Reliability-Based Approach for the Determination of the Required Compressive Strength of Concrete in Mix Design

Transcription

1 Reliability-Based Approah for the Determination of the Required Compressive Strength of Conrete in Mix Design Nader M Okasha To ite this version: Nader M Okasha. Reliability-Based Approah for the Determination of the Required Compressive Strength of Conrete in Mix Design. International Journal of Engineering and Information Systems (IJEAIS), 2017, 1 (6), pp <hal > HAL Id: hal Submitted on 3 Sep 2017 HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Reliability-Based Approah for the Determination of the Required Compressive Strength of Conrete in Mix Design Nader M. Okasha Department of Civil Engineering, University of Hail. Saudi Arabia Abstrat: Conrete is reognized as the seond most onsumed produt in our modern life after water. The variability in onrete properties is inevitable. The onrete mix is designed for a ompressive strength that is different from, typially higher than, the value speified by the strutural designer. Ways to alulate the ompressive strength to be used in the mix design are provided in building and strutural odes. These ways are all based on riteria related purely and only to the statistial nature of the onrete prodution proess. However, what really matters is the impat of the onrete properties on the soundness of the struture the onrete is intended to be built with. Strutural reliability has never been used expliitly in onrete mix design. In this paper, an approah for the determination of the required ompressive strength of onrete in mix design, based on the strutural reliability of the reinfored onrete omponents being onstruted with this onrete, is proposed. In addition, the results will be shown to be used for the optimization of the statistial parameters for quality ontrol in the onrete mix proess. The approah presented in this paper provides a pratial platform to effiiently onsider plant-speifi variability the mix design proess. Keywords: mix design, onrete, struture, reliability, variability, optimization. 1. INTRODUCTION The presene of variability in the properties of onrete requires that speial measures are taken in the onrete mix design proess to ensure its quality. Statistial proedures are used in the mix design proess in order to provide assurane of satisfying the intended purposes of the designed onrete. One of the onrete properties that is given onsiderable attention is its ompressive strength. Conrete is reognized as the seond most onsumed produt in our modern life after water. The variability in onrete properties is inevitable. This variability has been widely studied in the literature (Aihouni 2012, Laungrungrong et al., 2010, Nowak and Szerszen 2003a,b, Ellingwood et al., 1980, Mirza et al., 1979). Some of these studies have onsidered ontrolling this variability as a riterion for the quality ontrol of the onrete prodution proess (Aihouni 2012, Laungrungrong et al., 2010). Compressive strength, in most ases, is the most suitable and effetive tool for the ontrol of onrete quality (ACI 314R 2011), even when ompressive strength is not the most important quality to be ontrolled beause testing of the ompressive strength gives the best refletion of the hange not only in the average onrete quality but also in variability and in testing error (Day 2006). In the onrete mix design proess, the material engineer is provided a speified strength by the strutural designer,. An optimum result of the onrete prodution is a bath with all tested speimens giving ompressive strength exatly equal to. Realistially, the tested strength of onrete samples will differ from, some lower than and some higher. If the materials engineer provides a material with an average strength equal to then half of the onrete will have ompressive strength less than (Mamlouk and Zaniewski 2011). In order to avoid suh undesirable outome, the materials engineer designs the onrete to have a required mean strength,, greater than. Ways to alulate the value of to be used in the mix design are provided in building and strutural odes and standards. El-Reedy (2013) has ompared the proedures for determining the required ompressive strength of onrete for the mix design proess in different international odes and standards. The methods provided in odes and standards are based on riteria related purely and only to the statistial nature of the onrete prodution proess. Under unertainty, strutural reliability theory and tools have proved to be powerful means of quantifying strutural safety. Some of the ode methods impliitly aim to ahieve a given strutural reliability level by providing parameters that define the required strength of the onrete, whih depend on the available studies in the speifi ountry onerning the variability on onrete material properties that vary from one ountry to another and even from one loation to another in the same ountry (El-Reedy 2013). Strutural reliability has never been used expliitly in the determination of. Doing so an be very benefiial for a onrete prodution plant that exhibits variability in prodution different from ode onsidered variability. In suh a ase, the mix design may be based on values tailored to fit its own prodution onditions while ahieving the required reliability levels for the strutural omponents being built. In this paper, an approah for the determination of the required onrete ompressive strength in mix design based on the strutural reliability of the reinfored onrete omponents being onstruted with this onrete is proposed. The approah is illustrated on examples of reinfored onrete olumns and beams. Intuitively speaking, the strutural reliability an be enhaned by either 172

3 inreasing the mean of the onrete ompressive strength or reduing its standard deviation or both. The ost of inreasing the mean may be different from that of reduing the standard deviation of the produed onrete. In this paper, also, the results of the strutural reliability analysis will be used to find an optimum set of values for the target mean and standard deviation of the onrete ompressive strength to be produed at the lowest ost. 2. VARIABILITY IN THE COMPRESSIVE STRENGTH OF CONCRETE Variations in the properties or proportions of the onrete ingredients, as well as variations in transporting, plaing, and ompation of the onrete, lead to variability in the strength of the finished onrete (Aihouni 2012, Wight and MaGregor 2011, Laungrongrong et al., 2010). Aording to the ASTM Standards C31 and C39, the standard test for measuring the strength of onrete involves a ompression test on ylinders 15 m in diameter and 30 m high after they are made and ured for 28 days. Any bath of onrete is produed based on a mix design aiming to ahieve a speified strength value,, determined by the strutural designer. An optimum result of the onrete prodution is a bath with all ylinders giving required ompressive strength,, exatly equal to. Realistially, the tested strength of onrete samples will differ from, some lower than and some higher. Conrete strength is believed to generally follow a normal distribution (Mirza et al., 1979). Arafa (1997) showed that ready-mixed onrete types are well modeled by the normal distributions whereas site-mixed onrete is well represented by the log-normal distribution with low mean-to-nominal ratio and high oeffiient of variation. For bathes of low variability, strength values of onrete will tend to luster near to the average value; that is, the histogram of test results is tall and narrow. As the variability in the onrete ompressive strength inreases, the spread in the data inreases and the normal distribution urve beomes lower and wider. The urve of this normal distribution is symmetrial about the mean value of the data,, whereas the standard deviation,, measures the dispersion of the data. The mean is alulated as (ACI 214R-11) where x i is the tested strength of ylinder i and n is the number of tested ylinders. The sample standard deviation is alulated as ACI 214R-11) (1) ( ) (2) The oeffiient of variation, V R, is used to desribe the degree of dispersion relative to the mean, and is alulated as (ACI 214R-11) (3) 3. ACI PROCEDURE FOR DETERMINING THE REQUIRED COMPRESSIVE STRENGTH OF CONCRETE The ACI 318M (2011) lays out a proedure for the determination of the required ompressive strength for mix design that is based on the riteria established by the ACI 214R (2011) and the ACI 301M (2010). This proedure starts with establishing a representing sample standard deviation. The alulation of the sample standard deviation depends on the number, nature and age of test reords available at the prodution faility. If a onrete prodution faility has at least 30 onseutive strength test reords of onrete produed from the same speified lass or within 7 MPa of, and these reords are not more than 24 months old, then Equation (2) is used to alulate the sample standard deviation. However, if the onrete prodution faility has two groups of onseutive tests reords totaling at least 30 tests produed from the same speified lass or within 7 MPa of, and these reords are not more than 24 months old, the sample standard deviation is alulated as (ACI 318M, 2011) ( ) ( ) (4) where = sample standard deviations alulated from two test reords, 1 and 2, respetively, and = number of tests in eah test reord, respetively. Table 1: Modifiation fator for sample standard deviation when less than 30 tests are available. Number of tests * Coeffiient of variation 173

4 or more 1.00 * Interpolate for intermediate numbers of tests. There may be a ase where a onrete prodution faility does not have reords of at least 30 onseutive strength tests or two groups of onseutive tests totaling at least 30 tests produed from the same speified lass or within 7 MPa of. In suh a ase, if the prodution faility has test reords not more than 24 months old based on 15 to 29 tests representing a single reord of onseutive tests that span a period of not less than 45 alendar days onseutive tests, a sample standard deviation needs to be established as the produt of the sample standard deviation alulated from Equation (3) or Equation (4) and a modifiation fator hosen from Table 1. One the sample standard deviation has been determined, the required mean ompressive strength, (ACI 318M, 2011) f 1.34 f r max f for f 35 MPa 0.9 f 2.33 for f 35 MPa, is alulated as follows (5) However, if a prodution faility does not have test reords that meet any of the onditions listed above, as follows (ACI 318M, 2011) f 7.0 for f 21 MPa f r max f 8.3 for 21 f 35 MPa (6) 1.1 f 5.0 for f 35 MPa, is determined 4. STRUCTURAL RELIABILITY Unertainties are present in the resistane of strutural omponents, whih are aused by the variability in the strutural materials and onstruted setion properties. Unertainties are also present in the loadings applied to these elements, espeially the live loads and environmental loads due to wind, snow or earthquakes (Wight and MaGregor 2011). The topi of strutural reliability (Thoft-Christensen and Baker 1982) offers a rational framework to quantify these unertainties mathematially. This topi ombines theories of probability, statistis and random proesses with priniples of strutural mehanis and forms the basis on whih modern strutural design and assessment odes are developed and alibrated. A safety margin, g, also known as the performane funtion, is defined as the differene between the resistane of a strutural omponent and the load effet it is subjeted to, and is given by g = R L (7) Beause of the presene of unertainties, R and L are treated as random variables. From Equation (7), g is also a random variable and its distribution is presented shematially in Figure 1. The value of zero separates the ombinations of R and L that represent the safety of a strutural omponent from those ombinations that represent its failure. The distribution of g is used to find the probability of failure of the strutural omponent. 174

5 Probability Density Funtion (PDF) International Journal of Engineering and Information Systems (IJEAIS) Fail Safe Figure 1. Shemati sketh of a typial probability distribution of the safety margin of a strutural member The probability of failure of a strutural omponent is the hane that a partiular ombination of R and L will give a negative value of g, i.e., the load effet exeeds the resistane. This probability is equal to the ratio of the area in the fail region under the urve of g to the total area under the urve in Figure 1 (whih is equal to 1.0). This an be expressed as (Ellingwood 2005) P ( t) P[ g 0] F ( x, t) f ( x, t) dx (8) f R L 0 where F R (x,t) is the instantaneous umulative probability distribution funtion of the resistane and f L (x,t) is the instantaneous probability density funtion of the load effet. The reliability of a strutural omponent may also be represented by the reliability index,. A typial assumption is that g is a Guassian random variable. Aordingly, the reliability index β an be obtained from the probability of failure P f by (Ang and Tang 1984) 1 (1 P f ) where Ф is the standard normal distribution funtion. Strutural reliability has evolved over the past few deades suh that numerous methods for the alulation of the reliability index and the orresponding probability of failure have beome well established. In this study, First Order Reliability Method (FORM) is used to alulate the reliability index (Hasofer and Lind 1974). The software CALREL is used for suh omputations (Liu et al. 1989). 5. RELIABILITY BASED STRUCTURAL DESIGN In order to aount for the unertainties in strutural design, safety fators are established in design odes that in one hand magnify the design load effet, and in the other hand redue the nominal resistane values. The general formula for deterministially ensuring strutural safety with load and resistane safety fators is R L n i i Safety Margin, g = R L where is the strength redution fator, R n is nominal resistane (strength), is the load fator for the i th load effet and L i is the i th load effet. Load and resistane fators have been alibrated in strutural odes using the onepts of strutural reliability whih (9) (10) 175

6 take into aount the variability in load and resistane and ensure an aeptable level of reliability, alled the target reliability, for the designed strutural element. An initial guess of the safety fators is made by solving an optimization problem where the objetive is to minimize the differene between the reliability for a strutural omponent onsidered and the target reliability designated for it. Then, the safety fators determined in this way are adjusted taking into aount urrent engineering judgment and tradition. 6. SIMPLE COMPONENT EXAMPLE Consider the fundamental strutural reliability ase with the linear performane funtion in Equation (1). Assume that this equation represents the limit state for the failure of the axially loaded tension element shown in Figure 2. If the random variables R and L are independent and normally distributed, the reliability index beomes (Ang and Tang 1984) R L 2 2 R L where R, L are the means of the resistane and load, respetively, and R, L are the standard deviations of the resistane and load, respetively. L (11) R Figure 2. A strutural element subjeted to a tensile axial load. The tension element in Figure 2 is assumed to be subjeted to the tensile load L that is normally distributed with a mean of 20 kn and standard deviation of 2 kn. A speified mean of the resistane is assumed as Rs = 25 kn. The purpose of this example is to determine the required mean of the resistane that leads to a target reliability t. This is intended to simplistially resemble the ase where a onrete prodution faility needs to determine the required mean ompressive strength. Sine the standard deviation is a funtion of the variability in the prodution of the resistane, and assuming that this variability is onstant for any mean strength, the resistane oeffiient of variation V R is related to the standard deviation R by Equation (3) as follows V (12) R R R The solution of this problem beomes one where R is the root of the equation 176

7 R L 2 2 V R R L 0 t (13) The root of Equation (13) is found by the Optimization Toolbox of MATLAB (MathWorks 2014a) for different values of V R and and t and the results are shown in Figure 3. The figure shows the ratio of the required mean of strength, R to the speified mean strength, Rs as the oeffiient of variation of the resistane V R is hanged for different values of target reliability indies, t. It is lear from the figure that the required mean strength is sensitive to both the value of the oeffiient of variation of the mean resistane in addition to the target reliability. Required Mean Strength, R /Speified Mean Strength, Rs t = 4.5 t = 4.0 t = 3.5 t = 3.0 t = Coeffiient of Variation, V R Figure 3. Variation of the ratio of the required mean strength, R, to the speified mean strength, Rs, as a funtion of the resistane oeffiient of variation, V R, for different target reliability indies, t, for the strutural element subjeted to a tensile axial load. 7. REQUIRED COMPRESSIVE CONCRETE STRENGTH FOR A COLUMN In this analysis, a short onentrially loaded retangular tied reinfored onrete olumn with gross ross setional area A g and reinfored with steel having a ross-setional area of A s and yield stress of f y is onsidered. The design axial ompressive strength of the olumn, P n, is alulated as (ACI 318M, 2011) Pn r Ag 0.85 f Ast ( f y 0.85 f ) (14) where r = 0.8 is the fator aounting for aidental eentriity in the onentrially designed tied olumn. In this analysis, an arbitrary short onentri tied reinfored olumn is designed aording to the ACI 318M-11 (2011). The olumn is designed whih a speified onrete strength of f ' = 35 MPa. The results of the design along with the assoiated statistial properties of the design variables are shown in Table 2 that are extrated from the literature (Nowak and Szerszen 177

8 2003a,b, Ellingwood et al. 1980, Mirza et al. 1979, Allen 1970, Cornell 1969, Ellingwood 1978, MaGregor et al. 1983, MaGregor 1976, 1983, Pier and Cornell 1973). Table 2: Design variables and the assoiated statistial properties of the reinfored onrete olumn. Deterministi Distribution Coeffiient of Quantity Symbol Unit Bias fator design value type variation Yield strength of steel f y MPa 420 Normal Width B mm 300 Normal Breadth H Mm 400 Normal Area of steel reinforement A s mm Normal Distributed dead load L D N/m 2 7 Normal Distributed live load L L N/m 2 2 Normal Tributary area A T m Normal The performane funtion for the ompressive strength limit state of a short onentri tied olumn onstruted with onrete of required ompressive strength is given by [ ( ) ( )] ( ) (15) The required ompressive strength,, is a normally distributed random variable with mean,, and oeffiient of variation, V R. In order to ensure the target reliability for the olumn, t, the following equation needs to be satisfied 1 (1 Pg [ 0]) 0 (16) t The target reliability index, t, is set to a value of 4.0 for this olumn. The onrete mix design needs values for the statistial parameters of the required mean strength,, that satisfies equation (16) for the target reliability, t. The standard deviation is a funtion of the variability in the prodution of the onrete, and assuming that this variability is onstant in a prodution faility for any mean strength, the resistane oeffiient of variation V R is related to the standard deviation by Equation (12). For a given oeffiient of variation, the problem beomes one of finding the root of Equation (16), that is the mean of. A losed form solution does not exist for this problem. The evaluation of P[g < 0] requires an iterative reliability tehnique. The tehnique used herein is FORM (Hasofer and Lind 1974). The software CALREL (Liu et al. 1989) is used for suh omputations. The iterative solution tehnique used for finding the root of Equation (16) is the MATLAB funtion fzero that uses a ombination of bisetion, seant, and inverse quadrati interpolation methods in the Optimization Toolbox in MATLAB (MathWorks 2014a). An interfae between MATLAB and CALREL is established where a program written by the author modifies the input file for CALREL, runs CALREL, and extrats the reliability analysis results of CALREL from the output file in eah iteration of the MATLAB root solving proess. This is done for different values of the oeffiient of variation, where for eah value of the oeffiient of variation, a required mean onrete ompressive strength is determined. The results are plot in Figure

9 Required Mean Strength, f r /Speified Mean Strength, f International Journal of Engineering and Information Systems (IJEAIS) Coeffiient of Variation, V R Figure 4. Variation of the ratio of the required mean strength,, to the speified mean strength,, as a funtion of the resistane oeffiient of variation, V R, for the reinfored onrete olumn. Figure 4 shows the ratio of the required mean ompressive strength,, to the speified mean strength,, as the effiient of variation of the resistane V R is hanged. Evidently, as the oeffiient of variation inreases, the ratio / inreases sine this inrease implies more variability and more unertainty in the properties of the onrete material. This result underlines the importane of onduting a onrete mix design with a required ompressive strength that takes into aount the unique variability in the reords of the onrete prodution faility. The results obtained by the strutural reliability-based analysis are ompared with those obtained by the ACI equations. Figure 5 shows the required mean of strength,, found from: (a) the strutural reliability-based design, (b) from Equation (5) for the ase where the onrete prodution faility has at least 30 onseutive strength test reords, () from Equation (5) for the ase where the onrete prodution faility has at least 15 onseutive strength test reords but less than 30, and (d) from Equation (6) for the ase where the onrete prodution faility does not have at least 15 onseutive strength test reords. Figure 5 shows that the reliability-based approah gives results that are less onservative than those of the ACI equations for low values of the oeffiient of variation, and more onservative for higher values of the oeffiient of variation. The results from all soures are about the same for moderate values of the oeffiients of variation. 179

10 Required Mean Strength, f r International Journal of Engineering and Information Systems (IJEAIS) Reliability-Based Design More than 15 Test Reords and Less than 30 Test Reords 50 Less than 15 Test Reords 35 Figure 5. Variation of the required mean strength,, as a funtion of resistane oeffiient of the variation, V R, for the tied reinfored onrete olumn ompared with the ACI method results. 8. REQUIRED COMPRESSIVE CONCRETE STRENGTH FOR A BEAM The design flexural strength of an under-reinfored onrete beam M n having a width b and reinfored with steel having a ross-setional A s loated at an effetive depth d is alulated as Af s y M n As f y d 17. f b Coeffiient of Variation, V R More than 30 Test Reords (17) The shear strength of this beam is given as A f d s v y Vn 0.17 f bd (18) where A v is the area of shear reinforement rossing a shear rak and s is the spaing between the shear reinforement stirrups. Table 3: Design variables and the assoiated statistial properties of the reinfored onrete beam. Deterministi Distribution Coeffiient of Quantity Symbol Unit Bias fator design value type variation Yield strength of steel f y MPa 420 Normal Width b mm 400 Normal Effetive depth d mm Normal Span L m 8000 Normal

11 Area of flexural reinforement Area of shear reinforement A s mm (325) Normal A v mm 2 79 (8) Normal Stirrup spaing s mm 300 Normal Distributed dead load L D N/m 2 7 Normal Distributed live load L L N/m 2 2 Normal Tributary width B T m 4 Normal In this analysis, an arbitrary simply supported beam setion at midspan is designed aording to the ACI 318M-11. The beam is designed with a speified onrete strength of f ' = 30 MPa. The results of the design along with the assoiated statistial properties of the design variables are shown in Table 3 that are extrated from the literature (Nowak and Szerszen 2003a,b, Ellingwood et al. 1980, Mirza et al. 1979, Allen 1970, Cornell 1969, Ellingwood 1978, MaGregor et al. 1983, MaGregor 1976, 1983, Pier and Cornell 1973). The performane funtion for the flexural failure limit state of a singly reinfored beam, g m, is given by 2 Af s y LD LL BT L gm As f y d 17. F 8 r b (19) and for the shear limit state, g v, is given by g v F ' A r vf yd LD LL BT L bd 6 s 2 (20) In order to ensure that the target reliability assoiated with the flexural failure limit state, tm, and the shear limit state tv are ahieved, the following equations need to be satisfied 1 (1 Pg [ 0]) 0 (21) m tm 1 (1 Pg [ 0]) 0 (22) v tv The target reliability indies, tm and tv are set to the values of 3.0 and 3.5, respetively. Eah of Equations (21) and (22) is separately turned into an equality and solved to find the required mean ompressive strength,, needed to satisfy the assoiated target reliabilities for a given oeffiient of variation. The larger value of obtained from solving both equations is onsidered for the mix design. The resistane oeffiient of variation V R is related to the standard deviation by Equation (12). The software CALREL is used for the evaluation of P[g m < 0] and P[g v < 0] by FORM. An interfae between MATLAB and CALREL is established by the program written by the author. The problem is solved for different values of the oeffiient of variation, where for eah value of the oeffiient of variation, a required mean onrete strength is determined. The results are plot in Figure

12 Required Mean Strength, f r /Speified Mean Strength, f International Journal of Engineering and Information Systems (IJEAIS) Coeffiient of Variation, V R Figure 6. Variation of the ratio of the required mean strength,, to the speified mean strength,, as a funtion of the resistane oeffiient of variation, V R, for the reinfored onrete beam. Figure 6 shows the ratio of the required mean strength,, to the speified mean strength,, as the effiient of variation of the resistane, V R, is hanged. Clearly, the ratio / is insensitive to the hange in the oeffiient of variation. This result is onsistent with the findings in Okasha and Aihouni (2015), where the reliability index was found in beams to be muh less sensitive to the hange in the oeffiient of variation of onrete than it was found to be in olumns. 9. OPTIMIZATION OF THE REQUIRED CONCRETE COMPRESSIVE STRENGTH STATISTICAL PARAMETERS Quality ontrol in the onstrution industry, partiularly in the onrete prodution industry, has beome an important topi for researhers and pratitioners in the past few years (Okasha and Aihouni 2015, Aihouni 2012, ACI 121R-08, Day 2006). Statistial tools are used exlusively to lassify the quality of onrete produed in ready-mixed onrete failities. Redution of the variability and unertainty in the onrete properties has been the main goal in most quality ontrol measures pursued or proposed thus far. Any onrete prodution faility has its unique variability in the reords of tests it has onduted over its past. This variability is typially represented by the oeffiient of variation of onrete ompressive strength. Due to this variability, the onrete produed must have a required ompressive strength higher than the speified ompressive strength as explained in this paper. In one hand, quality ontrol in onrete prodution aims to maintain the mean strength of the produed onrete to be as lose as possible to the required ompressive strength. On the other hand, quality ontrol may study ways to redue the variability, represented by the oeffiient of variation, in the produed onrete. Either approah has its own assoiated ost. In lak of a rational approah for the seletion, the deision of whih approah to pursue may be subjetive. Even if both parameters, i.e., the mean and oeffiient of variation of the strength, are aimed for ontrol, there is no lear indiation of how muh of eah needs to be improved. It is shown herein by a simple optimization problem, whih is based on the results of the strutural reliability-based mix design, that an optimum solution an be established, where the solution entails the magnitudes for eah of the mean and the oeffiient of variation of the onrete strength to be targeted at the least possible ost. 182

13 Consider again the results of Figure 4. If a prodution plant had the option to selet a value for the oeffiient of variation for the onrete strength while ahieving the target reliability, the required mean strength must take one of the values on the urve shown in the figure. Conversely speaking, if the plant had the option to selet a value for the required onrete mean strength while ahieving the target reliability, the oeffiient of variation must also take one of the values on the urve shown in the figure. Thus, the problem is to determine whih ombination of the required mean and the oeffiient of variation that satisfies the target reliability at minimum ost. In order to find a solution to this problem, the ost assoiated with produing a onrete with ompressive strength must be known. Eah prodution plant has its own unique ost funtion. For sake of illustration and maintaining generality of the approah, an example ost funtion is assumed where the ost of produing 1 m 3 of onrete with is a linear funtion of the value of, and is given in USD urreny by Cost of produing onrete with f 5(20 f) (23) r r In addition, the ost assoiated with ahieving a given oeffiient of variation, V R, must be known. It is also assumed that the ost of ahieving V R is a linear funtion of the value V R, and is given in USD urreny by Cost of ahieving V 400(1 2.5 V ) (24) R R Equations (23) and (24) an be replaed with any plant-speifi ost funtion while the same approah applies. Equations (23) and (24) are graphially shown in Figure 7. The main assumption in establishing the ost funtions is that the ost proportionally inreases with inreasing and with dereasing V R. The total ost required for produing 1 m 3 of onrete with a set of values of and V R is the sum of osts in Equations (23) and (24). The optimization problem an now be formulated as follows Find: and V R To Minimize: 5(20 f ) 400(1 2.5 V ) r R (25) Subjet to: 1 (1 Pg [ 0]) 0 t (26) Where: [ ( )] ( ) (27) In this optimization problem, the two design variables are linked. The value of depends on V R and is determined by solving the root finding problem in the equality onstraint in Equation (26). The ombinations of and V R that satisfy the equality onstraint form the feasible spae whih ontains the optimum solution. Aordingly, instead of solving this problem onsidering the two design variables as free variables, V R is onsidered as the only design variable in the problem. In eah iteration, the value of is determined by solving the root finding problem in the equality onstraint in Equation (26) and then the objetive funtion in Equation (25) is alulated. Hene, the ombination of and V R that gives the minimum total ost an be identified. 183

14 Cost of Ahieving V R ($/m 3 ) Cost of Produing Conrete with f r ' ($/m 3 ) International Journal of Engineering and Information Systems (IJEAIS) (a) Required Mean Strength, f r ' (MPa) (b) Coeffiient of Variation, V R Figure 7. Cost funtion for (a) produing onrete with and (b) ahieving V R. 184

15 Total Cost ($) International Journal of Engineering and Information Systems (IJEAIS) Required Mean Strength, f r ' (MPa) Optimum Solution Coeffiient of Variation, V R Figure 8. Optimization results for finding the best values of the required ompressive strength and oeffiient of variation to be targeted at a minimum total ost. The optimization problem an be solved using numerous available tehniques. Herein, it is solved using the Sequential Quadrati Programming method (SQP) in the Optimization Toolbox of MATLAB. The values of are determined during the optimization proess for eah optimization iteration using the fzero funtion. An interfae between MATLAB and CALREL is established where a program written by the author modifies the input file for CALREL, runs CALREL, and extrats the reliability analysis results of CALREL from the output file in eah iteration of the MATLAB root solving proess. Figure 8 shows a graphial presentation of the feasible spae of the ombination of values of and V R that satisfy the equality onstraint and the total ost of eah ombination for a target reliability index of 4.0 in the olumn ase previously onsidered. The optimum solution is identified to be the ombination where the required ompressive strength to be targeted is MPa with a oeffiient of variation of leading to a total ost of about $/m 3. This solution depends on the ost funtion assumed, the target reliability onsidered, the limit state of the strutural omponent and the statistial parameters used. One these inputs are aurately established for a given onrete prodution faility, an optimum solution an be found following the same proedure. 10. CONCLUSIONS This paper proposes a strutural reliability-based approah for the mix design of onrete. The main fous of this paper is on determining the required ompressive strength of the onrete in the mix design proess. The approah is based on the strutural reliability of the strutures the onrete is used for onstruting. It an be onluded from the results of this paper that the required ompressive onrete strength an be more aurately determined if the prodution faility s oeffiient of variation of the ompressive strength, the type of the strutural element for whih the onrete is used to onstrut and the target reliability are all onsidered. The approah presented in this paper provides a pratial platform to effiiently onsider these fators in the mix design proess. It was also found in this paper that the influene of the variability of onrete on the strutural reliability of beams under flexure or shear is relatively insignifiant. 185

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