ABSTRACT KEY WORDS

M. J. SULEWSKA

NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

ABSTRACT KEY WORDS

• laboratory tests, • density index, • uncertainty of laboratory measurement

Compaction quality of built-in layers in ground or embankments is most frequently determined on the basis of a density index. The paper presents a theoretical basis and evaluation of the total standard uncertainty of laboratory measurement of a density index according to new international regulations.

1. INTRODUCTION

In the present global market, there is a necessity to unify methods of expressing measurement results in order to create a basis for international comparisons. An international procedure for the characterization of the quality of measurement results has been agreed, and in 1993, the International Organization for Standardization (ISO) published ”Guide to the Expression of Uncertainty in Measurement” [1]. This document introduced a new method for the evaluation of uncertainty in measurement. In this study the new method of uncertainty evaluation was applied to measurement of the compaction degree of non-cohesive soils [2].

2. THEORETICAL BASIS OF THE NEW CLASSIFICATION OF UNCERTAINTY IN MEASUREMENT

Measurement can be defined as the parametric identification of a model [3]. Measurement is always inaccurate – the measured value differs from the true value because of the imperfection of

a researcher, the device, the testing method and the variability of the test conditions.Uncertainty in measurement (u) is a value which allows for the determination of the limits of an interval containing (with an assumed probability) the unknown true value of the measured quantity.The uncertainty of a measurement result (u) is composed of many components, which can be divided into two categories, depending on the method of calculating their values:• type A uncertainties – (uA) determined using statistical methods,

they correspond to uncertainties caused by accidental effects,• type B uncertainties – (uB) determined using other methods,

they correspond to uncertainties caused by systematical effects introduced by measurement devices.

Both types of uncertainty should be treated in the same way – as random uncertainties, and the basis for their evaluation is the evaluation of standard deviations σ.The following values are calculated in order to evaluate the total uncertainty in measurement:a) particular standard uncertainties:

ui = σi (1)

Maria Jolanta Sulewska

Ph.D. Maria Sulewska, Bialystok Technical University, 45E Wiejska St., 15-351 Białystok, Poland. Phone: 0048 85 746 96 11, Fax: 0048 85 742 23 45, E-mail: sulewska@pb.bialystok.pl

Research field: soil mechanics

2005/3 PAGES 39 – 44 RECEIVED 14. 2. 2005 ACCEPTED 18. 4. 2005

2005 SLOVAK UNIVERSITY OF TECHNOLOGY 39

sulewska_01.indd 39 2. 5. 2006 12:30:01

40 NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

2005/3 PAGES 39 — 44

b) the total standard uncertainty, which is the sum of many component standard uncertainties:

(2)

c) the total uncertainty:

uc = kαu = kα σ (3)

where the multiplier kα is the value of the standardized variable, depending on the assumed level of confidence α; for normal distribution, it equals kα=2 or kα=3, which corresponds to α=0.95 or α=0.99. The result of the measurement is given in the following form:

for α = ... (4)

3. MEASUREMENTS BURDENED BY DOMINATING TYPE A STANDARD UNCERTAINTY

Direct measurements

The type A standard uncertainty manifests itself in the statistical spread of results in a series of repeated tests made under the same conditions. The estimator of the standard deviation of the average value is a measure of type A standard uncertainty. In order to obtain the value of type A uncertainty, it is necessary to make the following calculations:a) the average (expected) value of measured variable X:

(5)

b) the experimental variance of the average value , which is a measure of the uncertainty in the measurement of a series of n results:

(6)

c) the experimental standard deviation of the average value , which is a measure of the accidental uncertainty of the average deviation value:

(7)

Indirect measurements

In indirect measurements, value Y is a function of ”j” quantities of particular, directly measured variables Xj:

Y = f(Xj) for j = 1, 2, ..., N (8)

For each directly measured variable Xj, a series of n measurements is conducted, and the average values and standard uncertainties uAj are determined. Then the following calculations are made:

a) the average value of the directly measured variable:

(9)

b) the standard total uncertainty for :

(10)

4. MEASUREMENTS BURDENED BY DOMINATING TYPE B STANDARD UNCERTAINTY

Direct measurements

Imperfections of measuring devices are most often the source of type B standard uncertainties. The limit value of error ∆g, which is determined by a class index or the accuracy of reading from a scale, informs about a measuring device error. Assuming that device errors have a monotonous distribution (i.e., these errors assume the values from the interval ±∆g with equal probability), the standard deviation of a monotonous distribution is the type B standard uncertainty. It can be calculated in succession [1,3]:a) variance:

(11)

b) type B standard uncertainty:

(12)

Indirect measurements

In the case of indirect measurements there is the necessity to consider more than one type B standard uncertainty while evaluating the total uncertainty. It is calculated in succession:a) the value of the quantity indirectly measured according to the equation (9),b) total standard deviation for Y value:

sulewska_01.indd 40 2. 5. 2006 12:30:02

2005/3 PAGES 39 — 44

41NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

(13)

5. MEASUREMENTS BURDENED BY TYPE AB STANDARD UNCERTAINTY

Direct measurements

Uncertainties of type A (random) and type B (device errors) may occur in direct measurements. Standard total uncertainty is calculated according to the equation:

(14)

Indirect measurements

In the case of indirect measurements for the function Y=f(Xj), where j=1,2,…, N, it is necessary to calculate:a) the value measured directly according to the equation (9),b) total standard uncertainty for ”j” particular quantities directly measured according to the equation (14):c) total standard uncertainty for the average value :

(15)

6. DEGREE OF COMPACTION OF SOIL

The degree of compaction of non-cohesive soil is calculated according to the following equation:

[-] (16)

where: emax – maximum void ratio, which is obtained by the most

loose arrangement of dry soil in a cylinder [-], emin – minimum void ratio, which is obtained by the com-

paction of soil in a cylinder using vibration forks [-], e – void ratio of soil in an embankment or in subsoil [-].

Void ratios are defined by the following equations:

[-] (17)

[-] (18)

[-] (19)

where ρs - density of solid particles [g/cm3], ρd min - dry density of solid particles in the loosest state [g/cm3], ρd max - dry density of solid particles in the densest state [g/cm3], ρd - dry density of solid particles in an embankment or

subsoil [g/cm3].

Density of solid particles - in practice it is accepted for non-cohesive mineral soils ρs=2,65 g/cm3. It has been assumed that soil density had been measured with the required accuracy [2]:

g/cm3.

The dry density of solid particles in the loosest state is calculated in the following way:

[g/cm3] (20) The dry density of solid particles in the densest state is calculated according to the following equations:

[g/cm3] (21)

ms = mst – mt [g] (22)

vc1 = vc – ∆vc [cm3] (23)

[cm3] (24)

[cm3] (25)

∆hc = gt – h’ [cm] (26)

The dry density of solid particles in an embankment or subsoil is calculated according to the equation:

[g/cm3] (27)

where ρ - bulk density of soil in an embankment or subsoil [g/cm3],

sulewska_01.indd 41 2. 5. 2006 12:30:04

42 NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

2005/3 PAGES 39 — 44

w - water content of soil in an embankment or subsoil [%].

The water content of soil is most frequently measured using the drying method [2], and it is calculated in the following way:

[%] (28)

mw = mmt1 – mst1 [g] (29)

ms1 = mst1 – mt1 [g] (30) The bulk density of soil is determined directly in the subsoil or an embankment, using, for example, the sand-cone method [4] (fig. 1):

(31)

(32)

Calibration of the volume density of calibrated sand ρpk in a measuring cylinder:

(33)

(34)

Mass calibration of calibrated sand filling a cone:

(35)

The measurement of the degree of compaction of non-cohesive soil ID using the sand-cone method is an indirect measurement, which is a function of several direct measurements:

(36)

Notations in the equations (16) ÷ (36): ms – mass of soil in a cylinder [g], mst – mass of the cylinder containing soil (dry, loose arrangement) [g], mt – mass of an empty cylinder [g], Vc – cylinder volume [cm3], dc – inner diameter of a cylinder [cm], hc – inner height of a cylinder [cm], Vc1 – soil volume after vibrations [cm3], ∆Vc – decrease of soil volume in a cylinder after vibrations [cm3], ∆hc – decrease of soil height in a cylinder after vibrations [cm], gt – thickness of a plunger [cm], h’ – distance between the top edge of the cylinder and the top surface of a plunger [cm], mw – mass of water in a soil specimen taken from the subsoil or an embankment [g], ms1 – mass of soil in a sample [g], mmt1 – mass of evaporating dish with wet soil [g], mst1 – mass of evaporating dish with dried soil [g], mt1 – mass of an empty evaporating dish [g], V1 – volume of a measuring cylinder used for sand calibration [cm3], D1 – diameter of a measuring cylinder used for sand calibration [cm], h1 – height of a measuring cylinder used for sand calibration, – mass of a container with calibrated sand before cone calibration [g], - mass of a container with calibrated sand left after cone calibration [g], - mass of a container with calibrated sand before sand calibration [g], - mass of a container with calibrated sand left after sand calibration [g], M1 – mass of a container with calibrated

Fig. 1. Sand-cone: a) diagram of soil volume density test in situ, b) diagram of the calibration of the volume density of calibrated sand, c) diagram of the mass calibration of calibrated sand filling a sand-cone

sulewska_01.indd 42 2. 5. 2006 12:30:06

2005/3 PAGES 39 — 44

43NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

sand before examining the hole [g], M2 – mass of a container with calibrated sand left after examining the hole [g], M – mass of soil taken from the hole [g], Vd – hole volume [cm3].

7. DESCRIPTION OF THE EXAMINATIONS

The examinations of the degree of compaction were conducted in a laboratory on one subsoil model of a thickness of 0.60 cm prepared in a box of the dimensions of 1.30 x 1.30 x 1.05 m from medium sand of a low humidity, uniformly grained and compacted. All the direct measurements were repeated five times. The distances between the points in which the bulk density was measured using the sand-cone method were in a range of 0.20-0.30 m.

8. EXPERIMENTAL AND CALCULATION RESULTS

Direct measurementsThe experimental results xi (where i = 1,...,5) of variables Xj (where:

j = 1,...18), limit device errors ∆gj and average values are given in Table 1. The variances and standard uncertainties of B type

were calculated. The standard uncertainties of A type were not calculated because some of the direct measurements were not repeated (mmt1, mst1, mt1, M2, M).

Indirect measurementsFor particular indirect measurements, the following values were calculated:a) the average values of particular indirect measurements according to equations (35)÷(16), taking into account the average values of the direct measurements,b) variances of particular indirect measurements c) experimental variances of the average values according to equation (6) for all the variables (except those which were not repeated: Vd, ms1, mw),d) total variances according to the following equation:

(37)

Table 1. The direct test results, variances and type B standard uncertainties of particular direct measurements

Variables xj (j = 1,...,18)

Direct measurements xi Limit device error ∆gj

Average j = 1,...,18

uBji = 1 i = 2 i = 3 i = 4 i = 5

1 mst [g] 2050.0 2063.2 2070.3 2071.0 2071.0 0.1 2065.1 0.0582 mt [g] 1286.0 1286.1 1285.9 1286.0 1286.0 0.1 1286.0 0.0583 dc [cm] 7.12 6.90 6.88 7.14 6.96 0.01 7.00 0.0064 hc [cm] 12.50 12.53 12.48 12.49 12.50 0.01 12.50 0.0065 gt [cm] 1.61 1.60 1.62 1.59 1.58 0.01 1.60 0.0066 h’ [cm] 0.30 0.29 0.32 0.29 0.30 0.01 0.30 0.0067 mmt1 [g] 105.84 73.49 92.62 87.33 92.04 0.01 90.26 0.0068 mst1 [g] 104.18 72.51 91.24 86.31 90.91 0.01 89.03 0.0069 mt1 [g] 56.53 44.69 50.11 53.66 54.96 0.01 51.99 0.00610 D1 [cm] 11.25 11.24 11.23 11.24 11.24 0.01 11.24 0.00611 h1 [cm] 10.06 10.02 9.96 9.99 10.02 0.01 10.01 0.00612 M1’ [g] 3000.0 3000.0 3000.0 3000.0 3000.0 0.1 3000.0 0.058

13 M2’ [g] 2421.0 2419.6 2422.0 2422.4 2420.7 0.1 2421.1 0.058

14 M1’ ’ [g] 3000.0 3000.0 3000.0 3000.0 3000.0 0.1 3000.0 0.058

15 M2’ ’ [g] 933.5 928.6 927.5 934.5 929.2 0.1 930.7 0.05816 M1 [g] 3000.0 3000.0 3000.0 3000.0 3000.0 0.1 3000.0 0.05817 M2 [g] 907.5 1007.1 838.1 845.0 993.0 0.1 918.1 0.05818 M [g] 1900.5 1747.2 1957.5 1968.3 1770.0 0.1 1868.7 0.058

sulewska_01.indd 43 2. 5. 2006 12:30:08

44 NEW ASSESSMENT OF THE MEASUREMENT OF A DENSITY INDEX

2005/3 PAGES 39 — 44

e) total standard uncertainties according to equation (14),f) total uncertainty ucj according to equation (3).

Table 2 gives a summary of the calculations of the total uncertainties (absolute and relative) of particular geotechnical parameters related to the degree of compaction of the non-cohesive soils. The result of the measurement of the degree of compaction can be written in the following manner:ID = 0.678±0.231 at α = 0.05, which means that the experimental results are in the range of 0.909 to 0.447 with a probability of 95%.

9. CONCLUSIONS

• This study presents a calculation procedure for the total uncertainty in the measurement of geotechnical parameters

related to examinations of a non-cohesive soil state on the basis of the degree of compaction.

• Measurements of geotechnical parameters are burdened by type A and type B uncertainties.

• For most direct measurements, type A uncertainty is much higher than type B.

• The measurement of the degree of compaction is burdened by a considerably high total uncertainty (about 34%); random uncertainties in the measurements of emin and emax make a large contribution.

• The measurement of the degree of compaction using a laboratory method seems to be imprecise and ambiguous in the light of uncertainty calculations.

AcknowledgementsThe paper is the result of research work financially supported by the Ministry of Scientific Research and Information Technology.

Tab. 2. Summary of particular types of uncertainties of the geotechnical parameters tested

ParameterAverage

Type B uncertainty

uB

Relative uncertainty

%

Type A uncertainty

uA

Relative uncertainty

%

Total standard

uncertainty uAB

Total uncertainty

uc

Relative uncertainty

%

ρ [g/cm3] 1.866 0.002 0.11 0.004 0.21 0.004 0.008 0.43w [%] 3.33 0.022 0.66 0.083 2.49 0.086 0.172 5.17ρd [g/cm3] 1.806 0.002 0.11 0.004 0.22 0.005 0.009 0.50ρdmax [g/cm3] 1.910 0.004 0.21 0.035 1.83 0.036 0.071 3.72ρdmin [g/cm3] 1.620 0.003 0.19 0.029 1.79 0.029 0.059 3.64e [-] 0.468 0.006 1.28 0.003 0.64 0.007 0.013 2.78emin [-] 0.388 0.006 1.55 0.026 6.70 0.027 0.053 13.66emax [-] 0.636 0.007 1.10 0.030 4.72 0.031 0.061 9.59ID [-] 0.678 0.030 4.43 0.112 16.52 0.116 0.231 34.07

REFERENCES

[1] Guide to the Expression of Uncertainty in Measurement. ISO 1993.

[2] PN-88/B-04481 Building soils. Laboratory tests. (in Polish: Grunty budowlane. Badania próbek gruntu)

[3] TURZENIECKA D.: Evaluation of uncertainties of

measurement results. (in Polish: Ocena niepewności wyniku pomiarów), Wydawnictwo Politechniki Poznańskiej, Poznań 1997.

[4] BN-77/8931-12 Determination of degree of soil compaction (in Polish: Oznaczanie wskaźnika zagęszczenia gruntu).

sulewska_01.indd 44 2. 5. 2006 12:30:09