Stresses Around Underground Excavations

Stresses around underground excavations

李 煜 舲

Tel: 6713rosalee@chu.edu.tw

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