Clause 2.8(1)P Clause 2.8(2) Clauses 2.8(3) Clause 2.8(4)P The design assumptions, data, methods of calculation and results of the checking of safety
and serviceability should be recorded in a Geotechnical Design Report (GDR) for all geotechnical designs, including small and relatively simple structures in straightforward ground conditions. The level of detail may vary greatly. For simple designs, a single-sheet report may be sufficient (Fig. 2.9). A checklist of items which should normally be included in the GDR is given in EN 1997-1. The two most important parts of the GDR are:
• the Ground Investigation Report (see Chapter 3 of this guide)
• a plan for appropriate supervision and monitoring (see Chapter 4 of this guide).
Clause 2.8(5) A checklist is given of the items to be covered in the plan for supervision and monitoring.
The observational method cannot be applied where a sudden collapse could occur without warning, such as when ground and the interaction between ground and structure are not sufficiently ductile (brittle behaviour).
Fig. 2.9. Single-sheet GDR. (After Simpson and Driscoll, 1998)
Example 2.1: selection of a characteristic value using statistical methods Description of the problem
This example illustrates how the characteristic value of the drained shear strength parametersϕk¢ and ck¢ can be selected from the sets of values obtained from triaxial tests.
The ultimate limit state of an embedded retaining wall is considered. This ultimate limit state involves a large volume of soil. The characteristic value of the shear strength parameters is then a cautious estimate of their mean value. In this example, it is assumed that the variations of the shear strength are random, without significant local weaker zones and without significant trend with depth.
The calculations are performed using the statistical formulae given in the appendix to this chapter; different cases are treated with the corresponding statistical method and their results are compared:
(1) There is no information available other than the results obtained from the triaxial test results on four local samples from boreholes located on the site (local sampling).
Complementary to the results, it is assumed that there is reliable knowledge of the variability of c¢ and tanϕ¢ through their coefficients of variation.
(2) Instead of analysing c¢ and tanϕ¢ separately, an analysis is performed on the shear resistance using t as a linear function of s¢: the characteristic mean value at the 95%
confidence level for a linear trend between t and s¢ is determined, from which the values of ck¢ and tan ϕk¢ can be obtained.
Note:
s¢ = (σv¢ + σh¢)/2 t = (σv¢ – σh¢)/2 sinϕk¢ = ∆t/∆s¢ c¢ = t(0)/cosϕ¢
Evaluation using results ofϕ and c from tests with local samples only (case ‘VX
unknown’)
As only the information obtained from the results of the triaxial test is used in this first calculation, the method outlined in the appendix to this chapter is applied for homogeneous soil without significant trend and the case ‘VXunknown’. Table 2.2 gives the mean value and the standard deviation of the values of c¢ and tanϕ¢ estimated from the four triaxial test results.
From Table 2.2 the characteristic value of tanϕ¢ is calculated using equation (D2.1).
For a characteristic value having a reliability of 95%, Table 2.5 in the appendix (on the basis of four tests, n = 4, and ‘VXunknown’) gives kn, mean= 1.18:
tanϕk¢ = (tan ϕ¢)mean(1 – kn, meanVtanϕ) tanϕk¢= 0.603 × (1 – 1.18 × 0.118) = 0.519 tanϕk¢ = 27.5°
Table 2.2. Mean values of c¢ and tanϕ¢, their standard deviation and coefficient of variation obtained from four triaxial test results
c¢ (kPa) ϕ¢ (°) tanϕ¢ (–)
Coefficient of variation Vc= 0.667 Vtanϕ= 0.118
The calculation for the characteristic value of ck¢ gives ck¢ = cmean(1 – kn, meanVc)
ck¢= 3.75 × (1 – 1.18 × 0.667) ck¢= 0.8 kPa
In Fig. 2.10 the corresponding Mohr’s envelope is presented together with the results of the triaxial tests (values of t and s¢ at failure).
Evaluation using results ofϕ and c assuming ‘VXknown’
It should be noted that the use of ‘VX known’ is new and has not yet gained general acceptance in the geotechnical engineering profession. The method is presented here because it may be of some help to the engineer; but it should be used with caution.
Schneider (1999), using results from Rethati (1998) and Lumb (1974), presents typical values for the coefficient of variation VXfor soil properties, which are generally valid and of which the magnitudes have been confirmed by many researchers world-wide. Typical values of VXfor the tangent of the angle of shearing resistance and for the cohesion range from 0.05 to 0.15 and 0.3 to 0.5, respectively. Schneider (1999) recommends average values of VX= 0.1 for the angle of internal friction and VX= 0.4 for cohesion. These values are used to select the characteristic mean values with a reliability of 95% for tanϕ¢
and c¢ from the four tests results, applying the case ‘VXknown’.
When four tests results are available (n = 4) Table 2.6 in the appendix gives:
kn, mean= 0.82 for ‘VXknown’. Using equation (D2.1) the characteristic value tanϕk¢ is tanϕk¢ = (tan ϕ¢)mean(1 – kn, meanVtanϕ)
tanϕk¢= 0.603 × (1 – 0.82 × 0.10) = 0.554 tanϕk¢ = 29°
Similarly, the characteristic value ck¢ can be calculated using equation (D2.1):
ck¢ = cmean(1 – kn, meanVc) ck¢= 3.75 × (1 – 0.82 × 0.4) ck¢= 2.5 kPa
0 50 100 150 200 250 300 350
0 200 400
s¢–t test results s¢–t characteristic j¢k and c¢k 'Vx known' j¢k and c¢k 'Vx unknown'
s¢ = (s¢v + s¢h)/2 t = (s¢v – s¢h)/2
600 800
Fig. 2.10. Results of triaxial tests and their statistical evaluations with respect to Mohr´s envelopes for the characteristic shear parameters
In Fig. 2.10 the corresponding Mohr’s envelope is presented together with the results of the triaxial tests (values of t and s¢ at failure).
Evaluation using the values of s and t at failure
In the analysis above, c¢ and tanϕ¢ have been analysed separately. However, one observes that they are negatively correlated, i.e. the larger c¢ values correspond to the smallerϕ¢, and vice versa. Analysing c¢ and tanϕ¢ separately is conservative. Starting from the assumption of a linear relation between effective normal stress s¢ and shear strength t, one can, alternatively, plot all s¢– t points obtained from the triaxial tests (see Fig. 2.10) and determine the characteristic value of the angle of shearing resistance,ϕk¢, by using the method for local sampling, with a linear trend of the shear strength t with s¢.
Equation (D2.3) permits the determination of the characteristic value tkas a function of s¢, and this is applied in Table 2.3, where the parameter z stands for s¢ and parameter Xk stands for tk. The factort(0 95n.– 2) is determined using Table 2.6 of the appendix, which, for n = 12 test results and r = n – 2 = 10, gives t(0 95n.–2) =1 812. Table 2.3 indicates the. parameter values for linear regression through the measured s¢ and t values (zero intercept and slope of the linear regression of t* for s¢), the value of the term s1(applying equation (D2.2)) and the characteristic value tkas a function of s¢. It should be noted that the relationship between s¢ and tkis slightly hyperbolic due to the non-linear term s1. The distance between the linear regression and the characteristic value is smallest in the middle of the stress interval and increases slightly towards its beginning and end.
The characteristic values of ck¢ and tan ϕk¢ may be deduced by linearizing the relation s¢–tk in the stress interval relevant for the problem. As sinϕ¢ = ∆t/∆s¢ and c¢ = t(s¢ = 0)/cosϕ¢, for the stress interval s¢ between 200 and 500 kPa (see Fig. 2.10):
tanϕk¢ = 0.5 tanϕk¢ = 30°
ta
n
ck¢ = 1 kPaTable 2.3. Statistical analysis of the values of t as a function of s¢ according to equations (D2.2) and (D2.3) in the appendix to this chapter
Test results Statistical analysis
200.0 105.0 107.0 3.1 5.6 101.4
225.0 130.0 119.8 3.0 5.4 114.4
240.0 121.0 127.4 2.9 5.3 122.1
250.0 134.0 132.5 2.9 5.3 127.2
400.0 231.0 208.9 3.8 6.9 202.0
420.0 201.0 219.1 4.0 7.3 211.8
450.0 229.0 234.3 4.4 8.0 226.3
600.0 312.0 310.7 6.7 12.1 298.6
Coefficients of linear regression:
Intercept: c¢= 5.2 kPa Slope: sinϕ¢= 0.51, ϕ¢= 30.5°
In the present example, the characteristic value is close to the linear regression line through all the s¢–t values. This is due to the fact that the results show quite small variability.
Discussion
A summary of the statistical evaluations of the characteristic values of the shear strength parametersϕk¢ and ck¢ is presented in Fig 2.10 and Table 2.4. The characteristic values based on local test results for ϕ¢ and c¢, without any further knowledge (case ‘VX
unknown’), are in this example close to the smallest values of the test results (see Table 2.2). This is due to the small number of samples and the rather large variability of the tests results, especially for c¢.
The introduction of knowledge of the coefficient of variation VX (case ‘VX known’) has significant influence on the calculated characteristic value. The introduction of complementary information is especially relevant when few test results are available or when the variability is rather large.
Schneider (1999) proposed the following simplified equation, based on the assumption that there is always certain knowledge of the coefficient of variation of the soil parameters:
Xk= Xmean– 0.5s
where Xmeanand the standard deviation s are deduced from values for the local samples.
Application of Schneider’s equation to the values of Table 2.2 yields results that are in close agreement with the values obtained for the case ‘VXknown’.
The analysis of c¢ and tanϕ¢ as independent variables is generally over-conservative for evaluating the characteristic shear resistance of the soil. When c¢ and tanϕ¢ are not correlated, advantage can be taken from an analysis where all results are treated together in an s¢–t linear relation.