Basic Rock Mechanics

 

This module is intended to provide a basic understanding of the engineering principles used in the analysis of rock materials when these materials are intended to be used for an engineering purpose.  These principles form the foundation of the science of rock mechanics and have been widely adopted and used.  At the introductory level provided here, the applicability of these principles should be easily grasped by each  student. 

These notes are, in part, from my course Geology for Engineers, GeoE 221, a first-level course in geology.  The course is presented as a systematic study of Earth and how natural geologic processes are applied in engineering practice.  This course is a freshman-level Geological Engineering course but is also taken by students of all grades from other disciplines such as Civil and Environmental Engineering and Mechanical Engineering.


Part I

I. Rock Basics

A. Engineering Uses:  Rock is used for engineering purposes in 2 primary ways:  

1. as a building material:  items such as cut stones, beams, support columns, decorative panels, etc.  Each student can envision examples where rock has been used in one of these ways.

2. as a foundation:  For example, on Manhattan Island, the skyscapers are founded on granite.  In Central Park, just a short distance to the south, there are no buildings over a few stories.  Why?  The bedrock under the Park consists of marine shale and metamorphic rocks that will not support the weight of a skyscraper.

Thus, knowing and understanding basic rock properties will allow structures to be founded correctly so the required support will be there.

B. Rock Measurements:  the physical characteristics of a rock mass are a fundamental geologic property and are extremely important to engineers.  Analytical data on theses characteristics are generally derived in 2 ways:

1. laboratory measures:  are generally referred to as 'rock properties' and are acquired using small samples taken from the field site and analyzed in a laboratory setting.

2. field-scale measures:  most often referred to as 'rock mass properties' and are descriptions of the bulk strength properties of the rock mass.  The nature of these properties are governed primarily by 'discontinuities', or planes of weakness, that are present in the rock mass.  Examples of discontinuities are fractures, bedding planes, faults, etc.  The measured distance between fractures, bedding planes, and other structural features are also important when collecting field-scale data.

C. Definitions:  From an engineering standpoint, there is a difference between a rock and a stone.  There is also a difference between soil and dirt.  'Rock' is used to denote the mass of material in-situ, it is a part of the bedrock and has not been moved or disturbed.  'Stone' is used to denote rock material that has been removed from its bedrock location.  In the same way, 'soil' refers to that material naturally in place and 'dirt' is most often used to define soil that has been removed and is, or has, been transported.

 

II. Rock Properties

Not all rock is the same and it must be treated differently in an engineering project.  There are 3 fundamental processes which form rock, igneous, metamorphic, and sedimentary processes.  Each of these basic rock types have inherent structural characteristics that define it strength and durability, and hence, its usefulness in an engineering situation.  It is strongly suggested that the student not familiar with these basic rock types review a basic Physical Geology textbook, or better yet, enroll in a fundamental geology course with a laboratory.

It is very important to assess some basic properties of rock before it is used as a building or founding material.  Some of the more important properties of rocks are:

 

A. Specific Gravity:  this term describes the weight of a volume of rock with respect to an equal volume of water, which weighs 1.0 gm/cm3. By weighing equal volumes of water and different rocks, a 'specific gravity' (SG) for that rock can be determined.  These experiments are conducted in a controlled laboratory using very specific guidelines so there are no unexpected variations.  After such work has been performed, typical SG's for common rock types are:

 

Shale: ~2.75
Granite: ~2.65
Sandstone: ~2.2
Basalt: ~2.65
Marble: ~2.7
Limestone: ~2.45
Steel: 7.85
Gold: ~14

The meaning of the SG is that it represents the factor increase in weight of the rock per unit volume over the same unit volume of water.  For example, if the unit weight of water happened to be 750 gm, the the weight of an equal volume of shale would be about 2.75 x 750, or 2060 gm.  Specific gravity is reported as a dimensionless number.

B. Mass Density:  this is derived by multiplying the specific gravity by the density of water, specified as 1000 kg/m3.  So, for the above examples, the mass densities would be:

 

Shale: 2.75 x 1000 kg/m3 = 2750 kg/m3
Granite: 2.65 x 1000 kg/m3 = 2650 kg/m3
Sandstone: 2.2 x 1000 kg/m3 = 2200 kg/m3
Basalt: 2.65 x 1000 kg/m3 = 2650 kg/m3
Marble: 2.7 x 1000 kg/m3 = 2700 kg/m3
Limestone: 2.45 x 1000 kg/m3 = 2450 kg/m3
Steel: 7.85 x 1000 kg/m3 = 7850 kg/m3
Gold: 14 x 1000 kg/m3 = 14,000 kg/m3

C. Unit Weight:  in construction situations in the US, it is often desirable to describe all materials as a 'unit' weight.  It is essentially the same as the mass density except it is calculated in English units, not metric, and so the standard reference unit is 1.0 ft3.  Again, this measure is made in relation to water which weights 62.4 lb/ft3.  So, the unit weight of the above samples are:

 

Shale: 2.75 x 62.4 lb/ft3 = 172 lb/ft3
Granite: 2.65 x 62.4 lb/ft3 = 165 lb/ft3
Sandstone: 2.2 x 62.4 lb/ft3 = 137 lb/ft3
Basalt: 2.65 x 62.4 lb/ft3 = 165 lb/ft3
Marble: 2.7 x 62.4 lb/ft3 = 168 lb/ft3
Limestone: 2.45 x 62.4 lb/ft3 = 153 lb/ft3
Steel: 7.85 x 62.4 lb/ft3 = 490 lb/ft3
Gold: 14 x 62.4 lb/ft3 =874 lb/ft3

D. Rock Strength:  is a measure of the strength of a rock mass when subjected to any one or a combination of three primary forces:

 

1. Compressive Stress:  this stress consists of two opposing forces acting on a rock which decreases the volume of the rock per unit area.

 

'Compressive strength' is the maximum force that can be applied to a rock sample without breaking it.  Units of stress are either reported in pounds per square inch (psi in English units) or Newtons per square meter (N/m2 in metric units).  1.0 Newton is equal to 1.0 Kg-m/s2 and is derived by multiplying the mass by the gravity force, 9.81m/s2.

Using this method, the force on the bottom of a 1.0 m3 block of granite due to gravity is:

 

2.65 x 1000 kg/m3 = 2600 kg (this is the mass of the block)
We know that F = ma, so F = (2600 kg) x (9.81 m/s2) = 2.55 x 104 kg-m/s2, or 2.55 x 104 N

This resulting force is acting on the total area at the bottom of the block, which is 1.0 m2, so the total force exerted by the 1.0 m3 block of granite is 2.55 x 104 N/m2.

           Metric units of stress are equal to Pascals (Pa), which are units of pressure.  The equality is: 1.0 N/m2 = 1.0 Pa.

 

Compressive strength is derived by dividing the force over the area upon which it acts and is specified as the Greek letter s.  The stress formula is given as:

 

s = P/A,  where P is the engineering way of expressing force, F.

For example, we wish to determine the compressive force on a 6.0 m3 (1m wide, 1m deep, 6m high) block of granite that has an applied load (force) of 2000 KN. Does this load exceed the compressive strength for granite?

 

Solution: Using the above formula, we find the stress on the block as force divided by area:

 

s = P/A = 2,000,000 N / 1.0 m2 = 2,000,000 N/m2 which is well below the compressive strength of granite which ranges upward from about 200 x 106 N/m2.

2. Tensile Strength:  rocks placed in tension will show a decrease in the total volume of the rock per unit area due to forces directed outward, opposite in action.

 

Tensile strength for a rock is usually much lower than its compressive strength, i.e., rocks are most likely to fail under tension well before they would fail under compression.  Thus, it is very important to know the stress regime a rock will be subjected to when used in an engineering project.  Most rock materials are never placed in a situation where tension is the primary force.

3. Shear Strength:  shearing action is caused by two forces acting in opposite directions along a plane of weakness (fracture, fault, bedding plane, etc.) that is inclined at some angle to the forces. The result is a force couple which effectively tears the material.

 

Rifting in tectonic environment is nothing more than a large shearing of the solid crust of the Earth where the actual rift itself is usually inclined at about 30o to the tension forces.  In the case of rifting, tension is generally supplied by the upwelling of mantle material below the crust.  The US Geological Survey has a web site demonstrating tectonic principles at this site.

 

E. Elasticity:  this property describes the ability of rock material to rebound to its original shape after an applied stress is relieved, or removed.  While under stress, rock material often deformes and when the load is removed, it is possible that not all of the deformation will, or can be, restored, particularly when the load was excessively heavy.  There are 2 ranges used to describe deformation of the rock:

 

1. elastic deformation:  occurs when all of the deformation caused by the stress is restored upon its release.

2. plastic deformation:  when stress that is below a critical threshold value is released, all of the deformation is restored.  However, if the applied stress exceeds the threshold value (which differs for various materials and rock types), permanent deformation results due to the load.  This means that when the load is removed, there is a permanent alteration to the original shape of the rock or material.  This may, or may not, be a critical concern in an engineering project.

F. Strain:  is a property that is somewhat related to elasticity.  Materials that are subjected to a load, whether it be compressive, tensile, or shear, will deform and either stretch or shrink in length.  This action is referred to as 'strain' and is described mathematically as:

 

e = DL/L,  where L is length and DL is the change in length.

This is a dimensionless number.  It is usually spoken as "...the strain on the marble column was determined to be 0.001 mm per mm."  It is a length divided by a length which is dimensionless.

 

The ratio between stress and strain is referred to as the 'Modulus of Elasticity', or Young's Modulus and is denoted as E.  Mathematically:

 

E = P/A   =  s
      DL/L     e

The last rock strength parameter we will explore is a property that describes the amount of lateral extension (strain) of a material that is under a vertical (axial) strain:

 

lateral strain   =   DB/B
axial strain          DL/L

where B is in terms of lateral dimensions.  This ratio is designated m, or 'Poisson's ratio'.  m varies in natural rock from between 0.1 to 0.5.  One example of the use of Poisson's ratio is in the analysis of the propagation of an energy wave generated by an earthquake.  This wave moves through solid rock and is, therefore, somewhat subjected to rock properties.  The speed of propagation, or wave velocity, is dependent upon the Poisson's ratio of the rock.  As rock type changes, wave velocity changes as a function of rock properties.




Web Links

A companion page on Merlot that discusses basics soil mechanics

The American Rock Mechanics Association, ARMA

The International Society of Rock Mechanics, ISRM