3.3.2.1  System rigidity

For the determination of the internal force variable, the contact pressure, which depends on the proportion of the structure stiffness to the stiffness of the subsoil, needs to be analyzed.

For limp spread foundations, the distribution of the contact pressure correspondents to the load distribution. For rigid foundations a nonlinear distribution of the contact pressure with high edge stresses arises (Figure 3.5). The differentiation between limp and rigid foundations is defined by the system rigidity K according to Kany, which is a value for the assessment of the interactions between the structure and the foundation (Equation 3.1). The differentiation is stated in Table 3.1 [16,21]. The system rigidity K is determined according to Equation 3.2. It is determined by the component height h, the length l, and the modulus of elasticity of the building material E_B, which is founded in the elastic isotropic half-space (Figure 3.6) [16–20]:

where:

Circular spread foundations with a component height h and a diameter d have a system rigidity K in accordance with

For the calculation of spread foundations, normally only the rigidity of the foundation component is used to consider the rigidity of the building. The rigidity of the rising construction is considered only in special cases.

For limp spread foundations (K < 0.001), the settlement at the characteristic point is the same as the settlement of a rigid spread foundation (Figure 3.7). The characteristic point for rectangular foundations is at 0.74 of the half-width outward from the center. For circular spread foundations, the characteristic point is at 0.845 of the radius outward from the center.

Regardless of the load position and size, rigid spread foundations keep their forms. The distribution of the contact pressure has a strong nonlinear behavior with big edge stresses (Figure 3.5).

For rigid spread foundations, single foundations, and strip foundations with a big thickness, the distribution of the contact pressure can be determined according to Boussinesq, or with the stress trapeze method [16]. Otherwise, more detailed studies or sufficient conservative assumptions that are “on the safe side” become necessary.

3.3.2.2  Distribution of the contact pressure under rigid foundations according to Boussinesq

Based on the assumption that the subsoil is modeled as an elastic isotropic half-space, in the year 1885, Boussinesq developed equations that can be used for rigid spread foundations in simple cases [17].

The distribution of the contact pressure under a rigid strip foundation with a width b is given by Equation 3.4 (Figure 3.8). For an eccentric load with an eccentricity e, Borowicka enhanced the following equations [22]:

For circular and rectangular rigid spread foundations, the distribution of the contact pressure can be determined using Figure 3.9.

At the edge of spread foundations, infinitely large stresses arise. Due to the ultimate bearing capacity, imposed by the shear strength of the subsoil, these peak stresses cannot occur. The subsoil plasticizes at the edges of the foundations and the stresses shift to the center of the foundations [23].

3.3.2.3  Stress trapeze method

The stress trapeze method is a statically defined method, and it is the oldest to determine the distribution of the contact pressure. The stress trapeze method is based on the beam theory of elastostatic principles.

The distribution of the contact pressure is determined by the equilibrium conditions ΣV and ΣM, without considering deformations of the building or the subsoil interactions, respectively. Subsoil is simplified with linear elastic behaviour for calculation. Even large edge stresses are theoretically possible. The detection of the reduction of stress peaks due to plasticization is not immediately possible. All considerations are based on the assumption of Bernoulli stating that the cross-sections remain plane.

The force V is the resultant of the applied load, self-weight and buoyancy force. The resultant of the forces and the contact pressures have the same line of influence and the same value but point in opposite directions. To determine the distribution of the contact pressure of an arbitrarily spread foundation, Equation 3.7 is used. For the axes of coordinates, an arbitrarily rectangular coordinate system is used, where the zero point corresponds with the center of gravity of the subface (Figure 3.10) [24].

If the x- and the y-axis are the main axes of the coordinate system, the centrifugal moment I_xy = 0. Equation 3.7 is simplified to the following Equation 3.8. If the resultant force V acts at the center of gravity of the subface, the torques M_x = M_y = 0. The result is a constant distribution of the contact pressure according to Equation 3.9.

If the eccentricity of the resulting forces V is too large, theoretically tensile stresses occur, which are not absorbed by the system subsoilsuperstructure. An open gap occurs. In this case, Equations 3.7 through 3.9 are not applicable, and the determination of the maximum contact pressure is performed according to the following equation in conjunction with Table 3.2:

3.3.2.4  Subgrade reaction modulus method

Historically, an interaction between soil and structure was taken into account for the first time by using the subgrade reaction modulus method. The prepared subgrade reaction in relation with the change of shape was formulated in the nineteenth century by Winkler [25]. It was created for the design of railway tracks.

According to Winkler, the elastic model of the subsoil, which is also called half-space of Winkler, is a spring model, where at any point the contact pressure σ₀ is proportional to the settlement s (Equation 3.11). The constant of proportionality k_s is called the subgrade reaction modulus. The subgrade reaction modulus can be interpreted as a spring due to the linear mechanical approach for the subsoil behavior (Figure 3.11). However, this method does not consider the interactions between the independent, free-movable vertical springs.

where:

Using the beam-bending theory, it is possible to describe the curve of the bending moment for an arbitrary, infinitely long and elastic strip foundation with the width b, which is founded on the half-space of Winkler.

The curve of the bending moment of the strip foundation with the bending stiffness E_b × I is given by

The double differentiation of Equation 3.12 results in

The action q(x) corresponds to the contact pressure σ₀(x), which can be described by

With the elastic length L given as

and the elimination of s(x), Equation 3.16 follows. For a large number of boundary conditions, Equation 3.16 can be solved. For an infinite long strip foundation, the distribution of the contact pressure σ₀(x), the distribution of the bending moment M(x), and the distribution of the shear forces result according to Equations 3.17 through 3.19.

The subgrade reaction modulus is not a soil parameter. It depends on the following parameters:

• Oedometric modulus of the subsoil
• Thickness of the compressible layer
• Dimensions of the spread foundation

The subgrade reaction modulus method does not take into account the influence of neighboring contact pressures. It is therefore mainly suitable for the calculation of slender, relatively limp spread foundations with large column distances. With the subgrade reaction modulus method, it is not possible to determine settlements beside the spread foundation (Figure 3.12).

3.3.2.5  Stiffness modulus method

The stiffness modulus method according to Ohde (1942) describes the soil– structure interaction more accurately than the subgrade reaction modulus method, because the influence of adjacent contact pressures is considered on the settlement of an arbitrary point of the spread foundation [19,26]. In the stiffness modulus method, the bending moment of the linear elastic modeled spread foundation is linked with the bending moment of the linear elastic, isotropic modeled settlement trough. The same deformations arise.

Figure 3.13 represents the distribution of the settlement of a spread foundation according to the stiffness modulus method.

In geotechnical engineering practice, spread foundations with complex load situations and geometric boundary conditions are normally examined using computer programs. For most cases, no closed solutions are available for the statically indeterminate system of equations.

The assumption of infinite elastic subsoil has the consequence that theoretically infinite large stress peaks result at the edge of the spread foundation. Due to the plasticizing effect of the subsoil, these stress peaks do not occur in reality. Powerful computer programs consider this basic soil mechanical behavior.

3.3.3.1  Analysis of safety against loss of balance because of overturning

Up to now, the analysis of safety against loss of balance because of overturning was done by applying the resultant of the forces into the second core width. That means that the lower surface of the spread foundation has only a small part with an open gap. This approach is described in [27,28]. Thus, a resulting force in the first core width creates a compressive stress over the entire lower surface of the spread foundation.

According to the current technical regulations, the analysis of safety against loss of balance because of overturning is based on a principle of the rigid body mechanics. The destabilizing and stabilizing forces are compared based on a fictional tilting edge at the edge of the spread foundation:

The design value of the destabilizing force is estimated according to Equation 3.21, and the design value of the stabilizing action is estimated according to Equation 3.22:

In reality, the position of the tilting edge depends on the rigidity and the shear strength of the subsoil. With a decreasing rigidity and decreasing shear strength, the tilting edge moves to the center of the lower surface of the spread foundation.

Therefore, this analysis itself is not sufficient. It was complemented by the analysis of the limitation of the open gap, which is defined for the serviceability limit state. According to [10], the resultant force of the permanent loads has to be applied into the first core width and the resultant force of the variable loads has to be applied into the second core width (Figure 3.21).

3.3.3.2  Analysis of safety against sliding

The analysis of safety against sliding (limit state GEO-2) is calculated according to Equation 3.23. The forces parallel to the lower surface of the spread foundation have to be smaller than the total resistance, consisting of slide resistance and passive earth pressure. If the passive earth pressure is considered, the serviceability limit state has to be verified regarding the horizontal displacements.

where: $\begin{array}{l}{\text{R}}_{\text{d}}=\frac{{\text{R}}_{\text{k}}}{{\gamma }_{\text{R,h}}}\hfill \\ {\text{R}}_{\text{p,d}}=\frac{{\text{R}}_{\text{p,k}}}{{\gamma }_{\text{R,h}}}\hfill \end{array}$

The sliding resistance is determined according to the three following cases:

• Sliding in the gap between the spread foundation and in the subjacent, fully consolidated subsoil:
  3.24 $${\text{R}}_{\text{d}}=\frac{{\text{V}}_{\text{k}}\cdot tan\delta }{{\gamma }_{\text{R,h}}}$$

  
  where:

  Vk	= characteristic value of the vertical loadings [kN]
  δ	= characteristic value of the angle of base friction [°]

• Sliding when the gap passes through the fully consolidated soil, for example, in the arrangement of a foundation cut-off:
  3.25 $${\text{R}}_{\text{d}}=\frac{{\text{V}}_{\text{k}}\cdot tan{\phi }^{\prime }+\text{A}\cdot {\text{c}}^{\prime }}{{\gamma }_{\text{R,h}}}$$

  

where:

• Sliding on water-saturated subsoil due to very quick loading:
  3.26 $${\text{R}}_{\text{d}}=\frac{\text{A}\cdot {\text{c}}_{\text{u}}}{{\gamma }_{\text{R,h}}}$$

  

where:

For spread foundations that are concreted in situ, the characteristic value of the angle of base friction δ is the same as the characteristic value of the friction angle φ′ of the soil. For prefabricated spread foundation elements the characteristic value of the angle of base friction δ should be set to 2/3 φ′. The characteristic value of the angle of base friction should be δ ≤ 35°.

A passive earth pressure can be considered if the spread foundation is deep enough. Due to horizontal deformations, the passive earth pressure should be limited to 50% of the possible passive earth pressure. Basically, it must be verified whether the passive earth pressure exists during all possible stages in the construction and the service phase of the foundation.

3.3.3.3  Analysis of safety against base failure

The analysis of safety against base failure is guaranteed if the design value of the bearing capacity R_d is bigger than the design value of the active force V_d. R_d is calculated according to Equation 3.27. The principle scheme of a bearing failure of a spread foundation is shown in Figure 3.14.

The resistance of the bearing capacity is determined by the soil properties (density, shear parameters), the dimensions and the embedment depth of the spread foundation. Detailed information can be found in the incidental standard [29,30]. The characteristic resistance of the bearing capacity R_n,k is calculated analytically with a trinomial equation, which is based on the moment equilibrium of the failure figure of the bearing capacity in ideal plastic, plane deformation state [31]. The trinomial equation of the bearing capacity considers the width b of the foundation, the embedment depth d of the foundation and the cohesion c′ of the subsoil. All three aspects have to be factorized with the bearing capacity factors N_b, N_d, and N_c:

where:

• N_b = N_b0 · v_b · i_b · λ_b · ξ_b
• N_d = N_d0 · v_d · i_d · λ_d · ξ_d
• N_c = N_c0 · v_c · i_c · λ_c · ξ_c

The density γ₁ describes the density of the subsoil above the foundation level. The density γ₂ describes the density of the subsoil underneath the foundation level. The bearing capacity factors N_b, N_d, and N_c consider the following boundary conditions:

• Basic values of the bearing capacity factors: N_b0, N_d0, N_c0
• Shape parameters: ν_b, ν_d, ν_c
• Parameter for load inclination: i_b, i_d, i_c
• Parameters for landscape inclination: λ_b, λ_d, λ_c
• Parameters for the base inclination: ξ_b, ξ_d, ξ_c

The parameters of the bearing capacity factors N_b0, N_d0, N_c0 depend on the angle of friction of the subsoil φ’ and are calculated according to Table 3.3.

The shape parameters ν_b, ν_d, ν_c take into account the geometric dimensions of the spread foundation. For the standard applicable geometry, the shape parameters are summarized in Table 3.4.

If eccentric forces have to be considered the base area has to be reduced. The resulting load has to be in the center of gravity. The reduced dimensions a′ and b′ are calculated according to Equations 3.29 and 3.30. Basically applied, is a > b and a′ > b′, respectively. For spread foundations with open parts, the external dimensions may be used for the analysis, if the open parts are not bigger than 20% of the whole base area.

where ${\text{m}}_{\text{a}}=\frac{2+\frac{{\text{a}}^{\prime }}{{\text{b}}^{\prime }}}{1+\frac{{\text{a}}^{\prime }}{{\text{b}}^{\prime }}}\text{ and }{\text{m}}_{\text{b}}=\frac{2+\frac{{\text{b}}^{\prime }}{{\text{a}}^{\prime }}}{1+\frac{{\text{b}}^{\prime }}{{\text{a}}^{\prime }}}$

Forces T_k that are parallel to the foundation level are considered by the parameters i_b, i_d, i_c for the load inclination. The definition of the angle of the load inclination is shown in Figure 3.15. The determination of the parameters for the load inclination is shown in Tables 3.5 and 3.6. The orientation of the acting forces is determined by the angle ω (Figure 3.16). For a strip foundation ω = 90°.

An inclination of the landscape is considered by the parameters λ_b, λ_d, λ_c for landscape inclination. The parameters depend on the slope inclination β. The slope inclination has to be less than the angle of friction φ′ of the subsoil and the longitudinal axis of the foundation has to be parallel to the slope edge. The determination of the parameters for landscape inclination is shown in Figure 3.17 and Table 3.7.

The base inclination is considered by the parameters ξ_b, ξ_d, ξ_c for the base inclination (Table 3.8), which depend on the angle of friction φ’ of the subsoil and the base inclination α of the spread foundation. The definition of the base inclination is shown in Figure 3.18. The angle of the base inclination α is positive, if the failure body forms out in the direction of the horizontal forces. The angle of the base inclination α is negative, if the failure body forms in the opposite direction. In cases of doubt, both failure bodies have to be investigated.

The direct application of the defined equations is only possible if the sliding surface is formed in one soil layer. For layered soil conditions, it is permissible to calculate with averaged soil parameters if the values of the individual angles of friction do not vary more than 5° of the arithmetic average. In this case, the individual soil parameters may be weighted according to their influence on the shear failure resistance. The weighting is as follows.

• The average density is related to the percentage of the individual layers in the cross-section area of the failure body
• The average angle of friction and the average cohesion are related to the percentage of the individual layers in the cross-section area of the failure body

Authoritative for the sliding surface is the average of the angle of friction φ. To determine whether the failure body has more than one layer, it is recommended to define the failure body according to Equation 3.32 through 3.38 (Figure 3.19). For simple cases (α = β = δ = 0), the Equations 3.39 through 3.42 have to be applied.

where: $\text{sin}{\epsilon }_1=-\frac{sin\text{β}}{sin\phi }$

where $\text{sin}{\epsilon }_2=-\frac{sin\delta }{sin\phi }$

For spread foundations at slopes, the foundation depth d′ (Equation 3.43) and the parameters λ_b, λ_d, λ_c for landscape inclination have to be considered (Figure 3.20). In addition, it is necessary to carry out a comparative calculation with β = 0 and d′ = d. The smaller resistance is the basis of the analysis of the bearing capacity regarding base failure.

3.3.3.4  Analysis of safety against buoyancy

The analysis of safety against buoyancy (limit state UPL) is performed using Equation 3.44. This equation is the proof that the net weight of the structure is big enough compared to the buoyant force of the water. Shear forces (friction forces at the side) can only be considered if the transmission of the forces is ensured. Acting shear forces T_k may be

where:

• Vertical component of the active earth pressure E_av,k on a retaining structure depending on the horizontal component of the active earth pressure E_ah,k as well as the angle of wall friction δ_a (Equation 3.45)
  3.45 $${\text{T}}_{\text{k}}={\text{η}}_{\text{z}}\cdot {\text{E}}_{\text{ah,k}}\cdot tan{\delta }_{\text{a}}$$

  
• Vertical component of the active earth pressure in a joint of the subsoil, for example, starting at the end of a horizontal spur in dependency to the horizontal component of the active earth pressure and the angle of friction φ′ of the subsoil:
  3.46 $${\text{T}}_{\text{k}}={\text{η}}_{\text{z}}\cdot {\text{E}}_{\text{ah,k}}\cdot tan{\phi }^{\prime }$$

  

The smallest possible horizontal earth pressure min E_ah,k has to be used. For the design situation BS-P and BS-T, the adjustment factor is η_z = 0.80. For the design situation BS-A, the adjustment factor is η_z = 0.90. Only in justified cases can cohesion be taken into account, but it has to be reduced by the adjustment factors. For permanent structures, it has to be determined that in design situation BS-A, the safety against buoyancy is given without any shear forces T_k.

3.3.3.5  Analysis of foundation rotation and limitation of the open gap

Generally, the serviceability limit states refer to absolute deformations and displacements as well as differential deformations. In special cases, for example, time-dependent material behavior displacement rates have to be considered.

For the analysis of foundation rotation and limitation of the open gap, the resultant of the dead loads has to be limited into the first core width, which means that an open gap does not occur. The first core width for rectangular spread foundations can be determined according to Equation 3.47. For circular spread foundations Equation 3.48 is used. Furthermore, it should be guaranteed that the resultant of the permanent loads and the variable loads are in the second core width, so an open gap cannot occur across the center line of the spread foundation. The second core width for rectangular layouts can be determined according to Equation 3.49. For circular spread foundations Equation 3.50 is used. Figure 3.21 shows the first and the second core width for a rectangular spread foundation.

For single and strip foundations, which are founded on medium-dense, non-cohesive soils and stiff cohesive soils, respectively, no incompatible distortions of the foundation can be expected if the acceptable eccentricity is observed.

The analysis of foundation rotation and limitation of the open gap is mandatory according to [10], if the analysis of safety against loss of balance because of overturning is carried out by using one edge of the spread foundation as a tilting edge.

3.3.3.6  Analysis of horizontal displacements

Generally, for spread foundations the analysis of horizontal displacement is observed, if:

• The analysis of safety against sliding is performed without considering a passive earth pressure.
• For medium-dense, non-cohesive soils and stiff cohesive soils, respectively, only two-thirds of the characteristic sliding resistance in the foundation level and not more than one-third of the characteristic earth pressure are considered.

If these arguments are not true, it is necessary to analyze the possible horizontal displacements. Permanent loads and variable loads, as well as infrequent or unique loads, have to be considered.

3.3.3.7  Analysis of settlements and differential settlements

The determinations for settlements of spread foundations are conducted in accordance with [32]. Normally, the influence depth of the contact pressure is between z = b and z = 2b.

Due to the complex interaction between the subsoil and the construction, it is difficult to provide information about the acceptable settlements or differential settlements for constructions [33]. Figure 3.22 indicates the damage factors for the angular distortion as a result of settlements [33–35].

Regarding the tilting of high-rise structures, the analysis of safety against inclination has to check that the occurring tilting is harmless for the structure [33]. The analysis for rectangular spread foundations is performed according to Equation 3.51. The analysis for circular spread foundations is performed according to Equation 3.52.

In Equations 3.51 and 3.52:

E_m = Modulus for the compressibility of the subsoil

h_s = Height of the center of gravity above the foundation level

f_r and f_y = Tilting coefficients

V_d = Design value of the vertical loads

More detailed information can be found in [33] and [36].

3.3.3.8  Simplified analysis of spread foundations in standard cases

The simplified analysis of spread foundations in standard cases consists of a simple comparison between the base resistance σ_R,d and the contact pressure σ_E,d (Equation 3.53). For spread foundations with the area A = a × b or A′ = a′ × b′, the analysis of safety against sliding and base failure, as well as the analysis of the serviceability limit state, can be applied in standard cases. These standard cases include:

• Horizontal lower surface of the foundation and an almost horizontal landscape and soil layers
• Sufficient strength of the subsoil into a depth of twice the width of the foundation below the foundation level (minimum 2 m)
• No regular dynamic or mainly dynamic loads; no pore water pressure in cohesive soils
• Passive earth pressure can only be applied if it is assured by constructive or other procedures
• The inclination of the resultant of the contact pressure observes the rule tanδ = H_k/V_k ≤ 0.2 (δ = inclination of the resultant of the contact pressure; H_k = characteristic horizontal forces; V_k = characteristic vertical forces)
• The admissible eccentricity of the resultant of the contact pressure is observed
• The analysis of safety against loss of balance because of overturning is observed

The design values of the contact pressure σ_R,d are based on the combined examination of the base failure and the settlements. If only the SLS is analyzed, the admissible contact pressure increases with an increasing width of the spread foundation. If only the ULS is analyzed, the admissible contact pressure decreases with an increasing width of the spread foundation. Figure 3.23 shows the two fundamental demands for an adequate analysis against base failure (ULS) and the analysis of the settlements (SLS). For foundation widths that are bigger than the width b_s, the acceptable contact pressure decreases because of settlements.

The design values of the contact pressure σ_R,d for simplified analysis of strip foundations are specified in tables. The tabular values can also be used for single foundations [10,37,38].

If the foundation level is more than 2 m below the surface on all sides, the tabular values can be raised. The raise can be 1.4 times the unloading due to the excavation below a depth of ≥2 m under the surface.

The settlement values in the tables refer to detached strip foundations with a central loading (no eccentricity). If eccentric loads occur, the serviceability has to be analyzed. For the application of the current table values, it is essential to notice that in earlier editions of these tables, characteristic values were given [10].

The simplified analysis of the ULS and SLS of strip foundations in non-cohesive soils considers the design situation BS-P. For the design situation BS-T the tabular values are “on the safe side”. The tabular values are applicable for vertical loads. Intermediate values may be interpolated linearly. For eccentric loads the tabular values may be extrapolated if width b′ < 0.50 m. There must be a distance between the lower surface of the foundation and the groundwater level. The distance must be bigger than the width b or b′ of the foundation. For the application of the tables for non-cohesive soils the requirements in Table 3.9 must be fulfilled. The short forms of the soil groups are explained in Table 3.10.

The coefficient of uniformity C_u describes the gradient of the grain size distribution in the area of passing fractions of 10% and 60% and it is determined according to Equation 3.54 [39]. According to [40], the compactness D describes whether the subsoil is loose, medium dense, or dense. The compactness D is determined by the porosity n, according to Equation 3.55. The compression ratio D_pr is the relation between the proctor density ρ_pr (density at optimal water content) and the dry density ρ_d [41]. The compression ratio is calculated using Equation 3.56.

For simplified analysis of strip foundations Table 3.11 shows the permissible design values of the contact pressure σ_R,d for non-cohesive soils taking into account an adequate safety against base failure. If the settlement has to be limited additionally, Table 3.12 has to be applied. For the purpose of Table 3.12, the settlements are limited to 1–2 cm.

The permissible design values of the contact pressure σ_R,d for strip foundations in non-cohesive soils with the minimum width b ≥ 0.50 m and the minimum embedment depth d ≥ 0.50 m can be increased as follows:

• Increase of the design values by 20% in Table 3.11 and 3.12, if single foundations have an aspect ratio a/b < 2 resp. a′/b′ < 2; for Table 3.11 it is only applied if the embedment depth d is bigger than 0.60 × b resp. 0.60 × b′

  Table 3.11   Design values σR,d for strip foundations in non-cohesive soils and sufficient safety against hydraulic failure with a vertical resultant of the contact pressure

  Smallest embedment depth of the foundation [m]	Design value of the contact pressure σR,d [kN/m2] in dependence of the foundation width b resp. b′
  	0.50 m	1.00 m	1.50 m	2.00 m	2.50 m	3.00 m
  0.50	280	420	560	700	700	700
  1.00	380	520	660	800	800	800
  1.50	480	620	760	900	900	900
  2.00	560	700	840	980	980	980
  For buildings with an embedment depth 0.30 m ≤ d ≤ 0.50 m and foundation width b resp. b′ ≥ 0.30 m	210

  Table 3.12   Design values σR,d for strip foundations in non-cohesive soils and limitation of the settlements to 1–2 cm with a vertical resultant of the contact pressure

  Smallest embedment depth of the foundation [m]	Design value of the contact pressure σR,d[kN/m2] in dependence of the foundation width b resp. b′
  	0.50 m	1.00 m	1.50 m	2.00 m	2.50 m	3.00 m
  0.50	280	420	460	390	350	310
  1.00	380	520	500	430	380	340
  1.50	480	620	550	480	410	360
  2.00	560	700	590	500	430	390
  For buildings with an embedment depth 0.30 m ≤ d ≤ 0.50 m and foundation width b resp. b′ ≥ 0.30 m	210

• Increase of the design values by 50% in Tables 3.11 and 3.12, if the subsoil complies the values in Table 3.13 into a depth of twice the width under the foundation level (minimum 2 m under foundation level)

The permissible design values of the contact pressure for strip foundations in non-cohesive soils in Table 3.11 (even increased and/or reduced due to horizontal loads) have to be reduced if groundwater has to be considered:

• Reduction of the design values by 40%, if the groundwater level is the same as the foundation level

  Table 3.13   Requirements for increasing the design values σR,d for non-cohesive soils

  Soil group according to DIN 18196	Coefficient of uniformity according to DIN 18196 Cu	Compactness according to DIN 18126 D	Compression ratio according to DIN 18127 DPr	Point resistance of the soil penetrometer qc [MN/m2]
  SE, GE, SU, GU, ST, GT	≤3	≥0.50	≥98%	≥15
  SE, SW, SI, GE GW, GT, SU, GU	>3	≥0.65	≥100%	≥15

• If the distance between the groundwater level and the foundation level is smaller than b or b′, it has to be interpolated between the reduced and the non-reduced design values σ_R,d
• Reduction of the design values by 40%, if the groundwater level is above the foundation level, provided that the embedment depth d ≥ 0.80 m and d ≥ b; a separate analysis is only necessary if both conditions are not true

The permissible design values of the contract pressure σ_R,d in Table 3.12 can only be used if the design values in Table 3.11 (even increased and/or reduced due to horizontal loads and/or groundwater) are bigger.

The permissible design values of the contact pressure σ_R,d for strip foundations in non-cohesive soils shown in Table 3.11 (even increased and/or reduced due to groundwater) need to be reduced for a combination of characteristic vertical (V_k) and horizontal (H_k) loads as follows:

• Reduction by the factor (1 − H_k/V_k) if H_k is parallel to the long side of the foundation and if the aspect ratio is a/b ≥ 2 resp. a′/b′ ≥ 2
• Reduction by the factor (1 − H_k/V_k)² in all other cases

The design values of the contact pressure shown in Table 3.12 can only be applied if the design values σ_R,d shown in Table 3.11 (even increased and/or reduced due to groundwater) are bigger.

The simplified analysis of the ULS and SLS of strip foundations in cohesive soils is for the design situation BS-P. For the design situation BS-T, the tabular values are “on the safe side.” The tabular values are applicable for vertical and inclined loads. Intermediate values may be interpolated linearly. The tables are given for different types of soil. The short forms of the soil groups are explained in Table 3.10. If the Tables 3.14 through 3.17 are used, settlements of 2–4 cm can be expected. In principle, the Tables 3.14 through 3.17 are only applicable for soil types with a granular structure that may not collapse suddenly.

The design values σ_R,d for strip foundations in cohesive soil shown in Tables 3.14 through 3.17 (even reduced due to foundation width b > 2 m) may be increased by 20% if the aspect ratio is a/b < 2 resp. a′/b′ < 2.

The design values σ_R,d for strip foundations in cohesive soil shown in Tables 3.14 through 3.17 (even increased due to the aspect ratio) have to be reduced by 10% per meter at foundation width b = 2–5 m. For foundations with a width b > 5 m the ULS and the SLS have to be checked separately according to the classic soil mechanical analysis.