Concrete cone failure

Concrete cone failure is a fracture-dominated failure mode of structural anchors in concrete subjected to tensile loading. Unlike ductile steel yielding, concrete cone failure is characterized by sudden, catastrophic loss of capacity with minimal warning. The failure mechanism is governed by linear elastic fracture mechanics (LEFM) and exhibits a pronounced size effect as described by Bažant's Size Effect Law, where the nominal stress at failure decreases as the structure size increases. The failure surface typically forms at an angle of approximately 35° from the anchor axis, creating a cone-shaped or pyramidal fracture extending to the concrete surface.^[1]

Concrete cone failure for headed stud showing characteristic 35° failure angle

Concrete Cone Model geometry with 3h_ef projected width^[1]

Mechanical models

ACI 349-85 (Historical 45° Cone Method)

The 45° Cone Method, codified in ACI 349-85, assumed a conical failure surface at 45° from the anchor axis with uniform tensile stress distribution equal to the concrete tensile strength. The concrete cone failure load ${\displaystyle N_{0}}$ was calculated as:^[1]

${\displaystyle N_{0}=f_{ct}\cdot A_{N}}$

where:

• ${\displaystyle f_{ct}}$ = direct tensile strength of concrete [psi or MPa]
• ${\displaystyle A_{N}=9h_{ef}^{2}}$ = projected area for square pyramidal failure surface (or ${\displaystyle \pi h_{ef}^{2}}$ for circular approximation)
• ${\displaystyle h_{ef}}$ = effective embedment depth [in or mm]

The capacity scales with ${\displaystyle h_{ef}^{2}}$, assuming the failure surface area is the primary determinant.

Limitations:^[1]

This model has been superseded by the Concrete Capacity Design (CCD) method due to fundamental deficiencies:

1. No size effect: Assumed constant nominal stress regardless of embedment depth, leading to unconservative predictions for ${\displaystyle h_{ef}>200}$ mm
2. Incorrect failure angle: Experimental observations revealed actual failure angles closer to 35°, not 45°
3. Plasticity assumption: Incorrectly assumed simultaneous yielding across the entire failure surface, whereas concrete fracture is progressive and governed by crack propagation

Concrete Capacity Design (CCD) Approach

The Concrete Capacity Design (CCD) Method, developed by Werner Fuchs, Rolf Eligehausen, and John E. Breen in 1995, was based on evaluation of over 1,200 anchor test results and fracture mechanics theory.^[1] It is now the basis for modern design codes including ACI 318 and Eurocode 2 (EN 1992-4).^[2][3]

Under tension loading, the failure surface inclination is approximately 35° to the concrete surface. The basic concrete breakout strength ${\displaystyle N_{b}}$ (or ${\displaystyle N_{0}}$) of a single anchor in uncracked concrete, unaffected by edge influences or overlapping cones of neighboring anchors, is given by:

ACI 318-19 Formulation:^[2]

${\displaystyle N_{b}=k_{c}{\sqrt {f'_{c}}}\,h_{ef}^{1.5}}$

where:

• ${\displaystyle k_{c}=24}$ for cast-in-place anchors (inch-pound units: lb, in, psi)
• ${\displaystyle k_{c}=17}$ for post-installed anchors (inch-pound units)
• SI units: ${\displaystyle k_{c}=13.9}$ (cast-in) or ${\displaystyle k_{c}=9.8}$ (post-installed) when using N, mm, MPa
• ${\displaystyle f'_{c}}$ = specified compressive strength of concrete (cylinder strength) [psi or MPa]
• ${\displaystyle h_{ef}}$ = effective embedment depth [in or mm]

[[#ref_The factor ${\displaystyle k_{c}}$ is strictly unit-dependent. Using inch-pound values (24 or 17) with SI units will produce results approximately 73% too high (conversion factor: 24/13.9 ≈ 1.73), leading to dangerously unconservative designs. Always verify dimensional consistency before calculation.|^]]

EN 1992-4:2018 Formulation:^[3]

${\displaystyle N_{Rk,c}^{0}=k_{cr}{\sqrt {f_{ck}}}\,h_{ef}^{1.5}}$

where:

• ${\displaystyle k_{cr}=11.0}$ for uncracked concrete (SI units: N, mm, MPa)
• ${\displaystyle k_{cr}=7.7}$ for cracked concrete (SI units)
• ${\displaystyle f_{ck}}$ = characteristic cylinder compressive strength of concrete [MPa]
• ${\displaystyle h_{ef}}$ = effective embedment depth [mm]

[[#ref_EN 1992-4:2018 transitioned from cube strength (${\displaystyle f_{ck,cube}}$) to cylinder strength (${\displaystyle f_{ck}}$) to harmonize with international practice. Earlier versions of EN 1992-4 used cube strength with approximate conversion: ${\displaystyle f_{ck}\approx 0.8\,f_{ck,cube}}$.|^]]

The 1.5 Exponent - Size Effect:

The exponent 1.5 (rather than 2.0 from pure geometric similarity) arises from Bažant's Size Effect Law and LEFM principles:^[4]

${\displaystyle N_{b}\propto h_{ef}^{2}\times h_{ef}^{-0.5}=h_{ef}^{1.5}}$

• Geometric component: Projected area ${\displaystyle A_{Nc}\propto h_{ef}^{2}}$
• Fracture mechanics component: Nominal stress at failure ${\displaystyle \sigma _{N}\propto h_{ef}^{-0.5}}$ (from fracture energy and elastic modulus considerations)
• Combined effect: Capacity ${\displaystyle N_{b}=A_{Nc}\times \sigma _{N}\propto h_{ef}^{1.5}}$

This relationship has been validated experimentally for embedment depths ranging from 50 mm to 750 mm.^[4]

Overlapping projected areas in case of anchor groups^[1]

Modification Factors:

Current codes apply reduction factors to the basic capacity ${\displaystyle N_{b}}$ to account for:^[2][3]

• Edge distance effects: ${\displaystyle \psi _{ed,N}}$ (ACI) or ${\displaystyle \psi _{s,N}}$ (EN) - reduced capacity when anchors are near concrete edges
• Eccentric loading: ${\displaystyle \psi _{ec,N}}$ - load not applied concentrically to anchor group centroid
• Cracked concrete: ${\displaystyle \psi _{c,N}}$ (ACI) or incorporated in ${\displaystyle k_{cr}}$ (EN)
• Group effects: Overlapping failure cones reduce capacity proportionally to the ratio ${\displaystyle A_{Nc}/A_{Nco}}$

Comparison of Design Methods

Comparison of 45° Cone Method vs. CCD Method

Feature	ACI 349-85 (45° Cone)	CCD Method (ACI 318, EN 1992-4)	Implication
Failure angle	45° (assumed)	~35° (experimental)	CCD reflects actual behavior
Capacity scaling	${\displaystyle h_{ef}^{2}}$ (area-based)	${\displaystyle h_{ef}^{1.5}}$ (LEFM)	CCD accounts for size effect
Shallow anchors (<100 mm)	May underestimate	Accurate	CCD less conservative
Deep anchors (>300 mm)	Overestimates (unsafe)	Accurate/conservative	Critical safety improvement
Experimental validation	Limited data	1,200+ tests	CCD extensively validated[1]

The CCD method provides accurate and conservative predictions across a wide range of embedment depths, whereas the 45° cone method becomes increasingly unconservative (unsafe) for ${\displaystyle h_{ef}>200}$ mm due to its neglect of the size effect.^[1]

Influence of Head Size

For anchors with large bearing plate or head diameters relative to embedment depth, the bearing pressure in the concrete under the head is reduced, resulting in increased load-carrying capacity beyond the basic CCD prediction. Research has shown that head size effects become significant when the head diameter exceeds approximately 3 times the anchor shaft diameter.^[5][6]

The increased capacity is attributed to:^[6]

• Reduced stress concentrations at the concrete bearing surface
• More favorable stress distribution into the surrounding concrete cone
• Decreased likelihood of bearing failure initiating cone fracture

Different modification factors have been proposed in technical literature to quantify this effect, though specific provisions vary between design codes.^{[citation needed]}

Cracked Concrete

Anchors installed in cracked concrete members exhibit significantly lower load-bearing capacity compared to uncracked conditions. The capacity reduction ranges from 20% to 40% depending on crack width:^[7]

Crack Width	Typical Capacity Reduction	Design Consideration
0.3 mm	20–25%	Moderate reduction
0.5 mm	30–40%	Substantial reduction
> 0.5 mm	> 40%	Special design provisions required

Physical Mechanisms:^[7]

The reduction is primarily due to:

1. Loss of hoop stress transfer: Cracks interrupt the radial confinement of the concrete cone, reducing the ability to resist tensile stresses
2. Reduced aggregate interlock: Normal and tangential stresses cannot be effectively transferred across the crack plane
3. Stress concentration: Cracks act as stress risers, causing failure initiation at lower applied loads

Design Provisions:

• ACI 318-19:^[2] The code assumes cracked concrete conditions unless the designer can demonstrate otherwise. The concrete condition factor is:
  • ${\displaystyle \psi _{c,N}=1.0}$ for cracked concrete (baseline)
  • ${\displaystyle \psi _{c,N}=1.25}$ for cast-in anchors in uncracked concrete
  • ${\displaystyle \psi _{c,N}=1.0}$ for post-installed anchors in uncracked concrete (no increase permitted)

• EN 1992-4:^[3] The code uses different characteristic values:
  • ${\displaystyle k_{cr}=7.7}$ for cracked concrete
  • ${\displaystyle k_{cr}=11.0}$ for uncracked concrete

• Seismic zones: Cracked concrete assumption is mandatory for anchor design in Seismic Design Categories C through F (SDC C-F).^[2]

Under seismic loading, cyclic crack opening and closing further degrades capacity through aggregate interlock loss and surface attrition. This necessitates an additional 0.75 reduction factor per ACI 318-19 Section 17.10.3.1 for concrete-controlled failure modes.^[8]

Seismic Loading

Under seismic loading, anchors experience cyclic crack opening and closing, which degrades concrete cone capacity through several mechanisms:^[8]

Aggregate Interlock Degradation:

During seismic cycling, crack faces undergo attrition—grinding against each other—which smooths surfaces and reduces friction and aggregate interlock. This progressive degradation occurs through two distinct physical processes:^[9]

1. Crushing of asperities: The microscopic peaks of the fracture surface are crushed under high contact stresses, producing fine concrete dust (gouge material)
2. Polishing phenomenon: The accumulated debris acts as a solid lubricant while aggregate edges are worn smooth, further reducing the friction coefficient at the crack interface. This is particularly detrimental to post-installed expansion anchors, which rely on friction between the expansion sleeve and borehole wall

Crack Width Cycling - The "Pumping" Mechanism:

Seismic cracks do not simply open and stay open; they "breathe" due to moment reversals in the structural frame. Each opening/closing cycle can ratchet the anchor outward slightly, accumulating displacement. This pumping effect progressively reduces effective embedment depth and damages the local concrete matrix through alternating crushing (during closing) and release (during opening).^[9]

Zero Interlock Threshold - The 0.8 mm Criterion:

Research has established that aggregate interlock becomes negligible at crack widths exceeding 0.8 mm (0.03 in). This zero interlock threshold represents the crack width where opposing crack faces are sufficiently separated that roughness elements no longer engage effectively, and load transfer shifts entirely to dowel action of the anchor steel. The 0.8 mm value corresponds to the expected crack width when reinforcing steel reaches yield strain during a Design Basis Earthquake.^[9] Inside plastic hinge zones, crack widths can exceed several millimeters, generally prohibiting anchor installation without specialized reinforcement designs.

ACI 318-19 Seismic Provisions:

For anchors in Seismic Design Categories C through F (SDC C-F), ACI 318-19 Section 17.10.3.1 mandates a 0.75 capacity reduction factor for concrete breakout:^[2][8]

${\displaystyle N_{seismic}=0.75\times N_{cb}}$

where ${\displaystyle N_{cb}}$ is the nominal concrete breakout strength.

This factor accounts for:^[2]

• Physical degradation of concrete in cycled cracks
• Uncertainty in crack width during dynamic loading
• Conservative margin for life-safety critical applications

^

EN 1992-4 Seismic Performance Categories:^[3]

European standards employ a performance-based qualification system rather than fixed reduction factors:

• Category C1 (Low seismicity): Qualification testing with crack cycling up to 0.5 mm, applicable to non-critical applications
• Category C2 (High seismicity): Rigorous qualification requiring crack cycling up to 0.8 mm (zero interlock threshold), mandatory for critical structural connections in high seismic zones

The design reduction factor (${\displaystyle \alpha _{seis}}$) is product-specific, derived from C1/C2 testing performance. Values typically range from 0.6 for expansion anchors to near 1.0 for undercut anchors, reflecting actual degradation resistance rather than a blanket safety factor.

Comparison of Code Philosophies:

ACI 318-19 vs. EN 1992-4: Seismic Design Philosophy

Aspect	ACI 318-19 (USA)	EN 1992-4 (Europe)	Technical Basis
Approach	Penalty Factor	Performance Category	ACI applies blanket reduction; EU requires qualification
Seismic Factor	0.75 for concrete modes	${\displaystyle \alpha _{seis}}$ from testing	ACI assumes generic degradation; EU uses product-specific values
Testing Standard	ACI 355.2	EAD 330232/330499	Both require simulated seismic crack cycling
Crack Width	0.5 mm (implicit)	0.8 mm (C2 category)	EU's 0.8 mm threshold based on Hoehler's research
Cast-in Anchors	Treated as baseline	Integrated design, optional qualification	ACI assumes robustness; EU requires proof for critical applications

ACI 318-19 manages seismic risk through a generalized 0.75 reduction factor applied to theoretical static strength. EN 1992-4 employs a performance-based system where anchors must demonstrate specific resilience through rigorous C1/C2 qualification testing.^[9]

Qualification Testing:

ACI 355.2 requires post-installed anchors intended for use in SDC C-F to undergo simulated seismic testing:^[10]

1. Anchor installed in hairline pre-crack
2. Sustained tension load applied
3. Crack cycled between 0.0 mm and 0.5 mm for 10 cycles
4. Pass criterion: Residual capacity ≥ 160% of sustained load

European C2 qualification employs more severe protocols with crack widths up to 0.8 mm to verify performance at the zero interlock threshold.^[3]

Recent research suggests the 0.75 factor may be conservative for concrete breakout but potentially unconservative for certain steel failure modes in combined tension and shear.^[8]

Sustained Loading

Under high sustained tension loads, time-dependent concrete fracture (analogous to tertiary creep rupture) progressively reduces capacity. Experimental investigations indicate substantial capacity degradation under long-term loading:^[11]

Sustained Load Level (% of ultimate)	Approximate Time to Failure[11]	Design Consideration
50%	> 50 years	Safe for typical design life
60–70%	~50 years	95% survival probability
80%	~5 years	Unacceptable for permanent structures
90%	~6 months	Imminent failure risk

Design Implication: For 50-year service life, sustained loads should not exceed 55–60% of short-term ultimate capacity to maintain acceptable reliability levels.^[11]

Supplementary Reinforcement

ACI 318 distinguishes between two types of reinforcement in anchorage zones, with fundamentally different design purposes and verification requirements:^[2]

Supplementary Reinforcement (Confinement)

Supplementary reinforcement consists of ties, stirrups, or hairpins present in the potential breakout zone but not specifically detailed to transfer the full anchor load. Its primary functions are:^[2]

• Restrain crack widths to maintain aggregate interlock
• Provide confinement to the concrete failure cone
• Convert purely brittle failure to more gradual failure with warning

Design Benefit: When supplementary reinforcement meeting the code-prescribed detailing requirements is present, the strength reduction factor may be increased (Condition A: ${\displaystyle \phi =0.75}$ vs. Condition B: ${\displaystyle \phi =0.70}$). However, this reinforcement does not replace the concrete breakout capacity verification—it acts as backup protection to prevent catastrophic sudden failure.^[2]

Anchor Reinforcement (Load Transfer)

When concrete cone capacity is insufficient to resist design loads, anchor reinforcement (also called ductile steel element) is designed to carry 100% of the tensile load. This transforms the governing failure mode from brittle concrete fracture to ductile steel yielding, significantly improving structural safety and is the preferred approach for high-seismic or heavy industrial applications.^[2]

Design Criteria

ACI 318-19 Approach:^[2]

Per Section 17.5.2.9, when anchor reinforcement is provided to develop the full factored load:

${\displaystyle N_{sa}=n\cdot A_{se}\cdot f_{ya}}$

where:

• ${\displaystyle n}$ = number of reinforcing bars effectively engaged
• ${\displaystyle A_{se}}$ = effective cross-sectional area of reinforcement [mm² or in²]
• ${\displaystyle f_{ya}}$ = specified yield strength of reinforcement [MPa or psi]

When the reinforcement capacity ${\displaystyle N_{sa}}$ equals or exceeds the required factored tensile load ${\displaystyle N_{ua}}$, and the reinforcement is fully developed on both sides of the potential failure surface, concrete breakout verification may be waived.^[2]

EN 1992-4 Approach:^[3]

${\displaystyle N_{Rd,re}=\sum {\frac {A_{s}\cdot f_{yk}}{\gamma _{Ms}}}}$

where:

• ${\displaystyle A_{s}}$ = area of one anchor reinforcing bar [mm²]
• ${\displaystyle f_{yk}}$ = characteristic yield strength of reinforcement [MPa]
• ${\displaystyle \gamma _{Ms}=1.15}$ = partial safety factor for steel

Development Length Challenge

A critical practical constraint arises from geometric limitations:^[2]

The reinforcing bars must achieve full development length on both sides of the potential 35° failure surface. For anchors with shallow embedment (${\displaystyle h_{ef}<150}$ mm), achieving adequate development length between the failure plane and the concrete surface is often geometrically impossible with straight bars.

Solutions:^[2]

• Hooked bars: Standard 90° or 180° hooks reduce required development length by 30–50%
• Headed reinforcing bars: Mechanical end anchorage eliminates development length requirements
• Increased embedment: Design for deeper embedment when reinforcement is critical
• Strut-and-tie modeling: Alternative rational design approach for complex geometries

Strut-and-Tie Modeling

For complex anchorage zones such as corbels, beam ledges, or slab edges, strut-and-tie modeling per ACI 318 Chapter 23 or EN 1992-1-1 Section 6.5 provides a rigorous alternative to empirical anchor design:^[2]

• Tension ties: Anchor reinforcement modeled as discrete tension members
• Compression struts: Concrete stress field modeled at approximately 35° (matching cone geometry)
• Nodal zones: Force transfer points designed for adequate bearing capacity (typically ${\displaystyle 0.85f'_{c}\times A_{bearing}}$)

Example Application:

For a corbel with anchor tension ${\displaystyle T=200}$ kN:

• Tension tie capacity: ${\displaystyle A_{s}\times f_{y}\geq 200}$ kN
• Compression strut angle: ${\displaystyle \theta \approx 35^{\circ }}$ (matching failure cone)
• Nodal zone bearing: ${\displaystyle 0.85f'_{c}\times A_{bearing}\geq T/\sin \theta }$

This method is particularly valuable when standard anchorage provisions do not apply or when multiple load paths interact.^[2]