Geomechanics & Ground Control

Pillar Design

Tributary-area pillar stress, the Obert–Duvall strength formula and the all-important factor of safety.

PART 1

Topic Breakdown & Traps

The Engineering Principle

In bord-and-pillar mining the pillars carry the weight of the overburden that the excavated bords no longer support. The tributary area theory loads each pillar with the cover stress acting over the full pillar-plus-opening area, so the pillar stress rises above the cover stress by the ratio of those areas. The pillar's strength depends on its width-to-height ratio — squatter pillars are stronger — captured by the Obert–Duvall formula. The factor of safety is simply strength divided by stress.

The Core Formula Matrix

Cover stress:

σ_{v} = γ H

.

Tributary-area pillar stress (square pillars, width

W

, opening

B

σ_{p} = σ_{v} \frac{( W + B ) ^{2}}{W ^{2}}

Obert–Duvall pillar strength:

S_{p} = S_{1} (0.778 + 0.222 \frac{W}{H})

S_{1}

= strength of a cubical specimen,

H

= pillar height.

Factor of safety:

F S = \frac{S _{p}}{σ _{p}}

The ‘IIT Traps’

⚠
Square the area ratio. Tributary stress scales as $(W + B)^{2} / W^{2}$ , not $(W + B) / W$ — it's an *area* ratio.
⚠
Width-to-height, not height-to-width. Obert–Duvall uses $W / H$ ; squat pillars ( $W / H > 1$ ) are stronger.
⚠
FS = strength/stress. Inverting it makes a safe pillar look unsafe.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

At a depth of

200 m

the overburden unit weight is

25 kN/m^{3}

. The vertical cover stress is:

Q2MEDIUM2 Marks · NAT

Square pillars of width

W = 10 m

are separated by bords of width

B = 5 m

at a depth giving a cover stress of

5 MPa

. The pillar stress by tributary-area theory is ______ MPa. (Round off to two decimal places.)

MPa

Q3HARD2 Marks · NAT

A pillar of width

W = 10 m

and height

H = 8 m

has a cubical-specimen strength

S_{1} = 30 MPa

and carries a pillar stress of

11.25 MPa

. Using the Obert–Duvall formula, the factor of safety is ______. (Round off to two decimal places.)