PSI - Issue 81

Olena Romashko-Maistruk et al. / Procedia Structural Integrity 81 (2026) 269–275

271

4. Results and discussions The construction of a universal model of deformation of concrete and reinforced concrete elements and structures is one of the most important tasks of the reinforced concrete theory. It is quite obvious that it should be based on a universal function of the concrete deformation diagram. This is what the fractional-linear function can be considered (CEB-FIP (1991); Fib (2012); Romashko and Romashko (2019b)):

1 ( 2) ( / ) ( / ) / 1 2 1 1 c c c c c c k           

k

f







,

(1)

с

ck

since it contains a complex integral characteristic that allows this diagram to "adapt" to any type of load and any strain rate of concrete in reinforced concrete elements and structures. As is known, the modulus of concrete deformation can be considered as a certain coefficient of proportionality between its stresses and strains. Under static loads with nonlinear growth of plastic deformations of concrete, the change in this proportionality can be described by the elastic-plastic coefficient:

,

(2)

k E E E  

f

1  

со

cf

со с

ck

where co E is the initial modulus of concrete elasticity at

0  c  ; ck f , 1 c  and cf E is the compressive strength of concrete, the

corresponding critical strains of concrete and the modulus of concrete deformation under static loads (Fig. 1). It is quite obvious that under the action of dynamic loads the elastic-plastic coefficient of compressed concrete can be described in a similar way:

,

(3)

cf d k E E E ,   со d

f

 

d ck d со с , 1,

where ck d f , , c d 1,  and cf d E , are the compressive strength of concrete, the corresponding critical strains of concrete, and the modulus of concrete deformation under dynamic loads (Fig. 1). The same applies to the elastic-plastic coefficient of compressed concrete under prolonged loads:

,

(4)

cf l k E E E ,   со l

f

 

l ck l со с , 1,

where ck l f , , c l 1,  and cf l E , are the compressive strength of concrete, the corresponding critical strains of concrete, and the modulus of concrete deformation under the action of long-term loads (Fig. 1).

 c

f ck,du

E cf,du

E cf,d

1

2

f ck,d

E cf

3

E cf,l

f ck

4

E cf,lu

f ck,lu f ck,l

5

 c1,l

 c1,d

 c

 c,du

 c1

 c,lu

Fig. 1. Main design parameters of the diagrams of compressed concrete deformation under load: 1 – instantaneous dynamic; 2 – dynamic; 3 – standardized short term; 4 – long-term; 5 – very long-term According to expressions (2) and (3), the relationship between the elastic-plastic coefficients of concrete under the action of dynamic and static loads can be expressed by the following dependence: