PLANE TRUSS

Diketahui:p1 = 23000 kgp2 = 32000 kg

L1 = 8m = 800 cmL2 = 8m = 800 cmH = 8m = 800 cm

Profil WF 200 x 200 x 12 x 12

E = 2,1 x 10⁶ = 2100000 kg/cm²A = 71,53cm²I = 4980 cm⁴

PENYELESAIAN

Titik KoordinatX Y

1 800 8002 800 03 0 04 1600 0

E = 2100000 Kg/cm²A = 71,53 cm²I = 4980 cm⁴

Structure Stiffnes M atrix

M EM BER 1

x (cm) y (cm)3 0 02 800 01 800 1 0 187766,25

Cos Sin EA/LM ember Titik Koordinat L (cm)

1 0 -1 0K = 0 0 0 0

-1 0 1 00 0 0 0

  1 0 -1 0K = 0 0 0 0 X 187766.25  -1 0 1 0  0 0 0 0

7 8 1 2

187766.25 0 -187766.25 0 7K = 0 0 0 0 8

-187766.25 0 187766.25 0 10 0 0 0 2

M EM BER 2

x (cm) y (cm)4 1600 02 800 0

L (cm) Cos Sin EA/L

2 800 -1 0 187766,25

M ember Titik Koordinat

1 0 -1 0K = 0 0 0 0

-1 0 1 00 0 0 0

1 0 -1 0K = 0 0 0 0 X 187766,25

-1 0 1 00 0 0 05 6 1 2

187766,25 0 -187766,25 0 5

K = 0 0 0 0 6-187766,25 0 187766,25 0 1

0 0 0 0 2

M EM BER 3

x (cm) y (cm)3 0 01 800 800

M ember Titik Koordinat L (cm) Cos Sin EA/L

3 1131,37085 0,71 0,71 132770,7887

  0,5 0,5 -0,5 -0,5K = 0,5 0,5 -0,5 -0,5  -0,5 -0,5 0,5 0,5  -0,5 -0,5 0,5 0,5   0,5 0,5 -0,5 -0,5K = 0,5 0,5 -0,5 -0,5 X 132770,7887  -0,5 -0,5 0,5 0,5

  -0,5 -0,5 0,5 0,57 8 3 4

66385,39433 66385,39433 -66385,39433 -66385,39433 7K = 66385,39433 66385,39433 -66385,39433 -66385,39433 8

-66385,39433 -66385,39433 66385,39433 66385,39433 3-66385,39433 -66385,39433 66385,39433 66385,39433 4

M ember 4

x (cm) y (cm)2 800 01 800 800

M ember Titik Koordinat L (cm) Cos Sin EA/L

4 800 0 1 187766,25

  0 0 0 0K = 0 1 0 -1  0 0 0 0  0 -1 0 1

  0 0 0 0K = 0 1 0 -1 X 187766,25

  0 0 0 0  0 -1 0 1  1 2 3 4

0 0 0 0 1K = 0 187766,25 0 -187766,25 2

0 0 0 0 30 -187766,25 0 187766,25 4

M ember 5

x (cm) y (cm)4 1600 01 800 800

M ember Titik Koordinat L (cm) Cos Sin EA/L

5 1131,37085 -0,71 0,71 132770,7887

  0,5 -0,5 -0,5 0,5K = -0,5 0,5 0,5 -0,5  -0,5 0,5 0,5 -0,5  0,5 -0,5 -0,5 0,5   0,5 -0,5 -0,5 0,5

K = -0,5 0,5 0,5 -0,5 X 132770,7887  -0,5 0,5 0,5 -0,5  0,5 -0,5 -0,5 0,5  5 6 3 4

66385,39433 -66385,39433 -66385,39433 66385,39433 5K = -66385,39433 0 66385,39433 -66385,39433 6

-66385,39433 0 66385,39433 -66385,39433 366385,39433 0 -66385,39433 66385,39433 4

M atriks Kekakuan Dalam Sistem Koordinat Global

M EM BER 1

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

0 0 1 00 0 0 1

1 0 0 0[TT] = 0 1 0 0

0 0 1 0

0 0 0 1

187766,25 0 -187766,25 0K = 0 0 0 0

-187766,25 0 187766,25 00 0 0 0

[K]= ( [T]T.[k] ). [T]

7 8 1 2[K] = 187766,25 0 -187766,25 0 187766,25 0 -187766,25 0 7

0 0 0 0 × [T] =====► [K] = 0 0 0 0 8-187766,25 0 187766,25 0 -187766,25 0 187766,25 0 1

0 0 0 0 0 0 0 0 2

M EM BER 2

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

0 0 -1 00 0 0 -1

-1 0 0 0[TT] = 0 -1 0 0

0 0 -1 00 0 0 -1

187766,25 0 -187766,25 0K = 0 0 0 0

-187766,25 0 187766,25 00 0 0 0

[K]= ( [T]T.[k] ). [T]

5 6 1 2-187766,25 0 187766,25 0 187766,25 0 -187766,25 0 5

[K] = 0 0 0 0 × [T] =====► [K] = 0 0 0 0 6187766,25 0 -187766,25 0 -187766,25 0 187766,25 0 1

0 0 0 0 0 0 0 0 2

M EM BER 3

0,71 0,71 0,00 0,00

[T] = -0,71 0,71 0,00 0,000,00 0,00 0,71 0,710,00 0,00 -0,71 0,71

0,71 -0,71 0,00 0,00[TT] = 0,71 0,71 0,00 0,00

0,00 0,00 0,71 -0,710,00 0,00 0,71 0,71

66385,39433 66385,39433 -66385,39433 -66385,39433K = 66385,39433 66385,39433 -66385,39433 -66385,39433

-66385,39433 -66385,39433 66385,39433 66385,39433-66385,39433 -66385,39433 66385,39433 66385,39433

[K]= ( [T]T.[k] ). [T]

7 8 3 42,18279E-11 0 0 0 1,54346E-11 1,54346E-11 0 0 7

[K] = 93883,125 93883,125 -93883,125 -93883,125 × [T] =====► [K] = 1,45519E-11 132770,7887 0 -132770,79 80 0 0 0 0 0 0 0 3

-93883,125 -93883,125 93883,125 93883,125 0 -132770,7887 0 132770,79 4

M ember 4

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

0 0 0 10 0 -1 0

0 -1 0 0[TT] = 1 0 0 0

0 0 0 -10 0 1 0

0 0 0 0K = 0 187766,25 0 -187766,25

0 0 0 00 -187766,25 0 187766,25

[K]= ( [T]T.[k] ). [T]

1 2 3 40 -187766,25 0 187766,25 187766,25 0 -187766,25 0 1

[K] = 0 0 0 0 × [T] =====► [K] = 0 0 0 0 20 187766,25 0 -187766,25 -187766,25 0 187766,25 0 30 0 0 0 0 0 0 0 4

M ember 5

-0,71 0,71 0,00 0,00[T] = -0,71 -0,71 0,00 0,00

0,00 0,00 -0,71 0,710,00 0,00 -0,71 -0,71

-0,71 -0,71 0,00 0,00[TT] = 0,71 -0,71 0,00 0,00

0,00 0,00 -0,71 -0,710,00 0,00 0,71 -0,71

66385,39433 -66385,39433 -66385,39433 66385,39433K = -66385,39433 0 66385,39433 -66385,39433

-66385,39433 0 66385,39433 -66385,3943366385,39433 0 -66385,39433 66385,39433

[K]= ( [T]T.[k] ). [T]

5 6 3 40 46941,5625 0 0 -33192,6972 -33192,69716 0 0 5

[K] = 93883,125 -46941,5625 -93883,125 93883,125 × [T] =====► [K] = -33192,6972 99578,09149 0 -132770,79 60 0 0 0 0 0 0 0 3

-93883,125 0 93883,125 -93883,125 66385,39433 -66385,39433 0 132770,79 4

Joint Load vector

Gaya Yang Bekerja Terhadap Kontruksi

0 1[P] = 0 2

32000 3-23000 4

M atriks Kekakuan Struktur Dari M asing-M asing M ember

[S1] = 187766,25 0 0 0 10 0 0 0 20 0 0 0 30 0 0 0 4

187766,25 0 0 0 1[S2] = 0 0 0 0 2

0 0 0 0 30 0 0 0 4

0 0 0 0 1[S3] = 0 0 0 0 2

0 0 66385,39433 66385,39433 3

0 0 66385,39433 66385,39 4

0 0 0 0 1[S4] = 0 187766,25 0 -187766,25 2

0 0 0 0 30 -187766,25 0 187766,25 4

0 0 0 0 1[S5] = 0 0 0 0 2

0 0 66385,39433 -66385,39433 30 0 -66385,39433 66385,39 4

M atriks Kekakuan Struktur Secara Keseluruhan

375532,5 0 0 0 1[S] = 0 187766,25 0 -187766,25 2

0 0 132770,7887 0 30 -187766,25 0 320537,04 4

Invers M atriks Struktur Secara Keseluruhan

0,000002663 0,000000000 0,000000000 0,000000000[S-1] = 0,000000000 0,000012858 0,000000000 0,000007532

0,000000000 0,000000000 0,000007532 0,0000000000,000000000 0,000007532 0,000000000 0,000007532

Perpindahan Struktur yang Timbul

[d] = [S-1]*[P]

0 1[d] = -0,173230876 2

0,241016871 3-0,173230876 4

CEK

375532,5 0 0 0 0 0P = Sd = 0 187766,25 0 -187766,25 x -0,173230876 P = 0

0 0 132770,7887 0 0,241016871 320000 -187766,25 0 320537,0387 -0,173230876 -23000

M ember End Displacement and End Forces

M EM BER 1

0 7[v1] = 0 8

0 1-0,173230876 2

u = T v

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

0 0 1 00 0 0 1

0 7[u1] = 0 8

0 1-0,173230876 2

Q = ku

187766,25 0 -187766,25 0 0 7k = 0 0 0 0 [Q1] = 0 8

-187766,25 0 187766,25 0 0 10 0 0 0 0 2

F = TT Q

1 0 0 0 0 7[TT] = 0 1 0 0 [F1] = 0 8

0 0 1 0 0 10 0 0 1 0 2

M EM BER 2

  0 5[v2] = 0 6

  0 1  -0,173230876 2

u = T v

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

0 0 -1 00 0 0 -1

0 5[u2] = 0 6

0 10,173230876 2

Q = ku

187766,25 0 -187766,25 0 0 5k = 0 0 0 0 [Q2] = 0 6

-187766,25 0 187766,25 0 0 10 0 0 0 0 2

F = TT Q

-1 0 0 0 0 5[TT] = 0 -1 0 0 [F2] = 0 6

0 0 -1 0 0 10 0 0 -1 0 2

M EM BER 3

0 7[v3] = 0 8

0,241016871 3-0,173230876 4

u = T v

0,707106781 0,707106781 0 0[T] = -0,707106781 0,707106781 0 0

0 0 0,707106781 0,7071067810 0 -0,707106781 0,707106781

0 7[u3] = 0 8

0,047931937 3-0,292917391 4

Q = ku

132770,7887 0 -132770,7887 0 -6363,961031 7k = 0 0 0 0 [Q3] = 0 8

-132770,7887 0 132770,7887 0 6363,961031 30 0 0 0 0 4

F = TT Q

0,707106781 -0,707106781 0 0[TT] = 0,707106781 0,707106781 0 0 -4500 7

0 0 0,707106781 -0,707106781 [F3] = -4500 80 0 0,707106781 0,707106781 4500 3

4500 4

M ember 4

0 1[v4] = -0,173230876 2

0,241016871 3-0,173230876 4

u = T v

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

0 0 0 10 0 -1 0

-0,173230876 1[u4] = 0 2

-0,173230876 3-0,241016871 4

Q = ku

187766,25 0 -187766,25 0 0 1k = 0 0 0 0 [Q4] = 0 2

-187766,25 0 187766,25 0 0 30 0 0 0 0 4

F = TT Q

0 -1 0 0 0 1[TT] = 1 0 0 0 [F4] = 0 2

0 0 0 -1 0 30 0 1 0 0 4

M ember 5

0 5[v5] = 0 6

0,241016871 3-0,173230876 4

u = T v

-0,707106781 0,707106781 0 0[T] = -0,707106781 -0,707106781 0 0

0 0 -0,707106781 0,7071067810 0 -0,707106781 -0,707106781

0,0000000 5[u5] = 0,0000000 6

-0,2929174 3-0,0479319 4

Q = ku

132770,7887 0 -132770,7887 0 38891 5k = 0 0 0 0 [Q5] = 0 6

-132770,7887 0 132770,7887 0 -38891 30 0 0 0 0 4

F = TT Q

-0,707106781 -0,707106781 0 0 -27500 5[TT] = 0,707106781 -0,707106781 0 0 [F5] = 27500 6

0 0 -0,707106781 -0,707106781 27500 30 0 0,707106781 -0,707106781 -27500 4

Equilibrium Check

Titik MemberΣ1 2 3 4 5

1 0 0 0 0 0 02 0 0 0 0 0 03 0 0 4500 0 27500 320004 0 0 4500 0 -27500 -230005 0 0 0 0 -27500 -275006 0 0 0 0 27500 275007 0 0 -4500 0 0 -45008 0 0 -4500 0 0 -4500ΣF 0 0 0 0 0 0

→ Fx = 0∑↑ Fy = 0∑

Support Reactions

-27500 5R = 27500 6

-4500 7-4500 8

Gambar Support Reactions

M ember End Displacement in Local Coordinat System