Design Approaches for Determining Ground Failure Resistance According to Eurocode 7 (EN 1997-1)

In Eurocode 7, there are three design approaches for determining the ground failure resistance.

• Approach 1

• Approach 2

• Approach 3

In this article, the approaches are compared on the model of a foundation plate with a column. The differences between the individual approaches lies in the partial safety factors that affect various influencing values. These include the actions or loading, soil parameters, and resistances. It is important to mention that these reductions or increases sometimes occur in the approaches in combination. Furthermore, the German National Annex describes special rules for the application of Approach 2, which is also known as Design Approach 2* or 2+. In the following text and in RFEM 6, the term 2* is used for this approach.

System of Foundation Plate with Column

Foundation plate

• Length: w_x = 2.50 m
• Width: w_y = 2,50 m
• Thickness: t = 1.00 m
• Embedment depth: D = 1.00 m
• Self-weight G_p,k = 156.25 kN with γ = 25 kN/m³

Column

• Length: c_x = 0.50 m
• Width: c_y = 0.50 m
• Height: h = 4.00 m
• Self-weight: G_c,k = 25 kN with γ = 25 kN/m³

Ground Parameters

• Friction angle: φ'_d = 32°
• Shear parameter for cohesion: c'_k = 15 kN/m²
• Density of soil next to the foundation plate: γ_1.k 20 kN/m³
• Bulk density of soil below the foundation slab: γ_2,k = 20 kN/m³

Load Case 1 – Permanent Loads

• Vertical: V_G,z,k = 975 kN

Including the self-weight of the column G_c,k = 25 kN and the foundation G_p,k = 156.25 kN, the sum of the permanent vertical loads is V_G,k,tot = 156.25 kN + 25 kN + 975 kN = 1,156.25 kN. The self-weight of the foundation is automatically taken into account with the self-weight of the structure as long as the "Active self-weight" check box is selected. If the self-weight is to be entered manually, it is necessary to define additional loads for the foundation.

Load Case 2 – Variable Loads

• Vertical: V_Q,z,k = 1,000 kN
• Horizontal: H_Q,x,k = 190 kN

Partial Safety Factors

The following table shows the partial safety factors according to EN 1997‑1, A.3.

Actions A	Symbol	A1	A2
Permanent loads	γG	1.35	1.00
Variable loads	γQ	1.50	1.30
Soil Parameters (Material M)	Symbol	M1	M2
Effective shear angles	γ'φ	1.00	1.25
Effective cohesion	γ'c	1.00	1.25
Specific Weight	γγ	1.00	1.00
Resistance R	Symbol	R1	R2	R3
Ground Failure	γR;v	1.00	1.40	1.00
Sliding	γR;h	1.00	1.40	1.00

Approach 1

This design approach uses two different sets of partial safety factors.

In the first combination 1-1, the partial safety factors A1, M1, and R1 are used, with A1 (γ_G = 1.35; γ_Q = 1.5) increasing the unfavorable actions on the foundation, M1 (γ'_φ = γ'_c = γ_γ = 1.00) not reducing the soil parameters, and R1 (γ_R;v = γ_R;h = 1.00) not reducing the resistances.

In the second combination 1-2, A2, M2, and R1 are used, with A2 (γ_G = 1.00; γ_Q = 1.30) increasing the actions less than A1, and M2 (γ'_φ = γ '_c = 1.25; γ γ_γ = 1.00) reducing the soil parameters by reducing the soil resistance to ground failure.

For the design, the calculation has to be performed with both sets of partial safety factors, and the set with the higher ratio is governing.

Approach 1 (Combination 1-1) According to EN 1997-1, 2.4.7.3.4.2

Calculation of Ground Failure Resistance

Eccentricity e_x of Effective Vertical Load in x-Direction

The design shear force with additional foundation loads V_z,+add as well as the value of the resulting design bending moment M_y,+add in the foundation base center are required to determine the eccentricity of effective vertical loads.

V_z,+add,d = γ_G ⋅ V_G,k + γ_Q ⋅ V_Q,k = 1.35 ⋅ 1,156.25 kN + 1.5 ⋅ 1,000 kN = 3,060.94 kN

H_Q,x,d = γ_Q ⋅ H_Q,x,k = 1.50 ⋅ 190 kN = 285 kN

M_y,+add,d = (t + h) ⋅ H_Q,x,d = (1.00 m + 4.00 m) ⋅ 285 kN = 1,425 kNm

e_x = -M_y,+add,d / V_z,+add,d = -1,425 kNm / 3,060.94 kN = -0.466 m

Effective Foundation Length, Width, and Base

The eccentric loading reduces the allowable foundation base.

w_x - 2 ⋅ |e_x| = 2,50 m - 2 ⋅ 0,466 m = 1,569 m

w_y - 2 ⋅ |e_y| = 2.50 m - 2 ⋅ 0.000 m = 2.500 m

Effective length: L' = max(w_x - 2 ⋅ |e_x|; w_y - 2 ⋅ |e_y|) = 2.500 m

Effective width: B' = min(w_x - 2 ⋅ |e_x|; w_y - 2 ⋅ |e_y|) = 1.569 m

Effective area: A' = L' ⋅ B' = 2.500 m ⋅ 1.569 m = 3.922 m²

Soil Parameters to Use

Friction angle: φ'_d = arctan(tan(φ'_k ) / γ'_φ ) = arctan(tan(32°) / 1.00) = 32°

Shear parameter for cohesion: c'_d = c'_k / γ'_c = 15 kN/m² / 1.00 = 15 kN/m²

Bulk density: γ_1d = γ_2d = γ_1k / γ_γ = γ_2k / γ_γ = 20 kN/m³ / 1.00 = 20 kN/m³

The friction angle φ' describes the angle at which the shear strength of a soil is reached by friction between the soil components. In contrast, cohesion c' refers to the ratio of the shear strength that results from internal binding forces between the soil components – regardless of the stress applied. Both parameters play a central role in determining the shear strength of a soil under various loading conditions. The weight of the soil next to the foundation slab is denoted by γ_1d, the weight of the soil below the foundation slab by γ_2d.

Bearing Capacity Coefficients

N_q = e^{π ⋅ tan(φ‘_d)} ⋅ tan²(45°+φ‘_d / 2) = e^{π ⋅ tan(32°)} ⋅ tan²(45° + 32° / 2) = 23.18

The factor N_q considers the load-bearing capacity due to the self-weight of the soil.

N_c = (N_q - 1) ⋅ cot(φ‘_d ) = (23.18 kN - 1) ⋅ cot(32°) = 35.49

The factor N_c takes into account the load-bearing capacity due to the cohesion of the soil.

N_γ = 2 ⋅ (N_q - 1) ⋅ tan(φ‘_d) = 2 ⋅ (23.18 kN - 1) ⋅ tan(32°) = 27.72 mit δ ≥ φ'_d / 2 (rough base)

The N_γ factor considers the bearing capacity due to the shear strength of the soil.

Inclination of Foundation Base

b_q = (1 - α ⋅ tan(φ'_d))² = (1 - 0)² = 1

b_c = b_q - (1 - b_q) / (N_c ⋅ tan(φ'_d)) = 1 - 0 = 1

b_γ = b_q = 1

In this example, the inclination of the base area α = 0°, and thus has no influence on the bearing resistance.

Shape Coefficients for Rectangular Cross-Sections

The formulas for other cross-sections can be found in Eurocode 1997‑1, D.4.

s_q = 1 + B' / L' ⋅ sin(φ’_d) = 1 + 1.569 m / 2.50 m ⋅ sin(32°) = 1.333

s_c = (s_q ⋅ N_q - 1) / (N_q - 1) = (1,333 ⋅ 23,18 - 1) / (23,18 - 1) = 1,348

s_γ = 1 - 0.3 ⋅ B' / L' = 1 - 0.3 ⋅ 1.569 m / 2.50 m = 0.812

Slope Coefficients

m = (2 + L' / B') / (1 + L' / B') ⋅ cos²(ω) + (2 + B' / L') / (1 + B' / L') ⋅ sin²(ω)
= 0 + (2 + 1.569 m / 2.500 m) / (1 + 1.569 m / 2.500 m) ⋅ sin²(90°) = 1.614

i_q = (1 - H_d / (V_d + A' ⋅ c'_d ⋅ cot(φ‘_d)))^m
= (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 15kN/m² ⋅ cot(32°)))^1.614 = 0.858

i_c = i_q - (1 - i_q) / (N_c ⋅ tan(φ'_d))
= 0.858 - (1 - 0.858) / (35.49 ⋅ tan(32°)) = 0.852

i_γ = (1 - H_d / (V_d + A' ⋅ c'_d ⋅ cot(φ'_d)))^m+1
= (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 15kN/m² ⋅ cot(32°)))^1.614+1 = 0.781

The slope coefficient depends on the angle ω.

Bearing Resistance

Influence of foundation depth (the soil adjacent to the foundation and additional loads):

σ_R,q = q'_d ⋅ N_q ⋅ b_q ⋅ s_q ⋅ i_q = 20 kN/m² ⋅ 23.18 ⋅ 1 ⋅ 1.333 ⋅ 0.858 = 530.14 kN/m² with q'_d = γ_1d ⋅ D

Influence of cohesion:

σ_R,c = c'_d ⋅ N_c ⋅ b_c ⋅ s_c ⋅ i_c = 15 kN/m² ⋅ 35.49 ⋅ 1 ⋅ 1.348 ⋅ 0.852 = 611.11 kN/m²

Influence of foundation width (the soil under the foundation):

σ_R,γ = 0.5 ⋅ γ'_d ⋅ B' ⋅ N_γ ⋅ b_γ ⋅ s_γ ⋅ i_γ = 0.5 ⋅ 20 kN/m³ ⋅ 1.569 m ⋅ 27.72 ⋅ 1 ⋅ 0.812 ⋅ 0.781 = 275.57 kN/m² with γ'_d = γ_2d

Allowable soil pressure:

σ_R,k = R_k / A' = σ_s,q + σ_s,c + σ_s,γ = 530.14 kN/m² + 611.11 kN/m² + 275.57 kN/m² = 1,416.83 kN/m²

σ_R,d = σ_s,k / γ_R;v = 1,416.83 kN/m² / 1.00 = 1,416.83 kN/m²

Existing soil pressure:

σ_E,d = V_d / A' = 3,060.94 kN / 3.922 m² = 780.40 kN/m²

Design ratio

η₁ = σ_E,d / σ_R,d = 780.40 kN/m² / 1,416.83 kN/m² = 0.551 ≤ 1

Approach 1 (Combination 1-2) According to EN 1997-1, 2.4.7.3.4.2

Calculation of Ground Failure Resistance

Eccentricity e_x of Effective Vertical Load in x-Direction

V_z,+add,d = 1.00 ⋅ 1,156.25 kN + 1.30 ⋅ 1,000 kN = 2,456.25 kN

H_Q,x,d = 1.30 ⋅ 190 kN = 247 kN

M_y,+add,d = (1.00 m + 4.00 m) ⋅ 247 kN = 1,235 kNm

e_x = -1,235 kNm / 2,456.25 kN = -0.503 m

Effective Foundation Length, Width, and Base

Effective length: L' = max⁡(2.500 m; 2.500 m - 2 ⋅ 0.503 m) = 2.500 m

Effective width: B' = min⁡(2.500 m; 2.500 m - 2 ⋅ 0.503 m) = 1.494 m

Effective area: A' = 2.500 m ⋅ 1.494 m = 3.736 m²

Soil Parameters to Use

Friction angle: φ'_d = arctan(tan(32°)/1.25) = 26.56°

Shear parameter for cohesion: c'_d = 15 kN/m²/ 1.25 = 12 kN/m²

Bulk density: γ_1d = γ_2d = 20 kN/m³ / 1.00 = 20 kN/m³

Bearing Capacity Coefficients

N_q = e^{π ⋅ tan(26.56°)} ⋅ tan²(45° + 26.56° / 2) = 12.59

N_c = (12.59 kN - 1) ⋅ cot(26.56°) = 23.18

N_γ = 2 ⋅ (12.59 kN - 1) ⋅ tan(26.56°) = 11.59 with δ ≥ φ'_d / 2 (rough base)

Inclination of Foundation Base

b_q = b_c = b_γ = 1 since α = 0°

Shape Coefficients for Rectangular Cross-Sections

s_q = 1 + 1.494 m / 2.500 m ⋅ sin(26.56°) = 1.267

s_c = (1,267 ⋅ 12,59-1) / (12,59 - 1) = 1,290

s_γ = 1 - 0.3 ⋅ 1.494 m / 2.500 m = 0.821

Slope Coefficients

m = 0 + (2 + 1.494 m / 2.500 m) / (1 + 1.494 m / 2.500 m) ⋅ sin²(90°) = 1.626

i_q = (1 - 247 kN / (2,456.25 kN + 3.736 m² ⋅ 12kN/m² ⋅ cot(26.56°)))^1.626 = 0.847

i_c = 0.847 - (1 - 0.847) / (12.59 ⋅ tan(26.56°)) = 0.834

i_γ = (1 - 247 kN / (2,456.25 kN + 3.736 m² ⋅ 12kN/m² ⋅ cot(26.56°)))^{1.626 + 1} = 0.765

Bearing Resistance

Influence of foundation depth (the soil adjacent to the foundation and additional loads):

σ_R,q = 20 kN/m² ⋅ 12.59 ⋅ 1 ⋅ 1.267 ⋅ 0.847 = 270.26 kN/m² with q'_d = γ_1d ⋅ D

Influence of cohesion:

σ_R,c = 12 kN/m² ⋅ 23.18 ⋅ 1 ⋅ 1.1290 ⋅ 0.834 = 299.31 kN/m²

Influence of foundation width (the soil under the foundation):

σ_R,γ = 0.5 ⋅ 20 kN/m³ ⋅ 1.494 m ⋅ 11.59 ⋅ 1 ⋅ 0.821 ⋅ 0.765 = 108.68 kN/m² with γ'_d = γ_2d

Allowable soil pressure:

σ_R,k = σ_R,d = 270.26 kN/m² + 299.31 kN/m² + 108.68 kN/m² = 678.25 kN/m²

Existing soil pressure:

σ_E,d = 2,456.25 kN / 3.736 m² = 657.45 kN/m²

Design ratio

η₂ = 657.45 kN/m² / 678.25 kN/m² = 0.969 ≤ 1

Design Approach 1

η = max(η₁; η₂) = max(0.551; 0.969) = 0.969 ≤ 1

Approach 2 According to EN 1997-1, 2.4.7.3.4.2

In this design approach, a set of partial safety factors A1, M1, and R2 is used, with A1 (γ_G = 1.35; γ_Q = 1.5) increasing the unfavorable actions on the foundation, and M1 (γ'_φ = γ '_c = γ_γ = 1.00) not reducing the soil parameters, but with R2 (γ_R;v = γ_R;h = 1.40) reducing the resistances.

Calculation of Ground Failure Resistance

Eccentricity e_x of Effective Vertical Load in x-Direction

V_z,+add,d = 1.35 ⋅ 1,156.25 kN + 1.50 ⋅ 1,000 kN = 3,060.94 kN

H_Q,x,d = 1.50 ⋅ 190 kN = 285 kN

M_y,+add,d = (1.00 m + 4.00 m) ⋅ 285 kN = 1,425 kNm

e_x = -1,425 kNm / 3,060.94 kN = -0.466 m

Effective Foundation Length, Width, and Base

Effective length: L' = max(2.500 m; 2.500 m - 2 ⋅ 0.466 m) = 2.500 m

Effective width: B' = min(2.500 m; 2.500 m - 2 ⋅ 0.466 m) = 1.569 m

Effective area: A' = 2.500 m ⋅ 1.569 m = 3.922 m²

Soil Parameters to Use

Friction angle: φ'_d = 32°

Shear parameter for cohesion: c'_d = 15 kN/m²

Bulk density: γ_1d = γ_2d = 20 kN/m³

Bearing Capacity Coefficients

N_q = e^{π ⋅ tan(32°)} ⋅ tan²(45° + 32° / 2) = 23.18

N_c = (23.18 kN - 1) ⋅ cot(32°) = 35.49

N_γ = 2 ⋅ (23.18 kN - 1) ⋅ tan(32°) = 27.72 with δ ≥ φ‘_d / 2 (rough base)

Inclination of Foundation Base

b_q = b_c = b_γ = 1 since α = 0°

Shape Coefficients for Rectangular Cross-Sections

s_q = 1 + 1.569 m / 2.500 m ⋅ sin(32°) = 1.333

s_c = (1,333 ⋅ 23,18-1) / (23,18-1) = 1,348

s_γ = 1 - 0.3 ⋅ 1.569 m / 2.500 m = 0.812

Slope Coefficients

m = 0 + (2 + 1.569 m / 2.500 m) / (1 + 1.569 m / 2.500 m) ⋅ sin²(90°) = 1.614

i_q = (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 12kN/m² ⋅ cot(32°) ))^1.614 = 0.858

i_c = 0.858 - (1 - 0.858) / (23.18 ⋅ tan(32°)) = 0.852

i_γ = (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 12kN/m² ⋅ cot(32°)))^{1.614 + 1} = 0.781

Bearing Resistance

Influence of foundation depth (the soil adjacent to the foundation and additional loads):

σ_R,q = 20 kN/m² ⋅ 23.18 ⋅ 1 ⋅ 1.333 ⋅ 0.858 = 530.14 kN/m² with q‘_d = γ_1d ⋅ D

Influence of cohesion:

σ_R,c = 15 kN/m² ⋅ 35.49 ⋅ 1 ⋅ 1.1290 ⋅ 0.852 = 611.11 kN/m²

Influence of foundation width (the soil under the foundation):

σ_R,γ = 0.5 ⋅ 20 kN/m³ ⋅ 1.569 m ⋅ 27.72 ⋅ 1 ⋅ 0.812 ⋅ 0.781 = 275.57 kN/m² with γ‘_d = γ_2d

Allowable soil pressure:

σ_R,k = σ_R,d = 530.14 kN/m² + 611.11 kN/m² + 275.57 kN/m² = 1,416.83 kN/m²

σ_R,d = 1,416.83 kN/m² / 1.40 = 1,012.02 kN/m²

Existing soil pressure:

σ_E,d = 3,060.94 kN / 3.922 m² = 780.40 kN/m²

Design Approach 2

η = 780.40 kN/m² / 1,012.02 kN/m² = 0.771 ≤ 1

Approach 2* According to EN 1997-1, 2.4.7.3.4.2

In this design approach, a set of partial safety factors A1, M1, and R2 is used, with A1 (γ_G = 1.35; γ_Q = 1.5) increasing the unfavorable actions on the foundation, and M1 (γ'_φ = γ '_c = γ_γ = 1.00) not reducing the soil parameters, but with R2 (γ_R;v = γ_R;h = 1.40) reducing the resistances.

The eccentricity of the resultant and the inclination factors are not determined with the design values of actions as in Approach 2, but with characteristic actions. In most cases, this leads to smaller eccentricities and thus to a larger effective area, whereby the allowable soil pressure is higher than in Approach 2.

Calculation of Ground Failure Resistance

Eccentricity e_x of Effective Vertical Load in x-Direction

In contrast to the other approaches, this approach uses the characteristic values of the vertical load with additional foundation loads V_z,+add,k as well as the characteristic value of the resulting design bending moment in the foundation base center M_y,+add,k to determine the existing eccentricity.

V_z,+add,k = 1,156.25 kN + 1,000 kN = 2,156.25 kN

H_Q,x,k = 1.50 ⋅ 190 kN = 190 kN

_My,+add,k = (1.00 m + 4.00 m) ⋅ 190 kN = 950 kNm

e_x = -950 kNm / 2,156.25 kN = -0.441 m

Effective Foundation Length, Width, and Base

Effective length: L' = max(2.500 m; 2.500 m - 2 ⋅ 0.441 m) = 2.500 m

Effective width: B' = min(2.500 m; 2.500 m - 2 ⋅ 0.441 m) = 1.619 m

Effective area: A' = 2.500 m ⋅ 1.619 m = 4.047 m²

Soil Parameters to Use

Friction angle: φ'_d = 32°

Shear parameter for cohesion: c'_d = 15 kN/m²

Bulk density: γ_1d = γ_2d = 20 kN/m³

Bearing Capacity Coefficients

N_q = e^{π ⋅ tan(32°)} ⋅ tan²(45° + 32° / 2) = 23.18

N_c = (23.18 kN - 1) ⋅ cot(32°) = 35.49

N_γ = 2 ⋅ (23.18 kN - 1) ⋅ tan(32°) = 27.72 with δ ≥ φ‘_d / 2 (rough base)

Inclination of Foundation Base

b_q = b_c = b_γ = 1 since α = 0°

Shape Coefficients for Rectangular Cross-Sections

s_q = 1 + 1.619 m / 2.500 m ⋅ sin(32°) = 1.343

s_c = (1,343 ⋅ 23,18 - 1) / (23,18 - 1) = 1,359

s_γ = 1 - 0.3 ⋅ 1.619 m / 2.500 m = 0.806

Slope Coefficients

m = (2 + 1.619 m / 2.500 m) / (1 + 1.619 m / 2.500 m) ⋅ sin²(90°) = 1.607

i_q = (1 - 190 kN / (2,156.25 kN + 4.047 m² ⋅ 12kN/m² ⋅ cot(32°)))^1.607 = 0.868

i_c = 0.868 - (1 - 0.868) / (23.18 ⋅ tan(32°)) = 0.862

i_γ = (1 - 190 kN / (2,156.25 kN + 4.047 m² ⋅ 12kN/m² ⋅ cot(32°)))^{1.607 + 1} = 0.795

Bearing Resistance

Influence of foundation depth (the soil adjacent to the foundation and additional loads):

σ_R,q = 20 kN/m² ⋅ 23.18 ⋅ 1 ⋅ 1.343 ⋅ 0.868 = 540.42 kN/m² with q'_d = γ_1d ⋅ D

Influence of cohesion:

σ_R,c = 15 kN/m² ⋅ 35.49 ⋅ 1 ⋅ 1.348 ⋅ 0.862 = 623.50 kN/m²

Influence of foundation width (the soil under the foundation):

σ_R,γ = 0.5 ⋅ 20 kN/m³ ⋅ 1.619 m ⋅ 27.72 ⋅ 1 ⋅ 0.806 ⋅ 0.795 = 287.33 kN/m² with γ'_d = γ_2d

Allowable soil pressure:

σ_R,k = 540.42 kN/m² + 623.50 kN/m² + 287.33 kN/m² = 1,451.25 kN/m²

σ_R,d = 1,451.25 kN/m² / 1.40 =1,036.61 kN/m²

Existing soil pressure:

V_z,+add,d = 1.35 ⋅ 1,156.25 kN + 1.50 ⋅ 1,000 kN = 3,060.94 kN

σ_E,d = 3,060.94 kN / 4.047 m² = 756.33 kN/m²

Design Approach 2*

η = 756.33 kN/m² / 1,036.61 kN/m² = 0.730 ≤ 1

Approach 3 According to EN 1997-1, 2.4.7.3.4.2

In this design approach, a set of partial safety factors A1 and A2, M2, and R3 is used. For the actions from the structure, the partial safety factors from the data set A1 (γ_G = 1.35; γ_Q = 1.50) are used, while the geometric actions should be increased with the data set A2 (γ_G = 1.00; γ_Q = 1.30). In addition, the soil properties are reduced by M2 (γ'_φ = γ'_c = 1.25; γ_γ=1.00). R3 (γ_R;v = γ_R;h = 1.00) does not reduce the resistances.

Calculation of Ground Failure Resistance

Eccentricity e_x of Effective Vertical Load in x-Direction

V_z,+add,d = 1.35 ⋅ 1,156.25 kN + 1.50 ⋅ 1,000 kN = 3,060.94 kN

H_Q,x,d = 1.50 ⋅ 190 kN = 285 kN

_My,+add = (1.00 m + 4.00 m) ⋅ 285 kN = 1,425 kNm

e_x = -1,425 kNm / 3,060.94 kN = -0.466 m

Effective Foundation Length, Width, and Base

Effective length: L' = max(2.500 m; 2.500 m - 2 ⋅ 0.466 m) = 2.500 m

Effective width: B' = min(2.500 m; 2.500 m - 2 ⋅ 0.466 m) = 1.569 m

Effective area: A' = 2.500 m ⋅ 1.569 m = 3.922 m²

Soil Parameters to Use

Friction angle: φ'_d = arctan(tan(32°) / 1.25) = 26.56°

Shear parameter for cohesion: c'_d = 15 kN/m²/ 1.25 = 12 kN/m²

Bulk density: γ_1d = γ_2d = 20 kN/m³

Bearing Capacity Coefficients

N_q = e^{π ⋅ tan(26.56°)} ⋅ tan²(45° + 26.56° / 2) = 12.59

N_c = (12.59 kN - 1) ⋅ cot(26.56°) = 23.18

N_γ = 2 ⋅ (12.59 kN - 1) ⋅ tan(26.56°) = 11.59 with δ ≥ φ‘_d / 2 (rough base)

Inclination of Foundation Base

b_q = b_c = b_γ = 1 since α = 0°

Shape Coefficients for Rectangular Cross-Sections

s_q = 1 + 1.569 m / 2.500 m ⋅ sin(26,56°) = 1.281

s_c = (1,281 ⋅ 12,59 - 1) / (12,59 - 1) = 1,305

s_γ = 1-0.3 ⋅ 1.569 m / 2.500 m = 0.812

Slope Coefficients

m = (2 + 1.569 m / 2.500 m)/ (1 + 1.569 m / 2.500 m) ⋅ sin²(90°) = 1.614

i_q = (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 12kN/m² ⋅ cot(26.56°)))^1.614 = 0.858

i_c = 0.858 - (1 - 0.858) / (12.59 ⋅ tan(26.56°) ) = 0.846

i_γ = (1 - 285 kN / (3,060.94 kN + 3.922 m² ⋅ 12kN/m² ⋅ cot(26.56°)))^{1.614 + 1} = 0.781

Bearing Resistance

Influence of foundation depth (the soil adjacent to the foundation and additional loads):

σ_R,q = 20 kN/m² ⋅ 12.59 ⋅ 1 ⋅ 1.281 ⋅ 0.858 = 276.70 kN/m² with q'_d = γ_1d ⋅ D

Influence of cohesion:

σ_R,c = 12 kN/m² ⋅ 23,18 ⋅ 1 ⋅ 1,305 ⋅ 0,846 = 307,07 kN/m²

Influence of foundation width (the soil under the foundation):

σ_R,γ = 0.5 ⋅ 20 kN/m³ ⋅ 1.569 m ⋅ 11.59 ⋅ 1 ⋅ 0.812 ⋅ 0.781 = 115.19 kN/m² with γ'_d = γ_2d

Allowable soil pressure:

σ_R,k = σ_R,d = 276.70 kN/m² + 307.07 kN/m² + 115.19 kN/m² = 698.95 kN/m²

Existing soil pressure:

σ_E,d = 3,060.94 kN / 3.922 m² = 780.40 kN/m²

Design Approach 3

η = 780.40 kN/m² / 698.95 kN/m² = 1.117 ≥ 1

Comparison of Design Checks

The differences in the design process of Approaches 1 (Combination 1-1), 1 (Combination 1-2), 2, 2*, and 3 are mainly in the partial safety factors. The following table clearly shows the effects of the different safety concepts.

	Symbol	Unit	Approach
			1-1	1-2	2	2*	3
Partial Factors (Actions A)	A	[−]	1	2	1	1	1 (2) 1)A
	γGAAAA	[−]	1.35	1.00	1.35	1.35	1.35 (1.00)
	γQ	[−]	1.50	1.30	1.50	1.50	1.50 (1.30)
Vertical Load in z	VG,z+add,k	kN	1156.25
	VQ,z	kN	1000
	∑Vz+add,k	kN	2156.25
	Vz+add,d	kN	3060.94	2,456.25	3060.94	3060.94	3060.94
Horizontal Load in x	HQ,x,k	kN	190
	HQ,x,d	kN	285	247	285	285	285
Loads for Calculation of Resistances	Vz	kN	3060.94	2,456.25	3060.94	2,156.25 2)	3060.94
	Hx	kN	285	247	285	1902)	285
Design Moment in Foundation Base	My,x+add	kNm	1425	1235	1425	950	1425
Eccentricity in x	ex	m	-0.466	-0.503	-0.466	-0.441	-0.466
Effective Length	L'	m	2.500
Effective Width	B'	m	1.569	1.494	1.569	1.619	1.569
Effective Area	A'	m²	3.922	3.736	3.922	4.047	3.922
Partial Safety Factors (Material M)	M	[−]	1	2	1	1	2
	γ‘φ	[−]	1.00	1.25	1.00	1.00	1.25
	γ‘c	[−]	1.00	1.25	1.00	1.00	1.25
	γγ	[−]	1.00	1.00	1.00	1.00	1.00
Friction Angle	φ'k	°	32
	φ'd	°	32	26.56	32	32	26.56
Cohesion	c’k	kN/m²	15
	c'd	kN/m²	15	12	15	15	12
Specific Weight	γ1,k= γ2,k	kN/m³	20
	γ1,d= γ2,d	kN/m³	20
Bearing Capacity Coefficient	Nq	[−]	23.18	12.59	23.18	23.18	12.59
	Nc	[−]	35.49	23.18	35.49	35.49	23.18
	Nγ	[−]	27.72	11.59	27.72	27.72	11.59
Shape Coefficient for Rectangular Cross-Sections	sq	[−]	1.333	1.267	1.333	1.343	1.281
	sc	[−]	1.348	1.290	1.348	1.359	1.305
	sγ	[−]	0.812	0.821	0.812	0.806	0.812
	m	[−]	1.614	1.626	1.614	1.607	1.614
Slope Coefficients	iq	[−]	0.858	0.847	0.858	0.868	0.858
	ic	[−]	0.852	0.834	0.852	0.862	0.846
	iγ	[−]	0.781	0.765	0.781	0.795	0.781
Stress Foundation Depth	σR,q	kN/m²	530.14	270.26	530.14	540.42	276.70
Stress Cohesion	σR,c	kN/m²	611.11	299.31	611.11	623.50	307.07
Stress Foundation Depth	σR,γ	kN/m²	275.57	108.68	275.57	287.33	115.19
Partial Factors (Resistance R)	R	[−]	1	1	2	2	3
	γR;v	[−]	1.00	1.00	1.40	1.40	1.00
Allowable Soil Pressure	σR,k	kN/m²	1,416.83	678.25	1,416.25	1,451.25	698.95
	σR,d	kN/m²	1,416.83	678.25	1,012.02	1,036.61	698.95
Existing Soil Pressure	σE,d	kN/m²	780.40	657.45	780.40	756.33	780.40
Design ratio	η	[−]	0.551	0.969	0.771	0.730	1.117
			0.969
1) In the case of actions from the supporting structure, the partial safety factors from the data set A1 are used, whereas geometric actions have to be increased with the data set A2.
2) The characteristic actions are used to determine the resistance in Approach 2*.

Conclusion

Zusammenfassend lässt sich feststellen, dass die Verfahren aus EN 1997-1 unterschiedliche Sicherheitsniveaus und ökonomische Effizienz bieten.

Approach 1 is characterized by the fact that it requires two combinations with different safety concepts. This allows for a differentiated view of the safety requirements and ensures that the combination with the higher design ratio is governing. Combination 1-1 increases the actions on the foundation, whereas Combination 1-2 reduces the material properties.

Approach 2 and Approach 2* simplify the design compared to Approach 1, because each uses one data set only. This data set increases the actions and reduces the resistances without reducing the soil parameters. When calculating the soil bearing resistance, the position (eccentricity) and the load inclination of the resultant are important input values. Approach 2 uses the design values for this, while Approach 2* works with the characteristic actions, which leads to a greater bearing resistance in Approach 2*.

Approach 3 usually leads to particularly conservative results, because the data set of partial safety factors used here increases the actions and reduces the soil parameters, without reducing the resistances. This approach usually provides the highest level of safety.