PSI - Issue 81

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

312

However, in these studies we will be more interested in the calculation of internal forces arising in reinforced concrete elements under the influence of dynamic loading. In simplified calculations, it could be performed on the basis of CEB-FIP (1991), EN 1992-1-1 (2004), fib (2012) with correction of the design characteristics of concrete and reinforcement. But the correction dependencies in the specified standards are purely empirical in nature, since they were formed on the basis of only experimental studies of the main physical and mechanical characteristics of materials depending on their deformation rate (Rüsch (1960); Rasch (1962); Wesche and Krause (1972); Hjorth (1976); Bischoff and Perry (1991); Ross et al. (1995); Malvar (1998); Zhang et al. (2012)). These dependencies have no analytical justification. The same applies to relatively new studies of the dynamic characteristics of concrete (Ahmed (2015)). And the initial recommendations for determining the dynamic increase factors of reinforcement CEB Bulletin 187 (1988) are not included in either CEB-FIP (1991) or fib (2012). Based on the above, we will direct our research to the development of a simplified express methodology for predicting the strength of flexural reinforced concrete elements under the action of dynamic loads of different intensities. We will focus on the normal sections of these elements in order to focus on the significance of the influence of the dynamic increase factors of concrete and reinforcement on the strength. We will use the analytical justification of the specified dynamic increase factors of concrete, carried out by Romashko-Maistruk and Romashko (2025) when modeling the deformation processes of reinforced concrete elements using the well-known law of conservation of specific potential energy of deformation regardless of the loading mode (Romashko and Romashko (2019a)). The versatility of any calculation method depends on the assumptions on which it is based. Therefore, the proposed express method for calculating reinforced concrete elements under dynamic loads will be based on the well-known system of the following relations (Romashko and Romashko (2019b)): static ) ε , , ε ε ), N f( ε , ε f( , ε M c ct s c ct s = = ;

  

1/

( , s c r f    = ; , ) ct

geometric

(2)

) ct

σ

) ε f( c

σ

) ε f(

ct σ =

ε f(

c =

s =

physical (materials state)

,

,

.

s

and supplement it with the fractional-linear element stiffness function:

1/

r

M

o u D M r D D = − = /(1/ )

− −  (

D D 2

)





,

(3)

o

u

1/

r

M

u

u

where o D and u D - respectively, are the initial and ultimate (at the moment of exhaustion of strength) stiffness of the cross section of the reinforced concrete element; r 1/ and u r 1/ - respectively, the current curvature and the curvature of the element in the limit state; M and u M - respectively, the current and ultimate force in the reinforced concrete rod (strength). The analytical validity of this function is due to the fact that it results not only in a universal state diagram of a reinforced concrete element (Romashko and Romashko (2017)):

2

1/ D r M  −

2/(1/ )) (1/ ) ((1/ ) /(1/ )) r r r r u 



o

u

M

=

,

(4)

1 ( +

/ D M

−

o u

u

but also, the universal stress-strain diagram of concrete, included in well-known regulatory documents (CEB-FIP (1991); EN 1992 1-1 (2004); DSTU B V.2.6-156:2010 (2011); fib (2012)):

2

1 ( 2) ( / k + −  −  /   ) ( / 1 c c

) )

k

c c c c    

1 1

f



=



,

(5)

с

ck

where ck f and 1 c  are the compressive strength of concrete and the corresponding critical strains of concrete under static loads; k is the coefficient of elastoplasticity of compressed concrete; c  is the current values of relative strains of compressed concrete. To expand the scope of application of system (2), we will supplement it with dynamic increase factors of materials: c ck d ck f f DIF / , = , ctk d ctk ct f f DIF / , = , y d y y f f DIF / , = , (6) where ck d f , , ctk d f , , y d f , - respectively, the characteristic values of the compressive and tensile strength of concrete and the characteristic values of the strength of reinforcement beyond its yield point under dynamic loads. In particular, the dynamic increase factor of compressed concrete is proposed to be taken according to the studies of Romashko Maistruk and Romashko (2024b):