Method to Develop Target Levels of Reliability for Design Using LRFD

Transcription

1 Method to Develop Target Levels of Reliability for Design Using LRFD by Daniel R. Huaco Doctoral Researcher University of Missouri Department of Civil and Environmental Engineering E2509 Lafferre Hall Columbia, MO Ph Fx John J. Bowders, PE William A Davison Professor of Civil Engineering University of Missouri Department of Civil and Environmental Engineering E2509 Lafferre Hall Columbia, Missouri Ph bowdersj@missouri.edu J. Erik Loehr, PE James C. Dowell Associate Professor of Civil University of Missouri Department of Civil and Environmental Engineering E2509 Lafferre Hall Columbia, Missouri Ph loehrj@missouri.edu Submitted to the Transportation Research Board for presentation and publication Word Count: 5, Tables and Figures = 7,338 words Transportation Research Board 91th Annual Meeting January, 2012 Washington, D.C.

2 D.R.Huaco, J.J.Bowders and J.E.Loehr 2 ABSTRACT Target levels of reliability for civil engineering designs are normally established as a matter of policy by specification committees or agency leadership. Target values are generally selected with some balance of consideration among perceived costs in a general sense, the consequences of failure, as well as general historical performance information. Deliberations regarding target levels of reliability seldom explicitly consider the incremental costs that are required to increase the reliability of structures and seldom involve explicit calculations to guide or inform such decisions. An approach to establish target levels of reliability for design of bridge foundations at both strength and serviceability limit states is presented in this paper. The approach combines consideration of socially acceptable risk with economic considerations that seek to minimize the total cost associated with the foundations. Socially acceptable risk is generally represented through so-called FN curves, which describe socially acceptable relations between frequency of failure (F) and the consequences of failure (N), or some other undesired consequence. The economic optimization involves minimization of total foundation costs through evaluation of the costs of potential consequences of failure and incremental costs required to increase the reliability of the foundations.

3 D.R.Huaco, J.J.Bowders and J.E.Loehr 3 Method to Develop Target Levels of Reliability for Design Using LRFD INTRODUCTION Historically, engineers have compensated for the variability and uncertainty of bridge foundation design parameters by using experience and subjective judgment. New design approaches are evolving that allow designers to achieve more rational engineering designs with more consistent levels of reliability. One such approach is the Load and Resistance Factor Design (LRFD) method, which provides the potential to more explicitly address the uncertainties and variabilities involved using procedures from probability theory to achieve a prescribed level of safety (1). Until the early 1990 s, geotechnical engineers exclusively used the Allowable Stress Design (ASD) method that collectively accounts for the uncertainties of all design loads and resistances in a single factor of safety. In ASD, load combinations are treated without considering the probability of both a higher-than-expected load and a lower-than-expected strength occurring at the same time and place (2). LRFD provides the capability to separately account for uncertainty in the loads and the resistances by applying different load or resistance factors for each parameter. The load and resistance factors can be calibrated using actual performance statistics allowing designers to achieve uniform and consistent levels of reliability in both super structure and substructure designs. Normally the target level of reliability, which is also expressed as the probability of failure or using a reliability index, is established by an AASHTO specification committee (3) or is chosen as a function of the variability of loads and resistances. In the AASHTO specifications of 2004 (4), the design probability of failure is established to be one ten thousandth or (for geotechnical applications a target probability of failure of is often adopted). Alternatively the design target is a quantity called reliability index (β) which is related to the probability of failure. For the probability of failure of , β equals 3.57 for load and resistance lognormal distributions, and 3.72 for normal distributions. The disadvantage of establishing a single target level of safety for all structures is that it does not consider the increase in cost required to achieve this reliability. For some structures, the increase in cost could be as much as the cost of the structure itself designed for a reasonable lower level of reliability. In some cases, the level of reliability for design is chosen as a function of the variability of the loads and resistances. For example, according to the Kansas Department of Transportation, a β factor of 2.5 may be appropriate for conditions of where the uncertainty is reduced (5). This approach goes against the purpose of using factors to compensate for the variability or uncertainty of loads and resistances. Alternatively to the current practice, target levels of reliability can be established based on combined consideration of socially acceptable risk and economic optimization. Socially acceptable risk is generally represented through FN curves, which describe socially acceptable relations between frequency of failure (F) and some undesired consequence such as the number of lives lost (N). Economic optimization involves minimization of the total costs associated with construction and operation of bridges through evaluation of the potential costs of failure or unacceptable performance and costs required to reduce the likelihood of occurrence. The total cost of an infrastructure (life cycle cost) is expressed as a function based on the concept of the expected monetary value. The economic optimization analysis includes the mathematical

4 D.R.Huaco, J.J.Bowders and J.E.Loehr 4 minimization of the total cost function and, in the present work, geotechnical probabilistic analysis of the likelihood of failure of bridge foundations. CRITERIA TO ESTABLISH TARGET LEVELS OF RELIABILITY The approach proposed to identify target levels of reliability for design of bridge foundations at both strength and serviceability limit states is based on the combination of socially acceptable risk with economic considerations that seek to minimize the total cost associated with the foundations. Both societal y and economical probabilities failure are identified and compared in an FN chart similarly to that sketched in Figure Annual Probability of "Failure" Societal Economical Number of Fatalities, N Figure 1. Schematic of an FN chart comparing social and economic levels of acceptable risk (or probability of failure, P). The tolerability limits for socially acceptable levels of risk are controversial and in addition societal risk criteria should not be used in a prescriptive mode [but] should be regarded as no more than indicators or guidelines (6). FN curves are produced to invoke criteria by which to decide whether the risks in the system represented by the FN curve are tolerable or not. Such criteria are sometimes called Societal risk criteria. The most obvious type of criterion is a line on the FN chart. If a system s FN curve lies wholly below the criterion line, the system is regarded as tolerable, but if any part of the FN curve crosses above the criterion line, the system is regarded as intolerable. Safety measures to lower the FN curve may then be required (7). Governments and agencies around the world are developing FN charts to identify acceptable levels of risk from a political and/or social perspective. These types of plots show the relationship between the probability (frequency) of fatal events per year and the number of human lives lost. Figure 2 shows a graphic that is commonly presented to demonstrate what many people consider to be acceptable risk associated with several different activities or industries. The chart provides some general guidance on the accepted average annual risk posed by a variety of

5 D.R.Huaco, J.J.Bowders and J.E.Loehr 5 traditional civil facilities and other large structures or projects (8). The location of the civil facilities bubbles are empirically based on historical observations. Figure 2. Relationship between annual probability of failure (F) and lives lost (N) (expressed in terms of $ lost or lives lost) for common civil facilities (9). Two of the most influential FN charts are produced by the Hong Kong Government s Planning Department to be used by Potentially Hazardous Installations (PHI) and the Australian National Committee on Large Dams (ANCOLD) that emphasizes the importance of planning systematic dam safety programs. The importance of ANCOLD guidelines is that it provides a greater level of guidance than the majority of other guidelines. The concept of tolerability adopted by ANCOLD is essentially international, also being applied in 1993 by the United States Bureau of Reclamation (USBR) and is strongly influenced by the Health and Safety Executive (HSE) and endorsed by the UK Treasury (10). The FN chart shown in Figure 3 displays the ANCOLD (11) and the Hong Kong societal risk guidelines (12) overlapping on Baecher and Christian s traditional civil facilities graphic shown in Figure 2. The overlap shows good agreement between the agencies and the historical performance from Baecher and Christian.

6 Prob of failure per year (Poptimum) D.R.Huaco, J.J.Bowders and J.E.Loehr 6 Fatalities, N E+00 1E-01 1E-02 1E-03 Marginally Accepted Mine Pit Slopes Accepted Foundations 1E-04 Dams 1E-05 1E-06 1E-07 ANCOLD Commercial Aviation Hong Kong 1E-08 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 1E+12 Failure/repair cost, dollars (X) Figure 3. FN chart showing average annual risks posed by a variety of traditional civil facilities, large structures and the ANCOLD s and Hong Kong s societal risk guidelines. Curves on the FN charts define regions or levels of risks that are generally dependent on society s acceptability for the loss of life. FN charts also assist in analyzing the practicability, from an operational or financial perspective, of taking measures (where measures are available) to reduce the level of risk of a civil work. The consequences of failure in FN charts are normally expressed in terms of the number of losses of human lives. However, it is possible to assign a statistical value to human lives; therefore the consequences of FN charts can instead be expressed in terms of monetary value. In this case, the FN curves constitute boundaries between regions on the plot that define the acceptance or non-acceptance of probabilities of failure for a potential monetary amount of loss. From an economic perspective, an FN curve (risk curve) can constitute a curve established by a family of points that represent the optimum probability of failure for a given monetary loss. Life-Cycle Cost The life-cycle cost of a bridge is the summation of initial cost, the cost of maintenance and possibly the cost of upgrade. The concept of the life-cycle cost (total cost) is important for making economic decisions. If the initial cost of a bridge is large, the probability of repair or the consequence due to failure is expected to be small. On the other hand, if the initial cost is small, the probability of having regular repairs or having larger consequence cost may be expected along the life-cycle of the infrastructure. This implies that there is a tradeoff between the initial cost and the probability of incurring a consequence cost (failure cost) of an infrastructure along its life-cycle.

7 D.R.Huaco, J.J.Bowders and J.E.Loehr 7 Substructures and foundations constitute major components of bridges, and can represent more than half of the total bridge cost (13). The method of analysis and design is important because it can influence the structural performance and project budget. Expected Monetary Value In decision theory, the expected monetary value, EMV (also denoted as E), is a measure of the value or utility expected to result from a given strategy (decision), equal to the sum of the initial investment (cost) of a civil work plus the probability of an incurrence times the value of the consequence (gain or loss). In the case of civil works, consequences can be classified as recurring maintenance costs (vegetation control, drainage maintenance, etc) or unexpected maintenance cost (repair of slides, settlement or total failure and complete replacement, etc.). The term consequence is used as the costs associated with a future failure. In general, consequence can include human injuries or other less tangible things like legal liability or political consequences such as the loss of faith by the traveling public. In this paper, consequences are expressed in terms of dollars to make the evaluations convenient. The mathematical expression shown in Equation 1 represents the expected monetary value (E) of a civil work. ( ) (1) Where, E = expected monetary value A = initial cost of the civil work P = the probability of failure X = consequence cost of failure T = the cost associated with no failure or recurring maintenance cost. Since geotechnical infrastructures involve piles, drilled shafts, etc. which does not typically require recurring maintenance, the cost of maintenance, T in the equation is considered negligible. The simplified mathematical expression to denote the expected monetary value is displayed as Equation 2. The expected monetary value is now an equation with three independent variables (A, P and X). The initial cost (A) and the probability of failure (P) are inversely related, meaning that if the initial cost increases, it is assumed that the probability of failure decreases or if the initial cost decreases, the probability of failure increases. In the case of geotechnical infrastructures, if the foundation increases in cost due to an increase in size, the probability of poor performance (settlement or collapse) is assumed to decrease. The relations or functions between the initial costs of the geotechnical infrastructure and the probability of failure are shown in a future section. The function that relates the initial cost, A with the probability of failure is of the form displayed in Equation 3. (2) ( ) (3)

8 D.R.Huaco, J.J.Bowders and J.E.Loehr 8 Where, A = the initial cost of the foundation, b = the slope factor, P = the probability of failure, and d = the vertical intercept at P = 1.0 The values of variables b and d are obtained through probabilistic analyses and are considered unique and constants for each type of geotechnical infrastructure. Therefore, the expected monetary value equation is reduced to be a function of the probability, P the consequence cost, X and a couple of known constants. Minimum Expected Value By relating the initial cost (A) and the probability of failure (P), the expected monetary value is simplified to an equation of two variables. This function represents a surface in space in which the axes are the expected monetary value (E), the probability of failure (P) and the consequence cost (X). The surface has the shape of an open parabolic channel that flows upwards and out of the E-P plane (Figure 4). E P X Figure 4 Graphical representation of the relation between the expected monetary value (E), the probability of failure (P) and the consequence cost (X). An upwards parabolic curve is observed on any plane parallel to the E-P plane that intersects the X axis. By definition, this parabolic curve has a vertex or a point that is a minimum on that plane. This point represents the minimum expected value (less expensive lifecycle cost of an infrastructure) for a selected value of X. The coordinates of the minimum point are calculated in the next section.

9 D.R.Huaco, J.J.Bowders and J.E.Loehr 9 Derivation of the Optimum Probability Function and FN Curve In Figure 4, the curve that results from intersecting the E-P-X surface with a plane parallel to the E-P plane has a point that is a minimum (minimum expected value). The abscissa of the point is P and the ordinate is E. The coordinates of the minimum point are calculated by mathematically minimizing the function that defines E. The abscissa P of the minimum point is denoted as the optimum probability (P opt ) because at that value, the expected value (E) is a minimum. From Equation 2, the value of the initial cost (A) is replaced by the function of (A) shown in Equation 3. The expected monetary equation now has the following form (Equation 4): ( ) (4) The minimum value of this equation occurs when its tangent (slope) is zero. This can be calculated by taking the derivative of the expected monetary value (E) with respect to the probability of failure (P). The probability (P) at which the expected monetary value is minimum is defined as the optimum probability of failure (P opt ) and is denoted as follows (Equation 5). The value of the optimum probability (P opt ) will be different for different expected monetary value curves which are generated by assigning different values to the consequence (X) (Figure 5). It is possible to use different values of consequence because the expected monetary value of the geotechnical infrastructure (bridge foundation) includes as a consequence, the cost of failure of the entire bridge (X). Although the initial cost is related to the foundation only, the consequence depends on the cost or repair of the entire bridge. The target level of risk is plotted in an FN chart. The level of risk is plotted as a continuous curve that is generated by a family of points that represent the optimum probabilities of failure (P opt ) of an infrastructure for different consequences (X). (5)

10 D.R.Huaco, J.J.Bowders and J.E.Loehr , ,000 Expected monetary value - dollars 100,000 75,000 50,000 25, ,000 X = 1E8 X = 1E7 X = 1E6 X = 1E5 X = 1E4-50,000 1E-07 1E-05 1E-03 1E-01 1E+01 Optimum probability Figure 5. Expected monetary value curves in which the minimum (optimum) probability of failure (P opt ) is identified for different values of consequence cost X. Probability of Failure-Cost Relations The probability of failure (P) or the probability of an unsatisfactory performance of a foundation decreases when increasing the certainty of soil and material parameter values, when increasing the quality and/or size of the foundation or when improving the design (type) of the foundation. All these options increase the cost of the foundation. The probability of failure-cost relation (P-cost) is a function that relates the probability of failure and the initial cost of the infrastructure. The slope of this function represents the cost required to decrease the probability of failure. The term probability of failure is an event that does not necessarily only describe the chance of catastrophe. The behavior could constitute unsatisfactory performance but not catastrophic failure (14). The Corps of Engineers uses the term probability of unsatisfactory performance to describe probability of failure (15). The probabilities of failure were established through reliability analyses. Reliability as used in reliability theory is the probability of an event occurring, or the probability of a positive outcome (14). Reliability calculations provided a means to evaluate the combined effects of multiple design parameter uncertainties and variability. There are numerous methods of employing reliability theory to evaluate reliability of geotechnical infrastructure. Taylor s Series method was used to calculate probabilities of failure considering the combined effect of the variability of the input parameters. The Taylor series method is a procedure to compute the standard deviation and/or the coefficient of variation of the factors of safety for the strength (capacity) limit or service limits. The use of the method requires previous knowledge of the mean and standard deviation of all parameters. Distributions of factors of safety values for strength limit and settlement values for service limits can be

11 D.R.Huaco, J.J.Bowders and J.E.Loehr 11 developed using the mean and standard deviations values computed using Taylor s Series method. Distributions of factors of safety values for bridge foundation strength and service limits are assumed to be lognormal. The magnitude of the probability of failure is established by computing the area under the distribution curve less than unity (1.0) and the probability of exceeding a selected service limit was established by computing the area under the settlement value distribution curve larger than the select service limit. The initial cost, A of foundations depends of the foundation type, size and the cost of materials. Probabilities of failure-cost curves are developed by plotting the probability and cost pairs on semi-log graphs. Probability values are plotted in a log scale while the costs are plotted on an arithmetic scale. The probabilities of failure-cost functions can be established using regression analysis considering a logarithmic regression type. The functions that best fit the data and can predict values of probability of failure and costs are displayed on the graphs. The function reported has a linear form with the independent parameter, probability of failure (abscissa), expressed in a natural log scale instead of a logarithmic scale (to base 10). The independent constant, (d) which is an arithmetic value, represents the intersection of the function curve with a vertical line that passes through the probability of failure value equal to unity (P = 1.0). The term with the independent variable (probability of failure, P) is affected in Equation 3 by a factor (b) which is not the true slope (m) of the linear function. The factor b is acting on the natural log value of the probability of failure instead of the logarithmic value of the probability of failure on a semi log graph. The value of the slope factor b can be expressed in terms of the true function slope, m. Consider the following linear equation in a semi log graph (Equation 6): Where, A = the foundation initial cost, m = the slope of the linear function, P = the probability of failure, and d = the vertical intercept at P = 1.0 ( ) (6) A relation between the slope factor b and the slope m is established by comparing the terms with the independent variable P of Equation 3 and Equation 6. The relation between the slope factor b and the slope m is shown in Equation 7. ( ) (7) Therefore quantitatively, the value of the slope factor b is about 40 percent of the value of the true slope of the probability of failure-cost function in a semi-log plane. The optimum economic probability of failure can be expressed in terms of slope factor b or in terms of the true slope m of the function.

12 D.R.Huaco, J.J.Bowders and J.E.Loehr 12 The location of the risk curves in the FN chart depends of the value of the slope factor b which is obtained from the probability of failure-cost function. The slope factor b of the probability of failure-cost function is an essential parameter to define, understand and compare functions. This factor represents the cost required to decrease the probability of failure of a bridge foundation by one order of magnitude. A large slope factor b would indicate that it is very expensive to decrease the probability of failure by one order of magnitude. The locations of these risk curves in the FN charts are evaluated with respect to their proximity to socially acceptable risk boundaries and regions. Development, Analysis and Calibration of Risk Curves FN charts are graphical representations of social acceptability for the loss of lives. The risk curves proposed by world government safety agencies on FN charts are boundaries that define regions or levels of risk acceptable to society. They are not curves that define the economically optimum acceptable risk. However, it is in this chart of regions of social acceptability that the optimum economical risk curves are plotted. The optimal economic risk curves do not define regions of economic acceptability. These curves represent a family of points of probabilities that are economically optimum for different levels of consequence. Considering that economic risk curves are dependent on the same parameters as socially acceptable risk (i.e. consequence (X) and probability of failure (P)), they can be plotted on FN charts overlapping socially acceptable regions. Optimum economic risk curves can be developed for several bridge foundations types such as pile groups, drilled shafts and spread footings. The region in the FN chart that occupies the envelope of the curves constitutes an optimum economic zone. It is within the region that the optimum probabilities of failure are established. Figure 6 shows the shaded region of an optimum economic risk curve envelope developed for the design of bridge foundations considering the strength limit.

13 Prob of failure per year (Poptimum) D.R.Huaco, J.J.Bowders and J.E.Loehr 13 Fatalities, N E+00 1E-01 1E-02 1E-03 Mine Pit Slopes Foundations 1E-04 Dams 1E-05 Estimated US Dams 1E-06 Commercial Aviation 1E-07 1E-08 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 1E+12 Failure/repair cost, dollars (X) Figure 6. Probability of failure (risk) step function generated within foundation risk curve envelope according to the consequence of bridge failure. The target levels of reliability for geotechnical infrastructures are established by identifying within the optimum economic risk zones, probabilities of failure for different ranges of bridge failure costs (X). With the use of engineering judgment, a probability of failure can be selected for a specific range of failure costs which is normally associated with a specific size or category of bridge. In some cases, the target probability of failure is selected to be smaller to avoid public discomfort when designing for bridge service limits (settlements). Example of Probability vs. Costs Function for Drilled Shafts The probability of failure-cost function for drilled shaft strength limit state is developed using reliability theory. Factors of safety values are calculated using Equation 8. Factors of safety distribution curves are developed with mean and standard deviation values calculated using the Taylor s Series method. The values of probability of failure are calculated from the distribution curves while cost values are estimated based on Missouri Department of transportation (MoDOT) Pay Item reports. (8)

14 Drilled shaft cost, dollars, (A) D.R.Huaco, J.J.Bowders and J.E.Loehr 14 Where, F.S. = the factor of safety α = skin resistance coefficient S u = undrained shear strength D = drilled shaft diameter L S = drilled shaft length Q w = working load. A factor of safety distribution curve is developed for several drilled shaft sizes. Each factor of safety distribution curve generates a probability of failure data point. Depending on the size, each drilled shaft is also associated with a cost. Pairs of probability of failure and cost are plotted on a semi-log graph as shown in Figure ,000 90,000 80,000 70,000 60,000 50,000 y = -17,067ln(x) - 12,615 40,000 30,000 20,000 10, E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 Probability of failure, (P) Figure 7. Probability of failure-cost function for drilled shafts. Using Excel s regression functions, a logarithmic type trend line for the probability of failure-cost points are generated. The interval of interest ranges between 1 in a hundred (1x10-2 ) and 1 in a million (1x10-6 ) probability of failure. The slope factor b of the function is -17,067. The negative sign denotes an inverse correlation. The probability of failure decreases as the costs increases. Similarly, a probability of failure-cost function is developed for a pile group limit state using Equation 9. (9) Where, F.S. = the factor of safety F y = the pile yield strength

15 Cost pile group, 50 ft long, dollars, (A) D.R.Huaco, J.J.Bowders and J.E.Loehr 15 A = Cross sectional area of pile n = number of piles in the pile group, and Q w = working load. The probability of failure-cost function generated for pile group strength is shown in Figure 8. The slope factor b of the function is -1,352. Both drilled shafts and pile group risk curves fall within the region of social acceptability for civil works. This slope is smaller in magnitude than the slope generated for drilled shaft (b = -17,067). Risk curves of drilled shafts are located at higher risk levels than pile group risk curves. Drilled shafts slope factors b are larger in magnitude than pile group factors therefore it is more costly to decrease the probability of failure of drilled shafts. The target level of probability of failure for the design of drilled shafts from an economic perspective is larger than for pile groups. 35,000 30,000 25,000 20,000 15,000 10,000 A = -1,352 ln(p) + 17,043 5, E-08 1E-07 1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 Probability of elastic failure, (P) Figure 8. Probability of failure-initial cost function for pile groups. Risk curves are developed using both drilled shafts and pile group slope factors (Figure 9). The target levels of reliability for bridge foundations are established by identifying within the area between both risk curves, the probabilities of failure for different bridge failure costs values X. With the use of engineering judgment a probability of failure is selected for a specific range of failure cost that can be associated with size or category of a bridge. Graphically, the selection of consequence costs and target probabilities appear as step functions within the optimum economic risk zone.

16 Prob of failure per year (Poptimum) D.R.Huaco, J.J.Bowders and J.E.Loehr 16 Fatalities, N E+00 1E-01 1E-02 Mine Pit Slopes Marginally Accepted Accepted Foundations 1E-03 1E-04 Dams 1E-05 1E-06 1E-07 ANCOLD Commercial Aviation Hong Kong 1E-08 1E+4 1E+5 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 1E+12 Failure/repair cost, dollars (X) Figure 9. Probability of failure (risk) step function generated within foundation risk curve envelope according to the consequence of bridge failure. The location of the risk curve in the FN chart depends on magnitudes of the optimum probability, P opt (which depends on the slope factor b) and the consequences of failure cost, X (Equation 5). If the numerical magnitude of cost would change dramatically (for example if using a different currency), then the risk curve would shift towards the right or left of the original risk curve depending on if there is an increase or decrease of the costs respectively. In this case, the target probabilities value would not change. The change would be of the range of interest of the consequence cost, X in the FN chart that would shift to the right or to the left along with the horizontal displacement of the risk curve. In summary, the process to establish the suggested target probabilities of failure consisted in developing step functions in FN charts between the intersection of the ranges of consequence costs of geotechnical infrastructures and the region located between upper and lower risk curves for each level of failure (i.e. strength and three service limits). The upper and lower risk curves boundaries are developed considering the envelope of slope factor b (upper and lower extreme values) of all foundation types for each level of failure. Refinement of the location of the step function to establish the suggested probability of failure is based on judgment taking into account the societal and political acceptance of failure. Finally, the selected target probabilities of failure are useful along with the knowledge of the variability and/or uncertainty of the resistance parameters to establish resistance factors in the LRFD method.

17 D.R.Huaco, J.J.Bowders and J.E.Loehr 17 CONCLUSIONS Target levels of reliability for the design of geotechnical infrastructures can be established based on a combination of cost analysis and societal acceptance of risk. The levels of reliability obtained from the optimization of cost and risk provides the opportunity to decrease the costs of an infrastructure while designing to a consistent known level of safety. The engineering procedure is robust and can be applied to develop target levels of reliability for other civil works. ACKNOWLEDGEMENTS The authors of this paper would like to thank the Missouri Department of Transportation (MoDOT) for providing funding and information for the project. We would also like to thank Dr. Carmen Chicone, professor of the Department of Mathematics of the University of Missouri for his invaluable contributions and recommendations to this project. REFERENCES 1. Orr, T. L. (2005). Proceedings of the International Workshop on the Evaluation of Eurocode 7. ERTC 10 of the International Society for Soil Mechanics and Geotechnical Engineering and Department of Civil, Structural and Environmental Engineering, Trinity College, Dublin. March 31 and April 1, Kulicki, J.M., Prucz, Z., Clancy, C.M., Mertz, D.R., Nowak, A.S. (2007). Updating the Calibration Report for AASHTO LRFD Code. Final Report. Project No. NCHRP 20-7/186. National Cooperative Highway Research Program. Transportation Research Board (TRB). 3. Chang, Nien-Yin. (2006). Report No. CDOT-DTD-R CDOT Foundation Design Practice and LRFD Strategic Plan. Colorado Department of transportation Reseach Branch. February AASHTO (2004). AASHTO LRFD Bridge Design Specifications, Customary U.S. units, Third edition, 2004, American Association of State Highway & Transportation Officials. 5. KDOT (2008). Kansas Department of Transportation Design Manual. Bridge Section, Volume III (LRFD). Version 9/08 6. Ball, D.J., Floyd, P.J. (1998). Societal Risks, Final Report. Report Commissioned by the Health and Safety Executive, United Kingdom. 7. Evans, A.W., (2003). Transport Fatal Accidents and FN Curves Research Report 073. HSE Books, ISBN Baecher, G.B. (1982a). Simplified Geotechnical Data Analysis, Reliability Theory and Its Application in Structural and Soil Engineering. The Hague: Martinus Nijhoff Publishers. 9. Baecher, G.B., J.T. Christian (2003). Reliability and Statistics in Geotechnical Engineering, Chichester, England ; Hoboken, NJ : Wiley. 10. HM Treasury United Kingdom (1996). The Setting of Safety Standards, a report by an interdepartamental group and esternal advisers, June. 11. ANCOLD (1994). "Guidelines on Risk Assessment 1994." Australia-New Zealand Committee on Large Dams. Guidelines on Dam Safety Management.

18 D.R.Huaco, J.J.Bowders and J.E.Loehr Hong Kong Government Planning Department (1994). "Hong Kong Planning Standards and Guidelines, Chapter 11, Potentially Hazardous Installations." Hong Kong. 13. Xanthakos, P.P. (1996). Bridge Substructure and Foundation Design. Prentice Hall. 1 st edition. ISBN-10: ISBN-13: Duncan, J.M., Navin, M., Patterson, K. (1999). Manual for Geotechnical Engineering Reliability Calculations. Virginia Polytechnic Institute and State University. 15. U.S. Army Corps of Engineers (1998). Risk Based Analysis in Geotechnical Engineering for Support of Planning Studies, Engineering Circular No , Department of the Army U.S. Army Corps of Engineers, Washington D.C., 27 February (Available online at