Cable bolt calibration tool

A hybrid bolt calibration tool has been developed to help calibrate the hybrid bolt material properties.

The calibration tool is a simplification of the full 3DEC calculation. The tool was created to give a fast approximate solution to the hybrid bolt behavior under simplified conditions. A model consisting of two blocks is subjected to pull apart and pure shear tests. Force-displacement curves are reported for a given set of parameters. These force-displacement curves can be compared with laboratory experiments to calibrate material properties.

The tool works by solving the hybrid bolt structural element formulation in implicit form. The blocks are assumed to be infinitely stiff, a static solution is found for a given block displacement. The tool consists of two parts: a set of sliders on the left and a plot window on the right. Moving the sliders on the left updates the plots on the right. The top plot shows the cable axial force and the grout shear force along the cable. The top plot is symmetric because the joint separating the blocks is always in the middle. The horizontal lines show the cable axial yield strength and grout shear strength.

The middle plots shows both the cable displacement and cable strain. The horizontal line is the cable axial rupture strain.

The lower plot shows the force-displacement curves for a pull and shear test. Note that in a typical pullout test the bolt collar is pulled from the rock face; in this tool two nominal blocks of equal size are pulled apart. To convert a calibration tool result into an equivalent pullout test divide the displacement by two (for the pull test only). The blue dot in the lower plot show the displacement at which the upper two plots are generated. Move the displacement slider on the left side to show the cable forces and displacements at a different block displacement. The displacement slider also moves the blue dot but does not affect the lower plot.

Below the sliders are buttons to reset the position of the sliders to the default fully grouted rebar bolt properties. Another button copies a 3DEC property command to the Windows clipboard with the current property values.

The radio buttons on the bottom right of the left side change the upper two plots between pull apart and shear. When the pull test item is active (which is the default) the top plots show the forces, displacements and strains corresponding to a pull test displacement. Similarly, when the shear test item is active the upper plots correspond to a shear test with the given displacement. Note that when toggling this control the blue point jumps from the pull out curve to the shear curve in the lower plot.

1 Calibration procedure.

Before calibration a structural element node spacing length (slen) parameter should be selected. This length should be less than the characteristic block size in the rock mass under study. In the present study a constant spacing length of 10 cm is used.

The pull test behavior is calibrated first. The initial linear region of the pull test is controlled by the cable Young's modulus, cable area and grout stiffness. Larger moduli, area or stiffness make this region of the force-displacement curve steeper.

The deviation from linear before yield is controlled by the grout strength. The system loses modulus as the grout begins to fail away from the joint and into the blocks. Decreasing the grout strength results in a greater deviation from linear. Decreasing the grout strength too much results in pull out occurring before yield.

The cable yield strength input simply controls the maximum force a cable element can sustain. The cable rupture strain parameter directly controls the displacement where rupture occurs.

The shear test behavior has components of the pull test behavior so it is calibrated last. The initial linear region is primarily controlled by the dowel stiffness input. A larger dowel stiffness gives a steeper initial linear region. The shear-force at which the shear-force deviates from linear is given by the dowel yield strength input. The gradual increase in shear-force with increasing shear displacement is controlled by the joint friction angle and the cable Young's modulus. Typically these two parameters do not need to be changed. The maximum shear force the support can sustain is controlled by a combination of the cable axial yield strength and the dowel yield strength. In shear tests of the hybrid bolt the dowel typically ruptures before the cable ruptures from axial strain. Although, either could happen first depending on the parameters. The shear displacement at which shear rupture occurs is typically controlled by the dowel rupture strain.

Some of the parameters are involved in more than one feature of the force displacement curves which can make finding an exact match difficult. When calibrating hybrid bolt material properties it is conservative to error on the side of higher stiffness, lower strength and shorter rupture distances.

2 Calibration tool formulation

This document describes a tool to calibrate 3DEC cable structural element properties. A model consisting of two blocks is subjected to pull apart and pure shear tests. Force-displacement curves are reported for a given set of parameters. First some parameters are defined.

A	Bolt cross sectional area	201e-6 m²
E	Cable Youngs modulus	1.4e11 Pa
slen	Element length	0.1 m
dlen	Dowel length	0.1 m
K_bond	Grout stiffness	3e8 N/m/m
S_bond	Grout strength	2.8e5 N/m
N	(half) number of nodes	3
Yield	Cable yield strength	180e3 N
r	Cable rupture strain	0.2
ds	Dowel shear stiffness	1e7 N/m
dy	Dowel shear strength	62.8e3 N
dr	Shear rupture strain	0.41
θ	Joint friction angle	40 deg

Some additional constants are defined: \(c1 = E A/slen\), \(c2 = K_{bond}\ slen\) and \(c3= S_{bond}\ slen\). The problem geometry is

The blocks b0 and b1 are rigid. Each node is connected to adjacent nodes via an axial spring and attached to the host block via a shear spring with a shear dash pot. A single dowel spring and dash-pot is placed between the blocks which opposes displacement in the z direction but not in the x direction. b0 is fixed and b1 is moved in the positive x direction for a pull test and in the negative z direction for a shear test. The displacements of the nodes are written as dn0, dn1, dn2, dn3, dn4, and dn5. The displacements of the blocks are written as db0 and db1.

First the problem of finding the node displacements and block forces given the displacement of block 1 (db1) is addressed. This is done first for a cable and grout with infinite strength. Grout slip, cable pull out and cable yield are added subsequently. Finally, the behavior in shear displacement is described.

2.1 Elastic node displacement solution

The force on the axial springs depends on the relative displacement of the attached nodes. Tension is taken as positive.

\[ f_{s0} = c1\ (dn1-dn0) \]

\[ f_{s1} = c1\ (dn2-dn1) \]

the forces in the shear springs depends on the displacement of nodes and blocks.

\[ f_{ss0} = c2\ (dn0-db0) \]

\[ f_{ss3} = -c2\ (dn3-db1) \]

If the blocks and nodes are not moving the balance of force on each node is zero. For example, the balance of forces on n2 includes axial springs s1, and s2 and shear spring ss2. Axial spring s1 pulls n2 in the negative x direction, axial spring s2 pulls n2 in the positive x direction and shear spring ss2 pulls n2 in the negative x direction. So the sum of forces on n2 is

\[ f_{n2} = f_{s2} - f_{s1} - f_{ss2} = 0 \]

substituting the spring forces,

\[ c1\ (dn3-dn2) - c1\ (dn2-dn1) - c2\ (dn2-db0) = 0 \]

grouping terms by the node displacements,

\[ dn1 \ (c1) + dn2 \ (-2c1-c2) + dn3\ (c1) = - db0\ c2 \]

This can be done for each node, the force balance for node 3 is

\[ dn2\ (c1) + dn3\ (-2c1+c2) + dn4\ (c1) = - db1\ c2 \]

This can be written as a linear system \(Ax=b\) where \(A\) is a stiffness matrix, \(x\) is the node displacement vector and \(b\) is the node force.

\begin{equation} \begin{pmatrix} -c1-c2 & c1 &&&& \\ c1 & -2c1-c2 & c1 &&& \\ & c1 & -2c1-c2 & c1 && \\ && c1 & -2c1-c2 & c1 & \\ &&& c1 & -2c1-c2 & c1 \\ &&&& c1 & -c1-c2 \\ \end{pmatrix} \begin{pmatrix} dn0 \\ dn1 \\ dn2 \\ dn3 \\ dn4 \\ dn5 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \\ -db1\ c2 \\ -db1\ c2 \\ -db1\ c2 \end{pmatrix} \end{equation}

Solving this linear system for \(x\) gives the axial node displacements. The displacement of block 1 (db1) is the boundary condition, this term occurs in the forcing vector \(b\).

2.2 Grout slip

The above is only for an elastic grout and cable. The actual grout shear force is,

\begin{equation} f_{ss0} = \left\{ \begin{array}{ll} c2\ (dn0-db0) & \mathrm{if} \ c2\ (dn0-db0) < c3 \\ c3 & \mathrm{else} \\ \end{array} \right. \end{equation}

An iterative approach is used to resolve the grout slip. Nodes adjacent to the inner block boundary (n2 and n3) will slip first as the forces are highest here. Equation 1 is first solved assuming elastic (infinite strength) grout. If the grout shear force of the inner most nodes (n2 and n3) exceeds the grout shear strength the grout in these nodes is allowed to slip and equation 1 is solved again. In this new solution if the grout shear force in nodes n1 and n4 exceeds the grout strength they are allowed to slip and equation 1 is solved again etc.

To account for grout slip in equation 1 \(c2\) is set to zero in the corresponding row of the matrix A and vector \(b\). In the corresponding row in vector \(b\) a constant force of \(c3\) is added to account for the residual grout strength.

2.3 Cable pull out

A special case occurs when all the grout yields and the cable pulls out. Following the procedure above the matrix A becomes singular. The node displacements can be calculated directly in this case.

During pull out the force carried by spring i is:

\begin{equation} f_{si} = \left\{ \begin{array}{ll} c3\ (i+1) & \mathrm{if} \ i < N/2 \\ c3\ N & \mathrm{if} \ i=N \\ c3\ (N-i-1) & \mathrm{if} \ i > N/2 \\ \end{array} \right. \end{equation}

From these spring forces, the displacements of the nodes can be calculated by starting with the inner most node.

2.4 Cable yield

The axial force in the cable will always be the highest in the central element in this geometry. As displacement proceeds one of two things will happen: the cable will pull out or the cable will yield. It is possible to determine which will occur by looking at the force of the central axial spring (f_s(N-1)) in the pull out state. If this force is greater than the yield strength the cable will yield before pull out can occur.

\begin{equation} f_{s0} = \left\{ \begin{array}{ll} c1\ (dn1-dn0) & \mathrm{if} \ c1\ (dn1 -dn0) < yield \\ yield & \mathrm{else} \\ \end{array} \right. \end{equation}

The cable also has a rupture strain. To match the 3DEC large strain calculation the axial strain is \(\epsilon=db1/(slen+db1)\). If the axial spring ruptures the total block force is zero.

2.5 Force displacement curve in pull test

We want the force displacement curve resulting from moving block 1 in the positive x direction. Block 1 feels the cable via the forces in the shear springs attached to nodes in block 1. The total force on block 1 is

\[ f_{b1_x} = - \sum_{i=N/2}^{N-1} f_{ssi} = f_{s(N-1)} \]

which is equal to the force in the central axial spring.

2.6 Shear displacement

Consider that b1 is displaced in the negative z direction a distance db1_z. Three components of force oppose this displacement:

The vertical component of cable axial force f_ca
Additional frictional force induced by the cable f_fric
The dowel spring component f_d

The shear displacement of block 1 causes axial stretching of the cable. A displacement of block 1 in the z direction gives an equivalent horizontal block displacement of \(db1^*\)

\[ db1^* = \sqrt{db1_z^2 + slen^2}-slen \]

The cable axial force (f^*_{b1_x}) is determined using this equivalent displacement. The vertical component of this cable axial force is,

\begin{equation} f_{ca} = f^*_{b1_x}\ \cos \left(\arctan \left(slen/db1_z\right)\right) \end{equation}

This formulation over-predicts this vertical component of cable axial force. As a work around this component of force is scaled by a factor of 1/4 which brings the results into good agreement with 3DEC. The reason for the discrepancy is likely due to the assumption of infinite block stiffness in this formulation. The 3DEC blocks are elastic and the structural element nodes are connected to the block grid points. Some further development could give a better estimation of this force component.

The friction force is

\begin{equation} f_{fric} = f^*_{b1_x} \ \tan(\theta) \end{equation}

The dowel spring and dash-pot force is:

\begin{equation} f_{d} = \left\{ \begin{array}{ll} db1_z \ ds & \mathrm{if}\ db1_z \ ds \ < dy \\ dy & \mathrm{else} \\ \end{array} \right. \end{equation}

The dowel also has a rupture strain. If the dowel exceeds the rupture strain the total block force is zero.

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