200 -158400 V7844800 -211200 U8-211200 1208533 V8

#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 0

Page 37: Mekanika Rekayasa 4 ( Metode Matriks )

#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! 500#VALUE! #VALUE! 0#VALUE! #VALUE! 0#VALUE! #VALUE! -1000

Page 38: Mekanika Rekayasa 4 ( Metode Matriks )

#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

0 0 #VALUE! #VALUE!0 0 #VALUE! = #VALUE!

211200 -158400 #VALUE! #VALUE!#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

Page 39: Mekanika Rekayasa 4 ( Metode Matriks )

Reaksi Perletakan

#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

F1 -937500 0 0 0 0 0 0 -480000 -360000 0 0 0 0 #VALUE! #VALUE!G1 = 0 0 0 0 0 0 0 -360000 -270000 0 0 0 0 #VALUE! = #VALUE!G5 0 0 0 0 0 0 -360000 0 0 0 0 360000 -270000 #VALUE! #VALUE!

#VALUE!#VALUE!#VALUE!#VALUE!#VALUE!

Page 40: Mekanika Rekayasa 4 ( Metode Matriks )

E = 250000 MpaA = 15000 cm2

Batang 1L = 3000 mmEA/L = 1250000sin α = 0cos α = 1

Matriks kekakuan1 0 -1 0 1250000 0 -1250000 0

= 1250000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1250000 0 1250000 00 0 0 0 0 0 0 0

Matriks transformasi1 0 0 0 1 0 0 0

[T] = 0 1 0 0 0 1 0 00 0 1 0 0 0 1 00 0 0 1 0 0 0 1

1250000 0 -1250000 0= 0 0 0 0

-1250000 0 1250000 00 0 0 0

u1 v1 u2 v21250000 0 -1250000 0 u1

[Kg] = [T] = 0 0 0 0 v1-1250000 0 1250000 0 u2

0 0 0 0 v2

Batang 2L = 3000 mmEA/L = 1250000sin α = 0cos α = 1

Matriks kekakuan1 0 -1 0 1250000 0 -1250000 0

= 1250000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1250000 0 1250000 00 0 0 0 0 0 0 0

Matriks transformasi1 0 0 0 1 0 0 0 1250000 0 -1250000 0

[T] = 0 1 0 0 0 1 0 0 = 0 0 0 00 0 1 0 0 0 1 0 -1250000 0 1250000 00 0 0 1 0 0 0 1 0 0 0 0

[KI,1]

[TT] =

[TT].[K1]

[TT] [K1]

[KI,2]

[TT] = [TT].[K2]

Page 41: Mekanika Rekayasa 4 ( Metode Matriks )

u2 v2 u3 v31250000 0 -1250000 0 u2

[Kg] = [T] = 0 0 0 0 v2-1250000 0 1250000 0 u3

0 0 0 0 v3

Batang 3L = 3000 mmEA/L = 1250000sin α = 0cos α = 1

Matriks kekakuan1 0 -1 0 1250000 0 -1250000 0

= 1250000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1250000 0 1250000 00 0 0 0 0 0 0 0

Matriks transformasi1 0 0 0 1 0 0 0

[T] = 0 1 0 0 0 1 0 00 0 1 0 0 0 1 00 0 0 1 0 0 0 1

1250000 0 -1250000 0= 0 0 0 0

-1250000 0 1250000 00 0 0 0

u3 v3 u4 v41250000 0 -1250000 0 u3

[Kg] = [T] = 0 0 0 0 v3-1250000 0 1250000 0 u4

0 0 0 0 v4

Batang 4L = 5000 mmEA/L = 750000sin α = 0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 750000 0 -750000 0

= 750000 0 0 0 0 = 0 0 0 0-1 0 1 0 -750000 0 750000 0

[TT] [K2]

[KI,3]

[TT] =

[TT].[K3]

[TT] [K3]

[KI,4]

Page 42: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 0 0 0 0

Matriks transformasi0.6 0.8 0 0 0.6 -0.8 0 0

[T] = -0.8 0.6 0 0 0.8 0.6 0 00 0 0.6 0.8 0 0 0.6 -0.80 0 -0.8 0.6 0 0 0.8 0.6

450000 0 -450000 0= 600000 0 -600000 0

-450000 0 450000 0-600000 0 600000 0

u1 v1 u5 v5270000 360000 -270000 -360000 u1

[Kg] = [T] = 360000 480000 -360000 -480000 v1-270000 -360000 270000 360000 u5-360000 -480000 360000 480000 v5

Batang 5L = 4000 mmEA/L = 937500sin α = 1cos α = 0

Matriks kekakuan1 0 -1 0 937500 0 -937500 0

= 937500 0 0 0 0 = 0 0 0 0-1 0 1 0 -937500 0 937500 00 0 0 0 0 0 0 0

Matriks transformasi0 1 0 0 0 -1 0 0

[T] = -1 0 0 0 1 0 0 00 0 0 1 0 0 0 -10 0 -1 0 0 0 1 0

0 0 0 0= 937500 0 -937500 0

0 0 0 0-937500 0 937500 0

u2 v2 u5 v50 0 0 0 u2

[Kg] = [T] = 0 937500 0 -937500 v20 0 0 0 u50 -937500 0 937500 v5

Batang 6L = 2500 mmEA/L = 1500000

[TT] =

[TT].[K4]

[TT] [K4]

[KI,5]

[TT] =

[TT].[K5]

[TT] [K5]

Page 43: Mekanika Rekayasa 4 ( Metode Matriks )

sin α = 0.8cos α = 0.6 Matriks kekakuan

1 0 -1 0 1500000 0 -1500000 0= 1500000 0 0 0 0 = 0 0 0 0

-1 0 1 0 -1500000 0 1500000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 0.8 0 0 0.6 -0.8 0 0

[T] = -0.8 0.6 0 0 0.8 0.6 0 00 0 0.6 0.8 0 0 0.6 -0.80 0 -0.8 0.6 0 0 0.8 0.6

900000 0 -900000 0= 1200000 0 -1200000 0

-900000 0 900000 0-1200000 0 1200000 0

u2 v2 u6 v6540000 720000 -540000 -720000 u2

[Kg] = [T] = 720000 960000 -720000 -960000 v2-540000 -720000 540000 720000 u6-720000 -960000 720000 960000 v6

Batang 7L = 2500 mmEA/L = 1500000sin α = -0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 1500000 0 -1500000 0

= 1500000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1500000 0 1500000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 -0.8 0 0 0.6 0.8 0 0

[T] = 0.8 0.6 0 0 -0.8 0.6 0 00 0 0.6 -0.8 0 0 0.6 0.80 0 0.8 0.6 0 0 -0.8 0.6

900000 0 -900000 0= -1200000 0 1200000 0

-900000 0 900000 01200000 0 -1200000 0

u6 v6 u3 v3540000 -720000 -540000 720000 u6

[KI,6]

[TT] =

[TT].[K6]

[TT] [K6]

[KI,7]

[TT] =

[TT].[K7]

Page 44: Mekanika Rekayasa 4 ( Metode Matriks )

[Kg] = [T] = -720000 960000 720000 -960000 v6-540000 720000 540000 -720000 u3720000 -960000 -720000 960000 v3

Batang 8L = 4000 mmEA/L = 937500sin α = 1cos α = 0

Matriks kekakuan1 0 -1 0 937500 0 -937500 0

= 937500 0 0 0 0 = 0 0 0 0-1 0 1 0 -937500 0 937500 00 0 0 0 0 0 0 0

Matriks transformasi0 1 0 0 0 -1 0 0

[T] = -1 0 0 0 1 0 0 00 0 0 1 0 0 0 -10 0 -1 0 0 0 1 0

0 0 0 0= 937500 0 -937500 0

0 0 0 0-937500 0 937500 0

u3 v3 u7 v70 0 0 0 u3

[Kg] = [T] = 0 937500 0 -937500 v30 0 0 0 u70 -937500 0 937500 v7

Batang 9L = 5000 mmEA/L = 750000sin α = -0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 750000 0 -750000 0

= 750000 0 0 0 0 = 0 0 0 0-1 0 1 0 -750000 0 750000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 -0.8 0 0 0.6 0.8 0 0

[T] = 0.8 0.6 0 0 -0.8 0.6 0 00 0 0.6 -0.8 0 0 0.6 0.80 0 0.8 0.6 0 0 -0.8 0.6

[TT] [K7]

[KI,8]

[TT] =

[TT].[K8]

[TT] [K8]

[KI,9]

[TT] =

Page 45: Mekanika Rekayasa 4 ( Metode Matriks )

450000 0 -450000 0= -600000 0 600000 0

-450000 0 450000 0600000 0 -600000 0

u7 v7 u4 v4270000 -360000 -270000 360000 u7

[Kg] = [T] = -360000 480000 360000 -480000 v7-270000 360000 270000 -360000 u4360000 -480000 -360000 480000 v4

Batang 10L = 2500 mmEA/L = 1500000sin α = -0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 1500000 0 -1500000 0

= 1500000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1500000 0 1500000 00 0 0 0 0 0 0 0

Matriks transformasi0.6 -0.8 0 0 0.6 0.8 0 0

[T] = 0.8 0.6 0 0 -0.8 0.6 0 00 0 0.6 -0.8 0 0 0.6 0.80 0 0.8 0.6 0 0 -0.8 0.6

900000 0 -900000 0= -1200000 0 1200000 0

-900000 0 900000 01200000 0 -1200000 0

u5 v5 u6 v6540000 -720000 -540000 720000 u5

[Kg] = [T] = -720000 960000 720000 -960000 v5-540000 720000 540000 -720000 u6720000 -960000 -720000 960000 v6

Batang 11L = 2500 mmEA/L = 1500000sin α = 0.8cos α = 0.6

Matriks kekakuan1 0 -1 0 1500000 0 -1500000 0

= 1500000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1500000 0 1500000 00 0 0 0 0 0 0 0

[TT].[K9]

[TT] [K9]

[KI,10]

[TT] =

[TT].[K10]

[TT] [K10]

[KI,11]

Page 46: Mekanika Rekayasa 4 ( Metode Matriks )

Matriks transformasi0.6 0.8 0 0 0.6 -0.8 0 0

[T] = -0.8 0.6 0 0 0.8 0.6 0 00 0 0.6 0.8 0 0 0.6 -0.80 0 -0.8 0.6 0 0 0.8 0.6

900000 0 -900000 0= 1200000 0 -1200000 0

-900000 0 900000 0-1200000 0 1200000 0

u6 v6 u7 v7540000 720000 -540000 -720000 u6

[Kg] = [T] = 720000 960000 -720000 -960000 v6-540000 -720000 540000 720000 u7-720000 -960000 720000 960000 v7

Batang 12L = 3000 mmEA/L = 1250000sin α = 0cos α = 1

Matriks kekakuan1 0 -1 0 1250000 0 -1250000 0

= 1250000 0 0 0 0 = 0 0 0 0-1 0 1 0 -1250000 0 1250000 00 0 0 0 0 0 0 0

Matriks transformasi1 0 0 0 1 0 0 0

[T] = 0 1 0 0 0 1 0 00 0 1 0 0 0 1 00 0 0 1 0 0 0 1

1250000 0 -1250000 0= 0 0 0 0

-1250000 0 1250000 00 0 0 0

u5 v5 u7 v71250000 0 -1250000 0 u5

[Kg] = [T] = 0 0 0 0 v5-1250000 0 1250000 0 u7

0 0 0 0 v7

Overall Stiffnessu1 v1 u2 v2 u3 v3 u4 v4 u5 v5 u6 v6 u7 v7

1 F1 1520000 360000 -1250000 0 -270000 -360000 u1 0

[TT] =

[TT].[K11]

[TT] [K11]

[KI,12]

[TT] =

[TT].[K12]

[TT] [K12]

Page 47: Mekanika Rekayasa 4 ( Metode Matriks )

2 G1 360000 480000 0 0 -360000 -480000 v1 03 0 -1250000 0 3040000 720000 -1250000 0 0 0 -540000 -720000 u2 U24 0 0 0 720000 1897500 0 0 0 -937500 -720000 -960000 v2 V25 0 -1250000 0 3040000 -720000 -1250000 0 -540000 720000 0 0 u3 U36 0 0 0 -720000 1897500 0 0 720000 -960000 0 -937500 v3 V37 0 = -1250000 0 1520000 -360000 -270000 360000 u4 U48 G4 0 0 -360000 480000 360000 -480000 v4 09 300 -270000 -360000 0 0 2060000 -360000 -540000 720000 -1250000 0 u5 U5

10 0 -360000 -480000 0 -937500 -360000 2377500 720000 -960000 0 0 v5 V511 0 -540000 -720000 -540000 720000 -540000 720000 2160000 0 -540000 -720000 u6 U612 0 -720000 -960000 720000 -960000 720000 -960000 0 3840000 -720000 -960000 v6 V613 0 0 0 -270000 360000 -1250000 0 -540000 -720000 2060000 360000 u7 U714 -700 0 -937500 360000 -480000 0 0 -720000 -960000 360000 2377500 v7 V7

1 2 3 4 5 6 7 8 9 10 11 12 13 14Rearrangement

3 0 3040000 720000 -1250000 0 0 0 0 -540000 -720000 0 0 -1250000 0 0 U24 0 720000 1897500 0 0 0 0 -937500 -720000 -960000 0 0 0 0 0 V25 0 -1250000 0 3040000 -720000 -1250000 0 0 -540000 720000 0 0 0 0 0 U36 0 0 0 -720000 1897500 0 0 0 720000 -960000 0 -937500 0 0 0 V37 0 0 0 -1250000 0 1520000 0 0 0 0 -270000 360000 0 0 -360000 U49 300 0 0 0 0 0 2060000 -360000 -540000 720000 -1250000 0 -270000 -360000 0 U5

10 0 = 0 -937500 0 0 0 -360000 2377500 720000 -960000 0 0 -360000 -480000 0 V511 0 -540000 -720000 -540000 720000 0 -540000 720000 2160000 0 -540000 -720000 0 0 0 U612 0 -720000 -960000 720000 -960000 0 720000 -960000 0 3840000 -720000 -960000 0 0 0 V613 0 0 0 0 0 -270000 -1250000 0 -540000 -720000 2060000 360000 0 0 360000 U714 -700 0 0 0 -937500 360000 0 0 -720000 -960000 360000 2377500 0 0 -480000 V71 F1 -1250000 0 0 0 0 -270000 -360000 0 0 0 0 1520000 360000 0 02 G1 0 0 0 0 0 -360000 -480000 0 0 0 0 360000 480000 0 08 G4 0 0 0 0 -360000 0 0 0 0 360000 -480000 0 0 480000 0

3 4 5 6 7 9 10 11 12 13 14 1 2 8Unknown Displacement and Reaction

0 3040000 720000 -1250000 0 0 0 0 -540000 -720000 0 0 U20 720000 1897500 0 0 0 0 -937500 -720000 -960000 0 0 V20 -1250000 0 3040000 -720000 -1250000 0 0 -540000 720000 0 0 U30 0 0 -720000 1897500 0 0 0 720000 -960000 0 -937500 V30 0 0 -1250000 0 1520000 0 0 0 0 -270000 360000 U4

300 = 0 0 0 0 0 2060000 -360000 -540000 720000 -1250000 0 U50 0 -937500 0 0 0 -360000 2377500 720000 -960000 0 0 V50 -540000 -720000 -540000 720000 0 -540000 720000 2160000 0 -540000 -720000 U60 -720000 -960000 720000 -960000 0 720000 -960000 0 3840000 -720000 -960000 V60 0 0 0 0 -270000 -1250000 0 -540000 -720000 2060000 360000 U7

-700 0 0 0 -937500 360000 0 0 -720000 -960000 360000 2377500 V7

U2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 U2 #VALUE!V2 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 V2 #VALUE!U3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 U3 #VALUE!V3 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 V3 #VALUE!U4 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 > U4 #VALUE!

Page 48: Mekanika Rekayasa 4 ( Metode Matriks )

U5 = #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 300 U5 = #VALUE!V5 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 V5 #VALUE!U6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 U6 #VALUE!V6 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 V6 #VALUE!U7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0 U7 #VALUE!V7 #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! -700 V7 #VALUE!

Gaya BatangBatang 1

f1g1 =f2g2

f1 1 0 0 0 1250000 0 -1250000 0 0g1 = 0 1 0 0 0 0 0 0 0f2 0 0 1 0 -1250000 0 1250000 0 #VALUE!g2 0 0 0 1 0 0 0 0 #VALUE!

1250000 0 -1250000 0 00 0 0 0 0

-1250000 0 1250000 0 #VALUE!0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f2 #VALUE!g2 #VALUE!

Batang 2f2 1 0 0 0 1250000 0 -1250000 0 #VALUE!g2 = 0 1 0 0 0 0 0 0 #VALUE!f3 0 0 1 0 -1250000 0 1250000 0 #VALUE!g3 0 0 0 1 0 0 0 0 #VALUE!

1250000 0 -1250000 0 #VALUE!0 0 0 0 #VALUE!

-1250000 0 1250000 0 #VALUE!0 0 0 0 #VALUE!

[T1] [Kg1] [U1]

Page 49: Mekanika Rekayasa 4 ( Metode Matriks )

f2 #VALUE!g2 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 3f3 1 0 0 0 1250000 0 -1250000 0 #VALUE!g3 = 0 1 0 0 0 0 0 0 #VALUE!f4 0 0 1 0 -1250000 0 1250000 0 #VALUE!g4 0 0 0 1 0 0 0 0 #VALUE!

1250000 0 -1250000 0 #VALUE!0 0 0 0 #VALUE!

-1250000 0 1250000 0 #VALUE!0 0 0 0 #VALUE!

f3 #VALUE!g3 = #VALUE!f4 #VALUE!g4 #VALUE!

Batang 4f1 0.6 0.8 0 0 270000 360000 -270000 -360000 0g1 = -0.8 0.6 0 0 360000 480000 -360000 -480000 0f5 0 0 0.6 0.8 -270000 -360000 270000 360000 #VALUE!g5 0 0 -0.8 0.6 -360000 -480000 360000 480000 #VALUE!

450000 600000 -450000 -600000 00 0 0 0 0

-450000 -600000 450000 600000 #VALUE!0 0 0 0 #VALUE!

f1 #VALUE!g1 = #VALUE!f5 #VALUE!g5 #VALUE!

Batang 5f2 0 1 0 0 0 0 0 0 #VALUE!g2 = -1 0 0 0 0 937500 0 -937500 #VALUE!f5 0 0 0 1 0 0 0 0 #VALUE!g5 0 0 -1 0 0 -937500 0 937500 #VALUE!

0 937500 0 -937500 #VALUE!0 0 0 0 #VALUE!0 -937500 0 937500 #VALUE!

Page 50: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 #VALUE!

f2 #VALUE!g2 = #VALUE!f5 #VALUE!g5 #VALUE!

Batang 6f2 0.6 0.8 0 0 540000 720000 -540000 -720000 #VALUE!g2 = -0.8 0.6 0 0 720000 960000 -720000 -960000 #VALUE!f6 0 0 0.6 0.8 -540000 -720000 540000 720000 #VALUE!g6 0 0 -0.8 0.6 -720000 -960000 720000 960000 #VALUE!

900000 1200000 -900000 -1200000 #VALUE! f2 #VALUE!0 0 0 0 #VALUE! > g2 = #VALUE!

-900000 -1200000 900000 1200000 #VALUE! f6 #VALUE!0 0 0 0 #VALUE! g6 #VALUE!

Batang 7f6 0.6 -0.8 0 0 540000 -720000 -540000 720000 #VALUE!g6 = 0.8 0.6 0 0 -720000 960000 720000 -960000 #VALUE!f3 0 0 0.6 -0.8 -540000 720000 540000 -720000 #VALUE!g3 0 0 0.8 0.6 720000 -960000 -720000 960000 #VALUE!

900000 -1200000 -900000 1200000 #VALUE!0 0 0 0 #VALUE!

-900000 1200000 900000 -1200000 #VALUE!0 0 0 0 #VALUE!

f6 #VALUE!g6 = #VALUE!f3 #VALUE!g3 #VALUE!

Batang 8f3 0 1 0 0 0 0 0 0 #VALUE!g3 = -1 0 0 0 0 937500 0 -937500 #VALUE!f7 0 0 0 1 0 0 0 0 #VALUE!g7 0 0 -1 0 0 -937500 0 937500 #VALUE!

0 937500 0 -937500 #VALUE!0 0 0 0 #VALUE!0 -937500 0 937500 #VALUE!0 0 0 0 #VALUE!

Page 51: Mekanika Rekayasa 4 ( Metode Matriks )

f3 #VALUE!g3 = #VALUE!f7 #VALUE!g7 #VALUE!

Batang 9f7 0.6 -0.8 0 0 270000 -360000 -270000 360000 #VALUE!g7 = 0.8 0.6 0 0 -360000 480000 360000 -480000 #VALUE!f4 0 0 0.6 -0.8 -270000 360000 270000 -360000 #VALUE!g4 0 0 0.8 0.6 360000 -480000 -360000 480000 0

450000 -600000 -450000 600000 #VALUE!0 0 0 0 #VALUE!

-450000 600000 450000 -600000 #VALUE!0 0 0 0 0

f7 #VALUE!g7 = #VALUE!f4 #VALUE!g4 #VALUE!

Batang 10 f5 0.6 -0.8 0 0 540000 -720000 -540000 720000 #VALUE!g5 = 0.8 0.6 0 0 -720000 960000 720000 -960000 #VALUE!f6 0 0 0.6 -0.8 -540000 720000 540000 -720000 #VALUE!g6 0 0 0.8 0.6 720000 -960000 -720000 960000 #VALUE!

900000 -1200000 -900000 1200000 #VALUE!0 0 0 0 #VALUE!

-900000 1200000 900000 -1200000 #VALUE!0 0 0 0 #VALUE!

f5 #VALUE!g5 = #VALUE!f6 #VALUE!g6 #VALUE!

Batang 11f6 0.6 0.8 0 0 540000 720000 -540000 -720000 #VALUE!g6 = -0.8 0.6 0 0 720000 960000 -720000 -960000 #VALUE!f7 0 0 0.6 0.8 -540000 -720000 540000 720000 #VALUE!g7 0 0 -0.8 0.6 -720000 -960000 720000 960000 #VALUE!

900000 1200000 -900000 -1200000 #VALUE!0 0 0 0 #VALUE!

-900000 -1200000 900000 1200000 #VALUE!0 0 0 0 #VALUE!

f6 #VALUE!g6 = #VALUE!f7 #VALUE!

Page 52: Mekanika Rekayasa 4 ( Metode Matriks )

g7 #VALUE!

Batang 12f5 1 0 0 0 1250000 0 -1250000 0 #VALUE!g5 = 0 1 0 0 0 0 0 0 #VALUE!f7 0 0 1 0 -1250000 0 1250000 0 #VALUE!g7 0 0 0 1 0 0 0 0 #VALUE!

1250000 0 -1250000 0 #VALUE!0 0 0 0 #VALUE!

-1250000 0 1250000 0 #VALUE!0 0 0 0 #VALUE!

f5 #VALUE!g5 = #VALUE!f7 #VALUE!g7 #VALUE!

Reaksi Perletakan

#VALUE!#VALUE!#VALUE!#VALUE!

F1 -1250000 0 0 0 0 -270000 -360000 0 0 0 0 #VALUE! #VALUE!G1 = 0 0 0 0 0 -360000 -480000 0 0 0 0 #VALUE! = #VALUE!G4 0 0 0 0 -360000 0 0 0 0 360000 -480000 #VALUE! #VALUE!

#VALUE!#VALUE!#VALUE!#VALUE!

Hasil Komparasi Matriks dan SAP2000

Batang Matriks SAP20001 #VALUE! 583.332 #VALUE! 6253 #VALUE! 666.674 #VALUE! -138.895 #VALUE! 55.56

Page 53: Mekanika Rekayasa 4 ( Metode Matriks )

6 #VALUE! -69.447 #VALUE! 69.448 #VALUE! -55.569 #VALUE! -1111.11

10 #VALUE! 69.4411 #VALUE! -69.4412 #VALUE! -625

Page 54: Mekanika Rekayasa 4 ( Metode Matriks )

E = 21000A = 600I = 10000

Matrik kekakuan lokal batang1L = 500 cm = 1.25E+08

sin 270 = -1cos270 = 0

(K1) = 25200 0 0 -25200 0

0 20.16 5040 0 -20.16

0 5040 1680000 0 -5040

-25200 0 0 25200 0

0 -20.16 -5040 0 20.16

0 5040 840000 0 -5040

0 -1 0 0 0 01 0 0 0 0 0

= 0 0 1 0 0 00 0 0 0 -1 00 0 0 1 0 00 0 0 0 0 1

0 -1 0 0 0 01 0 0 0 0 00 0 1 0 0 00 0 0 0 -1 00 0 0 1 0 00 0 0 0 0 1

0 -20.16 -5040 0 20.16 -5040(Kg) = 25200 0 0 -25200 0 0

0 5040 1680000 0 -5040 840000

Kn/cm²cm²cm⁴

[T(1)]

[T(1)]T =

Page 55: Mekanika Rekayasa 4 ( Metode Matriks )

0 20.16 5040 0 -20.16 5040-25200 0 0 25200 0 0

0 5040 840000 0 -5040 1680000

u1 v1 θ1 u2 v2 θ2(Kg1) = 20.16 0 -5040 -20.16 0 -5040

0 25200 0 0 -25200 0-5040 0 1680000 5040 0 840000-20.16 0 5040 20.16 0 5040

0 -25200 0 0 25200 0-5040 0 840000 5040 0 1680000

Matrik kekakuan lokal batang 2L = 500 cm

= 1.25E+08 = 250000

(K) = 25200 0 0 -25200 0 00 20.16 5040 0 -20.16 50400 5040 1680000 0 -5040 840000

-25200 0 0 25200 0 00 -20.16 -5040 0 20.16 -50400 5040 840000 0 -5040 1680000

u2 v2 θ2 u3 v3 θ3(Kg2) = 25200 0 0 -25200 0 0

0 20.16 5040 0 -20.16 50400 5040 1680000 0 -5040 840000

-25200 0 0 25200 0 00 -20.16 -5040 0 20.16 -50400 5040 840000 0 -5040 1680000

Matrik kekakuan lokal batang 3L = 300 cm

= 27000000 = 90000sin 270 = -1cos 270 = 0

L³ L²

L³ L²

Page 56: Mekanika Rekayasa 4 ( Metode Matriks )

(K3) = 42000 0 0 -42000 0 00 93.33333 14000 0 -93.333333 140000 14000 2800000 0 -14000 1400000

-42000 0 0 42000 0 00 -93.33333 -14000 0 93.333333 -140000 14000 1400000 0 -14000 2800000

(T) = 0 -1 0 0 0 01 0 0 0 0 00 0 1 0 0 00 0 0 0 -1 00 0 0 1 0 00 0 0 0 0 1

0 1 0 0 0 0-1 0 0 0 0 00 0 1 0 0 00 0 0 0 1 00 0 0 -1 0 00 0 0 0 0 1

(Kg3) = 0 93.33333 14000 0 -93.333333 14000-42000 0 0 42000 0 0

0 14000 2800000 0 -14000 14000000 -93.33333 -14000 0 93.333333 -14000

42000 0 0 -42000 0 00 14000 1400000 0 -14000 2800000

u3 v3 θ3 u4 v4 θ4(Kg3) = 93.333333 0 14000 -93.33333 0 14000

0 42000 0 0 -42000 014000 0 2800000 -14000 0 1400000

-93.333333 0 -14000 93.33333 0 -140000 -42000 0 0 42000 0

14000 0 1400000 -14000 0 2800000

[T(1)]T =

Page 57: Mekanika Rekayasa 4 ( Metode Matriks )

overal stiffness

u1 v1 θ1 u2 v2 θ220.16 0 -5040 -20.16 0 -5040

0 25200 0 0 -25200 0-5040 0 1680000 5040 0 840000-20.16 0 5040 25220.16 0 5040

0 -25200 0 0 25220.16 5040-5040 0 840000 5040 5040 3360000

0 0 0 -25200 0 00 0 0 0 -20.16 -50400 0 0 0 5040 8400000 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

Page 58: Mekanika Rekayasa 4 ( Metode Matriks )

= 250000

0 0 0 0 0

5040 0 0

840000 0 4EI/L 0 2EI/L

0 0 0 0 0

-5040 0 0

1680000 0 2EI/L 0 4EI/L

0 0 0 0

0 0 0 0

0 0 1 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 1

0 1 0 0 0 0X -1 0 0 0 0 0

0 0 1 0 0 0

AE/L1 -AE/L1

12EI/L3 6EI/L2 -12EI/L3 6EI/L2

6EI/L2 -6EI/L2

-AE/L1 AE/L1

-12EI/L3 -6EI/L2 12EI/L3 -6EI/L2

6EI/L2 -6EI/L2

u21 v2

1 q21 u1

1 v11 q1

1

cos a sin a

-sin a cos a

cos a sin a-sin a cos a

Page 59: Mekanika Rekayasa 4 ( Metode Matriks )

0 0 0 0 1 00 0 0 -1 0 00 0 0 0 0 1

u1v1θ1u2v2θ2

u2v2θ2u3v3θ3

Page 60: Mekanika Rekayasa 4 ( Metode Matriks )

0 -1 0 0 0 01 0 0 0 0 0

X 0 0 1 0 0 00 0 0 0 -1 00 0 0 1 0 00 0 0 0 0 1

u3v3θ3u4v4θ4

Page 61: Mekanika Rekayasa 4 ( Metode Matriks )

u3 v3 θ3 u4 v4 θ40 0 0 0 0 0 u10 0 0 0 0 0 v10 0 0 0 0 0 θ1

-25200 0 0 0 0 0 u20 -20.16 5040 0 0 0 v20 -5040 840000 0 0 0 θ2

25293.333333333 0 14000 -93.33333 0 14000 u30 42020.16 -5040 0 -42000 0 v3

14000 -5040 4480000 -14000 0 1400000 θ3-93.333333333333 0 -14000 93.33333 0 -14000 u4

0 -42000 0 0 42000 0 v414000 0 1400000 -14000 0 2800000 θ4

Page 62: Mekanika Rekayasa 4 ( Metode Matriks )

u21

v21

q21

u11

v11

q11

Page 63: Mekanika Rekayasa 4 ( Metode Matriks )