PSI - Issue 6

V.A. Meleshko / Procedia Structural Integrity 6 (2017) 115–121 Meleshko V.A./ Structural Integrity Procedia 00 (2017) 000–000

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Appendix A. Programs for stiffness matrix of rod and bending moment

KLN x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ( ) T ← :=

k 2 11 ∈ .. for

k

k

  ,

x m 1 − x m + 2 T k ⋅

  

  

1 1 T k ⋅

 

∑ =

∑ =

0.1 ⋅

0.1 ⋅

←

←

integ1 k 1 −

integ2 k 1 −

m

2

m

2

in1 k ← N11 in1 k ← N12 in2 k , ← integ1 k 1 − ← in2 k ,

integ2 k 1 −

in1 m 1 − in1 m + ( ) 2

in2 m 1 − in2 m + ( ) 2

k

k

  

  

  

  

∑ =

∑ =

0.1 ⋅

0.1 ⋅

←

,

←

N21

N22

m

2

m

2

1 − 0

1 − 1 0    ⋅

N12 − N22 −

N12 − N22 −

N11 N21

N11 N21

K14 K24

K11 K21

  

  

  

  

  

  

  

  

  

  

  

←

⋅

,

←

1 −

K13 K11 ← − K12 K14 , ← K23 K21 , ← − K32 K23 , ← K31 K13 , ← K41 K14 , ← K42 K24 ← K43 K21 , ← − K34 K43 , ← K33 K11 , ← K44 K14 1 ⋅ K24 − , ← K22 K44 , ←

K11 K21 K31 K41

K12 K22 K32 K42

K13 K23 K33 K43

K14 K24 K34 K44

   

   

←

KLN

KLN

0 1

0 2

1 0

2 3

0 0

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

0.1 0.1

  

  

  

  

∆ M MI :=

←

x , ←

v 1 2 ∈ .. for

KLN ←

KL v

j 3 ∈ for i 1 3 ∈ .. for

0 ← K2exp i j , ,

←

0

K1exp i j ,

m 3 4 ∈ .. for

n 1 2 , 4 , ∈ for m 1 2 , 4 , ∈ for ,

n 3 4 ∈ .. for

MI 1 m , ← MI 1 n , ←

MI 2 m , ← MI 2 n , ←

i

i

j

j

KL 1 ( )

KL 2 ( )

←

+

←

+

K1exp i j ,

K2exp i j ,

K1exp i j ,

K2exp i j ,

m n ,

m n ,

KS K1exp K2exp + ← US KS 1 − PS ⋅ , ←

v 1 2 ∈ .. for

0 0 US 1 US 2 ( ) T

US 1 US 2 0 US 3 ( ) T

←

,

←

U 1

U

2

U U 1 U 2 ( ) T ←

P v KL v U v ⋅ , ←

←

∆ M

v 1 2 ∈ .. for

k 1 11 ∈ .. for

k

x v k , P v ( ) ⋅

P v ( )

∑ =

←

−

∆ M1 v k ,

 

 

1

2

k

1

∆ M1

∆ M

KLN – local stiffness matrix of the rod at time step; ∆ M – increment of bending moment; MI – index matrix.