RC Lining Design Calculator – Schleiss Method for Pressure Tunnels and Shafts

Schleiss introduced a comprehensive mathematical framework for the interaction between tunnel lining and surrounding rock mass in pressure tunnels and shafts in his PhD thesis (Schleiss, 1986). Building on this foundation, he went on in the early 1990s -while serving as Head of the Hydraulic Structures Section at Elektrowatt Engineering Ltd. (now AFRY) in Zurich - to develop one of the most widely recognised methods for the design of reinforced concrete (RC) linings for pressure tunnels and shafts. He subsequently presented and refined this method through a series of publications and practical engineering applications. His most frequently cited paper on the subject is linked below:

(https://www.researchgate.net/profile/Anton-Schleiss/publication/37445734_Design_of_reinforced_concrete_linings_of_pressure_tunnels_and_shafts/links/54f6cb990cf21d8b8a5d9696/Design-of-reinforced-concrete-linings-of-pressure-tunnels-and-shafts.pdf).

In this approach, reinforced concrete linings in pressure tunnels and shafts are treated as permeable systems. Under internal pressure, cracks are assumed to develop progressively in stages, increasing in number while decreasing in spacing. These cracks permit seepage, which redistributes pressure between the lining and the surrounding rock mass. The design therefore needs to consider this coupled hydraulic–mechanical behaviour to ensure that crack widths, reinforcement stresses, and seepage remain within acceptable limits.

There are two practical critiques of the Schleiss method (please refer to Prof. Anton Schleiss’s response to these critiques at the end of this post). First, the method assumes that successive cracking can occur, meaning that second, third, and subsequent cracks may form between earlier cracks due to bond action between the reinforcement and concrete. In this mechanism, the reinforcement transfers stress back into the concrete between previously formed cracks, potentially causing the tensile stress in the concrete to exceed its tensile strength and initiate new cracks. However, this assumption does not fully account for the significant reduction in lining stiffness after each cracking event. As the lining becomes less stiff, it is less likely to sustain internal pressure up to the same level reached before cracking, and a greater share of the load is transferred to the surrounding rock mass. As a result, further opening of existing cracks may be more likely than the formation of new cracks. Therefore, Schleiss method can underestimate the crack width which an important design item for RC lining design.

Second, the Schleiss method assesses cracking by comparing the induced tensile stress with the tensile strength of concrete. However, the tensile strength of concrete is well known to be highly variable and dependent on the concrete compressive strength at the time of loading. Therefore, estimating tensile strength solely from compressive strength may either overestimate or underestimate the actual tensile strength at the time of loading.

Despite above legitimate shortcomings, Schleiss method is common in industry then I developed the following calculation suite which includes design tool for pressure tunnels above groundwater level, pressure tunnel below groundwater and vertical shafts.

Response provided by Prof. Anton Schleiss to above mentioned comments:  

“

1)        My method for the design of reinforced concrete linings for pressure tunnels and shafts is described in detail in Chapter 7.4 of my book “Design of Pressure Tunnels and Pressure Shafts for Hydropower Plants” giving besides detailed calculation procedure also structural recommendations. The design method is based on a radial-symmetrical behavior of the lining-rock system assuming homogeneous and isotropic rock mass characteristics regarding deformability and permeability. Under this assumption the uncracked concrete lining transmits, besides the seepage pressure due to its permeability, only a mechanical compressive stress to the rock mass which reduces proportionally with the thickness of the lining. Thus, no bending or shear stresses exist which are influenced by the lining stiffness, and which would occur only in the case of anisotropic rock mass behavior. As soon as the lining is cracked, a significant part of the internal water pressure is transmitted through the cracks outside of the lining. Again, the concrete lining between the cracks transmits only a mechanical compressive stress which is independent from the crack spacing and small compared to the acting seepage water pressure outside of the cracked concrete lining. Thus, the lining-rock system considered in my method isn’t based on the lining stiffness. Furthermore, regarding the load sharing between the reinforced concrete lining and the rock mass, the reinforcement is smeared to an equivalent thickness of a steel liner. However, when the distribution of the cracks in the lining is determined, the stress pattern in the reinforcement is assumed to be parabolic based on experimental tests with reinforced beams under tension. Thus, smearing the steel liner is assuming even a smaller stiffness as the reinforcement itself in the concrete liner would have. Therefore, the crack width is rather underestimated. It may be concluded that my method doesn’t consider any stiffness of the reinforced cracked concrete lining since the lining transmits only compressive stresses to the rock mass. After lining cracking these compressive stresses, independent from the crack spacing, are significantly smaller than the seepage pressure transmitted through the cracks to the rock mass. 

2)        Regarding the tensile strength of the concrete, it is clear that it is influenced by the concrete quality characterized by the concrete compressive strength. For practical application the tensile strength of pressure tunnel concrete linings is typically in the range of 1 to 2 N/mm². However, when estimating the width and spacing of cracks width under tension loading due to rock mass deformation, my method considers the compressive strength of the concrete which directly influences the steel-concrete bond stress and therefore the crack width and spacing.

Note 1: The percentage of reinforcement in the case of very thick concrete linings (> 0.50 m) has to be based only on the concrete part which is directly influenced by the ring reinforcement layers. The concrete outside can be considered as filling concrete. 

Note 2: It should be noted that achieving the crack distribution predicted by my design method requires careful initial filling of the waterway, as described in Chapter 11 of the above-mentioned book (https://www.researchgate.net/publication/402751529_Design_of_Pressure_Tunnels_and_Pressure_Shafts_for_Hydropower_Plants).”