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Technical Manual for Design and Construction of Road Tunnels - Civil Elements
Chapter 10 - Tunnel Lining
This chapter covers considerations for the structural design, detailing and construction of tunnel linings for highway tunnels focusing on mined or bored tunnels. Tunnel linings are structural systems installed after excavation to provide ground support, to maintain the tunnel opening, to limit the inflow of ground water, to support appurtenances and to provide a base for the final finished exposed surface of the tunnel. Tunnel linings can be used for initial stabilization of the excavation, permanent ground support or a combination of both. The materials for tunnel linings covered in this chapter are cast-in-place concrete lining (Figure 10-1), precast segmental concrete lining (Figure 10-2), steel plate linings (Figure 10-3), and shotcrete lining (Figure 10-4). Uses, design procedures, detailing and installation are covered in subsequent sections of this chapter. The final finishes are not specifically addressed.
Figure 10-1 Cumberland Gap Tunnel
Cast-in-place concrete linings are generally installed some time after the initial ground support. Cast-in-place concrete linings are used in both soft ground and hard rock tunnels and can be constructed of either reinforced or plain concrete. Cast-in-place concrete linings can take on any geometric shape, with the shape being determined by the use, mining method and ground conditions.
Precast concrete linings are used as both initial and final ground support (Figure 10-2). Segments in the shape of circular arcs are precast and assembled inside the shield of a tunnel boring machine to form a ring. If necessary they can be used in a two pass system as only the initial ground support. Initial Support Segments for a two pass system are often lightly reinforced and rough cast. The second pass or final lining typically is cast-in-place concrete. Precast concrete linings can also be used in a one pass system where the segments provide both the initial and final ground support. One pass precast segmental concrete linings are cast to strict tolerances and are provided with gaskets and bolted together to reduce the inflow of water into the tunnel.
Figure 10-2 Precast Segmental Lining
Steel plate linings (liner plates) are a type of segmental construction where steel plates are fabricated into arcs that typically are assembled inside the shield of a tunnel boring machine to form a ring. The steel plate lining may form the initial and final ground support. The segments are provided with gaskets to limit the inflow of ground water into the tunnel. Steel plates are also used in lieu of lagging where steel ribs are used as the initial ground support. With the advent of precast concrete segments, liner plates are not used as much as previously.
Figure 10-3 Baltimore Metro Steel Plate Lining
As discussed in Chapter 9, shotcrete is a pneumatically applied concrete that is used frequently as an initial support but now with the advances in shotcrete technology permanent shotcrete lining is designed and constructed in conjunction with sequential excavation method ( SEM ) tunneling (Chapter 9). One of the first applications of final shotcrete lining in the United States is at Lehigh Tunnel No. 2 of Pennsylvania Turnpike. Shotcrete can take on a variety of compositions as discussed in Chapters 9 and 16. It can be applied over the exposed ground, reinforcing steel, welded wire fabric or lattice girders. It can be used in conjunction with rock bolts and dowels, it can contain steel or plastic fibers and it can be composed of a variety of mixes. It is applied in layers to achieve the desired thickness. Chapter 16 addresses using shotcrete for concrete lining repairs.
Figure 10-4 New Lehigh Tunnel on Pennsylvania Turnpike Constructed with Final Shotcrete Lining
Cross passages and refuge areas are usually mined by hand after the main tunnel is excavated. These areas, due to their unique shape and small areas are typically lined with cast-in-place concrete. There is insufficient quantity involved in the lining of these features to make prefabricated linings economic.
10.1.1 Load and Resistance factor Design (LRFD)
The design of tunnel linings, with the exception of steel tunnel lining plates, is not addressed in standard design codes. This chapter is intended to establish procedures for the design of tunnel linings utilizing the American Association of State Highways and Transportation Officials (AASHTO) LRFD Bridge Design Specifications, current edition.
LRFD is a design philosophy that takes into account the variability in the prediction of loads and the variability in the behavior of structural elements. It is an extension of the load factor design methodology that has been in use for a number of years. This chapter is intended to assist the designer in the application of the LRFD specifications to tunnel lining design and to provide for a uniform interpretation of the AASHTO LRFD specification as it applies to tunnel linings.
10.2 Design Considerations
10.2.1 Lining Stiffness and Deformation
Tunnel linings are structural systems, but differ from other structural systems in that their interaction with the surrounding ground is an integral aspect of their behavior, stability and overall load carrying capacity. The loss or lack of the support provided by the surrounding ground can lead to failure of the lining. The ability of the lining to deform under load is a function of the relative stiffnesses of the lining and the surrounding ground. Frequently, a tunnel lining is more flexible than the surrounding ground. This flexibility allows the lining to deform as the surrounding ground deforms during and after tunnel excavation. This deformation allows the surrounding ground to mobilize strength and stabilize. The tunnel lining deformation allows the moments in the tunnel lining to redistribute such that the main load inside the lining is thrust or axial load. The most efficient tunnel lining is one that has high flexibility and ductility.
A tunnel lining maintains its stability and load carrying capacity through contact with the surrounding ground. As load is applied to one portion of the lining, the lining begins to deform and in so doing, develops passive pressure along other portions of the lining. This passive pressure prevents the lining from buckling or collapsing. Ductility in the lining allows for the creation of "hinges" at points of high moment that relieve the moments so that the primary load action is axial force. This ductility is provided for in concrete by the formation of cracks in the concrete. Under reinforcing or no reinforcing help promote the initiation of the cracks. The joints in segmental concrete linings also provide ductility. In steel plate linings, the negligible bending stiffness of the steel plates and the inherent ductility of steel allow for the creation of similar hinges.
10.2.2 Constructability Issues
Each tunnel is unique. Ground conditions, tunneling means and methods, loading conditions, tunnel dimensions and construction materials all vary from tunnel to tunnel. Each tunnel must be assessed on its own merits to identify issues that should be considered during design such that construction is feasible. Some common elements that should be considered are as follows:
Materials: Selection of tunnel lining materials should be made to facilitate transportation and handling of the materials in the limited space inside a tunnel. Pieces should be small and easily handled. Piece lengths should be checked to ensure that they can negotiate the horizontal and vertical geometry of the tunnel. Materials should be nontoxic and nonflammable.
Details: Detailing should be performed to facilitate ease of construction. For example, sloping construction joints in cast-in-place concrete linings can eliminate the difficulty associated with building a bulkhead against an irregular excavated surface.
Procedures: Construction procedures should be specified that are appropriate for conditions encountered in the tunnel; conditions that are often moist or wet, sometimes even with flowing water. Allow means and methods that do not block off portions of the tunnel for significant periods of time. The entire length of the tunnel should be available as much as practical.
Tunnels are expensive and are constructed for long term use. Many existing tunnels in the United States have been in use for well over one hundred years with no end in sight to their service lives. Having a tunnel out of service for an extended period of time can result in great economic loss. As such, details and materials should be selected that can withstand the conditions encountered in underground structures. All structures, including tunnels require inspection, periodic maintenance and repair. Chapter 17 discusses tunnel inspection, maintenance, and rehabilitation. Nonetheless, detailing should be such that anticipated maintenance is simplified and long term durability is maximized.
Highway tunnels can also be exposed to extreme events such as fires resulting from incidents inside the tunnel. Tunnel lining design should consider the effects of a fire on the lining. The lining should be able to withstand the heat of the fire for some period of time without loss of structural integrity. The length of time required will be a function of the intensity of the anticipated fire and the response time for emergency personnel capable of fighting the fire. The tunnel lining should also sustain as little damage as possible so that the tunnel can go back into service as soon as possible. Protection from fire can be gained from concrete cover, tunnel finishes and the inclusion of plastic fibers in concrete mixes.
10.2.4 High Density Concrete
High density concrete is produced by using very finely ground cement and/or substituting various materials such as fly ash or blast furnace slag for cement. The cementitious content of high density concrete is very high. The high cement content makes handling difficult under ideal conditions. Complicated mixes with multiple admixtures and careful water monitoring are required to keep the concrete in a plastic state long enough to be placed in forms. High cement content will result in high heat of hydration. Proper curing of these materials is essential to produce a quality end product. Improper or incomplete curing can be the cause of severe cracking due to shrinkage. Shrinkage cracks can reduce the effectiveness of the product, affect its durability and potentially make it unusable.
High density concrete, however, can be beneficial in many tunnel applications. It can limit the inflow of water and provide significant protection against chemical attack. High density concrete has low heat conductivity which is beneficial in a fire. High density concrete should be used in conjunction with careful inspection and strict enforcement of specifications during construction.
10.2.5 Corrosion Protection
Corrosion is associated with steel products embedded in the concrete and otherwise used in tunnel applications. Ground water, ground chemicals, leaks, vehicular exhaust, dissimilar metals, deicing chemicals, wash water, detergents, iron eating bacteria and stray currents are all sources of corrosion in metals. Each of these and any other aspect that is unique to the tunnel under consideration must be evaluated during the design phase. Corrosion protection methods designed to combat the source of corrosion should be incorporated into the design.
Corrosion protection can take the form of coatings such as epoxies, powder coatings, paint or galvanizing. Insulation can be installed between dissimilar metals and sources of stray currents. High density concrete can provide protection for reinforcing steel. Coatings on concrete can minimize the infiltration of water, a component of almost all corrosion processes. Tunnel finishes can also protect the tunnel structural elements from attack by the various sources of corrosion.
Cathodic protection uses sacrificial material to protect the primary material from corrosion. In highly corrosive environments, an electrical current is induced in the materials to force corrosion to occur in the sacrificial material. These systems are highly effective when properly designed, installed and maintained. Sacrificial elements must be replaced and electrical supply equipment serviced regularly. Cathodic protection also requires a reliable long term source of electricity and adds to the maintenance and operation costs of the tunnel.
Increased concrete cover over reinforcing steel is an effective means of protecting reinforcing steel from corrosion. Increasing the concrete cover, however will also increase the thickness of the lining. The increased thickness will result in a larger excavation which will increase the overall cost of the tunnel. The use of increased concrete cover should be evaluated in terms of the overall cost of the tunnel compared to the benefit derived.
10.2.6 Lining Joints
Joints in linings are required to facilitate construction. Cast-in-place concrete requires construction joints. Construction joints can be sloped or formed. Segmental linings constructed from concrete or steel can have either bolted or unbolted joints. Unbolted joints are used in both gasketed and ungasketed concrete segments. Steel liner plates are bolted. More detailed information on the advantages and disadvantages of joints is provided in subsequent sections of this chapter.
Joints in linings also provide relief from stresses induced by movements due to temperature changes. Cast-in-place linings should have contraction joints every 30 feet and expansion joints every 120 feet. Expansion joints should also be used where cut and cover portions of the tunnel transition to the mined portion. Segmental concrete linings do not require contraction joints and require expansion joints only at the cut and cover interface.
10.3 Structural Design
Structural design will be governed by the latest AASHTO LRFD Bridge Design Specifications. The AASHTO specifications do not cover structural plain concrete which is frequently used in tunnel lining construction. This chapter will provide design procedures based on the AASHTO specifications for structural plain concrete. These procedures can be found in section 10.4 Cast-in-Place Concrete.
The loads to be considered in the design of structures along with how to combine the loads are given in Section 3 of the LRFD specifications. Section 3 of the LRFD specification divides loads into two categories: Permanent Loads and Transient Loads. Paragraph 3.3.2 "Load and Load Designation" of the LRFD specifications defines the following permanent loads that are applicable to the design of mined tunnel linings:
DC = Dead Load: This load comprises the self weight of the structural components as well as the loads associated with nonstructural attachments. Nonstructural attachments can be signs, lighting fixtures, signals, architectural finishes, waterproofing, etc. Typical unit weights for common building materials are given in Table 3.5.1-1 of the AASHTO LRFD specifications. Actual weights for other items should be calculated based on their composition and configuration.
DW = Dead Load: This load comprises the self weight of wearing surfaces and utilities. Utilities in tunnels can include power lines, drainage pipes, communication lines, water supply lines, etc. Wearing surfaces can be asphalt or concrete. Dead loads of wearing surfaces and utilities should be calculated based on the actual size and configuration of these items.
EH = Horizontal Earth Pressure Load. The information required to calculate this load are derived by the geotechnical data developed during the subsurface investigation program. The methods used in determining earth loads on mined tunnel linings are described in Chapters 6 and 7 of this manual.
ES = Earth surcharge load. This is the vertical earth load due to fill over the structure that was placed above the original ground line. It is recommended that a minimum surcharge load of 400 psf be used in the design of tunnels. If there is a potential for future development adjacent to the tunnel structure, the surcharge from the actual development should be used in the design of the structure. In lieu of a well defined loading, it is recommended that a minimum value of 1000 psf be used when future development is a possibility.
EV = Vertical earth pressure. The methods used in determining earth loads on mined tunnel linings are described in Chapters 6 and 7 of this manual.
Paragraph 3.3.2 "Load and Load Designation" of the LRFD specifications defines the following transient loads that are applicable to the design of mined tunnel linings:
CR = Creep.
CT = Vehicular Collision Force: This load would be applied to individual components of the tunnel structure that could be damaged by vehicular collision. Typically, tunnel linings are protected by redirecting barriers so that this load need be considered only under usual circumstances. It is preferable to detail tunnel structural components and appurtenances so that they are not subject to damage from vehicular impact.
EQ = Earthquake. This load should be applied to the tunnel lining as appropriate for the seismic zone for the tunnel. Other extreme event loadings such as explosive blast should be considered. The scope of this manual does not include the calculation of or design for seismic and blast loads, however, the designer must be aware that extereme event loads should be accounted for in the design of the tunnel lining.
IM = Vehicle dynamic load allowance: This load is applied to the roadway slabs of mined tunnels. This load can also be transmitted to a tunnel lining through the ground surface when the tunnel is under a highway, railroad or runway. Usually a mined tunnel is too far below the surface to have this transmitted to the structure. However, this load may be a consideration near the interface between the cut and cover approaches and the mined tunnel section. An equation for the calculation of this load is given in paragraph 18.104.22.168 of the AASHTO LRFD specifications.
LL = Vehicular Live Load: This load is applied to the roadway slabs of mined tunnels. This load can also be transmitted to a tunnel lining through the ground surface when the tunnel is under a highway, railroad or runway. Usually a mined tunnel is too far below the surface to have this loads from the surface transmitted to the structure however, this load may be a consideration near the interface between the cut and cover approaches and the mined tunnel section.. Guidance for the distribution of live loads to buried structures can be found in paragraphs 3.6.1 of the AASHTO LRFD specifications.
LS = Live Load Surcharge: This load is applied to the lining of tunnels that are constructed under other roadways, rail lines, runways or other facilities that carry moving vehicles. This is a uniformly distributed load that simulates the distribution of wheel loads through the earth fill. Usually a mined tunnel is too far below the surface to have this loads from the surface transmitted to the structure, however, this load may be a consideration near the interface between the cut and cover approaches and the mined tunnel section.
PL = Pedestrian Live load. Pedestrian are typically not permitted in highway tunnels, however, there are areas where maintenance and inspection personnel will need access. Areas such as ventilation ducts when transverse ventilation is used, plenums above false ceilings, and safety walks. These loads are transmitted to the lining through the supporting members for the described features.
SH = Shrinkage. Cut and cover tunnel structural elements usually are relatively massive. As such, shrinkage can be a problem. This load should be accounted for in the design or the structure should be detailed to minimize or eliminate it.
TU = Uniform Temperature. This load is used primarily to size expansion joints in the structure. If movement is permitted at the expansion joints, no additional loading need be applied to the structure. Since the structure is very stiff in the primary direction of thermal movement, the effects of the friction force resulting from thermal movement can be neglected in the design.
WA = Water load. This load represents the hydrostatic pressure expected outside the tunnel structure. Mined tunnels are usually detailed to be watertight without provisions for relieving the hydrostatic pressure. As such, the tunnel lining is subject to hydrostatic pressure. Hydrostatic pressure acts normal to the surface of the tunnel. It should be assumed that water will develop full hydrostatic pressure on the tunnel when no relief mechanism is used. The calculation of this load should take into account the specific gravity of the groundwater which can be saline near salt water. Both maximum and minimum hydrostatic loads should be used for structural calculations. For the purpose of design, the hydrostatic pressures assumed to be applied to underground structures should ignore pore pressure relief obtained by any seepage into the structures unless an appropriately designed pressure relief system is installed and maintained. Two groundwater levels should be considered: normal (observed maximum groundwater level) and extreme, 3 ft (1 m) above the 200-year flood level. The buoyancy force should be carefully evaluated to ensure that the applied dead load effect is larger than the applied buoyancy effect. Calculations for buoyancy should be based on minimum characteristic material densities and maximum water density. The total uplift force is equal to the weight of water displaced. Friction effects (the theoretical force required to dislodge the wedge of material over the tunnel) of overlying natural materials and backfill should not be taken into account, however the weight of soil and water over the tunnel should be used to calculate the resisting forces. When a relief system is included, the functioning of the relief system is evaluated to determine the hydrostatic pressure to be applied to the tunnel.
Some of the loads shown in paragraph 3.3.2 of the LRFD specifications are not shown above because they are not applicable to the design of mined highway tunnels as described below.
DD = Downdrag:This load comprises the vertical force applied to the exterior of the lining that can result from the subsidence of the surrounding soil due to the subsidence of the in-situ soil below the bottom of the tunnel. This load would not apply to mined tunnels since it requires subsidence or settlement of the material below the bottom of the structure to engage the downdrag force of the lining. For the typical highway tunnel, the overall weight of the structure is usually less than the soil it is replacing. As such, unless backfill in excess of the original ground elevation is paced over the tunnel or a structure is constructed over the tunnel, settlement will not be an issue for mined tunnels.
BR = Vehicular Breaking Force: This load would be applied only under special conditions where the detailing of the structure requires consideration of this load. Under typical designs, this force is resisted by the mass of the roadway slab and need not be considered in design.
CE = Vehicular centrifugal force: This load would be applied only under special conditions where the detailing of the structure requires consideration of this load. Under typical designs, this force is resisted by the mass of the roadway slab and need not be considered in design.
CV = Vessel Collision Force is not applicable since it would only be applied to immersed tube tunnels. Immersed tube tunnels are a specialized form of cut and cover tunnel and are covered separately in Chapter 12 of this manual.
EL = Accumulated locked-in force effects resulting from the construction process including secondary forces from post tensioning.
FR = Friction. As stated above, the structure is very stiff in the direction of thermal movement. Thermal movement is the source of the friction force. In a typical tunnel, the effects of friction can be neglected.
IC = Ice load. Since the tunnel is not subjected to stream flow nor exposed to the weather in a manner that could result in an accumulation of ice, this load is not used in cut and cover tunnel design.
SE = Settlement. For the typical highway tunnel, the overall weight of the structure is usually less than the soil it is replacing. As such, unless backfill in excess of the original ground elevation is paced over the tunnel or a structure is constructed over the tunnel, settlement will not be an issue for cut and cover tunnels. If settlement is anticipated due to poor subsurface conditions or due to the addition of load onto the structure or changing ground conditions along the length of the tunnel, it is recommended that a deep foundation (piles or drilled shafts) be used to support the structure. Ground settlements are difficult to predict and are best eliminated by the use of deep foundations.
TG = Temperature Gradient. This load should be examined on case by case basis depending on the local climate and seasonal variations in average temperatures. Typically due to the relative thin members used in tunnel linings, this load is not used. Paragraph 4.6.6 of the AASHTO LRFD specifications provides guidance on calculating this load. Note that paragraph C3.12.3 of the AASHTO LRFD Specifications allows the use of engineering judgment to determine if this load need be considered in the design of the structure.
WL = Wind on live load. The tunnel structure is not exposed to the environment, so it will not be subjected to wind loads.
WS = Wind load on structure. The tunnel structure is not exposed to the environment, so it will not be subjected to wind loads.
Section 3 of the LRFD specifications provides guidance on the methods to be used in the computations of these loads. The design example (Appendix G) shows the calculations involved in computing these loads.
10.3.2 Load Combinations
The AASHTO Specification defines four limit states; service, fatigue and fracture, strength, and extreme event). Each of these limits states contain several load combinations. These limit states and load combinations were developed for loadings that are typically encountered by highway bridges. Many of the loadings that bridges are subjected to are not applicable to tunnel linings. Loads such as wind, stream flow, vessel impact and fatigue do not occur in mined tunnels. The unique conditions under which tunnels operate allow for eliminating many of the loading conditions used for bridges. Tunnels should be designed for the following load combinations.
The loads described above should be factored and combined in accordance with the LRFD specification and applied to the tunnel lining. These load combinations are given in Table 3.4.1-1 of the AASHTO specifications. The recommended load cases for the design of linings for mined highway tunnels are given in Table 10-1.
|Load Comb. Limit State||DC||DW||EH*EV#||ES||LL, IM, LS, CT, PL||WA||TU, CR, SH||TG|
|Extreme Event I||1.25||0.90||1.50||0.65||1.35||0.90||1.50||0.75||γiEQP+||1.00||N/A||N/A||N/A|
* The load factors shown are for at-rest earth pressure. At-rest earth pressure should be used for all conditions of design of cut and cover tunnel structures.
# The load factors shown are for rigid frames. All cut and cover tunnel structures are considered rigid frames.
+ This load factor is determined on a project specific basis (refer to Chapter 13 Seismic Considerations).
When developing the loads to be applied to the structure, each possible combination of load factors should be developed.
10.3.3 Design Criteria
Historically there have been three basic methods used in the design of structures:
- Service load or allowable stress design which treats each load on the structure equally in terms of its probability of occurrence at the stated value. The factor of safety for this method is built into the material's ability to withstand the loading.
- Load factor design accounts for the potential variability of loads by applying varying load factors to each load type. The resistance of the maximum capacity of the structural member is reduced by a strength reduction factor and the calculated resistance of the structural member must equal or exceed the applied load.
- Load and resistance factor design takes into account the statistical variation of both the strength of the structural member and of the magnitude of the applied loads.
- The fundamental LRFD equation can be found in paragraph 22.214.171.124 of the AASHTO specification. This equation is:
|ΣηiγiQi ≤ fRn = Rr||10-1|
(AASHTO Equation 126.96.36.199-1)
In this equation, η is a load modifier relating to the ductility, redundancy and operation importance of the feature being designed. The load modifier ηi is comprised of three components;
|ηD =||a factor relating to ductility = 1.0 for tunnel linings constructed with conventional details and designed in accordance with the AASHTO LRFD specification.|
|ηR =||a factor relating to redundancy = 1.0 for mined tunnel linings.|
|ηI =||a factor relating to the importance of the structure = 1.05 for tunnel design. Tunnels usually are important major links in regional transportation systems. The loss of a tunnel will usually cause major disruption to the flow of traffic, hence the high importance factor.|
γi is a load factor applied to the force effects (Qi) acting on the member being designed. Values for γi can be found in Table 10.1 above.
Rr is the calculated factored resistance of the member or connection.
f is a resistance factor applied to the nominal resistance of the member (Rn) being designed. The resistance factors are given in the AASHTO LRFD specifications for each material in the section that covers the specific material. Specifically, Section 5 of the AASHTO LRFD specifications covers Concrete Structures and in general, the resistance factors to be used in concrete design can be found there. These values are as follow.
For Reinforced Concrete Linings:
f = 0.90 for flexure
f = 0.90 for shear
f = 0.70 for bearing on concrete
Since tunnel linings will experience axial loads, the resistance factor for compression must be defined. The value of f for compression can be found in Section 188.8.131.52.1 of the AASHTO LRFD specification as:
f = 0.75 for axial compression
Structural steel is covered in Section 6 of the AASHTO LRFD specification. Paragraph 184.108.40.206 gives the following values for steel resistance factors:
For Structural Steel Members:
ff = 1.00 for flexure
fv = 1.00 for shear
fc = 0.90 for axial compression for plain steel and composite members
Chapter 12 of the AASHTO specifications addresses the design of tunnel linings constructed from steel lining plate. Table 12.5.5-1 provides the following additional resistance factors to be used in the design of steel lining plate:
f = 1.00 for minimum wall area and buckling
f = 1.00 for minimum longitudinal seam strength
For Plain Concrete Members: Un-reinforced concrete is also referred to as plain concrete. The AASHTO provisions do not address plain concrete. The following design procedures should be followed for structural plain concrete.
Calculate the moment capacity on the compression face of the lining as follows:
|fMnC = f0.85 fc'S||10-2|
MnC = The nominal resistance of the compression face of the concrete
f = 0.55 for plain concrete
fc' = 28 day compressive strength of the concrete
S = The section modulus of the lining section based on the gross uncracked section
Calculate the moment capacity on the tension face of the lining as follows:
MnT = The nominal resistance of the tension face of the concrete
f = 0.55 for plain concrete
fc' = 28 day compressive strength of the concrete
S = The section modulus of the lining section
Calculate the compressive strength of the lining as follows:
|fPC = f0.6fc'A||10-4|
PC = The nominal resistance of lining in compression
f = 0.55 for plain concrete
fc' = 28 day compressive strength of the concrete
A = The cross sectional area of the lining section
Check the compression face as follows:
|QA/fPC + QM/fMnC λ 1||10-5|
QA = The axial load force effect modified by the appropriate factors
QM = The moment force effect modified by the appropriate factors
Calculate the tension strength of the lining as follows:
|fPT = 5f (fc')1/2||10-6|
PT = The nominal resistance of lining in tension
f = 0.55 for plain concrete
fc' = 28 day compressive strength of the concrete
Check the tension face as follows:
|QM/S - QA/A λ fPT||10-7|
Where the values of the variables are described above.
The shear strength of the lining is calculated as follows:
|fVn = f1.33(fc')1/2bwh||10-8|
Vn = The nominal resistance of lining in shear
f = 0.55 for plain concrete
fc' = 28 day compressive strength of the concrete
bw = the length if tunnel lining under design
h = the design thickness of the tunnel lining
This design method is adapted for LRFD from the provisions for structural plain concrete from the American Concrete Institute's Building Requirements for Structural Concrete ( ACI 318).
10.3.4 Structural Analysis
Structural analysis of tunnel linings has been a subject of numerous papers and theories. Great disparity of opinion exists on the accuracy and usefulness of these analyses. However, some rational method must be adopted to determine a lining's ability to maintain the excavated opening of a tunnel. Some widely accepted methods are described in this section.
Beam Spring Models A general purpose structural analysis program can be used to model the soil structure interaction. This method is known as the beam spring model. The computer model is constructed by placing a joint or node at points along the centroid of the lining. These nodes are joined by straight beam members that approximate the lining shape by a series of chords. When constructing this type of model, the chord lengths should be approximately the same as the lining thickness for the radii that can be expected in highway tunnels. Chord members that are too long can produce fictitious moments and chord members that are too short can result in computational difficulties because of the very small angles subtended by short members. A subtended angle dimension of approximately 60/R, where R is the radius of the tunnel in feet, will generally produce acceptable results. Properties such as cross sectional area and moment of inertia should be entered to accurately depict the real behavior of the lining. Since the compressive forces are generally large enough to have compression over the entire thickness of the lining, the area and moment of inertia are calculated using the gross, uncracked dimensions of the lining. In rock tunnels, overbreak will result in a lining thickness larger than the design thickness. The design thickness is used in the analysis. This type of model is useful in analyzing all geometric shapes.
The surrounding ground is modeled by placing a spring support at each joint. Springs can be placed in the radial and tangential directions. The tangential springs offer little value in the analysis and an unnecessary complication to the model. The numerical value of the spring constant at each support is calculated from the modulus of subgrade reaction of the surrounding ground multiplied by the tributary length of lining on each side of the spring. Many ground conditions can be encountered within the length of a single tunnel. Parametric studies that vary the ground conditions and the spring constants should be performed to determine the worst case scenario for the lining.
Loads are applied to the model and the displacement at each joint is checked. For joints that move away from the center of the tunnel into the ground, the spring is left active. When the joint displacement is toward the center of the tunnel, the spring is removed or made inactive. This process in repeated until all displacements match the spring condition (active or inactive) at that joint. Once the model converges, the moments, thrusts and shears are used to design the lining.
If the model reveals that the lining is beyond its capacity, making the lining thicker or stiffer will not alleviate the problem. In fact, stiffening the lining will cause it to attract more moment and it will likely continue to fail. The lining must be made to be more flexible. This can be accomplished by making the lining thinner, which may not work. The primary load action on the lining is axial load or thrust. If the lining is close to its capacity under this load action, then thinning will not work. Modeling lining flexibility such that the moments are relieved may show the lining to be adequate. This is what happens in reality. One way to model this phenomenon is to install full or partial hinges in the lining at points of theoretical high moment. The hinge can be modeled to accept as much moment as the lining can support or it can be modeled as a full hinge with no moment capacity. In reality, the lining is performing somewhere in between these two extremes. Analyzing both conditions will bracket the lining behavior and provide a reasonable assurance that the lining can support the loads.
Three Dimensional Models The model described above is usually a two dimensional model that represents a single foot along the length of the tunnel. More sophisticated models are required when large penetrations of the lining or intersecting tunnels are being analyzed. To model these conditions, a three dimensional finite element model is used. The model is constructed in a similar manner to the two dimensional model, with finite elements used to connect the nodes and create the three dimensional model. The modeling parameters described above hold true for this type of model also. The model should extend a minimum of one tunnel diameter beyond the feature being investigated on each side of the feature.
It has been argued that this model does not account for the nonlinearity of the surrounding ground, particularly in soft ground, nor does it account for the variation of ground movement with time. Careful development of loading diagrams and spring constants for this model can bracket the actual behavior of the surrounding ground. This will provide results that are comparable to more sophisticated analysis methods. It should be noted that this method of analysis typically over estimates the bending moment in the lining.
Empirical Method for Soft Ground For circular tunnels in soft ground, the validity of the beam spring model has been highly criticized. The beam spring model described above assumes the soil to be a homogenous elastic material when in fact it is often non-homogenous and the behavior is plastic rather than elastic. Plastic deformations of the soil take place and the lining "goes along for the ride", that is, the stiffness of the lining is incapable of resisting the soil deformations. Since the lining is typically more flexible than the surrounding soil, it distorts as the soil displaces and the lining's flexibility allows it to shed moments to the point where it is acting almost entirely in compression. Since the lining is not completely flexible, some residual moment remains in the lining. This moment is accounted for by assigning an arbitrary change in radius and calculating the theoretical moment resulting from this change in radius. Using this method, the thrust in the tunnel lining is calculated by the formula:
|T = wR||10-9|
T = the thrust in the tunnel lining
w = the earth pressure at the spring line of the tunnel due to all load sources
R = the radius of the tunnel
The percentage of radius change to be used is a function of the type of soil. Values for this percentage estimated by Birger Schmidt are shown in Table 10-2.
|Soil Type||ΔR/R - Range|
|Stiff to Hard Clays||0.15 - 0.40%|
|Soft Clays or Silts||0.25 - 0.75%|
|Dense or Cohesive Soils, Most Residual Soils||0.05 - 0.25%|
|Loose Sands||0.10 - 0.35%|
1. Add 0.1 to 0.3 percent for tunnels in compressed air, depending on air pressure.
2. Add appropriate distortion for effects such as passing neighbor tunnel.
3. Values assume reasonable care in construction, and standard excavation and lining methods.
The resulting bending moment in the lining is calculated using the following formula:
|M = 3EI/R x ΔR/R||10-10|
M = the calculated bending moment
R = radius to the centroid of the lining
ΔR = tunnel radius change
E = modulus of elasticity of the lining material
I = effective moment of inertia of the lining section
The effective moment of inertia can be calculated for precast segmental linings using the following formula:
|Ie = Ij + I(4/n)2||10-11|
Ie = The effective moment of inertia
Ij = The joint moment of inertia (conservative taken as zero)
I = The moment of inertia of the gross lining section
n = The number of joints in the lining ring
This formula was developed by Muir Wood
The moment of inertia for the uncracked section should be used for cast-in-place concrete linings.
This method should be used in conjunction with any other analysis for round tunnels in soft ground as verification. The method described above can be used for both concrete and steel segmental linings. It is recommended that steel lining plate also be checked using the provisions of Section 12.7 of the AASHTO specifications for wall resistance and resistance to buckling.
Numerical Methods Commercial software is also available to model both the lining and the surrounding ground as a continuum utilizing a three dimensional finite element or finite difference approach. FLAC3D is a finite difference based continuum analysis program, where the domain (ground) is assumed to be a homogeneous media. The structural elements (beam or shell elements) can be used to model the tunnel lining. Using interface elements between the lining elements and the surrounding ground, rock-lining interaction including slip can be simulated.
If the ground contains predominant weak planes and those are continuous and oriented unfavorably to the excavation, then the analysis should consider incorporating specific characteristics of these weak planes. In this case, mechanical stiffness (force/displacement characteristics) of the discontinuities may be much different from those of intact rock. Then, a discrete element method (DEM) can be considered to solve this type of problem. 3DEC is a commercially available program for this type of analysis. Unlike continuum analysis, the DEM permits a large deformation and finite strain analysis of an ensemble of deformable (or rigid) bodies (intact rock blocks) which interact through deformable, frictional contacts (rock joints).
It is greatly task dependent whether a continuum (FLAC3D) or discrete analysis (3DEC) is adequate. If the ground is soil, the FLAC3D is adequate. If the ground is jointed rockmass and the joints are predominant in rock-lining interaction, 3DEC should be utilized. These programs can be used to calibrate and verify beam spring models, and vice versa.
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