PSI - Issue 6

Author name / Structural Integrity Procedia 00 (2017) 000 – 000

2

Pavel A. Akimov et al. / Procedia Structural Integrity 6 (2017) 182–189

183

Peer-review under responsibility of the MCM 2017 organizers.

Keywords: Discrete-continual finite element method; finite element method; two-dimensional theory of elasticity; multipoint boundary problem

1. Formulation of the problem and approximation Let’s consider multipoint boundary problem of two-dimensional theory of elasticity, particularly static analysis of deep beam. Several elements of corresponding notation system are presented at Fig. 1.

Fig. 1. Static analysis of deep beam (sample of multipoint boundary problem).

Let’s  be domain occupied by structure,

{ ( , ) : 0 1 2 x x

, 0     1 x l x l 2 1

or

}

 

2

 1 1      b n k

{ ( , ) : 0 1 2 x x

1 x l

b x x x  

,

}

 

 

k ,

;

(1)

b

k

k

k

2, 1 k 

1,

2,

2

Let (parameter 1 l can be piecewise-constant). In accordance with the method of extended domain, proposed by Professor Alexander B. Zolotov [1], the given domain is embordered by extended one of arbitrary shape, particularly elementary. Let , 1, 2, ..., 1   b k n k  are extended subdomains, embordering subdomains , 1, 2, ..., 1     b k k n k  , const l k  1, for b b k x x x 2, 1 2 2,    . In general case k const l l x   ( ) 1 2 1

~

 1 1   b n k

x x {( , ) : 0 1 2

b x l x x x , 1 1 k 2,

b     } 2, 1 2 k

k 



 

,

.

(2)





k 

k

Without loss of generality we suppose piecewise constancy of physical and geometrical parameters of one group of subdomains from (2) along coordinate 2 x (“ basic ” dimension). It is necessary to note that physical and geometrical parameters of structure can be changed arbitrarily along 1 x . Thus, it is recommended to use DCFEM for approximation of these subdomains (discrete-continual design model is introduced).