Methodology for Obtaining Optimal Sleeve Friction and Friction Ratio Estimates from CPT Data ()
1. Introduction
Cone penetration testing (CPT) is a widely used and extensively researched geotechnical engineering in-situ test [1] [2] [3] [4] [5] for mapping soil profiles and assessing soil properties. CPT has significantly replaced the traditional methods of geotechnical site investigations such as sampling and drilling due to it being economical, repeatable, and relatively fast. The cone penetrometer has electronic sensors to measure penetration resistance at the tip and friction in the shaft (friction sleeve) during penetration. A CPT probe equipped with a pore-water pressure sensor is called a piezo-cone (CPTU cones). Figure 1 [6] illustrates the dimensions of the most commonly utilized penetrometers. Figure 2 [7] outlines the equations for obtaining sleeve friction and tip resistance where corrections are made for measured pore water pressures and differences in area (e.g., tip net area ratio and end area sleeve).
One of the main applications of CPT is the identification of soil type and the determination of soil stratigraphy. This soil classification facilitates grouping soils according to their engineering behavior (i.e., Soil Behavior Type (SBT)) and is conventionally carried out in the laboratory where borehole samples are analyzed and classified. CPT soil classification is made by empirically relating measured qc, fs and u values to type of soil in SBT charts. A number of classification methods have been utilized to predict soil type from either CPT or/both
Figure 1. Standard 10 cm2 and 15 cm2 penetrometers [6].
Figure 2. Determination of total cone resistance and total sleeve friction [7].
CPTu data. A very popular SBT chart was generated by Robertson et al. [2]. Robertson et al. [2] SBT chart is based on qt and friction ratio, Rf [where: Rf = (fs/qc)100%]. The SBT chart developed by Robertson et al. [2] identifies 12 soil types and is illustrated in Figure 3. For accurate CPT/CPTU soil classifications it is of paramount importance that cone bearing measurements of qc and fs with minimal distortions and added measurement errors are obtained. Unfortunately, both cone bearing and sleeve friction measurements obtain smoothed/averaged estimates of the true values.
The focus of the work outlined in this paper was to develop an optimal estimation algorithm for obtaining accurate sleeve friction and friction ratio estimates. Accurate cone bearing, sleeve friction and friction ratio estimates are of paramount importance when estimating soil behavior type from CPT date. The fc optimal filter estimation technique (so-called OSFE-IFM algorithm) is subsequently outlined along with a very challenging test bed simulation. For completeness the qcHMM algorithm is also subsequently summarized.
2. Mathematical Background
2.1. Cone Penetration Testing Cone Bearing Sleeve Friction Model
As previously outlined, CPT soil classifications are carried out by utilizing SBT charts where measured qc and fs values are empirically related to type of soil. The popular SBT chart developed by Robertson et al. [2] is based on qt and friction ratio, Rf. Measured cone bearing and sleeve values are blurred/averaged. It is required to apply optimal estimation algorithms so that the effect of blurring/averaging is minimized.
Figure 3. SBT chart by Robertson et al. [2] based on CPT cone resistance, qt, and friction ratio, Rf (where Rf = (fs/qc)100%).
2.2. Cone Bearing Model
The cone tip resistance measured at a particular depth is affected by the values above and below the depth of interest which results in an averaging or blurring of the true values (qv) values [8] [9] [10] [11]. This phenomenon is especially of concern when mapping thin soil layers which is critical for liquefaction assessment. Mathematically the measured cone tip resistance qc is described as [9] [10] [11].
(1)
where
d: the cone depth
dc: the cone tip diameter
Δ: the qc sampling rate
qc(d): the measured cone penetration tip resistance
qv(d): the true cone penetration tip resistance
wc(d): the qv(d) averaging function
v(d): additive noise, generally taken to be white with a Gaussian pdf
In Equation (1) it assumed that wc averages qt over 60 cone diameters centered at the cone tip. Boulanger and DeJong [8] outline how to calculate wc below (after correcting the equation for w1 [9] ):
(2a)
(2b)
(2c)
where
w1: accounts for the relative influence of any soil decreasing with increasing distance from the cone tip.
w2: adjusts the relative influence that soils away from the cone tip will have on the penetration resistance based on whether those soils are stronger or weaker.
: the depth relative to the cone tip normalized by the cone diameter.
: the normalized depth relative to the cone tip where w1 = 0.5C1.
C1: equal to unity for points below the cone tip, and linearly reduces to a value of 0.5 for points located more than 4 cone diameters above the cone tip.
mz: exponent that adjusts the variation of w1 with
.
mq: exponent that adjusts the variation of w2 with
.
Boulanger and DeJong [9] provide a thorough outline and review on the setting of the parameters given in Equation (2) based upon extensive research and modelling. In general terms, soils in front of the cone tip have a greater influence on penetration resistance than the soils behind the cone tip. In the subsequently outlined test bed simulations the parameters in Equation (2) are set identical to those outlined by Boulanger and DeJong. In this case, exponents mq = 2 and mz = 3.
Baziw and Verbeek [9] [10] [11] developed an algorithm to optimally obtain true qv cone bearing estimates from blurred measurements qc. The initial algorithm developed by Baziw and Verbeek [9] [10] (the so called qcHMM-IFM) combined a Bayesian recursive estimation (BRE) Hidden Markov Model (HMM) filter with Iterative Forward Modelling (IFM) parameter estimation in a smoother formulation. In recent modifications and enhancements of the qcHMM [11] it was possible to drop the IFM portion of the algorithm. This was predominantly accomplished by refining the HMM filter parameters.
2.3. Sleeve Friction Model
In CPT, sleeve friction is the measure of the average skin friction as the probe is advanced through the soil. Figure 4 outlines typical sleeve friction resistance and distribution generated by an algorithm (ABAQUS) which implements a Finite Element Model (FEM) designed for modelling large displacements such as those generated during CPT [12]. Figure 4(a) illustrates typical FEM sleeve friction resistance at the center of the sleeve. The high frequency fluctuations shown in Figure 4(a) are a type of measurement noise generated by the FEM algorithm due to the mesh size, the contact interface, and the parameter of soil and steel.
Figure 4(b) illustrates the FEM distribution of resistance along the length of the sleeve. In Figure 4(b) the sleeve friction close to cone tip is nearly 0 MPa and gradually increases to the uniform value of 0.029 MPa at approximately 30 mm from the bottom of the shaft for the case ϕ = 34˚ and
= 0.05 MPa. Susila and Hryciw [12] state that non-uniform sleeve friction distribution has been confirmed by Kiousis et al. [13]. Kiousis et al. state that there is a very thin separation between soil and cone shaft for approximately 35 mm above the upper end of the cone tip.
The sleeve friction distribution illustrated in Figure 4(b) can be thought of as a Sleeve Friction Weighting Function (SFWF) where various values of sleeve friction along the shaft (due to varying soils) are weighted to give a final measured value assumed to occur at the center of the shaft. The distribution illustrated in Figure 4(b) is mathematically approximated by Equation (3). Figure 5 illustrates the implementation of Equation (3).
Figure 4. (a) FEM typical sleeve friction of cone sleeve during penetration. (b) FEM typical distribution of friction resistance along the sleeve (ϕ = 34˚ and
= 0.05 MPa) [12].
Figure 5. Mathematically modelling the sleeve distribution illustrated in Figure 4(b).
(3a)
(3b)
where
= the distance from bottom of sleeve
SFWF = Sleeve Friction Weighting Function
The weighting of the true sleeve friction values by the SFWF coupled with blurring of the qv values can result in significant distortions in the calculated friction ratio Rf. Figure 6 illustrates a simulation of cone bearing, sleeve friction and friction ratio (it assumed that both qc and fc have been corrected for pore pressure). In Figure 6, the true values of qv, fv and Rfv are red traces while the corresponding measured values are the black traces. Table 1 outlines the corresponding soil behavior types (based on the SBT chart of Figure 3) for the test bed simulation illustrated in Figure 6.
The sleeve friction measurements ft were generated from the true sleeve friction values fv by implementing Equation (4) outlined below.
Figure 6. (a) Simulated cone bearing data qt (measured—black trace) and qv (true—red trace). (b) Simulated sleeve friction data ft (measured—black trace) and fv fv (true—red trace). (c) Simulated cone friction ratio Rft = 100 * ft/qt (measured—black trace) and Rfv = 100 * fv/qv (true—red trace).
Table 1. Corresponding SBTs for test bed simulation illustrated in Figure 6.
(4)
where
Δ: sleeve friction sampling rate
L: sleeve friction shaft length weaker
L*: L/Δ
l: l/2
l*: l/Δ
When implementing Equation (4), Δ is initially set to a 1 mm sampling rate. The simulated date sets are then obtained by extracting data from the 1 mm sampling rate data sets at the user specified rate. This is done so that the true in-situ measurement conditions are simulated.
As is illustrated in Figure 6(c) there are significant distortions in the simulated measured friction ratios based upon the measured cone bearing and sleeve friction values. This leads to uncertainties in soil classifications. This paper outlines an optimal sleeve friction estimation algorithm. The sleeve friction optimal estimation implemented in conjunction with the qcHMM algorithm facilitates obtaining accurate soil classification estimates.
3. OSFE-IFM Algorithm
The fv optimal filter estimation technique is referred to as the OSFE-IFM algorithm. The OSFE-IFM algorithm utilizes a posteriori information from the qcHMM algorithm and implements Iteration Forward Modelling (IFM).
3.1. OSFE-IFM Algorithm Formulation
The OSFE-HMM algorithm utilizes a posteriori information from the qcHMM algorithm so that the solution space is reduced. The qcHMM algorithm facilities quantifying the soil layering (i.e., layer interfaces). This soil layering information is inputted into the OSFE-HMM algorithm. Soil layering can readily be quantified based upon estimated qv values. Figure 7 illustrates Figure 6(a) where the soil layers (L1 to LN) are identified by blue lines and were determined from the output from the qcHMM algorithm (red lines). Each of these soil layers has an associated sleeve friction values fv1 to fvN which needs to be estimated.
The second component of the OSFE-HMM algorithm implements Iterative Forward Modelling (IFM) to estimate the sleeve friction values fv1 to fvN. IFM is a parameter estimation technique which is based upon iteratively adjusting the parameters until a user specified cost function is minimized. The desired parameter estimates are defined as those which minimize the user specified cost function. The IFM technique which is utilized within the OSFE-HMM algorithm is the downhill simplex method (DSM) originally developed by Nelder and Mead
Figure 7. Illustration Figure 6(a) with estimated soil layers identified by blue lines.
[14]. The DSM in multidimensions has the important property of not requiring derivatives of function evaluations and it can minimize nonlinear-functions of more than one independent variable. A simplex defines the most elementary geometric figure of a given dimension: a line in one dimension, the triangle in two dimensions, the tetrahedron in three, etc.; therefore, in an N-dimensional space, the simplex is a geometric figure that consists of N + 1 fully interconnected vertices. The DSM has been used in a variety of scientific applications such as obtaining seismic source locations [15] and blind seismic deconvolution [16].
The DSM starts at N + 1 vertices that form the initial simplex. The initial simplex vertices are chosen so that the simplex occupies a good portion of the solution space. In addition, it is also required that a scalar cost function be specified at each vertex of the simplex. The DSM searches for the minimum of the costs function by taking a series of steps, each time moving a point in the simplex away from where the cost function is largest. The simplex moves in space by variously reflecting, expanding, contracting, or shrinking. The simplex size is continuously changed and mostly diminished, so that finally it is small enough to contain the minimum with the desired accuracy.
For the OSFE-HMM algorithm, the IFM cost function to be minimized is the RMS difference between the measured sleeve friction values and synthetic sleeve friction measurements generated by implementing Equation (4) with the estimated sleeve friction values fv1 to fvN used as input. As with the test bed simulation, when implementing Equation (4) in OSFE-HMM algorithm, Δ is initially set to a 1 mm sampling rate. The synthetic sleeve friction measurements are then obtained by extracting data from the 1 mm sampling rate data sets at the user specified rate. This is done so that the true in-situ measurement conditions are replicated.
3.2. OSFE-IFM Test Bed Example
The performance of the OSFE-IFM algorithm was evaluated by processing the challenging test bed simulation illustrated in Figure 6. Figure 6 illustrates a highly variable CPT profile where it assumed that both the measured qc and fc have been corrected for pore pressure. In Figure 6, the true values of qv, fv and Rfv are red traces while the corresponding measured values are the black traces.
Figure 8 illustrates the output from the qcHMM algorithm. In Figure 8, it is shown the test bed specified true qv values (red line), derived measured qt values (black line) and estimated
values from the qcHMM algorithm (blue line). As is illustrated in Figure 8, the estimated
values are nearly identical to the true qv values.
Figure 9 illustrates the output from the OSFE-IFM algorithm after processing the measured ft sleeve values shown in Figure 6(b). In Figure 9, it is shown the test bed specified true fv values (red line), derived measured ft values (black line) and estimated
values from the OSFE-IFM algorithm (blue line). As is illustrated in Figure 9, the estimated
values are very close to the true fv values.
Figure 8. Specified qv values (red line), measured qt values (black line) and estimated
values obtained from implementing the qcHMM algorithm (blue line).
Figure 9. Specified fv values (red line), measured ft values (black line) and estimated
values obtained from implementing the OSFE-IFM algorithm (blue line).
Figure 10 illustrates the friction ratio output obtained from implementation of the qcHMM and OSFE-IFM algorithms where
values are derived from
and
estimates. In Figure 10 the test bed specified Rfv values, measured
Figure 10. Specified Rfv values (red line), measured Rft values (black line) and estimated
values obtained from implementing the qcHMM and OSFE-IFM algorithm (blue line).
Rft values and estimated
values are identified by red, black and blue lines, respectively. As is illustrated in Figure 10, the estimated
values are very close to the true Rfv values.
4. Conclusion
The cone penetration test (CPT) records cone bearing (qc), sleeve friction (fc) and dynamic pore pressure (u) with depth. A popular method to estimate soil type from CPT qc, fc and u measurements is the Soil Behavior Type (SBT) chart. The SBT plots cone resistance vs friction ratio, Rf [where: Rf = (fs/qc)100%]. There are distortions in the CPT qc and fs measurements which can result in significant erroneous SBT plots. The qcHMM algorithm was developed to address the qc blurring/averaging. The sleeve friction measurements are also averaged along the cone sleeve shaft. This paper has outlined an algorithm (so called OSFE-HMM algorithm) which utilizes a posteriori information from the cone bearing qcHMM estimation algorithm (i.e., soil layering interfaces) and implements iteration forward modelling for obtaining optimal estimates of sleeve friction values. A challenging test bed simulation outlined in this paper has clearly shown that the OSFE-HMM algorithm can be implemented so that optimal sleeve friction estimates are obtained from measured values. Implementation of the qcHMM and OSFE-IFM algorithms facilitates obtaining optimal friction ratio estimates. Accurate estimates of qc, fc and Rf are paramount for identifying soil behavior types from CPT data.