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

**Geometry**

Total length of sheet pile provided H_{pile} = **14500** mm

Number of different types of soil N_{s} = **2**

Retained height of soil d_{ret} = **3500** mm

Depth of unplanned excavation d_{ex} = **500** mm

Total retained height d_{s} = **4000** mm

Angle of retained slope β = **0.0** deg

Depth from ground level to top of water table retained side d_{w} = **1500** mm

Depth from ground level to top of water table retaining side; d_{wp} = **4000** mm

**Loading**

Variable surcharge p_{o,Q} = **10.0** kN/m^{2}

**Soil layer 1**

Characteristic shearing resistance angle ϕ’_{k,s1} = **30.0** deg

Characteristic wall friction angle δ_{k,s1} = **20.0** deg

Moist density of soil γ_{m,s1} = **15.0** kN/m^{3}

Characteristic saturated density of retained soil γ_{s,s1} = **17.0** kN/m^{3}

Height of soil 1 h_{1} = **8500** mm

**Soil layer 2**

Characteristic shearing resistance angle f’_{k,s2} = **27.0** deg

Characteristic wall friction angle d_{k,s2} = **16.0** deg

Moist density of soil γ_{m,s2} = **16.0** kN/m^{3}

Characteristic saturated density of retained soil γ_{s,s2} = **19.0** kN/m^{3}

Height of soil 2 h_{2} = **7000** mm

**Partial factors on actions – Section A.3.1 – Combination 1**

Permanent unfavourable action γ_{G} = **1.35**

Permanent favourable action γ_{G,f} = **1.00**

Variable unfavourable action γ_{Q} = **1.50**

Angle of shearing resistance γ_{f’} = **1.00**

Weight density γ_{g} = **1.00**

**Design soil properties – soil 1**

Design effective shearing resistance angle ϕ’_{d} = tan^{-1}[tan(ϕ’_{k})/γ_{f’}] = **30.0** deg

Design wall friction angle δ_{d} = tan^{-1}[tan(ϕ_{k})/γ_{f’}] = **20.0** deg

Design moist density of retained soil γ_{m.d1} = γ_{m}/γ_{γ} = **15.0** kN/m^{3}

Design saturated density of retained soil γ_{s.d1} = γ_{s}/γ_{γ} = **17.0** kN/m^{3}

Design buoyant density of retained soil γ_{d.d1} = γ_{s.d1} – γ_{w} = **7.2** kN/m^{3}

Active pressure using Coulomb theory K_{a1} = sin(α + ϕ’_{d})^{2} / (sin(α)^{2} × sin(α – δ_{d}) × (1 + √(sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d} – β)/(sin(α – δ_{d}) ´ sin(α + β))))^{2}) = **0.297**

Passive pressure using Coulomb theory K_{p1} = sin(90 – ϕ’_{d})^{2} / (sin(90 + δ_{d}) × [1 – √[sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d}) / (sin(90 + δ_{d}))]]^{2}) = **6.105**

**Design soil properties – soil 2 **

Design effective shearing resistance angle ϕ’_{d} = tan^{-1}(tan(ϕ’_{k}) /γ_{f’}) = **27.0** deg

Design wall friction angle δ_{d} = tan_{-1}(tan(δ_{k})/γ_{f’}) = **16.0** deg

Design moist density of retained soil γ_{m.d2} = γ_{m}/γ_{γ} = **16.0** kN/m^{3}

Design saturated density of retained soil γ_{s.d2} = γ_{s} /γ_{γ} = **19.0** kN/m^{3}

Design buoyant density of retained soil γ_{d.d2} = γ_{s.d2} – γ_{w} = **9.2** kN/m^{3}

Active pressure using Coulomb theory K_{a2} = sin(α + ϕ’_{d})^{2} / (sin(α)^{2} × sin(α – δ_{d}) × (1 + √(sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d} – β) / (sin(α – δ_{d}) × sin(α + β))))^{2}) = **0.336**

Passive pressure using Coulomb theory K_{p2} = sin(90 – ϕ’_{d})^{2} / (sin(90 + δ_{d}) × [1 – √[sin(ϕ’_{d} + δ_{d}) × sin(f’_{d}) / (sin(90 + δ_{d}))]]^{2}) = **4.416**

**Overburden on active side**

Overburden at 0 mm below GL in soil 1; OB’_{a11} = p_{o,Q} × γ_{Q} = **15.0** kN/m^{2}

Overburden at 1500 mm below GL in soil 1; OB’_{a21} = γ_{G} × γ_{m.d1} × h_{a1} + OB’_{a11} = **45.4** kN/m^{2}

Overburden at 4000 mm below GL in soil 1; OB’_{a31} = γ_{G} × γ_{d.d1} × h_{a2} + OB’_{a21} = **69.6** kN/m^{2}

Overburden at 8500 mm below GL in soil 1; OB’_{a41} = γ_{G} × γ_{d.d1} × h_{a3} + OB’_{a31} = **113.3** kN/m^{2}

Overburden at 8500 mm below GL in soil 2; OB’_{a42} = γ_{G} × γ_{d.d1} × h_{a3} + OB’_{a31} = **113.3** kN/m^{2}

Overburden at 11544 mm below GL in soil 2; OB’_{a51} = γ_{G} × γ_{d.d2} × h_{a4} + OB’_{a42} = **151.1** kN/m^{2}

**Overburden on passive side**

Overburden at 4000 mm below GL in soil 1; OB’_{p31} = 0 kN/m^{2} = **0.0** kN/m^{2}

Overburden at 8500 mm below GL in soil 1; OB’_{p41} = γ_{G,f} × γ_{d.d1} × h_{p3} + OB’_{p31} = **32.4** kN/m^{2}

Overburden at 8500 mm below GL in soil 2; OB’_{p42} = γ_{G,f} × γ_{d.d1} × h_{p3} + OB’_{p31} = **32.4** kN/m^{2}

Overburden at 11544 mm below GL in soil 2; OB’_{p51} = γ_{G,f} × γ_{d.d2} × h_{p4} + OB’_{p42} = **60.3** kN/m^{2}

**Pressure on active side**

Active at 0 mm below GL in soil 1; p’_{a11} = K_{a1} × OB’_{a11} = **4.5** kN/m^{2}

Active at 1500 mm below GL in soil 1; p’_{a21} = K_{a1} × OB’_{a21} = **13.5** kN/m^{2}

Active at 4000 mm below GL in soil 1; p’_{a31} = K_{a1} × OB’_{a31} + γ_{γ} × γ_{w} × (d_{L3} – d_{w}) = **53.8** kN/m^{2}

Active at 8500 mm below GL in soil 1; p’_{a41} = K_{a1} × OB’_{a41} + γ_{γ} × γ_{w} × (d_{L4} – d_{w}) = **126.4** kN/m^{2}

Active at 8500 mm below GL in soil 2; p’_{a42} = K_{a2} × OB’_{a42} + γ_{γ} × γ_{w} × (d_{L4} – d_{w}) = **130.8** kN/m^{2}

Active at 11544 mm below GL in soil 2; p’_{a51} = K_{a2} × OB’_{a51} + γ_{γ} × γ_{w} × (d_{L5} – d_{w}) = **183.8** kN/m^{2}

**Pressure on passive side**

Passive at 4000 mm below GL in soil 1; p’_{p31} = K_{p1} × OB’_{p31} + γ_{G,f} × γ_{w} × (d_{L3} – max(d_{s}, d_{w})) = **0.0** kN/m^{2}

Passive at 8500 mm below GL in soil 1; p’_{p41} = K_{p1} × OB’_{p41} + γ_{G,f} × γ_{w} × (d_{L4} – max(d_{s}, d_{w})) = **241.7** kN/m^{2}

Passive at 8500 mm below GL in soil 2; p’_{p42} = K_{p2} × OB’_{p42} + γ_{G,f} × γ_{w} × (d_{L4} – max(d_{s}, d_{w})) = **187.0** kN/m^{2}

Passive at 11544 mm below GL in soil 2; p’_{p51} = K_{p2} × OB’_{p51} + γ_{G,f} × γ_{w} × (d_{L5} – max(d_{s}, d_{w})) = **340.4** kN/m^{2}

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

Active moment about 11544 mm

Moment level 1;M_{a11} = 0.5 × p’_{a11} × h_{a1} × ((H – d_{L2}) + 2/3 × h_{a1}) = **36.9** kNm/m

Moment level 1; M_{a12} = 0.5 × p’_{a21} × h_{a1} × ((H – d_{L2}) + 1/3 × h_{a1}) = **106.7** kNm/m

Moment level 2; M_{a21} = 0.5 × p’_{a21} × h_{a2} × ((H – d_{L3}) + 2/3 × h_{a2}) = **155.3** kNm/m

Moment level 2; M_{a22} = 0.5 × p’_{a31} × h_{a2} × ((H – d_{L3}) + 1/3 × h_{a2}) = **563.5** kNm/m

Moment level 3; M_{a31} = 0.5 × p’_{a31} × h_{a3} × ((H – d_{L4}) + 2/3 × h_{a3}) = **731.8** kNm/m

Moment level 3; M_{a32} = 0.5 × p’_{a41} × h_{a3} × ((H – d_{L4}) + 1/3 × h_{a3}) = **1292.3** kNm/m

Moment level 4; M_{a41} = 0.5 × p’_{a42} × h_{a4} × ((H – d_{L5}) + 2/3 × h_{a4}) = **404.0** kNm/m

Moment level 4; M_{a42} = 0.5 × p’_{a51} × h_{a4} × ((H – d_{L5}) + 1/3 × h_{a4}) = **283.8** kNm/m

Passive moment about 11544 mm

Moment level 3; M_{p31} = 0.5 × p’_{p31} × h_{p3} × ((H – d_{L4}) + 2/3 × h_{p3}) = **0.0** kNm/m

Moment level 3; M_{p32} = 0.5 × p’_{p41} × h_{p3} × ((H – d_{L4}) + 1/3 × h_{p3}) = **2471.0** kNm/m

Moment level 4; M_{p41} = 0.5 × p’_{p42} × h_{p4} × ((H – d_{L5}) + 2/3 × h_{p4}) = **577.6** kNm/m

Moment level 4; M_{p42} = 0.5 × p’_{p51} × h_{p4} × ((H – d_{L5}) + 1/3 × h_{p4}) = **525.7** kNm/m

Total moments about 11544 mm

Total active moment; SM_{a} = **3574.5** kNm/m

Total passive moment; SM_{p} = **3574.5** kNm/m

**Required pile length**

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

Depth of equal pressure; d_{contra} = **5432** mm

Add 20% below this point; d_{e_add} = 1.2 × (H – d_{contra}) = **7334** mm

Minimum required pile length; H_{total} = d_{contra} + d_{e_add} = **12766** mm

**Pass** – *Provided length of sheet pile greater than minimum required length of pile*

**Pile capacity (EN1993-5)**

Maximum moment in pile (from analysis); M_{pile }= max(abs(M_{min}), abs(M_{max})) / 1m = **547.0** kNm/m

Maximum shear force in pile (from analysis); V_{pile }= **364.7** kN/m

Nominal yield strength of pile; f_{y_pile} = **355** N/mm^{2}

Name of sheet pile; Arcelor PU(18)

Classification of pile; 2

Plastic modulus of pile; W_{pl.y} = **2134** cm^{3}/m

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

Width of section; c = h / sin(α_{pile}) = **510** mm

Thickness of web; t_{w} = s = **9.0** mm

ε = √(235/f_{y_pile}) = **0.814**

c/t_{w} = 56.6 = 69.6ε < 72ε*PASS* *– Shear buckling of web within limits*

**Bending** **2**

Interlock reduction factor (cl.5.2.2); β_{B} = **1**

Design bending resistance (eqn.5.2);

M_{c,Rd} = W_{pl.y} × f_{y_pile} × β_{B} / γ_{M0} = **757.6** kNm/m** PASS** –

*Moment capacity exceeds moment in pile*

**Shear**

Projected shear area of web (eqn.5.6); A_{v} = s × (h – t) = **3769** mm^{2}

Design shear resistance (eqn.5.5); V_{pl,Rd} = A_{v} × f_{y_pile} / (√(3) × γ_{M0}) / b = **1287.6** kN/m**PASS** – *Shear capacity exceeds shear in pile*

**Partial factors on actions – Section A.3.1 – Combination 2**

Permanent unfavourable action; γ_{G} = **1.00**

Permanent favourable action; γ_{G,f} = **1.00**

Variable unfavourable action; γ_{Q} = **1.30**

Angle of shearing resistance; γ_{ϕ’} = **1.25**

Weight density; γ_{γ} = **1.00**

**Design soil properties – soil 1**

Design effective shearing resistance angle; ϕ’_{d} = tan^{-1}(tan(ϕ’_{k})/γ_{ϕ’}) = **24.8** deg

Design wall friction angle; δ_{d} = tan^{-1}(tan(δ_{k})/γ_{ϕ’}) = **16.2** deg

Design moist density of retained soil; γ_{m.d1} = γ_{m}/γ_{γ} = **15.0** kN/m^{3}

Design saturated density of retained soil; γ_{s.d1} = γ_{s}/γ_{γ} = **17.0** kN/m^{3}

Design buoyant density of retained soil; γ_{d.d1} = γ_{s.d1} – γ_{w} = **7.2** kN/m^{3}

Active pressure using Coulomb theory; K_{a1} = sin(α + ϕ’_{d})^{2} / (sin(α)^{2} × sin(α – δ_{d}) × (1 + √(sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d} – β)/(sin(α – δ_{d}) ´ sin(α + β))))^{2}) = **0.364**

Passive pressure using Coulomb theory; K_{p1} = sin(90 – ϕ’_{d})^{2} / (sin(90 + δ_{d}) × [1 – √[sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d}) / (sin(90 + δ_{d}))]]^{2}) = **3.977**

**Design soil properties – soil 2**

Design effective shearing resistance angle; ϕ’_{d2} = tan^{-1}(tan(ϕ’_{k})/γ_{ϕ’}) = **22.2** deg

Design wall friction angle; δ_{d2} = tan^{-1}(tan(δ_{k})/γ_{ϕ’}) = **12.9** deg

Design moist density of retained soil; γ_{m.d2} = γ_{m}/γ_{γ} = **16.0** kN/m^{3}

Design saturated density of retained soil; γ_{s.d2} = γ_{s}/γ_{γ} = **19.0** kN/m^{3}

Design buoyant density of retained soil; γ_{d.d2} = γ_{s.d2} – γ_{w} = **9.2** kN/m^{3}

Active pressure using Coulomb theory; K_{a2} = sin(α + ϕ’_{d})^{2} / (sin(α)^{2} × sin(α – δ_{d}) × (1 + √(sin(ϕ’_{d} + δ_{d}) × sin(ϕ’_{d} – β) / (sin(α – δ_{d}) × sin(α + β))))^{2}) = **0.406**

Passive pressure using Coulomb theory; K_{p2} = sin(90 – ϕ’_{d})^{2} / (sin(90 + δ_{d}) × [1 – √[sin(ϕ’_{d} + δ_{d}) × sin(f’_{d}) / (sin(90 + δ_{d}))]]^{2}) = **3.154**

**Overburden on active side**

Overburden at 0 mm below GL in soil 1; OB’_{a11} = p_{o,Q} × γ_{Q} = **13.0** kN/m^{2}

Overburden at 1500 mm below GL in soil 1; OB’_{a21} = γ_{G} × γ_{m.d1} × h_{a1} + OB’_{a11} = **35.5** kN/m^{2}

Overburden at 4000 mm below GL in soil 1; OB’_{a31} = γ_{G} × γ_{d.d1} × h_{a2} + OB’_{a21} = **53.5** kN/m^{2}

Overburden at 8500 mm below GL in soil 1; OB’_{a41} = γ_{G} × γ_{d.d1} × h_{a3} + OB’_{a31} = **85.8** kN/m^{2}

Overburden at 8500 mm below GL in soil 2; OB’_{a42} = γ_{G} × γ_{d.d1} × h_{a3} + OB’_{a31} = **85.8** kN/m^{2}

Overburden at 12532 mm below GL in soil 2;OB’_{a51} = γ_{G} × γ_{d.d2} × h_{a4} + OB’_{a42} = **122.9** kN/m^{2}

**Overburden on passive side**

Overburden at 4000 mm below GL in soil 1; OB’_{p31} = 0 kN/m^{2} = **0.0** kN/m^{2}

Overburden at 8500 mm below GL in soil 1; OB’_{p41} = γ_{G,f} × γ_{d.d1} × h_{p3} + OB’_{p31} = **32.4** kN/m^{2}

Overburden at 8500 mm below GL in soil 2; OB’_{p42} = γ_{G,f} × γ_{d.d1} × h_{p3} + OB’_{p31} = **32.4** kN/m^{2}

Overburden at 12532 mm below GL in soil 2;OB’_{p51} = γ_{G,f} × γ_{d.d2} × h_{p4} + OB’_{p42} = **69.4** kN/m^{2}

**Pressure on active side**

Active at 0 mm below GL in soil 1; p’_{a11} = K_{a1} × OB’_{a11} = **4.7** kN/m^{2}

Active at 1500 mm below GL in soil 1; p’_{a21} = K_{a1} × OB’_{a21} = **12.9** kN/m^{2}

Active at 4000 mm below GL in soil 1; p’_{a31} = K_{a1} × OB’_{a31} + γ_{G} × γ_{w} × (d_{L3} – d_{w}) = **44.0** kN/m^{2}

Active at 8500 mm below GL in soil 1; p’_{a41} = K_{a1} × OB’_{a41} + γ_{G} × γ_{w} × (d_{L4} – d_{w}) = **99.9** kN/m^{2}

Active at 8500 mm below GL in soil 2; p’_{a42} = K_{a2} × OB’_{a42} + γ_{G} × γ_{w} × (d_{L4} – d_{w}) = **103.5** kN/m^{2}

Active at 12532 mm below GL in soil 2; p’_{a51} = K_{a2} × OB’_{a51} + γ_{G} × γ_{w} × (d_{L5} – d_{w}) = **158.1** kN/m^{2}

**Pressure on passive side**

Passive at 4000 mm below GL in soil 1; p’_{p31} = K_{p1} × OB’_{p31} + γ_{G,f} × γ_{w} × (d_{L3} – max(d_{s}, d_{w})) = **0.0** kN/m^{2}

Passive at 8500 mm below GL in soil 1; p’_{p41} = K_{p1} × OB’_{p41} + γ_{G,f} × γ_{w} × (d_{L4} – max(d_{s}, d_{w})) = **172.8** kN/m^{2}

Passive at 8500 mm below GL in soil 2;p’_{p42} = K_{p2} × OB’_{p42} + γ_{G,f} × γ_{w} × (d_{L4} – max(d_{s}, d_{w})) = **146.2** kN/m^{2}

Passive at 12532 mm below GL in soil 2; p’_{p51} = K_{p2} × OB’_{p51} + γ_{G,f} × γ_{w} × (d_{L5} – max(d_{s}, d_{w})) = **302.7** kN/m^{2}

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

Active moment about 12533 mm

Moment level 1; M_{a11} = 0.5 × p’_{a11} × h_{a1} × ((H – d_{L2}) + 2/3 × h_{a1}) = **42.7** kNm/m

Moment level 1; M_{a12} = 0.5 × p’_{a21} × h_{a1} × ((H – d_{L2}) + 1/3 × h_{a1}) = **111.8** kNm/m

Moment level 2; M_{a21} = 0.5 × p’_{a21} × h_{a2} × ((H – d_{L3}) + 2/3 × h_{a2}) = **164.8** kNm/m

Moment level 2; M_{a22} = 0.5 × p’_{a31} × h_{a2} × ((H – d_{L3}) + 1/3 × h_{a2}) = **515.1** kNm/m

Moment level 3; M_{a31} = 0.5 × p’_{a31} × h_{a3} × ((H – d_{L4}) + 2/3 × h_{a3}) = **696.2** kNm/m

Moment level 3; M_{a32} = 0.5 × p’_{a41} × h_{a3} × ((H – d_{L4}) + 1/3 × h_{a3}) = **1244.0** kNm/m

Moment level 4; M_{a41} = 0.5 × p’_{a42} × h_{a4} × ((H – d_{L5}) + 2/3 × h_{a4}) = **561.3** kNm/m

Moment level 4; M_{a42} = 0.5 × p’_{a51} × h_{a4} × ((H – d_{L5}) + 1/3 × h_{a4}) = **428.7** kNm/m

Passive moment about 12533 mm

Moment level 3; M_{p31} = 0.5 × p’_{p31} × h_{p3} × ((H – d_{L4}) + 2/3 × h_{p3}) = **0.0** kNm/m

Moment level 3; M_{p32} = 0.5 × p’_{p41} × h_{p3} × ((H – d_{L4}) + 1/3 × h_{p3}) = **2151.5** kNm/m

Moment level 4; M_{p41} = 0.5 × p’_{p42} × h_{p4} × ((H – d_{L5}) + 2/3 × h_{p4}) = **792.7** kNm/m

Moment level 4; M_{p42} = 0.5 × p’_{p51} × h_{p4} × ((H – d_{L5}) + 1/3 × h_{p4}) = **820.5** kNm/m

Total moments about 12533 mm

Total active moment; SM_{a} = **3763.9** kNm/m

Total passive moment; SM_{p} = **3763.7** kNm/m

**Required pile length**

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

Depth of equal pressure; d_{contra} = **5694** mm

Add 20% below this point; d_{e_add} = 1.2 × (H – d_{contra}) = **8207** mm

Minimum required pile length; H_{total} = d_{contra} + d_{e_add} = **13901** mm

** PASS **– Provided length of sheet pile greater than minimum required length of pile

**Pile capacity (EN1993-5)**

Maximum moment in pile (from analysis); M_{pile }= max(abs(M_{min}), abs(M_{max})) / 1m = **549.1** kNm/m

Maximum shear force in pile (from analysis); V_{pile }= **358.1** kN/m

Nominal yield strength of pile; f_{y_pile} = **355** N/mm^{2}

Name of pile; Arcelor PU(18)

Classification of pile; 2

Plastic modulus of pile; W_{pl.y} = **2134** cm^{3}/m

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

Width of section; c = h / sin(a_{pile}) = **510** mm

Thickness of web; t_{w} = s = **9.0** mm

ε = √(235/f_{y_pile})= **0.814**

c / t_{w} = 56.6 = 69.6ε < 72ε

** PASS** –

*Shear buckling of web within limits*

**Bending**

Interlock reduction factor (cl.5.2.2); β_{B} = **1**

Design bending resistance (eqn.5.2);M_{c,Rd} = W_{pl.y} × f_{y_pile} × β_{B} / γ_{M0} = **757.6** kNm/m

** PASS** –

*Moment capacity exceeds moment in pile*

**Shear**

Projected shear area of web (eqn.5.6); A_{v} = s × (h – t) = **3769** mm^{2}

Design shear resistance (eqn.5.5); V_{pl,Rd} = A_{v} × f_{y_pile} / (√(3) × γ_{M0}) / b = **1287.6** kN/m

** PASS** –

*Shear capacity exceeds shear in pile*

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