# A New Method to Measure and Calculate Tri-axial Static Strain Change Based on Relative Displacements Between Points for Rock Mass and Structure

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## Introduction

All rock excavations by means of natural gravity caving, mechanical excavation, or blasting cause redistribution of the static stress and strain in the remaining rock mass. It is critical to measure the change of the strain state. Such a measurement can serve as quantification of the effectiveness of an excavation method or a blast in terms of minimized damage to the remaining rock or structure.

Due to the discontinuous nature of the rock mass with local joints and cracks, small strain gauges placed at a spot in a rock surface cannot provide representative measurements for the strain in the rock mass on an engineering scale (e.g., a few meters) around the measurement point. Such measurement with strain gauges is not useful for the rock mass. At present, there is no method available to calculate strain changes over a large area of a rock slope or an underground structure. New measurement technologies developed over the last few decades, such as GPS, radar systems,...

## Keywords

Strain change Blast damage Piecewise average 3D strain Survey points Rock structures Displacement gradient Laser scanning## List of Symbols

*x*_{A,}*y*_{A},*z*_{A}Coordinates of the point

*A*(m)*u*_{A,}*v*_{A},*w*_{A}Displacement components in

*x*,*y*, and*z*directions of point*A*(m)- Δ
*u*, Δ*v*, Δ*w* Relative displacement in

*x*,*y*, and*z*directions between two points (m)- \(\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}\)
Gradients in

*x*,*y*, and*z*directions of the displacement component in*x*direction- \(\varepsilon_{ij}\)
Strain component and where

*i*,*j*=*x*,*y*,*z*(dimensionless)- \(\gamma_{ {\rm max} }\)
Maximum shear strain (dimensionless)

*e*Extensional strain (dimensionless)

- \(\varepsilon_{i}\)
Principal strain and where

*i*= 1, 2, 3 (dimensionless)- \(\eta_{ij}\)
Finite strain component and where

*i*,*j*=*x*,*y*,*z*(dimensionless)*α*,*β*,*γ*Directional cosines of a principal strain relative to

*x-*,*y-*, and*z*-axes (dimensionless)- SVD
Singular value decomposition method

*A**M*×*N*Matrix*M*≥*N*- U
*M*×*N*Column-orthogonal matrix- W
*N*×*N*Diagonal matrix with positive or zero elements- V
Transpose of

*N*×*N*orthogonal matrix**A**Unknown vector for the displacement gradients

**B**Row vector established from

*M*relative displacement components*Χ*Error residual for the least-square minimization

## Notes

### Acknowledgements

Orica Chile Colleagues Francis Fritis, Roberto Veragua, Natalia Ortega, Nuria Salvador, Rolando Fuentes, and Stephen Jeric supported the field setup and data collections. They should be greatly acknowledged.

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