Figure 6-12: Illustration of the selection of resistance factors for structural Strength Limit State of

86
6.7.2.2
Fixity for Equivalent Cantilever Length for Buckling
Some structures supported on deep foundations and subjected to axial and lateral loads will include freestanding foundation lengths i.e., partially embedded and partially unsupported piles or drilled shaft lengths. An example of such a case is abridge pier over open water, with an unsupported pile/shaft length from the mudline to the base of the pile cap or column, subjected to lateral loads from water, wind, ice, vessel impacts, etc. A pile in such a condition can be analyzed as an equivalent cantilever attached to a fixed base at a particular depth (Davisson and Robinson 1965; Greimann and Wolde-Tinsea 1988;
Abendroth and Greimann 1989; and, Abendroth et al. 1989). This type of analysis is appropriate fora preliminary analysis for buckling of unsupported lengths of the pile. However, as discussed in Chapter 11, p-y analyses should be used in final design. Figure a) shows such a casein which L
u
is the length of unsupported pile and L

is the length of the embedded section of the pile. For such a condition, a flexural analysis may control the design of the foundation element. Such a design is indeterminate and often difficult to solve without simplifying assumptions, especially if the pile head is attached to a girder or cap. From a structural analysis standpoint, it is convenient to simplify the analysis by assuming the pile is fixed at some depth below the ground surface (Davisson and Robinson 1965). ab Partially embedded pile (b) Equivalent pile attached to a fixed base
Figure 6-13: Illustration of a partially embedded pile and equivalent cantilever pile (from Davisson
and Robinson 1965).
Figure b) shows the equivalent cantilever model of overall length, L
e
,
with a length below the ground to an equivalent fixed base, L
s
. The depth, L
s
,
is considered the depth to fixity for the equivalent cantilever model. This model is assumed to behave the same from a structural standpoint as the original model
(Davisson and Robinson 1965).

87 The length of the equivalent cantilever, L
e
, is given by (Abendroth and Greimann 1989):
𝐿𝐿
𝑒𝑒
= 𝐿𝐿
𝑢𝑢
+ Equation 6-20) Procedures in Davisson and Robinson (1965) only apply to long foundation elements, meeting the criteria For Clays: Equation 6-21)
𝐿𝐿
𝑒𝑒
�𝐸𝐸
𝑝𝑝
𝐼𝐼
𝑤𝑤
𝐸𝐸
𝑠𝑠
4
> 4 For Sands Equation 6-22)
𝐿𝐿
𝑒𝑒
�𝐸𝐸
𝑝𝑝
𝐼𝐼
𝑤𝑤
𝑚𝑚
ℎ
5
> 4 Where
L
e
=
Length of the equivalent cantilever (ft).
E
p
= Elastic modulus of the pile/shaft (ksi).
I
w

= Weak axis moment of inertia of the pile/shaft (ft.
E
s
= Soil modulus for clays = 0.465 S
u
(ksi).
n
h

= Rate of increase of soil modulus with depth for sands. For piles that do not meet the criteria in Equations 6-21 and 6-22, piles will behave as a rigid member rather than along flexible member, and the equivalent fixed pile conditions will not apply (Davisson and Robinson 1965). As long as criteria in Equations 6-21 and 6-22 is met, the depth to fixity below the ground, L
s
,
can be estimated as d
f
for preliminary purposes (Equations 6-18 and 6-19). Equations 6-18 and 6-19 are for an assumed loading condition of axial load only, with the shaft/pile assumed to be fixed at both ends. It is noted that these equations give depth to fixity from the ground line the unbraced length, L
u
,
must be determined by the designer considering the boundary conditions at the top of the pile. Other pile tip and loading conditions are addressed in Davisson and Robinson (1965). The pile spacing in this analysis has an effect on the soil modulus. For pile spacing of three times the pile width, the effective soil modulus should be reduced to 25 percent of the value fora single pile. Fora pile spacing of 8 times the pile width, no reduction is necessary. For spacing between 3 and 8 times the pile width, interpolation between these limits indicated above can be used (AASHTO 2014).

88 Once the actual pile conditions (Figure a) have been converted to an equivalent fixed pile condition Figure b, then the structural design is relatively straightforward. The lower boundary, at L
s
,
is fixed in the structural analysis. If a frame analysis is performed, the deflection of the pile head, moment distribution, shear distribution, and axial loads at the pile head will be close to those for the original design condition. However, the moment at the depth of the fixed based, L
s
, will be greater for this equivalent model compared to the actual condition. To analyze the embedded section of the pile, the loads at the pile head should be determined from the frame analysis and converted to loads at the ground surface through structural analysis. The loads at the ground surface can then be used to design the embedded portion of the pile using the methods previously described for analysis of a single deep foundation element (Davisson and Robinson 1965). Some practitioners may elect to use the moment at the depth of the fixed base from this approach as the design moment for the internal structural analysis of the deep foundation for simplicity (it avoids performing another analysis) and conservatism (because it is greater than the actual condition. However, since this approach introduces unnecessary and unquantified conservatism, which may lead to an uneconomical design, it is not recommended practice. The equivalent cantilever approach should only be used for preliminary estimates of required embedment lengths the equivalent cantilever method is only applicable fora specific condition (laterally loaded deep foundations with unsupported lengths above the ground surface. Note that, with this method, the equivalent cantilever length is a function of pile and soil stiffness, and is not dependent on the applied loads. This allows an initial estimate of pile length for lateral loading requirements to be developed if structural loads are not yet available. However, once structural loads are available, the recommended approach is to use the p-y method to determine the required pile embedment length and the internal shear and moments.
6.7.2.3
Recommendations Regarding Fixity
In summary, the following guidelines are recommended for the use of depth to fixity in the design of deep foundation elements Fixity is defined as the depth of a deep foundation element at which both lateral deflection and the slope of the deflected element are zero. Depth to fixity is applicable to structural analyses. Procedures are presented in AASHTO (2014) Section 6.15.2 to determine resistance factors above and below the depth to fixity. Depth to fixity can be initially estimated using Equations 6-18 and 6-19, but should be verified in final design by p-y analyses. The p-y analyses also provide a more reliable estimate of lateral displacement at the top of the pile/shaft. For foundation elements embedded in rock, a depth of fixity of half the foundation element diameter below the top of rock can be assumed. The equivalent cantilever approach, applicable for long foundation elements with unsupported lengths above the ground surface, should only be used for preliminary design.
•
P-y analyses should be used in final design to verify foundation element length, buckling, and lateral stability. The total embedment of the foundation element is typically greater than the depth to fixity.

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