Sachpazis cantilever retaining wall analysis & design (en1997-1-2004)

GEODOMISI Ltd. - Dr. Costas Sachpazis Civil & Geotechnical Engineering Consulting Company for

Structural Engineering, Soil Mechanics, Rock Mechanics, Foundation

Engineering & Retaining Structures. Tel.: (+30) 210 5238127, 210 5711263 - Fax.:+30 210 5711461 - Mobile: (+30)

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Project: Retaining wall Analysis & Design, In accordance with EN1997-1:2004 incorporating Corrigendum dated February 2009 and the recommended values.

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Civil & Geotechnical Engineering Sheet no./rev. 1

Calc. by

Dr. C. Sachpazis

Date

04/04/2014

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RETAINING WALL ANALYSIS (EN1997-1:2004)

In accordance with EN1997-1:2004 incorporating Corrigendum dated

February 2009 and the recommended values

Retaining wall details

Stem type; Cantilever with inclined front face

Stem height; hstem = 4000 mm

Stem thickness; tstem = 450 mm

Slope length to front of stem; lslf = 100 mm

Angle to rear face of stem; α = 90 deg

Angle to front face of stem; αf = 88.6 deg

Stem density; γstem = 25 kN/m3

Toe length; ltoe = 1000 mm

Heel length; lheel = 4000 mm

Base thickness; tbase = 450 mm

Base density; γbase = 25 kN/m3

Height of retained soil; hret = 3000 mm

Angle of soil surface; β = 15 deg

Depth of cover; dcover = 500 mm

Height of water; hwater = 300 mm

Water density; γw = 9.8 kN/m3

Retained soil properties

Soil type; Medium dense well graded sand and gravel

Moist density; γmr = 20 kN/m3

Saturated density; γsr = 22.3 kN/m3

Characteristic effective shear resistance angle; φ'r.k = 25 deg

Characteristic wall friction angle; δr.k = 16 deg

Base soil properties

Moist density; γmb = 20 kN/m3

Characteristic cohesion; c'b.k = 5 kN/m2

Characteristic adhesion; ab.k = 5 kN/m2

Characteristic effective shear resistance angle; φ'b.k = 25 deg

Characteristic wall friction angle; δb.k = 20 deg

Characteristic base friction angle; δbb.k = 25 deg

Loading details

Variable surcharge load; SurchargeQ = 10 kN/m2




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Calculate retaining wall geometry

Base length; lbase = ltoe + lslf + tstem + lheel = 5550 mm

Saturated soil height; hsat = hwater + dcover = 800 mm

Moist soil height; hmoist = hret - hwater = 2700 mm

Length of surcharge load; lsur = lheel = 4000 mm




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- Distance to vertical component; xsur_v = lbase - lheel / 2 = 3550 mm

Effective height of wall; heff = hbase + dcover + hret + lsur × tan(β) = 5022 mm

- Distance to horizontal component; xsur_h = heff / 2 = 2511 mm

Area of wall stem; Astem = hstem × (tstem + lslf / 2) = 2 m2

- Distance to vertical component; xstem = (hstem × tstem × (ltoe + lslf + tstem / 2) + hstem × lslf

/ 2 × (ltoe + 2 × lslf / 3)) / Astem = 1299 mm

Area of wall base; Abase = lbase × tbase = 2.498 m2

- Distance to vertical component; xbase = lbase / 2 = 2775 mm

Area of saturated soil; Asat = hsat × lheel = 3.2 m2

- Distance to vertical component; xsat_v = lbase - (hsat × lheel2 / 2) / Asat = 3550 mm

- Distance to horizontal component; xsat_h = (hsat + hbase) / 3 = 417 mm

Area of water; Awater = hsat × lheel = 3.2 m2

- Distance to vertical component; xwater_v = lbase - (hsat × lheel2 / 2) / Asat = 3550 mm

- Distance to horizontal component; xwater_h = (hsat + hbase) / 3 = 417 mm

Area of moist soil; Amoist = hmoist × lheel + tan(β) × lheel2 / 2 = 12.944 m

2

- Distance to vertical component; xmoist_v = lbase - (hmoist × lheel2 / 2 + tan(β) × lheel

3 / 6) /

Amoist = 3660 mm

- Distance to horizontal component; xmoist_h = ((heff - hsat - hbase) × (tbase + hsat + (heff - hsat

- hbase) / 3) / 2 + (hsat + tbase)2/2) / (hsat + tbase + (heff -

hsat - hbase) / 2) = 1757 mm

Area of base soil; Apass = dcover × (ltoe + lslf × dcover / (2 × hstem)) = 0.503

m2

- Distance to vertical component; xpass_v = lbase - (dcover × ltoe × (lbase - ltoe / 2) + lslf ×

dcover2 / (2 × hstem) × (lbase - ltoe - lslf × dcover / (3 ×

hstem))) / Apass = 503 mm

- Distance to horizontal component; xpass_h = (dcover + hbase) / 3 = 317 mm

Area of excavated base soil; Aexc = hpass × (ltoe + lslf × hpass / (2 × hstem)) = 0.503

m2

- Distance to vertical component; xexc_v = lbase - (hpass × ltoe × (lbase - ltoe / 2) + lslf ×

hpass2 / (2 × hstem) × (lbase - ltoe - lslf × hpass / (3 ×

hstem))) / Aexc = 503 mm

- Distance to horizontal component; xexc_h = (hpass + hbase) / 3 = 317 mm

Partial factors on actions - Table A.3 - Combination 1

Permanent unfavourable action; γG = 1.35

Permanent favourable action; γGf = 1.00

Variable unfavourable action; γQ = 1.50

Variable favourable action; γQf = 0.00

Partial factors for soil parameters – Table A.4 - Combination 1

Angle of shearing resistance; γφ' = 1.00




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Effective cohesion; γc' = 1.00

Weight density; γγ = 1.00


Design effective shear resistance angle; φ'r.d = atan(tan(φ'r.k) / γφ') = 25 deg

Design wall friction angle; δr.d = atan(tan(δr.k) / γφ') = 16 deg


Design effective shear resistance angle; φ'b.d = atan(tan(φ'b.k) / γφ') = 25 deg

Design wall friction angle; δb.d = atan(tan(δb.k) / γφ') = 20 deg

Design base friction angle; δbb.d = atan(tan(δbb.k) / γφ') = 25 deg

Design effective cohesion; c'b.d = c'b.k / γc' = 5 kN/m2

Design adhesion; ab.d = ab.k / γc' = 5 kN/m2

Using Coulomb theory

Active pressure coefficient; KA = sin(α + φ'r.d)2 / (sin(α)2 × sin(α - δr.d) × [1 +

√[sin(φ'r.d + δr.d) × sin(φ'r.d - β) / (sin(α - δr.d) × sin(α +

β))]]2) = 0.469

Passive pressure coefficient; KP = sin(αf - φ'b.d)2 / (sin(αf)

2 × sin(αf + δb.d) × [1 -

√[sin(φ'b.d + δb.d) × sin(φ'b.d) / (sin(αf + δb.d) ×

sin(αf))]]2) = 4.403

Sliding check

Vertical forces on wall

Wall stem; Fstem = γGf × Astem × γstem = 50 kN/m

Wall base; Fbase = γGf × Abase × γbase = 62.4 kN/m

Saturated retained soil; Fsat_v = γGf × Asat × (γsr - γw) = 39.8 kN/m

Water; Fwater_v = γGf × Awater × γw = 31.4 kN/m

Moist retained soil; Fmoist_v = γGf × Amoist × γmr = 258.9 kN/m

Base soil; Fexc_v = γGf × Aexc × γmb = 10.1 kN/m

Total; Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fexc_v +

Fwater_v = 452.6 kN/m

Horizontal forces on wall

Surcharge load; Fsur_h = KA × cos(δr.d) × γQ × SurchargeQ × heff =

33.9 kN/m

Saturated retained soil; Fsat_h = γG × KA × cos(δr.d) × (γsr - γw) × (hsat + hbase)2

/ 2 = 5.9 kN/m

Water; Fwater_h = γG × γw × (hwater + dcover + hbase)2 / 2 = 10.3

kN/m




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Moist retained soil; Fmoist_h = γG × KA × cos(δr.d) × γmr × ((heff - hsat -

hbase)2 / 2 + (heff - hsat - hbase) × (hsat + hbase)) = 143.9

kN/m

Total; Ftotal_h = Fsat_h + Fmoist_h + Fwater_h + Fsur_h = 194.1

kN/m

Check stability against sliding

Base soil resistance; Fexc_h = γGf × KP × cos(δb.d) × γmb × (hpass + hbase)2 / 2

= 37.3 kN/m

Base friction; Ffriction = ab.d × b + Ftotal_v × tan(δbb.d) = 216 kN/m

Resistance to sliding; Frest = Fexc_h + Ffriction = 253.4 kN/m

Factor of safety; FoSsl = Frest / Ftotal_h = 1.306

PASS - Resistance to sliding is greater than sliding force

Overturning check












33.9 kN/m


/ 2 = 5.9 kN/m


kN/m



kN/m

Base soil; Fexc_h = -γGf × KP × cos(δb.d) × γmb × (hpass + hbase)2 /

2 = -37.3 kN/m

Total; Ftotal_h = Fsat_h + Fmoist_h + Fexc_h + Fwater_h + Fsur_h =

156.7 kN/m

Overturning moments on wall

Surcharge load; Msur_OT = Fsur_h × xsur_h = 85.2 kNm/m




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Saturated retained soil; Msat_OT = Fsat_h × xsat_h = 2.5 kNm/m

Water; Mwater_OT = Fwater_h × xwater_h = 4.3 kNm/m

Moist retained soil; Mmoist_OT = Fmoist_h × xmoist_h = 252.8 kNm/m

Total; Mtotal_OT = Msat_OT + Mmoist_OT + Mwater_OT + Msur_OT =

344.8 kNm/m

Restoring moments on wall

Wall stem; Mstem_R = Fstem × xstem = 65 kNm/m

Wall base; Mbase_R = Fbase × xbase = 173.3 kNm/m

Saturated retained soil; Msat_R = Fsat_v × xsat_v = 141.3 kNm/m

Water; Mwater_R = Fwater_v × xwater_v = 111.4 kNm/m

Moist retained soil; Mmoist_R = Fmoist_v × xmoist_v = 947.6 kNm/m

Base soil; Mexc_R = Fexc_v × xexc_v - Fexc_h × xexc_h = 16.9 kNm/m

Total; Mtotal_R = Mstem_R + Mbase_R + Msat_R + Mmoist_R +

Mexc_R + Mwater_R = 1455.4 kNm/m

Check stability against overturning

Factor of safety; FoSot = Mtotal_R / Mtotal_OT = 4.222

PASS - Maximum restoring moment is greater than overturning moment

Bearing pressure check


Wall stem; Fstem = γG × Astem × γstem = 67.5 kN/m

Wall base; Fbase = γG × Abase × γbase = 84.3 kN/m

Surcharge load; Fsur_v = γQ × SurchargeQ × lheel = 60 kN/m

Saturated retained soil; Fsat_v = γG × Asat × (γsr - γw) = 53.7 kN/m

Water; Fwater_v = γG × Awater × γw = 42.4 kN/m

Moist retained soil; Fmoist_v = γG × Amoist × γmr = 349.5 kN/m

Base soil; Fpass_v = γG × Apass × γmb = 13.6 kN/m

Total; Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fpass_v +

Fwater_v + Fsur_v = 671 kN/m



33.9 kN/m


/ 2 = 5.9 kN/m


kN/m



kN/m




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Date

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Base soil; Fpass_h = -γGf × KP × cos(δb.d) × γmb × (dcover + hbase)2 /

2 = -37.3 kN/m

Total; Ftotal_h = max(Fsat_h + Fmoist_h + Fpass_h + Fwater_h +

Fsur_h - (ab.d × b + Ftotal_v × tan(δbb.d)), 0 kN/m) = 0

kN/m

Moments on wall

Wall stem; Mstem = Fstem × xstem = 87.7 kNm/m

Wall base; Mbase = Fbase × xbase = 233.9 kNm/m

Surcharge load; Msur = Fsur_v × xsur_v - Fsur_h × xsur_h = 127.8 kNm/m

Saturated retained soil; Msat = Fsat_v × xsat_v - Fsat_h × xsat_h = 188.3 kNm/m

Water; Mwater = Fwater_v × xwater_v - Fwater_h × xwater_h = 146.1

kNm/m

Moist retained soil; Mmoist = Fmoist_v × xmoist_v - Fmoist_h × xmoist_h = 1026.4

kNm/m

Base soil; Mpass = Fpass_v × xpass_v - Fpass_h × xpass_h = 18.7

kNm/m

Total; Mtotal = Mstem + Mbase + Msat + Mmoist + Mpass + Mwater

+ Msur = 1829 kNm/m

Check bearing pressure

Distance to reaction; x = Mtotal / Ftotal_v = 2726 mm

Eccentricity of reaction; e = x - lbase / 2 = -49 mm

Loaded length of base; lload = 2 × x = 5452 mm

Bearing pressure at toe; qtoe = Ftotal_v / lload = 123.1 kN/m2

Bearing pressure at heel; qheel = 0 kN/m2

Effective overburden pressure; q = (tbase + dcover) × γmb - (tbase + dcover + hwater) × γw =

6.7 kN/m2

Design effective overburden pressure; q' = q / γγ = 6.7 kN/m2

Bearing resistance factors; Nq = Exp(π × tan(φ'b.d)) × (tan(45 deg + φ'b.d / 2))2 =

10.662

Nc = (Nq - 1) × cot(φ'b.d) = 20.721

Nγ = 2 × (Nq - 1) × tan(φ'b.d) = 9.011

Foundation shape factors; sq = 1

sγ = 1

sc = 1

Load inclination factors; H = Ftotal_h = 0 kN/m

V = Ftotal_v = 671 kN/m

m = 2

iq = [1 - H / (V + lload × c'b.d × cot(φ'b.d))]m = 1

iγ = [1 - H / (V + lload × c'b.d × cot(φ'b.d))](m + 1)

= 1




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ic = iq - (1 - iq) / (Nc × tan(φ'b.d)) = 1

Net ultimate bearing capacity; nf = c'b.d × Nc × sc × ic + q' × Nq × sq × iq + 0.5 × (γmb

- γw) × lload × Nγ × sγ × iγ = 425.7 kN/m2

Factor of safety; FoSbp = nf / max(qtoe, qheel) = 3.459

PASS - Allowable bearing pressure exceeds maximum applied bearing pressure

Partial factors on actions - Table A.3 - Combination 2

Permanent unfavourable action; γG = 1.00

Permanent favourable action; γGf = 1.00

Variable unfavourable action; γQ = 1.30

Variable favourable action; γQf = 0.00

Partial factors for soil parameters – Table A.4 - Combination 2

Angle of shearing resistance; γφ' = 1.25

Effective cohesion; γc' = 1.25

Weight density; γγ = 1.00


Design effective shear resistance angle; φ'r.d = atan(tan(φ'r.k) / γφ') = 20.5 deg

Design wall friction angle; δr.d = atan(tan(δr.k) / γφ') = 12.9 deg


Design effective shear resistance angle; φ'b.d = atan(tan(φ'b.k) / γφ') = 20.5 deg

Design wall friction angle; δb.d = atan(tan(δb.k) / γφ') = 16.2 deg

Design base friction angle; δbb.d = atan(tan(δbb.k) / γφ') = 20.5 deg

Design effective cohesion; c'b.d = c'b.k / γc' = 4 kN/m2

Design adhesion; ab.d = ab.k / γc' = 4 kN/m2

Using Coulomb theory

Active pressure coefficient; KA = sin(α + φ'r.d)2 / (sin(α)2 × sin(α - δr.d) × [1 +

√[sin(φ'r.d + δr.d) × sin(φ'r.d - β) / (sin(α - δr.d) × sin(α +

β))]]2) = 0.590

Passive pressure coefficient; KP = sin(αf - φ'b.d)2 / (sin(αf)

2 × sin(αf + δb.d) × [1 -

√[sin(φ'b.d + δb.d) × sin(φ'b.d) / (sin(αf + δb.d) ×

sin(αf))]]2) = 3.111

Sliding check









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37.5 kN/m


/ 2 = 5.6 kN/m


kN/m


hbase)2 / 2 + (heff - hsat - hbase) × (hsat + hbase)) = 136

kN/m

Total; Ftotal_h = Fsat_h + Fmoist_h + Fwater_h + Fsur_h = 186.8

kN/m

Check stability against sliding

Base soil resistance; Fexc_h = γGf × KP × cos(δb.d) × γmb × (hpass + hbase)2 / 2

= 27 kN/m

Base friction; Ffriction = ab.d × b + Ftotal_v × tan(δbb.d) = 172.8 kN/m

Resistance to sliding; Frest = Fexc_h + Ffriction = 199.8 kN/m

Factor of safety; FoSsl = Frest / Ftotal_h = 1.07

PASS - Resistance to sliding is greater than sliding force

Overturning check












37.5 kN/m


/ 2 = 5.6 kN/m




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kN/m



kN/m

Base soil; Fexc_h = -γGf × KP × cos(δb.d) × γmb × (hpass + hbase)2 /

2 = -27 kN/m

Total; Ftotal_h = Fsat_h + Fmoist_h + Fexc_h + Fwater_h + Fsur_h =

159.8 kN/m

Overturning moments on wall

Surcharge load; Msur_OT = Fsur_h × xsur_h = 94.2 kNm/m

Saturated retained soil; Msat_OT = Fsat_h × xsat_h = 2.3 kNm/m

Water; Mwater_OT = Fwater_h × xwater_h = 3.2 kNm/m

Moist retained soil; Mmoist_OT = Fmoist_h × xmoist_h = 238.9 kNm/m

Total; Mtotal_OT = Msat_OT + Mmoist_OT + Mwater_OT + Msur_OT =

338.7 kNm/m

Restoring moments on wall

Wall stem; Mstem_R = Fstem × xstem = 65 kNm/m

Wall base; Mbase_R = Fbase × xbase = 173.3 kNm/m

Saturated retained soil; Msat_R = Fsat_v × xsat_v = 141.3 kNm/m

Water; Mwater_R = Fwater_v × xwater_v = 111.4 kNm/m

Moist retained soil; Mmoist_R = Fmoist_v × xmoist_v = 947.6 kNm/m

Base soil; Mexc_R = Fexc_v × xexc_v - Fexc_h × xexc_h = 13.6 kNm/m

Total; Mtotal_R = Mstem_R + Mbase_R + Msat_R + Mmoist_R +

Mexc_R + Mwater_R = 1452.2 kNm/m

Check stability against overturning

Factor of safety; FoSot = Mtotal_R / Mtotal_OT = 4.288

PASS - Maximum restoring moment is greater than overturning moment

Bearing pressure check


Wall stem; Fstem = γG × Astem × γstem = 50 kN/m

Wall base; Fbase = γG × Abase × γbase = 62.4 kN/m

Surcharge load; Fsur_v = γQ × SurchargeQ × lheel = 52 kN/m

Saturated retained soil; Fsat_v = γG × Asat × (γsr - γw) = 39.8 kN/m

Water; Fwater_v = γG × Awater × γw = 31.4 kN/m

Moist retained soil; Fmoist_v = γG × Amoist × γmr = 258.9 kN/m

Base soil; Fpass_v = γG × Apass × γmb = 10.1 kN/m




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Total; Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fpass_v +

Fwater_v + Fsur_v = 504.6 kN/m



37.5 kN/m


/ 2 = 5.6 kN/m


kN/m



kN/m

Base soil; Fpass_h = -γGf × KP × cos(δb.d) × γmb × (dcover + hbase)2 /

2 = -27 kN/m

Total; Ftotal_h = max(Fsat_h + Fmoist_h + Fpass_h + Fwater_h +

Fsur_h - (ab.d × b + Ftotal_v × tan(δbb.d)), 0 kN/m) = 0

kN/m

Moments on wall

Wall stem; Mstem = Fstem × xstem = 65 kNm/m

Wall base; Mbase = Fbase × xbase = 173.3 kNm/m

Surcharge load; Msur = Fsur_v × xsur_v - Fsur_h × xsur_h = 90.4 kNm/m

Saturated retained soil; Msat = Fsat_v × xsat_v - Fsat_h × xsat_h = 139 kNm/m

Water; Mwater = Fwater_v × xwater_v - Fwater_h × xwater_h = 108.2

kNm/m

Moist retained soil; Mmoist = Fmoist_v × xmoist_v - Fmoist_h × xmoist_h = 708.7

kNm/m

Base soil; Mpass = Fpass_v × xpass_v - Fpass_h × xpass_h = 13.6

kNm/m

Total; Mtotal = Mstem + Mbase + Msat + Mmoist + Mpass + Mwater

+ Msur = 1298.1 kNm/m

Check bearing pressure

Distance to reaction; x = Mtotal / Ftotal_v = 2573 mm

Eccentricity of reaction; e = x - lbase / 2 = -202 mm

Loaded length of base; lload = 2 × x = 5145 mm

Bearing pressure at toe; qtoe = Ftotal_v / lload = 98.1 kN/m2

Bearing pressure at heel; qheel = 0 kN/m2

Effective overburden pressure; q = (tbase + dcover) × γmb - (tbase + dcover + hwater) × γw =

6.7 kN/m2

Design effective overburden pressure; q' = q / γγ = 6.7 kN/m2




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Bearing resistance factors; Nq = Exp(π × tan(φ'b.d)) × (tan(45 deg + φ'b.d / 2))2 =

6.698

Nc = (Nq - 1) × cot(φ'b.d) = 15.273

Nγ = 2 × (Nq - 1) × tan(φ'b.d) = 4.251

Foundation shape factors; sq = 1

sγ = 1

sc = 1

Load inclination factors; H = Ftotal_h = 0 kN/m

V = Ftotal_v = 504.6 kN/m

m = 2

iq = [1 - H / (V + lload × c'b.d × cot(φ'b.d))]m = 1

iγ = [1 - H / (V + lload × c'b.d × cot(φ'b.d))](m + 1)

= 1

ic = iq - (1 - iq) / (Nc × tan(φ'b.d)) = 1

Net ultimate bearing capacity; nf = c'b.d × Nc × sc × ic + q' × Nq × sq × iq + 0.5 × (γmb

- γw) × lload × Nγ × sγ × iγ = 217.7 kN/m2

Factor of safety; FoSbp = nf / max(qtoe, qheel) = 2.22

PASS - Allowable bearing pressure exceeds maximum applied bearing pressure

RETAINING WALL DESIGN

In accordance with EN1992-1-1:2004 incorporating Corrigendum dated January

2008 and the recommended values

Concrete details - Table 3.1 - Strength and deformation characteristics for

concrete

Concrete strength class; C20/25

Characteristic compressive cylinder strength; fck = 20 N/mm2

Characteristic compressive cube strength; fck,cube = 25 N/mm2

Mean value of compressive cylinder strength; fcm = fck + 8 N/mm2 = 28 N/mm

2

Mean value of axial tensile strength; fctm = 0.3 N/mm2 × (fck / 1 N/mm

2)2/3 = 2.2 N/mm

2

5% fractile of axial tensile strength; fctk,0.05 = 0.7 × fctm = 1.5 N/mm2

Secant modulus of elasticity of concrete; Ecm = 22 kN/mm2 × (fcm / 10 N/mm

2)0.3 = 29962

N/mm2

Partial factor for concrete - Table 2.1N; γC = 1.50

Compressive strength coefficient - cl.3.1.6(1); αcc = 1.00

Design compressive concrete strength - exp.3.15; fcd = αcc × fck / γC = 13.3 N/mm2

Maximum aggregate size; hagg = 20 mm

Reinforcement details

Characteristic yield strength of reinforcement; fyk = 500 N/mm2

Modulus of elasticity of reinforcement; Es = 200000 N/mm2




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Partial factor for reinforcing steel - Table 2.1N; γS = 1.15

Design yield strength of reinforcement; fyd = fyk / γS = 435 N/mm2

Cover to reinforcement

Front face of stem; csf = 40 mm

Rear face of stem; csr = 50 mm

Top face of base; cbt = 50 mm

Bottom face of base; cbb = 75 mm

Check stem design at base of stem

Depth of section; h = 550 mm

Rectangular section in flexure - Section 6.1

Design bending moment combination 1; M = 235.8 kNm/m

Depth to tension reinforcement; d = h - csr - φsr / 2 = 488 mm

K = M / (d2 × fck) = 0.050

K' = 0.196

K' > K - No compression reinforcement is required

Lever arm; z = min(0.5 + 0.5 × (1 – 3.53 × K)0.5, 0.95) × d =

463 mm

Depth of neutral axis; x = 2.5 × (d – z) = 61 mm

Area of tension reinforcement required; Asr.req = M / (fyd × z) = 1171 mm2/m

Tension reinforcement provided; 25 dia.bars @ 200 c/c

Area of tension reinforcement provided; Asr.prov = π × φsr2 / (4 × ssr) = 2454 mm

2/m

Minimum area of reinforcement - exp.9.1N; Asr.min = max(0.26 × fctm / fyk, 0.0013) × d = 634

mm2/m

Maximum area of reinforcement - cl.9.2.1.1(3); Asr.max = 0.04 × h = 22000 mm2/m

max(Asr.req, Asr.min) / Asr.prov = 0.477

PASS - Area of reinforcement provided is greater than area of reinforcement required

Crack control - Section 7.3

Limiting crack width; wmax = 0.3 mm

Variable load factor - EN1990 – Table A1.1; ψ2 = 0.3

Serviceability bending moment; Msls = 152.3 kNm/m

Tensile stress in reinforcement; σs = Msls / (Asr.prov × z) = 134 N/mm2

Load duration; Long term

Load duration factor; kt = 0.4

Effective area of concrete in tension; Ac.eff = min(2.5 × (h - d), (h – x) / 3, h / 2) = 156250

mm2/m

Mean value of concrete tensile strength; fct.eff = fctm = 2.2 N/mm2

Reinforcement ratio; ρp.eff = Asr.prov / Ac.eff = 0.016

Modular ratio; αe = Es / Ecm = 6.675

Bond property coefficient; k1 = 0.8




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Strain distribution coefficient; k2 = 0.5

k3 = 3.4

k4 = 0.425

Maximum crack spacing - exp.7.11; sr.max = k3 × csr + k1 × k2 × k4 × φsr / ρp.eff = 441 mm

Maximum crack width - exp.7.8; wk = sr.max × max(σs – kt × (fct.eff / ρp.eff) × (1 + αe ×

ρp.eff), 0.6 × σs) / Es

wk = 0.177 mm

wk / wmax = 0.59

PASS - Maximum crack width is less than limiting crack width

Rectangular section in shear - Section 6.2

Design shear force; V = 153.5 kN/m

CRd,c = 0.18 / γC = 0.120

k = min(1 + √(200 mm / d), 2) = 1.641

Longitudinal reinforcement ratio; ρl = min(Asr.prov / d, 0.02) = 0.005

vmin = 0.035 N1/2/mm × k3/2 × fck

0.5 = 0.329 N/mm

2

Design shear resistance - exp.6.2a & 6.2b; VRd.c = max(CRd.c × k × (100 N2/mm

4 × ρl × fck)

1/3,

vmin) × d

VRd.c = 207.2 kN/m

V / VRd.c = 0.741

PASS - Design shear resistance exceeds design shear force

Horizontal reinforcement parallel to face of stem - Section 9.6

Minimum area of reinforcement – cl.9.6.3(1); Asx.req = max(0.25 × Asr.prov, 0.001 × (tstem + lslf)) =

614 mm2/m

Maximum spacing of reinforcement – cl.9.6.3(2); ssx_max = 400 mm

Transverse reinforcement provided; 16 dia.bars @ 200 c/c

Area of transverse reinforcement provided; Asx.prov = π × φsx2 / (4 × ssx) = 1005 mm

2/m


Check base design at toe




Depth to tension reinforcement; d = h - cbb - φbb / 2 = 367 mm

K = M / (d2 × fck) = 0.018

K' = 0.196


Lever arm; z = min(0.5 + 0.5 × (1 – 3.53 × K)0.5, 0.95) × d =

349 mm


Area of tension reinforcement required; Abb.req = M / (fyd × z) = 323 mm2/m




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Area of tension reinforcement provided; Abb.prov = π × φbb2 / (4 × sbb) = 1005 mm

2/m

Minimum area of reinforcement - exp.9.1N; Abb.min = max(0.26 × fctm / fyk, 0.0013) × d = 477

mm2/m

Maximum area of reinforcement - cl.9.2.1.1(3); Abb.max = 0.04 × h = 18000 mm2/m

max(Abb.req, Abb.min) / Abb.prov = 0.475






Tensile stress in reinforcement; σs = Msls / (Abb.prov × z) = 100.8 N/mm2




mm2/m


Reinforcement ratio; ρp.eff = Abb.prov / Ac.eff = 0.007




k3 = 3.4

k4 = 0.425

Maximum crack spacing - exp.7.11; sr.max = k3 × cbb + k1 × k2 × k4 × φbb / ρp.eff = 619 mm


ρp.eff), 0.6 × σs) / Es

wk = 0.187 mm

wk / wmax = 0.625




CRd,c = 0.18 / γC = 0.120

k = min(1 + √(200 mm / d), 2) = 1.738

Longitudinal reinforcement ratio; ρl = min(Abb.prov / d, 0.02) = 0.003

vmin = 0.035 N1/2/mm × k3/2 × fck

0.5 = 0.359 N/mm

2


4 × ρl × fck)

1/3,

vmin) × d

VRd.c = 134.9 kN/m

V / VRd.c = 0.722





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Check base design at heel




Depth to tension reinforcement; d = h - cbt - φbt / 2 = 388 mm

K = M / (d2 × fck) = 0.079

K' = 0.196


Lever arm; z = min(0.5 + 0.5 × (1 – 3.53 × K)0.5, 0.95) × d =

358 mm


Area of tension reinforcement required; Abt.req = M / (fyd × z) = 1529 mm2/m


Area of tension reinforcement provided; Abt.prov = π × φbt2 / (4 × sbt) = 2454 mm

2/m

Minimum area of reinforcement - exp.9.1N; Abt.min = max(0.26 × fctm / fyk, 0.0013) × d = 504

mm2/m

Maximum area of reinforcement - cl.9.2.1.1(3); Abt.max = 0.04 × h = 18000 mm2/m

max(Abt.req, Abt.min) / Abt.prov = 0.623






Tensile stress in reinforcement; σs = Msls / (Abt.prov × z) = 122.5 N/mm2




mm2/m


Reinforcement ratio; ρp.eff = Abt.prov / Ac.eff = 0.020




k3 = 3.4

k4 = 0.425

Maximum crack spacing - exp.7.11; sr.max = k3 × cbt + k1 × k2 × k4 × φbt / ρp.eff = 387 mm


ρp.eff), 0.6 × σs) / Es

wk = 0.142 mm




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wk / wmax = 0.475




CRd,c = 0.18 / γC = 0.120

k = min(1 + √(200 mm / d), 2) = 1.718

Longitudinal reinforcement ratio; ρl = min(Abt.prov / d, 0.02) = 0.006

vmin = 0.035 N1/2/mm × k3/2 × fck

0.5 = 0.353 N/mm

2


4 × ρl × fck)

1/3,

vmin) × d

VRd.c = 186.3 kN/m

V / VRd.c = 0.483


Secondary transverse reinforcement to base - Section 9.3

Minimum area of reinforcement – cl.9.3.1.1(2); Abx.req = 0.2 × Abt.prov = 491 mm2/m

Maximum spacing of reinforcement – cl.9.3.1.1(3); sbx_max = 450 mm

Transverse reinforcement provided; 16 dia.bars @ 300 c/c

Area of transverse reinforcement provided; Abx.prov = π × φbx2 / (4 × sbx) = 670 mm

2/m





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Documents

Sachpazis cantilever retaining wall analysis & design (en1997-1-2004)