Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17

> @ V e M r S S dV 1 1 ¬ U ³ > @ > @ T

(3-15)

> @ V e K r S D S dV 2 2 ¬ ³ > @ > @ > @ T

(3-16)

^ ` ^ '

` u w u w u w h h h h hm hm a a b b c c , , , , ,

h , m = 1,2,3,4

(3-17)

^ ` ^ ` T r z F F F ,

(3-18)

Matrices > @ S 1

and > @ S 2

are derived as follows:

N N 0¬ } } ‡ } ‡ } } } 4 1 4 0¬ 0¬ ¬¬ 0¬ 0¬

> @ N N S 1 ª «

(3-19)

1

0¬

¬

1

4

11 0¬¬ } < } < } º » ‡ } ‡ } < } < ¼ 44 0¬¬ 0¬ 0¬¬ 0¬¬¬ 0¬¬¬

1

4

11

44

... ‡ ‡

N N ...

0 ...

0

0 ...

0

ª « « « « «¬

x

x

x

x

1,

1

1,

4,

... ‡ ‡

N N ...

0 ...

0

0 ...

y

y

y

y

1,

4,

1,

4,

> @ S N N

(3-20)

2

r ‡ ‡ ‡ ‡ ‡ ‡ r 1 4 ... ... 0 ... . / .. /

...

0 ...

0

0

r

r

1/

4/

N N N N ... ...

y

y

x

x

y

y

x

x

1,

4

1

,

4,

1,

4,

,

1,

4,

}

0¬¬

0¬¬ ¬¬

< } <

º » » » » »

x

x

11,

44,

¬

< } <

}

0¬¬

0¬¬ ¬¬

y

y

11,

44,

<

<

}

0¬¬

0¬¬ ¬

}

44

11

r

r

< } <

< } <

x

x

11,

11,

»¼

y

y

11,

44,

The material stiffness matrix > @ D for axisymmetric problems is defined as below [22].

Q

Q

Q

1

¬¬¬¬¬¬¬

0 0

ª « « « « ¬

º » » » » ¼

Q

Q

Q

1

¬¬¬¬¬¬

E

> @ D

(3-21)

Q

1

0

Q

Q

¬¬¬¬¬¬¬¬

1 1 2 Q

Q

0 1 2 / 2 Q

0 0

We use the Newmark method to solve equations of motion. The Newmark family is the most widely used family of direct methods for solving the equation of motion which consists of the following equations [23]. > @ ^ ` > @ ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` ^ ` n n n n n n n n n n n n M K F t t t t t 1 1 1 2 2 1 1 1 1 1 1 2 1 ] ] J J ' ' § · ' ' ' ' ' ' ' ' ¨ ¸ © ¹ ' ' ' ' ' ' (3-22)

171