SAFETY OF RAC-BEAMS DESIGNED ACCORDING TO EUROCODE 2 PROVISIONS

ABSTRACT. When recycled materials are used as concrete aggregates, one has to regard the larger scattering in the concrete characteristics. In this article, probabilistic calculations are presented for typical R/C beams with application of different scattering degrees ON the concrete compressive strength. The worsening influence of the larger scattering in this parameter on the structure safety, leads to a proposed increase of the characteristic values of recycled aggregate concrete, when compared to conventional concrete.

Keywords: Recycled Aggregate Concrete (RAC), Scattering values, Safety, Safety factors, Probabilistic calculations, Concrete compressive strength, Eurocode 2 provisions.

Professor Dr.-Ing. Johann-Dietrich Wörner, obtained his PhD degree in Earthquake Engineering from Darmstadt University of Technology (TUD), in 1985. After teaching for four years as a Professor of Concrete Engineering he was appointed as Professor for Structural Engineering in 1994 at TUD. Since 1995, he is a President at the Darmstadt University of Technology. Professor Wörner's main fields in research are new materials, such as glass, composite materials, plastics, SIMCON, non-linear dynamics and recycling of materials. His objective is to emphasize the connections within all steps of engineering design, from modelling to optimization.

Ir Pieter Moerland, obtained his degree in Civil Engineering (Ir) in 1993 from the Delt University of Technology. During 1993 - 1997, he worked on researches dealing with 3-D R/C structures under seismic action and probabilistic design at Darmstadt University of Technology. Currently, he is working on static calculations at Arup GmbH in Düsseldorf for the PhD.

The use of recycled materials as concrete aggregates is an attempt to reduce the constant request for natural resources in the building industry. Moreover, the need for waste storage space will be reduced when aggregates can be reused. Recycled aggregates can be obtained from demolished structures, such as buildings, roads, pavements etc. Due to such a variety of sources, recycled aggregate can consist of particles with largely varying characteristics.

For example, a typical recycled aggregate can comprise differently aged concrete and clay based brick parts, all these have a significant effect on many of the properties. Selection of the recycled material types according to their sources leads to a smaller variation of aggregate characteristics within the new concrete, but causes organisational and logistical difficulties and this can make the product economically less attractive.

The heterogeneous composition of the aggregate materials is reflected in a relatively large scattering in concrete properties, when compared with conventional concrete. Especially compressive strength, tension strength, Young's modulus, creep and shrinkage show larger scattering values. The application of Recycled Aggregate Concrete (RAC) can lead to a reduction of the safety of a structure, when no special measures are taken with regard to this larger scattering.

In this paper, the effect of the larger scattering in concrete compressive strength fc on the safety of a structure was investigated. This was done using a range of typical R/C sections, designed for ULS, according to the currently valid European building codes Eurocode 1 (actions) [1] and Eurocode 2 (concrete structures) [2]. Based on the results. Obtained from probabilistic calculations in these sections, recommendations for the modification of the required characteristic distribution values for RAC are given.

The structure reliability, resulting from the application of these safety factors, can be calculated by probabilistic methods. These methods are based on the failure function Z, typically expressed by

In eq.(1) R represents the structure capacity and S the loading on the structure. R and S are represented by their distribution functions.

As R is a function of a number of material and/or geometric variables, the distribution functions of these variables result in an overall distribution function of R. This can be complicated if R is an implicit function of its variables, as is the case for R/C sections. In this study the influence of scattering in concrete compressive strength f_c and steel tension strength on the structure safety is investigated. For both properties, according to [3], usually a logarithmic normal distribution can be assumed.

Depending on the concrete class aimed at, the mean value f_cm and the characteristic value f_ck (95% probability of exceeding this value in a sample test) for the compressive strength are defined. lt is not to be expected that concrete classes higher than C20/25 can be realized when recycled aggregates are used, so that this commonly applied class is used here. [2] prescribes a f_cm-value of 28 N/mm² and a f_ck-value of 20 N/mm² for C20/25. This leads to a coefficient of variation V_c, equal to 17.42%.

For the commonly used steel quality S500 the and -values are defined 560 and 30 N/mm² respectively. These values result in a coefficient of variation V_y of 5.36%. For RAC there is not yet consistent experimental data which gives a guideline for a realistic choice of the distribution function of f_c available. A large joint research project is currently running in Germany in order to provide this information. In this study a number of idealised distributions which are likely to occur, is investigated.

The loading memher S comprises a combination of the different loading types. The applicable distributions for each specific loading type can be simply summed. The determination of appropriate distribution types and characteristic distribution values has been a topic of extensive investigations. In [4] and [5] it is recommended to use normal distributions for both live and dead loads. According to these studies the code value g for dead load can act as mean value. A coefficient of variation V_g of 10% is thought reasonable for this loading type. Concerning live load, the code value q should represent the value with a crossing probability of 2%. Its mean value is approximated by division of the latter value by a factor 1.824. V_q is taken equal to 40% here.

The structure or structure part safety can be expressed in terms of the probability of failure.

(2)

Herein fz represents the distribution of the failure function. A common representation of P_f is the reliability index

(3)

in which and represent the standard deviation and the mean value of Z, respectively. ß and Pf are related through the standard normal distribution: P_f is obtained by integration of this distribution.

(4)

In tables for standard normal distribution the integration in eq. (4) is performed for a wide range of ß-values. The available probabilistic methods for approximation of the integral in eq.(2) are based on Monte-Carlo-simulation or the more widely accepted FORM/SORM (First/Second Order Reliability Method) calculations.

Monte-Carlo-simulation is based on generation of a large number N of Z-values and counting the failure cases N_f, thus resulting in

(5)

The variables in Z are generated from their respective distributions by a random generator. The main disadvantage of this method is the time consumption. Especially for the required small P_f-values, a large number of simulations is necessary for achieving accurate results. Importance sampling (also: tail approximation) is a modified Monte-Carlo-method in which the number of required simulations is reduced by estimation of the shape of the distribution function of Z and calculation of ß according to eq. (3).

The much faster FORM is based on an analytical procedure. In the case of n stochastic design variables X_n, the failure boundary Z(X₁,..,X_n)=0 is dividing the n-dimensional space into a safe and an unsafe part. The distribution function f_z is cut by the failure boundary. FORM transforms f_z into a standard normal n-dimensional distribution and linearises the failure boundary (after transformation) in the point on Z(X₁,..,X_n)=0, having the highest f_z value, the so called design point (X₁*,..,X_n*). The finding of the design point requires iteration.

The reliability index ß equals the distance of the design point to the origin in the standard normal n-dimensional space. For a more theoretical elaboration of these methods is referred to text books, e.g. [6]. The program used in the calculations offers the possibility to use all discussed methods. In this study FORM is used for the parametric studies. These FORM results are exemplary controlled by the more imaginative Monte-Carlo-methods.

The failure function Z can be defined for ULS and SLS criteria. The values for ß, required by Eurocode 1, depend on the criteria used, the age of the structure and the classification of the structure in terms of structure type (bridge or building). Concerning the structure age, distinction is made between new structures and structures dose to the planned maximum age. In this study the ULS criteria for a new structure is investigated. The required ß-value for this case equals 4.7, which corresponds with a P_f value equal to 10⁶/year

Figure 1

Investigated R/C section. Dimensions in [m]

In order to cover a representative range of R/C structures, a typical R/C section subject to bending, is chosen. The reinforcement ratio p in this section is varied within the limits 0.1- 3.0%. Eurocode 2 allows a maximum p-value of 4%, but the interval chosen is considered to represent practical limits. The section dimensions and geometry are shown in Figure 1. For each p-value the failure probability is computed with the failure function

(6)

In eq.(6) M_S represents the scattering acting moment on the section and M_R represents the scattering section capacity resulting from the scattering material values, defined in the previous section. Failure of a R/C section can be either caused by exceeding the ultimate steel tensile strain _su(20 ⁰/₀₀) or by exceeding the ultimate concrete compressive strain _cu (3.5 ⁰/₀₀). Formally, EC2 does not put a limitation on the steel strain, but the 20 ⁰/₀₀ is the usual basis for design adds.

The failure mechanism can be guessed based on the reinforcement ratio in the section, but can not be exactly determined without an iterative procedure. Especially when scattering in f_c and f_y is present, there exists a p-range for which both failure mechanisms can occur. Therefore the section capacity is calculated by means of iteration in a subroutine in a structural reliability program [7].

The starting point in the failure probability calculation is the determination of the "code section capacity" M_Sds by using the material design values f_cd and f_yd prescribed by [2]

(7)

and

(8)

In [2] the expression for f_cd includes an additional multiplication factor 0.85, which accounts for the long term effects creep and shrinkage. This factor is inappropriate here as these effects are not considered in the determination of the f_c-distribution. The latter can be determined shortly after erection of a structure and does not include long term effects. Hence, application of the factor 0.85 would pretend a too large structure safety which starts to decrease practically immediately after removing the structure formwork. The distribution of M_S is dependent on the loading type and is related to M_Sds as described in the following. Three different loading types are investigated. The first type is 100% dead load, resulting in a M_S,g-distribution

(9)

In the second (theoretical) type 100% live bad, the bad distribution is represented by

(10)

For the third fall a mixed bonding type with an arbitrarily chosen g/q ratio of 1 is taken. Keeping in mind that q and g are characteristic boading values (prescribed by the building code), one can derive that M_S,g+q is represented by the combination

(11)

In designing RAC structures, one would like to adopt the usual design procedure without being required to use a new set of safety factors. In this way the designer does not have to bother about the aggregate source in his planned structure. For this reason the goal here is to arrive at required characteristic and mean values for a "RAC C20/25", which is to be treated in the design as a usual C20/25. These values have to be present in a sample check at the concrete plant or at the building site in order to have it approved as this quality.

The f_c distributions investigated are listed in Table 2.

Table 2

Investigated concrete distributions

Distribution type 1 is the reference C20/25 case. Distribution types 2 and 3 show the same mean value. The higher degree of scattering is realised by lower characteristic values. Distributions 4 and 5 represent cases with a characteristic value equal to the reference concrete. The larger scatter is obtained here by increasing the mean values. lt is evident that the first 2 distributions will result in lower safeties when compared to the reference concrete. lt is worth regarding these because if the ß-values found are not lower than the required 4.7, one could consider not to demand larger mean values for the material.

The last 2 presented distributions show the same degree of scatter (in terms of _c) as the previously discussed ones. However, they are modified with respect to both mean and characteristic values. The required characteristic value is increased by 2 [N/mm²] and the two larger scattering degrees are realised by means of differently enlarged f_cm-values. The distributions investigated here, are cases which could weIl represent the realistic shape of the f_c distribution of RAC.

A comparison of the effect of the loading type on the reliability index ß is shown in Figure 2. The reliability of the structure is varying with the applied reinforcement ratio p. Before entering in discussing the development of the reliability index, Figure 3 is shown in order to give an impression of the values of M_Sds, M_R, M_sg and M_sq for the p-range investigated. The respective distances of these values relative to each other, form the basis of ß. In order to show the scattering in the capacity function M_R both and -are shown.

The R-values are estimated by setting M_S in eq. (6) equal to zero. The different shapes of _R and M_Sds, i.e. the different positions of the tangent transition point are due to the different stress block shapes in the section for the unfactored and factored values for f_c. For reinforcement ratios, larger than approximately 1.4%, V_R increases rather quickly, due to the more important role concrete plays in the failure mechanism. On the loading side S, one can see the lower mean values and larger standard deviations for q-Ioads, compared to g-loads.

For the 2 ultimate cases (the 100% types) the steel failure mechanism does not produce large differences in ß. This difference is larger in the concrete failure range. Over the entire range, the 100% dead load case is the least safe case, so that this loading type will be taken in the following calculations. The fact that the two ultimate cases do not produce the boundary curves for the combined cases may seem confusing. The explanation is that the distribution of the allowable loading over g and q results in lower absolute values in the standard deviations on the loading side. ß can be estimated (s. eq. (3)) from following expressions for and :

In the 100% dead load case, a small value of is combined with a small value of . For the 100% live load case, both values are higher. Both combinations lead to comparable ß values, as is confirmed by Figure 2. For the g=q case, the lower absolute _S,i values play a negligible role in the _z value, keeping it small compared to the former 2 cases. takes values which are between those cases. This combination leads to the high ß-values for the g=q case, observed in Figure 2.