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FAILURE ENVELOPE OF EMBEDDED WALLS IN CLAY SUBJECTED TO COMBINED
HORIZONTAL
LOAD AND MOMENT Suraparb Keawsawasvong1, Boonchai Ukritchon2
1Graduate Student, Department of Civil Engineering,
Chulalongkorn University, Bangkok 10330, Thailand,
[email protected]
2Associate Professor, Department of Civil Engineering,
Chulalongkorn University, Bangkok 10330, Thailand,
[email protected]
Abstract At present, design processes of structures have become
complicated such as those in offshore engineering or large bridge
structures, etc. Generally, piles used in supporting those
structures carry loads not only the vertical load direction, but
also the horizontal direction arising from wave forces in the sea,
wind loading or forces of earthquake actions. Those forces may
result in damages and failures of structures if considering only
the vertical direction. This research presents a study of undrained
stability of embedded wall in clay subjected to combined horizontal
load and moment. The 2D plane strain condition is considered in
this analysis. The objective of this research is to determine the
failure envelope of embedded wall subjected to horizontal load and
moment. The studied parameters include untrained shear strength of
the clay layer, the thickness of the wall (D), and wall embedded
length (L). The results of analyses are presented in terms of
dimensionless graph between normalized horizontal load, moment, and
ratio of wall length to thickness (L/D). Finite element software,
PLAXIS was used in this analysis. The embedded wall is set to have
property of linear elastic material without failure consideration,
while the clay is modeled as the Mohr-Coulomb material. The
analyses consider ratio of wall length to thickness, ranging from
5-80 for wide practical applications. Series of failure envelopes
were determined for each value of L/D. Results of this research can
be applied in design and analysis of embedded walls, where their
results are more accurate. In addition, they provide preliminary
calculations for the similar loading cases of single pile in 3D
problems.
Keywords: Numerical analysis, Combined loading, Plane strain,
Finite element
Introduction In designing pile foundations of complex structures
such as foundations of offshore structures or large structures such
as bridges or high-rise buildings, it is not valid to consider
loading of pile foundations of those structures to exist only the
vertical load direction. Instead, loading considerations should
include horizontal load direction as well as overturning moment in
order to model the most realistic state. This is because in reality
forces acting on those structures may include wave forces, wind
loadings, or dynamics forces from earthquake actions. Such forces
can cause horizontal load and moment acting on the top of
individual pile. In the past, several research works presented
calculations of ultimate lateral pile resistance, where comparisons
of those methods were made in terms of advantages and disadvantages
by Ruigrok (2010) [10] and Reese (2007) [11].
One of the most popular methods for analyzing ultimate lateral
load on the pile is the method proposed by Blum (1932) [3] and
Broms (1964, 1965) [1], [2]. Even though those two methods can be
used to determine ultimate lateral resistance of pile, they are
different in theoretical background in modeling lateral resistance
of soil using simple geometrical earth pressure distribution. As a
result, results of calculations may be incorrect and not
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accurate enough. In addition, Blum's method does not use input
parameter of undrained shear strength in the calculation such that
its modeling does not follow the actual situation when applying for
the analysis case of cohesive soils.
At present, the analytical tools are more advanced that those in
the past. The finite element method has become popular in analyzing
ultimate load resistance of pile. Research works by Chaudry (1994)
[8], Klar (2008) [4], and Zhang (2011) [5] were based on the finite
element analyses in calculations of pile under lateral load.
However, their results were not presented in terms of dimensionless
chart between ultimate load and length or the graph of failure
envelope. One of examples of failure envelope studies include the
works by Ukritchon et al. (1998) [7] involving with limit state of
strip footing, not pile.
The above mentioned works did not consider the combined loading
of horizontal force and moment. In contrast, Ukritchon (1998) [9]
and Huang (2007) [6] applied finite element limit analyses in
determining the curve of failure envelope for the lateral force and
moment of embedded wall. The characteristics of failure envelope
for different types of loadings are shown in Figure 1b. Their
results were based on the modeling of wall as plate element and did
not consider the effect of wall thickness. However, the embedded
wall structures have the finite thickness in actual conditions.
Thus, the effect of wall thickness should be incorporated into the
study and it is the objective of this paper.
This research presents the dimensionless graph between ultimate
lateral load and length of embedded wall when subjected to
horizontal force and moment. In addition, the results of this
research also include the failure envelope for the general case of
embedded wall acted by horizontal force and moment. The two
dimensional finite element analyses, PLAXIS (Brinkgreve, 2002)
[12], are used in this study. In particular, the embedded walls are
modeled with solid elements with its finite thickness dimension as
shown in Figure 1a, not using plate element like previous research
works. As a result, its modeling corresponds to a more realistic
case of embedded wall in the field. The next section explains
important modeling issues and analysis details of this
research.
(a) (b)
Figure 1. (a) Problem geometry of combined lateral load and
moment, (b) Loading conditions
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Method of analysis
Model for analysis This research uses the commercial finite
element software, PLAXIS 2D (Brinkgreve, 2002) [12]. The two
dimensional plane strain geometry is used to model the embedded
wall in the clay layer. The clay model is the Mohr-Coulomb
material, where the undrained shear strength profile is homogeneous
and isotropic. The undrained condition of the clay is modeled by
the undrained type B, where state of stresses is effective stress,
the stiffness and Poisson's ratio are effective types. But, the
undrained shear strength, su is total stress, where the friction
angle, = 0 and the dilatancy angle, = 0 in the analysis. The soil
has the total unit weight of , and the effective Young's modulus,
E/su = 200. It should be noted that the input value of soil unit
weight does not have any effect to the limit load of this problem
since the analysis condition is undrained.
The embedded wall is modeled as elastic material of the
non-porous type. Its properties correspond to reinforced concrete
wall, where the unit weight = 24 kN/m3, Poisson's ratio = 0.21 and
Young's modulus = 2.545 107 kPa. Because of elastic material
modeling of embedded wall, there is no failure of the wall in the
analysis. This assumption is realistic in actual design practice.
The embedded wall must be designed to have enough thickness and
reinforcement in resisting shear and bending modes such that the
failure of the system is governed by the failure of the soil before
the failure of the wall happens. The embedded wall has the length
of L, and thickness of D.
The boundary condition of this problem corresponds to a typical
pattern used in finite element analysis of geotechnical
engineering. The bottom boundary plane is defined as zero movements
for both horizontal and vertical directions. The left and right
boundary planes are defined as zero horizontal movement, while it
only vertical movement is allowed, as shown in Figure 2.
The interaction between the clay and the wall is modeled using
soil-structure interface. These interface elements are modeled
around both sides of the embedded wall and at its base, as shown in
Figure 2. The interface roughness between the clay and the wall is
controlled by Rinter, which is the soil-wall adhesion factor. In
this analysis, Rinter has the value of 0.67, which is the typical
value for most soil-structure interface. Thus, according to the
Mohr-Coulomb material, the undrained shear strength for the
interface element, ci = 0.67su. In addition, for embedded wall
subjected to horizontal force and moment, there is a possibility
that separation can happen behind the interface between the clay
and the wall. Thus, the condition of no-tension of effective stress
is also applied at those interface elements.
The embedded wall is loaded at the top with the horizontal
force, H and the moment, M. The results of analyses are presented
in terms of relationship between dimensionless parameters of
failure horizontal load, H/suD and embedded length ratio, L/D. The
analyses consider several ratios of L/D, ranging from 5-80. The
larger value of L/D, the more slender of the embedded wall. For
each case of analysis, the thickness of the wall, D is changed
while the length of the wall remains constant, giving rise to
different values of L/D.
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(a) (b) Figure 2. Numerical models of finite element analysis,
(a) Schematic model
(b) Geometry model
Mesh Model Figure 3 shows a typical mesh used in the finite
element analysis of embedded wall subjected to horizontal load and
moment at its top. Triangular types of solid elements are used for
modeling both clay and wall. There are 15 nodes for each triangular
element, corresponding to the cubic strain element type. In
addition, very fine mesh distributions are employed in order to
obtain accurate result of limit state.
Figure 3. Typical mesh used in the finite element analysis with
15 nodes and 12 stress
points
Results According to the modeling section of embedded wall as
described earlier, results of present study are compared with
previous research works for different aspects of analyses as
follows.
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Comparisons with Bromss method (1964, 1965) [1], [2] Figure 4
shows a model of free head standing embedded wall proposed by
Bromss method (1964, 1965) [1], [2]. The wall is extended above the
ground surface by the eccentricity distance of e. Thus, when the
parameter, e is divided by the length of the wall, L, additional
dimensionless parameter, e/D becomes apparent, which is also one of
normalized parameters used in Bromss method. The study compares
ultimate lateral resistance of the present analyses with those of
Broms, for the case of e/D = 0, 1, 8, and 16, as shown in Figure 5.
It should be noted that the problem of free standing embedded wall
loaded by purely horizontal force with the eccentricity distance,
e, is statically equivalent to that of embedded wall subjected by
horizontal force and moment applied at its top, where e=M/H.
In comparing with Bromss method, the ultimate lateral resistance
of plane strain finite element analyses based on one unit length
out-of-plane must be multiplied with the wall thickness in order to
obtain the full load of the pile, assuming that the pile geometry
is square or circular. It can be seen from Figure 5 that for all
cases of L/D, the ultimate lateral resistance from Bromss method is
significantly higher than about 1.5-1.7 times that from finite
element analyses, particularly for very large values of L/D.
However, the difference between those analyses seems to be much
smaller for the case of smaller ratio of L/D.
Figure 4. Problem geometry of pile in Bromss (1964, 1965) [1],
[2] model
(a) (b)
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(c) (d)
Figure 5. Comparison of ultimate lateral resistance between the
finite element analysis and Broms (1965) [2], (a) e/D = 0, (b) e/D
= 1, (c) e/D = 8, (d) e/D = 16
Purely Lateral load or Purely Applied Moment Figures 6-13 show
examples of predicted failure mechanisms from finite element
analyses. Figure 6-9 correspond to the case of purely lateral load
of embedded wall, while Figure 10-13 correspond to the case of
purely applied moment of embedded wall. For each case, the failure
results include deformed mesh, total increment vector, and
incremental shear strain contour. Comparisons are made for three
cases of wall thickness ratio, L/D = 5, 40, and 80. The case of L/D
= 5 corresponds to a relatively thick wall, while the case of L/D =
80 corresponds to a relatively slender wall. For purely horizontal
load (Figure 6-9), the failure mechanism of the embedded wall
happens such that the wall rotates about some point near its tip
without translation movement of the wall. Near the ground surface,
the front side fails in the passive state condition, while the back
side fails in the active mode. On the other hand, for purely
applied moment (Figure 10-13), the embedded wall fails by wall
rotation at about the mid point of the wall without translation
movement of the wall. The failure zone of passive and active modes
of purely applied moment is much smaller than that of purely
horizontal load.
Figure 6. Deformed mesh for purely lateral load, where L/D = 5,
40, 80
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Figure 7. Total increment vectors for purely lateral load, where
L/D = 5, 40, 80
Figure 8. Incremental shear contours for purely lateral load,
where L/D = 5, 40, 80
Figure 9. Plastic points of for purely lateral load, where L/D =
5, 40, 80
Figure 10. Deformed mesh for purely applied moment, where L/D =
5, 40, 80
Figure 11. Total increment vectors for purely applied moment,
where L/D = 5, 40, 80
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Figure 12. Incremental shear strain contours for purely applied
moment, where L/D = 5, 40, 80
Figure 13. Plastic points for purely applied moment, where L/D =
5, 40, 80 Figure 14 shows the results of purely horizontal ultimate
load, H/suD or purely
ultimate moment, M/suLD for the case of plane strain condition
without considering scaling effect to the actual pile geometry. It
can be seen that the selected dimensionless terms, H/suD and M/suLD
give the best result, yielding the linear relationship between
those values and L/D. Thus, mathematical equation for predicting
those values has much simpler form because of linear
relationship.
(a) (b)
Figure 14. Normalized limit state solutions, (a) purely lateral
load, (b) purely applied moment
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Combined Lateral load and Moment In order to include the
complete loading state of embedded wall, this section presents the
case of embedded wall subjected to combined horizontal load and
moment. Figure 15-18 show the results of failure mechanism for the
case of L/D = 5, 40, and 80. For all cases, the ratios of applied
moment to horizontal load are constant, namely M/HL = 1/30. The
failure mechanism of this case is similar that that of pure
horizontal load. This is because the applied moment ratio, M/HL is
small.
Figure 15. Deformed mesh of combined lateral load and moment,
where L/D = 5, 40, 80, and M/HL = 1/30
Figure 16. Total increment vectors of combined lateral load and
moment, where L/D = 5, 40, 80, and M/HL = 1/30
Figure 17. Incremental shear strain contour of combined lateral
load and moment, where L/D = 5, 40, 80, and M/HL = 1/30
Figure 18. Plastic points of combined lateral load and moment,
where L/D = 5, 40, 80, and M/HL = 1/30
The failure envelope of embedded wall subjected to combined
lateral load and moment is obtained by analyzing two different
cases, as shown in Figure 1b. The first case corresponds to the
case where lateral load and moment produce overturning to the
same
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direction, labeled as I in the first quadrant. The second case
corresponds to the case where lateral load and moment produce
overturning to the opposite direction, labeled as II in the second
quadrant. The remaining of the graph in the third and fourth
quadrants is obtained from the symmetry of the problem. In
particular, the result of first quadrant is equal to that of the
third quadrant, while that of the fourth quadrant is equal to that
of the second one. The failure envelope is plotted as a function of
ultimate lateral resistance, H/suD and ultimate moment resistance,
M/suLD, as shown in Figure 19. Figure 19a, 19b, and 19c show the
case of embedded wall ratios, L/D = 5, 40, and 80, while Figure 19d
compares all failure envelopes for different values of L/D. It can
be seen that each failure envelope has the form of elliptical
shape, which is also rotated about 3/4 from the positive horizontal
axis. However, the rotated ellipse is not symmetrical at its
rotated major and minor axes. The non-symmetry of the ellipse
happens at both ends of the major axis, resulting in distorted form
of rotated ellipse. However, it can be seen that the distorted and
rotated ellipse does hold convexity condition, where the classical
concept of failure envelope is still valid. The size of rotated
ellipse is controlled by the ratio of L/D. The higher the ratio of
L/D, the larger the size of the rotated ellipse.
(a) (b)
(c) (d) Figure 19. Failure envelope for combined lateral load
and moment, (a) L/D = 5, (b) L/D =
40, (c) L/D = 80, (d) Comparisons for all ratios of L/D = 5, 20,
40, 60, 80
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Comparison of Failure envelope with Ukritchon (1998) [7] Figure
20 compares the failure envelope from the finite element analysis
of the present study and that of Ukritchon (1998) [7]. It should be
noted that the results of Ukritchon (1998) are based on the lower
(LB) and upper bound (UB) finite element limit analysis and do not
consider the effect of wall thickness. In comparing results, the
failure envelopes has to be replotted and normalized only by the
wall length, L, giving rise to dimensionless parameters as H/suL
and M/suL2. It can be seen that the failure envelope of the two
results has similar form of rotated nonsymmetrical ellipse, while
their size are significantly different. The failure envelope
reported by Ukritchon (1998) is much smaller than that of the
present study because of different modeling in no-tension condition
of soil-structure interface element. For results of Ukritchon
(1998), no-tension condition of interface element is modeled such
that the total normal stress is always compressive. But, the
present study models no-tension condition of interface element such
that the effective normal stress is always compressive.
Figure 20. Comparison of failure envelope between the present
study and Ukritchon (1998)
Conclusions This research presents the use of two dimensional
plane strain finite element model in analyzing ultimate state of
embedded wall subjected to combined horizontal load and moment.
Unlike previous research works in the past, the present study
considers the effect of wall thickness, which corresponds to a more
realistic case in the field. The results of analyses are presented
in terms of dimensionless variables, namely 1) embedded length
thickness ratio, L/D; 2) normalized ultimate lateral resistance,
H/suD; and 3) normalized ultimate moment resistance, M/suLD. The
ratio of L/D is modeled to have different ranges from 5-80. The
studies include analysis of the limit state for purely lateral
load, purely applied moment, and combined lateral and moment. The
studied results can be used in analyzing and designing embedded
wall for plane strain condition or actual pile geometry. However,
the latter case requires simple scaling from plane strain solutions
to obtain the limit state of actual pile. Comparisons of the latter
case show that the ultimate lateral resistance scaled from the
plane strain condition is much smaller than that of Broms (1965)
[2], which indicates that the results give much conservative
calculations. In addition, the failure envelope of H-M has the form
of rotated ellipse with distortion at both ends. The
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shape pattern of the present study is similar to that of
Ukritchon (1998) [7], but the latter has smaller size due to
different modeling of no-tension condition at soil-structure
interface.
Acknowledgment The first author would like to thank the
department of civil engineering, Chulalongkorn University for
financial support of his graduate there, where he received
the-100-year university celebration graduate research grant from
the department. This allows him to pursue the master study for his
interest in course works and research in geotechnical engineering,
Chulalongkorn University.
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foundation, division 91 (3), 77-99, 1965. [2] B.B. Broms,
Lateral resistance of piles in cohesive soils, Journal of the
soil
mechanics and foundation, division 90 (2), 27-63, 1964. [3] H.
Blum, Wirtschaftliche dalbenformen und deren berechnung,
Bautechnik, Heft 5,
1932.
[4] A. Klar, M. F. Randolph, Upper-bound and load-displacement
solution for laterally loaded piles in clays based on energy
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[5] L. Zhang, Nonlinear analysis of laterally loaded rigid piles
in cohesive soil, International journal for numerical and
analytical method in geomechanics 2013, 37:201-220, 2011.
[6] M. Huang, Q. Huang, Ultimate lateral resistance of sheet
pile walls by numerical lower bound analysis, Chinese Journal of
Geotechnical Engineering 2007, Vol. 29, Issue (7), 988-994,
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[7] B. Ukritchon, A.J. Whittle, S.W. Sloan, Undrained limit
analysis for combined loading of strip footings on clay, Journal of
Geotechnical and Geoenvironmental Engineering, ASCE, 124(1):
265276, 1998.
[8] A.R. Chaudhry. Static pile-soil-pile Interaction in Offshore
pile groups. Thesis (PhD), University of Oxford, England, 1994.
[9] B. Ukritchon. Application of Numerical limit analyses for
Undrained stability problems in clay. Thesis (PhD), Massachusetts
Institute of technology, USA, 1998.
[10] J.A.T. Ruigrok. Laterally loaded piles models and
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[11] L.C. Reese, W.F. Van Impe, Single piles and pile groups
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[12] R.B.J. Brinkgreve, et al., Plaxis 2D Version 8 Manual, A.A.
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