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have been validated through a series of parametric numerical
analyses. The applications of these simple design charts to
vehicular/transit tunnels also have been successfully applied
in real world projects in the past, particularly for deep tunnels
surrounded by relatively homogeneous ground.
There are, however, differences between vehicular/transit
tunnels and buried culverts and pipes. For example, tunnel
structures are generally of large dimensions and typically have
much greater structural stiffness than that of culverts and pipe
Figure 9-10. Simple frame analysis of structures. In addition, culverts and pipes are generally buried
racking deformations. at shallow depths where the simplified procedure developed
for deep tunnels may not necessarily be directly applicable.
To address the issues discussed above, numerical analysis
similar numerical analysis using parameters that are repre- using finite element/finite difference procedures was per-
sentative of highway culvert structures. formed for a wide range of parameters representative of actual
As indicated in Figure 9-9, if Frec = 1, the structure is con- culvert properties and geometries (that is, for flexible as well
sidered to have the same racking stiffness as the surrounding as rigid culverts). In addition, the parametric analysis included
ground, and therefore the racking distortion of the structure the construction condition in terms of burial depth. The
is about the same as that of the ground in the free field. When analysis, assumptions, and results are presented in the follow-
Frec is approaching zero, representing a perfectly rigid structure, ing sections.
the structure does not rack regardless of the distortion of the
ground in the free field. For Frec > 1.0 the structure becomes
flexible relative to the ground, and the racking distortion will 9.6.1 Types of Structures and Other
be magnified in comparison to the shear distortion of the Parameters Used in Evaluation
ground in the free field. This magnification effect is not caused The various parameters studied in this analysis are sum-
by the effect of dynamic amplification. Rather it is attributed marized in Table 9-1.
to the fact that the ground has a cavity in it as opposed to the
free field condition.
Step 5: Determine the racking deformation of the structure 9.6.2 Model Assumptions and Results
(s) using the following relationship: Six sets of parametric analyses were conducted. Assump-
s = Rrec free-field (9-16) tions made and results from these analyses are summarized
in the following sections.
Step 6: The seismic demand in terms of internal forces (M,
T, and V) as well as material strains can be calculated by im-
9.6.2.1 Parametric Analysis--Set 1
posing s upon the structure in a frame analysis as depicted
in Figure 9-10. Model Assumptions--Set 1. The parametric analysis--
It should be noted that the methodology developed above Set 1 (the Reference Set) started with a 10-foot diameter cor-
was intended to address the incremental effects due to earth- rugated steel pipe (or an equivalent liner plate lining) and a
quake-induced transient ground deformation only. The seis- 10-foot diameter precast concrete pipe to represent a flexible
mic effects of transient racking/ovaling deformations on cul- and a rigid culvert structure, respectively. Specific properties
verts and pipes must be considered additional to the normal used for these two different types of culvert structures are pre-
load effects from surcharge, pavement, and wheel loads, and sented in Table 9-2.
then compared to the various failure criteria considered rel- The soil profile used for Set 1 parametric analysis was as-
evant for the type of culvert structure in question. sumed to be a homogeneous deep (100-foot thick) soil de-
posit overlying a rigid base (for example, base rock). The as-
sumed Young's modulus and Poisson's ratio are Em = 3,000
9.6 Parametric and
psi and m = 0.3, respectively. It is recognized that this is an
Verification Analysis
ideal representation of actual conditions; however, these con-
Section 9.5 presents rational ovaling and racking analysis ditions provide a good basis for making comparison in para-
procedures robust enough to treat various types of buried metric analysis.
conduit structures. Some simple design charts have also been To account for the effects of shallow soil cover, five cases of
developed to facilitate the design process. These design charts varying embedment depths were analyzed for each culvert

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Table 9-1. Parameters used in the parametric analysis.
Parameters Descriptions
Structure Types FLEXIBLE CULVERTS:
Corrugated Aluminum Pipe
Corrugated Steel Pipe
Corrugated HDPE Pipe
RIGID CULVERTS:
Reinforced Concrete Pipe
Reinforced Concrete Box Type
Burial Depths 5d, 3d, 2d, 1d, 0.5d,
("d" represents the diameter of a circular pipe or the height of a box concrete
culvert)
Cross Section Circular
Geometry Types Square Box
Rectangular Box
Square 3-sided
Rectangular 3-sided
Diameters of Circular 5 feet (Medium Diameter)
Culverts 10 feet (Large Diameter)
Wall Stiffness of FLEXIBLE CULVERTS:
Circular Culverts 4
I=0.00007256 ft /ft, E= 2.9E+07 psi (Steel)
4
I=0.00001168 ft /ft, E= 1.0E+07 psi (Aluminum)
4
I=0.0005787 ft /ft, E= 1.1E+05 psi (HDPE)
Size Dimensions of 10 feet x 10 feet: Square Box and Square 3-sided
Box Culverts 10 feet x 20 feet: Rectangular Box and Rectangular 3-sided
Wall Stiffness of RIGID CULVERTS:
Box Culverts 4
I=0.025 ft /ft, t=0.67 ft, E= 4.0E+06 psi (Concrete)
4
I=0.2 ft /ft, t=1.33 ft, E= 4.0E+06 psi (Concrete)
Properties of E=3,000 psi (Firm Ground)
Surrounding E=7,500 psi (Very Stiff Ground)
Ground*
Total Unit Weight = 120 psf
* Note: The Young's Modulus values used in this study are for parametric analysis purposes only.
Table 9-2. Parametric Analysis Set 1--culvert lining properties (Reference Set).
Rigid Culvert Flexible Culvert
Culvert Properties (Concrete Pipe) (Corrugated Steel Pipe)
Culvert Diameter, ft 10 10
2
Young's Modulus, E/(1-v ), used in
2-D Plane Strain Condition, psi 4.0E+06 2.9E+07
0.00007256 ft4/ft
4
Moment of Inertia I, ft /ft 0.025 ft4/ft 4
(=1.505 in /ft)
Sectional Area A, ft2 per ft 0.67 0.02
2
EI (lb-ft per ft) 1.44E+07 3.03E+05
AE (lb per ft) 3.86E+08 8.35E+07
Poisson's Ratio 0.3 0.3
Note: Ground condition (firm ground with Em = 3000 psi, m = 0.3).

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Table 9-3. Analyses performed for variable embedment depths.
Soil Cover Culvert Diameter Embedment Depth
Cases Analyzed H (feet) d (feet) Ratio, H/d
Case 1 50 10 5
Case 2 30 10 3
Case 3 20 10 2
Case 4 10 10 1
Case 5 5 10 0.5
Case 6 2 10 0.2
type (that is, the flexible type and the rigid type). The six cases Results of Analysis--Set 1. Figures 9-19 and 9-20 show
of embedment depths are listed in Table 9-3. examples of culvert lining response in terms of lining
Figures 9-11 through 9-15 show the finite difference meshes thrust/hoop forces and bending moments, respectively. Ex-
(using computer program FLAC) used for the parametric amples presented in Figures 9-19 and 9-20 are for the flexi-
analysis accounting for the variable culvert embedment depths. ble culvert under the Case 1 conditions (that is, with a soil
Figure 9-16 graphically defines the "Embedment Depth cover of 50 feet deep). As indicated, the maximum response
Ratio" cited in Table 9-3. Figure 9-17 shows the culvert lining (that is, the most vulnerable locations) occurs at the knee-
modeled as continuous beam elements in the finite difference, and-shoulder locations around the lining, consistent with
soil-structural interaction analysis. the generally observed damage/damage mechanism for
The entire soil-structure system was subjected to an artifi- buried pipes/culverts (as well as circular tunnels) during
cially applied pseudo-horizontal acceleration of 0.3g (accelera- major earthquakes in the past (refer to the mechanism sketch
tion of gravity), simulating earthquake-induced vertically prop- depicted in Figure 9-2).
agating shear waves. As a result, lateral shear displacement in the Using the lining information presented in Table 9-2 and
soil overburden will occur. A simple, uniform pseudo accelera- the soil properties of the surrounding ground (that is, Em =
tion and a simple, uniform soil profile (with a uniform soil stiff- 3,000 psi, m = 0.3), the compressibility ratio (C) and flexibil-
ness modulus) were assumed for simplicity and are desirable in ity ratio (F) for the two culverts were calculated using Equa-
parametric analysis. Figure 9-18 presents the resulting lateral tion (9-5) and Equation (9-6), respectively. Their values are
soil displacement profile under lateral acceleration of 0.3g. presented in Table 9-4. The results of the analysis in terms of
Figure 9-11. Case 1 finite difference mesh Figure 9-12. Case 2 finite difference mesh
(soil cover = 50 feet). (soil cover = 30 feet).

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Figure 9-13. Case 3 finite difference mesh Figure 9-15. Case 5 finite difference mesh
(soil cover = 20 feet). (soil cover = 5 feet).
lining deformations (diameter changes) are presented in about 15 percent to 20 percent. This result is consistent
Tables 9-5 and 9-6. with previous studies as discussed in Section 9.5.1.
From these analyses the following observations were made:
The data contained in Table 9-6 is graphically presented in
· Flexible culverts experience greater deformation than the Figure 9-21. As seen, the flexible culvert deforms significantly
ground deformation in the free-field for both full-slip and more than the free field because its flexibility ratio (F = 22.6)
no-slip cases. is significantly greater than 1.0, suggesting the ground is much
· Rigid culverts experience less deformation than the ground stiffer than the lining. For the rigid culvert with F = 0.482 < 1.0,
deformation in the free-field for both full-slip and no-slip the lining is stiffer than the ground and therefore deforms less
cases. than the free-field.
· The full-slip condition gives more conservative values of Figure 9-22 shows the effects of culvert embedment depth on
lining deflections (DEQ) than the nonslip condition by the lining deformations, expressed by the ratios of the lining to
free-field deformation. It can be seen that the ratios of the lin-
ing to free-field deformation remained almost unchanged for
an embedment ratio of 1.0 or greater. When the embedment
Figure 9-14. Case 4 finite difference mesh
(soil cover = 10 feet). Figure 9-16. Definition of embedment depth ratio.

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vert than for the flexible culvert. The thrust ratio presented in
Figure 9-23 is defined as the maximum lining thrust obtained
from the finite difference analysis normalized to that derived
using the close-form solutions in Equations (9-11) and (9-12)
(for the no-slip interface condition). As indicated, the theo-
retical close-form solution somewhat overestimates the lining
thrust/hoop forces when the culvert is buried at shallow depth.
For a rigid culvert, the overestimation is no more than 15 per-
cent. For a flexible culvert the overestimation is negligible. The
figure also shows that the effect of embedment is negligible
when the embedment ratio is greater than about 3 or 4.
The embedment effects on bending response are illustrated
in Figure 9-24. Based on the results from the analysis, it ap-
pears that the potential for overestimation of bending de-
mand would occur for rigid types of culvert structures buried
at shallow depths by as much as 30 to 35 percent. Figure 9-24
also suggests that the effects of embedment depth on bending
response are insignificant when the embedment depth ratio
is greater than about 3.
Figure 9-17. Culvert beam element number.
It should be noted that the main reason for the overesti-
mation in thrust and bending forces is that the maximum
ratio is less than 1.0, the ratio of the actual culvert diameter free-field ground shearing strain used in calculating the close-
change to the free-field deformation decreases gradually. form solutions (Equation 9-11 and Equation 9-12) is the
The culvert embedment depth, however, showed some ef- maximum shearing strain that occurs at the culvert invert
fects on the thrust/hoop force and bending response of the (instead of the average free-field shearing strain within the
lining, as indicated in Figures 9-23 and 9-24. The embedment culvert depth). These results suggest that the maximum free-
effect on the thrust response is more obvious for the rigid cul- field ground strain is on the safe side.
Figure 9-18. Soil deformations subjected to pseudo lateral acceleration of 0.3g.

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Figure 9-19. Culvert lining thrust/hoop force distribution (for flexible culvert in Set 1,
Case 1 geometry).
Figure 9-20. Culvert lining bending moment distribution (for flexible culvert in Set 1,
Case 1 geometry).

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Table 9-4. Culvert lining compressibility and flexibility used in analysis.
Rigid Culvert Flexible Culvert
Properties (Concrete Pipe) (Corrugated Steel Pipe)
Compressibility Ratio, C 0.011 0.05
Flexibility Ratio, F 0.482 22.6
Table 9-5. Free-field ground strain and diameter change.
Closed-Form Free-Field Ground
Free-Field Maximum Ground Shear Diameter Change Using Eq. 9-3
Case No. Strain (from FLAC Analysis) D=0.5*D* max
(Embedment Ratio) max (feet)
Case 1 (H/d=5) 0.0129 0.065
Case 2 (H/d=3) 0.0085 0.043
Case 3 (H/d=2) 0.0064 0.032
Case 4 (H/d=1) 0.004 0.02
Case 5 (H/d=0.5) 0.003 0.015
Case 6 (H/d=0.2) 0.0022 0.011
Note: The maximum free-field ground shearing strain is the maximum shearing strain that could occur within the full depth
of the culvert (that is, from the crown to the invert). In the pseudo-static FLAC analysis, the maximum ground shearing
strains occur at the invert in all cases.
Table 9-6. Culvert diameter change--effect of interface slippage condition.
Culvert Diameter
Culvert Diameter Change (ft) Change (ft) for No-Slip
Case No. for Full-Slip Interface Using Interface Using FLAC Diameter Change Ratio
(Embedment Ratio) Eq. 9-7 Analysis for No-Slip to Full-Slip
For Flexible Culvert
Case 1 (H/d=5) 0.169 0.129 0.77
Case 2 (H/d=3) 0.111 0.082 0.74
Case 3 (H/d=2) 0.084 0.059 0.70
Case 4 (H/d=1) 0.052 0.036 0.68
Case 5 (H/d=0.5) 0.039 0.024 0.62
Case 6 (H/d=0.2) 0.029 0.018 0.62
For Rigid Culvert
Case 1 (H/d=5) 0.042 0.034 0.80
Case 2 (H/d=3) 0.028 0.021 0.77
Case 3 (H/d=2) 0.021 0.015 0.72
Case 4 (H/d=1) 0.013 0.009 0.67
Case 5 (H/d=0.5) 0.010 0.006 0.57
Case 6 (H/d=0.2) 0.007 0.004 0.51

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Figure 9-21. Culvert deformations versus free-field
deformations.
Figure 9-23. Embedment effects on culvert maximum
thrust/hoop forces.
Additional Parametric Analysis and Results. Additional
parametric analyses included (1) different circular culvert/
pipe sizes; (2) different culvert/pipe material, such as corru- rigid culvert. Similarly for the flexible culvert, C and F were
gated aluminum and HDPE pipes; (3) different soil stiffness; reduced from 0.05 to 0.025 and from 22.6 to 2.856, respec-
(4) square and rectangular shape culverts (constructed with tively (see Table 9-7).
reinforced concrete; (5) 3-sided flat roof rectangular concrete
culverts; and (6) different culvert/pipe wall stiffness. These Results of Analyses--Set 2. Figures 9-25 through 9-27
additional analyses were used to further verify that with some present the results of FLAC analysis. Compared to results
modifications, the close-form solutions developed for deep from Set 1 analysis (refer to Figures 9-22 through 9-24), the
circular bored tunnels and rectangular cut-and-cover tunnels Set 2 results indicated that:
(refer to Section 9.5) also can be used for circular and rectan-
gular culvert structures. · The ratios of the actual culvert deformation to free-field
ground deformation were significantly reduced, reflecting
9.6.2.2 Parametric Analysis--Set 2 the effect of higher culvert lining stiffness because of the re-
Model Assumptions--Set 2. Assumptions and parame-
duced culvert diameter.
· The bending and thrust force response of the smaller
ters used in parametric analysis Set 2 are the same as those
used in Set 1 (the Reference Case) except (1) the culvert di- 5-foot diameter culvert, when normalized to the close-
ameter was reduced from 10 feet to 5 feet; (2) the total soil form solutions, show similar trends to that of the larger
profile depth has been reduced from 100 feet to 50 feet; and culvert (10-foot diameter). Based on results in Figure 9-26,
(3) the culvert embedment depth was halved in each respec- when the burial depth is small, the close-form solutions
tive case to maintain the same embedment ratio (H/d). Be- (using the conservative maximum free-field ground strain
cause of this reduction in culvert size the resulting compress- value at the culvert invert elevation) tend to overestimate
ibility ratio (C) was reduced from 0.011 to 0.005 and the the thrust response by up to about 20 percent for the flex-
flexibility ratio (F) was reduced from 0.482 to 0.061 for the ible culvert. For the rigid culvert the overestimation is
greater than about 30 percent at very shallow burial depth.
Figure 9-22. Ratios of culvert deformations versus Figure 9-24. Embedment effects on culvert maximum
free-field deformations. bending moments.

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Table 9-7. Parametric analysis set 2--culvert lining properties.
Rigid Culvert Flexible Culvert
Culvert Properties (Concrete Pipe) (Corrugated Steel Pipe)
Culvert Diameter, ft 5 5
2
Young's Modulus, E/(1-v ), psi 4.0E+06 2.9E+07
4
Moment of Inertia, ft /ft 0.025 0.00007256
2
Sectional Area, ft per ft 0.67 0.02
2
EI (lb-ft per ft) 1.44E+07 3.03E+05
AE (lb, per ft) 3.86E+08 8.35E+07
Poisson's Ratio 0.3 0.3
Compressibility, C 0.005 0.025
Flexibility Ratio, F 0.061 2.856
Note: Ground condition (firm ground with Em = 3000 psi, m = 0.3).
The effect of shallow embedment depth on bending shows This suggests that the analytical methodology and procedure
similar trends to the thrust response (refer to Figure 9-27). previously presented in Section 9.5 provide a robust ap-
proach to accounting for the soil-structure interaction effect
9.6.2.3 Parametric Analysis--Set 3 in evaluating the seismic behavior of culverts with varying
characteristics.
Model Assumptions--Set 3. In this set of analyses the as-
sumptions and parameters are the same as those used in Set
1 (the Reference Case) except (1) the flexible culvert was
changed from corrugated steel pipe to corrugated aluminum
pipe (with lower bending and compression stiffness com-
pared to the steel pipe); and (2) the rigid concrete pipe was
made even more rigid by increasing its wall thickness from
0.67 feet to 1.33 feet. The resulting compressibility ratio and
flexibility ratio, along with other lining properties are pre-
sented in Table 9-8.
Results of Analyses--Set 3. Results from the analysis are
shown in Figures 9-28 through 9-30. As indicated, the results
are following the same trend as shown in results from Sets 1
and 2 analysis, even though a much more flexible culvert and Figure 9-26. Embedment effects on culvert maximum
a much more rigid culvert were used in this set of analysis. thrust/hoop forces (parametric analysis--Set 2).
Figure 9-25. Ratios of culvert deformations versus Figure 9-27. Embedment effects on culvert maximum
free-field deformations (parametric analysis--Set 2). bending moments (parametric analysis--Set 2).

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Table 9-8. Parametric analysis set 3--culvert lining properties.
Rigid Culvert Flexible Culvert
Culvert Properties (Concrete Pipe) Aluminum CMP
Culvert Diameter, ft 10 10
2
Young's Modulus, E/(1-v ), psi 4.0E+06 1.0E+07
4
Moment of Inertia, ft /ft 0.2 0.00001168
2
Sectional Area, ft per ft 1.333 0.01125
2
EI (lb-ft per ft) 1.152E+08 1.682E+04
AE (lb, per ft) 7.678E+08 1.62E+07
Poisson's Ratio 0.3 0.3
Compressibility, C 0.005 0.256
Flexibility Ratio, F 0.060 411.7
Note: Ground condition (firm ground with Em = 3,000 psi, m = 0.3).
9.6.2.4 Parametric Analysis--Set 4 meric conduits are being used with increasing frequency,
and polymers, especially high density polyethylene, are
Model Assumptions--Set 4. Only one type of lining was
likely to be the material of choice for many drainage appli-
analyzed in this set of analysis. The lining modeled in this
cations in the future. The typical properties of the HDPE
analysis is a 5-foot diameter corrugated HDPE pipe. The
material are presented in Table 9-9. Young's modulus of
reason for selecting HDPE in this analysis is because poly-
110,000 psi is appropriate for short term loading effects on
HDPE pipe. Poisson's ratio of HDPE pipe is estimated to be
about 0.45.
Results of Analyses--Set 4. Figures 9-31 through 9-33
present the results of the HDPE culvert analysis. As indi-
cated, the seismic behavior of the HDPE pipe also can be
predicted reasonably well using the analytical procedure
presented in Section 9-5. Like in other cases, if necessary,
some adjustments may be made to correct the overestima-
tion of thrust forces and bending moments when the pipe is
buried at a very shallow depth. For conservative design pur-
poses, however, it is recommended that no force reduction
be made.
Figure 9-28. Ratios of culvert deformations versus
free-field deformations (parametric analysis--Set 3).
Figure 9-29. Embedment effects on culvert maximum Figure 9-30. Embedment effects on culvert maximum
thrust/hoop forces (parametric analysis--Set 3). bending moments (parametric analysis--Set 3).

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Table 9-9. Parametric analysis set 4--culvert lining properties.
Flexible Culvert
Culvert Properties (Corrugated HDPE)
Culvert Diameter, ft 5
2
Young's Modulus, E/(1-v ), psi 1.1E+05
4
Moment of Inertia, ft per ft 0.0005787
2
Sectional Area, ft per ft 0.0448
2
EI (lb-ft per ft) 9.17E+03
AE (lb, per ft) 7.10E+05
Poisson's Ratio 0.45
Compressibility, C 2.927
Flexibility Ratio, F 94.424
Note: Ground condition (firm ground with Em = 3,000 psi, m = 0.3).
9.6.2.5 Parametric Analysis--Set 5 and 9-10). However, the soil stiffness has been increased from
Em = 3,000 psi (firm ground) to Em = 7,500 psi (very stiff ground).
Model Assumptions--Set 5. In this set of parametric
The entire soil profile was assumed to be homogeneous. The
analysis, the culvert lining properties used are identical to
soil overburden thickness (100-foot thick) and other condi-
those assumed in Set 1 (the Reference Case, refer to Tables 9-2
tions are the same as those in Set 1.
Results of Analyses--Set 5. The calculated compressibil-
ity and flexibility ratios also are included in Table 9-10. Be-
cause of the increased ground stiffness, the flexibility ratio for
the rigid culvert was computed to be 1.217, slightly greater
than 1.0. This suggests that the ovaling stiffness of the ground
is only slightly greater than the ovaling stiffness of the rigid
culvert. Based on the discussions presented in Section 9-4,
when the flexibility ratio is close to 1.0, the ovaling deforma-
tion of the lining should be about the same as that of the sur-
rounding ground.
Results from the FLAC analysis in Figure 9-34 show that
for the rigid culvert the ratio of the culvert deformation to the
Figure 9-31. Ratios of culvert deformations versus ground deformation is very close to 1.0, verifying the validity
free-field deformations (parametric analysis--Set 4).
Figure 9-32. Embedment effects on culvert maximum Figure 9-33. Embedment effects on culvert maximum
thrust/hoop forces (parametric analysis--Set 4). bending moments (parametric analysis--Set 4).

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Table 9-10. Parametric analysis set 5--very stiff ground condition.
Rigid Culvert Flexible Culvert
Culvert Properties (Concrete Pipe) (Corrugated Steel Pipe)
Culvert Diameter, ft 10 10
2
Young's Modulus, E/(1-v ), psi 4.0E+06 2.9E+07
4
Moment of Inertia, ft /ft 0.025 0.00007256
2
Sectional Area, ft per ft 0.67 0.02
EI (lb-ft2 per ft) 1.44E+07 3.03E+05
AE (lb, per ft) 3.86E+08 8.35E+07
Poisson's Ratio 0.3 0.3
Compressibility, C 0.027 0.127
Flexibility Ratio, F 1.217 57.122
Note: ground condition (very stiff ground with Em = 7,500 psi, m = 0.3).
of the analytical solutions discussed in Section 9-4. Figures 9-35 · Young's Modulus, E/(1 - 2) = 4.0E+06 psi
and 9-36 display similar results (normalized thrust forces and · Poisson's Ratio, = 0.3
bending moments) presented in other parametric analysis cases · Thickness, t = 0.67 ft
even though the ground stiffness was significantly changed · Moment of Inertia, I = 0.025 ft4/ft
(from Em = 3,000 psi to Em = 7,500 psi).
Five sets of parametric analyses have been performed con-
sidering the following combinations of variables: (1) culvert
9.6.2.6 Parametric Analysis--Set 6
sizes; (2) culvert sectional configurations; (3) soil stiffness;
Model Assumptions--Set 6. The parametric analyses dis- and (4) culvert burial depths. Table 9-11 below summarize
cussed thus far focused on the ovaling behavior of culverts. In specific parameters used in each case of analysis.
this section, a series of parametric analysis is performed for the The main purpose of this parametric analysis is to verify
rectangular and square shaped culverts. These culverts are as- that the rectangular flexibility ratio (Frec) developed in Equa-
sumed to be constructed with reinforced concrete. The sizes tion (9-14), Frec = (Gm / Ks) (w/h), is a proper representa-
and geometry of these concrete box culverts are graphically tion of the relative stiffness between the culvert's racking
presented in Figure 9-37. stiffness and the ground's racking stiffness. By using Frec, it
The concrete lining was modeled as continuous beam ele- is possible to accurately estimate the actual racking defor-
ments in the finite difference, soil-structural interaction mation of the culvert as long as the free-field ground defor-
analysis having the following properties: mation (free-field) is known.
Figure 9-34. Ratios of culvert deformations versus Figure 9-35. Embedment effects on culvert maximum
free-field deformations (parametric analysis--Set 5). thrust/hoop forces (parametric analysis--Set 5).

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analysis. In this FLAC analysis, the culvert structure is in-
cluded in the soil deposit model and subject to the same
pseudo-horizontal acceleration used in the free-field
FLAC analysis mentioned in Step 1 above. Note that since
s is related to free-field directly through Rrec, the racking
ratio, the comparison therefore also can be made between
the manually calculated Rrec = [2Frec/(1+Frec)], and Rrec
computed from the FLAC analysis.
4. If the manually estimated racking deformations (or the
Rrec values) are comparable to those computed by the soil-
structure interaction FLAC analysis, then the simplified
procedure developed in Section 9.5.2 can be considered to
Figure 9-36. Embedment effects on culvert maximum be validated.
bending moments (parametric analysis--Set 5).
Results of Analyses--Set 6. Based on the results from
the FLAC analysis (from both the free-field analysis run
The verification procedure is:
and the soil-structure interaction analysis run), the free-
field racking deformations and the actual culvert racking
1. Determine the free-field racking deformation of the
deformations were obtained. Ratios of the culvert to free-
ground (free-field). This was achieved in this analysis by ap-
plying a pseudo-horizontal acceleration in the entire free- field racking deformations are plotted for all five cases (for
field soil deposit in the FLAC analysis. Note that at this five different burial depths in each case) in Figures 9-38
time the FLAC model is a free-field soil deposit model that through 9-42. Based on the data presented in these figures,
does not contain the culvert structure in it. The resulting it appears that burial depth does not have significant influ-
free-field racking deformations then can be directly read ence on the racking deformation ratio for the rectangular
out from the output of the FLAC analysis. type of rigid culverts.
2. Given free-field, the racking deformation of the culvert can In the meantime, the structural racking stiffness (Ks) of the
be manually estimated by using the simple relationship culvert structure in each case was determined by a simple frame
presented in Equation (9-16), s = Rrec free-field. analysis based on the properties of the culvert structure; the re-
3. The manually estimated racking deformation derived sults are presented in Table 9-12. Then the rectangular flexibil-
above then is compared to the actual racking deformation ity ratio (Frec) was calculated using Equation (9-14), and results
of the culvert from the soil-structure interaction FLAC also presented in Table 9-12 for each case.
Figure 9-37. Various concrete box culvert sectional shapes and sizes used in
the parametric analysis--Set 6.

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Table 9-11. Soil and structure parameters used in the analysis.
Structural Configurations and Soil Properties
Case 1 10' x 10' Square Box, in Firm Ground (E m = 3,000 psi, m = 0.3)
Case 2 10' x 10' Square Box, in Very Stiff Ground (Em = 7,500 psi, m = 0.3)
Case 3 10' x 20' Rectangular Box, in Firm Ground (Em = 3,000 psi, m = 0.3)
Case 4 10' x 10' Square 3-Sided, in Very Stiff Ground (Em = 7,500 psi, m = 0.3)
Case 5 10' x 20' Rectangular 3-Sided, in Very Stiff Ground (Em = 7,500 psi, m = 0.3)
Note: For each case, the effects of culvert embedment depth (of 50 feet, 30 feet, 20 feet, 10 feet, and 5 feet, measured from
ground surface to top of the culvert roof) were studied.
The results show that for Case 1 the relative racking stiff- derived in Section 9-5. For Cases 2 through 5, the flexibility
ness of the ground to the structure is about 1.0, suggesting ratios are all greater than 1.0, suggesting that the structure
that the structure would rack in conformance with the free- would deform more than the ground in the free-field, and re-
field racking deformation in the ground. The results pre- sults shown in Figures 9-39 through 9-42 support this theory.
sented in Figure 9-38 show clearly that the FLAC calculated Figure 9-43 plots the racking ratio as a function of the flex-
racking deformations are about the same as the free-field de- ibility ratio based on the results obtained from the FLAC
formations, validating the definition of flexibility ratio (Frec) analysis and then compares them with the recommended
Figure 9-38. Racking ratios from FLAC analysis-- Figure 9-40. Racking ratios from FLAC analysis--
Case 1. Case 3.
Figure 9-39. Racking ratios from FLAC analysis-- Figure 9-41. Racking ratios from FLAC analysis--
Case 2. Case 4.