Shear strength design formulation for high-strength concrete beams: correcting the partition between concrete and stirrup contributions and the stirrup-yielding failure assumption

Design codes for reinforced concrete (ACI 318, BS 8110, Eurocode 2, NZS 3101, IS 456) compute the shear resistance of slender beams as the sum of a concrete contribution and a stirrup contribution, with formulas empirically calibrated on normal-strength concrete and, for the stirrup term, resting on the Ritter–Mörsch truss premise that stirrups yield before the web concrete crushes. Experiments on high-strength concrete (HSC) beams — including the source study, which found the codes underestimating the concrete contribution by about 40% and overestimating the stirrup contribution by about 45%, with failure by concrete crushing before stirrup yielding — show that this framework does not extrapolate to HSC. The problem is to develop and validate a shear strength design formulation for HSC beams, with and without transverse reinforcement, that removes these systematic component-level biases, represents the actual governing failure mechanism (reduced aggregate interlock across smooth cracks that cut through the aggregate, diagonal-crack penetration into the compression zone, and crushing before stirrup yield), and is demonstrably reliable across the experimental parameter space, so that it can replace empirically extrapolated code formulas. Post-2019 work documents persistent code misprediction and ongoing but incomplete refinement (ACI 318-19 recalibration, fib Model Code 2010 physically based formulations, continued Eurocode truss-model corrections), and no accepted resolution exists.

Background

In a reinforced concrete (RC) beam, the shear force (the internal force acting perpendicular to the beam axis, produced by transverse loads) is resisted by a combination of mechanisms: the compressed concrete above the tip of an inclined crack, the shearing resistance of the crack faces themselves, the dowel action of the longitudinal bars, and — when present — transverse reinforcement called stirrups, which are closed bars encircling the longitudinal steel and crossing potential diagonal cracks. Design codes idealize the total nominal shear resistance as a sum

$V_R = V_c + V_s,$

where $V_c$ is the 'concrete contribution' (all mechanisms other than the stirrups) and $V_s$ is the 'stirrup contribution'. In the American ACI 318 code the concrete term for members without axial force is essentially $V_c \approx 0.17\lambda\sqrt{f'_c}\,b_w d$ (with $f'_c$ the cylinder compressive strength, $b_w$ the web width, $d$ the effective depth, and $\lambda$ a correction for lightweight concrete), and the stirrup term is $V_s = (A_v f_{yt}/s)d$ (stirrup area $A_v$, yield strength $f_{yt}$, spacing $s$), derived from the Ritter–Mörsch truss analogy: the beam is idealized as a truss whose diagonal concrete struts are inclined at 45 degrees and whose vertical ties are the stirrups, with the implicit sequence that stirrups yield before the concrete of the web crushes. Eurocode 2 generalizes this to a variable strut inclination, $V_s = (A_v/s)z f_{ywd}\cot\theta$ with $1 \le \cot\theta \le 2.5$, bounded by a web-crushing check $V_{Rd,\max}$; the British BS 8110, New Zealand NZS 3101 and Indian IS 456 codes follow the same broad $V_c + V_s$ decomposition. All of these $V_c$ expressions were empirically fitted to tests of normal-strength concrete (NSC), typically with $f'_c$ below about 40–50 MPa.

High-strength concrete (HSC; compressive strength from roughly 50 up to 100 MPa — the source chapter, following ACI 318, counts concrete above 41 MPa, and comparative studies treat 55–90 MPa) changes the physics behind both terms. Because the cement matrix becomes stronger than the aggregate particles, diagonal cracks cut straight through the aggregates instead of following their boundaries; the resulting smooth crack faces provide much less aggregate interlock (the shear transfer that rough crack faces carry by mechanical interlocking of asperities), while the matrix itself gains compressive and tensile strength and bonds better with the steel. The consequence is a different mix of resisting mechanisms and a different failure sequence than in NSC.

The source chapter (Tahenni and Lecompte, 2019, an open-access book chapter) tested six small HSC beams ($100 \times 150$ mm cross-section, $f_c \approx 65$ MPa, shear-span to effective-depth ratio $a/d \approx 2.2$–$2.3$, i.e. slender beams governed by shear), half without and half with stirrups, monitoring crack development with digital image correlation (DIC — a full-field optical technique that tracks the deformation of a sprayed speckle pattern on the beam surface and yields crack openings, spacings and concrete strains at every load step). Comparing measured capacities with the five code predictions, they found the codes underestimate the concrete contribution by about 40% on average (Eurocode 2 best at about 24% low, IS 456 worst at about 65% low) and overestimate the stirrup contribution by about 45% on average; in every beam with stirrups the failure was crushing of the compression-zone concrete after full penetration of the diagonal crack, while the stirrups had not yielded — a direct contradiction of the Ritter–Mörsch sequence on which the stirrup term rests.

Two lines of later work define the state of the art against which this problem must be judged. First, general stress-field and crack-based theories — the modified compression field theory (which treats cracked concrete as a continuum with average stresses and strains related by material laws) and the critical shear crack theory (which ties shear strength to the width of the critical inclined crack) — underpin the fib Model Code 2010 shear provisions, formulated with graded 'levels of approximation' (successively simpler versions trading accuracy for ease of use) for members with and without shear reinforcement. Second, the empirical code framework itself has been repeatedly recalibrated, most prominently in ACI 318-19, whose revised one-way shear provisions make $V_c$ depend on the longitudinal reinforcement ratio and on a strain-dependent size-effect factor (reflecting the systematic loss of relative capacity with increasing member depth) and cap the beneficial use of $\sqrt{f'_c}$ above 70 MPa when no minimum shear reinforcement is provided. What no accepted formulation yet provides is a shear design model for HSC whose two components are separately faithful to the observed mechanisms and whose total predictions are demonstrably unbiased and safe over the tested HSC parameter range; database-scale studies up to 2022 continue to find most available code and literature equations systematically over- or under-predicting HSC beam capacities.

Problem Statement

Develop, and validate against experimental evidence, a shear strength design formulation for slender reinforced concrete beams made of high-strength concrete (compressive strength from roughly 50 up to 100 MPa), with and without transverse reinforcement, that corrects the defects documented in the source study and its successors. The formulation must satisfy, jointly and in service of one objective — a code-adoptable, mechanically faithful model of HSC shear resistance — the following linked requirements.

1. Component partition. It must provide separate, physically grounded expressions for the concrete contribution $V_c$ and the stirrup contribution $V_s$ whose predicted split is consistent with experimental evidence, removing the systematic misestimation documented for the empirical code formulas (underestimation of the concrete contribution and overestimation of the stirrup contribution in the tested HSC configurations). Because the two contributions interact at ultimate and cannot be measured separately in a failing beam, consistency of the split is assessed through the indirect routes the experiments do supply — as in the source study, where the beams tested without transverse reinforcement ($V_s = 0$) give the concrete contribution directly from their total capacities and the totals of the otherwise comparable stirrup-reinforced beams then fix the stirrup contribution by difference — together with the observed cracking and failure-mode evidence, rather than through a direct measurement of each contribution.

2. Failure-mechanism fidelity. It must represent the actual governing mechanism in HSC: diagonal cracks that cut through the aggregate particles with consequently reduced aggregate interlock, penetration of the critical diagonal crack into the compression zone, and crushing of the compression-zone concrete before yielding of the stirrups where that sequence is observed — so that the predicted failure mode is consistent with what is observed in HSC beam shear tests, in place of the Ritter–Mörsch assumption of stirrup yielding prior to concrete crushing.

3. Demonstrated reliability over the required parameter range. The accuracy of its total predicted capacity must be demonstrated against experimental HSC beam shear test evidence spanning at least the required parameter range — concrete strength from roughly 50 up to 100 MPa including both ends, a range of shear-span/effective-depth ratios, a range of longitudinal and transverse reinforcement ratios, and at least two member depths so as to exercise the size effect — reporting the bias and dispersion of the test-to-predicted capacity ratio — for example its mean value and coefficient of variation — and the assessment must include test data beyond any data used to calibrate the formulation.

The problem is solved when such a formulation — analytical, semi-empirical or mechanistic — is available and its validation demonstrates total predictions that are unbiased, in the sense that the mean test-to-predicted capacity ratio is close to unity, or conservatively safe, in the sense that they do not exceed the measured capacities, together with mechanism fidelity across the required parameter range. Arbitrarily large conservatism is not an alternative route to this criterion: a formulation whose predicted component split departs systematically from the experimentally inferred partition, in either direction, reintroduces the very component-level bias that requirement 1 asks to remove. The problem is not solved by a recalibrated regression that reproduces one laboratory's capacities without the mechanism content, nor by a mechanistic narrative without validation against experimental evidence over the required parameter range. No particular modelling family is prescribed, and new experimental work is not itself a deliverable: where existing data are sparse over the required parameter range, closing that gap is a means toward the single stated objective, not an independent question.

References

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