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n Introductionn Components of stressn Tow dimensional state of stressn In situ state of stressn Stress distributions around
single excavationsn Stress around a circular
excavation
n Calculation of stresses around other excavation shapes
n Stresses around multiple excavations
n Three-dimensional stress problems
n Stress shadowsn Influence of gravity
Introduction
Components of stressn Surface forcesn Surface tractionsn Stress at a pointn Transformation equationsn Principal planesn Planes stressn Plane strain conditions
Surface forces
Shear stress and Normal stress
τ⊥∥
Three dimension
Symmetrical condition
Stress transformation
Tow dimensional state of stress
Two dimension
Principal stresses
Principal plan
One dimensional compression
ν = - εx / εz = - εy / εz
Plan stress (σy = 0)
Shear strain
Plan strain (εy = 0)
In situ state of stressn Terzaghi and Richart’sapproach
n Heim’s rulen Results of in situ stress measurements
Initial stress ( Ko condition)
Initial stress
Vertical stress ( Overburden pressure )
Horizontal stress ( Lateral pressure )
Stress distributions around single excavations
n The streamline analogy for principal stress trajectories
Stress around a circular excavationn Stress at the excavation boundaryn Stresses remote from the
excavation boundaryn Axes of symmetryn Stresses independent of elastic
constantsn Stresses independent of size of
excavation n Principal stress contours
Analytical methods of mine design
n Parameter studies
n Physical model
n Photoelastic method
Principles of classical stress analysis
n Timoshenko and Goodier (1970)
n Prager (1957)
n Poulos and Davis (1974)
Closed-form solutions for simple excavation shapes
n Circular excavation
Superposition of stress filed
Initial stress Imposed state of traction
The differential equations of equilibrium in two dimensions for zero body force are
Airy (1862)
For plane strainconditions 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 thick-walled 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
One may take
Yields
For the axisymmetric problem :
For a circular hole with a traction-free surface, in a medium subject to a uniaxial stress pxx 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
n Elliptical excavation
Circular excavation in a biaxial stress filedPyy = p and pxx = Kp
Kirsch (1898)
For interior r = a
For θ = 0 and r →∞, far-field stresses
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 Average pillar stressesn Influence of pillar shape
Three-dimensional pillar stress problems
Stress shadows
Influence of gravity
