Big Dog Mining

Underground Pillar Design

What the pillars are carrying, what they can carry, and the ratio between the two, plus what you give up in recovery to get it.
Underground room-and-pillar layouts only: a regular array of pillars on one horizon. It is not a barrier pillar or panel design, it does not model abutment loading from an adjacent gob, it is not a numerical stress model, and it says nothing about the back between the pillars. Bolts hold the back up; this page is about what holds the mine up. A competent person signs the layout.

Step 1: The layout

Pillar width against room span decides both the load and the recovery.

Units ?

Some formulas are valid in one system only. The page checks.

Pillar shape ?

Square grid, or long parallel ribs.

Pillar width ?

Short horizontal dimension.

m Pillar length ?

Equal to width if square.

Room span ?

The opening between pillars.

m Mining height ?

Floor to back. Also the pillar height.

Step 2: Depth and stress

In-situ vertical stress is just the weight of what is above you.

Depth of cover ?

Surface to the mining horizon.

m Overburden rock ?

Fills in the density below.

Overburden density yours ?

Average over the whole cover.

t/m³

Step 3: Pillar strength

Eight published formulas, and they do not agree. Pick deliberately.

Strength formula ?

Each was fitted to one district and material.

Cube strength σ_S ?

A cubical pillar, not a core sample. Far weaker than lab strength.

MPa Required factor of safety ?

The formulas give a MEAN strength. 1.0 means half fail.

Step 4: What the pillars look like

A number is a prediction. This is an observation, and it outranks the number.

Observed pillar condition ?

What you can see underground. Compare it to the number.

The formulas were fitted to coal. The handbook is explicit that the statistical verification behind these constants concerns coal, and that there has been no similar verification for hard rock or even for softer minerals like trona, salt, oil shale or borates. Applying them outside coal is what the industry does, and it is still an extrapolation. The observed condition above is the check that costs nothing.

pillar stress

pillar strength

factor of safety

Step	Value	Where it comes from

The layout, and what each pillar carries

Factor of safety against pillar width

The eight published strength formulas

Source	a	b	α	β	Fitted to
Bunting (1911)	0.7	0.3	n/a	n/a	Pennsylvania anthracite
Obert and Duvall (1967)	0.78	0.22	n/a	n/a	Laboratory rock and coal
Bieniawski (1968)	0.64	0.36	n/a	n/a	South Africa coal
Skelly et al. (1977)	0.78	0.22	n/a	n/a	West Virginia coal
Mark-Bieniawski (Mark 1999)	n/a	n/a	n/a	n/a	Coal, and the only form that uses pillar length
Greenwald et al. (1939)	n/a	n/a	0.5	0.83	Pittsburgh seam mines English only
Holland (1964)	n/a	n/a	0.5	1	U.S. coal mines English only
Salamon and Munro (1967)	n/a	n/a	0.46	0.66	South Africa coal mines metric only
Hardy and Agapito (1977)	n/a	n/a	0.6	0.95	U.S. oil shale mines English only

SME Mining Engineering Handbook, Table 13.1-1, p. 1329, after Zipf 2001. Two general forms are printed with it: σ_P = σ_S(a + b·W/H) (Eq. 13.1-12) and σ_P = K·W^α/H^β (Eq. 13.1-13). Obert-Duvall and Bieniawski follow the first; Salamon-Munro and Holland follow the second.

The daggers are not decoration. Three of the power-law formulas are printed “use with English units only” and one “use with metric units only”. A power law is not dimensionless, so running one in the wrong system does not scale. It gives a different answer with no sign that anything happened. This page refuses rather than converts.

A conflict worth knowing about. Rock Mechanics for Underground Mining p. 379 gives Salamon and Munro as α = −0.66, β = 0.46, the same two numbers as the table above, with the letters swapped and a sign changed, because it writes the formula in a different form. Take α from one book and β from the other and the exponents land on the wrong variables. This page uses the SME convention throughout.

Rating a pillar by looking at it

Rating	Condition
1	No indication of stress-induced fracturing. Intact pillars.
2	Spalling on pillar corners; minor spalling of pillar walls. Fractures subparallel to the walls and short relative to pillar height.
3	Increased corner spalling. Fractures on pillar walls more numerous and continuous, lengths increasing but less than pillar height.
4	Continuous, subparallel, open fractures along pillar walls. Early diagonal fractures, the start of hourglassing. Fracture lengths over half the pillar height.
5	Continuous, subparallel, open fractures along pillar walls. Well-developed diagonal fractures, classic hourglassing. Fracture lengths over half the pillar height.

SME Mining Engineering Handbook, p. 1331, after Carmack et al. 2001. Hourglassing is the shape a pillar takes when the walls have spalled away and the load has migrated to a smaller core, which is a late stage and not a warning.

Published pillar widths by depth and mining height

Depth of cover, m
30.5	21
38.1	25
45.7	29	32
53.3	33	36	39	41	43
60.9	36	40	43	46	48	50	53
68.6	40	44	48	51	54	56	58
76.2	43	49	53	56	60	62	65
83.8	47	53	57	61	65	68	71
91.4	50	56	62	66	71	74	78
99.1	53	60	66	71	76	80	84
106.7	57	64	71	76	82	86	90
114.3	60	68	75	81	87	92	97
121.9	63	72	79	86	92	98	103

SME Mining Engineering Handbook, p. 1028, adapted from Vandergrift et al. 2004. Read down for depth of cover and across for mining height; the cell is a pillar width in feet. Mining heights run 0.9 to 2.7 m, so this is a COAL table and it is here as a sanity check on the shape of an answer rather than as a design for a hard-rock heading. The blanks are blank on the printed page: not every depth and height combination is covered, and this page does not interpolate into them.

What this tool does not know. The tributary area method assumes vertical loading only, that each pillar carries the full column above its share of the area, and that the load is spread evenly across the pillar. The handbook notes that none of the last one is true: stress concentrates at the pillar corners. It knows nothing about horizontal stress, about an abutment load from an adjacent mined-out panel, about a pillar sitting on a weak floor, or about time: pillars that stand for a decade can fail in the eleventh year. It does not design barrier pillars, and it does not check the back. Every published value has been checked against the printed page.

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Worked example

Worked with the values this page loads with, so the method can be checked against the cited source without entering anything. Change any input above and the calculator recomputes; the numbers below stay as a fixed reference case.

23.26 MPa
37.92 MPa
1.63

Pillar	11.0 m × 11.0 m	your input
Room span	8.0 m	your input
Pillar pitch, center to center	19.0 m	pillar + room
Extraction ratio	66.5%	1 − pillar area ÷ pitch area
Overburden density	2.65 t/m³	your input
Depth of cover	300.0 m	your input
= in-situ vertical stress Sv	7.80 MPa	ρ g h, derived here rather than taken as a gradient
Tributary load factor	× 2.98	(pitch ÷ width)², square grid
= pillar stress	23.26 MPa	tributary area, SME Ref Hbk 2nd ed. p. 331
Strength formula	Obert and Duvall (1967)	Laboratory rock and coal, Table 13.1-1 p. 1329
Cube strength σS	30.00 MPa	your input; NOT intact laboratory strength
Pillar width to height W/H	2.20	the shape term every formula turns on
= pillar strength σP	37.92 MPa	σS(0.78 + 0.22·W/H), Eq. 13.1-12
= factor of safety	1.63	strength ÷ stress
Target	1.50	your input

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Estimating aid only. This is not an engineering design and not a substitute for the judgment of a competent or qualified person. Verify every result against the cited source and your own site conditions. Full terms and disclaimer.