The document compares Winkler and elastic continuum solutions for modeling soil-pipe interaction due to tunneling. It finds that Vesic's subgrade modulus, commonly used in Winkler solutions, may not adequately model this problem as the loading distributions are different between the two methods. The elastic continuum solution is preferred as it always provides a conservative estimate and is more accurate. An alternative expression for the subgrade modulus in Winkler solutions is suggested that better matches results from the elastic continuum solution.

1. Soil-pipe Interaction due to Tunneling April 2009
Winkler Elastic Continuum Solutions
Soil–pipe interaction due to tunneling: comparison between
Winkler and elastic continuum solutions
George M. Iskander* - Ramy H. Gabr**
Introduction:
The aim of this paper is to study the effect of tunneling on existing buried pipelines
[Fig.1] by comparing between an elastic continuum solution and a closed-form Winkler
solution with Vesic subgrade modulus.
Tunneling process generates soil settlement deforming the pipe where it suffers
additional bending depends on the distribution of settlement at the pipeline level and the
relative stiffness between the pipe and the surrounding soil.
Conventionally, this problem is solved using Winkler-based models where an
appropriate subgrade modulus is assumed for both linear elastic and non-linear analyses.
In linear elastic analysis the subgrade modulus is usually determined by means of
Vesic’s expression that allows a beam on a Winkler foundation to exhibit similar
displacements and moments to that of a beam on an elastic half-space when loaded with
the same load.
To validate the comparison results, assumptions are given for both solutions (refer to
the original paper for assumptions (a) to (f))
Elastic Continuum Solution:
The pipe behavior and the soil continuum displacement are represented in [Eq.
(2)&(3)], and after introducing compatibility requirements, elastic continuum solution is
obtained in [Eq.(8)]
Constructing the components of [Eq.(8)] uses point loading however; it does not
satisfy displacement at the point of loading.
To solve this inadequacy, a reference displacement value for that point was
considered [Fig.2], where displacement at certain point due to uniform load equal to
average displacement for that load due to equivalent concentrated load at that point.
Normalized solution is proposed which describe maximum sagging bending moments
[Fig.3], where Smax and i (Inflection point) are the maim factors affecting the relation.
[Fig.4] shows when R increase, the normalized Bending moment decrease which
highlights the overestimation of bending moments when assuming the pipe follows the
curvature of the soil.
Closed-Form Solution of the Winkler Problem:
The pipe behavior is represented in [Eq.9] and illustrated mechanically in [Fig.5].The
infinite Winkler beam consider a concentrated load P to creates bending moment at
distance t [Eq.(10)], the continuous loading due to the soil trough settlement can be
replaced by an infinite number of infinitesimal concentrated loads dP(x) [Eq.(11)].
The maximum sagging moment and the moment due to the infinitesimal loads is
normalized afterthought [Eq.(13)to(15)] and a closed-form solution shall be used in [Eq.
(16)].
.
*Professor, Ain Shams University, Geotechnical Engineering Department.
Department of civil engineering
Ain shams university
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2. Soil-pipe Interaction due to Tunneling April 2009
Winkler Elastic Continuum Solutions
**Student, M.Sc, Structural Engineer.
[Fig.6] shows a comparison of the above solution with the numerical values derived
by Attewell that fit their solution with the current closed-form solution.
The Validity of Vesic’s Subgrade Modulus for Soil–Pipe Interaction:
Attewell suggest the use of the Vesic equation for the subgrade modulus (K∞ ) [Eq.
(17)] which physically means if this subgrade modulus is used to define the maximum
moment in an infinite beam under a concentrated load, this moment will be [Eq.(18)] for
the Winkler solution and [Eq.(19)] for the elastic continuum solution.
But [Eq.(19)] refers to a beam resting on the surface of an infinite half-space, thus for
the buried pipes, Attewell suggested taking K=2K∞ which corresponds to the case where
the pipe is buried at infinite depth which is also considered conservative.
Practically it is more likely to be between K∞ & 2K∞ .
Vesic derivation for [Eq.(17)] based on both Winkler and elastic continuum are
loaded by the same external loads. However, in this case, the tunnel effect may be
represented by a force distribution along the pipe that relates to the greenfield settlement.
These force distributions are not generally the same in both Winkler and elastic
continuum solution, and hence the Vesic expression might not necessarily be adequate
for this case.
[Fig.7] shows comparison between the normalized bending moment resulting from
the continuum elastic analysis and the Winkler solution using Vesic’s expression. It can
be shown that when i/ro increases the two solutions close to each other and tend to be
practically identical. However, when i/ro decreases, significant differences are
recognized due to:
1-The simple beam theory would not be accurate for a case where the extent of
deformation is comparable to the pipe radius.
2-Assumption (c) above may not be justifiable because the pipe is located close to the
tunnel, and the diameter of the pipe is similar to or bigger than that of the tunnel.
Then the comparison between the two models may be considered irrelevant unless the
true solution is known.
Because the Winkler method underestimates the solution [Fig.7], continuum solution
is preferred as it always provides conservative estimation and accurate solutions.
An Alternative Analogue for Winkler Solution:
In the Winkler system the normalized bending moment is a function of ºI, whereas in
the elastic continuum it was found to be a function of R, where a relation is given
between them [Eq.(20)].
[Fig.8] shows the comparison between the Winkler solution with the subgrade
modulus of [Eq.(20) and the continuum solution as before [Eq.(16)] showing a good
agreement between the two solutions.
Department of civil engineering
Ain shams university
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3. Soil-pipe Interaction due to Tunneling April 2009
Winkler Elastic Continuum Solutions
It should be noted that the current analysis is based on a linear elastic soil. In reality
soil non-linearity will be involved. However for small displacement, where elastic
behavior dominates, the elastic continuum solution is still valid.
Conclusions:
The problem of tunneling effects on existing pipelines was solved using an elastic
continuum solution and a closed-form solution for the Winkler solution using Vesic’s
subgrade modulus which was found -after comparing the solutions- to be not necessarily
adequate for this problem.
It was noted that the significant difference between the two models was observed in a
region where both models may possibly become inadequate because of potential violation
of the model assumptions, and that a different approach should be considered for solving
the problem.
In the comparison between the proposed continuum and Winkler models, it is
practically safer to use the proposed continuum method rather than Winkler solution, as it
will either be closer to the accurate solution or at least more conservative than Winkler
solution.
Vesic’s spring coefficient would not generally give identical results to that of a
continuum solution because Vesic’s expression is ideal when Winkler and the continuum
systems are loaded by identical external loads, but in the case, same settlement trough
does not necessarily result in the same loads.
For the current case an alternative expression for the subgrade modulus was
suggested to create similar maximum bending moment in the Winkler and elastic
continuum systems.
Department of civil engineering
Ain shams university
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