The challenge begins with the ground itself. Deep urban excavations rarely pass through uniform soil. Instead, a wall might thread through soft clay sandwiched between dense sands and stiff silts, and at each interface the earth pressure pushing on the structure can change abruptly. Soft layers tend to let the wall bulge outward and concentrate bending moments, while hard layers push back with strong constraint reactions. Traditional design models, which typically assume a smooth, linearly distributed pressure, simply cannot capture these jumps, and the discrepancies between calculated and observed deformation grow most pronounced precisely where projects are deepest and most demanding.

A second blind spot concerns the walls themselves. To resist enormous pressures at depth, engineers build ever thicker diaphragm walls, and thick structures do not behave the way classical beam theory assumes. Conventional models idealise the retaining wall as an Euler-Bernoulli beam, which accounts only for bending-induced rotation and ignores shear deformation, the internal sliding of one cross-section past another. For slender walls in shallow pits this simplification is harmless, but for the massive walls of deep excavations it introduces real error. Earlier work on shear effects had focused on buckling and boundary conditions rather than on the coupled bending-shear response of braced retaining structures in layered ground.

The research team, led by Jun Guan, Zejia Wu and Yanbin Fu, built a model that tackles both problems at once. Above the excavation level, the wall is loaded by an active earth pressure that depends on the wall’s own displacement, so the pressure redistributes as the wall moves, a feedback loop that conventional approaches freeze out. Below the excavation level, the embedded portion of the wall is treated as a Timoshenko beam resting on a Winkler elastic foundation, with soil resistance described by the widely used m-method, in which the ground’s reaction stiffness grows linearly with depth. Internal struts are represented as elastic supports whose reaction depends on their pre-applied force and stiffness, and the wall is sliced into computational segments at every strut location, soil-layer interface and current excavation level so that each segment sits within a single soil layer with a definite set of parameters.

Assembling these ingredients yields a fourth-order nonhomogeneous linear differential equation with variable coefficients for the wall’s horizontal displacement, an equation far too unwieldy for closed-form treatment by standard techniques. Rather than resorting to finite element modelling, which demands fresh meshing and re-analysis for every new project and excavation stage, the researchers turned to a power series method. They expressed the deflection of each wall segment as an infinite series, substituted it into the governing equation, and matched coefficients of like powers of the depth coordinate to derive a constructive analytical solution. Boundary conditions at the wall ends, together with deformation compatibility and force equilibrium between adjacent segments, pin down the unknown coefficients, and the same procedure is applied below the excavation level with the appropriate foundation reaction terms.

A crucial theoretical step was proving that the resulting series actually converges. Using the ratio test, the team showed that the limit of the ratio of successive terms falls below one, guaranteeing convergence, and they demonstrated that a ten-term expansion already satisfies engineering accuracy requirements. This matters practically: once the governing equations and solution are established, engineers can analyse different excavation stages, stratigraphic conditions and boundary conditions simply by updating input parameters, with no repeated mesh generation or staged numerical modelling, making the method especially convenient for the multi-scenario and parametric studies that dominate real design workflows.

To test the model, the researchers turned to an ultra-deep excavation in Hangzhou, a rectangular pit 22.40 metres long and 10.80 metres wide, retained by a cast-in-place diaphragm wall 800 millimetres thick and embedded to a depth of 60.37 metres, with its toe socketed into weathered rock. Excavation proceeded in eleven stages down to a final depth of 46.2 metres, threading through eleven distinct strata ranging from fill and sandy silts to a very soft muddy clay, rounded gravel and argillaceous siltstone. The team validated their predictions against measured wall displacements during the final four excavation stages, when the wall was working hardest.

The results were striking. In stage 8, the measured maximum wall displacement was 43.14 millimetres; the new model predicted 45.76 millimetres, an error of 6.07 percent, while a conventional Euler-Bernoulli benchmark predicted 40.60 millimetres, off by 12.7 percent. In stage 9 the new model erred by only 4.67 percent against the benchmark’s 11.9 percent; in stage 10 by 5.02 percent against 12.9 percent; and in stage 11 by 4.03 percent against 11.4 percent. Even more impressive was the prediction of where the maximum displacement occurs, the depth that most concerns designers deciding where to place struts. The new model located the critical depth within two metres of the measured value in every stage, and in stage 10 it was off by just 0.07 metres, whereas the benchmark consistently placed the bulge several metres too deep.

The parametric studies added further insight. Comparing simulations at excavation depths from 31 to 37 metres, the team found that shear deformation matters most in the segment between the lowest strut and the excavation surface, where it can account for up to 9.6 percent of total displacement when the ratio of that segment’s length to the wall thickness falls between roughly 0.85 and 4.17. Outside that range the shear effect fades to near insignificance, meaning the classical model remains a useful simplification for overall trends but must be supplemented in the critical zone. A separate comparison showed why accounting for layered ground is essential: when the model was reduced to assume homogeneous, linearly distributed earth pressure, it placed the maximum displacement depth 10 percent away from the field measurement, while the full heterogeneous model was within 2 percent. A scenario in which the properties of the soft sixth soil layer were modified shifted the maximum displacement magnitude by 12.13 percent and moved its depth upward by 8.03 percent, underscoring how sensitive deep walls are to local stratigraphic changes.

The authors are careful to note the model’s scope. It addresses monotonic inward wall movement during normal staged excavation and does not yet explicitly capture reversed wall movement, unloading-reloading behaviour, soil-structure separation or the progressive mobilisation of ultimate passive resistance, and the independent effects of individual soil parameters such as unit weight, cohesion and friction angle call for further controlled-variable study. Even so, the combination of displacement-dependent earth pressure, layered-strata handling and Timoshenko shear coupling, delivered as a fast, convergent analytical formula rather than a bespoke numerical model, offers deep-urban projects something they have lacked: a rigorous first-pass tool that tells engineers not just how far a wall will move, but exactly where it will move most, and therefore where to concentrate the bracing that keeps a city’s underground ambitions safe.

Subject of Research: Analytical prediction of lateral displacement of retaining structures in deep urban excavations

Article Title: Analytical formulation for lateral displacement of retaining structures in deep underground space and its application

Article References: Guan, J., Wu, Z., Zhang, B., Gu, J., Chen, F., Yelv, G., Zhou, Y., & Fu, Y. (2026). Analytical formulation for lateral displacement of retaining structures in deep underground space and its application. Results in Engineering, 32, Article 113205. https://doi.org/10.1016/j.rineng.2026.113205

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113205

Keywords: deep excavation, retaining structures, Timoshenko beam, earth pressure, soil-structure interaction, diaphragm wall, power series solution, shear deformation, underground space, geotechnical engineering, Hangzhou case study, deformation prediction

The research, led by Charlotte Mabbs with Lloyd Smith, both of Washington State University, addresses a gap that has persisted in sports ball mechanics for years. While the behavior of balls in head-on, or normal, impacts is routinely measured and modeled, oblique impacts are considerably more complicated. During such an impact, a ball can either slide across the surface — if it comes in at a shallow angle or the friction between ball and surface is low — or it can grip the surface, momentarily bringing its contact patch to a halt. When the ball grips, frictional forces stretch and shear the compliant cover and core, storing elastic energy that is later released as rotation. Capturing this transition between sliding and gripping has proven a stubborn challenge for previous simulations of tennis balls, soccer balls, and golf balls.

Earlier models typically relied on a constant coefficient of friction, treating the resistance between ball and surface as a single fixed value throughout the collision. Those models could reproduce sliding behavior or gripping behavior, but not both. A tennis ball study that achieved good agreement compared its simulation to only one impact condition, leaving its general validity uncertain. Other investigations varied the friction coefficient numerically under fixed conditions without experimental validation at all. The Washington State team took a different route: they implemented what they call a temporal friction model, in which the friction coefficient evolves during contact, transitioning between independently measured static and dynamic values depending on the relative sliding velocity between ball and surface.

To build the model, the researchers first needed to characterize the softball itself. They studied adult fastpitch softballs with a circumference of 306 millimeters and a mass of 0.2 kilograms, constructed with a rigid polyurethane foam core surrounded by a thin leather cover stitched with raised seams. Upon impact, a softball dissipates roughly 75 percent of its energy, so the material model had to capture severe energy loss. Using the explicit finite element solver LS-DYNA, the team employed a non-linear viscoelastic foam material model governed by a high-speed stress-strain loading curve, with parameters controlling hysteresis and energy dissipation tuned until simulated normal impacts at 21.4 and 30.6 meters per second matched measured stiffness and coefficient of restitution within 4 percent of laboratory results.

One of the study’s most striking findings concerns the ball’s moment of inertia — a measure of how its mass is distributed around its center. A homogeneous sphere, the standard simplification in sports ball modeling, underestimated the measured moment of inertia by 9.1 percent because the dense leather cover and seams push mass toward the outside of the ball. That seemingly small discrepancy had outsized consequences: the homogeneous model overpredicted the final angular velocity of a sliding impact by about 13 percent. By adding a thin shell of massless-stiffness elements to the ball’s radius and adjusting densities to match the measured inertia, the researchers brought the angular velocity error down to just 3 percent. For balls with seams — softballs, baseballs, cricket balls — the lesson is clear: assuming a uniform sphere is not good enough when rotation is at stake.

The experimental half of the study was equally ambitious. The team fired softballs from a pneumatic cannon at a steel plate across a wide envelope of conditions: speeds from 20.1 to 63.5 meters per second, spin rates up to 117 radians per second, and impact angles from 14 to 80 degrees. A triaxial load sensor recorded normal and shear forces during contact at 150 kilohertz, while high-speed cameras filming at up to 14,100 frames per second tracked the ball’s position and rotation through the roughly 1.5-millisecond collision. Ball rotation was computed by detecting and matching distinctive features on a randomly patterned leather cover frame by frame. Between every shot, the steel plate was cleaned with 1000-grit sandpaper to keep friction conditions consistent.

Friction measurements fed directly into the model. Sliding impacts — those in which the ball skids through contact — yielded a dynamic friction coefficient of 0.360, while an inclined plane test using a panel of leather removed from an actual softball gave a static coefficient of 0.625. The dynamic value carried a relatively large uncertainty of about 22 percent, consistent with the scatter reported in prior dynamic friction measurements on other balls. The static value aligned well with published engineering data for leather against metal, which typically cites values around 0.6. The temporal friction model blended these two values with an exponential decay governed by a transition parameter, tuned to match representative sliding and gripping impacts and then validated against the full range of angles and speeds.

The validation results were emphatic. Compared with a constant friction model using the dynamic coefficient, the temporal friction model reduced the mean normalized root-mean-square error in predicted angular velocity during contact by 29 percent, and by 81 percent compared with a constant friction model based on the static coefficient. Crucially, it was the first friction formulation for any sports ball to describe both sliding and gripping behavior simultaneously. In gripping impacts, the simulated friction coefficient lingered near the dynamic value for only about 10 percent of the contact duration before climbing rapidly to the static value as the ball’s contact patch came to rest; in sliding impacts, the coefficient stayed near the dynamic value for nearly half the impact. Predicted peak normal forces came within 2.5 percent of experiment, and tangential forces within 6.1 percent.

The model also reproduced the distinctive energy landscape of oblique impacts. As impact angle decreases from vertical, more of the ball’s incoming kinetic energy is converted into transverse motion and rotation, with rotational energy peaking at the shallowest angles at which the ball still grips the surface. The simulation correctly captured the inflection point — between 25 and 30 degrees — below which the ball slides through contact rather than gripping. Interestingly, the frictional force did not substantially reverse during contact, unlike the dramatic reversals seen in highly elastic superballs, a difference the researchers attribute to the softball’s prodigious energy dissipation. One residual discrepancy remained: the simulated frictional force peaked slightly earlier than measured, by roughly 0.1 to 0.16 milliseconds. Tests on a coverless ball, with the leather stripped away, largely eliminated the timing gap, suggesting the thin cover — only 10 percent of the ball’s volume — measurably influences shear response, perhaps through slip at the core-cover interface or the cover’s own compliance.

The implications extend beyond softball. Because softballs are simple in construction compared with the layered pills, yarn windings, and seams of baseballs and cricket balls, the inhomogeneity effects documented here are likely even more pronounced in those sports. The work also marks the first dynamic measurement of friction coefficients for a solid sports ball at speeds representative of actual play, and the first controlled laboratory experiments on softball oblique impacts of any kind — previous on-field studies of softball-bat collisions had reported lower tangential restitution values, consistent with the greater energy dissipation expected when a compliant, curved bat is involved. For governing bodies, equipment designers, and modelers of ball flight, the message is that both the velocity-dependent nature of friction and the true mass distribution of the ball must be respected. As the authors conclude, ball inhomogeneity and temporal friction are not refinements but necessities for accurately modeling how solid sports balls shear, grip, and spin when they meet the ground.

Subject of Research: Finite element modeling and experimental validation of oblique, frictional impacts of softballs

Article Title: Finite element modeling of oblique impacts of softballs

Article References: Mabbs, C., & Smith, L. (2026). Finite element modeling of oblique impacts of softballs. Sports Engineering, 29(2), Article 31. https://doi.org/10.1007/s12283-026-00564-5

Image Credits: AI Generated

DOI: 10.1007/s12283-026-00564-5

Keywords: softball, finite element model, oblique impact, friction, spin, sports engineering, coefficient of restitution, moment of inertia, LS-DYNA, sliding and gripping, shear deformation, sports biomechanics