Issue 50

K. Meftah et alii, Frattura ed Integrità Strutturale, 50 (2019) 276-285; DOI: 10.3221/IGF-ESIS.50.23

T

T a a D

a







d

d

(20)

A D 

By substituting the expression of the plastic multiplier d  into Eq. (15), the elasto-plastic tangent modulus is derived as:

T D D A D  a a

 

ep D D

(21)

T a a

The incremental stress-strain relationship is given as:   0 D d d        

0 ep

f

f

   

  

(22)

f

s    

s    d

d D 

 

s

 but not of the transverse shear stresses s

 , the direct

For Mindlin plate, yield function F is assumed to be function of f

D always remain elastic [1, 16, 17].

stresses associated with flexure only hence s

F INITE ELEMENT FORMULATION

T

he Mindlin-Reissner theory takes the shear deformation into account by decoupling the rotation of the plate cross section from the slope of the deformed mid-surface and the displacement field requires C 0 continuity only. Then the displacement fields (the transverse displacement w and two rotations  x ,  y ) are described by the same order of shape functions as follows:

w        

i w     xi       yi  

N

0 0

    

    

i

n



  d

x  

N

(23)

0

0

i

y    



i

1

N

0 0

i

The bending and shear strain-displacement relationships are given as:

n

n

1    i

1    i



. B d 

s 

. B d  si

;

(24)

f

fi

i

i

with



  

i    N x 

0

0



N N

    

      

i



0



i

    ; si

N

x B N  

i





B

(25)

0 0



fi

i y N N y x    

 

   

i

N

0



i

y 



i



0

The tangential stiffness matrix can be written as follows:





T

      T K B D B B D B dA                T f ep f s s

(26)

A

280