please help me with this question

have to do it with Excel Microsoft.

The book we use is Mechanics of Composite Materials, Robert M. Jones, Taylor and Francis, 1999, most recent edition

2. do the specific example of layups

2a (0,0,0,0,90,90,90,90,0,0,0,0,90,90,90,90)

Question, is the above the same as (0,90,0,90) if so, why

(90,90,90,90,0,0,0,0,90,90,90,90,0,0,0,0)

Question, is the above the same as (90,0,90,0) if so, why

2b (0,45,90,-45,0,45,90,-45)symm this means total of 16 layers

2c for case of total of only four layers, computer by hand, and also by computer

And make sure your hand calculations agrees with your computer

calculations for Aijnon, Bijnon, Dij non

(22, -17,-43, -36) total of four layers

(32,-45,-23,11) total of four layers

(0,0,0,0) total of four layer

2d do the special case of isotropic orthotropic layers

(0,0,0,0) but using E1/E2= 1.0

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Some explanation

2. do the specific example of layups

2a (0,0,0,0,90,90,90,90,0,0,0,0,90,90,90,90)

Question, is the above the same as (0,90,0,90) if so, why

(90,90,90,90,0,0,0,0,90,90,90,90,0,0,0,0)

Question, is the above the same as (90,0,90,0) if so, why

2b (0,45,90,-45,0,45,90,-45)symm this means total of 16 layers

2c for case of total of only four layers, computer by hand, and also by computer

And make sure your hand calculations agrees with your computer

calculations for Aijnon, Bijnon, Dij non

(22, -17,-43, -36) total of four layers

(32,-45,-23,11) total of four layers

(0,0,0,0) total of four layer

2d do the special case of isotropic orthotropic layers

(0,0,0,0) but using E1/E2= 1.0

3. read chapter 16,William D. Callister book Materials Science and Engineering

Do problems, 16.2, 16.6, 16.9, 16.10

Typical example,

For 16 layer, we have the weighting factors,

The layer location Weighting factor of Aij due to z_k - z_k-1 is,

(1,1,1,1, 1,1, 1,1, 1,1, 1,1,1,1,1,1)

The layer location Weighting factor of Bij due to z_k² - z_k-1² is,

(-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15)

The layer location Weighting factor of Dij due to z_k³ - z_k-1³ is,

(169, 127, 91, 61, 37, 19, 7,1, 1, 7, 19, 37, 61, 91, 127, 169)

For 4 layer laminate, we have the weighting factors

The layer location Weighting factor of Aij due to z_k - z_k-1 is,

(1,1,1,1,)

The layer location Weighting factor of Bij due to z_k² - z_k-1² is,

( -3, -1, 1, 3)

The layer location Weighting factor of Aij due to z_k³ - z_k-1³ is,

(7, 1, 1, 7 )

Then, for the four layer composites, we have

E₁/E₂=15, poisson₁₂= 0.25 computer possion21

The first thing is to find, Q11non, Q12non, !16non, Q22non, Q26non, Q66non

Because each layer is orthotropic (meaning each layer consists of all long fibers in the same direction)

Page 72,

Q11non= (1/E₂) (Q11) = Q₁₁ /E₂ = (E₁/E₂)/ (1-poisson₁₂*poisson ₂₁)

Q12non= (1/E₂) (Q11) = Q₁₂ /E₂ = (E₂/E₂) poisson 12/ (1-poisson₁₂ * poisson ₂₁)

Q22non= (1/E₂) (Q11) = Q₂₂ /E₂ = (E₂/E₂) { 1/ (1-poisson₁₂ * poisson₂₁) }

Q16non= (1/E₂) (Q11) = Q₁₆ /E₂ = 0

Q26non= (1/E₂) (Q26) = Q₂₆ /E₂ = 0

Q66non= (1/E₂) (Q66) = G₁₂ /E₂

From page 85,

G₁₂ = E₁/ { 2 (1+poisson₁₂) }

Q66non = (E₁/E₂) / { 2 (1+poisson₁₂) }

For the first layer (k=1), second layer (k=2), third layer (k=3), fourth layer (k=N=4)

From equation 2.85,page 77

We now specialize to our four layer laminate (theta1,theta2,theta3,theta4)= (22, -17,-43, -36)

Q11bar non = Q11non C⁴ + 2(Q12non+2Q66non) S²C²+ Q22 S⁴

Q12bar non = (Q11non +Q22non-4 Q66non) S²C²+ Q12non (S⁴+ C⁴ )

Q22bar non = Q11non S⁴ + 2(Q12non+2Q66non) S²C²+ Q22non C⁴

Q16bar non = (Q11non- Q12non- 2Q66non) SC³ + (Q12non-Q22non+2Q66non) S³C

Q26bar non = (Q11non- Q12non- 2Q66non) S³C + (Q12non-Q22non+2Q66non) SC³

Q66bar non = (Q11non+Q22non-2Q12non-2Q66non) S²C²+ Q66non (S4+C⁴)

So we now deal with first layer, k=1, where

theta1 = 22 degree

Compute C = cos(22 degree), S= sin (22 degree) then computer C⁴, S⁴, C²S², CS³, C³S,

So we now deal with second layer, k=2, where theta2 = -17 degree

Compute C = cos(-17 degree), S= sin (-17 degree) then computer C⁴, S⁴, C²S², CS³, C³S,

So we now deal with third layer, k=3, where theta1 = -43 degree

Compute C = cos(-43 degree), S= sin (-43 degree) then computer C⁴, S⁴, C²S², CS³, C³S,

So we now deal with fourth layer, k=4, where theta1 = -36 degree

Compute C = cos(-36 degree), S= sin (-36 degree) then computer C⁴, S⁴, C²S², CS³, C³S,

Similarly, for theta2=

From page 198, we have Aij, Bij, Dij

So that summation means sum over k=1 to k=N

Aij = summation { Qijbar (kth layer) (z_k-z_k-1) }

Bij = (1/2) summation { Qijbar (kth layer) (z_k² - z_k-1²) }

Dij = (1/3) summation { Qijbar (kth layer) (z_k³ - z_k-1³) }

We write the above in non-dimension form, total thickness is t, individual thickness of each layer is h_e

so that t= (N)(he)

Aijnon = Aij/(E₂ Nh_e) = (1/N) summation { Qijbar non (kth layer) (z_k-z_k-1) (l/h_e) }

Bijnon = Bij /(E₂ N²h_e²) = (1/2) (1/N²) summation { Qijbar non(kth layer) (z_k² - z_k-1²) (1/h_e²) }}

Dijnon = Dij /(E₂ N³h_e³) = (1/3) (1/N³) summation { Qijbar non(kth layer) (z_k³ - z_k-1³) (1/h_e³) }}

So for the Aijnon we now specialize into the case of four layer laminate

(theta1,theta2,theta3,theta4)= (22, -17,-43, -36)

Aijnon = Aij/ (E₂ Nh_e)

= (1/N) { Qijbar non (1st layer) (z₁-z_o) (l/h_e) + Qijbar non(2^nd layer) (z₂-z₁) (l/h_e)

+ Qijbar non(3^rd layer) (z₃-z₂) (l/h_e) + Qijbar non (4^th layer) (z₄-z₃) (l/h_e) }

Where we know the location weighting factor for Aij is, (1,1,1,1)

The above simplified to

Aijnon = Aij/(E₂ Nh_e) { Qijbar non(22 degree) (1) + Qijbar non(-17 degree) (1)

+ Qijbar non(-43 degree) + Qijbar non (-36 degree).(1) }

Then setting ij= (11,12,22,16,26,66) we then find A11non, A12non, A22non, A16non, A26non, A66non

Bij non = Bij/ (E₂N²h_e²)

= (1/2) (1/N) { Qijbar non (1st layer) (z₁²-z_o²) (l/h_e²) + Qijbar non(2^nd layer) (z₂²-z₁²) (l/h_e²)

+ Qijbar non(3^rd layer) (z₃²-z₂²) (l/h_e²) + Qijbar non (4^th layer) (z₄²-z₃²) (l/h_e²) }

We know

The layer location Weighting factor of Aij due to z_k - z_k-1 is, (1,1,1,1,)

The layer location Weighting factor of Bij due to z_k² - z_k-1² is, ( -3, -1, 1, 3)

The layer location Weighting factor of Bij due to z_k³ - z_k-1³ is, (7, 1, 1, 7 )

Then specializing we have,

Bij non = Bij/ (E₂N²h_e²)

= (1/2) (1/N²) { Qijbar non (22 degree) (-3) + Qijbar non(-17 degree) (-1)

+ Qijbar non(-43 degree) (1) + Qijbar non (-36 degree) (3) }

Then setting ij= (11,12,22,16,26,66) we then find B11non, B12non, B22non, B16non, B26non, B66non

Finally, we have

Dij non = Dij/ (E₂N²h_e²)

= (1/3) (1/N³) { Qijbar non (1st layer) (z₁³-z_o³) (l/h_e³) + Qijbar non(2^nd layer) (z₂³-z₁³) (l/h_e³)

+ Qijbar non(3^rd layer) (z₃³-z₂³) (l/h_e³) + Qijbar non (4^th layer) (z₄³-z₃³) (l/h_e³) }

Finally,

Dij non = Bij/ (E₂N³h_e³)

= (1/3) (1/N³) { Qijbar non (22 degree) (7) + Qijbar non(2^nd layer) (1)

+ Qijbar non(3^rd layer) (1)+ Qijbar non (4^th layer) (7) }

Then setting ij= (11,12,22,16,26,66) we then find D11non, D12non, D22non, D16non, D26non, D66non

For the 16 layers composites

The layer location Weighting factor of Aij due to z_k - z_k-1 is,

(1,1,1,1, 1,1, 1,1, 1,1, 1,1,1,1,1,1) these means the location are all assigned equal weight

Aijnon = Aij/(E₂ Nh_e)

{ Qijbar non(theta₁) (1) + Qijbar non(theta2) (1) + Qijbar non(theta₃) + Qijbar non (theta₄).(1)

+ Qijbar non(theta₅) (1) + Qijbar non(theta₆) (1) + Qijbar non(theta₇) (1)+ Qijbar non (theta₈).(1)

+ Qijbar non(theta₉) (1) + Qijbar non(theta₁₀) (1) + Qijbar non(theta₁₁) (1)+ Qijbar non (theta₁₂).(1)

+ Qijbar non(theta₁₃) (1) + Qijbar non(theta₁₄) (1) + Qijbar non(theta₁₅) + Qijbar non (theta₁₆).(1) }

Then setting ij= (11,12,22,16,26,66) we then find A11non, A12non, A22non, A16non, A26non, A66non

Remember t is total thickness of all the layers, he is thickness of each individual layer, so that t = (N)(h_e)

Where h_e is the thickness of only one layer, we assume all layers are of equal thickness

The layer location Weighting factor of Bij due to (z_k² - z_k-1²)/(N²h_e²) is,

(-15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15) this means the location of layer are assigned

Such that the outer layer has higher weighting factor than the inner layer

Bijnon = Bij/(E₂ Nh_e)

= (1/2) { Qijbar non(theta₁) (-15) + Qijbar non(theta₂) (-13) + Qijbar non(theta₃) (-11) + Qijbar non(theta₄) (-9)

+ Qijbar non(theta₅) (-7) + Qijbar non(theta₆) (-5) + Qijbar non(theta₇) (-3)+ Qijbar non (theta₈).(-1)

+ Qijbar non(theta₉) (1) + Qijbar non(theta₁₀) (3) + Qijbar non(theta₁₁) (5)+ Qijbar non (theta₁₂).(7)

+ Qijbar non(theta₁₃) (9) + Qijbar non(theta₁₄) (11) + Qijbar non(theta₁₅) (13)+ Qijbar non (theta₁₆).(15) }

Then setting ij= (11,12,22,16,26,66) we then find B11non, B12non, B22non, B16non, B26non, B66non

The layer location Weighting factor of Dij due to (z_k³ - z_k-1³)/(h_e³) is,

(169, 127, 91, 61, 37, 19, 7,1, 1, 7, 19, 37, 61, 91, 127, 169) this means the location of layer are assigned

Such that the outer layer has significantly higher weighting factor than the inner layer

Dijnon = Dij/(E₂ N³h_e³)

= (1/3) { Qijbar non(theta₁) (169) + Qijbar non(theta₂) (127) + Qijbar non(theta₃) (91) + Qijbar non(theta₄) (61)

+ Qijbar non(theta₅) (37) + Qijbar non(theta₆) (19) + Qijbar non(theta₇) (7)+ Qijbar non (theta₈).(1)

+ Qijbar non(theta₉) (1) + Qijbar non(theta₁₀) (7) + Qijbar non(theta₁₁) (19)+ Qijbar non (theta₁₂).(37)

+ Qijbar non(theta₁₃) 61) + Qijbar non(theta₁₄) (91) + Qijbar non(theta₁₅) (127)+ Qijbar non (theta₁₆).(169) }

Then setting ij= (11,12,22,16,26,66) we then find D11non, D12non, D22non, D16non, D26non, D66non

We can easily specialized to (0,45,90,-45,0,45,90,-45)symm

This means

(theta₁,theta₂,theta₃,theta₄, theta₅,theta₆,theta₇,theta₈, theta₉,theta₁₀,theta₁₁,theta₁₂, theta₁₃,theta₁₄,theta₁₅,theta₁₆)

So that theta₁= 0^o, theta₂ = 45^o, theta₃= 90^o , theta₄ = -45^o, theta₅=0^o, theta₆=45^o, theta₇=90^o, theta₈ = - 45^o,

Theta₉= - 45^o, theta₁₀= 90^o, theta₁₁= 45^o, theta₁₂= 0^o, theta₁₃= - 45^o, theta₁₄= 90^o, theta₁₅= 45^o, theta₁₆=0^o

Remember for each theta, computer S⁴, C⁴, S³C, SC³, S²C²