9.13 Fixed Earth-Support Method for Penetration into Sandy Soil

Transcription

1 476 Chapter 9: Sheet Pile Walls 9.13 Fixed Earth-Support Method for Penetration into y Soil When using the fixed earth support method, we assume that the toe of the pile is restrained from rotating, as shown in Figure 9.28a. In the fixed earth support solution, a simplified method called the equivalent beam solution is generally used to calculate L 3 and, thus, D. The development of the equivalent beam method is generally attributed to Blum (1931). In order to understand this method, compare the sheet pile to a loaded cantilever beam RSTU, as shown in Figure Note that the support at T for the beam is equivalent to the anchor load reaction (F) on the sheet pile (Figure 9.28). It can be seen that the point S of the beam RSTU is the inflection point of the elastic line of the beam, which is equivalent to point I in Figure It the beam is cut at S and a free support (reaction P s ) is provided at that point, the bending moment diagram for portion STU of the beam will remain unchanged. This beam STU will be equivalent to the section STU of the beam RSTU. The force P shown in Figure 9.28a at I will be equivalent to the reaction P s on the beam (Figure 9.29). The following is an approximate procedure for the design of an anchored sheet-pile wall (Cornfield, 1975). Refer to Figure Step 1. Determine L 5, which is a function of the soil friction angle below the dredge line, from the following: (deg) L 5 L 1 1 L A Water table L 1 O l 1 Anchor C l 2 F γ, φ 1 Deflected shape of sheet pile L 2 z γ sat φ L 5 L 3 I E 2 J D P L 5 F H D B (a) Pressure diagram G (b) Moment diagram FIGURE 9.28 Fixed earth support method for penetration of sandy soil

2 9.13 Fixed Earth-Support Method for Penetration into y Soil 477 R P s S Beam T U Moment diagram Figure 9.29 Equivalent cantilever beam concept Step 2. Step 3. Step 4. Step 5. Calculate the span of the equivalent beam as l 2 L 2 L 5 L. Calculate the total load of the span, W. This is the area of the pressure diagram between O and I. Calculate the maximum moment, M max, as WL /8. Calculate P by taking the moment about O, or Step 6. Pr 5 1 (moment of area ACDJI about Or) Lr Calculate D as (9.82) Step 7. 6Pr D 5 L Å (K p 2 K a )gr (9.83) Calculate the anchor force per unit length, F, by taking the moment about I, or F 5 1 Lr (moment of area ACDJI about I) Example 9.9 Consider the anchored sheet-pile structure described in Example 9.5. Using the equivalent beam method described in Section 9.13, determine a. Maximum moment b. Theoretical depth of penetration c. Anchor force per unit length of the structure Solution Part a Determination of L 5 : For 30, L 5 L 1 1 L

3 478 Chapter 9: Sheet Pile Walls L Net Pressure Diagram: From Example 9.5, K, K p 3, 16 kn/m 3 a 5 1 3, 9.69 kn/m 3, kn/m 2, kn/m 2. The net active pressure at a depth L 5 below the dredge line can be calculated as 2 (K p K a )L (9.69)( )(0.73) 17.1 kn/m 2 The net pressure diagram from z 0 to z L 1 L 2 L 5 is shown in Figure Maximum Moment: L W 5 a 1 2 b ( )(1.52) 1 a1 b (6.1)( ) 2 Part b kn/m Ll 2 L 2 L m M max 5 WLr 8 1 a 1 b (0.73)( ) 2 5 (197.2)(8.35) 8 Pr 5 1 (moment of area ACDJI about O ) Lr kn? m>m A 8.16 kn/m 2 O 1.53 m = l 1 F l 2 = 1.52 m kn/m 2 C 6.1 m = L 2 P I L 5 = 0.73 m 17.1 kn/m 2 J kn/m 2 D FIGURE 9.30

4 9.14 Field Observations for Anchor Sheet Pile Walls (16.27)(3.05) (16.27)(6.1) P H 1 1 (6.1)( ) ( ) X 3 (0.73) kn/m From Eq. (9.83) D 5 L Å Part c Taking the moment about I (Figure 9.30) F 5 1 a E 2 1 a 1 2 c Approximate 6Pr (K p 2 K a )gr (6)(114.48) Å ( )(9.69) m (16.27)(3.05) a b 1 (16.27)(6.1) a b 6.1 b (6.1)( ) a b 1 a1 b ( )(0.73) a b U kn/m c Approximate 9.14 Field Observations for Anchor Sheet Pile Walls In the preceding sections, large factors of safety were used for the depth of penetration, D. In most cases, designers use smaller magnitudes of soil friction angle,, thereby ensuring a built-in factor of safety for the active earth pressure. This procedure is followed primarily because of the uncertainties involved in predicting the actual earth pressure to which a sheet-pile wall in the field will be subjected. In addition, Casagrande (1973) observed that, if the soil behind the sheet-pile wall has grain sizes that are predominantly smaller than those of coarse sand, the active earth pressure after construction sometimes increases to an at-rest earth-pressure condition. Such an increase causes a large increase in the anchor force, F. The following two case histories are given by Casagrande (1973). Bulkhead of Pier C Long Beach Harbor, California (1949) A typical cross section of the Pier C bulkhead of the Long Beach harbor is shown in Figure Except for a rockfill dike constructed with 76 mm (3 in.) maximum-size quarry wastes, the backfill of the sheet-pile wall consisted of fine sand. Figure 9.32 shows the

5 480 Chapter 9: Sheet Pile Walls m Tie rod m 76 mm dia. Fine sand hydraulic fill 3.05 m 1V: 0.58H Mean low water level MZ 38 Steel sheet pile 1V: 1.5 H El.0 Rock dike 76 mm maximum size m Fine sand m 0 Scale 10 m Figure 9.31 Pier C bulkhead Long Beach harbor (Adapted after Casagrande, 1973) variation of the lateral earth pressure between May 24, 1949 (the day construction was completed) and August 6, On May 24, the lateral earth pressure reached an active state, as shown in Figure 9.32a, due to the wall yielding. Between May 24 and June 3, the anchor resisted further yielding and the lateral earth pressure increased to the at-rest state (Figure 9.32b). However, the flexibility of the sheet piles ultimately resulted in a gradual decrease in the lateral earth-pressure distribution on the sheet piles (see Figure 9.32c). May 24 June 3 August m m m MN/m 2 MN/m 2 MN/m m 3.05 m 3.05 m m m m (a) (b) (c) kn/m 2 Pressure scale Figure 9.32 Measured stresses at Station Pier C bulkhead, Long Beach (Adapted after Casagrande, 1973)

6 9.14 Field Observations for Anchor Sheet Pile Walls 481 With time, the stress on the tie rods for the anchor increased as shown in the following table. Stress on anchor Date tie rod (MN/m 2 ) May 24, June 3, June 11, July 12, August 6, These observations show that the magnitude of the active earth pressure may vary with time and depend greatly on the flexibility of the sheet piles. Also, the actual variations in the lateral earth-pressure diagram may not be identical to those used for design. Bulkhead Toledo, Ohio (1961) A typical cross section of a Toledo bulkhead completed in 1961 is shown in Figure The foundation soil was primarily fine to medium sand, but the dredge line did cut into highly overconsolidated clay. Figure 9.33 also shows the actual measured values of bending moment along the sheet-pile wall. Casagrande (1973) used the Rankine active earthpressure distribution to calculate the maximum bending moment according to the free earth support method with and without Rowe s moment reduction. Design method Free earth support method Free earth support method with Rowe s moment reduction Maximum predicted bending moment, M max kn-m 78.6 kn-m Top of fill 0 Scale 2 Tie rod kn-m 4 81 kn-m 6 65 kn-m May kn-m Depth (m) 205 kn-m Dredge line Figure 9.33 Bending moment from straingage measurements at test location 3, Toledo bulkhead (Adapted after Casagrande, 1973)

7 482 Chapter 9: Sheet Pile Walls Comparisons of these magnitudes of M max with those actually observed show that the field values are substantially larger. The reason probably is that the backfill was primarily fine sand and the measured active earth-pressure distribution was larger than that predicted theoretically Free Earth Support Method for Penetration of Clay Figure 9.34 shows an anchored sheet-pile wall penetrating a clay soil and with a granular soil backfill. The diagram of pressure distribution above the dredge line is similar to that shown in Figure From Eq. (9.42), the net pressure distribution below the dredge line (from z 5 L 1 1 L 2 to z 5 L 1 1 L 2 1 D) is s 6 5 4c 2 (gl 1 1grL 2 ) For static equilibrium, the sum of the forces in the horizontal direction is P 1 2s 6 D 5 F (9.84) where P 1 5 area of the pressure diagram ACD F 5 anchor force per unit length of the sheet pile wall A l 1 L 1 O F Water level 1 C l 2,, L 2 z sat, P 1 z 1 Dredge line E 2 D Clay D Clay sat = 0 c F 6 B Figure 9.34 Anchored sheet-pile wall penetrating clay

8 9.15 Free Earth Support Method for Penetration of Clay 483 Again, taking the moment about Or produces P 1 (L 1 1 L 2 2 l 1 2 z 1 ) 2s 6 D l 2 1 L 2 1 D Simplification yields s 6 D 2 1 2s 6 D(L 1 1 L 2 2 l 1 ) 2 2P 1 (L 1 1 L 2 2 l 1 2 z 1 ) 5 0 (9.85) Equation (9.85) gives the theoretical depth of penetration, D. As in Section 9.9, the maximum moment in this case occurs at a depth L 1, z, L 1 1 L 2. The depth of zero shear (and thus the maximum moment) may be determined from Eq. (9.69). A moment reduction technique similar to that in Section 9.11 for anchored sheet piles penetrating into clay has also been developed by Rowe (1952, 1957). This technique is presented in Figure 9.35, in which the following notation is used: 1. The stability number is c S n (gl 1 1grL 2 ) (9.86) where c 5 undrained cohesion (f 50). For the definition of g, gr, L 1, and L 2, see Figure The nondimensional wall height is L 1 1 L 2 a5 L 1 1 L 2 1 D actual (9.87) 3. The flexibility number is r [see Eq. (9.74)] 4. M d 5 design moment M max 5 maximum theoretical moment The procedure for moment reduction, using Figure 9.35, is as follows: Step 1. Obtain Hr 5 L 1 1 L 2 1 D actual. Step 2. Determine a5(l 1 1 L 2 )>Hr. Step 3. Determine S n [from Eq. (9.86)]. Step 4. For the magnitudes of a and S n obtained in Steps 2 and 3, determine M d >M max for various values of log r from Figure 9.35, and plot M d >M max against log r. Step 5. Follow Steps 1 through 9 as outlined for the case of moment reduction of sheet-pile walls penetrating granular soil. (See Section 9.11.)

9 484 Chapter 9: Sheet Pile Walls 1.0 Log = M d M max 0.6 = Log = M d M max 0.6 = Log = Figure 9.35 Plot of M d >M max against stability number for sheet-pile wall penetrating clay (From Rowe, P. W. (1957). Sheet Pile Walls in Clay, Proceedings, Institute of Civil Engineers, Vol. 7, pp ) M d M max = Stability number, S n Example 9.10 In Figure 9.34, let L and Also, let g517 kn>m m, L m, l m., g fr 5 35, and c 5 41 kn>m 2 sat 5 20 kn>m 3,. a. Determine the theoretical depth of embedment of the sheet-pile wall. b. Calculate the anchor force per unit length of the wall. Solution Part a We have K a 5 tan fr 2 5 tan

10 9.15 Free Earth Support Method for Penetration of Clay 485 and From the pressure diagram in Figure 9.36, and K p 5 tan fr 2 5 tan sr 1 5gL 1 K a 5 (17)(3)(0.271) kn>m 2 sr 2 5 (gl 1 1grL 2 )K a 5 3(17)(3) 1 ( )(6)4(0.271) kn>m 2 P 1 5 areas >2(3)(13.82) 1 (13.82)(6) 1 1>2( )(6) kn>m (20.73) (82.92) (49.71) 6 3 z From Eq. (9.85), m s 6 D 2 1 2s 6 D(L 1 1 L 2 2 l 1 ) 2 2P 1 (L 1 1 L 2 2 l 1 2 z 1 ) 5 0 s 6 5 4c 2 (gl 1 1grL 2 ) 5 (4)(41) 2 3(17)(3) 1 ( )(6) kn>m 2 So, (51.86)D 2 1 (2)(51.86)(D)( ) 2 (2)(153.36)( ) 5 0 L 1 3 m 1 l m l m kn/m 2 L 2 6 m kn/m kn/m m D Figure 9.36 Free earth support method, sheet pile penetrating into clay

11 486 Chapter 9: Sheet Pile Walls or D D Hence, D < 1.6 m Part b From Eq. (9.84), F 5 P 1 2s 6 D (51.86)(1.6) kn,m 9.16 Anchors Sections 9.9 through 9.15 gave an analysis of anchored sheet-pile walls and discussed how to obtain the force F per unit length of the sheet-pile wall that has to be sustained by the anchors. The current section covers in more detail the various types of anchor generally used and the procedures for evaluating their ultimate holding capacities. The general types of anchor used in sheet-pile walls are as follows: 1. Anchor plates and beams (deadman) 2. Tie backs 3. Vertical anchor piles 4. Anchor beams supported by batter (compression and tension) piles Anchor plates and beams are generally made of cast concrete blocks. (See Figure 9.37a.) The anchors are attached to the sheet pile by tie-rods. A wale is placed at the front or back face of a sheet pile for the purpose of conveniently attaching the tie-rod to the wall. To protect the tie rod from corrosion, it is generally coated with paint or asphaltic materials. In the construction of tiebacks, bars or cables are placed in predrilled holes (see Figure 9.37b) with concrete grout (cables are commonly high-strength, prestressed steel tendons). Figures 9.37c and 9.37d show a vertical anchor pile and an anchor beam with batter piles. Placement of Anchors The resistance offered by anchor plates and beams is derived primarily from the passive force of the soil located in front of them. Figure 9.37a, in which AB is the sheet-pile wall, shows the best location for maximum efficiency of an anchor plate. If the anchor is placed inside wedge ABC, which is the Rankine active zone, it would not provide any resistance to failure. Alternatively, the anchor could be placed in zone CFEH. Note that line DFG is the slip line for the Rankine passive pressure. If part of the passive wedge is located inside the active wedge ABC, full passive resistance of the anchor cannot be realized upon

12 9.16 Anchors /2 45 /2 D A Groundwater table F C G 45 /2 I Anchor plate H or beam E Sheet pile Anchor plate or beam Wale Tie rod B Section (a) Plan 45 /2 45 /2 45 /2 Groundwater table Tie rod Groundwater table Anchor pile Tie rod or cable (b) Concrete grout (c) Groundwater table Tie rod 45 /2 Anchor beam Compression pile (d) Tension pile Figure 9.37 Various types of anchoring for sheet-pile walls: (a) anchor plate or beam; (b) tieback; (c) vertical anchor pile; (d) anchor beam with batter piles failure of the sheet-pile wall. However, if the anchor is placed in zone ICH, the Rankine passive zone in front of the anchor slab or plate is located completely outside the Rankine active zone ABC. In this case, full passive resistance from the anchor can be realized. Figures 9.37b, 9.37c, and 9.37d also show the proper locations for the placement of tiebacks, vertical anchor piles, and anchor beams supported by batter piles.

13 488 Chapter 9: Sheet Pile Walls 9.17 Holding Capacity of Anchor Plates in Semi-Empirical Method Ovesen and Stromann (1972) proposed a semi-empirical method for determining the ultimate resistance of anchors in sand. Their calculations, made in three steps, are carried out as follows: Step 1. Basic Case. Determine the depth of embedment, H. Assume that the anchor slab has height H and is continuous (i.e., B 5 length of anchor slab perpendicular to the cross section 5`), as shown in Figure 9.38, in which the following notation is used: P p 5 passive force per unit length of anchor P a 5 active force per unit length of anchor fr 5 effective soil friction angle dr 5 friction angle between anchor slab and soil Pr ult 5 ultimate resistance per unit length of anchor W 5 effective weight per unit length of anchor slab Also, Pr ult gh2 K p cos dr 2 P a cos fr gh2 K p cos dr gh2 K a cos fr gh2 (K p cos dr 2 K a cos fr) (9.88) where K a 5 active pressure coefficient with dr 5fr (see Figure 9.39a) K p 5 passive pressure coefficient To obtain K p cos dr, first calculate K p sin dr 5 W 1 P a sin fr 5 W 1 1 2gH 2 K a sin fr 1 2gH 2 1 2gH 2 (9.89) 45 /2 45 /2 P a H P p P ult Figure 9.38 Basic case: continuous vertical anchor in granular soil

14 9.17 Holding Capacity of Anchor Plates in P a 0.3 Arc of log spiral K a Soil friction angle, (deg) (a) K p cos = K p sin (b) 4 5 Figure 9.39 (a) Variation of K a for dr 5fr, (b) variation of K p cos dr with K p sin dr (Based on Ovesen and Stromann, 1972) Step 2. Then use the magnitude of K p sin dr obtained from Eq. (9.89) to estimate the magnitude of K p cos dr from the plots given in Figure 9.39b. Strip Case. Determine the actual height h of the anchor to be constructed. If a continuous anchor (i.e., an anchor for which B 5`) of height h is placed in the soil so that its depth of embedment is H, as shown in Figure 9.40, the ultimate resistance per unit length is

15 490 Chapter 9: Sheet Pile Walls H h P us Figure 9.40 Strip case: vertical anchor Pr us 5 C C ov 1 1 C ov 1 H h S Pr ult c Eq (9.90) Step 3. where Pr us 5 ultimate resistance for the strip case C ov 5 19 for dense sand and 14 for loose sand Actual Case. In practice, the anchor plates are placed in a row with centerto-center spacing Sr, as shown in Figure 9.41a. The ultimate resistance of each anchor is H h S B S (a) Dense sand (B e B)/(H + h) Loose sand (S B)/(H h) (b) Figure 9.41 (a) Actual case for row of anchors; (b) variation of (B e 2 B)>(H 1 h) with (Sr 2 B)>(H 1 h) (Based on Ovesen and Stromann, 1972)

16 9.17 Holding Capacity of Anchor Plates in 491 P ult 5 Pr us B e (9.91) where B e 5 equivalent length. The equivalent length is a function of Sr, B, H, and h. Figure 9.41b shows a plot of (B e 2 B)>(H 1 h) against (Sr 2 B)>(H 1 h) for the cases of loose and dense sand. With known values of Sr, B, H, and h, the value of B e can be calculated and used in Eq. (9.91) to obtain P ult. Stress Characteristic Solution Neely, Stuart, and Graham (1973) proposed a stress characteristic solution for anchor pullout resistance using the equivalent free surface concept. Figure 9.42 shows the assumed failure surface for a strip anchor. In this figure, OX is the equivalent free surface. The shear stress (s o ) mobilized along OX can be given as m 5 s o s o 9 tan f 9 (9.92) where m shear stress mobilization factor o effective normal stress along OX Using this analysis, the ultimate resistance (P ult ) of an anchor (length B and height h) can be given as P ult M q ( h 2 )BF s (9.93) where M q force coefficient F s shape factor effective unit weight of soil The variations of M q for m 0 and 1 are shown in Figure For conservative design, M q with m 0 may be used. The shape factor (F s ) determined experimentally is shown in Figure 9.44 as a function of B/h and H/h. X H h O o so P ult m = so o tan Figure 9.42 Assumed failure surface in soil for stress characteristic solution

17 492 Chapter 9: Sheet Pile Walls M q (log scale) = 45 = 40 5 = 35 = 30 m = 0 m = H / h 4 5 Figure 9.43 Variation of M q with H/h and (After Neeley et al., With permission from ASCE.) 2.5 B/h = Shape factor, F s H / h 4 5 Figure 9.44 Variation of shape factor with H/h and B/h (After Neeley et al., With permission from ASCE.) Empirical Correlation Based on Model Tests Ghaly (1997) used the results of 104 laboratory tests, 15 centrifugal model tests, and 9 field tests to propose an empirical correlation for the ultimate resistance of single anchors. The correlation can be written as P ult tan fr H2 A 0.28 gah (9.94) where A 5 area of the anchor 5 Bh.

18 9.17 Holding Capacity of Anchor Plates in 493 Ghaly also used the model test results of Das and Seeley (1975) to develop a load displacement relationship for single anchors. The relationship can be given as P P ult u H 0.3 (9.95) where u 5 horizontal displacement of the anchor at a load level P. Equations (9.94) and (9.95) apply to single anchors (i.e., anchors for which Sr>B 5`). For all practical purposes, when Sr>B < 2 the anchors behave as single anchors. Factor of Safety for Anchor Plates The allowable resistance per anchor plate may be given as where FS 5 factor of safety. Generally, a factor of safety of 2 is suggested when the method of Ovesen and Stromann is used. A factor of safety of 3 is suggested for calculated by Eq. (9.94). Spacing of Anchor Plates P all 5 P ult FS The center-to-center spacing of anchors, Sr, may be obtained from P ult Sr 5 P all F where F 5 force per unit length of the sheet pile. Example 9.11 Refer to Figure 9.41a. Given: B h 0.4 m, S1.2 m, H 1m, kn/m 3, and 35. Determine the ultimate resistance for each anchor plate. The anchor plates are made of concrete and have thicknesses of 0.15 m. Solution From Figure 9.39a for 35, the magnitude of K a is about From Eq. (9.89), W Ht concrete (1 m)(0.15 m)(23.5 kn/m 3 ) kn/m K p sin d9 5 W 1 1 > 2 gh 2 K a sin f9 1 > 2 gh (0.5)(16.51)(1)2 (0.26)(sin 35) (0.5)(16.51)(1)

19 494 Chapter 9: Sheet Pile Walls From Figure 9.39b with 35 and K p sin 0.576, the value of K p cos is about 4.5. Now, using Eq. (9.88), P ult H 2 (K p cos K a cos ) ( )(16.51)(1) 2 [4.5 (0.26)(cos 35)] kn/m In order to calculate P us, let us assume the sand to be loose. So, C ov in Eq. (9.90) is equal to 14. Hence, For (SB)/(H h) and loose sand, Figure 9.41b yields So Hence, from Eq. (9.91) Pr us 5 D C ov C ov 1 a H T Pr ult 5 D h b 14 1 a 1 T kn>m 0.4 b Sr 2 B H 1 h B e (0.229)(H h) B (0.229)(1 0.4) B e 2 B H 2 h P ult P us B e (32.17)(0.72) kn Example 9.12 Refer to a single anchor given in Example 9.11 using the stress characteristic solution. Estimate the ultimate anchor resistance. Use m 0 in Figure Solution Given: B h 0.4 m and H 1m. Thus, H h 5 1 m 0.4 m B h m 0.4 m 5 1

20 9.19 Ultimate Resistance of Tiebacks 495 From Eq. (9.93), P ult M q h 2 BF s From Figure 9.43, with 35 and H/h 2.5, M q Also, from Figure 9.44, with H/h 2.5 and B/h 1, F s 1.8. Hence, P ult (18.2)(16.51)(0.4) 2 (0.4)(1.8) kn Example 9.13 Solve Example Problem 9.12 using Eq. (9.94). Solution From Eq. (9.94), P ult tan f 9aH2 A b gah H 1m A Bh ( ) 0.16 m 2 P ult tan 35 c (1) d 0.28 (16.51)(0.16)(1) < kn 9.18 Holding Capacity of Anchor Plates in Clay ( f50 Condition) Relatively few studies have been conducted on the ultimate resistance of anchor plates in clayey soils ( f50). Mackenzie (1955) and Tschebotarioff (1973) identified the nature of variation of the ultimate resistance of strip anchors and beams as a function of H, h, and c (undrained cohesion based on f50) in a nondimensional form based on laboratory model test results. This is shown in the form of a nondimensional plot in Figure 9.45 ( P ult >hbc versus H>h) and can be used to estimate the ultimate resistance of anchor plates in saturated clay ( f50) Ultimate Resistance of Tiebacks According to Figure 9.46, the ultimate resistance offered by a tieback in sand is P ult 5pdlsr o K tan fr (9.96)

21 496 Chapter 9: Sheet Pile Walls P ult hbc H h Figure 9.45 Experimental variation of with H>h for plate anchors in clay hbc (Based on Mackenzie (1955) and Tschebotarioff (1973)) P ult z l d Figure 9.46 Parameters for defining the ultimate resistance of tiebacks where fr 5 effective angle of friction of soil sr o 5 average effective vertical stress ( 5gz in dry sand) K 5 earth pressure coefficient The magnitude of K can be taken to be equal to the earth pressure coefficient at rest (K o ) if the concrete grout is placed under pressure (Littlejohn, 1970). The lower limit of K can be taken to be equal to the Rankine active earth pressure coefficient. In clays, the ultimate resistance of tiebacks may be approximated as P ult 5pdlc a (9.97) where c a 5 adhesion.

22 Problems 497 The value of c a may be approximated as 3c u (where c u 5 undrained cohesion). A factor of safety of 1.5 to 2 may be used over the ultimate resistance to obtain the allowable resistance offered by each tieback. Problems L 5 3 m, L 5 6 m, g517.3 kn>m 3, 5 35 ; 9.1 Figure P9.1 shows a cantilever sheet pile wall penetrating a granular soil. Here, L m, L m, g516.1 kn>m 3, g sat kn>m 3, and fr a. What is the theoretical depth of embedment, D? b. For a 30% increase in D, what should be the total length of the sheet piles? c. Determine the theoretical maximum moment of the sheet pile. 9.2 Redo Problem 9.1 with the following: g sat kn>m 3, and fr Refer to Figure Given: L 5 3 m, g516.7 kn>m 3, and fr Calculate the theoretical depth of penetration, D, and the maximum moment. g, and and c 5 29 kn>m 2 sat kn>m 3 fr 5 30,. 9.4 Refer to Figure P9.4, for which L m, L m, g515.7 kn>m 3, a. What is the theoretical depth of embedment, D? b. Increase D by 40%. What length of sheet piles is needed? c. Determine the theoretical maximum moment in the sheet pile. 9.5 Refer to Figure Given: L 5 4 m; for sand, g516 kn>m 3 ; fr and, for clay, g sat kn>m 3 and c 5 45 kn>m 2. Determine the theoretical value of D and the maximum moment. 9.6 An anchored sheet pile bulkhead is shown in Figure P9.6. Let L m, L, g517 kn>m 3, g sat 5 19 kn>m m, l m, and fr a. Calculate the theoretical value of the depth of embedment, D. b. Draw the pressure distribution diagram. c. Determine the anchor force per unit length of the wall. Use the free earth-support method. 2 Water table L 1 c 0 L 2 sat c 0 Dredge line D sat c 0 Figure P9.1

23 498 Chapter 9: Sheet Pile Walls Water table L 1 c 0 L 2 sat c 0 D Clay c 0 Figure P9.4 Water table L 1 l 1 Anchor c 0 L 2 sat c 0 D sat c 0 Figure P In Problem 9.6, assume that D actual 5 1.3D theory. a. Determining the theoretical maximum moment. b. Using Rowe s moment reduction technique, choose a sheet pile section. Take E MN>m 2 and s all 5 210,000 kn>m Refer to Figure P9.6. Given: L m, L m, l 1 5 l m, 5 16 kn/m 3, sat kn/m 3, and fr Use the charts presented in Section 9.10 and determine: a. Theoretical depth of penetration b. Anchor force per unit length c. Maximum moment in the sheet pile. 9.9 Refer to Figure P9.6, for which L 5 18 kn>m m, L m, l m, g, g sat kn>m 3, and fr Use the computational diagram method (Section 9.12) to determine D, F, and. Assume that C and R M max

24 Problems An anchored sheet-pile bulkhead is shown in Figure P9.10. Let L g516 kn>m 3, g sat kn>m m, L m, l m,, fr 5 32, and c 5 27 kn>m 2. a. Determine the theoretical depth of embedment, D. b. Calculate the anchor force per unit length of the sheet-pile wall. Use the free earth support method In Figure 9.41a, for the anchor slab in sand, H m, h m, B m, Sr m, fr 5 30, and g517.3 kn>m 3. The anchor plates are made of concrete and have a thickness of 76 mm. Using Ovesen and Stromann s method, calculate the ultimate holding capacity of each anchor. Take g concrete kn>m A single anchor slab is shown in Figure P9.12. Here, H m, h m, g517 kn>m 3, and fr Calculate the ultimate holding capacity of the anchor slab if the width B is (a) 0.3 m, (b) 0.6 m, and (c) m. (Note: center-to-center spacing, Sr 5`.) Use the empirical correlation given in Section 9.17 [Eq. (9.94)] Repeat Problem 9.12 using Eq. (9.93). Use m 5 0 in Figure Water table L 1 l 1 Anchor c 0 L 2 sat c 0 D Clay c 0 Figure P9.10 H c 0 h P ult Figure P9.12

25 500 Chapter 9: Sheet Pile Walls References BLUM, H. (1931) Einspannungsverhältnisse bei Bohlwerken, W. Ernst und Sohn, Berlin, Germany. CASAGRANDE, L. (1973). Comments on Conventional Design of Retaining Structures, Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM2, pp CORNFIELD, G. M. (1975). Sheet Pile Structures, in Foundation Engineering Handbook, ed. H. F. Wintercorn and H. Y. Fang, Van Nostrand Reinhold, New York, pp DAS, B. M., and SEELEY, G. R. (1975). Load Displacement Relationships for Vertical Anchor Plates, Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 101, No, GT7, pp GHALY, A. M. (1997). Load Displacement Prediction for Horizontally Loaded Vertical Plates. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 1, pp HAGERTY, D. J., and NOFAL, M. M. (1992). Design Aids: Anchored Bulkheads in, Canadian Geotechnical Journal, Vol. 29, No. 5, pp LITTLEJOHN, G. S. (1970). Soil Anchors, Proceedings, Conference on Ground Engineering, Institute of Civil Engineers, London, pp MACKENZIE, T. R. (1955). Strength of Deadman Anchors in Clay, M.S. Thesis, Princeton University, Princeton, N. J. NATARAJ, M. S., and HOADLEY, P. G. (1984). Design of Anchored Bulkheads in, Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 110, No. GT4, pp NEELEY, W. J., STUART, J. G., and GRAHAM, J. (1973). Failure Loads of Vertical Anchor Plates is, Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM9, pp OVESEN, N. K., and STROMANN, H. (1972). Design Methods for Vertical Anchor Slabs in, Proceedings, Specialty Conference on Performance of Earth and Earth-Supported Structures. American Society of Civil Engineers, Vol. 2.1, pp ROWE, P. W. (1952). Anchored Sheet Pile Walls, Proceedings, Institute of Civil Engineers, Vol. 1, Part 1, pp ROWE, P. W. (1957). Sheet Pile Walls in Clay, Proceedings, Institute of Civil Engineers, Vol. 7, pp TSCHEBOTARIOFF, G. P. (1973). Foundations, Retaining and Earth Structures, 2nd ed., McGraw-Hill, New York. TSINKER, G. P. (1983). Anchored Street Pile Bulkheads: Design Practice, Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 109, No. GT8, pp