<< Previous | Contents | Next >> |

## Technical Manual for Design and Construction of Road Tunnels - Civil Elements

### Appendix E - Analytical Closed Form Solutions

#### E.1 Analytical Elastic Closed Form Solutions for Rock Tunnels

As discussed in Section 6.6.2, the state of stress due to tunnel excavation can be calculated from analytical solutions or using numerical analysis. Kirsch's elastic closed form solution is one of the commonly used analytical solutions and is presented in Figure E-1. The closed form solution is restricted to simple geometries and material models, and therefore often of limited practical value. However, the solution is considered to be a good tool for a "sanity check" of the results obtained from numerical analyses.

**Figure E-1 Kirsch's Elastic Solution (Kirsch, 1898)**

Section 6.6.2 also describes other common analytical solutions proposed by Hoek et al. (1995), Bischoff and Smart (1977), and Brady & Brown (1985).

Analytical solutions to calculate support stiffness and maximum support pressure for concrete/shotcrete, steel sets, and ungrouted mechanically or chemically anchored rock bolts/cables are summarized in Table E-1.

Support System | Support stiffness (K) and maximum support pressure (P_{max}) |
---|---|

Concrete /Shotcrete lining | |

Blocked steel sets | |

Ungrouted mechanically or chemically anchored rock bolts or cables |

**NOTATION:** K = support stiffness; P_{max} = maximum support pressure; E_{c} = Young's modulus of concrete; t_{c} = lining thickness; r_{i} = internal tunnel radius; σ_{cc} = uniaxial compressive strength of concrete or shotcrete; W = flange width of steel set and side length of square block; X = depth of section of steel set; A_{s} = cross section area of steel set; I_{s} = second moment of area of steel set; E_{s} = Young's modulus of steel; σ_{ys} = yield strength of steel; S = steel set spacing along the tunnel axis; θ= half angle between blocking points in radians; t_{B} = thickness of block; E_{B} = Young's modulus of block material; l = free bolt or cable length; d_{b} = bolt diameter or equivalent cable diameter; E_{b} = Young's modulus of bolt or cable; T_{bf} = ultimate failure load in pull-out test; s_{c} = circumferential bolt spacing; s_{l} = longitudinal bolt spacing; Q = load-deformation constant for anchor and head.

#### E.2 Analytical Elastic Closed Form Solutions for Ground Support Interaction

Analytical solutions for ground-support interaction for a tunnel in soil are available in the literatures. The solutions are based on two dimensional, plane strain, linear elasticity assumptions in which the lining is assumed to be placed deep and in contact with the ground (no gap), i.e., the solutions do not allow for a gap to occur between the support system and ground. The background information for the common closed form models are presented in Appendix B of the FHWA Tunnel Design Guidelines (2004) which is reproduced here in Section E.3 for convenience.

Early analytical solutions by Burns and Richard (1964), Dar and Bates (1974), and Hoeg (1968) were derived for the overpressure loading, while solutions by Morgan (1961), Muir Wood (1975), Curtis (1976), Rankin, Ghaboussi and Hendron (1978), and Einstein et al. (1980) were for excavation loading. Solutions are available for the full slip and no slip conditions at the ground-lining interface. Appendix E present the available published analytical solutions in Table E-2. A sample analysis is presented in Table E-3 to illustrate the applications of various closed-form solutions for a 22ft diameter circular tunnel with 1.5 ft thick concrete lining. The tunnel is located at 105 ft deep from the ground surface to springline and groundwater table is located 10 ft below the ground surface. Details of input parameters are shown in Table E-3a. The calculated lining loads from various analytical solutions are presented in Table E-3b.

Analytical Solutions | Thrust | Moment | |
---|---|---|---|

Wu & Penzien (1997) | Relaxation | crown = P Springline = P P P P | =(-1/4)×(1-k |

Overburden | Crown = P Springline = P P P P | =(-1/4)×(1-k | |

Einstein & Schwartz (1979) | Excavation full slip | Crown = [0.5×(1+k springline = [0.5×(1+k | =-0.5×(1-k |

Excavation no slip | Thrust at Crown Thrust at Springline | =-[0.25×(1-k | |

a a a β = ((F + 6) × (1-v b C = [(d/2) × E F = (d/2) | |||

Peck, Hendron & Moharaz (1972) overburden | Thrust at Crown Thrust at Springline | =(1-k_{0}) × b_{2} × γ_{s} × h × (d/2)^{2}/6 | |

b a | |||

Ranken, Ghaboussi and Hendron (1978) | Overpressure (no slip) | Thrust at Crown Thrust at Springline | Moment at Crown Moment at Springline |

Overpressure (full slip) | Thrust at Crown Thrust at Springline | Moment at Crown Moment at Springline | |

Excavation (No Slip) | Thrust at Crown Thrust at Springline | Moment at Crown Moment at Springline | |

Excavation (Full Slip) | Thrust at Crown Thrust at Springline | Moment at Crown Moment at Springline | |

L J /[((3-2×v N /[((3-2×v L J N L J /[((3-2×v N /[((3-2×v L J N | |||

Muir & Wood (1975) Excavation | Thrust at Crown Thrust at Springline | = (σ η = (Φ | |

Full Slip | k = (1-v β = (E η = (Φ Φ λ = (3×E Q Q I = (w R σ σ | ||

Curtis (1976) | Excavation Full Slip | Thrust at Crown Thrust at Springline N | = -{1/2×(σ_{v}-σ_{H})×η^{2}×(d/2)^{2}]×(3-4×v_{m})/(5-6×v_{m}+4×Q_{2}) |

Excavation No Slip | Thrust at Crown
=N Thrust at Springline N N | = -[1/4×(σ_{v}-σ_{H})×η^{2}×(d/2)^{2}]/[1+Q_{2}×(3-2×v_{m})] | |

k=(1-v β = (E η = (Φ Φ λ = (3×E Q Q I = (w |

**Notation:**

- v
_{m}: Poisson's ration for ground - v
_{L}: Poisson's ration for Liner - E
_{m}: Young's Modulus for ground - E
_{L}: Young's Modulus for Liner - t
_{L}: Thickness of Liner - w
_{L}: Width of Liner

- A
_{L}: Cross-Sectional Area of Liner - γ
_{m}: Ground Unit Weight - γ
_{w}: Water Unit Weight - d: Diameter of Tunnel
- I: Moment Inertia per Unit Length
- C: Compressibility

- F: Flexibility
- k
_{0}: Coefficient of Lat. Earth Pressure - h: Depth to Springline
- h
_{w}: Depth from Water Table - R
_{s}: Stiffness Factor - S: Surcharge

**Table E-3 Sample Concrete Lining Load Calculation for a 22-ft Diameter Circular Tunnel in Soil**

(a) Input Data

Lining Properties | Ground Properties | ||
---|---|---|---|

Width = | 5 ft | Elastic Modulus, E_{m} = | 2.03E+06 lb/ft^2 |

Thickness, t = | 1.500 ft | Poissson's Ratio, n_{m} = | 0.41 |

Compressive Strength Concrete, f'_{c} = | 5000 psi | ||

Elastic Modulus, E_{l} = | 5.80E+08 lb/ft^2 | Soil Unit Weight, g = | 130 lb/ft^3 |

External Diameter (OD) = | 22 ft | Water Unit Weight, g_{w} = | 62.4 lb/ft^3 |

Poisson's Ratio, n_{l} = | 0.25 | ||

Number of Joints = | 0 | ||

Determine Thrusts and Moments for: | |||

Depth to Springline = | 105 ft | ||

Depth from water table = | 95 ft | ||

Coeff. Lateral Pressure, K_{0} = | 0.7 |

(b) Concrete Lining Loads Calculated from Various Analytical Solutions

Analytical Solutions | Thrust at Crown/ft | Thrust at Springline/ft | Moment/ft | |
---|---|---|---|---|

Wu & PenzienRelaxation Overburden | -129698 -131020 | -132731 -136283 | -15165 -26316 | |

Einstein & SchwartzExcavation Full Slip Excavation No Slip | 97536 108108 | 153444 142872 | -54264 -50176 | |

Peck, Hendron, & MoharazOverburden | 139515 | 156634 | -94164 | |

Ranken, Ghaboussi, & HendronOverpressure Case 1 (no slip) Case 2 (full slip) Excavation Case 3 (no slip) Case 4 (full slip) | 117912 139514 108105 120554 | 178237 156635 142869 130420 | Crown -84545 -91640 -48037 -52125 | Springline 89593 96688 52315 56403 |

Muir-WoodExcavation Full Slip | 124377 | 137264 | -18055 | |

CurtisExcavation Full Slip Excavation No Slip | 132119 125095 | 138192 145216 | -25644 -23690 |

#### Appendix B.1 - Elastic Closed Form Models for Ground-Lining Interaction

The source document for Appendices B.1 and B.2 is: *Guidelines for Tunnel Lining Design* by the Technical Committee on Tunnel Lining Design ofthe Underground Technology Research Council, edited by T.D. O'Rourke (1984), and reproduced here, for convenience.

Several closed form models for ground-lining interaction have been developed on the basis of elastic ground and lining properties. Although the models are limited by assumptions of elasticity and specific conditions of loading, they nonetheless possess several attractive features, including their relative simplicity, sensitivity to significant ground and support characteristics, and ability to represent the mechanics of ground-lining interaction. The models are useful for eva 1 uating the variation in lining response to changes in soil, rock, and structural material properties, in-situ stresses, and lining dimensions. However, considerable judgment must be exercised by the tunnel designer in applying these models. Their chief value lies in their ability to place bounding conditions on performance and thereby supplement the many practical considerations of tunnel operation, construction infl uence, and variation in ground conditions discussed in the main body of this work.

Some special characteristics of elastic closed form models are discussed by Schmidt (1984).

A.I Background

Most elastic closed form models are based on the assumption that the ground is an infinite, elastic, homogeneous, isotropic medium. The interaction between the ground and a circular elastic, thin walled lining is assumed to occur under plane strain conditions. The models involve either full slip or no slip conditions along the ground-lining interface.

In some models (Muir Wood, 1975; Curtis, 1976), equations have been developed for interface conditions that involve a shear strength between that of full and no slip conditions. The magnitude of the vertical stress is assumed equal to the product of the soil unit weight, y, and the depth to the longitudinal centerline of the tunnel, H. The increased stress from crown to invert is not considered so that the solutions are appropriate for deep tunnels. Finite element analyses by Ranken, Ghaboussi, and Hendron (1978) and a review of analytical work by Einstein and Schwartz (1979) indicate that tunnels are sufficiently deep for application of the elastic solutions when HID is greater than about 1.5, where D is the outside diameter of the tunnel.

The elastic models can be divided into two categories according to the conditions of in-situ stress that prevail when the lining is installed and loaded. Work by Morgan (1961), Muir Wood (1975), Curtis' (1976), Ranken, Ghaboussi, and Hendron (1978), and Einstein and Schwartz (1979) has been based on lining response within a stressed ground mass.

This condition is commonly referred to as excavation loading. Work by Burns and Richard (1964), Hoeg (1968), Peck, Hendron, and Mohraz (1972), Dar and Bates (1974), and Mohraz, et al. (1975) has been based on lining response in a ground mass subjected to an externally applied pressure.

This condition is commonly referred to as overpressure loading.

Overpressure loading imp 1 ies that the 1 ining is installed before external loads are applied. This assumption is suitable for simulating the effects of external blasting and the placement offill above a previously constructed tunnel. Models developed on the basis of overpressure loading do not simulate the most frequently encountered situation in which the lining is constructed in soil or rock subjected to in-situ stresses. In general, models based on overpressure loading resul t in higher values of thrust and moment compared to those based on excavation loading.

A.2 Analytical Results

The analytical results derived from the work of Ranken, Ghaboussi, and Hendron (1978) for excavation loading are used in this appendix to show how moments and thrusts vary as a function of the relative stiffuess between the ground and lining. The conditions of in-situ stress assumed in the model are illustrated in Figure A.l, where the vertical stress is defined as previously mentioned and the horizontal stress is defmed as the product of the coefficient of earth pressure at rest, K_{o}, and the vertical stress. It is not possible to install a lining without some relief of in-situ stresses. The amount of stress relief will depend on the characteristics of the excavation and support process and is particularly sensitive to the distance support is installed behind the excavated face. The model therefore represents a limiting condition of restraint against inward ground movement.

It is convenient to summarize the analytical results in dimensionless form. Accordingly, the dimensionless moment, or moment coefficient is given by M/(γHR^{2}) where M is the moment per unit length of tunnel, y is the ground unit weight, H is the depth to the tunnel center line, and R is the external lining radius. Similarly, the thrust coefficient is given by T/(γHR), where T is the thrust per unit length of tunnel. The dimensionless parameters that reflect the relative stiffuess between the ground and lining are referred to as the flexibility ratio, F, and the compressibility ratio, C.

The flexibility ratio is a measure of the flexural stiffuess of the ground to that of the lining. Assuming a rectangular cross-section of the lining, the flexibility ratio is defined as

F = (E_{m} / E_{1} (R/t)^{3} [(2(1- v_{1}^{2}))/(1 + v_{m})] (A.1)

in which E_{m} is the modulus of the surrounding medium, or ground, E_{1} is the modulus of the lining, t is the lining thickness, and v_{1} and v_{m} are the Poisson ratios ofthe lining and ground, respectively.

The compressibility ratio is a measure of the extensional stiffuess of the ground to that of the lining. Assuming a rectangular cross-section of the tunnel lining, the compressibility ratio is defmed as

C = (E_{m}/E_{1}) (R/t) [(1 - v_{1}^{2})/((1 + v_{m}) (1 - 2 v_{m}))] (Equation A.2)

**Figure A.1 Stresses and Lining Geometry for Elastic Closed Form Models of Ground-Lining Interaction**

It should be pointed out that slightly different expressions for the fl exibility and compressibi 1 ity ratios have been used by others (e.g. Muir Wood, 1975; Einstein and Schwartz, 1979). As v_{m} approaches 0.5 in Eq. A.2, as would be the case for a fully saturated clay, the value of C approaches infinity. Einstein and Schwartz (1979) point out that this trend can be conceptually misleading, and have derived an alternative expression on the basis of slightly different assumptions.

Figure A.2 shows the maximum moment coefficient plotted as a function ofF pertaining to K_{o} = 0.5 and 2.0 for full and no slip conditions. The plots represent absolute values of the moment, which achieves a maximum at the crown, springline, and invert of the tunnel. The moment coefficient diminishes rapidly as F increases to about 20. Thereafter, there is little variation in moment as the relative stiffuess between ground and lining increases. The plots pertain to C = 0.4 and v_{m}= 0.4.

Because neither of these parameters has a significant influence on moment, the figure may be used as a good approximation of the relationship for other values of C and v_{m} generally encountered in practice.

The thrust coefficient does not vary significantly as a function ofF for values ofF greater than about 3. However, the thrust decreases substantially with increased C as shown in Figure A.3. This figure was developed for K_{o} = 0.5 and 2.0, F = 10, and v_{m} = 0.4 under full and noslip conditions. The highest thrust occurs generally in the crown and invert, with thrusts being more pronounced for no slip as opposed to full-slip conditions. The thrust can be affected significantly by v_{m}. Although not shown, the curves in Figure A.2 would be displaced upward for v_{m}>

**Figure A.2 - Maximum Moment Coefficient as a Function of the Flexibility Ratio**

**Figure A.3 - Thrust Coefficient as a Function of the Compressibility Ratio**

Figures A.2 and A.3 are instructive as indicators of the qualitative behavior of flexible tunnel linings. It should, however, be recognized that quantitative values for analysis of specific cases depend considerably on the value assigned to the at-rest earth pressure coefficient, K_{o}, which must generally be estimated on the basis of relatively crude characterizations of actual site conditions. In sandy soils of geologically recent origin with relatively high internal friction, K_{o} may approximate 0.5. In overconsolidated clays, K_{o} will often exceed 1.0. In rocks that have been subject to complex geological processes, K_{o} may be extremely variable. Additional comp I ications arise because the excavation process tends to relieve in-situ stresses adjacent to the tunnel lining. As a consequence, the lining may be subjected to a stress state significantly less than that based on the assumption of at-rest horizontal stresses and full overburden pressure.

A.3 Applications

The equations, on which Figures A.2 and A.3 are based, were developed for linear elastic linings. Concrete linings, however, are characterized by significant nonlinear stress-strain behavior. Structural failure of a concrete lining results from crushing on the compressive face, and the load bearing capacity of the lining may significantly exceed the structural bending capacity of the section.

Linear elastic models may be biased to a reI atively low assessment of the lining capacity because they tend to emphasize the bending capacity of the section.

The lining designer should recognize this bias. In Appendix B.2, the nonlinear response of a concrete lining is considered and compared with the response modeled by the linear elastic solutions.

There are many factors in addition to the effects of nonlinearity that the designer must consider. Concrete creep and the use of segmental linings may lead to an increase in the relative stiffness between the ground and lining. The relief of in-situ stresses during excavation may cause substantial reductions in pressure relative to those inferred by excavation loading. The actual ground loads may not be distributed continously along the lining, but may be concentrated at specific locations as would be the case for gravity loads in jointed rock and soil where significant loosening is permitted. Moreover, loads from shove jacks and contact grouting as well as those associated with future construction may be more critical than the loads from ground-lining interaction.

Careful evaluation of the many factors affecting lining response requires judgment. Linear elastic models supp I ement judgment. As discussed previously, the models are appropriate1y used when they bracket the limiting conditions of performance and point out trends in lining response as a result of variations of important parameters.

#### Appendix B.2 - Linear Response of Concrete Linings

As discussed in Appendix B.1, concrete linings are characterized by nonlinear stress-strain behavior so that linear elastic models may lead to results that are not consistent with actual performance. It is useful, therefore, to understand how linings are influenced by nonlinear characteristics. The moment-thrust diagram provides a means of comparing linear and nonlinear responses under similar conditions of loading and relative stiffness between the ground and concrete lining. This appendix provides a brief discussion of moment-thrust diagrams and summarizes analytical results showing the differences between lining performance modeled with linear and nonlinear concrete properties.

B.l Moment-Thrust Interaction Diagrams

When the thrust and moment around the lining have been calculated, it is necessary to evaluate these quantities in comparison with allowable values. Normally, it is only necessary to make this comparison at locations where one of the quantities is maximum or where there is an abrupt change in the lining section. Moment and thrust interact strongly, so it is customary to check these quantities together by using the moment thrust (M -T) interaction diagram to represent the allowable combination. The MT interaction diagram can be drawn for each section of the lining and depends only on the section dimensions and material properties.

One way to obtain a M-T interaction diagram is to use the procedure of the ACI Code (ACI Committee 318, 1983) in which the combinations of moment and thrust, which cause failure of the section under unconfined conditions, are computed and shown on a diagram in which thrust and moment are the axes. A typical M-T diagram for one section of a tunnel lining is shown in Figure B.l. This diagram may represent all the lining sections if they have constant dimensions and composition, or several such diagrams may be used to represent different lining sections.

To determine whether the section for which the M-T diagram in Figure B.1 is adequate, the moment and thrust combination obtained in the analysis should be plotted on the diagram as shown. The ACI Code procedure for constructing the diagram provides for capacity reduction factors as a safety measure to cover uncertainties in material properties, determination of section resistance, and the difference between concrete strength from cylinder tests and the structure. If the moment and thrust combination lies inside the diagram, the section is adequate. If it lies outside the diagram, the section is not adequate. The loads on the lining may be multiplied by a load factor to give the moment and thrust combination an additional margin of safety.

B.2 Linear and Nonlinear Response

Figure B.l shows the difference that would be obtained between linear and nonlinear analyses for a lining section composed of reinforced concrete. In the figure, the moment-thrust paths are plotted for two different conditions of relative stiffness between the ground and lining. The nonlinear and linear paths, which intersect the interaction diagram below the balance point, pertain to a flexibility ratio less than that for the paths that intersect above the balance point. Each path is the locus of moment and thrust combinations corresponding to a given type of loading. As discussed in Chapter 3 and Appendix A, the loading and attendant ground-lining interaction may be modeled by means of excavation, ov~rpressure, or gravity loading.

When linear analyses are performed, the material stressstrain response must follow a linear relationship even though the actual stresses carried by the 1 ining may be well above the analytical values. Linear analyses are usually used to design above ground structures, with the under standing that linear assumptions are conservative. The error resulting from using linear analysis for a tunnel lining will be more pronounced than for an above ground structure because the confmement and greater indeterminacy of the underground structure provide more opportunity for moment redistribution.

As the nonlinear moment-thrust path in Figure B.1 intersects the interaction diagram below the balance point, the concrete cracks and the eccentricity decreases resulting in a higher value of thrust (point 2) than would be obtained in the linear analysis (point 1). The section has additional capacity even after the moment-thrust path has reached the envelope, and the thrust continues to increase even though the moment capacity drops off (point 3). Above the balance point, the thrust capacity calculated by nonlinear analysis will be closer to that calculated by linear analysis, as evidenced by comparing the percentage difference between points 4 and 5 with that of points 1 and 3.

A key aspect of the lining response, which is shown by nonlinear analysis, is that the concrete tunnel lining does not fail by excessive moment. It fails by thrust which is affected indirectly by moment.

Figure B.2 helps illustrate the general conditions summarized in Figure B.l by means of a specific example. The figure shows the moment thrust interaction diagram for a 9-in. (230 mm)-thick concrete lining section. A one-foot length (305 mm)of a continuous lining with no reinforcing steel is considered. Also shown on the graph are moment thrust paths for the crown obtained from analyses of an l8-ft (5.5 m)-diameter circular lining with the same cross-section as that used to draw the interaction diagram. A uniform gravity load was applied across the tunnel diameter as shown in the figure. Nonlinear geometric and material properties of the lining were modeled, as described by Paul, et al. (1983). The analyses were performed using a beam-spring simulation in. which the ratio of the tangential to radial spring stiffness was one fourth. Analyses were performed with spring stiffness corresponding to moduli of the surrounding medium of 111,000 and 1,850,000 psi (770 and 12,800 MN/m2), representing soft and medium hard rock. The increased capacity associated with increased stiffness of the media illustrated by the nearly two-fold difference in maximum thrust for the two cases. When the moment and thrust are below the balance point, the thrust capacity from nonlinear analysis exceeds that from linear analysis by four times. When the moment-thrust paths intersect the M-T diagram above the balance point, the difference in maximum thrust between the linear and nonlinear analyses is only about 10 percent.

It should be emphasized that nonlinear analysis is subject to VIrtually all constraints that apply for linear models. As discussed in Appendix A, there are many additional factors the designer must consider, covering variations in material properties, ground loading, and construction methods. Nevertheless, nonlinear analysis provides insight regarding the manner in which the concrete lining deforms and shares load with the surrounding ground. The results ofnonlinear modeling may be especially useful for moment and thrust combinations below the balance point of the interaction diagram, where 1 inear eva1 uations tend to underestimate the load carrying capacity by a significant margin.

**Figure B.1 -- General Moment-Thrust Diagram for a Reinforced Concrete Lining with Linear and Nonlinear Moment-Thrust Paths.**

**Figure B.2 - Moment-Thrust Paths for an Unreinforced Concrete Lining in Rock.**

<< Previous | Contents | Next >> |