Geomechanics & Ground Control

Stress Around Openings

In-situ cover stress and the Kirsch tangential stresses at the boundary of a circular opening — where roof and sidewall stresses concentrate.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Before mining, the rock carries a vertical (cover) stress

σ_{v} = γ H

from the weight of the overburden, plus a horizontal stress

σ_{h}

. Excavating an opening forces this stress to flow *around* the void, concentrating the tangential stress at the boundary. For a circular opening the Kirsch solution gives the boundary tangential stress: highest at the sidewall and lowest (sometimes *tensile*) at the roof and floor. Tensile boundary stress is the warning sign for roof slabbing.

The Core Formula Matrix

Vertical cover stress:

σ_{v} = γ H

(

γ

= unit weight,

H

= depth).

Hydrostatic field (

σ_{h} = σ_{v} = p

): boundary tangential stress is uniform,

σ_{θ} = 2 p

.

Biaxial field, circular opening (Kirsch, boundary):

σ_{θ, sidewall} = 3 σ_{v} - σ_{h}, σ_{θ, roof} = 3 σ_{h} - σ_{v}

The ‘IIT Traps’

⚠
Sidewall vs roof formulae are mirror images. Sidewall $= 3 σ_{v} - σ_{h}$ ; roof $= 3 σ_{h} - σ_{v}$ . Swapping them sends the concentration to the wrong wall.
⚠
Roof stress can go negative (tensile). When $σ_{v} > 3 σ_{h}$ the roof tangential stress is tensile — don't clip it to zero.
⚠
Unit weight in kN/m³, depth in m ⇒ stress in kPa. $25 \times 200 = 5000 kPa = 5 MPa$ ; keep the factor of 1000 straight.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

A circular opening sits in a hydrostatic stress field of

10 MPa

. The tangential stress at its boundary is:

Q2MEDIUM2 Marks · NAT

A circular roadway lies in a field with vertical stress

σ_{v} = 10 MPa

and horizontal stress

σ_{h} = 2 MPa

. The tangential stress at the sidewall is ______ MPa. (Round off to two decimal places.)

MPa

Q3HARD2 Marks · NAT

For the same field (

σ_{v} = 10 MPa

σ_{h} = 2 MPa

), the tangential stress at the roof of the circular opening is ______ MPa (a negative value indicates tension). (Round off to two decimal places.)

MPa