Theory

Content

• Analyses in program Ground Loss
• Analysis of subsidence trough
• Volume loss
• Recommended values of parameters for volume loss analysis
• Classical theory
• Analysis for layered subsoil
• Shape of subsidence trough
• Coefficient of calculation of inflection point
• Subsidence trough with several excavations
• Analysis of subsidence trough at a depth
• Calculation of other variables
• Analysis of failure of buildings
• Tensile cracks
• Gradient damage
• Relative deflection
• Failure of a section of building

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

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

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 u_a follows from:

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

Recommended values of parameters for volume loss analysis

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:

where:
σ_v - verall stress along excavation axis
σ_t - excavation lining resistance (if lining is installed)
S_n - 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.

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:

where:
σ_z - geostatic stress in center of excavation
K_r - coefficient of pressure at rest of cohesive soil

The roof u_a and the bottom u_b deformations of excavation follow from:

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:

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:

where:
Z - depth of center point of excavation
r - excavation radius
u_a - 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 S_int 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

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:

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

where:
L_int - length of subsidence trough along interfaces above excavation
S_int - settlement of respective interface

Solution after Fazekas

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

where:
L - length of subsidence trough
L_int - length of subsidence trough along interfaces above excavation
S_int - settlement of respective interface

Solution after Peck

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

where:
L_int - length of subsidence trough along interfaces above excavation
S_int - settlement of respective interface
L_inf - 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:

where:
S_i - settlement at point with coordinate x_i
S_max - maximum terrain settlement
L_inf - 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:

where:
S_i - settlement at point with coordinate x_i
S_max - maximum terrain settlement
L - reach of subsidence trough

Coefficient of calculation of inflection point

When the classical methods are used the inputted coefficient k_inf allows the determination of the inflection point location based on L_inf=L/k_inf. In this case the coefficient k_inf 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 k_inf 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 S_max at a terrain surface and the displacement of roof excavation u_a 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

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

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:

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:

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

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	–	–	–