Spread foundations
6.3. Direct method: ULS design
6.3.1. Bearing resistance
Clause 6.5.2.1(1)P The fundamental ULS requirement is represented by the inequality
Vd£ Rd (6.1)
where Vd is the ULS design load normal to the foundation and Rd is the design bearing resistance of the foundation against loads normal to it (Fig. 6.2). Rdmay be calculated using analytical or semi-empirical models. Vdincludes the weight of the foundation and of any backfill material (to be considered as a ‘structural action’) placed on top of it. Earth pressures on structural elements above the foundation level are geotechnical actions and are also included in Vdwhere relevant.
The basic inequality Vd£ Rdhas to be checked for the recommended partial factors for persistent and transient situations in Annex A (Table A.1 for partial factors on actions or the effects of actions; Tables A.2 and A.5 for partial factors on soil parameters and resistances).
For accidental situations, the partial factors are usually put equal to 1.0. It should be noted that the recommended values for the partial factors given in Annex A are established for analytical methods in all three Design Approaches and for the semi-empirical method in DA-2; thus semi-empirical methods in DA-1 and DA-3 may need the application of model factors according to clause 2.4.7.1(6).
EN 1997-1 recommends that water pressures not caused by the foundation loads are included explicitly in calculations. For drained conditions, it suggests that water pressures are included as actions. A typical situation is illustrated in Fig. 6.3. It is consistent with the
FV,FHM
W2
W1
AH
AV
Vd
Rd P
Fig. 6.2. Example of actions on the footing. Vdis the component, normal to the foundation, of the design values of the following actions (the subscript d is omitted for clarity):
H, M and V, which are structural actions A and P, which are earth pressures W1and W2,, which are weights of backfill weight of the footing
Clause 2.4.2(9)P definition of actions to treat all water pressures in drained conditions as actions, since they
are known at the start of the calculation. This implies that the resistance is calculated in terms of effective stress. The question may arise: how to apply the partial factors to the weight of a submerged or partially submerged structure? The force due to water pressure acting on the underside of the foundation acts so as to reduce the value of Vd, and may then be considered as ‘favourable’, while the (total) weight of the foundation is unfavourable.
Physically, however, it is the submerged weight (total weight minus upward force of water) which has to be sustained by the soil, for which the resistance is then expressed in terms of effective forces; the same single partial factor may be applied to the sum of these actions.
This is illustrated in Fig. 6.3. The design values of the actions due to the weight of the submerged footing and to the backfill become the design values of their effective weights.
Action factors of 1.0 (DA-1 Combination 2) and 1.35 (DA-1 Combination 1, and DA-2 and DA-3) are applied to the effective weight of the submerged footing and backfill if they are unfavourable. This is illustrated in Example 6.1. Action factors of 1.0 apply for all Design Approaches if the effective weight of the foundation is favourable.
The calculations for undrained conditions are shown in Fig. 6.4 and in Example 6.1.
A method similar to that outlined in Fig. 6.4 applies in the semi-empirical calculation model given in Annex E, which is expressed in terms of a total stress (independent of whether the soil is in a drained or an undrained condition). Example 6.3 illustrates this.
For the design of structural members, water pressures may be unfavourable (e.g. for rafts or closed caissons forming boxes), and the action factor for unfavourable structural actions should be applied to the water pressures (see Section 6.7).
d1 g1
g2 d2
d3
Rd
U2 U2
F¢1 U1 F¢1 U1
W S
Fig. 6.3. Actions on a footing with hydrostatic water pressure. (After Simpson and Driscoll, 1998.) Ab= area of footing base
Ac= area of column cross-section
γ1= total weight density of soil above the water table γ2= total weight density of soil below the water table S = action from the superstructure
W = weight of footing
U1, U2= forces due to water pressure
F1= effect of the action of backfill on the foundation
= (γ1d1+γ2d2)(Ab– Ac)
= F1¢+ U1
= (γ1d1+ (γ2–γw)d2+γwd2)(Ab– Ac) where
F1¢ = effective effect of the action of the backfill on the foundation U2=γw(d2+ d3)Ab
Vdmust be matched by design resistance Rd, which in this case is an effective force
Analytical calculation
Clause 6.5.2.2(1) The sample calculation model given in Annex D is widely recognized. Annex D gives an idea of the degree of accuracy and conservatism that is appropriate. The recommended values for the partial factors of Annex A (Tables A.1, A.2 and A.5) have been calibrated to the method in Annex D.
Annex D is marked as ‘informative’. The designer may, therefore, use other methods that may be specified in the National Annex.
Clauses 6.5.2.2(4) to 6.5.2.2(6)
The calculation model of Annex D applies to a single soil layer. Caution is advised for layered soils or ground containing discontinuities. Paragraph (5) may be over-conservative or inappropriate in several situations. More refined calculation models may be employed, e.g.
for punching failure, for squeezing, or for undrained shear resistance increasing with depth.
The partial factors to be used depend on which Design Approach is adopted.
Basic input parameters for the analytical calculations are characteristic values of soil shear strength (ck¢, ϕk¢ in drained conditions; cu, kin undrained conditions) and weight density (γk
andγk¢). Their characteristic value should be selected to account for the variability of the ground relative to the size of the foundation and the stiffness of the structure:
• For a large, spread foundation, the characteristic value of the shear resistance parameters will usually be a cautious estimate of the mean value under the foundation. When a building is supported on several footings, the characteristic value should account for the spatial variability of the shear strength parameters over the footprint of the building and for the stiffness of the supported structure.
Rd
U1 F¢ U1
W
S 1
F1¢
Fig. 6.4. Footing with hydrostatic water pressure in granular soil over undrained clay. (After Simpson and Driscoll, 1998.)
Arrows show hydrostatic water pressures (vertical components only). Those acting on the footing are included in Vdwhilst those alongside the footing augment Rd. In this (undrained) case, Rdis a total force, acting on the base, which includes both the hydrostatic pore pressures and excess pore pressures due to undrained shearing.
The action on the ground, Vd, which is to be supported by the undrained resistance Rd, is given by Vd= S + W + F1= S + W + (F1¢ + U1)
F1¢ is defined in Fig. 6.3.
A possible alternative approach is to include the hydrostatic water pressures in the clay as part of Vd, as for the drained case; these act upwards on the base of the footing, reducing the value of Vdas in Fig. 6.3. The excess pore pressures generated at the base of the footing due to shearing remain part of Rd, however. This is consistent with the definition of actions in clause 2.4.2(2)P, since the hydrostatic component is known, but the excess pore pressure is unknown at the start (and end) of the calculation
• Local weak spots, leading to significantly lower bearing resistance, should be identified when selecting the characteristic values. If no such weak spots exist, the characteristic shear strength may be a cautious estimate of the mean values over the footprint of the structure. If weak spots exist and the supported structure is not stiff enough to transfer loads from footings at weak spots to footings at stronger spots, the characteristic value of the shear strength parameters may be a cautious estimate of the mean or lower values under each footing separately.
When selecting the characteristic value of ground shear strength parameters for the design of a small footing, attention should be paid to failure surfaces which may develop preferentially along points of weakness in the ground. Should this possibility be likely, then the selection of the characteristic value should be governed by the lower values of the ground strength parameters.
The characteristic value of the unit weight of the soil should be a cautious estimate of its mean value.
The effective overburden pressure at the level of the foundation, q¢, is a cautious estimate of its mean value in the vicinity of the footing. It should account for unfavourable water levels and the adverse effect on bearing resistance of a reduction in overburden through any removal of soil.
Clause 2.4.7.3.4.2 Some features to note when applying an analytical calculation model using DA-1
For persistent and transient situations, the design has to be checked for both Combinations 1 and 2:
Combination 1: A1 ‘+’ M1 ‘+’ R1 Combination 2: A2 ‘+’ M2 ‘+’ R1
The partial factors are applied at the source, i.e. on the actions (for recommended values, see Table A.1) and on the material shear strength parameters c¢ and tanϕ¢ or cu (for recommended values, see Table A.2). For spread foundations, the factors on the resistance are equal to 1.0 (see Table A.5).
Combination 2 usually determines the dimensions of the foundation, except in situations where a horizontal variable load, the value of which is large compared with the permanent vertical load, leads to a large overturning moment on the foundation. Such a load combination may lead to a very eccentric design value of the resulting action. It is advisable, in a first step, to calculate the size of the foundation for the partial factors of Combination 2, and then to check, in a second step, that the resulting size fulfils the ULS requirements for the partial factors of Combination 1.
Step 1 – sizing of foundation using Combination 2: A2 ‘+’ M2 ‘+’ R1. The design values of the structural actions on the foundation are obtained by applying the factors in set A2 of Table A.3 (recommended values are γF= 1.0 on unfavourable, permanent loads and γQ= 1.3 on unfavourable, variable loads; but these may be changed in the National Annex).
Design values of geotechnical actions (e.g. active pressure) are obtained by applying the factors of set M2 of Table A.4 to the shear strength parameters, and the action factors of set A2 of Table A.3 to unfavourable, permanent geotechnical actions (recommended value is γF= 1.0) and to the unfavourable, variable geotechnical actions (recommended value:
γQ= 1.3).
The design values of the soil shear strength parameters are obtained by applying factors larger than 1.0 to the characteristic values, as in set M2 of Table A.4 (recommended values areγϕ =1.25,γc¢= 1.25 andγcu= 1.4; but these may be changed in the National Annex). The partial factor on the resistanceγRvis 1.0 according to set R1 of Table A.5.
A partial material factor of 1.0 is applied to the characteristic (effective) weight density of the ground when calculating the overburden pressure as part of the bearing resistance.
The design values of the actions and of the shear strength parameters permit the calculation of:
• The design values of the components of the resulting actions normal (Vd) and parallel (Hd) to the foundation.
• The eccentricities eB, d and eL, d of the resulting actions and hence the effective dimensions B¢ and L¢ of the foundation.
• The bearing resistance factors Nq(ϕd), Nγ(ϕd) and Nc(ϕd).
• The inclination factors iq, iγand icas functions of the design values of the shear strength parameters and of Hdand of Vd.
• The shape factors sq, sγand scand the base inclination factors bq, bγand bc.
• The design value of the bearing resistance, from Rd/A¢.
Usually, several iterations will be needed to obtain the optimal foundation size to fulfil the ULS requirement.
Step 2 – checking the size of the foundation found in step 1 for the sets of partial factors of Combination 1: A1 ‘+’ M1 ‘+’ R1. The design values of the actions on the foundation are obtained by applying the factors according to Table A.3, set A1 (recommended values are:
γF= 1.35 on unfavourable, permanent, loads andγQ= 1.5 on unfavourable, variable loads;
but they may be changed in the National Annex). Design values of geotechnical actions (e.g.
active pressure) are obtained by applying these action factors to the characteristic values of the geotechnical actions. The design values of the soil shear strength parameters are equal to their characteristic values (with partial factors equal to 1.0, Table A.4, set M1). The partial factor on the resistanceγRvis 1.0 according to Table A.5.
It should be noted that a partial action factor equal to 1.0 is applied to the permanent water pressures, and a partial material factor equal to 1.0 is applied to the characteristic (effective) weight density of the ground, when calculating the (effective) overburden pressure.
Since the size of the foundation is usually determined by Combination 2, the calculation for Combination 1 is reduced to a simple check that the size of the footing fulfils the requirement of Combination 1.
Notes.
(1) For vertical or nearly vertical loads, it is often obvious that Combination 1 is not relevant for determining the foundation dimensions. Calculations for step 2 are not then necessary.
(2) Permanent actions may be favourable when, for example, they are acting in combination with large variable actions. In such cases, it is necessary to perform two calculations for Combination 1: one in which the permanent, vertical action is considered to be favourable, and hence acquires an action factor of 1.0, and another one in which the permanent vertical action is considered to be unfavourable, and hence acquires an action factor of 1.35. The first case is denoted by ‘Vfavourable’ and the second by ‘Vunfavourable’ in the examples.
In Combination 1, the design value of the resistance is equal to its characteristic value when the load is vertical and is somewhat lower when the load is inclined.
Clause 2.4.7.3.4.3 Some features when applying an analytical calculation model using DA-2