ACI 533.5R-2020 Guide for Precast Concrete Tunnel Segments.
6.1.2 Beam-spring method—Using the beam-spring (also known as bedded beam) method endorsed by AASHTO DCRT-1, JSCE 2007, and ÖVBB 2011, the lining can be modeled in the cross-sectional plane as a series of beam elements that span between the longitudinal joints of the segments. As shown in Fig. 6.1.2(a), the interaction between the ground and the lining is modeled in two-dimensional domain using translational springs in the radial and tangen- tial directions. When modeling in three-dimensional domain, interaction of lining with the ground in longitudinal direc- tion of the tunnel is also modeled by longitudinal springs accordingly. Because the lining and ground are represented by a series of beams and springs, this method is referred to as the beam-spring or bedded-beam method. In the U.S. tunnel industry, the stifness of the springs is generally calculated using formulas recommended by USACE EM 1110-2- 2901 (Eq. (6.1.1a)). Furthermore, various two-dimensional approaches are used to evaluate efects of the segment joints, including solid ring models with fully bending rigidity, solid ring models with reduced bending rigidity (Muir Wood 1975), ring models with multiple hinged joints, and ring models with rotational springs. However, two-dimensional models cannot be used to represent circumferential joints or staggered arrangement of segments between rings.
6.1.3 Finite element method and discrete element method simulations—In soft ground, loose rock, and partially homo- geneous solid rock, ÖVBB 2011 and AFTES-WG7 recom- mend using FEM and fnite diference method (FDM) to calculate the forces in the tunnel lining. The discrete element method (DEM), as shown in Fig. 6.1.3a, is generally consid- ered more appropriate for tunnels in fractured rock. Recom- mended engineering properties for analysis of segmental lining in the rock formations includes properties of intact rock such as unit weight; modulus of elasticity; unconfned compressive strength (UCS); internal friction angle; tensile strength; and properties of discontinuities such as joint spacing, joint apparent dip direction, and joint apparent dip. Other required discontinuities parameters for a DEM modeling, as shown in Table 6.1.3 for a typical tunnel boring machine (TBM)-bored tunnel in rock, includes peak joint friction angle, peak joint cohesion, residual joint friction angle, residual joint cohesion, joint normal stifness, joint shear stifness, E int /E mass , geological strength index (GSI), and M i , which is a material constant for the intact rock based on Hoek-Brown failure criterion (Hoek and Brown 2018). In rock tunneling, a two-dimensional approach is generally sufcient for continuous linear structures that do not contain sudden changes in cross-sectional geometry or high concen Fig. 6.1.3a—DEM model and developed internal forces along the lining perimeter as a result of DEM analysis on large-diameter tunnel excavation in fractured rock: (a) axial forces; and (b) bending moments.ACI 533.5R pdf download.