Manual 2 – Reliability equation - FWPA · Manual 2 – Reliability equation PROJECT NUMBER:...

Manual 2 – Rel iabi l i ty equation

PROJECT NUMBER: PN07.1052

August 2007

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USP2007/039

MANUAL NO. 2

Reliability Equations

R.H. Leicester, M. Nguyen and C-H. Wang

April 2008

This report has been prepared for Forest & Wood Products Australia (FWPA).

Please address all enquiries to:

Urban Systems Program

CSIRO Sustainable Ecosystems

P.O. Box 56, Highett, Victoria 3190

Manual No.2: Reliability Equations 2

© 2008 CSIRO

To the extent permitted by law, all rights are reserved and no part of this publication covered by

copyright may be reproduced or copied in any form without acknowledgment of this reference source.


Contents

EXECUTIVE SUMMARY ........................................................................................................ 4

1. INTRODUCTION ............................................................................................................ 5

2. STATISTICAL PARAMETERS ...................................................................................... 6

3. EXACT EQUATION FOR THE DURABILITY FACTOR ............................................ 8

4. APPROXIMATE EQUATIONS FOR THE DURABILITY FACTOR .......................... 9

5. PERIMETER DECAY ROUND SECTION, BENDING STRENGTH ........................ 10

6. INTERNAL DECAY, ROUND SECTION, BENDING STRENGTH .......................... 11

7. PERIMETER DECAY, RECTANGULAR SECTION, BENDING STRENGTH ........ 12

8. TENSION STRENGTH FOR RECTANGULAR SECTION UNDER SINGLE-EDGE

CORROSION ................................................................................................................. 13

9. COMPARISONS BETWEEN THE APPROXIMATE AND EXACT SOLUTIONS .. 14

10. ALPHA VALUES TO BE USED IN DESIGN ......................................................... 31

11. RECOMMENDATIONS FOR DESIGN ....................................................................... 32

REFERENCES ......................................................................................................................... 33


Executive Summary

The purpose of this Manual is to present a procedure for implementing the results of a

service life analysis via a simple computational procedure. The procedure chosen is to

assume that the impact of an environmental attack is equivalent to an effective loss of

cross-section

The effective loss of Section is based on an estimation of deff, the effective depth of attack

by decay fungi or corrosion. The value of deff is evaluated from

deff = d (1 + Vd)

where d is the estimated mean depth of decay or corrosion, Vd is the coefficient of

variation that is a measure of the uncertainty in d and is a parameter that depends on the

type of structural element and the degree of reliability required. The coefficient of

variation Vd needs to take into account both the variability of the structural member and

the uncertainties of the estimate of decay attack.

The derivation of the parameter is based on a simple first order reliability analysis and

is related to the choice of a reliability index β. A process of trial and error is used to find

the appropriate value of that gives the correct structural answer for a given structural

element, attack pattern and choice of β.

Results are given for a few common cases of members having square, flat or round cross-

sections, and for the case of both perimeter and internal attack.


1. Introduction

We consider a structural element for a given design life. We then make the assumption that

(a) the design load is the same, no matter what the length of the life and (b) that the design

strength is a constant value and is equal to the strength that exists at the end of the design life.

These are both conservative assumptions.

The “exact” value of the design strength will be taken to be derived according to a

simplified approximation procedure used by Ravindra and Galambos (1978). We then derive

an “approximate” design procedure by assuming that the loss of cross-section is given by deff

defined by

deff = d (1 + Vd) (1)

where d is the estimated mean depth of decay or corrosion, Vd is the coefficient of

variation that is a measure of the uncertainty in d and is a parameter that is chosen to obtain

a fit between the exact and approximate solutions. Having removed the effective loss of cross

section, then the residual section is checked to see whether it has the appropriate load capacity

according to the normal structural design rules given in AS 1720.


2. STATISTICAL PARAMETERS

We write the load capacity R of a structural element to be

R = g(d) × f (2)

Where d is the depth of decay or corrosion, g(d) is a geometrical function of d, and f

is the ultimate strength of the material that has not been attacked by decay, corrosion

etc.

To a first approximation we can take the mean value of load capacity R and variance

of strength R to be given by Ang and Tang (2007)

R = g( d ) f (3)

R d

d d ,f f d d ,f f

R R

d ff

22

2 2 2 (4)

Hence the coefficient of variation of load capacity VR is given approximately by

R RV / R22

dur fV V2 2 (5)

where Vf is the initial coefficient of variation of the load capacity for material that has

not been attacked by decay, corrosion etc, and Vdur is given by


dur d

d d

gV V d / g d

d (6)


3. EXACT EQUATION FOR THE

DURABILITY FACTOR

The durability factor Kdurability will be defined by

Kdurability = Rdesign / Rdesign,0 (7)

where Rdesign denotes the design load capacity and Rdesign,0 denotes the design load

capacity if the material is not attacked by decay, corrosion etc.

Based on a first order reliability analysis by Ravindra and Galambos (1978), the

value of Rdesign will be taken to be given by

Rdesign = 0.9R exp (−0.6VR) (8)

where design denotes a reliability index.

Hence equations (1), (7) and (8) lead to

Kdurability = [g d /g(0)] [exp −0.6 (VR − Vf)] (9)

where the function g( ) is given by equation (1) and VR is given by equation (5).


4. APPROXIMATE EQUATIONS FOR THE

DURABILITY FACTOR

An approximate equation for the durability factor, denoted by Kdurability.approx can be

taken to be given by

Kdurability.approx = g(deff) / g(0) (10)

Where the function g(..) is defined by equation (1) and deff is an effective depth of

decay defined by

deff = d [1 + Vd] (11)

where is a factor that depends on the choice of the reliability index . In the

following we will investigate the choice of α that will produce a good match of

Kdurability.approx and Kdurability . To do this we first we need to derive the equations (1)

and (6) for particular structural cases.


5. PERIMETER DECAY ROUND SECTION,

BENDING STRENGTH

Consider a pole of diameter D and a depth of circumferential decay d.

Then

g(d) = (/32) (D − 2d)3 (12)

and hence

Vdur =

ddV

D d

6

2 (13)

d


6. INTERNAL DECAY, ROUND SECTION,

BENDING STRENGTH

g(d) = (/32) (D3 − 8d

3) (14)

Vdur =

dd V

D d

3

3 3

24

8 (15)

D 2d


7. PERIMETER DECAY, RECTANGULAR

SECTION, BENDING STRENGTH

g(d) = (d − 2D) (D − 2d)2/6 (16)

dur dv v d D d D d D d / D d D d2 2

2 2 4 2 2 2 2 (17)

B = D

D d

d


8. TENSION STRENGTH FOR

RECTANGULAR SECTION UNDER

SINGLE-EDGE CORROSION

g(d) = (D − d) (18)

Hence

Vdur = Vd d / (D − d ) (19)

d

D


9. COMPARISONS BETWEEN THE

APPROXIMATE AND EXACT

SOLUTIONS

SQUARE MEMBERS IN BENDING (perimeter decay)

EFFECT OF beta

RECTANGULAR MEMBER (perimeter decay)

beta = 0.5 Vd = 2

alpha = 0.2 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean



beta = 2 Vd = 2

alpha = 0.8 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


beta = 1 Vd = 2

alpha = 0.4 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean



beta = 3 Vd = 2

alpha = 1 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


SQUARE MEMBERS IN BENDING (Perimeter decay)

EFFECT OF CHANGING Vd


beta = 2 Vd = 1

alpha = 0.8 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


beta = 2 Vd = 3

alpha = 0.8 Vf = 0.2

gamma = D/B = 1

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


RECTANGULAR MEMBERS IN BENDING, B/D =10, (Perimeter corrosion)

Effect of beta


beta = 0.5 Vd = 3

alpha = 0.2 Vf = 0.2

gamma = B/D = 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


beta = 1 Vd = 3

alpha = 0.4 Vf = 0.2

gamma = B/D = 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean



beta = 2 Vd = 3

alpha = 0.8 Vf = 0.2

gamma = B/D = 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


beta = 3 Vd = 3

alpha = 1 Vf = 0.2

gamma = B/D = 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean



beta = 4 Vd = 3

alpha = 1.2 Vf = 0.2

gamma = B/D= 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2

d/D

R/R

o

K exact

K approx

R mean


CIRCULAR MEMBERS IN BENDING (perimeter decay)

EFFECT OF Beta

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P Q

ROUND SECTIONS (perimeter decay)

beta = 4 Vd = 3

alpha = 1.2 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2

d/D = mean depth of decay

R/R

o

K exact

K approx

R mean

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P Q


beta = 3 Vd = 3

alpha = 1 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P Q


beta = 2 Vd = 3

alpha = 0.8 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P Q


beta = 1 Vd = 3

alpha = 0.4 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P Q


beta = 0.5 Vd = 3

alpha = 0.2 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean


ROUND SECTIONS IN BENDING (perimeter decay)

EFFECT of Vd

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

I J K L M N O P


beta = 2 Vd = 1

alpha = 0.8 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

I J K L M N O P


beta = 2 Vd = 3

alpha = 0.8 Vf = 0.2

0

0.5

1

0 0.05 0.1 0.15 0.2


R/R

o

K exact

K approx

R mean


ROUND SECTIONS IN BENDING (central decay)

EFFECT OF Beta

ROUND SECTIONS (internal decay)

beta = 0.5 Vd = 1

alpha = 0.2 Vf = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

d/D

R/R

o

K exact

K approx

R mean


beta = 1 Vd = 1

alpha = 0.3 Vf = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

d/D

R/R

o

K exact

K approx

R mean



beta = 2 Vd = 1

alpha = 0.4 Vf = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

d/D

R/R

o

K exact

K approx

R mean


beta = 3 Vd = 1

alpha = 0.5 Vf = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

d/D

R/R

o

K exact

K approx

R mean



beta = 4 Vd = 1

alpha = 0.6 Vf = 0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6

d/D

R/R

o

K exact

K approx

R mean


PLATE SECTION IN TENSION (decay on only one edge)

EFFECT OF Beta

PLATE SECTION IN TENSION (only one surface decayed)

beta = 0.5 Vd = 4

alpha = 0.05 Vf = 3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

d/D

R/R

o

K exact

K approx

R mean


beta = 1 Vd = 4

alpha = 0.1 Vf = 3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

d/D

R/R

o

K exact

K approx

R mean



beta = 2 Vd = 4

alpha = 0.15 Vf = 3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

d/D

R/R

o

K exact

K approx

R mean


beta = 3 Vd = 4

alpha = 0.2 Vf = 3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

d/D

R/R

o

K exact

K approx

R mean



beta = 4 Vd = 4

alpha = 0.25 Vf = 3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5

d/D

R/R

o

K exact

K approx

R mean


10. ALPHA VALUES TO BE USED IN

DESIGN

The following fitted values of alpha have been derived with the assumption that Vf = 0.2, and

also Vd = 3. The data shows that in the range Vd = 1-3 there is no measurable effect on the

fitting parameter α. The following table summarises the values of α obtained by trial and

error.

Safety

Index β

Fitted α values

Square

member,

bending

strength,

perimeter

decay,

Plate

member,

bending

strength,

perimeter

corrosion

(B/D = 10)

Plate

member,

tension

strength,

single edge

corrosion

(B/D = ∞)

Round

member,

bending

strength,

perimeter

decay

Round

member,

bending

strength,

centre decay

β = 0.5 0.2 0.2 0.05 0.2 0.2

β = 1 0.4 0.4 0.10 0.4 0.3

β = 2 0.8 0.8 0.15 0.8 0.4

β = 3 1.0 1.0 0.20 1.0 0.5

β = 4 1.2 1.2 0.25 1.2 0.6

As mentioned earlier the procedure used is conservative. Also it has been suggested that

failure due to decay or corrosion does not occur with out warning and so does not warrant the

full level of reliability that would normally be used in structural design in cases where

durability is not involved. Hence we would suggest that an α-factor corresponding to safety

indices β of 1 and 2 be used for low and normal consequence of failure structural elements

respectively.


11. RECOMMENDATIONS FOR DESIGN

For the cases of attack by decay fungi, marine borer or corrosive factors, use the models

developed within the Design for Durability project to obtain parameters for d, the depth of the

loss in cross-section due to either biological or corrosion attack for a chosen design life Ldesign.

Then use equation (1) to estimate deff , the effect loss of section defined by

deff = d (1 + Vd) (20)

where d is the mean loss of strength and Vd is the uncertainty defined by

Vd

2 = Vdur

2 + VM

2

in which Vdur is the uncertainty observed in the data of the prediction model, and VM is the uncertainty in the model itself. Typically a value of VM = 0.5 would be appropriate. A useful check on Vd is to take a look at the data obtained in the “reality” checks. For the case of tension members corroding on one surface only, the value of α to be used is 0.15. For all other types of members the value of α to be used is 0.8 and 0.4 for normal and low consequence of failure elements respectively. Although the effect of durability on design stiffness has not been discussed herein, it is probably appropriate to use α = 0 (ie use the mean estimate) except when a structure is highly sensitive to serviceability characteristics. Although the recommendations given here are probably about correct, it would be highly desirable that design values of α should be re-evaluated on the basis of more sophisticated reliability studies that take into consideration the time-varying properties of both strength and loads.


REFERENCES

Ang, A.H.S and Tang, W.H. (2007) Probability concepts in engineering. Emphasis on

applications to civil and environmental engineering. John Wiley and Sons.

Ravindra, M. K. and Galambos, T. V. (1978) Load and resistance factor design for steel.

Journal of the Structural Division Proc. Of ASCE 104, ST9, Sept 1331-1354.

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