master 2021/6/7 13:34 page 1 #1

(1)

z x y

My

M_x σ_zdA y x

G y

My

Mx

x

z x y

τzxdA y x

G y

τyzdA

xC

y_C

C Mt

M_t

Qy

Qx

Q_y

Q_x N

N

C G

N = Z

A

σzzdA Qy =

Z

A

τyzdA Q_x =

Z

A

τ_zxdA

Mx≡ M(G,x) = Z

A

σzzydA My ≡ M(G,y)=−

Z

A

σzzxdA Mt≡ M(C,z)=

Z

A

[τyz(x− xC)− τzx(y− yC)] dA

(2)

a b

F

R

F

R y x C G,C

τsz

C G

x y Qx

c

F

R Qx

N y

x

G C

G l M_t

M_y Mx

(a) (b)

(c) (d)

a b ˆ

v

A

B D

l z

F

F R

R C

G l F a

F l Rb

(e) a b

Q_y z

(3)

⊥ k¹ k2

P P⁰

u⁰_⊥=−u⊥

u⁰_k1= u_k1

u⁰_k2= u_k2

θ_⊥⁰ = θ_⊥

θ⁰_k2=−θk2

θ⁰_k1=−θk1

P

P⁰ u⁰_⊥= u_⊥

u⁰_k1=−uk1

u⁰_k2=−uk2

θ_⊥⁰ =−θ⊥

θ_k2⁰ = θ_k2 θ_k1⁰ = θ_k1

⊥ k¹ k2

∗

∗ F_⊥⁰ =−F⊥

F_k1⁰ = F_k1

F_k2⁰ = F_k2

C_⊥⁰ = C_⊥

C_k2⁰ =−Ck2

C_k1⁰ =−Ck1

F_⊥⁰ = F_⊥

F_k1⁰ =−Fk1

F_k2⁰ =−Fk2

C_⊥⁰ =−C⊥

C_k2⁰ = C_k2 C_k1⁰ = C_k1

†

Figure 1: An overview of symmetrical and skew-symmetrical (general- ized) loading and displacements.

(4)

G,x z y

ρx

∆z

v(z) θ

θ + ∆θ

z ∆θ y

∆z

∆θ

ρx+ y cy∆z

(a) (b) (c)

G,y z

x

ρy

∆z

u(z)

φ

φ + ∆φ

∆φ

z x

∆z ρy− x

bx∆z

(d) (e) (f)

∆φ

_z = a + bx + cy (1)

dθ dz = 1

ρx

, θ =−dv

dz, d²v

dz² =− 1 ρx

(2) dφ

dz = 1 ρy

, φ = +du

dz, d²u dz² = + 1

ρy

(3)

σz = Ezz = Ez

¯

− 1 ρy

x + 1 ρx

y

(4)

N = ZZ

A

EzzdA = EA¯ (5) Mx =

ZZ

A

EzzydA = EJxx

1

ρ_x − EJ^xy 1

ρ_y (6)

My =− ZZ

A

EzzxdA =−EJxy

1 ρx

+ EJyy

1 ρy

(7)

(5)

z x y

My

Mx

σzdA y x

G

∆z s y

My

Mx

x

EA = ZZ

A

E_z(x, y) dA (8)

EJxx = ZZ

A

Ez(x, y)yy dA (9)

EJxy = ZZ

A

Ez(x, y)yx dA (10) EJyy =

ZZ

A

Ez(x, y)xx dA (11)

¯

= N

EA. (12)

1 ρx

= M_xEJ_yy+ M_yEJ_xy EJ_xxEJ_yy− EJ²xy

(13) 1

ρ_y = MxEJxy + MyEJxx

EJxxEJyy− EJ²xy

(14) 1

ρ_eq = s 1

ρ²_x + 1

ρ²_y (15)

σ_z= E_z_z (16)

= αM + βM + γN (17)

(6)

α x, y, E_z, EJ∗∗

= E_z(x, y) −EJ^xyx + EJyyy EJxxEJyy− EJ²xy

(18)

β x, y, Ez, EJ∗∗

= Ez(x, y)−EJxxx + EJxyy EJ_xxEJ_yy− EJ²xy

(19) γ x, y, E_z, EA

= E_z(x, y) 1

EA. (20)

(xN, yN)≡ ¯eρ²_xρ_y

ρ²_x+ ρ²_y,− ¯eρxρ²_y ρ²_x+ ρ²_y

!

;

ˆ n_k =q

ρ²_x+ ρ²_y

1 ρ_x, 1

ρ_y

,

ˆ n⊥=q

ρ²_x+ ρ²_y

−1 ρ_y, 1

ρ_x

,

λ

Mx

My

=

"₁

ρx

1 ρy

#

= 1

EJ_xxEJ_yy− EJ²xy

EJyy EJxy

EJ_xy EJ_xx

| {z }

[EJ ]

Mx

My

(21)

(7)

z x

y

σz|P+ ^∂σ_∂z^z_Pdz dxdy σz|Pdxdy

P

τzx|P + ^∂τ_∂x^zx_Pdx dydz

τzx|Pdydz

τyz|P+ ^∂τ_∂y^yz

Pdy dzdx τyz|Pdzdx

P ≡ (x, y, z) dP ≡ (dx, dy, dz) qzdxdydz

P + dP

S_y = dM_x

dz , S_x=−dMy

dz , (22)

dσz

dz = α x, y, Ez, EJ∗∗

Sy− β x, y, E^z, EJ∗∗

Sx (23)

dτzx

dx +dτyz

dy +dσz

dz + q_z = 0 (24)

¯ τ_zit =

Z

A^∗

dσ_z

dz dA, (25)

¯ τzi= 1

t Z

t

τzidr (26)

¯ τ_zit =

Z

A^∗

yS_y Jxx

+xS_x Jyy

dA = y¯^∗A^∗ Jxx

S_y+x¯^∗A^∗ Jyy

S_x, (27)

¯

τ_zit = q_zi= Z s

0

Z t/2

−t/2

dσ_z

dz drdς ≈ Z s

0

dσ_z dz

_r=0tdς. (28)

¯

τzi(s)t(s) = q(s) = Z s

a

dσz

dz tdς + ¯τzi(a)t(a)

| {z }

qA

. (29)

τ (s) = S1

f (s) + S2

f (s) + τ f (s) + τ f (s) (30)

(8)

G

dσz

D dz

t A^∗

¯ τzi

Mx

My

Mx+ dMx

My+ dMy

Sx

Sy Sy

z dσz= 0

¯

τzi τ¯zi

t A^∗ t A^∗

A

B

(a) (b) (c) (d)

S₁û S₂û τ_Aû τ_Bû

≡ [1 stress unit ] f;S1(s) f;S2(s) f;A(s) f;B(s)

(9)

∆U = Z

s

τ² 2Gsz

t∆zds (31)

∂∆U

∂ ¯τi

= ¯δit∆z (32)

(10)

K_t= 4A² H ₁

tdl (33)

τmax= Mt

2tminA (34)

.

K_T ≈ 1 3

Z l 0

t³(s)ds (35)

KT ≈ 1 3

X

i

lit³_i (36)

τ_max= M_tt_max KT

(37)

(11)

qi = ∂U

∂Q_i

dU dl = 1

2





 N Mx

My

Q_x Qy

Mt







>





a_1,1 g_1,2 g_1,3 i_1,4 i_1,5 i_1,6 0 b2,2 e2,3 i2,4 i2,5 i2,6

0 0 b3,3 i3,4 i3,5 i3,6

0 0 0 c_4,4 f_4,5 h_4,6 0 0 0 0 c5,5 h5,6

0 0 0 0 0 d6,6











 N Mx

My

Q_x Qy

Mt







, (38)

a1,1= 1

EA {b^2,2, b3,3, e2,3} = {Jyy, Jxx, 2Jxy} E J_xxJ_yy− Jxy²

d6,6= 1

GK_t {c^4,4, c5,5, f4,5} ={χx, χy, χxy} , GA

(12)

u_P = u + z (1 + ˇ_z) cos θ

p1− sin²φ sin²θsin φ v_P = v− z (1 + ˇz) cos φ

p1− sin²φ sin²θsin θ

w_P = w + z (1 + ˇ_z) cos φ cos θ

p1− sin²φ sin²θ − 1

! ,

ˇ

z(z) = 1 z

Z z 0

zdς

= 1 z

Z z

0 − ν

1− ν (_x+ _y) dς,

uP = u + zφ (39)

v_P = v− zθ (40)

w_P = w. (41)

∂w

∂x = ¯γzx− φ (42)

∂w

∂y = ¯γyz+ θ (43)

x= ∂uP

∂x = ∂u

∂x + z∂φ

∂x (44)

_y = ∂v_P

∂y = ∂v

∂y − z∂θ

∂y (45)

γ_xy = ∂u_P

∂y + ∂v_P

∂x (46)

=

∂u

∂y +∂v

∂x

+ z

+∂φ

∂y − ∂θ

∂x

(47)

e =





∂u

∂x∂v

∂y

∂u

∂y +^∂v_∂x



 =



 ex

e_y gxy



 ≡ Q (48)

(13)

κ =





+^∂φ_∂x

−^∂θ_∂y +^∂φ_∂y −^∂θ_∂x



 =



 κx

κy

κ_xy



 (49)

_P≡ = e + z κ . (50)



 σx

σ_y τxy



 = σ = D = D e + z D κ , (51)

D = E

1− ν²





1 ν 0 ν 1 0 0 0 ^1−ν₂



 , (52)

_z =− ν

1− ν (_x+ _y) . (53)

q =



 qx

qy

q_xy



 =Z

h

σ dz

= Z

h

D dz

| {z }

a

e + Z

h

D zdz

| {z }

b

κ (54)

q_z =

q_xz q_yz

q_xz = Z

h

τ_zxdz q_yz= Z

h

τ_yzdz. (55)

m =



 m_x m_y mxy



 =Z

h

σ zdz

= Z

h

D zdz

| {z }

b ≡ b^T

e + Z

h

D z²dz

| {z }

c

κ . (56)

q m

=

a b

b^T c

e κ

(57)

(14)

υ^†= 1 2

q m

^> e κ

(58)

= 1 2

e κ

^>

a b

b^T c

e κ

. (59)

a = h D b = 0 c = h³

12D , γz =

γ¯yz

¯ γzx

q_z =

qxz

q_yz

υ^‡= 1

2q^>_z γz = 1

2qxzγ¯xz+1

2qyzγ¯yz. (60) υ^‡= 1

2γ^>_z χ Z

h

G dz

| {z }

Γ

γz (61)

qz = Γ γz. (62)

G = E

2 (1 + ν)

1 0 0 1

,

τ_zx(z) =− Z z

−^h

2+o

∂σ_x

∂x +∂τxy

∂y dz (63)

τ_yz(z) =− Z z

−^h₂+o

∂τ_xy

∂x +∂σ_y

∂y dz. (64)

D₁₂₃ =





E1

1−ν12ν21

ν21E1

1−ν12ν21 0

ν12E2

1−ν12ν21

E2

1−ν12ν21 0

0 0 G₁₂



 (65)



 σ1

σ2

τ₁₂



 = T1



 σx

σy

τ_xy







 1

2

γ₁₂



 = T2



 x

y

γ_xy



 (66)

(15)

T₁ =



 m² n² 2mn n² m² −2mn

−mn mn m²− n²



 (67)

T₂ =



 m² n² mn

n² m² −mn

−2mn 2mn m²− n²



 (68)

m = cos(α) n = sin(α) (69)

T⁻¹₁ (+α) = T₁(−α) T⁻¹₂ (+α) = T₂(−α) (70)

σ = D D ≡ Dxyz= T⁻¹₁ D123T2 (71)

G =

n²Gz1+ m²G2z mnGz1− mnG2z

mnG_z1− mnG2z m²G_z1+ n²G_2z

.

k^∗_x= 12F l

Ebh³ (72)

mx= m^∗_x my= 0 κx= k_x^∗ κy =−νkx^∗,

mx= m^∗_x my= νm^∗_x κx= 1− ν²

k^∗_x κy= 0.

g(y)≥ 0 (73)

f (y)≥ 0 (74)

g(y)· f(y) = 0, (75)

(16)

f (ξ, η)^def= X

i

N_i(ξ, η)f_i (76)

N_i(ξ, η)^def= 1

4(1± ξ) (1 ± η) , (77)

∂f

∂ξ =

f2− f1

2

| {z }

[∆f /∆ξ]₁₂

1− η 2

| {z }

N1+N2

+

f3− f4

2

| {z }

[∆f /∆ξ]₄₃

1 + η 2

| {z }

N4+N3

= aη + b (78)

∂f

∂η =

f4− f¹ 2

1− ξ 2

+

f3− f² 2

1 + ξ 2

= cξ + d. (79)

f (ξ, η) =

N1(ξ, η) · · · Ni(ξ, η) · · · Nn(ξ, η)





 f₁

... fi

... fn







= N (ξ, η) f , (80)

x ξ

= m ξ

= X4 i=1

N_i ξ

x_i, (81)

m ξ

=

x(ξ, η) y(ξ, η)

x(ξ, η) = X4

i=1

Ni(ξ, η)xi y(ξ, η) = X4 i=1

Ni(ξ, η)yi.

f (ξ, η)^def= X

i

N_i(ξ, η)f_i (82)

"_∂f

∂ξ

∂f

∂η

#

=

"_∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η

#

| {z }

J^>(ξ,η; xi)

"_∂f

∂f∂x

∂y

#

(83)

(17)

"_∂f

∂ξ∂f

∂η

#

=X

i

"_∂N

i

∂N∂ξi

∂η

#

f_i. (84)

J^>(ξ, η) =

"_∂x

∂ξ

∂y

∂ξ

∂x

∂η

∂y

∂η

#

(85)

=X

i

"_∂N

i

∂ξ 0

∂Ni

∂η 0

# xi+

"

0 ^∂N_∂ξⁱ 0 ^∂N_∂ηⁱ

# yi

!

(86)

"_∂f

∂x∂f

∂y

#

= J^>−1

"

. . . ^∂N_∂ξⁱ . . . . . . ^∂N_∂ηⁱ . . .

#



 ... fi

...





 (87)

= J^>−1

"_{∂ N}

∂ξ

∂ N

∂η

#

| {z }

L (ξ,η; xi), or just L (ξ,η)

f (88)

Z 1

−1

f (ξ)dξ≈ Xn

i=1

f (ξ_i)w_i; (89)

p(ξ)^def= a_mξ^m+ a_m−1ξ^m−1+ . . . + a₁ξ + a₀ Z 1

−1

p(ξ)dξ = Xm j=0

(−1)^j + 1 j + 1 aj

r (a_j, (ξ_i, w_i))^def= Xn

i=1

p(ξ_i)w_i− Z 1

−1

p(ξ)dξ (90)

∂r (aj, (ξi, wi))

∂a_j = 0, j = 0 . . . m (91) Z _b

a

g(x)dx = Z ₁

−1

g (m(ξ))dm dξ dξ≈

Xn i=1

g (m(ξ_i)) dm dξ

ξ=ξi

w_i. (92)

m(x) =

1− ξ 2

| {z } a +

1 + ξ 2

| {z } b.

(18)

dm

dξ = dN1

dξ a + dN2

dξ b = b− a 2 Z b

a

g(x)dx≈ b− a 2

Xn i=1

g

b + a

2 +b− a 2 ξi

wi. (93) Z 1

−1

Z 1

−1

f (ξ, η) dξdη≈ Xp

i=1

Xq j=1

f (ξ_i, η_j) w_iw_j (94) Z 1

−1

Z 1

−1

f (ξ, η) dξdη≈ Xpq l=1

f ξ_l

w_l (95)

ξ_l= (ξ_i, η_j) , w_l = w_iw_j, l = 1 . . . pq (96)

dAxy = 1 2!

1 x (ξP , ηP ) y (ξP , ηP ) 1 x (ξ_P + dξ, η_P ) y (ξ_P + dξ, η_P ) 1 x (ξ_P + dξ, η_P + dη) y (ξ_P + dξ, η_P + dη) +

+ 1 2!

1 x (ξP + dξ, ηP + dη) y (ξP + dξ, ηP + dη) 1 x (ξP , ηP + dη) y (ξP , ηP + dη) 1 x (ξ_P , η_P ) y (ξ_P , η_P )

. (97)

A = 1 2!

1 x₁ y₁ 1 x2 y2

1 x3 y3

, H = 1 n!

1 x1

1 x₂ ... ... 1 x_n+1

(98)

dAxy ≈ 1 2!

1 x y

1 x + x,ξdξ y + y,ξdξ 1 x + x_,ξdξ + x,ηdη y + y_,ξdξ + y,ηdη

+

+ 1 2!

1 x + x_,ξdξ + x_,ηdη y + y_,ξdξ + y_,ηdη 1 x + x,ηdη y + y,ηdη

1 x y

dAxy = 1 2

1 x y

0 x,ξ y,ξ

0 x,η y,η

dξdη + 1 2

0 x_,ξ y_,ξ 0 x,η y,η

1 x y

dξdη

(19)

dAxy = 1 2

x,ξ y,ξ

x_,η y_,η

dξdη + 1 2

x,ξ y,ξ

x_,η y_,η dξdη

dAxy =

x,ξ y,ξ

x,η y,η

| {z }

|^J^T^(ξ^P^,η^P^{; x , y )}|

dAξη (99)

ZZ

Axy

g(x, y)dA_xy = Z 1

−1

Z 1

−1

g (x (ξ, η) , y (ξ, η))|J(ξ, η)| dξdη, (100) ZZ

Axy

g( x )dAxy ≈ Xpq

l=1

g x ξl J(ξl)

wl (101)

(20)





X(ξ, η) Y (ξ, η) Z(ξ, η)



 = Xn i=1

N_i(ξ, η)



 X_i

Y_i Zi



 ,



 x(ξ, η) y(ξ, η) z(ξ, η)



 = Xn

i=1

N_i(ξ, η)



 x_i y_i zi





(102)



 u(ξ, η) v(ξ, η) w(ξ, η)



 = X4

i=1

Ni(ξ, η)



 u_i v_i wi



 (103)



 θ(ξ, η) φ(ξ, η) ψ(ξ, η)



 = X4

i=1

N_i(ξ, η)



 θ_i φ_i ψi



 (104)

u =





 ... u_i

...





 v =





 ... v_i

...





 w =





 ... w_i

...







θ =





 ... θi

...





 φ =





 ... φi

...





 ψ =





 ... ψi

...







u(ξ, η) = N (ξ, η) u v(ξ, η) = N (ξ, η) v d^>=

u^> v^> w^> θ^> φ^> ψ^>

(105)

_∂u

∂x∂u

∂y

= J⁰−1

"

. . . ^∂N_∂ξⁱ . . . . . . ^∂N_∂ηⁱ . . .

#

| {z }

L (ξ,η; xi) or just L (ξ,η)





 ... u_i

...





 (106)







∂u

∂x∂u

∂y

∂v

∂x∂v

∂y





=

L (ξ, η) 0 0 L (ξ, η)

| {z }

Q (ξ,η)

u v

(107)

(21)







∂θ

∂x∂θ

∂φ∂y

∂φ∂x

∂y





= Q (ξ, η)

θ φ

(108)



 e_x e_y gxy



 =





+1 0 0 0

0 0 0 +1

0 +1 +1 0





| {z }

H⁰







∂u

∂x∂u

∂y

∂v

∂x∂v

∂y





= H⁰Q (ξ, η)

u v

(109)



 κ_x κy

κxy



 =





0 0 +1 0

0 −1 0 0

−1 0 0 +1





| {z }

H⁰⁰







∂θ

∂x∂θ

∂y

∂φ

∂x∂φ

∂y





= H⁰⁰Q (ξ, η)

θ φ

(110)

e =h

H⁰Q (ξ, η) 0 0 0 0i

| {z }

Be(ξ,η)

d (111)

κ =h

0 0 0 H⁰⁰Q (ξ, η) 0i

| {z }

Bκ(ξ,η)

d . (112)

(ξ, η, z) = B_e(ξ, η) + z B_κ(ξ, η)

d ; (113)

¯γzx

¯ γ_yz

= L (ξ, η) w +

0 + N (ξ, η)

− N (ξ, η) 0

θ φ

, (114)

γ¯_zx

¯ γyz

=

0 0 L (ξ, η) 0

− N (ξ, η)

N (ξ, η)

0 0

| {z }

Bγ(ξ,η)

d (115)

d^>=

u^> v^> w^> θ^> φ^> ψ^>

(116)

G^>=

U^> V^> W^> Θ^> Φ^> Ψ^>

(117)

(22)

δ Υe= δ d^>G . (118) σ = D (z) Be(ξ, η) + Bκ(ξ, η)z

d (119)

δ = Be(ξ, η) + Bκ(ξ, η)z

δ d (120)

q = a B_e(ξ, η) + b B_κ(ξ, η)

d (121)

m = b^>Be(ξ, η) + c Bκ(ξ, η)

d , (122)

δ e = Be(ξ, η) δ d (123) δ κ = Bκ(ξ, η) δ d , (124)

δ Υ^†_i = ZZ

A

Z

h

δ ^>σ dzdA

= ZZ

A

Z

h

Be+ Bκz δ d>

D Be+ Bκz

d dzdA

= δ d^>

ZZ

A

Z

h

B^>_e + B^>_κz

D Be+ Bκz dzdA

d

= δ d^>K^†d , (125)

δ Υ^†_i = ZZ

A

δ e^>q + δ κ^>m dA

= δ d^>

ZZ

A

B^>_e a Be+ b Bκ

+ B^>_κ b Be+ c Bκ

dA

d

= δ d^>Kσd , (126)

a , b , c

= Z

h

D

1, z, z² dz, ZZZ

Ω

g(ξ, η, x, y, z)dΩ = (127)

= Z +1

−1

Z +1

−1

Z +^h₂+o

−^h₂+o

g(ξ, η, x(ξ, η), y(ξ, η), z)dz

J (ξ, η) dξdη,

(23)

δ Υ^‡_i = ZZ

A

δ γ^>_z qzdA

= δ d^>

h

ZZ

A

B^>_γΓ B_γdA

d

= δ d^>K^‡d . (128)

δ Υ_i= δ Υ^†_i + δ Υ^‡_i

= δ d^>

K^†+ K^‡ d

= δ d^>K d . (129)

δ d^>G = δ Υe= δ Υi= δ d^>K d , ∀ δ d , (130)

G = K d ; (131)

(24)

α 2

y< 0

y> 0

α 2 α 2

α 2

-a +a

-b +b

x y

x= −^αy_2a

y= z= ν^αy_2a γxy = 0 exact solution, pure in-plane bending

x= −^αy_2a

y= 0, z=_1−ν^ν ^αy_2a γxy = −^αx_2a

four noded, isop. element, in-plane trapezoidal mode

γxy= ^α₂ γxy= 0 γxy= −^α₂ Cb

Ciso4

C_b = ¹⁺

1−ν 2 (^a_b)²

1−ν²

≈ 1.48 if ν = 0.3,^a_b = 1

α 2

u =

1 + _1−ν^ν²2

_Eα2y²

8a² +^Gα_8a²2^x²

u = ^α²_8a^Ey2²

undeformed domain a)

b)

c)

d)

residual (spurious) shear deformation;

overall error assessment:

(25)

γ_{z ˆ}₄₁ z

z

˜ γ_{z ˆ}₁₂ 1

1

˜ w₄

w₁

w1

˜ w₂

α α

β

˜¯ γ_yz,n1

˜¯ γ_zx,n1 x

y

n1 n2

n3 n4

12ˆ 41ˆ

x y (a)

(b) β

β α

α

(26)

S (ξ, η, z) =



. . . uˆi(ξ, η, z) . . . . . . vˆi(ξ, η, z) . . . . . . wˆ_i(ξ, η, z) . . .



 (132)

u (ξ, η, z) = S (ξ, η, z) d . (133)

˙u (ξ, η, z) = S (ξ, η, z) ˙d (134) E_kin= 1

2 ZZZ

Ω

˙u^> ˙u ρdΩ (135) E_kin= 1

2 ZZZ

Ω

hS ˙di>h S ˙di

ρdΩ, (136)

E_kin = 1

2 ˙d^>ZZZ

Ω

S^>S ρdΩ

˙d = 1

2 ˙d^>M ˙d . (137) M =

ZZZ

Ω

S^>S ρdΩ, (138)

˙d^>G = dE_kin dt =d

dt

1

2 ˙d^>M ˙d

=1 2

d¨^>M ˙d + ˙d^>M ¨d

= ˙d^>M ¨d .

G = M ¨d (139)

δ u (ξ, η, z) = S (ξ, η, z)δ d , (140)

δ d^>F = ZZZ

Ω

(δ u )^> p dΩ

= ZZZ

Ω

S δ d>

p dΩ

= δ d^>

ZZZ

Ω

S^>p dΩ,

F = ZZZ

Ω

S^>p dΩ (141)

(27)

ˆı ˆ

 ˆ ˆı

ˆk



g1

g3 g2

g5

g4

g6 g9

g8

g7

θe1n1ˆıe1

we1n2kˆe1

e1 e3

e2 e4

we1n2ˆke1

θe1n1ˆıe1

= ug2ˆıg2+ vg2ˆg2+ wg2kˆg2

= θg1ˆıg1+ ϕg1ˆg1+ ψg1kˆg1

kˆ ˆıg∗

kˆ_g∗

ˆ

_g∗

ˆ ˆı ˆk

 ˆı

ˆ

 ˆk

Gej = Kejdej (142)

dgl =





 u_gl v_gl wgl

θ_gl ϕ_gl ψgl







. (143)

d^>_g =

d^>_g1 d^>_g2 . . . d^>_gl . . . d^>_gn

(144) F^>_g =

F^>_g1 F^>_g2 . . . F^>_gl . . . F^>_gn

; (145)

R^>_g =

R^>_g1 R^>_g2 . . . R^>_gl . . . R^>_gn

(146)

we1n2 =hˆke1, ˆıg2iug2+hˆke1, ˆg2ivg2+hˆke1, ˆkg2iwg2 (147) θe1n1 =hˆıe1, ˆıg1iθg1+hˆıe1, ˆg1iφg1+hˆıe1, ˆkg1iψg1 (148)

P_e1

10,7=hˆke1, ˆı_g2i P_e1

13,4=hˆıe1, ˆı_g1i

P_e1

10,8=hˆke1, ˆ_g2i P_e1

13,5=hˆıe1, ˆ_g1i

P

=hˆk , ˆk i P

=hˆı , ˆk i,

(28)

node X Y Z

g1 −ac 0 +a

g2 0 +as +a

g3 +ac 0 +a

g4 −ac 0 0

g5 0 +as 0

g6 +ac 0 0

g7 −ac 0 −a

g8 0 +as −a

g9 +ac 0 −a

Uni

Vni

Wni

Θni

Φni

Ψni

uni vniwniθniϕniψni

i = 1 . . . 4

n1 n2 n3 n4

e1 g1 g2 g5 g4

e2 g2 g3 g6 g5

e3 g4 g5 g8 g7

e4 g5 g6 g9 g8

(29)

ue1ni

ve1ni

we1ni

θe1ni

ϕe1ni

ψe1ni

d_g2

d_g1 d_g3 d_g4 d_g5 d_g6 d_g7 d_g8 d_g9

ue2ni

ve2ni

we2ni

θe2ni

ϕe2ni

ψe2ni

d_g2

ue3ni

ve3ni

we3ni

θe3ni

ϕe3ni

ψe3ni

d_g2

ue4ni

ve4ni

we4ni

θe4ni

ϕe4ni

ψe4ni

d_g2

d_g1 d_g3 d_g4 d_g5 d_g6 d_g7 d_g8 d_g9 Pe1

Pe2

Pe3

Pe4

(30)

d_g1d_g2d_g3d_g4d_g5d_g6d_g7d_g8d_g9 F_g1

F_g2 F_g3 F_g4 F_g5 F_g6 F_g7 F_g8 F_g9

symm b

(a) (b)

(c) (d)

dej = Pejdg, ∀j. (149) Gej= KejPejdg, ∀j; (150) δ d^>_g G_g←ej = P_ejδ d_g>

G_ej, ∀ δ dg (151) G_g←ej = P^>_ejG_ej (152) G_g←ej = P^>_ejK_ejP_ejd_g; (153)

G_g =X

j

G_g←ej =





 X

j

P^>_ejK_ejP_ej

| {z }

Kg←ej





 d^g = K_gd_g, (154)

(31)

b_ej = (i_max− imin+ 1) l, (155) b = max

ej b_ej (156)

F_g =X

j

P^>_ejF_ej; (157)

(32)

d1

d2

d3

1:3

0.2 mm

0 d1+ 1 d2+ 0 d3= 0.2 3 d1− 0 d2+ 1 d3= 0 I:

(a) II:

(b)

d1

d2

d3

R1

R2

R3

II

I I ∩ II

(a) configuration space (b) reaction space k α_I

span (α_I, α_II),

⊥ II, k α_II

⊥ I, k α_I

⊥ α_I

⊥ α_II k I ∩ II

I ∩ II k αII ⊥ (I ∩ II)

X

i

αjidi = α^>_j d = βj

α^>_I =

3 0 1

βI= 0 α^>_II=

0 1 0

β_II= 0.2 X

i

α_jid_i= α^>_j d = β_j, j = 1 . . . m (158)

L^>d = β . (159)

L^>δ d = 0 , (160)

R =− L ` , (161)

(33)

R^j =−





 ... αji

...





 `^j (162)

K d = F + R . (163)

K d + L ` = F

K L L^> 0

d

`

=

F β

, (164)

1

2d^>K d − d^>F + `^> L^>d − β

, (165)

1

2d^>K d − d^>F L^>d − β = 0

R =− L `^∗. dk=X

i6=k

−αji

α_jk

di+

βj

α_jk

, j = 1 . . . m (166)

r t

=

E_R E_T

| {z }

E

d d =

E^>_R E^>_T

| {z }

E^>≡ E⁻¹

r t

,

r = ERd t = ETd ,

t_j =

n−mX

h=1

λ_jhr_h+ δ_j, j = 1 . . . m (167)

t = λ r + δ , (168)

d =

E^>

I λ

r +

E^>

0 δ

= Λ r + ∆ ; (169)

(34)

=

di 1

rh

0

dk λkh= 0

+

0

∆k= a

dk λkh ∆k

retained dof.

tied dof.

dk = a fix.disp. kind

general di= rh

d = Λ r + ∆

h

i

k

δ d = Λ δ r = Λ1δr1+ Λ2δr2+ . . . + Λn−mδrn−m (170) h Λh, Ri = 0 h = 1 . . . n − m, (171) or, equivalently,

Λ^>R = 0 . (172)

K Λ r + ∆

= F + R (173)

K Λ r = F − K ∆

+ R , (174)

Λ^>K Λ

| {z }

K_R

r = Λ^> F − K ∆

| {z }

F_R

+ Λ^>R

| {z }

=0

, (175)

KRr = FR (176)

d^∗= Λ r^∗+ ∆ ; (177)

R^∗ = K Λ r^∗+ ∆

− F . (178)

d^∗_ej = Pejd^∗. (179)

e = B^e_ej(ξ, η) d^∗_ej κ = B^κ_ej(ξ, η) d^∗_ej (180)

= B^e_ej(ξ, η) + B^κ_ej(ξ, η)z

d^∗_ej. (181) γz= B^γ_ej(ξ, η) d^∗_ej. (182)