Sachpazis: Sloped rear face retaining wall example

Job Ref.

Project: Sloped rear face retaining wall Analysis & Design, In accordance with
EN1997-1:2004 incorporating Corrigendum dated February 2009 and the
recommended values
Section
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) 6936425722 & (+44) 7585939944, costas@sachpazis.info

Sheet no./rev. 1

Civil & Geotechnical Engineering

GEODOMISI Ltd. - Dr. Costas Sachpazis

Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

RETAINING WALL ANALYSIS
In accordance with EN1997-1:2004 incorporating Corrigendum dated February
2009 and the recommended values
Retaining wall details
Stem type;

Cantilever with inclined rear face

Stem height;

hstem = 4700 mm

Stem thickness;

tstem = 300 mm

Slope length to rear of stem;

lslr = 400 mm

Angle to rear face of stem;

α = atan(hstem / lslr) = 85.1 deg

Stem density;

γstem = 25 kN/m

3

Toe length;

ltoe = 3000 mm

Heel length;

lheel = 2500 mm

Base thickness;

tbase = 550 mm

Key position;

pkey = 5700 mm

Key depth;

dkey = 500 mm

Key thickness;

tkey = 500 mm

Base density;

γbase = 25 kN/m

Height of retained soil;

hret = 4000 mm

3

Angle of soil surface;

β = 10 deg

Depth of cover;

dcover = 700 mm

Depth of excavation;

dexc = 700 mm

Height of water;

hwater = 500 mm

Job Ref.

recommended values
Section

Calc. by


Dr. C. Sachpazis

Water density;

Date

Chk'd by
Date

22/12/2013
γw = 9.8 kN/m

3

Retained soil properties
Soil type;

Organic clay

Moist density;

γmr = 15 kN/m

Saturated density;

γsr = 16 kN/m

Characteristic effective shear resistance angle;

φ'r.k = 29 deg

Characteristic wall friction angle;

δr.k = 16 deg

3

3

Base soil properties
Soil type;

Organic clay

Moist density;

γmb = 17 kN/m

Characteristic cohesion;

c'b.k = 55 kN/m

3
2

2

Characteristic adhesion;

ab.k = 52 kN/m

Characteristic effective shear resistance angle;

φ'b.k = 26 deg

Characteristic wall friction angle;

δb.k = 15 deg

Characteristic base friction angle;

δbb.k = 26 deg

Loading details
Variable surcharge load;

Sheet no./rev. 1



SurchargeQ = 10 kN/m

2

App'd by

Date

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

22/12/2013

Chk'd by
Date

App'd by

Date

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by


Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

Calculate retaining wall geometry
Base length;

lbase = ltoe + tstem + lslr + lheel = 6200 mm

Base height;

hbase = tbase + dkey = 1050 mm

Saturated soil height;

hsat = hwater + dcover = 1200 mm

Moist soil height;

hmoist = hret - hwater = 3500 mm

Length of surcharge load;

lsur = (lheel + lslr × hsoil / hstem) = 2900 mm

- Distance to vertical component;

xsur_v = lbase - (lheel + lslr × hsoil / hstem) / 2 = 4750 mm

Effective height of wall;

heff = hbase + dcover + hret + lsur × tan(β) = 6261 mm

- Distance to horizontal component;

xsur_h = heff / 2 - dkey = 2631 mm

Area of wall stem;

Astem = hstem × (tstem + lslr / 2) = 2.35 m

2


xstem = (hstem × tstem × (ltoe + tstem / 2) + hstem × lslr / 2 × (ltoe + tstem + lslr / 3)) / Astem = 3263 mm

Area of wall base;

Abase = lbase × tbase + dkey × tkey = 3.66 m


xbase = (lbase × tbase / 2 + dkey × tkey × (pkey + tkey / 2)) / Abase = 3295 mm

Area of saturated soil;

Asat = hsat × (lheel + lslr × hsat / (2 × hstem)) = 3.061 m


xsat_v = lbase - (hsat × lheel / 2 + lslr × hsat / (2 × hstem) × (lheel + lslr × hsat / (3 × hstem))) / Asat = 4924 mm


xsat_h = (hsat + hbase) / 3 - dkey = 250 mm

Area of water;

Awater = hsat × (lheel + lslr × hsat / (2 × hstem)) = 3.061 m

2

2

2

2

2

2

2
lheel

2


xwater_v = lbase - (hsat ×

/ 2 + lslr × hsat / (2 × hstem) × (lheel + lslr × hsat / (3 × hstem))) / Asat = 4924 mm


xwater_h = (hsat + hbase) / 3 - dkey = 250 mm

Area of moist soil;

Amoist = (hret - hwater) × (lheel + lslr × (hmoist + 2 × hsat) / (2 × hstem)) + tan(β) × (lheel + lslr × hsoil / hstem) / 2 =

2

2

10.37 m

2

2

xmoist_v = lbase - (hmoist × (lheel + lslr × hsat / hstem) / 2 + lslr × hmoist / (2 × hstem) × ((lheel + lslr × hsat / hstem) + lslr ×
3

hmoist / (3 × hstem)) + tan(β) × (lheel + lslr × hsoil / hstem) / 6) / Amoist = 4852 mm

xmoist_h = ((heff - hsat - hbase) × (tbase + hsat + (heff - hsat - hbase) / 3) / 2 + (hsat + hbase) × ((hsat + hbase)/2 - dkey)) /
(hsat + hbase + (heff - hsat - hbase) / 2) = 1785 mm

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by


Dr. C. Sachpazis

Area of base soil;

Date

Chk'd by
Date

22/12/2013

App'd by

Date

2

Apass = dcover × ltoe = 2.1 m


xpass_v = lbase - (dcover × ltoe× (lbase - ltoe / 2)) / Apass = 1500 mm


xpass_h = (dcover + hbase) / 3- dkey = 83 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

Effective cohesion;

γc' = 1.00

Weight density;

γγ = 1.00

Design effective shear resistance angle;

φ'r.d = atan(tan(φ'r.k) / γφ') = 29 deg

Design wall friction angle;

δr.d = atan(tan(δr.k) / γφ') = 16 deg


φ'b.d = atan(tan(φ'b.k) / γφ') = 26 deg


δb.d = atan(tan(δb.k) / γφ') = 15 deg

Design base friction angle;

δbb.d = atan(tan(δbb.k) / γφ') = 26 deg

Design effective cohesion;

c'b.d = c'b.k / γc' = 55 kN/m

Design adhesion;

2

2

ab.d = ab.k / γc' = 52 kN/m

Using Coulomb theory
Active pressure coefficient;

2

2

2

KA = sin(α + φ'r.d) / (sin(α) × sin(α - δr.d) × [1 + √[sin(φ'r.d + δr.d) × sin(φ'r.d - β) / (sin(α - δr.d) × sin(α + β))]] )
= 0.400

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Passive pressure coefficient;

Date

Chk'd by
Date

22/12/2013

App'd by

2

Date
2

KP = sin(90 - φ'b.d) / (sin(90 + δb.d) × [1 - √[sin(φ'b.d + δb.d) × sin(φ'b.d) / (sin(90 + δb.d))]] ) = 4.051

Sliding check
Vertical forces on wall
Wall stem;

Fstem = γGf × Astem × γstem = 58.8 kN/m

Wall base;

Fbase = γGf × Abase × γbase = 91.5 kN/m

Saturated retained soil;

Fsat_v = γGf × Asat × (γsr - γw) = 18.9 kN/m

Water;

Fwater_v = γGf × Awater × γw = 30 kN/m

Moist retained soil;

Fmoist_v = γGf × Amoist × γmr = 155.6 kN/m

Total;

Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fwater_v = 354.8 kN/m

Horizontal forces on wall
Surcharge load;

Fsur_h = KA × cos(δr.d + (90 - α)) × γQ × SurchargeQ × heff = 35.1 kN/m


Fsat_h = γG × KA × cos(δr.d + (90 - α)) × (γsr - γw) × (hsat + hbase) / 2 = 7.9 kN/m

Water;

Fwater_h = γG × γw × (hwater + dcover + hbase) / 2 = 33.5 kN/m


Fmoist_h = γG × KA × cos(δr.d + (90 - α)) × γmr × ((heff - hsat - hbase) / 2 + (heff - hsat - hbase) × (hsat + hbase)) =

2

2

2

129.3 kN/m
Total;

Ftotal_h = Fsat_h + Fmoist_h + Fwater_h + Fsur_h = 205.9 kN/m

Check stability against sliding
Base soil resistance;

2

Fexc_h = γGf × KP × cos(δb.d) × γmb × (hpass + hbase) / 2 = 36.7 kN/m

Base friction;

Ffriction = ab.d × b + Ftotal_v × tan(δbb.d) = 225 kN/m

Resistance to sliding;

Frest = Fexc_h + Ffriction = 261.7 kN/m

Factor of safety;

FoSsl = Frest / Ftotal_h = 1.271
PASS - Resistance to sliding is greater than sliding force

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

Overturning check
Overturning moments on wall
Surcharge load;

Msur_OT = Fsur_h × xsur_h = 92.4 kNm/m


Msat_OT = Fsat_h × xsat_h = 2 kNm/m

Water;

Mwater_OT = Fwater_h × xwater_h = 8.4 kNm/m


Mmoist_OT = Fmoist_h × xmoist_h = 230.9 kNm/m

Base soil;

Mexc_OT = -Fexc_h × xexc_h = 5.5 kNm/m

Total;

Mtotal_OT = Msat_OT + Mmoist_OT + Mexc_OT + Mwater_OT + Msur_OT = 339.2 kNm/m

Restoring moments on wall
Wall stem;

Mstem_R = Fstem × xstem = 191.7 kNm/m

Wall base;

Mbase_R = Fbase × xbase = 301.5 kNm/m


Msat_R = Fsat_v × xsat_v = 93.3 kNm/m

Water;

Mwater_R = Fwater_v × xwater_v = 147.9 kNm/m


Mmoist_R = Fmoist_v × xmoist_v = 754.8 kNm/m

Total;

Mtotal_R = Mstem_R + Mbase_R + Msat_R + Mmoist_R + Mwater_R = 1489.2 kNm/m

Check stability against overturning
Factor of safety;

FoSot = Mtotal_R / Mtotal_OT = 4.39
PASS - Maximum restoring moment is greater than overturning moment

Bearing pressure check
Wall stem;

Fstem = γG × Astem × γstem = 79.3 kN/m

Wall base;

Fbase = γG × Abase × γbase = 123.5 kN/m

Surcharge load;

Fsur_v = γQ × SurchargeQ × (lheel + lslr × hsoil / hstem) = 43.5 kN/m


Fsat_v = γG × Asat × (γsr - γw) = 25.6 kN/m

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Water;

Fwater_v = γG × Awater × γw = 40.5 kN/m


Fmoist_v = γG × Amoist × γmr = 210 kN/m

Base soil;

Fpass_v = γG × Apass × γmb = 48.2 kN/m

Total;

Date

Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fpass_v + Fwater_v + Fsur_v = 570.7 kN/m

Surcharge load;




2

2

Water;




2

129.3 kN/m
2

Base soil;

Fpass_h = -γGf × KP × cos(δb.d) × γmb × (dcover + hbase) / 2 = -101.9 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 = 258.8 kNm/m

Wall base;

Mbase = Fbase × xbase = 407 kNm/m

Surcharge load;

Msur = Fsur_v × xsur_v - Fsur_h × xsur_h = 114.2 kNm/m


Msat = Fsat_v × xsat_v - Fsat_h × xsat_h = 124 kNm/m

Water;

Mwater = Fwater_v × xwater_v - Fwater_h × xwater_h = 191.3 kNm/m


Mmoist = Fmoist_v × xmoist_v - Fmoist_h × xmoist_h = 788.1 kNm/m

Base soil;

Mpass = Fpass_v × xpass_v - Fpass_h × xpass_h = 80.8 kNm/m

Total;

Mtotal = Mstem + Mbase + Msat + Mmoist + Mpass + Mwater + Msur = 1964.1 kNm/m

Check bearing pressure
Distance to reaction;

x = Mtotal / Ftotal_v = 3442 mm

Eccentricity of reaction;

e = x - lbase / 2 = 342 mm

Loaded length of base;

lload = 2 × (lbase - x) = 5516 mm

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

2

Bearing pressure at toe;

qtoe = 0 kN/m

Bearing pressure at heel;

qheel = Ftotal_v / lload = 103.4 kN/m

Effective overburden pressure;

q = (tbase + dcover) × γmb - (tbase + dcover + hwater) × γw = 4.1 kN/m

2
2

2

Design effective overburden pressure;

q' = q / γγ = 4.1 kN/m

Bearing resistance factors;

Nq = Exp(π × tan(φ'b.d)) × (tan(45 deg + φ'b.d / 2)) = 11.854

2

Nc = (Nq - 1) × cot(φ'b.d) = 22.254
Nγ = 2 × (Nq - 1) × tan(φ'b.d) = 10.588
Foundation shape factors;

sq = 1
sγ = 1
sc = 1

Load inclination factors;

H = Ftotal_h = 0 kN/m
V = Ftotal_v = 570.7 kN/m
m=2
m

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

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

=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 = 1482.4 kN/m

Factor of safety;

2

FoSbp = nf / max(qtoe, qheel) = 14.329
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

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

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


φ'r.d = atan(tan(φ'r.k) / γφ') = 23.9 deg


δr.d = atan(tan(δr.k) / γφ') = 12.9 deg


φ'b.d = atan(tan(φ'b.k) / γφ') = 21.3 deg


δb.d = atan(tan(δb.k) / γφ') = 12.1 deg

Design base friction angle;

δbb.d = atan(tan(δbb.k) / γφ') = 21.3 deg

Design effective cohesion;

c'b.d = c'b.k / γc' = 44 kN/m

Design adhesion;

ab.d = ab.k / γc' = 41.6 kN/m

2
2

Using Coulomb theory
Active pressure coefficient;

2

2

2

KA = sin(α + φ'r.d) / (sin(α) × sin(α - δr.d) × [1 + √[sin(φ'r.d + δr.d) × sin(φ'r.d - β) / (sin(α - δr.d) × sin(α + β))]] )
= 0.489

Passive pressure coefficient;

2

2

KP = sin(90 - φ'b.d) / (sin(90 + δb.d) × [1 - √[sin(φ'b.d + δb.d) × sin(φ'b.d) / (sin(90 + δb.d))]] ) = 2.961

Sliding check
Wall stem;

Fstem = γGf × Astem × γstem = 58.8 kN/m

Wall base;

Fbase = γGf × Abase × γbase = 91.5 kN/m


Fsat_v = γGf × Asat × (γsr - γw) = 18.9 kN/m

Water;

Fwater_v = γGf × Awater × γw = 30 kN/m


Fmoist_v = γGf × Amoist × γmr = 155.6 kN/m

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Total;

Date

Chk'd by
Date

22/12/2013

App'd by

Date

Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fwater_v = 354.8 kN/m

Surcharge load;




Water;




2

2

2

119.3 kN/m
Total;

Ftotal_h = Fsat_h + Fmoist_h + Fwater_h + Fsur_h = 189.3 kN/m

Check stability against sliding
Base soil resistance;

2

Fexc_h = γGf × KP × cos(δb.d) × γmb × (hpass + hbase) / 2 = 27.1 kN/m

Base friction;

Ffriction = ab.d × b + Ftotal_v × tan(δbb.d) = 180 kN/m

Resistance to sliding;

Frest = Fexc_h + Ffriction = 207.2 kN/m

Factor of safety;

FoSsl = Frest / Ftotal_h = 1.094
PASS - Resistance to sliding is greater than sliding force

Overturning check
Overturning moments on wall
Surcharge load;

Msur_OT = Fsur_h × xsur_h = 99.8 kNm/m


Msat_OT = Fsat_h × xsat_h = 1.8 kNm/m

Water;

Mwater_OT = Fwater_h × xwater_h = 6.2 kNm/m


Mmoist_OT = Fmoist_h × xmoist_h = 213 kNm/m

Base soil;

Mexc_OT = -Fexc_h × xexc_h = 4.1 kNm/m

Total;

Mtotal_OT = Msat_OT + Mmoist_OT + Mexc_OT + Mwater_OT + Msur_OT = 324.8 kNm/m

Restoring moments on wall
Wall stem;

Mstem_R = Fstem × xstem = 191.7 kNm/m

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

Wall base;

App'd by

Date

Mbase_R = Fbase × xbase = 301.5 kNm/m


Msat_R = Fsat_v × xsat_v = 93.3 kNm/m

Water;

Mwater_R = Fwater_v × xwater_v = 147.9 kNm/m


Mmoist_R = Fmoist_v × xmoist_v = 754.8 kNm/m

Total;

Mtotal_R = Mstem_R + Mbase_R + Msat_R + Mmoist_R + Mwater_R = 1489.2 kNm/m

Check stability against overturning
Factor of safety;

FoSot = Mtotal_R / Mtotal_OT = 4.584
PASS - Maximum restoring moment is greater than overturning moment

Bearing pressure check
Wall stem;

Fstem = γG × Astem × γstem = 58.8 kN/m

Wall base;

Fbase = γG × Abase × γbase = 91.5 kN/m

Surcharge load;

Fsur_v = γQ × SurchargeQ × (lheel + lslr × hsoil / hstem) = 37.7 kN/m


Fsat_v = γG × Asat × (γsr - γw) = 18.9 kN/m

Water;

Fwater_v = γG × Awater × γw = 30 kN/m


Fmoist_v = γG × Amoist × γmr = 155.6 kN/m

Base soil;

Fpass_v = γG × Apass × γmb = 35.7 kN/m

Total;

Ftotal_v = Fstem + Fbase + Fsat_v + Fmoist_v + Fpass_v + Fwater_v + Fsur_v = 428.2 kN/m

Surcharge load;




Water;




2

2

2

119.3 kN/m

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

2

Base soil;

Fpass_h = -γGf × KP × cos(δb.d) × γmb × (dcover + hbase) / 2 = -75.4 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 = 191.7 kNm/m

Wall base;

Mbase = Fbase × xbase = 301.5 kNm/m

Surcharge load;

Msur = Fsur_v × xsur_v - Fsur_h × xsur_h = 79.3 kNm/m


Msat = Fsat_v × xsat_v - Fsat_h × xsat_h = 91.5 kNm/m

Water;

Mwater = Fwater_v × xwater_v - Fwater_h × xwater_h = 141.7 kNm/m


Mmoist = Fmoist_v × xmoist_v - Fmoist_h × xmoist_h = 541.8 kNm/m

Base soil;

Mpass = Fpass_v × xpass_v - Fpass_h × xpass_h = 59.8 kNm/m

Total;

Mtotal = Mstem + Mbase + Msat + Mmoist + Mpass + Mwater + Msur = 1407.3 kNm/m

Check bearing pressure
Distance to reaction;

x = Mtotal / Ftotal_v = 3287 mm

Eccentricity of reaction;

e = x - lbase / 2 = 187 mm

Loaded length of base;

lload = 2 × (lbase - x) = 5826 mm

Bearing pressure at toe;

qtoe = 0 kN/m

Bearing pressure at heel;

qheel = Ftotal_v / lload = 73.5 kN/m

Effective overburden pressure;

q = (tbase + dcover) × γmb - (tbase + dcover + hwater) × γw = 4.1 kN/m

Design effective overburden pressure;

q' = q / γγ = 4.1 kN/m

Bearing resistance factors;

Nq = Exp(π × tan(φ'b.d)) × (tan(45 deg + φ'b.d / 2)) = 7.298

2
2

2
2

Nc = (Nq - 1) × cot(φ'b.d) = 16.141
Nγ = 2 × (Nq - 1) × tan(φ'b.d) = 4.915
Foundation shape factors;

sq = 1
sγ = 1
sc = 1

2

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Load inclination factors;

Date

Chk'd by
Date

22/12/2013

App'd by

Date

H = Ftotal_h = 0 kN/m
V = Ftotal_v = 428.2 kN/m
m=2
m

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

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

=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 = 843 kN/m

Factor of safety;

2

FoSbp = nf / max(qtoe, qheel) = 11.471
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;

C32/40

Characteristic compressive cylinder strength;

fck = 32 N/mm

Characteristic compressive cube strength;

fck,cube = 40 N/mm

Mean value of compressive cylinder strength;

fcm = fck + 8 N/mm = 40 N/mm

Mean value of axial tensile strength;

fctm = 0.3 N/mm × (fck / 1 N/mm )

2
2
2

2

2
2 2/3

= 3.0 N/mm

2

2

5% fractile of axial tensile strength;

fctk,0.05 = 0.7 × fctm = 2.1 N/mm

Secant modulus of elasticity of concrete;

Ecm = 22 kN/mm × (fcm / 10 N/mm )

Partial factor for concrete - Table 2.1N;

γC = 1.50

2

2 0.3

2

= 33346 N/mm

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

Compressive strength coefficient - cl.3.1.6(1);

fcd = αcc × fck / γC = 21.3 N/mm

Maximum aggregate size;

Date

αcc = 1.00

Design compressive concrete strength - exp.3.15;

App'd by

hagg = 20 mm

2

Reinforcement details
2

Characteristic yield strength of reinforcement;

fyk = 500 N/mm

Modulus of elasticity of reinforcement;

Es = 200000 N/mm

Partial factor for reinforcing steel - Table 2.1N;

γS = 1.15

Design yield strength of reinforcement;

fyd = fyk / γS = 435 N/mm

2

2

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 = 700 mm

Rectangular section in flexure - Section 6.1
Design bending moment combination 1;

M = 243.2 kNm/m

Depth to tension reinforcement;

d = h - csr - φsr / 2 = 640 mm
2

K = M / (d × fck) = 0.019
K' = 0.196
K' > K - No compression reinforcement is required
0.5

Lever arm;

z = min(0.5 + 0.5 × (1 – 3.53 × K) , 0.95) × d = 608 mm

Depth of neutral axis;

x = 2.5 × (d – z) = 80 mm

Area of tension reinforcement required;

Asr.req = M / (fyd × z) = 920 mm /m

Tension reinforcement provided;

20 dia.bars @ 200 c/c

2

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013
2

App'd by

Date

2

Area of tension reinforcement provided;

Asr.prov = π × φsr / (4 × ssr) = 1571 mm /m

Minimum area of reinforcement - exp.9.1N;

Asr.min = max(0.26 × fctm / fyk, 0.0013) × d = 1006 mm /m

Maximum area of reinforcement - cl.9.2.1.1(3);

2

2

Asr.max = 0.04 × h = 28000 mm /m
max(Asr.req, Asr.min) / Asr.prov = 0.641
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 = 146.7 kNm/m

Tensile stress in reinforcement;

σs = Msls / (Asr.prov × z) = 153.6 N/mm

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) = 150000 mm /m

2

2

2

Mean value of concrete tensile strength;

fct.eff = fctm = 3.0 N/mm

Reinforcement ratio;

ρp.eff = Asr.prov / Ac.eff = 0.010

Modular ratio;

αe = Es / Ecm = 5.998

Bond property coefficient;

k1 = 0.8

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 = 495 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 = 0.228 mm
wk / wmax = 0.76
PASS - Maximum crack width is less than limiting crack width

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

Rectangular section in shear - Section 6.2
Design shear force;

V = 135.6 kN/m
CRd,c = 0.18 / γC = 0.120
k = min(1 + √(200 mm / d), 2) = 1.559

Longitudinal reinforcement ratio;

ρl = min(Asr.prov / d, 0.02) = 0.002
1/2

vmin = 0.035 N /mm × k
Design shear resistance - exp.6.2a & 6.2b;

3/2

0.5

× fck
2

2

= 0.385 N/mm
4

1/3

VRd.c = max(CRd.c × k × (100 N /mm × ρl × fck) , vmin) × d
VRd.c = 246.7 kN/m
V / VRd.c = 0.550
PASS - Design shear resistance exceeds design shear force

Horizontal reinforcement parallel to face of stem - Section 9.6
2

Minimum area of reinforcement – cl.9.6.3(1);

Asx.req = max(0.25 × Asr.prov, 0.001 × (tstem + lslr)) = 700 mm /m

Maximum spacing of reinforcement – cl.9.6.3(2);

ssx_max = 400 mm

Transverse reinforcement provided;


Area of transverse reinforcement provided;

Asx.prov = π × φsx / (4 × ssx) = 1005 mm /m

2

2


Check base design
Depth of section;

h = 550 mm


M = 138.1 kNm/m


d = h - cbb - φbb / 2 = 465 mm
2

K = M / (d × fck) = 0.020
K' = 0.196
Lever arm;

0.5

z = min(0.5 + 0.5 × (1 – 3.53 × K) , 0.95) × d = 442 mm

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013


Abb.req = M / (fyd × z) = 719 mm /m




Abb.prov = π × φbb / (4 × sbb) = 1571 mm /m


Abb.min = max(0.26 × fctm / fyk, 0.0013) × d = 731 mm /m


Date

x = 2.5 × (d – z) = 58 mm


App'd by

Abb.max = 0.04 × h = 22000 mm /m

2

2

2

2

2

max(Abb.req, Abb.min) / Abb.prov = 0.465


wmax = 0.3 mm


ψ2 = 0.3


Msls = 85.4 kNm/m


σs = Msls / (Abb.prov × z) = 123 N/mm

2

Load duration;

Long term


kt = 0.4






ρp.eff = Abb.prov / Ac.eff = 0.010

2

2

Modular ratio;

αe = Es / Ecm = 5.998


k1 = 0.8


k2 = 0.5
k3 = 3.4
k4 = 0.425


sr.max = k3 × cbb + k1 × k2 × k4 × φbb / ρp.eff = 610 mm


wk = 0.225 mm

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date

wk / wmax = 0.75

Design shear force;

V = 135.7 kN/m
CRd,c = 0.18 / γC = 0.120
k = min(1 + √(200 mm / d), 2) = 1.656


ρl = min(Abb.prov / d, 0.02) = 0.003
1/2

vmin = 0.035 N /mm × k

3/2

0.5

2

× fck

= 0.422 N/mm

2

4

1/3

VRd.c = 204.3 kN/m
V / VRd.c = 0.664


M = 126.1 kNm/m


d = h - cbt - φbt / 2 = 492 mm
2

K = M / (d × fck) = 0.016
K' = 0.196
Lever arm;

0.5

z = min(0.5 + 0.5 × (1 – 3.53 × K) , 0.95) × d = 467 mm


x = 2.5 × (d – z) = 62 mm


Abt.req = M / (fyd × z) = 621 mm /m

2




Abt.prov = π × φbt / (4 × sbt) = 1005 mm /m


Abt.min = max(0.26 × fctm / fyk, 0.0013) × d = 774 mm /m


2

2

2

2

Abt.max = 0.04 × h = 22000 mm /m
max(Abt.req, Abt.min) / Abt.prov = 0.77

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date



wmax = 0.3 mm


ψ2 = 0.3


Msls = 68.9 kNm/m


σs = Msls / (Abt.prov × z) = 146.7 N/mm

Load duration;

Long term


kt = 0.4



2

2

2




ρp.eff = Abt.prov / Ac.eff = 0.007

Modular ratio;

αe = Es / Ecm = 5.998


k1 = 0.8


k2 = 0.5
k3 = 3.4
k4 = 0.425


sr.max = k3 × cbt + k1 × k2 × k4 × φbt / ρp.eff = 562 mm


wk = 0.247 mm
wk / wmax = 0.825

Design shear force;

V = 90.4 kN/m
CRd,c = 0.18 / γC = 0.120
k = min(1 + √(200 mm / d), 2) = 1.638


ρl = min(Abt.prov / d, 0.02) = 0.002

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013
1/2

vmin = 0.035 N /mm × k

3/2

0.5

× fck
2

App'd by

Date

2

= 0.415 N/mm
4

1/3

VRd.c = 204.1 kN/m
V / VRd.c = 0.443

Check key design
Depth of section;

h = 500 mm


M = 7.1 kNm/m


d = h - cbb - φk / 2 = 417 mm
2

K = M / (d × fck) = 0.001
K' = 0.196
0.5

Lever arm;

z = min(0.5 + 0.5 × (1 – 3.53 × K) , 0.95) × d = 396 mm


x = 2.5 × (d – z) = 52 mm


Ak.req = M / (fyd × z) = 41 mm /m

2




Ak.prov = π × φk / (4 × sk) = 1005 mm /m


Ak.min = max(0.26 × fctm / fyk, 0.0013) × d = 656 mm /m


Ak.max = 0.04 × h = 20000 mm /m

2

2

2

2

max(Ak.req, Ak.min) / Ak.prov = 0.652


wmax = 0.3 mm


ψ2 = 0.3

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013


Date

Msls = 7.1 kNm/m


App'd by

σs = Msls / (Ak.prov × z) = 17.7 N/mm

2

Load duration;

Long term


kt = 0.4



2

2




ρp.eff = Ak.prov / Ac.eff = 0.007

Modular ratio;

αe = Es / Ecm = 5.998


k1 = 0.8


k2 = 0.5
k3 = 3.4
k4 = 0.425


sr.max = k3 × cbb + k1 × k2 × k4 × φk / ρp.eff = 659 mm


wk = 0.035 mm
wk / wmax = 0.117

Design shear force;

V = 26 kN/m
CRd,c = 0.18 / γC = 0.120
k = min(1 + √(200 mm / d), 2) = 1.693


ρl = min(Ak.prov / d, 0.02) = 0.002
1/2

vmin = 0.035 N /mm × k

3/2

0.5

× fck
2

2

= 0.436 N/mm
4

1/3

VRd.c = 181.8 kN/m
V / VRd.c = 0.143

Job Ref.

recommended values
Section

Sheet no./rev. 1



Calc. by

Dr. C. Sachpazis

Date

Chk'd by
Date

22/12/2013

App'd by

Date


Secondary transverse reinforcement to base - Section 9.3
Minimum area of reinforcement – cl.9.3.1.1(2);

2

Abx.req = 0.2 × Abb.prov = 314 mm /m

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


Area of transverse reinforcement provided;

Abx.prov = π × φbx / (4 × sbx) = 565 mm /m

2

2


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