TA7 W34 no.REMR­CS-24 c.3

U -CE-CPr~otth United Stat• Govwnment

REPAIR, EVALUATION, MAINTENANCE, AND REHABILITATION RESEARCH PROGRAM

TECHNICAL REPORT REMR-CS-24

RELIABILITY OF

STEEL CIVIL WORKS STRUCTURES

by

Paul F. Mlakar, Sasson Toussi, Frank W. Kearney, and Dawn White

U.S. Army Construction Engineering

Research Laboratory

DEPARTMENT OF THE ARMY US Army Corps of Engineers

PO Box 4005, Champaign, Illinois 61824-4005

September 1989 Final · Report

Approved tor Pamlic Release; Dislribution Unlimited

RESEARCH LIB RY US ARMY ENGINEER WATERWAYS

EXPERIMENT STAT\C 1 VICKSBURG, MISStSSlPPI

Prepared tor DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000

Under Civil Works Research Work Unit 31301

UNCLASSIFIED SECUKI I Y CLASSIFICATION Of- THIS PAGE

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Reliability of Steel Civil Works Structures (U)

12 PEPS(;NAL AUTHOR(S}

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Final FROM_ _ _ _ TO 1989 September 68

'" ';\ IPP; tr\H'NTARY NOTA ~ION A report ot the Concrete and Steel Structures Problem Area of the Repair, Evaluation, Maintenance, and Rehabilitation (REMR) Research program. Available [rom National Technical Information Service, 5285 Port Roval Road Springfield, VA 22161

1 7 COSATI CODES 18 SUBJECT TERMS (Contmue on reverse If neceuary and tdentdy by block number)

~ ~IE=-~ 1 C:iRO~P-i -- REMR Pile structures SUB GROUP ·-- Cjvil engineering ,_ __ L? . -- --+----_Q_2 -

i ' Maintenance manag~ment ''I ,'\f1) :;.<til. r \Con;r~u+' w' re11!'rse d necenary and tdenttfy by blo(k number)

The reliability function of a structure--the probability that the

stucture wi 11 perform adequately--represents a condition index that l s useful

tn evaluating the structure's condition. A generally applicable, three-step

procedure has been developed for calculating structural reliability ustng

existing information about a part i cuI a' structure and the evaluator ' s confi-

dence 1n this information. The structure's margin of safety lS approximated

by a normal distribution and its moments are evaluated from potnt estimates.

It lS suggested that the three-step procedure be used to evaluate the safety (Cor.t 1 d -~

.'() c)l) • ..{If'( I 11()~, '\'JAIL A Ill u T Y 0 ~ ,1-fl) T fl A C T 21 AB'lTRACT StCUK:fY lLA)):I-ICATION

0 'JN•; ,1)),f •f ~) ( d\il .MIT I D [X] ) ;.\rJl t

.!.'J :'-~;. '\.;, t l)~ ilt )f'• lNS!HLE •NLJIV1DIJAL

Diane P~ Mann DO FORM 1473, 8·1 r,'AR

AS RP r 0 [)fl( IJ)[R) UNCLASSIFIED .'.'b fELEPHUM' (ln<lude Area Code) I lie OFFICE )Y'\i180L

(217)373-7223 CECER-IMT 8l f\•)R t>C•t•on m.-~y hP u~Pd unt1l t>~hduSt••O

:.1· ••th,.., ··d•'"l"' "'' ,_,IJ~"'"t" ·,;(, ''•TY (lA))'Fi(ATI(ll< ·)/ '"'') :Ot.•;t

---- t!NCLASS-'(FIEb- -

UNCLASSIFIED

BLOCK l 9. (Con t 'd)

of existing sheet pile and other metal structures. The method has been applied to three sheet pile structures representative of those maintained by the U.S. Army Corps of Engineers.

UNCLASSIFIED

PREFACE

The study reported here was authorized by Headquarters, U.S. Army Corps of

Engineers (HQUSACE), as part of the Concrete and Steel Structures Problem Area

of the Repair, Evaluation, Maintenace, and Rehabilitation (REMR) Research

Program. The work was performed under Work Unit 3301, "Structures Damage

Index to Determine Remaining Life and Reliability of Metal Structures." Mr.

Frank Kearney was Principal Investigator for this Work Unit. Dr. Tony C. Liu

(CECW-ED) was the REMR Technical Monitor for this work.

Mr. Jesse A. Pfeiffer, Jr. (CERD-C) is the REMR Coordinator at the

Directorate of Research and Development, HQUSACE; Mr. John R. Mikel and Mr.

James Crews (CECW-OM) and Dr. Tony C. Liu (CECW-ED) serve as the REMR Overview

Committee; Mr. WilLiam F. McCleese (CEWES-SC-A), U.S. Army Engineer Waterways

Experiment Station (WES), is the REMR Program Manager. Mr. J. McDonald

(CEWES-SC-R) is the Problem Area Leader.

This study was conducted by the Structures Division of JAYCOR for the

Engineering and Materials Division (EM) of the U.S. Army Construction Engin­

eering Research Laboratory (USACERL), during the period April 1985 to

September 1985. Appreciation is expressed for the assistance of Dr. Dawn

White, Mr. Barry Hare, and Dr. Anthony Kao (all of USACERL) and Ms. Caroline

Cummins, Ms. Susan Lanier, and Mr. William J. Flathau (all of JAYCOR). COL

Carl 0. Magnell ts Commander and Director of USACERL, and Dr. L. R. Shaffer 1s

Technical Director. Dr. Robert Quattrone ts chief of USACERL-EM.

1

CONTENTS

PREFACE

METRIC CONVERSION FACTORS

PART I: INTRODUCTION • •••••••••••••••••••••••••••••••••••••••••

Background • •.•••...••••••••••....••.••....•••............• Purpose •••...••.•.•••.••.•..•.•.••. • • •..•• • • ....... • • • • • • · Approach • ..•••••••.•••.••.••.....•••.••.•••......••.. Mode of Technology Transfer •••••••••••••••••••••••••.

PART II: METHODOLOGY • •••••••••••••••••••••••••••••••

Reliability as an Index of Condition ••••••••••••

Deterministic Randomness of Probabilistic

Mode 1 •••••••••••••••••••••••••• Variables ••••••••• Calculation •••••••.

Step 1. Step 2. Step 3. Example Calculation ••••••••••••••••••••••••

Summary • ••••...•••••••••••••••••••.••.••••••••••••• Field Applications •••••••••••••••••••••••••••••••••

PART III: O'BRIEN LOCK AND DAM.

General Information ••• Interlock Strength ••••

Load ••••••••

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlock Interlock Reliability ......................•.......... Shear Rupture in Cross Wall •••••••••••.•••••••••• Tension Rupture of Tie Rod ••••••••••••••• Buckling of Sheet Piles •••••••••••••••••• Ana 1 ys is .................................... .

PART IV: POINT PLEASANT CANAL •••••••••••••••••••••••••••••••••

Structural Description •••••••••••••••••••••• Load Analysis .............................. . Strength Analysis ••••••••••••••••••••••••••••• Reliability of Sheet Pile Bulkhead •••••••••• Analysis .............................................. .

Page

1

4

5

5 8 8 8

9

9

10 12 14 17

18 18

19

19 20 24 26 27 31 34 36

38

38 39 42 43 44

PART V: PORT STRUCTURES IN ISRAEL............................. 45

Corrosion of Steel Sheet Piles.......................... 45 Probabilistic Analysis ••••••••••••••••••••••••• •,•....... 45 Analysis................................................ 48

PART VI: SUMMARY AND RECOMMENDATIONS •••••••••••••••••••• 51

REFERENCES. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 52

APPENDIX A:

APPENDIX B:

APPENDIX c:

MICROCOMPUTER PROGRAM FOR ESTIMATING THE RELIABILITY OF EXISTING STRUCTURES •••••••••

VARIABLE LIST •••••••••••••••••••••••••••••••••••

GLOSSARY OF STATISTICAL TERMS •••••••••••••••••••••

2

Al

Bl

Cl

No.

1 2 3

4 5

LIST OF TABLES

Values of Parameters for Miter Girder Example •••••••••••••••• Safety Margins for Miter Girder Example ••..•••••.•••••••.•••• Random Variables Affecting Reliability of Interlocking

System ••••••••••••••••••••••••••••••••••••••••••••••••••••• Random Variables Affecting Reliability of Tie Rods •.••••••••• Random Variables Affecting Reliability of Sheet Piles

54 54

55 55

Against Buckling........................................... 55 6 Random Variables Affecting Reliability of Point Pleasant

Bulkhead................................................... 56

LIST OF FIGURES

No.

1 Frequency of steel structures by type........................ 6 2 Structurally critical elements of steel sheet pile

structures .....• ,........................................... 7 3 Miter gates: horizontally framed typical girder data •••••••• 11 4 Reliability function of normalized safety margin ••••••••••••• 16 5 Sheet pile interlocks........................................ 21 6 Measured distribution of interlock strength •••••••••••..••••• 23 7 Cross section of cofferdam ••••••••••••••••••••••••••••••••••• 25 8 Estimated condition of interlocking system ••••..••••••••••••• 27 9 Shear analysis in cross wall •••••••••••••••••••.••••••••••••• 28

10 Estimated condition of cross wall shear resistance ••••••••••• 31 11 Horizontal section of sheet piling of guide wall ••••••••.•••. 32 12 Guide wall................................................... 32 13 Estimated condition of tie rod •••.••••••••••••••••••••••••••• 34 14 Estimated condition of system against local buckling ••••••••• 36 15 Typical cross section of sheet piling bulkhead •.••••••••••••• 39 16 Simplified equivalent beam method for the design of

anchored bulkhead in clean sand •••••••••••••••••••••••••••• 40 17 Sheet pile analysis ...............•••........................ 41 18 Estimated condition of Point Pleasant bulkhead sheet piles ••• 44 19 Distribution of loads and rate of corrosion on sheet pile

bulkheadt northern wharf in Kishon ••••••••••••••••••••••••• 46 20 Service life of Larssen IV sheet piles, t = 0.58 in.

(14.8 mm), with various ~ates of corrosion ••••••••••••••••• 47 21 Reliability of sheet piles; Mmax' underwater zone •••••••••••• 49 22 Reliability of sheet piles (1/2 M and tidal zone) ••••••••• SO max C1 Normal distribution ••••••••.••••••••••••••••••••••••••••••••• C2

3

METRIC CONVERSION FACTORS

Non-SI units of measurement used in this report can be converted to SI

(metric) units using the following table.

Multi ell feet

inches

pounds (force) per square inch

pounds per cubic inch

square inches

square feet

kip (1000 lb load)

Bl 0.3048

0.00254

6894.757

27679.91

6.4516

0.09290304

456.6

4

To Obtain

metres

metres

pascals

grams per liter

square centimetres

square metres

kilograms

ASSESSING THE RELIABILITY OF STEEL CIVIL WORKS STRUCTURES

PART I: INTRODUCTION

Background

1. A maJor mtsston of the U.S. Army Corps of Engineers (USAGE) is

maintaining the operational efficiency of various navigation, flood control,

and hydroelectric power projects. The importance of this mission increases

with time as older projects deteriorate and few new projects are authorized.

Of particular concern are steel structures. A recent study (JAYCOR 1985)

documented that USAGE Civil Works projects include many structurally sig­

nificant steel features such as bridges, outlet gates, spillway gates, lock

gates and valves, sheet piles, hydropower gates, and penstock liners (Figure

1). The study further estimated that each year these steel structures experi­

ence about 44 different problems to varying degrees.

2. Research has shown that a relatively significant fraction of the

problems occur with steel sheet piling. Deterministic analyses of this

structural feature have been conducted by the U.S. Army Construction Engineer­

ing Research Laboratory (USA-CERL) (Kearney 1986). That study of sheet pile

structures identified the elements that could fail (Figure 2).

3. A structural condition index would make it easier to allocate public

funds where they are most needed to solve these problems and avert failures.

Such an index should indicate the composite structural integrity and the

overall operational condition of a particular feature, and reflect the extent,

severity, and type of deterioration. It should also take into account the

effects of loads imposed on the structure. The reliability function can

provide such a cond-ition ind-ex. ffy quantifying changes rn resi-stance and

loading, this function indicates the probability that a structure will perform

adequately under certain conditions. Also, as used in this report, it pro­

vides more information than a deterministic analysis because it shows the

gradual deterioration of condition over time.

5

., Q. ... 0 u ., s::.

>. .0

'0 • a- 0 ... • Q. 0 ., • ... ;:J -u ;:J ... -en -0 ., 00

~ c ., u ... ., a..

100

80

60

40

20

Bridge Lock Spillway Gate

Sheet pile Outlet Gate

Figure 1. Frequency of steel structures by type (after JAYCOR 1985)

Hydropower

STEEL SHEET PILE STRUCTURES

STEEL SHEET PILE WALL

CELLULAR COFFER DAMS (INDIVIDUAL l

CELLULAR COFFER DAMS (CONNECTED l

SHEET PILE {BULKHEAD) SECTION

WALE

TIE ROD

BULK HEAD ANCHOR

~FENDER CAP BACKFILL

SHEET PILE SECTION

CONCRETE CAP GRAVEL/ ROCK INFILL COFFERDAM SHEET PILE SECTION

CROSS WALL SHEET PILE SECTION

FENDER

CONCRETE CAP GRAVEL/ ROCK INFILL

~WEB -----FLANGE

INTERLOCKS ~ WALE SECT ION ~CONNECTION

ROD -c:::::: TURNBUCKLES CONNECTION

---ANCHOR -=::::::::::::: ARM OR P LATE """"'~::==:---FENDER PILE

CONNECTION

WEB

~ INTERLOCKS -==-===== ARMOR PLATE

CONNECTION

WEB

~INTERLOCK WEB

~INTERLOCK ---ARMOR PLATE -- CONNECTION

Figure 2. Structurally critical elements of steel sheet pile structures

7

Purpose

4. The purpose of this report is to describe the development of a

structural reliability function and illustrate its use to assess the condition

of USACE steel structures.

Approach

5. A general, probabilistic procedure for evaluating a structure's

reliability and for using that as an index of its integrity is described. The procedure is then applied to three types of sheet pile structures which are

representative of Corps-maintained facilities.

Mode of Technology Transfer

6. It is recommended that the results of this work be transferred to

the field through presentation at Repair, Evaluation, Maintenance, and

Rehabilitation (REMR) meetings and inclusion in the REMR notebooks as short

technical notes.

8

PART II: METHODOLOGY

Reliability as an Index of Condition

7. This chapter describes a general procedure for estimating the

reliability of existing CE structures. The reliability of a structure is the

probability that, when operating under stated environmental conditions, it

will perform its intended function adequately for a specified interval of time

(Kapur and Lamberson 1977). The goal of reliability analysis is to find the

likelihood that the load exceeds the resistance. Thus, each calculation has

two distinct parts for each mode of failure: stress (load) analysis and

strength (resistance) analysis. This reliability function provides all the

characteristics desirable for an index of a structure's present condition. As

a mathematical "probability," it 1s a numerically ordered measure, with unity

representing the best condition (best performance) and zero representing the

worst condition (no performance). "Stated environmental conditions" expli­

citly incorporate the effects of imposed service loadings that were not

contemplated during design. The "specified interval of time" is necessary for

quantifying any deterioration in strength capacity that a structure experi­

ences. rinally, the "intended function" makes it possible to address manage­

able operation and maintenance problems.

8. Probabilistic methods will be most useful in evaluating existing

structures, rather than in designing new ones, because they are better for

handling uncertainties. For design, the Corps of Engineers has traditionally

used deterministic rather than probabilistic methods. For example, in

designing a lock chamber with sheet pile cells as walls, uncertainties will

arise about operational loadings and material strengths. A prudent designer

uses deterministic, conservative values, possibly overdesigning the feature.

The additional cost incurred by overdesigning is usually only a small fraction

of the total construction cost. On the other hand, when evaluating a struc­

ture's need for rehabilitation, the same uncertainties will arise as in

design, along with others arising from the evaluator's inability to make

certain measurements and inspect certain parts. However, in rehabilitation,

the costs of uncertainty are much greater. Not only is the actual work

9

costly, but there is the additional cost of loss of function, which should be

avoided if possible. Overly conservative estimates may result in costly,

unnecessary repairs, but overly optimistic estimates may result in failure to

perform critical repairs, perhaps with drastic consequences. A probabilistic

evaluation method, with its more accurate accounting of uncertainties, can

enable evaluators to estimate the reliability of structures with greater

confidence.

9. A three-step method has been developed for calculating the reliabil­

ity of a structure.

a. Select a deterministic design model for the relevant mode of failure.

b. Quantify the certainty about the variables in this model.

c. Perform a series of design-like analyses and combine their results to estimate reliability.

This procedure resembles a traditional design procedure as much as possible.

The modifications explicitly incorporate the engineer's certainty about the

values of key parameters. The details of the procedure have been developed to

yield a mathematically justified estimate for use by those with minimal

experience 1n probability theory.

10. To illustrate the procedure, consider the hypothetical, horizontally­

framed miter gate girder shown in Figure 3. If the girder has corroded

appreciably on its upstream flange at midspan, the reliability of this simple

component will be estimated as an index of the girder's structural integrity.

Step 1. Deterministic Model

11. In the first step-of estimating reliability, the engineer must

select a deterministic model for the mode of failure threatening a particular

-exi-sting ·s true tur-e. -In -mo-st cases-, .this model _can he the same analytical

procedure used to design the structure for this contingency. On the simple

extreme, it can be an equilibrium equation written from a free-body diagram of

a statically determinate component. On the other hand, the model could be a

comprehensive stress analysis of finite elements for a complex, indeterminate

system. As in the design situation, it is best to use the simplest model that

adequately incorporates the essential factors of the failure mode. Later, for

estimating the reliability of the structure, it is convenient to formulate

10

FACE OF RECESS

CURRENT 1-LOCK

R

W.l.

FACE OF LOCK WALL

Figure 3. Miter gates: horizontally framed typical girder data

w N•-2

w R I • R 2 • R • 2 SIN e p 1 • R COS 6 • N COT 8

W. L. • WORK LINE

N. A. • NEUTRAL AXIS

this model as a margin of safety, MS, which is the difference between the

resistance of the component and the effect of the loading:

MS = Resistance - Load.

12. In the example shown in Figure 3, the mode of failure is taken to be

the yielding of the girder, which represents a manageable failure. It is not

an issue of structural safety, since the ductility of the girder and the load

redistributing ability of the skin plate and the intercostals provide a

reserve capacity. As given in Engineer Manual 1110-2-2703 (HQUSACE 1984), the

axial stress, fa' and the flexural stress, fbl' in the upstream flange at

midspan are:

f w (!:! e + t) = cot a A 2

(1)

w(t - a) L2 2 2 fbl = [- - La cot e + (t - a) - a ] 2I 4

(2)

1n which A and I represent the area and moment of inertia of the girder and

the other variables are defined in Figure 3. These equations follow from an

equilibrium analysis of the girder as half of a symmetric three-hinged arch

and are the same ones used by the Corps of Engineers in designing new miter

gates. For this example, the margin of safety is:

(3)

1n which fy 1s the yield strength of the girder material. ,

Step 2. Randomness of Variables

13. In the second step, the engineer quantifies his or her uncertainty

about the actual values of the deterministic model's variables. As in the

design process, some of these may be known with great confidence and can be

assigned a single value. Others may be less precisely defined and should be

12

considered random variables. For them, a mean and a standard deviation are

assigned. Those two parameters are precisely defined in elementary probabil­

ity texts (Benjamin and Cornell 1970). The mean measures the central tendency

of the variable or the value about which scatter can be expected if repeated

observations of the phenomenon are made. The standard deviation, which has

the same physical units as the mean, measures the dispersion of repeated

observations abput the mean value.

14. The engineer can draw upon extensive sources of information to assign

values for these parameters. In some cases, a sample of repeated observations

having useful descriptive statistics may have been collected. Often, data

collected for similar problems may apply. An experienced eng1neer can

quantify his or her expert judgment by selecting the mean and the standard

deviation for random variables. This step is different from the design

situation in which a single conservative value is assigned to uncertain

factors. However, in evaluating existing structures, it is important to

quantify what is known about variables by using means, and to quantify what ts

not known about them by using standard deviations.

15. In the example shown in Figure 3, the geometry of the girder is pre­

sumed to be known with confidence. Accordingly, the values for these para­

meters will be deterministically assigned, as given in Table 1. These numbers

correspond to an example design (Granade 1980), except for the moment of

inertia, I, which is reduced by 30 percent to reflect hypothetical corrosion

of the upstream flange. The yield stress of the structural steel, fy' 1s

assumed to be random. It is reasonable to expect that the findings of a

general study (American Iron and Steel Institute 1978) of this factor's varia­

bility apply to the example outlined here. Accordingly, the mean and standard

deviation of f are assigned as indicated in Table 1. Note that the mean y

value is greater than the 36,000 psi normally used 1n design. The girder

loading is also taken to be random. Judgment is used to assign its mean value

to be 150 percent of the hydrostatic value, reflecting the effects of ice,

mud, and live loads. The uncertainty about this factor is quantified 1n a

standard deviation of 20 percent of the mean.

13

Step 3. Probabilistic Calculation

16. In the last step of the reliability estimation, the engineer first

computes approximations for the mean and the standard deviation of the margin

of safety. A further approximation of reliability is then calculated from

these values. The details of this step should be meaningful to a design

engineer who has only a basic understanding of probability. Their rational

basis is summarized in this section, and supporting references are cited for

engineers having more extensive experience in reliability.

17. Having considered some of the input variables to the functional

expression for the margin of safety, MS, to be random, it follows that MS is

itself a random variable. Convenient point estimates of its mean and standard

deviation can be found from those of the random inputs using the following

procedure (through Eq 11), given by Rosenblueth (1975). First consider the

safety margin MS(X) to be a function of only a single generic input random

variable, X, with associated mean, X, and standard deviation, ax, values. By

approximating the actual distribution of X with a discrete one having the same

mean and standard deviation, it follows that the mean,MS, and standard

deviation, oMS' of MS are:

MS + MS MS + = 2 (4)

= . MS+ - MS 0

MS 2 (5)

where:

(6)

(7)

For this single input variable, one deterministically evaluates the MS at

levels one standard deviation above and below the mean of the input. The mean

and standard deviation of MS are then calculated from the sum and difference

14

of these results using Equations 4 and s. More generally, if MS (Xl' Xz, ... Xn) 1s a function of n independent input variables:

MS MS1

MS 2 MS n

= ms ms ms ms

where ms is the margin of safety calculated using the mean value of each

variable:

x > n

(8)

(10)

and MSi and oMS. are computed from Equations 4 and 5 as if Xi were the 1

only random variable and the others were equal to their mean values. For this

general case, 2n + 1 design-like evaluations of the margin of safety are

required.

18. With these estimates of MS and eMS' a convenient assumption can be

made about the form of the MS distribution in order to complete the estimation

of reliability. Since the margin of safety is the difference between resis­

tance and load which, tn turn, often involves sums of random variables, there

is some justification to approximate MS with a normal distribution (Benjamin

and Cornell 1970). Since the structural evaluations of operational interest

do not involve extremely rare events, in most cases the reliabilities calcu­

lated will not be extremely sensitive to the form of this distribution. The

assumption of a normal dis t ri but ion has t~1o other advantages: it is fami 1 iar

even to those new to probability, and- standard taotes and- approximating

functions are available. Finally, it follows that the reliability is given

by:

R = Probability (MS > 0) = 1 - F (- MS ) 0

MS (11)

tn which F() is the standard normal cumulative distribution function (Benjamin

and Cornell 1970).

15

19. Figure 4 shows this reliability function. Note that reliability

depends on the mean and the standard deviation of the margin of safety through

their ratio, MS/aMs• As MS/aMS approaches infinity, R approaches 1, indicat­

ing a perfect condition. Conversely, large negative values of the ratio cause

R to approach the worst condition R = 0. Interestingly, a zero safety margtn

corresponds to a SO percent reliability, reflecting the uncertainty in this

random variable.

20. In summary, to complete this final step the engtneer performs 2n + 1

design-like, deterministic calculations of the safety margin in accordance

with Equations 6, 7, and 10. The results are then used in Equations 8 and 9

to estimate the mean and standard deviation of the random safety margin.

z 0 ~ u z ~ ~

> ~

J m 4 J w ~

~

ID

0.8

0.6

0~

0.04---~.C~--------~---r--~~--~--~--~----~--,---~

-3 -2 0

MS/~ MS

2

Figure 4. Reliability function of normalized safey margin

3

Reliability, which indexes structural condition on a zero to one scale,

follows from Equation 11. Note that this sequence is identical irrespective

of the deterministic model selected in Step 1. It applies for a comprehen­

sive, finite element stress analysis as well as for a statically determinate

equilibrium equation. Appendix A provides a microcomputer program to imple­

ment this procedure.

Example Calculation

21. For the example given 1n Figure 3, n = 2 variables have been con­

sidered random, fy and w. Accordingly, five deterministic evaluations of

Equation 3 are required, as summarized in Table 2. The estimates of MS and

0 MS' considering only fy to be random, are from Equations 4 and 5 and

second and third rows of Table 2:

MS - MS 4271 + 13071 + MS

1 2 2 8671 psi

MS - MS 13071 - 4271 + 4400 psi 0

MS 2 2 1

Considering only w to be random, the last two rows of Table 2 give:

14856 + 2485 2

14856 - 2485 2

8671 psi

6186 ps.t

the

Substitution of these results and ms from the first row of Table 2, into Eq 8

leads to:

(1.000) (1.000)

from which MS 8671 psi. In turn, Equation 9 becomes:

(

0 Ms)2

1 + 8671

0MS 8208 psi

17

which can be solved for oMS = 8216 psi. Finally, from Equation 11 R = 1 - F

(-1.056) = O.BS, using the standard normal tables given in Kapur and Lamberson

(1977) and Benjamin and Cornell (1970). Because R is high on the probability

scale of 0 to 1, but slightly less than 1, this index warns of the likelihood

of the girder yielding, but indicates that it might be acceptable to defer

repair of the deterioration until the next dewatering if one is scheduled

soon. However, further corrosion of the flange would change the index and the

interpretation.

Summary

22. A three-step procedure can be used to evaluate the reliability of an

existing structure as a condition index of its integrity. In general, the

steps are as follows:

a. Write deterministic equations relating the variables of the appropriate mode of. failure.

b. Assign to each of these variables a mean and a standard dev­iation that reflect the knowledge and uncertainty for the particular structure being considered.

c. Calculate point estimates for the mean and the standard devia­tion of the safety margin, and compute reliability assuming a normal distribution.

Field Applications

23. Chapters 3 through 5 describe how these calculations were applied to

three sheet pile structures representative of Corps facilities: O'Brien Lock

and Dam in Chicago, the Point Pleasant Canal in New Jersey, and port struc-

-ture-s -.i:n -I-srael. -E-ach o-f -these -st-ructu-res -h-ad -been -studi-ed pr-evi-ously. Those

results were compared to the results of the probabilistic analyses.

18

PART III: O'BRIEN LOCK AND DAM

24. In 1979, an inspection of the O'Brien Lock and Dam, located on the

Illinois waterway in Chicago, revealed excessive corrosion of sheet piling. A

deterministic analysis of the problem {Kearney 1986) concluded that the struc­

ture should have a service life of 83 years. For this study, the O'Brien Lock

and Dam has been used as an example of how probabilistic analysis can be

employed to calculate service life.

General Information

25. The O'Brien Lock and Dam, a cellular cofferdam, presents a complex

problem because the factors affecting its service life are many and subtle.

Cellular cofferdams are double-wall retaining structures constructed of

interlocking steel sheet piles. The piles form adjacent cells that are filled

with soil or rock fragments {Lacroix, Esrig, and Luscher 1970). In analyzing

cofferdams, one needs to consider not only the structure's individual charac­

teristics, but also the interactions among soil, rock, water, weather, steel,

and concrete. A dam's shape and location also make it an individual system of

unique characteristics: cofferdams may be shaped 1n a circle, diaphragm, or

cloverleaf; sheet piling walls can be constructed on dry land, in fresh water,

or in seawater. However, all cofferdams have one factur in common: any

interlock in a single sheet pile cell is a possible failure location, and

unlike most conventional structures, there is little redundancy in the system.

26. The following types of cofferdam failures have been observed and

reported:

a. Sliding on the base.

b. Bank sliding.

c. Flooding of a cell.

d. Erosion in streams--scour and loss of cell fill.

e. Boiling of land cofferdams.

f. Shearing of silt and clay fill.

g. Impact from a barge.

h. Shear rupture of cross wall webs.

19

1. Improper construction.

1• Deficient interlocking.

k. Faulty sheet pile connections.

1. Interlock failure near connecting sheet piles.

m. Corrosion.

27. In order to present clear examples, only sound structures were

considered; this study is not concerned with failures resulting from faulty

materials, design, or construction. Using soil mechanics theory, Terzaghi

(1945) provides a sound basis for evaluating completed cofferdam structures,

to assure that they are safe from tilting, sliding, overturning, or shearing

vertically.

28. Corrosion of the steel sheet piles is a problem that becomes signifi­

cant as a cofferdam ages. Many investigators have shown that corrosion of

buried piles is negligible (Young 1981). On the other hand, Kearney's study

of the O'Brien Lock and Dam (1986) indicates excesstve corrosion of steel

sheet piles, especially in the region of coarse gravels. The present study

reflects the need to consider corrosion as a major factor 1n cofferdam

stability. Therefore, failures caused by corrosion were the focus of this

analysis: interlock failure, rupture of the crosswalls, tensile rupture of

anchorage systems, and buckling of sheet piles.

29. For each of these modes of failure the stress (load) and strength

(resistance) were calculated using the pro~abilistic methods of Part II.

Interlock Strength

30. The walls of a cellular cofferdam are subjected to an axial load (hoop

tensiori) that can damage the sheet pTles by rupturing their webs. This ten-

sion is also transmitted to the curved interlocking portions. These pieces

may bend until sufficient deformation damages the sheet piles by pulling the

claws apart. Another possible mode of failure is the simultaneous tensile

rupture of the thumbs (shanks) of the two bulbs (Figure 5). Because each

interlock is vulnerable, it is realistic to analyze a single sheet pile rather

than a group of them.

20

(a) Free body diagram of interlocking forces

(b) Exaggerated illustration of tensile yielding of the shanks

Figure 5. Sheet pile interlocks

31. Kearney (1986) presents the following deterministic model for evalua­

ting interlock strength. In summary, the interlock strength, Nu, is found

from:

in which:

N1 = force per unit length between interlock finger and thumb

N2 = force per unit length between interlocked thumbs

(12)

These internal forces are obtained from the following two equations (Kearney

1986):

where:

2 e3ou (-1 +/t 2 N1 = + t

2 /4e 3

4(N2e 2 - N1e 1) 2 (N1+ N2)2

= tl au a u

t 1 = thickness of web (initially 0.375 in.)

t 2 =thickness of thumb (initially 0.35 in.)

el = eccentricity of Nl (0.48 in.)

e2 = eccentricity of N2 (0.48 in.)

e3 = eccentricity of finger (0.56 in.)

au = yield strength of sheet pile {60,000 psi)

(all as shown in Figure 5).

2 ) (13)

(14)

Using these numbers, Nu is determined by using Equation 12. This value of Nu

is superseded by t 1au (hoop tension for web failure), if the latter value is

lower. In turn, both of these values are superseded by 2t 2au (hoop tension

for thumb rupture) if this is the lowest value.

22

32. The variability in N , interlock strength, is shown by the histogram u

tn Figure 6, which is based on a large number of tests by Kay (1975). It 1s

obvious that the histogram is skewed. The truncation at the upper end 1s

associated with web failures rather than interlock failures, and it is likely

that a greater degree of symmetry would have been produced had there been no

web failures. There is some justification for fitting a normal distribution

to an interlock histogram (Kay 1975). Kay has indicated that a normal

distribution with a coefficient of variati.on (ratio of standard deviation to

mean) of 0.11 will provide a good representation of the interlock strength.

Therefore, a deterministic analysis of strength, as shown above, results in

the mean value of the interlock strength; then a value of 0.11 can be taken

for the coefficient of variation.

)­u z LLJ 5 :;::::)

0 LLJ a:: LL.

20 25

INTERLOCK STRENGTH- kips per inch

Figure 6. Measured distribution of interlock strength (Kay 1975)

23

30

1 '

33. To obtain the interlock strength for a given amount of corrosion, it

1s assumed that t 1 and t 2 are reduced by the same amount of corrosion, £.

Although corrosion may also affect the distances e 1, e2 and e3 , the effect of

changes in the eccentricities on the interlock strength is not as great as

that of the eroded thicknesses (Bower 1973). Therefore, it is reasonable to

neglect the effect of corrosion on the eccentricities.

Interlock Load

34. The calculation of the interlock load, N, begins with the equation for

the hoop tension in a cylinder (from membrane theory)

N R q(x) (15)

where R ts the radius of the circular cell and q(x) denotes the difference

between the internal and external pressures on the cylindrical wall. Figure 7

illustrates a cross section of the cofferdam. This figure indicates that

there is no danger of bursting the cylindrical shells on the land side of the

land wall, since the sheet piles there are completely buried. For the chamber

side of the land wall, the maximum tension occurs at the bottom of the

chamber:

N = R[K y(H- s) + (K y 1 + y )(s -d) - y (h- d)] A A w w

where:

N = interlock load

R = radius of the clrcular cell {316 in.)

KA = coefficient of horizontal earth pressure (variable)

y = total unit weight of the cell fill (variable)

H = elevation of the top of the cell (504 in.)

s = elevation of the pore water inside the cell (variable)

y' = submerged unit weight of the cell fill = Y - Yw

Yw = unit weight of the water

d = elevation of the bottom of the chamber (198 in.)

h elevation of the water in the ~hamber (198 in., dewatered)

24

(16)

N VI

h = -4

d = -141

River Side

I

H = i 7

s = .. 3

h = -4

Chamber Side

d=-18.5 I

. )( Y ~J\:X XX :XX.~

El -351

b = 276" -··~

(a) Cross sectipn of river wall

_H = t7

El. -1.5 I

h = -4

d = -18.51

~

w

El. -351

(b) Cross section of land wall

Figure 7. Cross section of cofferdam

35. For the probabilistic method a mean and standard deviation must be selected for each of the items marked "variable" on the above list. A great deal of uncertainty is associated with the coefficient of horizontal earth pressure, KA. Some authorities (Tennessee Valley Authority 1957) have

recommended the use of the Rankine active earth pressure coefficient. Terzaghi (1945) objects to the use of that coefficient, recommending instead a value in the range of 0.4 to 0.5 for sand and gravel fills. Kay (1976) proposes a normal distribution of mean value of 0.4 and standard deviation of 0.125. For this problem, a mean of 0.45 and a standard deviation of 0.125 are used 1n this report due to the uncertainty regarding the type of fill.

36. The unit weight value, y, of the backfill material 1s another con­cern. Usually, a weighted average of the total unit weight of the fill above the phreatic surface and the submerged unit weight of the fill below this surface is used (Kay 1976). Kay has proposed that this parameter will be

described by a normal distribution of a standard deviation of 0.08 Yw' 1n which Yw is the unit weight of water. The mean value is chosen based on available information and the engineer's judgment. For this calculation,

y = 0.0723 lb/cu in.

37. The elevation of the pore water inside the cell, s, is another

uncertain parameter. For this, a mean value of s = 468 in., as described by Kearney (1986), and a standard deviation of 20 in. are used. The hoop tension therefore depends on three parameters with means and standard deviations as

indicated in Table 3.

Interlock Reliability

38. Considering the marg1n of safety, MS, by which Nu exceeds N,

MS = Nu - N (17)

the interlock failure occurs when MS ~ 0.

Nu and N, MS is also a random variable. the reliability function is:

Because of the randomness of both

From its definition (Equation 11),

R(£) = P[MS > 0] (18)

where £ denotes the amount of corros1on and R(£) shows how reliable the struc­ture is for that amount of corrosion. T~is function, which has been computed

26

using the technique described in Part II, is presented in Figure 8. This

figure also presents reliability as a function of time, t, using Lacroix's

assumption that £ = 0.0025 in./yr (Lacroix, Esrig, and Luscher 1970). t

Shear Rupture 1n Cross Wall

39. A cross wall may become so corroded that it will rupture in shear.

Since the hoop tension is transmitted to the Y-pile at the junctions between

the cylindrical walls and the cross walls, horizontal tension will be devel­

oped in the cross walls (Figures 9a and 9b). The combination of a cross wall

and the segments of the corresponding cylindrical walls act as an I-beam to

>­... _J

m <l _J LLJ a::

TIME, YEARS

20 40 60 80 100 120 I. 0 ......----~------r-----.---...:::--r------r------,

-----+-------+-------

0.4 -----------1------~..r-------~

0.2-+---

0.00 0.05 0.10 0.15 0.20 0.25 0.30

e, in. (CORROSION RATE= 0.0025 in/yr.)

Figure 8. Estimated condition of interlocking system, O'Brien Lock and Dam

27

R

b

CELLULAR COFFERDAM

( 0)

~ N' • 2N COS a

TRANSMITTED TENSION TO CROSS WALL

(b)

I -5 -I ~

N' ~· .......--s

CROSS WALL (d)

MOHR CIRCLE ( e )

-~-

b

-~-

SEGMENT OF A COFFERDAM REPRESENTED AS AN I- BEAM

( c )

Smax

~ ORIENTATION OF Smax

( f )

STRESS IN AN ELEMENT OF A WEB OF CROSS WALL. MOHR'S CIRCLE, Smax

Figure 9. Shear anal~sis in cross wall

28

resist the shear developed by the lateral pressures on both sides of the land

wall (Figure 9c). Consequently, an element of the cross wall near the Junc­

tion with the Y-pile is subjected to shear, S, and tension, N' (Figure 9d).

By the Mohr's Circle, the maximum shear (the ''load'') in the cross wall is:

(19)

N' 1s the tension transmitted to the cross wall and 1s expressed as:

N' 2N cos a (20)

where N is the hoop tension and a is as shown in Figure 9a.

40. This shear is not carried by the cross wall alone; however, due to the

lack of a reliable theory to estimate how much of the shear is resisted by the

fill, it is assumed that all of it is carried by the cross wall. For the

I-beams with heavy flanges, the shearing stress is nearly constant on a cross

section of the web of an I-beam:

S - P/b (21)

where b is shown in Figure 9c, and the definition of P follows.

41. The land wall of the lock carries greater shear than the r1ver wall

because of the pressure of the backfill. The largest total shear, P, occurs

at the bottom of the lock chamber and is determined from:

where p is the net lateral pressure acting on the crosswall. At the bottom of

the chamber, the following expression can be substituted for p:

p = KA y (H - s) + (KA y' + yw) (s - x), d < x < h (23)

29

With this substitution the total shear is g1ven by Equation 24. The uncer­

tainties of the variables in this equation are the same as for Equation 18.

L 2 L 2 + - KAy(H - s) - - y (h - d) 2 2 w (24)

In Equations 22 through 24:

KA coefficient of ho~izontal earth pressure (variable)

y = total unit weight of soil (variable)

L = average distance beLweEon CC0SS walls (316 in. )

H = elevation of the top of the cell fill (50!~ in.)

s = elevation of the pore '.>later (variable)

d = elevation of the botto:n of the chamber (198 in.)

y I submerged unit weight of the soil = y - y w

Yw =unit weight of the water (0.0361 lb/cu in.)

x = an arbitrary elevation

h =elevation of the water inside the chamber (198 in., dewatered).

Now the maxtmum shear (Equation 21) can be calculated.

42. The shear strength of the steel ts taken as O.Sau with a 10 percent

coefficient of variation, where a is the yield strength of steel (vari-u

able). The shear strength (resistance) ts then determined from:

S = 0.5 a (t - e) u y 0

1n which t 0 ts the initial thickness of the web in a cross wall (t0

=

0.375 in.), and E denotes the amount of corroston.

(25)

43. As with the reliability of the interlock, the margin of safety for

shear failure of the cross walls is:

(26)

30

and the reliability of this structual element is:

R(E) = P(MS > 0).

Figure 10 illustrates this reliability function, developed for the shear

strength of cross walls.

Tension Rupture of Tie Rod

(27)

44. A guide wall is formed by interlocking Z-shaped sheet piles driven

into clay and glacial till. The wall is partially supported by a system

consisting of wales and tie rods (Figures 11 and 12). The tie rods are

20 1.0

-

0.8

-0.6

>-t-:J m <(

...J 0.4

w a:

0.2

.

0.0 T I I I

40

I

TIME, YEARS

60

I f I I I

80 100 120

1\ \ \

1\ \

I I f I I • I I

0.00 005 0.10 0.15 0.20 0.25 0.30

E, in. (CORROSION RATE= 0.0025 in./yr.)

Figure 10. Estimated condition of cross wall shear resistance, O'Brien Lock and Dam

31

'-----------+..._,, ________ ---

Tie Rod

Figure 11. Horizontal section of sheet piling of guide wall

Fl---.. r::;::::_-------11 x•h

x=O

Figure 12. c"uide wall

32

several yards apart. One bay of the wall between successive tie rods 1s

considered in Equation 28. The tension, P, in a tie rod must balance the

forces on one bay of the wall. This force has been derived from Equation 25

1n Kearney (1986) and is defined as:

p = L 2 2 2 [KAy(H - s ) + (K y' + y )s

2 A w

The remainder of the variables are defined as follows:

L =distance between two adjacent rods (84 in.)

KA coefficient of lateral earth pressure (variable)

H =elevation of top of wall (552 in.)

s = elevation of backfill saturated water (variable)

h = elevation of water in r1ver (variable)

a= elevation of wale (444 in.)

K 1s as described 1n Equation 29.

3 3 3 3 - 3 -K y(H - s ) - (KAy' + y )s + y h + Ky'd +

A w w 2 2

3KAya(H - s )

2 2 2 + 3(KAy + y )as - 3y ah - 3Ky'ad

w w

The rod's strength 1s calculated from:

P = A a u u

1T (D - ~::)2 = 4

a u

= 0

(28)

(29)

(30)

where D 1s the diameter of the rod (D = 3 in.), t:: 1s the amount of corrosion,

and a 1s the ultimate tensile strength of the steel. The reliability u

function is defined as:

R(s) = P(MS > 0) = P(P - P > 0) u

(31)

Table 4 lists the random variables affecting the reliability function of the

tie rods. Figure 13 illustrates the reliability function. Although the tie

rods corroded less than the sheet piles themselves (Kearney 1986), the lower

33

I

I I I I

TIME, YEARS

1.0 100 200 300 400 500 600 700

>- 0.6 ------·-----

1-_J

m <t _J

0.4 w a:

0.2

o.o~--~~~~~~~~~~-r~~~~-r~~~r4~-~~

0.00 0.25 0.50 0.75 1.00 1.25 150 1.75

E, in. (CORROSION RATE • 0.0025 in./yr.)

Figure 13. Estimated conditions of tie rod, O'Brien Lock and Dam

horizontal scale of Figure 13 presents reliability conservatively by us1ng the

sheet pile corrosion rate.

Bu~kling of Sheet Piles

45. Local buckling of the sheet piling is not a disastrous problem; how­

ever, it will form a plastic hinge in the system. Formation of a plastic

hinge reduces the degree of redundancy of the sheet piling system. The manner

in which this reduction endangers the stability of the sheet piling is beyond

the scope of this study; however, the effect of corros1.on on buckling

34

stability is considered. A deterministic formulation has been developed for

this problem (Kearney 1986). The loading 1s defined as:

1 3 1 - 3 + 6 yw(h- x) + 6 Ky'(d- x) (32)

where x 1s an arbitrary elevation, P is computed from Eq 28, and K ts computed

from Eq 29. When solved for the point of maximum moment, x is as follows:

On the other hand, the resisting moment 1s determined by:

where:

=

=

E =

t =

2 a w

1- ---L----2SE{t - e:)

2 I - Ke:

c

yield point of the steel

18 1n. (Figure 11)

modulus of elasticity

initial thickness (0.5 in.)

I =moment of inertia {18.37 cu in.)

Ke: =the effect of corrosion on the moment of inertia (55 cu in.)

C = 5.5 in. {Figure 11)

Defining the reliability function as:

R(e:) = P(MR - Mmax > 0)

(33)

(34)

(35)

and ustng Table 5 for the values of random variables yields Figure 14, which

shows the effect of corrosion on the stability of the sheet piles.

35

>­r-...J

m

1.0

-

0.8

-

0.6

TIME, YEARS

20 40 60 80 100 -\

\

~ ~-r---

<l 0.4 --~------ -·

...J w 0:: -

~- -·-~ -

\ 0.2

- i'--._ I I I I I I I I I ' I I I I I I 0.0

0.00 0.05 0.10 0.15 0.20 0.25

E, in. (CORROSION RATE • 0.0025 in./yr.)

Figure 14. Estimated condition of system against local buckling, O'Brien Lock and Dam

Analysis

120

0.30

46. The probaoiTistic analysis reveal"S i:trcrt -t-he stru-c·tur€ will perform

safely for 70 years (the worst case, predicted from the interlock reliability

curve). Over the next 12 years, the ability of the interlocking systems and

the cross walls to withstand shear stresses will gradually decrease. Pro­

tective measures are recommended. Although the local buckling of the sheet

pile plates indicates an earlier destabilization of the system, it is not

considered indicative of system failure. Due to the ductility of the steel

plates and the redistribution of the moments (stresses), a plastic analysis

36

would be needed to understand the nature of this problem and its conse­

quences. These findings are consistent with the deterministic service life

assessment in Kearney (1986). However, the probabilistic findings are

believed to be more useful since they estimate gradual deterioration as an

explicit function of corrosion and time.

37

PART IV: ·POINT PLEASANT CANAL

47. The Point Pleasant Canal, located in the community of Point Pleasant

Borough, NJ, extends about 9000 ft between the Manasquan River and Barnegat

Bay. The canal bulkhead was extensively surveyed during 1983-1984 to deter­

mlne the extent of corrosion (Department of the Army, 1984). Although the

survey indicated that the canal was generally in good shape, it was recom­

mended that "action should be taken to prevent further corrosion." Surveyors

also predicted that based on a permissible wear of 33 percent at a max1mum

corrosion rate of 4 mils per year, the structure should be usable for at least

30 more years.

48. A reliability analysis leads to a more useful conclusion, despite the

fact that the only available descriptive structural and geotechnical informa­

tion of this canal bulkhead was that contained in the survey documentation.

The current study investigated the effect of corrosion on the bending capacity

(strength) of the system. A deterministic model that best characterized the

canal bulkhead was then chosen and a probabilistic model of parameter varia­

tion was developed as described in Part II.

Structural Description

49. The canal, which is about 150 ft wide, is constructed of var1ous sheet

pile sections (American Society of Metals standard sections Z-27, DA-27, and

MA-22) with a nominal thickness of 0.375 in. It has a wood sheet wall that

has been cut off slightly above the canal bottom. The steel piling 1s about 8

ft above the mudline on the water side, and the fill matecial is on the land

side. The fill and the mud materials are both sand. The mudline is about 3

ft below the water along the sides of the canal, and high water is generally

30 1n. below the tops of the sheets. Tide varies along the length of the

canal, but it is in the range of 2.5 ft. Figure 15 shows a cross section of

the sheet pile bulkhead.

38

Top Q! Bul~heQ__d __

Supporting Pile Connection

--~I ___ __:!_§'-~ --------- --

EI. t5 5

High W:Jter Level El. +4.0 -- \] ___ ~----------~

Low Water Leve I --~ El +I s'

~'--'-- -1.5'

Top of

/ /

Existing Timber Sheet Piling

____ __/

Tension Pile 10 HP 4 2

Water Level

Figure 15. Typical cross section of sheet piling bulkhead

Load Analysis

50. Sheet pile bulkheads are one of the most popular types of water-soil

supporting systems, although corrosion of the sheet piles seriously affects

the bulkheads' bending moment capacity. However, the key question is how the

actual soil load distribution is developed. Figure 16 illustrates the lateral

earth pressure diagram proposed for the flexible anchored bulkheads in clean

sand, and corresponds to the results of the model test with flexible anchored

bulkheads conducted at Princeton University (Tschebotarioff 1955). This

experiment showed that full restraint of the lower portion of the bulkhead was

effective when the ratio D/H (depth [buried] to height) equaled 0.43, where

39

------!- W. L.

T .. 0 H

·- - "'-- i.. - jt -

Figure 16. Simplified equivalent beam method for the design of anchored bulkhead in clean sand (Kearney 1986)

the point of contraflexure approximately corresponded to the dredge-line

elevation for normal backfilling conditions. Figure 17a illustrates the

assumed deflected shape of the bulkhead for which the point of contraflexure

corresponds to some elevation below the mudline. Figure 17b shows simplified

equivalent beam for the analysis of the anchored bulkhead in the sand back­

fill. The anchor pull, Ap' can be determined from the condition that the sum

of the moments of forces about the fictitious hinge is equal to zero:

(36)

In Equation 36, y refers ~o the total and y' to the submerged unit weight of

the granular backfill; all the other symbols are as shown in Figure 17. The

active lateral earth pressure coefficient, KA, is computed by:

b = (1 - f'H) 0.33f" (37)

where b is the distance from the top of the bulkhead to the point of tieback

attachment, and H is the height of the bulkhead (Figure 17a). The coefficient

f" expresses the effect of wall friction on the reduction of the active earth

pressure. The coefficient f' accounts for uncertainty with regard to the

relative importance of the passive earth pressure above the anchor level and

40

I h •3.75'

VA~~~~~fON ______,:==!J!..,.'=--2-0.,..1-( l-~r--::~---'L--

\E-BACK POINT

Ap~____,.

Y. Q--

MUD LINE

1 I \ \

--------; \ \

H •VARIABLE w

-------

a~jlf11'-POINT OF CONTRAFLEXURE _/ KAYh KAY'(H-h)

FICTITIOUS BEAM

(a) Assumed deflected shape of (b) Distributed lateral earth-and-sheet pile bulkhead water pressure on fictitious beam

Figure 17. Sheet pile analysis

the tensile strength of the sand layer that has been saturated by the capil­

lary action above the water level. Because of these considerations and the

large degree of uncertainty associated with the backfill material (it is not a

clean sand), 1n these calculations KA has a mean value of 0.4 and a standard

deviation of 0.125.

51. The one common factor for all sheet pile bulkhead structures is that

the maximum bending moment occurs in the underwater zone. Considering this

factor and measuring down from the water level (on the land side), the loca­

tion of zero shear, z, is determined from:

A 1 2 1 2 - K yhz - 1

KAy'z 2 (38) + Y a + y az - 2 KAyh 0

p 2 w w A 2 or

1 2 (KAyh - y a)z +

1 2 1 2 2 K y'z + 2 KAyh 2 Y a - A = 0 (39)

A w w p

41

The value of z obtained from Equation 39 ts used to determine the max1mum

moment:

'(40)

Equation 40 reveals that the maximum moment depends on five random variables:

H, y, y', KA' and a. Table 6 reflects the mean and standard deviations used

for these variables.

Strength Analysis

52. Initially, three modes of failure under the loading described above

were considered: flexural yielding, plate buckling, and tensile rupture of

the anchorage pile. Preliminary analyses indicated that the latter two were

less important than the first. Accordingly, the maximum bending based on the

applied load is compared with the permissible strepgth bending of the sheet

piles:

a S u

(41)

where MR is the maximum allowed moment, S is the section modulus and au is the

ultimate tensile strength of steel. It is believed that, based on the yield

point of steel and with a sa_fety factor of 1.4, this stress is sufficient to

take care of vibration and other unfavorable effects where sheet piling is

used in clean sand (Tschebotarioff 1955). However, many earth structures have

a safety factor of 1.5 against rupture of soil. This increase in permissible

stress results from the ductility of the piling material. The sheet piling 1s

a DA-27, for which the mean section modulus is 10.725 cu in./ft. However,

because of corrosion of the steel sheet piling, the thickness diminishes

gradually with time. At some time, the thickness of a sheet pile at a

critical section is reduced by an amount, E, due to corrosion; the thickness

42

at that time 1s t0

- e:, where t0

is the initial thickness. The mean section

modulus of the pile per unit width is then:

S = S - Ke: 0

(42)

where S is the section modulus of the sheet pile, S0

1s the initial value of

the section modulus, and the reduction factor, K, 1s a constant which can be

determined exactly from the shape of the cross section. K can also be

calculated approximately from the properties of the cross section. The result

is:

K :::: 28.6 sq tn. (43)

Consequently, the mean bending capacity, MR, of this sheet pile 1s determined

from:

(44)

Due to the approximation involved 1n determining K, a variation of 0.2 lS

chosen for MR, while a mean value of 24,000 ps1 with a standard deviation of

480 psi lS used for a u . The mean value of MR lS calculated from Equation 41.

Reliability of Sheet Pile Bulkhead

53. The reliability of the sheet pile bulkheads 1s investigated by consid­

ering a margin of safety, MS, which is defined as:

(45)

Failure occurs when this margin of safety becomes negative. MS is a random

variable because both MR and M are random variables. The reliability function

is defined as:

R(e:) = P[MS>O] (46)

43

This function depends on s1x random variables whose mean values and standard

variations are as shown in Table 6. Figure 18 illustrates the reliability

function for the Point Pleasant Canal bulkheads, for different amounts of

corros1on.

Analysis

54. The analysis estimates that the Point Pleasant bulkhead should serve

50 years without any significant deterioration. Given the previous, deter­

ministic estimate of 30 years (Department of the Army, 1984), this new result

suggests the possibility of cost avoidance.

1.0

-0.8

->-..... 0.6 ...J

m -<t ...J

~ 0.4

-

0.2

-

0.0 I I ' 0.0

25.0

TIME, YEARS

50.0 75.0

\ \

\ \ \ \

I I I I I I ' ' I I I I

0.1 0.2 0.3

S, in. (CORROSION RATE= 0.004 in./yr.)

100.0

I

0.4

Figure 18. Estimated condition of Point Pleasant bulkhead sheet piles

44

PART V: PORT STRUCTURES IN ISRAEL

Corrosion of Steel Sheet Piles

55. Buslov (1983) has developed and proposed an analysis of the rema1n1ng

service life of sheet piles, applying his approach to the P~rt structures of

Israel that were inspected for corrosion between 1976 and 1~81. Unlike con­

ventional design practice, which recommends the use of a co~rosion allowance

based only on structural considerations, he offers the addi~ion of a detailed

analysis of the actual character of corrosion to the design process. In his

approach, the sheet pile section 1s chosen according to the maximum bending

load and resistance; however, this section must contain som~ corrosion allow­

ance. In the area of active corrosion (mostly in the tidal and splash zone),

the existing service life ts determined by considering thre~ factors: rate of

corrosion, actual bending moments, and flange thickness, which had already

been determined based on the value of the maximum bending m~ment. To illus­

trate his technique, he considers a corrosion allowance of ~.25 mm (0.09 in.)

based on a 30-year life span, or in general, corrosion of 0,075 mm/yr. This

general corrosion, plus the maximum bending moment, were us~d during design to

choose the sheet pile section. He then considers the upper portions of the

section where the corrosion is rather high (Figure 19) and ~etermines the

potential service life for several corrosion rates. Figure 20 contains the

result of his calculations for the sheet pile bulkhead at Ktshon (Larssen IV n

Section; t = 0.58 in.). 0

Probabilistic Analysis

56. The current research considers the same sheet pile bulkhead in Kishon,

which consists of a concrete deck on concrete piles, fronti~g a sheet pile

retaining wall (Buslov 1983). Two cases are considered: c~rrosion of the

sheet pile under the water (corrosion is relatively low but moments are rather

large) and corros1on in the upper sections of the sheet pil~s (corrosion is

usually high but bending moments are rather small). The sh~et pile cross

section is a Larssen IVn. The corrosion rate is considered to be at least 0.4

45

I

0.00

-1.70'

-3o.oo·

/

ZONE OF MAXIMUM CORROSION DAMAGE

._--------+~~ RATE OF GENERAL CORROSION

Jj I ON OUTSIDE FLANGES

I' v 0.4 4 M. MAX.___,(

I II I II I II I II

•• ,, ., II

0.60M.MAX& I

0.70 M. MAX

M. MAX.

~-

BENDING MOMENT DIAGRAM

L \

I I

0 0.1 0.2 0.3 0.4 0.5

RATE OF CORROSION, mm I YR. NOTE: 1 mm • 40 mi Is

Figure 19. Distribution of loads and rate of corrosion on sheet pile bulkhead, northern wharf in Kishon

(after Buslov 1983)

46

80

0. 2 mm/year

60

en ... 0 G)

>-.. 40 en ., > ... G) en ., a: ., 0

> ... G) U)

20

I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Moments Ratio: M /Mmax Note: I mm = 40 mils

Figure 20. Service life of Larssen IVn sheet piles, t = 0.58 1n. (14.8 mm), with various rates of corrosion

(after Buslov 1983)

47

mm/yr within the tidal zone (upper portions of section) and 0.07 mm/yr for the

underwater zone. The maximum bending moment is expected to occur in the

underwater zone, while the moments in the upper portions of the wall, where

corrosion is particularly active, seldom exceed O.SMmax (Buslov 1983}. In

both cases, the margin of safety is expressed in terms of the bending moment

loading, M1 , and the resistance, MR:

The mean, M1 , 1s taken to be the value reported by Buslov. The corresponding

standard deviation is taken to b~ 30 percent of this value (o1 = 0.3 H1 ) based

on experience with O'Brien Lock and Dam and Point Pleasant Canal. The mean

bending moment resistance is expressed as:

M = a S(€) R u

1n which a indicates the yield stress, and S(€) reflects the effects of u

corros1on on the section modulus. As in two earlier case studies, oM = R

0.1 MR. The reliability 1s then estimated as a function of E using the

general procedure discussed in Part II.

(48)

57. Figure 21 illustrates the result, which 1s based on the max1mum bend­

log moment and on corrosion of the underwater zone (low corrosion). The lower

horizontal scale of the same figure relates reliability as a function of time

to corrosion of 0.07 mm/yr. Figure 22 shows the reliability of the same

structure but based on corrosion of the tidal zone and a 50 percent reduction

in the applied moment. The lower abscissa relates reliability to time and to

a higher corrosion rate of 0.4 mm/yr.

Analysis

58. This case study is handicapped by the lack of access to all the data

available to Buslov. Nevertheless, the procedure used provides a rational

evaluation of the structure's reliability. Examination of Figures 21 and 22

48

indicates that the tidal zone is more likely to be the location of maintenance

problems, even though the underwater zone has a larger load. This conclusion

is consistent with that of Buslov. The reliability levels of these figures

quantify the conservatism of the deterministic method he used.

TIME, YEARS

20 40 60 80 100 120 '140

0.6 >-..... -' -m <X 0.4 -' w 0.::

0.2

o.o-r-.~~~~+-.--r~-,--~T-~~-r-4~------~~

0.0

Figure 21.

0.1 0.2 0.3

€, in. (CORROSION RATE• 0.07 in./yr.)

Reliability of sheet piles; M , underwater zone, Kishon max

49

0.4

TIME, YEARS

10 20 30

>-.... 0.6 ...J

m ex ...J w 0.4

1.0 I I I

.............._

1\ -

- \ - \

----------·-

0.8

a; . ~ - ~

0.2

0.0 -r I T I J ( I ( I I I I I I I I I I I I I I f

0.0 0.1 0.2 0.3 0.4 0.5 0.6

£,in. (CORROSION RATE • 0.4mm/yr.)

Figure 22. Reliability of sheet piles (1/2 Mmax and tidal zone), Kishon

50

PART VI: SUMMARY AND RECOMMENDATIONS

59. The reliability function of an existing structure serves as an

index of its structural condition. It explicitly reflects changes in resis­

tance that have accumulated during service as well as changes in loading

imposed by operational conditions.

60. This report has documented the development of a straightforward,

generally applicable, three-step procedure for evaluating structural relia­

bility. The margin of safety is assumed to be normally distributed, and point

estimates are used to calculate the moments; these are standard probabilistic

methods. Its computational details resemble familiar design procedures and

are thus attractive to practicing civil engineers.

61. The procedure has been applied to three different types of sheet

pile structures representative of those maintained by the Corps. Emphasis has

been placed on using the available and familiar formulation but in a proba­

bilistic manner. The results indicate that rehabilitative cost avoidances may

be achieved if the procedure is used in lieu of traditional deterministic

practices.

62. The three-step procedure developed herein is recommended for use 1n

evaluating the safety of existing sheet pile and other metal structures oper­

ated and maintained by the Corps of Engineers. The reliability of structures

as calculated by this procedure should be used as a component of the condition

index of the information system under development to manage REMR activities in

Corps field offices.

63. The procedure could be adapted to the particular characteristics of

concrete structures to establish a c_on.sl..st..ent- e_valnative_ condi-tio-n- ind-ex-- fo-r

these features.

51

REFERENCES

American Iron and Steel Institute. 1978. "Proposed Criteria for Load and Resistance Factor Design of Steel Building Structures," AISI Bulletin No. 27.

Benjamin, J. R., and Cornell, C. A. 1970. Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill Book Company.

Bhattacharyya, Gouri K., and Johnson, Richard A. 1988. Statistical Concepts and Methods, John Wiley & Sons.

Bower, J. E. 1973. "Predicted Pull Out Strength of Sheet Piling Interlocks," Proceedings of the American Society of Civil Engineers, Vol 99, No. SM10.

Buslov, V. 1983. "Corrosion of Steel Sheet Piles in Port Structures," Journal of Waterway, Port, Coastal and Ocean Engineering, American Society of Civil Engineers, Vol 109, No. 3, ASCE, pp 273-295.

Department of the Army. 1984. Corrosion Survey Along Point Pleasant Canal Bulkhead.

Granade, C. J., Jr. 1980. User's Guide: Computer Program fo~ Structural Design of Miter Gates (CIMITER), Instruction Report K-80 (Interim Report), U.S. Army Engineer District, Mobile, AL.

Headquarters, U.S. Army Corps of Engineers, Department of the Army. 1984 (Feb). "Engineering and Design, Lock Gates and Operating Equipment," Engineer Manual 1110-2-2703.

JAYCOR. 1985 (Apr). "Services to Provide a Report on Reliability of Civil Works Metal Structures," Proposal No. 8650-04, JAYCOR, Structures Division, Vicksburg, MS.

Kapur, K. C. and Lamberson, L. R. 1977. Reliability 1n Engineering Design, John Wiley & Sons.

Kay, J. Neil. 1975. "I~terlock Tension in Steel Sheet Piling Interlocks," Journal of Structural Division, American Society of Civil Engineers, Vol 101, No. ST10, ASCE, pp 2093-2101.

l--g]6. -'t-sheet tri-te 1nter-lock -rension; Probabilistic Design, 11

Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol 102, No. GTS, ASCE, pp 411-423.

Kearney, F. 1986. sis of Corrodin Sheet Pile Structures O'Brien Lock and Dam, Technical Report M-86 12, Army Construction Engineering Research Laboratory, Champaign, IL.

Lacroix, Yves, Esrig, Melvin I., and Luscher, Ulrich. 1970. "Design, Construction and Performance of Cellular Cofferdams," Proceedings of the ASCE Specialty Conference on Lateral Stresses and Earth Retaining Structures, ASCE, pp 271-328.

52

Rosenblueth, E. 1975. "Point Estimates for Probability Moments," Proceedings

of the National Academy of Science, Vol 72, No. 10, pp 3812-3814.

Tennessee Valley Au~hority. 1957. ''Steel Sheet Piling Cellular Cofferdams on

Rock," TVA Technical Monograph No. 75, Vol. 1, Tennessee Valley Authority.

Terzaghi, K. 1945. "Stability and Stiffness of Cellular Cofferdams," Transac­

tions of the American Society of Civil Engineers, Vol 110, No. 2253, pp 1083-

1202.

Tschebotarioff, Gregory P. 1955. Soil Mechanics, Foundations, and Earth

Structures, McGraw-Hill Book Company.

Young, F. E. 1981. Piles and Foundations, Thomas Telford Limited, London,

England.

53

Table 1

Values of Parameters for Miter Girder Example

Deterministic Variables

Variable Value

a 41.32 tn.

A 76.81 sq tn.

d 67.88 sq in.

I 31,700 in. 4

L 740 1n.

t 63.88 in.

Cot 0 3

Random Variables

Variable Mean Standard Deviation

39,600 psi 4,400 psi

w 1,000 lb/in. 200 lb/in.

Table 2

Safety Margins for Miter Girder Example

_f ~ MS p .. li lb/in. .2!.!:.

x1 = 39,600 x2 = 1,000 ms = 8,671

X - a = 35,200 1,000 MS 1+ = 4,271 1 xl

x1+ a = 44,000 1,000 MS1_ = 13,071

xl 39,600 x2+ a = 800 MS2+ = 14,856

x2 39,600 X - a = 1,200 MS

2_ = 2,485

2 x2

54

y,

Variable

KA

h' 1n.

Table 3

Random Variables Affecting Reliability of Interlocking System

Variable Mean Standard Deviation

KA 0.45 0.125

s ' tn. 468 20

lb/cu in. 0.0723 0.08 y = w 0.0029

Table 4

Random Variables Affecting Reliability of Tie Rods

Mean Standard Deviation

0.45 0.129

444 20

y' lb/cu in. 0.0723 0.0029

s ' tn.

R

a psi u,

Variable

E, psi

oy' ps1

y, lb/cu ft

s, 1n.

h, tn.

520

1.4

60,000

Table 5

Random Variables Affecting Reliability of Sheet Piles Against Buckling

Mean

30,000,000

36,000

0.0723

0.45

1.4

520

444

55

20

0.333

6000

Standard Deviation

3,000,000

3,600

0.0029

0.129

0.333

20

20

Variable

H, ft

y, lb/cu ft

y', lb/cu ft

a, ft

aall' psl

~·:see Figure 17.

Table 6

Random Variables Affecting Reliability of Point Pleasant Bulkhead

Mean

10.0

0.4

125.0

70.0

0.0

24,000

( Eq 41)

Standard Deviation

1. 5.':

1.25:t:

480:j::j:

0.2 x mean

**Kay, J. Neal, "Sheet Pile Interlock Tension; Probabilistic Design," Journal

of Geotechnical Engineering, American Society of Civil Engineers, Vol 102,

No. GTS (ASCE, 1976), pp 411-423 :j:Corrosion Survey Along Point Pleasant Canal Bulkhead (Department of the

Army, 1984). :j::j:Tschebotarioff, Gregory P., Soil Mechanics, Foundations, and Earth

Structures (McGraw-Hill Bqok Company, 1955).

56

APPENDIX A: MICROCOMPUTER PROGRAM FOR ESTIMATING THE RELIABILITY

OF EXISTING STRUCTURES

This appendix provides a microcomputer program for estimating the relia­bility of existing structures. The program is written in BASIC for an IBM PC or compatible computer. Translation onto other hardware or into other soft­ware is straightforward. The main program in lines 10 through 500 implements the general procedure. The subroutine in lines 1000 through 1130 specifically addresses the safety margin for the tutorial miter girder example. The data statements in lines 2000 through 2020 contain the input values for the ex­ample. Other problems can be solved by replacing the subroutine and the data statements. The execution of the program for the tutorial example follows the program listing.

Al

10 REM RELIABIL.BAS 20 REM Paul Mlakar,PhD,PE 30 REM JAYCOR 40 REM 2732 Washington St 50 REM Vicksburg,MS 39180· 60 REM <601) 634-6361 70 REM 21 Aug 85 80 REM IBMPCXT PROGRAM ESTIMATES RELIABILITY USING J650-85-003/1321 94 ' INITIALIZE AND INPUT 100 CLEAR 130 READ F:t,N 140 DIM X <N> ,M<N> ,S<N> ,X$<N> 145 READ X:i-<0> 150 FOR I=1 TO N 160 READ X$< I>, M <I> , S (I) 170 X <I) =M <I> 180 NEXT I 190 GOSUB 1000 200 YB=Y. 210 'COMPUTE EACH INDEPENDENT RANDOM VARIABLE'S CONTRIBUTION USING PEPM 220 CLS 230 KEY OFF 240 PRINT "RELIABIL.BAS",DATE$,TIME:t 250 PRINT 260 PRINT F$ 270 PRINT 280 PRINT "Xi","MXi","SXi","MYi","VYi" 290 M<0>=1 300 S<0>=1 310 FOR I=1 TO N 320 X<I>=M<I>-S<I> 330 GOSUB 1000 340 YM=Y 350 XCI>=MCI>+S<I> 360 GOSUB 1000 370 YP=Y 380 X C I > =M < I> 390 MY=CYM+YP)/2/YB

-400 VY=<YP-YM)/(YP+YM> 410 PRINT X$ ( I > , M < I > , S (I > , MY, VY 420 M<O>=M(0)iii'1Y 430 S<O>=S<O>*Cl+VY*VY> 440 NEXT I 450 'COMPUTE TOTAL RESULT OF ALL RANDOM VARIABLES FOR Y=MS 460 PRINT. 470 M<O>=M<O>*YB 480 S<O>=SQRCS<0>-1> 482 'COMFVTE RELIABILITY ASSUMING NORMALITY 484 Z=l!S<O> 485 S<O>=M(O)*S<O> 486 R=1-.5/(1+.196854*Z+.115194*Z*Z+.000344*ZA3+.019527*ZA4)A4 488 IF MCO><O THEN LET R=1-R 49(1 PRINT "Y", "1'1Y", "SY", "YM", "F(" 495 PRINT Xt<O>,M<O>,S<O>,YB,R 500 END

A2

...

1000 'SUBROUTINE RETURNS Y=MS FOR MITER GIRDER EXAMPLE 1010 ,:.i=41. 32 1020 AR=76.81 1030 D==67.88 1 040 I Nc-.:31 700 1 0~50 L:::: 7 i~(l 1060 T=6:3. 88 L ')"70 CCJ:::3 1080 FY==X<l> 1090 W:X(2) 1100 FA=W/AR•<LI2*CO+T> 1110 FB=W•<T-A>/2~IN•<L~L/4-L*A*CO+CT-A>A2-A*A> 1120 ·'f=FY--FA-FB 1130 F~ETURN 2000 DATA "Miter- G1r-der- E:-:ample" ,2, "MS" 2010 DATA "fy psi",39600,4400 2020 DATA "w lb/in",1C>00,200

A3

RUN RELIABIL.BAS 08-21-1987

Miter Girder E~ample

Xi 1"1X i fy psi .39600 w lblin 1 (H)i)

( MY ,.IS 8670.556 0

11:58:27

SXi MYi VYi 4400 1 .5074647 200 1 -.7134364

SY YM R 8214.58 8670~554 .854171

~

A4

a

I

w, L, e,t

MS

X

X

X

MS

ms

R

APPENDIX B: VARIABLE LIST

axial

flexural stress

area of girder

moment of inertia of girder

Figure 3

margin of safety

arbitrary elevation

generic input random variable (Eq 4 through 10)

mean of X

standard deviation of X j •

mean of margin of safety

standard deviation of margin of safety

margin of safety calculated ustng mean value of each

variable

reliability (Eq 11) radius of circular cell (Eq 15) (O'Brien: 316 in.)

interlock strength

B1

a u

N

d

--s

h

force/unit length between interlock finger and thumb

force/unit length between interlocked thumbs

thickness of web (O'Brien: initially 0.375 in.)

thickness of thumb (O'Brien initially 0.35 in.)

thickness of finger (O'Brien: initially 0.50 in.)

eccentricity of N1 (O'Brien: 0.48 in.)

eccentricity of N2 (O'Brien: 0.48 in.)

eccentricity of finger (O'Brien: 0.56 in.)

yield strength of sheet pile (O'Brien: 60,000 psi)

hoop tension (interlock load)

difference between internal and external pressures on

the cylindrical wall

elevation of the bottom of the chamber (O'Brien:

198 in.)·

-el--evat-ion -of -the -pot"e -Wat-er inside -t-he -e-ell ·(variable)

(Eq 16)

elevation of backfil saturated water (variable)

(Eq 28)

elevation of the water in the chamber (O'Brien:

198 in., dewatered)

B2

H

y

y'

e:

t

s

. N'

a

F

p

b

p

elevation of the top of the fill (O'Brien: 504 in.) (Eq 16) elevation of top of wall (O'Brien: 552 in.) (Eq 28)

total unit weight of the cell fill (variable)

submerged unit weight of the cell fill = y - Yw

unit weight of water (0.0361 lb/cu in.)

coefficient of horizontal earth pressure (variable)

the amount of corrosion

time

shearing stress

maximum shearing stress

tension transmitted to the cross wall

Figure 9a

Shear force in a cross wall bottom of a chamber probability (Eq 18)

tension in a tie rod (Eq 28)

Figure 9c distance top of bulk head to point of tie back attachment (Eq 37)

net lateral pressure acting on the cross wall

B3

X

L

a

A

a . y

a u

D

w

t

E

I

an arbitrary elevation

average distance between cross walls (O'Brien:

316 in.) (Eq 22, 23, 24)

distance between two adjacent tie rods (O'Brien:

84 in.) (Eq 28)

transverse sheer (variable) (Eq 25)

yields point of steel (Eq 34)

shear strength

initial thickness (O'Brien: 0.375 in.)

elevation of wale (O'Brien: 444 in.)

strength of tie rod

cross-sectional area of tie rod

ultimate tensile stress of the steel

diameter of the rod

loading moment, local buckling of sheet piles

maximum allowed

Fi~ure 11 (O'Brien: 18 in.)

initial thickness (O'Brien: 0.5 in.)

modulus of elasticity

moment of inertia (O'Brien: 18.37 cu in.) ~

B4

c

K e:

the effect of corrosion on moment of inertia O'Brien: 55 cu in.)

Figure 11 (O'Brien: 5.5 in.)

Ap anchor pull

H, b, h, L, a Figure 17 (Eq 37)

f" effect of wall friction on reduction of active earth pressure

f'

z

M

s

K

S(e:)

uncertainty in relative importance of passive earth ·pressure above anchor, and tensile strength of sand saturated by capillary action above water level.

location of zero shear

maximum moment

section modulus

initial value of section modulus

reduction factor

reflects the effects of corrosion on section modulus

B5

APPENDIX C: GLOSSARY OF STATISTICAL TERMS

CUMULATIVE DISTRIBUTION FUNCTION: The ·cumulative distribution function 1s denoted F(x), and has the properties:

(1)

(2)

(3)

Lim F(x) = 1; x•~

Lim F(x) = 0 x•~

F(x) is a nondecreasing function.

MEAN: The sample mean or average of a set of n measurements x 1 , x 2 , ••• ,xn is the sum of these measurements divided by n. The mean is denoted by x, which is expressed operationally

X =

n I x.

i=l 1

n

NORMAL DISTRIBUTION: A normal distribution has a bell-shaped density

(x-p)2

l e

2n a

.with a mean= panda standard deviation= a.

The probability of the interval extending

(1) one standard deviation each side of the mean: P[p - a < x < p + a] = 0.683

(2) two standard deviations on each side of the mean: P[p - 2a < x < p + 2a] = 0.954

(3) three standard deviations on each side of the mean: P[p - 3a < x < p + 3a] = 0.997

C1

1.1 - 11 1.1 1.1 + o 1.1 + 2o

Figure C1. Normal distribution. (Source: Statistical Concepts and Methods, G. K. Bhattacharyya and R. A. Johnson, © 1988, John Wiley & Sons.)

Used with permission

POINT ESTIMATE: A statistic intended for estimating a parameter e is called a

point estimator, denoted e. An estimator e is purely a function of

the sample observations, and its value can be readily computed once

the sample 1s observed. The numerical value is called a point

estimate from a particular sample.

RANDOM VARIABLE: A random variable x is a numerical valued function defined

on a sample space. In other words, a number X(e), providing a measure

of the characteristic of interest, is assigned to each simple event,

e, in the sample space.

STANDARD DEVIATION: The sam~le standard deviation is the square root of the

variance and has the same units as the mean

s = variance =

C2

n

I - 2 . 1(x.- x) 1= 1

n - 1

VARIANCE: The sample variance, s, is the variation of individual data points about the mean. The dispersion of a set of n measurements x 1 , x 2 , ••• ,

xn is defined as

n I - 2 .

1(x.- x)

2 1= 1 s = ~~---n - 1

CJ

.. ·•