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Analyses in program Ground Loss

Analyses performed in the program "Ground Loss" can be divided into the following groups:

  • analysis of the shape of subsidence trough above excavations
  • analysis of building damage

The failure analysis of building is based on the shape of subsidence trough.

Analysis of subsidence trough

The analysis of subsidence trough consists of several sequential steps:

  • determination of the maximum settlement and dimensions of subsidence trough for individual excavations
  • back calculation of the shape and dimensions of subsidence trough providing it is calculated at a given depth below the terrain surface.
  • determination of the overall shape of subsidence trough for more excavations
  • post-processing of other variables (horizontal deformation, slope)

The analysis of maximum settlement and dimensions of subsidence trough can be carried out using either the theory of volume loss or the classical theories (Peck, Fazekas, Limanov).

Volume loss

The volume loss method is a semi-empirical method based partially on theoretical grounds. The method introduces, although indirectly, the basic parameters of excavation into the analysis (these include mechanical parameters of a medium, technological effects of excavation, excavation lining etc) using 2 comprehensive parameters (coefficient k for determination of inflection point and a percentage of volume loss VL). These parameters uniquely define the shape of subsidence trough and are determined empirically from years of experience.

Settlement expressed in terms volumes
Settlement expressed in terms volumes

The maximum settlement Smax, and location of inflection point Linf are provided by the following expressions:

Expression

Expression

where:
A - excavation area
Z - depth of center point of excavation
k - coefficient to calculate inflection point (material constant)
VL - percentage of volume loss

The roof deformation ua follows from:

Expression

where:
r - excavation radius
VL - percentage of volume loss

Data needed for the determination of subsidence trough using the volume loss method:

Coefficient to calculate inflection point k
Soil or rock k
cohesionless soil 0,3
normaly consolidated clay 0,5
overconsolidated clay 0,6-0,7
clay slate 0,6-0,8
quartzite 0,8-0,9
Percentage of volume loss VL
Technology VL
TBM 0,5-1
Sequential excavation method 0,8-1,5

Several relationships were also derived to determine the value of lost volume VL based on stability ratio N defined by Broms and Bennermarkem:

Expression

where:
σv - verall stress along excavation axis
σt - excavation lining resistance (if lining is installed)
Sn - undrained stiffness of clay

For N < 2 the soil/rock in the vicinity of excavation is assumed elastic and stable. For N ∈ < 2,4 local plastic zones begin to develop in the vicinity of excavation, for N ∈ < 4,6 a large plastic zone develops around excavation and for N = 6 the loss of stability of tunnel face occurs. Figure shows the dependence of stability ration and lost volume VL.

Graph

Classical theory

Convergence analysis of an excavation and calculation of the maximum settlement in a homogeneous body are the same for all classical theories. The subsidence trough analyses then differ depending on the assumed theory (Peck, Fazekas, Limanov).

When calculating settlement the program first determines the radial loading of a circular excavation as:

Expression

where:
σz - geostatic stress in center of excavation
Kr - coefficient of pressure at rest of cohesive soil

The roof ua and the bottom ub deformations of excavation follow from:

Expression

Expression

where:
Z - depth of center point of excavation
r - excavation radius
E - modulus of elasticity of rock/soil in vicinity of excavation
ν - Poisson's number of rock/soil in vicinity of excavation

The maximum terrain settlement and the length of subsidence trough are determined as follows:

Expression

Expression

where:
Z - depth of center point of excavation
r - excavation radius
E - modulus of elasticity of rock/soil in vicinity of excavation
ν - Poisson's number of rock/soil in vicinity of excavation

When the tunnel roof displacement is prescribed the maximum settlement is provided by the following expression:

Expression

where:
Z - depth of center point of excavation
r - excavation radius
ua - tunnel roof displacement
ν - Poisson's number of rock/soil in vicinity of excavation

Analysis for layered subsoil

When determining a settlement of layered subsoil the program first calculates the settlement at the interface between the first layer above excavation and other layers of overburden Sint and determines the length of subsidence trough along layers interfaces. In this case the approach complies with the one used for a homogeneous soil.

Next (as shown in Figure) the program determines the length of subsidence trough L at the terrain surface.

Analysis of settlement for layered subsoil
Analysis of settlement for layered subsoil

The next computation differs depending on the selected analysis theory:

Solution after Limanov

Limanov described the horizontal displacement above excavation with the help of lost area F:

Expression

where:
L - length of subsidence trough
F - volume loss of soil per 1 m run determined from:

Expression

where:
Lint - length of subsidence trough along interfaces above excavation
Sint - settlement of respective interface

Solution after Fazekas

Fazekas described the horizontal displacement above excavation using the following expression:

Expression

where:
L - length of subsidence trough
Lint - length of subsidence trough along interfaces above excavation
Sint - settlement of respective interface

Solution after Peck

Peck described the horizontal displacement above excavation using the following expression:

Expression

where:
Lint - length of subsidence trough along interfaces above excavation
Sint - settlement of respective interface
Linf - distance of inflection point of subsidence trough from excavation axis at terrain surface

Shape of subsidence trough

The program offers two particular shapes of subsidence troughs – according to Gauss or Aversin.

Curve based on Gauss

A number of studies carried out both in the USA and Great Britain proved that the transverse shape of subsidence trough can be well approximated using the Gauss function. This assumption then allows us to determine the horizontal displacement at a distance x from the vertical axis of symmetry as:

Expression

where:
Si - settlement at point with coordinate xi
Smax - maximum terrain settlement
Linf - distance of inflection point

Curve based on Aversin

Aversin derived, based on visual inspection and measurements of underground structures in Russia, the following expression for the shape of subsidence trough:

Expression

where:
Si - settlement at point with coordinate xi
Smax - maximum terrain settlement
L - reach of subsidence trough

Coefficient of calculation of inflection point

When the classical methods are used the inputted coefficient kinf allows the determination of the inflection point location based on Linf=L/kinf. In this case the coefficient kinf represents a very important input parameter strongly influencing the shape and slope of subsidence trough. Its value depends on the average soil or rock, respectively, in overburden – literature offers the values of kinf in the range 2,1 - 4,0.

Based on a series of FEM calculations the following values are recommended:

gravel soil G1-G3 kinf = 3,5
sand and gravel soil S1-S5,G4,G5, rocks R5-R6    kinf = 3,0
fine-grained soil F1-F4 kinf = 2,5
fine-grained soil F5-F8 kinf = 2,1

The coefficient for calculation of inflection point is inputted in the frame "Project".

Subsidence trough with several excavations

The principal of superposition is used when calculating the settlement caused by structured or multiple excavations. Based on input parameters the program first determines subsidence troughs and horizontal displacements for individual excavations. The overall subsidence trough is determined subsequently.

Other variables, horizontal strain and gradient of subsidence trough, are post-processed from the overall subsidence trough.

Analysis of subsidence trough at a depth

A linear interpolation between the maximal value of the settlement Smax at a terrain surface and the displacement of roof excavation ua is used to calculate the maximum settlement S at a depth h below the terrain surface in a homogeneous body.

Analysis of subsidence trough at a depth
Analysis of subsidence trough at a depth

The width of subsidence trough at an overburden l is provided by:

Expression

where:
L - length of subsidence trough at terrain surface
r - excavation radius
Z - depth of center point
z - analysis depth

The values l and S are then used to determine the shape of subsidence trough in overburden above an excavation.

Calculation of other variables

A vertical settlement is accompanied by the evolution of horizontal displacements which may cause damage to nearby buildings. The horizontal displacement can be derived from the vertical settlement providing the resulting displacement vectors are directed into the center of excavation. In such a case the horizontal displacement of the soil is provided by the following equation:

Expression

where:
x - distance of point x from axis of excavation
s(x) - settlement at point x
Z - depth of center point of excavation

The horizontal displacements are determined in a differential way along the x axis and in the transverse direction they can be expressed using the following equation:

Expression

where:
x - distance of point x from axis of excavation
s(x) - settlement at point x
Z - depth of center point of excavation
Linf - distance of inflection point

Analysis of failure of buildings

The program first determines the shape and dimensions of subsidence trough and then performs analysis of their influence on buildings.

The program offers four types of analysis:

  • determination of tensile cracks
  • determination of gradient damage
  • determination of a relative deflection of buildings (hogging, sagging)
  • analysis of the inputted section of a building

Tensile cracks

One of the causes responsible for the damage of buildings is the horizontal tensile strain. The program highlights individual parts of a building with a color pattern that corresponds to a given class of damage. The maximum value of tensile strain is provided in the text output.

The program offers predefined zones of damage for masonry buildings. These values can be modified in the frame "Settings". Considerable experience with a number of tunnels excavated below build-up areas allowed for elaborating the relationship between the shape of subsidence trough and damage of buildings to such precision that based on this it is now possible to estimate an extent of compensations for possible damage caused by excavation with accuracy acceptable for both preparation of contractual documents and for contractors preparing proposals for excavation of tunnels.

Recommended values for masonry buildings from one to six floors are given in the following table.

Horizontal strains (per mille)
Proportional h.s. (per mille) Damage Description
0.2 – 0.5 Microcracks Microcracks
0.5 – 0.75 Little damage - superficial Cracks in plaster
0.75 – 1.0 Little damage Small cracks in walls
1.0 – 1.8 Medium damage, functional Cracks in walls, problems with windows and doors
1.8 – Large damage Wide open cracks in bearing walls and beams

Gradient damage

One of the causes leading to the damage of buildings is the slope subsidence trough. The program highlights individual parts of a building with a color pattern that corresponds to a given class of damage. The maximum value of tensile strain is provided in the text output.

The program offers predefined zones of damage for masonry buildings. These values can be modified in the frame "Settings". Considerable experience with a number of tunnels excavated below build-up areas allowed for elaborating the relationship between the shape of subsidence trough and damage of buildings to such precision that based on this it is now possible to estimate an extent of compensations for possible damage caused by excavation with accuracy acceptable for both preparation of contractual documents and for contractors preparing proposals for excavation of tunnels.

Recommended values for masonry buildings from one to six floors are given in the following table.

Gradient
Gradient Damage Description
1:1200 - 800 Microcracks Microcracks
1:800 - 500 Little damage - superficial Cracks in plaster
1:500 - 300 Little damage Small cracks in walls
1:300 - 150 Medium damage, functional Cracks in walls, problems with windows and doors
1:150 - 0 Large damage Wide open cracks in bearing walls and beams

Relative deflection

Definition of the term relative deflection is evident from the figure. The program searches regions on buildings with the maximum relative deflection both upwards and downwards. Clearly, from the damage of building point of view the most critical is the relative deflection upwards leading to "tensile opening" of building.

Relative deflection
Relative deflection

Verification of the maximum relative deflection is left to the user – the following tables list the ultimate values recommended by literature.

Type of
structure
Type of damage Ultimate relative deflection Δ/l
Burland and Wroth Meyerhof Polshin a Tokar ČSN 73 1001
Unreinforced
bearing walls
Cracks in walls For L/H = 1 - 0.0004
For L/H = 5 - 0.0008
0.0004 0.0004 0.0015
Cracks in bearing structures For L/H = 1 - 0.0002
For L/H = 5 - 0.0004

Failure of a section of building

In a given section the program determines the following variables:

  • maximum tensile strain
  • maximum gradient
  • maximum relative deflection
  • relative gradient between inputted points of a building

Evaluation of the analyzed section is left to the user – the following tables list the recommended ultimate values of relative rotation and deflection.

Type of structure Type of damage Ultimate relative gradient
Skempton Meyerhof Polshin a Tokar Bjerrum ČSN 73 1001
Frame structures and reinforced bearing walls Structural 1/150 1/250 1/200 1/150  
Cracks in walls 1/300 1/500 1/500 1/500 1/500
Type of
structure
Type of damage Ultimate relative deflection Δ/l
Burland and Wroth Meyerhof Polshin a Tokar ČSN 73 1001
Unreinforced
bearing walls
Cracks in walls For L/H = 1 - 0.0004
For L/H = 5 - 0.0008
0.0004 0.0004 0.0015
Cracks in bearing structures For L/H = 1 - 0.0002
For L/H = 5 - 0.0004


Ground loss problem

Cracks in a masonry building due to ground movements

About author

The author, Radko Bucek MSc., Ph.D., is a distinguished expert in the field of underground structures. He gained his abundant experience while working with world-known consultant companies such as Golder Associates, SG-Geotechnika and D2-Consult. Recently he works as the Chief Tunnel Engineer in Mott MacDonald Prague.

In cooperation with company Fine, developer of civil engineering software, the original program has been considerably improved and implemented into professional geotechnical software package GEO.

References

Logo Consult

The software was used by Austrian company D2-Consult during preparation of underground line „Budapest metro line 4” in Budapest to predict ground settlement and its effects on surrounding buildings.
 

Logo Ko-Ka

Since 2005 the program has been continuously used by company KO-KA for designing collectors in Prague.
 

Mott MacDonald

Recently the software has been used by company Mott MacDonald, Prague. The author of the program works there as the chief engineer in branch of underground structures.