5.1. General Observations
The reliability index β is given as a function of the reinforcement ratio ρl and the effective depth d. The partial safety factors γT vary between 1.0 and 1.5 so that the required safety level can be easily identified. To this end, the level at which these factors lead to a target reliability index β equal to 3.8 was determined, considering a reference period of 50 years.
Figure 7 symbolically illustrates the results of the parametric studies. The curve in
Figure 7 is divided into three states: A, B and C. The reliability index is approximately
β ≈ 3.9 (state A) for reinforcement ratios up to
ρl ≈ 2.0‰. Then, the reliability index decreases approximately linearly until it reaches its minimum of
β ≈ 2.9 (state B) for a reinforcement ratio of approximately
ρl ≈ 4.75‰. Afterwards, the reliability index slightly increases again (state C).
In state A, the textile reinforcement governs the bending load capacity. Both parameters—the design value of the textile tensile strength ftd in the ultimate limit state and the textile stress ft* at the design point x*—govern the load-bearing capacity. In state C, the reverse occurs, and the design concrete compressive strength fcd in the ultimate limit state and the concrete compressive stress fc* at the design point x* govern the load-bearing capacity. In state B, reflecting a textile failure, the design value of the textile tensile strength ftd in the ultimate limit state governs the load-bearing capacity. However, the results show that concrete failure occurs with the same longitudinal reinforcement ratio. The reason for this is that the tensile strength of the reinforcement is underestimated in the calculation model due to the applied partial safety factor γT in the ultimate limit state. In fact, the load-bearing capacity of the textiles is higher than the capacity of the concrete in state B. As the reinforcement ratio increases, the reversed conditions occur at failure load. The probability that a concrete failure occurs increases.
In general, at any reinforcement ratio ρl, the cause of failure can be the concrete and the reinforcement. However, at low ρl values, the current concrete compressive strength is not likely to be fully used. In this case, the textile reinforcement is governing in the event of failure. At high ρl values, reversed conditions occur, and the concrete strength is usually critical for a failure event.
In the following sections, the influence of selected parameters on the probability of failure is presented. In order to illustrate this influence, only the parameter being examined varies in each case. A detailed description of the calculations can be found in [
9].
5.2. Comparison of the Methods for Determining Reliability
To show the equivalence of the methods FORM, MC and MC-IS, the calculated probabilities of failure (or reliability indices
β) are compared for an AR-glass and a carbon textile reinforcement (
γT = 1.5) (
Figure 8). Comparable reliability indices are obtained regardless of the calculation method used.
The reliability indices determined according to the MC and MC-IS are constantly about Δ
β ≈ 0.1 higher or lower over the entire range of the curve than when using the FORM. This difference was also observed in the example calculations of Rackwitz [
29]. In the calculations carried out by Ricker [
19], the differences were partly Δ
β ≈ 0.4. It can be assumed that with the MC, the probabilities of failure can be more accurately calculated if a sufficient amount of simulations are conducted. However, since in the example addressed in this study the differences are small, they can be neglected. Based on these considerations, a FORM analysis is sufficient to determine the reliability of textile-reinforced concrete components subject to bending loads. For this reason, only the results of the parametric studies determined with the FORM are shown below.
5.3. Influence of the Stress-Strain Curve of Concrete
The influence of the assumed concrete stress-strain curve on the reliability index is shown in
Figure 9. The three states A, B and C can be observed in both approaches. In state A, it can be seen that the reliability indices
β are similar for the same
γT values, independently of the material model used for the concrete. A small difference is observed in state C. In this case, the safety level is slightly higher when a linear stress-strain curve is applied than when a rectangular stress block is applied. The governing difference is the range of concrete failure (state C), which in the case of a rectangular stress-strain curve starts at a higher reinforcement ratio. This is due to the larger resulting concrete compressive zone force
Fc, which occurs when a rectangular stress block is considered. As a consequence, concrete failure only occurs for high reinforcement ratios (see [
23]).
The results show that the calculated bending capacities of the concrete can be sufficiently approximated by a linear and by a rectangular stress-strain curve [
23]. Thus, in the parametric studies shown below, only the results determined under the assumption of a linear stress-strain curve for the concrete are presented.
5.4. Influence of the Reinforcement Type
The influence of the reinforcement type on the reliability index was investigated using specimens reinforced with AR-glass and a carbon textile reinforcement. In the calculations, the type of reinforcement was taken into account by different parameters for the tensile strength
ft and the Young’s modulus
Et. The results are shown in
Figure 10.
According to
Figure 10, the shapes of the reliability index curves are similar, independently of the reinforcement type. Only the transition from state A to B occurs for the AR-glass textile reinforcement at a higher reinforcement ratio, which is due to the different tensile strengths. In comparison to AR-glass textile reinforcement, carbon textile reinforcement can withstand significantly larger tensile force
Ft with the same reinforcement ratios
ρl presupposed. Thus, to fulfil the equilibrium of forces with the same reinforcement ratio, a greater concrete pressure zone force
Fc is required for carbon textile reinforcement than for AR-glass textile reinforcement. Since the same concrete compressive strength
fc is assumed for both types of reinforcement, concrete failure occurs in specimens with carbon reinforcement at a lower reinforcement ratio.
In state A, a partial safety factor γT equal to 1.3 for both types of reinforcement and a partial safety factor γC equal to 1.5 for the concrete are sufficient to achieve the desired safety level of βtarget equal to 3.8. For a reinforcement ratio of ρl ≈ 3.5‰, the reliability index calculated goes below the threshold of 3.8 by about Δβ = 0.1. Due to the governing concrete failure, the safety level can then no longer be significantly increased by applying a larger partial safety factor γT for the textile reinforcement. From the authors’ point of view, the undercutting of the target index of 0.1 is still acceptable. In state C, the reliability index is initially slightly below the target value of β equal to 3.8; however, the value increases for a higher reinforcement ratio. As a result, the calculated reliability indices rapidly reach the threshold value of 3.8. The assumed partial safety factor for the textile reinforcement (1.0 ≤ γT ≤ 1.5) has no influence on the reliability indices calculated, as the different curves in state C have the same value due to the governing concrete failure.
The sensitivity factors for each parameter depending on the reinforcement ratio
ρl are shown in
Figure 11. These factors describe the influence of the different parameters on the reliability index
β. The sensitivity factors are similar for both types of textiles. In all the three states, it is noticeable that the live load
q has the greatest influence on the reliability index due to its wide scattering. At the same time, it can be observed that the textile tensile strength
ft is a governing factor in state A, while the most influential factor in state C is the concrete compressive strength
fc. In total, the sensitivity factor of the impact side
αE, which is composed by the sensitivity factor for the model uncertainty of the loading models
θE and the sensitivity factors for the dead load
g and the live load
q, has a value of approximately 0.9 in state A and approximately 0.8 in state C. Hence, these values exceed the recommendation in EN 1990 (EC0) [
21], where
αE,EC0 is equal to 0.7. This value is proposed in the code for semi-probabilistic calculations, in which the resistance side can be considered decoupled from the impact side by specifying corresponding constant weighting factors.
On the resistance side,
αR is approximately 0.5 (state A) or
αR is approximately 0.6 (state C). These values are slightly lower than the recommended values in EN 1990 [
21] where
αR,EC0 is equal to 0.8. However, the values in the code for
αE,EC0 and
αR,EC0 can never be achieved in a fully probabilistic calculation since the squared sum of
αE,EC02 +
αR,EC02 is 1.13, which is higher than 1.0. Thus, for certain conditions, the sensitivity factors specified in the code for a semi-probabilistic calculation are on the safe side.
The calculations presented in this section show that the same partial safety factor for textile reinforcement is needed for both AR-glass and carbon textiles to achieve the required safety level. Therefore, the type of reinforcement has almost no influence on
γT. This result was observed for all the parameter variations [
9].
5.5. Influence of the Effective Depth d
The influence of the effective depth
d on the reliability index
β was investigated for a carbon textile reinforcement with effective depths of
d = 15 mm and
d = 60 mm (
Figure 12 and
Figure 13).
The effective depth
d has an influence on the progression of the reliability index curves both in state A and in state C. Subsequently, the effective depth
d is 60 mm; to achieve an appropriate safety level, it is sufficient to apply a partial safety factor
γT equal to 1.3 and a partial safety factor
γC equal to 1.5 in states A and C, respectively. For the smallest effective depth, i.e.,
d = 15 mm, and for the selected partial safety factors
γT equal to 1.3 and
γC equal to 1.5, the required reliability index
βtarget of 3.8 cannot be ensured. These results can be explained by the large influence of the effective depth
d scattering on the reliability index
β, which is proven by the sensitivity factors (
Figure 13). These observations are valid for the application of a linear or a rectangular stress-strain curve for the concrete analysis [
9].
In the case of very thin structural members, even a slight deviation of the reinforcement from the correct position can lead to a large reduction of the bending capacity. For this reason, in order to ensure a sufficient safety level with a constant partial safety factor
γT independent of the effective depth, the authors recommend using a reduction factor for small effective depth, taken from the old German code DIN 1045(88) [
30]. DIN 1045(88) proposes to increase the internal forces with a factor of 15/(
d + 8) for structural members with effective depths smaller than 70 mm. This approach can be adopted for thin textile-reinforced members. Since the probabilistic calculations show that the required safety level can be ensured for effective depths
d greater than 60 mm (
Figure 12a), it is sufficient to apply a reduction for effective depths
d ≤ 60 mm. With Equations (17) and (18), an effective depth
deff can be determined, which can be used to calculate the required flexural textile reinforcement.
Figure 14 shows the influence of
d and
deff on the reliability index. The required safety level is ensured with the effective depth
deff for the three states A, B and C. In contrast, in the calculation of the reliability indices, the threshold
βtarget of 3.8 cannot be achieved when the effective depth d is not reduced. Consequently, the required safety level can be ensured while retaining the common partial safety factor for concrete
γC equal to 1.5.