ABSTRACT: This paper presents a theoretical analysis of the effect of shape and size on strength of geologic materials. The results indicate that the shape effect is caused by the development of confining pressure at the interfaces. The linearization of the obtained formula is of the same form determined by experiments. The size effect is attributed to the fact that the geologic materials contain discontinuities in the form of fractures, bedding planes, and cleats. The maximum reduction factor for the size effect is the one-sixth power of the volume of a cubic test specimen. The results of this work are particularly useful in determining the strength of coal pillars in underground mines.


This paper presents a theoretical formulation for determining the breaking strength of a geological material such as coal. The formulas are derived with the use of the equilibrium condition and the Mohr-Columb yield criterion. A linearization of the obtained formula is identical to the shape correction formula developed on the basis of experiments.

The compressive strength test on rock and coal is not conducted in a simple state of compression. The friction between the coal and test platens or roof and floor formations induces a stress perpendicular to the compressive load. The induced stress acts as a confining stress, resulting in an increase in strength, which is dependent on the specimen geometry and the friction. The effect of the induced stress on the material breaking strength is generally known as the shape factor. For a geological material, the strength increase due to the presence of the induced confining stress is utilized in the design of mine pillars and in interpreting the test result because it cannot be neglected as stated by various researchers (Brady, et al., 1975; Shorey, 1988).

This paper also discusses the cause of the reduction in the strength as the size of test specimen on mine pillar increases. The phenomenon, known as the size effect, is attributed to flaws contained in geologic material.


A mine pillar can support an overburden pressure exceeding the so-called "compressive" strength of the pillar. For example, some coal mines operate at a depth of 1000 meters even though the "compressive" strength of the coal tested in a laboratory is of the order of 10 to 20 MPa. It is believed that this increase in the supporting capability is due to the friction between a mine pillar and roof and floor rock.

The stresses acting on an element of a rectangular pillar or a test specimen are shown in Figure 1. To derive an equilibrium equation, the horizontal stress, q, is distributed uniformly across the height. The assumption is valid for relatively small deformations and a chubby test specimen or mine pillar. The equilibrium condition for the horizontal component can be written as follows:

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