Issue 59

H.A. Mobaraki et alii, Frattura ed Integrità Strutturale, 59 (2022) 198-211; DOI: 10.3221/IGF-ESIS.59.15

F INITE E LEMENT S OLUTION

I

n order to obtain numerical results, we propose a higher-order plate element as shown in Fig. 2. The element has 9 nodes and each node has 5 degrees of freedom including the axial displacement 0 u , lateral displacements 0 0 , v w , and independent rotations   , x y . To obtain the generalized displacement corresponding to each degree of freedom inside an element, the Lagrange interpolation is used. This can be stated as:        0 , u u x y N d        0 , v v x y N d        0 , w w x y N d (12)            , x x x y N d

 y x y ,



  

    



N d

y

where   d is the element nodal displacement vector and   u N ,   v N ,   w N ,

     y N are the shape function

  

x N , and

matrices, defined as:     u N N 1





N

N

0 0 0 0

0 0 0 0

0 0 0 0

2

9

    v N N 0





N

N

0 0 0 0

0 0 0

0

0 0 0

1

2

9

    w N





N

N

N

0 0

0 0 0 0

0 0

0 0

0 0

(13)

1

2

9





      x N



N

N

N

0 0 0

0 0 0 0

0

0 0 0

0

1

2

9





      y N



N

N

N

0 0 0 0

0 0 0

0

0 0 0 0

1

2

9

i N functions are  

where the







         1 1 2 1 1 2 1 N              2 4 1 1 2 1 N             3 2 1 1 2 1 N               4 4 1 2 1 1 N            5 16 1 1 N

(14)

202