Effect of combined translation and torsion on undrained uplift capacity of plate anchors: plastic limit analysis (PLA) solution Hamid Reza Nouri1, Giovanna Biscontin2 and Charles P. Aubeny, M. ASCE3 ABSTRACT Uplift capacity of plate anchors have been the focus of numerous studies, since anchor plates are designed for pull-out in normal operating conditions. However, the response of plate anchors under six-degrees-of-freedom loading caused during extreme loading conditions is poorly understood. The purpose of this study is to propose a simple yet sufficiently accurate analytical solution to investigate the behavior of plate anchor under combined in-plane translation and torsion and to evaluate its effect on the plate uplift bearing capacity. To this end, a modified plastic limit analysis (PLA) approach is introduced and compared with limit equilibrium (LE) and simplified upper bound baseline solutions. The proposed method is verified with three dimensional finite element (3D-FE). The variables considered in this study include plate aspect ratio, plate thickness, as well as load direction and eccentricity. Results of analytical solutions indicate the insensitivity of the “shape” of the shear-torsion yield envelope to plate thickness. This finding facilitates the use of simplified yet reasonable yield envelope for infinitely thin plate obtained from simplified PLA approach for other plate thicknesses. The “size” of the failure envelope (controlled by pure torsional and translational capacity) could be predicted fairly accurately by PLA and LE methods. Combination of these analytical methods offers a simple yet reasonably accurate solution to describe shear-torsion response of anchor plate. The obtained shear-torsion yield envelope is then fitted in the generalized six-degrees-of-freedom yield surface which describes the reducing effect of moment, torsion, and planar forces on the uplift capacity of plate. KEY WORDS: plate anchor, torsional and translational bearing capacity, plastic limit analysis, limit equilibrium, finite element analysis, yield envelope. 1 Senior Geotechnical Engineer, Shannon and Wilson, Inc., 400 North 34th Street Suite 100, P.O. Box 300303 Seattle, Washington, USA 98103. Ph.: +1 (206) 695-6924, E-mail: Nouri.hreza@gmail.com; hrn@shanwil.com 2 University Lecturer in Geotechnical Engineering, University of Cambridge, Department of Engineering, Schofield Centre, High Cross, Madingley Road, Cambridge, UK CB3 0EL. Ph.: +44 (1223) 768-044, E-mail: gb479@cam.ac.uk 3 Professor, Texas A&M University, Civil Engineering CE/TTI, 3136 TAMU, College Station, TX 77843-3136, USA. Ph.: (979) 845-4478, E-mail: caubeny@civil.tamu.edu INTRODUCTION 1 In recent years the growing trend in searching and developing hydrocarbon reserves has forced 2 offshore oil and gas industry into deep and ultra deep water. It becomes increasingly difficult and 3 costly to keep the natural period of traditional fixed jacket or gravity-based structures below the 4 dominant frequency of the sea wave spectrum in water depths exceeding 500 m (Aubeny et al. 5 2001). Thus, floating structures anchored to the seabed using catenary or taut-wire moorings has 6 taken the place of conventional platforms (Aubeny et al. 2001). Plate anchors are increasingly 7 being used to moor large floating offshore structures in deep and ultra deep water. 8 Plate anchors are installed to impart substantial uplift loading in normal operating conditions. 9 However, they could be subjected to six-degrees-of-freedom general loading as well. The recent 10 hurricane events in the Gulf of Mexico resulted in partial failure and drift of 17 deepwater 11 Mobile Offshore Drilling Units (MODUs) in hurricanes Ivan, Katrina, and Rita. The drift of the 12 platform causes a significant change in the orientation and amount of the resultant force, which 13 appears as a substantial out-of-plane force acting on the anchor of the intact line. This could lead 14 to substantial reduction in uplift capacity of the plate, failure of the anchor, and catastrophic 15 damages to adjacent oil and gas infrastructure through colliding with other exploration and 16 production platforms and rupturing the network of oil and gas pipelines by dragging anchors of 17 the failed line on the seabed. Importance of understanding the plate anchor response under out-18 of-plane loading is not limited to design considerations under extreme loading conditions. It is a 19 frequent practice for operators to work on two close wells in the same field. Instead of relocating 20 the drilling unit and its mooring system, it is more efficient to winch between wells on one 21 mooring system without removing and reinstalling anchors. If the wells are far enough, anchors 22 will be subjected to out-of-plane loading. 23 The undrained uplift bearing capacity of deeply embedded plates has been the focus of the 24 majority of previous studies, while the plate bearing capacity under general loading conditions 25 has received considerably less attention. The purpose of this study is to develop adequately 26 accurate yet simple analytical solutions to quantify the decreasing effect of translation and 27 torsion on pull-out bearing capacity of plate anchors. These simplified approaches are more 28 attractive solutions for practical purposes compared to expensive numerical simulation. 29 This study idealizes the geometry of the plate anchor as a rectangular plate of length L, width W 30 and thickness t subjected to eccentric loading (Fig. 1): L is taken as the shorter dimension, 31 aligned with the x-axis. Also assuming a homogeneous condition for soil, any spatial variation of 32 soil parameters is neglected in this study (i.e. constant undrained shear strength). In general, any 33 eccentricity angle ψ or load angle γ may occur. First, a number of simple baseline solutions for 34 pure translation (e = 0 or γ = 90°), and pure torsion (γ ≠ 90°and e = ∞) are derived from limit 35 equilibrium (LE). For general conditions of eccentricity and load orientation for infinitely thin 36 plate, a semi-analytical upper-bound limit analysis based approach is proposed. For a plate of 37 finite thickness under general eccentric load, a more powerful method is required. To this end, 38 the upper-bound plastic limit analysis (PLA) proposed by Yang et al. (2010) for infinitely thin 39 plates is adopted and modified to better predict the bearing capacity for plates with finite 40 thickness. The accuracy of the baseline solutions and PLA approach are evaluated through 41 comparisons to three-dimensional finite element solutions. The proposed analytical solutions are 42 used to develop a simplified approach to incorporate the shear-torsion interaction equation in the 43 generalized yield envelope for plate anchor under six-degrees-of-freedom loading condition. The 44 complete generalized yield surface is used to evaluate the uplift capacity of anchor plates under 45 combined eccentric translational/planar forces and out-of-plane moments. 46 Figure 1- Schematic of eccentrically loaded plate BACKGROUND 47 The majority of previous studies concerning the bearing capacity of plates are focused on 48 undrained uplift capacity and limited literature is available on the bearing capacity of plate 49 anchors subjected to general loading condition. O’Neill et al. (2003) used two-dimensional finite 50 element (2D-FE) analysis to investigate the behavior of rectangular and wedge-shaped strip 51 anchors subjected to combined translational, vertical, and rotational loadings. They also 52 developed plastic upper bound solutions to evaluate plane strain ultimate parallel, normal, and 53 rotational plate capacity factors to validate the FE results. They produced FE based yield 54 surfaces and developed an approach to predict the kinematics and trajectory of drag anchor 55 during the installation. Murff et al. (2005) used 2D-FE to develop yield loci for plates of 56 different thickness and roughness. Yang et al. (2010) employed 3D-FE, as well as plastic upper 57 bound limit analysis, to study the behavior of infinitely thin plates subjected to six-degrees-of-58 freedom loading. They also introduced a plasticity solution to determine yield loci of an 59 infinitely thin plate under combined translation-torsion. They developed equations to calculate 60 pure sliding and torsional bearing capacity. Nouri (2013) conducted 3D-FE analysis to evaluate 61 the response of plate anchors with finite thickness under in-plane two-way translation and 62 torsion. Their 3D-FE results indicated the insensitivity of the shape of yield envelope (i.e. plate 63 interaction response under combined loading) to thickness variations of square and rectangular 64 anchors. 65 Yang et al. (2010) Upper Bound PLA Solution: Overview 66 Yang et al. (2010) developed an upper bound plastic limit analysis (PLA) formulation for deeply 67 embedded square and rectangular plate anchors under combination of co-planar sliding and 68 torsion. They assumed deep embedment for the plate anchor, which ensures “no separation” of 69 undrained cohesive soil in contact with the foundation. The full attachment of the foundation and 70 soil complies with the normality concept (Tan, 1990) which is essential to apply the limit 71 analysis approach in soils (Chen and Liu, 1990). This assumption seems valid since due to the 72 low permeability of clays in combination with a high loading rate, suction will be generated on 73 faces of the plate (Wang et al. 2010). 74 In their approach a plate is assumed subjected to a force, F, in the plane parallel to the plate 75 faces, passing through its center (Fig. 2). If the plate is subjected to a virtual rotation rate, β , 76 around point O(xo, yo), F could be calculated by equating the work done by external force, F, and 77 the total rate of energy dissipation in the vicinity of plate failure: 78 ( ) ( )cos sin s e f o f o D DF x x y yφ φ β + =  − + −     (1) where the total rate of energy dissipation is equal to the rate of energy dissipated by sliding along 79 the top and bottom surfaces of the plate ( sD ) in addition to the rate of energy dissipated by the 80 soil resistance on the plate edges ( eD ) (see Appendix A), and ϕ=ψ+γ is the angle between line of 81 action of the external force and y-axis. The minimum value for the force, F, can be determined 82 by optimizing the kinematics of the plate rigid body motions with respect to the coordinates of 83 the rotation center of the rotating plate, O(xo, yo). The pure sliding (shear loading) occurs when 84 the line of action of the force, F, is passing through the center of the plate and pure torsion 85 occurs when the eccentricity of the line of action of the force, F, with respect to center of the 86 plate is infinity (e.cos(γ)= ∞). The rate of energy dissipated along an arbitrary element of dx by 87 dy on top or bottom surfaces of the anchor plate with center at (x, y) and distance of 88 ( ) ( )2 2 20 0( , ) ( / 2)R x y x x y y t= − + − + from the center of rotation (Fig. 2(a)), is 89 ( , )s udD s R x y dxdyα β=  which can be integrated over the plate area (see Appendix A). The 90 undrained shear strength, su, along top and bottom plate surfaces is assumed to be fully 91 developed in Yang et al. (2010) formulation. The adhesion factor (α) for fully bonded plate and 92 soil is assumed to be unity. The second dissipation term ( eD ) due to the resistance of the soil on 93 the four ends of the plate could be resolved into sliding (Fst) and normal (Fnt) components (Fig. 94 2(b)). As these terms are dependent on the plate thickness, they vanish for infinitely thin plates. 95 Yang et al. (2010) assumes that both the normal (Fnt) and sliding (Fst) soil resistances on plate 96 ends are fully mobilized at failure. Thus, no interaction is assumed between normal and 97 tangential forces along the plate edges. 98 (a) (b) Figure 2- Upper bound PLA mechanism for shear-torsion loading of a plate anchor: (a) plan view of the plate and mechanism; (b) Detail D-1: isometric view of the plate edge and acting normal (Fnt) and tangential (Fst) forces Assuming an arbitrary element of t (plate thickness) by dl (= dx and dy for edges parallel to x and 99 y axis respectively) on each edges of the plate (Fig. 2), the sliding and normal components are 100 calculated (i.e. . . .st udF s t dlα= and . . .nt e udF N s t dl= ). Ne is a simple plane strain bearing 101 capacity factor for each element to determine the normal component and is assumed equal to 102 Ne=7.5 (O’Neill et al. 2003). The adhesion factor (α) of unity is also adopted for fully attached 103 plate and soil. Total values of normal and tangential portions of soil resistance on the plate edges 104 are determined by integrating along each plate edge (see Appendix A). 105 By substituting the dissipation terms in Eq. 1 and canceling out the angular velocity, β , an 106 expression is obtained in terms of xo and yo for external force, F. A least upper bound is obtained 107 by minimizing F with respect to the rotation coordinates (xo, yo). 108 Issues with the existing Yang et al. (2010) PLA Solution 109 Tables 1 and 2 summarize the results of the Yang et al. (2010) PLA solutions as well as the 3D-110 FE developed by Nouri (2013) for the ultimate shear and torsion bearing capacity of the square 111 and rectangular (W/L = 2) plates of various thicknesses (t/L = 0, 1/20, 1/14, 1/10, and 1/7). The 112 Yang et al. (2010) PLA solution developed by Yang et al. (2010) seem to reasonably predict the 113 ultimate shear bearing capacity (Ns,max) regarding its satisfactory agreement with the 3D-FE 114 results for plate of different geometry. However, the unconservative over-prediction of the 3D-115 FE results for maximum torsional resistance (Nt,max) by the PLA solution and the growing 116 difference for the thicker plates suggest the inaccuracy of the current PLA formulation to predict 117 torsional resistance for plates of larger thickness. 118 Fig. 3 plots present the PLA and 3D-FE derived normalized shear-torsion interaction curve for 119 rectangular (W/L=2) plates. Comparison of Hx/Hx,max - T/Tmax yield envelopes in Fig. 3(a) 120 indicates the unconservative tendency of the PLA approach to over-predict the bearing capacity 121 by 20-30%. The over-prediction trend is also repeated in PLA derived Hy/Hy,max - T/Tmax 122 interaction curves for rectangular and square plates (Figs. 3(b) and 4, respectively). PLA 123 envelopes show a vertical non-interacting portion for low values of torsional resistance 124 (T/Tmax<0.3-0.4), which is not observed in 3D-FE envelopes. This vertical section indicates that 125 the PLA approach predicts no change of the shear resistance by the plate torsion which ends in 126 overestimating the bearing capacity values. This non-interacting vertical portion in the PLA 127 envelopes shortens for the plates of smaller thickness and eventually vanishes for infinitely thin 128 plate (t =0), since the resistance for thinner plates is solely controlled by the energy dissipated on 129 the top and bottom surfaces of the plate. This could also confirm the inaccuracy of simplifying 130 assumption on full mobilization of shear and normal forces acting on the edges of the plate. 131 Fig. 5 also presents PLA and 3D-FE normalized interaction curves for anchor plate under two-132 way in-plane translation (Hx/Hx,max - Hy/Hy,max) for square and rectangular (W/L = 2) plates with 133 thicknesses of t = L/20 and L/7. PLA unconservative tendency to over-estimate the shear bearing 134 capacity is repeated similar to shear-torsion yield envelops. The non-interacting segment appears 135 again at both ends of the yield envelope where maximum shear resistance is expected to diminish 136 due to the shear resistance mobilization in another perpendicular direction according to 3D-FE 137 results. This unconservative trend is more pronounced for plate anchors of smaller aspect ratio 138 and higher thicknesses, as the yield envelope for rectangular plate of t = L/20 nearly tracks 3D-139 FE data points. 140 Approaching the results of Yang et al. (2010) PLA and 3D-FE yield envelopes and ultimate 141 torsional resistance for smaller thicknesses suggests that Yang’s assumption on the negligible 142 interaction of sliding (Fst) and normal (Fnt) forces on the edges of the plate at soil failure 143 condition is not accurate for plates of finite thickness as also suggested by Yang et al. (2010). 144 (a) (b) Figure 3- Comparison of Yang et al. (2010) PLA and 3D-FE predictions for rectangular (W/L=2) plates under combined: (a) shearx-torsion; (b) sheary-torsion Figure 4- Comparisons of the Yang et al. (2010) PLA and 3D-FE predictions for square (W/L=1) plate under combined shear-torsion Figure 5- Comparisons of the Yang et al. (2010) PLA and 3D-FE predictions for square and rectangular (W/L=2) plates under coplanar shearx - sheary As shown in Tables 1 and 2, this assumption could be valid for plates subjected to one-way pure 145 translation where the edges are under pure normal or shear forces, but when the plate is subjected 146 to torsion or combined two-way in-plane translation, there will be a combination of shear and 147 normal reaction forces acting on the plate edges. Neglecting the interaction and assuming full 148 mobilization of both sliding and normal components may result in estimating higher dissipated 149 energy due to soil sliding and normal resistance on the plate edges and over-predicting the 150 overall plate resistance especially for thicker plates. Thus, the insensitivity of 3D-FE derived 151 yield surfaces to plate thickness (see Figs. 3, 4, and 5) is not predicted by the current upper 152 bound solution. Modification of the assumption for interaction of sliding-normal forces on the 153 plate edges could be incorporated in the existing PLA solution to improve the current 154 formulation. The modified PLA solution will be proposed in the following sections of this study. 155 Table 1. Comparison of PLA and FE results for the ultimate shear force (Ns,max) and torsion moment (Nt,max) bearing capacity factors for square plate Mode of Plate Loading Plate thickness (t/L) Finite Element Values Upper Bound Solution Difference of FE and Yang et al. (2010) (%) Difference of FE and Current study (%) Yang et al. (2010) Current (modified) Shear, Ns,max 0 2 2 2 0.0 1/20 2.90 2.85 2.85 -1.7 1/14 3.30 3.21 3.21 -2.7 1/10 3.67 3.70 3.70 0.8 1/7 4.37 4.43 4.43 1.4 Torsion, Nt,max 0 0.765 0.76 0.76 -0.7 -0.7 1/20 1.14 1.24 1.16 8.8 1.8 1/14 1.26 1.44 1.34 14.3 6.3 1/10 1.34 1.71 1.57 27.6 17.2 1/7 1.47 2.12 1.90 44.2 29.3 156 Table 2. Comparison of PLA and FE results for the ultimate shear force (Nsx,max, Nsy,max) and torsion moment (Nt,max) bearing capacity factors for rectangular plate (W/L=2) Mode of Plate Loading Plate thickness (t/L) Finite Element Values Upper Bound Solution Difference of FE and Yang et al. (2010) (%) Difference of FE and Current study Yang et al. (2010) Current (modified) Shear, Nsx,max 0 2 2 2 0.0 1/20 2.84 2.80 2.80 -1.4 1/14 3.21 3.14 3.14 -2.2 1/10 3.55 3.60 3.60 1.4 1/7 4.20 4.29 4.29 2.1 Shear, Nsy,max 0 2 2 2 0.0 1/20 2.50 2.48 2.48 -0.8 1/14 2.72 2.68 2.68 -1.5 1/10 2.93 2.95 2.95 0.7 1/7 3.32 3.36 3.36 1.2 Torsion, Nt,max 0 1.19 1.19 1.19 0.0 0.0 1/20 1.66 1.75 1.67 5.4 0.6 1/14 1.84 2 1.89 8.7 2.7 1/10 1.98 2.32 2.18 17.2 10.1 1/7 2.27 2.81 2.59 23.8 14.1 ANALYTICAL BASELINE SOLUTIONS 157 This study also introduces a number of convenient analytical and semi-analytical solutions which 158 can provide useful reference solutions to evaluate the modified PLA and FEA approaches. 159 Pure Sliding and Rotation: Limit Equilibrium Approach 160 Nouri et al. (2014) proposed a limit equilibrium (LE) approach to estimate the pure sliding and 161 torsional resistance for shallow foundations. This study uses the same general LE methodology 162 and failure mechanism to derive bearing capacity factors for pure sliding in the x and y directions 163 for a deeply embedded plate anchor of thickness t, in a homogeneous soil with an adhesion factor 164 α at the soil-plate interface. Following expressions are developed to calculate bearing capacity 165 from the assumed stress distribution and failure mechanism in Fig. 6: 166 x max , 2 2sx max e u H L tN N s LW W L α α = = + +   (2) 167 ymax , 2 2sy max e u H L tN N s LW W L α α = = + +   (3) 168 where su is the soil undrained shear strength, L and W are the smaller and larger sides of the 169 plate, and Ne is the simple plane strain bearing capacity factor equal to 7.5 (O’Neill et al. 2003). 170 The adhesion factor, α, for the fully bonded condition is assumed equal to unity. Note that the 171 results of two and three dimensional sliding resistance are typically normalized by us L and 172 us WL respectively. The above expression for pure x-shear capacity factor, Nsx,max, for a strip plate 173 (L/W ≈ 0) reduces to Ns,max=2(1+Ne.t/L) as proposed by O’Neill et al. (2003) for ultimate parallel 174 plate capacity in 2D plane strain condition. 175 Translation Torsion Figure 6- Assumed stress distribution of plate pure sliding and torsion for limit equilibrium solution The plate aspect ratio (W/L) does not influence the pure shear capacity of the plate anchor under 176 x-sliding, where Nsx (Eq. 2) yields to 4.29 and 4.15 for rectangular (W/L=2) and plane strain strip 177 plate (W/L = 20) of t = L/7 (about 3% different). Evidently, this difference becomes even less for 178 plates of smaller thickness, where in the extreme condition the Nsx becomes independent of W/L 179 NeSu NeSu αSu αSu L W Hxmax Tmax αSu: Shear on all sides NeSu: Bearing on all sides L W for infinitely thin plate (Nsx =2 for square and rectangular plates). 180 For the special case of pure torsion with zero thickness, t = 0, the following LE solution provides 181 a convenient benchmark for ultimate torsional resistance Tmax0 (the subscript ‘0’ referring to zero 182 thickness) to compare with the numerical calculations: 183 /2 /2 2 2 0 /2 /2 2 W L max u W L T s x y dxdy − − = +∫ ∫ (4) 184 Results of the above LE based solution for pure torsion also conforms with Yang et al. (2010) 185 PLA approach where they propose the following closed form solution: 186 2 max 0 0 2 2 2 / ( ) sin cos( / ) /tan tan 6 cos 4 2 6 sin 2 t max u o o o o o o N T s WL W L L Wln lnθ θ θ θπα α θ θ = =          + + + −                   (5) 187 where θo = tan-1(L/W). Note that the results of two and three dimensional moment resistance are 188 typically normalized by 2us L and 2 us WL respectively. 189 For the general case of a plate with finite thickness, we used a limit equilibrium based approach 190 by summing the torsion resistance on the edges of a rotating plate (Fig. 6). We adopted the 191 simplifying assumption of no interaction effects between bearing (normal) and tangential 192 resistance acting on the plate edges (i.e. full mobilization of these components as shown in Fig. 193 6) which yielded the following expression for additional torsional resistance from plate edges: 194 max 1 2 2t e f e W L tN C N L W L α  ∆ = + +     (6) 195 A correction factor, Cf, is included in Eq. 6 to emphasize that some adjustment for interaction 196 effects on plate edges is needed. The total torsional resistance is the sum of Eqs 5 and 6, Ntmax0 + 197 ∆Ntmaxe. This closed-form expression portrays the variables affecting torsional capacity and, as 198 will be seen, offers a simple calibration expression to match the finite element solutions. 199 Combined loading 200 For plate of zero thickness (t/L = 0) an upper bound virtual work analysis provides a useful check 201 for the Yang et al. (2010) PLA and FEA. Nouri et al. (2014) originally introduced this approach 202 to evaluate the resistance of surface foundations (i.e. embedment of zero) under one-way 203 eccentric parallel loading [Fig. 7(b)]. This study generalizes the same approach for a plate of 204 zero thickness subjected to an eccentric translational loading [Fig. 7(a)]: A horizontal load H is 205 applied at a distance eCos(γ) from the center of the plate, with an eccentricity angle of ψ, load 206 angle of γ, and an associated motion about a center of rotation located a distance ρ from the 207 center. Equating external virtual work, W , to internal energy dissipation leads to: 208 ( ) DH eCosρ γ β = +   (7) 209 where β is a virtual angular velocity. This relationship is written based on the assumption that 210 the line perpendicular to external force, H, through center of rotation is passing through the 211 center of the plate. This assumption has been proved to be valid for infinitely thin plate (t/L = 0) 212 through PLA parametric analysis. The validity of this assumption could also be evaluated 213 through assuming the center of rotation in a random location and developing the formulation. 214 However, taking advantage of the PLA proved assumption will keep the solution simple which is 215 the main purpose of developing the baseline formulation. 216 The rate of internal energy dissipation D is the soil resistance times the local velocity integrated 217 over the plate area: 218 ( ) ( ) /2 /2 2 2 /2 /2 2 W L u W L D s x Cos y Sin dxdyβ ρ ψ γ ρ ψ γ − − = + + + + +      ∫ ∫ (8) 219 A least upper bound collapse load is obtained by minimizing H in Eq. 7 with respect to ρ and 220 setting it equal to zero, with the assumption of constant e, ψ, and γ values, which leads to: 221 ρβ ∂ ∂ = DH   1 (9) 222 For the case of zero plate thickness the collapse load then becomes equal to: 223 ( ) ( ) ( ) ( ) /2 /2 2 2/2 /2 2 W L opt u W L opt opt xCos ySin H s dxdy x Cos y Sin ρ ψ γ ψ γ ρ ψ γ ρ ψ γ − − + + + + =    + + + + +    ∫ ∫ (10) 224 where ρopt is the distance to the optimal center of rotation corresponding to a least upper bound. 225 For any arbitrary value of ρopt, D and H could be derived through numerical evaluation of 226 integrals in Eqs. 8 and 10. The eccentricity, e, corresponding to the arbitrary value of ρopt, is 227 obtained from Eq. 7 with known D , H, and ρopt. Parametric study is also possible with 228 evaluating the equations for a sweep of ρopt values. 229 For a special case of eccentricity and loading directions aligned with either the major or minor 230 axis [e.g. eccentricity angle of ψ=0.0° and γ=0.0° as shown in Fig. 7(b)] Eqs. 7, 8, and 10 could 231 be reduced to a more concise form (Nouri et al. 2014): 232 ( ) DH eρ β = +   (11) 233 /2 /2 2 2 /2 /2 2 ( ) W L u W L D s x y dxdyβ ρ − − = + +∫ ∫ (12) 234 /2 /2 2 2/2 /2 2 ( ) W L opt u W L opt x H s dxdy x y ρ ρ− − + = + + ∫ ∫ (13) 235 Although the analytical evaluation of the double integral in Eq. 12 to calculate D generates a 236 long expression, this equation could be reduced to a single integral (Appendix B) appropriate for 237 the simple design spreadsheet calculations. In addition, analytical integration of Eq. 13 results in 238 a closed-form expression for H: 239 2 21 2 1 2 1 2 1 2 1 2 2 2 1 1 2 2 2 2 / 2 / 22 ln ln ( ) / 2 / 2 / 2 / 2 / 4 / 4 u opt opt b W b WH s a a W b b b W b W a L a L b a W b a W ρ ρ  + + = − + −  − −  = + = − = + = + (14) 240 Contrary to the PLA approach, the proposed method requires no search or optimization 241 procedure since the final solution directly relates the eccentricity, e, to the distance to the optimal 242 center of rotation, ρopt, and thus the equivalent external force H=H(ρopt) (Eq.14). This feature 243 also eliminates the complications in optimization procedure in locating the absolute minimum 244 which could be mistaken by the greater local minimums. Regardless of the numerical integration 245 required to evaluate the rate of energy dissipation, the final solution in this method is derived 246 through closed-form expressions easy to implement in spreadsheets. Thus, it can provide a 247 simple robust tool for routine design calculations. 248 Conducting the analysis for a sweep of ρ values generates a predicted reduction in load capacity 249 (H/Hmax) versus eccentricity e for a plate under planar eccentric load aligned with the major or 250 minor plate axis. As shown in Fig. 8, shear resistance for loading in the x and y directions show a 251 significant reduction. Square plates, W/L = 1, experience the greatest decrease in H resistance, 252 with eccentricity as low as e/L = 0.1 reducing load capacity by more than 5% and eccentricity e/L 253 = 0.5 (load application at the edge of the plate) reducing capacity by more than 40%. With 254 increasing aspect ratio W/L the plate anchor becomes progressively more resistant to eccentric 255 loading. The predictions also show that the reduction in capacity is always the greatest for 256 loading in the x direction; i.e., for load orientations normal to the long axis of the plate. Note that 257 torsion is equal to T=H. eCos(γ). Results of this formulation will be referred as PLAt=0, hereafter. 258 (a) (b) Figure 7- Model for virtual work analysis for combined sliding-torsion: (a) generalized condition; (b) eccentricity and loading directions aligned with the axis Figure 8- Reduction of shear capacity with eccentricity for square and rectangular plates of zero thickness Modified PLA solution 259 To improve the Yang et al. (2010) PLA formulation and incorporate the shear-normal force 260 interaction along the edges of the plate, the dissipation term that accounts for the soil resistance 261 along the plate edges, eD , should be revised. Other terms of dissipation (i.e. energy dissipation 262 along the top and bottom surfaces of the plate, sD ) and external work remains unchanged in the 263 revised formulation. 264 The assumed angular velocity, β , of the plate rotating about point O(xo, yo) in Fig. 2, prescribes 265 the relative increments of displacement in the normal and parallel directions with respect to the 266 plate edges as follows (based on the assumption of rigid plate): 267 For AB and CD (sides parallel to x axis): 0 0 n s u x x u y y β β  = −  = −   (15-a) For AD and BC (sides parallel to y axis): 0 0 n s u y y u x x β β  = −  = −   (15-b) where nu and su are displacement increments normal and parallel to the plate vertical end faces, 268 respectively. A general relationship for the sliding-normal interaction curve, f (Nnt, Nst), is 269 adopted as follows: 270 ,max ,max ( , ) 1 0 m n nt st nt st nt st N Nf N N N N     = + − =           (16) where Nnt = Fnt /(su×t×dl) and Nst = Fst /(su×t×dl) are non-dimensional forms of Fnt and Fst 271 (respectively normal and shear forces on the plate edges in Fig. 2) acting on arbitrary element of 272 dimensions t (plate thickness) by dl (= dx and dy for sides parallel to x and y axis respectively) on 273 the plate edges as shown in Fig. 2. Nnt,max =Ne which is the plane strain bearing capacity factor, is 274 typically equal to 7.5 (O’Neill et al. 2003). The maximum sliding force Nst,max is equal to 275 adhesion factor α and assumed to be unity for the plate fully bonded to the soil. The assumption 276 of no detachment is necessary for deriving a limit analysis solution with an associated flow rule, 277 but the adhesion factor could take any value between zero and unity. Here the assumption of α=1 278 is consistent with the fully bonded condition. The center of area of the element on the edge of the 279 plate with coordinates (x,y) is assumed to be the representative point of the element. As 280 discussed by Prager (1959) the loads and plastic displacements associated with yielding of the 281 foundation are treated as generalized plastic strains. As the upper bound solution is obtained for 282 failure conditions, this study assumes that the increments of elastic displacement are negligible 283 when compared to the plastic increments. Therefore, assuming an associated flow rule, 284 increments of total displacements of the anchor plate in directions normal and tangential to the 285 plate edges at point (x, y) at the failure condition could be calculated as follows: 286 1 ,max 1 ,max ( , ) ( ) ( , ) ( ) m p nt st nt nt nt m nt nt n p nt st st st st n st st f N N mNu u N N f N N nNu u N N λ λ λ λ − − ∂ = = = ∂ ∂ = = = ∂     (17) By dividing the above two expressions and substituting nu and su from Eqs. (15-a) and (15-b) 287 for each plate edge, ntu , stu , λ and β are eliminated and expression in terms of Nnt and Nst are 288 obtained as follows: 289 For AB and CD (sides parallel to x axis): 1 ,max0 1 0 ,max nm stnt n m st nt Nx x Nm y y n N N − −  −   =      −     (18-a) For AD and BC (sides parallel to y axis): 1 ,max0 1 0 ,max nm stnt n m st nt Ny y Nm x x n N N − −  −   =      −     (18-b) Going through the details to implement the sliding-normal force interaction relationship in the 290 plasticity solution indicates that in order to obtain a “closed from” of the upper bound solution, m 291 should be equal to n (m=n). For simplicity a circular relationship for the sliding-normal 292 interaction curve is adopted (m=n=2): 293 For AB and CD (sides parallel to x axis): 2 ,max0 0 ,max stnt st nt Nx x N y y N N  −   =    −    (19-a) For AD and BC (sides parallel to y axis): 2 ,max0 0 ,max stnt st nt Ny y N x x N N  −   =    −    (19-b) Therefore, having two equations, (19-a) or (19-b) and the yield surface, Eq. (16), the two 294 unknowns, Nnt and Nst, could be determined: 295 For AB and CD (sides parallel to x axis): 0.52 ,max 0 ,max ,max 0 2 ,max 0 ,max 0 . 1ntst st st nt nt st st N x x N N N y y N x x N N N y y −   −  = +   −         −  =       −    (20-a) For AD and BC (sides parallel to y axis): 0.52 ,max 0 ,max ,max 0 2 ,max 0 ,max 0 . 1ntst st st nt nt st st N y y N N N x x N y y N N N x x −   −  = +   −         −  =       −    (20-b) Having Nnt and Nst in terms of element position (x, y) and point of rotation (x0, y0), the increment 296 of energy dissipation rate, e.g. for a representative element on BC, could be obtained as follows: 297 ( ) ( ), 0 0 0 0β β= − + − = − + −  e BC st nt u st ntdD F x x F y y dy ts N x x N y y dy (21) Where x is constant for side BC of the plate, i.e. x=L/2. Therefore, the total rate of dissipated 298 energy along the BC side is calculated by integrating the above expression: 299 /2 /2 , , 0 0 /2 /2 2 β − −   = = − + −    ∫ ∫   W W e BC e BC u st nt W W LD dD s N x N y y tdy (22) Following the same procedure, the rate of energy dissipation for the other edges of the plate can 300 be derived: 301 /2 , 0 0 /2 2 β −   = − + −    ∫  L e AB u st nt L WD s N y N x x tdx (23-a) /2 , 0 0 /2 2 L e CD u st nt L WD s N y N x x tdxβ −   = + + −    ∫  (23-b) /2 , 0 0 /2 2 W e AD u st nt W LD s N x N y y tdyβ −   = + + −    ∫  (23-c) Where Nst and Nnt are substituted from Eqs. (20-a) and (20-b). Thus, the rate of energy 302 dissipation due to soil resistance on the plate edges is determined: 303 , , , ,e e AB e CD e BC e ADD D D D D= + + +     (24) The other terms of dissipation and external work remain unchanged in the revised formulation. 304 The basis of the Yang and modified PLA solutions are summarized in Appendix A. 305 306 COMPARISON TO BASELINE AND PLA SOLUTIONS 307 The finite element studies are now compared to the baseline and PLA solutions through 308 comparing the two basic components defining the yield envelope: 309 1- The “size” of the yield envelope quantified by the predicted maximum resistance under 310 pure translational (Hx,max, Hy,max) and rotational (Tmax) loading. 311 2- The “shape” of the yield envelope defined by the mathematical model f (Hx /Hx,max, Hy 312 /Hy,max, T /Tmax) = 0 which is curve fitted to the yield surface, and represented by 313 interaction factors (i.e. hx, hy, h, and mz): 314 ,max ,max max ,max ,max max ( , , ) 1 0 hhyhx mz y yx x x y x y H HH HT Tf H H T H H T       = + + − =               (25) The mathematical relationship in Eq. 25 represents the ellipsoid interaction or yield surface in 315 the Hx/Hx,max - Hy/Hy,max - T/Tmax normalized space. 316 Size of the Yield Envelope: Pure Translation and Rotation 317 FEA, LE, and modified PLA shear and torsional bearing capacity factors are compared in Fig. 9. 318 The FEA maximum shear resistance, Ns,xmax and Ns,ymax, increases linearly with increasing plate 319 thickness, t, similar to predictions from LE (Eqs. 2 and 3). FEA shear bearing factors are also in 320 good agreement with LE predictions. Although the LE solutions increasingly over-predict the 321 FEA values with increasing plate thickness, the difference does not exceed 2.8% which occurs 322 for the thickest square plate (t/L= 1/7). Note that LE and PLA methods generate the same 323 solutions for the ultimate shear capacity. 324 Figure 9- Normalized maximum resistance for pure translation and torsion As for the torsional resistance, Nt,max, the FEA solution does not show a linear trend predicted by 325 both the LE (Eqs. 5 and 6) and modified PLA formulations (Table 1 and 2; Fig. 9). With plate 326 thickness increasing, the uncorrected (Cf = 1) LE values gradually exceed the FEA solution, by 327 up to 44% for square plate of t/L = 1/7. It can be shown that an interaction factor Cf = 0.67 in Eq. 328 6 will make the limit equilibrium predictions a good fit for the finite element solutions for both 329 the square and rectangular plates. Likewise, modified PLA tends to overestimate the increase in 330 torsional capacity due to plate thickness. However, taking into account the interaction effect of 331 the normal and tangential forces acting on the plate edges brings the modified PLA predictions 332 closer to FEA values compared to Yang et al. (2010) PLA as compared in Tables 1 and 2 (e.g. 333 maximum difference of 29% compared to 44% for square plate of t/L = 1/7). Over-estimation of 334 the capacity by the modified PLA with increasing thickness is also expected since the PLA 335 formulation is based on “upper bound” limit analysis solution. 336 Shape of Yield Envelope: Combined Loading 337 Figs. 10 and 11 compare the FEA shear-torsion interaction envelopes to modified PLA and the 338 PLA virtual work solution for infinitely thin plate (i.e. PLAt=0, Eqs. 7-14). Load capacities are 339 presented in normalized form, H/Hmax versus T/Tmax, for rectangular (Fig. 10) and square plates 340 (Fig. 11). All the FEA predictions for the plate thickness range of t = L/7 to L/20 are essentially 341 lying on a single curve indicating that the shape of the yield envelopes is independent of the plate 342 thickness. A noticeable improvement is observed in modified PLA predictions for interaction 343 response (Figs. 10 and 11) when compared with the Yang et al. (2010) PLA predictions (Figs. 3-344 5). Although the modified PLA values are slightly unconservative especially for thicker plates, 345 the modified PLA predictions are generally less sensitive to the plate thickness compared to 346 Yang et al. (2010) approach and also in overall good agreement with the FEA solutions. 347 The insensitivity of the FEA interaction response with respect to variations in plate thickness and 348 close agreement to the FEA yield envelopes, introduces the PLAt=0 solution as a realistic 349 portrayal of the yield envelope for non-zero thickness (Fig. 11). However, except for the 350 rectangular plate under x-shear and torsion, the PLAt=0 predictions tend to be slightly on a 351 conservative side. 352 Fig. 12 shows the modified PLA and FEA interaction diagrams for the plate under co-planar 353 translation (i.e. non-eccentric loading with varying load directions γ). With some tendency to be 354 on an unconservative side, especially for thicker plates, the modified PLA predictions are in 355 good agreement with the FEA solutions. The minimal influence of the plate thickness on the 356 shape of FEA derived yield envelopes is better captured by the modified PLA solutions 357 compared to the Yang PLA approach (Fig. 5). In general, the modified PLA offers markedly 358 improved predictions compared to Yang et al. (2010) PLA (Fig. 5) for plate under co-planar 359 shear. For the case of zero thickness, the yield envelope obtained from the PLAt=0 solution, will 360 be circular for both square and rectangular plates. The theoretical basis is easy to explain: for 361 plate with area of base = A, thickness = zero, and soil undrained shear strength = su the sliding 362 resistance will be constant and equal to Hmax = 2Asu, regardless of the direction of sliding for 363 Tresca yield criterion. This value actually represents the radius of the circular yield locus in the 364 Hx -Hy space as shown in Fig. 12. This circular interaction relationship in Hx -Hy space applies 365 for a theoretically infinitely thin plate regardless of the shape of geometric base. Fig. 12 indicates 366 that the plate thickness induces a small but noticeable departure from a circular envelope in both 367 PLA and FEA solutions. This departure is greater for thicker plates with lower aspect ratio and 368 more pronounced in modified PLA predictions compared to FEA results. A simplifying 369 assumption of a circular Hx/Hx,max -Hy/Hy,max yield envelope (i.e. hx=hy=2) therefore generates 370 reasonably realistic predictions, albeit underestimating the FEA yield envelope by about 13%. 371 Adopting interaction coefficients of hx=hy=2.5 in Eq. 25 generates a more realistic yield 372 envelope which tracks the FEA derived solutions. 373 (a) (b) Figure 10- Analytical and FEA yield envelopes for rectangular plate: (a) shearx-torsion interaction; (b) sheary-torsion interaction Figure 11- Analytical and FEA yield envelopes for shear-torsion interaction in square plate Figure 12- Analytical and FEA yield envelopes for shearx-sheary, square and rectangular plates ECCENTRICITY AND LOAD CAPACITY REDUCTION 374 One-directional shear-torsion (Hx-T and Hy-T) and zero load angle (γ=0) 375 The reduction effect of torsion on plate sliding capacity and the combined loading effects could 376 also be portrayed through the effect of eccentricity on the load capacity. Figs. 13-16 illustrate the 377 sliding load reduction versus the eccentricity for square and rectangular plates of W/L = 2, 4, and 378 8 with thickness of t = L/7. Both sliding load capacity and eccentricity are expressed in 379 dimensionless form of N=H/suLW and e/L, respectively. The sliding load factor (N=H/suLW) is 380 not normalized by the maximum pure capacity in Figs. 13-16, which makes the capacity 381 reduction curves convenient to estimate both for the magnitude of shear resistance as well as the 382 reduction/interaction effect. As discussed previously, the insensitivity of the shape of FEA 383 sliding-torsion yield envelope and fairly accurate and simple baseline solutions to predict size 384 and shape of the yield locus, provides the opportunity to develop a simplified approach to predict 385 the shear-torsion plate capacity. The PLA virtual work baseline solution for zero thickness 386 (PLAt=0) plate under combined loading (Fig. 8) was adopted in combination with the limit 387 equilibrium derived equations for pure translational load capacity (Hmax using Eqs. 2 and 3 which 388 proved to be fairly accurate), to generate the simplified solution in Figs. 13-16. 389 The FEA predictions for the square and rectangular (W/L=2) plates are compared with the 390 simplified and modified PLA solutions in Figs. 13 and 14, while Figs. 15 and 16 just include the 391 results of analytical simplified and modified PLA predictions for plates of W/L=4 and 8. Figs. 392 14-16 also evaluate the effect of eccentricity angle (ψ) for the rectangular plates sliding in x and 393 y directions (ψ = 90° and 0° when γ = 0°) with various eccentricities. 394 Figure 13- Reduction effect of eccentricity on sliding load capacity for square plate of t = L /7 Figure 14- Reduction effect of eccentricity on sliding load capacity for rectangular (W/L=2) plate of t = L /7 Figure 15- Reduction effect of eccentricity on sliding load capacity for rectangular (W/L=4) plate of t = L /7 Figure 16- Reduction effect of eccentricity on sliding load capacity for rectangular (W/L=8) plate of t = L /7 As discussed in the previous sections, in most cases both simplified and modified PLA solutions 395 over-estimate the FEA predictions with increasing plate thickness except for the simplified 396 solution for rectangular plate under y-sliding (Fig. 14). For rectangular plates (i.e. W/L=2, 4 and 397 8 in Figs. 14-16) under eccentric x-sliding the PLA and simplified approaches are essentially 398 yielding the same values. As for the plate bearing capacity under y-sliding, the modified PLA 399 predictions show an unconservative trend compared to FEA solutions (Fig. 14), while the 400 simplified solution is on a slightly conservative side. 401 In general, the modified PLA approach provides a useful tool with minimal computational effort 402 and slightly unconservative yet fairly satisfactory predictions which is expected regarding its 403 upper bound solution and tendency to overestimate the bearing resistance for increased plate 404 thickness. However, the application of simplified method is also advisable, considering the 405 simple and convenient formulation to implement in spreadsheet applications and fairly accurate 406 results. Furthermore, there is no optimization procedure involved in the simplified solution to 407 search for a least upper bound collapse load which offers the simplified solution an advantage 408 over the modified PLA. Note that modified and PLAt=0 approaches generate exactly the same 409 yield envelopes for the infinitely thin plates. 410 Co-planar shear-torsion (Hx-Hy-T) 411 Figs. 13-16 propose the load capacity for plates under eccentric shear load aligned with the 412 minor and major axis of the plate (i.e. ψ = 0° or 90° with γ = 0°); thus, the effect of load angle, γ, 413 and other eccentricity angles (ψ) or in other words the general Hx-Hy-T combination is not 414 evaluated. Impact of eccentricity angle (ψ) was partially evaluated in Figs. 13-16. Examining the 415 effect of load angle is also clearly significant since any increase in γ (with constant e and ψ) will 416 reduce the torsional load, so that for γ = 90° the torsion yields to zero. Evaluating the effect of 417 load angle, γ, is possible through modified PLA solution (Eqs. 15-24). 418 The simplified solution is also applicable using the generalized PLAt=0 approach (Eqs. 7-10) for 419 infinitely thin plate. The simplified solution also requires the ultimate shear resistance, Hmax, 420 which is estimated based on a simple formulation summarized in Appendix C. 421 422 PLATE UPLIFT CAPACITY UNDER GENERALIZED LOADING 423 Up to this point, the bearing capacity of plate anchor subjected to combined co-planar shear load 424 and torsion has been studied. Now a question arises that how the combination of shear-torsion 425 loading affects the uplift bearing capacity of the plate anchor. 426 Throughout the paper, it was well discussed that the plate thickness does not influence the shape 427 of shear-torsion or two-way shear yield envelope in the normalized load space. We also 428 examined the shape of the normalized yield envelopes for normal-shear forces, normal force-429 moment, and shear force-moment in the API/Deepstar 2D-FE study (Andersen et al. 2004; Murff 430 et al. 2005) for plates of t=L/20 to L/7. Our evaluations indicated a marginal impact of the plate 431 thickness on the shape of normalized yield surface. Therefore, application of generalized 432 interaction envelope for infinity thin plate could be extended to plate under six degrees of 433 freedom loading. The following mathematical model is adopted to describe the shape of yield 434 envelope for the plate anchor under general loading: 435 ,max ,max ,max ,max ,max max 1/ ,max ,max ,max ,max ,max max ( , , , , , ) 1 0 y yn x x n x y x y pshmy hyn mx hx mz y yn x x n x y x y M HF M H Tf F M M H H T M HF M H T F M M H H T =                 + + + + + − =                                    (26) 436 (a) (b) (c) Figure 17- Yield envelopes for plane strain strip plates of thickness t=L/20 to L/7 (Andersen et al. 2003): (a) normal-moment; (b) normal-shear; (c) moment-shear where Fn, Hx, Hy = normal, x-shear, y- shear forces, Mx, My, and T: x-moment, y-moment, and 437 torsion components of the generalized combined loading at failure, Fn,max, Hx,max, Hy,max, Mx,max, 438 My,max, and Tmax are the corresponding maximum values, and eight constants of n, mx, my, hx, hy, 439 mz, s, and p are the interaction factors. 440 Regarding the insensitivity of the shape of normalized yield surface to plate thickness, we 441 adopted the same interaction factors proposed in Yang et al. (2010) for the infinitely thin plate. 442 Interaction factors which define the shape of yield envelope are summarized in Table 3. These 443 factors could be used for square and rectangular (W/L=2) plates with any arbitrary plate thickness; 444 For other aspect ratios, Interaction factors and maximum capacity values should be evaluated. 445 Calculation of the bearing capacity for each degree-of-freedom using Eq. 26, requires all the six 446 maximum capacity factors for the desired plate thickness and aspect ratio (rather than available 447 values) to be estimated. The interaction factors for the shear-torsion yield envelope (hx, hy, and 448 mz) are refined regarding the present study. 449 Table 3. Interaction factors for generalized yield envelope 450 Factors Square Rectangular (W/L=2) n 3.26 3.20 mx 1.91 1.86 my 1.91 2.47 hx 2.50 2.50 hy 2.50 2.50 h 0.70 0.75 mz 1.75 ( ) 51.15 1.23 cos φ  + s 3.87 3.75/mz p 1.56 1.93 Notes: 1/ ,max ,max ,max ,max ,max max 1 0 pshmy hyn mx hx mz y yn x x n x y x y M HF M H Tf F M M H H T                 = + + + + + − =                                    ϕ: the angle between line of action of the external force and y-axis (ϕ=ψ+γ; Fig. 1) The following relationships and recommendations also can be applied to calculate the maximum 451 capacity values for every six degrees of freedom loading. Note that the weight of plate anchor for 452 typical thickness values available in industry has a negligible effect on the capacity of the anchors: 453 1- Uplift or normal ultimate capacity factor (Fn,max): 454 The uplift or normal ultimate capacity of the plane strain strip plate anchor could be estimated 455 using the upper-bound derived relationship proposed by O’Neill et al. (2003): 456 ,max ,max 1(3 2) 2 2 + = = + + +   n n u F tN Ls L απ α (27) 457 where α is the plate-soil adhesion factor. Table 4 also summarizes the predictions for uplift 458 capacity factor using other effective analytical and numerical approaches for 2D plane stain and 459 3D circular, square, and rectangular plates of different thicknesses. 460 Table 4. Maximum pure capacity factor for deeply embedded plate anchor N or m al (u pl ift ) b ea rin g ca pa ci ty (N n, m ax ) 2D plane strain Lower bound PLA (Rowe, 1978) t/L=0; W/L>6: 10.28 2D plane strain finite element (Elkhatib and Randolph, 2005) t/L=1/20; W/L>6: 11.62 t/L=1/7; W/L>6: 11.93 3D upper and lower bound PLA; exact solution (Martin and Randolph, 2001) t/L=0; Circular1: 13.11 3D large deformation finite element (LDFE) (Wang et al. 2009) t/L=1/20; Square (W/L=1): 13.17 t/L=1/20; W/L=2: 12.35 t/L=1/20; W/L=4: 11.28 t/L=1/20; W/L=6: 10.95 M om en t b ea rin g ca pa ci ty (N M ,m ax ) 2D finite element, smooth (α=0.0) and rough (α=1.0) strip plates (Elkhatib and Randolph, 2005) t/L = 1/7: 1.44 and 1.63 for α=0.0 and 1.0 t/L = 1/20: 1.41 and 1.59 for α=0.0 and 1.0 3D upper bound PLA (Yang et al. 2010) t/L=0; Circular: 2 3max ,max 1.9u mM R s Nπ= ⇒ = Calibrated 3D finite element (Yang et al. 2010) t/L=0; Square (W/L=1): NMx,max= NMy, max=1.9 t/L=0; W/L=2: NMx, max=2.15; NMy, max=1.70 Note: 1- For other plate thicknesses, the pull-out capacity factor for square shaped anchor could be used with reasonable accuracy to calculate capacity factor for circular plates with similar plate area. All the capacity factors are correct for deeply embedded plates (i.e. H/L or H/D ≥ 6 where H is 461 the embedment depth, L is the plate shorter length, and D is the diameter of circular anchor). 462 Adopting this assumption means that the failure mechanism is localized around the plate and 463 does not extend to the surface and therefore the bearing capacity is not affected by the effective 464 overburden pressure and soil weight. Thus, the soil is assumed to be a weightless material in the 465 analysis. Also the plate and soil are fully bonded (i.e. no breakaway or separation occurs between 466 plate and soil at the time of failure). Appropriate predictions are provided by Song et al. (2008) 467 and Wang et al. (2010) to evaluate the plate pullout capacity under breakaway/separation 468 condition or when the overburden pressure is influential on the anchor capacity. 469 2- Moment ultimate capacity factor (Mx,max, My,max): 470 O’Neill et al. (2003) used the plane strain upper bound plasticity solution to evaluate the 471 maximum moment capacity factor for a strip plate: 472 2 max ,max 2 12M u M tN L s L π   = = +       (28) 473 The solution could be generalized for square and rectangular plates of finite thickness to 474 calculate moment factors about x and y plate axis using the generalized 2D collapse mechanism 475 in 3D space (Chi 2010): 476 2 2 ,max ,max 2 11 1 1 2 3 π            = = + + +                         x Mx u M t W W tN WL s W L L W (29) 477 2 2 ,max ,max 2 1 1 12 3 y My u M t L tN WL s L W L π        = = + + +                 (30) 478 Note that the equation for NMx,max does not provide a good estimates as the W/L increases, 479 probably due to the adopted failure mechanism, while the equation for NMy,max works more 480 effectively better, especially for greater W/L. For strip plate (i.e. W/L→∞ or L/W→0) Eq. (30) 481 yields the O'Neill et al (2003). 482 Some FE and PLA derived solutions for moment capacity factor are also included in Table 4 for 483 2D plane stain and 3D circular, square, and rectangular plates of different thicknesses. 484 3- Parallel or shear ultimate capacity factor (Nsx,max, Nsy,max): 485 As discussed previously, the limit equilibrium equations (Eqs. 2-3) provide simple yet fairly 486 accurate predictions. 487 4- Torsional ultimate capacity factor (Nt,max): 488 The closed form solution developed based on limit equilibrium solution (Eqs. 5-6) with factor of 489 Cf=0.67 provides a fairly accurate and simple estimation for the torsional maximum capacity. 490 CONCLUSIONS 491 The focus of this study is to offer a modified upper bound plastic limit analysis (PLA) as well as 492 a simplified baseline solution based on the limit equilibrium and virtual work approaches for 493 general conditions of shear-torsional loading. This study also evaluates the effect of eccentric 494 shear forces on the uplift capacity of a plate anchor under six-degrees-of-freedom generalized 495 loading. This study indicates the following: 496 1. Eccentricity reduces the plate shear and consequently uplift capacities. The reduction 497 begins to become significant (>5% of reduction in shear capacity) in square plates at 498 eccentricity levels of e>0.1L (Fig. 8). For perspective, an eccentricity e/L = 0.5 - a load 499 applied at the edge of a square plate - reduces shear capacity by more than 40%. 500 2. Plates of higher aspect ratio (W/L) are less susceptible to uplift capacity reduction due to 501 planar eccentric loading. Reduction in uplift capacity due to eccentricity of in-plane shear 502 loading is always more significant for shear loading parallel to the short axis of the plate. 503 3. The proposed baseline solutions to calculate the pure shear capacity, limit equilibrium 504 Eqs. 2 and 3, offer fairly accurate predictions for plate under pure shear loading (Fig. 9). 505 The limit equilibrium derived Eq. 5, to estimate pure torsional capacity for theoretical 506 infinitely thin plate (t = 0), is accurate as well. For plates of finite thickness (t > 0), Eq. 6 507 offers the increase in torsional capacity induced by plate thickness which can provide 508 reasonably accurate solutions if used in combination with a correction factor Cf = 0.67. 509 4. The proposed modifications in PLA formulation for plate under combined two-way 510 translation-torsion, significantly improves the Yang et al. (2010) PLA predictions both in 511 terms of “size” (i.e. pure capacity values) as well as the “shape” (i.e. interaction response) 512 of the two-way shear and shear-torsion yield envelopes. Predictions of the proposed 513 modified PLA approach for the shape of shear-torsion yield envelope in normalized load 514 space generally agree well with FEA predictions (Figs. 10-12). However, the method 515 generally over-predicts the increase in pure torsional resistance associated with increased 516 plate thickness, t (Tables 1 and 2; Fig. 9, and Figs. 13-16). Application of the modified 517 PLA approach is therefore advisable when the obtained normalized yield envelope is 518 used in combination with the FEA derived or corrected pure torsional capacity (Eqs. 5 519 and 6) and pure shear capacity (Eqs. 2-3). 520 5. The general insensitivity of the shape of the shear-torsion yield loci to plate thickness t 521 (Figs. 10-12) offers the opportunity to develop a simplified analysis in which fairly 522 accurate analytical expressions for shear resistance (Eqs. 2 and 3) are used in conjunction 523 with a virtual work analysis for zero thickness plate (PLAt=0: Eqs. 6-10) to predict the 524 reduction in capacity due to eccentricity. The method is fairly robust, as it requires no 525 find a least upper bound. Predictions from the simplified method provide reasonable 526 conservative estimates of the reduction in y-shear resistance induced by eccentricity, 527 however similar to modified PLA approach the reduced x-shear capacity factors will be 528 over-predicted by the simplified approach, thus the results should be used with caution. 529 6. Reviewing the normalized yield envelope for plate anchor under other combinations of 530 loading indicates the insensitivity of the normalized yield surface for a plate under 531 generalized six-degrees-of-freedom loading to the plate thickness (Figs. 10-12, Fig. 17). 532 Thus, to evaluate the uplift capacity reduction of a plate under any load combinations, the 533 normalized yield surface for an infinitely thin plate (Eq. 17, Table 3) could be used in 534 combination with the ultimate capacity factors available in the current study and literature 535 (Table 4). 536 Appendix A. Energy Dissipation Rate Terms for PLA (Fig. 2) 537 Ref. Mechanism Location Equation Y an g et a l. (2 01 0) fo rm ul at io n Slip at the edges of the plate Sides AD and BC /2 ( , ) 0 0 0 /2 2 / 2 / 2 W e AD BC u e W D s N y y L x L x tdyβ α α − =  − + − + +  ∫  Sides AB and CD /2 ( , ) 0 0 0 /2 2 / 2 / 2 L e AB CD u e L D s N x x W y W y tdxβ α α − =  − + − + +  ∫  Slip at the top and base of plate ABCD Base and top /2 /2 /2 /2 2 ( , ) W L s u W L D s R x y dxdyα β − − = ∫ ∫  Cu rre nt st ud y (M od ifi ed P LA ) Slip at the edges of the plate Sides AD and BC /2 , 0 0 /2 /2 , 0 0 /2 2 2 β β − −   = − + −      = + + −    ∫ ∫   W e BC u st nt W W e AD u st nt W LD s N x N y y tdy LD s N x N y y tdy 0.52 ,max 0 ,max ,max 0 2 ,max 0 ,max 0 . 1 −   −  = +   −         −  =       −    nt st st st nt nt st st N x x N N N y y N x x N N N y y where: x = L/2 for BC x = -L/2 for AD Sides AB and CD /2 , 0 0 /2 /2 , 0 0 /2 2 2 β β − −   = − + −      = + + −    ∫ ∫   L e AB u st nt L L e CD u st nt L WD s N y N x x tdx WD s N y N x x tdx 0.52 ,max 0 ,max ,max 0 2 ,max 0 ,max 0 . 1ntst st st nt nt st st N y y N N N x x N y y N N N x x −   −  = +   −         −  =       −    where: y = W/2 for AB y = -W/2 for CD Slip at the top and base of plate ABCD Base and top /2 /2 /2 /2 2 ( , ) W L s u W L D s R x y dxdyα β − − = ∫ ∫  [Same as Yang et al. (2010)] Notes: (x0, y0): coordinates of the center of rotation (Fig. 2) (xf, yf): coordinates of the application point for the external force H (Fig. 2) su: undrained shear strength of the soil L and W: Length and width of the plate anchor parallel to x and y axis respectively α: adhesion factor β : virtual rate of rotation Ne: plane strain bearing capacity factor equal to 7.5 ( ) ( ) ( )2 2 20 0 ( / 2, ): x x y y tR x y − + − + Nst: normalized shear force acting along the plate edge Nnt: normalized force acting perpendicular to the plate edge Nst,max: maximum normalized shear force acting along the plate edge Nnt,max: maximum normalized force acting perpendicular to the plate edge y constant for the sides AB (y=W/2) and CD (y=-W/2) x constant for the sides BC (x=L/2) and AD (x=-L/2) APPENDIX B. Integration of dissipation rate for d=0 analysis 538 Analytical evaluation of the inner integral in Eq. 12 is possible to permit a single numerical 539 integration. The resulting expression is: 540 /2 2 1 1 1 1 2 2/2 2 2 1 2 2 2 1 1 2 2 2 2 2 ( ) ln / 2 / 2 W u W opt opt a cD s a c a c y dy a c a L a L c a y c a y β ρ ρ −  + = − +  +  = + = − = + = + ∫ (B-1) 541 Eq. B-1 can be integrated between the limits -W/2 to W/2 using classical numerical integration 542 formulas. 543 APPENDIX C. Estimation of Hmax for the simplified solution 544 For a plate anchor subjected to horizontal load of H (Fig. 1), these relationships are valid: 545 ( ) ( ) ,max ,max 1 0 .sin .cos hyhx yx x y x y HH H H H H H H φ φ    + − =          = = (C-1) 546 where ϕ is the angle between line H and y-axis (i.e. ϕ=ψ+γ in Fig. 7). Hx,max and Hy,max are 547 estimated by the limit equilibrium derived equations for pure translational load capacity using 548 Eqs. 2 and 3 which proved to be fairly accurate. As discussed previously in the paper, application 549 of the FE derived hx=hy=2.5 offers a fairly accurate estimate for the shape of Nsx- Nsy yield 550 envelope. Therefore, H for the simplified solution is evaluated using the following equation: 551 ( ) ( ) 0.42.52.5 ,max ,max sin cos x y H H H φ φ −      = +            (C-2) 552 REFERENCES Andersen, K.A., Murff, J.D. & Randolph, M.F. (2004). “Deepwater Anchor Design Practice- Vertically Loaded Drag Anchors”, Phase II Report to API/Deepstar JIP, Volume III. Aubeny, CP, Murff, JD and Roesset JM (2001). “Geotechnical issues in deep and ultra deep waters,” International Journal of Geomechanics. Vol. 1, No. 2, pp. 225-247 Chen, WF, and Liu, XL (1990). Limit analysis and soil plasticity. Elsevier Publishing Co., Amsterdam, The Netherlands. Chi, Ch (2010). Plastic limit analysis of offshore foundation and anchor. Doctoral dissertation, Texas A&M University. Elkhatib, S, and Randolph, MF (2005). “The effect of interface friction on the performance of drag-in plate anchors,” Proc. Int. Symp. On Frontiers in Offshore Geotechnics, IS-FOG05, Perth, pp. 171-177. Martin, CM, and Randolph, MF (2001). “Applications of the lower and upper bound theorems of plasticity to collapse of circular foundations,” Proc. 10th Int. Conf. Int. Association of Computer Methods and Advances in Geomechnics, Tucson, 2, pp. 1417-1428. Murff, JD, Aubeny, CP, Yang, M (2010). “The effect of torsion on the sliding resistance of rectangular foundations,” Proc. 2nd Int. Symp. on Frontiers in Offshore Geotechnics (ISFOG), Perth, Australia, pp. 439-444. Murff, JD, Randolph, MF, Elkhatib, S, Kolk, HJ, Ruinen, RM, Strom, PJ, and Thorne, CP (2005). “Vertically loaded plate anchors for deepwater applications,” Proc. Int. Symp. on Frontiers in Offshore Geotechnics: ISFOG 2005, Perth, Australia, pp. 31-48. Nouri, H, Biscontin, G, Aubeny, C (2014). “Undrained bearing capacity of shallow foundations under combined sliding and torsion,” ASCE Journal of Geotechnical and Geoenvironmetal Engineering. Vol. 140, No. 8. Nouri, H (2013). Numerical Methods in Offshore Geotechnics: Applications to Submarine Landslides and Anchor Plates. Doctoral dissertation, Texas A&M University. O’Neill, MP, Bransby, MF, and Randolph, MF (2003). “Drag anchor fluke-soil interaction in clays,” Canadian Geotechnical Journal, vol. 40, pp. 78-94. Prager, W (1959). An Introduction to Theory of Plasticity. Addison Wesley: Reading, MA. Rowe, RK. (1978). “Soil-structure interaction analysis and its application to the prediction of anchor behavior,” PhD thesis, University of Sydney, Sydney, Australia. Song, Z, Hu, Y, and Randolph, MF (2008). “Numerical simulation of vertical pullout of plate anchors in clay,” ASCE Journal of Geotechnical and Geoenvironmenal Engineering. Vol. 134, No. 6, pp. 866-875. Tan, F (1990). Centrifuge and theoretical modelling of conical footings on sand. Ph.D. thesis, The University of Cambridge, Cambridge, U.K. Wang, D, Hu, Y, and Randolph, MF (2010). “Three-dimensional large deformation finite element analysis of plate anchors in uniform clay,” ASCE Journal of Geotechnical and Geoenvironmenal Engineering. Vol. 136, No. 2, pp. 355-365. Yang, M, Murff, JD, and Aubeny, CP (2010). “Undrained Capacity of Plate Anchors under General Loading,” ASCE Journal of Geotechnical and Geoenvironmental Engineering. Vol. 136, No. 10, pp. 1383-1393.