National Academies Press: OpenBook
« Previous: Summary
Page 3
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 3
Page 4
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 4
Page 5
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 5
Page 6
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 6
Page 7
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 7
Page 8
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 8
Page 9
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 9
Page 10
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 10
Page 11
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 11
Page 12
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 12
Page 13
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 13
Page 14
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 14
Page 15
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 15
Page 16
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 16
Page 17
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 17
Page 18
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 18
Page 19
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 19
Page 20
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 20
Page 21
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 21
Page 22
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 22
Page 23
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 23
Page 24
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 24
Page 25
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 25
Page 26
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 26
Page 27
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 27
Page 28
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 28
Page 29
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 29
Page 30
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 30
Page 31
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 31
Page 32
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 32
Page 33
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 33
Page 34
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 34
Page 35
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 35
Page 36
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 36
Page 37
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 37
Page 38
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 38
Page 39
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 39
Page 40
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 40
Page 41
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 41
Page 42
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 42
Page 43
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 43
Page 44
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 44
Page 45
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 45
Page 46
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 46
Page 47
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 47
Page 48
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 48
Page 49
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 49
Page 50
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 50
Page 51
Suggested Citation:"Chapter 1 - Background." National Academies of Sciences, Engineering, and Medicine. 2010. LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures. Washington, DC: The National Academies Press. doi: 10.17226/14381.
×
Page 51

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

31.1 Research Objectives NCHRP Project 24-31, “LRFD Design Specifications for Shallow Foundations” was initiated with the objective to “develop recommended changes to Section 10 of the AASHTO LRFD Bridge Design Specifications for the strength limit state design of shallow foundations.” The current AASHTO specifications, as well as other existing codes employing reliability-based design (RBD) principles, were calibrated using a combination of reliability theory, fitting to allow- able stress design (ASD) (also called working stress design [WSD]), and engineering judgment. The main challenges of the project were, therefore, the compilation of large, high- quality databases and the development of a procedural and data management framework that would enable load and resis- tance factor design (LRFD) parameter evaluation and future updates. Meeting these challenges required the following specific objectives: 1. Establish the state of practice in bridge shallow founda- tions design and construction. 2. Define the ultimate limit states (ULSs) for individual and combined loading of shallow foundations under expected bridge loading conditions. 3. Build databases of shallow foundation performance under vertical, lateral, and moment loading conditions. 4. Establish methods for the various limit state predictions and assess their uncertainty via databases, model analyses, parametric studies, and the probabilistic approach when required. 5. Develop a procedure for calibrating resistance factors for the identified ULS. 6. Establish factors and procedures. 7. Modify AASHTO’s specifications based on the above findings. 1.2 Engineering Design Methodologies 1.2.1 Working Stress Design The WSD method, also called ASD, has been used in civil engineering since the early 1800s. Under WSD, the design loads (Q), which consist of the actual forces estimated to be applied to the structure (or a particular element of the struc- ture), are compared to the nominal resistance, or strength (Rn) through a factor of safety (FS): where Q = design load, Qall = allowable design load, Rn = nominal resistance of the element or the structure, and Qult = ultimate geotechnical foundation resistance. The Standard Specifications for Highway Bridges (AASHTO, 1997), based on common practice, presents the traditional fac- tors of safety used in conjunction with different levels of control in analysis and construction. Although engineering experience over a lengthy period of time resulted in adequate factors of safety, their source, reliability, and performance had remained mostly unknown. The factors of safety do not necessarily con- sider the bias, in particular, the conservatism (i.e., under- prediction) of the analysis methods; hence, the validity of their assumed effect on the economics of design is questionable. 1.2.2 Limit State Design Demand for more economical design and attempts to improve structural safety have resulted in the re-examination of the entire design process over the past 50 years. The design Q Q R FS Q FS all n ult≤ = = ( )1 C H A P T E R 1 Background

of a structure needs to ensure that while being economically viable it will suit the intended purpose during its working life. Limit state (LS) is a condition beyond which the structure (i.e., a bridge in the relevant case), or a component, fails to fulfill in some way the intended purpose for which it was designed. Limit state design (LSD) comes to meet the require- ments for safety, serviceability, and economy. LSD most often refers, therefore, to two types of limit states: the ULS, which deals with the strength (maximum loading capacity) of the structure, and the serviceability limit state (SLS), which deals with the functionality and service requirements of a structure to ensure adequate performance under expected conditions (these can be, for example, under normal expected loads or extreme events like impact, an earthquake, etc.). The ULS design of a structure and its components (e.g., a column or shallow foundation) depends upon the predicted loads and the capacity of the component to resist them (i.e., resistance). Both loads and resistance have various sources and levels of uncertainty. Engineering design has historically compensated for these uncertainties by using experience and subjective judgment. The new approach that has evolved aims to quantify these uncertainties and achieve more rational en- gineering designs with consistent levels of reliability. These uncertainties can be quantified using probability-based meth- ods resulting for example with the LRFD format, which allows the separation of uncertainties in loading from uncertainties in resistance, and the use of procedures from probability theory to assure a prescribed margin of safety. The same principles used in LRFD for ULS can be applied to the SLS, substituting the capacity resistance of the component with a serviceability limit, such as a quantified displacement, crack, deflection or vibration. Since failure under the SLS will not lead to collapse, the prescribed margin of safety can be smaller, i.e., the SLS can tolerate a higher probability of “failure” (i.e., exceedance of the criterion) compared with that for the ULS. 1.2.3 Geotechnical and AASHTO Perspective The LSD and LRFD methods are becoming the standard methods for modern-day geotechnical design codes. In Europe (CEN, 2004; DIN EN 1997-1, 2008 including the National Annex, 1 draft 2009), Canada (Becker, 2003), China (Zhang, 2003), Japan (Honjo et al., 2000; Okahara et al., 2003), the United States (Kulhawy and Phoon, 2002; Withiam, 2003; Paikowsky et al., 2004), and elsewhere, major geotechnical design codes are switching from ASD (or WSD) to LSD and LRFD. A variation of LRFD was first adopted by AASHTO for the design of certain types of bridge superstructures in 1977 under a design procedure known as Load Factor Design (LFD). AASHTO LRFD Bridge Design Specifications was published in 1994 based on NCHRP Project 12-33. From 1994 to 2006, the AASHTO LRFD specifications applied to geotechnical engineering utilized the work performed by Barker et al. (1991). This code was mostly based on an adaptation of WSD to LRFD and only marginally addressed the SLS. Continuous attempts have been made since then to improve the scientific basis on which the specifications were developed, including NCHRP Project 20-7 (Task 88), NCHRP Projects 12-35 and 12-55 for earth pressures and retaining walls, NCHRP Project 12-24 for soil-nailing, and NCHRP Project 24-17 that calibrated for the first time the LRFD parameters for deep foundations based on extensive databases of deep foundation testing (Paikowsky et al., 2004). NCHRP Project 12-66 addresses the needs of SLS in design of bridge foundations. The project’s approach has required developing serviceability criteria for bridges based on foundation performance, defining methods for the eval- uation of foundation displacements and establishing their uncertainty, and calibrating the resistance factors assigned for the use of these methods based on the established SLS and target reliability. The backbone of the study has been the development of databases to establish the uncertainty of the methods used to evaluate the horizontal and vertical dis- placements of foundations. Of the various AASHTO studies related to LRFD calibration and implementation, one important component remained deficient and that was the ULS of shallow foundations. The topic is problematic because the ULS of coupled loading is not easily identified, and the current specifications (AASHTO, 2008), although providing the theoretical estimation of the bearing resistance of soil (Section 10.6.3.1), contain specific language to exclude inclination factors (C10.6.3.1.2a), noting that the specified resistance factors are limited, varying for all conditions between φ = 0.45 to φ = 0.50. The combination of the foundation loads in the ULS frame- work is quite complex and needs to be addressed systemati- cally either via the existing nominal resistance calculation pro- viding safety limits and appropriate resistance factors and/or a new methodology directly applicable to the evaluation of the ULS under the desired load combinations. This issue is further explored in Section 1.6. 1.3 Load and Resistance Factor Design 1.3.1 Principles The intent of LRFD is to separate uncertainties in load- ing from uncertainties in resistance and then to use proce- dures from probability theory to ensure a prescribed margin of safety. Sections 1.3 and 1.4 outline the principles of the methodology and present the common techniques used for its implementation. Figure 1 shows probability density functions (PDFs) for load effect (Q) and resistance (R). “Load effect” is the load cal- 4

Figure 1. An illustration of PDFs for load effect and resistance. R, Q f R (R ), f Q(Q ) Resistance (R) mQ Rn Load Effect (Q) mR Qn FS mR mQ culated to act on a particular element (e.g., a specific shallow foundation), and the resistance is its bearing load capacity. In geotechnical engineering problems, loads are usually bet- ter known than are resistances, so the Q typically has smaller variability than the R; that is, it has a smaller coefficient of variation (COV), hence a narrower PDF. In LRFD, partial safety factors are applied separately to the load effect and to the resistance. Load effects are increased by multiplying characteristic (or nominal) values by load factors (γ); resistance (strength) is reduced by multiplying nominal values by resistance factors (φ). Using this approach, the fac- tored (i.e., reduced) resistance of a component must be larger than a linear combination of the factored (i.e., increased) load effects. The nominal values (e.g., the nominal resistance, Rn, and the nominal load, Qn) are those calculated by the specific calibrated design method and the loading conditions, respec- tively, and are not necessarily the means (i.e., the mean loads, mQ, or mean resistance, mR of Figure 1). For example, Rn is the predicted value for a specific analyzed foundation, obtained by using Vesic´ ’s bearing capacity calculation, while mR is the mean possible predictions for that foundation considering the various uncertainties associated with that calculation. This principle for the strength limit state is expressed in the AASHTO LRFD Bridge Design Specifications (AASHTO, 1994, 1997, 2001, 2006, 2007, 2008) in the following way: R R Qr n i i i= ≥∑φ η γ ( )2 where the nominal (ultimate) resistance (Rn) multiplied by a resistance factor (φ) becomes the factored resistance (Rr), which must be greater than or equal to the summation of loads (Qi) multiplied by corresponding load factors (γi) and a modifier (ηi). where ηi are factors to account for effects of ductility (ηD), redundancy (ηR), and operational importance (ηI). Based on considerations ranging from case histories to exist- ing design practice, a prescribed value is chosen for probability of failure. Then, for a given component design (when applying resistance and load factors), the actual probability for a fail- ure (the probability that the factored loads exceed the factored resistances) should be equal to or smaller than the prescribed value. In foundation practice, the factors applied to load effects are typically transferred from structural codes, and then resis- tance factors are specifically calculated to provide the pre- scribed probability of failure. The importance of uncertainty consideration regarding the resistance and the design process is illustrated in Figure 1. In this figure, the central factor of safety is FS — = mR/mQ, whereas the nominal factor of safety is FSn = Rn/Qn. The mean factor of safety is the mean of the ratio R/Q and is not equal to the ratio of the means. Consider what happens if the uncertainty in resistance is increased, and thus the PDF broadened, as suggested by the dashed curve. The mean resistance for this curve (which may represent the result of another predictive method) remains unchanged, but the variation (i.e., un- certainty) is increased. Both distributions have the same mean factor of safety one uses in WSD, but utilizing the dis- tribution with the higher variation will require the applica- tion of a smaller resistance factor in order to achieve the same prescribed probability of failure to both methods. The limit state function g corresponds to the margin of safety, i.e., the subtraction of the load from the resistance such that (referring to Figure 2a): For areas in which g < 0, the designed element or structure is unsafe because the load exceeds the resistance. The proba- bility of failure, therefore, is expressed as the probability (P) for that condition: In calculating the prescribed probability of failure (pf), a derived probability density function is calculated for the margin of safety g(R,Q) (refer to Figure 2a), and reliability is expressed using the “reliability index,” β. Referring to Figure 2b, p P gf = <( )0 5( ) g R Q= − ( )4 η η η ηi D R l= ≥ 0 95 3. ( ) 5

the reliability index is the number of standard deviations of the derived PDF of g, separating the mean safety margin from the nominal failure value of g being zero: where mg and σg are the mean and standard deviation of the safety margin defined in the limit state function Equation 4, respectively. The relationship between the reliability index (β) and the probability of failure (pf) for the case in which both R and Q follow normal distributions can be obtained based on Equa- tion 6 as the following: where Φ is the error function defined as The relationship between β and pf is provided in Table 1. The relationships in Table 1 remain valid as long as the assumption is that the reliability index (β) follows a normal distribution. As the performance of the physical behavior of engineer- ing systems usually cannot obtain negative values (load and resistance), it is better described by a lognormal distribution. The margin of safety is taken as log R − log Q, when the resis- tances and load effects follow lognormal distributions. Thus, the limit state function becomes the following: u du− ⎡ ⎣⎢ ⎤ ⎦⎥2 2 exp . Φ z z( ) = −∞ ∫ 1 2π e pf = −( )Φ β ( )7 β σ σ σ= = −( ) +m m mg g R Q Q R2 2 6( ) If R and Q follow lognormal distributions, log R and log Q follow normal distributions, thus the safety margin, g, follows a normal distribution. As such, the relationship obtained in Equation 7 is still valid to calculate the failure probability. Figure 2b illustrates the limit state function, g, for normal dis- tributed resistance and load, the defined reliability index, β (also termed target reliability, βT), and the probability of fail- g R Q R Q= ( )− ( ) = ( )ln ln ln ( )8 6 Table 1. Relationship between reliability index and probability of failure. Reliability index Probability of failure pβ f 1.0 0.159 1.2 0.115 1.4 0.0808 1.6 0.0548 1.8 0.0359 2.0 0.0228 2.2 0.0139 2.4 0.00820 2.6 0.00466 2.8 0.00256 3.0 0.00135 3.2 6.87 E-4 3.4 3.37 E-4 3.6 1.59 E-4 3.8 7.23 E-5 4.0 3.16 E-5 0 1 2 3 R, Q 0 1 2 3 4 Pr ob ab ili ty d en sit y fu nc tio n mR mQ mg (=mR−mQ) Resistance (R) Load effect (Q) Qn Rn Performance (g) g < 0 (fa ilu re ) -0.5 0 0.5 1 1.5 g(R,Q) = R− Q mg(=mR−mQ) Performance (g) βσg Fa ilu re re gi on a re a = p f f(g) (a) (b) Figure 2. An illustration of probability density function for (a) load, resistance, and performance function and (b) the performance function (g(R,Q)) demonstrating the margin of safety (pf) and its relation to the reliability index,  (g = standard deviation of g).

ure, pf. For lognormal distributions, these relations will relate to the function g = ln(R/Q) as explained above. The values provided in Table 1 are based on series expan- sion and can be obtained by a spreadsheet (e.g., NORMSDIST in Excel) or standard mathematical tables related to the stan- dard normal probability distribution function. It should be noted, however, that previous AASHTO LRFD calibrations and publications for geotechnical engineering, notably Barker et al. (1991) and Withiam et al. (1998), have used an approx- imation relationship proposed by Rosenblueth and Esteva (1972), which greatly errs for β < 2.5, the typical zone of inter- est in ULS design calibration (β = 2 to 3) and errs even more in the zone of interest for SLS calibrations (β < 2.0). For lognormal distributions of load and resistance one can show (e.g., Phoon et al., 1995) that Equation 6 becomes the following: where mQN, mRN = the mean of the natural logarithm of the load and the resistance, σQN, σRN = the standard deviations of the natural log- arithm of the load and the resistance, and mQ, mR, = the simple means and the coefficients of COVQ, COVR variation for the load and the resistance of the normal distributions. These values can be transformed from the lognormal distri- bution using the following expressions for the load and similar ones for the resistance: 1.3.2 The Calibration Process The problem facing the LRFD analysis in the calibration process is to determine the load factor (γ) and the resistance factor (φ) such that the distributions of R and Q will answer to the requirements of a specified β. In other words, the γ and φ described in Figure 3 need to answer to the prescribed tar- get reliability (i.e., a predetermined probability of failure) described in Equation 9. Several solutions are available and are described below, including the recommended procedure for the research reported herein (see Section 1.3.5). and m mQN Q QN= ( )−ln . ( )0 5 112σ σQN QCOV2 21 10= +( )ln ( ) β σ σ = − + = ( ) +( ) + m m m m COV COV RN QN QN RN R Q Q R 2 2 2 21 1ln ( )⎢⎣ ⎥⎦ +( ) +( )[ ]ln ( )1 1 92 2COV COVR Q 1.3.3 First Order Second Moment The First Order Second Moment (FOSM) method of cali- bration was proposed originally by Cornell (1969) and is based on the following. For a limit state function g(m): where m1 and σi = the means and standard deviations of the basic variables (design parameters); χi, i = 1,2, . . . , n; g +i = mi +Δmi, and g −i = mi −Δmi for small increments Δmi; and Δxi is a small change in the basic variable value, xi. Practically, the FOSM method was used by Barker et al. (1991) to develop closed-form solutions for the calibration of the geotechnical resistance factors (φ) that appeared in the previous AASHTO LRFD specifications. variance or σ σ χ g g ii n i ig 2 2 1 2 13≈ ∂ ∂ ⎛ ⎝⎜ ⎞ ⎠⎟ ≈ = + ∑ i ( ) − ⎛ ⎝⎜ ⎞ ⎠⎟ − = ∑ gi ii n iΔχ σ 2 1 2i mean . . . ,m g m m m mg n≈ ( )1 2 3 12, , , ( ) 7 1 2 3 R, Q 0 1 2 3 4 Pr ob ab ili ty d en sit y fu nc tio n mR Resistance (R) Load effect (Q) Qn Rn mQ RnQnγ φ Figure 3. An illustration of the LRFD factors determination and application (typically  >– 1,  <– 1) relevant to the zone in which load is greater than resistance (Q > R).

where λR = resistance bias factor, mean ratio of measured resistance over predicted resistance; COVQ = coefficient of variation of the load; COVR = coefficient of variation of the resistance; and βT = target reliability index. When just dead and live loads are considered, Equation 14 can be rewritten as where γD, γL = dead and live load factors, QD/QL = dead to live load ratio, λQD, λQL = dead and live load bias factors, COVQD = coefficient of variation for dead load, and COVQL = coefficient of variation for live load. The probabilistic characteristics of the foundation loads are assumed to be those used by AASHTO for the superstruc- ture (Nowak, 1999); thus γD, γL, λQD and λQL are fixed, and a resistance factor can be calculated for a resistance distribution (λR, COVR) for a range of dead load to live load ratios. 1.3.4 First Order Reliability Method LRFD for structural design has evolved beyond FOSM to the more invariant First Order Reliability Method (FORM) approach (e.g., Ellingwood et al., 1980; Galambos and Ravindra, 1978), while geotechnical applications have lagged behind (Meyerhof, 1994). In order to be consistent with the previous structural code calibration and the load factors to which it leads, the calibration of resistance factors for deep foundations in NCHRP Project 24-17 used the same method- ology (Paikowsky et al., 2004). The LRFD partial safety fac- tors were calibrated using FORM as developed by Hasofer and Lind (1974). FORM can be used to assess the reliability of a component with respect to specified limit states and provides a means for calculating partial safety factors φ and γi for resis- tance and loads, respectively, against a target reliability level, β. FORM requires only first and second moment information on resistances and loads (i.e., means and variances) and an as- sumption of distribution shape (e.g., normal, lognormal, φ λ γ γ λ = + ⎛⎝⎜ ⎞⎠⎟ + + + R D D L L QD QL R Q Q Q COV COV COV 1 1 2 2 2 D D L QL T R QD Q Q COV COV C + ⎛⎝⎜ ⎞⎠⎟ +( )[ + + λ βexp ln 1 1 2 2 OVQD2 15 ( )] ⎧ ⎨⎪ ⎩⎪ ⎫ ⎬⎪ ⎭⎪ ( ) φ λ γ β= ( ) + + +( ) ∑R i i Q R Q T R Q COV COV m COV 1 1 1 2 2 2exp ln 1 14 2+( )[ ]{ }COVQ ( ) etc.). The calibration process is presented in Figure 4 and detailed by Paikowsky et al. (2004). Each limit state (ultimate or serviceability) can be repre- sented by a performance function of the form: in which X = (X1, X2, . . . , Xn) is a vector of basic random variables of strengths and loads. The performance function g(X), often called the limit state function, relates random vari- ables to either the strength or serviceability limit state. The limit is defined as g(X) = 0, implying failure when g(X) ≤ 0 (but strictly g(X) < 0) (see Figures 2 and 4). Referring to Figure 4, the reliability index, β, is the distance from the origin (in stan- dard normal space transformed from the space of the basic random variables) to the failure surface at the most probable point on that surface, that is, at the point on g(X) = 0 at which the joint probability density function of X is greatest. This is sometimes called the design point, and is found by an itera- tive solution procedure (Thoft-Christensen and Baker, 1982). This relationship can also be used to back calculate represen- tative values of the reliability index, β, from current design practice. The computational steps for determining β using FORM are provided by Paikowsky et al. (2004). In developing code provisions, it is necessary to follow current design practice to ensure consistent levels of reliabil- ity over different evaluation methods (e.g., pile resistance or displacement). Calibrations of existing design codes are needed to make the new design formats as simple as possible and to put them in a form that is familiar to designers. For a given reliability index, β, and probability distributions for resis- tance and load effects, the partial safety factors determined by the FORM approach may differ with failure mode. For this rea- son, calibration of the calculated partial safety factors (PSFs) is important in order to maintain the same values for all loads at different failure modes. In the case of geotechnical codes, the calibration of resistance factors is performed for a set of load factors already specific in the structural code. Thus, the load factors are fixed. A simplified algorithm was used in NCHRP Project 24-17 to determine resistance factors: 1. For a given value of the reliability index, β, probability distributions and moments of the load variables, and the coefficient of variation for the resistance, compute mean resistance, mR, using FORM. 2. With the mean value for R computed in Step 1, the PSF, φ, is revised as where mLi and mR are the mean values of the load and strength variables, respectively, and γi, i = 1, 2, . . . , n, are the given set of load factors. φ γ = = ∑ i Li i n R m m 1 17( ) g X g X X Xn( ) = ( )1 2 16, , . . . ( ), 8

A comparison between resistance factors obtained using FORM and resistance factors using FOSM for 160 calibra- tions of axial pile capacity prediction methods is presented in Figure 5. The data in Figure 5 suggest that FORM results in resistance factors that are consistently higher than those obtained by FOSM. As a rule of thumb, FORM provided resis- tance factors for deep foundations approximately 10% higher than those obtained by FOSM. The practical conclusions that can be obtained from the observed data are that first evalua- tion of data can be done by the simplified closed-form FOSM approach and the obtained resistance factors are on the low side (safe) for the resistance distributions obtained in the NCHRP 24-17 project (Paikowsky et al., 2004). 1.3.5 Monte Carlo Simulation Monte Carlo Simulation (MCS) has become AASHTO’s pre- ferred calibration tool and is recommended for all AASHTO- related calibrations. MCS is a powerful tool for determining the failure probability numerically, without the use of closed- form solutions such as those given by Equations 14 and 15. The objective of MCS is the numerical integration of the expression for failure probability, as given by the following equation: where I is an indicator function which is equal to 1 for gi ≤ 0, i.e., when the resulting limit state is in the failure region, and equal to 0 for gi > 0, when the resulting limit state is in the safe region. N is the number of simulations carried out. As N→∞, the mean of the estimated failure probability using Equation 18 can be shown to be equal to the actual failure probability (Rubinstein, 1981). Code calibration in its ideal format is accomplished in an iterative process by assuming agreeable load (γ) and resistance (φ) factors and determining the resultant reliability index, β. When the desired target reliability index, βT, is achieved, an acceptable set of load and resistance factors has been deter- mined. One unique set of load and resistance factors does not exist; different sets of factors can achieve the same target reli- ability index (Kulicki et al., 2007). The MCS process is simple and can be carried out as follows: • Identify basic design variables and their distributions. Load is assumed to be normally distributed. • Generate N number of random samples for each design vari- able based on its distributions, i.e., using the reported statis- tics of load and resistance and computer-generated random numbers. • Evaluate the limit state function N times by taking a set of the design variables generated above and count the number for which the indicator function is equal to 1. p P g N I gf i i N = ≤( ) = ≤[ ] = ∑0 1 0 18 1 ( ) 9 Notes: ST = Structural MCS = Monte Carlo Simulation μ = mean g(x) = performance function of the limit state = limit state function g(x) = 0 = limit defining failure for g(x) < 0 gL(x) = linearized performance function Definition of Failure Define Limit States Level: Ultimate & Serviceability Define Statistical Characteristics of Basic Random Variables Resistance Load Determine Model Uncertainty for Strength (from database) Determine Load Uncertainties from Superstructure to Foundation (from ST code) MCS or Probability Calculation to Get Statistical Properties of Scalar R Reliability Assessment Back-calculated Beta vs Load Ratio Curves in Practice Assign Target Betas Review Target Betas in the Literature and Practice Calculate Load and Resistance Factors Select Load and Resistance Factors Adjust for Mean/Nominal Parameters Case Study Designs for Comparison Q R Failure Region g(x)=0 gL(x)=0 Contours of fRS=fx(x) Safe Region μS μR Figure 4. Resistance factor analysis flow chart using FORM (Ayyub and Assakkaf, 1999; Ayyub et al., 2000; Hasofer and Lind, 1974).

• If the sum of the indicator function is Nf, i.e., the limit state function was gi ≤ 0 (in the failure region) for Nf num- ber of times out of the total of N simulations carried out, then the failure probability, pf, can be directly obtained as the ratio Nf /N. Using the MCS process, the resistance factor can be calcu- lated based on the fact that to attain a target failure probabil- ity of pfT, NfT (Number of samples to obtain target failure at the limit states) of the limit state must fall in the failure region. Since in the present geotechnical engineering LRFD only one resistance factor is used while keeping the load factors constant, a suitable choice for the resistance factor would shift the limit state function so that Nf T samples fall in the failure region. The resistance factor derived in this study using MCS is based on this concept. Kulicki et al. (2007) made several observations regarding the process outlined above: 1. The solution is only as good as the modeling of the distri- bution of load and resistance. For example, if the load is not correctly modeled or the actual resistance varies from the modeled distribution, the solution is not accurate. In other words, if the statistical parameters are not well defined, the solution is equally inaccurate. 2. If both the distribution of load and resistance are assumed to be normally or lognormally distributed, a MCS using these assumptions should theoretically produce the same results as the closed-form solutions. 3. The power of the MCS is its ability to use varying distribu- tions for load and resistance. In summary, refinement in the calibration should be pur- sued, not refinement of the process used to calculate the reli- ability index. The MCS, as discussed above, is quite adequate and understandable to the practicing engineer. Refinement should be sought in the determination of the statistical param- eters of the various components of force effect and resistance and using the load distributions available for the structural analysis; this means focusing on the statistical parameters of the resistance. 1.4 Format for Design Factor Development 1.4.1 General AASHTO development and implementation of LSD and LRFD have been driven primarily by the objectives of achiev- ing a uniform design philosophy for bridge structural and geotechnical engineering thereby obtaining a more consis- tent and rational framework of risk management in geotech- nical engineering. Section 1.3 detailed the principles of LRFD and described the calibration process. The philosophies of attaining this 10 y = 1.1267x 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 - 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 R es ist an ce fa ct or s u sin g FO R M Resistance factors using FOSM Driven Piles Static Analysis Driven Piles Dynamic Analysis Drilled Shafts FOSM = FORM Linear No. of points = 160 Mean = 1.148 Std. Dev. = 0.039 Figure 5. Comparison of resistance factors obtained using FOSM versus those obtained using FORM for a target reliability of   2.33 (Paikowsky et al., 2004).

calibration, however, vary widely: values are chosen based on a range of already available parameters, based on expert opinion, based on comprehensive resistance calibration, or using the material factor approach. A previous effort to cal- ibrate the ULS of deep foundations concentrated on com- prehensive calibration of the resistance models as an integral entity (Paikowsky et al., 2004). This philosophy was based on the fact that in contrast to other engineering disciplines (e.g., structural analysis), the model uncertainty in geotech- nical engineering is dominant. The specifications provide an ideal framework for prescribed comprehensive methodology and, hence, direct calibration of the entire methodology, when possible, results in highly accurate LRFD as demonstrated in the following sub-sections. This approach was followed by and large in the development of the SLS (NCHRP Project 12-66) and is followed (when possible) in this study as well. The calibration of shallow foundations for ULS has, however, more complex aspects that cannot be (at present time) cal- ibrated directly. Hence, Section 1.4.2 (based primarily on Honjo and Amatya, 2005) is provided as a background to the diverse approach of the current research. 1.4.2 Material and Resistance Factor Approach Some of the key issues in developing sound geotechnical design codes based on LSD and LRFD are definition of char- acteristic values and determination of partial factors together with the formats of design verification (Simpson and Driscoll, 1998; Orr, 2002; Honjo and Kusakabe, 2002; Kulhawy and Phoon, 2002). The characteristic values of the design param- eters are conveniently defined as their mean values. The approach concerning design factor development for- mats can be summarized as whether one should take a material factor approach (MFA) or a resistance factor approach (RFA). In MFA, partial factors are directly applied to the character- istic values of materials in the design calculation, whereas in RFA, a resistance factor is applied to the resulting resistance cal- culated using the characteristic values of materials. One of the modifications of RFA is a multiple resistance factor approach (MRFA) where several resistance factors are employed to be applied to relatively large masses of calculated resistances. The advantage of MRFA is claimed to be that it ensures a more consistent safety margin in design compared with RFA (Phoon et al. 1995, 2000; Kulhawy and Phoon, 2002). In gen- eral, MFA originated in Europe whereas RFA originated in North America. However, both approaches are now used inter- changeably worldwide; for example, the “German approach” to EC7 coincides with RFA while Eurocode 7 allows several design approaches (both MFA and RFA), and the member state can define their preference in their National Annex to the EC7. 1.4.3 Code Calibrations A procedure to rationally determine partial factors in the design verification formulas based on reliability analysis is termed “code calibration.” Section 1.3.2 and the details in Sections 1.3.3, 1.3.4, and 1.3.5 presented the analytical mean- ing of the calibration in the LRFD methodology. One of the best known and most important studies in this area is by Ellingwood et al. (1982) in which load and resistance factors were determined based on a reliability analysis using FORM. Since then, a reasonable number of code calibration studies have been carried out in structural engineering (e.g., Nowak, 1999). However, rational code calibration studies for geo- technical engineering codes have only begun to be undertaken in the past decade or so (Barker et al. 1991; Phoon et al., 1995; Honjo et al., 2002; Paikowsky et al., 2004). Barker et al. (1991) proposed resistance factors for the AASHTO bridge foundation code published in 1994 (AASHTO, 1994). The calibration was based on FOSM but used back-calculation from factors of safety and introduced a significant number of engineering judgments in deter- mining the factors along a not-so-clearly described process. Based on the difficulties encountered in using the work of Barker et al. (1991), the partial factors for deep foundations in the AASHTO specification were revised by Paikowsky et al. (2004). In Paikowsky et al. (2004), a large database was devel- oped and used in a directly calibrated model (an RFA approach together with a reliability analysis by FORM) to determine the resistance factors. The SLS calibration (NCHRP Project 12-66) was developed in a similar approach, using MCS to determine the factors. Examples from both studies are provided in Sec- tions 1.4.4 and 1.4.5. Phoon et al. (1995, 2000) carried out cal- ibration of the factors for transmission line structure founda- tions based on MRFA by reliability analysis. Some simplified design formats were employed, and factors were adjusted until the target reliability index was reached. Kobayashi et al. (2003) have calibrated resistance factors for building foundations for the Architectural Institute of Japan (AIJ) limit state design building code (AIJ, 2002). This code provides a set of load and resistance factors for all aspects of building design in a unified format. FORM was used for the reliability analysis, and MRFA was the adopted format of design verification as far as the foun- dation design was concerned. 1.4.4 Example of Code Calibrations—ULS The capacity of the comprehensive direct model calibra- tion resistance factor approach is demonstrated. Large data- bases of pile static load tests were compiled, and the static and dynamic pile capacities of various design methods were com- pared with the nominal strength obtained from the static load test. The geotechnical parameter variability was minimized 11

(indirectly) by adhering to a given consistent procedure in soil parameters selection (e.g., NSPT [Number of Blows in a Stan- dard Penetration Test] correction and friction angle correla- tions), as well as load test interpretation (e.g., establishing the uncertainty in Davisson’s criterion for capacity determina- tion and then using it consistently). Two examples for such large calibrations are presented in Figures 6 and 7 for given specific dynamic and static pile capacity prediction methods, respectively (Paikowsky et al., 2004). Further subcategorization of the analyses led to detailed resistance factor recommendations based on pile type, soil type, and analysis method combinations. Adherence to the uncertainty of each combination as developed from the data- base and consistent calibrations led to a range of resistance factors (see, for example, Table 25 of NCHRP Report 507, Paikowsky et al., 2004). Recent versions of the specifications (AASHTO, 2006, 2008) avoided the detailed calibrations and presented one “simplified” resistance factor (φ= 0.45) for static analysis of piles, along with one design method (Nordlund/ Thurman). The first large LRFD bridge design project in New England (including superstructure and substructure) based on AASHTO 2006 specifications is currently under construction. A large static load test program preceded the design. Identifiable details are not provided, but Tables 2 and 3 present the capacity eval- uation for two dynamically and statically tested piles (Class A prediction, submitted by the project consultant, Dr. Samuel Paikowsky, about one month before testing) using the cal- ibrated resistance factors for the specific pile/soil/analysis method combination versus the “simplified” AASHTO version of the resistance factor. In both cases, the calculated factored capacity using the “simplified” resistance factor exceeded the unfactored and factored measured resistance (by the load test) in a dangerous way, while the use of the calibrated resistance fac- tors led to consistent and prudent design. The anticipated sub- structure additional cost has increased by 100% (in comparison to its original estimate based on the AASHTO specifications), exceeded $100 million (at the time of the load test program), and delayed the project 1 year. The power of the comprehen- sive, direct RFA calibration based on databases versus arbitrary 12 0 0.5 1.5 2 2.5 Ratio of Static Load Test Results over the Pile Capacity Prediction using the CAPWAP method 0 5 10 15 20 25 30 35 40 45 50 55 60 N um be r o f P ile C as es 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 R el at iv e Fr eq ue nc ylog normal distribution mlnx = 0.233 σlnx = 0.387 normal distribution mx = 1.368 >31 σx = 0.620 Figure 6. Histogram and frequency distributions for all (377 cases) measured over dynamically (CAPWAP) calculated pile-capacities in PD/LT2000 (Paikowsky et al., 2004). 0 0.5 1 1.5 2 2.5 3 Ratio of Static Load Test Results over the Pile Capacity Prediction using the α-API/Nordlund/Thurman design method 0 2 4 6 8 10 12 14 16 18 20 22 N um be r o f P ile C as es 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 R el at iv e Fr eq u en cy log normal distribution mlnx = -0.293 σlnx = 0.494 normal distribution mx = 0.835 σx = 0.387 Figure 7. Histogram and frequency distribution of measured over statically calculated pile capacities for 146 cases of all pile types (concrete, pipe, H) in mixed soil (Paikowsky et al., 2004).

assignments of resistance factors is clearly demonstrated in the first significant case of its use in New England. 1.4.5 Example of Code Calibrations—SLS The factors associated with the SLS were evaluated under NCHRP Project 12-66. Following the development of ser- viceability criteria for bridges (Paikowsky, 2005; Paikowsky and Lu, 2006), large databases of foundation performance were accumulated and analyzed for direct RFA calibrations (Paikowsky et al., 2009a, 2009b). Examples of databases examining the performance of displacement analyses of shallow foundations are presented in Figures 8 and 9 for the AASHTO (2008) and Schmertmann et al. (1978) settlement analysis methods, respectively. These robust analysis results allow direct calibration of resistance factors for applied loads 13 Table 2. H Pile—summary (14  177, penetration = 112 ft). Static: Static Pile Capacity Combinations: Analysis combination Estimated capacity (Rn) (kips) NCHRP 507 resistance factor for H piles in sand ( ) Factored resistance (Rr) NCHRP 507 resistance factor for H piles in mixed soils ( ) Factored resistance (Rr) AASHTO LRFD specifications 2006 resistance factor ( ) Factored resistance (Rr) -Method/Thurman (Steel Only) 894 268 179 -Method/Thurman (Box Area) 1,076 0.30 323 0.20 215 Not specified Nordlund/Thurman (Steel Only) 841 379 252 379 Nordlund/Thurman (Box Area) 1,023 0.45 460 0.35 307 0.45 460 FHWA Driven Ver. 1.2 (Steel Only) 845 FHWA Driven Ver. 1.2 (Box Area) 1,032 Notes: 1. Resistance Factors taken from the resistance factors for redundant structures listed in Table 25 of NCHRP Report 507 (Paikowsky et al., 2004). Recommended range for preliminary design. Reference: Static Pile Capacity and Resistance Factors for Pile Load Test Program, GTR report submitted to Haley and Aldrich, Inc. (H&A), June 21, 2006 (Paikowsky, Thibodeau, and Griffin). Note: Above DRIVEN values were obtained by inserting the friction values and unit weights directly into DRIVEN, limiting the friction angle to 36 . Dynamic: Sakonnet River Bridge Test Pile Program Portsmouth, RI—Summary of Dynamic Measurement Predictions and Factored Resistance (H Piles) 1Values represent EOD predictions and average of all BOR predictions. 2All factors taken from NCHRP Report 507 (Paikowsky et al., 2004) 3Only factors for BOR CAPWAP appear in AASHTO (2006) specifications and are marked by shaded cells Reference: Pile Capacity Based on Dynamic Testing and Resistance Factors for Pile Load Test Program, GTR report submitted to H&A, July 17, 2006 (based on earlier submittals of data and analyses) (Paikowsky, Chernauskas, and Hart). Static Load Test Load Test Capacity (Davisson’s Criterion): Qu = 378 kips at 0.68 in Resistance Factors NCHRP Report 507 and AASHTO Specifications: = 0.55 (1 test pile large site variability) = 0.70 (1 pile medium site variability) Factored Resistance: Rr = 208 to 265 kips Reference: Load Test Results presented and analyzed by H&A. Energy approach CAPWAP Pile type Time of driving EA1 (kips) 2 Rr (kips) CAP1 (kips) 2 Rr (kips) EOD 481 0.55 265 310 0.65 202 H BOR 606 0.40 242 434 0.65 2823

for a given SLS criterion (displacement). The data in Figures 8 and 9 are related to the following: 1 ft (0.30 m) ≤ B ≤ 28 ft (8.53 m), Bavg = 8 ft (2.44 m), 1.0 ≤ L/B ≤ 6.79, L/Bavg = 1.55, 25.2 ksf (1,205 kPa) ≤ qmax ≤ 177.9 ksf (8,520 kPa) for which B and L are the footing width and length, respectively, and qmax is the maximum stress applied to the foundations under the measured displacement. 1.4.6 Perspective of Shallow Foundations ULS Calibration The preceding sections have outlined the available for- mats of factor development and a powerful implementa- tion via robust databases. The established RFA was utilized in two extensive studies: one related to the ULS of deep foun- dations (NCHRP Project 24-17) and one related to the SLS of all foundations (NCHRP Project 12-66). The complexity of the ULS of shallow foundations (to be discussed in the next section) requires a multifaceted approach in which combinations of calibrations are utilized for obtaining the desired factors. The method of approach is presented in Chapter 2 of this report. Mutiple approaches are needed for the ULS of shallow foundations because of the following: 1. The capacity of shallow foundations on granular soils under centric vertical load is calculated via a relatively simple model (the bearing capacity model without cohesion-related factors, modified by shape and depth factors only). This type of foundation and loading is commonly tested and, hence can be calibrated using a large database (the database is presented in Section 3.2). 2. Determination of the capacity of shallow foundations under combined loading conditions requires a multiparameter model. The differentiation between favorable and unfavor- 14 Table 3. 42-in Pipe Pile—summary (diam. = 42 in, wall thickness (w.t.) = 1 in, 2-in tip, penetration = 64 ft). Static: Static Pile Capacity Combinations: Assumed Displaced Soil Volume Based on Uniform Wall Thickness (1.0 in) Notes: 1. Resistance Factors taken from the resistance factors for redundant structures listed in Table 25 of NCHRP Report 507 (Paikowsky et al., 2004). 2. Tip resistance for steel only included 2-in. wall thickness accounting for the driving shoe. Recommended range for preliminary design soil plug only. Reference: Static Pile Capacity and Resistance Factors for Pile Load Test Program, GTR report submitted to Haley and Aldrich, Inc. (H&A), June 21, 2006 (Paikowsky, Thibodeau, and Griffin). Static Load Test (Open Pipe Pile) Load Test Capacity (Davisson’s Criterion): Qu = 320 kips at 0.52 in Resistance Factors NCHRP Report 507 and AASHTO Specifications: = 0.55 (1 pile large site variability) = 0.70 (1 pile medium site variability) Factored Resistance: Rr = 176 to 224 kips Reference: Load Test Results presented and analyzed by H&A. Analysis combination Estimated capacity (Rn) (kips) NCHRP 507 resistance factor for pipe piles in sand ( ) Factored resistance (Rr) NCHRP 507 resistance factor for pipe piles in mixed soils ( ) Factored resistance (Rr) AASHTO LRFD specifications 2006 resistance factor ( ) Factored resistance (Rr) -Method/Thurman (Steel Only) 924 324 231 - -Method/Thurman (30% Tip Area) 984 345 246 - -Method/Thurman (50% Tip Area) 1,084 380 271 - -Method/Thurman (70% Tip Area) 1,184 415 296 - -Method/Thurman (100% Tip Area, plugged) 1,335 0.35 467 0.25 334 Not specified - Nordlund/Thurman (Steel Only) 690 379 241 310 Nordlund/Thurman (30% Tip Area) 750 412 262 337 Nordlund/Thurman (50% Tip Area) 850 467 297 382 Nordlund/Thurman (70% Tip Area) 950 522 332 427 Nordlund/Thurman (100% Tip Area, plugged) 1,101 0.55 605 0.35 385 0.45 495

able loading conditions is quite complex due to coupled loads and resistances. ULS under combined loading requires both an attempt to calibrate the existing methodology and an examination of a different approach for design, as described in Section 1.5. 3. The capacity of shallow foundations on rock under all types of loading is highly dependent on the relative scale of the foundation width to the rock discontinuity spacing and on the nature of the rock and its discontinuities. No established bearing capacity theory exists for these cases. The calibra- tion of such cases, both for ULS and SLS (not included in NCHRP Project 12-66), requires therefore establishing models, using sophisticated analysis methods for evaluating both strength and serviceability, and performing a proba- bility evaluation of incomplete data and calibration. 1.5 Bearing Capacity of Shallow Foundations 1.5.1 Basic Formulation Buismann (1940) and Terzaghi (1943) adopted the solution for metal punching proposed by Prandtl (1920, 1921) to the foundation bearing capacity problem. They defined a three- term bearing capacity equation by the superposition of the effects of soil cohesion, soil surcharge, and weight of soil, respectively. For a general case of centric vertical loading of a rigid strip footing (plain strain problem) on a cohesive- frictional soil surface with a uniform surcharge of q, the ulti- mate bearing capacity (qu) is given as the following: where c = soil cohesion; γ = unit weight of the soil beneath the foundation; B = footing width; q = overburden pressure at the level of the footing base; and Nc, Nq, and Nγ are bearing capacity factors for cohesion, overburden, and self-weight of soil, respectively. For weightless soil (γ = 0), Prandtl (1920) and Reissner (1924) developed the following formulas for Nc and Nq: N Nc q f= −( )1 20cot ( )φ q cN qN BNu c q= + +( )1 2 19γ γ ( ) 15 0 2 4 6 8 10 12 14 16 18 20 N um be r o f C as es 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.17 0.20 0.22 R el at iv e Fr eq ue nc y 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Bias (Measured Load/Calculated Load) normal distribution log normal distribution λ = 0.674 σx = 0.916 λ = 2.840 σx = 2.265 AASHTO Analysis Δ = 0.25 in n = 85 load tests 0.5 1.0 1.5 2.0 2.5 3.0 Foundation Settlement (in) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 λ, COV AASHTO Bias (λ) COV (a) (b) Figure 8. (a) Histogram and frequency distributions of measured over calculated loads for a settlement ( = 0.25 in) using AASHTO’s analysis method for 85 shallow foundation cases, and (b) variation of the bias ( ) and uncertainty in the ratio between measured to calculated loads for shallow foundations on granular soils under displacements ranging from 0.25 to 3.00 in.

where φf = friction angle. The bearing capacity factor Nc is sometimes credited to Caquot and Kérisel (1953). These formulas are exact closed-form solutions based on Prandtl’s assumption of rupture surfaces (see Figure 10) in which the downward movement of the active wedge (I) is resisted by the shear resistance along the slip surfaces CDE (along the transi- Nq f f = ( ) +⎛⎝⎜ ⎞ ⎠⎟exp tan tan ( )π φ φ 2 45 2 21° 16 0 5 10 15 20 25 N um be r o f C as es 0.00 0.05 0.10 0.15 0.20 0.25 0.30 R el at iv e Fr eq ue nc y 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Bias (Measured Load/Calculated Load) normal distribution log normal distribution λ = -0.324 σx = 0.745 λ = 0.950 σx = 0.734 Schmertmann Analysis (1970) Δ = 0.25 in n = 81 load tests 0 5 10 15 20 25 30 N um be r o f C as es 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 R el at iv e Fr eq ue nc y Bias (Measured Load/Calculated Load) normal distribution log normal distribution λ = 0.068 σx = 0.630 λ = 1.296 σx = 0.818 Schmertmann et al. Analysis (1978) = 0.25 in n = 81 load tests (a) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Foundation Settlement (in) 0.0 0.5 1.0 1.5 2.0 λ, COV (b) Schmertmann 1970 & Schmertmann et al. 1978 Bias (λ) 1970 COV 1970 Bias (λ) 1978 COV 1978 Bias (λ) Das COV Das Figure 9. (a) Histogram and frequency distributions of measured over calculated loads for a settlement ( = 0.25 in) using Schmertmann (1970) and Schmertmann et al. (1978) analysis methods for 81 shallow foundation case, and (b) variation of the bias ( ) and uncertainty in the ratio between measured to calculated loads for shallow foundations on granular soils under displacements ranging from 0.25 to 3.00 in. Q0 L = q A C D EBI II III Figure 10. Assumed rupture surfaces by Prandtl (1920, 1921).

tion zone [II] and passive wedge [III]) and the overburden pressure, q. 1.5.2 The Factor N 1.5.2.1 Nγ Formulations A closed-form analytical solution for the bearing capacity problem including the effects of the unit weight of the soil beneath the footing via the factor Nγ is not possible. Different solutions for Nγ were developed based on empirical relations, analytical derivations, or numerical analyses. Some of these solutions are listed below and are presented in Figure 11 for comparison. 1.5.2.2 Formulas Based on Empirical Relations Formulas based on empirical relations are the following: Meyerhof (1963): N Nq fγ φ= −( ) ( )1 1 4 22tan . ( ) Muhs and Weiss (1969) and Muhs (1971) adapted by Eurocode 7 (2005) and DIN 4017 (2006): Brinch Hansen (1970): Steenfelt (1977): Gudehus (1981): Ingra and Baecher (1983) for footings with L/B = 6: Ingra and Baecher (1983) for square footings: 1.5.2.3 Formulas Based on Analytical Derivations Formulas based on analytical derivations are the following: Vesic´ (1973): Chen (1975): Michalowski (1997) for a rough footing base: Zhu et al. (2001): 1.5.2.4 Formulas Based on Numerical Analyses There is one formula based on numerical analyses: Bolton and Lau (1993): N Nq fγ φ= −( ) ( )1 1 5 33tan . ( ) N Nq fγ φ= +( ) ( )2 1 1 07 32tan . ( ) N f fγ φ φ= +( )exp . . tan tan ( )0 66 5 11 31 N Nq fγ φ= +( ) +( )2 1 45 2 30tan ( ) N Nq fγ φ= +( )2 1 29tan ( ) N fγ φ= − +( )exp . . ( )2 046 0 173 28 N fγ φ= − +( )exp . . ( )1 646 0 173 27 N fγ φ= ( )( )−exp . tan ( ).5 19 1 261 5 N f fγ φ φ= + ( )− ( )(0 08705 0 3231 2 0 04836 22. . sin . sin ) ⎛⎝⎜ ⎞⎠⎟ −⎡⎣⎢ ⎤ ⎦⎥Nq fexp tan ( ) π φ 2 1 25 N Nq fγ φ= −( )1 5 1 24. tan ( ) N Nq fγ φ= −( )2 1 23tan ( ) 17 0 5 10 15 20 25 30 35 40 45 50 Friction angle, φf (deg) 0.01 0.1 1 10 100 1000 B ea rin g ca pa ci ty fa ct or , N γ Vesic (1973) Meyerhof (1963) Brinch Hansen (1970) Chen (1975) Ingra & Baecher (1983) L/B=6 Ingra & Baecher (1983) L/B=1 EC7 (2005) Michalowski (1997) Bolton & Lau (1993) Zhu et al. (2001) Gudehus (1981) Steenfelt (1977) Figure 11. Bearing capacity factor N versus friction angle (f ) according to different proposals.

1.5.3 General Bearing Capacity Formulation The basic equation by Terzaghi has been modified to ac- count for the effects of the shape of the footing, load inclina- tion, load eccentricity, and shear strength of the embedment depth on the ultimate bearing capacity. Some of these modi- fications were incorporated originally by Meyerhof (1953) and then further enhanced by Meyerhoff (1963), Brinch Hansen (1961, 1970), and Vesic´ (1973, 1975) to give what is known as the General Bearing Capacity Equation: where si = shape factors, ii = load inclination factors, di = depth factors, and B′ = is the effective (i.e., functional) width of the footing considering the effect of load eccentricity (see Equa- tion 35). Various approaches for the calculation of these factors including evaluation and critical review are presented in the following sections. 1.5.4 Eccentricity The effect of eccentric loading on the bearing capacity is usually accounted for via Meyerhof ’s (1953) effective area consideration. Bearing capacity is calculated for the footings’ effective dimensions by the following: where M, MB and ML = the moments loading in L and B direc- tions, respectively; V = the total vertical load; and eL and eB = load eccentricities along footing length (L) and footing width (B), respectively. In contrast, other approaches describe the decrease in the bearing capacity with the increase in the eccentricity of the load using reduction factors. These factors indicate the ratio of the average ultimate bearing capacity under eccentric load- ing, qu, to that under the centric vertical loading, qu,centric. The formulas are mostly based on small-scale model tests on cohe- sionless soils without embedment, i.e., embedment depth of the foundation (Df) = 0 and c = 0. Some approaches are spec- ified below, and their evaluations are presented in Figure 12. The approaches are the following: ′ = − ′ = − = = L L e B B e e M V e M V L B B L L B 2 2 35 i i with and ( ) q cN s d i qN s d i B N s d iu c c c c q q q q= + +( ) ′1 2 34γ γ γ γ γ ( ) Meyerhof (1953): Giraudet (1965): Ticof (1977): Bowles (1996): Paolucci and Pecker (1997): Ingra and Baecher (1983): q q e B e B u u centric, . . (= − ⎛⎝⎜ ⎞⎠⎟ + ⎛⎝⎜ ⎞⎠⎟1 3 5 3 03 41 2 ) q q e B e B u u centric, . . . ( )= − ⎛⎝⎜ ⎞⎠⎟ <1 0 5 0 3 40 1 8 for q q e B e B u u centric, . ( )= − < <1 0 0 3 39for q q e B u u centric, . ( )= − ⎛⎝⎜ ⎞⎠⎟1 1 9 38 2 q q e B u u centric, exp ( )= − ⎛⎝⎜ ⎞⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟12 37 2 q q e B u u centric, ( )= − ⎛⎝⎜ ⎞⎠⎟1 2 36 2 18 0 0.1 0.2 0.3 0.4 0.5 Load eccentricity to footing width ratio, e/B 0 0.2 0.4 0.6 0.8 1 q u / q u , ce n tr ic Meyerhof (1953) Giraudet (1965) Ticof (1977) Ingra & Baecher (1983) Bowles (1996) Paolucci & Pecker (1997) Gottardi & Butterfield (1993) Perau (1995) Figure 12. Reduction factors for shallow foundations under vertical-eccentric load.

Gottardi and Butterfield (1993): Perau (1995, 1997): Figure 12 presents the ratio of eccentric to centric load capac- ity versus the ratio of load eccentricity to the smaller foot- ing width (B) of a strip footing. From the figure, it can be seen that the influence of load eccentricity in the approaches of Meyerhof (1953), Ticof (1977), and Ingra and Baecher (1983) is very similar. The curve according to Bowles (1996) shows a different progression, beyond an eccentricity of e/B = 0.1. Here, the decrease of the bearing capacity is less pronounced as com- pared to the three aforementioned approaches. In contrast, the approach by Giraudet (1965) shows a completely different progression and a much smaller reduction of bearing capacity for smaller load eccentricities. One cannot derive conclusions regarding the validity of the different approaches based on this figure alone. For example, it seems that Meyerhof’s (1953) approach leads to a greater bearing capacity; however, this is not entirely so. The change in the shape factors because of the change in the footing size, as effective width and effective length, must be considered as well. Figure 13 shows some of the reviewed approaches together with experimental results cited by Perau (1995). It can be seen that the three selected equations (Meyerhof, 1953; Ticof, 1977; and Ingra and Baecher, 1983) represent a lower boundary of the experimental results. 1.5.5 Shape Factors The effect of a foundation shape other than a strip footing (plain strain condition) has to be considered with foundation shape factors. A footing is theoretically defined as a strip foot- ing for the length to width ratios of L/B > 10. Practically, foun- dations possessing the ratio of L/B > 5 already behave as strip footings (Vesic, 1975). Due to the difficulties in obtaining mathematical solutions that consider the effect of a founda- tion shape, semi-empirical approaches have been formulated. Various relations proposed for shape factors, si, are listed in Table 4. For eccentrically loaded footings, the effective foot- ing dimensions B′ and L′ have to be used to compute the shape factors (e.g., AASHTO, 2007; EC 7, 2005). The presented shape factors in Table 4 are empirical except for the expressions by Zhu and Michalowski (2005) that have been derived from numerical simulations. For example, to determine the shape factor, sγ, footings with different length to q q e B u u centric, . ( )= −1 2 5 43 q q e B u u centric, . ( )= −1 0 36 42 width (L/B) ratios under centric vertical loading and without embedment have been modeled and analyzed. Figures 14 and 15 present the numerical values of the afore- mentioned shape factors sγ and sq, respectively, versus the foundation width to length ratios, B/L. Due to the fact that the bearing capacity of Equation 19 was developed for strip footings assuming plain strain conditions, the values of the shape factors approach unity for long footings (as B/L → 0). Practically, the value of sγ is within the range of 1 ± 0.05 for L/B ≥ 6.7 (B/L ≤ 0.15), and the value of sq is within the same range for L/B ≥ 10.0 (B/L ≤ 0.10) for most cases. For footings with dimension ratios close to unity (approach- ing equidimension), the deviations of the shape factors from the unity proposed by different authors show that very careful consideration is required in the choice of the shape factors. The values suggested by Meyerhof (1963) for sγ (see Figure 14) are always greater than unity and increase with the decrease in the width to length ratio (B/L). In contrast, the values cal- culated with other equations decrease below unity as the ratio increases. The reason for this is that Meyerhof’s (1963) val- ues of Nγ for a strip footing (B/L → 0) are smaller than those for a circle (B/L = 1), and the bearing capacities for the footing 19 0 0.1 0.2 0.3 0.4 0.5 Load eccentricity to footing width ratio, e/B 0 0.2 0.4 0.6 0.8 1 q u / q u , ce n tr ic Meyerhof (1953) Ticof (1977) Ingra & Baecher (1983) Perau (1995) Experimental results (n=61) Figure 13. Reduction factors for shallow foundations under vertical-eccentric load compared with test results from different authors as presented by Perau (1995). The experimental results presented are from Ramelot and Vanderperre (1950) as cited by Döerken (1969) for B/L = 1; Meyerhof (1953) for B/L = 1, 1/6, and 6; Schultze (1952) for B/L = 2; Das (1981) for B/L = 1/3; Giraudet (1965) for B/L = 1/3.5; and Eastwood (1955) for B/L = 1/1.8, 1/2.25, and 1/3.

with width to length ratios between B/L → 0 and B/L = 1 are linearly interpolated values. Hence, a consistent set of equa- tions for the bearing capacity factors and their modifications by the same author are recommended for use in the bearing capacity calculation. In summary, the foundation shape (vary- ing between strip to equidimensional footing) and hence, the shape factor have an important influence on the ultimate bear- ing capacity. 1.5.6 Depth Factors If the foundation is placed with a certain embedment depth, Df, below the ground surface, the bearing capacity is affected in two ways: one, by the overburden pressure, q = γ  Df, and two, via the shear strength of the soil above the base level. Table 5 presents typically used expressions of the depth factors. Figure 16 presents the values of the depth factor dq versus the friction angle for the different expressions pro- vided in Table 5. In contrast to the factors proposed by Meyerhof (1963), the depth factor dq according to Brinch Hansen (1970) decreases with the increase in the soil friction angle. The depth factors proposed by Brinch Hansen are greater than those proposed by Meyerhof. The depth factors listed in AASHTO (2007) are also shown in Figure 16. These values lie between the expressions proposed by Meyerhof and Brinch Hansen. 1.5.7 Load Inclination Factors An inclination in the applied load always results in a reduced bearing capacity, often of a considerable magnitude (Brinch Hansen, 1970). Meyerhof (1953) suggested that the vertical component of the bearing capacity under a load inclined at an angle α to the vertical is obtained using the following inclina- tion factors: These expressions were modified by Meyerhof and Koumoto (1987) and presented for cases of footings on a sand surface, i fγ α φ= −( )1 452 ( ) i ic q= = − °( )1 90 442α ( ) 20 Table 4. Shape factors proposed by different authors. ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ Reference Footing base shape sc sq s Rectangle 1 q c NB L N 1 tan f B L 1 0.4 B L D e B ee r ( 19 70 ) a s m o di fie d by V es ic (19 73 ) Circle and Square 1 q c N N 1 tan f 0.6 Rectangle 1 1 q q q s N N 1 sin f B L 1 0.3 B L EC 7 (2 00 5) an d D IN 4 01 7 (20 06 ) Circle and Square 1 1 q q q s N N 1 sin f 0.7 M ey er ho f ( 19 63 ) Rectangle 1 0.1 p B K L 1 ; for 0 1 0.1 ( / ); for 10 f p f K B L 2 1 0.1 ; tan 45 2 p f p B K L K Pe ra u (19 95 , 1 99 7) Rectangle Not applicable 2 1 1.6 tan / 1 f B L B L 1 1 B L Zh u an d M ic ha lo w sk i (20 05 ) Rectangle Not applicable Not applicable 2 2 1.5 1 (0.6sin 0.25) / for 30 ; 1 (1.3sin 0.5)( / ) exp( / ) for 30 f f f f B L L B L B

21 0 0.2 0.4 0.6 0.8 1 Footing side ratio, B/L 0.2 0.4 0.6 0.8 1 1.2 Sh ap e fa ct or , s γ Vesic (1973) Din 4017 (2006) Perau (1995) Meyerhof (1963) φf=25° Meyerhof (1963) φf=37° Zhu and Michalowski (2005) φf=25° Zhu and Michalowski (2005) φf=37° Figure 14. Shape factor s proposed by different authors versus footing side ratio, B/L. 0 0.2 0.4 0.6 0.8 1 Footing side ratio, B/L 1 1.2 1.4 1.6 1.8 2 Sh ap e fa ct or , s q Vesic (1973) φf=25° Vesic (1973) φf=37° DIN 4017 (2006) φf=25° DIN 4017 (2006) φf=37° Perau (1995) φf=25° Perau (1995) φf=37° Figure 15. Shape factor sq proposed by different authors versus footing side ratio, B/L.

when the embedment ratio (Df /B) is unity, and for footings on a clay surface, as shown in Equations 46 through 48. Assum- ing that a footing with a perfectly rough base on a sand surface starts to slide when the load inclination angle to the vertical is approximately equal to the soil’s friction angle, the follow- ing expression was proposed: For the case of footings with an embedment ratio equal to 1 in a soil with a friction angle greater than 30°, the inclina- tion factor was expressed as the following: i D B cf fγ α α φ= −( ) > ′ = =cos sin , ( )1 30 1 0 47for ,° i D B c f fγ α α φ= − ⎛ ⎝⎜ ⎞ ⎠⎟ ′ = =cos sin sin , ( )1 0 0 46for For footings on the surface of clay: where ca = adhesion between the clay and the base of the footing. Muhs and Weiss (1969) suggested, based on DEGEBO (Deutsche Forschungsgesellschaft für Bodenmechanik) tests with large-scale models of shallow footings on sands, that there is a distinct difference between load inclination effects when the inclination is in the direction of the longer side, L, and when the inclination is in the direction of the shorter side, B. Thus, the direction of load inclination as well as the i caγ α α α α = −( ) = = −( cos sin cos . sin 1 0 1 0 81 for ) = =for undrained shear strength of the c ca n clay ( )48 22 Table 5. Depth factors proposed by different authors. Author dc dq d Meyerhof (1963) 1 0.2 f c p D d K B 1 0.1 for 10 1 for 0 f q p f f D d K B qd d 2 / 1: 1 2 tan 1 sin / f q f f f D B d D B Brinch Hansen (1970) and Vesic (1973) 1 tan 1 1 q c q C f q q q d d d N d d N 2 / 1: 1 2 tan 1 sin tan / f q f f f D B d arc D B 1 where 2tan (45 / 2)p fK ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 28 30 32 34 36 38 40 42 44 Soil friction angle, φf (deg) 1 1.1 1.2 1.3 1.4 D ep th fa ct or , d q Brinch Hansen (1970) Df/B=1 Brinch Hansen (1970) Df/B=2 Brinch Hansen (1970) Df/B=4 Meyerhof (1963) Df/B=1 Meyerhof (1963) Df/B=2 Meyerhof (1963) Df/B=4 AASHTO (2007) Df/B=1 AASHTO (2007) Df/B=2 AASHTO (2007) Df/B=4 Figure 16. Depth factor dq proposed by different sources versus soil friction angle, f.

ratio B/L affect the inclination factor. Brinch Hansen (1970) incorporated the inclination effects as Vesic (1975) proposed the factors in the following forms: where H and V = the horizontal and vertical components of the applied inclined load, P (see Figure 17); θ = the projected direction of the load in the plane of the footing, measured from the side of length L in degrees; A′ = the effective area of the footing; c = soil cohesion; and L′ and B′ are as defined in Equation 35. Figures 18 and 19 are graphical presentations of Equa- tions 49 through 53 for load inclination factors iq and iγ, respectively. n L B L B B L B = + ′ ′( ) + ′ ′( ) ⎡ ⎣⎢ ⎤ ⎦⎥ + + ′ ′( ) + ′ ′ 2 1 2 1 2cos θ L( ) ⎡ ⎣⎢ ⎤ ⎦⎥sin ( ) 2 53θ i H V A c f n γ φ= − + ′( ) ⎛ ⎝⎜ ⎞ ⎠⎟ + 1 52 1 cot ( ) i H V A c q f n = − + ′( ) ⎛ ⎝⎜ ⎞ ⎠⎟1 51cot ( )φ i H V A c f γ φ= − + ′( ) ⎛ ⎝⎜ ⎞ ⎠⎟1 0 7 50 5 . cot ( ) i H V A c q f = − + ′( ) ⎛ ⎝⎜ ⎞ ⎠⎟1 0 5 49 5 . cot ( )φ The inclination factor ic results from Caquot’s theorem of corresponding stress states (De Beer and Ladanyi 1961 and Vesic´ 1973 as cited by Vesic´ 1975): where iq is given by Equation 51. i i i N i i N i c q q c f q q q f= − − = − − − > 1 1 1 0 tan ( )φ φfor 54a c c f nH A c N = − ′ =1 0for 54bφ ( ) 23 P H θ V L B Figure 17. Inclined load without eccentricity and the projected direction, . 0 0.2 0.4 0.6 0.8 1 Load ratio, H/V 0 0.2 0.4 0.6 0.8 1 Lo ad in cl in at io n fa ct or , i q Brinch Hansen (1970) Meyerhof (1953) φf=35° Vesic (1975) L/B =1 (θ=90) Vesic (1975) L/B =5 (θ=90) Figure 18. Load inclination factor iq versus load ratio, H/V, for c = 0, f = 35°, and any Df /B. 0 0.2 0.4 0.6 0.8 1 Load ratio, H/V 0 0.2 0.4 0.6 0.8 1 Lo ad in cl in at io n fa ct or , i γ Brinch Hansen (1970) Meyerhof (1953) φf=35° Meyerhof & Koumoto (1987) Df/B=0, φf=35° Vesic (1975) L/B =1 (θ=90) Vesic (1975) L/B =5 (θ=90) Figure 19. Load inclination factor i versus load ratio, H/V, for c = 0, f = 35°, and Df /B = 0.

Reduction factors for the case of a load inclination related to the case of centrically and vertically loaded footings can be found in Ticof (1977), Ingra and Baecher (1983), and Gottardi and Butterfield (1993) (see Figure 20). These expressions were determined based on model foundation test results on sand without embedment and as such are valid for the case of Df = 0, c = 0: Ticof (1977): Ingra and Baecher (1983): Gottardi and Butterfield (1993): 1.6 An Alternative Approach and Method of Analysis for Limit State Design of Shallow Foundations 1.6.1 Some Aspects of Stability and Safety of Shallow Foundations 1.6.1.1 Bearing Capacity and Sliding Limit States Geotechnical resistances such as the bearing capacity of shal- low foundations are entirely load dependent. The application q q H V u u centric, 57= −1 0 48. ( ) i q q H V H V u u centric, 5= − ⎛⎝⎜ ⎞⎠⎟ + ⎛⎝⎜ ⎞⎠⎟1 2 41 1 36 2 . . ( 6) q q H V u u centric, 55= − ⎛⎝⎜ ⎞⎠⎟1 1 36 2 . ( ) of LRFD in cases of foundations under complex loading is, therefore, difficult as there is no strict separation between load and resistance. Furthermore, it is not always clear whether a load should be classified as favorable or unfavorable, which may have consequences for the calibration of safety factors. The difficulty in classification applies especially to the influ- ence of the vertical load on the bearing capacity. In order to illustrate this problem, Figure 21a shows the bearing capacity limit state and sliding limit state of a shallow foundation under inclined loading as a function of vertical and horizontal loads. In this so-called interaction diagram, the sliding limit state is illustrated as a simple straight line with an inclination tan δS representing the soil foundation interfacial shear resistance accounting for the roughness of the footing’s base. The bearing capacity limit state is a closed curve in this illustration. The interaction diagram depicts the well-known physical phenomenon that the occurrence of hor- izontal loads reduces the bearing capacity of a shallow foun- dation, which is described by the inclination factors used in the traditional bearing capacity equation. A similar diagram can be generated for eccentric vertical loading or in the three- dimensional space for eccentric and inclined loading. As the inclination factors depend on the characteristic load inclination Hk/Vk, the bearing capacity calculation implies a radial load path, which is the same for loading and resistance as indicated in Figure 21(a). However, only the vertical com- ponents of load and resistance are compared within the proof of stability. On the other hand, the sliding resistance calcula- tion is based on the assumption of a step-like load path. For a given vertical load, the associated horizontal resistance is calculated, which itself is compared to the horizontal load component. The distances between design loads (Hd or Vd, respectively) and design resistances (Rh,d or Rv,d, respectively) in Figure 21(a) represent the actual degree of mobilization. In Figure 21(b), bearing capacity limit state and sliding limit state are referred to the maximum vertical resistance, Vmax (i.e., under centric vertical loading only). Hence, the diagram shows the pure interaction of the load components without any other influences on the bearing capacity. In this illustration it is shown that a maximum horizontal load, Hmax, can be applied for V/Vmax ≈ 0.42. Let us now consider a certain horizontal load, H < Hmax. For this case, a mini- mum vertical load (min V) is required to carry the horizon- tal load. This means the load inclination is limited and the limit is provided either by the bearing capacity limit state or by the sliding limit state, whichever is more restrictive. With increasing vertical load, the resultant load inclination de- creases and, hence, the bearing capacity of the system in- creases. However, because of the convex shape of the bear- ing capacity limit state, the degree of mobilization increases if V/Vmax > 0.42, so the magnitude of the applicable vertical load is limited as well (max V). 24 0 0.2 0.4 0.6 0.8 Load ratio, H/V 0 0.2 0.4 0.6 0.8 1 q u / q u , ce n tr ic Ticof (1977) Ingra & Baecher (1983) Gottardi & Butterfield (1993) Figure 20. Reduction factors for shallow foundations under inclined loading (c = 0, Df = 0).

1.6.1.2 Favorable and Unfavorable Load Actions Now consider a given vertical load, e.g., the foundation dead load, VG. In the ULS (i.e., the condition in which the bearing capacity is fully mobilized), this load is associated with one spe- cific horizontal load. A larger horizontal load can only be ap- plied if the vertical load is increased simultaneously, e.g., by increasing the dead weight applied to the footing. The vertical load acts favorably because an increase in the vertical load re- sults in the possible increase of the horizontal load. These rela- tionships are, however, valid only for V/Vmax < 0.42. Larger vertical loads (VG + ΔVG) act unfavorably because they reduce the maximum allowable horizontal load. In this situation, an ar- bitrary increase in the dead load applied to the footing would be counterproductive because it does not help to improve the per- formance of the system in resisting horizontal loads. These complex interrelations demonstrate that the role of the vertical load component is not unique. Hence, within the standard design procedure it is difficult to classify the vertical load as a favorable or unfavorable load. The use of the presented simple interaction diagrams may help, however, to better understand the complex interaction of the load components (Lesny, 2006). 1.6.1.3 Example The favorable and unfavorable actions may affect the safety of the system as demonstrated by the following example of a vertical breakwater (Lesny and Kisse, 2004; Lesny, 2006). The breakwater is a structure supported by a strip footing of width BC founded on sand and subjected to vertical, horizontal, and moment loading (see Figure 22). The basic parameters of the system are (Lesny et al., 2000; Oumeraci et al., 2001) Caisson: Bc = 17.5 m, hc = 23 m Crushed stone layer: φf = 44.2°, γ ′(effective unit weight) = 10.4 kN/m3, tanδs = 0.5 Subsoil: φf = 38.2°, γ ′= 10.2 kN/m3 Water depth at still water level: hs = 15.5 m 25 0 Rk,hd,h /RR )V(R kk,hkH dH kV dV Rk,vd,v /RR )V/H(R kkk,v maxV V H s sliding resistance bearing resistance 0 V favorable V unfavorable 1H 2H maxH 1Vmin 2Vmin 1Vmax 2Vmax maxV/V GV ~0.42 GG VV maxV/H 1 (a) Definition of load paths (b) Effect of vertical load on resistance sliding resistance bearing resistanceradial load path steplike load path Notes: H resultant horizontal load Hmax max. horizontal load that can be carried by the system Hd design value of horizontal load Hk/Vk load inclination (characteristic values) Rv,d design value of bearing capacity Rv,k characteristic value of bearing resistance (capacity) Rh,d design value of sliding resistance Rh,k characteristic sliding resistance R resistance factor s base friction angle V resultant vertical load min V min. vertical load required for a certain applied horizontal load max V max. vertical load possible for certain applied horizontal load VG permanent vertical load VG additional permanent vertical load Vk characteristic vertical load Vd design value of vertical load Vmax bearing capacity under pure vertical loading, i.e., max. vertical load that can be carried by the system V/Vmax degree of utilization of maximum vertical load Figure 21. Influence of load components on bearing resistance and sliding resistance utilizing interaction diagram.

Figure 23 depicts the bearing capacity limit state and the sliding limit state of the breakwater for a fixed eccentricity of ek/B = 0.12 in the V-H plane. We assume a fictitious characteristic loading mainly due to dead weight and wave loading of HQ,k (horizontal fictitious characteristic loading due to dead weight) = 2.55 MN VG,k (vertical fictitious characteristic loading due to wave) = 15 MN The factored design loads below were developed assum- ing vertical and horizontal load factors of γG = 1.35 and 1.00 for unfavorable and favorable permanent action, respectively, and γQ = 1.50 and 1.0 for unfavorable and favorable variable action, respectively. The factor γG is applied to the vertical loads only, and the factor γQ is applied to the horizontal loads. The horizontal and vertical factored design loads are the following: HQ,d = 3.82 MN VG,d = 15 MN (V favorable) VG,d = 20.3 MN (V unfavorable) The safety of the system may be expressed here by the avail- able resistance factor resulting from the characteristic resis- tance divided by the associated design load: γR = Rk/Ld. Hence, the safety for the sliding limit state is γR,h = Rh,k/HQ,d = 2.0. For the bearing capacity limit state, the safety is γR,v = Rv,k/VG,d = 2.7 if V is favorable, but only γR,v = 2.0 if V is unfavorable. Under both conditions, the safety of the system seems to be sufficient. These results do not represent, however, the actual safety of the foundation. In the interaction diagram of Figure 23, the actual safety is described by the closest distance of the loading to the resistance of the foundation as indicated by the arrows. Additional load components acting along this path are most hazardous. If arbitrary load paths are possible, only additional load components acting within the circles sketched in Fig- ure 23 are admissible. Such a critical load situation is not arti- ficial; it may occur if the wave height is higher than assumed for design, resulting in an increase of the horizontal load along with a decrease of the vertical load due to uplift forces. The actual safety can be determined with the help of the fac- tored design load vector → Q = [VG,d; HQ,d] and the additional load vector Δ → Q in the V-H plane, which coincides with the radius of the circles in Figure 23 (Butterfield, 1993). For the design load components given above ( → Q) the maximum additional load- ing is limited by the sliding limit state and amounts to ΔQ = 3.30 MN (V favorable) or ΔQ = 5.68 MN (V unfavorable), re- spectively. Thus, the actual safety of the system is the following: The actual safety in both cases is considerably smaller than the one calculated previously using the regular design proce- γ R Q Q Q V V = +( ) =Δ 1 21 1 28 . . favorable unfavorable{ ( )58 26 Ch rh sh h CB Ce )t(S )t(Fu SWLwave trough wave crest )t(Fh subsoil uplift rotation failure sliding along the base crushed rubble mound bearing resistance failure in subsoil bearing resistance failure in rubble mound GF Notes: hs water depth at still water level (SWL) hr height of rubble mound h* height of wave crest hC height of breakwater BC width of breakwater eC eccentricity of dead weight FG of breakwater Fh(t) horizontal wave load Fu(t) resultant of uplift water pressure due to wave loading S(t) seepage force in rubble mound Figure 22. Breakwater, wave loading, and failure modes.

dure. However, the safety for V when assumed to be unfavor- able is greater than when V is favorable, as indicated also by the longer arrow in Figure 23. Not only is this result contra- dictory to the result of the regular safety calculation, but it is also inconsistent with the classification of V as unfavorable to begin with because this load actually improved the safety of the system. The reason for these inconsistencies can be found in the convex shape of the resultant resistance. As a consequence, the safety of the system depends on the load path. This may be crit- ical for design situations with large variable loads, especially if the vertical load is small. 1.6.1.4 Conclusions and Alternative Solution The example given in Section 1.6.1.3 clearly demonstrates that the assumption of certain load paths within traditional design procedures may lead to a misinterpretation of the safety level. Hence, for the calibration of resistance factors, possi- ble load combinations and the associated load paths have to be identified in advance for evaluation of their significance to the bearing capacity. For this purpose, the use of an interaction diagram for visualization and better understanding is helpful and may be necessary. This problem can also be solved with an alternative design method, which directly considers the interaction of the differ- ent load components without assuming specific load paths. This method is based on a consistent definition of the ULS of a shallow foundation by a unique limit state equation without the need for distinguishing between different failure modes. This model can also be extended to analyze the deformations of the foundation within the SLS. Such a model is introduced in the following section. 1.6.1.5 Note Concerning References of Related Work The concept of an interaction diagram to describe the ULS of a shallow foundation was introduced by Butterfield and Ticof (1979). This concept was later utilized by Nova and Montrasio (1991), Montrasio and Nova (1997), Gottardi and Butterfield (1993, 1995), Martin and Houlsby (2000, 2001), and others. However, this work focused on the calculation of displacements and rotations dealing essentially with forces and moments act- ing on a single plane (one-way, inclined-eccentric loading). As a result, the failure condition played a minor role and was established by a pure curve fitting only. Work on arbitrary loading conditions (two-way lateral, eccentric, and torsional loading) was first developed by Lesny (2001) with the result- ing influence parameters related to physical factors rather than curve fitting (see also Lesny and Richwien, 2002, and Lesny et al. 2002). Lesny used earlier experimental work conducted 27 0 10000 20000 30000 40000 50000 60000 70000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 V [k N ] H [kN] fictitious characteristic loading fictitious design loading (V favorable) fictitious design loading (V unfavorable) bearing resistance Rv,k sliding resistance Rh,k safety: γR,v = Rv,k/Vd = 2.7 safety: γR,h = Rh,k/Hd = 2.0 Notes: H resultant horizontal load Hd design value of resultant horizontal load Rh,k characteristic value of sliding resistance Rv,k characteristic value of bearing resistance V resultant vertical load V γ γ d design value of resultant vertical load R,h available resistance factor for sliding R,v available resistance factor for bearing resistance safety: γR,v = Rv,k/Vd = 2.0 Figure 23. Interaction diagram for stability analysis of a vertical breakwater.

by Perau (1995, 1997) at the Institute of Soil Mechanics and Foundation Engineering at the University of Duisburg-Essen (UDE), Essen, Germany. Recently, Byrne and Houlsby (2005) and Bienen et al. (2006) presented experimental work on shal- low foundations on sand under arbitrary loading condi- tions as well. In this work, the failure or yield condition is defined by so-called swipe tests, in which the load path followed closely the failure or yield surface in the interaction diagram. However, the use of these data for the research project reported on herein (NCHRP Project 24-31) may be limited as the tests remain close to but below failure. In other words, failure loads for definite loading conditions are not directly available. 1.6.2 Alternative Design Method for Shallow Foundations 1.6.2.1 Overview The alternative design method includes two components. The first component is a failure condition that describes the ULS of a shallow foundation without the need to distinguish between different failure modes. The second component is a displacement rule that reflects the complete load-displacement relation within the SLS before the system reaches its ULS. The failure condition can be used independently of the dis- placement rule and may be combined with other methods for settlement analysis. It has been developed for foundations on granular soils with and without embedment, whereas the dis- placement rule is currently developed for foundations with- out embedment only. Please note that in the general definition of the failure condition and the displacement rule the notation of the load components is different from the notation used pre- viously. An in-depth discussion of the subject and the normal- ization concept validation via small-scale testing is presented in Appendix A. For more information, refer to Kisse and Lesny (2007) and Kisse (2008). 1.6.2.2 Failure Condition In the general case, a single footing is loaded by a verti- cal load, F1, horizontal load components F2 and F3, a torsional moment, M1, and bending moment components, M2 and M3 (see Figure 24). The load components are summarized in the load vector → Q: For the basic case of a footing on non-cohesive soil without embedment, the geometry of the footing described by the side ratio (b – = b2/b3), weight (γ), the soil’s shear strength (tan φf), and a quantity describing the roughness of the footing base (μs) have to be considered as well (see Figure 24). With these input parameters, the failure condition of the general form is defined by the following expression: In Equation 61, all load components are referred to as F10, which is the resistance of a footing under pure vertical loading. This quantity is calculated using the traditional bearing capac- F F a F M a b b F M M a 2 2 3 2 1 10 2 1 2 2 2 3 10 2 2 2 3 2+ ( ) + +( )( ) + + 3 2 10 2 1 10 1 10 1 0 b F F F F F ( ) − − ⎛⎝⎜ ⎞⎠⎟ = α ( )61 F Q b f s  , , , , 60γ φ μtan ( )( ) = 0  Q F F F M M MT = [ ]1 2 3 1 2 3 ( )59 28 3b 1x 2F 3M 1M 2M1 F 2x 3F 2b d 3x g γ, φf Notes: b2, b3 length of the footing referred to x2-, x3- axis d embedment depth F1 vertical load F2, F3 horizontal load (referred to x2-, x3- axis) M1, M2, M φf γ 3 torsional and bending moments (referred to x1-, x2-, x3- axis) unit weight of soil angle of internal friction Figure 24. Geometry and loading.

ity formulae. The advantage of the formulation described in Equation 61 is that the complex interaction of the load com- ponents is considered directly without using reduction fac- tors or the concept of the effective foundation area. Other influences on the bearing capacity are included in F10. It should be noted that as F10 is the bearing capacity under vertical- centric loading only, the uncertainties of the calculation method are reduced to the bearing capacity factors and the shape and depth factors (if required) of the traditional bearing capacity calculations. Thus, no inclination factors or use of effective area are necessary. The use of such factors and the concept of effec- tive area were the cause for difficulties in establishing the degree of conservatism and hence a source of ambivalent application of LRFD facing the existing AASHTO 2008 specifications. In an interaction diagram like the ones in Figures 21 or 23, the failure condition spans a failure surface, which is the outer boundary of the admissible loading. The parameters a1,2,3 gov- ern the inclination of this failure surface for small vertical load- ing where the limit states of sliding and the restriction of the eccentricity to 1⁄3 of the foundation width have previously been relevant (see Figure 25). These limit states are integrated by defining the parameters a1,2,3 and α according to Equation 62: The limit state uplift is already included in Equation 61 because only positive vertical loads are admissible. The param- eters provided in Equation 62 have been derived from an analysis of numerous small-scale model tests conducted at the Institute of Soil Mechanics and Foundation Engineering at UDE. Figures 26 and 27 show the failure condition compared with the model test results for various load combinations. a e a a s f f 1 3 2 3 2 0 098 0 4 = ( ) ( ) = = −( )π μ φ π φtan . . tan , , 2 1 3, 62α = . ( ) In the case of footings embedded in the soil, the failure condition according to Equation 61 needs to be extended if the shearing resistance in the embedment zone is taken into account: In Equation 63b, F1,min is the bearing capacity due to pure vertical tension loading resulting from the shearing resistance in the embedment zone, which may be carefully calculated using an earth pressure model. F10 can be determined using the traditional bearing capacity equation taking into account depth factors provided by Brinch Hansen (1970). The increas- ing capacity for horizontal and moment loading is considered by the parameters ai according to Equation 64, which requires additional verification at this stage: a eS k f f 1 32 1 2 2 = ( ) ( )( ) + −( ) −( )π μ φ π φ π φ , tantan tan f f d be a a ( ) −( )⎡⎣⎢ ⎤⎦⎥ = = −( )1 0 098 0 2 2 3 tan . ( ) . φ 64 42 1 0 5 1 1 3 2 + − −( )( )⎡⎣⎢ ⎤⎦⎥ = . exp . π α d b with 63b , , f F F F F z = − 1 10 1 101 min min ( ) F F a F M a b b F M M a 2 2 3 2 1 10 2 1 2 2 2 3 10 2 2 2 3 2+ ( ) + +( )( ) + + 3 2 10 2 1 10 1 1 1 1 1 b F f F F f f F F z z z ( ) − +( ) −⎡⎣⎢ ⎤ ⎦⎥ − +( ) 0 0− ⎛⎝⎜ ⎞⎠⎟⎡⎣⎢ ⎤ ⎦⎥ =fz α ( )63a 29 or ψ δS ψ δS 2b/e.admψ = arctan 232 bMorF232 bMorF F1 uplift bearing resistance sliding resistance or restriction of eccentricity 232 bMorF232 bMorF F1 admissible: F < 0 ultimate limit state: F = 0 Notes: b2 length of the footing referred to x2- axis F failure condition F1 vertical load F2 horizontal load (referred to x2- axis) M3 bending moment (referred to x3- axis) base friction angle angle defined by allowable eccentricity 2 arctan b eadm , adm e usually is b2/3 Figure 25. Isolated limit states (left) and failure condition (right).

30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00 0.04 0.08 0.12 0.16 0.20 0,24 0,28 )1995,Perau ;1986,Nacke&Nendza(33.0b )1993,Haubrichs(5.0b )1995,Perau(1b )1993,Haubrichs(2b )1995,Perau(3b )1995,Perau(5b (Eq. 61)conditionfailure10 1 F F 102 FF 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00 0.04 0.08 0.12 0.16 0.20 33.0b 1b 3b 5b (Eq. 61)conditionfailure 10 1 F F 103 FM 2b e .adm Notes: F1 vertical load F2 horizontal load (referred to x2- axis) b b2/b3 side ratio F10 bearing capacity under pure vertical loading, i.e., maximum vertical load that can be carried by the system M3 bending moment Figure 26. Failure condition for inclined loading (left) and eccentric loading (right) versus failure loads from small-scale model tests. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.00 0.04 0.08 0.12 0.16 0.20 0,24 0,28 0,32 0,36 0,40 )1995,Perau(1b,MMFF 2121 )1995,Perau(1b,MMFF 3121 )1993,Haubrichs(2b,MMFF 3131 )1993,Haubrichs(2b,MMFF 2131 )1995,Perau(1b,MMMFF 32121 condition (Eq. 61)failure 10 1 F F 2 1023 2 3 2 2 2 10322 2 1 2 101 2 3 2 2 Fba MM Fbba M Fa FF Notes: b2, b3 length of the footing (referred to x2-, x3- axis) b b2/b3 side ratio F1 vertical load F2, F3 horizontal load (referred to x2-, x3- axis) F10 bearing capacity under pure vertical loading, i.e., max. vertical load that can be carried by the system M1, M2, M3 torsional and bending moments (referred to x1-, x2-, x3- axis) a1, a2, a3 parameters of failure condition according to (Eq. 61) Figure 27. Failure condition for general loading versus failure loads from small-scale model tests. where μS,k = value of characteristic roughness of the foundation base. 1.6.2.3 Displacement Rule The displacements and rotations of the foundation due to arbitrary loading inside the failure surface are described by the displacement rule. The displacements ui and rotations ωi are summarized in a displacement vector: Due to the complex interaction of load components, dis- placements, and rotations, the displacement rule is formulated using the well-known strain hardening plasticity theory with  u u u uT = [ ]1 2 3 1 2 3ω ω ω ( )65

isotropic hardening (e.g., Zienkiewicz, 2005). Hence, displace- ments and rotations are calculated according to Equation 66, assuming that all deformations are plastic: The components of the displacement rule are a yield sur- face described by the yield condition, F, which is derived from the failure condition equation (Equation 61): with the parameters a1,2,3 of Equation 62, a plastic potential, G: and a hardening function, H: In Equation 68, Fb is the hardening function and c1, c2, and c3 are the parameters of the plastic potential. The yield surface according to Equation 67 expands due to isotropic hardening until the failure surface defined by Equa- tion 61 is reached (see Figure 28). Thus, the parameters ci and H F F F u G Fa a = − ∂ ∂ ∂ ∂ ∂ ∂ i i 1 1 ( )69 G Q F F F c F M c b b F M b b b  ,( ) = +( ) + +( )( ) + 2 2 3 2 1 2 1 2 2 2 3 2 2 2 3 3 2 3 2 3 2 2 1 11 c b F M c b F F F F F b b b b ( ) + ( ) − − ⎛⎝⎜ ⎞⎠⎟ β⎡ ⎣⎢ ⎤ ⎦⎥ = 2 0 ( )68 F Q F F F a F M a b b F M a a a  ,( ) = +( ) + +( )( ) + 2 2 3 2 1 2 1 2 2 2 3 2 2 2 3 3 2 3 2 3 2 2 1 11 a b F M a b F F F F F a a a a a ( ) + ( ) − − ⎛⎝⎜ ⎞⎠⎟ ⎡ ⎣⎢ ⎤ ⎦⎥ = 2 0 ( )67 du H F Q G Q dQ T    = ∂ ∂ ⎛ ⎝⎜ ⎞ ⎠⎟ ∂ ∂ 1 ( )66 β in Equation 68 have to be determined as functions of ai and α, respectively. The expansion of the yield surface depends mainly on the vertical displacement, u1, which itself depends on the degree of mobilization of the maximum resistance, F10. Hence, it is sufficient to define the hardening parameter, Fa, in Equation 67 as a function of these two quantities accord- ing to the following: Many hardening laws (e.g., Nova and Montrasio, 1991) require small-scale model tests under centric vertical load- ing to determine the hardening parameter. Since this is not convenient for practical applications, the initial and final stiff- ness of the corresponding load-displacement curve, k0 and kf, respectively, may be determined using a method proposed by Mayne and Poulos (2001) in which the soil stiffness can be determined by any standard procedure. Figure 29 shows the results of the proposed model applied to the example breakwater of Figure 22. Safety factors are not applied here. On the left side of Figure 29 the failure condi- tion and the loading in the F1 − F2 plane and in the F1 − M3/BC plane are shown. Obviously, the stability of the breakwater is governed by the high horizontal loading. Only an increase in the vertical loading (i.e., of the breakwater weight) would lead to a sufficient safety. The right side of Figure 29 shows the ver- tical and horizontal displacements of the breakwater depend- ing on the corresponding load components, F1 and F2. How- ever, due to some conservative assumptions made in the current version of the proposed model, a breakwater width of 21.0 m instead of 17.5 m was required to reach stability. 1.6.2.4 Implementation of a Safety Concept To implement a safety concept for the ULS based on load and resistance factors, the bearing capacity and loading for the characteristic input parameters shall be considered first. The bearing capacity defined by the failure condition is qualita- tively shown again in the interaction diagram of Figure 30. Each load combination to be checked marks a point in the interaction diagram. Connecting all load points provides a polygon in the interaction diagram (see Figure 30). It can be shown that the corners of this polygon are represented by load combinations, which either consider live loads to the full extent or neglect them. Because of the convexity of the failure condi- tion, it is sufficient to prove only these load combinations. To get the design failure condition, Fd, the parameters ai in Equations 61 and 62 are divided by the required resis- tance factor γR,i . Additionally, a resistance factor also has to be adapted to F10. This procedure means that in practice the fail- ure surface shrinks as depicted in Figure 30. F F k u k u F k u a f f = +( ) − − + ⎛ ⎝⎜ ⎞ ⎠⎟ ⎧⎨⎩ ⎫⎬⎭10 1 0 1 10 1 1 exp ( )70 31 32 b/M 2F 1F yield surface failure surface Notes: b3 length of the footing (referred to x3-axis) F1 vertical load F2 horizontal load M2 bending moment (referred to x2-axis) Figure 28. Isotropic expansion of the yield surface in the loading space.

32 F2 [kN/m] F1 [kN/m] M3/BC kN/m Notes: Bc width of breakwater F1 vertical load F2 horizontal load (referred to x3-axis) M3 torsional and bending moments (referred to x1-, x2-, x3-axis) u1 vertical displacements (settlements) u2 horizontal displacements Figure 29. Failure condition (left) and load displacement curves (right) for the example breakwater. 101 F/F )Fb(M;F/F 1023102 combinations Lk characteristic load 0,...L,tan,bF fk combinations Ld design load 0,...,L,tan,bF Rfd Notes: b2 length of the footing (referred to x2-axis) b b2/b3 side ratio Fk characteristic failure condition Fd design failure condition F1 vertical load F2 horizontal load M3 bending moment F10 bearing capacity under pure vertical loading, i.e., maximum vertical load that can be carried by the system kL characteristic load combination dL design load combination R resistance factor f angle of internal friction φ φ γ φ γ Figure 30. Illustration of the safety concept principle. The resistance factors are no longer distinguished according to different limit states but according to the possible load inter- actions. So at least resistance factors for pure vertical loading, inclined loading, torsional loading, and eccentric loading may be defined: γR,pure vertical for F10 (pure vertical loading) γR,horizontal for a1 (inclined loading) γR,torsional for a2 (torsional loading) γR,eccentric for a3 (eccentric loading) The case of inclined and eccentric load combinations may result in a coupled interaction of the resistance factors. These cases, like other aspects of this concept, require fur- ther analysis. The application of load factors means that load components are reduced if they work favorably and are increased if they work unfavorably regarding the bear- ing capacity of the foundation (considering the aspects that were discussed earlier). This may cause displacements and distortions of the load polygon in the interaction diagram.

For simplicity, it is assumed here that the load polygon in Figure 30 expands evenly. Finally, the foundation stability is verified, if it can be shown that where → Ld is one design load combination of the set of all design load combinations, Ld, which need to be checked. If the inequal- ity (Equation 71) is fulfilled, all design load combinations are located inside the design failure surface. 1.7 Bearing Capacity of Shallow Foundations on Rock 1.7.1 Overview The bearing capacity of foundations founded on rock masses depends mostly on the ratio of joint spacing to foun- dation width, as well as intact and rock mass qualities like joint orientation, joint condition (open or closed), rock type, and intact and mass rock strengths. Failure modes may con- sist of a combination of modes, some of which include bear- ing capacity failure. Limited review of the bearing capacity of foundations on rock, as well as the relationships among bear- ing capacity mechanisms, unconfined compressive strength (qu), and other rock parameters is presented. Emphasis is placed on classifications and parameters already specified by AASHTO and methods of analysis utilized in this study for bearing capacity calibrations. Loads on foundation elements are limited by the structural strength, the ultimate (geotechnical) limit state (strength), and the load associated with the serviceability limit state. The relationships among these limits when applied to founda- tions on rock are often vastly different than when they are applied to shallow foundations on soil. For typical concrete strengths in use today, the strength of the concrete member is significantly less than the bearing capacity of many rock masses. The structural design of the foundation element will dictate, therefore, the minimum element size and, conse- quently, the maximum contact stress on the rock. In other loading conditions—such as intensely loaded pile tips, con- centrated loads of steel supports in tunnels, or the bearing capacity of highly fractured or softer homogeneous rocks (such as shale and sandstone)—the foundation’s geotechnical limit state (bearing capacity) can be critical. While settlement (i.e., serviceability) is often the limit that controls the design load of shallow foundations on soil, for many rocks the load re- quired to develop common acceptable settlement limits well exceeds the bearing capacity values. As such, both settlement ∀ ( ) < ∈   Ld Ld R i dF L. . . 71, , ( ),γ 0 and capacity are important to quantify for the design of shal- low foundations on both soil and rock. The research herein addresses, however, only the bearing capacity (i.e., the ULS of shallow foundations). 1.7.2 Failure Mechanisms of Foundations on Rock Failure of foundations on rock may occur as the result of one of several mechanisms, as shown in Figure 31 (Franklin and Dusseault, 1989). The failure modes are described by the Canadian Foundation Engineering Manual (Canadian Geo- technical Society, 2006) in the following way: 1. Bearing capacity failures occur when soil foundations are overloaded (see Figures 31a and b). Such failures, although uncommon, may occur beneath heavily loaded footings on weak clay shales. 2. Consolidation failures, common in weathered rocks, occur where the footing is placed within the weathered profile (see Figures 31c and e). In this case, unweathered rock core-stones are pushed downward under the footing load because of a combination of low shear strength along clay- coated lateral joints and voids or compressible fillings in the horizontal joints. 3. A punching failure (see Figure 31d) may occur where the foundation rock comprises a porous rock type, such as shale, tuff, and porous limestone (chalk). The mechanism includes elastic distortion of the solid framework between the voids and the crushing of the rock where it is locally highly stressed (Sowers and Sowers, 1970). Following such a failure, the grains are in much closer contact. Continued leaching and weathering will weaken these rock types, result- ing in further consolidation with time. 4. Slope failure may be induced by foundation loading of the ground surface adjacent to a depression or slope (see Fig- ure 31f). In this case, the stress induced by the foundation is sufficient to overcome the strength of the slope material. 5. Subsidence of the ground surface may result from collapse of strata undercut by subsurface voids. Such voids may be natural or induced by mining. Natural voids can be formed by solution weathering of gypsum or rock salt and are com- monly encountered in limestone terrain (see Figure 31g). When weathering is focused along intersecting vertical joints, a chimney-like opening called a pipe is formed, which may extend from the base of the soil overburden to a depth of many tens of meters. When pipes are covered by granular soils, the finer silt and sand components can wash downward into the pipes, leaving a coarse sand and gravel arch of limited stability, which may subsequently collapse (see Figure 31h). 33

1.7.3 Bearing Capacity Failure Mechanisms Out of the various aforementioned possible failures of foundations on rock, this research is focused on those as- sociated with bearing capacity mechanisms. The mecha- nism of potential failure in jointed rocks depends mostly on the size of the loaded area relative to the joint spacing, joint opening, and the location of the load. Figure 32 (a through c) shows three simple possible analyses associated with the ratio of foundation width to joint spacing and the joint conditions. 1. Closed Spaced Open Joints: Figure 32a illustrates the condition where the joint spacing, s, is a fraction of B, and the joints are open. The foundation is supported by un- confined rock columns; hence, the ultimate bearing 34 (a) (b) (c) (d) (e) (f) (g) (h) Rigid Plastic Rigid Porous Cavity Figure 31. Mechanisms of foundation failure from Franklin and Dusseault (1989), adapted from Sowers (1979): (a) Prandtl-type shearing in weak rock, (b) shearing with superimposed brittle crust, (c) compression of weathered joints, (d) compres- sion and punching of porous rock underlying a rigid crust, (e) breaking of pinnacles from a weathered rock surface, (f) slope failure caused by superimposed loading, (g) collapse of a shallow cave, and (h) sinkhole caused by soil erosion into solution cavities (Canadian Geotechnical Society, 2006).

35 (a) Close, open joints, S < B: Unconfined Compressions (b) Close, closed joints, S < B: Compression Zones (after Bell) (c) Wide joints, S > B: Splitting (after Meyerhof, Bishoni) (d) Thick rigid layer over weak compressible layer: Flexure Failure (e) Thin rigid layer over weak compressible layer: Punching Failure S B Compression zones SB Compression zone B Comp. zone S Split Rigid Weak compressible H B B HRigid Weak compressible Shear Figure 32. Bearing capacity failure modes of rock (based on Sowers, 1979). Prism I Prism II I-I =qq + γ B 2 tanθ 3-II I-II = qoσ −σ σ τ σσ Figure 33. Mohr Circle analysis of bearing capacity based on straight-line failure planes and prismatic zones of triaxial compression and shear (based on Sowers, 1979). capacity approaches the sum of the unconfined compres- sive strengths of each of the rock prisms. Because all rock columns do not have the same rigidity, some will fail be- fore others reach their ultimate strength; hence, the total capacity is somewhat less than the sum of the prism strengths. 2. Closed Spaced Joints in Contact: The Bell-Terzaghi analy- sis is shown in Figure 32b. When s < B and the joints are closed so that pressure can be transmitted across them without movement, the rock mass is essentially treated as a continuum, and the bearing capacity can be evalu- ated in the way shown in Figure 33 in which the major principal stress of Prism II (σ1-II) is equal to the embed- ment confining stresses qo, and the minor principal stress of Prism II (σ3-II) is equal to the major principal stress of Prism I (σ1-I) such that the bearing capacity is the major principal stress of Prism I and is expressed in Equation 72: where c is cohesion, and φf is friction angle of the rock mass. 3. Wide Joints: If the joint spacing is much greater than the foundation width, s >> B (see Figure 32c), the proposed failure mechanism is a cone-shaped zone forming below the foundation that splits the block of rock formed by the joints. Equation 73 can be used to approximate the bearing capacity assuming that the load is centered on the joint block and little pressure is transmitted across the joints: q JcNult cr≈ ( )73 q cult f = +⎛⎝ ⎞⎠2 45 2tan ( )φ 72

where s = the spacing between a pair of vertical open dis- continuities, φf = the friction angle of intact rock, and Nφ = the bearing capacity factor given by: 4. Thick and Thin Rigid Rock Layer over Weak Compress- ible Layer: As shown in Figures 31d, 32d, and 32e, depend- ing on the ratio H/B and S/B and on the flexural strength of the rock stratum, two forms of failure occur when the rock formation consists of an extensive hard seam under- lain by a weak compressible stratum. If the H/B ratio is large and the flexural strength is small, the rock failure occurs by flexure (see Figure 32d). If the H/B ratio is small, punch- ing is more likely (see Figure 32e). The same analysis can also be used for designs with hard rock layers over voids. Bearing capacity calculations for flexural or punching fail- ure are proposed by Lo and Hefny (2001) and by ASCE (Zhang and Einstein, 1998; Bishoni, 1968; Kulhawy, 1978). 1.7.4 The Canadian Foundation Engineering Manual The bearing capacity methods for foundations on rock proposed by the Canadian Foundation Engineering Manual (Canadian Geotechnical Society, 2006) are described to be suitable for all ranges of rock quality, noting that the design bearing pressure is generally for SLSs not exceeding 25 mm (1 in.) settlement. The Canadian Foundation Engineering Manual (Canadian Geotechnical Society, 2006) considers a N f φ φ = + ⎛ ⎝⎜ ⎞ ⎠⎟tan ( )2 45 2 77 36 0 5 10 15 20 S/B 0 50 100 150 Ncr 50 45 40 35 30 25 20 For square foundations shape correction = 0.85 φ = 0 2 4 6 8 10 12 H/B 0 0.5 1 J (a) bearing capacity factors for circular foundation on jointed rock, with S/B > 1 and H/B > 8. (b) correction factor J for rock layer thickness, H S B q0 = cNcrJ H Figure 34. Bearing capacity factors for rock splitting (based on Bishoni, 1968). For continuous strip foundations: where B and L = width and length of the footing, respectively; J = a correction factor dependent upon the thick- ness of the foundation rock below the footing and the width of the footing; and Ncr = bearing capacity factor. Based on laboratory test results and the Ncr solution by Bishoni (1968), J is estimated by the following: where H is the average spacing between a pair of horizon- tal discontinuities. Values of Ncr derived from models for splitting failure depend on the s/B ratio and φf, which will be discussed later. The values for square footings are 85% of the circular. Graphical solutions for the bearing capacity factor (Ncr) and correction factor (J) by Bishoni (1968) are provided in Figures 34a and 34b, respectively. The bearing capacity fac- tor (Ncr) is given by Goodman (1980): N N N s B N N cr f= + ( )⎛⎝⎜ ⎞⎠⎟ − ⎛ ⎝⎜ ⎞ ⎠⎟ − 2 1 1 12φ φ φ φ φcot cot ( )φ φf N( )+ 2 12 76 H B J> =5 1 ( )75b H B J H B ≤ = +5 0 12 0 4. . ( )75a q JcN L B ult cr = + ⎛⎝⎜ ⎞⎠⎟2 2 0 18. . ( )74

soil or rock debris), and for a foundation width greater than 300 mm (1 ft). For sedimentary rocks, the strata must be hor- izontal or nearly so. The bearing-pressure coefficient, Ksp, as given in Figure 35, takes into account the size effect and the presence of discon- tinuities and includes a nominal safety factor of 3 against the lower-bound bearing capacity of the rock foundation. The factor of safety against general bearing failure (ULSs) may be up to ten times higher. For a more detailed explanation, the Canadian Foundation Engineering Manual (Canadian Geo- technical Society, 2006) refers to Ladanyi et al. (1974) and Franklin and Gruspier (1983) who discuss a special case of foundations on shale. It is often useful to estimate a bearing pressure for preliminary design on the basis of the material de- scription. Such values must be verified or treated with cau- tion for final design. Table 7 presents presumed preliminary design bearing pressure for different types of soils and rocks. 1.7.5 Goodman (1989) The considered mode of failure is shown in Figures 36a through 36c, in which a laterally expanding zone of crushed rock under a strip footing induces radial cracking of the rock on either side (Goodman, 1989). The strength of the crushed rock under the footing is described by the lower failure enve- lope (curve for Region A) in Figure 37, while the strength of the less fractured neighboring rock is being described by the upper curve in the same figure (curve for Region B). The largest horizontal confining pressure that can be mobilized to support the rock beneath the footing (Region A in Figure 37) is ph, de- termined as the unconfined compressive strength of the adja- cent rock (Region B of Figure 37). This pressure determines the lower limit of Mohr’s circle tangent to the strength envelope of the crushed rock under the footing. Triaxial compression tests 37 Table 6. Coefficients of discontinuity spacing, Ksp (Canadian Geotechnical Society, 2006). Discontinuity spacing Description Distance m (ft) Ksp Moderately close 0.3 to 1 (1 to 3) 0.1 Wide 1 to 3 (3 to 10) 0.25 Very wide > 3 (> 10) 0.4 0 0.4 0.8 1.2 1.6 2 Ratio c/B 0 0.1 0.2 0.3 0.4 0.5 V al ue o f K sp 0.020 0.010 0.005 0.002 0.001 δ/c = 0 3 10 1+300 sp c BK c + = c = spacing of discontinuities δ = aperture of discontinuities B = footing width Valid for 0.05 < c/B < 2.0 0 < δ/c < 0.02 Figure 35. Bearing pressure coefficient (Ksp) (based on Canadian Geotechnical Society, 2006). rock to be sound when the spacing of discontinuities is in excess of 0.3 m (1 ft). When the rock is sound, the strength of the rock foundation is commonly in excess of the design requirements provided the discontinuities are closed and are favorably oriented with respect to the applied forces, i.e., the rock surface is perpendicular to the foundation, the load has no tangential component, and the rock mass has no open discon- tinuities. Under such conditions, the design bearing pressure may be estimated from the following approximate relation: where qa = design bearing pressure; qu-core = average unconfined compressive strength of rock (as determined from ASTM D2938); and Ksp = an empirical coefficient, which includes a factor of safety of 3 (in terms of WSD) and ranges from 0.1 to 0.4 (see Table 6 and Figure 35). The factors influencing the magnitude of the coefficient are shown graphically in Figure 35. The relationship given in Fig- ure 35 is valid for a rock mass with spacing of discontinuities greater than 300 mm (1 ft), aperture of discontinuities less than 5 mm (0.2 in.) (or less than 25 mm [1 in.] if filled with q K qa sp u core= × − ( )78

on broken rock can define the latter strength envelope, and thus the bearing capacity can be found (Goodman, 1989). Examination of Figure 37 leads to the conclusion that the bearing capacity of a homogeneous, discontinuous rock mass cannot be less than the unconfined compressive strength of the rock mass around the footing, and this can be taken as the lower bound. If the rock mass has a constant angle of in- ternal friction (φf) and unconfined compressive strength (qu) (Mohr-Coulomb material), the mechanism described in Fig- ure 37 establishes the bearing capacity as where Nφ is calculated using Equation (77). Figure 38 depicts a footing resting on a portion of a single joint block created by orthogonal vertical joints each spaced distance s. Such a condition might arise, for example, in weath- ered granite (Goodman, 1989). If the footing width (B) is equal q q Nult u= +( )φ 1 ( )79 to the joint spacing (s), the rock foundation can be compared to a column whose strength under axial load should be approx- imately equal to the unconfined compressive strength (qu). If the footing contacts a smaller proportion of the joint block, the bearing capacity increases toward the maximum value consis- tent with the bearing capacity of homogeneous, discontinuous rock, obtained with the construction of the Mohr-Coulomb failure envelopes described in Figure 37 or from Equation 79, which takes into account the friction angle (φf) of the homo- geneous discontinuous rock. This problem was studied by Bishoni (1968), who assumed that some load is transferred laterally across joints. Modifying this boundary condition for an open-jointed rock mass in which lateral stress trans- fer is zero, yields q q N N S B ult u N N = − ⎛⎝⎜ ⎞⎠⎟ − ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ ⎧ ⎨⎪ −( ) 1 1 1 1 φ φ φ φ ⎩⎪ ⎫ ⎬⎪ ⎭⎪ ( )80 38 Table 7. Presumed preliminary design bearing pressure (Canadian Geotechnical Society, 2006). Group Types and conditions of rocks Strength of rock material Preliminary design bearing pressure (5) kPa (ksf) Remarks Massive igneous and metamorphic rocks (granite, diorite, basalt, gneiss) in sound condition (2) High-very high 10,000 (200) These values are based on the assumption that the foundations are carried down to unweathered rock. Foliated metamorphic rocks (slate, schist) in sound condition(1) (2) Medium-high 3,000 (60) Not applicable Sedimentary rocks: cemented shale, siltstone, sandstone, limestone without cavities, thoroughly cemented in conglomerates, all in sound condition(1) (2) Medium-high 1,000–4,000 (20–80) Not applicable Compaction shale and other argillaceous rocks in sound condition (2)(4) Low-medium 500–1,000 (10–20) 1,000 (20) Not applicable Broken rocks of any kind with moderately close spacing of discontinuities (0.3 m [11.8 in]) or greater), except argillaceous rocks (shale), limestone, sandstone, shale with closely spaced bedding Not applicable (See note 3) Not applicable Rocks Heavily shattered or weathered rocks Not applicable (See note 3) Not applicable Notes: 1. The above values for sedimentary or foliated rocks apply where the strata or the foliation are level or nearly so, and, then, only if the area has ample lateral support. Tilted strata and their relation to nearby slopes or excavations should be assessed by a person knowledgeable in this field of work. 2. Sound rock conditions allow minor cracks at spacing not closer than 1 m (39.37 in). 3. To be assessed by examination in-situ, including test loading if necessary. 4. These rocks are apt to swell on release of stress, and on exposure to water they are apt to soften and swell. 5. The above values are preliminary estimates only and may need to be adjusted upwards or downwards in a specific case. No consideration has been made for the depth of embedment of the foundation. Reference should be made to other parts of the Manual when using this table.

Pτ σh qf Strength of Rock Mass (region B) Strength of Rock Mass (region A) qf Strip footing Ph B A B Figure 37. Analysis of bearing capacity on rock (based on Goodman, 1989). Comparing the results of Goodman’s (1989) computa- tions with Equations 79 and 80 shows that open joints reduce the bearing capacity only when the ratio S/B is in the range from 1 to 5. The bearing capacity of footings on rock with open joints increases with increasing φf for any of the S/B ratios rang- ing from 1 to 5. 1.7.6 Carter and Kulhawy (1988) Carter and Kulhawy (1988) suggested that the Hoek and Brown strength criterion for jointed rock masses (Hoek and Brown, 1980, see also Section 1.8.2.4) can be used in the eval- uation of bearing capacity. The curved strength envelope for jointed rock mass can be expressed as where σ1 = major principal effective stress, σ3 = minor principal effective stress, qu = uniaxial compressive strength of the intact rock. s and m = empirically determined strength parameters for the rock mass, which are to some degree anal- ogous to c and φf of the Mohr-Coulomb failure criterion. Carter and Kulhawy (1988) suggested that an analysis of the bearing capacity of a rock mass obeying this criterion can be made using the same approximate technique as used in the Bell (1915) solution. The details of this approach are described in Figure 39. A lower bound to the failure load was calculated by finding a stress field that satisfies both equilibrium and the failure criterion. For a strip footing, the rock mass beneath the foundation may be divided into two zones with homoge- neous stress conditions at failure throughout each, as shown in Figure 39. The vertical stress in Zone I is assumed to be zero, while the horizontal stress is equal to the uniaxial com- pressive strength of the rock mass, given by Equation 81 as s0.5qu. For equilibrium, continuity of the horizontal stress σ σ σ1 3 3 2 0 5 = + +( )mq squ u . ( )81 39 Cracking Crushing (a) (b) Wedging Punching (c) (d) Shear (e) Figure 36. Modes of failure of a footing on rock including development of failure through crack propagation and crushing beneath the footing (a-c), punching through collapse of voids (d), and shear failure (e) (based on Goodman, 1989). B s Figure 38. Footing on rock with open, vertical joints (based on Goodman, 1989).

across the interface must be maintained and therefore the bearing capacity of the strip footing may be evaluated from Equation 81 (with σ3 = s0.5qu) as In an errata to Carter and Kulhawy (1988), Equation (82a) was modified to the following: A similar approach to the bearing capacity analysis of a strip footing was proposed by Carter and Kulhawy (1988) to be used for a circular foundation with an interface between the two zones that was a cylindrical surface of the same diam- eter as the foundation. In this axisymmetric case, the radial stress transmitted across the cylindrical surface at the point of collapse of the foundation may be greater than qu , without necessarily violating either radial equilibrium or the failure cri- terion. However, because of the uncertainty of this value, the radial stress at the interface is also assumed to be qu for the case of a circular foundation. Therefore, the predicted (lower bound) bearing capacity is given by Equations 82a and 82b. The m and s constants are determined by the rock type and the conditions of the rock mass, and selecting an appropriate category is easier if either the Rock Mass Rating (RMR) sys- tem or the Geological Strength Index (GSI) classification data are available as outlined below. Both bearing capacity formu- lations expressed in Equations 82a and 82b were investigated in this study. s s q s m s s qult u= + +( )( )0 5. ( )82b q m s qult u= +( ) ( )82a 1.8 Rock Classification and Properties 1.8.1 Overview A rock mass comprises blocks of intact rock that are sep- arated by discontinuities such as cleavage, bedding planes, joints, and faults. Table 8 provides a summary of rock mass discontinuity definitions and characteristics. These naturally formed discontinuities create weakness surfaces within the rock mass, thereby reducing the material strength. As previ- ously discussed, the influence of the discontinuities upon the material strength depends upon the scale of the foundation relative to the position and frequency of the discontinuities (Canadian Foundation Geotechnical Society, 2006). This section provides a short review of rock mass classi- fication/characterization systems and rock properties that are relevant to the methods selected for bearing capacity evaluation. Methods allowing engineering classification of rock mass are reviewed including the Rock Mass index (RMi) system, RMR system and the Hoek-Brown GSI. 1.8.2 Engineering Rock Mass Classification 1.8.2.1 Classification Methods A number of classification systems have been developed to provide the basis for engineering characterization of rock masses. A comprehensive overview of this subject is pro- vided by Hoek et al. (1995). Most of the classification sys- tems incorporating various parameters were derived from civil engineering case histories in which all components of the engineering geological parameters of the rock mass were considered (Wickham et al., 1972; Bieniawski, 1973, 1979, 1989; Barton et al., 1974). More recently, the systems have been modified to account for the conditions affecting rock mass stability in underground mining. While no single clas- sification system has been developed for or applied to foun- dation design, the type of information collected for the two more common civil engineering classification schemes—the Q system (Barton et al., 1974), used in tunnel design, and RMR (Bieniawski, 1989), used in tunnel and foundation design—are often considered. These techniques have been applied to empirical design situations, where previous expe- rience greatly affects the design of the excavation in the rock mass. Table 9 outlines the many classification systems and their uses. Detailed descriptions of the different systems and the engineering properties associated with them are beyond the scope of this work and are restricted to the methods relevant to the current research. The two most commonly used rock mass classification systems today are RMR, developed by Bieniawski (1973) and 40 Figure 39. Lower bound solution for bearing capacity (Carter and Kulhawy, 1988).

strength, joint distance, and ground water condition. It has often been suggested that when using rock classification schemes—such as the RQD, RMR, and Q-system—only the natural discontinuities, which are of geological or geo- morphic origin, should be taken into account. However, it is often difficult, if not impossible, to judge whether a discon- tinuity is natural or artificial after activities such as drilling, blasting, and excavation. 1.8.2.2 Rock Quality Designation (RQD) In 1964, D. U. Deere introduced an index to assess rock qual- ity quantitatively called RQD. RQD is a core recovery percent- age that is associated with the number of fractures and the amount of softening in the rock mass that is observed from the drill cores. Only the intact pieces with a length greater than 100 mm (4 in.) are summed and divided by the total length of the core run (Deere, 1968). RQD Length of core pieces cm total core = ≥∑ 10 length 100 % ( )( ) 83 41 Table 8. Rock mass discontinuity descriptions (Hunt, 1986). Discontinuity Definition Characteristics Fracture A separation in the rock mass, a break. Signifies joints, faults, slickensides, foliations, and cleavage. Joint A fracture or crack in rock not accompanied by dislocation. Most common defect encountered. Present in most formations in some geometric pattern related to rock type and stress field. Open joints allow free movement of water, increasing decomposition rate of mass. Tight joints resist weathering and the mass decomposes uniformly. Fault A fracture along which there has been an observable amount of displacement. Fault zones usually consist of crushed and sheared rock through which water can move relatively freely, increasing weathering. Faults generally occur as parallel to sub-parallel sets of fractures along which movement has taken place to a greater or lesser degree. Slickenside A smooth often striated surface produced on rock by movement along a fault or a subsidiary fracture. Shiny, polished surfaces with striations. Often the weakest elements in a mass, since strength is often near residual. Foliation Plane Continuous foliation surface results from orientation of mineral grains during metamorphism. Can be present as open joints or merely orientations without openings. Strength and deformation relate to the orientation of applied stress to the foliations. Cleavage The quality of a crystallized substance or rock of splitting along definite planes. A fragment obtained by splitting along preferred planes of weakness, e.g., diamond. Bedding Plane Any of the division planes which separate the individual strata or beds in sedimentary or stratified. Often are zones containing weak materials such as lignite or montmorillonite clays. Mylonite A fine-grained laminated rock formed by the shifting of rock layers along faults. Fine-grained rock formed in shear zones. Cavities Openings in soluble rocks resulting from groundwater movement or in igneous rocks from gas pockets. In limestone, range from caverns to tubes. In rhyolite and other igneous rocks, range from voids of various sizes to tubes. adopted by the South African Council of Scientific and Indus- trial Research (CSIR), and the Norwegian Geotechnical Insti- tute index (NGI-index or Q-system) (Barton et al., 1974). Both classification systems include Rock Quality Designation (RQD). In this study, the RMR geomechanics classification system was adopted because (1) the overwhelming majority of states evaluate RQD and utilize the RMR system (this infor- mation is based on a questionnaire presented in Chapter 3) and (2) it was favored by the available rock property data of the case histories. The Geological Strength Index (GSI), based on the RMR system and the tables from the latest ver- sions of the Hoek-Brown failure criterion (e.g., Hoek et al., 2002), was used. The systems presented in this report and utilized in the calibration (1) give a numerical value (have a numerical form), (2) present a result that can be used to determine/ estimate the strength, (3) have been successfully used in the past, and (4) are applicable to hard rock masses. The param- eters included in the classification systems resulting in a numerical value are presented in Table 10. The most com- monly used parameters are the intact rock strength, joint

Table 9. Major rock classification/characterization systems (Edelbro, 2004, modified after Palström, 1995). Name of classification Author and first version Country of origin Application Form and type 1 Remarks Rock Load Theory Terzaghi, 1946 USA Tunnels with steel supports Descriptive F, Behavior F, Functional T Unsuitable for modern tunneling Stand Up Time Lauffer, 1958 Austria Tunneling Descriptive F, General T Conservative New Austrian Tunneling Method (NATM) Rabcewicz, 1964/65 and 1975 Austria Tunneling in incompetent (overstressed) ground Descriptive F, Behavioristic F, Tunneling concept Utilized in squeezing ground conditions Rock Quality Designation (RQD) Deere et al., 1966 USA Core logging tunneling Numerical F, General T Sensitive to orientation effects. In Deere, 1968 A Recommended Rock Classification for Rock Mechanical Purposes Coates and Patching, 1968 For input in rock mechanics Descriptive F, General T The Unified Classification of Soils and Rocks Deere et al., 1966 USA Based on particles and blocks for communication Descriptive F, General T In Deere and Deere, 1988 Rock Structure Rating (RSR) Concept2 Wickham et al., 1972 USA Tunnels with steel supports Numerical F, Functional T Not useful with steel fiber shotcrete Rock Mass Rating (RMR)-System, Council of Scientific and Industrial Research (CSIR) Bieniawski, 1974 SouthAfrica Tunnels, mines, foundations, etc. Numerical F, Functional T Unpublished base case records Q-System Barton et al., 1974 Norway Tunnels, large chambers Numerical F, Functional T Mining RMR (MRMR) Laubscher, 1975 Mining Numerical F, Functional T In Laubscher, 1977 The Typological Classification Matula and Holzer, 1978 For use in communication Descriptive F, General T 3The Unified Rock Classification System (URCS) Williamson, 1980 USA For use in communication Descriptive F, General T In Williamson, 1984 Basic Geotechnical Description (BGD) ISRM, 1981 For general use Descriptive F, General T Rock Mass Strength (RMS) Stille et al., 1982 Sweden Numerical F, Functional T Modified RMR Modified Basic RMR (MBR) Cummings et al., 1982 Mining Numerical F, Functional T Simplified Rock Mass Rating Brook and Dharmaratne, 1985 Mines and tunnels Numerical F, Functional T Modified RMR and MRMR Slope Mass Rating (SMR) Romana, 1985 Spain Slopes Numerical F, Functional T Ramamurthy/Arora Ramamurthy and Arora, 1993 India For intact and jointed rocks Numerical F, Functional T Modified Deere and Miller approach Geological Strength Index (GSI) Hoek et al., 1995 Mines, tunnels Numerical F, Functional T Rock Mass Number (N) Goel et al., 1995 India Numerical F, Functional T Stress-free Q-system Rock Mass Index (RMi) Palmström, 1995 Norway Rock engineering communication, characterization Numerical F, Functional T 1Descriptive F = Descriptive Form: the input to the system is mainly based on descriptions. Numerical F = Numerical Form: the input parameters are given numerical ratings according to their character. Behavioristic F = Behavioristic Form: the input is based on the behavior of the rock mass in tunnel. General T = General Type: the system is worked out to serve as a general characterization. Functional T = Functional Type: the system is structured for a special application (for example, for rock support) (Palmström, 1995). 2RSR was a forerunner to the RMR system, although they both give numerical ratings to the input parameters and summarize them to a total value connected to the suggested support. 3The Unified Rock Classification System (URCS) is associated with Casagrande’s classification system for soils in 1948.

RQD is used as a standard quantity in drill core logging, and its greatest value is perhaps its simplicity, low cost, and quick determination. RQD is simply a measurement of the percentage of “good” rock recovered from an interval of a borehole. The procedure for measuring RQD is illustrated in Figure 40. The recommended procedure for measuring the core length is to measure it along the centerline of the core. Core breaks caused by the drilling process should be fitted together and counted as one piece. The relationship between the numerical value of RQD and the engineering quality of the rock mass as proposed by Deere (1968) is given in Table 11. When no cores are available, one can estimate RQD from relevant information, for instance, joint spacing (Brady and Brown, 1985). Priest and Hudson (1976) found that an esti- mate of RQD could be obtained from joint spacing (λ [number of joints per meter]) measurements made on an exposure by using the following: For λ = 6 to 16 joints/meter, the following simplified equa- tion can be used (Priest and Hudson, 1976): Equations 84 and 85 are probably the simplest ways of determining RQD, when no cores are available. Palmström RQD = − +3 68 110 4. . ( )λ 85 RQD e= +( )−100 0 1 10 1. . ( )λ λ 84 (1982) presented the relationship between Jv and RQD in a clay free rock mass along a tunnel as the following: where Jv is the volumetric joint count and the sum of the num- ber of joints per unit length for all joint sets in a clay-free rock mass. For Jv < 4.5, RQD = 100. The RQD is not scale dependent and is not a good measure of the rock mass quality in the case of a rock mass with joint spacing near 100 mm. If the spacing between continuous joints is 105 mm (core length), the RQD value will be 100%. If the spacing between continuous joints is 95 mm, the RQD value will be 0%. For large-sized tunnels, RQD is of question- able value. It is, as mentioned by Douglas and Mostyn (1999), unlikely that all defects found in the boreholes would be of significance to the rock mass stability. 1.8.2.3 Rock Mass Rating (RMR) In 1973, Bieniawski introduced RMR as a basis for geo- mechanics classification. The rating system was based on Bieniawski’s experience in shallow tunnels in sedimentary rocks. Originally, the RMR system involved 49 unpublished case histories. Since then, the classification system has under- gone several significant changes. In 1974, there was a reduc- tion of parameters from eight to six, and, in 1975, there was an adjustment of ratings and a reduction of recommended support requirements. RQD Jv= −115 3 3. ( )86 Table 10. Parameters included in different classification systems resulting in a numerical value (Edelbro, 2004). Parameters RQD RSR RMR Q MRMR RMS MBR SRMR* SMR **RAC GSI N RMi Block size – – – – – – X – – – – – X Joint orientation – – X – – – X – – – – – X Number of joint sets – – – X – X – – – – – X X Joint length – – – – – – – – – – – – X Joint spacing X X X X X X X X X X X X X Joint strength – X X X X X X X X X X X X Rock type – X – – – – – – – – – – – State of stress – – – X X – X – – – – – – Groundwater condition – X X X X X X X X – – X – Strength of intact rock – – X X X X X X X X X X X Blast damage – – – – X – X X – – X – – *SRMR = Simplified Rock Mass Rating * *RAC - Ramamurthy and Arora Classification 43

44 Table 11. Correlation between RQD and rock mass quality (Deere, 1968). RQD % Rock quality < 25 Very Poor 25–50 Poor 50–75 Fair 75–90 Good 90–100 Excellent In 1976, a modification of rating class boundaries (as a result of 64 new case histories) to even multiples of 20 took place, and, in 1979, there was an adoption of the International Society for Rock Mechanics (ISRM) rock mass description. The newest version of RMR is from 1989, when Bieniawski published guidelines for selecting rock reinforcement. In this version, Bieniawski suggested that the user could interpolate the RMR values between different classes and not just use dis- crete values. Therefore, it is important to state which version is used when RMR values are quoted. When applying this classification system, one divides the rock mass into a num- ber of structural regions and classifies each region separately. The RMR system uses six parameters, which are rated. The ratings are added to obtain a total RMR value. The six param- eters are the following: 1. Unconfined compressive strength of intact rock material (qu), 2. RQD, 3. Joint or discontinuity spacing (s), 4. Joint condition, 5. Ground water condition, and 6. Joint orientation. The first five parameters represent the RMR basic parameters (RMRbasic) in the classification system. The sixth parameter is treated separately because the influence of discontinuity ori- entations depends upon the engineering application. Each of these parameters is given a rating that symbolizes the RQD. The first five parameters of all the ratings are algebraically summed and can be adjusted, depending on the joint and tunnel orientation, by the sixth parameter as shown in Equa- tions 87a and 87b. RMR RMR adjustment for joint orientatbasic= + ion 87a( ) Figure 40. Procedure for measurement and calculation of rock quality designation (Sabatini et al., 2002).

The final RMR value is grouped into five rock mass classes (see Table 12 and the relevant Table 10.4.6.4-3 in the AASHTO [2008] specifications). The various parameters in the system are not equally important for the overall classification of the rock mass, since they have been given different ratings. Higher RMA indicates better rock mass condition/quality. The RMR system is very simple to use, and the classification parameters are easily obtained from either borehole data or underground mapping. Most of the applications of RMR have been in the field of tunneling, but RMR has also been applied in the stabil- ity analysis of slopes and shallow foundations, caverns, and dif- ferent mining openings. 1.8.2.4 Geological Strength Index (GSI) Hoek et al. (1995) introduced the GSI as a complement to their generalized rock failure criterion and as a way to esti- mate the material constants s, a, and mb in the Hoek-Brown failure criterion. GSI estimates the reduction in rock mass strength for different geological conditions. The GSI has been updated for weak rock masses several times (1998, 2000, and 2001) (Hoek et al., 2002). The aim of the GSI system is to determine the properties of the undisturbed rock mass. For disturbed rock masses, compensation must be made for the lower GSI values obtained from such locations. The strength of the rock mass depends on factors such as the shear strength of the surfaces of the blocks defined by disconti- nuities, their continuous length, and their alignment relative to the load direction (Wyllie, 1992). If the loads are great enough to extend fractures and break intact rock or if the rock mass can dilate, resulting in loss of interlock between the blocks, then the rock mass strength may be diminished significantly from that of the in situ rock. Where foundations contain poten- tially unstable blocks that may slide from the foundation, the shear strength parameters of the discontinuities should be used in design, rather than the rock mass strength. RMR parameters 87bbasic = + + + +( )∑ 1 2 3 4 5 ( ) If rock masses contain many discontinuities or are heavily jointed with discontinuities having similar strength charac- teristics, they can be treated as an isotropic continuum, and their strength can be estimated using methods based on a con- tinuum approach. The strength and deformation properties of jointed rock masses can, therefore, be estimated using the Hoek-Brown failure criterion (Hoek and Brown, 1997) from three parameters (Hoek and Marinos, 2000; Marinos and Hoek, 2001): • The unconfined compressive strength of the intact rock elements contained within the rock mass. • A constant, mi, which defines the frictional characteristics of the component minerals within each intact rock element. • The GSI, which relates the properties of the intact rock elements to those of the overall rock mass (see Table 13) (Canadian Geotechnical Society, 2006). The generalized Hoek-Brown failure criterion is defined as the following: where σ′1 and σ′3 = the principal effective stresses at failure; qu = the unconfined compressive strength of the intact rock pieces; mb = the value of the Hoek-Brown constant m for the rock mass, and mi = the Hoek-Brown constant for the intact rock (see Table 14) (Canadian Geotechnical Society, 2006); and s and a = constants that depend upon the rock mass characteristics. For GSI > 25, a = 0.5, and For GSI < 25, s = 0, and a GSI = −0 65 200 . . s GSI = −⎛⎝⎜ ⎞⎠⎟exp . 100 9 m m GSI b = −⎛⎝⎜ ⎞⎠⎟1 100 28 exp ; ′ = ′ + ′ + ⎛ ⎝⎜ ⎞ ⎠⎟σ σ σ 1 3 3q m q su b u a ( )88 45 Table 12. Meaning of rock mass classes and rock mass classes determined from total ratings (Bieniawski, 1978). Parameter/properties of rock mass Rock mass rating (rock class) Ratings 100–81 80–61 60–41 40–21 <20 Classification of rock mass Very Good Good Fair Poor Very Poor Average stand-up time 10 years for 15 m span 6 months for 8 m span 1 week for 5 m span 10 hours for 2.5 m span 30 minutes for 1 m span Cohesion of the rock mass kPa (ksf) > 400 ( > 90) 300–400 (67.44–90) 200–300 (45–67.44) 100–200 (22.48–45) < 100 (< 22.48) Friction angle of the rock mass > 45o 35o–45o 25o–35o 15o–25o < 15o

Table 13. GSI estimates for rock masses (Hoek and Marinos, 2000). D ec re as in g In te rl oc ki ng o f R oc k Pi ec es 10 20 30 40 50 60 70 80 Decreasing Surface Quality Geological Strength Index From the letter codes describing the structure and surface of the rock mass, select the appropriate box in this chart. Estimate the average value of the geological strength index (GSI) from the contours. Do not attempt to be too precise, i.e., quoting a range of GSI from 36 to 42 is more realistic than stating that GSI=38. STRUCTURE V ER Y G O O D V er y ro ug h, fr es h un w ea th er ed su rfa ce s G O O D R ou gh , s lig ht ly w ea th er ed , i ro n sta in ed su rfa ce s Fa ir Sm oo th , m od er at el y w ea th er ed o r a lte re d su rfa ce s PO O R Sl ic ke ns id ed , h ig hl y w ea th er ed su rfa ce s w ith co m pa ct c oa tin gs o r f ill in gs o f a ng ul ar fr ag m en ts V ER Y P O O R Sl ic ke ns id ed , h ig hl y w ea th er ed su rfa ce s w ith so ft cl ay c oa tin gs o r f ill in gs BLOCKY – very well interlocked undisturbed rock ma ss consisting of cubical blocks form ed by three orthogonal discontinuity sets. VERY BLOCKY – interlocked, partially disturbed rock mass with mu ltifaceted angular blocks form ed by four or mo re discontinuity sets. BLOCKY/DISTURBE D – folded and/or faulted with angular blocks form ed by ma ny intersecting discontinuity sets. DISINTEGRATED – poorly interlocked, heavily broken rock mass with a mi xture of angular and rounded rock pieces. The Hoek-Brown constant (mi) can be determined from triaxial testing of core samples using the procedure discussed by Hoek et al. (1995) or can be determined from the values given in Table 14 (Canadian Geotechnical Society, 2006). Most of the values provided in Table 14 have been derived from triaxial testing on intact core samples. The ranges of values shown reflect the natural variability in the strength of earth materials and depend upon the accuracy of the lithological description of the rock. For example, Marinos and Hoek (2001) note that the term “granite” describes a clearly defined rock type that exhibits very similar mechanical characteristics, independent of origin. As a result, mi for granite is defined as 32±3. On the other hand, volcanic breccia is not very precise in terms of mineral composition, with the result that mi is given as 19±5, denoting a higher level of uncertainty (Canadian Geo- technical Society, 2006). The ranges of values depend upon the granularity and interlocking of the crystal structure. Higher values are associated with tightly interlocked and more fric- tional characteristics. 1.8.3 Current AASHTO (2008) Practice The strength of intact rock material is determined using the results of unconfined compression tests on intact rock cores, splitting tensile tests on intact rock cores, or point load strength tests on intact specimens of rock. The rock is classi- fied using the RMR system as described in Table 15. For each of the five parameters in Table 15, the relative rating based on the ranges of values provided is to be evaluated. The RMR is 46

determined as the sum of all five relative ratings. The RMR should be adjusted in accordance with the criteria in Table 16. The rock classification should be determined in accordance with Table 17. Emphasis is placed on visual assessment of the rock and the rock mass because of the importance of the discontinuities in rock. The geomechanics classification can be used to estimate the value of GSI for cases where RMR is greater than 23, as follows: where RMR89 = RMR according to Bieniawski (1989) as pre- sented in Table 17. For RMR89 values less than 23, the mod- ified Tunneling Quality Index (Q′) is used to estimate the value of GSI: ′ = ×Q RQD J J Jn r a ( )90 GSI RMR= −89 5 89( ) where Jn = number of sets of discontinuities, Jr = roughness of discontinuities, and Ja = discontinuity condition and infilling. Table 18 gives the values of the parameters used to evalu- ate Q′ in Equation 90. The determination of the shear strength of fractured rock masses is essential in foundation design analyses. The Hoek and Brown criteria can be used to evaluate the shear strength of fractured rock masses in which the shear strength is represented as a curved envelope that is a function of the unconfined compressive strength of the intact rock, qu, and two dimensionless constants, m and s. The values of m and s as defined in Table 19 should be used. The shear strength of the rock mass should be determined using the method GSI Qe= ′ +9 44 91log ( ) 47 Table 14. Values of the Hoek-Brown Constant (mi) for intact rock by rock group (Marinos and Hoek, 2001). Clastic Conglomerate Breccia 1 Sandstone (17±4) Silstone (7±2) Greywack e (18±3) Claystone (4±2) Shale (6±2) Marl (7±2 ) Carbonates Crystalline Lim estone (12±3) Spartic Lim estone (10±2) Micritic Lim estone (9±2) Dolom ite (9±3) Evaporites Gypsum (8±2) Anhydrite (12±2) Sedimentary Non-clastic Organic Chalk (7±2 ) Non-foliated Marble (9±3) Hornfels (19±4) Meta Sandstone (19±3) Quartzite (20±3) Slightly foliated Migmatite (29±3) Amphibolite (26±6) Gneiss (28±5) Metamorphic Foliated 2 Schist (12±3) Phyllite (7±3) Slate (7±4) Light Granite (32±3 ) Granodi orite (29±3) Diorite (25±5) Plutonic Dark Gabbro (27±3 ) Norite (20±5) Dolerite (16±5) Hypabyssal Porphyry (20±5) Diabase (15±5) Peridotite (25±5) Lava Ryolite (25±5) Andesite (25±5) Dacite (25±3) Basalt (25±5) Igneous Volcanic Pyroclastic Agglom erate (19±3) Breccia (19±5) Tuff (13±5) Notes: Values in parentheses are estimates. 1 Conglomerates and breccias may have a wide range of values, depending on the nature of the cementing material and the degree of cementation. Values range between those of sandstone and those of fine-grained sediments. 2 These values are for intact rock specimens tested normal to bedding or foliation. Values of mi will be significantly different if failure occurs along a weakness plane.

Table 15. Geomechanics classification of rock masses (AASHTO, 2008, Table 10.4.6.4-1). PARAMETER RANGES OF VALUES Point load strength index >175 ksf 85–175 ksf 45–85 ksf 20–45 ksf For this low range, unconfined co mp ressive test is preferred Strength of intact rock ma terial Unconfined co mp ressive strength >4,320 ksf 2,160– 4,320 ksf 1,080– 2,160 ksf 520– 1,080 ksf 215– 520 ksf 70–215 ksf 20–70 ksf 1 Relative Rating 15 12 7 4 2 1 0 Drill core quality RQD 90% to 100% 75% to 90% 50% to 75% 25% to 50% <25% 2 Relative Rating 20 17 13 8 3 Spacing of joints >10 ft 3–10 ft 1–3 ft 2 in–1 ft <2 in 3 Relative Rating 30 25 20 10 5 Condition of joints Very rough surfaces Not continuous No separation Hard joint wall rock Slightly rough surfaces Separation <0.05 in Hard joint wall rock Slightly rough surfaces Separation <0.05 in Soft joint wall rock Slicken- sided surfaces or Gouge <0.2 in thick or Joints open 0.05–0.2 in Continuous joints Soft gouge >0.2 in thick or Joints open >0.2 in Continuous joints 4 Relative Rating 25 20 12 6 0 Inflow per 30 ft tunnel length None <400 gal/hr 400–2,000 gal/hr >2,000 gal/hr Ratio = joint water pressure/ ma jor principal stress 0 0.0–0.2 0.2–0.5 >0.5 Ground water conditions (use one of the three evaluation criteria as appropriate to the me thod of exploration) General Conditions Co mp letely Dry Moist only (interstitial water) Water under m oderate pressure Severe water problem s 5 Relative Rating 10 7 4 0 Table 16. Geomechanics rating adjustment for joint orientations (AASHTO, 2008, Table 10.4.6.4-2). Strike and dip orientations of joints Very favorable Favorable Fair Unfavorabl e Very unfavorable Tunnels 0 –2 –5 –10 12 Foundations 0 –2 –7 –15 25 Ratings Slopes 0 –5 –25 –50 60 48

developed by Hoek (1983) and Hoek and Brown (1988, 1997) as follows: where τ = the shear strength of the rock mass (ksf), qu = average unconfined compressive strength of rock core (ksf), m, s = constants from Table 19, σ′n = effective normal stress (ksf), and φ′i = the instantaneous friction angle of the rock mass (degrees): When a major discontinuity with a significant thickness of infilling is to be investigated, the shear strength is governed by the strength of the infilling material and the past and expected future displacement of the discontinuity. The elastic modulus of a rock mass (Em) is taken as the lesser of the intact modulus h m sq m q n u u = + ′ +( ) 1 16 3 2 σ ′ = + ⎛⎝⎜ ⎞⎠⎟⎡⎣⎢ ⎤ ⎦⎥ − − − φi h htan cos . sin1 2 1 3 24 30 0 33 − ⎧⎨⎩ ⎫⎬⎭ − 1 1 2 τ φ φ= ′ − ′( )cot cos ( )i i um q 8 92 of a sample of rock core (Ei) or the modulus determined from one of the following equations: where Em = elastic modulus of the rock mass (ksi), Em ≤ Ei, Ei = elastic modulus of intact rock from tests (ksi), and RMR = rock mass rating. or where Em is the elastic modulus of the rock mass (ksi), and Em/Ei is a reduction factor based on RQD determined from Table 20 (dim.). For critical or large structures, determination of rock mass modulus (Em) using in situ tests may be warranted. It is extremely important to use the elastic modulus of the rock mass for computation of displacements of rock materials under applied loads. Use of the intact modulus will result in unrealistic and unconservative estimates. Poisson’s ratio for E E E Em m i i= ⎛⎝⎜ ⎞⎠⎟ ( )94 Em RMR = ⎛ ⎝⎜ ⎞ ⎠⎟ − 145 10 93 10 40 ( ) 49 Table 18. Joint parameters used to determine Q’ (Barton et al., 1974). 1. No. of sets of discontinuities = Jn 3. Discontinuity condition and infilling = Ja Massive 0.5 3.1 Unfilled cases One set Healed 0.75 Two sets Stained, no alteration Three sets Silty or sandy coating Four or more sets Clay coating Crushed rock 20 3.2 Filled discontinuities Sand or crushed rock infill 2. Roughness of Discontinuities = Jr Stiff clay infilling < 5 mm Noncontinuous joints 4 Soft clay infill < 5 mm thick Rough, wavy 3 Swelling clay < 5 mm Smooth, wavy 2 Stiff clay infill > 5 mm thick Rough, planar 1.5 Soft clay infill > 5 mm thick Smooth, planar 1 Swelling clay > 5 mm Slick and planar 0.5 Filled discontinuities 1 Note: Add + 1 if mean joint spacing > 3 m. 2 4 9 15 1 3 4 4 6 8 12 10 15 20 Table 17. Geomechanics rock mass classes determined from total ratings (AASHTO, 2008, Table 10.4.6.4-3). RMR rating 100–81 80–61 60–41 40–21 <20 Class No. I II III IV V Description Very good rock Good rock Fair rock Poor rock Very poor rock

Table 19. Approximate relationship between rock mass quality and material constants used in defining nonlinear strength (Hoek and Brown, 1988; AASHTO, 2008, Table 10.4.6.4-4). Rock type A = Carbonate rocks with well developed crystal cleavage— dol omite, limestone, and marble B = Lithified argrillaceous rocks— mudstone, siltstone, shale, and slate (normal to cleavage) C = Arenaceous rocks with strong crystals and poorly developed crystal cleavage— sandstone and quartzite D = Fine grained polyminerallic igneous crystalline rocks— andesite, dolerite, diabase, and rhyolite E = Coarse-grained polyminerallic igneous and metamorphic crystalline rocks— amphibolite, gabbro, gneiss, granite, norite, quartz-diorite Rock quality Constants A B C D E INTACT ROCK SAMPLES Laboratory size specim ens free from discontinuities. CSIR rating: RM R = 100 m s 7.00 1.00 10.00 1.00 15.00 1.00 17.00 1.00 25.00 1.00 VERY GOOD QUALITY ROCK MASS Tightly interlocking undisturbed rock with unweathered joints at 3–10 ft. CSIR rating: RM R = 85 m s 2.40 0.082 3.43 0.082 5.14 0.082 5.82 0.082 8.567 0.082 GOOD QUALITY ROCK MASS Fresh to slightly weathered rock, slightly disturbed with joints at 3– 10 ft. CSIR rating: RM R = 65 m s 0.575 0.0029 3 0.821 0.0029 3 1.231 0.0029 3 1.395 0.00293 2.052 0.00293 FAIR QUALITY ROCK MASS Several sets of m oderately weathered joints spaced at 1–3 ft. CSIR rating: RM R = 44 m s 0.128 0.0000 9 0.183 0.0000 9 0.275 0.0000 9 0.311 0.00009 0.458 0.00009 POOR QUALITY ROCK MASS Num erous weathered joints at 2 to 12 in; some gouge. Clean com pacted waste rock. CSIR rating: RM R = 23 m s 0.029 3 x 10 -6 0.041 3 x 10 -6 0.061 3 x 10 -6 0.069 3 x 10 -6 0.102 3 x 10 -6 VERY POOR QUALITY ROCK MASS Nu me rous heavily weathered joints spaced < 2 in with gouge. Waste rock with fines. CSIR rating: RM R = 3 m s 0.007 1 x1 0 -7 0.010 1 x1 0 -7 0.015 1 x1 0 -7 0.017 1 x1 0 -7 0.025 1 x1 0 -7 50 rock is determined from tests on intact rock core. Where tests on rock core are not practical, Poisson’s ratio may be esti- mated from Table 21. 1.8.4 Summary A common way of determining the rock mass strength is by using a failure criterion. The existing rock mass failure criteria are stress dependent and often include one or sev- eral parameters that describe the rock mass properties. These parameters are usually based on classification or char- acterization systems. The unconfined compressive strength, block size and shape, joint strength, and a scale factor are the most important parameters that should be used when esti- mating the rock mass strength. Based on findings, selected systems and criteria have been discussed in this chapter. These include RMR, GSI, and the Hoek-Brown criterion. GSI is similar to RMR, but incorporates newer versions of Bieniawski’s original system (Bieniawski 1976, 1989). The Hoek-Brown criterion is the most widely used failure criterion for estimating the strength of jointed rock masses despite its lack of a theoretical basis and the limited amount of exper- imental data that went into the first development of the cri- terion (Sjöberg, 1997).

51 Table 20. Estimation of Em based on RQD (O’Neill and Reese, 1999; AASHTO, 2008, Table 10.4.6.5-1). Em/EiRQD (percent) Closed joints Open joints 100 1.00 0.60 70 0.70 0.10 50 0.15 0.10 20 0.05 0.05 Table 21. Summary of Poisson’s Ratio for intact rock (AASHTO, 2008, Table C10.4.6.5-2, modified after Kulhawy, 1978). Poisson's Ratio, Rock type No. of values No. of rock types Minimum Maximum Mean Standard deviation Granite 22 22 0.39 0.09 0.2 0.08 Gabbro 3 3 0.2 0.16 0.18 0.02 Diabase 6 6 0.38 0.2 0.29 0.06 Basalt 11 11 0.32 0.16 0.23 0.05 Quartzite 6 6 0.22 0.08 0.14 0.05 Marble 5 5 0.4 0.17 0.28 0.08 Gneiss 11 11 0.4 0.09 0.22 0.09 Schist 12 11 0.31 0.02 0.12 0.08 Sandstone 12 9 0.46 0.08 0.2 0.11 Siltstone 3 3 0.23 0.09 0.18 0.06 Shale 3 3 0.18 0.03 0.09 0.06 Limestone 19 19 0.33 0.12 0.23 0.06 Dolostone 5 5 0.35 0.14 0.29 0.08

Next: Chapter 2 - Research Approach »
LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures Get This Book
×
MyNAP members save 10% online.
Login or Register to save!
Download Free PDF

TRB’s National Cooperative Highway Research Program (NCHRP) Report 651: LRFD Design and Construction of Shallow Foundations for Highway Bridge Structures explores recommended changes to Section 10 of the American Association of State Highway and Transportation Officials’ Load Resistance Factor Design Bridge Design Specifications for the strength limit state design of shallow foundations.

Appendixes A through H for NCHRP Report 651 are available online.

Appendix A: Alternative Model Background

Appendix B: Findings—State of Practice, Serviceability and Databases

Appendix C: Questionnaire Summary

Appendix D: UML-GTR ShalFound07 Database

Appendix E: UML-GTR RockFound07 Database

Appendix F: Shallow Foundations Modes of Failure and Failure Criteria

Appendix G: Bias Calculation Examples

Appendix H: Design Examples

  1. ×

    Welcome to OpenBook!

    You're looking at OpenBook, NAP.edu's online reading room since 1999. Based on feedback from you, our users, we've made some improvements that make it easier than ever to read thousands of publications on our website.

    Do you want to take a quick tour of the OpenBook's features?

    No Thanks Take a Tour »
  2. ×

    Show this book's table of contents, where you can jump to any chapter by name.

    « Back Next »
  3. ×

    ...or use these buttons to go back to the previous chapter or skip to the next one.

    « Back Next »
  4. ×

    Jump up to the previous page or down to the next one. Also, you can type in a page number and press Enter to go directly to that page in the book.

    « Back Next »
  5. ×

    To search the entire text of this book, type in your search term here and press Enter.

    « Back Next »
  6. ×

    Share a link to this book page on your preferred social network or via email.

    « Back Next »
  7. ×

    View our suggested citation for this chapter.

    « Back Next »
  8. ×

    Ready to take your reading offline? Click here to buy this book in print or download it as a free PDF, if available.

    « Back Next »
Stay Connected!