Introduction Vibro replacement is an accepted method for subsoil improvement, whereby large columns of coarse backfill material are installed in the ground using special vibrators .The bearing behaviour of this composite system, consisting of stone columns and soil, cannot be reliably determined by simple investigation methods. Theoretically, however, the efficiency of vibro replacement can be reliably evaluated. Probably, the most commonly applied method is that developed by Priebe  and .
However, questions concerning the application of this method to extremely soft soils, 'floating' columns and for the proof of safety against slope or embankment failure were not discussed in previous publicaions.
Extremely soft soils Preliminary remarks In Germany, the first guidelines for the application of deep vibratory compaction techniques in general and for the vibro replacement technique in particular were published in 1976 by the German institution Forschungsgesellschaft f³r das Straflenwesen . Within this, the application of vibro replacement was restricted to soils with an undrained shear strength, c u, of at least 15kN/m 2to25kN/m 2.
Comment Vibro replacement suffers from this very conservative value. Although the technique had already grown out of its infancy by the time the guidelines were published, there were still some uncertainties over both design and performance. Taking this into account, some caution was still justified.
However since then design methods and, to a greater extent, equipment have been improved to allow stone columns to be installed safely and to perform reliably, even in very soft soils. In such cases soil improvement is achieved by drainage rather than by reinforcement.
In fact, even soils with an undrained shear strength as low as c u= 5kN/m 2(ie soils which in their natural state can scarcely allow a person to walk on them) can be treated and improved by vibro replacement. This is not just based on sporadic experiences in Germany, where extremely soft soils do not play an important role in structural engineering, but moreover on extensive application on highway projects in South East Asia .
It goes without saying that the treatment of extremely soft soils requires utmost care and the suitability of the method to a project has to be carefully examined. But it is not justified to cling to formerly defined values based on different circumstances.
Finally, the design method described here may be particularly suitable for extremely soft soils as it hardly allows a serious misjudgement. This is because it does not deliver absolute improvement values but only a relative improvement value which always refers to the original state of the soil.
'Floating' columns Preliminary remarks The starting point of Priebe's design method is the examination of a unit cell as part of an unlimited load area on an unlimited column grid.The evaluation, simplified by the following three restricting conditions, gives the basic improvement value, n 0. l The bulk density of column and soil is ignored l The column material is incompressible l The column is founded on a rigid layer The second step of the method considers the compressibility of the column material, ie the second restricting condition is abandoned. The result is the reduced improvement value, n 1.The third step refers to the influence of the bulk densities of column and soil which is considered by the depth factor, f d.The product of f dand n 1gives the final improvement value n 2which - at least - is recommended for settlement calculations.
With regard to the last restricting condition, this was not considered in  because this condition is met in most practical cases. However, it is not met when using 'floating' columns. Here additional evaluations are required. The following formulae, their context and application are subject to the same terms as in .
General aspects Vibro replacement rightly stands for ground improvement. Although the installed stone columns of coarse backfill material present stiffer structural members, they still depend on the support of the surrounding soil. This interaction is the reason that, for example in the case of vibro replacement for the foundation of a road embankment, there is hardly ever any danger of column punching. Some waviness at foundation level will certainly occur but will - if at all - hardly be noticeable at the embankment surface.
A similar situation occurs for stone columns ending above a soft layer which is not suitable to bear concentrated loads. At the boundary between the columns and the underlying soil there is a balancing of stress and strain. This balancing takes place more or less in both zones, ie in the upper treated soil as well as in the soil below.
To approach the 'floating' case of columns in poor soil the obvious thing to do would be to assume that the balancing takes place solely either in the upper treated zone or in the untreated zone below.
Balance of stress in the upper treated zone In this case it is assumed that the treated depth can be divided into a zone with full effectiveness of the columns and a transitional zone. Above the transitional zone the stress within the unit cell is distributed in the way derived from normal calculations, ie with the proportional load on the column, m9. At the bottom of the transitional zone, ie at the end of the column, there exists, according to the assumption, a uniform stress p, if disregarding the unit weight of column and soil (In  m9 was reduced in comparison to m).That means that a difference, DP, has to be transferred from the column to the soil.
DP = p EPSILON A EPSILON (m' - A C/A) = p EPSILON A EPSILON [(n - 1)/n - A C/A] The differential load, DP, has to be transferred from the column to the soil by shear resistances R.
These consist of cohesion, c S, and friction, F, in the soil.
R = p EPSILON dia CEPSILON (c S+ F) dia C= column diameter The friction results from the lateral support of the soil and its friction angle, w S.It is far too conservative to ascertain lateral support from the proportional external load on the soil between the columns, p S, only. On the other hand, it is questionable whether - at the predetermined value of K=1 - additionally the total weight of the soil should be considered.
However, the height of the transitional zone amounts to h = DP/R and, on the condition of a linearly decreasing improvement, the settlement of this zone has to be calculated with a reduced average improvement factor of n'=(1+n)/2.
Balance of stress in the untreated underlying zone In this case it is assumed that the load on the column is transferred down the entire depth of the treated layer.
Thus, logically, the column is pressing the underlying soil with its increased load and causes a depression. Only in the widest sense this effect may be described as punching.
As already indicated, failure-like processes similar to those experienced with sinking piles are not possible with vibro replacement because of the transfer of load to the surrounding soil.Therefore, the depression caused by a single column in the underlying soil can be compared with the settlement of a circular footing and calculated accordingly.
It is sufficient to calculate this settlement with the total proportional load of the column and not with the difference in pressure between the column and the soil, but then to only use the depth where the pressure has decreased by load distribution to the value of p.
Assessment of the approaches Settlement calculations are generally performed using the deformation moduli or compression index and initial void ratio only. Consequently, the first approach is disadvantageous, for the simple reason that more or less vague values with regard to shear resistance also have to be considered.
As already mentioned, the balance in stress and strain takes place in zones both above and below the bottom of the stone columns. If either of the approaches is used, the obtained settlements will always turn out to be too high which is - even in the sense of being intentionally on the safe side - not always acceptable.
In the first case of the assumed balancing solely in the upper, improved zone the inaccuracy increases with the growing stiffness of the underlying stratum. In the second case, assuming balancing solely in the lower, untreated zone, inaccuracy increases with growing initial stiffness of the upper treated soil.
In the second case, however, the limitation of inaccuracy is relatively simple: the punching has to be limited to the very value which the upper treated strata - without improvement - will allow.This is because the punching into the lower soil is coupled with an equal vertical compression of upper soil.
s' P= s PEPSILON s 0/ (s P+ s 0) s P= calculated punching value s 0= settlement of the treated layers without improvement s' P= reduced punching value Consequently, it is simpler and more accurate to use the second approach, applying the reduction procedure. The author has calculated settlement this way using the GRETA program - software developed by the author not only for vibro replacement but extended to nearly all kinds of vertical foundation members.
Thus the entire settlement of vibro replacement projects consists of three parts:
l Settlement of the treated soil l Additional settlement by punching into the layer directly below the treated soil.
l Settlement of all layers below the treatment depth s = s u+ s' P+ s lsu = calculated settlement of upper treated soil (eg Priebe's method) s l= calculated settlement of lower layers Embankment or slope failure Preliminary remarks Excessive building settlement may cause substantial structural damage. Fortunately, such damage can often be avoided as there is advance warning of problems in most cases.
The situation is completely different with regard to stability failure.Generally, there is no warning and thus no chance for damage limitation. Here, calculations have to be on the safe side. Therefore, for economic reasons the calculation should be as reliable as possible.
Unfortunately, in cases of ground improvement by vibro replacement, frequently the proof of sufficient safety is unknowingly based on overestimating the improvement achieved by the treatment.
This problem was not dealt with in .
Explanation According to Priebe's design method, the shear values of improved ground are not averaged proportionally to the cross-section of the stone columns and the soil in between but proportionally to their loads. That means the high friction value of stone columns does not only contribute according to their cross section but also according to their load. However, this is valid for external loads on the columns only after installation.
If now, for example to prove the safety against failure of an embankment slope, the shear values averaged this way are thoughtlessly assigned to the sliding surface in treated ground, this results in an inaccuracy on the unsafe side. This may be substantial, particularly where there are deep sliding surfaces.
The fact is, because the approach would implicitly require that not only the deposited load of the embankment, but also the entire load of the improved soil above the sliding surface, is mainly borne by the columns which, at best, could be anticipated in exceptional cases only.
However, to avoid the described inaccuracy it is unjustified to average - right from the beginning - the shear values in proportion to the cross-section of the columns and the surrounding soil. If the proof of safety is based - as usual - on a plane system, two approaches which provide sufficient safety may be taken into account: either to apply adjusted loads or to correct the averaged shear values.
As to the following formulae and their context the same terms as in  apply.
Adjusted loads This approach requires the slice method to be applied to prove safety against embankment or slope failure. Columns and soil are then transformed into equal area strips forming individual slices, each with their attributed properties (ie their own characteristic unaltered shear values).
In this case, the load and weight of the embankment is distributed on the column and soil slices according to the Priebe method. For that, the portion of the slice load which lies above the improved stratum has only to be multiplied by corresponding factors for the column and soil slices respectively.
fc= m 1' EPSILON 1/(A c/A) fs= (1 - m 1') EPSILON 1/(1 - A c/A) This approach, with different loads on column and soil slices, is plausible and relatively accurate, if the simplifications and approximations in the determination of the values m 1' are disregarded.
Admittedly, the input of slices for the calculation may be a bit time consuming, all depending on the program which will be used.
Corrected shear values The corrected shear values approach does not necessarily depend on column and soil slices. It is based on the assumption of averaged shear values of the improved stratum, determined on the basis of a reduced load on the columns. It is recommended to calculate the proportional load on the columns, m 1'', from the ratio between the load and weight of the embankment, Q c, and the total load and weight of the sliding mass, Q.
m1'' = A c/A + m 1' - A c/A EPSILON Q' c/Q With this reduced proportional load on the columns the shear values can be averaged as usual.
Even if this method does not require the sliding body to be subdivided into slices and by no means into slices depending on the grid of columns, it is advisable to calculate with slices - bringing it close to the method of adjusted loads - and consequently is more accurate.
Ground failure Though - with reservations - ground failure beneath foundations is comparable with embankment or slope failure, usually safety is proved rather differently. It is not necessary to go into detail here as  outlines how it can be proved in case of vibro replacement.
With regard to overestimating the degree of improvement using vibro replacement, principally the same situation as for embankment or slope failure applies, but the possible inaccuracy in the procedure outlined in  is far less.
This is because a considerable portion of the failure line outside of the foundation is in any case calculated with the initial shear values of the treated soil.The remainder of the sliding mass beneath the foundation does not play such an important role - in comparison to load and weight of the foundation itself - to justify any exaggerated accuracy.
Heinz Priebe is founder of GEOStat. He worked previously at Keller Grundbau, Germany, for 35 years, specialising in ground imrpovement techniques and the design of vibro replacement.
References  Kirsch K (1993). Die Baugrundverbesserung mit Tiefenr³ttlern, 40 Jahre Spezialtiefbau:
1953-1993. Festschrift, Werner-Verlag, D³sseldorf.
 Priebe HJ (1995). The design of vibroreplacement, Ground Engineering, December 1995, pp31-37.
 Priebe HJ (1995). Die Bemessung von R³ttelstopfverdichtungen, Bautechnik 72, H3.
 Forschungsgesellschaft f³r das Straflenwesen: Merkblatt f³r die Untergrundverbesserung durch Tiefenr³ttler (1979).
 Raju VR (1997).The behaviour of very soft cohesive soils improved by vibro replacement.Ground improvement conference, London, 1997.