Analysis of the behavior of structures under the effect of progressive rupture of a cavity

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Geo-mechanical characteristics of scale model soils_

Characteristics	Unit	Pulverulent soil	Coherent soil
Young's modulus (E)	MPa	50–100	50–100
Friction angle (φ)	°	26	28–30
Cohesion (c)	KPa	≈ 0	≈200
Poisson's ratio (ν)	/	0.3	0.3
Density (ρ)	kg/m3	2200	2200

Properties of structural elements_

Parameters	Name	Unit	Value
Type of behavior	Material type	-	Elastoplastic
Normal stiffness	EA	kN/m	132000
Flexural rigidity	EI	KNm2/m	4389
Equivalent thickness	d	m	0.632
Weight	w	KN/m/m	10
Poisson's ratio	ν	-	0.35

Summary of conducted research studie_

Author	Year	Objective	Type
Nakai et al.	1997	Investigate the effect of 3D and expansion on ground movements during tunnel excavation	Experimental
Dyne	1998	Analyze the different parameters: the opening of the cavity, the width of the cavity, and the height of the covering	Experimental 2D scale model
Burd et al.	2000	Study soil-structure interaction during tunneling under masonry structures and analysis	Numerical MEF-OXFEM
Laefer	2001	Study the damage to structures on shallow foundations subject to soil movements induced by excavation	Experimental (a small-scale model of 1/10th).
Mahamma	2002	Study the soil-structure interaction phenomena during the collapse of a mine gallery. The collapse of the mine gallery was modeled by successive sinking of a cylinder along the axis of propagation of the rupture	Experimental
Shanin et al.	2004	The study of the effect of ground movements and their mechanical behavior during tunnel excavation.	Experimental trap model
Boumalla	2005	Vary a number of parameters such as the opening of the cavity, the height of the cover, the rate of initiation of a melt, or the subsidence of the ground	Experimental
Sung et al.	2006	Analyze the settlements and ground pressure at the surface due to the tunnel in the cases without and with the foundation structure in the vicinity.	Experimental
Castro et al.	2007	Study the “block caving” mining method, not the movements that occur on the surface of the land	Experimental large-scale 3D model
Trueman et al.	2008
Lee & Bassett	2007	Simulate the deformation of the tunnel by changing its diameter, to investigate the behavior of existing foundations located near the tunnel	Experimental
Kikumoto et al.	2009
Caudron	2007	Characterize the influence of soil-structure interaction during the formation of a sinkhole	Experimental and numerical
Deck and Anirudth	2010	To investigate the phenomenon of soil-structure interaction due to mine subsidence, taking into account the influence of length, rigidity of the structure, mechanical properties of the soil, and intensity of subsidence.	Numerical 2D model CESAR LCPC
Boramy Hor	2012	Simulate ground movements and their consequences on the surface.	Experimental/numerical 3D physical model
Al Heib et al.	2013	Understanding sinkhole consequences on masonry structures using a large small-scale physical modeling. The paper presents the main results of the small-scale physical model designed to study the consequences of subsidence on structures. Present the transfer of movements from the soil to the structure. The objective is to understand and then to predict the real behavior and the damage of structures on subsidence areas.	Experimental
Nghiem et al.	2014	Physical model for damage prediction in structures due to underground excavations: a small-scale physical model (1/40 scale factor on the dimensions) under normal gravity. It has been designed for developing and validating experimentally new methods of prediction of damages to masonry structures induced by subsidence (generally resulting from underground excavations of tunnels and mines)	Experimental
Keawsawasvong	2021	Limit analysis solutions for spherical cavities in sandy soils under overloading. An investigation on the stability of spherical cavities in sandy soils under overloading at the ground surface is carried out in this study. By using finite element limit analysis, a spherical cavity is numerically simulated under an axisymmetric condition, and the lower and upper bound solutions of the stability of spherical cavities can be obtained	Numerical
Yongyao et al.	2023	A numerical simulation study on the evolutionary characteristics of the damage process of karst soil cavity under positive pressure effect	Numerical
Keba and Isobe	2024	Bearing capacity of a shallow foundation above the soil with a cavity based on a rigid plastic finite element method. Based on the rigid plastic finite element method (RPFEM), this study investigates the performance of the footing on the soil with a cavity. The RPFEM is used in plane strain conditions and necessitates only a few materials to predict the bearing capacity: the unit weight of the soil, the cohesion, the shear resistance angle, and the dilation angle	Numerical

List of scale factors_

Symbol	Scale factor concerned	Dimension	Value
L*	Length of reference	L	1/40
x*	Coordinates	L	1/40
E*	Modulus of elasticity	ML−1 t−2	3/40
ρ*	Density	ML−3	3
g*	Acceleration of gravity	Lt−2	1
F*	External punctual force	MLt−2	3/64000
p*	Superficial force	ML−1 t−2	3/40
U*	Displacement	L	1/40
σ*	Constraint	ML−1 t−2	3/40
γ*	Inertia acceleration	Lt−2	1

Geo-mechanical characteristics of different materials (Caudron, 2007)_

Layer	Materials	E (MPa)	υ	Rtraction (MPa)	Cohesion (MPa)	φ (°)
8	Marls	70	0.25–0.30	0.30	0.80	28
2 and 5	Stones	100	0.25–0.30	0.30	0.80	29
6	Clay sand	130	0.25–0.30	0.20	1.2	30
3 and 9	Limestone	20	0.25–0.30	0.80	2.00	31
7	Stones	200	0.25–0.30	01	1.00	35
4	Marls	50	0.25–0.30	0.1	0.20	26
1	Stones	50	0.25–0.30	0.20	0.40	27

List of similarity laws_

Number	Similarity law	Meaning of scale factors
1	x*/L*=1	Equality of coordinates relative to length scale
2	U*/L*=1	Equality of displacements relative to length scale
3	U0*/L*=1	Equality of displacements at origin relative to length scale
4	g*/γ*=1	Equality of acceleration scale to gravity scale
5	E*L*2/F*=1	Conservation of the ratio of elasticity modulus scale by length squared to force scale
6	Y*t*2/L*=1	Identity of acceleration and length scales as time cannot be altered
7	P*L*2/F*=1	Conservation of the ratio of pressure scale times length squared to force scale
8	(σ0*L*2)/F*=1	Conservation of the ratio of stress scales times length squared to force scale
9	(ρ*γ*L3*)/F*=1	Conservation of the ratio between scales of quantities determining inertia force relative to force scale

Structure characteristics in real size and scale model_

Characteristics	Values
	Real model	Scale model
Module (MPa)	33000	2475
Section (m2)	0.04	25×10−6
Inertia (m4)	1.33×10−4	52×10−12
Loading (kPa)	10	0.75

Empirical formulas for determining i (Dolzhenko, 2002)_

Authors	Proposed expression	Soil type	Calculated i value
Atkinson & Potts. (1977)	i = 0.25(1.5C + D)	Dense sands with surcharge	3.65 m
Oteo & Sagaseta. (1982)	i = 0.525H + 0.42R	Granular soils	5.67 m
Dyer et al. (1986)	i = 0.29H	Loose to medium dense sand	2.60 m
Al Abram (1998)	i = 0.15H + 0.5D	Analogical soil	3.60 m

Soil properties_

Parameters	Name	Unit	Pulverulent soil	Coherent soil	Air
Material model	Model	-	Mohr-Coulomb	Mohr-Coulomb	Mohr-Coulomb
Material type	Type	-	Drained	Drained	Drained
Soil unit weight above phreatic level	γunsat	kN/m3	17	20	5
Soil unit weight below phreatic level	γsat	kN/m3	19	22	5
Permeability in horizontal direction	kx	m/day	1	0	1
Permeability in vertical direction	ky	m/day	1	0	1
Young's modulus	E	kN/m2	100000	100000	5
Poisson's ratio	ν	-	0.3	0.3	0.1
Cohesion	c	kN/m2	2	200	1
Friction angle	φ	°	26	26	5
Dilatancy angle	ψ	°	7	9	1
Strength reduction factor interne	Rinter	-	1	1	1