ayman okeil, ph.d., pe tuna ulger, ph.d. ahmed elshoura ... · louisiana has ~2600 culverts in its...

Ayman M. Okeil, Ph.D., P.E.

Steve Cai, Ph.D., P.E.

Louisiana State University1

Ayman Okeil, Ph.D., PE

Tuna Ulger, Ph.D.

Ahmed Elshoura, Ph.D. Student

Department of Civil and Environmental Engineering

Louisiana State University

2018 Louisiana Transportation Conference

Baton Rouge, LA (February 27, 2018)

Louisiana has ~2600 culverts in its inventory (NBI).

Load rating of these culverts reveals that some have

a RF < 1.0.

Louisiana is not alone in this situation. Many states

have similar culvert rating challenges.

NCHRP Project 15-54 (Active – July 2015).

Louisiana’s challenge is mainly in cast-in-place (CIP)

reinforced concrete box culverts.

This promoted the initiation of this project, which is

funded by the Louisiana Transportation Research

Center (LTRC).

4

In Louisiana, detailing in old standards is an issue.

Standard details produce corner joints that lack ability to

transfer moments ⇒ a clear hinge is always assumed.

CIP-RC culverts appear to be performing well even if

traditional analyses show otherwise.

8 cast-in-place box culverts (selected by LA-DOTD)

were load tested.

2 barrels (1 exterior, 1 interior) are instrumented.

48 sensors are used at critical locations.

• Strain – strain sensors (36)

• Displacement – LVDT (8)

• Rotations – tilt meters (4)

3 different load paths.

Loaded truck speed ~ 5 mph (no-dynamic effect).

Developed and calibrated FE models.

Locations of Selected Culverts

#1#6

#2

#3

#4

#5

#7

#8

No. Of

Opening

Size

(ft x ft)

Skewness

(degree)tslab (in) twal (in)

Fill Height

(ft)

Road

Pavement

1 4 7x7 90.0 8.5 7.0 2.00 Asphalt

2 5 6x6 90.0 8.0 6.0 7.00 Asphalt

3 3 8x8 86.7 9.0 8.0 1.08 Asphalt

4 2 12x12 90.0 12.0 12.0 2.24 Asphalt

5 3 7x7 45.0 8.5 7.0 4.14 Conc.

6 3 5x4 45.0 7.5 6.0 1.78 Asphalt

7 3 6x6 60.0 8.0 6.0 1.08 Conc.

8 4 8x8 90.0 9.0 8.0 1.60 Asphalt

A typical instrumentation Plan – 45° Skew

Corner Tiltmeters Midspan Strain Gages

and LVDTs

Sensor positions before the load test Load position at predefined load path

Current condition assessment Weighing truck axles

Detailed test notes documenting each field test were added taken:

• date

• time

• truck configuration

• load path

• test runs

• issues

• …

• FE models were studied in SAP 2000

and STAAD PRO

• 3D Model is considered (more

realistic and provides better load

distribution)

• 3D Shell and 3D Solid elements

were investigated.

• Shell elements were chosen as they

provide:

• Direct access the nodal solutions

• Faster FE model updates

• Lesser run time and

• Lesser post processing time

• Input: field measurements and

available project information

• Initial assumptions ( ERC , partially rigid

slab-to-wall connections)

Plate offset (If FE package does not

specify plate offset rigid assumption is

required)

• walls (tw / 2) and slabs (ts / 2)

Rigid haunch (partial length and

culvert specific)

• k = 1/2, 2/3, 1 etc.

Rigid Concrete

Area

Top slab clear span length in FE Model Rigid zone definition

rigid

shell

elements

shell

elements

Concrete

rotational

springs

Modeling Corner Details

Developed FE models for all tested culverts

WallsBottom

Slab

Top

Slab

Edge

BeamsRigid

Joints

Tire contact are: 10in x 20in(AASHTO LRFD 3.6.1.2.5)

If, depth of fill, H ≤ 2ft(AASHTO LRFD 4.6.2.10)

• Case 1: Traffic travel // span

Eperp = 96 + 1.44*S (4.6.2.10.2-1)

Espan = LT + LLDF*H (4.6.2.10.2-2)

If depth of fill, H > 2ft(AASHTO LRFD 3.6.1.2.6)

• Case 1: Traffic travel // span

Eperp = 10in + LLDF*H

Espan = 20in + LLDF*H

Overlapping pressures distributed

uniformly over encompassing area. Espan

Eperp

P

Travel

Direction

10inx20in

H

Road

Surface

Top Slab

2) Soil spring model

Backfill soil springs, Ksi,wall , ( Clough &

Duncan 1991)

Foundation models;

• Roller supports (Uz=0) or

• Subgrade soil spring model

(Compression only)

Ks,f ( compression only, Bowles

1996 )

1) Roller support model

Subgrade springs

Ba

ckfill

sp

rin

gs

Rotational

springs

Soil Modeling

Culvert 8

FH = 1.60 ft

Culvert 4

FH = 2.24 ft

Culvert 5

FH = 4.14 ft

Standard vs in-situ culvert dimensions (slab thickness)

Assumed vs cored concrete strength

Soil spring stiffness (iterative model adjustment)

Cracked, Icr , vs. Uncracked, Ig .

Rigid vs pinned joint definition

• Rotational springs, Kθ, are inserted at wall/haunch and

slab/haunch joints

• Kθ values varied along the transverse direction

Rotational springs (Kθ)x

y

• Error estimator – (Two-Norm)

• All sensors above certain magnitude under

each load path provide one error

• Error = average(Errorpath1 + Errorpath2 +

Errorpath3)

Max

readings

( yexp, yFE)

Active zones for each load path

𝐸𝑟𝑟𝑜𝑟 =

𝑁𝑙𝑝

𝑛𝑙𝑝,𝑖 𝑦𝑒𝑥𝑝 − 𝑦𝐹𝐸

2

𝑦𝑒𝑥𝑝2

∗1

𝑛𝑙𝑝,𝑖∗1

𝑁𝑙𝑝

Nlp = number of load path

nlp,i = number of sensors in active load pathyexp = average of max N load test data pointyFE = average of max N FE analysis data point

Culvert #1 • 4 Barrels – 7 ft x 7 ft Opening

• Fill depth = 2.2 ft

• Length = 33 ft – 9 in.

• Econc = 3,200 ksi

• twall = 7 in. , twall = 8.5 in.

• Backfill Springs: Ks,wall = ∞ ↔ Ks,wall = 0

(medium ~ dense soil) Ks,wall = 60~192 pci;

• Connection rigidity: Kθ,slab(int) , Kθ,slab(ext) , Kθ,wall(int) , Kθ,wall(ext) = ∞

Kθ,slab(int) , Kθ,slab(ext), , Kθ,wall(int) , Kθ,wall(ext) = 0

• Soil springs: Ks,found Uz = 0, under the walls

Ks,found 150~250 pci ;

11.2 18.6 18.612.2 ft 4.6 ft

Test Truck GVW = 48.25 kips

x

y

Backfill Soil Spings:

Ks,wall = ∞ ↔ Ks,wall = 0 ( Kθ,slab & Kθ,wall = ∞ )

15 % difference in Uz direction between ‘semi rigid’ and ‘free stiffness’

Maximum exterior wall deflection in Ux = -0.000739 in when Ks,wall = 0

Conclusion:

• Backfill soil springs resist rotation but that is not the case for experiments. Soil-

structure interaction was not included in future models

Subgrade Springs:

Ks varied ↔ Case 1) 150 pci, 2) 250 pci, 3) rigid , and 4) rigid (under wall only)

Demand forces at exterior slab are compared

Conclusion:

• Maximum variation is about 4% for moments and deflections. Future models are

calibrated using rigid subgrade (Case 3) assumption due to increased run time (x3) for

each load path compared to the Cases 1 and 2.

Case / ForcesA

(kip/in)

M

(kip-in/in)

V

(kip/in)

1) k = 150 pci 0.0075 3.1737 0.0367

2) k = 250 pci 0.0051 3.1402 0.0359

3) k = rigid 0.0077 3.0504 0.0356

4) k = rigid

(only walls)0.0078 3.0497 0.0356

Case Δ (in)

1) k = 150 pci -0.003140

2) k = 250 pci -0.003180

3) k = rigid -0.003276

4) k = rigid (only walls) -0.003274

5) Experiment -0.002830

Rotational Springs (Wall):

It can be assumed that the slab cracks first at the corner around

slab/wall connections; therefore, moments are partially released at that

cracked location.

This is also true for single layer reinforced wall sections at the exterior

walls.

Wall rotations are already less than the experimental rotations without

backfill soil stiffness. Releasing the rotational wall stiffness would

additionally reduced the exterior wall rotations.

In conclusion, the rotational wall springs are assumed fixed at exterior

and interior slab/wall connections in future models which significantly

reduced the multiple combination of possible stiffness iterations in FE

models.

• Kθ,wall(int) , Kθ,wall(ext) = ∞

Rotational Springs (Slab) :

Rotational spring stiffness were extracted from global stiffness matrix

(Rigid multiplication factor = 1)

Kθ,slab(ext) = Kθ,global x R (reduction factor ~ 0-10%)

Kθ,slab(int) = Kθ,global x R (reduction factor ~ 40 -100%)

Other considered cases:

• Single layer reinforcement ↔ hinge at exterior slab rotational

spring locations

Kθ,slab(ext) = 0

• Double layer reinforcement ↔ full continuity at interior slab

rotational spring locations

Kθ,slab(int) = ∞

Multiple iterations were attempted to minimize the error between

experiment and FE model

Case 1 – Nominal

Ec (ksi) 3000

tslab (in) 8.50

twall (in) 7

Kθ,slab(ext) (k/θ/in) 0

Kθ,slab(int) (k/θ/in) fixed

Error %113

Case 2 – Cored Conc. P

Ec (ksi) 4573

tslab (in) +10% 9.35

twall (in) 7

Kθ,slab(ext) (k/θ/in) 0


Error %19.6

Case 3

Ec (ksi) 4573

tslab (in) +10% 9.35

twall (in) 7

Kθ,slab(ext)

(k/θ/in)

fixed


Error %17.2

Case 4

Ec (ksi) 4573

tslab (in) +15% 9.80

twall (in) 7

Kθ,slab(ext)

(k/θ/in)

10000


Error %15.6

Typical Culvert test and Sensor

Locations

Travel

Direction

(Load Path)

11.2 18.6 18.612.2 ft 4.6 ft

Test Truck GVW = 48.25 kips

Modified Properties Cases 2, 3 & 4Nominal (Case 1) vs Modified Properties

Case 2 &3

Strain readings at exterior top slab mid section

Modified Properties Case 2, 3 & 4

Displacements at exterior top slab mid section Typical rotational response at wall and slab

Modified Properties Case 2, 3 & 4

Typical slab corner strain responses

Vehicle TypeGVW

(kips)

HL-93 (INV) N/A

HL-93 (OPR) N/A

LADV-11(INV) N/A

LA Type 3 41.0

LA Type 3S2 73.0

Type 3-3 80.0

LA Type 6 80.0

LA Type 8 88.0

NRL 80.0

SU4 54.0

SU5 62.0

SU6 69.5

SU7 77.5

1 2 3 4 5

6

7 9

8

Critical sections for max demand forces

Force Section

M_max (+) 2,5

M_min (-) 1,3,4

V_max 1,2,3

A_max 6,7,8,9

Concrete Box Culvert Load Rating Equation

𝑅𝐹 =𝐶 ± 𝛾𝐷𝐶𝐷𝐶 ± 𝛾𝐷𝑊𝐷𝑊 ± 𝛾𝐸𝑉𝐸𝑉 ± 𝛾𝐸𝐻𝐸𝐻 ± 𝛾𝐸𝑆𝐸𝑆

𝛾𝐿𝐿 ∗ 𝐿𝐿 + 𝐼𝑀 ± 𝛾𝐿𝐿 ∗ 𝐿𝑆

Strain Responses

Each culvert is analyzed using the calibrated model considering all design and

rating trucks.

-5 0 10 20

[ft]

Load Positions

-5 0 10 20

[ft]

Load Positions

Displacement Responses

Moment (+)

(HL-93 live load envelope)

Moment (-)


Shear (±)


M (+)max M (-) min V (±) max/min

Loads Types M(kip-in/in) - V(kip/in) - A (kip/in) at critical sections

Load Type GW

(kips)

2 3 1 3 5 4 4 6 7 8 9

M (+) M (-) V V M (+) M (-) V A A A A

DC dead - 0.494 0.152 -0.502 0.019 -0.034 0.285 -0.464 0.028 -0.043 -0.055 -0.104

DW dead - 0.246 -0.020 -0.204 0.013 -0.017 0.177 -0.214 0.015 -0.019 -0.018 -0.047

EV dead - 0.749 -0.019 -0.609 0.036 -0.050 0.521 -0.641 0.045 -0.057 -0.055 -0.140

EH dead - -0.506 -1.198 0.353 0.018 0.018 0.112 0.262 -0.004 -0.020 -0.021 0.023

ES dead - -0.054 -0.123 0.032 0.002 0.002 0.011 0.024 0.000 -0.002 -0.002 0.002

LS live - -0.268 -0.616 0.159 0.009 0.009 0.053 0.122 -0.002 -0.010 -0.010 0.011

HL-93 live NA 2.286 -1.410 0.149 -0.162 2.168 -1.212 0.157 -0.157 -0.128 -0.189 -0.161

HL-93 TD live NA 2.014 -1.323 0.136 -0.146 1.889 -1.232 0.142 -0.157 -0.129 -0.246 -0.209

LA Type 3 live 41 1.144 -0.855 0.080 -0.087 1.095 -0.805 0.085 -0.095 -0.077 -0.140 -0.117

LA Type 3-S2 live 73 1.163 -0.924 0.081 -0.090 1.098 -0.913 0.086 -0.093 -0.075 -0.141 -0.118

LA Type 6 live 80 1.400 -1.038 0.097 -0.105 1.305 -0.854 0.102 -0.106 -0.086 -0.162 -0.135

LA Type 8 live 80 1.370 -0.997 0.092 -0.103 1.301 -0.843 0.097 -0.097 -0.078 -0.161 -0.133

Type 3-3 live 80 1.062 -0.733 0.069 -0.080 1.018 -0.637 0.073 -0.076 -0.062 -0.122 -0.101

NRL live 80 1.236 -0.934 0.085 -0.093 1.101 -0.887 0.090 -0.096 -0.078 -0.171 -0.144

SU 4 live 54 1.249 -0.831 0.085 -0.092 1.202 -0.704 0.090 -0.101 -0.082 -0.160 -0.134

SU 5 live 62 1.229 -0.887 0.085 -0.092 1.182 -0.793 0.090 -0.101 -0.082 -0.162 -0.136

SU 6 live 69.5 1.229 -0.887 0.085 -0.092 1.143 -0.821 0.090 -0.101 -0.082 -0.172 -0.145

SU 7 live 77.5 1.229 -0.902 0.085 -0.092 1.121 -0.849 0.090 -0.100 -0.081 -0.173 -0.145

Culvert #1 un-factored dead and live loads at critical sections for each load type

HL-93 (INV) 2 3 1 3 5 4 4 6 7 8 9

Load Factor M (+) M (-) V V M (+) M (-) V A A A A

γ_DC 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25

γ_DW 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50

γ_EV 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30

γ_EH 0.50 0.50 1.35 0.50 1.35 0.50 0.50 1.35 1.35 0.50 0.50

γ_ES 0.50 0.50 1.50 0.50 1.50 0.50 0.50 1.50 1.50 0.50 0.50

γ_LS 0.00 0.00 1.75 0.00 1.75 0.00 0.00 1.75 1.75 0.00 0.00

γ_LL 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75

γ_DC*DC 0.62 -0.63 0.02 -0.04 0.36 -0.58 0.04 -0.05 -0.07 -0.13 -0.15

γ_DW*DW 0.37 -0.31 0.02 -0.03 0.27 -0.32 0.02 -0.03 -0.03 -0.07 -0.07

γ_EV*EV 0.97 -0.79 0.05 -0.06 0.68 -0.83 0.06 -0.07 -0.07 -0.18 -0.18

γ_EH*EH -0.25 0.18 0.02 0.01 0.15 0.13 0.00 -0.03 -0.03 0.01 0.01

γ_ES*ES -0.03 0.02 0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.00

γ_LS*LS 0.00 0.00 0.02 0.00 0.09 0.00 0.00 -0.02 -0.02 0.00 0.00

γ_LL*LL 5.95 -3.67 0.39 -0.42 5.65 -3.21 0.41 -0.41 -0.34 -0.64 -0.54

φcφsφRn 14.80 -16.12 0.96 -0.96 14.80 -16.12 0.96 -37.04 -37.04 -37.04 -37.04

RF 2.20 3.97 2.08 1.98 2.32 4.53 2.07 86.31 103.78 57.24 67.47

Culvert #1: Load factors, factored loads and rating factors at critical sections for

HL-93 (inventory level)

HL-93 (OPR) 2 3 1 3 5 4 4 6 7 8 9

Load Factor M (+) M (-) V V M (+) M (-) V A A A A

γ_DC 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25

γ_DW 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50

γ_EV 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30 1.30

γ_EH 0.50 0.50 1.35 0.50 1.35 0.50 0.50 1.35 1.35 0.50 0.50

γ_ES 0.50 0.50 1.50 0.50 1.50 0.50 0.50 1.50 1.50 0.50 0.50

γ_LS 0.00 0.00 1.75 0.00 1.75 0.00 0.00 1.75 1.75 0.00 0.00

γ_LL 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35 1.35

γ_DC*DC 0.62 -0.63 0.02 -0.04 0.36 -0.58 0.04 -0.05 -0.07 -0.13 -0.15

γ_DW*DW 0.37 -0.31 0.02 -0.03 0.27 -0.32 0.02 -0.03 -0.03 -0.07 -0.07

γ_EV*EV 0.97 -0.79 0.05 -0.06 0.68 -0.83 0.06 -0.07 -0.07 -0.18 -0.18

γ_EH*EH -0.25 0.18 0.02 0.01 0.15 0.13 0.00 -0.03 -0.03 0.01 0.01

γ_ES*ES -0.03 0.02 0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.00 0.00

γ_LS*LS 0.00 0.00 0.02 0.00 0.09 0.00 0.00 -0.02 -0.02 0.00 0.00

γ_LL*LL 4.59 -2.83 0.30 -0.33 4.36 -2.47 0.31 -0.32 -0.26 -0.49 -0.42

φcφsφRn 14.80 -16.12 0.96 -0.96 14.80 -16.12 0.96 -37.04 -37.04 -37.04 -37.04

RF 2.86 5.15 2.67 2.57 3.00 5.87 2.68 110.57 132.51 74.21 87.46

Culvert #1: Load factors, factored loads and rating factors at critical sections for

HL-93 (operational level)

Vehicle TypeGVW

(kips)

Rating

Factor

Section

NumberSection Location

HL-93 (INV) N/A 1.98 3 Exterior Top Slab Corner Shear

HL-93 (OPR) N/A 2.57 3 Exterior Top Slab Corner Shear

LA Type 3 41.0 3.85 3 Exterior Top Slab Corner Shear

LA Type 3S2 73.0 3.74 3 Exterior Top Slab Corner Shear

Type 3-3 80.0 3.19 3 Exterior Top Slab Corner Shear



NRL 80.0 3.61 3 Exterior Top Slab Corner Shear

SU4 54.0 3.67 3 Exterior Top Slab Corner Shear




Average Fill Height = 2.20 ft; Wearing Surface = Asphalt

Culvert #1 Minimum Load Rating Factors and Sections


Vehicle TypeGVW

(kips)

Rating

Factor

Section








LA Type 8 88.0 9.69 2 Exterior Top Slab Center Moment


SU4 54.0 8.45 2 Exterior Top Slab Center Moment






Vehicle TypeGVW

(kips)

Rating

Factor

Section
















Vehicle TypeGVW

(kips)

Rating

Factor

Section






Type 3-3 80.0 5.69 2 Exterior Top Slab Center Moment








Average Fill Height = 4.14 ft; Wearing Surface = Concrete


Vehicle TypeGVW

(kips)

Rating

Factor

Section
















Vehicle TypeGVW

(kips)

Rating

Factor

Section


HL-93 (INV) N/A 1.61 2 Exterior Topslab Midspan Moment

HL-93 (OPR) N/A 2.07 2 Exterior Topslab Midspan Moment

LA Type 3 41.0 3.40 2 Exterior Topslab Midspan Moment







SU5 62.0 3.21 2 Exterior Topslab Midspan Moment



Average Fill Height = 1.08; Wearing Surface = Concrete


Vehicle TypeGVW

(kips)

Rating

Factor

Section


HL-93 (INV) N/A 1.36 2 Exterior Top Slab Center Moment

HL-93 (OPR) N/A 1.76 2 Exterior Top Slab Center Moment


LA Type 3S2 73.0 2.81 2 Exterior Top Slab Center Moment

Type 3-3 80.0 2.44 2 Exterior Top Slab Center Moment



NRL 80.0 2.99 2 Exterior Top Slab Center Moment





Average Fill Height = 1.60; Wearing Surface = Asphalt

Culvert

Fill

Height

(ft)

Span

Length

(ft)

Rating Factor

HL-93 (INV) HL-93 (OPR) Legal (Truck Type)

1 2.20 7 1.98 2.57 3.19 (Type 3-3)

2 7.00 6 4.57 5.93 7.42 (Type 3-3)

3 1.08 8 1.12 1.40 1.60 (LA Type 6)

4 2.24 12 1.39 1.81 2.14 (NRL)

5 4.14 7 2.15 2.79 5.69 (Type 3-3)

6 1.78 5 2.57 3.29 4.12 (Type 3-3)

7 1.08 6 1.61 2.07 2.55 (LA Type 6)

8 1.60 8 1.36 1.76 2.42 (LA Type 6)

Results of minimum rating factors show that the following legal trucks

produce the minimum rating factor (RF)

• Sometimes, available software cannot always capture detailed

features presented earlier.

• Conducting a refined 3D analysis is not going to be always possible.

• Alternatively, we can adjust load levels in 2D models to account for

some of the aforementioned features.

p1

p2

L

Idealized Model

Mmid'

'

'

'

xcr1

xcr2

Rigid Connections

p1

p2

L

Idealized Model

Mmid

Idealized

Hinge

• To equate the moment from the simplified models.

𝑀𝑚𝑖𝑑 =𝑝1𝐿

2

8+

𝑝2 − 𝑝1 𝐿2

16𝑀𝑚𝑖𝑑

′ =𝑝1′𝐿′

2

8+

𝑝2′ − 𝑝1

′ 𝐿′2

16

• Charts can be developed for a factor, 𝛼, to reduce pressures on

exterior walls.

0%

10%

20%

30%

40%

50%

60%

4 6 8 10 12

a=

M' m

id/M

mid

Nominal Box Size (ft)

tslab = 10 in.tslab = 9 in.tslab = 8 in.tslab = 7 in.tslab = 6 in.

k1 = 2/3k2 = 0

p1

p2

L

Idealized Model

Mmid

Idealized

Hinge𝛼

𝛼

𝛼

Goal:

• A similar approach can be used

for top slab loads.

• Harder to develop charts because

of the connectivity to interior walls.

• Can be done on a case by case

basis.

Idealized Model

p1'

Idealized

Hinge

L

k ,intW

p

Idealized Model

xcr1

Rigid Connections

L

k ,intW''

p

Calibrated FE models show that AASHTO live load

distribution is conservative for fill height H ≤ 2 ft

(produces higher strain and displacements than the

experimental results). This confirms statement in

AASHTO Commentary C4.6.2.10.2 that “Distribution

widths for positive and negative moments are wider”.

Culvert responses are sensitive to the assumed

rotational slab springs. So, for Old Louisiana Standard

Culvert Details:

• Kθ,slab(ext) can be conservatively assumed to be clear hinges;

i.e., equal to 0. However, it should be calibrated to mimic

actual behavior.

• Kθ,slab(int) can be modeled with full continuity i.e., equal to ∞.

Shear controls the rating for the first culvert. Interesting!

The authors gratefully acknowledge the financial support provided by the Louisiana Transportation Research Center (LTRC Project No. 16-3ST) and Louisiana Department of Transportation and Development (SOI: DOTLT100018).

The input and support from the following individuals is also gratefully acknowledged:

• Paul Fossier (LADOTD)

• Arthur D’Andrea (LADOTD)

• Dana Feng (LADOTD)

• Ching Tsai (LADOTD)

• Qiming Chen (LADOTD)

• Bill King (LADOTD)

• Walid Alaywan (LTRC)

• Amar Raghavendra (LTRC)

• Ashton Pantico (BDI)

• Brice Carpenter (BDI)

• Marco Canales (LSU)

55

ayman okeil, ph.d., pe tuna ulger, ph.d. ahmed elshoura ... · louisiana has ~2600 culverts in its...

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