Influence of geometry and material properties on the axial vibration of a rock bolt

A continuous dynamic model for the axial vibration of a rock bolt system is presented. The model comprises three sections: the fixed length, bonded into the rock, the free length, which is not coupled to the rock, and the protruding length, which extends beyond the rock. The head assembly is modelled as a discrete mass and a spring, and a further discrete mass is included, representing a testing device that can be attached to the protruding end. Each section is modelled as a continuous elastic rod governed by the wave equation, with suitable compatibility conditions applied between the sections and boundary conditions, which also account for the effect of the discrete components, applied at the ends. Solutions in non-dimensional form are substituted into the boundary conditions to allow the natural frequencies to be calculated, and it is shown that two possible solutions for the mode shapes can be used for the fixed length—an exponential solution or the classical sinusoidal solution—depending on the stiffness of the grout relative to that of the bar. The conditions for which the two solutions are valid are developed, and changes in the frequency ratio with changes in length ratio, and the stiffness ratios of the grout and the anchor head relative to the stiffness of the fixed length of the anchorage are examined. Generally, the state of a bolt after installation is unknown and this does not provide proper assurance of the safety of the structure for which the bolts are used. The model provides a viable tool for helping to assess the condition of the bolt by using the natural frequencies associated with areas of the bolt of particular interest, e.g. the free length. The results show how the changes in the stiffness and/or length ratios affect the dynamics associated with fixed length of the bolt and the quality of the bonding installation. A case study is presented showing how the model can be used effectively to interpret real data.