Concrete, the physical properties of which are given in Table 2 below, is an artificial stone obtained from a correctly dosed mixture (that is to say in well-determined proportions) of binder, water, aggregates (large and small) and possibly additives. This mixture, when wet, is called fresh concrete. After hydration, it hardens to give a very resistant stone called concrete. It is mainly used for its resistance in compression.

According to Table 3 below, in a volume of concrete, the different components are in different proportions.

The characteristic strengths of concrete are: compressive strength f_c28 (main characteristic determined by a standardized laboratory test); the tensile strength

f_t28 and the shear (sliding) strength τ_bj.

During its hardening, the concrete takes, little by little, its resistance and at the end of the 28^th day, this resistance reaches its characteristic value, designating the class of concrete. The resistance of concrete increases according to a logarithmic law of formula (2) below:

f c j = log ( j ) log ( 7 ) ⋅ f c 28 ≅ 0.685 f c 28 ⋅ log ( j + 1 ) (2)

where, j is the age of the concrete, in days (d ≤ 28 days) and f_c28 is the strength of the concrete at 28 days of age, expressed in MPa.

The characteristic compressive strength of concrete at age “j”, when “j” is very large, can be determined by the following formula (3):

f c j = 1.1 f c 28 for j > 28 (3)

The tensile strength of concrete (f_tj) is determined by relation (4) below:

f t j = 0.6 + 0.06 f c 28 (4)

For calculations in RC, the tensile strength of concrete is neglected.

The shear strength τ b j is calculated by formula (5) below:

τ b j = k c ⋅ f t j (5)

where, k_c = (1.5 - 2.0) is a coefficient depending on the class of the concrete.

The deformations of concrete are: shrinkage (of the order of 5 × 10⁻⁴), swelling (often neglected) and creep.

Considered as a reinforcing element, steel is an alloy of iron (approximately 98%) and carbon (up to 2.0%) containing a very low percentage of impurities (originating from mining or formed during manufacturing) as well as adjuvant (up to 10% sometimes) to improve its qualities [40] .

The steels used in reinforced concrete structures are: smooth rounds (RL, in French), which have no roughness on the surface and high adhesion steels (HA), having roughness to improve the steel-concrete adhesion. Concrete steels have different grades which correspond to their qualities of elasticity limit (varies between 200 and 500 MPa) and resistance [41] .

The density of steel is 7.85 t/m³ (2 to 3 times greater than that of concrete). Thanks to its high mechanical resistance, it is considered the lightest construction material (2 times lighter than wood and 10 times lighter than ordinary concrete). Its coefficient of thermal conductivity is 58.35 W/m∙˚C (40 times that of concrete). At 200˚C, its modulus of elasticity E decreases and at 600˚C already, it changes to the plastic state [42] .

The main mechanical characteristics of steel are its mechanical resistance and its deformability, which depend on its composition and the technology of its manufacture. The strain diagrams of steel in tension and in compression are identical; they are symmetrical with respect to the origin O. These diagrams make it possible to determine the mechanical characteristics of steels (the elastic limit Fe and the resistance limit Frup), the characteristic unit strains of elasticity ε e l , of rupture ε r u p and the modulus of elasticity E [43] .

In general, a load-bearing reinforced concrete element is a complex system made up of several components (concrete, steel, and interface) whose behaviours are very different from each other. The behaviour of the structure depends essentially not only on the behaviour of concrete and steel but also on that of the steel-concrete interface.

By definition, the interface can be considered to be an area of concrete that surrounds the steel, and which does not a priori have any specified shape. Its shape depends directly on that of steel. The behaviour of the interface is represented by the adhesion which is a complex phenomenon and makes it possible to ensure a bond between two surfaces in contact and to transmit a stress exerted according to the contact surface of the two materials.

In the sense of reinforced concrete, once the concrete is subjected to external loads, the loading of the reinforcements is necessary for the normal functioning of the structure. Thus, this role is performed by the interface. The latter must allow total transmission of forces without slipping of the reinforcements in the concrete sheath before its deterioration. Once the deterioration of the interface has taken place, it causes a loss of adhesion which then leads to a change in the behaviour of steel and concrete [44] . It is therefore necessary to have a good understanding of the behaviour of the interface during structural calculations and during the execution of the work.

The investigation of the behaviour of the interface presented by Lutz and Gergely made it possible to conclude that the transmission of the force between the concrete and the steel takes place by the action of the following three physical phenomena: A physicochemical adhesion, directly linked to the composition of each of them and the way they are set up; a mechanical interaction, this mechanical action is particularly due to the presence of ribs along the reinforcing RCr and friction between the two contact surfaces [45] . These physical phenomena depend on various parameters: the dimensions of the structure, the nature of concrete, reinforcement and its characteristics [46] .

For ribbed steels, mechanical interaction is the main mechanism governing the behaviour of the interface. On the contrary, physic-chemical adhesion and friction are the mechanisms which dominate the behaviour of the interface in the case of smooth steels [47] [48] [49] [50] . Under the action of these physical phenomena and according to Caquot’s theory, two types of stresses can develop at the interface: Shear stresses generated by sliding along the axis of the RCr. If the axial force increases, these stresses can cause the concrete to break along conical surfaces inclined at an angle of 45˚ with the axis of the RCr and radial stresses perpendicular to the axis of the RCr (Perchat [51] , Tepfers [52] , Tilantera and Rechardt [53] ). The essential value characterizing the steel-concrete bond is its shear strength. This data makes it possible to determine the resistance of a steel cast in concrete, as well as to define an anchoring length necessary to maintain the link. The ultimate adhesion stress f_bd, also called τ_max in work relating to adhesion (Equation (5)), is calculated experimentally according to the following expression:

τ max = F max / ( π ⋅ ∅ ⋅ l b ) (6)

with, F_max: maximum force obtained experimentally; ∅ : diameter of the steeland; l_b: length anchored in the concrete.

The regulatory ultimate limit value of the adhesion stress is noted τ_su and is valid according to the BAEL 91 rules:

τ s u = 0.6 ⋅ φ s ⋅ f t j (7)

φ_s: The sealing coefficient relating to steel (equal to 1.5 for HA and 1 for RL).

For HA reinforcements, the calculation value of the ultimate bond strength f_bd can be taken according to Eurocode 2, part 8.4.2 equal to:

f b d = 2.25 ⋅ n 1 ⋅ n 2 ⋅ f c t d (8)

with, n₁ is a coefficient linked to the conditions of adhesion: n₁ = 1, when the conditions of adhesion are “good” and n₁ = 0.7 in the other cases; n₂ is a coefficient linked to the diameter of the RCr and is equal to the max ((1.32 – φ)/100; 1); f_ctd: the design tensile strength of concrete as indicated in EC2 in part 3.1.6 [54] .

As a reminder, the tests which make it possible to study the behaviour of the steel-concrete interface are: The pull-out type test or pull-out test, the push-in type test and the double tie rod [55] .

The intrinsic behaviour of the interface at the local level is influenced by various external factors and parameters (limiting conditions, loads, sizing) and/or internal parameters (characteristics of materials: concrete, steel, adhesion). These parameters can occur at several scales and at different stages of interface degradation.

According to several documentary studies, the main parameters that influence the behaviour of the interface are: The geometrical characteristics of the steel, the characteristics of the concrete, the type of loading and the confinement. Other so-called secondary parameters are: Concrete placement, corrosion and age of concrete [56] [57] [58] [59] [60] .

When we talk about the cracking of reinforced concrete, we must necessarily talk about the degradation of the steel-concrete bond, since this one produces radial micro-cracks which can evolve towards different failure mechanisms, and in certain cases, they will be able to cause the fall of the resistance of the integrated system of the three components: steel-bond-concrete. Depending on these three components, failure can occur: either in concrete (longitudinal cracks), or in steel (plasticization of the reinforcement), or in the connection (cylindrical cracks) [61] .

To show the corrosion degradation of High Adhesion (HA) steel reinforcements, we cut out pieces of steel of about 30 cm that were attacked by rust. They were taken from reinforced concrete posts and beams intended for waste reception or recycling (Figure 2). The diameters of these uncorroded samples are respectively, from left to right, 6 mm, 8 mm and 12 mm (Figure 3).

The study was conducted at the Civil Engineering Laboratory of the National High School of Public Works (NHSPW) in N’Djamena, Chad. The latitude, longitude and altitude are respectively 12.132965; 15.0540224 and 298 meters. According to the Google Maps application, consulted on April 25, 2018 at 10:00 a.m., the coordinates of the site are: 12˚07'58.7"N 15˚03'14.5"E.

Samples should be cleaned using the drill hammer, half-round, medium-sized file and wire brush. Indeed, the hacksaw is used to cut steels into pieces of about 30 cm each; the pitting hammer removes the surface layer of rust; the medium-soft file is used to remove the resistant layer of rust; the wire brush, to clean the sample and the calliper, to measure the diameter of steels with the surface layer of rust and without this layer.

1) Sample collection

For our work, we took samples from all sources, the origin and age of which we do not know, since they were taken from broken load-bearing structures intended for the recycling center. The objective here is to determine the impact of corrosion on steel and therefore on its ability to resist, the rate of reduction of which could be assessed.

2) Measurement of the rough rusty section

The samples being well cleaned, it is a question of measuring the new diameters of the parts most affected by corrosion using the calliper. You will first need to know how to read the measurements made with a calliper. The reading example is given below.

Vernier scale 1/10^th

a) The zero of the vernier is opposite a graduation of the ruler.

Read on the ruler the number of mm before the zero of the vernier; here 9 mm.

b) The zero of the vernier is not opposite a graduation of the ruler.

Locate the vernier scale, tenths of an mm, which is best aligned, here 0.60 mm.

We finally read: 9 + 0.60 = 9.60 mm.

3) Removal of the rust layer

With the pitting hammer, we had to tap on the rusty sample to remove all the rusty layer, then we filed down the resistant layers of the rust with the medium-soft file and finally we used the wire brush to clean sample. Thus, we have samples of which the parts most affected by corrosion are revealed to be measured with the calliper.

4) Measurement of the net section most affected by rust

Using the caliper, the most damaged (degraded) sections are measured, as it is this section that is considered to be the least resistant part during a bending force of the structure, for example. These operations are presented in Figure 4 for HA6, Figure 5 for HA8 and Figure 6 for HA 12. All measured values are grouped in Table 4.

5) Calculation of the corrosion rate (T_c)

It is defined by the following formula [62] :

T c = ∅ n c − ∅ c ∅ n c (9)

with, ∅ n c is the uncorroded diameter and ∅ c is the corroded diameter

6) Modelling of degradation

Wagner’s theory of oxidation kinetics (Equation 10) was used to model the progression of the corrosion layer over time (Figure 7). Considering the constant k = 0.2, we could theoretically plot the Wagner function under Matlab.

Y = k ⋅ t (10)

with, Y: oxide thickness in µm; t: time in s and k: kinetic constant in µm/s^1/2.

The oxide curve progresses and stabilizes after a while, limited by an asymptote

defined by the maximum thickness. This is because the oxide layer behaves after a while, like a shield for steel.

Sample HA6 is a 30 cm section of high-tack steel 6mm in diameter (Figure 2) which has been corroded (Figure 3). The entire rust layer was removed and the new diameter of the most degraded part could be measured with a calliper (Figure 4). The value read is 3 mm. Formula (9) above made it possible to calculate a corrosion rate of 50% reported in Table 4.

In Figure 8 established by Mangat and Elgalf, which gives the different behavioural curves of HA10 steels with various experimental and numerical corrosion rates, we notice that for a corrosion rate of 7.5%, the load-carrying capacity at break dropped by half (1/2) for a deflection of about 3 mm. This implies that for a corrosion rate of 50% determined on the HA6, the ultimate limit can be assumed to be negligible (i.e. the loss of the initial bearing capacity of the steel is considered equal to 100%, i.e. 4/4 of the initial ultimate resistance). Therefore, a column or beam having reinforcement with such a corrosion rate runs a great risk of bending failure.

Applying the same reasoning for the HA6 steel reinforcement sample and considering a corrosion rate for the HA8 sample (Figure 5) equal to 23.75%, the ultimate strength of this steel will decrease significantly by at most ¾ of its initial ultimate strength with even higher risks of rupture for the reinforcements and therefore for the reinforced concrete.

Applying the same reasoning for the third HA12 sample (Figure 6) with a corrosion rate of 5%, the ultimate strength of this corroded steel will decrease by at most 1/4 of its initial ultimate strength.

By a logarithmic regression of the data RC on the results of this study, the trend curve of the loss of the bearing capacity as a function of the corrosion rate is presented in Figure 9. Above 50% corrosion rate, we can consider that the steel reinforcements no longer contribute to the resistance of an element of the structure. It emerges from this work that the corrosion rate of reinforcement is low when its nominal diameter is large and that the loss of the bearing capacity of a structural element is important when the corrosion rate is high.

Thus, the mathematical model of the loss of bearing capacity as a function of the corrosion rate can be written as below (11):

Y = 0.3013 ln ( x ) + 1.2063 (11)

with, Y: the loss of bearing capacity as a fraction of the initial ultimate strength of the steel considered; ln: Natural logarithm and x: the corrosion rate in%.

Note that the coefficient of determination R² = 0.9714 is close to 1.

In short, the loss of the bearing capacity of a structural member is an increasing

function of the rate of corrosion degradation. However, the study also shows that corrosion degradation rates are high for small reinforcement diameters. So, to guarantee a long-lasting life for a reinforced concrete structure, this study shows that we must avoid the excessive use of small diameter reinforcements. Also, this study shows that for a rational and durable operation of a complex steel-concrete element, it is not only necessary to avoid the corrosion of the steel in the concrete but it is also necessary an irreproachable adhesion allowing a good transmission of forces between concrete and steel.

Finally, at a given moment in the state of corrosion of the reinforcements in the concrete, when the tensile or bending stresses are greater than the bearing capacity or design limit resistance of an element of the structure, the cracks appears and the rupture is evident.

According to the diagram of Figure 1, we want to specify that the present study relates exclusively to the loss of the steel section at the local level by corrosion to estimate the reduction in the bearing capacity at the general level. To this end, the studies of the degradation of the interface by corrosion (cause of the loss of adhesion) and of the volumetric expansion by corrosion (cause of the disappearance of the ribs) which was briefly approached and which was not developed during this experiment is the subject of studies in perspective.

During the experiment, we found that at any given time t, the progression of the corrosion layer varied depending on the diameters of the reinforcement samples tested. Normally, according to our understanding, two reinforcements of different diameters made of the same high adhesion material, used under the same environmental and stress conditions must corrode in the same way therefore have an equal thickness of the corrosion layer. We were unable to identify this contradiction and believe that this is certainly due to the multitude and complexity of certain causes of corrosion announced in paragraph 2.2.