Geotechnical and Geological Engineering

, Volume 32, Issue 2, pp 577–586 | Cite as

Analytical Expression for Vertical Stress within an Inclined Mine Stope with Non-parallel Walls

  • Ching Hung Ting
  • Nagaratnam Sivakugan
  • Wayne Read
  • Sanjay Kumar ShuklaEmail author
Original Paper


Arching is a phenomenon that occurs in many situations in geotechnical engineering. When underground mine stopes are backfilled, a significant fraction of the self-weight of the backfill is carried by the side walls. As a result, the vertical stress at the bottom of the stope is significantly less than its overburden pressure. Few analytical expressions published in the literature can be used to determine the vertical stresses of stope with parallel walls. The objective of this paper is to extend the analytical solution previously developed by the authors to long plane-strain stopes with non-parallel walls with both slopes leaning to the same side. Different combinations of wall inclination are examined using the new analytical expression developed. To validate the analysis, the proposed results are compared with numerical model results. The results show that the proposed analytical expression is capable of estimating the vertical stress within mine stopes when the inclination of the hangingwall to the horizontal (α) is less than that of footwall (β). An important behavioural trend for the stress distribution is observed, where with the same overburden pressure and base width, the stress magnitude experienced by fill material significantly varies depending on the wall inclination.


Arching Backfills Mine stopes Non-parallel walls 

List of symbols


Wall inclination at hangingwall to the horizontal


Wall inclination to the horizontal (for parallel wall) or

Wall inclination at footwall (for non-parallel wall)


Interfacial friction angle between the fill and the wall


Friction angle of backfill


Unit weight of the fill


Dilation angle


Poisson’s ratio


Density of material


Modified cohesion of the fill for an inclined stope

ζ1, ζ2

Modified fill cohesion for hangingwall and footwall, respectively


Shear stress along the wall

τ1, τ2

Shear stress acting at hangingwall and footwall, respectively


Normal stress at the wall at depth z

σn1, σn2

Normal stress on hangingwall and footwall, respectively


Lateral stress at depth z


Vertical stress at depth z


Cohesion of the fill


Surcharge on top of fill material

x1, x2, x3, x4

Horizontal distance between toe of the stope to edge of differential element at hangingwall and footwall


Fill depth measured from top of the backfill


The depth of onset of the inward curve of (z vs σ z ) for case α > β


Area of the differential element


Stope width at the base of backfill


Stope width at the top of backfill


Young’s modulus


Total height of backfill or stope


Lateral stress ratio


At rest lateral stress ratio or at rest earth pressure coefficient


Active lateral stress ratio or active earth pressure coefficient


Modified lateral stress ratio for inclined stope

K1, K2

Modified lateral stress ratio for hangingwall and footwall, respectively

L1, L2

Stope width of differential element of dz

S1, S2

Shear force acting along hangingwall and footwall, respectively

N1, N2

Normal force acting on hangingwall and footwall, respectively


Internal vertical force


Self-weight of the backfill


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ching Hung Ting
    • 1
  • Nagaratnam Sivakugan
    • 2
  • Wayne Read
    • 3
  • Sanjay Kumar Shukla
    • 4
    Email author
  1. 1.Advanced GeomechanicsPerthAustralia
  2. 2.Discipline of Civil and Environmental Engineering, School of Engineering and Physical SciencesJames Cook UniversityTownsvilleAustralia
  3. 3.School of Engineering and Physical SciencesJames Cook UniversityTownsvilleAustralia
  4. 4.Discipline of Civil Engineering, School of EngineeringEdith Cowan UniversityPerthAustralia

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