# Worked Example | Analysis and Design of Steel Sheet Pile Wall (EN 1997-1)

This article contains a solved example of the analysis and design of steel sheet pile walls in accordance with BS EN1997-1:2004 â€“ Code of Practice for Geotechnical design and the UK National Annex.

Geometry
Total length of sheet pile provided Hpile = 14500 mm
Number of different types of soil Ns = 2
Retained height of soil dret = 3500 mm
Depth of unplanned excavation dex = 500 mm
Total retained height ds = 4000 mm
Angle of retained slope Î² = 0.0 deg
Depth from ground level to top of water table retained side dw = 1500 mm
Depth from ground level to top of water table retaining side;Â  dwp = 4000 mm

Variable surcharge po,Q = 10.0 kN/m2

Soil layer 1
Characteristic shearing resistance angle Ï•â€™k,s1 = 30.0 deg
Characteristic wall friction angleÂ  Î´k,s1 = 20.0 deg
Moist density of soil Î³m,s1 = 15.0 kN/m3
Characteristic saturated density of retained soil Î³s,s1 = 17.0 kN/m3
Height of soil 1Â  h1 = 8500 mm

Soil layer 2
Characteristic shearing resistance angle fâ€™k,s2 = 27.0 deg
Characteristic wall friction angle dk,s2 = 16.0 deg
Moist density of soil Î³m,s2 = 16.0 kN/m3
Characteristic saturated density of retained soil Î³s,s2 = 19.0 kN/m3
Height of soil 2 h2 = 7000 mm

Partial factors on actions â€“ Section A.3.1 â€“ Combination 1
Permanent unfavourable action Î³G = 1.35
Permanent favourable action Î³G,f = 1.00
Variable unfavourable action Î³Q = 1.50
Angle of shearing resistance Î³Ï•â€™ = 1.00
Weight density Î³g = 1.00

Design soil properties â€“ soil 1
Design effective shearing resistance angle Ï•â€™d = tan-1[tan(Ï•â€™k)/Î³Ï•â€™] = 30.0 deg
Design wall friction angleÂ  Î´d = tan-1[tan(Ï•k)/Î³Ï•â€™] = 20.0 deg
Design moist density of retained soil Î³m.d1 = Î³m/Î³Î³ = 15.0 kN/m3
Design saturated density of retained soil Î³s.d1 = Î³s/Î³Î³ = 17.0 kN/m3
Design buoyant density of retained soil Î³d.d1 = Î³s.d1 â€“ Î³w = 7.2 kN/m3

Active pressure using Coulomb theory Ka1 = sin(Î± + Ï•â€™d)2 / (sin(Î±)2 Ã— sin(Î± â€“ Î´d) Ã— (1 + âˆš(sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d â€“ Î²)/(sin(Î± â€“ Î´d) Ã— sin(Î± + Î²))))2) = 0.297

Passive pressure using Coulomb theory Kp1 = sin(90 â€“ Ï•â€™d)2 / (sin(90 + Î´d) Ã— [1 â€“ âˆš[sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d) / (sin(90 + Î´d))]]2) = 6.105

Design soil properties â€“ soil 2
Design effective shearing resistance angle Ï•â€™d = tan-1(tan(Ï•â€™k) /Î³Ï•â€™) = 27.0 deg
Design wall friction angleÂ  Î´d = tan-1(tan(Î´k)/Î³Ï•â€™) = 16.0 deg
Design moist density of retained soil Î³m.d2 = Î³m/Î³Î³ = 16.0 kN/m3
Design saturated density of retained soil Î³s.d2 = Î³s /Î³Î³ = 19.0 kN/m3
Design buoyant density of retained soil Î³d.d2 = Î³s.d2 â€“ Î³w = 9.2 kN/m3

Active pressure using Coulomb theory Ka2 = sin(Î± + Ï•â€™d)2 / (sin(Î±)2 Ã— sin(Î± â€“ Î´d) Ã— (1 + âˆš(sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d â€“ Î²) / (sin(Î± â€“ Î´d) Ã— sin(Î± + Î²))))2) = 0.336

Passive pressure using Coulomb theory Kp2 = sin(90 â€“ Ï•â€™d)2 / (sin(90 + Î´d) Ã— [1 â€“ âˆš[sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d) / (sin(90 + Î´d))]]2) = 4.416

Overburden on the active side
Overburden at 0 mm below GL in soil 1; OBâ€™a11 = po,Q Ã— Î³Q = 15.0 kN/m2
Overburden at 1500 mm below GL in soil 1; OBâ€™a21 = Î³G Ã— Î³m.d1 Ã— ha1 + OBâ€™a11 = 45.4 kN/m2
Overburden at 4000 mm below GL in soil 1; OBâ€™a31 = Î³G Ã— Î³d.d1 Ã— ha2 + OBâ€™a21 = 69.6 kN/m2
Overburden at 8500 mm below GL in soil 1; OBâ€™a41 = Î³G Ã— Î³d.d1 Ã— ha3 + OBâ€™a31 = 113.3 kN/m2
Overburden at 8500 mm below GL in soil 2; OBâ€™a42 = Î³G Ã— Î³d.d1 Ã— ha3 + OBâ€™a31 = 113.3 kN/m2
Overburden at 11544 mm below GL in soil 2; OBâ€™a51 = Î³G Ã— Î³d.d2 Ã— ha4 + OBâ€™a42 = 151.1 kN/m2

Overburden on the passive side
Overburden at 4000 mm below GL in soil 1; OBâ€™p31 = 0 kN/m2 = 0.0 kN/m2
Overburden at 8500 mm below GL in soil 1; OBâ€™p41 = Î³G,f Ã— Î³d.d1 Ã— hp3 + OBâ€™p31 = 32.4 kN/m2
Overburden at 8500 mm below GL in soil 2;Â OBâ€™p42 = Î³G,f Ã— Î³d.d1 Ã— hp3 + OBâ€™p31 = 32.4 kN/m2
Overburden at 11544 mm below GL in soil 2;Â OBâ€™p51 = Î³G,f Ã— Î³d.d2 Ã— hp4 + OBâ€™p42 = 60.3 kN/m2

Pressure on the active side
Active at 0 mm below GL in soil 1;Â pâ€™a11 = Ka1 Ã— OBâ€™a11 = 4.5 kN/m2
Active at 1500 mm below GL in soil 1; pâ€™a21 = Ka1 Ã— OBâ€™a21 = 13.5 kN/m2
Active at 4000 mm below GL in soil 1; pâ€™a31 = Ka1 Ã— OBâ€™a31 + Î³Î³ Ã— Î³w Ã— (dL3 â€“ dw) = 53.8 kN/m2
Active at 8500 mm below GL in soil 1; pâ€™a41 = Ka1 Ã— OBâ€™a41 + Î³Î³ Ã— Î³w Ã— (dL4 â€“ dw) = 126.4 kN/m2
Active at 8500 mm below GL in soil 2; pâ€™a42 = Ka2 Ã— OBâ€™a42 + Î³Î³ Ã— Î³w Ã— (dL4 â€“ dw) = 130.8 kN/m2
Active at 11544 mm below GL in soil 2; pâ€™a51 = Ka2 Ã— OBâ€™a51 + Î³Î³ Ã— Î³w Ã— (dL5 â€“ dw) = 183.8 kN/m2

Pressure on the passive side
Passive at 4000 mm below GL in soil 1; pâ€™p31 = Kp1 Ã— OBâ€™p31 + Î³G,f Ã— Î³w Ã— (dL3 â€“ max(ds, dw)) = 0.0 kN/m2
Passive at 8500 mm below GL in soil 1; pâ€™p41 = Kp1 Ã— OBâ€™p41 + Î³G,f Ã— Î³w Ã— (dL4 â€“ max(ds, dw)) = 241.7 kN/m2
Passive at 8500 mm below GL in soil 2; pâ€™p42 = Kp2 Ã— OBâ€™p42 + Î³G,f Ã— Î³w Ã— (dL4 â€“ max(ds, dw)) = 187.0 kN/m2
Passive at 11544 mm below GL in soil 2; pâ€™p51 = Kp2 Ã— OBâ€™p51 + Î³G,f Ã— Î³w Ã— (dL5 â€“ max(ds, dw)) = 340.4 kN/m2

By iteration the depth at which the active moments equal the passive moments has been determined as 11544 mm as follows:-

Moment level 1;Ma11 = 0.5 Ã— pâ€™a11 Ã— ha1 Ã— ((H â€“ dL2) + 2/3 Ã— ha1) = 36.9 kNm/m
Moment level 1; Ma12 = 0.5 Ã— pâ€™a21 Ã— ha1 Ã— ((H â€“ dL2) + 1/3 Ã— ha1) = 106.7 kNm/m
Moment level 2; Ma21 = 0.5 Ã— pâ€™a21 Ã— ha2 Ã— ((H â€“ dL3) + 2/3 Ã— ha2) = 155.3 kNm/m
Moment level 2; Ma22 = 0.5 Ã— pâ€™a31 Ã— ha2 Ã— ((H â€“ dL3) + 1/3 Ã— ha2) = 563.5 kNm/m
Moment level 3; Ma31 = 0.5 Ã— pâ€™a31 Ã— ha3 Ã— ((H â€“ dL4) + 2/3 Ã— ha3) = 731.8 kNm/m
Moment level 3; Ma32 = 0.5 Ã— pâ€™a41 Ã— ha3 Ã— ((H â€“ dL4) + 1/3 Ã— ha3) = 1292.3 kNm/m
Moment level 4;Â Ma41 = 0.5 Ã— pâ€™a42 Ã— ha4 Ã— ((H â€“ dL5) + 2/3 Ã— ha4) = 404.0 kNm/m
Moment level 4;Â Ma42 = 0.5 Ã— pâ€™a51 Ã— ha4 Ã— ((H â€“ dL5) + 1/3 Ã— ha4) = 283.8 kNm/m

Moment level 3; Mp31 = 0.5 Ã— pâ€™p31 Ã— hp3 Ã— ((H â€“ dL4) + 2/3 Ã— hp3) = 0.0 kNm/m
Moment level 3; Mp32 = 0.5 Ã— pâ€™p41 Ã— hp3 Ã— ((H â€“ dL4) + 1/3 Ã— hp3) = 2471.0 kNm/m
Moment level 4; Mp41 = 0.5 Ã— pâ€™p42 Ã— hp4 Ã— ((H â€“ dL5) + 2/3 Ã— hp4) = 577.6 kNm/m
Moment level 4; Mp42 = 0.5 Ã— pâ€™p51 Ã— hp4 Ã— ((H â€“ dL5) + 1/3 Ã— hp4) = 525.7 kNm/m

Total active moment; SMa = 3574.5 kNm/m
Total passive moment;Â SMp = 3574.5 kNm/m

Required pile length
Length of pile required to balance moments; H = 11544 mm

Depth of equal pressure; dcontra = 5432 mm
Add 20% below this point;Â de_add = 1.2 Ã— (H â€“ dcontra) = 7334 mm

Minimum required pile length; Htotal = dcontra + de_add = 12766 mm

Pass â€“ Provided length of sheet pile greater than the minimum required length of the pile

Pile capacity (EN1993-5)
Maximum moment in pile (from analysis); Mpile = max(abs(Mmin), abs(Mmax)) / 1m = 547.0 kNm/m
Maximum shear force in pile (from analysis); Vpile = 364.7 kN/m
Nominal yield strength of pile; fy_pile = 355 N/mm2
Name of sheet pile;Â  Arcelor PU(18)
Classification of pile; 2
Plastic modulus of pile;Â Wpl.y = 2134 cm3/m

Shear buckling of web (cl.5.2.2(6))
Width of section; c = h / sin(Î±pile) = 510 mm
Thickness of web;Â  tw = s = 9.0 mm
Îµ = âˆš(235/fy_pile) = 0.814
c/tw = 56.6 = 69.6Îµ < 72Îµ
PASS â€“ Shear buckling of web within limits

Bending 2
Interlock reduction factor (cl.5.2.2); Î²B = 1
Design bending resistance (eqn.5.2);Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Mc,Rd = Wpl.y Ã— fy_pile Ã— Î²B / Î³M0 = 757.6 kNm/m
PASS â€“ Moment capacity exceeds moment in pile

Shear
Projected shear area of web (eqn.5.6); Av = s Ã— (h â€“ t) = 3769 mm2
Design shear resistance (eqn.5.5); Vpl,Rd = Av Ã— fy_pile / (âˆš(3) Ã— Î³M0) / b = 1287.6 kN/m
PASS â€“ Shear capacity exceeds shear in pile

Partial factors on actions â€“ Section A.3.1 â€“ Combination 2

Permanent unfavourable action;Â Â Î³G = 1.00
Permanent favourable action; Î³G,f = 1.00
Variable unfavourable action; Î³Q = 1.30
Angle of shearing resistance; Î³Ï•â€™ = 1.25
Weight density; Î³Î³ = 1.00

Design soil properties â€“ soil 1
Design effective shearing resistance angle; Ï•â€™d = tan-1(tan(Ï•â€™k)/Î³Ï•â€™) = 24.8 deg
Design wall friction angle;Â  Î´d = tan-1(tan(Î´k)/Î³Ï•â€™) = 16.2 deg
Design moist density of retained soil; Î³m.d1 = Î³m/Î³Î³ = 15.0 kN/m3
Design saturated density of retained soil; Î³s.d1 = Î³s/Î³Î³ = 17.0 kN/m3
Design buoyant density of retained soil;Â Î³d.d1 = Î³s.d1 â€“ Î³w = 7.2 kN/m3

Active pressure using Coulomb theory; Ka1 = sin(Î± + Ï•â€™d)2 / (sin(Î±)2 Ã— sin(Î± â€“ Î´d) Ã— (1 + âˆš(sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d â€“ Î²)/(sin(Î± â€“ Î´d) Â´ sin(Î± + Î²))))2) = 0.364

Passive pressure using Coulomb theory;Â Kp1 = sin(90 â€“ Ï•â€™d)2 / (sin(90 + Î´d) Ã— [1 â€“ âˆš[sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d) / (sin(90 + Î´d))]]2) = 3.977

Design soil properties â€“ soil 2

Design effective shearing resistance angle;Â Ï•â€™d2 = tan-1(tan(Ï•â€™k)/Î³Ï•â€™) = 22.2 deg
Design wall friction angle;Â Î´d2 = tan-1(tan(Î´k)/Î³Ï•â€™) = 12.9 deg
Design moist density of retained soil; Î³m.d2 = Î³m/Î³Î³ = 16.0 kN/m3
Design saturated density of retained soil; Î³s.d2 = Î³s/Î³Î³ = 19.0 kN/m3
Design buoyant density of retained soil;Â Î³d.d2 = Î³s.d2 â€“ Î³w = 9.2 kN/m3

Active pressure using Coulomb theory; Ka2 = sin(Î± + Ï•â€™d)2 / (sin(Î±)2 Ã— sin(Î± â€“ Î´d) Ã— (1 + âˆš(sin(Ï•â€™d + Î´d) Ã— sin(Ï•â€™d â€“ Î²) / (sin(Î± â€“ Î´d) Ã— sin(Î± + Î²))))2) = 0.406

Passive pressure using Coulomb theory; Kp2 = sin(90 â€“ Ï•â€™d)2 / (sin(90 + Î´d) Ã— [1 â€“ âˆš[sin(Ï•â€™d + Î´d) Ã— sin(fâ€™d) / (sin(90 + Î´d))]]2) = 3.154

Overburden on the active side
Overburden at 0 mm below GL in soil 1; OBâ€™a11 = po,Q Ã— Î³Q = 13.0 kN/m2
Overburden at 1500 mm below GL in soil 1;Â  OBâ€™a21 = Î³G Ã— Î³m.d1 Ã— ha1 + OBâ€™a11 = 35.5 kN/m2
Overburden at 4000 mm below GL in soil 1; OBâ€™a31 = Î³G Ã— Î³d.d1 Ã— ha2 + OBâ€™a21 = 53.5 kN/m2
Overburden at 8500 mm below GL in soil 1; OBâ€™a41 = Î³G Ã— Î³d.d1 Ã— ha3 + OBâ€™a31 = 85.8 kN/m2
Overburden at 8500 mm below GL in soil 2; OBâ€™a42 = Î³G Ã— Î³d.d1 Ã— ha3 + OBâ€™a31 = 85.8 kN/m2
Overburden at 12532 mm below GL in soil 2;OBâ€™a51 = Î³G Ã— Î³d.d2 Ã— ha4 + OBâ€™a42 = 122.9 kN/m2

Overburden on the passive side
Overburden at 4000 mm below GL in soil 1; OBâ€™p31 = 0 kN/m2 = 0.0 kN/m2
Overburden at 8500 mm below GL in soil 1;Â OBâ€™p41 = Î³G,f Ã— Î³d.d1 Ã— hp3 + OBâ€™p31 = 32.4 kN/m2
Overburden at 8500 mm below GL in soil 2;Â OBâ€™p42 = Î³G,f Ã— Î³d.d1 Ã— hp3 + OBâ€™p31 = 32.4 kN/m2
Overburden at 12532 mm below GL in soil 2;OBâ€™p51 = Î³G,f Ã— Î³d.d2 Ã— hp4 + OBâ€™p42 = 69.4 kN/m2

Pressure on the active side

Active at 0 mm below GL in soil 1; pâ€™a11 = Ka1 Ã— OBâ€™a11 = 4.7 kN/m2
Active at 1500 mm below GL in soil 1; pâ€™a21 = Ka1 Ã— OBâ€™a21 = 12.9 kN/m2
Active at 4000 mm below GL in soil 1; pâ€™a31 = Ka1 Ã— OBâ€™a31 + Î³G Ã— Î³w Ã— (dL3 â€“ dw) = 44.0 kN/m2
Active at 8500 mm below GL in soil 1; pâ€™a41 = Ka1 Ã— OBâ€™a41 + Î³G Ã— Î³w Ã— (dL4 â€“ dw) = 99.9 kN/m2
Active at 8500 mm below GL in soil 2; pâ€™a42 = Ka2 Ã— OBâ€™a42 + Î³G Ã— Î³w Ã— (dL4 â€“ dw) = 103.5 kN/m2
Active at 12532 mm below GL in soil 2; pâ€™a51 = Ka2 Ã— OBâ€™a51 + Î³G Ã— Î³w Ã— (dL5 â€“ dw) = 158.1 kN/m2

Pressure on the passive side

Passive at 4000 mm below GL in soil 1; pâ€™p31 = Kp1 Ã— OBâ€™p31 + Î³G,f Ã— Î³w Ã— (dL3 â€“ max(ds, dw)) = 0.0 kN/m2
Passive at 8500 mm below GL in soil 1;Â pâ€™p41 = Kp1 Ã— OBâ€™p41 + Î³G,f Ã— Î³w Ã— (dL4 â€“ max(ds, dw)) = 172.8 kN/m2
Passive at 8500 mm below GL in soil 2;pâ€™p42 = Kp2 Ã— OBâ€™p42 + Î³G,f Ã— Î³w Ã— (dL4 â€“ max(ds, dw)) = 146.2 kN/m2
Passive at 12532 mm below GL in soil 2; pâ€™p51 = Kp2 Ã— OBâ€™p51 + Î³G,f Ã— Î³w Ã— (dL5 â€“ max(ds, dw)) = 302.7 kN/m2

By iteration the depth at which the active moments equal the passive moments has been determined as 12533 mm as follows:-

Moment level 1;Â Ma11 = 0.5 Ã— pâ€™a11 Ã— ha1 Ã— ((H â€“ dL2) + 2/3 Ã— ha1) = 42.7 kNm/m
Moment level 1; Ma12 = 0.5 Ã— pâ€™a21 Ã— ha1 Ã— ((H â€“ dL2) + 1/3 Ã— ha1) = 111.8 kNm/m
Moment level 2; Ma21 = 0.5 Ã— pâ€™a21 Ã— ha2 Ã— ((H â€“ dL3) + 2/3 Ã— ha2) = 164.8 kNm/m
Moment level 2; Ma22 = 0.5 Ã— pâ€™a31 Ã— ha2 Ã— ((H â€“ dL3) + 1/3 Ã— ha2) = 515.1 kNm/m
Moment level 3; Ma31 = 0.5 Ã— pâ€™a31 Ã— ha3 Ã— ((H â€“ dL4) + 2/3 Ã— ha3) = 696.2 kNm/m
Moment level 3; Ma32 = 0.5 Ã— pâ€™a41 Ã— ha3 Ã— ((H â€“ dL4) + 1/3 Ã— ha3) = 1244.0 kNm/m
Moment level 4; Ma41 = 0.5 Ã— pâ€™a42 Ã— ha4 Ã— ((H â€“ dL5) + 2/3 Ã— ha4) = 561.3 kNm/m
Moment level 4; Ma42 = 0.5 Ã— pâ€™a51 Ã— ha4 Ã— ((H â€“ dL5) + 1/3 Ã— ha4) = 428.7 kNm/m

Moment level 3;Â Mp31 = 0.5 Ã— pâ€™p31 Ã— hp3 Ã— ((H â€“ dL4) + 2/3 Ã— hp3) = 0.0 kNm/m
Moment level 3; Mp32 = 0.5 Ã— pâ€™p41 Ã— hp3 Ã— ((H â€“ dL4) + 1/3 Ã— hp3) = 2151.5 kNm/m
Moment level 4; Mp41 = 0.5 Ã— pâ€™p42 Ã— hp4 Ã— ((H â€“ dL5) + 2/3 Ã— hp4) = 792.7 kNm/m
Moment level 4; Mp42 = 0.5 Ã— pâ€™p51 Ã— hp4 Ã— ((H â€“ dL5) + 1/3 Ã— hp4) = 820.5 kNm/m

Total active moment; SMa = 3763.9 kNm/m
Total passive moment; SMp = 3763.7 kNm/m

Required pile length
Length of pile required to balance moments; H = 12533 mm

Depth of equal pressure; dcontra = 5694 mm
Add 20% below this point; de_add = 1.2 Ã— (H â€“ dcontra) = 8207 mm
Minimum required pile length; Htotal = dcontra + de_add = 13901 mm

PASS â€“ Provided length of sheet pile greater than the minimum required length of pile

Pile capacity (EN1993-5)

Maximum moment in pile (from analysis);Â Mpile = max(abs(Mmin), abs(Mmax)) / 1m = 549.1 kNm/m
Maximum shear force in pile (from analysis); Vpile = 358.1 kN/m
Nominal yield strength of pile; fy_pile = 355 N/mm2
Name of pile;Â  Arcelor PU(18)
Classification of pile; 2
Plastic modulus of pile; Wpl.y = 2134 cm3/m

Shear buckling of web (cl.5.2.2(6))

Width of section; c = h / sin(apile) = 510 mm
Thickness of web; tw = s = 9.0 mm
Îµ = âˆš(235/fy_pile)= 0.814
c / tw = 56.6 = 69.6Îµ < 72Îµ

PASS â€“ Shear buckling of web within limits

Bending
Interlock reduction factor (cl.5.2.2); Î²B = 1
Design bending resistance (eqn.5.2);Mc,Rd = Wpl.y Ã— fy_pile Ã— Î²B / Î³M0 = 757.6 kNm/m

PASS â€“ Moment capacity exceeds moment in pile

Shear
Projected shear area of web (eqn.5.6); Av = s Ã— (h â€“ t) = 3769 mm2
Design shear resistance (eqn.5.5);Â Vpl,Rd = Av Ã— fy_pile / (âˆš(3) Ã— Î³M0) / b = 1287.6 kN/m

PASS â€“ Shear capacity exceeds shear in the pile