PSI - Issue 81

Olena Romashko-Maistruk et al. / Procedia Structural Integrity 81 (2026) 310–315

313

2 / ))/9) ((1 log(     +

, / cu c ck d ck f DIF f DIF = =

,

(7)

s

5 1 10 − − = s

where s   - the maximum strain rate of compressed concrete under the action of static (quasi-static) loads, cu DIF - the limiting value of the dynamic increase factor of compressed concrete under the action of a dynamic load close to instantaneous 3 1 10 − = s m   s   ;

2

(

)

   

   

2

2

2 1

1

2 1

k

k

k

− 

−

−

  

  

,

(8)

/

− ln( 1) k

DIF

f

f

=

=

− +

−

(

)

(

)

, ck du ck

cu

2

2

k

k

k

−

−

obtained analytically using the hypothesis of invariance and independence from the loading mode of the potential energy of fracture or ultimate concrete deformation (Romashko and Romashko-Maistruk (2022)). It is also proposed to supplement system (2) with the function of the elastic-plastic coefficient of concrete according to Romashko-Maistruk and Romashko (2024a): 2 / ) /18) / )) (log( / )/9) /5 (9 log( 1 ( 1) (log(             m m m d k k k  +  − = + −  , (9)

5 1 10 − − = s

; m   -

where k - coefficient of elastoplastic of compressed concrete under static (quasi-static) loads at a strain rate the strain rate of compressed concrete under the action of instantaneous dynamic loading, limited by the

s  

3 1 10 − = s

,

m  

and the critical strain function of compressed concrete:

DIF k k c d d c с / 1 1,   =   .

(10)

3. Results and discussions To evaluate the proposed express methodology, calculations of the strength of normal sections of flexural reinforced concrete elements with different reinforcement at different strain rates of concrete were carried out. In this case, the dynamic increase factors of reinforcement (DIF y ) were determined according to the methods of CEB Bulletin 187 (1988) and Malvar (1998). The initial data for calculating the flexural element were as follows: cross-sectional dimensions b*h= 282*300 mm; reinforcement characteristics f yk =632 MPa, E s =2.1  10 ⁶ MPa; concrete characteristics f ck =40 MPa, E c =3.85  10 ⁶ MPa. The cross-sectional area of the working reinforcement was taken to be: A s =466 mm² in weakly reinforced elements, A s =2520 mm² and A s =3825 in heavily reinforced elements. The reinforcement ratios of the specified elements were µ s =0.006, µ s =0.033 and µ s =0.05, respectively. Based on the results of these calculations, the effect of the strain rate on the strength of normal sections of flexural reinforced concrete elements (M d /M), on the dynamic increase factors of concrete (DIF c ) and reinforcement (DIF y ) was analyzed. It has been established that under dynamic loads of any intensity, it is the reinforcement that plays a decisive role in ensuring the strength of reinforced concrete elements.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

b)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

a)

Relative valuesM d /M, DIF c , DIF y

Relative valuesM d /M, DIF c , DIF y

DIFc DIFy Md/M

DIFc DIFy Md/M

-8

-6

-4

-2

0

2

4

-8

-6

-4

-2

0

2

4

log έ

log έ

Fig. 1. The influence of the dynamic increase factors of tensioned reinforcement DIF y according to CEB Bulletin 187 (1988) (a) and according to Malvar (1998) (b), together with the dynamic increase factor of compressed concrete DIF c according to expression (7), on the relative strength M d /M of lightly reinforced (A s =466 mm 2 ) elements subjected to bending at different strain rates.