Page 1 TUGAS ANALISIS PANEL BETON BERTULANG Diketahui : DOF = 4 fc' = 30 MPa fy = 400 MPa E beton = 30000 MPa E baja = 200000 MPa t = 300 mm v beton = 0.2 v baja = 0.3 Nodal Koordinat : x1 = 0 mm y1 = 0 mm x2 = 3000 mm y2 = 0 mm x3 = 3000 mm y3 = 2000 mm x4 = 0 mm y4 = 2000 mm Ditanya : 1). Gaya pada tulangan. Jawaban : *) Bagi panel menjadi 2 bagian segitiga : *) Bagian baja tulangan 2 1 2). {σ} principal dan {ɛ} principal. 3). Berapakah nilai α, dimana α=(σ2/σ1) ρ = 1,25% ρ = 1,25% ρ = 0.5% 1 2 1 2 3 4 5 6 3 4 5 6 1 2 2 1 4 5 4 1 2 Node 4 (0,2) Node 1 (0,0) Node 3 (3,2) Node 2 (3,0)
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Page 1
TUGAS ANALISIS PANEL BETON BERTULANGDiketahui :DOF = 4fc' = 30 MPa fy = 400 MPa E beton = 30000 MPa E baja = 200000 MPa t = 300 mmv beton = 0.2v baja = 0.3
Nodal Koordinat :x1 = 0 mm y1 = 0 mmx2 = 3000 mm y2 = 0 mmx3 = 3000 mm y3 = 2000 mmx4 = 0 mm y4 = 2000 mm
Ditanya :1). Gaya pada tulangan.
Jawaban :*) Bagi panel menjadi 2 bagian segitiga :
*) Bagian baja tulangan
2
1
2). {σ} principal dan {ɛ} principal.3). Berapakah nilai α, dimana α=(σ2/σ1)
ρ = 1,25%ρ = 1,25%
ρ = 0.5%
1
21
2
3
4
5
6
3
4
56
1
2
2
1
4
5 4
1
2
Node 4 (0,2)
Node 1 (0,0)
Node 3 (3,2)
Node 2 (3,0)
Page 2
**) Menyusun Matriks Kekakuan Baja Tulangan*) Menentukan Perhitungan Luas (A) Untuk Tulangan
1.25%0.50%
A1 = 11250 mm2A2 = 3000 mm2
*) Kekakuan Lokal Tulangan[K1] lokal = 750000 0 -750000 0
tul1&2 0 0 0 0-750000 0 750000 0
0 0 0 0
[K2] lokal = 300000 0 -300000 0tul2 0 0 0 0
-300000 0 300000 00 0 0 0
*) Definisikan Matriks [T][T1] tul1 = 6.12574E-17 1 0 0
-1 6.12574227E-17 0 00 0 6.1257423E-17 10 0 -1 6.12574227E-17
[T2] tul 2 = 1 0 0 00 1 0 00 0 1 00 0 0 1
[T3] tul 3 = 6.12574E-17 -1 0 01 6.12574227E-17 0 00 0 6.1257423E-17 -10 0 1 6.12574227E-17
[T1]T = 6.12574E-17 -1 0 0tul1 1 6.12574227E-17 0 0
0 0 6.1257423E-17 -10 0 1 6.12574227E-17
[T2]T = 1 0 0 0tul2 0 1 0 0
0 0 1 00 0 0 1
[T3]T = 6.12574E-17 1 0 0tul3 -1 6.12574227E-17 0 0
0 0 6.1257423E-17 10 0 -1 6.12574227E-17
ρ1 =ρ2 =
Page 3
*) Matriks Kekakuan Tulangan Global [K] [K] = [T]T . [K]lokal . [T]
1 2 3 4[K] tul1 = 2.81435E-27 4.59430671E-11 -2.814354E-27 -4.5943067E-11 1
4.59431E-11 750000 -4.594307E-11 -750000 2-2.8144E-27 -4.5943067E-11 2.8143539E-27 4.59430671E-11 3-4.5943E-11 -750000 4.5943067E-11 750000 4
1 2 3 4[K] tul2 = 300000 0 -300000 0 1
0 0 0 0 2-300000 0 300000 0 3
0 0 0 0 4
1 2 3 4[K] tul3 = 2.81435E-27 -4.5943067E-11 -2.814354E-27 4.59430671E-11 1
-4.5943E-11 750000 4.5943067E-11 -750000 2-2.8144E-27 4.59430671E-11 2.8143539E-27 -4.5943067E-11 34.59431E-11 -750000 -4.594307E-11 750000 4
*) Menyusun Matriks Global Tulangan Gabungan [K]tul5 6 7 8
[K] tul = 300000 4.59430671E-11 -300000 0 54.59431E-11 750000 0 0 6
-300000 0 300000 -4.5943067E-11 70 0 -4.594307E-11 750000 8
**) Menyusun Matriks Kekakuan Panel Beton*) Menyusun Matrik Fungsi Koordinat Panel Beton
MATRIKS A :[A1] = 1 x1 y1 0 0
0 0 0 1 x11 x2 y2 0 00 0 0 1 x21 x3 y3 0 00 0 0 1 x3
1 0 0 0 00 0 0 1 01 3000 0 0 00 0 0 1 30001 3000 2000 0 00 0 0 1 3000
Page 4
[A2] = 1 x1 y1 0 00 0 0 1 x11 x3 y3 0 00 0 0 1 x31 x4 y4 0 00 0 0 1 x4
1 0 0 0 00 0 0 1 01 3000 2000 0 00 0 0 1 30001 0 2000 0 00 0 0 1 0
INVERS MATRIK A :[A1]-1 = 1 0 0 0 0
-0.000333 0 0.000333 0 00 0 -0.000500 0 0.00050 1 0 0 00 -0.000333 0 0.000333 00 0 0 -0.000500 0
[A2]-1 = 1 0 0 0 00.000000 0 0.000333 0 -0.000333
-0.0005000 0 0.000000 0 0.00050 1 0 0 00 0.000000 0 0.000333 00 -0.000500 0 0.000000 0
MATRIKS G :[G]= 0 1 0 0 0
0 0 0 0 00 0 1 0 1
MATRIKS B :[B1] = -0.00033333 0 0.00033333333 0 0
0.000000 0 0.000000 -0.000500 00 -0.00033333333 -0.000500 0.000333333333 0.0005
[B2] = 0 0 0.00033333333 0 -0.0003333330.000000 -0.0005 0.000000 0 0.000000
-0.0005000 0 0.000000 0.000333333333 0.0005
[B1]T = -0.00033333 0 00.000000 0 -0.000333
0.000333333 0 -0.0005000 -0.0005 0.000333333330 0 0.00050 0.0005 0
[B2]T = 0 0 -0.00050.000000 -0.0005 0.000000
0.000333333 0 0.0000000 0 0.00033333333
-0.00033333 0 0.00050 0.0005 -0.0003333333
Page 5
*) Elemen Beton Linier ElastikG = 12500
[D] = 1.041666667 30000 6000 06000 30000 00 0 12000
[D] = 31250 6250 06250 31250 00 0 12500
*) Menyusun Matriks Kekakuan Panel Beton Global [K] 1 2 3 4 5
[K] pan1 = 3125000 0 -3125000 937500 00.000000 1250000 1875000.00 -1250000.00 -1875000-3125000 1875000 5937500.000 -2812500 -2812500937500 -1250000 -2812500 8281250 1875000
0.000000 -1875000 -2812500.00 1875000.000 2812500-937500 0 937500.00 -7031250 0
1 2 3 4 5[K] pan2 = 2812500 0 0 -1875000 -2812500
0.000000 7031250 -937500.00 0.000000 9375000 -937500 3125000.00 0 -3125000
-1875000 0 0 1250000 1875000-2812500.0 937500 -3125000.0 1875000.000 59375001875000 -7031250 937500.00 -1250000 -2812500
*) Menyusun Gabungan Matriks Kekakuan Panel Beton Global [K] 5 6 7 8
[K] pan = 5937500.00 0.000000 -3125000 937500 50.000000 8281250 1875000 -1250000 6
-3125000.0 1875000.00 5937500 -2812500 7937500.00 -1250000 -2812500 8281250 8
*) Assembly Gabungan Matriks Kekakuan Struktur Global [K]str5 6 7 8
[K] global = 6237500.0 0.00 -3425000.00 937500.00 50.00 9031250.00 1875000.00 -1250000.00 6
-3425000.0 1875000.00 6237500.00 -2812500.00 7937500.00 -1250000.00 -2812500.00 9031250.00 8
[K]-1 global = 0.000000 0.000000 1.4722276E-07 1.71806608E-08 50.000000 0 -5.020067E-08 4.13991826E-09 60.000000 0.000000 2.8644994E-07 6.69750337E-08 70.000000 4.13991826E-09 6.6975034E-08 1.3037346E-07 8
*) Mencari Deformasi Struktur 1.3072228E+27Beban F = 0,1 . Ag . fc' 7.6498051E-28
= 2700000 N
{x} struktur = [K]-1 global . {P}
{P} = 00
27000000
{x} struktur = 0.397501442 U5
Page 6
-0.13554182 U60.773414832 U70.180832591 U8
Page 7
{δ} beton 1= 0 {δ} beton 1= 00 00 0.397501442373 U50 -0.13554182112 U6
0.397501442 U5 0.773414832367 U7-0.13554182 U6 0.180832590994 U8
*) Regangan Principal Panel Beton{ɛ} 1 = 0 = ɛxx {ɛ} 2 = -0.00012530446 = ɛxx
-6.7771E-05 = ɛyy 9.04162955E-05 = ɛyy0.000198751 0.000281249279
Panel Beton 1= 0.0001706
= -0.0002383
Panel Beton 2= 0.0003025
= -0.0003374
Panel Beton 1σxx = -0.42356819098σyy -2.11784095492τxy 2.484384014829
= 1.48745496421
= -4.02886411012
σ2σ1
= -0.36919959
Panel Beton 2σxx = -3.35066263225σyy 2.042356338458τxy 3.515615985171
= 4.532549328126
= -5.84085562192
σ2σ1
= -0.77600777
*) Mencari Regangan {ɛ} Panel Beton{ɛ} = [B] {δ}
= ɣxy = ɣxy
*) ɛ max principal
ɛ1 max =
ɛ2 max =
ɛ1 max =
ɛ2 max =
σ1 max =
σ2 max =
*) Check apakah σ1 max dan σ2 max masih di dalam biaxial diagram!α =
σ1 max =
σ2 max =
*) Check apakah σ1 max dan σ2 max masih di dalam biaxial diagram!α =
( + 𝜎𝑥𝑥)/2+√(𝜎𝑦𝑦 〖 ( − )𝜎𝑥𝑥 𝜎𝑦𝑦 〗 ^2/2+ )𝜏𝑥𝑦( + 𝜎𝑥𝑥)/2−√(𝜎𝑦𝑦 〖 ( − )𝜎𝑥𝑥 𝜎𝑦𝑦 〗 ^2/2+ )𝜏𝑥𝑦
( + )/2+√(𝜀𝑥𝑥 𝜀𝑦𝑦 〖 ( − )𝜀𝑥𝑥 𝜀𝑦𝑦 〗
^2/2+( 〖𝛾𝑥𝑦〗 ^2 ))
( + 𝜎𝑥𝑥)/2+√(𝜎𝑦𝑦 〖 ( − )𝜎𝑥𝑥 𝜎𝑦𝑦 〗 ^2/2+ )𝜏𝑥𝑦( + 𝜎𝑥𝑥)/2−√(𝜎𝑦𝑦 〖 ( − )𝜎𝑥𝑥 𝜎𝑦𝑦 〗 ^2/2+ )𝜏𝑥𝑦
( + )/2−√(𝜀𝑥𝑥 𝜀𝑦𝑦 〖 ( − )𝜀𝑥𝑥 𝜀𝑦𝑦 〗
^2/2+( 〖𝛾𝑥𝑦〗 ^2 ))
( + )/2+√(𝜀𝑥𝑥 𝜀𝑦𝑦 〖 ( − )𝜀𝑥𝑥 𝜀𝑦𝑦 〗
^2/2+( 〖𝛾𝑥𝑦〗 ^2 ))( + )/2−√(𝜀𝑥𝑥 𝜀𝑦𝑦 〖 ( − )𝜀𝑥𝑥 𝜀𝑦𝑦 〗
^2/2+( 〖𝛾𝑥𝑦〗 ^2 ))
Page 8
*) Menghitung Gaya Pada TulanganTulangan 1
{x} global tul 1 member = 00
0.7734 U70.1808 U8
{x} lokal tul 1 member = [T] . {x} global member= 0
00.18083-0.77341
[P] member = [K] lokal member . {x} lokal member= -135624.443245
0135624.44325
0.00000
Tulangan 2{x} global tul 2 member = 0.773414832367 U7
0.180832590994 U80.397501442373 U5-0.13554182112 U6
{x} lokal tul 2 member = [T] . {x} global member= 0.7734148323670.180832590994
0.39750-0.13554
[P] member = [K] lokal member . {x} lokal member= 112774.0169982
0-112774.01700
0.00000
Tulangan 3{x} global tul 3 member = 0.397501442373 U5
-0.13554182112 U600
{x} lokal tul 3 member = [T] . {x} global member= -0.13554182112-0.39750144237
0.000000.00000
[P] member = [K] lokal member . {x} lokal member= -101656.365836
0101656.36584
0.00000
Page 9
KESIMPULAN :
1). Didapatkan Gaya Aksial Pada Baja TulanganTulangan 1 : 135624.44 NTulangan 2 : 112774.017 NTulangan 3 : 101656.37 N
Panel Beton 1σ1 max = 2.178411872σ2 max = -4.9535638
α = σ2σ1
= -0.43976659 Panel Beton 2
σ1 max = 4.169733974σ2 max = -5.04076706
α = σ2σ1
= -0.827202273). Apakah struktur beton mengalamai damage ataukah undamage? Panel Beton 1
σ1 = 0.072613729|fc| σ2 = -0.16511879
|fc| Panel Beton 2
σ1 = 0.138991132|fc| σ2 = -0.16802557
|fc|
2). {σ} principal dan {ɛ} principal
→ Struktur Beton Sudah mengalami Damage
Page 10
3
3
4
2
1
4
3
Page 11
3
Page 12
0y10y20y3
00000
2000
Page 13
0y10y30y4
000
20000
2000
00000
0.0005
0000
-0.0003330.0005
010
00.0005
0
00.0005
-0.000333333
Page 14
6-937500 1
0 2937500 3
-7031250 40 5
7031250 6
61875000 1-7031250 2937500 3
-1250000 4-2812500 58281250 6
Page 15
TUGAS PERILAKU STRUKTUR BETON
Diketahui :DOF = 4fc' = 30 MPa fy = 400 MPa E beton = 30000 MPa E baja = 200000 MPa t = 300 mmv beton = 0.3 , v baja = 0.2
Nodal Koordinat :x1 = 0 mm y1 = 0x2 = 3000 mm y2 = 0x3 = 3000 mm y3 = 3000x4 = 0 mm y4 = 3000
Ditanya :1). Gaya pada tulangan.
Jawaban :*) Bagi panel menjadi 2 bagian segitiga :
*) Bagian baja tulangan
*) Menentukan Perhitungan Luas (A) Untuk Tulangan
2). {σ} principal dan {ɛ} principal.3). Berapakah nilai α, dimana α=(σ2/σ1)
ρ = 1,25%ρ = 1,25%
ρ = 0.5%
1
21
2
4
5
6
3
4
56
1
2
II
I
4
52
1.25%0.50%
A1 = 11250 mm2A2 = 4500 mm2
*) Kekakuan Lokal Tulangan[K1] lokal = 750000 0 -750000
tul1&2 0 0 0-750000 0 750000
0 0 0
[K2] lokal = 300000 0 -300000tul2 0 0 0
-300000 0 3000000 0 0
*) Definisikan Matriks [T][T1] tul1 = 6.125742E-17 1 0 0
-1 6.1257423E-17 0 00 0 6.12574227E-17 10 0 -1 6.125742E-17
[T2] tul 2 = 1 0 0 00 1 0 00 0 1 00 0 0 1
[T3] tul 3 = 6.125742E-17 -1 0 01 6.1257423E-17 0 00 0 6.12574227E-17 -10 0 1 6.125742E-17
[T1]T = 6.125742E-17 -1 0 0tul1 1 6.1257423E-17 0 0
0 0 6.12574227E-17 -10 0 1 6.125742E-17
[T2]T = 1 0 0 0tul2 0 1 0 0
0 0 1 00 0 0 1
[T3]T = 6.125742E-17 1 0 0tul3 -1 6.1257423E-17 0 0
0 0 6.12574227E-17 10 0 -1 6.125742E-17
*) Matriks Kekakuan Tulangan Global [K]
ρ1 =ρ2 =
[K] = [T]T . [K]lokal . [T]
[K] tul1 = 2.814354E-27 4.5943067E-11 -2.8143539E-27 -4.59431E-114.594307E-11 750000 -4.5943067E-11 -750000-2.81435E-27 -4.594307E-11 2.81435388E-27 4.594307E-11-4.59431E-11 -750000 4.59430671E-11 750000
[K] tul2 = 300000 0 -300000 00 0 0 0
-300000 0 300000 00 0 0 0
[K] tul3 = 2.814354E-27 -4.594307E-11 -2.8143539E-27 4.594307E-11-4.59431E-11 750000 4.59430671E-11 -750000-2.81435E-27 4.5943067E-11 2.81435388E-27 -4.59431E-114.594307E-11 -750000 -4.5943067E-11 750000
*) Menyusun Matriks Kekakuan Panel Beton
*) Menyusun Matrik Fungsi Koordinat Panel Beton
MATRIKS A :[A1] = 1 x1 y1 0
0 0 0 11 x2 y2 00 0 0 11 x3 y3 00 0 0 1
1 0 0 00 0 0 11 3000 0 00 0 0 11 3000 3000 00 0 0 1
[A2] = 1 x1 y1 00 0 0 11 x3 y3 00 0 0 11 x4 y4 00 0 0 1
1 0 0 00 0 0 11 3000 3000 00 0 0 11 0 3000 0
0 0 0 1
INVERS MATRIK A :[A1]-1 = 1 0 0 0
-0.000333 0 0.000333 00 0 -0.000333 00 1 0 00 -0.000333 0 0.0003330 0 0 -0.000333
[A2]-1 = 1 0 0 00.000000 0 0.000333 0
-0.0003333 0 0.000000 00 1 0 00 0.000000 0 0.0003330 -0.000333 0 0.000000
MATRIKS G :[G]= 0 1 0 0
0 0 0 00 0 1 0
MATRIKS B :[B1] = -0.000333333 0 0.000333333333 0
0.000000 0 0.000000 -0.0003330 -0.0003333333 -0.000333 0.0003333333
[B2] = 0 0 0.000333333333 00.000000 -0.0003333333 0.000000 0
-0.0003333 0 0.000000 0.0003333333
[B1]T = -0.000333333 0 00.000000 0 -0.000333
0.0003333333 0 -0.0003330 -0.0003333333 0.0003333333330 0 0.0003333333330 0.00033333333 0
[B2]T = -0.0003333333 00 0.0000000 -0.000333
*) Elemen Beton Linier ElastikG = 6.541078E-06
[D] = 1.0989010989 30000 9000 0
9000 30000 00 0 5.952381E-06
[D] = 32967.032967 9890.10989011 09890.1098901 32967.032967 0
0 0 6.54107797E-06
*) Matriks Kekakuan Panel Beton Global [K] 1 2 3 4
[K] pan1 = 4945054.9451 0 -4945054.94505 1483516.48350.000000 0.0009811617 0.000981 -0.000981
-4945054.945 0.0009811617 4945054.946036 -1483516.4841483516.4835 -0.0009811617 -1483516.4845 4945054.946
0.000000 -0.0009811617 -0.000981 0.000981-1483516.484 0 1483516.483516 -4945054.945
1 2 3 4[K] pan2 = #VALUE! #VALUE! #VALUE! #VALUE!
#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!
*) Assembly Gabungan Matriks Kekakuan Panel Beton Global [K] 5 6 7 8
[K] global = 0.000981 -0.000981 #VALUE! #VALUE!0.000000 4945055 #VALUE! #VALUE!0.000000 0.000000 #VALUE! #VALUE!0.000000 0 #VALUE! #VALUE!
[K]-1 global = #VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!
*) Mencari Deformasi Struktur #VALUE!Beban F = 0,1 . Ag . fc' #VALUE!
= 27000000 N
{x} struktur = [K]-1 global . {P}
{P} = 00
270000000
{x} struktur = #VALUE!
#VALUE!#VALUE!#VALUE!
*) Mencari Regangan {ɛ} Panel Beton
(0,3) (3,3)4 3
1 2(0,0) (3,0)
mmmmmmmm
3
4
53
4
1
2
2
1
4
3
0000
0000
0 0x1 y10 0x2 y20 0x3 y3
0 00 00 0
3000 00 0
3000 3000
0 0x1 y10 0x3 y30 0x4 y4
0 00 00 0
3000 30000 0
0 3000
0 00 0
0.0003333333 00 00 00 0.0003333333
0 0-0.000333 0
0.0003333333 00 00 -0.0003330 0.0003333333
0 00 11 0
0 00 0.0003333333
0.0003333333 0
-0.000333333 00.000000 0.0003333333
0.0003333333 -0.000333333
5 60 -1483516.484 1 4945055 0 -4945055 1483516
-0.000981162 0 2 0 0.000981 0.000981 -0.00098-0.000981162 1483516.4835 3 -4945055 0.000981 4945055 -14835160.0009811617 -4945054.945 4 1483516 -0.00098 -1483516 49450550.0009811617 0 5 0 -0.00098 -0.00098 0.000981
0 4945054.9451 6 -1483516 0 1483516 -49450550 0 0 0
5 6 0 0 0 0#VALUE! #VALUE! 1 1 2 3 4#VALUE! #VALUE! 2#VALUE! #VALUE! 3 0 4945055 0 0#VALUE! #VALUE! 4 0.000000 0 0 0#VALUE! #VALUE! 5 0 0 0 0#VALUE! #VALUE! 6 0 0 0 0
0.000981 0 0 0#VALUE! #VALUE! 0 0#VALUE! #VALUE! 0 0
5 #VALUE! #VALUE! 0 06 1 2 3 478 4945055 4945055 -4945055 1483516
0 0.000981 0.000981 -0.000985 -4945055 0.000981 4945055 -14835166 1483516 -0.00098 -1483516 49450557 0.000981 -0.00098 -0.00098 0.0009818 #VALUE! #VALUE! 1483516 -4945055
#VALUE! #VALUE! 0 0#VALUE! #VALUE! 0 0
1 2 3 4
0 -1483516 0 0 1-0.00098 0 0 0 2-0.00098 1483516 0 0 30.000981 -4945055 0 0 40.000981 0 0 0 5
0 4945055 0 0 60 0 0 0 70 0 0 0 85 6 7 8
-1483516 #VALUE! #VALUE! #VALUE! 10 #VALUE! #VALUE! #VALUE! 20 0 0 0 30 0 0 0 4
0.000000 #VALUE! #VALUE! #VALUE! 5#VALUE! #VALUE! #VALUE! #VALUE! 6#VALUE! #VALUE! #VALUE! #VALUE! 7#VALUE! #VALUE! #VALUE! #VALUE! 8
5 6 7 8
-1483516 #VALUE! #VALUE! #VALUE!-0.00098 #VALUE! #VALUE! #VALUE!-0.00098 1483516 0 00.000981 -4945055 0 00.000981 #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!#VALUE! #VALUE! #VALUE! #VALUE!
5 6 7 8
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