Dinamika - Portal 4 Lantai

Dinamika - Portal 4 Lantai

Beton ɣ = 2400 = 0.0024Fc' = 25 MpaE =

= 23500 Mpa= 235000

g = 9.81 Kg/s²

= 70 cm B₁ = 50 cm= 70 cm = 60 cm= 70 cm B₂ = 40 cm= 70 cm = 60 cm= 60 cm B₃ = 30 cm= 70 cm = 50 cm= 50 cm B₄ = 20 cm= 70 cm = 30 cm

= 450 cm X₁ = 1100 cm= 400 cm X₂ = 900 cm= 350 cm X₃ = 600 cm= 300 cm X₄ = 300 cm

Tentukan bentuk -bentuk ragam struktur portal 4 lantai pada gambar diatas.Dengan menggunakan Metode :a. Stodolab. Holzer

Kg/mᵌ Kg/cmᵌ

4700√Fc'

Kg/cm²

K₁

K₂

K₃

K₄

L₁L₂L₃L₄


Penyelesaian :

Inersia yang terjadi :I₁ = 1/12 x b x hᵌ

= 2000833I₂ = 1/12 x b x hᵌ

= 2000833I₃ = 1/12 x b x hᵌ

= 1715000I₄ = 1/12 x b x hᵌ

= 1429167

Kekakuan tiap batang := 12EI + 12EI + 3EI + 12EI + 3EI

= 61918.79 + 61919 + 15480 + 61919 + 15480= 216716= 12EI + 12EI + 12EI + 12EI

= 88161.72 + 88162 + 88162 + 88162= 352647= 12EI + 12EI + 12EI

= 112800 + 112800 + 112800= 338400= 12EI + 12EI

= 149269 + 149269= 298537

Massa tiap batang :M₁ =

= 807.3394M₂ =

= 528.440M₃ =

= 220.1835M₄ =

= 44.037

cm⁴

cm⁴

cm⁴

cm⁴

K₁Lᵌ Lᵌ Lᵌ Lᵌ Lᵌ

K₂Lᵌ Lᵌ Lᵌ Lᵌ

K₃Lᵌ Lᵌ Lᵌ

K₄Lᵌ Lᵌ

W₁ / g

W₂ / g

W₃ / g

W₄ / g


Matrik Kekakuan :

0 0 298537K = K 0 = K -298537

0 00 0 0

Matrik Fleksibiltas :

F = 1.375E-05 1.041E-05 7.450E-06 4.614E-061.041E-05 1.041E-05 7.450E-06 4.614E-067.450E-06 7.450E-06 7.450E-06 4.614E-064.614E-06 4.614E-06 4.614E-06 4.614E-06

= 1 7.1250 5.3899 3.8591 2.3902518000 K 5.3899 5.3899 3.8591 2.3902

3.8591 3.8591 3.8591 2.39022.3902 2.3902 2.3902 2.3902

Matrik Massa :

0 0 0 44.037M = m 0 0 0 = M 0.000

0 0 0 0.0000 0 0 0.000

0.0227083 0 0 0M' = 1 0 0.0045417 0 0

m 0 0 0.0018924 00 0 0 0.0012386

Matrik Dinamis :

D = m 7.1250 5.3899 3.8591 2.3902 x 44.037518000 K 5.3899 5.3899 3.8591 2.3902 0.000

3.8591 3.8591 3.8591 2.3902 0.0002.3902 2.3902 2.3902 2.3902 0.000

= m 313.761 1186.756 2039.314 1929.725518000 K 237.351 1186.756 2039.314 1929.725

169.943 849.714 2039.314 1929.725105.258 526.289 1263.093 1929.725

= m 0.0006 0.0023 0.0039 0.0037k 0.0005 0.0023 0.0039 0.0037

0.0003 0.0016 0.0039 0.0037

K₄ -K₄-K₄ K₄ + K₃ -K₃

-K₃ K₃ + K₂ -K₂-K₂ K₂ + K₁

K⁻ᶦ

M₄M₃

M₂M₁


0.0002 0.0010 0.0024 0.0037


-298537 0 0636937 -338400 0

-338400 691046.88 -352646.90 -352646.9 569362.65

0.000 0.000 0.000220.183 0.000 0.000

0.000 528.440 0.0000.000 0.000 807.339

0.000 0.000 0.000220.183 0.000 0.000

0.000 528.440 0.0000.000 0.000 807.339


Bentuk ragam dan frekuensi natural dari Mode 1 :1 x = D x

ω²Ambil harga permisalan untuk permulaan x

1x = 1

11

Iterasi 10.0006 0.0023 0.0039 0.0037 x 1.0

= 0.0005 0.0023 0.0039 0.0037 1.00.0003 0.0016 0.0039 0.0037 1.00.0002 0.0010 0.0024 0.0037 1.0

0.0106 1= m 0.0104 = 0,0106 m 0.9860

k 0.0096 k 0.91210.0074 0.6992

Iterasi 20.0006 0.0023 0.0039 0.0037 x 1.0000

= 0.0005 0.0023 0.0039 0.0037 0.98600.0003 0.0016 0.0039 0.0037 0.91210.0002 0.0010 0.0024 0.0037 0.6992

0.0091 1= m 0.0089 = 0,0091 m 0.9837

k 0.0081 k 0.89850.0060 0.6660

Iterasi 30.0006 0.0023 0.0039 0.0037 x 1.0000

= 0.0005 0.0023 0.0039 0.0037 0.98370.0003 0.0016 0.0039 0.0037 0.89850.0002 0.0010 0.0024 0.0037 0.6660

0.0089 1= m 0.0087 = 0,0089 m 0.9834

k 0.0080 k 0.89660.0059 0.6617

Iterasi 40.0006 0.0023 0.0039 0.0037 x 1.0000

= 0.0005 0.0023 0.0039 0.0037 0.98340.0003 0.0016 0.0039 0.0037 0.89660.0002 0.0010 0.0024 0.0037 0.6617

0.0089 1= m 0.0087 = 0,0089 m 0.9833

k 0.0079 k 0.89640.0059 0.6612


Iterasi 50.0006 0.0023 0.0039 0.0037 x 1.0000

= 0.0005 0.0023 0.0039 0.0037 0.98330.0003 0.0016 0.0039 0.0037 0.89640.0002 0.0010 0.0024 0.0037 0.6612

0.0089 1= m 0.0087 = 0,0089 m 0.9833

k 0.0079 k 0.89630.0059 0.6611

Iterasi 60.0006 0.0023 0.0039 0.0037 x 1.0000

= 0.0005 0.0023 0.0039 0.0037 0.98330.0003 0.0016 0.0039 0.0037 0.89630.0002 0.0010 0.0024 0.0037 0.6611

0.0089 1= m 0.0087 = 0,0089 m 0.9833

k 0.0079 k 0.89630.0059 0.6611

Y X15 112 0.983332698.5 0.896334454.5 0.661084030 0

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

16

MODE 1

MODE 1


Bentuk ragam dan frekuensi natural dari Mode 4 :1 x = E' x

ω²E' = M' K

E' = k 0.02271 0.00000 0.00000 0.00000 x 298537m 0.00000 0.00454 0.00000 0.00000 -298537

0.00000 0.00000 0.00189 0.00000 00.00000 0.00000 0.00000 0.00124 0

6779 -6779.279 0 0= k -1355.856 2892.7557 -1536.9 0

m 0 -640.375 1307.7102 -667.33520 0 -436.8012 705.23328

Ambil harga permisalan permulaan x-1

x = 1-11

Iterasi 16779 -6779 0 0 x -1

= -1356 2893 -1537 0 10 -640 1308 -667 -10 0 -437 705 1

-13558.56 1= k 5785.5114 = -13558,6 k -0.4267

m -2615.42 m 0.19291142.0345 -0.0842

Iterasi 26779 -6779 0 0 x 1

= -1356 2893 -1537 0 -0.42670 -640 1308 -667 0.19290 0 -437 705 -0.0842

9672.0343 1= k -2886.676 = 9672,034 k -0.2985

m 581.71594 m 0.0601-143.6598 -0.0149

Iterasi 36779 -6779 0 0 x 1

= -1356 2893 -1537 0 -0.29850 -640 1308 -667 0.06010 0 -437 705 -0.0149

8802.5943 1= k -2311.651 = 8802,594 k -0.2626

m 279.68678 m 0.0318


-36.74593 -0.0042


-298537 0 0636937 -338400 0

-338400 691046.88 -352646.90 -352646.9 569362.65


Iterasi 46779 -6779 0 0 x 1.0000

= -1355.86 2893 -1537 0 -0.26260 -640 1308 -667 0.03180 0 -437 705 -0.0042

8559.5864 1= k -2164.355 = 8559,586 k -0.2529

m 212.50494 m 0.0248-16.82254 -0.002

Iterasi 56779 -6779 0 0 x 1.000

= -1356 2893 -1537 0 -0.2530 -640 1308 -667 0.0250 0 -437 705 -0.002

8493.4696 1= k -2125.466 = 8493,47 k -0.250

m 195.70106 m 0.023-12.23029 -0.001

Iterasi 66779 -6779 0 0 x 1.000

= -1356 2893 -1537 0 -0.2500 -640 1308 -667 0.0230 0 -437 705 -0.001

8475.7736 1= k -2115.172 = 8475,774 k -0.250

m 191.34437 m 0.023-11.08 -0.001

Y X15 112 -0.2508.5 0.0234.5 -0.0010 0

-2 0 20

2

4

6

8

10

12

14

16

MODE 4

MODE 2


Bentuk ragam dan frekuensi natural dari mode antara atau mode 2 :( dengan menggunakan nilai x dari mode 1 )

= = 1 4.91670 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0

= 1 0 0 0 - 1 4.91670 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0

= 0 -4.916697 -10.75647 -12.12130 1 0 00 0 1 00 0 0 1

= 0.0006 0.0023 0.0039 0.0037 x 0.0000m 0.0005 0.0023 0.0039 0.0037 0.0000k 0.0003 0.0016 0.0039 0.0037 0.0000

0.0002 0.0010 0.0024 0.0037 0.0000

= 0 -0.0007 -0.0026 -0.0036m 0 0.0000 -0.0010 -0.0018k 0 0.0000 0.0004 -0.0003

0 0.0000 0.0003 0.0013

Persamaan Iterasi yang terjadi :1 x = x

ω²

1Permisalan nilai x : 1

11

Iterasi 10 -0.0007 -0.0026 -0.0036 x 1

= 0 0.0000 -0.0010 -0.0018 10 0.0000 0.0004 -0.0003 10 0.0000 0.0003 0.0013 1

-0.0069 1= m -0.0028 = -0,0069m 0.4043

k 0.0002 k -0.02670.0015 -0.2226

S₀ M₄ X₄ M₃ X₃ M₂ X₂ M₁ X₁M₄ X₄ M₄ X₄ M₄ X₄ M₄ X₄

S₁

D₂

D₂

-2 0 20

2

4

6

8

10

12

14

16

MODE 4

MODE 2


10.7565 12.12130 00 00 0

10.7565 12.12130 00 00 0

-4.9167 -10.7565 -12.12131.0000 0.0000 0.00000.0000 1.0000 0.00000.0000 0.0000 1.0000


Iterasi 20.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.40430.0003 0.0016 0.0039 0.0037 -0.02670.0002 0.0010 0.0024 0.0037 -0.2226

0.0006 1= m 0.0005 = 0,0006m 0.7531

k 0.0001 k 0.0951-0.0003 -0.4693

Iterasi 30.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.75310.0003 0.0016 0.0039 0.0037 0.09510.0002 0.0010 0.0024 0.0037 -0.4693

0.0010 1= m 0.0008 = 0,0010m 0.8459

k 0.0002 k 0.1981-0.0005 -0.5725

Iterasi 40.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.84590.0003 0.0016 0.0039 0.0037 0.19810.0002 0.0010 0.0024 0.0037 -0.5725

0.0012 1= m 0.0010 = 0,0012m 0.8761

k 0.0004 k 0.3047-0.0006 -0.4929

Iterasi 50.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.87610.0003 0.0016 0.0039 0.0037 0.30470.0002 0.0010 0.0024 0.0037 -0.4929

0.0020 1= m 0.0018 = 0,0020m 0.9254

k 0.0011 k 0.57100.0000 0.0000

Iterasi 60.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.92540.0003 0.0016 0.0039 0.0037 0.57100.0002 0.0010 0.0024 0.0037 0.0000

0.0050 1= m 0.0048 = 0,0050m 0.9703

k 0.0041 k 0.82310.0025 0.5098


Iterasi 70.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.97030.0003 0.0016 0.0039 0.0037 0.82310.0002 0.0010 0.0024 0.0037 0.5098

0.0080 1= m 0.0078 = 0,0080m 0.9815

k 0.0071 k 0.88590.0051 0.6394

Iterasi 80.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.98150.0003 0.0016 0.0039 0.0037 0.88590.0002 0.0010 0.0024 0.0037 0.6394

0.0087 1= m 0.0086 = 0,0087m 0.9831

k 0.0078 k 0.89500.0057 0.6583

Iterasi 90.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.98310.0003 0.0016 0.0039 0.0037 0.89500.0002 0.0010 0.0024 0.0037 0.6583

0.0088 1= m 0.0087 = 0,0088m 0.9833

k 0.0079 k 0.89620.0058 0.6607

Iterasi 100.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.98330.0003 0.0016 0.0039 0.0037 0.89620.0002 0.0010 0.0024 0.0037 0.6607

0.0088 1= m 0.0087 = 0,0088m 0.9833

k 0.0079 k 0.89630.0058 0.6610

Iterasi 110.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.98330.0003 0.0016 0.0039 0.0037 0.89630.0002 0.0010 0.0024 0.0037 0.6610

0.0088 1= m 0.0087 = 0,0088m 0.9833

k 0.0079 k 0.89630.0059 0.6611


Iterasi 120.0006 0.0023 0.0039 0.0037 x 1

= 0.0005 0.0023 0.0039 0.0037 0.98330.0003 0.0016 0.0039 0.0037 0.89630.0002 0.0010 0.0024 0.0037 0.6611

0.0089 1= m 0.0087 = 0,0089m 0.9833

k 0.0079 k 0.89630.0059 0.6611

Y X15 112 0.98338.5 0.89634.5 0.66110 0

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

16

MODE 2

MODE 2


Bentuk ragam dan frekuensi natural dari mode antara atau mode 3 :( dengan menggunakan nilai x dari mode 2 )

= = 1 4.91670 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0

= 0 -4.916697 -10.75647 -12.12134 - 1 4.91670 1 0 0 0 00 0 1 0 0 00 0 0 1 0 0

= -1 -9.83336 -21.51244 -24.241080 1 0 00 0 1 00 0 0 1

= -1 -9.83336 -21.51244 -24.24108 x 00 1 0 0 m 00 0 1 0 k 00 0 0 1 0

= 0 -0.00069 -0.00257 -0.00359m 0 0.00004 -0.00099 -0.00183k 0 0.00003 0.00041 -0.00025

0 0.00002 0.00025 0.00126

Persamaan Iterasi yang terjadi :1 x = x

ω²

-1Permisalan nilai x : 1

-1-1

Iterasi 10 -0.0007 -0.0026 -0.0036 x -1.0000

= 0 0.0000 -0.0010 -0.0018 1.00000 0.0000 0.0004 -0.0003 -1.00000 0.0000 0.0003 0.0013 -1.0000

0.0055 1= m 0.0029 = 0,0014m 0.5220

k -0.0001 k -0.0236-0.0015 -0.2736

S₁' M₄ X₄ M₃ X₃ M₂ X₂ M₁ X₁M₄ X₄ M₄ X₄ M₄ X₄ M₄ X₄

S₂

D₃

D₃

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

16

MODE 2

MODE 2


10.7560 12.11970 00 00 0

10.7560 12.11970 00 00 0

-0.00069 -0.00258 -0.003620.00004 -0.00099 -0.001830.00003 0.00041 -0.000250.00002 0.00025 0.00126


Iterasi 20 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.52200 0.00003 0.00041 -0.00025 -0.02360 0.00002 0.00025 0.00126 -0.2736

0.0007 1= m 0.0005 = -0,0006m 0.7937

k 0.0001 k 0.1072-0.0003 -0.5000

Iterasi 30 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.79370 0.00003 0.00041 -0.00025 0.10720 0.00002 0.00025 0.00126 -0.5000

0.0010 1= m 0.0008 = 0,0018m 0.8594

k 0.0002 k 0.1959-0.0006 -0.6055

Iterasi 40 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.85940 0.00003 0.00041 -0.00025 0.19590 0.00002 0.00025 0.00126 -0.6055

0.0011 1= m 0.0009 = 0,0013m 0.8748

k 0.0003 k 0.2364-0.0007 -0.6477

Iterasi 50 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.87480 0.00003 0.00041 -0.00025 0.23640 0.00002 0.00025 0.00126 -0.6477

0.0011 1= m 0.0010 = 0,0012m 0.8796

k 0.0003 k 0.2533-0.0007 -0.6646

Iterasi 60 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.87960 0.00003 0.00041 -0.00025 0.25330 0.00002 0.00025 0.00126 -0.6646

0.0011 1= m 0.0010 = 0,0012m 0.8813

k 0.0003 k 0.2601-0.0008 -0.6713


Iterasi 70 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88130 0.00003 0.00041 -0.00025 0.26010 0.00002 0.00025 0.00126 -0.6713

0.0011 1= m 0.0010 = 0,0011m 0.8820

k 0.0003 k 0.2628-0.0008 -0.6740

Iterasi 80 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88200 0.00003 0.00041 -0.00025 0.26280 0.00002 0.00025 0.00126 -0.6740

0.0011 1= m 0.0010 = 0,0011m 0.8822

k 0.0003 k 0.2638-0.0008 -0.6750

Iterasi 90 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88220 0.00003 0.00041 -0.00025 0.26380 0.00002 0.00025 0.00126 -0.6750

0.0011 1= m 0.0010 = 0,0011m 0.8823

k 0.0003 k 0.2643-0.0008 -0.6754

Iterasi 100 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88230 0.00003 0.00041 -0.00025 0.26430 0.00002 0.00025 0.00126 -0.6754

0.0011 1= m 0.0010 = 0,0011m 0.8824

k 0.0003 k 0.2644-0.0008 -0.6756

Iterasi 110 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88240 0.00003 0.00041 -0.00025 0.26440 0.00002 0.00025 0.00126 -0.6756

0.0011 1= m 0.0010 = 0,0011m 0.8824

k 0.0003 k 0.2645-0.0008 -0.6757


Iterasi 120 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88240 0.00003 0.00041 -0.00025 0.26450 0.00002 0.00025 0.00126 -0.6757

0.0011 1= m 0.0010 = 0,0011m 0.8824

k 0.0003 k 0.2645-0.0008 -0.6757

Iterasi 130 -0.00069 -0.00257 -0.00359 x 1.0000

= 0 0.00004 -0.00099 -0.00183 0.88240 0.00003 0.00041 -0.00025 0.26450 0.00002 0.00025 0.00126 -0.6757

0.0011 1= m 0.0010 = 0,0011m 0.8824

k 0.0003 k 0.2645-0.0008 -0.6757

Y X15 112 0.88248.5 0.26454.5 -0.67570 0

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

16

MODE 3

MODE 3


-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

16

MODE 3

MODE 3

Mode 1 :

ω² = 112.9923 k/m

mass no. m m ω² ɸ m ω² ɸ Σ m ω² ɸ 1/k 1/k Σ m ω² ɸ4 44.0367 4975.809 1 4975.809 4975.809 3.349668E-06 0.01673 220.1835 24879.04 0.983333 24464.38 29440.18 2.955083E-06 0.08702 528.4404 59709.7 0.896334 53519.86 82960.05 2.835698E-06 0.23521 807.3394 91223.16 0.661084 60306.17 143266.2 4.614339E-06 0.6611

0.0000

Mode 2 :

ω² = 112.9935 k/m

mass no. m m ω² ɸ m ω² ɸ Σ m ω² ɸ 1/k 1/k Σ m ω² ɸ4 44.0367 4975.863 1 4975.863 4975.863 3.349668E-06 0.01673 220.1835 24879.31 0.983333 24464.64 29440.5 2.955083E-06 0.08702 528.4404 59710.35 0.896333 53520.39 82960.89 2.835698E-06 0.23531 807.3394 91224.15 0.661082 60306.63 143267.5 4.614339E-06 0.6611

0.0000

Mode 3 :

ω² = 876.2404 k/m

mass no. m m ω² ɸ m ω² ɸ Σ m ω² ɸ 1/k 1/k Σ m ω² ɸ4 44.0367 38586.73 1 38586.73 38586.73 3.349668E-06 0.12933 220.1835 192933.7 0.8824 170245.6 208832.4 2.955083E-06 0.61712 528.4404 463040.8 0.2645 122484.2 331316.6 2.835698E-06 0.93951 807.3394 707423.5 -0.6757 -478016 -146700 4.614339E-06 -0.6769

0.0012

Mode 3 :

ω² = 8475.774 k/m

mass no. m m ω² ɸ m ω² ɸ Σ m ω² ɸ 1/k 1/k Σ m ω² ɸ4 44.0367 373245.1 1 373245.1 373245.1 3.349668E-06 1.25023 220.1835 1866225 -0.249555 -465726 -92480.83 2.955083E-06 -0.27332 528.4404 4478941 0.022575 101114.1 8633.261 2.835698E-06 0.02451 807.3394 6842826 -0.001307 -8945.324 -312.0629 4.614339E-06 -0.0014

0.0001

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Dinamika - Portal 4 Lantai