Design of Sheet Pile Walls UsingTRULINE Composite Wall Sections Design Methods and Examples A Report Presented to Formtech Enterprises, Inc./ TRULINE 126 Ben Burton Circle Bogart, Georgia 30622 by Ensoft, Inc. 3003 Howard Lane Austin, Texas 78728 UPDATE2Includes1,2&4RebarJanuary 13, 2014 iiTable of Contents List of Figures ................................................................................................................................ ivList of Tables .................................................................................................................................. vIntroduction ..................................................................................................................................... 1Recommended Structural Capacities of Truline Series 800 and Series 1200 Composite Wall Sections ........................................................................................................................................... 1Moment Capacity and Bending Stiffness ................................................................................... 1Shear Capacity ............................................................................................................................ 5Comparison of Unfilled and Concrete-Filled Sections ................................................................... 8Analysis and Design of Cantilever Sheet Pile Walls ...................................................................... 9Design Example of Cantilever Sheet Pile Wall in Cohesionless Soil ...................................... 12Compute Wall Pressures Acting on Wall ............................................................................. 12Depth of Penetration for Design ........................................................................................... 14Compute Location of Zero Shear Force ................................................................................ 14Compute Maximum Moment Developed in Wall ................................................................ 14Note Regarding Passive Earth Pressure Coefficient ............................................................. 15Design Example of Cantilever Sheet Pile Wall in Cohesive Soil ............................................. 15Analysis and Design of Anchored Sheet Pile Walls ..................................................................... 18Free Earth Support Method ....................................................................................................... 18Design Example of Anchored Sheet Pile Wall Using Free Earth Support Method Granular Soil ............................................................................................................................................ 20Compute Distribution of Earth Pressures Acting on Wall .................................................... 20Compute Depth of Point of Zero Net Pressure ..................................................................... 21Compute Force Resultants of Pressures Acting on Wall ...................................................... 21 iiiCompute Sum of Moments ................................................................................................... 21Solve for D1 and Select Value of D ...................................................................................... 21Compute Tension Force in Tieback ...................................................................................... 22Compute Location of Point of Maximum Moment .............................................................. 22Compute Maximum Moment in Wall Section ...................................................................... 22Design Example Using PYWall 3.0.............................................................................................. 23Compute Earth Pressure Distribution Acting on Wall .............................................................. 24Compute Equivalent Spring Constant for Tieback ................................................................... 24Computed Results ..................................................................................................................... 25Optimization of Design ............................................................................................................. 27References ..................................................................................................................................... 30Appendix A Earth Pressure Coefficients for Wall Friction and Sloping Backfill ..................... 31 ivList of Figures Figure 1 Earth Pressures Acting on a Cantilever Sheet Pile Wall ............................................. 10Figure 2 Resultant Earth Pressure Diagram for Cantilever Sheet Pile in Cohesionless Soil ..... 11Figure 3 Example Problem for Sheet Pile Wall in Cohesionless Soil ....................................... 13Figure 4 Earth Pressures for Cantilever Sheet Pile Wall in Cohesive Soil ................................ 16Figure 5 Earth Pressure Distributions Used in Design of Anchored Sheet Piling by Free Earth Support Method for Cohesionless and Cohesive Soils ....................................... 19Figure 6 Earth Pressure Distributions Used in Design Example for Anchored Sheet Pile Wall in Cohesionless Soil ............................................................................................. 20Figure 7 Example Problem for PYWall ..................................................................................... 24Figure 8 Lateral Deflection vs. Depth Below Pile Head ........................................................... 26Figure 9 Bending Moment vs. Depth Below Pile Head............................................................. 26Figure 10 Shear Force vs. Depth Below Pile Head ................................................................... 27Figure 11 Comparison of Computed Wall Deflections ............................................................. 28Figure 12 Comparison of Computed Wall Moments ................................................................. 29Figure 13 Comparison of Computed Wall Shear Forces ........................................................... 29Figure A1 Active and Passive Earth Pressure Coefficients with Wall Friction and Sloping Backfill (from NAVFAC ,1986 and Caquot and Kerisel, 1948) .................... 31 vList of Tables Table 1 Allowable Moment Capacity and Effective Bending Stiffnessof Gravel-filled Truline Sections .............................................................................................................. 2Table 2 Series 800 Factored Moment Capacities in-lbs/ft width - Reinforced Sections 1 Rebar ............................................................................................................................ 3Table 3 Series 800 Factored Moment Capacities in-lbs/ft width - Reinforced Sections 2 Rebar ............................................................................................................................ 3 Table 4 Series 800 Factored Moment Capacities in-lbs/ft width - Reinforced Sections 4 Rebar ............................................................................................................................ 3 Table 5 Series 1200 Factored Moment Capacity, in-lbs/ft width - Reinforced Sections 4 Rebar ............................................................................................................................ 4Table 6 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 1 Rebar .............. 4 Table 7 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 2 Rebar .............. 4 Table 8 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 4 Rebar .............. 5Table 9 Series 1200 Bending Stiffness, lb-in2/ft width - Reinforced Sections 4 Rebar ........... 5Table 10 Allowable Shear Capacity of Gravel-filled Truline Sections ....................................... 6Table 11 Series 800 Shear Capacity/ft width ............................................................................... 6Table 12 Series 1200 Shear Capacity/ft width ............................................................................. 6Table 13 Shear Capacity of Steel Reinforcement in Series 1200 Section for s =4.2 inches .............................................................................................................................. 7Table 14 Ratio of Concrete-filled to Unfilled Moment Capacities for Series 800 Doubly Reinforced Sections ........................................................................................................ 8Table 15 Ratio of Concrete-filled to Unfilled Moment Capacities for Series 1200 Doubly Reinforced Sections ........................................................................................... 8Table 16 Ratio of Concrete-filled to Unfilled Bending Stiffness for Series 800 Sections .......... 9Table 17 Ratio of Concrete-filled to Unfilled Bending Stiffness for Series 1200 Sections ........................................................................................................................... 9Table 18 Approximate Values for Required Depths for Cantilever Sheet Pile Walls in Cohesionless Soil .......................................................................................................... 10Table 19 Sum of Moment Computations ................................................................................... 21Table 20 Example Soil Properties for Example Using PYWall ................................................ 23Table 21 Results Computed by PYWall 3.0 for Example Problem ............................................25 1Design of Sheet Pile Walls Using Truline Composite Wall Sections Design Methods and Examples Introduction This report presents the recommended structural capacities for Truline composite wall sections and a summary of design methods commonly used to design cantilevered and anchored sheet pile structures. Several design examples are provided to illustrate the application of these design methods for sheet pile walls using the Truline Series 800 and 1200 composite wall sections.Recommended Structural Capacities of Truline Series 800 and Series 1200 Composite Wall Sections Moment Capacity and Bending Stiffness The tables presented in this section are the recommended values for factored structural moment capacities and effective bending stiffness of Truline Series 800 and 1200 PVC sheet pile sections filled with reinforced concrete. The factored moment capacities were determined from nonlinear moment vs. curvature behavior computed using LPile 2012. The nominal moment capacities were determined when the maximum compressive strain in concrete reached 0.003 in/in. The reported ultimate (factored) moment capacities were computed by multiplying the nominal moment capacity by a strength reduction factor, |, of 0.65. The reported bending stiffnesses are for moment levels equal to the ultimate moment capacity and are for cracked sections. The choice of a strength reduction factor of 0.65 is conservative. This is a value typically used for structural sections with combined flexure and axial load (see Chapter 10 in ACI 318-08). The rational for selection of this value is the similarity in construction methods for placement of concrete into the Truline sections to that used for drilled shaft foundations. In construction, concrete is placed from the top of section by free-fall if placed in the dry or by tremie or pumping if placed in the wet. Typically, vibration cannot be used due to the depth of 2concrete placement. In addition, no opportunity exists for direct visual inspection because the construction is not above ground and formwork is not used. A reasonable argument can be made to raise the strength reduction factor to 0.90 if the wall sections are constructed above ground and moved into place. Presumably, in this type of construction, the use of vibration to consolidate concrete will be possible and no placement of concrete will be made in the wet. An evaluation of the analyses used to evaluate moment capacity and bending stiffness found that the moment capacity of all sections were controlled by tension reinforcement, thus permitting use of a strength reduction factor of 0.90 (see ACI 318-08 section R9.3.2.2). Values of moment capacity and bending stiffness were computed for a range of concrete compressive strengths and reinforcement options. All values are reported in values per foot width of section. Tables 1 through 5 present moment capacities of the Truline sections for gravel-filled sections (Table 1) and for concrete-filled sections with various reinforcement and concrete compressive strength options (Tables 2 through 5). Table 1 Allowable Moment Capacity and Effective Bending Stiffnessof Gravel-filled Truline Sections Truline Section Allowable Moment Capacity,* in-lb/ft Bending Stiffness, EI, lb-in2/ft Series 80053,12425,080,000 Series 1200134,532158,080,000. *Based on full scale performance test by Architectural Testing, Inc. Report #70174.01-122-44, not theoretical calculations.. Tables 6 through 9 present bending stiffnesses of the Truline sections for concrete-filled sections with various reinforcement and concrete compressive strength options. 3 Table 2 Series 800 Factored Moment Capacities in-lbs/ft width Reinforced Sections 1 Rebar Bar Size Concrete Compressive Strength, f'cpsi/ft width 3,0003,5004,0004,5005,000 No. 441,30042,50043,40044,20044,700 No. 543,10044,60045,60046,40047,100 No. 652,30056,10058,10059,60060,800 No. 756,90063,50069,10073,40075,800 No. 859,80067,60074,70081,00086,600 No. 962,30070,50078,30085,60092,400 No. 1065,00073,80082,00089,90097,500 No. 1167,50076,80085,30093,600101,600 No. 1473,20082,90092,300101,400110,000 Table 3 Series 800 Factored Moment Capacities in-lbs/ft width - Reinforced Sections 2 Rebar Bar Size Concrete Compressive Strength, f'cpsi/ft width 3,0003,5004,0004,5005,000 No. 464,50065,90066,80067,50068,000 No. 586,10088,90091,00092,60093,900 No. 6104,500112,000116,000118,900121,300 No. 7113,800127,000138,200146,600151,500 No. 8119,700135,100149,300162,000173,100 No. 9124,600141,100156,700171,300184,900 No. 10129,900147,400164,000179,900195,000 No. 11135,000153,100170,600187,200203,200 No. 14146,500165,900184,600202,700220,100 Table 4 Series 800 Factored Moment Capacities in-lbs/ft width - Reinforced Sections 4 Rebar Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 495,900100,900104,900108,100109,900 No. 5127,100132,800138,100143,100147,700 No. 6161,200167,900174,100179,800185,300 No. 7201,600209,100216,200222,800229,000 No. 8248,200256,600264,500271,900278,900 4Table 5 Series 1200 Factored Moment Capacity, in-lbs/ft width - Reinforced Sections 4 Rebar Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 5159,300164,400168,800171,900173,800 No. 6205,900212,300218,200223,600228,400 No. 7261,500268,800275,600281,800287,800 No. 8325,900334,100341,800348,900355,700 No. 9395,800404,900413,300421,300428,700 No. 10484,600494,600503,900512,700521,000 No. 11579,200589,900600,100609,600618,600 Table 6 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 1 Rebar Bar Size Concrete Compressive Strength, f'cpsi 3,0003,5004,0004,5005,000 No. 464,920,00067,370,00069,380,00071,080,00072,560,000 No. 568,430,00071,100,00073,310,00075,170,00076,790,000 No. 681,460,00084,950,00088,160,00090,900,00093,280,000 No. 796,350,000100,210,000103,770,000107,100,000110,350,000 No. 8111,380,000115,960,000120,110,000123,910,000127,450,000 No. 9125,280,000130,780,000135,620,000140,090,000144,140,000 No. 10140,620,000147,160,000153,050,000158,290,000163,130,000 No. 11154,500,000162,020,000168,900,000175,050,000180,630,000 No. 14181,660,000191,420,000200,140,000208,040,000215,310,000 Table 7 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 2 Rebar Bar Size Concrete Compressive Strength, f'cpsi 3,0003,5004,0004,5005,000 No. 4109,000,000112,300,000115,000,000117,400,000119,400,000 No. 5136,700,000142,000,000146,400,000150,100,000153,300,000 No. 6162,800,000169,800,000176,200,000181,600,000186,400,000 No. 7192,600,000200,300,000207,400,000214,100,000220,600,000 No. 8222,700,000231,800,000240,100,000247,700,000254,800,000 No. 9250,500,000261,500,000271,200,000280,100,000288,200,000 No. 10281,200,000294,400,000306,100,000316,500,000326,200,000 No. 11309,000,000324,200,000337,800,000350,100,000361,200,000 No. 14363,300,000382,800,000400,300,000416,100,000430,600,000 5 Table 8 Series 800 Bending Stiffness, lb-in2/ft width - Reinforced Sections 4 Rebar Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 4234,820,000239,000,000242,900,000246,235,000249,500,000 No. 5309,785,000317,402,000323,775,000330,000,000334,400,000 No. 6383,022,000394,125,000404,646,000413,000,000420,000,000 No. 7459,000,000475,658,000489,700,000501,500,000511,750,000 No. 8537,000,000558,815,000576,993,000592,869,000606,500,000 Table 9 Series 1200 Bending Stiffness, lb-in2/ft width - Reinforced Sections 4 Rebar Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 5693,000,000703,000,000709,000,000719,000,000726,000,000 No. 6891,000,000907,000,000921,000,000933,000,000943,000,000 No. 71,119,000,0001,135,000,0001,155,000,0001,172,000,0001,186,000,000 No. 81,352,000,0001,383,000,0001,409,000,0001,432,000,0001,452,000,000 No. 91,593,000,0001,633,000,0001,667,000,0001,696,000,0001,722,000,000 No. 101,881,000,0001,931,000,0001,974,000,0002,011,000,0002,044,000,000 No. 112,167,000,0002,228,000,0002,279,000,0002,324,000,0002,364,000,000 Shear Capacity The structural capacity in shear is computed as the sum of the shear capacity of the Truline sheet pile section and the reinforced concrete. Thus, the nominal shear capacity, Vn, can be expressed as F s c nV V V V + + =Where: Vc is the shear strength provided by the concrete, Vs is the shear strength provided by the steel shear reinforcement, and VF is the shear strength provided by the Truline sheet piling. The allowable shear force for the Truline sections is shown in Table 10. 6Table 10 Allowable Shear Capacity of Gravel-filled Truline Sections Truline SectionAllowable Shear Capacity,* lb/ft Series 8006,313 Series 120017,689 * All pile sections must be filled with gravel to ensure the web is fully supported and the shear load is transferred from flange to flange by the fill material. Shear load must be applied by continuous beam or whaler on the face of the wall. For concrete members subjected to shear and flexure only, the shear strength provided by the concrete is (Eq. 11-3 from ACI 318-08): d b f Vw c c' = 2Where: =1.0 for normal weight concrete, bw is the width of the section, and d is the distance from the extreme compression edge to the centroid of the tension reinforcement.The shear capacities of sections without steel shear reinforcement are shown in Tables 11 and 12. Table 11 Series 800 Shear Capacity/ft width Vc +VF Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 lb/cell5,4405,6205,7905,9506,100 lb/ft10,88011,24011,58011,90012,200 Table 12 Series 1200 Shear Capacity/ft width Vc +VF Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 lb/cell18,34018,74019,12019,46019,800 lb/ft24,45025,00025,50025,95026,400 Additional shear strength can be provided by the addition of shear reinforcement steel to the section. The shear strength of the steel shear reinforcement is computed using sd f AVyt vs = 7Where: Av is the area of shear reinforcement within spacing s, fyt is the yield stress of the shear reinforcement, up to a maximum of 60,000 psi, and d is distance from the compression edge of the concrete to the centroid of the tension reinforcement. The spacing s is limited to a maximum of d/2 by ACI 318-08 Section 11.4.5.1. This means s will be no greater than 4.4 inches for No. 5 bars down to 4.2 inches for No. 11 bars. ACI 318-08 Section 11.4.6.1 requires a minimum shear reinforcement, Av,min, when Vu exceeds 0.5 | Vc. for sections greater than 10 inches in thickness. A Series 1200 section falls under this minimum reinforcement requirement. The minimum shear reinforcement is computed using (ACI 318-08 Eq. 11-13) ytwytwc vfs bfs bf A5075 . 0min ,> ' =Where: bw is the width of a section or 8.56 inches for an individual cell in a Series 1200 section. Here the value of bw is assumed equal to the width of an individual cell in a Truline wall section. The shear capacity of steel reinforcement is presented in Table 13. Table 13 Shear Capacity of Steel Reinforcement in Series 1200 Section for s =4.2 inches Bar SizeNo. 3No. 4 Vs, lb/cell3,0005,440 Vs, lb/ft4,0007,253 The ultimate (factored) shear capacity is computed using a strength reduction factor, |, of 0.75 (see ACI 318-08 Section 9.3.2.3). ( )F s c n uV V V V V + + = s | |As an example, the factored shear capacity of a Series 1200 wall, with fc =4,000 psi, Grade 60, No. 9 vertical reinforcement, and Grade 60, No. 3 shear reinforcement is ( ) lb/ft 100 , 22 lb/ft 000 , 4 lb/ft 500 , 25 75 . 0 = + =nV | 8Comparison of Unfilled and Concrete-Filled Sections The increase in moment capacity and bending stiffness achieved by filling the sections with reinforced concrete varies in proportion to the concrete compressive strength and amount of steel reinforcement. The ratios of increase for moment capacity and bending stiffness are presented in Tables 14 to 17.The increase in bending stiffness is much larger than the increase in moment capacity for all sections. The increase in moment capacity for concrete-filled walls allows higher wall sections to be constructed. The increase in bending stiffness for concrete-filled walls results in lower lateral deflections of the walls. Table 14 Ratio of Concrete-filled to Unfilled Moment Capacities for Series 800 Doubly Reinforced Sections Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 41.811.901.972.032.07 No. 52.392.502.602.692.78 No. 63.033.163.283.383.49 No. 73.793.944.074.194.31 No. 84.674.834.985.125.25 Table 15 Ratio of Concrete-filled to Unfilled Moment Capacities for Series 1200 Doubly Reinforced Sections Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 51.181.221.251.281.29 No. 61.531.581.621.661.70 No. 71.942.002.052.092.14 No. 82.422.482.542.592.64 No. 92.943.013.073.133.19 No. 103.603.683.753.813.87 No. 114.314.384.464.534.60 9Table 16 Ratio of Concrete-filled to Unfilled Bending Stiffness for Series 800 Sections Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 49.369.539.699.829.95 No. 512.3512.6612.9113.1613.33 No. 615.2715.7116.1316.4716.75 No. 718.3018.9719.5320.0020.40 No. 821.4122.2823.0123.6424.18 Table 17 Ratio of Concrete-filled to Unfilled Bending Stiffness for Series 1200 Sections Bar Size Concrete Compressive Strength, f'c psi 3,0003,5004,0004,5005,000 No. 57.387.497.557.667.73 No. 69.499.669.819.9410.05 No. 711.9212.0912.3112.4912.64 No. 814.4014.7315.0115.2615.47 No. 916.9717.4017.7618.0718.35 No. 1020.0420.5721.0321.4321.78 No. 1123.0923.7424.2824.7625.19 Analysis and Design of Cantilever Sheet Pile Walls A cantilever sheet pile wall consists of sheet piling driven deep enough into the ground to become a fixed, vertical cantilever. In a limit equilibrium analysis, this type of wall is supported by a lateral earth pressure equal to the difference between the active and passive earth pressures above and below a point of rotation. This type of wall is only suitable for walls of moderate height. In a limit equilibrium analysis, the earth pressure assumed to be acting on a cantilever sheet pile wall in cohesive soil is that shown in Figure 1. The earth pressure acting on the wall is assumed as the difference between the passive and active earth pressures. The wall is assumed not to move laterally at the pivot point.The distributions of earth pressures are different for cohesive and cohesionless soils. In addition, it is possible for the earth pressures in cohesive soils to vary with time, particularly when excavations are made in front of the wall. 10The analysis procedure for cantilever sheet pile walls in granular, cohesionless soils is as follows: (pppa) (pppa) (pppa) (pppa) Figure 1 Earth Pressures Acting on a Cantilever Sheet Pile Wall (1)Assume a trial depth of penetration, D. A starting value can be obtained from Table 18. Table 18 Approximate Values for Required Depths for Cantilever Sheet Pile Walls in Cohesionless Soil SPT Blowcount, NSPT, blows/ft Relative Density, DrDepth of Penetration 0-4Very loose2.0 H 5-10Loose1.5 H 11-30Medium1.25 H 31-50Dense1.0 H Over 50Very dense0.75 H H =height above the dredge line. (2)Determine the active and passive lateral earth pressure distributions on both sides of the wall. 11A1A2ABCO1E F JTOP OF GROUNDDREDGE LINEHDa pK D H DK ) ( + pK D H ) ( + a pDK K D H + ) (pDK aK D H ) ( + aDK E'P1P2P3A1A2ABCO1E F JTOP OF GROUNDDREDGE LINEHDa pK D H DK ) ( + pK D H ) ( + a pDK K D H + ) (pDK aK D H ) ( + aDK E'P1P2P3 Figure 2 Resultant Earth Pressure Diagram for Cantilever Sheet Pile in Cohesionless Soil (3)Determine the depth of wall needed to achieve static equilibrium for forces acting in the horizontal direction. 0 02 2 1= ECJ FBA A EAF F F .....................................................(1) Solve Equation 1 for the distance Z. For uniform cohesionless soil: ( )( )( ) D H K KD H K D KZa pa p22 2+ + =................................................... (2) 12Take moments about the tip of the wall at point F and check if the sum of moments is equal to zero. Revise the depth of penetration D until convergence (sum of moments equal to zero) is achieved. Add 20 to 40 percent to the calculated depth of penetration. This results in a factor of safety of approximately 1.5 to 2.0. Alternatively, one may use a reduced value of the passive earth pressure coefficient for design. A typical value is 50 to 75 percent of the maximum passive resistance. Compute the maximum bending moment developed in the wall prior to increasing the depth by 20 to 40 percent. The lateral displacement can be estimated by assuming that the wall is fixed at a depth of D and loaded by a triangular load equal to the actual applied active loading. The lateral movement at any distance y below the top of the wall is computed by( )5 4 524 560 + = y yEIPto................................................(3) Design Example of Cantilever Sheet Pile Wall in Cohesionless Soil The dimensions and soil properties of the example problem for a sheet pile wall in cohesionless soil is shown in Figure 3. The procedure for analyzing the wall is as follows. Compute Wall Pressures Acting on Wall The pressures in units of psf acting at points A1, A2, B, C, E, and Jshown in Figure 2 are: ( )( ) . 092 , 5 2 . 2220 . 374 2 . 22262 . 17 0 . 3740 . 374 ) 271 . 0 )( 12 )( 115 (11 21+ = + ' = = ' =+ = ' + == = =D HK K K D pD p K K D pD DK p pHK pp a p JA a p Ea A Aa A From statics0 =hFor( ) ( ) ( ) 02 2 22 2 1 1 21= + + + + +Dp pzp pDp p HpA E J E A A A 13Solving for z: BH =12 ftD =12.628 ftP1P2P3yzxPoint of Zero Shear Force =115 pcf =65 pcf| =35 deg.Ka=0.271Kp=3.690BH =12 ftD =12.628 ftP1P2P3yzxPoint of Zero Shear Force =115 pcf =65 pcf| =35 deg.Ka=0.271Kp=3.690 Figure 3 Example Problem for Sheet Pile Wall in Cohesionless Soil ( )J EA A Ep pHp D p pz =1 1 Summing moments about the bottom point on the wall: ( ) ( ) ( )6 6 6 2 3021 2222 21 1 21Dp pDp pzp pDpHD Hp MA A A E J E A A + + + + +|.|

\|+ = = The above equations can be solved by trial and error by assuming a value for D, computing z, and computing the sum of moments, varying D until the computed sum of moments is zero. Alternatively, the equations can be programmed in an electronic spreadsheet program and using the Goal Seek tool to obtain a solution. The solution for the above problem finds ft 082 . 2ft 628 . 12==zD 14Depth of Penetration for Design The depth of penetration is determined by increasing the value of D by 20 to 40 percent. In this case, D =15.2 to 17.7 ft. Use D =17 ft. Compute Location of Zero Shear Force The point of zero shear force requires the computation of the depths x (depth of zero shear force) and y (depth of zero net pressure on wall). The depth of zero net pressure is computed using ( )ft 683 . 1) 271 . 0 690 . 3 ( 650 . 3741== '=a pAK Kpy Compute Maximum Moment Developed in Wall The force resultants P1, P2, and P3 are computed using: ( )( )( )( ) lb 633 , 2ft 868 . 4) 271 . 0 690 . 3 ( 653 . 389 244 , 2 ( 2 2lb 3 . 389 ) 683 . 1 )( 0 . 374 (lb 244 , 2 ) 12 )( 0 . 374 (22132 12213 2 1211 212211 211= ' ==+= '+= ' = = += = == = =x K K PK KP Pxx K K P P Py p PH p Pa pa pa pAAThe maximum moment is computed using ft 550 . 10313 3 2 2 1 1 max=|.|

\|+ = + =yHP P P M lbs - in 800 , 260lbs - ft 730 , 21) 623 . 1 )( 633 , 2 ( ) 99 . 5 )( 3 . 389 ( ) 550 . 10 )( 244 , 2 (ft 623 . 13ft 990 . 532max32== + == ==|.|

\|+ =Mxxy 15Consulting Table 5, any of the sections with No. 7 or larger reinforcement are suitable for this wall. Note Regarding Passive Earth Pressure Coefficient In the above design example, the passive earth pressure coefficient was computed assuming that the wall is vertical and frictionless and that the ground surface behind the wall is horizontal. For these conditions, the computation of the passive earth pressure coefficient using Rankine theory is both appropriate and conservative.It is common practice when designing steel sheet pile walls to consider the effect of wall friction on the passive earth pressure coefficient. When doing so, many designers use the coefficients developed by Caquot and Kerisel (1948). These coefficients are illustrated in Figure A1 of the Appendix of this report and are reported in NAVFAC DM 7.02 (1986) in Figure 6 on page 7.2-67. This manual is available in PDF format for download from http://portal.tugraz.at/portal/page/portal/Files/i2210/files/eng_geol/NAVFAC_DM7_02.pdf. Design Example of Cantilever Sheet Pile Wall in Cohesive Soil Cantilever sheet pile structures are typically used for small walls. The analysis that follows was originally developed by Blum (1931) and is for under short-term loading conditions. For long-term loading conditions, the analysis is made using the analysis for cantilever sheet pile walls in cohesionless soils using the fully drained shearing properties of the soil presented above. The earth pressures acting on a cantilever sheet pile wall in cohesive soils are shown in Figure 4. The dimensions and soil properties for this design example are: H =14 ft, =120 pcf, ' =60 pcf, c =500 psf Compute the following quantities: H0 =2c/ =8.333 ft H H0 =5.67 ft H 2 c =680 psf 4c H =320 psf 4c + H =3,680 psf 16HDecH20 =H ce + 4H ce 4( ) c H Z pe p4 + = DREDGE LINEc 4zc H 2 c 2HDecH20 =H ce + 4H ce 4( ) c H Z pe p4 + = DREDGE LINEc 4zc H 2 c 2 Figure 4 Earth Pressures for Cantilever Sheet Pile Wall in Cohesive Soil Sum horizontal forces: ( ) ( ) 0 42820 21= + D H cczH H c H Solve for z ( ) ( )( )cH H c H H c Dz82 4 20 = Sum moments about tip of wall ( )( ) ( ) 02468322 200 21= + |.|

\| + =DH ccz H HD H H c H M 17Computation strategy: 1.Assume a value for D. 2.Calculate z. 3.Calculate sum of moments. 4.Repeat until convergence (sum of moments) is achieved. Alternatively, one may solve the quadratic equation (sum of moments equation) for D. The solution for the above given data: z =1.300 ft D =14.147 ft The solution values are quite sensitive to the input values. If the value of cohesion is varied plus of minus 5 percent the values of H =22.532 ft for c =475 psf and H =9.75 ft for c =525 psf. Similar sensitivity is found for slight variations in unit weight and geometry. Selection of Wall Section The selection of the wall section is made based on the moment developed in the wall. The design moment is computed at the dredge line. For the above conditions, the moment is ( ) ( )( )( ) ( )lb - in 700 , 43lb - ft 640 , 3) ft 1 ( psf 680 ft 67 . 5ft 1 226120 61=== = c H H H M Checking the level of moment developed in the wall against the allowable moment capacity values shown in Table 2, a Series 800 section with a concrete compressive strength of 3,500 psi and reinforced with (1) No. 5 bar will be adequate for this application for short-term, undrained conditions. 18Analysis and Design of Anchored Sheet Pile WallsA number of design methodologies are used to design anchored sheet pile walls. The USS Steel Sheet Pile Design Manual provides the details on the free earth support method, Rowes moment reduction method, the fixed earth support method (equivalent beam method), graphical methods, and design using the Danish rules. Each of these methods is a hand computation method that assumes relatively simple soil profiles and do not require use of a computer program. For complicated problems, including staged construction, the computer program PYWall from Ensoft, Inc. may be considered for use. PYWall considers the nonlinear lateral load-transfer properties of the soil and may consider multiple levels of tiebacks, struts, and braces. Additional information about PYWall may be obtained from www.ensoftinc.com.Free Earth Support Method Anchored walls are supported by passive resistance at the toe of the wall and the anchor tie rods at the top of the wall. Wall heights may extend up to 25 feet, depending on local soil conditions. A procedure for the free earth support method is the following:1.Compute the active and passive lateral pressured using appropriate coefficients of lateral earth pressures. (See Figure 5) 2.Calculate the weight of overburden and surcharge loads at the dredge level, ' H.3.Calculate the point of zero pressure using a pap pK Hy'= .............................................................(4) 19ayHwH1D1(pppa)D1pppae H KaLowWaterTCohesionless Soile H Ka H1' KaPa KaayHwH1D1(pppa)D1pppae H KaLowWaterTCohesionless Soile H Ka H1' KaPa Ka HwH1De HPaLowWaterTL2queHpaHtCohesive SoilHwH1De HPaLowWaterTL2queHpaHtCohesive Soil Figure 5 Earth Pressure Distributions Used in Design of Anchored Sheet Piling by Free Earth Support Method for Cohesionless and Cohesive Soils 4.Calculate Pa, the resultant force of the earth pressure above point a, and its distance, L, below the tie rod elevation. Static equilibrium is attained by making the wall deep enough that the moment due to the net passive pressure will balance the moment due to the resultant active force, Pa. Sum moments about the tie rod level. 0 ) ( ) ( ) )( (1 3221 21= + + =D y H D p p P L Mt a p a ............................. (5) Solving for D1; usually a trial and error solution is used. Alternatively, the equation can be solved using the Goal Seek option in an electronic spreadsheet program. Compute the tension in the tie rod by 21 21) ( D p p P Ta p a =The maximum bending moment occurs at the point of zero shear force in the wall below the tie rod elevation. 20Select the appropriate sheet pile section for the maximum moment developed. Add 20 to 40 percent to D1 to provide a margin of safety or divide the passive resistance force Pp by a factor of safety of 1.5 to 2.0. Design Example of Anchored Sheet Pile Wall Using Free Earth Support Method Granular SoilCompute Distribution of Earth Pressures Acting on Wall The earth pressures values at points B, C1, C2, and E (see Figure 6) are: psf 0 . 149 ) 271 . 0 )( 5 )( 110 (1= = =a BK H p psf 4 . 360 ) 271 . 0 )( 13 ( 60 0 . 1491= + = + =a w B CK H P p psf 2 . 368 271 . 0 )] 13 ( 60 ) 5 ( 110 [ ] ' [1 2= + = + =a w e CK H H p 1 1 18 . 216 ) 271 . 0 613 . 3 ( 65 ) ( ' D D D K K pa p E= = = 1245Sand Backfill =110 pcf =60 pcf| =35 deg.Ka=0.271Medium Sand =65 pcf| =34.5 deg.Ka=0.277Kp=3.61ABC1C2DEH1=5 ftHw=13 ftHt=13.5 ftDD14.5 ftyH =18 ftT31245Sand Backfill =110 pcf =60 pcf| =35 deg.Ka=0.271Medium Sand =65 pcf| =34.5 deg.Ka=0.277Kp=3.61ABC1C2DEH1=5 ftHw=13 ftHt=13.5 ftDD14.5 ftyH =18 ftT3 Figure 6 Earth Pressure Distributions Used in Design Example for Anchored Sheet Pile Wall in Cohesionless Soil 21Compute Depth of Point of Zero Net Pressure The depth of zero net pressure below the dredge line is computed by ft 662 . 1) 271 . 0 614 . 3 ( 654 . 360) ( '2===a pCK Kpy Compute Force Resultants of Pressures Acting on Wall The resultant forces acting on the wall are: 21 1 1 211 215212 214211 2132211 2114 . 108 ) 8 . 216 )( (lb 0 . 306 ) 662 . 1 )( 2 . 368 )( (lb . 374 , 1 ) 0 . 149 4 . 360 )( 13 )( ( ) (lb . 938 , 1 ) 0 . 149 )( 13 (lb 6 . 372 ) 0 . 149 )( 5 )( (D D D D p Py p Pp p H Pp H Pp H PECB C wB wB= = == = == = == = == = = Compute Sum of Moments Equate the sum of moments acting about the tieback to zero and solve for D1. Table 19 Sum of Moment Computations ForceForce, lbsArm, ftMoment, ft-lbs P1372.6-1.167-435 P21,938.7.0013,563 P31,374.9.16712,594 P4306.014.6084,470 P5 214 . 108 D 3 / 2 662 . 1 5 . 131D + +21 128 . 72 644 , 1 D D Total. 192 , 30 . 644 , 1 28 . 722131+ D D Solve for D1 and Select Value of D Solve for D1 by trial and error or by spreadsheet solution using the Goal Seek function. ft 62 . 5 662 . 1 956 . 3ft 956 . 311= + = + ==y D DD 22Evaluate P5 using value of D1. lb 696 , 1 ) 956 . 3 ( 4 . 108 4 . 1082 21 5 = = = D PTo provide a margin of safety, increase D by 20 to 40 percent (6.74 to 7.86 ft). Use D =7.5 ft for the design. Compute Tension Force in Tieback h lb/ft widt 294 , 2696 , 1 0 . 306 374 , 1 937 , 1 6 . 3725 4 3 2 1 = + + + =+ + + + = P P P P P T For sizing of tieback, increase by 33%: Use 3,340 lb/ft width for design. The total force in an individual tieback will depend on the lateral spacing between tiebacks. Prior to designing the tieback, it is necessary for the geotechnical engineer to perform a slope stability analysis to determine the position of the potential slip surface. The anchor block for the tieback must be located beyond the slip surface in order for the tieback to perform as designed. Compute Location of Point of Maximum Moment The location of the maximum moment, x, is below the water level at location of zero shear force. In this case, it is possible to write a quadratic equation using the horizontal tieback forces. r below wate ft732 . 80 921 , 1 0 . 149 13 . 8'221 1 1== ++ + +xx xx K x p P Ta b Compute Maximum Moment in Wall Section The maximum moment is computed at the point of zero shear force. The maximum moment developed in the wall is 23 ( ) ( ) ( )kips - in 8 . 117lbs - in 117,800lb - ft 9,8153ft) (1 '2 331 1 21211 max = = = + |.|

\|+ =w t a BH H x TxKxp xHP M There are two acceptable reinforcing designs, using the 800 series, for the wall in this example.Eithera 2 bar design with No. 7(or larger) rebar as seen in Table 3, or alternately a 4 bar design with No. 5(or larger) rebar as seen in Table 4 are acceptable for this wall section. Design Example Using PYWall 3.0 Computer PYWall 3.0 from Ensoft (www.ensoftinc.com) considers soil-structure interaction by using a generalized beam-column model and analyzes the behavior of a flexible retaining wall or soldier-pile wall with or without tiebacks or bracing systems. Unlike the limit equilibrium analysis methods discussed previously, PYWall solves the nonlinear differential equation for a beam column that ensures compatibility of displacements of the wall and resistance forces exerted by the soil. In addition, PYWall can include the force versus deformation behavior of the tiebacks used for anchored walls. The output from PYWall includes computation and graphs of lateral deflection, bending moment, and shear force versus wall elevation. The following is a design example for the 20-ft high wall shown in Figure 7.The soil properties for this example problem are shown in Table 20. Table 20 Example Soil Properties for Example Using PYWall Soil Type Depth Range, ft Total Unit Weight Cohesionc50k, pci Stiff Clay w/o Free Water 023 ft 0276 in. 124 pcf 0.0718 pci 1,750 psf 12.15 psi 0.007600 Stiff Clay w/o Free Water 2360 ft 276720 in. 124 pcf 0.0718 pci 3,000 psf 20.83 psi 0.0051,000 2420 ft35 ft10 ft5 ft5 ft20 ft35 ft10 ft5 ft5 ft Figure 7 Example Problem for PYWall Compute Earth Pressure Distribution Acting on Wall The earth pressure distribution in stiff clay is based on the recommendations of Article 46 in Terzaghi, Peck, and Mesri (1996) and is illustrated in Figure 7. In this example, the maximum earth pressure for a 12-inch wide section of wall is set equal to 0.35 H w =(0.35) (0.0718 pci) (240 in) (12 in) =72.4 lb/in. The earth pressure distribution varies from zero at the top of wall to the peak value at one-quarter of the wall height and from the peak value at three-quarters of the wall height to zero at the bottom of the exposed face of the wall. Compute Equivalent Spring Constant for Tieback The tieback anchor will be modeled as a linear spring based on the extension of the tieback anchor bar attached to a unyielding anchor block. The spring constant is equal to tiebacktiebackLsAEk =..........................................................(6) Where A =cross sectional area of tieback anchor bar, E =Youngs modulus, L =length of anchor bar, and stieback =horizontal spacing between tieback anchor bars. For a No. 9 bar, 25 ft long, and a spacing of 6 ft, the anchor spring constant for a 12-inch wide section of wall is (1.00 in2)(29,000,000 psi)(12 in.)/((300 in.)(72 in.)) =16,110 lb/inch. 25Computed Results The input value for bending stiffness is for a 12-inch wide vertical section, as listed in Tables 6-9 for the two different Truline sections and different reinforcing options. In a PYWall analysis, the moment developed is related to the bending stiffness of the wall. In general, the wall deflection will decrease as the bending stiffness increases. So, it is necessary to try different reinforcement options to determine the most economical wall section to use. A set of computed results is shown in Table 21. Table 21 Results Computed by PYWall 3.0 for Example Problem Wall Reinforcement EI, lb-in2 Mult, in-lbs Mmax, in-lbs Mmax/Mult Maximum Deflection Anchor Force No. 5709,000,000168,800365,0002.1623.2905,600 No. 6921,000,000218,200367,0001.6822.6005,610 No. 71,155,000,000275,600368,0001.3352.1305,620 No. 81,409,000,000341,800369,0001.0801.8005,630 No. 91,667,000,000413,300370,0000.8951.5605,640 No. 101,974,000,000503,900367,0000.7281.3505,390 No. 112,279,000,000600,100371,0000.6181.2005,650 An examination of these results finds that a Series 1200 wall section with a concrete compressive strength of 4,000 psi and reinforced with No. 9 bars is the minimum section that will work for this example problem. The following figures are graphs of lateral deflection, bending moment and shear force versus depth below the top of wall for the Series 1200 wall section with a concrete compressive strength of 4,000 psi and reinforced with No. 9 bars.The lateral deflection profile is shown in Figure 8. The peak deflection is 1.56 inches and is developed at an elevation approximately 10 feet below the top of the wall. The bending moment profile is shown in Figure 9. The peak moment occurs at an elevation approximately 10 feet below the top of the wall. This is the same location as for the peak lateral deflection. 2605101520253035-0.5 0 0.5 1 1.5 2Depth in FeetLateral Deflection, Inches05101520253035-0.5 0 0.5 1 1.5 2Depth in FeetLateral Deflection, Inches Figure 8 Lateral Deflection vs. Depth Below Pile Head 05101520253035-400,000 -200,000 0 200,000 400,000Depth in FeetBending Moment, in-lbs05101520253035-400,000 -200,000 0 200,000 400,000Depth in FeetBending Moment, in-lbs Figure 9 Bending Moment vs. Depth Below Pile Head 27The shear force profile is shown in Figure 10. There are two locations of high shear force in the wall and one location of high shear at the connection point for the tieback anchor. The two high shear force values are 7,390 lbs just above the dredge line (20 ft) and the maximum value of 8,710 lbs at 25.5 feet below the top of the wall.The shear force due to the tieback is 5,640 lb/ft width. The actual force developed in the tieback will depend on the horizontal spacing between tiebacks. For example, if the tiebacks are spaced 6 ft apart on centers, the force in an individual tieback is 33,840 lbs. 05101520253035-10,000 -5,000 0 5,000 10,000Depth in FeetShear Force, lbs05101520253035-10,000 -5,000 0 5,000 10,000Depth in FeetShear Force, lbs Figure 10 Shear Force vs. Depth Below Pile Head Optimization of Design It is possible to optimize the design once the section and reinforcement are selected to provide adequate strength and acceptable lateral deflection. Typically, one may shorten the length of the pile sections either if the analysis of moment and deflection finds at portion of the wall with minimal values, or one may elect not to place concrete and reinforcing steel below the depth and use gravel-filled sections where the moment developed in the wall is lower than the allowable moment capacities for gravel-filled walls presented in Table 1. 28The figures 11, 12 and 13 compare the results of the first analysis, shown in blue, to an optimized analysis, shown in green, in which the length of the wall sections was reduced from 35 to 30 feet and a gravel-filled section was used from 25 to 30 feet. For the optimized design, maximum wall deflection increased from 1.55 to 1.64 inches. The maximum mobilized moment increased from 369,000 to 379,000 in-lbs. The maximum shear force increased from 9,560 to 10,800 lbs. 05101520253035-0.5 0 0.5 1 1.5 2Depth in FeetLateral Deflection, Inches05101520253035-0.5 0 0.5 1 1.5 2Depth in FeetLateral Deflection, Inches Figure 11 Comparison of Computed Wall Deflections In performing the optimization of design, the moment versus depth diagram was examined to determine the depth below which a gravel-filled section would not be over-loaded and the depth below which moments are negligible. The optimized design changes were made to the numerical model and the wall was re-analyzed. The changes in wall performance were judged to be acceptable. 2905101520253035-400,000 -200,000 0 200,000 400,000Depth in FeetBending Moment, in-lbs05101520253035-400,000 -200,000 0 200,000 400,000Depth in FeetBending Moment, in-lbs Figure 12 Comparison of Computed Wall Moments 05101520253035-15000 -10000 -5000 0 5000 10000Depth in FeetShear Force, lbs05101520253035-15000 -10000 -5000 0 5000 10000Depth in FeetShear Force, lbs Figure 13 Comparison of Computed Wall Shear Forces 30References Blum, H. (1931). Einspannungsverhltnisse bei Bolhwerken. Wilh. Ernst und Sohn, Berlin. Caquot, A., and Kerisel, J . (1948). Tables for the Calculation of Passive Pressure, Active Pressure and Bearing Capacity of Foundations, Transl. from French by M. A. Bec, Gauthier-Villars, Paris, 120 p. Clayton, C. R. I., and Milititsky, J . (1986). Earth Pressure and Earth Retaining Structures, Surrey University Press, 300 p. Naval Facilities Engineering Command, (1986). NAVFAC Design Manual 7.2, Foundations and Earth Structures, US Government Printing Office. Terzaghi, K., Peck, R. B., and Mesri, G. (1996). Soil Mechanics in Engineering Practice, 3rd Edition, Wiley, New York, 549 p. United States Steel Corporation (1975). USS Steel Sheet Piling Design Manual, 132 p. Wang, S.-T. (2007). Computer Program PYWall, Version, 3.0, Ensoft, Inc., Austin, Texas, www.ensoftinc.com. 31Appendix A Earth Pressure Coefficients for Wall Friction and Sloping Backfill Figure A1 Active and Passive Earth Pressure Coefficients with Wall Friction and Sloping Backfill (from NAVFAC, 1986 and Caquot and Kerisel, 1948)