Shear design remains a critical aspect of reinforced concrete (RC) structural engineering. Unlike flexural failure, which often exhibits significant ductility and warning signs, shear failure can be sudden and brittle, potentially leading to catastrophic collapse if not adequately addressed.1 Consequently, building codes provide detailed provisions for calculating shear capacity and designing appropriate reinforcement. The American Concrete Institute's Building Code Requirements for Structural Concrete (ACI 318) is the governing standard for concrete design in the United States and is influential globally.
For decades, the fundamental approach to calculating the concrete contribution to shear strength (Vc) in ACI 318 remained largely unchanged, relying on empirical equations developed based on testing primarily smaller-scale specimens available over 50 years ago.3 However, the 2019 edition, ACI 318-19, introduced significant updates to these shear provisions, marking a notable departure from past practice.4 A central element of this update is the explicit incorporation of a size effect factor, denoted as λs.
This factor directly addresses a growing body of research and experimental evidence indicating that the nominal shear strength of concrete members does not increase in direct proportion to their depth.6 In fact, for deeper members, particularly those without significant shear reinforcement, the shear stress capacity tends to decrease. The introduction of λs represents a significant philosophical shift in the ACI code, moving away from purely empirical or simplified models towards incorporating principles of fracture mechanics to address known limitations and improve the safety and reliability of shear design, especially for large-scale structures. Previous code editions, lacking this explicit size correction, could lead to unconservative designs for large beams and thick slabs, a concern substantiated by extensive database analysis and theoretical work, notably by Professor Zdeněk Bažant and ACI Committee 446.3 The adoption of λs, grounded in Bažant's Size Effect Law derived from fracture mechanics principles, aims to rectify this discrepancy, making the code more theoretically sound and enhancing reliability across the full range of member sizes.10 This change, however, necessitates that engineers understand and apply new calculation procedures and grasp the underlying mechanics driving the size effect phenomenon.11
This report provides a comprehensive technical explanation of the ACI 318-19 shear provisions related to the size effect. It details the specific code equations and their application, delves into the underlying scientific rationale rooted in fracture mechanics and Bažant's Size Effect Law, explores the mechanics of shear transfer and how member size influences it, discusses the practical implications for structural design, and offers brief comparisons with the preceding ACI 318-14 standard and other relevant international codes like Eurocode 2 and CSA A23.3.
A significant change in ACI 318-19 was the consolidation and refinement of the equations used to calculate the nominal shear strength provided by concrete, Vc, particularly for non-prestressed members.4 The code moved away from the multiple, sometimes complex, equations present in ACI 318-14 towards a more unified framework presented in Table 22.5.5.1. A key aspect of this new framework is that the specific equation used for Vc now depends explicitly on whether the provided area of shear reinforcement (Av) meets or exceeds the minimum required area (Av,min) as specified in ACI 318-19 Section 9.6.3.9
Vc Calculation per Table 22.5.5.1
For non-prestressed members subject to shear, ACI 318-19 Table 22.5.5.1 provides three distinct equations for calculating Vc 16:
In these equations:
Size Effect Factor (λs)
The size effect factor, λs, is defined in ACI 318-19 Equation 22.5.5.1.3 11:
This factor is always less than or equal to 1.0. It begins to reduce the calculated Vc (when using Equation (c)) for members with an effective depth d greater than 10 inches (approximately 250 mm).7 The deeper the member, the smaller the value of λs, and thus the lower the calculated concrete shear contribution.
Applicability
The size effect factor λs is not limited to one-way shear in beams. It is also applied in the calculation of two-way (punching) shear strength (vc) for slabs and footings, as per ACI 318-19 Section 22.6.6 Furthermore, it applies to the calculation of concrete strut capacity (fce) in strut-and-tie models (STM) designed according to Chapter 23, but only if the minimum reinforcement requirements of Section 23.7 are not met.6 An important exception exists for isolated and combined footings designed under Chapter 13; ACI 318-19 Section 13.2.6.2 explicitly permits neglecting the size effect factor (λs=1.0) for calculating shear strength in these foundation elements.20
Table 1 provides a comparative summary of the Vc calculation approaches in ACI 318-14 and ACI 318-19 for non-prestressed members.
Table 1: Comparison of Vc Equations for Non-Prestressed Members (ACI 318-14 vs. ACI 318-19)
Code Version | Condition / Application | Vc Equation | Key Parameters Influencing Vc |
ACI 318-14 | Simplified Method (General Use) | Vc=2λfc′bwd | fc′, bw, d, λ |
ACI 318-14 | Detailed Method (Optional, or if Nu present) | Vc=(1.9λfc′+2500ρwMuVud)bwd≤3.5λfc′bwd | fc′, bw, d, λ, ρw, Vu/Mu, Nu (implicitly) |
ACI 318-19 | Av≥Av,min (Eq. 22.5.5.1(a)) | Vc=(2λfc′+6AgNu)bwd | fc′, bw, d, λ, Nu |
ACI 318-19 | Av≥Av,min (Eq. 22.5.5.1(b)) | Vc=(8λ(ρw)1/3fc′+6AgNu)bwd | fc′, bw, d, λ, ρw, Nu |
ACI 318-19 | Av<Av,min (Eq. 22.5.5.1(c)) | Vc=(8λsλ(ρw)1/3fc′+6AgNu)bwd | fc′, bw, d, λ, ρw, Nu, λs |
Note: | Max(Vc,a, Vc,b) permitted when Av≥Av,min | Vc also limited by 5λfc′bwd (ACI 22.5.5.1.1) |
The dependence of the Vc calculation method on the provided shear reinforcement (Av vs Av,min) is a significant conceptual change. It implicitly recognizes that the presence of minimum shear reinforcement alters the shear transfer mechanisms within the concrete member. In members without stirrups (or with less than minimum), shear failure is often dictated by the formation and propagation of a critical diagonal crack. The primary mechanisms resisting this failure are shear stresses in the uncracked compression zone, aggregate interlock across the crack faces, and dowel action of the longitudinal bars.24 As member depth increases, diagonal cracks tend to widen for a given stress level, which reduces the effectiveness of aggregate interlock and exacerbates the size effect phenomenon.24 The λs factor in Equation (c) is intended to account for this reduction in shear capacity for deeper members lacking adequate shear reinforcement.16
Conversely, when at least minimum shear reinforcement (Av≥Av,min) is present, these stirrups intersect potential diagonal cracks. This serves multiple purposes: it directly contributes to shear resistance (Vs) via a truss mechanism, it helps control crack widths, and it enhances confinement.25 By limiting crack widths, stirrups help maintain the effectiveness of aggregate interlock and dowel action, thereby potentially preserving or even enhancing the concrete's contribution (Vc) and mitigating the size effect.25 ACI 318-19 reflects this by permitting the use of Equations (a) or (b) – which yield potentially higher Vc values and do not include the λs reduction factor – when minimum stirrups are provided.16 While research indicates that stirrups reduce the size effect, they may not eliminate it entirely, especially in very deep members.25
The introduction of the size effect factor (λs) in ACI 318-19 was not an arbitrary adjustment but was grounded in decades of research into the fundamental mechanics of concrete failure, particularly the principles of fracture mechanics applied to quasibrittle materials.
Limitations of Prior Models
Classical theories of structural mechanics, including plasticity theory and the empirical models underlying shear provisions in ACI codes prior to 318-19, generally assumed that the nominal shear strength (shear force divided by a characteristic area, like bwd) was either independent of the member size or scaled linearly with depth.3 However, extensive experimental testing, particularly on larger-scale beams and slabs, consistently demonstrated that this assumption was incorrect. The actual shear strength per unit area tends to decrease as the member depth increases – a phenomenon termed the "size effect".3 This discrepancy meant that older code provisions could become unconservative for large members, potentially compromising structural safety.10
Introduction to Quasibrittle Fracture Mechanics
Concrete is classified as a quasibrittle material, meaning its failure behavior lies between that of perfectly brittle materials (like glass) and ductile materials (like mild steel).11 Unlike ductile materials that yield extensively before failure, or brittle materials that fail abruptly when a stress limit is reached, the failure of quasibrittle materials is governed by the formation and propagation of cracks, which requires energy.11 A key concept in quasibrittle fracture is the fracture process zone (FPZ), a region of microcracking and material damage that develops ahead of the tip of a visible, macroscopic crack.12 The size and characteristics of this zone, relative to the overall structure size, play a critical role in determining the failure load and the extent of the size effect.
Bažant's Size Effect Law (SEL)
The pioneering work of Professor Zdeněk P. Bažant at Northwestern University was instrumental in developing a theoretical framework to explain and quantify the size effect in concrete and other quasibrittle materials.11 Bažant distinguished between two primary types of size effect:
Bažant's Size Effect Law (SEL) provides a mathematical description of this energetic size effect. For structures failing after significant stable crack growth (typical for concrete), the law describes a transition between two asymptotic regimes:
A general approximate form of Bažant's SEL that captures this transition is 12:
$σN=1+d/d0Bft′$or, expressed in terms of shear strength (vc):vc=1+d/d0v0
where:
Adoption Rationale in ACI 318-19
The adoption of the size effect factor in ACI 318-19 was the culmination of sustained efforts by several ACI technical committees, including ACI 318 Subcommittee E (Shear and Torsion), ACI-ASCE Committee 445 (Shear and Torsion), and ACI Committee 446 (Fracture Mechanics).10 These committees recognized the safety concerns associated with the older shear provisions, particularly their potential unconservatism for large members without shear reinforcement.10
The specific formula for λs adopted in ACI 318-19 Eq. 22.5.5.1.3:
λs=1+d/102≤1.0(d in inches)
is a direct application of Bažant's SEL, corresponding to the general form C(1+d/d0)−1/2 with the constants C=2 and d0=10 inches chosen by the committee based on calibration against experimental data and reliability analyses.11 The decision was supported by extensive analysis of shear test databases, notably the one compiled by ACI-ASCE Committee 445, which demonstrated that the new provisions incorporating the size effect offer significantly improved accuracy and reliability compared to the ACI 318-14 equations across the full range of member sizes.3
The considerable time lag – 35 years – between Bažant's initial proposal of applying fracture mechanics to shear design in 1984 11 and its formal adoption in ACI 318-19 is noteworthy. This delay likely reflects several factors inherent in the evolution of building codes. Incorporating complex theoretical concepts like fracture mechanics into codes intended for practical design requires not only robust scientific validation but also overcoming potential barriers related to the perceived complexity compared to established empirical methods. Furthermore, the lack of widespread, in-depth fracture mechanics education within undergraduate civil engineering curricula may have contributed to slower acceptance among practitioners.11 The eventual adoption was ultimately driven by the persistent efforts of researchers and committees, the accumulation of overwhelming experimental evidence highlighting the shortcomings of older methods for large structures, and the development of a relatively simple, calibrated factor (λs) that could be readily implemented within the existing code framework.10 This experience underscores that major code advancements often require a confluence of scientific progress, extensive data validation, persistent advocacy, and addressing practical implementation challenges.
Understanding why shear strength decreases relative to member size requires examining the fundamental mechanisms by which shear forces are transferred across potential failure planes in reinforced concrete members, particularly those without shear reinforcement.
Shear Transfer Mechanisms
In a cracked RC beam without stirrups, the applied shear force is resisted through a combination of mechanisms:
Crack Propagation and Fracture Modes
While diagonal shear cracks typically initiate due to principal tensile stresses exceeding the concrete's tensile strength, the ultimate failure often involves the propagation of these cracks under a combination of tensile and shear stresses (mixed-mode fracture). Research, including experiments conducted by Bažant and colleagues, has shown that under certain loading conditions designed to induce shear, cracks can propagate predominantly in Mode II (sliding mode), parallel or sub-parallel to the direction of maximum shear stress, rather than purely in Mode I (opening mode) perpendicular to the maximum principal tensile stress.15 These experiments, using notched beams loaded to create high shear zones, confirmed the existence of shear fracture as a distinct failure mechanism in concrete, governed by energy release principles.15 Microscopically, this Mode II macroscopic crack likely forms from the coalescence of many small, inclined tensile microcracks within the fracture process zone.15
Influence of Member Depth (Size Effect Mechanics)
Increasing the effective depth (d) of a member, while keeping other parameters like reinforcement ratio constant, influences these shear transfer mechanisms and promotes the size effect:
Role of Aggregate Size (da)
The maximum size of the coarse aggregate (da) acts as a fundamental material length scale that influences concrete's fracture behavior.13 It affects:
R-Curves and Fracture Energy
For quasibrittle materials like concrete, the energy required to propagate a crack is not constant but increases as the crack extends. This phenomenon is described by the R-curve (Resistance curve), which plots the fracture resistance (energy per unit area, R) as a function of crack extension (Δa).15 The increasing resistance is attributed to mechanisms developing in the wake of the crack tip, such as aggregate bridging and frictional interlock along the crack faces. The size effect observed in geometrically similar specimens of varying sizes can be mathematically linked to the material's R-curve; essentially, the size effect law represents the envelope of failure conditions determined by the R-curve for different structure sizes.15
Bažant's research determined the Mode II (shear) fracture energy (GfII) from size effect tests and found it to be substantially higher – approximately 25 times – than the Mode I (tensile) fracture energy (GfI) for the same concrete.15 This significant difference highlights the inadequacy of applying purely tensile fracture concepts to shear failure. The much larger energy dissipation in Mode II is attributed primarily to the frictional work done by aggregate interlock as the crack faces slide relative to each other, a mechanism absent in pure Mode I opening.29 However, as noted earlier, the effectiveness of this energy-dissipating interlock mechanism diminishes in larger members due to wider cracks, contributing to the size effect where the overall nominal shear strength decreases despite the inherently high shear fracture energy.24
The introduction of the size effect factor (λs) in ACI 318-19 has significant practical consequences for the design of reinforced concrete structures, particularly for members where shear is a critical consideration and minimal or no shear reinforcement is provided.
Reduced Vc for Deep Beams and Thick Slabs
The most direct impact of λs is the reduction in the calculated nominal shear strength provided by concrete (Vc) for members with an effective depth (d) exceeding 10 inches (250 mm) when designed using Equation (c) from Table 22.5.5.1 (i.e., when Av<Av,min).6 This reduction can be substantial, with industry reports suggesting decreases of 30% to 40% in some cases, particularly for sections with low longitudinal reinforcement ratios (ρw).7 The members most affected by this reduction include:
Impact on Shear Reinforcement
The reduction in the calculated ϕVc (where ϕ is the strength reduction factor, typically 0.75 for shear) has direct implications for shear reinforcement requirements. If the factored shear force (Vu) exceeds the reduced ϕVc, shear reinforcement designed to resist the excess shear (Vs=(Vu−ϕVc)/ϕ) must be provided, even in cases where ACI 318-14 might not have required it.7 For members already requiring shear reinforcement, the lower Vc contribution means a larger portion of the total shear must be resisted by the steel (Vs), potentially necessitating:
This can lead to increased complexity in reinforcement detailing and potentially contribute to reinforcement congestion, making concrete placement more difficult.18
Illustrative Numerical Comparison
To quantify the impact of λs, consider the calculation of Vc for a normalweight (λ=1.0) concrete beam (fc′=4000 psi, bw=12 inches) with 1% longitudinal reinforcement (ρw=0.01) and no axial load (Nu=0), assuming Av<Av,min. We compare the ACI 318-19 Vc (using Eq. 22.5.5.1(c)) with the simplified ACI 318-14 Vc (using 2λfc′bwd) for various effective depths (d).
Table 2: Numerical Comparison of Vc for Varying Depths (Av<Av,min, fc′=4 ksi, ρw=1%, bw=12 in)
Effective Depth d (in) | λs | Vc,14 (Simplified) (kips) | Vc,19 (Eq. c) (kips) | % Reduction from ACI 318-14 (Simplified) |
10 | 1.000 | 15.2 | 10.9 | 28.3% |
15 | 0.894 | 22.8 | 14.6 | 36.0% |
20 | 0.816 | 30.4 | 17.8 | 41.4% |
30 | 0.707 | 45.5 | 23.1 | 49.2% |
40 | 0.632 | 60.7 | 27.5 | 54.7% |
Note: The reduction is relative to the ACI 318-14 simplified equation. The ACI 318-14 detailed equation could yield different results.
Table 2 clearly illustrates the significant reduction in calculated Vc introduced by the λs factor as member depth increases. For a 40-inch deep beam under these assumptions, the ACI 318-19 Vc (Eq. c) is less than half of the value calculated using the simplified ACI 318-14 equation. Even at a depth of 10 inches, where λs equals 1.0, the ACI 318-19 Eq. (c) yields a lower Vc than the ACI 318-14 simplified equation because it incorporates the (ρw)1/3 term (which is less than 1 for ρw<100%) and uses a coefficient of 8 instead of 2 (but 8×(0.01)1/3≈1.72, which is less than 2).
Industry Impact and Economic Consequences
The changes, particularly the λs factor, have drawn considerable attention and some concern from the construction industry, especially the precast concrete sector.7 Manufacturers have reported that the reduced shear capacity necessitates thicker slab sections (e.g., for double tees, flat slabs used in parking structures or buildings) which can require costly upgrades to existing casting forms and potentially increase material usage.7 For buried structures like precast manhole tops or vault lids, the industry has questioned the necessity of the λs reduction, citing decades of successful performance of elements designed under previous codes (e.g., 12-inch thick slabs) that might now calculate as deficient under ACI 318-19.7 This discrepancy between the new code calculations and historical field performance for certain established product types has led some engineers in the precast industry to explore alternative design standards, such as the AASHTO LRFD Bridge Design Specifications, which may have different shear provisions perceived as less penalizing for these specific applications.7
Potential economic consequences of the ACI 318-19 shear provisions include:
While these changes aim to enhance safety and reliability based on improved understanding of shear behavior, they introduce a potential conflict for specific structure types where historical performance under older codes seemed adequate. This highlights the ongoing challenge of balancing theoretical advancements and safety margins with established practices and economic considerations in code development.
The challenge of accurately modeling concrete shear strength, including the size effect, is addressed differently by various international building codes. Comparing the ACI 318-19 approach with those of Eurocode 2 (EC2) and the Canadian Standard CSA A23.3 provides valuable context.
Eurocode 2 (EN 1992-1-1)
CSA A23.3-19 (Canadian Standard)
Comparison Summary
Table 3 summarizes the different approaches to handling size effect in one-way shear.
Table 3: Comparison of Size Effect Treatment in Major Concrete Codes (One-Way Shear)
Code | Approach (Members w/o Stirrups, Av<Av,min) | Approach (Members w/ Stirrups, Av≥Av,min) | Key Factor/Mechanism for Size Effect |
ACI 318-19 | Empirical Vc eq. based on ρw, Nu, fc′, modified by explicit Size Effect Factor λs=2/(1+d/10) (based on Bažant's SEL). | Empirical Vc eq. based on ρw, Nu, fc′ (larger of two options). No explicit λs factor applied (implicitly assumes stirrups mitigate size effect). Vn=ϕ(Vc+Vs). | Explicit λs factor (for Av<Av,min); Based on Fracture Mechanics (Bažant's SEL). |
EC2 | Empirical VRd,c eq. based on ρl, fck, σcp, modified by explicit Size Effect Factor k=1+200/d. | Variable Angle Truss Model (VATM). Shear resistance based on VRd,s (stirrup yield) and VRd,max (strut crush). No explicit size effect factor in VATM equations. Size implicitly considered via lever arm z≈0.9d and potentially choice of θ. No Vc term used with VATM. | Explicit k factor (for Av<Av,min); Empirical basis. Implicit geometric effect (z) in VATM. |
CSA A23.3-19 | General Method (MCFT): Vc calculated using factor β, which depends on strain ϵx and crack spacing szθ (related to d and aggregate size). | General Method (MCFT): Vc calculated using β (dependent on ϵx,szθ), Vs calculated based on stirrups and angle θ (dependent on ϵx). Vr=Vc+Vs+Vp. Size effect incorporated via β and θ calculations. | Size effect integrated within MCFT calculations via strain (ϵx) and crack spacing (szθ) influencing β and θ. For STM (deep beams), strut strength fcu modified based on strain ϵ1. Based on Compression Field Theory. |
The diversity seen in Table 3 highlights the ongoing evolution and debate surrounding the best way to model concrete shear. ACI 318-19's adoption of an explicit factor based on Bažant's SEL represents a direct incorporation of fracture mechanics principles. EC2 uses an empirical factor for simpler cases and relies on the VATM (which has its own theoretical basis but is less explicit about size effect) for members with stirrups. CSA A23.3 leans heavily on the MCFT, which attempts to integrate size effect more fundamentally through strain and crack spacing considerations. Each approach reflects different research trajectories and balances theoretical rigor, empirical calibration, and practical usability in distinct ways.27
The introduction of the size effect factor and the revised shear provisions in ACI 318-19 has spurred further research, evaluation, and discussion within the structural engineering community.
Reliability and Strength Reduction Factor (ϕ)
A key aspect of code development is ensuring adequate and consistent structural reliability across various design scenarios. Research efforts have focused on evaluating the reliability implications of the ACI 318-19 shear provisions.10 Studies employing Monte Carlo simulations and utilizing updated statistical databases for material properties and experimental results (like the ACI 445 database) have generally concluded that the ACI 318-19 provisions represent a significant improvement in accuracy and reliability compared to previous versions, particularly by addressing the unconservatism for large members without shear reinforcement.10
However, these reliability analyses have also revealed nuances. A notable study by Aguilar et al. (2024) found that while reliability improved overall, it was not uniform across all member types and sizes.10 Their findings suggest that for beams with at least minimum shear reinforcement (Av≥Av,min) and for small-to-medium sized members (d≤33 inches) without shear reinforcement (Av<Av,min), the reliability achieved might justify an increase in the shear strength reduction factor (ϕ) from the current 0.75 to 0.80. Conversely, for large members (d>33 inches) without shear reinforcement, the calculated reliability indices remained below desirable target values (e.g., target β=3.0), indicating that even with the λs factor, the current ϕ=0.75 should not be increased for these specific cases.10 This finding implies that the current ACI 318-19 provisions, while improved, may not achieve perfectly uniform reliability. The suggestion of potentially using different ϕ factors based on member size and the presence of shear reinforcement points towards the complexity of shear behavior and could foreshadow future code refinements, perhaps involving more sophisticated reliability-based formats, to achieve more consistent safety levels across all design situations.
Critiques and Alternative Models
While the adoption of Bažant's SEL represents a significant advancement, ongoing academic discussion continues to explore potential limitations and refinements.12 Some critiques or areas of further investigation include:
Areas for Future Research
Despite the progress made in ACI 318-19, several areas warrant further investigation:
The incorporation of the size effect factor (λs) into the shear design provisions of ACI 318-19 represents a watershed moment in the evolution of concrete building codes in the United States. It signifies a move away from purely empirical formulations towards a design approach more firmly rooted in the principles of fracture mechanics, acknowledging the quasibrittle nature of concrete and the experimentally verified phenomenon that nominal shear strength does not scale linearly with member size.
Recap: ACI 318-19 introduces the factor λs=2/(1+d/10) (for d in inches), based directly on Bažant's Size Effect Law. This factor reduces the calculated concrete shear contribution (Vc) for members deeper than 10 inches when minimal or no shear reinforcement is present (Av<Av,min). The code provides a revised framework for calculating Vc (Table 22.5.5.1), making the calculation dependent on the amount of shear reinforcement provided relative to the minimum required.
Significance: This change addresses a known deficiency in previous code editions, which could be unconservative for large beams and thick slabs designed without shear reinforcement. By incorporating size effect, ACI 318-19 aims to provide more consistent reliability and improved safety across the full spectrum of member sizes encountered in modern construction. It marks a significant step in integrating fundamental material mechanics into practical design standards.
Practical Impact: For structural engineers, the ACI 318-19 shear provisions necessitate adopting new calculation procedures for Vc. The primary consequence is a potentially significant reduction in the calculated shear capacity provided by concrete for deeper members (e.g., deep beams, thick slabs, foundations) that lack minimum shear reinforcement. This may lead to requirements for providing shear reinforcement where none was previously needed, or increasing the amount (closer spacing or larger bars) of shear reinforcement in members already requiring it. These changes can impact reinforcement detailing, potentially increasing congestion, and may have economic implications related to material quantities (concrete and steel) and labor costs. Industry feedback, particularly from the precast sector, highlights concerns about the impact on established designs and manufacturing processes.
Call to Understanding: The shift towards a more mechanics-based approach for shear design in ACI 318-19 underscores the importance for practicing engineers and researchers to not only master the new calculation procedures but also to understand the underlying physical principles. A thorough grasp of fracture mechanics, the size effect phenomenon, and the mechanics of shear transfer is essential for correctly interpreting and applying the code provisions, ensuring the design of safe, reliable, and efficient concrete structures in the face of evolving engineering knowledge and construction practices.