Journal of Marine Science and Engineering Article The Bearing Capacity Evaluations of a Spread Footing on Single Thick Stratum or Two-Layered Cohesive Soils Chao-Ming Chi * and Zheng-Shan Lin Department of Civil Engineering, Feng Chia University, Taichung 4072, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-4-2451-7250 (ext. 3120) Received: 6 September 2020; Accepted: 24 October 2020; Published: 29 October 2020 Abstract: Nowadays many countries plan to increase the percentages of renewable energies by developing offshore wind power. Due to the large sizes of offshore foundations, such as spudcan footings of jack-up barges or pile anchors of wind turbines, the affected soil depth range caused by the foundation under loading can be relatively deep, so the affected range may include a single thick layer or stratified soils. This paper utilizes limit analysis and FLAC numerical simulation to investigate the bearing capacity of a footing on single thick stratum or two-layered cohesive soils. Under nature deposition condition, the undrained shear strength of most cohesive soil approximately increases linearly as the depth increases. The closed-form upper bound solutions of fully rough or fully smooth footings on thick cohesive soils are provided, for the purpose of fast evaluations in practical engineering, and the outcomes are within the results from the FLAC simulation and slip circle method. The problems of punch-through shear failure or soil squeezing could be critical for two-layered soils under some conditions, and the associated bearing factors and the failure mechanisms from different methods are demonstrated and discussed in the article. Keywords: offshore wind power; jack-up barges; bearing capacity of foundations; cohesive soil squeezing; punch-through shear failure; FLAC numerical simulation 1. Introduction The ultimate bearing capacity of strip foundations on a homogeneous and isotropic soil layer is generally estimated by using Terzaghi’s [1] expression. The exact solution of the bearing capacity factor, Nc = p + 2, for perfectly rough surface footing resting on homogeneous and isotropic undrained cohesive soils is proposed by Prandtl [2] and for perfectly smooth footing is estimated by Hill [3]. However, in reality, the undrained shear strength profile of cohesive soils usually does not satisfy the homogeneous and isotropic conditions. Under nature deposition conditions, the strength profile of cohesive soils might vary with depth. For example, the self-weight of the soil causes a decrease in void ratio with depth, and this decrease often causes an approximately linear increase in strength with depth, particularly in normally-consolidated clays (Figure1)[4,5]. There are several on-going offshore wind farm constructions in Taiwan Strait. The in-situ soil conditions of Taiwan offshore wind farm locations are complicated, and it is common that the affected soil depth range caused by the foundation contains different soil types strata, such as clay, silt or sand layers. Because the silty soils exhibit the undrained behavior during shear, they could be regarded as cohesive soils in the analysis of foundation bearing capacity [6]. Brown and Meyerhof [7], Merifield et al. [8], J. Mar. Sci. Eng. 2020, 8, 853; doi:10.3390/jmse8110853 www.mdpi.com/journal/jmse J. Mar. Sci. Eng. 2020, 8, 853 2 of 19 and SNAME [9] indicate that there are at least three basically different foundation failure mechanisms that should be considered in layered cohesive soils: (I) General shear failure occurs if the soil strengths of the following layers do not vary significantly; (II) Squeezing should be considered if the footing is placed on a soft cohesive soil layer overlying a strong layer; (III) Punch-through failure must be of particular note if the footing is placed on a strong layer that overlies a weak layer. To estimate the ultimate bearing capacity of strip footings resting on non-homogeneous cohesive soils, Reddy and Srinivasan [10], Brown and Meyerhof [7], Davis and Booker [11], Chen [12], and Merifield et al. [8] developed different analysis models and methods. Davis and Booker [11] provided the exact solutions of bearing capacity of foundations on cohesive soils with the undrained shear strength increasing with the depth (Figure1) based on the method of characteristics. Brown and Meyerhof [7] proposed semi-empirical bearing capacity factors for layered clay by conducting a series of model tests. Merifield et al. [8] applied numerical limit analysis to evaluate the bearing capacity for layered clay and Reddy and Srinivasan [10] and Chen [12] calculated upper bound solutions assuming a simple circular failure surface (Figure2). The plastic collapse load of the foundations can be solved by the upper bound method or the lower bound method, and the upper bound method is much easier to apply than the lower bound method and widely used. To study the ultimate bearing capacity of a strip footing on the surface of a cohesive soil with strength increasing with depth and stratified cohesive soils, this paper applies the upper bound method and FLAC numerical simulation. The geometric (top layer thickness H and footing width B) and soil strength (top layer undrained shear strength Su,top and bottom layer undrained shear strength Su,bot) parameters used in FLAC analysis cover most of the possible ranges. Brown and Meyerhof [7], Meyerhof and Hanna [13], and Merifield et al. [8] indicate the bearing capacity of a foundation on two-layered cohesive soils is the function of the normalized layer thickness (H/B) and strength ratio (Su,bot/Su,top). Furthermore, the bearing capacity is not affected by subsequent layers for a weaker layer overlying a stronger layer system (Su,bot > Su,top) when H/B ≥ 0.7, or stronger layer overlying softer layer system (Su,bot < Su,top) when H/B ≥ 3.0 according to the laboratory testing study by Brown and Meyerhof [7]. In summary, the properties applied in FLAC numerical simulation, 0.125 ≤ H/B ≤ 2.0 and 0.2 ≤ Su,bot/Su,top ≤ 2.0, cover most problems of practical interest. Figure 1. Soil profile of the strength increasing with depth linearly. J. Mar. Sci. Eng. 2020, 8, 853 3 of 19 Figure 2. Two-layered cohesive soil strength profile. 2. Background Reddy and Srinivasan [10] and Chen [12] employed the slip circle method (Figure3) to compute the ultimate bearing capacity (qult) of a strip footing on two-layered cohesive soils by Equation (1) r 2 qult ( B ) Nc = = r × (2q + 2nq1) (1) Su,top ( B )sinq − 0.5 where B is width of foundation, r is the radius of the slip circle, q is an angle, n is the relative strength and could be given by S n = u,bot − 1 (2) Su,top and q1 could be expressed as H q = cos−1 cosq + (3) 1 r where H is the distance from the bottom of the footing to the interface of the two cohesive soil layers. Figure 3. Sketch of failure surface of two-layered slip circle method (modified from [10]). For a least upper bound value of Equation (1) corresponding to the circle, the following conditions must be satisfied 8 ¶Nc <> = 0 ¶q (4) ¶N :> c = 0 ¶r J. Mar. Sci. Eng. 2020, 8, 853 4 of 19 The bearing capacity factors, Nc, of layered cohesive soils could be computed by Equations (1) and (4), and the values of Nc with various normalized layered thicknesses (H/B) and the relative strength (n) are given in Figure4a. It can be seen from this figure that if H/B remains a constant value, the value of Nc increases as the relative strength (n) increases until reaching a critical relative strength (ncritical). For example, the critical relative strength is 0.07 for H/B = 0.5 and the Nc value would be a plateau value after n ≥ ncritical. On the other hand, the critical relative strength decreases as H/B increases. When H/B is 0.66, the critical relative strength is 0.0 or Su,bot/Su,top = 1.0 and the plateau value of Nc is 5.52. It can be inferred that the bearing capacity would not be affected by the subsequent layer after n ≥ ncritical. Additionally, the value of Nc for the n < 0 side should be less or equal to 5.52, the bearing factor of the foundation on the top stronger layer, and it decreases to 5.52 for the n > 0 side once the normalized layered thickness is greater than 0.66. Therefore, all the curves in Figure4a should be bounded and the corrected results are shown in Figure4b. It can be further seen in this figure that the foundation failure mechanism changes with varying n value. For instance, as n value increases for H/B = 0.25, it could be inferred that the foundation failure mode transfers from punch-through shear failure, general shear failure, to squeezing while the value of Nc reaches the plateau value (7.97). (a) Nc Uncorrected (b) Nc Corrected Figure 4. Bearing capacity factor Nc of various relative strength and normalized layered thickness for strip footings. Considering only single cohesive stratum with undrained shear strength varying linearly with depth (Figure1) can be given by Su = Su0 + kz (5) the expression for Nc is shown as the following equation r 2 qult ( B ) kB r Nc = = r × 2q + 2 × × (sinq − qcosq) (6) Su0 ( B )sinq − 0.5 Su0 B where Su0 is the soil strength at the surface, and k is the rate of increase in undrained strength with depth. 3. Numerical Verification and Comparison to Traditional Limit Analysis This paper applies the upper bound method and FLAC numerical simulation to analyze the ultimate bearing capacity of a rigid strip footing resting on: (I) a cohesive soil layer with increasing undrained shear J.