Joint Properties

Joint properties are conventionally derived from laboratory testing (e.g., triaxial and direct shear tests). These tests can produce physical properties for joint friction angle, cohesion, dilation angle and tensile strength, as well as joint normal and shear stiffnesses. The joint cohesion and friction angle correspond to the parameters in the Coulomb strength criterion.

Values for normal and shear stiffnesses for rock joints typically can range from roughly 10 to 100 MPa/m for joints with soft clay in-filling, to over 100 GPa/m for tight joints in granite and basalt. Published data on stiffness properties for rock joints are limited; summaries of data can be found in Kulhawy (1975), Rosso (1976) and Bandis et al. (1983).

Approximate stiffness values can be back-calculated from information on the deformability and joint structure in the jointed rock mass and the deformability of the intact rock. If the jointed rock mass is assumed to have the same deformational response as an equivalent elastic continuum, then relations between jointed rock properties and equivalent continuum properties can be derived.

For uniaxial loading of rock containing a single set of uniformly spaced joints oriented normal to the direction of loading, the following relation applies:

(1)\[\begin{split}\frac{1}{E_m}=\frac{1}{E_r}+\frac{1}{k_ns} \\ \\\end{split}\]

or

\[k_n=\frac{E_mE_r}{s(E_r-E_m)}\]

where:	\(E_m\)	=   rock mass Young’s modulus;
	\(E_r\)	=   intact rock Young’s modulus;
	\(k_n\)	=   joint normal stiffness; and
	\(s\)	=   joint spacing.

A similar expression can be derived for joint shear stiffness:

(2)\[k_s=\frac{G_mG_r}{s(G_r-G_M)}\]

where:	\(G_m\)	=   rock mass shear modulus;
	\(G_r\)	=   intact rock shear modulus; and
	\(k_s\)	=   joint shear stiffness.

The equivalent continuum assumption, when extended to three orthogonal joint sets, produces the following relations:

(3)\[\begin{split}\begin{align} E_i&=\left(\frac{1}{E_r}+\frac{1}{s_ik_{ni}}\right)^{-1} &(i=1, 2, 3) \\ \\ G_{ij}&=\left(\frac{1}{G_r}+\frac{1}{s_ik_{si}}+\frac{1}{s_jk_{sj}}\right)^{-1} &(i,j=1,2,3) \end{align}\end{split}\]

Several expressions have been derived for two- and three-dimensional characterizations and multiple joint sets. References for these derivations can be found in Singh (1973), Gerrard (1982(a) and (b)) and Fossum (1985).

There is a limit to the maximum joint stiffnesses that are reasonable to use in a 3DEC model. If the physical normal and shear stiffnesses are less than ten times the equivalent stiffness of adjacent zones, then there is no problem with using physical values. If the ratio is more than ten, the solution time will be significantly longer than for the case in which the ratio is limited to ten, without much change in the behavior of the system. Serious consideration should be given to reducing supplied values of normal and shear stiffnesses to improve solution efficiency usinbg the following relation:

(4)\[k_n < 10\left[max\left[\frac{K+4/3G}{∆z_{min}}\right]\right]\]

where \(K\) and \(G\) are the bulk and shear moduli of the block material and \(\Delta z_{min}\) is the smallest dimension of the zone adjoining the joint in the normal direction.

There may also be problems with block interpenetration if the normal stiffness, kn, is very low. A rough estimate should be made of the joint normal displacement that would result from the application of typical stresses in the system (\(u = \sigma/k_n\)). This displacement should be small compared to a typical zone size. If it is greater than, say, 10% of an adjacent zone size, then either there is an error in one of the numbers or the stiffness should be increased.

Published strength properties are more readily available for joints than for stiffness properties. Summaries can be found, for example, in Jaeger and Cook (1969), Kulhawy (1975) and Barton (1976). Friction angles can vary from less than 10º for smooth joints in weak rock (such as tuff), to over 50º for rough joints in hard rock (such as granite). Joint cohesion can range from zero cohesion to values approaching the compressive strength of the surrounding rock.

It is important to recognize that joint properties measured in the laboratory typically are not representative of those for real joints in the field. Scale dependence of joint properties is a major question in rock mechanics. Often, the only way to guide the choice of appropriate parameters is by comparison to similar joint properties derived from field tests; however, field test observations are extremely limited. Some results are reported by Kulhawy (1975).