Ground Reaction Curve Calculator

Engineering Tools

Convergence–confinement analysis for circular tunnels: ground reaction, longitudinal deformation profile and support characteristic curves, with the equilibrium point and factor of safety. Runs in your browser — your inputs never leave this device. See how it compares with published examples.

Checked against published examples

Every value in the “This tool” column is computed by the same code that runs the calculator above. A value agrees when it is within the stated tolerance: the precision of the published figure, or 5 % where the source gives values read from charts.

Ground reaction curve, Mohr–Coulomb

Duncan Fama (1993) · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Rock-mass strength σcm (MPa)	4.53	4.533	Agrees
Slope k	2.28	2.283	Agrees
Critical pressure p_cr (MPa)	2.88	2.884	Agrees
Maximum convergence u_max (m)	0.0427	0.04265	Agrees
Plastic radius r_p (m)	7.96	7.962	Agrees
r_p / R	1.592	1.5924	Agrees

Hoek (2012), Figure 4: R = 5 m, p0 = 7 MPa, c = 1.5 MPa, φ = 23°, E = 1800 MPa, ν = 0.3.

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Longitudinal deformation profile

Vlachopoulos & Diederichs (2009) · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Convergence at the face (m)	0.011197	0.0111965	Agrees
Convergence 5 m behind the face (m)	0.0304	0.03039	Agrees

Hoek (2012), Figures 6–7.

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Support characteristic curves and factor of safety

Hoek (2012), eqs. 12–19 · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Shotcrete p_max (MPa)	2.037	2.0370	Agrees
Shotcrete stiffness K (MPa/m)	387	387.0	Agrees
Shotcrete yield displacement (m)	0.03565	0.035653	Agrees
Shotcrete factor of safety	3.4	3.35	Agrees
Bolts p_max (MPa)	0.354	0.3540	Agrees
Bolts stiffness K (MPa/m)	63	62.6	Agrees
Bolts yield displacement (m)	0.03604	0.036040	Agrees
Bolts factor of safetyThe note's text quotes 2.875; its Figure 13 gives 1.0.	1.0	1.02	Agrees
Steel sets p_max (MPa)	1.47	1.470	Agrees
Steel sets stiffness K (MPa/m)The published spreadsheet uses E·A/(s·R); the note's own eq. 15 gives E·A/(s·R²) = 248 MPa/m, which this tool follows.	1242	248.4	Differs — see note
Steel sets factor of safetyDiffers only because of the stiffness above.	2.1	2.62	Differs — see note
Steel sets FS with the published K = 1242 MPa/m	2.1	2.15	Agrees

Hoek (2012), Figure 13: support installed 5 m behind the face; each support analysed on its own.
The published steel-set area A = 0.0300 m² is used as given.

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Ground reaction at p_i = 5 MPa, Hoek–Brown (a = 0.5)

Carranza-Torres & Fairhurst (1999) · Fairhurst, C. & Carranza-Torres, C. (2002). Closing the circle. Proc. 50th Annual Geotechnical Engineering Conference, University of Minnesota, 21–84. (opens in a new tab)

Quantity	Published	This tool	Result
Scaled in-situ stress S₀	0.6	0.59	Agrees
Scaled internal pressure Pᵢ	0.1	0.10	Agrees
Scaled critical pressure P_cr	0.32	0.311	Agrees
Critical pressure p_cr (MPa)	16	15.8	Agrees
r_p / R	1.64	1.623	Agrees
Plastic radius r_p (m)	6.26	6.201	Agrees
Scaled convergence u·2G/(R(p₀ − p_cr))	3.1	3.07	Agrees
Convergence u (m)	0.08	0.083	Agrees

R = 3.82 m, p0 = 30 MPa, σci = 30 MPa, m_b = 1.7, s = 0.0039, a = 0.5, ψ = 0, G = 1 GPa, ν = 0.25.
Published values were read from dimensionless charts, so the tolerance is 5 %.

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Hoek–Brown input via equivalent Mohr–Coulomb

Duncan Fama (1993) + Hoek et al. (2002) · Rocscience (2004). RocSupport Tutorial Manual. (opens in a new tab)

Quantity	Published	This tool	Result
Plastic radius r_p (m)	13.8	13.78	Agrees
Convergence u_max / R (%)	2.0	2.03	Agrees

Example 1: R = 6 m, p0 = 1.62 MPa, σci = 7 MPa, mi = 10, GSI = 15, D = 0, E = 353 MPa (Hoek et al. 2002, eq. 11).
ν = 0.3 is RocSupport's default; the tutorial does not state it.

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Generalised Hoek–Brown (a = 0.553)

Carranza-Torres (2004) · Rocscience (2004). RocSupport Tutorial Manual. (opens in a new tab)

Quantity	Published	This tool	Result
Plastic radius r_p (m)	26.3	26.35	Agrees
Convergence u_max / R (%)	13.1	13.11	Agrees

Example 2: R = 5 m, p0 = 2.02 MPa, σci = 4 MPa, mi = 12, GSI = 17, D = 0, E = 299 MPa.
ν = 0.3 and ψ = 0 are RocSupport's defaults; the tutorial does not state them.

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Method, assumptions and limits

What the calculator does

Ground reaction curve — the wall convergence of a circular tunnel as the internal support pressure drops from the in-situ stress to zero. The primary method is the elasto-plastic solution of Carranza-Torres (2004) for the generalised Hoek–Brown criterion, with a dilation angle; for Mohr–Coulomb the same formulation gives the classical small-strain solution. Displacements are integrated numerically. The Duncan Fama (1993) solution, as presented by Hoek (2012), is available for comparison; with Hoek–Brown input it uses equivalent Mohr–Coulomb parameters (Hoek et al. 2002).
Longitudinal deformation profile — how much of the final convergence has already happened at a given distance from the face, after Vlachopoulos & Diederichs (2009), Panet (1995) or the empirical fit used by Hoek (1999). It sets the convergence at which the support starts to carry load.
Support characteristic curves — elastic–perfectly plastic responses of a closed shotcrete or concrete ring, steel sets in contact with the rock and ungrouted anchored bolts (Hoek 2012). Combined support adds the stiffnesses and yields when its first component yields (Hoek & Brown 1980). RocSupport instead adds the component capacities, so its combined capacity and factor of safety are higher.
Equilibrium and factor of safety — where the support curve meets the ground reaction curve; the factor of safety is the support capacity divided by the pressure at equilibrium. The squeezing class follows Hoek & Marinos (2000).

Assumptions and limits

Circular tunnel, equal in-situ stresses in all directions, plane strain and small strains.
Homogeneous, isotropic, elastic–perfectly plastic rock mass: no strain softening, no time-dependent behaviour.
Support is a uniform, closed ring or pattern with no bending; real linings are loaded unevenly and carry less.
The deformation profiles come from unsupported tunnels; installing stiff support changes them.
These are first estimates for learning and sensitivity studies. Tunnel design needs site-specific investigation, numerical analysis and independent checking.

Your data

The calculation runs entirely in your browser. Your inputs are kept in the page address after the # sign, so you can bookmark or share them; browsers do not send that part of the address to our server.

References

Carranza-Torres, C. (2004). Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 41 (Suppl. 1), 629–639. doi.org/10.1016/j.ijrmms.2004.03.111 (opens in a new tab)
Carranza-Torres, C. & Fairhurst, C. (1999). The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 36(6), 777–809. doi.org/10.1016/S0148-9062(99)00047-9 (opens in a new tab)
Carranza-Torres, C. & Fairhurst, C. (2000). Application of the convergence–confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure criterion. Tunnelling and Underground Space Technology 15(2), 187–213. doi.org/10.1016/S0886-7798(00)00046-8 (opens in a new tab)
Chern, J.C., Shiao, F.Y. & Yu, C.W. (1998). An empirical safety criterion for tunnel construction. Proc. Regional Symposium on Sedimentary Rock Engineering, Taipei, 222–227.
Duncan Fama, M.E. (1993). Numerical modelling of yield zones in weak rocks. In Hudson, J.A. (ed.) Comprehensive Rock Engineering 2, 49–75. Pergamon.
Fairhurst, C. & Carranza-Torres, C. (2002). Closing the circle — some comments on design procedures for tunnel supports in rock. Proc. 50th Annual Geotechnical Engineering Conference, University of Minnesota, 21–84. itasca-downloads.s3.amazonaws.com/documents/offices/International/Fairhurst-Files/Closing_the_Circle.pdf (opens in a new tab)
Hoek, E. (1999). Support for very weak rock associated with faults and shear zones. Proc. Int. Symp. on Rock Support and Reinforcement Practice in Mining, Kalgoorlie.
Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. static.rocscience.cloud/assets/verification-and-theory/RocSupport/Rock-support-interaction-analysis-for-tunnels-2-April-2012.pdf (opens in a new tab)
Hoek, E. Practical Rock Engineering, chapter “Tunnels in weak rock”. Rocscience. static.rocscience.cloud/assets/resources/learning/hoek/13.-Tunnels-in-weak-rock_2023-06-30-120753_ftju.pdf (opens in a new tab)
Hoek, E. & Brown, E.T. (1980). Underground Excavations in Rock. Institution of Mining and Metallurgy, London.
Hoek, E., Carranza-Torres, C. & Corkum, B. (2002). Hoek–Brown failure criterion — 2002 edition. Proc. NARMS-TAC, Toronto, 1, 267–273. www.rocscience.com/assets/resources/learning/hoek/Hoek-Brown-Failure-Criterion-2002.pdf (opens in a new tab)
Hoek, E. & Diederichs, M.S. (2006). Empirical estimation of rock mass modulus. Int. J. Rock Mech. Min. Sci. 43(2), 203–215. doi.org/10.1016/j.ijrmms.2005.06.005 (opens in a new tab)
Hoek, E. & Marinos, P. (2000). Predicting tunnel squeezing problems in weak heterogeneous rock masses. Tunnels and Tunnelling International 32(11), 45–51; 32(12), 33–36.
Panet, M. (1995). Calcul des tunnels par la méthode de convergence–confinement. Presses de l’ENPC, Paris.
Rocscience (2004). RocSupport Tutorial Manual, Examples 1–2. www.rocscience.com/downloads/rocsupport/RocSupport%20Tutorial.pdf (opens in a new tab)
Salençon, J. (1969). Contraction quasi-statique d’une cavité à symétrie sphérique ou cylindrique dans un milieu élastoplastique. Annales des Ponts et Chaussées 4, 231–236.
Vlachopoulos, N. & Diederichs, M.S. (2009). Improved longitudinal displacement profiles for convergence confinement analysis of deep tunnels. Rock Mech. Rock Eng. 42(2), 131–146. doi.org/10.1007/s00603-009-0176-4 (opens in a new tab)

Educational aid, not certified design software. The convergence–confinement method assumes a circular tunnel, equal in-situ stresses in all directions, a homogeneous elastic–perfectly plastic rock mass and uniformly distributed support (no bending). For a non-circular section, results for an equivalent circular radius are only a rough guide. Use it to learn and to explore sensitivities; check every result independently before using it in design — terms of use.

Checked against published examples

Ground reaction curve, Mohr–Coulomb

Duncan Fama (1993) · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Rock-mass strength σcm (MPa)	4.53	4.533	Agrees
Slope k	2.28	2.283	Agrees
Critical pressure p_cr (MPa)	2.88	2.884	Agrees
Maximum convergence u_max (m)	0.0427	0.04265	Agrees
Plastic radius r_p (m)	7.96	7.962	Agrees
r_p / R	1.592	1.5924	Agrees

Hoek (2012), Figure 4: R = 5 m, p0 = 7 MPa, c = 1.5 MPa, φ = 23°, E = 1800 MPa, ν = 0.3.

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Longitudinal deformation profile

Vlachopoulos & Diederichs (2009) · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Convergence at the face (m)	0.011197	0.0111965	Agrees
Convergence 5 m behind the face (m)	0.0304	0.03039	Agrees

Hoek (2012), Figures 6–7.

Load this example in the calculator

Support characteristic curves and factor of safety

Hoek (2012), eqs. 12–19 · Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. (opens in a new tab)

Quantity	Published	This tool	Result
Shotcrete p_max (MPa)	2.037	2.0370	Agrees
Shotcrete stiffness K (MPa/m)	387	387.0	Agrees
Shotcrete yield displacement (m)	0.03565	0.035653	Agrees
Shotcrete factor of safety	3.4	3.35	Agrees
Bolts p_max (MPa)	0.354	0.3540	Agrees
Bolts stiffness K (MPa/m)	63	62.6	Agrees
Bolts yield displacement (m)	0.03604	0.036040	Agrees
Bolts factor of safetyThe note's text quotes 2.875; its Figure 13 gives 1.0.	1.0	1.02	Agrees
Steel sets p_max (MPa)	1.47	1.470	Agrees
Steel sets stiffness K (MPa/m)The published spreadsheet uses E·A/(s·R); the note's own eq. 15 gives E·A/(s·R²) = 248 MPa/m, which this tool follows.	1242	248.4	Differs — see note
Steel sets factor of safetyDiffers only because of the stiffness above.	2.1	2.62	Differs — see note
Steel sets FS with the published K = 1242 MPa/m	2.1	2.15	Agrees

Hoek (2012), Figure 13: support installed 5 m behind the face; each support analysed on its own.
The published steel-set area A = 0.0300 m² is used as given.

Load this example in the calculator

Ground reaction at p_i = 5 MPa, Hoek–Brown (a = 0.5)

Quantity	Published	This tool	Result
Scaled in-situ stress S₀	0.6	0.59	Agrees
Scaled internal pressure Pᵢ	0.1	0.10	Agrees
Scaled critical pressure P_cr	0.32	0.311	Agrees
Critical pressure p_cr (MPa)	16	15.8	Agrees
r_p / R	1.64	1.623	Agrees
Plastic radius r_p (m)	6.26	6.201	Agrees
Scaled convergence u·2G/(R(p₀ − p_cr))	3.1	3.07	Agrees
Convergence u (m)	0.08	0.083	Agrees

R = 3.82 m, p0 = 30 MPa, σci = 30 MPa, m_b = 1.7, s = 0.0039, a = 0.5, ψ = 0, G = 1 GPa, ν = 0.25.
Published values were read from dimensionless charts, so the tolerance is 5 %.

Load this example in the calculator

Hoek–Brown input via equivalent Mohr–Coulomb

Duncan Fama (1993) + Hoek et al. (2002) · Rocscience (2004). RocSupport Tutorial Manual. (opens in a new tab)

Quantity	Published	This tool	Result
Plastic radius r_p (m)	13.8	13.78	Agrees
Convergence u_max / R (%)	2.0	2.03	Agrees

Example 1: R = 6 m, p0 = 1.62 MPa, σci = 7 MPa, mi = 10, GSI = 15, D = 0, E = 353 MPa (Hoek et al. 2002, eq. 11).
ν = 0.3 is RocSupport's default; the tutorial does not state it.

Load this example in the calculator

Generalised Hoek–Brown (a = 0.553)

Carranza-Torres (2004) · Rocscience (2004). RocSupport Tutorial Manual. (opens in a new tab)

Quantity	Published	This tool	Result
Plastic radius r_p (m)	26.3	26.35	Agrees
Convergence u_max / R (%)	13.1	13.11	Agrees

Example 2: R = 5 m, p0 = 2.02 MPa, σci = 4 MPa, mi = 12, GSI = 17, D = 0, E = 299 MPa.
ν = 0.3 and ψ = 0 are RocSupport's defaults; the tutorial does not state them.

Load this example in the calculator

Method, assumptions and limits

What the calculator does

Ground reaction curve — the wall convergence of a circular tunnel as the internal support pressure drops from the in-situ stress to zero. The primary method is the elasto-plastic solution of Carranza-Torres (2004) for the generalised Hoek–Brown criterion, with a dilation angle; for Mohr–Coulomb the same formulation gives the classical small-strain solution. Displacements are integrated numerically. The Duncan Fama (1993) solution, as presented by Hoek (2012), is available for comparison; with Hoek–Brown input it uses equivalent Mohr–Coulomb parameters (Hoek et al. 2002).

Longitudinal deformation profile — how much of the final convergence has already happened at a given distance from the face, after Vlachopoulos & Diederichs (2009), Panet (1995) or the empirical fit used by Hoek (1999). It sets the convergence at which the support starts to carry load.

Support characteristic curves — elastic–perfectly plastic responses of a closed shotcrete or concrete ring, steel sets in contact with the rock and ungrouted anchored bolts (Hoek 2012). Combined support adds the stiffnesses and yields when its first component yields (Hoek & Brown 1980). RocSupport instead adds the component capacities, so its combined capacity and factor of safety are higher.

Equilibrium and factor of safety — where the support curve meets the ground reaction curve; the factor of safety is the support capacity divided by the pressure at equilibrium. The squeezing class follows Hoek & Marinos (2000).

Assumptions and limits

Circular tunnel, equal in-situ stresses in all directions, plane strain and small strains.

Homogeneous, isotropic, elastic–perfectly plastic rock mass: no strain softening, no time-dependent behaviour.

Support is a uniform, closed ring or pattern with no bending; real linings are loaded unevenly and carry less.

The deformation profiles come from unsupported tunnels; installing stiff support changes them.

These are first estimates for learning and sensitivity studies. Tunnel design needs site-specific investigation, numerical analysis and independent checking.

Your data

References

Carranza-Torres, C. (2004). Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 41 (Suppl. 1), 629–639. doi.org/10.1016/j.ijrmms.2004.03.111

Carranza-Torres, C. & Fairhurst, C. (1999). The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 36(6), 777–809. doi.org/10.1016/S0148-9062(99)00047-9

Carranza-Torres, C. & Fairhurst, C. (2000). Application of the convergence–confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure criterion. Tunnelling and Underground Space Technology 15(2), 187–213. doi.org/10.1016/S0886-7798(00)00046-8

Chern, J.C., Shiao, F.Y. & Yu, C.W. (1998). An empirical safety criterion for tunnel construction. Proc. Regional Symposium on Sedimentary Rock Engineering, Taipei, 222–227.

Duncan Fama, M.E. (1993). Numerical modelling of yield zones in weak rocks. In Hudson, J.A. (ed.) Comprehensive Rock Engineering 2, 49–75. Pergamon.

Fairhurst, C. & Carranza-Torres, C. (2002). Closing the circle — some comments on design procedures for tunnel supports in rock. Proc. 50th Annual Geotechnical Engineering Conference, University of Minnesota, 21–84. itasca-downloads.s3.amazonaws.com/documents/offices/International/Fairhurst-Files/Closing_the_Circle.pdf

Hoek, E. (1999). Support for very weak rock associated with faults and shear zones. Proc. Int. Symp. on Rock Support and Reinforcement Practice in Mining, Kalgoorlie.

Hoek, E. (2012). Rock-support interaction analysis for tunnels in weak rock masses. Rocscience course notes. static.rocscience.cloud/assets/verification-and-theory/RocSupport/Rock-support-interaction-analysis-for-tunnels-2-April-2012.pdf

Hoek, E. Practical Rock Engineering, chapter “Tunnels in weak rock”. Rocscience. static.rocscience.cloud/assets/resources/learning/hoek/13.-Tunnels-in-weak-rock_2023-06-30-120753_ftju.pdf

Hoek, E. & Brown, E.T. (1980). Underground Excavations in Rock. Institution of Mining and Metallurgy, London.

Hoek, E., Carranza-Torres, C. & Corkum, B. (2002). Hoek–Brown failure criterion — 2002 edition. Proc. NARMS-TAC, Toronto, 1, 267–273. www.rocscience.com/assets/resources/learning/hoek/Hoek-Brown-Failure-Criterion-2002.pdf

Hoek, E. & Diederichs, M.S. (2006). Empirical estimation of rock mass modulus. Int. J. Rock Mech. Min. Sci. 43(2), 203–215. doi.org/10.1016/j.ijrmms.2005.06.005

Hoek, E. & Marinos, P. (2000). Predicting tunnel squeezing problems in weak heterogeneous rock masses. Tunnels and Tunnelling International 32(11), 45–51; 32(12), 33–36.

Panet, M. (1995). Calcul des tunnels par la méthode de convergence–confinement. Presses de l’ENPC, Paris.

Rocscience (2004). RocSupport Tutorial Manual, Examples 1–2. www.rocscience.com/downloads/rocsupport/RocSupport%20Tutorial.pdf

Salençon, J. (1969). Contraction quasi-statique d’une cavité à symétrie sphérique ou cylindrique dans un milieu élastoplastique. Annales des Ponts et Chaussées 4, 231–236.

Vlachopoulos, N. & Diederichs, M.S. (2009). Improved longitudinal displacement profiles for convergence confinement analysis of deep tunnels. Rock Mech. Rock Eng. 42(2), 131–146. doi.org/10.1007/s00603-009-0176-4