New Equations Design Earthquake-Resistant Columns for Ductility and Shear at Once

When an earthquake strikes a reinforced concrete building, the columns carrying its weight must survive in two very different ways at the same time. They need to bend and deform without shattering, a property engineers call ductility, and they must resist shear, the slicing force that can snap a column clean through. For decades, design codes have treated these two demands with separate, largely independent procedures, leaving engineers to check one, then the other, and hope the two answers do not quietly undermine each other. A new study published in the Bulletin of Earthquake Engineering argues that this piecemeal approach is no longer necessary, and it backs that claim with a set of closed-form equations that handle both requirements in a single, transparent calculation.

The research, led by Giorgio Monti of Sapienza University of Rome together with Dario De Domenico of the University of Messina, Qingcong Zeng of Zhejiang University, and Giuseppe Quaranta, also of Sapienza, addresses a deceptively simple question: for a rectangular concrete column facing a known axial load, bending moment, and shear demand, exactly how much transverse reinforcement, the stirrups and cross-ties wrapped around the section, is required? The answer matters because transverse reinforcement does double duty. It confines the concrete core, allowing the section to keep deforming after the outer cover has spalled away, and it acts as the shear-resisting web of an internal truss. More stirrups generally mean more ductility and more shear strength, but the two demands scale differently with axial load, geometry, and the severity of the expected seismic cycling.

The heart of the new framework is a strain-based description of two limit states. The first, called Damage Limitation, marks the point at which the longitudinal tension steel first yields while the concrete cover is still intact. The second, Near Collapse, describes the section at the brink of failure, which can arrive by two distinct routes: rupture of the tension reinforcement, or crushing of the confined concrete core after the cover has spalled. By writing sectional equilibrium and strain compatibility in dimensionless form, the authors derive closed-form expressions for the concrete and steel strains at each of these states, and from those strains they compute the curvature ductility capacity, the ratio of the curvature at Near Collapse to the curvature at Damage Limitation.

What makes the formulation unusual is its branch-exact structure. Depending on whether the compression steel and the compression concrete are in their elastic or plastic ranges, the equilibrium equations resolve into four distinct constitutive branches, each with its own closed-form solution. The method identifies which branch is physically admissible for any given column and selects the corresponding strain solution, so the answer is not an approximation averaged across regimes but an exact solution of the idealized section model. The concrete is treated as a no-tension, bilinear material whose strength and ultimate strain are amplified by confinement, with amplification factors left in a general form so that designers can plug in the provisions of whatever code they follow, including the second-generation Eurocode 2 demonstrated in the paper.

Of course, an exact branch-by-branch solution is a verification tool, not a design shortcut. To make the method usable at a drawing board, the authors introduce compact approximations of the governing strains and then perform something mathematically ambitious: they invert the ductility relationship. Instead of calculating how much ductility a given column has, the equations directly output the transverse reinforcement ratio needed to achieve a prescribed curvature ductility demand. For the practically common case of symmetric longitudinal reinforcement, typical in seismic columns because earthquakes reverse the loading direction, this inversion collapses into a single closed-form expression involving the axial load ratio, the confinement effectiveness coefficients, and the target ductility.

Validation was carried out on two fronts. First, the compact strain expressions were compared against the branch-exact solutions across 57,767 admissible parameter combinations spanning realistic column sizes, reinforcement layouts, and axial load levels. The median absolute error was about one percent, and the vast majority of cases fell within five percent of the exact value, with the largest deviations confined to a small subset of cases governed by an elastic-elastic branch that the compact form does not reproduce. Second, and more decisively, the entire curvature formulation was tested against an independent fiber-section numerical analysis on a random sample of 1,000 mechanically admissible column sections. The branch-exact equations matched the numerical curvatures and ductilities with maximum relative errors below 0.14 percent, and the compact design version reproduced curvature ductility with a mean absolute relative error of about 2.1 percent, with nearly 86 percent of cases within five percent.

The second half of the framework tackles shear through a variable-angle truss model, enhanced by a machine-learning-assisted calibration published by some of the same authors in 2022. In this model, shear capacity is the smaller of two resistances: the force at which the stirrups yield, and the force at which the diagonal concrete strut between cracks crushes. The concrete contribution depends on the aspect ratio of the member, the axial stress, and crucially on the displacement ductility demand, because repeated inelastic cycling degrades the concrete’s ability to carry shear, an effect the model captures through an inverse-square dependence on ductility. The authors also added an extrapolation-control bound so that the machine-learned coefficients are not stretched beyond the experimental database that calibrated them, which included 119 rectangular column tests and 373 beam tests.

The payoff of combining the two models is an explicit map of which requirement governs. Because the final transverse reinforcement is simply the larger of the ductility-controlled and shear-controlled values, the designer can see analytically where the transition occurs, and the transition is anything but intuitive. Increasing the axial load makes the ductility problem worse: it reduces curvature ductility capacity, so more confinement steel is needed. Yet the same axial load, normalized over the gross section, strengthens the concrete shear strut and reduces the reinforcement demanded by shear. In the paper’s worked example, the two curves cross at an axial load ratio of roughly 0.25; below that value the design is shear-governed, above it the design is ductility-governed. No sequential checking procedure reveals this competition so cleanly.

The authors are careful about boundaries. The closed-form ductility inversion applies only within a formally defined domain where the assumed sequence of limit-state events holds, and the shear inversion is valid only within the calibration ranges of the underlying truss model, with strut inclination parameters between one and five. The formulation addresses solid rectangular sections under uniaxial bending and does not cover biaxial response, torsion, or the progressive deterioration of cover spalling during cyclic loading. The curvature ductility demand and the displacement ductility demand that feed the two halves of the procedure are distinct quantities, and the paper stresses that they must be derived consistently from the preceding structural analysis rather than through any universal conversion.

Even within those limits, the study offers something the field has lacked: a direct, mechanically interpretable design equation for the reinforcement that must simultaneously deliver deformation capacity and shear resistance in earthquake-exposed columns. Because every quantity is dimensionless, the influence of each parameter, axial load, confinement effectiveness, steel ratio, geometry, is visible in the algebra rather than buried in iteration. The authors point toward extending the framework to asymmetric reinforcement layouts, biaxial loading, and richer experimental validation, but the immediate message is already striking. Two requirements that codes have long kept in separate chapters can, with the right mathematics, be solved together, exactly, and on a single sheet of paper.

Subject of Research: Analytical design of transverse reinforcement in reinforced concrete columns for simultaneous ductility and shear requirements under seismic loading

Article Title: Simultaneous analytical design of ductility and shear capacity in reinforced concrete columns

Article References: Monti, G., De Domenico, D., Zeng, Q., & Quaranta, G. (2026). Simultaneous analytical design of ductility and shear capacity in reinforced concrete columns. Bulletin of Earthquake Engineering. https://doi.org/10.1007/s10518-026-02682-8

Image Credits: AI Generated

DOI: 10.1007/s10518-026-02682-8

Keywords: reinforced concrete, seismic design, ductility, shear capacity, transverse reinforcement, confinement, curvature ductility, columns, earthquake engineering, closed-form equations, structural analysis, Eurocode 2