Design of Spread Foundations on Rock Mass in the Second Generation of Eurocode 7
Abstract
1. Introduction
1.1. The Structural Eurocode System and Its Evolution to the Second Generation
- Restructuring the document to make it more consistent with other Eurocodes, easier to understand and navigate, more comprehensive in its technical coverage, and easier to add new topics.
- Improving guidance on selecting characteristic ground parameters and designing water pressures, application of numerical methods, rock engineering, and dynamic design.
- Improving ease of use by clarifying existing clauses, removing repetitions, and removing unnecessary information.
- Part 1—General rules (EN 1997-1 [2]). This part supplements EN 1990 [1] by setting out additional principles and requirements for the safety, serviceability, robustness, and durability of geotechnical structures. Design and verification are based on the partial factor method or other reliability-based methods, prescriptive rules, testing, or the observational method.
- Part 2—Ground properties (EN 1997-2 [3]). This part establishes rules for obtaining information about the ground at a site, necessary for the design and execution of geotechnical structures, including temporary geotechnical structures.
- Part 3—Geotechnical structures (EN 1997-3 [4]). This part establishes principles and requirements for the design and verification of: (i) permanent or temporary geotechnical structures (slopes, cuttings, embankments, shallow foundations, piled foundations, retaining structures, reinforced fill structures, soil nailed structures, and ground improvement); (ii) supporting elements (anchors, reinforcing elements in reinforced fill structures, soil nails, rock bolts and rock surface support, and ground improvement); (iii) groundwater control measures (reduction of hydraulic conductivity, dewatering and infiltration, and the use of impermeable barriers).
- Clear definitions of rock, rock masses, and discontinuities, with explicit consideration of the discontinuous nature of rock masses.
- Recognition that rock mass geometrical properties—such as discontinuity spacing and orientation—can be treated probabilistically.
- Recognition that ground properties must consider the geometrical properties of the discontinuities when they affect the mechanical behavior of the ground.
- Recognition that reliability-based methods may be used.
- Understanding that risk-informed approaches may apply in design situations where uncertainties or consequences are outside common ranges.
- Inclusion of rock mass classification systems.
- Inclusion of the physical and chemical properties relevant to rock.
- Acknowledgement of the potential need to determine in situ stress states, along with methods for their assessment.
- Introduction of specific failure envelopes for rock, rock masses, and discontinuities, along with specific aspects of rock and rock mass deformability.
- Identification of failure mechanisms specific to rock masses.
- Specification of partial factor values on ground properties for rock, rock masses, and discontinuities.
- Increased attention given to the use of the observational method and prescriptive rules in rock engineering design.
- Design considerations for rock anchors, rock bolts, and surface support systems.
- Guidance on groundwater control measures in rock engineering applications.
1.2. Application of the Second Generation of the Eurocodes to the Design of Spread Foundations
2. Design of Spread Foundations on Rock Mass in Eurocode 7
2.1. Basis of Limit State Design in Eurocode 7 in EN 1990 and EN 1997-1
2.1.1. General
2.1.2. Reliability-Based Design
2.1.3. Partial Factors in the Semi-Probabilistic Framework
2.1.4. Consideration of Geometrical Uncertainties
2.2. Rock Mass Behavior and Ground Properties in EN 1997-1 and EN 1997-2
2.3. Bearing Failure Limit State for Spread Foundations in EN 1997-3
2.4. Calculation Model for Spread Foundations on Continuous Rock
3. Case Study
3.1. Basic Assumptions
3.2. Ground Conditions
- Geological Strength Index: GSI = 40
- Uniaxial compressive strength of intact rock: σci = 5.0 MPa
- Non-dimensional material parameter: mi = 7
- Disturbance factor: D = 0
3.3. Methodologies Used for Safety Verification
4. Derivation of Basic Input Data
4.1. Representative Values of the “Equivalent” Mohr–Coulomb Parameters
4.2. Mean Values of the “Equivalent” Mohr–Coulomb Parameters
4.3. Mean Values of Actions
4.4. Other Sources of Uncertainty
4.4.1. Geometrical Properties
4.4.2. Bearing Resistance Model
- When expressing the bearing resistance in terms of Mohr–Coulomb parameters, which are obtained by linearizing the Hoek–Brown failure envelope [24] within a specific range of minor principal stresses, there is an error due to a second-order behavior that is being ignored. A small coefficient of variation of 1% was obtained when analysing the ratio between equivalent Mohr–Coulomb and Hoek–Brown models within the fitting range (Figure 3).
- The Hoek–Brown envelope itself is an empirical model obtained after curve fitting to testing data [34]. The standard deviations obtained could be considered a measurement of the Hoek-Brown model’s uncertainty, but such values are not given. A roughly estimate of 10% for the coefficient of variation can be assumed, given the coefficients of determination between 0.68 and 0.99 obtained for different rock materials [34].
- The wedge failure mechanism is an idealization of the true mechanism of bearing failure of spread foundations on rock mass. Other failure mechanisms have been proposed by other authors [35]. Because of that, there is a model uncertainty, which is difficult to quantify.
5. Safety Verification Against Bearing Failure
5.1. Deterministic Approach—Allowable Stress Design
5.2. Semi-Probabilistic Eurocode 7 Solution—Limit State Design
5.2.1. General
5.2.2. Using the “Equivalent” Mohr–Coulomb Failure Envelope
5.2.3. Using the Hoek–Brown Failure Envelope
5.3. Probabilistic Analysis—Limit State Design
6. Discussion
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
CEN | European Committee for Standardization |
ULS | Ultimate Limit State |
CC | Consequence Class |
SF | Global Safety Factor |
MFA | Material Factor Approach |
RFA | Resistance Factor Approach |
VC | Verification Case |
JRC | Joint Research Centre |
GSI | Geological Strength Index |
JCSS | Joint Committee on Structural Safety |
FORM | First-Order Reliability Method |
SORM | Second-Order Reliability Method |
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Consequence Class | 1-Year Reference Period β | 50-Year Reference Period | |
---|---|---|---|
β50 | Pf,50 | ||
CC3 | 5.2 | 4.3 | ~10−5 |
CC2 | 4.7 | 3.8 | ~10−4 |
CC1 | 4.2 | 3.3 | ~10−3 |
Partial Factors on | Symbol | Material Factor Approach, Either both (a) and (b) or only (c) | Resistance Factor Approach, Either (d) or (e) | |||
---|---|---|---|---|---|---|
(a) | (b) | (c) | (d) | (e) | ||
Actions, effects of actions | γF, γE | VC1 | VC3 | VC1 | VC1 | VC4 |
Ground properties | γM | M1 | M2 | M2 | Not factored | |
Bearing resistance | γRN | Not factored | 1.4 |
Action or Effect | Partial Factor γF and γE for Verification Cases | |||
---|---|---|---|---|
Type | Resulting Effect | Structural Resistance | Geotechnical Design | |
Verification Case | VC1 | VC3 | VC4 | |
Permanent action (Gk) | Unfavorable γG | 1.35 kF | 1.0 | Not applied |
Favorable γG,fav | 1.0 | 1.0 | ||
Variable action (Qk) | Unfavorable γQ | 1.5 kF | 1.3 | 1.5/1.35 |
Favorable γQ,fav | 0 | 0 | 0 | |
Effects of actions (E) | Unfavorable γE | Not applied | Not applied | 1.35 kF |
Favorable γE,fav | 1.0 |
Ground Property | Symbol | Set M1 | Set M2 | |
---|---|---|---|---|
Rock material and rock mass | Shear strength | γτr | 1.0 | 1.25 kM |
Unconfined compressive strength (for foundation purposes only) | γqu | 1.0 | 1.40 kM | |
Rock discontinuities | Shear strength | γτdis | 1.0 | 1.25 kM |
Coefficient of residual friction (when roughness component neglected) | γtanφdis,r | 1.0 | 1.10 kM |
Consequence Class | Description | Consequence Factor | ||
---|---|---|---|---|
Actions kF | Resistances kR | Material Properties kM | ||
CC3 | High | 1.1 | 1.1 | 1.1 |
CC2 | Normal | 1.0 | 1.0 | 1.0 |
CC1 | Low | 0.9 | 0.9 | 0.9 |
Rock Property | Symbol | Coefficient of Variation, Vx (%) |
---|---|---|
Weight density | γ | 5–10 |
Peak or residual effective cohesion | c′p or c′r | 30–50 |
Coefficient of friction | tan φ | 5–15 |
Shear strength at failure | τf | 15–25 |
Unconfined compressive strength | qu | 20–80 |
Quantity | Symbol | Mean Value | Units |
---|---|---|---|
Permanent vertical load | NG | 1100.0 | kN/m |
Variable vertical load | NQ | 217.8 | kN/m |
Variable moment | MQ | 87.1 | kNm/m |
Foundation width | B | 4.00 | m |
Cohesion | c′ | 123.3 | kPa |
Friction angle | φ′ | 44.7 | ° |
Quantity | Symbol | Expression | Value | Units |
---|---|---|---|---|
Normal load | N | NG + NQ | 1317.8 | kN/m |
Moment | M | MQ | 87.1 | kNm/m |
Eccentricity | eB | M/N | 0.07 | m |
Effective width | B′ | B – 2 × eB | 3.87 | m |
Major principal stress | σ1 | Equation (13) | 1541.3 | kPa |
Bearing resistance | RN | Equation (12) | 5961.4 | kN/m |
Safety factor | SF | Equation (28) | 4.52 |
Quantity | Symbol | Units | VC1 | VC3 | VC4 |
---|---|---|---|---|---|
Permanent vertical load | NG;d | kN/m | 1634 | 1100 | 1100 |
Variable vertical load | NQ;d | kN/m | 825 | 650 | 556 |
Variable moment | MQ;d | kNm/m | 330 | 260 | 222 |
Eccentricity | e′B | m | 0.13 | 0.15 | 0.09 |
Effective width | B′ | m | 3.63 | 3.60 | 3.72 |
Vertical load | Nd | kN/m | 2459 | 1750 | 2459 |
Quantity | Symbol | Units | MFA | MFA | MFA | RFA | RFA |
---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (d) | (e) | |||
Major principal stress | σ1 | kPa | 1017 | 680 | 761 | 1017 | 1017 |
Bearing resistance | RN;d | kN/m | 3692 | 2450 | 2765 | 2637 | 2701 |
Vertical load | Nd | kN/m | 2459 | 1750 | 2459 | 2459 | 2459 |
Utilization ratio—Nd/RN;d | 67% | 71% | 89% | 93% | 91% | ||
Overdesign factor—RN;d/Nd | 1.50 | 1.40 | 1.12 | 1.07 | 1.09 |
Partial Factor γτr | GSI | mi | σci (MPa) |
---|---|---|---|
1.00 (comb. (a)) | 40 | 7.00 | 5.00 |
1.25 kM (comb. (b)) | 32.98 | 5.12 | 3.36 |
1.25 (comb. (c)) | 35.26 | 5.52 | 4.00 |
Quantity | Symbol | Units | MFA | MFA | MFA | RFA | RFA |
---|---|---|---|---|---|---|---|
(a) | (b) | (c) | (d) | (e) | |||
Major principal stress | σ1 | kPa | 975 | 667 | 741 | 975 | 975 |
Bearing resistance | RN;d | kN/m | 3541 | 2402 | 2693 | 2530 | 2590 |
Vertical load | Nd | kN/m | 2459 | 1750 | 2459 | 2459 | 2459 |
Utilization ratio—Nd/RN;d | - | 69% | 73% | 91% | 97% | 95% | |
Overdesign factor—RN;d/Nd | - | 1.44 | 1.37 | 1.10 | 1.03 | 1.05 |
Random Variable | Symbol | Units | Distribution | Mean Value | Coefficient of Variation |
---|---|---|---|---|---|
Foundation width | B | m | Normal | 4.00 | 1.5% |
Permanent vertical load | G | kN/m | Normal | 1100.00 | 4% |
Variable vertical load | Q | kN/m | Gumbel | 217.76 | 50% |
Variable moment | M | kNm/m | Gumbel | 87.10 | 50% |
Cohesion | c′ | kPa | Log-normal | 123.30 | 40% |
Coefficient of friction | tan φ′ | Log-normal | 0.99 | 10% | |
Model uncertainty | θM | Normal | 1.00 | 15% |
B | G | Q | M | c′ | tan φ′ | θM |
---|---|---|---|---|---|---|
−0.03 | 0.06 | 0.18 | 0.03 | −0.23 | −0.23 | −0.92 |
Approach | Safety Indicator | Comb. | Mohr–Coulomb | Hoek–Brown |
---|---|---|---|---|
Deterministic | Global safety factor | 4.52 (>3.00) | ||
Semi-probabilistic | Overdesign factor | (a) and (b) | (1.50 and 1.40) 1.40 | (1.44 and 1.37) 1.37 |
(c) | 1.12 | 1.10 | ||
(d) | 1.07 | 1.03 | ||
(e) | 1.09 | 1.05 | ||
Probabilistic | Reliability index | 4.89 (>4.30) |
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Pereira, R.; Bogusz, W.; Lamas, L. Design of Spread Foundations on Rock Mass in the Second Generation of Eurocode 7. Geotechnics 2025, 5, 46. https://doi.org/10.3390/geotechnics5030046
Pereira R, Bogusz W, Lamas L. Design of Spread Foundations on Rock Mass in the Second Generation of Eurocode 7. Geotechnics. 2025; 5(3):46. https://doi.org/10.3390/geotechnics5030046
Chicago/Turabian StylePereira, Renato, Witold Bogusz, and Luís Lamas. 2025. "Design of Spread Foundations on Rock Mass in the Second Generation of Eurocode 7" Geotechnics 5, no. 3: 46. https://doi.org/10.3390/geotechnics5030046
APA StylePereira, R., Bogusz, W., & Lamas, L. (2025). Design of Spread Foundations on Rock Mass in the Second Generation of Eurocode 7. Geotechnics, 5(3), 46. https://doi.org/10.3390/geotechnics5030046