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Soil-pipe Interaction due to Tunneling April 2009Winkler 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.**Student, M.Sc, Structural Engineer.

Department of civil engineeringAin shams university

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[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/r increases the two solutions close to each other and tend to be practically identical. However, when i/r 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.


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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.


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