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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)
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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).
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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.
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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.
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Locations of Selected Culverts
#1#6
#2
#3
#4
#5
#7
#8
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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
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A typical instrumentation Plan – 45° Skew
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Corner Tiltmeters Midspan Strain Gages
and LVDTs
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Sensor positions before the load test Load position at predefined load path
Current condition assessment Weighing truck axles
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Detailed test notes documenting each field test were added taken:
• date
• time
• truck configuration
• load path
• test runs
• issues
• …
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• 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)
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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
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Developed FE models for all tested culverts
WallsBottom
Slab
Top
Slab
Edge
BeamsRigid
Joints
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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
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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
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Culvert 8
FH = 1.60 ft
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Culvert 4
FH = 2.24 ft
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Culvert 5
FH = 4.14 ft
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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
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• 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
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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
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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
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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
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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) = ∞
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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
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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
Kθ,slab(int) (k/θ/in) fixed
Error %19.6
Case 3
Ec (ksi) 4573
tslab (in) +10% 9.35
twall (in) 7
Kθ,slab(ext)
(k/θ/in)
fixed
Kθ,slab(int) (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
Kθ,slab(int) (k/θ/in) fixed
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
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Modified Properties Cases 2, 3 & 4Nominal (Case 1) vs Modified Properties
Case 2 &3
Strain readings at exterior top slab mid section
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Modified Properties Case 2, 3 & 4
Displacements at exterior top slab mid section Typical rotational response at wall and slab
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Modified Properties Case 2, 3 & 4
Typical slab corner strain responses
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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
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Strain Responses
Each culvert is analyzed using the calibrated model considering all design and
rating trucks.
-5 0 10 20
[ft]
Load Positions
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-5 0 10 20
[ft]
Load Positions
Displacement Responses
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Moment (+)
(HL-93 live load envelope)
Moment (-)
(HL-93 live load envelope)
Shear (±)
(HL-93 live load envelope)
M (+)max M (-) min V (±) max/min
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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
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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)
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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)
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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
LA Type 6 80.0 3.27 3 Exterior Top Slab Corner Shear
LA Type 8 88.0 4.21 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
SU5 62.0 3.66 3 Exterior Top Slab Corner Shear
SU6 69.5 3.66 3 Exterior Top Slab Corner Shear
SU7 77.5 3.66 3 Exterior Top Slab Corner Shear
Average Fill Height = 2.20 ft; Wearing Surface = Asphalt
Culvert #1 Minimum Load Rating Factors and Sections
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Culvert #2 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
HL-93 (INV) N/A 4.57 3 Exterior Top Slab Corner Shear
HL-93 (OPR) N/A 5.93 3 Exterior Top Slab Corner Shear
LA Type 3 41.0 10.68 3 Exterior Top Slab Corner Shear
LA Type 3S2 73.0 9.40 3 Exterior Top Slab Corner Shear
Type 3-3 80.0 7.42 3 Exterior Top Slab Corner Shear
LA Type 6 80.0 7.44 3 Exterior Top Slab Corner Shear
LA Type 8 88.0 9.69 2 Exterior Top Slab Center Moment
NRL 80.0 10.19 3 Exterior Top Slab Corner Shear
SU4 54.0 8.45 2 Exterior Top Slab Center Moment
SU5 62.0 8.44 2 Exterior Top Slab Center Moment
SU6 69.5 8.57 2 Exterior Top Slab Center Moment
SU7 77.5 9.29 2 Exterior Top Slab Center Moment
Average Fill Height = 7.00 ft; Wearing Surface = Asphalt
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Culvert #3 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
HL-93 (INV) N/A 1.12 1 Exterior Top Slab Corner Shear
HL-93 (OPR) N/A 1.40 1 Exterior Top Slab Corner Shear
LA Type 3 41.0 1.96 1 Exterior Top Slab Corner Shear
LA Type 3S2 73.0 1.87 1 Exterior Top Slab Corner Shear
Type 3-3 80.0 1.76 1 Exterior Top Slab Corner Shear
LA Type 6 80.0 1.60 1 Exterior Top Slab Corner Shear
LA Type 8 88.0 2.00 1 Exterior Top Slab Corner Shear
NRL 80.0 1.94 1 Exterior Top Slab Corner Shear
SU4 54.0 1.88 1 Exterior Top Slab Corner Shear
SU5 62.0 1.88 1 Exterior Top Slab Corner Shear
SU6 69.5 1.93 1 Exterior Top Slab Corner Shear
SU7 77.5 1.94 1 Exterior Top Slab Corner Shear
Average Fill Height = 1.08 ft; Wearing Surface = Asphalt
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Culvert #4 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
HL-93 (INV) N/A 1.39 3 Exterior Top Slab Corner Shear
HL-93 (OPR) N/A 1.81 3 Exterior Top Slab Corner Shear
LA Type 3 41.0 2.53 3 Exterior Top Slab Corner Shear
LA Type 3S2 73.0 2.27 3 Exterior Top Slab Corner Shear
Type 3-3 80.0 2.15 3 Exterior Top Slab Corner Shear
LA Type 6 80.0 2.16 3 Exterior Top Slab Corner Shear
LA Type 8 88.0 2.90 3 Exterior Top Slab Corner Shear
NRL 80.0 2.14 3 Exterior Top Slab Corner Shear
SU4 54.0 2.19 3 Exterior Top Slab Corner Shear
SU5 62.0 2.29 3 Exterior Top Slab Corner Shear
SU6 69.5 2.19 3 Exterior Top Slab Corner Shear
SU7 77.5 2.19 3 Exterior Top Slab Corner Shear
Average Fill Height = 2.24 ft; Wearing Surface = Asphalt
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Culvert #5 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
HL-93 (INV) N/A 2.15 3 Exterior Top Slab Corner Shear
HL-93 (OPR) N/A 2.79 3 Exterior Top Slab Corner Shear
LA Type 3 41.0 6.56 2 Exterior Top Slab Center Moment
LA Type 3S2 73.0 6.38 3 Exterior Top Slab Corner Shear
Type 3-3 80.0 5.69 2 Exterior Top Slab Center Moment
LA Type 6 80.0 5.73 2 Exterior Top Slab Center Moment
LA Type 8 88.0 7.42 2 Exterior Top Slab Center Moment
NRL 80.0 6.86 3 Exterior Top Slab Corner Shear
SU4 54.0 6.48 3 Exterior Top Slab Corner Shear
SU5 62.0 6.87 3 Exterior Top Slab Corner Shear
SU6 69.5 6.96 3 Exterior Top Slab Corner Shear
SU7 77.5 7.07 3 Exterior Top Slab Corner Shear
Average Fill Height = 4.14 ft; Wearing Surface = Concrete
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Culvert #6 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
HL-93 (INV) N/A 2.57 3 Exterior Top Slab Corner Shear
HL-93 (OPR) N/A 3.29 3 Exterior Top Slab Corner Shear
LA Type 3 41.0 4.79 3 Exterior Top Slab Corner Shear
LA Type 3S2 73.0 4.69 3 Exterior Top Slab Corner Shear
Type 3-3 80.0 4.12 3 Exterior Top Slab Corner Shear
LA Type 6 80.0 4.16 3 Exterior Top Slab Corner Shear
LA Type 8 88.0 5.25 3 Exterior Top Slab Corner Shear
NRL 80.0 4.25 3 Exterior Top Slab Corner Shear
SU4 54.0 4.31 3 Exterior Top Slab Corner Shear
SU5 62.0 4.35 3 Exterior Top Slab Corner Shear
SU6 69.5 4.25 3 Exterior Top Slab Corner Shear
SU7 77.5 4.25 3 Exterior Top Slab Corner Shear
Average Fill Height = 1.78 ft; Wearing Surface = Asphalt
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Culvert #7 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
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
LA Type 3S2 73.0 3.04 3 Exterior Top Slab Corner Shear
Type 3-3 80.0 2.63 3 Exterior Top Slab Corner Shear
LA Type 6 80.0 2.55 3 Exterior Top Slab Corner Shear
LA Type 8 88.0 3.47 3 Exterior Top Slab Corner Shear
NRL 80.0 3.28 3 Exterior Top Slab Corner Shear
SU4 54.0 2.74 3 Exterior Top Slab Corner Shear
SU5 62.0 3.21 2 Exterior Topslab Midspan Moment
SU6 69.5 3.15 3 Exterior Top Slab Corner Shear
SU7 77.5 3.24 3 Exterior Top Slab Corner Shear
Average Fill Height = 1.08; Wearing Surface = Concrete
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Culvert #8 Minimum Load Rating Factors and Sections
Vehicle TypeGVW
(kips)
Rating
Factor
Section
NumberSection Location
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 3 41.0 3.06 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
LA Type 6 80.0 2.42 2 Exterior Top Slab Center Moment
LA Type 8 88.0 3.04 2 Exterior Top Slab Center Moment
NRL 80.0 2.99 2 Exterior Top Slab Center Moment
SU4 54.0 2.85 2 Exterior Top Slab Center Moment
SU5 62.0 2.85 2 Exterior Top Slab Center Moment
SU6 69.5 2.87 2 Exterior Top Slab Center Moment
SU7 77.5 2.98 2 Exterior Top Slab Center Moment
Average Fill Height = 1.60; Wearing Surface = Asphalt
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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)
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• 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
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• 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:
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• 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
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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!
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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
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