The research team, led by Shi-ju Ma with colleagues including Kai-hang Han, Xiao-hua Bao, Jiann-Wen Woody Ju, Kai-rong Hong, and Xiang-sheng Chen, focused on a single deceptively simple parameter: the effective ratio of transverse bending stiffness, conventionally written as η. In the design method most widely used for shield tunnels—the equivalent stiffness ring model, sometimes called the modified conventional method—engineers pretend the segmental ring is a continuous hoop, but assign it a uniformly reduced bending stiffness of ηEI, where EI is the stiffness of the unjointed cross-section and η is a number between zero and one. Choosing that number is not an academic exercise. Set η too high and the calculation overestimates the structure’s capacity, producing designs so conservative that steel and concrete are wasted and project costs balloon. Set it too low and the tunnel lining may be under-designed, leaving too little reserve against the crushing pressures of water-saturated ground.

To pin down realistic values of η, the team built three physical models at a geometric scale of 1:30, based on the prototype of the Jinan Yellow River Tunnel in China. That real tunnel has an outer diameter of 15.2 meters, an inner diameter of 13.9 meters, segment thickness of 0.65 meters, and segment width of 2.0 meters, with each ring assembled in a so-called 9+1 configuration: nine standard B-blocks plus one wedge-shaped key segment, the K-block. The model rings shrank to an outer diameter of just over half a meter, with walls barely two centimeters thick. Getting the mechanics right at that scale demanded more than shrinking the geometry. Using dimensional analysis and the Buckingham π theorem, the researchers derived the full set of similarity criteria linking geometry, elastic modulus, stress, strain, and applied loads between model and prototype, then hunted for materials whose properties would satisfy those criteria simultaneously.

The material selection is one of the study’s most technically interesting aspects. The prototype segments are C60 reinforced concrete with an elastic modulus of 36 gigapascals; the prototype bolts are high-strength steel at 206 gigapascals. After comparing nine candidate polymers, the team chose high-density polyethylene (HDPE) for the model segments, with an elastic modulus of 1.175 gigapascals, which lands almost exactly on the 30-fold modulus reduction demanded by the similarity relations. The bolts posed a subtler problem, because joint behavior depends not on bolt stiffness alone but on the ratio of segment stiffness to the combined stiffness of all bolts crossing a joint. By enforcing that ratio—rather than the raw bolt properties—the researchers preserved what they call joint stiffness similarity. Trial calculations showed that polybutylene terephthalate (PBT) bolts of 4.0 millimeter diameter best replicated the transverse segment joints, while HDPE bolts of 3.0 millimeter diameter reproduced the longitudinal ring joints, with similarity indices held above 80 percent for bending, tension-compression, and shear.

Three tunnel configurations were tested. The first was a homogeneous ring: three continuous, unjointed rings bolted together longitudinally, representing the theoretical ideal in which joints contribute no stiffness reduction. The second used straight-jointed assembly, in which the segment joints line up continuously along the tunnel’s length. The third used staggered-jointed assembly, the arrangement actually favored in practice, where the joints of adjacent rings are offset by 64.29 degrees—five bolt-hole positions—so that each ring’s joints are bridged by the intact shoulders of its neighbors. Each model consisted of three rings, with the central ring instrumented, because the researchers wanted the measurements to capture both the transverse joint effects within a ring and the longitudinal interaction between neighboring rings that gives staggered joints their advantage.

Loading was applied through a self-developed rig in the spirit of the load-structure method: a vertical point load delivered to the tunnel crown via a steel cable and pulley guidance system, which the authors note mitigates the influence of the curved tunnel surface on the loading effect. Ten stages of loading were applied, each adding 20 newtons, for a total of 200 newtons. Eight digital dial indicators, spaced at 45-degree intervals around the central ring starting at the crown, tracked radial displacements throughout. The response was almost perfectly linear: as load increased, the vertical diameter of every ring shrank steadily while the horizontal diameter grew, and the deformation pattern—largest at crown and invert, next largest at the springlines, smallest along the diagonals—was consistent across all three models. What differed, dramatically, was the magnitude.

Under identical loads, the straight-jointed ring deformed the most, the staggered-jointed ring somewhat less, and the homogeneous ring the least. That ordering is precisely what joint theory predicts, but the quantitative gap is what matters for designers. The researchers defined the stiffness efficiency ratio as the ratio of deformation in the homogeneous ring to deformation in the assembled ring: the smaller the assembled ring’s deformation relative to the unjointed benchmark, the closer η approaches unity. Computed from vertical diameter changes, η ranged from 0.098 to 0.130 for straight-jointed assembly and from 0.210 to 0.696 for staggered-jointed assembly. Computed from horizontal diameter changes, the values were lower still: 0.068 to 0.096 for straight joints and 0.138 to 0.573 for staggered joints. In other words, a straight-jointed large-diameter ring may retain barely a tenth of the bending stiffness of its monolithic equivalent, and even the favorable staggered arrangement can fall below a quarter of it at low load levels.

Those numbers sit within the wide spread reported by earlier studies—prior model tests on smaller tunnels found η values from 0.03 to 0.80 depending on assembly and loading—but they are notable because they come from a model engineered to replicate a genuinely super-large-diameter prototype, complete with joint-stiffness similarity rather than crude stiffness reduction. The vertical-diameter-based values consistently exceeded the horizontal ones, a detail the authors highlight as a caution for anyone calibrating the modified conventional method: the direction in which convergence is measured changes the answer. The staggered ring’s advantage also has a clear mechanical explanation. Because the joints in one ring sit opposite the mid-span of segments in the adjacent ring, the bending moment reaching a longitudinal joint is smaller than the moment in the main cross-section, so the joint’s flexibility is recruited less often.

The practical implications cut in two directions. First, the results confirm that the homogeneous ring model, which ignores joints entirely, is a poor fit for large-diameter shield tunnels: it overestimates joint bending moments and drives unnecessarily conservative, costly designs. The equivalent stiffness ring model, armed with experimentally grounded η values like those measured here, offers a defensible middle path. Second, the study is candid about its limits. The tests used straight bolts rather than the inclined bolts employed in the real tunnel, applied a simply supported point load rather than distributed ground pressure, and did not reproduce the surrounding stratum’s confining resistance—so the reported η ranges should be treated as parametric references, to be validated against field monitoring or numerical simulation before direct extrapolation to a specific project.

Even with those caveats, the work adds a valuable data point to a global engineering conversation. As megacities push crossings under wider rivers and deeper seas, tunnel diameters keep growing, and the penalty for misjudging joint stiffness grows with them—both in financial terms and in safety margins. A half-meter polyethylene ring, loaded with weights on pulleys in a laboratory, may seem an unlikely hero for infrastructure worth billions. But by faithfully scaling the joint mechanics of a 15-meter giant, it delivers exactly the kind of number that tunnel designers have long needed: a measured, physically grounded answer to the question of how much stiffness a tunnel really loses when it is built from pieces rather than poured as one.

Subject of Research: Experimental determination of the effective transverse bending stiffness ratio of large-diameter shield tunnel linings using scaled model tests

Article Title: Scaled-down model test research on effective ratio of transverse bending stiffness for large-diameter shield tunnel

Article References: Ma, S.-J., Han, K.-H., Bao, X.-H., Ju, J.-W. W., Hong, K.-R., & Chen, X.-S. (2026). Scaled-down model test research on effective ratio of transverse bending stiffness for large-diameter shield tunnel. Results in Engineering, 32, Article 113141. https://doi.org/10.1016/j.rineng.2026.113141

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113141

Keywords: shield tunnel, segment lining, bending stiffness ratio, scaled model test, similarity theory, staggered joints, straight joints, equivalent stiffness ring model, Jinan Yellow River Tunnel, tunnel engineering, HDPE model segments, joint stiffness