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

2

1 0 0 -0.808= 0 1 0 0.001 0.600

0 0 2 0.028

-0.808

= 0.001 0.600

0.056

εx

εy MN/m2

gxy

6.01

= -2.51

7.27

-0.81

= 0.60

0.06

Tuesday,18 April 2023

σ1

σ2 MN/m2

τ12

εx

εy MN/m2

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

1

Zt1

2

3

Zb1 Zt2

Zb2 Zt3

Zb3

Middle

Zt1

Zb1 = Zt2

Zb2 = Zt3

Zb3