Annex B
Example Calculations for Pile Resistance Analysis and Verifications
Note: Some of the following examples were taken from [58]. Further examples can be found in [64].
Figure B1.1a shows a foundation situation with a pile of diameter D = 1,2 m and a permanent load F_{G,k}= 1,5 MN, in addition to a variable load F_{Q,rep,k} = 1,0 MN. Two static pile load tests were executed, the results of which are included in Figure B1.1b and Table B1.1 as R_{m1} and R_{m2}. The ultimate settlement is defined as s_{g} = s_{ult} = 0,1 · 120 cm = 12 cm using Eq. (5.1). Because the static pile load tests were only executed up to a settlement s = 10 cm, the ultimate settlement was extrapolated.
The analysis comprises the characteristic pile resistances in the ultimate limit state (ULS) for “flexible” and “stiff” piles after 5.2.2 and the characteristic boundary lines in the serviceability limit state (SLS) after 5.2.3, and the external capacity and serviceability for the specified pile load. An allowable
pile settlement allow. s_{k} = 2,0 cm is specified by the structural design for the serviceability limit state (SLS).
The static pile load tests can be taken from [154].
The characteristic, ultimate pile resistance R_{c,k} is given by the lesser of either the mean value (R_{c,m})_{mean} or the minimum value (R_{c,m})_{min} of the pile load test results using Equation (A4.1) as follows:
The correlation factors ξ_{1} and ξ_{2} depend on the number of static pile load tests performed and are selected after Annex A4. The correlation factors given there apply to “flexible” compression piles. If “stiff” compression piles are used, the correlation factors may be divided by 1,1, assuming that ξ_{1} does not become smaller than 1,0.
For the range of small pile settlements, after 6.4, Eq. (6.13), characteristic boundary lines were derived for the serviceability limit state analysis using the k values (based on [59]). For the present case k = 0,15 was adopted to relate to the average of the measured values.
Note: The method adopted here for the characteristic boundary lines in the service load range represents only one possible option. Other reasoned procedures are also possible.
Figure B1.2 and Table B1.1 show the results for the determination of R_{c,k}(SLS) and R_{c,k} = R_{c,k} (ULS).
Using the results from Table B1.1, the characteristic pile resistance R_{c,k} in the serviceability limit state (ULS) can be determined for “flexible” and “stiff” piles in Table B1.2.
The limit state condition after 6.2 and 6.3:
F_{c,d} ≤ R_{c,d}
must be adhered to for ultimate limit state (ULS) analysis.
When determining the pile resistances R(SLS) in the serviceability limit state, differentiation after 6.4 is required whether minor or major (adopted here) differential pile settlements are to be expected. To this end, the characteristic boundary curves in the service load range were derived from the recorded pile load test data in Table B1.1 and Figure B1.1b.
The specified allowable settlement (e.g. specified in the structural design) in the example is allow. s_{k} = 2 cm.
Using Figure B1.3a, serviceability is demonstrated via pile forces after 6.3, Eq. (6.11) as:
F_{d} (SLS) = F_{k} = 2,500 MN < R_{d} (SLS) = R_{k} (SLS) = 2,7 MN.
Analysis by comparing settlements in accordance with Figure B3.3b after 6.3, Eq. (6.12) results to:
exist. s_{k,max} = 1,8 cm < allow. s_{k} = 2,0 cm
Dynamic pile load tests were performed to determine the ultimate capacity of reinforced concrete displacement piles and then calibrated against a static load test from a different, but comparable site. Evaluation using the direct method (CASE equation) resulted in the following measured data for the compressive resistance of the ground against the pile in the ultimate limit state:
The respective characteristic ultimate limit state resistances for “flexible” and “stiff” compression piles shall be derived from the measured data of the dynamic pile load tests.
The characteristic pile resistance in the ultimate limit state (ULS) R_{c,k} is given by the lesser of the mean value (R_{c,m})_{mean} or the minimum value (R_{c,m})_{min} of the pile load test results using Equation (A4.3) as follows:
The correlation factors ξ_{5} and ξ_{6} to be adopted are given in Annex A4.2 and are derived from the base values of the correlation factors ξ_{0}._{5} and ξ_{0 6}, the increased value Δξ and the model factor η_{D}. The correlation factors ξ_{5} and ξ_{6 }may be divided by 1.1 for “stiff” compression piles.
The correlation factors ξ_{5} and ξ_{6} are provided in Table B2.1 and the determination of the characteristic compression pile resistances R_{c,k} in Table B2.2.
Figure B3.1 (example taken from DIN 4014:199003) summarises the information on soil type, ground strength and pile geometry necessary for the determination of the axial pile resistance R_{c,k}(s) based on empirical data.
The characteristic resistancesettlement curve shall be determined using the table data after 5.4.6 (Tables 5.12 to 5.15).
Note: Reference is made to the application principles and limitations in 5.4.3, in particular with regard to the upper table values. In the example presented here both the lower and the upper table values are used as examples (not as a rule).
The ultimate limit state skin friction values for the sand and the clay are given in Tables 5.13 and 5.15 in 5.4.6.2. By adopting the associated pile skin areas, the ultimate limit state pile shaft resistances R_{s,k,i} are provided in Table B3.1.
The settlement s_{sg}, in cm, is calculated as follows, adopting the ultimate limit state pile shaft resistance R_{s,k} in MN:
s_{Sg} = 0,50 · R_{s,k} + 0,50.
Using the figures from the example the pile head settlement is:
s_{sg} = 0,50 · 1,243 + 0,50 = 1,1 cm for the lower table values and
s_{sg} = 0,50 · 1,726 + 0,50 = 1,4 cm for the upper table values.
A mean soil strength is adopted in a region from 1 · D (0,9 m) above and 3 · D (3 · D = 2,70 m) below the pile base to determine R_{b,k}. For this zone a mean cone resistance q_{c,m} = 17,5 MN/m^{2} is shown in the penetration test diagram in Figure B3.1.
The pile base capacity can be calculated by adopting the figures from Table 5.12 in 5.4.6.2 and taking the previously determined value of q_{c,m} into consideration. Table B3.2 reproduces the calculated figures.
The pile resistances calculated from the pile base and pile shaft resistances are listed in Tables B3.3 and B3.4 as a function of the pile head settlement and are given for the lower and upper values. The settlement of the pile head for each value of the pile resistance R_{c,k} is given by the characteristic resistance settlement curve in Figure B3.2.
Figure B4.1 summarises the information on soil type, ground strength and pile geometry required to determine the axial pile resistance R_{c,k}(s) based on empirical data.
The characteristic resistancesettlement curve shall be determined using the table data after 5.4.4 (Tables 5.1 to 5.4).
Note: Reference is made to the application principles and limitations in 5.4.3, in particular with regard to the upper table values. In the example presented here both the lower and the upper table values are used as examples (not as a rule).
The empirical skin friction values in the zones of the loadbearing noncohesive soil and the weak cohesive soil are given by Tables 5.2 and 5.4 in Section 5.4.4. Together with the corresponding pile skin areas, taking the correlation factor for the skin area from Table 5.5 in Section 5.4.4 into consideration, the pile shaft resistance upon mobilisation of the ultimate limit state state R_{s,k} (s_{sg*}) is given in Table B4.1 and the pile shaft resistance R_{s,k} (s_{g}) at failure in Table B4.2.
Upon mobilisation of the failure state the settlement in cm for the skin friction s_{sg*}, adopting R_{s,k}(s_{sg*}) in MN, is determined using the following equation for the pile shaft resistance R_{s,k}(s_{sg*}):
s_{sg*} = 0,50 · R_{s,k}(s_{sg*}).
Using the figures from the example the pile head settlement is:
s_{sg*} = 0,50 · 0,715 = 0,4 cm 
for the lower table values and 

s_{sg* }= 0,50 · 1,002 = 0,5 cm 
for the upper table values. 
For determination of R_{b,k} a mean soil strength is adopted from 4 · D_{eq} below to 1 · D_{eq} above the pile base..
The equivalent pile diameter of a square prefabricated driven pile is determined using:
Using the dimensions of the example the equivalent pile diameter is:
The nominal value of the square pile base area in this case is:
The penetration test diagram in Figure B4.1 displays a mean characteristic cone resistance along the respective length of:
Using the figures in Table 5.1 of these Recommendations and referring to the previously determined value of q_{c,m} and the correlation factor for the pile base area in Table 5.5 (5.4.4), the pile base resistance can be calculated. Table B4.3 contains the determined numerical values.
Tables B4.4 and B4.5 contain the pile resistances calculated for the lower and upper values from the pile base and shaft resistances as a function of the pile head settlement. The settlement of the pile head for each value of the pile resistance R_{c,k} is given by the characteristic resistancesettlement curve in Figure B4.2.
Figure B5.1 summarises the information on soil type, ground strength and pile geometry required for determination of the axial pile resistance R_{c,k}(s) based on empirical data.
The characteristic resistancesettlement curve using the table data after 5.4.8.2 (Tables 5.24 and 5.25) shall be determined.
Note: Reference is made to the application principles and limitations in 5.4.3, in particular with regard to the upper table values. In the example presented here both the lower and the upper table values are used as examples (not as a rule).
The ultimate limit state skin friction values in the loadbearing, noncohesive soil are contained in Table 5.25 in 5.4.8.2. Adopting the associated pile skin areas, the ultimate pile shaft resistance R_{s,k }is provided in Table B5.1.
The settlement in cm for the skin friction s_{sg} with R_{s,k} in MN is determined using the following equation for the pile shaft resistance R_{s,k}:
Using the figures from the example the pile head settlement is:
s_{sg} = 0,50 · 0,316 + 0,5 = 0,7 cm  for the lower table values and 

s_{sg} = 0,50 · 0,430 + 0,5 = 0,7 cm  for the upper table values. 
Note: Loadbearing strata with q_{c} ≥ 7,5 MN/m^{2} are not considered if they are underlain by weak strata.
To determine R_{b,k} a mean soil strength q_{c.m} = 17,5 MN/m^{2} is adopted from 4 · D_{eq} below to 1 · D_{eq} above the pile base.
The nominal value of the pile base area in this case is:
Adopting the figures from Table 5.24 in 5.4.8.2 and taking the previously determined value of q_{c,m} into consideration, the pile base capacity can be calculated. Table B5.2 contains the determined figures.
The pile resistances calculated from the pile base and pile shaft resistances are given in Tables B5.3 and B5.4 as a function of the pile head settlement for the lower and upper values. The settlement of the pile head for each value of the pile resistance R_{c,k} is derived from the characteristic resistancesettlement curve in Figure B5.2.
Figure B6.1 summarises the tested or measured value R_{c,m} from a static load test for the resistancesettlement curve of a prefabricated reinforced concrete driven pile. It also contains the information on soil type, ground strength and pile geometry required to determine the axial pile resistance R_{c,k}(s) based on empirical data after 5.4.4.2.
For the comparison with the results of the static load test, the characteristic resistancesettlement curve shall be determined using the empirical data after 5.4.4.2 (Tables 5.1 to 5.4). In addition, the design resistance in the ultimate limit state (ULS), based on both static load tests and on empirical data, is determined.
Note: Comparison of the axial pile performance derived from a static pile load test with the empirical data after 5.4 requires that the pile skin friction is adopted over the entire length of the pile. The calculation of the resistancesettlement curve after 5.4.4.2 deviates from the previous procedure for determining characteristic axial pile resistance (see example B4) inasmuch as nonloadbearing or weak strata with q_{c} ≤ 7,5 MN/m^{2 }or c_{u,k} ≤ 60 kN/m^{2} (approximate extrapolation of table values) are taken into consideration.
Note: Reference is made to the application principles and limitations in 5.4.3, in particular with regard to the upper table values (not as a rule). In this example the upper table values are adopted to evaluate the static pile load tests.
The empirical skin friction values in the zone of the loadbearing noncohesive and the weak cohesive soil are given by Tables 5.2 and 5.4 in Section 5.4.4.2 and, together with the corresponding pile skin areas, taking the correlation factor for the skin area from Table 5.5 into consideration, the pile shaft resistance upon mobilisation of the ultimate limit state R_{s,k} (s_{sg*}) is given in Table B6.1 and the pile shaft resistance R_{s,k}(s_{g}) at failure in Table B6.2.
Upon mobilisation of the failure state the settlement in cm for the skin friction s_{sg*}, adopting R_{s,k}(s_{sg*}) in MN, is determined using the following equation for the pile shaft resistance R_{s,k}(s_{sg*}):
Using the figures from the example analysis the pile head settlement is:
For determination of R_{b,k} a mean soil strength is adopted in a region from 4 · D_{eq} below to 1 · D_{eq} above the pile base..
The equivalent pile diameter of a square prefabricated driven pile is determined using the following equation:
Using the dimensions of the example the equivalent pile diameter is:
The nominal value of the square pile base area in this case is:
The penetration test diagram in Figure B6.1 displays a mean characteristic cone resistance along the respective length of:
Using the figures given in Table 5.1 of 5.4.4.2 and taking the previously determined value of q_{c,m} and the correlation factor for the pile type as given in Table 5.5 into consideration, the pile base resistance can be calculated. Table B6.3 contains the determined figures.
The pile resistance calculated for the upper values from empirical data of the pile base and pile shaft resistance is given in Table B6.4 as a function of the pile head settlement. The settlement of the pile head for each value of the pile resistance R_{c,k} is derived from the characteristic resistancesettlement curve in Figure B6.2 as an upper empirical value R_{c,k}(s) and is listed for comparison with the tested or measured data from the load tests R_{c,m}(s).
The characteristic pile resistance R_{c,k} for the ultimate limit state (ULS) shall be derived from R_{c,m} using Equation (A4.1) and adopting the correlation factors ξ_{1} and ξ_{2} from Table A4.1.
Where only one pile load test is used, the correlation values are ξ_{1} = ξ_{2}. If the structural system is sufficiently stiff, “stiff” compression piles can be adopted by dividing the correlation factors by 1,1, whereby ξ_{1} may never become smaller than 1. The determined correlation factors are:
– ξ_{1} = ξ_{2} = 1,35 
for flexible systems; 

– ξ_{1} = ξ_{2} = 1,23 
for stiff systems. 
The characteristic pile resistances R_{c,k} are given by the measured pile resistance in the ultimate limit state (ULS) R_{c,m} = 3,408 MN after Table B6.4 using:
– R_{c,k} = 2,524 MN 
for flexible systems;  
– R_{c,k} = 2,771 MN 
for stiff systems. 
a) From empirical data
The design resistances in the ultimate limit state (ULS) based on empirical data R_{c,d} are determined by dividing the characteristic pile resistances R_{c,k} by the partial factor γ_{t} = 1,4 after the table in Annex A3.2.
The design values of the upper and lower table values are derived and summarised in Table B6.5. The lower table values are of comparative character only.
b) From the pile load test
To derive the design values of the pile resistances R_{c,d} in the ultimate limit state from the pile load test, the characteristic pile resistances R_{c,k} after B6.3
are divided by the partial factor γ_{t}= 1,1 from Table A3.2. The design values are given by:
– R_{c,d} = 2,524 MN/1,1 = 2,295 MN 
for flexible systems;  
– R_{c,d} = 2,771 MN/1,1 = 2,519 MN 
for stiff systems. 
The design values in the ultimate limit state (ULS) from empirical data and those from the static pile load test are compared in Figure B6.3.
Figure B7.1 summarises the information on soil type, ground strength and pile geometry required to determine the axial pile resistance R_{c,k}(s) based on empirical data after 5.4.5.3.
The pile base volume necessary to accept a characteristic action F_{k} based on empirical data is required for the preliminary design of a Franki pile. In this example, initially the lower empirical values after 5.4.5.3 are adopted, whereby the pile base is designed for a characteristic permanent action F_{G,k} = 1,20 MN and a characteristic variable action F_{Q,k} = 0,40 MN.
Analysis of the ultimate limit state (ULS, GEO2) must be performed for the necessary pile base volume determined in the preliminary design for the characteristic action, adopting the upper and lower empirical values, based on the actual driving energy expended during pile installation as shown in Figure B7.2.
To check the pile resistance based on empirical data after 5.4.5.3 the pile resistance in the ultimate limit state is compared to the static load test in Figure B7.2. To this end, the characteristic pile resistance must be determined using the upper table values and taking into consideration the nonloadbearing strata, and compared to the result of the static load test.
The empirical skin friction values in the zone of the loadbearing boulder clay from a depth of 0,80 m above driving depth and, adopting the associated pile skin areas, the ultimate pile shaft resistance R_{s,k} after Table B7.1, are given in Table 5.11 in 5.4.5.3.
Note: The nonloadbearing strata and the strata overlying them were not taken into consideration for the preliminary design. The empirical ultimate skin friction for cohesive soils after Table 5.11 can be used for the boulder clay.
The Franki pile is designed for a characteristic action F_{k} = F_{G,k} + F_{Q,k}. For Loading Case 1 this means that a characteristic axial pile resistance of:
must be verified. The necessary pile base resistance is given in Table B7.2.
A mean soil strength is adopted in the zone from 3 · D_{s} below to 2 · D_{s }above driving depth to determine R_{b,k}. A mean undrained shear strength c_{u,k} = 0,140 MN/m^{2} is given for this zone by the penetration test diagram in Figure B7.1.
Adopting the undrained shear strength c_{u,k} and the necessary characteristic pile base resistance R_{b,k} from Table B7.2 gives the pile base volume for the lower empirical values in Figure 5.10. The necessary base volume of the Franki pile for the lower empirical values is given in Table B7.2.
The ultimate pile shaft resistance after Table B7.1 is:
The driving energy for the final 2 m is given by the driving energy diagram in Figure B7.2 as:
The norm driving energy after 5.4.5.3, Table 5.9, for a Franki pile with D_{s} = 56,0 cm is:
and the norm driving energy ratio:
The pile base resistance R_{b,k} is given using the standard driving work ratio and the existing pile base volume after Table B7.2 from Figure 5.10:
The axial compression pile resistance in the ultimate limit state from empirical data is given by:
Note: Reference is made to the application principles and limitations in 5.4.3, in particular with regard to the upper empirical values (not as a rule).
The ultimate limit state pile shaft resistance using the upper values from Table 5.11 is given in Table B7.3.
Using the norm driving energy ratio:
and the existing pile base volume after Table B7.2, the pile base resistance R_{b,k }is given by Figure 5.11.
The axial compression pile resistance in the ultimate limit state from empirical data is:
The design value of the compressive pile resistance is given by:
and the design value of the action by:
Where γ_{t} = 1,40, γ_{G} = 1,35 and γ_{Q} = 1,50 for the DSP design situation.
must be adhered to for analysis of the ultimate limit state. The utilisation factors are given in Table B7.4.
The characteristic axial pile resistance based on empirical data from a static load test is examined.
Note: Comparison of the axial pile performance derived from a static pile load test with the empirical data after 5.4.5.3 requires that the pile skin friction is adopted over the entire length of the pile. The calculation of the pile resistance deviates from the previous procedure for determining characteristic axial pile resistance inasmuch as nonloadbearing or weak strata with q_{c} ≤ 7,5 MN/m^{2} or c_{u,k} ≤ 0,06 MN/m^{2} are taken into consideration. Reference is made also to the application principles and limitations in 5.4.3, in particular with regard to the upper table values (not as a rule). In this example the upper table values are adopted to evaluate the static pile load tests.
The empirical skin friction values in the zone of the loadbearing boulder clay from a depth of 0,80 m above driving depth are given in Tables 5.10 and 5.11 of these Recommendations and, adopting the associated pile skin areas, the ultimate pile shaft resistance R_{s,k }after Table B7.5.
Using the standard driving work ratio W = 0,71 after B7.3.2 and the pile base volume V = 1,05 m^{3} from Figure 5.11 the pile base resistance R_{b,k} is:
The axial pile resistance based on empirical data is:
Extrapolating the resistancesettlement curve from Figure B7.2 using the hyperbola method gives the axial pile resistance for an ultimate settlement sult = s_{g} = 0,1 · D_{s}:
The deviation in the calculated pile resistance from the measured value is ΔR_{1} =0,1%.
In this case the upper empirical data thus correspond well to the results of the static load test taken for reference.
Stratum  Soil properties 
Fill (sand)  
Soft stratum 
The ultimate and serviceability limit states shall be analysed for a prefabricated, square, reinforced concrete pile with an edge length a_{s} = 0,35 m and a permanent action F_{G,k} = 450 kN imposed by structural loads. The results of a static pile load test are available as shown in Figure B8.2 and Table B8.2. It is known from a settlement analysis that the soft stratum will settle by 5 cm below the assumed infinite fill. The settlements caused by the fill in the loadbearing ground can be ignored. In the serviceability limit state (SLS) a maximum pile head settlement allow. s_{k} = 0,5 cm is permitted, which in this case is compared to a lower bound of the characteristic RSC in the serviceability limit state. The pile is also assumed to be rigid.
The characteristic pile resistance in the ultimate limit state (ULS) R_{c,k} is given by the measured value for R_{c,m} from the load test using Eq. (A4.1) as follows:
where the correlation factor is ξ_{1} = 1,35 after Annex A.4. The correlation factors given there refer to “flexible” compression piles and are applied for single pile behaviour, as is assumed here.
For the range of small pile settlements, after 6.4, Eq. (6.13) characteristic boundary lines with values of k = 0,15 (based on [59]), were derived to form the basis for the serviceability limit state analysis, here applied in approximation to the measured values in the serviceability limit state.
Note: The method adopted here for the characteristic boundary lines in the service load range is only one possible option. Other reasoned procedures are also possible.
Figure B8.2 and Table B8.2 show the results determined for R_{c,k} (SLS) and R_{c,k} = R_{c,k} (ULS).
In the ultimate limit state (ULS), s_{ult} = s_{g} = 0,10 · D_{b} shall be adopted for the pile head settlement, if no other criteria are selected. An equivalent pile diameter D_{eq} = 39,5 cm = 0,395 m is given for the square pile.
The characteristic pile resistance in the ultimate limit state ULS in Table B8.2 is thus:
The characteristic pile resistance R_{c,k}(SLS) in the serviceability limit state for allow. s = 0,5 cm from Figure B8.2 and Table B8.2 is:
A settlement of exist. s_{k} ≈ 0,3 cm results from the characteristic RSC for the existing characteristic action F_{G,k} = 450 kN imposed on the pile by structural loads in the serviceability limit state (SLS).
Note: It is assumed in approximation that the pile resistances in the pile load test only result from the loadbearing soil and the strata above make no contribution.
In Figure B8.3 the pile settlements under the effect F_{G,k} for the ultimate and serviceability limit states are compared to the settlements of the soft stratum.
In the ultimate limit state (ULS) the neutral point is located 3,7 m below the ground surface and within the soft stratum. The characteristic actions from negative skin friction shown in Table B8.3 are determined using Eq. (4.1) for the soft stratum and Eq. (4.2) for the fill.
After determining the existing settlement s ≈ 0,3 cm for the serviceability limit state (SLS), the neutral point is located at a depth of 9,5 m below grade. This deviates from Figure B8.3 where the neutral point is determined for s ≈ 0,5 cm; the difference is however ignored. The characteristic actions from negative skin friction are determined in Table B8.4 for the serviceability limit state.
For analysis of the ultimate limit state the limit state condition must be adhered to:
The negative skin friction is adopted here as a permanent action analogous to 4.4.3 (1) in the DSP design situation.
In the serviceability limit state the limit state condition:
must be adhered to.
Note: The adopted pile settlements s(SLS) and s_{ult} = s_{g} resulting from structural loads are increased slightly by the characteristic pile effect from negative skin friction, in particular in terms of s(SLS). Iteration can normally be dispensed with, because the procedure shown here adopting s(SLS) ≈ 0,3 cm (without negative skin friction) is conservative and normally leads to an unfavourable characteristic pile effect F_{c,k} as a result of the large depth of the neutral point in the serviceability limit state.
For analysis of internal capacity (structural failure) the largest effect at the depth of the neutral point in the serviceability limit state is given by:
The internal pile design (structural failure of the pile material) in the ultimate limit state shall be done using this effect F_{k} = F_{G,k} + F_{n,k,SLS}.
The internal capacity of the pile shown in Figure B9.1 is analysed. The soil resistance and the effect on the pile must first be determined. The pile is then designed, adhering to the corresponding internal pile capacity analyses.
The following characteristic actions are taken into consideration:
Permanent vertical action:  F_{G,k} = 3,333 MN 

Variable vertical action:  F_{Q,rep} = 2,000 MN 

Permanent horizontal action:  H_{G,k} = 0,600 MN 

Variable horizontal action:  H_{Q,rep} = 0,400 MN. 
The system comprising the characteristic actions, the ground profile and the depthdependent subgrade modulus distribution is shown in Figure B9.1.
According to 6.3.2 the maximum mobilisable characteristic normal stress σ_{h,k }between the pile and the soil is limited by the characteristic passive earth pressure e_{ph,k} calculated for the plane state.
(B9.1)
The design value of the characteristic lateral normal stress resultant, referred to as the lateral soil resistance B_{h,d}, may not exceed the design value of the three dimensional passive earth pressure E^{r}ph,d for the embedment length of the pile as far as the pivot point (displacement zero point).
(B9.2)
Adherence to the conditions arising from Eq. (B9.1) and Eq. (B9.2) is to be checked prior to pile design.
Figure B9.2 b) shows the pile’s characteristic mobilised subgrade stress from the structural analysis of an elastically supported beam, adopting the subgrade reaction moduli from Figure B9.2 a). The distribution of the characteristic passive earth pressure is shown in Figure B9.2 c). It is clear from the superimposed subgrade stress and passive earth pressure in Figure B9.2 d) that the characteristic mobilised subgrade stresses σ_{h,k} in the upper region of the pile exceed the characteristic passive earth pressure e_{ph,k}. The approach for the subgrade reaction modulus distribution is modified such that Eq. (B9.1) is adhered to.
Figure B9.3 a) shows the iteratively determined subgrade reaction moduli, reduced in the upper pile region, which are subsequently adopted for further design. The corresponding characteristic mobilised subgrade stress is shown in Figure B9.3 b) and the characteristic passive earth pressure in Figure B9.3 c). The superimposed subgrade stress and passive earth pressure in Figure B9.3 d) show that the values of the characteristic mobilised normal stresses σ_{h,k} are now below the characteristic passive earth pressure e_{ph,k}. The condition in Eq. B9.1 is thus met.
While determining the action effects and stresses it must also be demonstrated that the resulting lateral soil resistance is smaller than the design value of the passive earth pressure as far as the pile pivot (zero displacement point).
The lateral soil resistance is given by integrating the mobilised subgrade stress from the pile head to the pivot (displacement zero point), located at an elevation of 13.47 m, using beam analysis. The characteristic soil resistance forces given by integrating the stress areas as shown in Figure B9.3 b) are:
The design value of the soil resistance is therefore:
(B9.3)
The threedimensional passive earth pressure as a result of the selfweight of the soil is calculated to DIN 4085:200710 as follows:
(B9.4)
Where:
D: Pile diameter
z: Depth in question.
For the elevation here of13,47 m:
The threedimensional passive earth pressure E^{r}_{ph,k} to the elevation of 13.47 m is given by the passive earth pressure distribution in the planar case as follows:
Applying the partial factor γ_{R,e} the design value of the threedimensional passive earth pressure is:
For analysis of the lateral soil resistance:
The threedimensional passive earth pressure is not exceeded.
The design values of the action effects are given by adopting the partial factors for the limit state in the structure and the ground (STR and GEO2) after the EC 71 Handbook [44], also see Annex A2. The actions are allocated to the permanent design situation DSP.
The design value of the normal force is:
(B9.6)
If the changeable load acts favourably, the design value is:
(B9.7)
The design value of the shear force is:
(B9.8)
The design value of the bending moment is:
(B9.8)
Figure B9.4 and Figure B9.5 show the distribution of action effect design values with reduced subgrade reaction modulus adopted for structural analyses.
The minimum concrete strength class to DIN 10451:200808 for exposure class XC2 with regard to foundation components is C16/20. DIN EN 1536:199906 envisages a concrete strength class between C20/25 and C30/37 for bored piles.
Adopted: C25/30 XC2
The nominal concrete cover c_{nom} to DIN 10451:200808 is given by:
(B9.9)
Where c_{min} is the minimum concrete cover to DIN EN 1536:199906 as a function of D:
Δc Allowance value to DIN 1045:200808 as a function of the exposure class:
Giving the nominal concrete cover:
To guarantee the bond to DIN 10451:200808 the condition:
must also be adhered to.
The partial factors of the materials for the ultimate limit state are adopted from DIN 10451:200808:
The design value of the concrete compressive strength is given in accordance with DIN 10451:200808 by taking the longterm impact on the compressive strength into consideration using the factor α (α = 0,85 for normal weight concrete):
(B9.10)
Where f_{ck} = 25,0 N/mm^{2}.
The design value of the yield strength of the reinforcement steel BSt 500 S (A) is given by:
(B9.11)
Where
Thus
The circular section is designed using the ω method with dimensionless coefficients.
Assuming a diameter of 12 mm for the transverse and 32 mm for the longitudinal reinforcement the centres d_{1} of the longitudinal reinforcement are at:
For the purposes of analysis the bending reinforcement centres are adopted at d_{1} = 0,1.h = 15 cm, i.e. d_{1}/h = 0,1.
The relative bending moment is given by:
(B9.12)
The relative normal force is given by:
(B9.13)
The corresponding, necessary degree of mechanical reinforcement is given by reading the diagram for d_{1}/h = 0,1 as:
The corresponding strains are:
The necessary longitudinal reinforcement is given by:
(B9.14)
The minimum longitudinal reinforcement to DIN EN 1536:199906, Table 4 is given as a function of the nominal pile diameter by:
(B9.16)
The minimum reinforcement in this case is not a governing factor.
The adopted longitudinal reinforcement is:
DIN EN 1536:199906 requires that a minimum spacing of 100 mm and a maximum spacing of 400 mm is adhered to for the longitudinal bars.
(B9.17)
The necessary minimum spacing is therefore adhered to.
The Civil Engineering Standards Committee (NABau) interpretation states that the smaller value of the crosssectional width between the reinforcement centroid (tension flange) and the pressure resultant (corresponds to the smallest width normal to the inner lever arm z) is used as the effective width for shear force design of circular crosssections.
Bending design gives:
Using the longitudinal reinforcement centres adopted in bending design the height of the compression zone is:
(B9.18)
Because the zero strain line lies within the crosssection, a stress block may be adopted in simplification for the distribution of the concrete compressive stress to DIN 1045:200808. The height of the stress block is given by:
(B9.19)
The pressure resultant lies in the centroid of a circle segment with an opening angle α_{c} of:
(B9.20)
The centroid distance from the circle centre is:
(B9.21)
In line with the interpretation of the Civil Engineering Standards Committee the true tensile reinforcement, i.e. the reinforcement within the tension zone, is adopted for shear force design. To determine the reinforcement centroid the reinforcement is assumed to be uniformly distributed along its centroid axis. The radius of the reinforcement centroid axis is:
(B9.22)
The centroid of the ring section with an opening angle α_{s} of:
(B9.23)
lies at a distance from the centre of the circle:
(B9.24)
This approximation deviates from the true location of the centroid of the steel tensile forces, because stress distribution in the reinforcement is not taken into consideration. However, the approach is conservative due to the larger true lever arm.
The smaller value of the crosssectional width is therefore at the height of the pressure resultant and is:
(B9.25)
The structural height is given by:
(B9.26)
The inner lever arm is:
(B9.27)
The design value of the shear capacity without shear reinforcement in the equivalent rectangle to DIN 1045:200808 is:
(B9.28)
Where
η_{1} = 1,0 for normal weight concrete
(negative compressive force) Giving:
Shear reinforcement is therefore required.
The limitation in the compression member inclination is:
(B9.29)
Where:
(B9.30)
giving:
The design value of the maximum shear capacity is:
(B9.31)
The necessary shear reinforcement is given by:
(B9.32)
However, this approach does not take the increased effect of the shear reinforcement due to radial thrust forces caused by the continuous support of the compression strut on the circular link (boiler pressure) into consideration. Therefore, [5] derives the shear resistances from the basic equations in DIN 10451, taking the thrust forces of the circular link and the nonuniform distribution of the shear force on the longitudinal reinforcement into consideration for axissymmetrical, longitudinally reinforced beams with shear reinforcement. If shear reinforcement is in helix format, the helix inclination α must also be taken into consideration. Deriving the tension strut strength for circular links leads to the design value given in DIN 10451:200808:
(B9.33)
supplemented by a factor α_{k} known as a the effectiveness factor, dependent only on dimensionless geometry parameters and a term for the helix inclination α. Due to the small range of values of 0,715 ≤ α_{k} ≤ π / 4 = 0,785 for practical applications, [5] proposes a simplified estimate of α_{k} = 0,75 .
The shear force component V_{Rd,c} to DIN 10451:200808 is adopted to determine the compression strut inclination, whereby the effective width can be estimated as b_{w} = A_{eff}/z ≈ 0,9 · D = 0,9 1,5 = 1,35 m. Using Eq. (B9.30):
Using Eq. (B9.29) the limitation in the compression strut inclination is:
The capacity of the compression strut is limited in accordance with DIN 10451:200808, whereby the effective width may be adopted as b_{w} = 2 · R_{S }≈ D = 1,5 m using the approach after [5]. Given an angle of inclination of the helix of a = 85° (cf. adopted reinforcement) the capacity of the compression strut is limited to:
(B9.34)
The necessary shear reinforcement is given by Eq. (B9.33):
In order to take into consideration the favourable impact of central normal compression forces, [5] divides the bearing resistance into individual components in a truss model with an additive concrete contribution, whereby the shear force transfer as a result of crack formation is ignored, such that by adopting V_{Rd,c} = 0
(B9.35)
In addition to the effect of the truss model, the resistance in the compression and failure process zone is adopted:
(B9.36)
The semiempirical approach was adopted by [5] for shear force resistance:
(B9.37)
where the term λ · N_{Ed} considers the favourable influence of central normal compression forces via a strutted frame. The inclination of the strutted frame λ = z_{C} /a corresponds to the change in the distance z_{N} between the centroid axis and the centroid of the compression zone over the length a. In simplification, the length a may be estimated as the distance between the point of acting of the shear force and the location of the maximum moment where a = 892160 = 732 cm. Hence:A normal compression force is adopted by applying the appropriate partial factor for favourable actions.
In contrast to the NABau interpretation, [5] adopts the reinforcement of the half crosssection in the tension zone in simplification for the degree of longitudinal reinforcement.
(B9.38)
Using b_{eff} = 0,9·D = 1,35m and the values previously determined in Equation (B9.37):
The required reinforcement in accordance with Equation (B9.36) is therefore:
In contrast to the NABau interpretation, the approach in accordance with [5] gives an approx. 16% lower shear force reinforcement. This is caused by the increase in shear capacity due to the strutted frame approach for the normal forces. If no, or only minor, compressive forces act on the structural element, the design approach in accordance with [5] returns more shear reinforcement than the NABau interpretation, because in this case the additional effect imposed by the circular reinforcement’s boiler pressure not covered by the DIN 10451:200808 design concept becomes governing. The approach after [5] is therefore recommended for design.
The minimum reinforcement to DIN 10451 is given by the base value for determining the minimum reinforcement ρ = 0,068% as:
(B9.39)
The minimum diameter of the transverse reinforcement in accordance with DIN EN 1536:199906 is:
(B9.40)
The minimum clear bar spacing in accordance with DIN EN 1536:199906 may not be smaller than for longitudinal bars at:
The minimum longitudinal spacing in accordance with DIN 10451:200808 for
(B9.41)
is:
(B9.42)
The following shear reinforcement is used:
Helix 2core dia.
Figure B10.1 shows a bridge abutment founded on 8 tubular steel piles. The lateral pressure on the piles as a result of the backfill must be determined, based on [35].
Boundary conditions:
The characteristic actions from lateral pressure are determined in line with 4.5, whereby the characteristic flow pressure p_{f,k} is determined using Eq. (4.5) and adopting the geometry of the pile group as shown in Figure 4.7.
The characteristic effect of the flow pressure P_{f, k}_{w}