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66 Table 24. Summary of database UML-GTR RockFound07 cases used for foundation capacity evaluation. No. No. of Location Foundation No. of Size range of rock Shape South type sites (ft) USA Canada Italy UK Australia Taiwan Japan Singapore Russia cases types Africa Shallow Square 4 0.07 < B < 23 Foundations 331 22 10 Circular 2 1 1 3 13 0 1 0 1 0 Bavg = 2.76 (D = 0) 29 Shallow Circular 0.23 < B < 3 Foundations 28 8 2 0 0 0 8 0 0 0 0 0 0 28 Bavg = 1.18 (D > 0) Rock Circular 0.33 < B < 9 61 49 14 19 4 1 0 21 1 0 1 0 2 Sockets 61 Bavg = 2.59 1 Three (3) cases had been omitted in the final statistics due to a clay seam in the rock. diameter (B) ranging from 0.33 ft to 9 ft with an average (Bavg) bearing capacity is taken as equal to this stress. Ultimate bearing of 2.59 ft. Table 24 presents a summary of the database case capacity is the stress at which sudden catastrophic settlement histories breakdown based on foundation type, embedment, of a foundation occurs. Bearing capacity and ultimate bear- sites, size and country. It can be inferred from Table 24 that most ing capacity define the ULS and differ only in the foundation of the shallow foundation and rock socket data were obtained response to load. Appendix F presents a review of foundation from load tests carried out in Australia and the United States, modes of failure and suggests that the terms "bearing capacity" respectively. and "ultimate bearing capacity" should be used interchangeably to define the maximum loading (capacity) of the ground, depending on the mode of failure. 3.3 Determination of the Measured Strength Limit State for Foundations Under 3.3.2 Failure (Ultimate Load) Criteria Vertical-Centric Loading 3.3.2.1 Overview--Shallow Foundations on Soils 3.3.1 Overview The strength limit state is a "failure" load or the ultimate The strength limit state of a foundation may address two capacity of the foundation. The bearing capacity (failure) can kinds of failure: (1) structural failure of the foundation material be estimated from the curve of vertical displacement of the itself and (2) bearing capacity failure of the supporting soils. footing against the applied load. A clear failure, known as a While both need to be examined, this research addresses the general failure, is indicated by an abrupt increase in settle- ULSs of the soil's failure. The ULS consists of exceeding the ment under a very small additional load. Most often, however load-carrying capacity of the ground supporting the founda- (other than for small scale plate load tests in dense soils), test tion, sliding, uplift, overturning, and loss of overall stability. load-settlement curves do not show clear indications of bear- In order to quantify the uncertainty of an analysis, one needs ing capacity failures. Depending on the mode of failure, a clear to find the ratio of the measured ("actual") capacity to the cal- peak or an asymptote value may not exist at all, and the failure culated capacity for a given case history. The measured strength or ultimate load capacity of the footing has to be interpreted. limit state (i.e., the capacity) of each case needs, therefore, to Appendix F provides categorization of failure modes fol- be identified. lowed by common failure criteria. The interpretation of the Depending on the footing displacements, one may define failure or ultimate load from a load test is made more complex (1) allowable bearing stress, (2) bearing capacity, (3) bearing by the fact that the soil type or state alone does not determine stress causing local shear failure, and (4) ultimate bearing the mode of failure (Vesic , 1975). For example, a footing on capacity (Lambe and Whitman, 1969). Allowable bearing stress very dense sand can also fail in punching shear if the foot- is the contact pressure for which the footing movements are ing is placed at a greater depth, or if loaded by a transient, within the permissible limits for safety against instability and dynamic load. The same footing will fail in punching shear functionality, hence defined by SLS. Bearing capacity is that if the very dense sand is underlain by a compressible stratum contact pressure at which settlements become very large and such as loose sand or soft clay. It is clear from the above dis- unpredictable because of shear failure. Bearing stress causing cussion that the failure load of a footing is clearly defined only local shear failure is the stress at which the first major non- for the case of general shear; for cases of local and punching linearity appears in a load-settlement curve, and generally the shear, it is often difficult to establish a unique failure load.

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67 Criteria proposed by different authors for the failure load 3.3.2.3 The Uncertainty in the Minimum Slope interpretation are presented in Appendix F, while only the Failure Criterion Interpretation selected criterion is presented in the following section. Such In order to examine the uncertainty in the method selected interpretation requires that the load test be carried to very large displacements, which constrains the availability of test for defining the bearing capacity of shallow foundations on soils, data, in particular for larger footing sizes. the following failure criteria (described in detail in Appendix F) were used to interpret the failure load from the load-settlement curves of footings subjected to centric vertical loading on 3.3.2.2 Minimum Slope Failure (Ultimate) granular soils (measured capacity): (a) minimum slope cri- Load Criteria, Vesic (1963) terion (Vesic , 1963), (b) limited settlement criterion of 0.1B Based on the load-settlement curves, a versatile ultimate load (Vesic, 1975), (c) log-log failure criterion (De Beer, 1967), and criterion is recommended to define the ultimate load at the (d) two-slope criterion (shape of curve). point where the slope of the load-settlement curve first reaches Examples F1 and F2 in Appendix F demonstrate the zero or a steady, minimum value. The interpreted ultimate application of the four examined criteria to the database loads for different tests are shown as black dots in Figure 53 for UML-GTR ShalFound07. The measured bearing capacity could soils with different relative densities, Dr . For footings on the be interpreted for 196 cases using the minimum slope criterion surface of, or embedded in, soils with higher relative densities, (Vesic, 1963) and 119 cases using the log-log failure criterion there is a higher possibility of failure in general shear mode, and (De Beer, 1967). Most of the footings failed before reaching a the failure load can be clearly identified for Test Number 61 in settlement of 10% of footing width (the limited settlement Figure 53. For footings in soils with lower relative densities, criterion of 0.1B [Vesic , 1975] could therefore only be applied however, the failure mode could be local shear or punching to 19 cases). A single "representative" value of the relevant shear, with the identified failure location being arbitrary at measured capacity was then assigned to each footing case. times (e.g., see Test Number 64). A semi-log scale plot with the This was done by taking an average of the measured capacities base pressure (or load) in logarithmic scale can be used as an al- interpreted using the minimum slope criterion, the limited ternative to the linear scale plot if it facilitates the identification settlement criterion of 0.1B (Vesic , 1975), the log-log fail- of the starting of minimum slope and hence the failure load. ure criterion, and the two-slope criterion (shape of curve). Figure 53. Ultimate load criterion based on minimum slope of load-settlement curve (Vesic , 1963).

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68 120 60 no. of data = 196 100 mean = 0.978 50 Relative frequency (%) No. of footing cases COV = 0.053 80 40 60 30 40 20 20 10 0 0 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Ratio of "representative" capacity to the capacity interpreted using minimum slope criterion Figure 54. Histogram for the ratio of representative measured capacity to Figure 55. Example of L1-L2 method interpreted capacity using the minimum for capacity of foundations on slope criterion for 196 footing cases in rocks showing the regions of the granular soils under centric vertical loading. load-displacement curve and interpreted limited loads (Hirany and Kulhawy, 1988). The statistics of the ratios of this representative value over the interpreted capacity using the minimum slope criterion curves, the L1-L2 method proposed by Hirany and Kulhawy and the log-log failure criterion were comparable with the (1988) was adopted. mean of the ratio for the minimum slope criterion being 0.98 A typical load-displacement curve for foundations on rock is versus that for the limited settlement criterion being 0.99. presented in Figure 55. Initially, linear elastic load-displacement Due to the simplicity and versatility of its application, the relations take place; the load defining the end of this region is minimum slope criterion was selected as the failure inter- interpreted as QL1. If a unique peak or asymptote in the curve pretation criterion to be used for all cases of footing, includ- exists, this asymptote or peak value is defined as QL2. There is ing those with combined loadings. Figure 54 shows the histo- a nonlinear transition between loads QL1 and QL2. If a linear gram for the ratio of the representative measured capacity to region exists after the transition, as in Figure 55, the load at the the interpreted capacity using the minimum slope criterion. start of the final linear region is defined as QL2. In either case, Figure 54 presents the uncertainty associated with the use of QL2 is the interpreted failure load. This criterion is similar to the the selected criterion, suggesting that the measured capacity aforementioned minimum slope failure proposed by Vesic interpreted using the minimum slope criterion has a slight for foundations in soil. The selection of the ultimate load using overprediction. this criterion is demonstrated in Example F3 of Appendix F using a case history from the UML-GTR RockFound07 data- base. It can be noted that the axes aspect ratios (scales of axes 3.3.3 Failure Criterion for Footings on Rock relative to each other) in the plot of the load-settlement curve The bearing capacity interpretation of loaded rock can changes the curve shape, and thus could affect the inter- become complex due to the presence of discontinuities in the pretation of the ultimate load capacity. However, unlike the rock mass. In a rock mass with vertical open discontinuities, interpretation of ultimate capacity from pile load tests, which where the discontinuity spacing is less than or equal to the utilizes the elastic compression line of the pile, there is no footing width, the likely failure mode is uniaxial compression generalization of what the scales of the axes should be relative of rock columns (Sowers, 1979). For a rock mass with closely to each other for the shallow foundation load tests. It can only spaced, closed discontinuities, the likely failure mode is the be said that depending on the shape of the load-settlement curve, general wedge occurring when the rock is normally intact. For a "favorable" axes aspect ratio needs to be fixed. This should a mass with vertical open discontinuities spaced wider than the be done on a case-by-case basis, using judgment, so that the footing width, the likely failure mode is splitting of the rock region of interest (e.g., if the minimum slope criterion is mass and is followed by a general shear failure. For the inter- used, the region where the change in the curve slope occurs) pretation of ultimate load capacities from the load-settlement is clear. The L1-L2 method was applied to all cases for which