LRFD Cast-in-Place Concrete Retaining Wall

August 10, 2011

James LuebkeWisconsin Department of Transportation

Wall ParametersWall Parameters

Height = 20’Height (exposed) = 16’Footing Cover = 4’ Footing Width = 10’Footing Thickness = 2.0’Supports Traffic

Soil Parameters (c’ = 0)φf = 30 degreesγf = 0.120 kcfφfd = 34 degreesγfd = 0.120 kcf

Wall Design Process

Non - Propriety Wall (WisDOT/Consultants Responsibility)Soil InvestigationPreliminary Geometry and Loads (Geotechnical)External Stability, Overall Stability and Settlement (Geotechnical)Wall Design (Structural)– Final Geometry and Loads– External Stability, Structural Resistance and Serviceability

Revise Overall Stability and Settlement?

LRFD CIP Concrete Wall Example Outline

Loads and Load CombinationsExternal Stability– Bearing Resistance– Sliding– Eccentricity

Structural ResistanceOverall Stability and Settlement

Loads– Horizontal Earth Pressure (EH)– Vertical Earth Pressure (EV)– Live Load Surcharge (LS)– Dead Load (DC)

Limit States– Service – Strength I

» Max/Min load factors » Live Load

Loads and Load Combinations

Strength Ia

Strength Ib

Live Load

Active Earth Pressure Resultant

Active Earth Pressure ResultantFT = ½ γf H2 ka = ½(0.120)(20)2(0.314) = 7.55 klf

Coefficient of Active Earth Pressure (Coulomb Theory)

FT

γf H ka

HEV

VEVδ’

φf = 30.0 °β = 0.0 °θ = 87.6 °δ = 21.0 °

Γ = 2.726Γ 1sin φ f δ+( ) sin φ f β−( )sin θ δ−( ) sin θ β+( )+

⎛⎜⎝

⎞⎟⎠

2

=

kasin θ φ f+( )2

Γ sin θ( )2 sin θ δ−( )= ka = 0.314

Live Load Surcharge Resultant

Live Load Surcharge ResultantFSUR = γf heq H ka = (0.120)(2)(20)(0.314) = 1.51 klf

Equivalent height of soil for live load surcharge (heq)

FSUR

γf heq ka

WisDOT policy item:γf heq = 240 psf (w/ Traffic Loads)γf heq = 100 psf (w/o Traffic Loads)

HLSδ’

Unfactored LoadsLoad Moment

Arm Moment LRFD Load TypeItem Description Value

(kip/ft) Item Value (ft) Item

Value (kip-ft/ft)

V1 Wall stem front batter 0.51 dv1 3.8 MV1 1.9 DCV2 Wall stem 2.70 dv2 4.4 MV2 11.8 DCV3 Wall stem back batter 1.01 dv3 5.1 MV3 5.2 DCV4 Wall footing 3.00 dv4 5.0 MV4 15.0 DCV7 Soil backfill 9.45 dv7 7.8 MV7 73.8 EVV8 Soil backfill 0.81 dv8 5.4 MV8 4.4 EVV9 Soil backfill 0.00 dv9 8.3 MV9 0.0 EVV10 Live load surcharge 1.23 dv10 7.4 MV10 9.1 LSV11 Active earth pressure 2.99 dv11 10.0 MV11 29.9 EH

Unfactored Vertical Forces and Moments

Unfactored Loads

Unfactored Horizontal Forces and Moments

Load Moment Arm Moment LRFD

Load TypeItem Description Value

(kip/ft) Item Value (ft) Item Value

(kip-ft/ft)

H1 Live load surcharge 1.39 dh1 10.0 MH1 13.9 LS

H2 Active earth force 6.93 dh2 6.7 MH2 46.2 EH

FT

H2δ’

δ’= 90-87.6+21

H1 = FSURcos (δ’)H2 = FTcos (δ’)

FSUR

H1δ’

H/2H/3

Factored Loads

Factored Forces and Moments

Load Combination γDC γEV γLS_V γLS_H γEH Application

Strength Ia 0.90 1.00 - 1.75 1.50 Sliding, Eccentricity

Strength Ib 1.25 1.35 1.75 1.75 1.50 Bearing, Wall Strength

Service I 1.00 1.00 1.00 1.00 1.00 Overall, Settlement & Crack Control

Load Combinations

Load CombinationVert. Loads

V (kips/ft)

MomM

(kip-

ents V

kip/ft)

Horiz. L

(kips/

oads H

ft)

MomM

(kip-k

ents H ip/ft)

Strength Ia 21.2 153.6 12.8 93.5Strength Ib 29.5 208.9 12.8 93.5Service I 21.7 151.2 8.3 60.0

V_Ia n 0.90VDC 1.00VEV+ 0.00 VLS+ 1.50 VEH+( )= V_Ia= 21.2 kips

External Stability

Bearing Resistance (Nominal)qn = cNcm + γfdDfNqmCwq + 0.5γfdB’NγmCwγ

c = Cohesion (ksf)Ncm = Cohesion term **Nqm = Surcharge term**Nγm = Unit weight term **Cwq = Groundwater correction factorCwγ = Groundwater correction factorB’ = Effective footing width

= 0 ksf= Ncscic= Nqsqdqiq= Nγsγiγ= 1.0 (Dw >1.5B+Df )= 1.0 (Dw >1.5B+Df )= B-2e

** Modified bearing capacity factor (includes shape factors, load inclination factors and depth correction factor Tables 10.6.3.1.2a-2,3,4 LRFD)

External Stability

Bearing Resistance (Nominal)qn = cNcm + γfdDfNqmCwq + 0.5γfdB’NγmCwγ

c = Cohesion (ksf)Ncm = Cohesion term **Nqm = Surcharge term**Nγm = Unit weight term **Cwq = Groundwater correction factorCwγ = Groundwater correction factorB’ = Effective footing width

= 0 ksf= 15.2 = 11.3 = 8.1= 1.0= 1.0= 7.8’

qn = 0 + (0.120)(4.0’)(11.3)(1.0) + 0.5(0.120)(7.8)(8.1)(1.0) = 9.21 ksf

External Stability

Bearing Resistance (Factored)

qn = 9.21 ksf Nominal bearing resistanceφb = 0.55 Bearing resistance factor(1)

qr = φb qn Factored bearing resistance

(1) LRFD Table 11.5.6-1 (Semi-Gravity Walls)

qr = (0.55)(9.21) = 5.07 ksf

External Stability

Bearing Capacity to Demand Ratio (CDR)

Load Combination Vert. Loads V (kips/ft)

Moments (kip-kip/f

MV t)

Horiz. Loa(kips/ft

ds H )

Moments (kip-kip/f

MH t)

Strength Ia 21.2 153.6 12.8 93.5Strength Ib 29.5 208.9 12.8 93.5Service I 21.7 151.2 8.3 60.0

qr = 5.07 ksfx = (ΣMr – ΣMo)/ ΣVe = B/2 - xσv = ΣV/(B-2e) = ΣV/B’

Factored bearing resistanceDist. from toe to load resultantWall eccentricityUltimate bearing stress

= (208.9 – 93.5)/29.5 = 3.91 ft= (10/2) – 4.25 = 1.09 ft= 29.3/[10.0-(2)(0.75)] = 3.78 ksf

CDRBRG = qr/σv = 5.07/3.78 = 1.34 > 1.0 ==> OK

External Stability

Limiting Eccentricity (CDR)

Load Combination Vert. Loads V (kips/ft)

Moments (kip-kip/f

MV t)

Horiz. Loa(kips/ft

ds H )

Moments (kip-kip/f

MH t)

Strength Ia 21.2 153.6 12.8 93.5Strength Ib 29.5 208.9 12.8 93.5Service I 21.7 151.2 8.3 60.0

emax = B/4x = (ΣMr – ΣMo)/ ΣVe = B/2 - x

= 10/4 = 2.50 ft= (153.6 – 93.5)/21.2 = 2.83 ft= (10/2) – 4.25 = 2.17 ft

CDRecc= emax/e = 2.50/2.17 = 1.15 > 1.0 ==> OK

External Stability

Sliding ResistanceRR = φτRτ + φepRep

RR = Factored sliding resistanceφτ = Shear resistance factorRτ = Nominal sliding resistanceφep = Passive pressure resistance factorRep = Nominal passive resistanceV = Total vertical force (Strength Ia)

= φτRτ + φepRep= 1.0 (Table 11.5.6-1) = V tan φfd= 0.5 (Table 10.5.5.2.2-1) = TBD= 21.2 kips

Rr = (1.0)(21.2)tan(34) + 0.5 Rep

External Stability

Sliding Resistance (Passive Resistance)RR = φτRτ + φepRep

kp = tan(45 + φfd/2)2

= tan(45 + 34/2)2 = 3.54 σH = kp γfd (y2+y1)/2

= (3.54)(0.120)(5+4)/2 = 1.91 ksfRep = σH (y2 – y1)

= (1.91)(5-4) = 1.91 kips

Rr = (14.3) + 0.5 (1.91) =15.3 kips

Rep

y

External Stability

Sliding Resistance (CDR)

Load Combination Vert. Loads V (kips/ft)

Moments (kip-kip/f

MV t)

Horiz. Loa(kips/ft

ds H )

Moments (kip-kip/f

MH t)

Strength Ia 21.2 153.6 12.8 93.5Strength Ib 29.5 208.9 12.8 93.5Service I 21.7 151.2 8.3 60.0

Rr = 15.3 kipsRu = 12.8 kips

Factored sliding resistanceFactored sliding force

CDRSliding = Rr/Ru = 15.3/12.8 = 1.19 > 1.0 ==> OK

Structural Resistance

Loads (Strength Ib & Service)Element Capacities & Checks

Structural Resistance

Loads (Strength Ib)– Heel– Toe– Stem

Critical Sections (Vu & Mu)

Structural Resistance

Element Capacities & Checks– Shear LRFD [5.8.3.3]

Vr > Vu

– Flexure LRFD [5.7.3.2]

Mr > Mu

– Min. Flexural Reinforcement LRFD [5.7.3.3.2] Mr > lesser of (1.2Mcr and 1.33Mu )

– Crack Control (Service Loads) LRFD [5.7.3.4]

smax > s– Temp. & Shrinkage Steel LRFD [5.10.8]

Structural Resistance

Element Capacities & Checks– Shear LRFD [5.8.3.3]

Vr = φvVn = φv min[ 0.0316β(f’c)1/2bdv , 0.25f’cbdv ]– Flexure LRFD [5.7.3.2]

Mr = φfMn = φfAsfy(ds-a/2)– Min. Flexural Reinforcement LRFD [5.7.3.3.2]

Mcr = Scfr = (Ig/yt)0.37f’c1/2

– Crack Control (Service Loads) LRFD [5.7.3.4]

smax = (700γe/βsfss)-2dc

– Temp. & Shrinkage Steel LRFD [5.10.8]

Overall Stability11.6.2.3 LRFD • Limiting equilibrium methods of analysis (FS = 1.857)• Service I Load Combination• φ=0.75• CDR = FS*φ > 1.0

CDROverall = FS*φ = (1.857)(0.75)= 1.39 > 1.0 ==> OK

SettlementHough’s Method10.6.2.4.2 LRFD– Settlement of Footings on Cohesionless Soils

1. Subdivide subsurface ( 2 x base width & 10’ layers)2. Determine corrected SPT values (N1)603. Use Figure 10.6.2.4.2-1 to determine Bearing Capacity

Index C’4. Calculate effective vertical stresses (σ’o) and increase in

stresses at mid-height of layers (Δσv)5. Determine layer settlement

6. Determine total settlement

SettlementBearing Capacity Index C’

Figure 10.6.2.4.2-1 LRFD

Sand & Gravel

N1 = Corrected SPT Value (N1)60= 30

C’ = Bearing Capacity Index= 100

Source: Figure 10.6.2.4.2-1 LRFD(AASHTO, 2010; modified after Hough, 1959)

Settlement

10’

10’

Layer 1

Layer 2

σv= Equivalent Vertical Stress (Service)= 2584 psf

σ’o = Dlayer 1 γfd= (4+10/2)(120) = 1080 psf

Δσv = σv [B’/(B’+a)]= 2584 [8.4/(8.4+5)]= 1620 psf

9’4’2:1 Method

Calculate Stresses

σv = ΣV/B’

Settlement

ΔH1 = Elastic settlement of Layer 1Hc = 120” C’ = 100σ’o = 1080 psfΔσv = 1620 psf

Determine Layer Settlement

ΔH1 = (120”)(1/100) log [(1080+1620)/1080]= 0.48”

Settlement

ΔH1 = 0.48”Hc = 120” C’ = 100σ’o = 2280 psfΔσv = 928 psf

Check Total Settlement

ΔH2 = (120”)(1/100) log [(2280+928)/2280]= 0.18”

Se = 0.48”+0.18”

Se = 0.66”

(Eqn. 10.6.2.4.2-3)

(Eqn. 10.6.2.4.2-2)

< 1.0” ==> OK

LRFD CIP Concrete Wall Example Summary

Loads and Load CombinationsExternal Stability– Bearing Resistance– Sliding– Eccentricity

Structural ResistanceOverall Stability and Settlement

Questions?

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