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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 socalled 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 wellknown 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 steplike 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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(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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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 VH 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 VH 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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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 (oneway, inclinedeccentric 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 (twoway 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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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 DuisburgEssen (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 socalled 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 2431) 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 noncohesive 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 loaddisplacement
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 indepth discussion of the subject and the normal which is the resistance of a footing under pure vertical loading.
ization concept validation via smallscale 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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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 smallscale 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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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
smallscale 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 wellknown 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
smallscale model tests.
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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 smallscale 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 loaddisplacement 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 x3axis) F2 horizontal load in Equations 61 and 62 are divided by the required resis
F1 vertical load M2 bending moment (referred to x2axis)
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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F1 [kN/m]
M3/BC kN/m F2 [kN/m]
Notes:
Bc width of breakwater M3 torsional and bending moments (referred to x1, x2, x3axis)
F1 vertical load u1 vertical displacements (settlements)
F2 horizontal load (referred to x3axis) 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 x2axis) 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.