Estimation on Measurement Uncertainty in

Estimation on Measurement Uncertainty in Determination of Compressive Strength of Hardened Concrete Test Method: BSEN 12390-3:2009 A.G. Piyal Aravinna Senior QC Officer Central Engineering Consultancy Bureau 11, Jawatta Road, Colombo 00500, Sri Lanka. Tel.: +94112505688, Fax: +94112598215, Email : [email protected]

1 CECBLS

Before This Exercise…….

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Before start this exercise, please refer the pervious lesson “Basic Concepts of Measurement Uncertainty”

2

Estimation Process of Uncertainty

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1 Specification 2 Identification of Sources of Uncertainty 3 Quantification of Uncertainty Component Changes of Apparatus/Equipment/Method

Re-Evaluate

4 Conversion to Standard Uncertainty (u)

Re-Evaluate

5 Calculation of Combined Uncertainty ( uc ) 6 Calculation of Effective Degree of Freedom (Veff) 7 Calculation of Coverage Factor (k ) 8 Calculation of Expanded Uncertainty (U) No Satisfaction 3

End

1 Specification

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Cube compressive strength (fc)

X

Y

X Y Z

LZ LY LX

fc= F/A where F = maximum applied force on cube, A = cross sectional area

fc = F/(LX,Ave × LY,Ave) where LX,Ave , LY,Ave are averages of three pairs of orthogonal dimensions of X and Y directions (lording surfaces). Parameters to be considered for the estimation on Measurement Uncertainty are F, Lx,Ave , LY,Ave. 4

Required Data and Their Accuracy

CECBLS

Data Orthogonal dimensions /mm

Z axis Z1

Z2

X axis LZ,Ave

X1 X2

X3

LX,Ave

Area /mm2

Y axis Xstd

Y1 Y2 Y3 LY,Ave

Max. Force (F)/ (kN)

Strength (fc)/ (N/mm2)

Ystd

Accuracy of dimensions - EN 12390-3:2009 (E), Annex B

Required accuracy of dimensions = 150 mm × 0.5% = 0.75 mm

Accuracy of load measurement

Accuracy of the maximum load (at failure) is required to measure nearest 1 kN

5

Data Recording Sheet

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6

How to Calculate ?

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Calculate the uncertainty of compressive strength of concrete cube using the data of sample No. SS521 Orthogonal dimensions /mm Sample No.

Z axis (LZ)

X axis (Lx)

Z1

Z2

Zm

X1

X2

SS521

150.5

150.4

150.45

150.8

151.9

Sig.Fig.

4

4+1=5

X3

Area (A) /mm2

Y axis (LYH) Lx,Ave

150.8 151.17 4+1=5

Lx,s

Y1

Y2

0.560

150.6

150.8

Y3

LY,Ave

LY,s

151.7 151.03 0.467 22831.2 4+1=5

Max. Strength Force (fc)/ (F)/ (N/mm2) (kN)

5+1=6

651. 3

28.51 3+1 = 4

7

2 Sources of Uncertainty Factors

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Source of Uncertainty

Remarks

Deviation of reading from nominal

Estimated by regular calibration within equip. specs. Not considered – maintained within recommendation Not considered – within recommendation Not considered – within recommendation Not considered – maintained within recommendation

Loading rate Loading Stiffness of machine machine Alignment of loading patens Operating temperature

8

2 Sources of Uncertainty… Factors

Source of Uncertainty

Accuracy of equipment ( Digital Vernier Calipers) Accuracy of equipment (Analogue Vernier Calipers) Vernier Resolution of observed Caliper output (Analogue/ (for digital Vernier Calipers) Digital) Repeatability of reading Operating temperature

CECBLS

Remarks Estimated by using regular calibration report From specification of analog venire caliper Estimated from equipment stability and resolution Estimated by regular in house operator verification Not considered – maintained within recommendation

9

2 Sources of Uncertainty ... Factors Factors

Source ofUncertainty Uncertainty Source of Inhomogeneity among Inhomogeneity among specimens specimens Dimensional variation of cube Dimensional variation of faces Test cube faces specimen Perpendicularity of cube face Test Roughness of loading Perpendicularity of surface cube specime face n Moisture condition of cubes Roughness of loading surface Moisture condition of cubes

CECBLS

Remarks Remarks Not considered – require repeated Not considered – require repeated testing testing Estimated by repeated measurements Estimated by repeated measurements Not considered – within recommendation Not – within recommendation Notconsidered considered – within Not considered – maintained within recommendation recommendation

Not considered – within recommendation Not considered – maintained within recommendation

10

2 Sources of Uncertainty …

CECBLS

In this example, analogue Vernier calipers is used for the length measurement of cube specimens.

Summery 1. Uncertainty of length measurements a) Accuracy of reading (of analog venire caliper) b) Repeatability of reading c) Dimensional variation of cube faces 2. Uncertainty of Force measurement: a) Deviation of reading from nominal 11

Sections Completed 1 Specification

CECBLS

√

2 Identification of Sources of Uncertainty

√

3 Quantification of Uncertainty Component Changes of Apparatus/Equipment/Method

Re-Evaluate


Re-Evaluate

5 Calculation of Combined Uncertainty ( uc ) 6 Calculation of Effective Degree of Freedom (Veff) 7 Calculation of Coverage Factor (k ) 8 Calculation of Expanded Uncertainty (U) No Satisfaction

End

These two component will be explained as a combined section. 12

Quantification of Uncertainty Component and Conversion to Standard Uncertainty

CECBLS

1 Uncertainty of length measurements a) Uncertainty of accuracy of reading (of analog venire caliper) Vernier reading of the measurement

149.80 (l1)

Zero error of Vernier reading

- 0.4 (l0)

The best estimate of the zero error, (l0) = 0.4 mm The Best estimate of the reading (l1) = 149.80 mm The best estimate of the length (l) = l1 - l0 = 149.80 - (- 0.4 ) = 150.2 mm 13

Quantification of Uncertainty Component and Conversion to Standard Uncertainty …

CECBLS

1 Uncertainty of length measurements … a) Uncertainty of accuracy of reading (of analog venire caliper).... Minimum reading (from the equipment manual) = ± 0.05 mm Uncertainty of reading = ± 0.05 mm Standard uncertainty of a length measurement [u(l1)] u(l1) = ± [(0.05/30.5] = 0.028868 mm (Assuming rectangular distribution)

Summery Uncertainty of reading

Type of evaluation

Minimum reading = ± 0.05 mm

Type A

Probability/ statistics applied

Degree of freedom

From Rectangular Distribution

Standard uncertainty, u(l1)

inﬁnite (∞)

0.0289 mm 14


Table 1

1 Uncertainty of length measurements … b) Uncertainty of repeatability of length measurements (Lr)

This can be determined from regular in-house operator verification using the same Vernier caliper (verified against gauge block of representative size). Thirty measurements of length of a same concrete cube is given in Table 1

Uncertainty of Lr = Std. deviation of Lr, [u(Lr,s)] = ± 0.098 Standard uncertainty of repeated reading [u(Lr)] Equation used for calculation u(Lr) = 0.018 mm

Summery

u s - Standard deviation of measurement n – Number of observations

Uncertainty Type of Probability/ Degree of Standard of reading evaluation statistics used freedom uncertainty, u(Lr) ± 0.098 mm

Type A

29

CECBLS

0.018 mm

Measurement No. (n) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Mean SD (s)

Value /mm 150.9 150.9 150.7 150.7 150.9 150.7 150.5 150.8 150.7 150.9 150.8 150.7 150.8 150.7 150.9 150.7 150.8 150.8 150.9 150.9 150.8 150.9 150.9 150.8 150.7 150.8 150.9 150.7 150.8 150.8 150.79 0.098

15


CECBLS

1 Uncertainty of length measurements … C) Dimensional variation of cube faces Orthogonal dimensions /mm Sample No.

Z axis (LZ)

X axis (Lx)

Z1

Z2

Zm

X1

X2

SS521

150.5

150.4

150.45

150.8

151.9

Sig.Fig.

4

4+1=5

X3

Area (A) /mm2

Y axis (LYH) Lx,Ave

150.8 151.17

Lx,s

Y1

Y2

0.560

150.6

150.8

Y3

LY,Ave

LY,s

151.7 151.03 0.467 22831.2

4+1=5

4+1=5

Max. Strength Force (fc)/ (F)/ (N/mm2) (kN)

5+1=6

651. 3

28.51 3+1 = 4

Uncertainty of readings of LX , (LX,s)

Uncertainty of reading of LX = Std. deviation of LX ,(LX,s) = 0.560 mm

Standard uncertainty of readings of LX, [u(LX)] Equation used for calculation n =3 u(LX) = 0.396 mm 16


CECBLS

1 Uncertainty of length measurements … C) Dimensional variation of cube faces … Area (A) /mm2

Orthogonal dimensions /mm

Sample No.

Z axis (LZ)

X axis (Lx)

Z1

Z2

Zm

X1

X2

SS521

150.5

150.4

150.45

150.8

151.9

Sig.Fig.

4

4+1=5

X3

Y axis (LYH) Lx,Ave

150.8 151.17

Lx,s

Y1

Y2

0.560

150.6

150.8

Y3

LY,Ave

LY,s

151.7 151.03 0.467 22831.2

4+1=5

4+1=5

Max. Strength Force (fc)/ (F)/ (N/mm2) (kN) 651.

5+1=6

3

28.51 3+1 = 4

Uncertainty of readings of LY (LY,s) Uncertainty of reading of LY = Std. deviation of LY (LY,s) = 0.467 mm

Standard Uncertainty of readings of LY , [u(LY)] u(LY) = 0.330 mm

Equation used for calculation,

n =3

Summery Uncertainty of reading LX,s

± 0.560 mm

LY,s

± 0.467 mm

Type of Probability/ Evaluation Statistics applied Type A

Degree of freedom

Standard uncertainty,

2

u(LX) = 0.396 mm u(LY) = 0.330 mm

17


CECBLS

1. Uncertainty of force measurement Deviation of reading from nominal This can be estimated from calibration report.

The Relative standard uncertainty [ur(y)] ur(y) = u(y)/|y|×100%

y - Measurement result of parameter y u(y) - the standard uncertainty of measurement

Standard uncertainty of force [u(F)] u(F) at 800 kN = 0.58/100 × 800 = 4.64 kN

Summery Uncertainty of reading ± 0.058 kN

n=2

Type of Evaluation Type B

source from calibration report

Degree of freedom 1

*This can be considered as a combined stranded uncertainty of F ,[uc(F)]

Standard uncertainty,u(F)* 4.64 kN 18


CECBLS

√


√


Re-Evaluate

√


Re-Evaluate

5 Calculation of Combined Uncertainty ( uc ) 6 Calculation of Effective Degree of Freedom (Veff) 7 Calculation of Coverage Factor (k ) 8 Calculation of Expanded Uncertainty (U) No Satisfaction

End

19

5 Calculation of Combined Standard Uncertainty (uc)

CECBLS

5.1 Combined Uncertainty of Venire readings, uc(La) The best estimate of the length l = l1 - l0 = 149.80 - (- 0.4 ) = 150.2 mm (from Slide 13) l0 = The best estimate of the zero error, l1 = The best estimate of the reading

As uncertainty of l1 and l0 affect the result of l, combine standard uncertainty should be calculated for l. uc(La)

1

u(L0)

uc(La) = [2 ×(0.02288)2 ] 0. 5 = 0.040825 mm

20

5 Calculation of Combined Standard Uncertainty (uc) …

CECBLS

5.2 Combined Uncertainty of length Accuracy of venire, Repeatability of reading, Dimensional variation among deferent locations (in perpendicular to direction) affect the uncertainty of average dimensions (Lx,Ave and LY,Ave). Therefore, combined standard uncertainty should be calculated for Lx,Ave and LY,Ave Uncertainties of; Venire Repeatability of readings [ 𝑐 𝑠 ] and Dimensional variations should be combined to calculate combine uncertainty of the Therefore,

The combined standard uncertainty of dimension measurement of Lx , uc(Lx) uc(Lx)

u(Lr)

𝑐

u(LX)

The combined standard uncertainty of dimension measurement of LY , uc(LY) uc(LY)

𝑎

u(Lr)

u(LY) 333 mm

21

5 Calculation of Combined Standard Uncertainty (uc) …

CECBLS

5.2 Combined standard Uncertainty of Area [uc(A)] Uncertainties of Lx,Ave and Ly,Ave should be combined to calculate

Method of propagation

A= (LX,Ave × LY,Ave )

Sensitivity coefficient of LY,Avg

Sensitivity coefficient of LX,Avg

,

=

,

,

,

=

,

,

333

𝟐 22

5 Calculation of Combined Standard Uncertainty ( uc ) … Uncertainties of area [uc(A)] and force [u(F)] should be combined to dentine the standard uncertainty of

CECBLS


fc= F/A

Sensitivity coefficient of F

Sensitivity coefficient of A

=

× .

N/mm2 23


CECBLS

√


√


Re-Evaluate

√


Re-Evaluate

5 Calculation of Combined Uncertainty ( uc )

√

6 Calculation of Effective Degree of Freedom (Veff) 7 Calculation of Coverage Factor (k ) 8 Calculation of Expanded Uncertainty (U) No Satisfaction

End

24

6 Calculation of Effective Degree of Freedom (Veff)

CECBLS

Expanded uncertainty (U) is calculated by equation given below

U = kpuc Where kp = coverage factor at level of confidence p, uc = combined uncertainty

The coverage factor (kp), is defined as below, of a t-distribution with Veff degrees of freedom

kp = t(1+p)/2 (Veff) where t(1+p)/2 = student t value at level of confidence p (from t distribution, one side), Veff = effective degree of freedom

Method of propagation to calculate Effective Degree of Freedom (Veff) where Ci = Sensitivity coefficient of Xi Vi = Coverage factor Xi u(Xi) = Uncertainty factor Xi

25

6 Calculation of Effective Degree of Freedom (Veff) … Effective degree of freedom of dimension measurement a) Effective degree of freedom of

𝐗, 𝐀𝐯𝐠 [

𝒆𝒇𝒇 𝑳𝑿,𝑨𝒗𝒈

]

Required data

(from previous calculations)


,

=

𝒄 𝒚 𝒆𝒇𝒇 𝑵 𝒊

𝒊

(

= uc(LX) =

398

=1

𝒊

𝒊

uc(La) = 0.041

𝒊

u(Lr) = 0.018

,

𝒆𝒇𝒇 𝑳𝑿,𝑨𝒗𝒈

CECBLS

(

(

(

= 0.396 mm

0.3984 = = 𝟐. 𝟎𝟒𝟎𝟕 1 × 0.041 1 × 0.018 1 × 0.396 + + ∞ 29 2

26

6 Calculation of Effective Degree of Freedom (Veff) … Effective degree of freedom of dimension measurement b) Effective degree of freedom of

𝐘, 𝐀𝐯𝐠

[

𝒆𝒇𝒇 𝑳𝒀,𝑨𝒗𝒈

]

Required data


Method of propagation ,


𝒊

=

𝒊

𝒊

CECBLS

= uc(LY) =

333

=1

𝒊

uc(La) = 0.041 , ,

(

(

(

,

u(Lr) = 0.018 (

= 0.330 mm

0.3334 = = 𝟐. 𝟎𝟕𝟑𝟕 1 × 0.041 1 × 0.018 1 × 0.330 + + ∞ 29 2

27

6 Calculation of Effective Degree of Freedom (Veff) … Effective degree of freedom of area [

]

CECBLS

Required data



=

,


𝒊

=

,

= 151.03

𝒊

𝒊

,

𝒊

=

,

,

=

,

= (

,

)=

uc(Lx)

(

,

)=

uc(LY)

4

151.03 875

333

333

, ,

28

6 Calculation of Effective Degree of Freedom (Veff) …

CECBLS

Required data

Effective degree of freedom of strength


𝐜

Method of propagation 𝑐 =


𝒊

𝜕𝑓 1 1 = = 𝜕𝐹 𝐴 22831.2

𝒊

𝒊

= 4.37997 × 10

𝒊

𝑐 =

= .

×

×

.

×

.

×

×

= −

=−

mm × .

= −1.24889 × 10

N mm

uc(F) = 4.640

N

.

.

29

Sections completed 1 Specification

CECBLS

√


√


Re-Evaluate

√


Re-Evaluate

5 Calculation of Combined Uncertainty ( uc )

√

6 Calculation of Effective Degree of Freedom (Veff)

√

7 Calculation of Coverage Factor (k ) 8 Calculation of Expanded Uncertainty (U) No Satisfaction

End

30

7 Calculating Coverage Factor (k) …

CECBLS

kp = t(1+p)/2 (Veff) where t(1+p) = student t value at level of confidence p (from t distribution), Veff = effective degree of freedom

Coverage Factor at truncated Veff You can calculate k using round off value of Veff Veff = , Therefore, Truncated Veff = 1 Coverage Factor at 95% confidence level (k0.95) = t0.975 (3)= 12.706

Coverage Factor at interpolated t-factor When Veff is not an integer, you can use Equation given below to estimate more accurate coverage factor 𝒑

𝒆𝒇𝒇

(𝟏 𝒑)/𝟐

𝒆𝒇𝒇

(𝟏 𝒑)/𝟐

Where n = integer of Veff k0.95 = (2 -

) t09.75 (1) + (

–1) t09.75 (2) = 0.96 × 12.706 + 0.04 × 4.303 = 12.3699

Coverage Factor at 95% confidence level (k0.95) = 12.37

31

8 Expanded Uncertainty (U)

CECBLS

U= k uc

Expanded uncertainty at level of confidence p, (Up)= kp uc = 12.37 ×

N/mm2

= 1.21 N/mm2

Reporting the results Cube compressive strength is 28.51 ± 1.21 N/mm2 at level of confidence of 95% (k=12.706). The cube compressive strength was tested in accordance with BSEN 12390-3:2009. The test method requires rounding to nearest 0.1N/mm2, without quoting the measurement uncertainty. 32

References

CECBLS

1.

Jeffery, G. H., Bassett, J., Mendham, J. Denney, R. C., Eds., (1989 ). Vogel’s textbook of quantitative chemical analysis, 5th ed. New York: John Wiley & Sons.

2.

International Organization for Standardization (2017). General requirements for the competence of testing and calibration laboratories (ISO/IEC 17025).

3.

Multi-Agency Radiological Laboratory Analytical Protocols Manual, USEPA, USDoD, USDOE, USDHS, USNRC, USFDA, USGS and USNIST, Volume I: Chapters 10 - 17 and Appendix F, July 2004.

4.

British stranded (2009). Testing hardened concrete: Compressive strength of test specimens (BS EN 12390-3:2009).

5.

A Guidance on Measurement Uncertainty for Civil Engineering and Mechanical Testing Laboratories, Technical Guide 2, SAC-SINGLAS Technical Guide, SPRING Singapore, 2 Bukit Merah Central, 159835,Singapore, 2nd Edition, June 2007. 33