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Reliability-based design of sheet pile structures
Reliability Engineering and System Safety 33 ( 1991) 215-230
i Reliability-Based Design of Sheet Pile Structures A d n a n A. Basma Department of Civil Engineering, Jordan University of Science and Technology, lrbid, Jordan (Received 12 January 1990; accepted 30 May 1990)
A BSTRA CT This paper concerns the use of the conventional Coulomb theo O' of lateral earth pressure against retaining structures with some empirical modifications in the design of cantilever sheet piles. The design procedure adopted basically involves the computation of the embedment depth and the maximum bending moment in the sheet pile. Design charts are developed for a wide range of soil types, water level heights and loading conditions. In order to portray the actual conditions on the sheet pile, the design ./'actors, namely the soil properties, height of water level and surcharge loads, are treated as random variables. Using the developed design charts and through a first-order Taylor's series expansion, the variation in the embedment depth as well as the maximum moment are evahtated and presented in a graphicalform. Thefinal design process entails the combination of the mean and variance of the design requirements based on a given probability of failure (reliability). Examples to illustrate the use of the reliability-based design method are presented.
INTRODUCTION Cantilever sheet piles are widely used in engineering practice as earth retaining structures. Generally speaking, they are composed of a single row of parallel embedded sheet piles. There are several types of such walls that are in common use: (a) wooden, (b) precast concrete and (c) steel sheet piles. Because of their resistance to high-driving stresses developed when being driven into hard soils, steel sheet piles are the most convenient. In the United States they are about 0-4-0.5 inches thick. European sections may be thinner or wider? '2 215 Reliability Engineering and System Safeo' 0951-8320/91/$03"50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
216
Adnan A. Basma
Sheet piles derive their support solely by passive pressure on the front of the embedment portion of the pile below the dredge line. The magnitude of the earth pressure exerted by the soil on the sheet pile structure is not unique. It depends on the physical properties of the soil and their variations, the deformation pattern of the soil structure, surcharge loads and the fluctuation of the water front level. While classical methods of designing soil structures were developed analytically, other methods, based on field observations and model test results 3-5 have received much attention. In addition, and due to the large variations encountered in soil properties and other natural phenomena, probabilistic techniques have also been growing in u s e . 6 - 9 Hence, for designing soil structures and, for the purposes of this paper, sheet piles, the use of conventional deterministic design would seem inappropriate. However, the inclusion of probabilistic concepts may provide a better and more viable design method. The general purpose of this work is to extend on the existing Coulomb method of lateral earth pressure in order to design sheet piles by incorporating the soil properties, water front level and the surcharge loads. The design is evaluated through a predetermined wall reliability. The procedure presented here therefore uses the concept of reliability (or probability of failure) rather than the classical method of arbitrarily assigning a design safety factor. GENERAL FRAMEWORK Using the Coulomb method, the lateral earth pressure on the sheet pile is determined with (1) granular soil (sand) and (2) cohesive soil (clay) below the dredge line. The backfill in both cases is considered to be granular soil. Through a mathematical analysis, regression equations are obtained to determine both the depth of penetration and the maximum moment on the wall for a wide range of soils, water levels and surcharge loads. The variation in the design, expressed as the variance, is determined by Taylor's series expansion about the mean. The final design process entails the combination of the mean and the variance, or standard deviation, of the embedment depth and the maximum moment with a predetermined reliability. MATHEMATICAL F O R M U L A T I O N
(a) Cantilever sheet pile in granular soils The simplest solution to this problem is to assume that the wall is subjected to active pressure on the backfill and passive below the dredge line. Figure
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I. Cantileversheet pile pressure diagram for (al granular soil and (b) cohesive soil.
l(a) shows the net overall pressure on the wall. In this figure the actual wall height, H', is artificially increased by h e = q/7, where q is the surcharge load and 7 is the unit weight of the backfill above the water level. The new wall height, H, used in the design is thus (H' + he), with the surcharge replaced by an equivalent depth of backfill soil. The embedment depth, D, is calculated by the following equation:
D=h+d where
h =PED/A PEP = pressure at point E on the wall A = 7'/K 7' = effective unit weight of the sand in the backfill
(1)
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Adnan A. Basma
K=Kp-K~ Kp = passive earth pressure coefficient, calculated with a wall-soil friction angle 6 = 0 ° Ka = active earth pressure coefficient, calculated with 6 = 0 ~ and d is calculated by the following fourth-degree equation: t°
d4 - d3p ff A - d2(8P/A) - d[~--£ (25A + p'p)] - (6P.:p'p + 4p2)/A2 = O Here P = area A C D F E B P
p
pp = h,rp + ( H - h,)7 rp - 7'hKp It is important to note that the values o f K, and Kp are calculated with 6 = 0 ° since this value produces the m a x i m u m K~ and the m i n i m u m Kp which control the design o f sheet piles. The m a x i m u m moment, Mmax, is calculated at a point where the shear on the wall is zero. This point (shown in Fig. 1 as K) is where the shaded area F K L is equal to P. The m a x i m u m m o m e n t is thus calculated by the following equation: ~ Mma x :
P(2 +
(2)
Z') - - ~[½7'2'2K]
where : ' = ( 2 P i g } , ' ) 1,2
Note that in the above calculations the granular soil is assumed to have an angle o f internal friction ~b and a cohesion c = 0.
(b) Cantilever sheet pile in cohesive soils The net pressure diagram for sheet piles driven into cohesive soils is shown in Fig. l(b). Following the analysis as above, the e m b e d m e n t depth, D, is calculated by the following equation:~° D2( 4 c - PED) - - 2 D P
P(12c5 + P) 2c + PED
where c = cohesion P = area A C D E B PEt) = pressure at point E to the right o f the wall
= 0
(3)
Reliability-based design of sheet pile structures
219
The maximum moment is calculated by the following equation:t° Mmax = P ( : ' + 5 ) - pME:'2/2
(4)
where PME = pressure at point E to the left of the wall ='= P/P,~E
In the above calculations the clay below the dredge line is assumed to have a cohesion c and a total unit weight 7v, with ~b= 0.
DESIGN EQUATIONS A N D CHARTS Utilizing the mathematical analysis just presented, a computer program was developed with eqns (1), (2), (3) and (4) to give design solutions for several (1) granular soils and (2) cohesive soils below the dredge line. Tables 1 and 2 list the various soil properties used for granular and cohesive soils, respectively. 12 The wall heights used in the design were 5, 10, 20, 30, 40, 50 and 60 feet, whereas the water front heights were 0 ÷, 0-2, 0.4, 0"6, 0.8 and 1.0H. The computer output provided all the possible design solutions for the embedment depth, D, and the maximum moment, Mm~x. To develop the design equations and due to the large number of data, a regression analysis was performed using the statistical package for social science (SPSS) program, ta The best-fit model was adopted as the design equation. The final results obtained follow. TABLE I Properties of Granular Soil Soil O'pe
dp (deg)
;,
(Ib/ft3) (kN/m 3)
Very loose sand
25
Loose sand
30
Medium sand
35
Dense sand
40
Very dense sand
45
90-110 (14.4-17.6) 100-!20 (16.0-19"2) 110-125 (17"6-20-0) ! 15-130 (18-4-20-8) 120-140 (19.2-22-4)
220
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Properties of Cohesive Soil Soil type
Soft clay Firm to soft clay Firm clay Stiff to firm clay Stiff clay Very stiff clay
(" {k ip3" _/72) (kPa)
)' {lb/~?3 ) (kN/m 3)
0-50 124) 0-75 136) 1.00 1481 2-00 061 3.00 ( 144t >4.00 (> 1 9 2 1
90-105 114-4-16-8) 95-I 15 115"2-18-4) 105-120 (16'8-19-2) 110-130 ( 17.6-20.8} 115-135 ( 18.4-2 I-6) 120-140 (19.2-22'4)
For sand below the dredge line
(5)
log :t = 0.344 + 0.212 log fl - 0"60 tan r 2 ----0"972 SEE = 0"113 and log Mm. x = - 1 " 0 1 + 0"46fl + 2"68 l o g H r 2 = 0"956 SEE = 2"987
tan¢
(6)
and for clay below the dredge line log :t = -0"46 + 0-38fl - 0"5 log c r 2 = 0"951 SEE = 0"191
(7)
and log Mm. x = 0"36 + 0"46fl + O ' 0 5 1 H - 0"115c r 2 = 0"965 SEE = 2'112 where ct = D / H fl -----h w / H
D= hw = H = = Mmax = c=
e m b e d m e n t depth below dredge line (feet) height o f water level from the top o f the wall (feet) corrected wall height (feet) angle o f internal friction (deg) m a x i m u m m o m e n t on the wall (k. ft/ft) cohesion (ksf)
(8)
Reliability-based design of sheet pile structures
221
It should be pointed out that eqns (5), (6), (7) and (8) are considered to be the mean value equations, and thus bars must be placed over the terms to indicate their means.
Variations in the design of sheet piles To determine the variation of x and Mmax, dependent variables, the variations of the design inputs, independent variables, must be assessed. Using eqns (5), (6), (7) and (8), and through a Taylor's series expansion about the mean, the variation of the dependent variables can be evaluated in terms of the variations of the independent variables. An explanation of this technique follows. Consider the following relation: (9)
Y = f ( X I, X 2, X 3 . . . . . X n)
In this equation Y is the dependent variable whereas Xt, X 2, X 3. . . . . Xn are the independent variables. Expanding the relation in eqn (9) in a Taylor's series about the mean and truncating at the linear terms will yield the following: ? = f ( X I , X2, X3 . . . . . X.) (I0) and
i=1
where bars are used over the terms to indicate their means and tr 2 to indicate their variance. It should be reiterated that eqn (11) is applicable to uncorrelated random variables, Xi. Furthermore, this method of estimating the mean and variance of random variables has proven to be effective (within _+10%) for actual values, especially when the independent random variables have a small coefficient of variation (CV_< 30%) and a well-behaved function near the mean. t4-t7 To evaluate the variance of~ and Mm.~x,eqn (11) is applied in eqns (5), (6), (7) and (8), and the results are as follows. For sand 0",,2
and
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~2um~x= 0-01 exp 2 [ 1 . 0 7 ~ - 2.3 tan 4q 5 ~
+0.22CV~
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(12)
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223
Reliability-based design of sheet pile structures
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(14)
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(15)
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224
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Reliability-based design of sheet pile structures
225
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clay below the dredge line. PROBABILISTIC DESIGN A N D RELIABILITY OF THE SHEET PILE WALL
Selecting the most economical and safe design is the goal of all engineers. Generally speaking, conventional methods of design, especially in the field of geotechnical engineering, tend to select arbitrary safety factors based on the mean value of the design parameters and the predicted risk involved. These safety factors are in many cases very high, thus producing uneconomical designs, or low, hence resulting in unsafe designs. However, and in reality, every measurable quantity varies and, therefore, strict reliance on their mean values would present a problem to the designer, and more so if these variations are high. In such cases probabilistic techniques become handy since they carefully evaluate each design input with great consideration given to their mean and variation, while deterministic designs rely on arbitrarily assigning a safety factor. The former uses the concept of the probability that a design will fail (or reliability) rather than the safety factor. In order to evaluate the reliability of a given design of a sheet pile, one must first assess the variations of the input variables. These variations are usually expressed by the mean and variance (or standard deviation). The reliability of the design is thus evaluated from an appropriate distribution. Conversely, the design satisfying a given reliability is evaluated from the probability distribution. Commonly used distributions include normal, lognormal, and extreme types I, II and III. '4"' s Ifthe interest in the design is the
226
Adnan A. Basma
average values, the normal and/or log-normal are usually used. On the other hand, if the interest is the extremes (maximum or minimum) in the design, an extreme type is used. The two parameters of interest in the design of sheet piles are (1) the embedment depth and (2) the maximum moment. Since a critical situation arises when the smallest value of an embedment depth is selected, then the most appropriate choice of distribution would be an extreme value for minima. Yet, another critical situation will arise when the largest maximum moment occurs on the wall. Thus, the use of an extreme value for maxima would seem reasonable to describe the maximum moment. Suffice it to point out that the most critical condition will be when the smallest embedment depth is selected with the maximum moment being largest on the wall. A low probability of failure (or high reliability) in this case will give the safest design. To evaluate the best design of a sheet pile with a predetermined reliability the following assumptions are made: (a) the design inputs are independent random variables; (b) the distribution of the embedment depth is type I minima whereas that for the maximum moment is type I maxima. The first assumption can be easily verified and with the argument just presented the second would seem viable. Additionally, the extreme type I distribution has been widely used in many structural designs. 1")'~5 The reliability R of a design can be defined as follows: (16)
R = P ( X < x)
or, in other words, the reliability R is the probability that the random variable Xis less than or equal to a given selected value x (the design value). Alternatively, the probability of failure pf = 1 - R. Observe that if the distribution of X is known then R = F(X) or the cumulative probability (see Fig. 10).
X~ x.]= F(x)
Fig. !0. Design reliability with known distribution.
Reliability-based design of sheet pile structures
227
Since the e m b e d m e n t depth, D, and thus ~t is a type I minima, then R~ = F(~) = 1 - exp { - e x p [a~(~ - u~)]j
(17)
where R~ = reliability o f a selected value o f ~t and thus D a, = 1.282/a~ u~ = 5 + 0.577/a~ and for the m a x i m u m m o m e n t with a type I maxima RM = F ( M ) = exp { - e x p [ - - a M ( M -- %)]}
(18)
where R u = reliability o f a selected design m a x i m u m m o m e n t
am = 1"282/au u m = .,V"I- 0 . 5 7 7 / %
Normalizing :t and M with respect to the mean and standard deviation, and substituting in eqns (17) and (18), yields R, = 1 - exp { - e x p [1.282(z, - 0.45)]}
(19)
RM = exp {--exp [-1"282(zM + 0"45)]}
(20)
and where -
=
-
"¢'2
zM = ( M - .Q')/o"M Figure 11 is a graphical presentation o f eqns (19) and (20). 100
''
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99h, 1.6
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4
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5
1
228
Adnan A. Basma
E X A M P L E S O N T H E P R O B A B I L I S T I C D E S I G N O F S H E E T PILE The following examples are presented to illustrate the use of the design charts.
Example 1: Sheet pile with sand below the dredge line. Consider the inputs below: Input
Variable
Mean
CV
a
a2
1
H'
--
--
--
2
~
15ft (4.56m) 352
0-12
4-2
17.6
3
7
--
--
--
4
q
0-17
61"2
5
h,,
1201b/ft 3 (19"2 k N / m 3) 360 Ib/ft z (17"3 kPa) 10 ft [3"04 m)
0'3
3
3 745.4 9
Calculations: h e = 360/120 = 3 ft. The variance o f h e can be evaluated by eqn (11), and a2e=3745-4/(120)2=0.26. H = H ' + h e = 1 8 f t , and since H ' is constant, therefore tr2 = 0"26 and CV H = (0x/-0~/18 = 0-028. fl = 10/18 = 0"56 and by eqn (11) trp = 0"167; thus CVa = 0-30. With the inputs calculated above use Figs 2 and 6 to determine the mean o f ~ and try: ~ = 0.75 and tr~ = 0"13. With a reliability R = 99%, -~ = 1.64 (Fig. 11), and the design value ~t = 0.75 + (1.64)(0.13)=0"963 and thus D = 0.963 x 18 = 17.3 ft (5"26 m). F r o m Figs 4 and 8 determine the mean and standard deviation of Mm~,x: M m , x = 8 0 k . f t / f t , X = 10000 and Y=0"2; thus O ' M m a x = ( 1 0 0 0 0 x 0"2)t/2----44.7 k. ft/ft. With a reliability o f R = 99%, z M= 3.14 (Fig. 11), and the design value Mm~x = 80 + (3.14X44.7) = 220.42 k. ft/ft (980.9 k N . m/m).
Example 2: Sheet pile with clay below the dredge line. Consider the same example as above but replacing input 2 by the following: Input
Variable
Mean
CV
tr
tr'-
2
c
1-5 ksf (72 kPa)
0"6
0-9
0-81
In a similar fashion use Figs 3 and 7 to determine the mean o f ~ and try: ~ = 0 - 4 5 and try= 0-20, and with R = 9 9 % the design value o f ~t= 0-45 + (1-64)(0.20) = 0.78 and thus D = 0.78 x 18 = 14-0 ft (4-27 m).
Reliability-based design of sheet pile structures
229
From Figs 5 and 9 Mm',x= 23 k. ft/ft, X = 90 and Y= 1"00, and thus (90 X 1"0) t/2 = 9.49. With R = 99% the design Mm.,x= 23 + (3"14X9-49) = 52"8 k. ft/ft (310-6 kN. m/m).
O'Mmax =
It should be noted that in the above examples a reliability of 99% (Pr = 10 -2) was used. This is a value that has been recommended by many researchers, t4"t5 especially in the field of geotechnical engineering. 6 However, in some cases a higher reliability value (99.9%, Pr = 10- 3) must be used. In any case, the author does not recommend a value less than 99% in the design of sheet piles. Furthermore, observe that in Example 1 the design ratio, ~t, is 0.963, which may be satisfactorily based on the charts but would be uneconomical and thus anchorage is recommended. Finally, it is important to point out that since the soil properties c, q~ and ~ are random variables so are parameters that are related to them, in particular Ka and Kp, and thus the soil lateral pressure on the wall. Furthermore, the centroid of force due to the soil lateral pressure changes, since the adj usted height of the wall, H, varies with the applied surcharge load, q.
SUMMARY This paper presented a means of designing sheet piles by the introduction of probabilistic techniques. While conventional design methods use the concept of the safety factor, probabilistic design uses reliability. The design process presented in this paper provides a better and more comprehensive approach to designing soil structures, in general, and sheet piles, in particular.
REFERENCES 1. American Institute of Steel Construction, Manual of Steel Construction, 8th edn, Chicago, 1981. 2. British Standards Institution, Foundations, Ciril Engineering Code of Practice, CP 2004, London, 1972. 3. Rowe, P. W., Theoretical and experimental analysis of sheet-pile walls. Proceedings of Institution of Civil Engineers, London, 4(I) (1955) 32-69. 4. Richards, F. E., Analysis of sheet-pile retaining walls. Transactions of American Society of Civil Engineers, 122 (1957) 1113-32. 5. Nataraj, M. S. & Hoadley, P. G., Design of anchored bulkheads in sands. American Society of Civil Engineers, Journal of Geotechnical Engineering, 110(4) (April 1984) 505-15. 6. Harr, M. E., Mechanics of Particulate Media: A Probabilistic Approach. McGraw-Hill, 1977.
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7. K rahn, J. & Fredlund, D. G., Variability in the engineering properties of natural soil deposits. In Proceedings of the 4th International Conference on Application oJ"Probability and Statistics and Probability in Soil and Structural Engineering, Italy, 1983, pp. 1017-29. 8. Rethati, L., Asymmetry in the distribution of soil properties and its elimination. In Proceedings of the 4th International Conference on Application of Probability and Statistics and Probability in Soil and Structural Engineering, Italy, 1983, pp. 1057-68. 9. McGuffy, V. J., lori, Z. K. & Athanasiou-Grivas, D., Statistical geotechnical properties of Iockport clay. Transportation Research Board Report TRR 809, ! 981. pp. 54-60. 10. Bowles, J. E., Foundation Analysis and Design, 4th edn. McGraw-Hill, 1988. 11. Das, B. M., Principles of Foundation Engineering. Brooks/Cole Engineering Division, 1984. 12. Winterkorn, H. F. & Fang Hsai-Yang, Foundation Engineering Handbook. Van Nostrand Reinhold Company, 1975. 13. Nie, N. H., Hull, C. H., Jenkins, G. M., Steinbrenner, K. & Bent, D. H., SPSS: Statistical Package for the Social Sciences, 2nd edn. McGraw-Hill, 1975. 14. Benjamin, J. R. & Cornell, C. A., Probabilitj, Statistics, and Decision for Ciril Enghleers. McGraw-Hill, 1970. 15. Ang, A. H. S. & Tang, W. H., Probability Concepts in Engineerhlg Planning and Design, Vols I and II. John Wiley, 1975. 16. Bennett, R. M., Comments on the first order ~'s. second order reliability analysis of series structures. Structural Safety Journal, 4 (1987) 241-2. 17. Dolinski, K., First order second moment approximation in reliability of structural systems: critical review and alternative approach. Structural Safety Journal 3 (1983) 21 !-31.
i Reliability-Based Design of Sheet Pile Structures A d n a n A. Basma Department of Civil Engineering, Jordan University of Science and Technology, lrbid, Jordan (Received 12 January 1990; accepted 30 May 1990)
A BSTRA CT This paper concerns the use of the conventional Coulomb theo O' of lateral earth pressure against retaining structures with some empirical modifications in the design of cantilever sheet piles. The design procedure adopted basically involves the computation of the embedment depth and the maximum bending moment in the sheet pile. Design charts are developed for a wide range of soil types, water level heights and loading conditions. In order to portray the actual conditions on the sheet pile, the design ./'actors, namely the soil properties, height of water level and surcharge loads, are treated as random variables. Using the developed design charts and through a first-order Taylor's series expansion, the variation in the embedment depth as well as the maximum moment are evahtated and presented in a graphicalform. Thefinal design process entails the combination of the mean and variance of the design requirements based on a given probability of failure (reliability). Examples to illustrate the use of the reliability-based design method are presented.
INTRODUCTION Cantilever sheet piles are widely used in engineering practice as earth retaining structures. Generally speaking, they are composed of a single row of parallel embedded sheet piles. There are several types of such walls that are in common use: (a) wooden, (b) precast concrete and (c) steel sheet piles. Because of their resistance to high-driving stresses developed when being driven into hard soils, steel sheet piles are the most convenient. In the United States they are about 0-4-0.5 inches thick. European sections may be thinner or wider? '2 215 Reliability Engineering and System Safeo' 0951-8320/91/$03"50 © 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
216
Adnan A. Basma
Sheet piles derive their support solely by passive pressure on the front of the embedment portion of the pile below the dredge line. The magnitude of the earth pressure exerted by the soil on the sheet pile structure is not unique. It depends on the physical properties of the soil and their variations, the deformation pattern of the soil structure, surcharge loads and the fluctuation of the water front level. While classical methods of designing soil structures were developed analytically, other methods, based on field observations and model test results 3-5 have received much attention. In addition, and due to the large variations encountered in soil properties and other natural phenomena, probabilistic techniques have also been growing in u s e . 6 - 9 Hence, for designing soil structures and, for the purposes of this paper, sheet piles, the use of conventional deterministic design would seem inappropriate. However, the inclusion of probabilistic concepts may provide a better and more viable design method. The general purpose of this work is to extend on the existing Coulomb method of lateral earth pressure in order to design sheet piles by incorporating the soil properties, water front level and the surcharge loads. The design is evaluated through a predetermined wall reliability. The procedure presented here therefore uses the concept of reliability (or probability of failure) rather than the classical method of arbitrarily assigning a design safety factor. GENERAL FRAMEWORK Using the Coulomb method, the lateral earth pressure on the sheet pile is determined with (1) granular soil (sand) and (2) cohesive soil (clay) below the dredge line. The backfill in both cases is considered to be granular soil. Through a mathematical analysis, regression equations are obtained to determine both the depth of penetration and the maximum moment on the wall for a wide range of soils, water levels and surcharge loads. The variation in the design, expressed as the variance, is determined by Taylor's series expansion about the mean. The final design process entails the combination of the mean and the variance, or standard deviation, of the embedment depth and the maximum moment with a predetermined reliability. MATHEMATICAL F O R M U L A T I O N
(a) Cantilever sheet pile in granular soils The simplest solution to this problem is to assume that the wall is subjected to active pressure on the backfill and passive below the dredge line. Figure
Reliability-based design of sheet pile structures
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q
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.
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~,~::~ ,
'I
t
~D
;! t
Clay ,,c
~
L . . . . .
-
N
~
J
-bFig.
I. Cantileversheet pile pressure diagram for (al granular soil and (b) cohesive soil.
l(a) shows the net overall pressure on the wall. In this figure the actual wall height, H', is artificially increased by h e = q/7, where q is the surcharge load and 7 is the unit weight of the backfill above the water level. The new wall height, H, used in the design is thus (H' + he), with the surcharge replaced by an equivalent depth of backfill soil. The embedment depth, D, is calculated by the following equation:
D=h+d where
h =PED/A PEP = pressure at point E on the wall A = 7'/K 7' = effective unit weight of the sand in the backfill
(1)
218
Adnan A. Basma
K=Kp-K~ Kp = passive earth pressure coefficient, calculated with a wall-soil friction angle 6 = 0 ° Ka = active earth pressure coefficient, calculated with 6 = 0 ~ and d is calculated by the following fourth-degree equation: t°
d4 - d3p ff A - d2(8P/A) - d[~--£ (25A + p'p)] - (6P.:p'p + 4p2)/A2 = O Here P = area A C D F E B P
p
pp = h,rp + ( H - h,)7 rp - 7'hKp It is important to note that the values o f K, and Kp are calculated with 6 = 0 ° since this value produces the m a x i m u m K~ and the m i n i m u m Kp which control the design o f sheet piles. The m a x i m u m moment, Mmax, is calculated at a point where the shear on the wall is zero. This point (shown in Fig. 1 as K) is where the shaded area F K L is equal to P. The m a x i m u m m o m e n t is thus calculated by the following equation: ~ Mma x :
P(2 +
(2)
Z') - - ~[½7'2'2K]
where : ' = ( 2 P i g } , ' ) 1,2
Note that in the above calculations the granular soil is assumed to have an angle o f internal friction ~b and a cohesion c = 0.
(b) Cantilever sheet pile in cohesive soils The net pressure diagram for sheet piles driven into cohesive soils is shown in Fig. l(b). Following the analysis as above, the e m b e d m e n t depth, D, is calculated by the following equation:~° D2( 4 c - PED) - - 2 D P
P(12c5 + P) 2c + PED
where c = cohesion P = area A C D E B PEt) = pressure at point E to the right o f the wall
= 0
(3)
Reliability-based design of sheet pile structures
219
The maximum moment is calculated by the following equation:t° Mmax = P ( : ' + 5 ) - pME:'2/2
(4)
where PME = pressure at point E to the left of the wall ='= P/P,~E
In the above calculations the clay below the dredge line is assumed to have a cohesion c and a total unit weight 7v, with ~b= 0.
DESIGN EQUATIONS A N D CHARTS Utilizing the mathematical analysis just presented, a computer program was developed with eqns (1), (2), (3) and (4) to give design solutions for several (1) granular soils and (2) cohesive soils below the dredge line. Tables 1 and 2 list the various soil properties used for granular and cohesive soils, respectively. 12 The wall heights used in the design were 5, 10, 20, 30, 40, 50 and 60 feet, whereas the water front heights were 0 ÷, 0-2, 0.4, 0"6, 0.8 and 1.0H. The computer output provided all the possible design solutions for the embedment depth, D, and the maximum moment, Mm~x. To develop the design equations and due to the large number of data, a regression analysis was performed using the statistical package for social science (SPSS) program, ta The best-fit model was adopted as the design equation. The final results obtained follow. TABLE I Properties of Granular Soil Soil O'pe
dp (deg)
;,
(Ib/ft3) (kN/m 3)
Very loose sand
25
Loose sand
30
Medium sand
35
Dense sand
40
Very dense sand
45
90-110 (14.4-17.6) 100-!20 (16.0-19"2) 110-125 (17"6-20-0) ! 15-130 (18-4-20-8) 120-140 (19.2-22-4)
220
Adrian .4. Basma TABLE 2
Properties of Cohesive Soil Soil type
Soft clay Firm to soft clay Firm clay Stiff to firm clay Stiff clay Very stiff clay
(" {k ip3" _/72) (kPa)
)' {lb/~?3 ) (kN/m 3)
0-50 124) 0-75 136) 1.00 1481 2-00 061 3.00 ( 144t >4.00 (> 1 9 2 1
90-105 114-4-16-8) 95-I 15 115"2-18-4) 105-120 (16'8-19-2) 110-130 ( 17.6-20.8} 115-135 ( 18.4-2 I-6) 120-140 (19.2-22'4)
For sand below the dredge line
(5)
log :t = 0.344 + 0.212 log fl - 0"60 tan r 2 ----0"972 SEE = 0"113 and log Mm. x = - 1 " 0 1 + 0"46fl + 2"68 l o g H r 2 = 0"956 SEE = 2"987
tan¢
(6)
and for clay below the dredge line log :t = -0"46 + 0-38fl - 0"5 log c r 2 = 0"951 SEE = 0"191
(7)
and log Mm. x = 0"36 + 0"46fl + O ' 0 5 1 H - 0"115c r 2 = 0"965 SEE = 2'112 where ct = D / H fl -----h w / H
D= hw = H = = Mmax = c=
e m b e d m e n t depth below dredge line (feet) height o f water level from the top o f the wall (feet) corrected wall height (feet) angle o f internal friction (deg) m a x i m u m m o m e n t on the wall (k. ft/ft) cohesion (ksf)
(8)
Reliability-based design of sheet pile structures
221
It should be pointed out that eqns (5), (6), (7) and (8) are considered to be the mean value equations, and thus bars must be placed over the terms to indicate their means.
Variations in the design of sheet piles To determine the variation of x and Mmax, dependent variables, the variations of the design inputs, independent variables, must be assessed. Using eqns (5), (6), (7) and (8), and through a Taylor's series expansion about the mean, the variation of the dependent variables can be evaluated in terms of the variations of the independent variables. An explanation of this technique follows. Consider the following relation: (9)
Y = f ( X I, X 2, X 3 . . . . . X n)
In this equation Y is the dependent variable whereas Xt, X 2, X 3. . . . . Xn are the independent variables. Expanding the relation in eqn (9) in a Taylor's series about the mean and truncating at the linear terms will yield the following: ? = f ( X I , X2, X3 . . . . . X.) (I0) and
i=1
where bars are used over the terms to indicate their means and tr 2 to indicate their variance. It should be reiterated that eqn (11) is applicable to uncorrelated random variables, Xi. Furthermore, this method of estimating the mean and variance of random variables has proven to be effective (within _+10%) for actual values, especially when the independent random variables have a small coefficient of variation (CV_< 30%) and a well-behaved function near the mean. t4-t7 To evaluate the variance of~ and Mm.~x,eqn (11) is applied in eqns (5), (6), (7) and (8), and the results are as follows. For sand 0",,2
and
-E
/~°'42exp2[-1-37tant~] 9-2
~2um~x= 0-01 exp 2 [ 1 . 0 7 ~ - 2.3 tan 4q 5 ~
+0.22CV~
+ a~ +
]
(12)
222
Adnan A. Basma 1.0-1.2 --25
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/ I
-30
I
0.5-
1
/
I
i.=
s~ ='35
"o
I0
14;
Im.
"40
0.1.
0.05
45
0.2
0.02-
Fig. 2. Nomograph to determine ~t with sand below the dredge line. 1.0-
-1.2 -0.5
~1.0
'1.0 -0.5 . . . .
..,,,.. - ] c l
0.5-
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d •2 . 0
:3.0 .0.2 L4.0
Fig. 3.
Nomograph to determine ~t with clay below the dredge line. H, ft 40
'
30
'
20
'
10
I
I
4
I I
I
I .... 10 =
f i
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a
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n
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i 5
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[ I 0
Fig. 4.
0
'~ L
m
i .... 10'
a h n u , [z 02
0,4
0.6
,
n , 0.8
J 1.0
Nomograph to determine the maximum moment, M, with sand below the dredge line.
223
Reliability-based design of sheet pile structures
40
30
l
,
H, ft
I
20
,
I
,
10 I :0.5
2.0
11111 i i i 10s
102
10T
3
"M, k.ft/ft
Fig. 5.
] J' h o s o'8 1!o
i , o 02'04
Nomograph to determine the maximum moment, M, with clay below the dredge line.
For clay g2 = (l/~) exp 2 [0.87ff] [0-095g~ -b 0"038 CV 2]
(14)
and g2Mmax= exp 2 [0.12H + 1.06fl- 0.264~] [0-075g 2 + 6g~ + 0-37g~]
(15)
where CV is the coefficient of variation = g/mean. Figures 2 to 9 show n o m o g r a p h i c solutions for eqns (5), (7), (6), (8), (12), (14), (13) and (15), respectively. 0"310"(1
.
y
[
-
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i
,
,
.
,
,
t
,
,
0-1 0.2 0.3 0.4 0-5
¢Vp
Fig. 6. Nomograph to determine the standard deviation of:t, a~, with sand below the dredge line.
224
Adrian A. Basma
-0,65 -0.6
i
"
CVc: 0.7
~,
o.s
0.3 0.1
/o ~"°I //~,\~
o~
.o.s
III ~
.0.4
//~2.0~
0
0-3
, i I I I I i [ 0°5 0.7
/~o °
o-~
0.2
0 Fig. 7. Nomograph to determine the standard deviation of:t, ~ , with clay below the dredge line.
X 101-
I--I.0
io 2_
103.
,
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4
110
.---
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~
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//
',
r, o' ,L o 0"4-- 20
Fig. 8.
Nomograph to determine the standard deviation of the maximum moment, au, with sand below the dredge line.
Reliability-based design of sheet pile structures
225
X Y
- 0.5
0.1 0.5
~-0
,'
~-o.4 1
i0.a
10'l
-1.0
rO
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.3
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-5 OH, ft e~
-6
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._-0 P
-4
7
2.0 Fig. 9. Nomograph to determinethe standard deviationof the maximummoment,aM,with
clay below the dredge line. PROBABILISTIC DESIGN A N D RELIABILITY OF THE SHEET PILE WALL
Selecting the most economical and safe design is the goal of all engineers. Generally speaking, conventional methods of design, especially in the field of geotechnical engineering, tend to select arbitrary safety factors based on the mean value of the design parameters and the predicted risk involved. These safety factors are in many cases very high, thus producing uneconomical designs, or low, hence resulting in unsafe designs. However, and in reality, every measurable quantity varies and, therefore, strict reliance on their mean values would present a problem to the designer, and more so if these variations are high. In such cases probabilistic techniques become handy since they carefully evaluate each design input with great consideration given to their mean and variation, while deterministic designs rely on arbitrarily assigning a safety factor. The former uses the concept of the probability that a design will fail (or reliability) rather than the safety factor. In order to evaluate the reliability of a given design of a sheet pile, one must first assess the variations of the input variables. These variations are usually expressed by the mean and variance (or standard deviation). The reliability of the design is thus evaluated from an appropriate distribution. Conversely, the design satisfying a given reliability is evaluated from the probability distribution. Commonly used distributions include normal, lognormal, and extreme types I, II and III. '4"' s Ifthe interest in the design is the
226
Adnan A. Basma
average values, the normal and/or log-normal are usually used. On the other hand, if the interest is the extremes (maximum or minimum) in the design, an extreme type is used. The two parameters of interest in the design of sheet piles are (1) the embedment depth and (2) the maximum moment. Since a critical situation arises when the smallest value of an embedment depth is selected, then the most appropriate choice of distribution would be an extreme value for minima. Yet, another critical situation will arise when the largest maximum moment occurs on the wall. Thus, the use of an extreme value for maxima would seem reasonable to describe the maximum moment. Suffice it to point out that the most critical condition will be when the smallest embedment depth is selected with the maximum moment being largest on the wall. A low probability of failure (or high reliability) in this case will give the safest design. To evaluate the best design of a sheet pile with a predetermined reliability the following assumptions are made: (a) the design inputs are independent random variables; (b) the distribution of the embedment depth is type I minima whereas that for the maximum moment is type I maxima. The first assumption can be easily verified and with the argument just presented the second would seem viable. Additionally, the extreme type I distribution has been widely used in many structural designs. 1")'~5 The reliability R of a design can be defined as follows: (16)
R = P ( X < x)
or, in other words, the reliability R is the probability that the random variable Xis less than or equal to a given selected value x (the design value). Alternatively, the probability of failure pf = 1 - R. Observe that if the distribution of X is known then R = F(X) or the cumulative probability (see Fig. 10).
X~ x.]= F(x)
Fig. !0. Design reliability with known distribution.
Reliability-based design of sheet pile structures
227
Since the e m b e d m e n t depth, D, and thus ~t is a type I minima, then R~ = F(~) = 1 - exp { - e x p [a~(~ - u~)]j
(17)
where R~ = reliability o f a selected value o f ~t and thus D a, = 1.282/a~ u~ = 5 + 0.577/a~ and for the m a x i m u m m o m e n t with a type I maxima RM = F ( M ) = exp { - e x p [ - - a M ( M -- %)]}
(18)
where R u = reliability o f a selected design m a x i m u m m o m e n t
am = 1"282/au u m = .,V"I- 0 . 5 7 7 / %
Normalizing :t and M with respect to the mean and standard deviation, and substituting in eqns (17) and (18), yields R, = 1 - exp { - e x p [1.282(z, - 0.45)]}
(19)
RM = exp {--exp [-1"282(zM + 0"45)]}
(20)
and where -
=
-
"¢'2
zM = ( M - .Q')/o"M Figure 11 is a graphical presentation o f eqns (19) and (20). 100
''
"I"
''l'
'''I'
' +'Ill
,I I' Ill I ' '''I'
,,i
99.5
99h, 1.6
1.1
1.8
Z~ Fig. 1 I.
~.g
3
4
ZM Reliability, R, for :, and zu.
5
1
228
Adnan A. Basma
E X A M P L E S O N T H E P R O B A B I L I S T I C D E S I G N O F S H E E T PILE The following examples are presented to illustrate the use of the design charts.
Example 1: Sheet pile with sand below the dredge line. Consider the inputs below: Input
Variable
Mean
CV
a
a2
1
H'
--
--
--
2
~
15ft (4.56m) 352
0-12
4-2
17.6
3
7
--
--
--
4
q
0-17
61"2
5
h,,
1201b/ft 3 (19"2 k N / m 3) 360 Ib/ft z (17"3 kPa) 10 ft [3"04 m)
0'3
3
3 745.4 9
Calculations: h e = 360/120 = 3 ft. The variance o f h e can be evaluated by eqn (11), and a2e=3745-4/(120)2=0.26. H = H ' + h e = 1 8 f t , and since H ' is constant, therefore tr2 = 0"26 and CV H = (0x/-0~/18 = 0-028. fl = 10/18 = 0"56 and by eqn (11) trp = 0"167; thus CVa = 0-30. With the inputs calculated above use Figs 2 and 6 to determine the mean o f ~ and try: ~ = 0.75 and tr~ = 0"13. With a reliability R = 99%, -~ = 1.64 (Fig. 11), and the design value ~t = 0.75 + (1.64)(0.13)=0"963 and thus D = 0.963 x 18 = 17.3 ft (5"26 m). F r o m Figs 4 and 8 determine the mean and standard deviation of Mm~,x: M m , x = 8 0 k . f t / f t , X = 10000 and Y=0"2; thus O ' M m a x = ( 1 0 0 0 0 x 0"2)t/2----44.7 k. ft/ft. With a reliability o f R = 99%, z M= 3.14 (Fig. 11), and the design value Mm~x = 80 + (3.14X44.7) = 220.42 k. ft/ft (980.9 k N . m/m).
Example 2: Sheet pile with clay below the dredge line. Consider the same example as above but replacing input 2 by the following: Input
Variable
Mean
CV
tr
tr'-
2
c
1-5 ksf (72 kPa)
0"6
0-9
0-81
In a similar fashion use Figs 3 and 7 to determine the mean o f ~ and try: ~ = 0 - 4 5 and try= 0-20, and with R = 9 9 % the design value o f ~t= 0-45 + (1-64)(0.20) = 0.78 and thus D = 0.78 x 18 = 14-0 ft (4-27 m).
Reliability-based design of sheet pile structures
229
From Figs 5 and 9 Mm',x= 23 k. ft/ft, X = 90 and Y= 1"00, and thus (90 X 1"0) t/2 = 9.49. With R = 99% the design Mm.,x= 23 + (3"14X9-49) = 52"8 k. ft/ft (310-6 kN. m/m).
O'Mmax =
It should be noted that in the above examples a reliability of 99% (Pr = 10 -2) was used. This is a value that has been recommended by many researchers, t4"t5 especially in the field of geotechnical engineering. 6 However, in some cases a higher reliability value (99.9%, Pr = 10- 3) must be used. In any case, the author does not recommend a value less than 99% in the design of sheet piles. Furthermore, observe that in Example 1 the design ratio, ~t, is 0.963, which may be satisfactorily based on the charts but would be uneconomical and thus anchorage is recommended. Finally, it is important to point out that since the soil properties c, q~ and ~ are random variables so are parameters that are related to them, in particular Ka and Kp, and thus the soil lateral pressure on the wall. Furthermore, the centroid of force due to the soil lateral pressure changes, since the adj usted height of the wall, H, varies with the applied surcharge load, q.
SUMMARY This paper presented a means of designing sheet piles by the introduction of probabilistic techniques. While conventional design methods use the concept of the safety factor, probabilistic design uses reliability. The design process presented in this paper provides a better and more comprehensive approach to designing soil structures, in general, and sheet piles, in particular.
REFERENCES 1. American Institute of Steel Construction, Manual of Steel Construction, 8th edn, Chicago, 1981. 2. British Standards Institution, Foundations, Ciril Engineering Code of Practice, CP 2004, London, 1972. 3. Rowe, P. W., Theoretical and experimental analysis of sheet-pile walls. Proceedings of Institution of Civil Engineers, London, 4(I) (1955) 32-69. 4. Richards, F. E., Analysis of sheet-pile retaining walls. Transactions of American Society of Civil Engineers, 122 (1957) 1113-32. 5. Nataraj, M. S. & Hoadley, P. G., Design of anchored bulkheads in sands. American Society of Civil Engineers, Journal of Geotechnical Engineering, 110(4) (April 1984) 505-15. 6. Harr, M. E., Mechanics of Particulate Media: A Probabilistic Approach. McGraw-Hill, 1977.
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Adnan A. Basma
7. K rahn, J. & Fredlund, D. G., Variability in the engineering properties of natural soil deposits. In Proceedings of the 4th International Conference on Application oJ"Probability and Statistics and Probability in Soil and Structural Engineering, Italy, 1983, pp. 1017-29. 8. Rethati, L., Asymmetry in the distribution of soil properties and its elimination. In Proceedings of the 4th International Conference on Application of Probability and Statistics and Probability in Soil and Structural Engineering, Italy, 1983, pp. 1057-68. 9. McGuffy, V. J., lori, Z. K. & Athanasiou-Grivas, D., Statistical geotechnical properties of Iockport clay. Transportation Research Board Report TRR 809, ! 981. pp. 54-60. 10. Bowles, J. E., Foundation Analysis and Design, 4th edn. McGraw-Hill, 1988. 11. Das, B. M., Principles of Foundation Engineering. Brooks/Cole Engineering Division, 1984. 12. Winterkorn, H. F. & Fang Hsai-Yang, Foundation Engineering Handbook. Van Nostrand Reinhold Company, 1975. 13. Nie, N. H., Hull, C. H., Jenkins, G. M., Steinbrenner, K. & Bent, D. H., SPSS: Statistical Package for the Social Sciences, 2nd edn. McGraw-Hill, 1975. 14. Benjamin, J. R. & Cornell, C. A., Probabilitj, Statistics, and Decision for Ciril Enghleers. McGraw-Hill, 1970. 15. Ang, A. H. S. & Tang, W. H., Probability Concepts in Engineerhlg Planning and Design, Vols I and II. John Wiley, 1975. 16. Bennett, R. M., Comments on the first order ~'s. second order reliability analysis of series structures. Structural Safety Journal, 4 (1987) 241-2. 17. Dolinski, K., First order second moment approximation in reliability of structural systems: critical review and alternative approach. Structural Safety Journal 3 (1983) 21 !-31.