The study, published in Case Studies in Construction Materials, was led by Yi Ding, with co-authors Xinru Wang, Man Xu, and Huiwen Tian. Their work addresses a deceptively simple question: how do you calculate the shear strength of a squat wall when part of it is made not of ordinary concrete, but of ultra-high performance concrete, or UHPC — a fiber-reinforced cementitious material that can reach compressive strengths above 130 megapascals, roughly double or triple that of conventional structural concrete? The answer, they found, requires rewriting one of the key equations that engineers use to size these walls, because the new material fundamentally changes how forces flow through the structure.

UHPC is not just stronger concrete in a loose sense. It is engineered through optimized particle packing — arranging fine sand, cement, and admixtures so that almost every void is filled — and through the addition of steel fibers that bridge cracks and hold the material together after it fractures. In the team’s earlier laboratory tests, shear walls in which UHPC was cast only in the boundary elements, the heavily loaded vertical zones at the edges of the wall, showed remarkable improvements. Compared with identical walls made entirely of normal-strength concrete, the UHPC-reinforced versions carried 6.79 to 20.64 percent higher peak loads, showed 2.54 to 24.11 percent greater ductility, and dissipated 44.88 to 57.65 percent more cumulative energy under simulated earthquake loading. Just as striking, the steel fibers restrained crack growth so effectively that the walls avoided the catastrophic spalling, bar buckling, and longitudinal bar fracture that rendered the conventional walls essentially unrepairable after testing.

But predicting how strong such a hybrid wall should be turned out to be a genuine theoretical challenge. The researchers first benchmarked three established calculation methods against their experimental data from thirteen full-scale wall specimens, which included rectangular and barbell-shaped cross-sections, UHPC distributed at various heights and widths, and axial load ratios of 0.25 and 0.45. The Chinese standard JGJ 3-2010 overestimated strength by an average factor of 1.51, the American standard ACI 318-19 by 1.56, and even the best-performing option — the softened strut-and-tie model developed by Hwang and colleagues — overshot by 1.38 on average. None of these methods was designed with a material as strong and as crack-resistant as UHPC in mind, and none accounted for how locally concentrating that material in the boundary elements redistributes the internal load paths.

To understand the failure mechanism directly, the team built highly refined three-dimensional finite element models in the LS-DYNA simulation environment. Concrete and UHPC were represented with the Karagozian and Case material model, a sophisticated constitutive framework that captures strength surfaces, damage evolution, and failure, with parameters specially calibrated for UHPC’s much higher strength. Steel reinforcement was modeled explicitly with beam elements, and the interface between UHPC and normal concrete was treated with shared nodes, reflecting the experimental observation that the two materials remained fully bonded without separation. After validating the modeling approach against independent low-rise wall tests by other researchers — achieving peak load prediction errors within 5 percent — the team turned the simulation loose on squat walls with an aspect ratio of 1, the regime where shear dominates.

The simulations revealed the failure sequence in vivid detail. At first cracking, tiny macroscopic fractures appeared precisely at the horizontal interface where the UHPC boundary element meets the normal-strength concrete web, marking the point where tensile strain first exceeded the material’s limit. As lateral drift increased, the outermost longitudinal bars in the boundary elements reached their yield stress — around 509 megapascals on the tension side and 501 on the compression side — while diagonal cracks spread through the web. By peak load, these diagonal cracks had carved the web into a lattice of small zones, and the concrete between them consolidated into a set of diagonal compression struts. The wall ultimately failed by crushing along its principal diagonal strut, exactly the mechanism that the strut-and-tie theory describes: steel bars acting as tension ties, concrete acting as compression struts, and the two working together in a self-equilibrating truss embedded within the wall.

That confirmation opened the door to the model’s central innovation. In the classical strut-and-tie framework, the depth of the diagonal compression strut is tied to the height of the compression zone at the base of the wall, and existing formulas for that height were derived from elastic theory for ordinary reinforced concrete columns. The researchers found that casting UHPC into the boundary elements shifts the compression zone significantly — but because squat walls are disturbed regions where the familiar assumption that plane sections remain plane breaks down, they could not simply adjust the flexural theory. Instead, they ran sixty parametric simulations, sweeping aspect ratios from 0.5 to 1.5, concrete strengths from C30 to C80, UHPC strengths from 120 to 180 megapascals, UHPC widths from 10 to 40 percent of the wall width, UHPC heights from 30 to 100 percent of the wall height, and axial load ratios from 0.05 to 0.45.

The parametric results distilled into a remarkably clean relationship. The height of the compression zone increases linearly with the normalized axial load — heavier vertical compression pushes the crushing zone deeper into the wall — but decreases with the width of the UHPC region following a power law, because the stronger material concentrates compression into a narrower band. The team encoded both effects in a single equation, bounded by sensible limits, and embedded it in a five-step design procedure that retains the familiar machinery of the softened strut-and-tie model: determine the geometry, select the dominant tie mechanism based on the strut’s inclination angle, compute the strut-and-tie coefficient that captures how distributed reinforcement enhances capacity, evaluate the softened concrete strength along the strut, and finally project the diagonal crushing capacity into the horizontal shear strength. Notably, the team deliberately kept the concrete softening coefficient conservative rather than increasing it for UHPC, even though studies of full-section UHPC walls suggest the material’s tensile toughness could justify a higher value — a choice made to avoid overestimating strength in a safety-critical application.

The validation was the most convincing part. Across sixty-eight walls — fifty-nine from the new simulations and nine from independent experimental programs by three other research groups — the proposed model achieved an average predicted-to-actual strength ratio of 1.03 with a standard deviation of 0.20. The existing methods fared considerably worse: JGJ 3-2010 averaged 1.19 with a standard deviation of 0.27, ACI 318-19 averaged 1.11 with 0.31, and the original Hwang model averaged 1.14 with 0.22. In other words, the new model is both nearly unbiased and the most consistent, which is precisely the combination a design engineer wants. The model did show characteristic blind spots: it underestimated precast walls in which UHPC also filled lap-splice zones and vertical columns, because those extra contributions lie outside its scope, and it overestimated two specimens that failed not by strut crushing but by fracture of longitudinal bars at a weak post-cast interface — a different failure mode entirely.

The authors are candid about these limits, noting that the model applies to cast-in-place walls where UHPC sits primarily in the boundary elements and works in reliable composite action with the surrounding normal concrete, and that further validation is needed for other UHPC configurations and interface conditions. Even so, the significance is hard to overstate. Squat shear walls are among the least forgiving elements in seismic design, and the ability to exploit UHPC’s extraordinary strength and crack-bridging fibers in exactly the zones where they matter most — without paying the cost of casting an entire wall in the premium material — could reshape how engineers protect tall buildings in earthquake-prone regions. A formula that turns laboratory performance into trustworthy numbers is the bridge between a promising material and a safer built environment, and this study has just laid a carefully calibrated plank across it.

Subject of Research: Shear strength prediction of squat UHPC-reinforced concrete shear walls under seismic loading

Article Title: A strut-and-tie model for predicting shear strength of squat shear walls reinforced with UHPC under seismic loading

Article References: Ding, Y., Wang, X., Xu, M., & Tian, H. (2026). A strut-and-tie model for predicting shear strength of squat shear walls reinforced with UHPC under seismic loading. Case Studies in Construction Materials, 25, Article e06575. https://doi.org/10.1016/j.cscm.2026.e06575

Image Credits: AI Generated

DOI: 10.1016/j.cscm.2026.e06575

Keywords: UHPC, shear walls, strut-and-tie model, seismic design, finite element analysis, squat walls, concrete structures, earthquake engineering, boundary elements, shear strength, structural engineering, LS-DYNA

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