Rock Mechanics and Rock Engineering

, Volume 52, Issue 1, pp 83–95 | Cite as

Examination of Mean Stress Calculation Approaches in Rock Mechanics

  • Ke GaoEmail author
  • John P. Harrison
Original Paper


The mean stress, as a fundamental statistical property of a group of stress data, is essential for stress variability characterisation. However, currently in rock mechanics, the mean stress is customarily and erroneously calculated by separately averaging the principal stress magnitudes and orientations. In order to draw the attention of our community to the appropriate approach for stress variability characterisation, here we compare the customary scalar/vector mean with that obtained by the mathematically rigorous tensorial approach—the Euclidean mean. Calculation of mean stress using both a small group of actual in situ stress measurement results and a large group of simulated stress data (obtained using the combined finite–discrete element method, FEMDEM) demonstrates that the two approaches yield different results. Further investigation of these results shows that the scalar/vector approach may yield non-unique and non-orthogonal mean principal stresses, and these may deviate significantly from those of the Euclidean mean. Our calculations and comparisons reveal that the scalar/vector approach is deficient because it processes the principal stress magnitudes and orientations separately as independent quantities and ignores the connection between them. Conversely, the tensorial approach appropriately averages the tensors that simultaneously carry not only the information of stress magnitudes and orientations, but also the inherent relations between them. Therefore, arbitrarily using scalar/vector mean stress of in situ stress measurements as input in further rock engineering analyses may yield significantly erroneous results. We advise that stress data should be statistically processed in a tensorial manner using tensors referred to a common Cartesian coordinate system.


Stress tensor Rock mass Mean stress Euclidean mean FEMDEM 



Combined finite–discrete element method

List of symbols


Young’s modulus of boundary plate


Young’s modulus of rock


Boundary loading in x direction


Boundary loading in y direction


ith stress tensor, i = 1, 2, …,n

\({{\mathbf{\bar {S}}}_{\text{E}}}\)

Euclidean mean stress tensor


Plunge of principal stress

\(\bar {\varphi }\)

Mean of \(\varphi\)


Trend of principal stress

\(\bar {\theta }\)

Mean of \(\theta\)


Normal component of stress tensor

\(\bar {\sigma }\)

Mean of \(\sigma\)

\({\sigma _1}\)

Major principal stress

\({\bar {\sigma }_1}\)

Mean of \({\sigma _1}\)

\({\sigma _2}\)

Intermediate principal stress

\({\bar {\sigma }_2}\)

Mean of \({\sigma _2}\)

\({\sigma _3}\)

Minor principal stress

\({\bar {\sigma }_3}\)

Mean of \({\sigma _3}\)


Shear component of stress tensor

\(\bar {\tau }\)

Mean of \(\tau\)



We acknowledge the support of the NSERC (Canada) Discovery Grant (no. 491006) and the University of Toronto. The very valuable suggestions and comments of our colleague Dr. Nezam Bozorgzadeh are highly appreciated.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringUniversity of TorontoTorontoCanada
  2. 2.GeophysicsLos Alamos National LaboratoryLos AlamosUSA

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