Stresses around underground excavations
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Introduction Components of stress Tow dimensional state of stress In situ state of stress Stress distributions around single excavations Stress around a circular excavation
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Calculation of stresses around other excavation shapes Stresses around multiple excavations Three-dimensional stress problems Stress shadows Influence of gravity
Introduction
Components of stress n
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Surface forces Surface tractions Stress at a point Transformation equations
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Principal planes Planes stress Plane strain conditions
Surface forces
Shear stress and Normal stress
Three dimension
τ
Symmetrical condition
Stress transformation
ow mens ona state o stress
Two dimension
Principal stresses
Principal plan
One dimensional compression
ν = - εx / ε z = - εy / εz
Plan stress (σy = 0)
Shear strain
Plan strain (εy = 0)
In situ state of stress n
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Terzaghi and Richart s approach Heim s rule Results of in situ stress measurements ’
Initial stress ( K o condition)
Initial stress
Vertical stress ( Overburden pressure
Horizontal stress ( Lateral pressure )
Stress distributions around single excavations n
The streamline analogy for principal stress trajectories
tress aroun a c rcu ar excavation n n
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Stress at the excavation boundary Stresses remote from the excavation boundary Axes of symmetry Stresses independent of elastic constants Stresses independent of size of excavation Principal stress contours
Analytical methods of mine design n
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Parameter studies Physical model
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Photoelastic method
Principles of classical stress analysis n
Timoshenko and Goodier (1970
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Prager (1957)
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Poulos and Davis (1974)
Closed-form solutions for simple excavation shapes n
Circular excavation
Superposition of stress filed Initial stress
Imposed state of tractio
The differential equations of equilibrium in two dimensions for zero body force are
Airy (1862)
For plane strain conditions and isotropic elasticity
The strain compatibility equation in two dimensions :
The stress distribution for isotropic elasticity is independent of the elastic properties of the medium
Airy stress function U(x,y):
Satisfy the equilibrium equations:
Biharmonic equation
A thickwalled cylinder of elastic material, subjected to interior pressure, pi and exterior pressure, po
Axisymmetric problem
General solution :
Uniqueness of displacement requires B= 0
Where
a and b are the inner and outer radii of the cylinder
Airy stress function may be expressed as the real part of two analytic function φ and χ of a complex variable z
The transformation between the rectangular Cartesian and Cylindrical Polar co-ordinates
Yields
One may take
For the axisymmetric problem :
For a circular hole with a traction-free surface, in a medium subject to a uniaxial stress p xx at infinity, the source function are
Boundary condition : σrr = σrθ = 0 at r=a, and σrr
pxx for θ = 0 and r
Stress components :
Orthogonal elliptical
Closed-form solutions for simple excavation shapes n
Circular excavation
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Elliptical excavation
Circular excavation in a biaxial stress filed Pyy = p and pxx = Kp
Kirsch (1898)
For interior r = a
For θ = 0 and r
, far-field stresse
Boundary stresses : Side wall
Crown
For Ko =0 For Ko = 1
Hydrostatic stress filed (Ko = 1)
Axes of symmetry
Elliptical Excavation Poulos and Davis (1974) Jaeger and Cook (1979) Bray (1977)
Bray (1977) solution :
The stresses components :
The boundary stress around an elliptical opening :
When the axes of the ellipse are oriented parallel to the filed stress directions :
Lamb (1956) : since q = W / H = a / b
Calculation of stresses around other excavation shapes n
Influence of excavation shape and orientation
Stresses around multiple excavations n
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Average pillar stresses Influence of pillar shape
Three-dimensional pillar stress problems
Stress shadows
Influence of gravity
