THE GENERALIZED MULTI-TERM EKM

whereV denotes per unit width volume of the angle-ply panel in they-direction. Substituting the expressions of strain components εx, εz, γxy, γyz and γzx from Eq. (4.5) into Eq. (4.7) yields

∫

t

∫

V

[δw{σz,z+τzx,x−(ρ+ ˆρ)¨u}+δv{τyz,z+τxy,x−(ρ+ ˆρ)¨v}+δu{τxz,z+σx,x−(ρ+ ˆρ) ¨w} +δσ_x{(p₁₁+ ˆp₁₁)σ_x−δ₁ξp¯₁₂−δ₁²ξ²p¯₁₂+ (p₁₆+ ˆp₁₆)τ_xy+ (p₁₃+ ˆp₁₃)σ_z−u_,x}

+δσz{(p31+ ˆp31)σx+p36τxy+p33σz−w,z}+δτyz{¯s44τyz+ (¯s45+ ˆs45)τzx−v,z} (4.7) +δτ_zx{(¯s₄₅+ ˆs₄₅)τ_yz+ (¯s₅₅+ ˆs₅₅)τ_zx−u_,z −w_,x}+δτ_xy{(p₆₁+ ˆp₆₁)σ_x+p₆₃σ_z

+(p₆₆+ ˆs₆₆)τ_xy−v_,x}]dV dt= 0, ∀ δu, δv, δw, δσ_i, δτ_ij

where ξ is dimensionless in-plane coordinate along x-direction and ζ^(k) is non-dimensional local thickness parameter for thekth layer (ζ^(k)= (z−zk−1)/t^(k)) which takes value 0 to 1 for each layer.

Panel is subjected to uniform distributed pressure (σ_z = -p₁, -p₂) at bottom and top surface and there is no shear stress (τ_zxand τ_yz) at bottom and top surface of panel. For perfect interlaminar bounding case, displacements (u,v,w) and transverse stresses (σ_z,τ_zx,τ_yz) need to satisfy following condition at thekth interface:

[(u, v, w, σ_z, τ_yz, τ_zx)|ζ=1]^(k) = [(u, v, w, σ_z, τ_yz, τ_zx)|ζ=0]^(k+1) (4.8) Alongx-axis AFG panel can have any type of support such as Simply supported (σx=v=w= 0), Clamped (u=v=w= 0), Free (σx =τxy =τxz = 0).

THE GENERALIZED MULTI-TERM EKM

boundary conditions. The functions gⁱ_l(ξ) are to be solved in this step. For this purpose, first variation δXl is obtained as,

δX_l=

∑n i=1

f_lⁱ(ξ)δg_lⁱcosωt, l= 1,2, . . .8. (4.10) Functions g_lⁱ are divided into two column vector G¯ and G.ˆ G¯ contains all the six variables that appear in the boundary and interface conditions along z-direction andGˆ contains remaining two variables:

G¯ = [g¹₁. . . gⁿ₁ g₂¹. . . g₂ⁿ g₃¹. . . g₃ⁿ g₅¹. . . g₅ⁿ g¹₇. . . gⁿ₇ g₈¹. . . g₈ⁿ]^T

Gˆ = [g¹₄. . . gⁿ₄ g₆¹. . . g₆ⁿ]^T (4.11) Equations (4.9) and (4.10) are substituted in Eq. (4.7), and perform integration alongx-axis. Since variation is arbitrary, the coefficient ofδgⁱ must vanish, yields the following set of ODEs,

M ¯G_,ζ =A¯^m(ω)G¯ +Aˆ^mGˆ +Q¯^m_p (4.12)

K^mGˆ =A˜^mG¯ +Q˜^m_p (4.13)

whereM,A¯^m,Aˆ^m,K^m andA˜^m are 6n×6n, 6n×6n, 6n×2n, 2n×2nand 2n×6nmatrices and A¯^m =A¯+A¯^v; Aˆ^m=Aˆ+Aˆ^v; K^m=K+K^v; A˜^m =A˜+A˜^v; Q˜^m_p =Q˜_p+Q˜^v_p;Q¯^m_p =Q¯_p+Q¯^v_p. The nonzero terms of these matrices are given as,

M_i1j1 =M_j6i6 =⟨f₈ⁱf₁^j⟩a, M_i2j2 =M_j5i5 =⟨f₇ⁱf₂^j⟩a, M_i3j3 =M_j4i4 =⟨f₅ⁱf₃^j⟩a

A¯_i1j3 = ⁻_a^t⟨f_3,ξ^j f₈ⁱ⟩a, A¯_i1j5 =t¯s₄₅⟨f₈ⁱf₇^j⟩a, A¯^v_i

1j5 =δ₂t¯s₄₅⟨ξf₈ⁱf₇^j⟩a

A¯i1j6 =t¯s55⟨f₈ⁱf₈^j⟩a, A¯^v_i

1j6 =δ2t¯s55⟨ξf₈ⁱf₈^j⟩a, A¯i2j5 =t¯s44⟨f₇ⁱf₇^j⟩a, A¯i2j6 =t¯s45⟨f₇ⁱf₈^j⟩a, A¯^v_i

2j6 =δ2t¯s45⟨ξf₇ⁱf₈^j⟩a, A¯i3j4 =tp33⟨f₅ⁱf₅^j⟩a

Aˆ_i3j1 =tp₃₁⟨f₅ⁱf₄^j⟩a, Aˆ^v_i

3j1 =δ₁tp₃₁⟨ξf₅ⁱf₄^j⟩a, Aˆ_i3j2 =tp₃₆⟨f₅ⁱf₆^j⟩a, A¯_i4j6 = ⁻_a^t⟨f₃ⁱf_8,ξ^j ⟩a Aˆ_i5j2 = ⁻_a^t⟨f₂ⁱf_6,ξ^j ⟩a, Aˆ_i6j1 = ⁻_a^t⟨f₁ⁱf_4,ξ^j ⟩a

Ki1j1 =p11⟨f₄ⁱf₄^j⟩a, K_i^v

1j1 =δ1(p11−p¯12)⟨ξf₄ⁱf₄^j⟩a−¯k11, Ki1j2 =p16⟨f₄ⁱf₆^j⟩a

¯k₁₁=δ²₁p¯₁₂⟨ξ²f₄ⁱf₄^j⟩a, K_i^v

1j2 =δ₁p₁₆⟨ξf₄ⁱf₆^j⟩a, K_i2j2 =p₆₆⟨f₆ⁱf₆^j⟩a

K_i^v

2j2 =δ₂¯s₆₆⟨ξf₆ⁱf₆^j⟩a, K_i2j1 =K_i1j2, K_i^v

2j1 =K_i^v

1j2

A˜_i1j1 = ¹_a⟨f₄ⁱf_1,ξ^j ⟩a, A˜_i1j4 =−p₁₃⟨f₄ⁱf₅^j⟩a, A˜^v_i

1j4 =−δ₁p₁₃⟨ξf₄ⁱf₅^j⟩a

A˜i2j2 = ¹_a⟨f₆ⁱf_2,ξ^j ⟩a, A˜i2j4 =−p63⟨f₆ⁱf₅^j⟩a, A¯i4j3 =−ρω²t⟨f₃ⁱf₃^j⟩a

(4.14)

A¯^v_i

4j3 =−δ_pω²t⟨ρ ξf₃ⁱf₃^j⟩a,A¯_i5j2 =−ρω²t⟨f₂ⁱf₂^j⟩a, A¯^v_i

5j2 =−δ_pω²t⟨ρ ξf₂ⁱf₂^j⟩a

A¯i6j1 =−ρω²t⟨f₁ⁱf₁^j⟩a, A¯^v_i

6j1 =−δpω²t⟨ρ ξf₁ⁱf₁^j⟩a

whereip = (p−1)n+iand jq= (q−1)n+j forp, q= 1,2, . . . ,8. Q¯p and Q˜p are load vectors of size 6nand 2n, respectively, whose non-zero terms are given by

Q¯_pi3 =tp₃₃⟨f₅ⁱ⟩a(p^k_a+ζtp_d), Q˜_pi1 =−p₁₃⟨f₄ⁱ⟩a(p^k_a+ζtp_d), Q¯_pi4 =−t⟨f₃ⁱ⟩ap_d Q˜^v_pi1 =−δ₁p₁₃⟨ξf₄ⁱ⟩a(p^k_a+ζtp_d), Q˜_pi2 =−p₆₃⟨f₆ⁱ⟩a(p^k_a+ζtp_d)

(4.15)

where⟨. . .⟩a=a∫₁

0(. . .)dξ, and p^k_a=p_a+p_dz_k. The functions f_lⁱ are known so the above elements of matrices are obtained in closed form.

Solving the system of algebraic equations (4.13) for Gˆ and substituting back the solution into Eq. (4.12) yields

G¯_,ζ =A(ω)G¯ +Q_p (4.16)

withA=M⁻¹[A¯^m+Aˆ^mK^m−1A˜^m] andQp=M⁻¹[Q¯^m_p+Aˆ^mK^m−1Q˜^m_p]. Equation (4.16) is system of first order differential equations of size 6n.

For static case, the elements of matrices which is function of natural frequencies (ω) and time (t) now becomes zero, ¯A_i4j3= ¯A_i5j2= ¯A_i6j1=0 and ¯A^v_i

4j3= ¯A^v_i

5j2= ¯A^v_i

6j1=0. Therefore, the coefficient of above system of first order ODEs Eq. (4.16) is not dependent on natural frequencies (ω). So the final form of ODEs along thickness direction, Eq. (4.16) now written as,

G¯_,ζ =A ¯G+Q_p (4.17)

This represent the set of first order non-homogenous ODEs having constant coefficient.

Similarly, for free vibration case the element corresponding to load vectorsQ¯p,Q¯^v_p,Q˜p and Q˜^v_p is equals to zero due to absence of any external applied load, ( ¯Q_pi3= ¯Q_pi4= ˜Q_pi1= ˜Q_pi2= ˜Q^v_pi1=0).

So, final system of ODEs, Eq. (4.16) now reduced to,

G¯,ζ =A(ω)G¯ (4.18)

Equation (4.18) is system of first order homogenous ODEs whose coefficients are functions of natural frequencies (ω). These set of ODEs, for static case Eq. (4.17) and for dynamic case Eq. (4.18), respectively are solved in closed form manners using same the technique as discussed in Chapter 2 and 3. This completely determines G(ζ). Now,¯ G(ζ) can be obtained by solving the algebraicˆ equation (4.13). In this way, first iteration step is completed now.

THE GENERALIZED MULTI-TERM EKM

4.2.2 Second Iteration Step

Nowg_lⁱ(ζ) is known from the first step and arbitrary variation is considered along thex-direction therefore variation for this case is written as:

δX_l=

∑n i=1

gⁱ_l(ζ)δf_lⁱcosωt, l=1,2,. . . ,8. (4.19) Similarly like first step, the functionf_lⁱ(ξ) is partition intoF¯ andF.ˆ F¯ contains all the six variables that appear in the boundary and interface conditions alongx-direction and Fˆ contains remaining two variables:

F¯ = [f₁¹. . . f₁ⁿ f₂¹. . . f₂ⁿ f₃¹. . . f₃ⁿ f₄¹. . . f₄ⁿ f₆¹. . . f₆ⁿ f₈¹. . . f₈ⁿ]^T

Fˆ = [f₅¹. . . f₅ⁿ f₇¹. . . f₇ⁿ]^T (4.20) Substituting Eq. (4.19) in Eq. (4.7), and perform integration along z-axis. Since variation is arbi- trary, the coefficient of δf_lⁱ must vanish, yields the following set of governing equations:

N¯F_,ξ=B¯^f(ξ, ω)F¯ +Bˆ^f(ξ)Fˆ+P¯^f_m(ξ) (4.21)

LˆF=B˜^f(ξ)F¯ +P˜_m (4.22)

whereB¯^f =B¯ +ξB¯^v+ξ²B¯^vv; Bˆ^f =Bˆ +ξBˆ^v ; P¯^f_m =P¯_m+ξP¯^v_m ; B˜^f =B˜ +ξB˜^v; andN,B¯^f, Bˆ^f,L and B˜^f are 6n×6n, 6n×6n, 6n×2n, 2n×2n and 2n×6n matrices, respectively and the nonzero terms of these matrices are given as,

N_i1j1 =N_j4i4 =⟨g₄ⁱg₁^j⟩h, N_i2j2 =N_j5i5 =⟨g₆ⁱg^j₂⟩h, N_i3j3 =N_j6i6 =⟨gⁱ₈g^j₃⟩h

B¯_i1j4 =p₁₁⟨gⁱ₄g₄^j⟩h, B¯_i^v

1j4 =δ₁(p₁₁−p¯₁₂)⟨g₄ⁱg₄^j⟩h, B¯_i^vv

1j4 =−δ₁²p¯₁₂⟨gⁱ₄g^j₄⟩h

Bˆi1j1 =p13⟨gⁱ₄g₅^j⟩h, Bˆ_i^v

1j1 =δ1p13⟨gⁱ₄g^j₅⟩h, B¯i1j5 =p16⟨g₄ⁱg₆^j⟩h, B¯^v_i

1j5 =δ1p16⟨g₄ⁱg^j₆⟩h, B¯i2j4 =p61⟨g₆ⁱg₄^j⟩h, B¯_i^v

2j4 =δ1p61⟨g₆ⁱg₄^j⟩h, B¯i2j5 =p66⟨gⁱ₆g₆^j⟩h, B¯_i^v

2j5 =δ2¯s66⟨g₆ⁱg₆^j⟩h, Bˆi2j1 =p63⟨g₆ⁱg₅^j⟩h

B¯_i3j1 =−⟨gⁱ₈^g

j 1,ζ

t ⟩h, Bˆ_i3j2 = ¯s₄₅⟨gⁱ₈g^j₇⟩h, Bˆ_i^v

3j2 =δ₂¯s₄₅⟨gⁱ₈g^j₇⟩h

B¯_i3j6 = ¯s₅₅⟨g₈ⁱg₈^j⟩h, B¯_i^v

3j6 =δ₂¯s₅₅⟨g₈ⁱg₈^j⟩h, B¯_i4j6 =−⟨g₁^ig

j 8,ζ

t ⟩h

Bˆ_i5j2 =−⟨gⁱ₂^g

j 7,ζ

t ⟩h, Bˆ_i6j1 =−⟨g₃^ig

j 5,ζ

t ⟩h, L_i1j1 =p₃₃⟨g₅ⁱg₅^j⟩h

Li2j2 = ¯s44⟨g₇ⁱg₇^j⟩h, B˜i1j3 =⟨gⁱ₅^g

j 3,ζ

t ⟩h, B˜i1j4 =−p31⟨gⁱ₅g^j₄⟩h

B˜^v_i

1j4 =−δ₁p₃₁⟨gⁱ₅g^j₄⟩h, B˜_i1j5 =−p₃₆⟨g₅ⁱg^j₆⟩h, B˜_i2j2 =⟨g₇^ig

j 2,ζ

t ⟩h

B˜_i2j6 =−s¯₄₅⟨gⁱ₇g^j₈⟩h, B˜_i^v

2j6 =−δ₂s¯₄₅⟨gⁱ₇g^j₈⟩h, B¯_i4j1 =−aρω²⟨g₁ⁱg₁^j⟩h

(4.23)

B¯_i^v

4j1 =−δ_paρω²ξ⟨g₁ⁱg₁^j⟩h,B¯_i5j2 =−aρω²⟨g₂ⁱg₂^j⟩h, B¯^v_i

5j2 =−δ_paρω²ξ⟨gⁱ₂g^j₂⟩h

B¯i6j3 =−ρaω²⟨g₃ⁱg₃^j⟩h, B¯_i^v

6j3 =−δpaρω²ξ⟨g₃ⁱg₃^j⟩h

Similalrly,P¯^f_m andP˜^f_m are 6n×1 and 2n×1 are load vectors and their non-zero elements are, P¯_mi1 =p₁₃⟨gⁱ₄(p^k_a+p_dtζ)⟩h, P¯_mi2 =p₆₃⟨g₆ⁱ(p^k_a+p_dtζ)⟩h, P¯_mi6 =−p_d⟨gⁱ₃⟩h

P¯_mi1^v =δ1p13⟨g₄ⁱ(p^k_a+pdtζ)⟩h, P˜_mi1 =−p33⟨gⁱ₅(p^k_a+pdtζ)⟩h

(4.24)

where ⟨. . .⟩h =∑_L

k=1t^(k)∫₁

0(. . .)^(k)dζ. Since gⁱ(ζ) are known in close form from previous step, so the above elements of matrices are obtained in closed form.

Rewrite the governing Eq. (4.21) and (4.22) equation as

N¯F_,ξ ={B(ω) +¯ ξB¯^v(ω) +ξ²B¯^vv}F¯+ (Bˆ +ξBˆ^v)Fˆ+ (P¯m+ξP¯^v_m) (4.25)

LˆF= (B˜ +ξB˜^v)F¯ +P˜_m (4.26)

Substituting Fˆ from Eq. (4.26) into Eq. (4.25) yields the following set of first order ODEs with varying coefficients forF:¯

F¯_,ξ ={B₀(ω) +ξB₁(ω) +ξ²B₂}F¯+P₀+ξP₁ (4.27) whereB0=N⁻¹(B¯ +BLˆ ⁻¹B);˜ B1=N⁻¹(B¯^v+BLˆ ⁻¹B˜^v+Bˆ^vL⁻¹B);˜

B₂ =N⁻¹(B¯^vv+Bˆ^vL⁻¹B˜^v);P₀ =N⁻¹(P¯_m+BLˆ ⁻¹P˜_m); P₁ =N⁻¹(P¯^v_m+Bˆ^vL⁻¹P˜_m)

For static case, the elements of matrices which depend on natural frequencies (ω) and time (t) becomes zero, ¯B_i4j1= ¯B_i5j2= ¯B_i6j3= ¯B_i^v

4j1= ¯B^v_i

5j2= ¯B_i^v

6j3=0. Therefore, the coefficients of final system of first order ODEs Eq. (4.27) is not now function of unknown natural frequencies (ω). So, final system of ODE Eq. (4.27) now expressed as,

F¯_,ξ ={B₀+ξ₁B₁+ξ²B₂}F¯ +P₀+ξP₁ (4.28) Above equation represents the set of first order non-homogeneous ODEs with variable coefficient (function ofξ).

Similarly, for the dynamic case (free vibration), all the elements in the load vectors (P¯^f_m and P˜^f_m) becomes zero due to absence of external applied load ( ¯P_mi1= ¯P_mi1^v = ¯P_mi2= ¯P_mi6= ˜P_mi1=0).

Therefore, load vectors P0 and P1 in Eq. (4.27) are now equals to zero. Therefore, the Eq. (4.27) reduced to,

F¯_,ξ1 ={B₀(ω) +ξ₁B₁(ω) +ξ₁²B₂}F¯ (4.29) Now, Eq. (4.29) is a system of simultaneous homogeneous first order ordinary differential equations (6n) with variable coefficients which is function ofξ-coordinate and also contains natural frequencies (ω).

NUMERICAL RESULTS AND DISCUSSIONS

The final approximate solution for present system of ODEs Eq. (4.28) and (4.29) are obtained by employing modified power series method as discussed in Chapter 2 and 3. Now F¯ is known functions which is substituting into Eq. (4.22) to solve the functions F. Now the second step isˆ completed and further, these two steps of thickness and in-plane directions are repeated to achieve the required level of accuracy.