reliability analysis for flexural capacity of...

Peer-reviewed by international ex-perts and accepted for publication by SEI Editorial Board

Paper received: July 13, 2015Paper accepted: January 17, 2016

Structural Engineering International Nr. 2/2016 Scientific Paper 121

Reliability Analysis for Flexural Capacity of Recycled Aggregate Concrete Beams Jianzhuang Xiao, Prof. Dr.; Kaijian Zhang, PhD candidate; Qinghai Xie, PhD candidate; Tongji Univ., Shanghai, China.

Contact: [email protected]

DOI: 10.2749/101686616X14555428758920

Abstract

Concrete made of recycled coarse aggregate (RCA) derived from multiple sources can have relative variation in mechanical properties compared with natural aggregate concrete (NAC), resulting in a lower reliability of a recycled aggregate concrete (RAC) beam than that of NAC beam. In this paper, a method is proposed to improve the reliability of beams made with RAC by increasing the reinforcement ratio. According to the equilibrium equations of beams sub-jected to bending moment, the increase in reinforcement ratio can be achieved by decreasing the RAC design strength. Same mean compressive strengths are assumed for RAC and NAC and four factors are considered including concrete grade, reinforcement grade, reinforcement ratio and the ratio of external loads in reliability analysis. Using NAC beams as a reference and keeping the target reli-ability index the same as that of RAC beams, the analysis results show that RAC design strength decreases with increase in coefficient of variation (CoV) of RAC compressive strength. When the CoV of RAC compressive strength is 20%, the RAC design strengths are 0.93 and 0.90 times of the NAC design strength with a 95% confidence level for grade C30 and C40 concrete, respectively.

Keywords: recycled aggregate concrete (RAC); beams in bending; strength vari-ation; reliability index; design strength .

Although previous studies indicate that the behaviour of RAC beam is similar to NAC beam, it is not reason-able to say components or structures made with RAC and NAC have a sim-ilar reliability. Sources and crushing procedure of RCA have a significant influence on its quality. The multiple or unknown RCA sources may lead to a higher variability in RAC strength than in NAC, further causing a higher failure probability of RAC compo-nents or structures. Attention has been paid to this issue. Studies in Ref. [12] investigated the reliability of RAC beams and reported that the reliabil-ity index of RAC beams can meet the requirements of the Chinese code13 if the reinforcement ratio or RAC com-pressive strength is increased. Model error has a remarkable influence on the reliability of RAC beams as well. A theoretical investigation was carried out in Ref. [14] to evaluate the influ-ence of the quality and quantity of recycled aggregates on the reliability of RAC elements. The results show that stricter design procedures are required for RAC to ensure the same reliability as that of conventional concrete.

Before applying and popularizing RAC in practical engineering, it is nec-essary to analyse the reliability of RAC components and to promote an effec-tive method of improving the reliabil-ity of RAC components. In this paper, the influence of these factors, such as the compressive strength of RAC, tensile strength of the reinforcement, the reinforcement ratio and external loads, on the reliability of RAC beams are investigated. Meanwhile, the RAC design strength is evaluated and determined to guarantee that the reli-ability of RAC beams will satisfy the specifications.

Sources of Variability

RCAs are generally reclaimed from constr uction and demolition wastes. The basic RCA engineering proper-ties including shape and texture, den-sity, water absorption, moisture content and strength characteristics vary con-siderably,15 which invariably produces

RAC and natural aggregate concrete (NAC) in their physical and mechani-cal properties, namely RAC has lower density, lower strength (under equal water : cement ratio) and an elastic modulus among others. The struc-tural behaviour of RAC has also been investigated. Ref. [5] reported that while RAC beams have larger defor-mations, 16 series of RAC and NAC beams have similar load-bearing capacity. The flexural performance and shear capacity of RAC beams are comparable with those of NAC beams and the general flexural theory and existing shear design methods are applicable to reinforced RAC beams.6,7 It is feasible to use RAC in reinforced concrete beams consider-ing that the flexural behaviour of RAC beams is satisfactory compared to the behaviour of NAC beams under both the service and ultimate loadings.8 RAC beams had comparable ultimate flexural strength and approximately 13% higher deflection compared with NAC beams.9 Moreover, the RAC beams possessed approximately 12% lower shear capacity compared with NAC beams10 and the shear capacity decreases with increase in replace-ment ratio of RCA.11

Introduction

A large amount of concrete waste from demolished structures is gen-erated annually and the amount is expected to increase in the future. Using recycled coarse aggregate (RCA) to replace part or all natural coarse aggregate (NCA) in concrete mixing is one of the effective meth-ods of reclaiming waste concrete. As it is an eco-friendly material, in recent years, many researchers have been engaged in exploring ways to improve the basic behaviour of recycled aggre-gate concrete (RAC). At the material level, most of the research findings have been extensively reviewed and summarized. These investigations were focused mainly on the property of interfacial transition zones (ITZ),1 physical and mechanical properties,2,3 and durab ility improvement of RAC.4 The above investigations indicated that some difference exists between

122 Scientific Paper Structural Engineering International Nr. 2/2016

strength is less than that of RAC com-pressive strength. The CoV of RAC strength is also presented in the two tables with limited tested data: the CoV of RAC compressive strength varies from approximately 3 to 20% whereas for tensile strength it varies from 6 to 30%. Table 3 presents the sta-tistical data of NAC strength according to Chinese codes.30,31

Table 3 shows that the standard devia-tion of NAC strength increases with the increase in concrete strength and the CoV of NAC strength with a lower grade concrete is larger than that with a higher grade concrete. Tables 1 and 2 illustrate that the CoV of RAC strength for some of the series is lower than the CoV of NAC strength obtained as per Chinese code. The reasons can be described as follows: (a) The CoV of NAC strength in Chinese code is a sta-tistical upper limit value, and the real CoV of NAC samples produced in lab may be smaller. (b) The single source of lab-produced RCA can partially account for the low CoV of strength for some of the RAC samples. If the RCA is from a single source, the par-ent concrete strength, the chemical composition and contaminants will be more stable than for RCA from multi-ple sources and the RAC strength will not vary significantly. (c) The number of RAC specimens is not quite enough compared with that of NAC. From the statistical viewpoint, more rea-sonable results will be obtained when the number of specimens is adequate. Based on these three explanations, it is reasonable to assume that the upper limit value for CoV of RAC strength is larger than that of NAC strength con-sidering the multiple sources of RCA and its contaminants.

Reliability Analysis of RAC Beam

The RAC strength is lower than that of NAC with identical water: cement ratio as mentioned earlier. It is reliable to produce RAC with the same strength as NAC by properly reducing the water: cement ratio in the mixing propor-tions.26,32 Thus in this paper, two hypoth-eses are assumed: (a) NAC and RAC have same mean compressive strength values; (b) the CoV of RAC is larger than that of NAC and it falls within an interval, which is from the CoV of NAC compressive strength to 0.2.

For reinforced concrete beams, the tar-get reliability index is different for duc-tile and brittle failure modes according

new construction materials with vary-ing quality.16 Processing level and qual-ity of the parent concrete play a very important role in the characteristics of RCAs.17 The chemical composition of RCAs will be considerably variable con-sidering the vast range of environments and conditions that parent concrete has been exposed to.16 Furthermore, the presence of contaminants will affect the variability of mechanical proper-ties of RAC. The percentages of bitu-minous and clay materials significantly affect the variability of compressive strength at 28 days.18

Figure 1 summarizes how the original RCA sources can influence the final mechanical properties of RAC based on previous research.

Figure 1 shows that the RCA strength, old mortar content, chemical compo-sition and contaminants are directly affected by the multiple sources of the RCA. Besides the RCA sources, the old mortar content is determined by crushing procedure as well.19 The greater content of old mortar is, the higher porosity will be, which results in higher water absorption and lesser density of RCAs. The size and shape of RCAs are mostly determined by RCA crushing procedures, which affect the RCA properties. The chemical com-position including sulphate, chloride, alkali, and so on also has an obvious

effect on RCA properties. A variety of contaminants such as asphalt, gypsum, metals, plastic, rubber, soil or wood20 can be found in RCAs from the dem-olition of existing structures. From Fig. 1 it can be concluded that the RCA properties and variation in con-taminant content cause the variability in RAC mechanical properties includ-ing compressive and tensile strength, elastic modulus and rupture strength.

Statistical Parameters of RAC Strength

In reliability analysis, as the reliability of RAC beams is directly related to the RAC strength, more attention should be paid to the statistical parameters of RAC strength. It is widely recog-nized that the RAC strength is lower than that of NAC for the same water: cement ratio.21–23 Table 1 lists the collected data of RAC compressive strength3,14,21,24–29 and Table 2 summa-rizes the RAC mean tensile strength data23,27,29 not considering the RCA sources and RAC strength, mix pro-portion and specimen type (prism and cylinder) in different countries.

From Tables 1 and 2, it can be observed that with the increase in replace-ment percentage of RCA in RAC, the strength of RAC decreases. The total quantity of data of RAC tensile

Fig. 1: Factors that influence the mechanical properties of RAC

Wood

Single source Different source

RCA sources

Crushing procedureParent concrete strength Old mortar content

Water absorption Density

Porosity factor

Sulphate

Chloride

Chemical composition

Alkali, etc

RCA properties

Mechanical properties of RAC

Contaminants

Asphalt Gypsum Metals Plastic Rubber Soil, etc

Copressive strengthTensile strengthElastic modulus Rupture strength, etc

RCA size and shape


to Chinese code.13 The target reliability indexes are 3.2 for beams subjected to bending moment.

The reliability analysis is performed considering four random variables: the compressive strength of concrete fc, the strength of reinforcement fy, the dead load G and live load Q. Concrete and reinforcement strengths are assumed to follow lognormal distributions.14 As for external loads, normal distribution and type I extreme value distribution have been assumed for dead load and live load, respectively.

Tables 4 and 5 show the statistical parameters of reinforcement and external loads.33

Assuming there is a linear relationship between the load and the load effect, the following equations can be drawn:

w = SQ,K/SG,K = QK/GK (1)

where, QK and GK are the characteris-tic values of live load to dead load; w is the ratio of QK to GK; SQ,K and SG,K are the characteristic load effects of live load and dead load. According to Table 5, the mean values of dead load and live load effects are 1.06 SG,K and 0.406 ωSG,K.

Flexural Capacity Analysis

In order to simplify the analysis, only beams with bottom reinforcement as longitudinal reinforcement are used to analyse the reliability of RAC beams. Based on the balance equation, the flexural capacity of concrete beams with bottom reinforcements subjected to bending moment can be drawn as follows:31

(2)

Dividing both sides of Eq. (2) with , then

(3)

(4)

The performance function of beams can be denoted as:

(5)

where, is the unit characteristic value of dead load effect which equals SG,K divided by . In this function,

Literatures Specimen numbers Mean value (MPa) CoV (%)

Denmark24 5–10% lower Small increase

Spain25 Slightly lower

China21 75 41.6 (0%) 8.27

100 41.5 (30%) 9.49

98 40.2 (50%) 9.70

98 36.5 (100%) 8.22

Spain26 24 44.0 (0%) 8.8

14 43.0 (25%) 9.3

43 41.5 (25%) 11.0

20 46.0 (50%) 9.3

14 46.5 (100%) 12.6

33 40.0 (100%) 16.7

Kuwait3 10 22.7 (0%) 4.18

10 32.3 (0%) 2.47

10 36.0 (0%) 2.33

10 46.0 (0%) 2.14

10 53.5 (0%) 1.87

10 20.3 (NM) 2.61

10 29.2 (NM) 3.27

10 32.2 (NM) 2.41

10 39.4 (NM) 2.16

10 46.5 (NM) 3.17

China27 83 18.48 (100%) 13.8

83 27.16 (100%) 15.5

83 43.65 (100%) 18.28

40 20.17 (100%) 7.73

40 31.09 (100%) 5.79

40 46.66 (100%) 6.88

Italy14 10 60.6 (0%) 6.4

10 61.1 (0%) 3.0

10 60.6 (0%) 7.4

10 66.2 (100%) 2.3

10 46.1 (100%) 4.5

10 45.7 (100%) 5.1

10 64.0 (100%) 2.5

10 38.1 (100%) 10.7

Portugal28 4 52.0 (0%) 3.3

4 52.0 (25%) 2.7

4 51.0 (50%) 2.4

4 51.0 (100%) 2.7

Italy29 66 (Total) 55.8 (0%) 4.4

47.2 (50%) 3.0

36.5 (100%) 4.1

63 (Total) 53.1 (0%) 3.8

50.6 (50%) 4.6

45.1 (100%) 4.3

Note: NM means not mentioned in the paper; the values in parentheses are RCAs replacement ratios, hereafter.

Table 1: Mean value and CoV of RAC compressive strength


References Specimen numbers Mean value (MPa) CoV (%)

Italy29 66 (Total) 3.56 (0%) 16.1

2.81 (50%) 14.5

2.5 (100%) 8.8

63 (Total) 4.79 (0%) 6.2

3.18 (50%) 7.0

4.06 (100%) 6.5

China27 83 1.39 (100%) 29.3

83 1.97 (100%) 24.7

83 2.65 (100%) 22.2

40 1.90 (100%) 16.0

40 2.31 (100%) 14.5

40 2.83 (100%) 20.9

United Arab Emirates23 10–25% lower

Table 2: Mean value and CoV of RAC tensile strength

Concrete grade

fcuk (MPa)

fcum (MPa)

r (MPa)

CoV (%)

fcm (MPa)

fck (MPa)

fc (MPa)

ftm (MPa)

ft (MPa)

C25 25 33.2 5 15 22.19 16.7 11.9 2.36 1.27

C30 30 38.2 5 13 25.61 20.1 14.3 2.55 1.43

C35 35 43.2 5 12 28.90 23.4 16.7 2.71 1.57

C40 40 48.2 5 10 32.31 26.8 19.1 2.89 1.71

C45 45 53.2 5 9 35.01 29.6 21.1 2.98 1.80

Table 3: Statistical data of NAC strength

Type of reinforcement fym (MPa) fyk(MPa) fy (MPa) r (MPa) CoV (%)

HRB335 380.2 335 300 21.7 5.7

HRB400 448.0 400 360 26.8 6.0

Table 4: Data of reinforcement strength

Type of loads Mean value/characteristic value CoV (%)

Dead load 1.060 7.0

Live load 0.406 29.2

Table 5: Statistical parameters of external loads

the concrete strength, reinforcement strength, dead load effects and live load effects are considered as variables. Also Ωp is the model error defined as the ratio of test results to theory results. Based on the pioneer research in Ref. [12], the mean value and CoV of Ωp are 1.1 and 0.04, respectively, for NAC beams. In this paper, the model error of RAC beam is assumed to be equal to that of NAC beam, considering the insufficient test results of RAC beams.

The reliability index of concrete beam subjected to moment (ductile failure) equals 3.2. Once the grades of NAC and reinforcement have been determined, the load effects caused by external loads (assuming w = 1.0) in the beam with a given reinforcement ratio can

be calculated. Keeping the load effects equal to NAC beams, the reliability index of RAC beam can be obtained by using the JC method (one kind of First-order Second-moment method) which was proposed in Ref. [34] and then rec-ommended by the Joint Committee on Structural Safety (the original source of the name “JC method”). As men-tioned before, the CoV of RAC com-pressive strength is larger than that of NAC although they have the same mean strength values. In this paper, the CoV of RAC compressive strength var-ies from c to 20% (c is the CoV of NAC strength). Figure 2 shows the reliability indexes of RAC beams.

HRB335-C30 indicates that HRB335 is selected as the reinforcement and

the mean compressive strength of RAC is the same as that of NAC with a concrete grade of C30. The minimum and maximum reinforcement ratios are determined by Eqs. (6) and (7) as follows:

(6)

(7)

where, xb is the balanced relative height of compression zone. When the concrete grade is not over C50, xb is respectively 0.550 and 0.518 for HRB335 and HRB400 according to the Chinese code.31

Table 6 gives the values of minimum (Min), medium (Med-1 and Med-2) and maximum (Max) reinforcement ratios for different types of beams.

Figure 2 demonstrates that the reliabil-ity index of the RAC beam decreases with the increase in CoV of RAC com-pressive strength (CoVfc

). Besides, the reinforcement ratio also has an obvi-ous influence on the reliability index. For instance, the reliability index of a beam with a minimum reinforcement ratio has almost no change with the increase in CoVfc

, whereas the reliabil-ity index of the beam with a maximum reinforcement ratio drops quickly from 3.2 to ~3.05, which means that the RAC beam is becoming easier to fail. This is because the CoV of flexural capacity of the RAC beam (CoVMu

) will increase rapidly when the rein-forcement ratio is higher. In order to prove the explanation, 100 000 groups of RAC (C30) compressive strength and reinforcement (HRB335) strength were stochastically generated accord-ing to their distribution type by using MATLAB software. Then Eq. (4) is used to calculate the flexural capac-ity of the RAC beam. The statistical parameters are shown in Fig. 3.

Mumax is the maximum flexural capac-ity of the RAC beam. Figure 3a shows that the flexural capacity of RAC beam increases with the increase in reinforcement ratio although the growth rate is decreasing. Figure 3b shows that the CoVMu

of RAC beam increases rapidly with the increase in CoVfc

when the reinforcement ratio of the RAC beam is high. Figure 3b also shows that CoVMu

is smaller than CoVfc

, which indicates that the variation of RAC strength has limited influence on the variation of flexural capacity with the help of reinforce-


Fig. 2: Reliability indexes of RAC beams: (a) HRB335-C30 and (b) HRB400-C40

3.06

3.08

3.10

3.12

3.14

3.16

3.18

3.20

3.22

HRB335-C30

3.04

3.06

3.08

3.10

3.12

3.14

3.16

3.18

3.20

3.22

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20CoVfc

0.10 0.12 0.14 0.16 0.18 0.20CoVfc

r = rmin

r = 1.02%

r = 1.82%r = rmax

HRB400-C40

r = rmin

bb

r = 1.06%

r = 1.90%r = rmax

(a) (b)

Reinforcement grade C30 C40

Min Med-1 Med-2 Max Min Med-1 Med-2 Max

HRB335 (%) 0.21 1.02 1.82 2.62 0.26 1.34 2.42 3.50

HRB400 (%) 0.20 0.82 1.44 2.06 0.21 1.06 1.90 2.75

Table 6: Reinforcement ratios of C30 and C40 concrete beams

Fig. 3: Statistical parameters of flexural capacity of RAC beams with different reinforcement ratios. (a) Mean value of flexural capacity and (b) CoV of flexural capacity

0 0.01 0.02 0.03 0.040

0.2

0.4

0.6

0.8

1.0

0.1

0.2

0.3

00.02

0.040.04

0.06

0.08

0.10

0.12

0.14

(a) (b)

0.06

0.07

0.08

0.09

0.10

0.11

0.12

CoV fc

CoV

Mu

Mu

/ Mu m

ax

rr

ments. When the reinforcement ratio is lower, the value of CoVMu

has almost no change with the increase in CoVfc

especially for the case of minimum reinforcement ratio, which means that the limited influence is more apparent when the reinforcement ratio is lower. This explains why the reliability of the RAC beam with a minimum reinforce-ment ratio has no obvious drop with the increase in CoVfc

.

In the case of maximum reinforcement ratio, considering that RAC and NAC beams have same resistances (mean values) and same load effects when

the reinforcement ratios are same, the reason why the reliability index of the RAC beam with a maximum reinforce-ment ratio drops is obviously the higher CoV of flexural capacity of the RAC beam. The physical explanation is that the increase in reinforcement ratio will cause the compressed part of the section to increase due to the balance equation; hence the contribution of concrete and consequently its CoV are higher.

Calculation of RAC Design Strength

Popularizing the application of RAC in civil engineering structures is a key

step to promote sustainable develop-ment. Thus, it is necessary to determine the design value of RAC strength in order to design the RAC components and structures. As discussed before, the reliability index of RAC beams is lower than the target value if the RAC mean strength is the same as that of NAC. One way to improve the reli-ability of RAC beam is to increase the reinforcement ratio. Another way to increase the reliability of RC beams is by maintaining the same reinforce-ment ratio and by using a concrete with slightly increased mechanical


Component type Load effect Load type v Reinforcement ratio Concrete grade Reinforcement grade

Beam Moment Dead load, Single live load

0.25,rmin, rmed–1,

C30, C40 HRB335, HRB4000.50,

1.00,rmed–2, rmax,

1.50

Table 7: Considered factors in determination of RAC design strength

Fig. 4: Flow chart for calculating design strength of RAC

Definition of factors: NAC compressivestrength, reinforcement strength, reinforcement

ratio, w, bt, and corresponding CoVs

Start

Determination of load effects for the given bt and flexural capacity MNu

Determination of MRu by substituting rR and fc in Eq. (4)

Yes

No

End

Calculation of original bRAC by replacing NAC with RAC withoutchanging other factors

Yes

No

Changing one of these factors:CoVfc, r, fcm, fym, w after

each iteration

fcRAC = fc

fc = fc – d

rR = r

r = r + d|bRAC – bt| ≤ e

|MRu – M

Nu | ≤ e

properties; these are subjects for dis-cussion in future studies.

There are many factors influencing the design strength of RAC based on reli-ability theory. In this paper, the consid-ered factors are listed in Table 7.

This paper determines the RAC design strength according to the following steps and the corresponding flow chart is displayed in Fig. 4.

1. Calculate the mean values of load effects that can be withstood by the NAC beam and the fl exural capacity of NAC beam (MN

u) for given factors, such as statistical parameters and distribution types of NAC compres-sive strength ( fc) and reinforcement strength ( fy), the CoV and distribu-tion type of load effects, the ratio

of characteristic live load to char-acteristic dead load (w), original reinforcement ratio ( r) and target reliability index (bt).

2. Use RAC to replace NAC and cal-culate the original reliability index (using Eq. (5)) of the RAC beam with the same reinforcement grade, reinforcement ratio and load effects as that of NAC beam.

3. Improve the reliability index of RAC beams by increasing the rein-forcement ratio and then the fi nal value can be obtained, which can make the reliability of RAC beam reach the target reliability index.

4. Assume the original RAC design strength equals that of NAC (fc) and then substitute the fi nal reinforce-ment ratio (r R) and fc in Eq. (4) to

determine the fl exural capacity of the RAC beam (MR

u).5. Decrease the RAC design strength

( fcRAC) until the fl exural capacity of

RAC beam equals that of the NAC beam which has been calculated in step (1).

6. Change one of these factors such as CoVfc

, original reinforcement ratio w, concrete strength and rein-forcement strength in each iteration (steps 1–5) to calculate the corre-sponding design strength of RAC.

This paper defines g b as the amplifi-cation factor of reinforcement ratio which is determined by the following equation:

(8)

According to the Chinese code,31 the characteristic value of concrete com-pressive strength fck which is defined by Eq. (9) has a 95% confidence level.

(9)

The partial coefficient of NAC is 1.4 and is determined by Eq. (10):

(10)

With the same defining principles of fck and g NAC, the partial coefficient of RAC g RAC can also be determine d, which is shown in Fig. 5.

Figure 5a indicates that the amplifi-cation factor of reinforcement ratio varies obviously with the increase in CoVfc

when the reinforcement ratio of beams is different. When the CoVfc is 20%, the reinforcement ratios of the RAC beams should be increased by ~0.1% and 3.2% for the minimum and maximum reinforcement ratios, respectively, to ensure that the reli-ability of the RAC beams will reach the target requirement. Figure 5b shows that the partial coefficient of RAC (g RAC) is lower than g NAC and decreases with the increase of CoVfc. Acc ording to Eq. (9), the higher CoVfc is, the lower characteristic compressive strength will be, which results in the lower g RAC. Figure 5c shows the design


Fig. 5: Calculated results of HRB335-C30 beam. (a) Amplification factor of reinforcement ratio, (b) Partial coefficient of RAC and (c) The ratio of RAC design strength fcRAC to NAC design strength fc

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.20

1.25

1.30

1.35

1.40

0.95

0.96

0.97

0.98

0.99

1.00

f c RA

C/f

c

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20CoVfc

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20CoVfc

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20CoVfc

r = rmin

r = 1.02%

r = 1.82%r = rmax

r = rmin

r = 1.02%

r = 1.82%r = rmax

r = rmin

r = 1.02%

r = 1.82%r = rmax

HRB335-C30

HRB335-C30

HRB335-C30

g b

g RA

C

(a)

(c)

(b)

Fig. 6: Calculated results of RAC design strength: (a) C30 and (b) C40

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.200.88

0.90

0.92

0.94

0.96

0.98

1.00

CoVfc

f c RA

C/f

c

f c RA

C/f

c

CoVfc

0.10 0.12 0.14 0.16 0.18 0.200.80

0.85

0.90

0.95

1.00

r = min

r = med-1

95% confidence interval


Lower limit

r = med-2

r = max

r = min

r = med-1



Lower limit

r = med-2

r = max

(a) (b)

strength of RAC, and the ratio of fcRAC

to fc decreases from 1.0 to ~0.95 with the increase in CoVfc

for the case of maximum reinforcement ratio. As mentioned earlier, in the case of maxi-mum reinforcement ratio, the higher CoV of compressive strength is, the higher CoV of flexural capacity will be and this further results in the higher

increase of reinforcement ratio which finally causes the lower design strength of the RAC.

Evaluation of RAC Design Strength

Above figures give the calculation results based on HRB335-C30 beam. To determine the final design strength of RAC, this section takes into account

factors listed in Table 7 and calcu-lates the design strength based on the flow chart shown in Fig. 4. There are 64 beams that need to be analysed regardless of the CoV of RAC com-pressive strength. Based on the calcu-lation results of 64 beams, Fig. 6 shows the corresponding design strengths for RAC.


In Fig. 6, the ratios of fcRAC to fc are divided into four groups based on four reinforcement ratios of RAC beams including minimum, maximum and two medium (med-1 and med-2) reinforcement ratios. When the CoVfc

is 20%, the lower value of RAC design strength is 0.90 fc for C30 and 0.84 fc for C40. The increase in CoVfc

results in the decrease of RAC design strength. For different w, it is found that a lower w causes a lower ratio of fcRAC to fc. For example, the RAC design strength increases from 0.88 fc to 0.95 fc while the corresponding w varies from 0.25 to 1.50 based on the analysis of HRB400-C40 beams.

Figure 6 also gives the fitting curves of data with different confidence lev-els including 95, 98 and 100% (lower limit). Similar to the definition of char-acteristic strength of concrete, the con-fidence level of RAC design strength is defined by the following equation:

(11)

where, fcmRAC is the mean design

strength of calculated data; l is the coef-ficient corresponding to confidence level, l = 1.645 for a 95% confidence level; and sRAC is the standard devia-tion of RAC compressive strength.

The fitting curves shown in Fig. 6 are transformation forms of RAC design strength (ratio of fcRAC

to fc), which is fitted by the following model.

(12)

The fitting coefficients for different confidence levels are listed in Table 8.

The R2 in Table 8 indicates that the fit-ting curves are in good agreement with the original calculated RAC design strengths. If the CoVfc is 20%, the design strength of C30 is 0.93 fc for the 95% confidence level, 0.92 fc for 98% confidence level and 0.90 fc for the lower limit. For C40, the correspond-ing design strengths are 0.90 fc, 0.88 fc and 0.85 fc. The descending rate of fit-ting curves increases with the growth of CoVfc, which should be considered in the determination of RAC design strength. Generally, the CoVfc is less

Fitting coeffi cients

C30 C40

95% 98% 100% 95% 98% 100%

a 1.0182 1.0209 1.0294 1.0079 1.0093 1.0167

b –29.008 –33.099 –43.500 –38.243 –43.848 –59.416

R2 0.9997 0.9997 0.9986 0.9988 0.9986 0.9939

Table 8: The fi tting coeffi cients for different confi dence levels

than 20% according to the statistical results shown in Table 1; if so, then the RAC design strength can be obtained easily by the fitting curves.

Conclusions

This paper analysed the variations in RAC properties and reliability of RAC beams subjected to bending moment. The following conclusions can be drawn:

1. The RCA source has a signifi cant infl uence on the mechanical proper-ties of RAC. Multiple sources cause the variability in RCA properties and the contents and types of con-taminants, which fi nally contributes to the variability in mechanical properties of RAC such as compres-sive strength and tensile strength.

2. The CoV varies from approximately 3% to 20% for RAC compressive strength, and from 6 to 30% for ten-sile strength according to the statis-tical analysis of data collected from published literatures.

3. The reliability index of RAC beam is lower than that of NAC beam owing to the higher CoV of RAC strength. To guarantee the reliability of RAC beams, the minimum and maxi-mum reinforcement ratios should be increased by about 0.1 and 3.5% respectively, compared with NAC beams.

4. The higher CoV of RAC strength has a relatively obvious infl uence on the reliability of RAC beams hav-ing maximum reinforcement ratios compared with RAC beams hav-ing minimum reinforcement ratio, because the increase in reinforce-ment ratio will cause an increase in the compressed part of the sec-tion based on the balance equation. Therefore, the contribution of con-crete and consequently its CoV are higher.

5. When the CoV of RAC compressive strength is 20%, the RAC design strengths are 0.93 and 0.90 times of NAC design strength for C30 and C40 respectively with a 95% confi -dence level to guarantee the reliabil-

ity of RAC beams. Other methods such as improving the mix strength of RAC need to be investigated in future studies.

Acknowledgement

The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (51325802).

Nomenclaturefcum Mean value of cubic

compressive strengthfcuk Characteristic value

of cubic compressive strength

fcm Mean value of axis compressive strength

fck Characteristic value of axis compressive strength

fc Design value of axis compressive strength

ftm Mean value of axis tensile strength

ft Design value of axis tensile strength

s Standard deviation of NAC strength

fym Mean value of rein-forcement strength

fyk Characteristic value of reinforcement strength

fy Design strength of reinforcement

Qk Characteristic value of live load

Gk Characteristic value of dead load

w Ratio of Qk to Gk

SQ,k Characteristic load effects of live load

SG,k Characteristic load effects of dead load

MU Flexural capacityMu Unit flexural capacityAs Reinforcement areah0 Effective height of

beam sectionb Width of beam sectiona1 Constant coefficient,

equals to 1.0r Reinforcement ratioS–

G,k Unit characteristic value of dead load effect which equals to SG,k divided by bh

20

Ωp Model error


xb Balanced relative height of compression zone

rmin Minimum reinforce-ment ratio

rmax Maximum reinforce-ment ratio

bt Target reliability index

r R Reinforcement ratio of RAC beam

MNu Unit flexural capacity

of NAC beamM

Ru Unit flexural capacity

of RAC beamMumax Maximum unit flex-

ural capacity of the RAC beam

CoVfc CoV of RAC com-pressive strength

CoVMu CoV of flexural capac-ity of RAC beam

g b Amplification factor of reinforcement ratio

g NAC Partial coefficient of NAC

g RAC Partial coefficient of RAC

fcRAC RAC design strengthfcmRAC Mean value of RAC

design strengthl Coefficient corre-

sponding to confi-dence level, l = 1.645 for 95% confidence level

s RAC Standard deviation of RAC compressive strength

a, b and R2 Fitting coefficients

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