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24 1 of LRFD in cases of foundations under complex loading is, Ticof (1977) therefore, difficult as there is no strict separation between load Ingra & Baecher (1983) 0.8 and resistance. Furthermore, it is not always clear whether Gottardi & Butterfield (1993) a load should be classified as favorable or unfavorable, which may have consequences for the calibration of safety factors. 0.6 The difficulty in classification applies especially to the influ- qu / qu,centric ence of the vertical load on the bearing capacity. 0.4 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 0.2 and horizontal loads. In this so-called interaction diagram, the sliding limit state is illustrated as a simple straight line 0 with an inclination tan S representing the soil foundation 0 0.2 0.4 0.6 0.8 interfacial shear resistance accounting for the roughness of Load ratio, H/V the footing's base. The bearing capacity limit state is a closed Figure 20. Reduction factors for shallow curve in this illustration. The interaction diagram depicts the foundations under inclined loading well-known physical phenomenon that the occurrence of hor- (c = 0, Df = 0). izontal loads reduces the bearing capacity of a shallow foun- dation, which is described by the inclination factors used in Reduction factors for the case of a load inclination related the traditional bearing capacity equation. A similar diagram to the case of centrically and vertically loaded footings can be can be generated for eccentric vertical loading or in the three- found in Ticof (1977), Ingra and Baecher (1983), and Gottardi dimensional space for eccentric and inclined loading. and Butterfield (1993) (see Figure 20). These expressions As the inclination factors depend on the characteristic load were determined based on model foundation test results on inclination Hk/Vk, the bearing capacity calculation implies a sand without embedment and as such are valid for the case radial load path, which is the same for loading and resistance of Df = 0, c = 0: as indicated in Figure 21(a). However, only the vertical com- ponents of load and resistance are compared within the proof Ticof (1977): of stability. On the other hand, the sliding resistance calcula- tion is based on the assumption of a step-like load path. For 2 qu H a given vertical load, the associated horizontal resistance is = 1 - 1.36 (55) qu , centric V calculated, which itself is compared to the horizontal load component. The distances between design loads (Hd or Vd, Ingra and Baecher (1983): respectively) and design resistances (Rh,d or Rv,d, respectively) 2 in Figure 21(a) represent the actual degree of mobilization. qu H H = 1 - 2.41 + 1.36 (56) In Figure 21(b), bearing capacity limit state and sliding qu , centric V V limit state are referred to the maximum vertical resistance, Vmax (i.e., under centric vertical loading only). Hence, the Gottardi and Butterfield (1993): diagram shows the pure interaction of the load components qu H without any other influences on the bearing capacity. In this = 1- (57) illustration it is shown that a maximum horizontal load, qu , centric 0.48 i V Hmax, can be applied for V/Vmax 0.42. Let us now consider 1.6 An Alternative Approach a certain horizontal load, H < Hmax. For this case, a mini- and Method of Analysis mum vertical load (min V) is required to carry the horizon- for Limit State Design tal load. This means the load inclination is limited and the of Shallow Foundations limit is provided either by the bearing capacity limit state or by the sliding limit state, whichever is more restrictive. With 1.6.1 Some Aspects of Stability and Safety increasing vertical load, the resultant load inclination de- of Shallow Foundations creases and, hence, the bearing capacity of the system in- creases. However, because of the convex shape of the bear- 1.6.1.1 Bearing Capacity and Sliding Limit States ing capacity limit state, the degree of mobilization increases Geotechnical resistances such as the bearing capacity of shal- if V/Vmax > 0.42, so the magnitude of the applicable vertical low foundations are entirely load dependent. The application load is limited as well (max V).

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25 (a) Definition of load paths (b) Effect of vertical load on resistance Hd Rh,d Rh,k / R H1 H2 Hmax H / Vmax Hk 0 Rh,k ( Vk ) H min V1 min V2 steplike load path VG Vk sliding resistance sliding resistance Vd 0 ~0.42 R v ,d R v ,k / R s VG VG radial load path bearing resistance R v,k (Hk / Vk ) max V1 max V2 Vmax 1 bearing resistance V V / Vmax V favorable V unfavorable Notes: H resultant horizontal load V resultant vertical load Hmax max. horizontal load that can be carried by the min V min. vertical load required for a certain applied horizontal system load Hd design value of horizontal load max V max. vertical load possible for certain applied horizontal load Hk/Vk load inclination (characteristic values) VG permanent vertical load Rv,d design value of bearing capacity VG additional permanent vertical load Rv,k characteristic value of bearing resistance Vk characteristic vertical load (capacity) Vd design value of vertical load Rh,d design value of sliding resistance Vmax bearing capacity under pure vertical loading, i.e., max. Rh,k characteristic sliding resistance vertical load that can be carried by the system R resistance factor V/Vmax degree of utilization of maximum vertical load s base friction angle Figure 21. Influence of load components on bearing resistance and sliding resistance utilizing interaction diagram. 1.6.1.2 Favorable and Unfavorable Load Actions favorable or unfavorable load. The use of the presented simple interaction diagrams may help, however, to better understand Now consider a given vertical load, e.g., the foundation dead the complex interaction of the load components (Lesny, 2006). 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- 1.6.1.3 Example plied if the vertical load is increased simultaneously, e.g., by The favorable and unfavorable actions may affect the safety increasing the dead weight applied to the footing. The vertical of the system as demonstrated by the following example of a load acts favorably because an increase in the vertical load re- vertical breakwater (Lesny and Kisse, 2004; Lesny, 2006). The sults in the possible increase of the horizontal load. These rela- breakwater is a structure supported by a strip footing of width tionships are, however, valid only for V/Vmax < 0.42. Larger BC founded on sand and subjected to vertical, horizontal, and vertical loads (VG + VG) act unfavorably because they reduce moment loading (see Figure 22). The basic parameters of the the maximum allowable horizontal load. In this situation, an ar- system are (Lesny et al., 2000; Oumeraci et al., 2001) bitrary increase in the dead load applied to the footing would be counterproductive because it does not help to improve the per- Caisson: Bc = 17.5 m, hc = 23 m formance of the system in resisting horizontal loads. These Crushed stone layer: f = 44.2, (effective unit weight) = complex interrelations demonstrate that the role of the vertical 10.4 kN/m3, tans = 0.5 load component is not unique. Hence, within the standard Subsoil: f = 38.2, = 10.2 kN/m3 design procedure it is difficult to classify the vertical load as a Water depth at still water level: hs = 15.5 m

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26 BC uplift wave crest rotation failure Fh ( t ) wave trough eC SWL h hC crushed FG rubble mound sliding along the base hs S(t) hr subsoil bearing resistance failure in rubble mound Fu ( t ) bearing resistance failure in subsoil Notes: hs water depth at still water level eC eccentricity of dead weight FG of breakwater (SWL) Fh(t) horizontal wave load hr height of rubble mound Fu(t) resultant of uplift water pressure due to wave loading h* height of wave crest S(t) seepage force in rubble mound hC height of breakwater BC width of breakwater Figure 22. Breakwater, wave loading, and failure modes. Figure 23 depicts the bearing capacity limit state and the the bearing capacity limit state, the safety is R,v = Rv,k/VG,d = 2.7 sliding limit state of the breakwater for a fixed eccentricity of if V is favorable, but only R,v = 2.0 if V is unfavorable. Under ek/B = 0.12 in the V-H plane. both conditions, the safety of the system seems to be sufficient. We assume a fictitious characteristic loading mainly due to These results do not represent, however, the actual safety dead weight and wave loading of of the foundation. In the interaction diagram of Figure 23, the actual safety is described by the closest distance of the loading HQ,k (horizontal fictitious characteristic loading due to dead to the resistance of the foundation as indicated by the arrows. weight) = 2.55 MN Additional load components acting along this path are most VG,k (vertical fictitious characteristic loading due to wave) hazardous. If arbitrary load paths are possible, only additional = 15 MN load components acting within the circles sketched in Fig- ure 23 are admissible. Such a critical load situation is not arti- The factored design loads below were developed assum- ficial; it may occur if the wave height is higher than assumed ing vertical and horizontal load factors of G = 1.35 and 1.00 for design, resulting in an increase of the horizontal load along for unfavorable and favorable permanent action, respectively, with a decrease of the vertical load due to uplift forces. and Q = 1.50 and 1.0 for unfavorable and favorable variable The actual safety can be determined with the help of the fac- action, respectively. The factor G is applied to the vertical tored design load vector Q = [VG,d ; HQ,d] and the additional load loads only, and the factor Q is applied to the horizontal vector Q in the V-H plane, which coincides with the radius of loads. The horizontal and vertical factored design loads are the circles in Figure 23 (Butterfield, 1993). For the design load the following: components given above (Q) the maximum additional load- ing is limited by the sliding limit state and amounts to Q = HQ,d = 3.82 MN 3.30 MN (V favorable) or Q = 5.68 MN (V unfavorable), re- VG,d = 15 MN (V favorable) spectively. Thus, the actual safety of the system is the following: VG,d = 20.3 MN (V unfavorable) The safety of the system may be expressed here by the avail- R = (Q + Q ) Q = { 1.21 1.28 V favorable V unfavorable (58) able resistance factor resulting from the characteristic resis- tance divided by the associated design load: R = Rk/Ld. Hence, The actual safety in both cases is considerably smaller than the safety for the sliding limit state is R,h = Rh,k /HQ,d = 2.0. For the one calculated previously using the regular design proce-

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

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28 by Perau (1995, 1997) at the Institute of Soil Mechanics and in Appendix A. For more information, refer to Kisse and Lesny Foundation Engineering at the University of Duisburg-Essen (2007) and Kisse (2008). (UDE), Essen, Germany. Recently, Byrne and Houlsby (2005) and Bienen et al. (2006) presented experimental work on shal- 1.6.2.2 Failure Condition low foundations on sand under arbitrary loading condi- tions as well. In this work, the failure or yield condition is In the general case, a single footing is loaded by a verti- defined by so-called swipe tests, in which the load path followed cal load, F1, horizontal load components F2 and F3, a torsional closely the failure or yield surface in the interaction diagram. moment, M1, and bending moment components, M2 and M3 However, the use of these data for the research project reported (see Figure 24). The load components are summarized in the on herein (NCHRP Project 24-31) may be limited as the tests load vector Q : remain close to but below failure. In other words, failure loads for definite loading conditions are not directly available. Q T = [ F1 F2 F3 M1 M 2 M 3 ] (59) For the basic case of a footing on non-cohesive soil without 1.6.2 Alternative Design Method embedment, the geometry of the footing described by the side for Shallow Foundations ratio (b = b2/b3), weight (), the soil's shear strength (tan f), 1.6.2.1 Overview and a quantity describing the roughness of the footing base (s) have to be considered as well (see Figure 24). With these The alternative design method includes two components. input parameters, the failure condition of the general form The first component is a failure condition that describes the ULS of a shallow foundation without the need to distinguish F (Q , b , , tan f , s ) = 0 (60) 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. is defined by the following expression: The failure condition can be used independently of the dis- placement rule and may be combined with other methods for 2 F2 + F3 2 2 M1 2 M2 + M3 2 + + - settlement analysis. It has been developed for foundations on (a1 F10 )2 (a2 (b2 + b3 ) F10 )2 (a3b2 F10 )2 granular soils with and without embedment, whereas the dis- placement rule is currently developed for foundations with- F1 F 1- 1 = 0 (61) out embedment only. Please note that in the general definition F10 F10 of the failure condition and the displacement rule the notation of the load components is different from the notation used pre- In Equation 61, all load components are referred to as F10, viously. An in-depth discussion of the subject and the normal- which is the resistance of a footing under pure vertical loading. ization concept validation via small-scale testing is presented This quantity is calculated using the traditional bearing capac- x2 F1 d M2 M1 F3 F2 M3 b2 x3 b3 g , f x1 Notes: b2, b3 length of the footing referred to x2-, x3- axis M1, M2, M3 torsional and bending moments (referred to d embedment depth x1-, x2-, x3- axis) F1 vertical load unit weight of soil F2, F3 horizontal load (referred to x2-, x3- axis) f angle of internal friction Figure 24. Geometry and loading.

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29 ity formulae. The advantage of the formulation described in In the case of footings embedded in the soil, the failure Equation 61 is that the complex interaction of the load com- condition according to Equation 61 needs to be extended if ponents is considered directly without using reduction fac- the shearing resistance in the embedment zone is taken into tors or the concept of the effective foundation area. Other account: influences on the bearing capacity are included in F10. It should be noted that as F10 is the bearing capacity under vertical- 2 F2 + F3 2 M2 2 + M3 M2 2 centric loading only, the uncertainties of the calculation method + 1 + are reduced to the bearing capacity factors and the shape and (a1 F10 ) 2 (a2 (b2 + b3 ) F10 ) 2 (a3b2 F10 )2 depth factors (if required) of the traditional bearing capacity F F calculations. Thus, no inclination factors or use of effective area - (1 + f z ) 1 - f z 1 - (1 + f z ) 1 - f z = 0 (63a) F10 F10 are necessary. The use of such factors and the concept of effec- tive area were the cause for difficulties in establishing the degree F1, min F10 of conservatism and hence a source of ambivalent application with fz = (63b) 1 - F1, min F10 of LRFD facing the existing AASHTO 2008 specifications. In an interaction diagram like the ones in Figures 21 or 23, In Equation 63b, F1,min is the bearing capacity due to pure the failure condition spans a failure surface, which is the outer vertical tension loading resulting from the shearing resistance boundary of the admissible loading. The parameters a1,2,3 gov- in the embedment zone, which may be carefully calculated ern the inclination of this failure surface for small vertical load- using an earth pressure model. F10 can be determined using ing where the limit states of sliding and the restriction of the the traditional bearing capacity equation taking into account eccentricity to 1/3 of the foundation width have previously been depth factors provided by Brinch Hansen (1970). The increas- relevant (see Figure 25). These limit states are integrated by ing capacity for horizontal and moment loading is considered defining the parameters a1,2,3 and according to Equation 62: by the parameters ai according to Equation 64, which requires additional verification at this stage: a1 = ( 2 ) s ( tan f ) e (- 3) tan f , a2 = 0.098, a3 = 0.42, = 1.3 (62) a1 = ( 2 ) ( tan )(e( ) ) S ,k f - 3 tan f The limit state uplift is already included in Equation 61 because only positive vertical loads are admissible. The param- ( ( 2) )( 1 + 2 - tan 1 - e ( f ) - tan f d b2 ) eters provided in Equation 62 have been derived from an a2 = 0.098 (64) 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 a3 = 0.42 1 + 0.5 1 - exp - d b2 ( ( )) with the model test results for various load combinations. = 1.3 uplift F2 or M 3 b 2 F2 or M3 b2 F2 or M 3 b 2 F2 or M3 b2 bearing resistance S or admissible: sliding resistance or F<0 restriction of eccentricity ultimate limit state: F=0 F1 F1 = arctan adm. e / b 2 Notes: b2 length of the footing referred to x2- axis M3 bending moment (referred to x3- axis) F failure condition S base friction angle F1 vertical load angle defined by allowable eccentricity F2 horizontal load (referred to x2- axis) arctan adm e , adm e usually is b2/3 b2 Figure 25. Isolated limit states (left) and failure condition (right).

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30 F2 F10 M 3 F10 0.00 0.04 0.08 0.12 0.16 0.20 0,24 0,28 0.00 0.04 0.08 0.12 0.16 0.20 b 0.33 ( Nendza & Nacke, 1986; 0 Perau, 1995) 0 0.1 b 0.5 (Haubrichs, 1993) 0.1 adm. e b2 0.2 b 1 (Perau, 1995) 0.2 0.3 b 2 (Haubrichs, 1993) 0.3 0.4 b 3 (Perau, 1995) 0.4 F1 F1 0.5 b 5 (Perau, 1995) 0.5 F10 F10 0.6 failure condition (Eq. 61) 0.6 b 0.33 0.7 0.7 b 1 0.8 0.8 b 3 b 5 0.9 0.9 failure condition (Eq. 61) 1 1 Notes: F1 vertical load F10 bearing capacity under pure vertical loading, i.e., F2 horizontal load maximum vertical load that can be carried by the system (referred to x2- axis) M3 bending moment b b2/b3 side ratio Figure 26. Failure condition for inclined loading (left) and eccentric loading (right) versus failure loads from small-scale model tests. where the displacement rule. The displacements ui and rotations i S,k = value of characteristic roughness of the foundation are summarized in a displacement vector: base. u T = [u1 u2 u3 1 2 3 ] (65) 1.6.2.3 Displacement Rule Due to the complex interaction of load components, dis- The displacements and rotations of the foundation due to placements, and rotations, the displacement rule is formulated arbitrary loading inside the failure surface are described by using the well-known strain hardening plasticity theory with 2 2 F2 F3 2 M1 M2 2 2 M3 a1 F10 2 a 2 b2 b 3 F10 2 a 3 b 2 F10 2 0.00 0.04 0.08 0.12 0.16 0.20 0,24F 0,28 0,32 0,36 0,40 1 F2 M1 M 2 , b 1 ( Perau, 1995) 0 F1 F2 M1 M 3 , b 1 ( Perau, 1995) 0.1 F1 F3 M1 M 3 , b 2 ( Haubrichs, 1993) 0.2 F1 F3 M1 M 2 , b 2 (Haubrichs, 1993) 0.3 F1 F2 M1 M 2 M 3 , b 1 ( Perau, 1995) 0.4 failure condition (Eq. 61) F1 0.5 F10 0.6 0.7 0.8 0.9 1 Notes: b2, b3 length of the footing (referred to x2-, x3- axis) F10 bearing capacity under pure vertical loading, i.e., max. b b2/b3 side ratio vertical load that can be carried by the system F1 vertical load M1, M2, M3 torsional and bending moments (referred to x1-, x2-, x3- F2, F3 horizontal load (referred to x2-, x3- axis) 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.

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31 isotropic hardening (e.g., Zienkiewicz, 2005). Hence, displace- in Equation 68 have to be determined as functions of ai and ments and rotations are calculated according to Equation 66, , respectively. The expansion of the yield surface depends assuming that all deformations are plastic: mainly on the vertical displacement, u1, which itself depends on the degree of mobilization of the maximum resistance, F10. T 1 F G Hence, it is sufficient to define the hardening parameter, Fa, du = dQ (66) H Q Q in Equation 67 as a function of these two quantities accord- ing to the following: The components of the displacement rule are a yield sur- -k0u1 face described by the yield condition, F, which is derived from Fa = ( F10 + k f u1 ) 1 - exp (70) the failure condition equation (Equation 61): F10 + k f u1 2 + F3 F2 2 2 2 2 F (Q , Fa ) = M1 M2 M3 Many hardening laws (e.g., Nova and Montrasio, 1991) + + + (a1 Fa ) 2 (a2 (b2 + b3 ) Fa ) 2 (a3b3 Fa ) 2 (a3b2 Fa ) 2 require small-scale model tests under centric vertical load- ing to determine the hardening parameter. Since this is not 2 F1 F1 a convenient for practical applications, the initial and final stiff- - 1- = 0 (67) Fa Fa ness of the corresponding load-displacement curve, k0 and kf , respectively, may be determined using a method proposed with the parameters a1,2,3 of Equation 62, a plastic potential, G: by Mayne and Poulos (2001) in which the soil stiffness can be determined by any standard procedure. 2 + F3 2 M12 2 Figure 29 shows the results of the proposed model applied G (Q , Fb ) = F2 M2 + + to the example breakwater of Figure 22. Safety factors are not (c1 Fb )2 (c2 (b2 + b3 ) Fb )2 (c3b3 Fb )2 applied here. On the left side of Figure 29 the failure condi- 2 M3 F F 2 tion and the loading in the F1 - F2 plane and in the F1 - M3/BC + - 1 1- 1 = 0 (68) plane are shown. Obviously, the stability of the breakwater is (c3b2 Fb )2 Fb Fb governed by the high horizontal loading. Only an increase in the vertical loading (i.e., of the breakwater weight) would lead and a hardening function, H: to a sufficient safety. The right side of Figure 29 shows the ver- tical and horizontal displacements of the breakwater depend- F Fa G H =- i i (69) ing on the corresponding load components, F1 and F2. How- Fa u1 F1 ever, due to some conservative assumptions made in the current version of the proposed model, a breakwater width of 21.0 m In Equation 68, Fb is the hardening function and c1, c2, and instead of 17.5 m was required to reach stability. 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- 1.6.2.4 Implementation of a Safety Concept tion 61 is reached (see Figure 28). Thus, the parameters ci and To implement a safety concept for the ULS based on load and resistance factors, the bearing capacity and loading for the M 2 / b3 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 failure surface interaction diagram. Connecting all load points provides a polygon in the interaction diagram (see Figure 30). It can be F2 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- yield surface tion, it is sufficient to prove only these load combinations. F1 To get the design failure condition, Fd , the parameters ai Notes: b3 length of the footing (referred to x3-axis) F2 horizontal load in Equations 61 and 62 are divided by the required resis- F1 vertical load M2 bending moment (referred to x2-axis) tance factor R,i . Additionally, a resistance factor also has to be Figure 28. Isotropic expansion of the yield surface in adapted to F10. This procedure means that in practice the fail- the loading space. ure surface shrinks as depicted in Figure 30.

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32 F1 [kN/m] M3/BC kN/m F2 [kN/m] Notes: Bc width of breakwater M3 torsional and bending moments (referred to x1-, x2-, x3-axis) F1 vertical load u1 vertical displacements (settlements) F2 horizontal load (referred to x3-axis) u2 horizontal displacements Figure 29. Failure condition (left) and load displacement curves (right) for the example breakwater. The resistance factors are no longer distinguished according The case of inclined and eccentric load combinations to different limit states but according to the possible load inter- may result in a coupled interaction of the resistance factors. actions. So at least resistance factors for pure vertical loading, These cases, like other aspects of this concept, require fur- inclined loading, torsional loading, and eccentric loading may ther analysis. The application of load factors means that be defined: load components are reduced if they work favorably and are increased if they work unfavorably regarding the bear- R,pure vertical for F10 (pure vertical loading) ing capacity of the foundation (considering the aspects that R,horizontal for a1 (inclined loading) R,torsional for a2 (torsional loading) were discussed earlier). This may cause displacements and R,eccentric for a3 (eccentric loading) distortions of the load polygon in the interaction diagram. characteristic load F2 / F10 ; M 3 (b 2 F10 ) combinations Lk design load combinations Ld Fd b, tan f , L, R ,... 0 Fk b, tan f , L,... 0 F1 / F10 Notes: b2 length of the footing (referred to x2-axis) F10 bearing capacity under pure vertical loading, i.e., b b2/b3 side ratio maximum vertical load that can be carried by the system Fk characteristic failure condition Lk characteristic load combination Fd design failure condition Ld design load combination F1 vertical load F2 horizontal load R resistance factor M3 bending moment f angle of internal friction Figure 30. Illustration of the safety concept principle.