Issue 38

I. N. Shardakov et alii, Frattura ed Integrità Strutturale, 38 (2016) 339-350; DOI: 10.3221/IGF-ESIS.38.44

Figure 1 : Four-point bending of reinforced concrete beam (a) and the calculation scheme of the beam with a crack (b) .

According to Fig. 1, the developed mathematical model describes all structural elements of the beam: concrete (1), steel reinforcement (2), and supporting elements (3). The problem is formulated by using the virtual displacement principle [23].

b

o

a

b

o

a

f f A A A A A A               . f

(1)

Boundary and initial conditions are specified as

u u u 1 2 3 ; 0,        x x L u u 1 2 3 0,

L

2

i         u t i x V V 1 2 0, 0, 1, 3;

u

(2)

i



t

0

t

0

   

x S ( ), 

n p t

n n

p

Boundary conditions imposed on free surfaces, contact lines between the beam and supporting elements and boundary between the steel reinforcement and concrete follow from the variational Eq. (1). The variation of the work of internal and external forces for the structural elements of the beam can be written as:

k

k k ij ij 





  

A

dV



bV

(3)

t 2

k



u

k

k

k

k





i



 



u dV p u dS   

A

f

i

i

i

2



b

b S p

V

Expressions (1)-(3) contain the following notation: A   – variation of the work of inertia and external forces; the superscripts b o a , , denote concrete, supporting elements and a reinforcement rods, respectively, the upper index k is used as “b” “ a” and “ o ” for different structural elements, i , j – integer indices taking values 1, 2, 3 in accordance with the axes of Cartesian coordinate system ( x x x 1 2 3 , , ); k i u – displacement vector components; k ij  , k ij  – strain and stress tensor components; k  – material density; k i p – components of the external force vector describing the impulse force applied along the normal n to the localized surface areas S p of the beam. – variation of the work of internal forces; f A 

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