Key Engineering Materials, ISSN: 1662-9795, Vol. 1011, pp 61-65

doi: 10.4028/p-pMy5i9

© 2025 The Author(s). Published by Trans Tech Publications Ltd, Switzerland.

Submitted:
2024-10-15

Revised:
2025-03-20

Accepted:
2025-03-20

Online:
2025-05-14

Minimum Reinforcement against Brittle Failure in Concrete Structures

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

Marta Słowik and Izabela Skrzypczak

¹Department of Structural Engineering, Faculty of Civil Engineering and Architecture, Lublin University of Technology, 40 Nadbystrzycka, 20-618 Lublin, Poland

Faculty of Civil and Environmental Engineering and Architecture, Rzeszow University of Technology, 12 Powstańców Warszawy, 35-959 Rzeszow, Poland

m.slowik@pollub.pl, izas@prz.edu.pl

Keywords: Concrete, Lightly Reinforced Concrete Member, Cracking Resistance, Minimum Reinforcement Ratio.

Abstract

The paper presents the rules for determining the minimum reinforcement in reinforced concrete structures that have been recommended in recent years and currently. It has been analyzed whether the reinforcement in structural members designed in different periods of time on the basis of various standard regulations regarding minimum reinforcement meets the conditions for protecting the structure against brittle failure. The necessary reinforcement has been determined using the method derived on the basis of nonlinear fracture mechanics of concrete. In the performed analysis the scale effect has been taken into account as well.

Introduction

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

Large concrete structures, including massive foundations, hydrotechnical structures and dams, are often designed as lightly reinforced structures. The reinforcement in this type of structures is designed to transfer stresses caused by restrained imposed deformations and protect against sudden, brittle failure. Changing environmental conditions during the working life of a building structure (heavy rains, long-term droughts) may cause landslides or uneven settlements of the structure. This leads to a change in the soil conditions, which in turn brings the risk of a safety loss of the structure. The use of an appropriate reinforcement can eliminate the risk of a sudden brittle failure of the structure. Particularly, this problem becomes important in lightly reinforced concrete structures where a low reinforcement ratio is used. Standard rules for dimensioning of concrete structures specify the minimum reinforcement ratio ( ), which is the lowest limit of reinforcement in reinforced concrete members. Concrete members with a smaller cross-section of reinforcement than the minimum ( ) are called lightly reinforced members. These members, in accordance with standard regulations, are not separated into the another group, but they are treated as unreinforced concrete members. As a consequence, the influence of reinforcing bars on the load-carrying capacity and crack resistance of lightly reinforced concrete structures is omitted.

The correct estimation of minimum reinforcement is of the paramount importance to protect the structural concrete member against brittle failure. The development of fracture mechanics, in particular non-linear models of tensile concrete, provides the basis for accurate determination of the minimum reinforcement. As the existing concrete structures were designed in different periods of time on the basis of different rules, it has been analyzed if the provided reinforcement gives the adequate safety against brittle failure.

Minimum Reinforcement - Review of Calculation Rules

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

The rules of determining the minimum reinforcement in reinforced concrete structures were changing during time. For example, in the Polish standard from 1976 [1] the minimum reinforcement ratio in flexural members was and in the next Polish edition of the standard from 1984 [2] the minimum reinforcement ratio was depended on the type of steel bars and was
for smooth bars and for deformed bars. After Poland accession to the European Union, Polish national construction codes were adapted to the Eurocodes. In the Polish regulations [3] which are based on Eurocode 2 the minimum reinforcement is calculated from the Equation (Eq. 1):

Minimum reinforcement is design to protect concrete structures against a sudden, brittle failure. Reinforcing steel bars should carry tensile stress after cracking and therefore load carrying capacity of reinforced concrete cross section should not be lower than cracking moment. The concept of comparing load capacity with cracking moment predominates nowadays but the problem connecting with the minimum reinforcement estimation lays in calculating the cracking moment in a proper way. Usually the cracking moment which is calculated on the basis of the mean tensile strength of concrete (see Eq. 2) gives too low values comparing with test results (Eq. 2):

When analyzing experimental results, for example [4], and numerical simulations, for instance [5], it can be concluded that more realistic calculations of cracking moment are obtained when using flexural tensile strength of concrete instead of axial tensile strength of concrete . The flexural tensile strength of concrete is higher than the axial tensile strength of concrete and the ratio of their values is described by the coefficient (Eq. 3):

When taking into account , the cracking moment for rectangular cross section of width and height is (Eq. 4):

The load carrying capacity resulting from reinforcement is (Eq. 5):

When comparing Eq. 4 with Eq. 5 and adopting an inner and we obtain (Eq. 6):

It can be observed that Eq. 6 is the same as Eq. 1 from Eurocode 2 when and . It indicates that the tensile stress in concrete at cracking has been assumed in Eurocode 2 as when estimating and thus the flexural tensile strength of concrete was applied in [3]: . However, the experimental results have shown that the relation between and depends on the member's dimensions and concrete properties. Therefore, the obligatory value of does not take into account the scale effect and the effect of concrete strength when calculating minimum reinforcement.

Minimum Reinforcement - Comparative Analysis

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

The application of nonlinear fracture models of tension concrete when analyzing cracking process in concrete members has shown that the relation between and varies over a wide range, as it was reported for example in . The tendency was observed that coefficient is higher in small members and in members made by high strength concrete. The numerical simulations performed by Hillerborg et al. [5] allowed to develop the coefficient depending on the bending member's height and the characteristic concrete length. The characteristic concrete length is the fracture parameter of concrete defined by nonlinear fracture mechanics (Eq. 7):

where: fracture energy of concrete and modulus of elasticity.
The relation between the coefficient and worked out in [5] is shown in Fig. 1. Some illustrative values of coefficient are presented in Tab. 1. They were read from Fig. 1 for three heights of the member's cross section: and two concrete compressive strength classes C12/15 and C30/37.

Fig. 1 Dependence of the coefficient on

Fig. 1. Dependence of the coefficient on

Table 1. Values of coefficient derived from Fig. 1

Concrete | Height of a cross section h [mm]
| 300 | 800 | 1300
C12/15 (lch=823 mm) | 1.60 | 1.37 | 1.26
C30/37 (lch=487 mm) | 1.45 | 1.25 | 1.20

A significant influence of the height of cross section on the coefficient can be observed when analyzing the values presented in Tab. 1 and the impact of concrete grade on is visible as well. Furthermore, when analyzing the scale effect at calculating from Eq. 6, the ratio of total height to effective depth should be taken into account. The ratio depends on the total height of cross section as well. The calculations of which were made for different member's heights with variable values of concrete cover (c), dimeter of stirrup ( ) and diameter of longitudinal steel bars ( ) are presented in Tab. 2.

Table 2. Values of

d=h-(c+ φstir+0.5φ)
Height of a cross section h [mm]

300
800
1300

c=20 [mm], φstir=6 [mm],
φ=20 [mm] d=h-36 [mm]
1.14
1.05
1.03

c=30mm, φstir=8 mm,
φ=28 [mm] d=h-57 [mm]
1.21
1.07
1.04

Minimum reinforcement ratio was calculated on the basis on Eq. 6 in case of flexural concrete member. In order to include the scale effect and the effect of concrete grade, the coefficient was read from Fig. 1. The calculations were performed for three different heights of cross section: 300; , two concrete compressive strength classes: C12/15, C30/37 and two types of steel bars: smooth bars of (steel grade S220), deformed bars of (steel grade S400). Smooth bars were considered in the analysis as in several existing lightly reinforced concrete structures such type of reinforcement has been used. The assumption was also made that . The results of the calculations are presented in Tab. 3, whereas, for performing the comparative analysis, the minimum reinforcement ratios derived from earlier Polish standard recommendations [1,2] and on the basis of the rules given in Eurocode 2 [3] are compiled in Tab. 4.

Table 3. Values of calculated from Eq. 6 with coefficient from Fig. 1 and

Concrete | Steel | Height of a cross section h [mm]
|  | 300 | 800 | 1300
C12/15 (lch=823 mm) | S220 | 0.0026 | 0.0019 | 0.0017
S400 | 0.0015 | 0.0011 | 0.0009
C30/37 (lch=487 mm) | S220 | 0.0048 | 0.0035 | 0.0031
S400 | 0.0026 | 0.0019 | 0.0017

Table 4. The comparison of derived from standard recommendations

Concrete | Steel | Height of a cross section h [mm]
|  | 300 | 800 | 1300
C12/15 (lch=823 mm) | S220 | 0.0026 | 0.0019 | 0.0017
S400 | 0.0015 | 0.0011 | 0.0009
C30/37 (lch=487 mm) | S220 | 0.0048 | 0.0035 | 0.0031
S400 | 0.0026 | 0.0019 | 0.0017

When juxtaposing derived on the basis of fracture mechanics from Tab. 3 with standard values presented in Tab. 4 the following conclusions have been drawn:

All standard recommendations for do not take into account the scale effect;

In previous Polish codes [1] and [2] the influence of concrete properties on was not included;

The largest differences between from Eq. 6 and determined on the basis of Polish codes [1] and [2] were obtained for relatively small members made of a higher strength concrete;

The similar values of obtained from standard rules like the ones derived from fracture mechanics were noticed only when was calculated on the basis of Eurocode 2 [3] in case of the member's height 800 mm .

Summary

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

Minimum reinforcement according to protection the concrete structure against dangerous, brittle failure should be determined from the condition of comparing the load capacity of the reinforced concrete section with the cracking moment. The formula which was derived on the basis of this assumption (see Eq. 6) is the most rational way of calculating the minimum reinforcement ratio. It
allows to take into account the scale effect and the effect of concrete strength. However, in order to obtain a proper value of reinforcement ratio, the coefficient which characterizes the relation between flexural tensile strength and axial tensile strength of concrete should be correctly determined. The coefficient varies over a wide range and is higher in small members and in members made by high strength concrete. The minimum reinforcement ratio determined based on Eq. 6 and using coefficient from Fig. 1 ( derived from numerical analysis reported in [5]) gives rational solution both in relatively small members as in massive structures.

Although the standard method of calculating recommended in Eurocode 2 is based on the condition of protecting the member against sudden failure after cracking, however the adoption of the tensile stress in concrete at cracking as has cased that he scale effect is not included when calculating minimum reinforcement from Eq. 1. The earlier Polish rules according to in several cases gave lower values than those resulting from fracture mechanics approach.

The performed analysis has shown that the provided reinforcement in the existing lightly reinforced concrete structures which were design on basis of different standard regulations in several cases cannot meet requirements for an effective protection the structure against brittle failure.

Acknowledgement

The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

The research leading to these results has received funding from the commissioned task entitled 'VIA CARPATIA Universities of Technology Network named after the President of the Republic of Poland Lech Kaczyński,' under the special purpose grant from the Minister of Education and Science, contract no. MEiN/2022/DPI/2575, MEiN/2022/DPI/2578 as part of the action 'ISKRA - building inter-university research teams.'

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The original version of this paper is available on https://www.scientific.net/KEM.1011.61.pdf

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