Load and resistance factor design represents a more rational approach by which the more significant uncertainties listed above (i.e., load and material resistance) can be incorporated quantitatively into the design process. As used in the AASHTO LRFD Specification (AASHTO, 1997a), the basic LRFD equation or relationship is defined by:
n i iQ R
∑
γ ≤φ (Eq. 2-2) where: iγ = Statistically-based load factor generally greater than one
i
Q = Load
n
R = Nominal (ultimate) resistance
φ = Statistically-based resistance factor generally less than one Application of Eq. 2-2 is illustrated in Figure 2-2.
Figure 2-2
LRFD Design Approach
When applying LRFD, the estimated magnitudes of the various types of load effects are multiplied by appropriate load factors to determine the factored load effects, and the estimated nominal (ultimate) resistance is multiplied by a resistance factor prescribed for the model used to estimate material resistance and the field and/or laboratory test methods used to develop the material properties. For simplicity, the nominal and factored loads and the nominal and factored resistances are shown in Figure 2-2 as unique values. Of course, the load and material resistances vary, such as shown in Figure 2-3.
Figure 2-3 shows a possible variation of load and resistance as a function of the frequency or probability of occurrence. If the peak values of load, Q, and resistance, R, are defined by their respective mean values, Q and R , the equivalent ASD factor of safety is the ratio of R to Q as shown in the figure, and the margin of safety is the difference between R and Q . However, the figure also points out that some potential for failure exists in the area where the distributions of Q and R overlap. Therefore, unless very high factors of safety are used, some probability of failure will always exist.
Figure 2-3
Variation of Load and Resistance
Figure 2-3 shows a possible variation of load and resistance as a function of the frequency or probability of occurrence. If the peak values of load, Q, and resistance, R, are defined by their respective mean values, Q and R , the equivalent ASD factor of safety is the ratio of R to Q as shown in the figure, and the margin of safety is the difference between R and Q . However, the figure also points out that some potential for failure exists in the area where the distributions of Q and R overlap. Therefore, unless very high factors of safety are used, some probability of failure will always exist.
Another factor which must be considered is the distribution of load and resistance. Figure 2-3 represents only one pair of distributions. Figure 2-4 shows that using the ratio of R to Q to define safety can be misleading. Two pair of load and resistance distributions are presented which have identical values of Q and R . The upper distribution for resistance is relatively small with steep flanks about R , whereas the lower distribution for resistance is broad with flat flanks, also about R . As a result, the area of overlap between the upper distributions of R and Q is small, representative of a small probability of failure. Conversely, the area of overlap between the lower distributions is large, representative of a greater probability of failure.
For substructure design, the majority of loads which must be supported are prescribed by the structural or bridge designer in the form of vehicle and other types of transient load types. Thus, with the exception of the type of soils used for backfill behind walls and abutments and around culverts, geotechnical engineers have only limited control over the load side of the relationship. However, on the resistance side, geotechnical engineers have the opportunity to control the extent and type of sampling and testing used to characterize a site, the procedures or models used for design, and the measures employed to monitor the construction processes.
Figure 2-4
Distribution of Load and Resistance
Reliability-based design (or LRFD) is the process by which the risks and uncertainties associated with the safety of a system are defined in mathematical terms. In applying the process, the uncertainties (or distributions) of load and resistance are assumed to be independent, random variables, and the design risk is defined by the probability of failure, pf. To evaluate pf, a single
probability density function is developed, as shown in Figure 2-5, which represent the combined distributions and uncertainties of Q and R.
The left side of Figure 2-5 shows typical distributions of load and resistance and the right side shows the combined distribution function for Q and R. For the pair of distributions to the left, failure is possible in the shaded area where the distributions overlap and the margin of safety is represented by the difference between R and Q . For the combined distribution, failure is possible in the shaded area to the left of the ordinate and the margin of safety is represented by the number of standard deviations, β, of the mean value of the combined distribution to the right of the ordinate. In LRFD parlance, β is referred to as the reliability index. Depending on the shape of original distributions (i.e., both normal, both lognormal or one normal and one lognormal), the probability of failure can be mathematically related to the reliability index. For lognormal distributions, the relationship between pf and β is presented in Table 2-1.
Figure 2-5 Reliability Index, β
Table 2-1
Relationship Between pf and β for Lognormal Distribution
Reliability Index β Probability of Failure pf 1.96 1:10 2.50 1:100 3.03 1:1 000 3.57 1:10 000 4.10 1:100 000 4.64 1:1 000 000
To implement the LRFD concept in a design code, it is necessary to select a minimum level of safety which must be achieved. One process is to conduct cost-benefit analyses to determine the total costs (i.e., sum of initial construction cost, maintenance costs and estimated costs of failures) as a function of the probability of failure. Using this approach, the target probability of failure would be the pf for
which costs are minimized. Alternatively, the target probability of failure could be established based on the failure rates estimated from actual case histories. However, the probability of failure for constructed facilities is not solely a function of uncertainties associated with the design process, and probably is at least an order of magnitude lower than the theoretical probability of failure. For general guidance, information such as that presented in Figure 2-6 regarding empirical rates of failure for civil engineering facilities can be used. For structure foundations, the annual probability of failure ranges from about 1:100 to 1:1000. In calibrating the LRFD Specification, values of pf
range approximately within these limits, depending on the foundation type and the level of redundancy available in the system (e.g., pile foundations are usually constructed in a group such that failure of a single pile does not imply failure of the group).
Figure 2-6
Empirical Rates of Failure for Civil Works Facilities
(Kulhawy, et al., 1995)
A limit state is a condition beyond which a structural component, such as a foundation or other bridge component, ceases to fulfill the function for which it is designed. The limit states which must be evaluated in the AASHTO LRFD Specification (AASHTO, 1997a) include:
y Service Limit State y Strength Limit State y Extreme Limit State y Fatigue Limit State
The Service Limit State represents structure performance under service load conditions. Examples for substructure design include settlement of a foundation or lateral displacement of a retaining wall. Another example of a Service Limit State condition is presented in Figure 2-7 which shows the rotation of a rocker bearing on an abutment caused by instability of the earth slope which supported the abutment.
Strength Limit States involve the total or partial collapse of the structure. Examples of Strength Limit States in geotechnical engineering include bearing capacity failure, sliding, and overall slope instability such as shown in Figure 2-8.
Figure 2-7
Example Condition for Service Limit State Evaluation
Figure 2-8
Example Condition for Strength Limit State Evaluation 2.4 LRFD Calibration
Calibration of the load and resistance factors is required to achieve the desired results when applying LRFD. Calibration procedures for selection of resistance factors are described fully in Chapters 3 and 7, but generally involve:
y Engineering judgment y Fitting to ASD
y Reliability theory
For the LRFD Specification, a combination of approaches was used in selecting resistance factors, φ, for design. In general, the resistance factors selected for most of the methods used for foundation design were developed principally using reliability-based calibration procedures where sufficient performance data were available (e.g., bearing resistance of footings and individual deep foundation elements). The value of φ chosen for a particular design procedure and limit state from a reliability- based calibration can take into account the:
y Variability of the soil and rock properties
y Reliability of the equations used for predicting resistance y Quality of the construction workmanship
y Extent of soil exploration y Consequence(s) of a failure
Where insufficient or no data were available to conduct a reliability-based calibration, resistance factors were selected primarily by fitting to ASD (e.g., eccentricity, anchored and MSE wall design) and judgment. Therefore, the principal benefit of LRFD, namely to achieve consistent levels of safety in component design, are not completely realized. The resistance factors incorporated in the LRFD Specification were checked by trial designs to confirm that the results are comparable to current ASD practice.
2.5 Summary
The incorporation of LRFD represents a significant step and major improvement in the processes of foundation, retaining wall and culvert design, as it permits design based primarily on a rational evaluation of performance reliability, rather than the judgment and experience of and individual designer (although the importance of these even in LRFD should not be minimized). As it is currently embodied in the AASHTO Specification, LRFD offers many advantages, primarily that it:
y Accounts separately for variability in load and resistance prediction
y Achieves more consistent levels of safety in structure and substructure design y Does not require knowledge of probability or reliability theory
On the other hand, LRFD does present a challenge to the practicing engineer, in that: y Implementation requires a change for engineers accustomed to ASD y Resistance factors vary with design methods and are not constant
y Rigorous calibration of load and resistance factors to meet individual situations requires availability of statistical data and probabilistic design algorithms
Subsequent chapters will provide additional information regarding the development and application of LRFD for substructure design.
CHAPTER 3
PRINCIPLES OF LIMIT STATES DESIGN