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Sebuah lamina dengan data :
θ = 66
= 20
= 6.5
= 7.2
= 0.7
= 0.1
= -6.5
= 10
= -1.7Tuesday,18 April 2023
Ditanyakan :
Penyelesaian :
pada sumbu prinsipil ke sumbu geometri
[T] =
0.165 0.835 0.743= 0.835 0.165 -0.743
-0.372 0.372 -0.669
0.165 0.835 -0.743
= 0.835 0.165 0.7430.372 -0.372 -0.669
o
E1 GN/m2
E2 GN/m2
G12 GN/m2
μ12
μ21
σx MN/m2
σy MN/m2
τxy MN/m2
- Tegangan awal orthotropis (σ1, σ2, τ12)
- Tegangan awal solitisasi (εx, εy, gxy)
1. Matrik Transformasi → Matriks yang menghubungkan tegangan dan regangan
cos2q sin2q 2 sin q cos q
sin2q cos2q -2 sin q cos q
-sin q cos q sin q cos q cos2q - sin2q
[T]-1
2. Matrik S → Menghubungkan matriks tegangan dan regangan
0
[S] = 0
0 0
0
= 0
0 0
0.050 -0.035 0= -0.035 0.154 0
0 0 0.139
23.79 5.41 0
= 5.41 7.73 00 0 7.20
3. Matrik R
1 0 0[R] = 0 1 0
0 0 2
1 0 0
= 0 1 00 0 0.5
4. Tegangan Orthotropis
=
= [T]
0.165 0.835 0.743 -6.5= 0.835 0.165 -0.743 x 10
-0.372 0.372 -0.669 -1.7
6.01
= -2.51
7.27
5. Regangan Orthotropis
S11 S12
S12 S22
S66
1/E1 -μ12/E1
-μ12/E1 1/E2
1/G12
[S]-1 m2/GN
[R]-1
σx σ1
σy [T]-1 σ2
τxy τ12
σ1 σx
σ2 σy
τ12 τxy
σ1
σ2 MN/m2
τ12
= [S]
0.050 -0.035 0 6.01= 0.001 -0.035 0.154 0 x -2.51
0 0 0.139 7.27
0.39
= 0.001 -0.60
1.01
6. Transformasi regangan orthotropi ke regangan solitisasi
=
2
1 0 0 0.39= 0 1 0 0.001 -0.60
0 0 0.5 1.01
0.388
= 0.001 -0.596
0.5052
=
2 2
0.165 0.835 -0.743 0.388= 0.835 0.165 0.743 0.001 -0.596
0.372 -0.372 -0.669 0.505
-0.808= 0.001 0.600
0.028
= [R]
ε1 σ1
ε2 σ2
g12 τ12
ε1
ε2 MN/m2
g12
ε1 ε1
ε2 [R]-1 ε2
g12 g12
ε1
ε2
g12
εx ε1
εy [T]-1 ε2
gxy g12
εx εx
εy εy
gxy gxy
Properties Input Data :
No. Layert Zt Zb θ
( mm ) ( mm ) ( mm ) ° ( Mpa ) ( Mpa ) ( Mpa )1 0.20 -0.45 -0.25 40 5000 3000 500 0.20 0.202 0.30 -0.25 0.05 42 4500 2800 400 0.18 0.113 0.40 0.05 0.45 44 4000 2400 300 0.16 0.10
Load Input data :Nx Ny Nxy Mx My Mxy
( N ) ( N ) ( N ) ( Nmm ) ( Nmm ) ( Nmm )46 8 2 36 6 1
Layer 1 Layer 2 Layer 3-0.45 -0.25 -0.25 0.05 0.05 0.45
Soliti
sasi
Tega
ngan -264.88 -98.81 -78.31 117.02 90.43 292.38
-36.76 -15.98 -14.03 12.85 14.75 52.50
-19.92 -2.36 2.46 15.69 5.33 2.20
Rega
ngan -0.137 -0.051 -0.051 0.077 0.077 0.249
0.079 0.028 0.028 -0.048 -0.048 -0.149
0.029 0.014 0.014 -0.008 -0.008 -0.038
Ort
hotr
opis
Tega
ngan -190.24 -66.91 -47.08 85.98 59.24 178.82
-111.40 -47.88 -45.26 43.89 45.95 166.05
108.87 40.37 32.22 -50.16 -37.63 -119.79
Rega
ngan -0.041 -0.015 -0.015 0.024 0.024 0.075
-0.017 -0.008 -0.008 0.006 0.006 0.025
E1 E2 G12 v12 v21
σx
σy
τxy
εx
εy
gxy
σ1
σ2
τ12
ε1
ε2
-0.20 -0.10 0.00 0.10 0.20 0.30
-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50
εsolitisasi
x
y
xy
-250 -200 -150 -100 -50 0 50 100 150 200 250
-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50
σorthotropis
1
2
12-1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60
-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50
εorthotropis
1
2
12
-300 -200 -100 0 100 200 300 400
-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.50
σsolitisasi
x
y
xy
Ort
hotr
opis
Rega
ngan
0.430 0.159 0.159 -0.248 -0.248 -0.791
1. Menghitung [Q] di setiap layer (Persamaan 2.61)
0 5208 625 0
0 = 1042 3125 0 Mpa
0 0 0 0 500
0 4591 514 0
0 = 505 2857 0 Mpa
0 0 0 0 400
0 4065 390 0
0 = 407 2439 0 Mpa
0 0 0 0 300
2sinθ cosθ
[T] = -2sinθ cosθ
-sinθ cosθ sinθ cosθ
0.587 0.413 0.985 0.587 0.413 -0.985
0.413 0.587 -0.985 → 0.413 0.587 0.985
-0.492 0.492 0.174 0.492 -0.492 0.174
0.552 0.448 0.995 0.552 0.448 -0.995
0.448 0.552 -0.995 → 0.448 0.552 0.995
-0.497 0.497 0.105 0.497 -0.497 0.105
0.517 0.483 0.999 0.517 0.483 -0.999
0.483 0.517 -0.999 → 0.483 0.517 0.999
-0.500 0.500 0.035 0.500 -0.500 0.035
1 0 0 1.0 0.0 0.0
[R] = 0 1 0 → 0.0 1.0 0.00 0 2 0.0 0.0 0.5
g12
Q11 Q12
[Q]1 = Q12 Q22
Q66
Q11 Q12
[Q]2 = Q12 Q22
Q66
Q11 Q12
[Q]3 = Q12 Q22
Q66
2. Menghitung [T] dan [T]-1 di setiap layer (Persamaan 2.71)
cos2θ sin2θ
sin2θ cos2θ
cos2θ - sin2θ
[T]1 = [T]-11 =
[T]2 = [T]-12 =
[T]3 = [T]-13 =
3. Menghitung [R] dan [R]-1 (Persamaan 2.72)
[R]-1 =
4. Menghitung [T]-T
[R] [T] [R]⁻¹
0.587 0.413 0.492
0.413 0.587 -0.492-0.985 0.985 0.174
0.552 0.448 0.497
0.448 0.552 -0.497-0.995 0.995 0.105
0.517 0.483 0.500
0.483 0.517 -0.500-0.999 0.999 0.035
5. Menghitung [Ǭ] (Persamaan 2.78)
[T]⁻¹ [Q]
3216 1929 815
2001 2854 416610 211 1631
2620 1704 554
1703 2439 303559 308 1594
2155 1524 450
1524 2098 371441 363 1425
6. Menghitung [A] [B] [D] (Persamaan 4.21)
2291 1506 509 = 1521 2142 323
466 280 1375
-88 -34 -29 = -39 -63 -1
-15 12 -19
[T]-T =
[T]-T1 =
[T]-T2 =
[T]-T3 =
[Ǭ] = [T]-T
[Ǭ]1 =
[Ǭ]2 =
[Ǭ]3 =
[Aij] = Ʃ [Ǭij] (zk - zk -1)
[Bij] = 0.5 Ʃ [Ǭij] (z2k - z2
(k -1))
160 104 37 = 106 148 23
32 18 93
7. Matriks [A] [B] [D] disusun menjadi matriks 6x6
2291 1506 509 -88 -34 -291521 2142 323 -39 -63 -1
[ABD] = 466 280 1375 -15 12 -19-88 -34 -29 160 104 37-39 -63 -1 106 148 23-15 12 -19 32 18 93
9E-04 -6E-04 -2E-04 6E-04 -5E-04 1E-04-6E-04 9E-04 2E-06 -5E-04 6E-04 -1E-04
-2E-04 9E-06 8E-04 7E-05 -2E-04 1E-046E-04 -5E-04 1E-04 1E-02 -9E-03 -3E-03-5E-04 6E-04 -1E-04 -9E-03 1E-02 4E-057E-05 -2E-04 1E-04 -2E-03 2E-04 1E-02
=
9E-04 -6E-04 -2E-04 6E-04 -5E-04 1E-04 46
-6E-04 9E-04 2E-06 -5E-04 6E-04 -1E-04 8
= -2E-04 9E-06 8E-04 7E-05 -2E-04 1E-04 x 2
6E-04 -5E-04 1E-04 1E-02 -9E-03 -3E-03 36
-5E-04 6E-04 -1E-04 -9E-03 1E-02 4E-05 6
7E-05 -2E-04 1E-04 -2E-03 2E-04 1E-02 1
0.0560
-0.0351
[Dij] = 0.333 Ʃ [Ǭij] (z3k - z3
(k -1))
8. Menghitung [A B D]⁻¹
[ABD]-1 =
9. Menghitung regangan (e0) dan kelengkungan (k) pada middle surface
εx0 Nx
εy0 Ny
gxy0 [ABD]-1 Nxy
kx Mx
ky My
kxy Mxy
εx0
εy0
gxy0
kx
ky
kxy
εx0
εy0
= -0.0044
0.4289
-0.2536
-0.0738
= + z
0.0560 0.4289
= -0.0351 + -0.45 -0.2536
-0.0044 -0.0738
-0.1370= 0.0790
0.0288
0.0560 0.4289
= -0.0351 + -0.25 -0.2536
-0.0044 -0.0738
-0.0512= 0.0283
0.0141
0.0560 0.4289
= -0.0351 + -0.25 -0.2536
-0.0044 -0.0738
-0.0512= 0.0283
0.0141
0.0560 0.4289
= -0.0351 + 0.05 -0.2536
gxy0
kx
ky
kxy
8. Mencari regangan (e) setiap layer (Persamaan 4.10)
εx εx0 kx
εy εy0 ky
gxy gxy0 kxy
a. Regangan (e) pada Layer 1
εx
εy
gxy Layer 1-zt
εx
εy
gxy Layer 1-zb
b. Regangan (e) pada Layer 2
εx
εy
gxy Layer 2-zt
εx
εy
-0.0044 -0.0738
0.0774= -0.0478
-0.0081
0.0560 0.4289
= -0.0351 + 0.05 -0.2536
-0.0044 -0.0738
0.0774= -0.0478
-0.0081
0.0560 0.4289
= -0.0351 + 0.45 -0.2536
-0.0044 -0.0738
0.2490= -0.1492
-0.0376
=
2
=
2
1.0 0.0 0.0 -0.1370= 0.0 1.0 0.0 x 0.0790
0.0 0.0 0.5 0.0288
gxy Layer 2-zb
c. Regangan (e) pada Layer 3
εx
εy
gxy Layer 3-zt
εx
εy
gxy Layer 3-zb
9. Mencari regangan (ε1, ε2 , g12 ) setiap layer (Persamaan 2.70 - 2.73)
εx εx
εy [R]-1 εy
gxy gxy
a. Regangan (ε1, ε2 , g12 ) layer 1-Zt
εx εx
εy [R]-1 εy
gxy gxy Layer 1-zt
Layer 1-zt
-0.1370
= 0.0790
0.0144
2
= [T]
2 2
0.587 0.413 0.492 -0.1370= 0.413 0.587 -0.492 x 0.0790
-0.985 0.985 0.174 0.0144
-0.0407
= -0.0174
0.2152
2
= [R]
2
1 0 0 -0.0407= 0 1 0 x -0.0174
0 0 2 0.2152
-0.041
= -0.017
0.430
=
2
εx
εy
gxy
Layer 1-zt
ε1 εx
ε2 εy
g12 gxy
Layer 1-zt Layer 1-zt
ε1
ε2
g12
Layer 1-zt
ε1 ε1
ε2 ε2
g12 Layer 1-zt g12
Layer 1-zt
ε1
ε2
g12 Layer 1-zt
b. Regangan (ε1, ε2 , g12 ) layer 1-Zb
εx εx
εy [R]-1 εy
gxy gxy Layer 1-zb
Layer 1-zb
1.0 0.0 0.0 -0.0512= 0.0 1.0 0.0 x 0.0283
0.0 0.0 0.5 0.0141
-0.0512
= 0.0283
0.0070
2
= [T]
2 2
0.587 0.413 0.492 -0.0512= 0.413 0.587 -0.492 x 0.0283
-0.985 0.985 0.174 0.0070
-0.0149
= -0.0080
0.0795
2
= [R]
2
1 0 0 -0.0149= 0 1 0 x -0.0080
0 0 2 0.0795
-0.0149
= -0.0080
0.1591
= [R]
εx
εy
gxy
Layer 1-zb
ε1 εx
ε2 εy
g12 gxy
Layer 1-zb Layer 1-zb
ε1
ε2
g12
Layer 1-zb
ε1 ε1
ε2 ε2
g12 Layer 1-zb g12
Layer 1-zb
ε1
ε2
g12 Layer 1-zb
c. Regangan (ε1, ε2 , g12 ) layer 2-Zt = layer 1-Zb
ε1 ε1
ε2 ε2
2
-0.0149
= -0.0080
0.1591
=
2
1.0 0.0 0.0 0.0774= 0.0 1.0 0.0 x -0.0478
0.0 0.0 0.5 -0.0081
0.0774
= -0.0478
-0.0040
2
= [T]
2 2
0.587 0.413 0.492 0.0774= 0.413 0.587 -0.492 x -0.0478
-0.985 0.985 0.174 -0.0040
0.0237
= 0.0059
-0.1240
2
= [R]
g12 Layer 2-zt g12
Layer 1-zb = Layer 2-zt
ε1
ε2
g12 Layer 2-zt
d. Regangan (ε1, ε2 , g12 ) layer 2-Zb
εx εx
εy [R]-1 εy
gxy gxy Layer 2-zb
Layer 2-zb
εx
εy
gxy
Layer 2-zb
ε1 εx
ε2 εy
g12 gxy
Layer 2-zb Layer 2-zb
ε1
ε2
g12
Layer 2-zb
ε1 ε1
ε2 ε2
g12 Layer 2-zb g12
2
1 0 0 0.0237= 0 1 0 x 0.0059
0 0 2 -0.1240
0.0237
= 0.0059
-0.2481
= [R]
2
0.0237
= 0.0059
-0.2481
=
2
1.0 0.0 0.0 0.2490= 0.0 1.0 0.0 x -0.1492
0.0 0.0 0.5 -0.0376
0.2490
= -0.1492
-0.0188
2
= [T]
2 2
Layer 2-zb
ε1
ε2
g12 Layer 2-zb
e. Regangan (ε1, ε2 , g12 ) layer 3-Zt = layer 2-Zb
ε1 ε1
ε2 ε2
g12 Layer 3-zt g12
Layer 2-zb
ε1
ε2
g12 Layer 3-zt
f. Regangan (ε1, ε2 , g12 ) layer 3-Zb
εx εx
εy [R]-1 εy
gxy gxy Layer 3-zb
Layer 3-zb
εx
εy
gxy
Layer 3-zb
ε1 εx
ε2 εy
g12 gxy
Layer 3-zb Layer 3-zb
0.587 0.413 0.492 0.2490= 0.413 0.587 -0.492 x -0.1492
-0.985 0.985 0.174 -0.0188
0.0752
= 0.0246
-0.3954
2
= [R]
2
1 0 0 0.0752= 0 1 0 x 0.0246
0 0 2 -0.3954
0.0752
= 0.0246
-0.7909
=
=
3216 1929 815 -0.1370 = 2001 2854 416 x 0.0790
610 211 1631 0.0288
-264.88
= -36.76
-19.92
ε1
ε2
g12
Layer 3-zb
ε1 ε1
ε2 ε2
g12 Layer 3-zb g12
Layer 3-zb
ε1
ε2
g12 Layer 3-zb
10.Mencari σx, σy, τxy (Persamaan 2.79)
σx εx
σy [Ǭ] εy
txy gxy
a. Mencari σx, σy, τxy Layer 1-zt
σx εx
σy [Ǭ]1 εy
txy Layer 1-zt gxy Layer 1-zt
σx
σy
txy Layer 1-zt
=
3216 1929 815 -0.0512 = 2001 2854 416 x 0.0283
610 211 1631 0.0141
-98.81
= -15.98
-2.36
=
2620 1704 554 -0.0512 = 1703 2439 303 x 0.0283
559 308 1594 0.0141
-78.31
= -14.03
2.46
=
2620 1704 554 0.0774 = 1703 2439 303 x -0.0478
559 308 1594 -0.0081
117.02
= 12.85
15.69
b. Mencari σx, σy, τxy Layer 1-Zb
σx εx
σy [Ǭ]1 εy
txy Layer 1-zb gxy Layer 1-zb
σx
σy
txy Layer 1-zb
c. Mencari σx, σy, τxy Layer 2-Zt
σx εx
σy [Ǭ]2 εy
txy Layer 2-zt gxy Layer 2-zt
σx
σy
txy Layer 2-zt
d. Mencari σx, σy, τxy Layer 2-Zb
σx εx
σy [Ǭ]2 εy
txy Layer 2-zb gxy Layer 2-zb
σx
σy
txy Layer 2-zb
=
2155 1524 450 0.0774 = 1524 2098 371 x -0.0478
441 363 1425 -0.0081
90.43
= 14.75
5.33
=
2155 1524 450 0.2490 = 1524 2098 371 x -0.1492
441 363 1425 -0.0376
292.38
= 52.50
2.20
= [T]
=
e. Mencari σx, σy, τxy Layer 3-Zt
σx εx
σy [Ǭ]3 εy
txy Layer 3-zt gxy Layer 3-zt
σx
σy
txy Layer 3-Zt
f. Mencari σx, σy, τxy Layer 3-Zb
σx εx
σy [Ǭ]3 εy
txy Layer 3-zb gxy Layer 3-Zb
σx
σy
txy Layer 3-Zb
11. Mencari σ1, σ2, τ12 (Persamaan 2.69)
σ1 σx
σ2 σy
t12 txy
a. Mencari σ1, σ2, τ12 Layer 1-zt
σ1 σx
σ2 [T]1 σy
t12 Layer 1-zt txy Layer 1-zt
0.587 0.413 0.985 -264.88= 0.413 0.587 -0.985 x -36.76
-0.492 0.492 0.174 -19.92
-190.24= -111.40
108.87
=
0.587 0.413 0.985 -98.81= 0.413 0.587 -0.985 x -15.98
-0.492 0.492 0.174 -2.36
-66.91= -47.88
40.37
-47.08
-45.26
= 32.22
0.552 0.448 0.995 -78.31= 0.448 0.552 -0.995 x -14.03
-0.497 0.497 0.105 2.46
-47.08= -45.26
32.22
85.9843.89
-50.16
=
0.552 0.448 0.995 117.02
b. Mencari σ1, σ2, τ12 Layer 1-Zb
σ1 σx
σ2 [T]1 σy
t12 Layer 1-zb txy Layer 1-zb
c. Mencari σ1, σ2, τ12 Layer 2-zt
σ1 σx
σ2 [T]2 σy
t12 Layer 2-zt txy Layer 2-zt
d. Mencari σ1, σ2, τ12 Layer 2-Zb
σ1 σx
σ2 [T]2 σy
t12 Layer 2-zb txy Layer 2-zb
= 0.448 0.552 -0.995 x 12.85-0.497 0.497 0.105 15.69
85.98= 43.89
-50.16
59.2445.95
-37.63
=
0.517 0.483 0.999 90.43= 0.483 0.517 -0.999 x 14.75
-0.500 0.500 0.035 5.33
59.24= 45.95
-37.63
178.82166.05
-119.79
=
0.517 0.483 0.999 292.38= 0.483 0.517 -0.999 x 52.50
-0.500 0.500 0.035 2.20
178.82= 166.05
-119.79
f. Mencari σ1, σ2, τ12 Layer 3-zt
σ1 σx
σ2 [T]3 σy
t12 Layer 3-zt txy Layer 3-zt
g. Mencari σ1, σ2, τ12 Layer 3-Zb
σ1 σx
σ2 [T]3 σy
t12 Layer 3-zb txy Layer 3-zb
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