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C

C=Pl1sin α−l2sin β

l1cos α−l2cos β

δh

AO

P C

δh

AO =P l3

1sin α

12EsI−Cl3

1cos α

12EsIsin α

P C h

Es

I I =bt3/12 t

E1U

δh

BO =P l3

2sin β

12EsI−Cl3

1cos β

12EsIsin β

δO=l2sin βl1cos α−l1sin αl2cos β

l1cos α−l2cos β

δh

1=δh

AO

δO

l1sin α+δh

BO

δO

l2sin β

=σ1Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

Est3(l1cos α−l2cos β)2

h

1=σ1Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

Est3(l1cos α−l2cos β)3

E1U=Est3(l1cos α−l2cos β)3

Lyl2

1l2

2(l1+l2) (cos αsin β−sin αcos β)2

E1U

Pm

i=1 Bij

σ1Bj

ˆ

E1j

=

m

X

i=1

σ1Bij

EI

1Uij

jth ˆ

E1j

ˆ

E1j=Bj

m

P

i=1

Bij

EI

1Uij

EI

1Uij

i j

E1eq

ˆ

E1j

σ1Lb =

n

X

j=1

σ1jLjb

Ljjth L=

n

P

j=1

Lj

b

n

E1eqL=

n

X

j=1

ˆ

E1jLj

E1eq

E1eq =1

L

n

X

j=1

BjLj

m

P

i=1

Bij

EI

1Uij

E1eq =Est3

L

n

X

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

l2

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

(l1ij cos αij −l2ij cos βij )2

σ2

E2U

δv

bCO =W l3

3cos2γ

12EsI

W=σ2b(l1cos α−l2cos β)v

E2U

M0=−W l3cos γ

2M0

M0

M0

OA =l2

l1+l2

M0M0

OB =l1

l1+l2

M0

l M

δ=Ml2

6EI φ=−M0

OAl1

6EsI

M0

OA M0

δv

rCO =l1l2l2

3cos2γ

12EsI(l1+l2)W

M0

1=W1l1cos α

2+l2

l1+l2

M0

M00

1=W1l1cos α

2−l2

l1+l2

M0

M0

2=−W2l2cos β

2−l1

l1+l2

M0

M00

2=−W2l2cos β

2+l1

l1+l2

M0

W1=−l2cos β

l1cos α−l2cos βW

W2=l1cos α

l1cos α−l2cos βW

δv

AO =W1l3

1cos2α

12EsI

δv

BO =W2l3

2cos2β

12EsI

δv

O=δAO(−l2cos β) + δB O (l1cos α)

l1cos α−l2cos β

δv

AO δv

BO

δv

O=l2

1l2

2cos2αcos2β(l1+l2)

12EsI(l1cos α−l2cos β)2W

δv

2=δv

bCO +δv

rCO +δv

O

=W

12EsIl2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

v

2=σ2(l1cos α−l2cos β)

Est3Lyl2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

E2U=Est3Ly

(l1cos α−l2cos β)l2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2−1

E2ULy

EI

2ULj

jth EI

2U

W Ly

AE2U

=W Lj

AEI

2U

A=Bij b

EI

2U=E2U

Lj

Ly

E2eq EI

2U

σ2

jth σ2

σ2Bb = m

X

i=1

σ2ij Bij !b

2ij

jth

ˆ

E2j2jBj=

m

X

i=1

EI

2U ij 2ij Bij

2ij =2ji= 1,2...m jth ˆ

E2j

jth

ˆ

E2j=

m

P

i=1

EI

2U ij Bij

Bj

∆2j

g

2L=

n

X

j=1

2jLj

g

22jjth

Ljjth

σ2L

E2eq

=

n

X

j=1

σ2Lj

ˆ

E2j

E2eq =L

n

P

j=1

LjBj

m

P

i=1

EI

2U ij Bij

E2eq =LEst3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1 l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2−1

G12Uτ

δs

OA =δs

OB

F1=cot α

cot α−cot βF

F2=−cot β

cot α−cot βF

Vs=1

cot α−cot βF

δs

OA =F1sin αl3

1

12EsI−Vscos αl3

1

12EsIsin α

s F1

G12U

Vsδs

OA =δs

OB = 0

τ

δs

bCO =F l3

3sin2γ

12EsI

F=τ b (l1cos α−l2cos β)Ms=F l3sin γ

2Ms

MsMs

1=l2

l1+l2

MsMs

2=l1

l1+l2

Ms

l

M δ =Ml2

6EI

φ=l1l2Ms

6EsI(l1+l2)

δs

rCO =φl3sin γ=l1l2l2

3sin2γF

12EsI(l1+l2)

δs

CO =δs

bCO +δs

rCO =F l2

3sin2γ

12EsIl3+l1l2

l1+l2

F

γs=τ(l1cos α−l2cos β)l2

3sin2γ

Est3Lyl3+l1l2

l1+l2

G12U=τ

γs

=Est3Ly

l2

3sin2γ(l1cos α−l2cos β)l3+l1l2

l1+l2

G12ULy

GI

12U

Ljjth

GI

12U

τ

G12U

Ly=τ

GI

12U

Lj

GI

12U=G12U

Lj

Ly

τ

G12eq τ jth

τBj=

m

X

i=1

τij Bij

ˆ

G12jγjBj=

m

X

i=1

GI

12U ij γij Bij

ˆ

G12jjth γjγij

jth jth GI

12Uij

i j

γj=γij i= 1,2, ..., m jth

ˆ

G12j=

m

P

i=1

GI

12U ij Bij

Bj

τ

n

γgL=

n

X

j=1

γjLj

γgG12

τ

G12eq

L=

n

X

j=1

τj

ˆ

G12j

Lj

τ=τj

G12eq =L

n

P

j=1

LjBj

m

P

i=1

GI

12U ij Bij

GI

12Uij

G12eq =LEst3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1 l2

3ij sin2γij l3ij +l1ij l2ij

l1ij +l2ij −1

ν12

ν12

ν12U

ν12U=−0

2

0

1

0

10

2

0

1(= h

1)

δ0

AO =−P l3

1sin α

12EsI−Cl3

1cos α

12EsIcos α

P C

δ0

BO =P l3

2sin β

12EsI−Cl3

1cos β

12EsIcos β

δ0

O=δ0

AO(−l2cos β) + δ0

BO (l1cos α)

l1cos α−l2cos β

=σ1l2

1l2

2Ly(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

0

2=σ1l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

ν12U=−cos αcos β(l1cos α−l2cos β)

(cos αsin β−sin αcos β)Ly

ν12ULy

νI

12U

Ljjth

νI

12U0

1Bij =0I

1Bij

0

2Ly=0I

2Lj(.)I

ν12eq =−1

L

n

X

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

(cos αij sin βij −sin αij cos βij )

cos αij cos βij

ν21

ν21

ν21U

ν21U=−00

1

00

2

00

100

2

00

2(= v

2)

00

1

δ00

AO =−W1l3

1cos αsin α

12EsI

δ00

BO =W2l3

2cos βsin β

12EsI

W1W2

δ00

1=δ00

AO

δO

l1sin α+δ00

BO

δO

l2sin β

=W l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

12EsI(l1cos α−l2cos β)2

δO

00

1=σ2l2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

Est3(l1cos α−l2cos β)2

ν21U=−Lyl2

1l2

2(l1+l2) cos αcos β(cos αsin β−sin αcos β)

(l1cos α−l2cos β)3l2

3cos2γl3+l1l2

l1+l2+l2

1l2

2(l1+l2) cos2αcos2β

(l1cos α−l2cos β)2

ν21ULy

g21

1

g21

1=1jj= 1,2, ..., n

ν21eq =L

n

P

j=1

Bj

m

P

i=1

νI

21U ij Bij

Ly

νI

21Uij

ν21eq =−L

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

l2

1ij l2

2ij (l1ij +l2ij ) cos αij cos βij (cos αij sin βij −sin αij cos βij )

(l1ij cos αij −l2ij cos βij )2l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2

βij >90◦

GA

L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦

i j

θ= 30◦

E∗

1

Es

=E∗

2

Es

= 2.3t

l3

E∗

1E∗

2

E1E2G12 ν12 ν21

E1GA =

Est

l3cos θ

(h

l+sin θ) sin2θ

E2GA =Est

l3(h

l+sin θ)

cos3θ

G12GA =

Est

l3(h

l+sin θ)

(h

l)2(1+2 h

l) cos θ

ν12GA =cos2θ

(h

l+sin θ) sin θ

ν21GA =(h

l+sin θ) sin θ

cos2θ

E1eq =Est3

L

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

l2

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

(l1ij cos αij −l2ij cos βij )2

E2eq =LEst3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij)

m

P

i=1 l2

3ij cos2γij l3ij +l1ijl2ij

l1ij+l2ij +l2

1ijl2

2ij(l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij−l2ij cos βij )2−1

G12eq =LEst3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij)

m

P

i=1 l2

3ij sin2γij l3ij +l1ijl2ij

l1ij+l2ij −1

ν12eq =−1

L

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

(cos αij sin βij −sin αij cos βij )

cos αij cos βij

ν21eq =−L

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij)

m

P

i=1

l2

1ijl2

2ij (l1ij +l2ij) cos αij cos βij (cos αij sin βij −sin αij cos βij )

(l1ij cos αij −l2ij cos βij)2l2

3ij cos2γij l3ij +l1ijl2ij

l1ij+l2ij +l2

1ijl2

2ij(l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij−l2ij cos βij )2

θ= 30◦

G∗

12

Es

= 0.57 t

l3

G∗

12

E∗

2ν∗

12 =E∗

1ν∗

21 ν∗

12 ν∗

21

ν∗

12 =ν∗

21 = 1 G=E/2(1 + ν)

E G ν

ith jth

E1Uij =Esij t3(l1ij cos αij −l2ij cos βij )3

Lyl2

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

Esij ith jth

E1eq =t3

L

n

X

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

l2

1ij l2

2ij (l1ij +l2ij ) (cos αij sin βij −sin αij cos βij )2

Esij ((l1ij cos αij −l2ij cos βij )2)

E2eq =Lt3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

Esij l2

3ij cos2γij l3ij +l1ij l2ij

l1ij +l2ij +l2

1ij l2

2ij (l1ij +l2ij ) cos2αij cos2βij

(l1ij cos αij −l2ij cos βij )2−1

G12eq =Lt3

n

P

j=1

m

P

i=1

(l1ij cos αij −l2ij cos βij )

m

P

i=1

Esij l2

3ij sin2γij l3ij +l1ij l2ij

l1ij +l2ij −1

L=n(h+lsin θ)l1ij =l2ij =l3ij =l αij =θ βij = 180◦−θ γij = 90◦i j

E1eq =κ1t

l3cos θ

(h

l+ sin θ) sin2θ

E2eq =κ2t

l3(h

l+ sin θ)

cos3θ

G12eq =κ2t

l3h

l+ sin θ

h

l2(1 + 2h

l) cos θ

h/l = 1 h/l = 1.5

h/l = 2

E1θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

E1θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

E2θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

E2θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

G12 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

G12 θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

ν12 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

ν12 θ= 45◦

h/l = 1 h/l = 1.5

h/l = 2

ν21 θ= 30◦

h/l = 1 h/l = 1.5

h/l = 2

ν21 θ= 45◦

E1E2

ν12 ν21 G12

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