PSI- Issue 9

I. Shardakov et al. / Procedia Structural Integrity 9 (2018) 199–206 Author name / Structural Integrity Procedia 00 (2018) 000–000

204

6

It should also be noted that density of the concrete specimen required for further calculations was determined by weighing and for a particular image it was equal to 2464 kg/m 3 . All experimentally obtained values of frequencies of free vibrations, logarithmic decrements of attenuation and density were used in the computational algorithm for determining the characteristics of concrete: elastic modules (E, G), Poisson's ratio and parameter  characterizing the dissipative properties of concrete. 4. Determination of the model parameters The proposed computational algorithm for determining the elastic and dissipative characteristics of concrete consists in generating an iterative sequence of solutions of the initial boundary value problem (1) - (4). At each iteration two problems, corresponding to the first and second loading schemes, are solved. At the first iteration ( 1 i  ) when solving the initial boundary value problems, the standard values of the elastic and dissipative characteristics of the concrete ( 0 9 30 10 Pa E   , 0 0.2   , 0 7 10 sec    ) were used as the initial approximation. In what follows, the superscript denotes the number of the iteration sequence. The results of the obtained solutions were used to construct vibrograms of the displacement velocities along the normal to the surface at the points of the specimen for which the experimental vibration diagrams were obtained (Fig. 3a and 6a). These vibrograms were processed with the algorithm identical to that used for the experimental vibrograms. As a result, the values of the natural vibration frequencies ( 1 long f , 1 rot f ) and the logarithmic damping decrements ( 1 long  , 1 rot  ) were obtained. We further assume that Young's modulus E plays a dominant role in determining the frequencies of free longitudinal vibrations of a concrete specimen (the first loading scheme), and the square of the eigenfrequency in the selected dominant mode is directly proportional to the stiffness. Then the next approximation of Young’s modulus can be obtained from the recurrent relation

exp 2

   

   

f f

long i long

.

(5)

1

i

, E i i

0,1, 2, ...,



E

N





Reasoning in the same way with respect to dissipative properties, we can assume that the logarithmic decrement is directly proportional to the parameter β. From this it follows that each subsequent approximation of the dissipative parameter can be determined from the recurrent relation

exp long i long       1 , i i i 

.

(6)

0,1, 2,...,

N

Further, we assume that the shear modulus G plays the dominant role in determining the frequencies of free flexural torsional vibrations of a concrete sample (the second loading scheme), and the square of the frequency for the distinguished dominant mode is directly proportional to the stiffness. Then the next approximation of the shear modulus can be obtained from the recurrence equation

   

   

exp 2

f

.

(7)

rot

1

i

, G i i

0,1, 2, ...,



G

N





i

f

rot

In accordance with the known relation, Nowatski (1975), Poisson's ratio in the   1 i  -th iteration is defined as follows:

1

i



E

1      . 1 1 i i

(8)

2

G