⚡ electrical engineering

A closed-Form Upper Bound on the Concrete Breakout Resistance of Anchor Groups in Rectangular Pedestals Under the EN 1992-4 Narrow-Member Rule

This paper derives a closed-form upper bound on the concrete breakout resistance for groups of headed bolts in rectangular pedestals under EN 1992-4, demonstrating that this resistance is capped by the pedestal's plan dimensions rather than embedment depth, and providing design charts and comparisons to ACI 318 to show that concrete breakout typically governs over steel strength in practical applications.

Original authors: Eghosasere Oluwaseyi Rowland-Lato

Published 2026-10-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Eghosasere Oluwaseyi Rowland-Lato

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a heavy piece of machinery, a tall tower, or a building column standing on a concrete block. To keep it from tipping over or being pulled away by strong winds, engineers bolt it down to that block using thick metal rods. These rods are embedded deep inside the concrete. For decades, engineers have relied on a set of rules to calculate how much weight these bolts can hold before the concrete around them shatters and breaks apart. The rules assume that if you make the concrete block deeper, the bolts can hold more weight. It seems logical: more concrete means a stronger grip. However, there is a hidden catch in these rules for blocks that are not very wide. If the block is narrow, the concrete can only hold so much weight, no matter how deep you bury the bolts. Once the bolts reach a certain depth, the strength stops growing. It hits a ceiling.

This paper investigates exactly where that ceiling is and what it means for the safety of structures. The author, an engineer from the University of Port Harcourt, looked at the specific mathematical rules used in Europe to design these connections. He asked a simple but critical question: for a concrete block of a given size, what is the absolute maximum weight it can hold before the concrete breaks, regardless of how many bolts are used or how strong those bolts are? The study focuses on rectangular blocks, which are common in construction, and examines groups of bolts arranged symmetrically. The goal was to find the theoretical limit of the concrete's strength, a limit that cannot be exceeded by simply adding more steel or making the bolts longer.

The research reveals a surprising truth about how these connections fail. The maximum weight a concrete block can support is determined entirely by the size of the block itself and the strength of the concrete, not by the bolts. Once the bolts are spaced in a specific, optimal pattern, adding more bolts or using stronger steel does not increase the load the concrete can carry. The concrete will break first, and it will break at the same weight whether there are two bolts or twenty. The study found that the best way to arrange the bolts to reach this maximum strength is to place the outermost bolts at a specific distance from the edges, effectively spreading them out to cover the widest possible area without hitting the sides of the block. If the block is a square, this arrangement is straightforward. If the block is a rectangle, the optimal spacing changes slightly depending on how long and narrow it is, but the principle remains the same: the concrete's capacity is fixed by its geometry.

The author calculated this maximum limit using a method called numerical optimization, which is like running thousands of computer simulations to test every possible arrangement of bolts. The results confirmed a precise mathematical formula for this upper limit. For a square block, the strongest arrangement occurs when the outer bolts are placed at the quarter points of the side length. For a rectangular block, the limit is slightly lower if the block is very narrow, but it follows a predictable pattern. The study also compared these European rules with similar rules used in the United States. It found that the American rules allow for a slightly higher weight limit, about eighteen percent more, but only for blocks that are not extremely deep. For deeper blocks, the difference grows.

Perhaps the most significant finding concerns the safety of the structure. Engineers often prefer a design where the metal bolts stretch and bend before the concrete breaks. This is called a ductile failure, and it gives people a warning sign before a collapse happens. If the concrete breaks suddenly, it is a brittle failure, which is dangerous. The paper shows that for the concrete to hold enough weight to let the bolts bend first, the concrete block needs to be enormous. For a typical set of four bolts used in construction, the block would need to be nearly two meters wide to allow the bolts to bend safely. Most real-world blocks are much smaller, often less than a meter wide. This means that in almost every practical situation, the concrete will break before the bolts bend. The only way to achieve the safer, bending failure mode in these smaller blocks is to add extra steel reinforcement inside the concrete, specifically designed to hold the load if the bolts start to stretch.

The study also provides a practical tool for designers. It offers a chart that tells engineers exactly how big a concrete block needs to be to hold a specific weight using only the concrete's strength. If a block is smaller than the chart suggests, the engineer knows immediately that they must add reinforcement, regardless of how strong the bolts are. This removes the guesswork from the early stages of design. The research confirms that while you can make the bolts stronger, you cannot make the concrete stronger than its geometric limit. The concrete has a maximum capacity, and once you hit it, the only way to go higher is to make the block bigger or to reinforce it with steel. This insight helps engineers design safer, more efficient structures by understanding the true limits of the materials they use.

Technical Summary: A Closed-Form Upper Bound on Concrete Breakout Resistance

Problem Statement
In structural engineering, tension loads from column bases, tower stanchions, and equipment supports are frequently anchored via cast-in headed bolts into concrete pedestals. These pedestals often have plan dimensions only slightly larger than the base plate (typically 500–1000 mm). Under the Eurocode 2 (EN 1992-4) and ACI 318-19 design frameworks, such members are classified as "narrow members." In these cases, the effective embedment depth ( $h_{ef}$ ) is capped by the narrow-member rule, which limits $h_{ef}$ based on the largest edge distance and bolt spacing rather than the actual bolt length.

This rule creates a design plateau: once the embedment is sufficient to trigger the narrow-member constraint, increasing the bolt length, diameter, or grade yields no additional concrete breakout resistance. While this plateau is implicit in the code equations, the literature lacked a closed-form expression defining the absolute maximum resistance achievable for a pedestal of a given size, regardless of the bolt layout. Consequently, designers could not easily determine the theoretical upper limit of tension a specific pedestal could carry without anchor reinforcement.

Methodology
The paper derives a closed-form upper bound for the design concrete breakout resistance ( $N_{Rd,c}$ ) of symmetric anchor groups in rectangular pedestals with sides $a \le b$ . The analysis relies on the following assumptions and steps:

Model Formulation: The study utilizes the Concrete Capacity Design (CCD) method as implemented in EN 1992-4. It assumes concentric tension loading, doubly symmetric groups, and sufficient member depth to prevent thickness-controlled failure.
Optimization: The derivation maximizes the resistance equation over all possible outer anchor spacings ( $g_x$ and $g_y$ ). The narrow-member rule defines the effective embedment as $h'_{ef} = \max(c_{max}/1.5, s_{max}/3)$ , where $s_{max}$ is interpreted as the spacing of the outermost anchors.
Analytical Derivation: The author analytically solves for the optimal spacing that minimizes $h'_{ef}$ while maximizing the projected area and edge factors. This is performed for both square ( $a=b$ ) and rectangular ( $a<b$ ) geometries.
Verification: The derived closed-form solutions are verified through:
- Numerical Optimization: A grid search (70x70) followed by Nelder-Mead refinement over anchor gauges for groups ranging from 1x1 to 4x4.
- Symbolic Algebra: Computer algebra systems confirmed the derivatives and simplifications.
- Comparison: The results are compared against ACI 318-19 provisions and existing test data ranges.

Key Contributions and Results
The paper establishes that there is a "ceiling" to concrete breakout resistance determined solely by the pedestal geometry and concrete strength, independent of the number, diameter, or grade of the bolts.

Closed-Form Upper Bound: For a rectangular pedestal with sides $a \le b$ and aspect ratio $r = a/b$ , the maximum resistance is:
$N_{Rd,c,max} = \frac{\sqrt{6}}{9} \frac{k_1 \sqrt{f_{ck}} a \sqrt{b} \psi(r)}{\gamma_{Mc}}$
Where $\psi(r) = 1$ if $r \ge 0.5$ , and $\psi(r) = 0.7 + 0.6r$ if $r < 0.5$ .
For a single central anchor, the bound is lower: $N_{Rd,c,max}^{(1)} = \frac{k_1 \sqrt{f_{ck}} b^{1.5} r(0.7 + 0.3r)}{31.5 \gamma_{Mc}}$ .
Optimal Layout: The maximum resistance is achieved when the outer anchors are spaced at $b/2$ along the long side. For square pedestals, this corresponds to placing outer anchors at the quarter points ( $g = a/2$ ). Any symmetric group (2x2, 3x3, etc.) with outer anchors at these positions attains the same maximum; adding more inner anchors does not increase the ceiling.
Geometric Efficiency: For a fixed plan area, a square pedestal provides the highest resistance. Elongating the pedestal reduces the ceiling proportionally to $r^{0.25}$ .
ACI 318-19 Comparison: The ACI 318-19 counterpart follows the same structural form. For optimal groups where the limited embedment is below 280 mm, the ACI ceiling is approximately 1.18 times higher than the EN 1992-4 value.
Ductility Implications: The study calculates the pedestal sizes required for the concrete breakout ceiling to exceed the steel strength of the bolts (ensuring ductile steel failure). The results indicate that for practical bolt sizes (e.g., four M24 grade 4.6 bolts in C30/37 concrete), the required pedestal side is approximately 1.6 m. Since typical pedestals are much smaller (e.g., 700 mm), concrete breakout invariably governs in practice. Achieving ductility via bolt selection alone is impossible in standard pedestals; it requires anchor reinforcement.

Significance and Claims
The paper claims to provide the first closed-form statement of the maximum concrete breakout resistance for narrow-member pedestals. Its primary significance lies in:

Design Clarity: It allows designers to immediately determine the maximum tension a pedestal can carry based solely on its dimensions and concrete grade, without iterative bolt layout optimization.
Realistic Expectations: It demonstrates that increasing bolt size or embedment beyond the narrow-member limit is futile for increasing concrete capacity.
Ductility Guidance: It quantifies the impracticality of achieving ductile failure modes (steel yielding before concrete breakout) in typical pedestal sizes, reinforcing the necessity of anchor reinforcement for ductile design.
Conservatism: The author notes that the derived ceiling represents the upper bound allowed by the code model, which is likely conservative compared to physical test results that may show higher resistance due to member thickness and surface reinforcement effects not fully credited in the standard model.

The paper concludes with a design chart (Eq. 7) to determine the required pedestal side for a given design tension, serving as a direct tool for preliminary design and the decision to implement anchor reinforcement.

Technical Summary: A Closed-Form Upper Bound on Concrete Breakout Resistance

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