Recommended Design Specifications for Live Load Distribution to Buried Structures (2010)

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92.1 Review and Evaluation of Relevant Experience 2.1.1 Soil Constitutive Models Numerous soil constitutive models have been developed and are available for finite-element analysis. Lade (2005) sum- marized widely available soil constitutive models. Each model has different capabilities and requires different experimental data for calibration. Predicting the response of buried structures to surface live loads in a finite-element analysis requires a soil constitutive model that captures culvert-soil interaction accurately. Research has been conducted with linear-elastic soil models (e.g., Moore and Brachman, 1994, and Fernando and Carter, 1998) with nonlinear models, including nonlinear elastic models, perfectly plastic models, and plastic models with hardening (e.g., Pang, 1999). For typical culvert analysis, which has been historically conducted in two dimensions, stress-dependent stiffness and shear failure have been found to be important characteristics of suitable soil models. The Duncan-Selig hyperbolic model (Duncan et al., 1980, and Selig, 1988) has such features and has been implemented in the finite-element programs CANDE (Musser, 1989) and SPIDA (Heger et al., 1985) to analyze soil-structure inter- action problems for culverts. Soil properties based on these models have been used to develop current AASHTO specifi- cations for reinforced concrete and thermoplastic pipe. The Duncan-Selig model consists of the hyperbolic Young’s mod- ulus model developed by Duncan et al. (1980) and the hyper- bolic bulk modulus developed by Selig (1988). As discussed below, the soil properties used with this model were devel- oped by Selig (1988). CANDE, developed by the FHWA, has been widely used to design culverts, but operates only in two dimensions. For ease of computation and to allow comparison with CANDE, the research team conducted preliminary analyses in two dimen- sions and then extended these models to three dimensions for a complete investigation of actual live load distribution. 3D modeling is computationally intensive, so it is impor- tant to select the computationally simplest soil model that can accurately capture culvert-soil interaction resulting from live load. The research team selected three soil models with vary- ing levels of sophistication for preliminary assessment: • Linear-elastic (representing the simplest possible model), • Mohr-Coulomb (linear-elastic model with post-failure plasticity), and • Hardening-soil (stress dependence plus plasticity, similar to Duncan-Selig). 2.1.2 Model Verification with Field Tests Initial investigation of culvert responses to live loads from 2D analyses with linear-elastic, Mohr-Coulomb, and hardening- soil models showed that responses from the Mohr-Coulomb and hardening-soil models were very close to each other whereas responses from the linear-elastic model were signifi- cantly different from other models (see Appendix A for the de- tailed treatment). As a result, the Mohr-Coulomb model was selected for use in the 3D analysis of field tests. Subsequent Panel comments suggested comparison of the Mohr-Coulomb and hardening-soil models in the 3D analysis as a confirmation of selection of an appropriate soil model. To compare culvert responses from these two soil models, 3D analyses were per- formed of a long-span metal arch from NCHRP Project 12-45 (McGrath et al., 2002) and HDPE pipe from the Minnesota DOT (MNDOT) study (McGrath et al., 2005). Method of Approach Soil-structure interaction analysis of culverts subjected to the surface live load is performed using Plaxis 3D Tunnel, C H A P T E R 2 Findings

Version 21 (Brinkgreve and Broere, 2004). Two structural models were selected as described above: (1) a long-span metal arch, Test 2, 3-ft cover (NCHRP Report 473); and (2) an HDPE pipe, Pipe Run 7, A-2 backfill, 2.8-ft cover (MNDOT study). These structures were analyzed with both Mohr-Coulomb and hardening-soil models, and structural responses were com- pared. In the metal arch model, backfill was assumed to have properties of SW85, and the soil above the crown of the arch was assumed to have properties of SW95. With the HDPE pipe model, backfill was assumed to have properties of ML95. The interface strength was assumed to be 50% of the strength of surrounding soil. Structures, finite-element models, live load tests, and material properties were described in detail in the lit- erature (McGrath et al., 2002, and McGrath et al., 2005). The results of the comparison of predictions from computer mod- els with data from actual field tests was often poor; extenuating circumstances are discussed in Section 2.1.3. Results Metal Arch in Test 2 with 3 ft Cover. Figure 2-1 compares vertical crown displacements and horizontal chord extensions along the culvert for the two cases of the metal arch analysis: the case with the Mohr-Coulomb model and the case with the hardening-soil model. Figure 2-2 compares thrusts and moments under the wheel load for the two cases. These figures also show displacements and forces measured in the field tests. Measured thrusts and moments are average values of measure- ments under the left and right wheels. Tables 2-1 through 2-3 also compare displacements and forces of the two cases. Differ- ences in moments and thrusts of the two soil models were insignificant. Displacements were slightly smaller with the hardening-soil model than the Mohr-Coulomb model: by 9% 10 1Plaxis 3D Tunnel is a finite-element package focused on the analysis of struc- tures in soil and rock materials. Special features include non-linear, time- dependent, and anisotropic behavior of the earth materials, multiphase problems, and soil-structure interaction. (a) Vertical crown (b) Horizontal chord Figure 2-1. Comparison of displacements between cases with Mohr-Coulomb and hardening-soil models (metal arch, Test 2, 3 ft cover). (a) Thrust (b) Moment Figure 2-2. Comparison of thrusts and moments under wheel between cases with Mohr-Coulomb and hardening-soil models (metal arch, Test 2, 3 ft cover).

11 Plaxis 3D (in.) Ratio:Plaxis 3D / Field Test Vertical or Horizontal Field Test (in.) Mohr- Coulomb Hardening- Soil Mohr- Coulomb Hardening- Soil Vertical crown displacement 0.45 0.79 0.72 1.74 1.58 Horizontal chord extension 0.25 0.40 0.34 1.60 1.34 Table 2-1. Summary of displacements under wheel (metal arch, Test 2, 3 ft cover). Plaxis 3D (kip/ft) Ratio: Plaxis 3D/Field Test Location Field Test (kip/ft) Mohr- Coulomb Mohr- Coulomb Hardening- Soil Hardening- Soil NS 0.69 3.12 3.54 4.54 5.15 NC 1.42 4.41 4.70 3.10 3.31 NH 9.07 5.92 6.27 0.65 0.69 CR 3.08 7.05 7.75 2.29 2.51 SH 5.84 5.80 6.18 0.99 1.06 SC 2.46 4.08 4.46 1.66 1.81 SS -0.34 2.75 3.25 -7.98 -9.44 NS = north springline; NC = north chord point at intersection of top plates and side plates; NH = north haunch, about halfway between NC and CR; CR = crown; SS = south springline; SC = south chord point at intersection of top plates and side plates; SH = south haunch, about halfway between SC and CR Table 2-2. Summary of thrusts under wheel (metal arch, Test 2, 3 ft cover). -0.03 -3.14 -3.61 94.73 108.90 -0.38 -2.75 -2.13 7.27 5.63 -5.24 -5.25 -5.42 1.10 1.03 2.04 13.71 13.38 6.74 6.57 -3.28 -6.43 -6.26 1.96 1.91 -1.97 -2.62 -1.78 1.33 0.91 0.07 -2.66 -3.31 -35.61 -44.31 Plaxis 3D (kip/ft) Ratio: Plaxis 3D/Field Test Location Field Test (kip/ft) Mohr- Coulomb Mohr- Coulomb Hardening- Soil Hardening- Soil NS NC NH CR SH SC SS See notes on Table 2 for definition of Location acronyms. Table 2-3. Summary of moments under wheel (metal arch, Test 2, 3 ft cover). for the vertical crown displacement and by 16% for the hori- zontal displacement. Therefore, displacement results were closer to the field measurements with the hardening-soil model than with the Mohr-Coulomb soil model in this case. HDPE Pipe with A2 Backfill and 2.8-ft Cover. For the HDPE pipe study, evaluating the Plaxis model predictions for the Mohr-Coulomb and hardening-soil models against the field data was accomplished by comparing deflections, and evaluating the force predictions. Figures 2-3 and 2-4 compare vertical crown displacements horizontal diameter extensions along the culvert for the two cases of HDPE pipe analysis: the case with the Mohr-Coulomb model and the case with the hardening-soil model. These figures also show displacements measured in the field tests. In the figures, P3 indicates a single axle centered over the pipe, P4 indicates tandem axles located symmetrically over the pipe, “heavy” indicates 24,000 lb axles and “light” indicates 18,000 lb axles. The dates of the tests (i.e., when the data were collected) are also provided. For ex- ample, Oct ’00 represents nearly the first loading after con- struction, May ’01 represents a time shortly after the winter frost had melted and the ground was soft, and Aug ’02 repre- sents data after numerous loading cycles at a dry time of the year. Figures 2-5 and 2-6 compare thrusts moments under the wheel load for the two cases. Tables 2-4 and 2-5 compare field displacements with the Plaxis model displacements, while Table 2-6 compares forces for the two cases. Calculated displacements were larger with the hardening- soil model than with the Mohr-Coulomb model: by about 30% for the vertical crown displacement and by about 55%

for the horizontal displacement. Therefore, displacement results from the Mohr-Coulomb model were closer to the measured displacements in the field tests in this particular case. Because of the larger displacements, moments and thrusts were also larger with the hardening-soil model. Thrusts from the hardening-soil model were up to 20% higher than those from the Mohr-Coulomb model, and moments were up to 44% higher. Softer soil responses obtained for the hardening-soil model can be explained by the lower stiffness of the hardening- soil model when compared with that of the Duncan-Selig model. By selecting higher stiffness values for hardening-soil properties of ML95, force results can be brought closer to those of the Mohr-Coulomb model. 2.1.3 Discussion During development and testing of computer models, soil models, and modeling methodology (described in Appendix A) the research team learned that soil-structure interaction analy- sis of buried culverts subjected to live loads with the linear- elastic soil model could produce significantly different struc- tural response than those with the Mohr-Coulomb soil model 12 (a) Heavy truck (b) Light truck Figure 2-3. Comparison of crown vertical displacements between cases with Mohr-Coulomb and hardening-soil models (HDPE pipe, A-2 soil, 2.8 ft cover). (a) Heavy truck (b) Light truck Figure 2-4. Comparison of horizontal diameter changes between cases with Mohr-Coulomb and hardening-soil models (HDPE pipe, A-2 soil, 2.8 ft cover).

and the hardening-soil model. Given that a difference between the linear-elastic model and the Mohr-Coulomb model is whether or not soil failure is modeled by plasticity, plasticity is one of the key characteristics of soil models for this project. The analyses presented here indicate some of the diffi- culties in predicting structural response of buried culverts subjected to live loads. The soil parameters currently used in design appear to yield soil behavior that is softer than achieved in the field tests. Given the variability of real-world soils and in-field compaction effort, this conservatism is jus- tified in design. Soil parameters could be developed just for the current study that match the soil test data, which in turn produce better estimates of live load response of buried cul- verts. However, the same question will arise—how should the parameters be modified for design of actual structures that will experience all of the variability noted? Given the success of the Duncan-Selig model and the Selig (1988) properties, it is appropriate to continue with conservative design parameters. In addition to the soil parameters of a specific soil type, other uncertainties in the field tests made matching field data in the analysis difficult. Backfill densities were reported as the density 13 (a) Heavy truck (b) Light truck Figure 2-5. Comparison of thrusts between cases with Mohr-Coulomb and hardening-soil models (HDPE pipe, A-2 soil, 2.8 ft cover). (a) Heavy truck (b) Light truck Figure 2-6. Comparison of moments between cases with Mohr-Coulomb and hardening-soil models (HDPE pipe, A-2 soil, 2.8 ft cover).

measured at the time of backfilling; however, there is consid- erable activity over the pipes after the backfilling is completed, and this activity likely densifies the soil. For example • In the long-span study, the soil surface was compacted with a large vibratory roller prior to the live load tests to ensure that the surface soil could carry the heavily loaded truck without significant rutting. This probably densified the clean gravel backfill. • In the MNDOT study, the backfill was overlaid with 8 in. of gravel and 4 in. of pavement. Again, the backfill over the top crown of the test pipes probably was densified prior to live load testing. • In the MNDOT study, after the construction was com- plete, there was still considerable variability in the data be- cause of seasonal differences, temperature variations and, perhaps, other parameters. Because of these circumstances, the overprediction of dis- placements and forces was expected in the soil models. The differences between the model predictions and the field data could be addressed by increasing soil properties until the pre- dictions match field data; however, this would be artificial because the soil properties used for the study have been vali- dated repeatedly over the last 20 years. The question then is whether or not to match specified field properties or anti- cipate densification through subsequent construction in order to match field test results. The use of specified soil properties was selected as being appropriately conservative. Any densifica- tion that occurs because of subsequent construction activities should provide additional safety. This approach was further justified given the variability of highway construction sites, potential variations in backfill type and density, and variations in depth of cover. 2.1.4 Recommended Soil Constitutive Model The preceding sections document the investigation of soil models for analysis of live load effects on buried structures. Based on these studies, a linearly elastic, perfectly plastic soil model, with a Mohr-Coulomb failure criterion, was selected. This selection offered the best mix of capturing the important aspects of soil behavior in transmitting live loads to structures and simplicity in modeling to allow the research team to complete the greatest number of analyses in the least amount of time. In implementing this soil model, the research team recommended that the elastic soil properties be selected based on depth of fill. This technique does not offer all of the bene- fits of the Duncan-Selig/hardening-soil models in capturing stress-dependent stiffness behavior of soil, but, for the pur- poses of a live load study, this technique appears to provide sufficient accuracy. 14 Ratio: Plaxis 3D / Field TestField Test (in.) Plaxis 3D (in.) Mohr-Coulomb Model Hardening Soil Model Truck Position Oct 00 May 01 Aug 02 Mohr- Coulomb Hardening Soil Oct 00 May 01 Aug 02 Oct 00 May 01 Aug 02 3 0.061 0.036 0.039 0.069 0.089 1.13 1.91 1.76 1.47 2.48 2.29 Heavy 4 0.065 0.040 0.039 0.076 0.103 1.18 1.91 1.96 1.59 2.58 2.65 3 0.049 0.018 0.022 0.053 0.068 1.07 2.92 2.39 1.38 3.75 3.07 Light 4 0.049 0.026 0.026 0.058 0.077 1.18 2.23 2.23 1.58 2.98 2.98 Table 2-4. Summary of vertical displacements under wheel (HDPE pipe, A-2 soil, 2.8 ft cover). 0.009 0.004 0.005 0.016 0.024 1.74 3.91 3.13 2.69 6.06 4.85 0.016 0.008 0.006 0.019 0.031 1.21 2.43 3.24 1.95 3.90 5.20 0.005 0.003 0.003 0.012 0.018 2.40 4.00 4.00 3.67 6.11 6.11 0.007 0.006 0.004 0.015 0.023 2.11 2.46 3.69 3.35 3.91 5.87 Ratio: Plaxis 3D / Field TestField Test (in.) Plaxis 3D (in.) Mohr-Coulomb Model Hardening Soil Model Truck Position Oct 00 May 01 Aug 02 Mohr- Coulomb Hardening Soil Oct 00 May 01 Aug 02 Oct 00 May 01 Aug 02 3 Heavy 4 3 Light 4 Table 2-5. Summary of horizontal chord extensions under wheel (HDPE pipe, A-2 soil, 2.8 ft cover). Thrust (lb/in.) Moment (lb-in./in.) Crown Peak Peak Positive Peak Negative Truck Position MC HS MC HS MC HS MC HS 3 19.5 24.6 28.7 32.1 15.7 20.5 -11.1 -15.4 Heavy 4 24.5 32.8 33.2 39.8 11.7 14.7 -11.5 -16.5 3 14.9 18.8 22.0 24.3 12.0 15.1 -8.6 -11.7 Light 4 18.5 24.9 25.2 30.2 8.8 10.5 -8.7 -12.4 Table 2-6. Summary of force results (HDPE pipe, A-2 soil, 2.8 ft cover).

Parameters for the soil model were those reported above based on the Selig (1988 and 1990) properties. The bulk mod- ulus values in Selig (1990) should be considered suitable for analysis when justified by data, but may not be a lower bound. The proposed properties proved somewhat conservative rel- ative to field data. Soil angles of friction vary depending on the stress state. In the modeling, one may choose whether the angle of friction depends on initial stresses (prior to live load) or on the stresses present because of the live load. This choice did not significantly affect the structural response of culverts to sur- face live loads. The research team also found that a smaller angle of friction resulted in greater structural responses. Given that soil under live loads is under higher stresses than the state before the live loads are applied, angle of friction of soil under live loads should be lower than for the soil before the live loads are applied. Although this change in the angle of fric- tion because of live loads does not affect the soil strength sig- nificantly, the research team believes that it is more accurate to use angles of friction under higher stresses than those based on only geostatic stress state. The research team used soil angles of friction at a reference confinement of 14.7 psi from Selig’s parameters (1988) at any depth in the 3D analysis, instead of variable angles of friction calculated from the stress state before the live load application. 2.2 Selection and Development of Refined Analytical Models This section describes the general characteristics of the software, soils, model dimensions, structures, live load, and modeling sequence. Examples of typical models are provided in Section 2.2.7. 2.2.1 Software All core analyses were done with FLAC3D, with supporting or quality control calculations done using SAP 2000, PLAXIS, ANSYS, and FLAC2D. FLAC3D is a continuum code used in analysis, testing, and design by geotechnical, civil, and mining engineers. The software uses an explicit finite difference for- mulation that can accommodate large displacements and strains and non-linear material behavior, even if yield or fail- ure occurs over a large area or if total collapse occurs. FLAC3D can model some complex behaviors not readily available in other codes, such as problems that consist of several stages, large displacements and strains, non-linear material behavior and unstable systems (even cases of yield/failure over large areas, or total collapse). (Chapter 2.1 of Appendix B has addi- tional information.) Built-in soil models include elastic, Mohr-Coulomb, and Cam-Clay, but all analyses reported here used the Mohr- Coulomb soil model (except for a thin, elastic layer used to pre- vent the live load from causing failure of the soil surface, and in some analyses, an elastic layer representing the pavement). All culvert structures modeled here used shell structural elements—isotropic shell elements were used for concrete, PVC, FRP and smooth metal culverts, and orthotropic shell elements were used for corrugated metal and profile wall cul- verts. FLAC3D’s built-in programming language, FISH, facil- itated preparing data sets and extracting/processing results. 2.2.2 Soil Properties Development of soil constitutive models and properties has been addressed in preceding sections (and in Appendix A). Major decisions include use of the Mohr-Coulomb soil, soil property variations with depth, and the use of soil friction angles representative of high-stress conditions. All models used one of four soil materials: well-graded or gravelly sand at 85% standard compaction (SW85), well- graded or gravelly sand at 95% standard compaction (SW95), inorganic silts and fine sands at 85% standard compaction (ML85), and inorganic clays at 85% standard compaction (CL85). Mohr-Coulomb soil parameters for SW95, SW85, ML85, and CL85 are provided in Tables 2-7 through 2-10. 2.2.3 Model Dimensions In general, model dimensions were larger for increasing cover depth and increasing culvert depth. In some instances, the research team initially used smaller model widths, observed 15 Depth Modulus of Elasticity E Poisson’s Ratio Angle of Friction Dilatation Angle Cohesion c (ft) (psi) (deg) (deg) (psi) 0 to 1 1,300 0.26 38.0 8.0 0.001 1 to 6 2,100 0.21 38.0 8.0 0.001 6 to 11 2,600 0.19 38.0 8.0 0.001 11 to 18 3,300 0.19 38.0 8.0 0.001 Table 2-7. Parameters for linear-elastic and Mohr-Coulomb models for SW85.

results indicating that the models were too narrow, and reran the models using greater width. Table 2-11 summarizes the model dimensions. 2.2.4 Culvert Structures Nine culvert structures were modeled: • Concrete arch, • Concrete pipe • Concrete box, • Corrugated metal pipe, • Corrugated metal arch, • Fiberglass-reinforced plastic pipe, • HDPE profile wall pipe, • PVC pipe, and • Smooth metal pipe. The solid cross-section culverts were all modeled as isotropic structures; the corrugated metal pipe, corrugated metal arch, and HDPE profile wall pipe were modeled as orthotropic structures. The properties used for the various culvert struc- tures are listed in Tables 2-12 through 2-16. Concrete box and concrete arch properties were calculated from the section thickness and concrete modulus of 4,030,000 psi (box) and 3,605,000 psi (arch). Uncracked concrete properties were used in the analyses. The rationale was twofold: (1) cracking was not expected and (2) if cracking did occur, the resulting softer structure would have lower, unconservative structural loads. All structures were modeled using three-node, planar-shell elements, either isotropic or orthotropic. The pipe stiffness2 of the various round pipe models varied by more than three orders of magnitude, from 10 lb/in/in for 96-in.-diameter smooth metal pipe, to 47,000 lb/in/in for 12-in.-diameter concrete pipe (See Figures 2-7 and 2-8). 16 Depth Modulus of Elasticity E Poisson’s Ratio Angle of Friction Dilatation Angle Cohesion c (ft) (psi) (deg) (deg) (psi) 0 to 1 1,600 0.40 48.0 18.0 0.001 1 to 5 4,100 0.29 48.0 18.0 0.001 5 to 10 6,000 0.24 48.0 18.0 0.001 10 to 18 8,600 0.23 48.0 18.0 0.001 Table 2-8. Parameters for linear-elastic and Mohr-Coulomb models for SW95. Depth Modulus of Elasticity E Poisson’s Ratio Angle of Friction Dilatation Angle Cohesion c (ft) (psi) (deg) (deg) (psi) 0 to 1 600 0.25 30.0 0.0 3.0 1 to 6 700 0.24 30.0 0.0 3.0 6 to 13 800 0.23 30.0 0.0 3.0 13 to 18 850 0.3 30.0 0.0 3.0 Table 2-9. Parameters for linear-elastic and Mohr-Coulomb models for ML85. Depth Modulus of Elasticity E Poisson’s Ratio Angle of Friction Dilatation Angle Cohesion c (ft) (psi) (deg) (deg) (psi) 0 to 1 100 0.33 18.0 0.0 6.0 1 to 7 250 0.29 18.0 0.0 6.0 7 to 14 400 0.28 18.0 0.0 6.0 14 to 18 600 0.25 18.0 0.0 6.0 Table 2-10. Parameters for linear-elastic and Mohr-Coulomb models for CL85. 2Pipe stiffness, PS: where E = modulus of elasticity, I = moment of inertia, r = mean pipe radius PS EI r = 0 149 3.

17 Culvert Type Culvert Dimensions Cover Depth (inches) Model Width (inches) Model Height (inches) Round Pipe 12” 12 24 48 96 36 54 108 216 42 54 78 126 Round Pipe 24” 12 24 48 96 144 72 72 108 216 72 73 84 108 156 204 Round Pipe 48” 12 24 48 96 144 144 144 144 144 144 138 150 174 198 222 Round Pipe 96” 12 24 48 96 144 288 288 288 288 288 252 264 288 336 384 Concrete Box 48” x 48” 12 24 48 96 144 144 144 144 138 150 174 222 Concrete Box 96” x 96” 12 24 48 96 288 288 288 288 252 264 288 336 Corrugated Metal Arch 20.1 ft x 9.1 ft 12 48 96 240 240 240 229 265 313 Corrugated Metal Arch 30.1 ft x 18.1 ft 12 48 96 360 360 360 444 456 528 Concrete Arch 25.4 ft x 10 ft 12 48 96 300 300 300 252 288 336 Concrete Arch 43.11 ft x 13.8 ft 12 48 96 528 528 528 352 364 436 Table 2-11. Summary of model dimensions. Pipe Dia. (in) OD (in) Wall Thickness (in) A (in 2 /in) I (in 4 /in) EA (lb/in) EI (lb-in 2 /in) 12 16 2 2 0.667 7,210,000 2,403,000 24 30 3 3 2.25 10,815,000 8,111,000 48 58 5 5 10.42 18,025,000 37,552,000 96 114 9 9 60.75 32,445,000 219,004,000 Table 2-12. Summary of isotropic structural properties for concrete pipe. Pipe Dia. (in) OD (in) Wall Thickness (in) A (in2/in) I (in4/in) EA (lb/in) EI (lb-in2/in) 24 24.75 0.375 0.375 0.00439 10,875,000 127,000 48 49.25 0.625 0.625 2.25 18,125,000 65,250,000 96 97.874 0.937 0.937 10.42 27,173,000 302,180,000 Table 2-13. Summary of isotropic structural properties for smooth metal pipe.

2.2.5 Live Load Analyses were conducted for dead load (soil loading only) and combined dead plus live load. The dead load response was subtracted from the combined response to determine the live only response. Dead loads (i.e., soil loads) were not fac- tored. Live loads were applied and factored as follows: where mmpf is the multiple presence factor (1.2) P is the wheel load magnitude (16,000 lb) IM is the dynamic load allowance H ≤ 8 H is the depth of cover from road surface to top of cul- vert, in. 33 1 0 125 12 − ⎡ ⎣⎢ ⎤ ⎦⎥ . , i H LL m IM Pmpf= + ⎡ ⎣⎢ ⎤ ⎦⎥1 100 1( ) 2.2.6 Modeling Sequence Three states of the model were analyzed and saved for each analysis conducted: • State 1 is the soil mass in equilibrium, with no culvert or live load. State 1 was achieved by creating the model grid, applying material properties to the soil materials, and plac- ing stresses in the grid. • State 2 (dead load) is the soil mass plus the culvert, in equi- librium. State 2 was achieved by excavating the soil (with no cycling of the model), installing the culvert in the soil, and then cycling to equilibrium. • State 3 (dead load plus live load) is State 2 plus application of the live load defined above. Unless noted otherwise, all numerical results presented in tables and figures in this report are for State 3 minus State 2 (i.e., dead plus live load minus dead load). All graphical out- 18 Pipe Dia.(in) OD (in) Wall Thickness (in) A (in2/in) I (in4/in) EA (lb/in) EI (lb-in2/in) 12 14.7 1.4 1.4 0.229 140,000 22,900 24 30 2.4 2.4 2.25 240,000 225,000 48 58 3.5 3.5 10.42 350,000 1,042,000 Table 2-14. Summary of isotropic structural properties for PVC pipe. Pipe Dia.(in) Profile Wall Thickness (in) Direction A (in 2 /in) I (in 4 /in) EA (lb/in) EI (lb-in 2 /in) 12 2"X1.07" 0.1984 Circumferential 0.1984 0.0305 19840 3050 Longitudinal 0.06984 0.0000726 6984 7.26 24 4"X2.16" 0.344 Circumferential 0.344 0.1997 34400 19970 Longitudinal 0.14108 0.00057 14108 57 48 5.98"X3.12" 0.48 Circumferential 0.48 0.634 48000 63400 Longitudinal 0.22481 0.00219 22481 219 60 8"X3.51" 0.541 Circumferential 0.5407 0.9 54070 90000 Longitudinal 0.25389 0.00299 25389 299 Table 2-15. Summary of orthotropic structural properties for HDPE profile wall pipe. Pipe Dia.(in) Corrugation Wall Thickness (in) Direction A (in2/in) I (in4/in) EA (lb/in) EI (lb- in2/in) 12 2-2/3"X1/2" 0.0598 Circumferential 0.0646 0.00189 1873400 54810 Longitudinal 0.000587 1.92E-05 17013 558 24 2-2/3"X1/2" 0.0747 Circumferential 0.0807 0.00239 2340300 69310 Longitudinal 0.001137 3.75E-05 32982 1088 48 2-2/3"X1/2" 0.1046 Circumferential 0.113 0.00343 3277000 99470 Longitudinal 0.003083 0.000103 89412 2987 96 3"X1" 0.1046 Circumferential 0.1308 0.01546 3793200 448340 Longitudinal 0.000706 0.000117 20464 3402 Table 2-16. Summary of orthotropic structural properties for corrugated metal pipe.

19 Figure 2-7. Pipe stiffness for the round culverts modelled. Figure 2-8. Location and intensity of live load, before factoring.

put from FLAC3D is for States 1, 2, or 3 (but not for example State 3 minus State 2). 2.2.7 Typical Soil and Structure Models It may not be possible to adequately illustrate “typical models” for 830 analyses ranging from 12-in.-diameter pipe to 43-ft arches, and for 12-in. to 12-ft cover. However, figures 20 Figure 2-9. Round pipe model for 24″ deep, 24″ diameter (shading indicates soil modulus). Figure 2-10. Round pipe model for 96″ deep, 96″ diameter (shading indicates soil modulus). Figure 2-12. Concrete arch model for 48″ deep, 25.4 ft  10 ft cross section (shading indicates soil modulus). in this section illustrate the soil zone meshes, culvert struc- tural meshes, and live load application. Figures 2-9 through 2-12 show typical soil meshes, where the different soil layers are indicated by shading. Figures 2-13 and 2-14 show typical structural meshes. Figure 2-15 shows a typi- cal deformed mesh, where the deformation represents grid dis- Figure 2-11. Concrete box model for 96″ deep, 48″  48″ cross section (shading indicates soil modulus).

developed to provide the data necessary to consider live load effects in the context of those existing design methods. Thus, the research team first reviews the existing simplified design criteria and then presents analysis of the computer model data in the context of those criteria. In analyzing the data to develop new simplified procedures, the research team focused on procedures that emphasize sim- plicity and an actual physical process by which live loads atten- uate with increasing depth of fill. The research team avoided simple curve fitting and, where some effects were considered minor, opted for simple equation forms broadly applicable to many types of culverts, rather than greater accuracy that would require the introduction of more coefficients. The research team took this approach because the goal of the project was simplified design procedures and because computer models are increasingly generally available for situations where greater precision is required. 2.3.1 Current Simplified Culvert Design This section reviews the limit states for culvert types included in the AASHTO specifications and the simplified procedures used to develop designs. Guidelines were devel- oped for analyzing all types of culverts in 2D finite-element 21 Figure 2-13. Typical structural model for a large round pipe (different shadings indicate the normal pressure on the culvert). Figure 2-14. Typical structural model for an arch (shading indicates the normal pressure on the culvert). Figure 2-15. Typical live load application, showing deformed grid (that represents vertical stress). Item Typical Small Model Typical Large Model Number of grid points 2040 186894 Number of zones 1740 170607 Number of structural elements 696 5194 Numbers of structural nodes 390 2862 Table 2-17. Range of mesh sizes. placement and the shading indicates vertical stress. Table 2-17 illustrates the range of mesh sizes. 2.3 Development of SDEs Current AASHTO procedures for simplified design of cul- verts vary with the type of pipe and the performance limits that must be evaluated. For example, simplified design of concrete culverts is based on the indirect method where a three-edge bearing load that produces an equivalent bending moment to the in-ground loads is determined, while corrugated metal pipe is evaluated solely on the basis of compressive thrust due to earth load. The proposed SDEs presented in this section were

programs. Design of culverts with finite-element analyses requires a pressure distribution at the ground surface that reduces the total applied load to account for the live load attenuation in the third dimension that is not modeled. This historically has been completed using AASHTO distributions to reduce the live load to the magnitude at the depth of the pipe crown. The pipe properties that affect pipe-soil interaction are the flexural stiffness, EI, and the axial stiffness, EA (see Table 2-18). Concrete, corrugated metal, and thermoplastic pipe all have different relative combinations of these parameters and thus, the behavior is different for each type of pipe: • Concrete pipe. Concrete pipe carries live (and earth) loads primarily with bending moments because of the high flex- ural stiffness. The high hoop stiffness also causes the pipe to have high contact pressure. • Corrugated metal pipe. Corrugated metal pipe has rela- tively low flexural stiffness that allows these pipes to deflect away from load. However, loads are resisted through the high axial stiffness, resulting in high contact pressures and high thrust forces. • Thermoplastic pipe. Thermoplastic pipes (and corrugated polyethylene in particular) have low flexural and low hoop stiffness. The parametric study showed that the pipe moves away from live loads both through flexure and circumfer- ential shortening. This extra motion of the pipe relative to both concrete and corrugated metal results in higher soil strains and, as a result, a significant portion of the load arches around the pipe. As a result of these three different behaviors, the develop- ment of the SDEs resulted in different expressions for each type of pipe. SDEs were first developed for reinforced concrete pipe and box sections, based on McGrath, Liepins, and Beaver (2005), which established distribution widths for box sections with 0 to 2 feet of soil cover. The 2005 study demonstrated that live loads on rigid structures distribute further longitudinally in culverts with longer spans. This result is logical given that a longer span can undergo larger deformations, allowing the re- distribution. A value of 0.06 times the span was added to the dis- tribution width to account for this effect. Given that the same effect was noted in more deeply buried box culverts, the use of the term is also applied at greater depths. A further benefit of this term is that the proposed distribution equations produce the same result at 2-ft depth, whether computed using the strip equations for depths of 2 feet or less or the proposed equations for depths of 2 feet or greater. 2.3.1.1 Standard Versus LRFD Specifications In developing the proposed design equations, the research team compared the proposed calculation procedures and the methods in the current AASHTO LRFD and Standard Specifi- cations. The comparisons revealed two significant differences between the specifications that must be addressed: • Multiple presence. For a single loaded lane, the LRFD Spec- ification includes a 20% increase in service load to account for the likelihood of overloaded trucks. However, because of a reduced load factor in the LRFD code, the factored loads in the two codes are approximately the same. The code com- parisons below are on the basis of service loads. To provide a common basis for comparison, loads computed using the Standard Specification are increased by 20%. • Impact. The LRFD Specification uses a linearly variable impact factor reducing from 1.33 at a 0 ft depth to 1.00 at a depth of 8 feet, while the Standard Specifications use a stepwise impact factor that decreases from 1.3 at 0 depth of fill to 1.0 at a depth of 3 feet. The two methods are com- pared in Figure 2-16, which shows that the LRFD Speci- fications are more conservative by about 15 to 20% at depths between 2 and 4 feet. This is addressed in the compar- 22 Stiffness Parameter Concrete Corrugated Metal Thermoplastic EI Stiff Flexible Flexible EA Stiff Stiff Flexible Table 2-18. Relative stiffness parameters for typical culvert pipe. Figure 2-16. Comparison of LRFD and standard specification impact factors.

isons below by increasing the calculated value from the Standard Code. 2.3.1.2 Concrete Culverts Concrete culverts fall into three categories: pipe, box sec- tions, and arches. Concrete box sections and arches are all designed with either finite-element analyses or computer analyses of a frame model subjected to an applied pressure distribution. Thus, a simplified design procedure for box and arch sections requires development of an appropriate pres- sure distribution. AASHTO allows design of concrete pipe by two methods: the direct design procedures of the SIDD method and the traditional indirect design procedures. The direct design procedures require a pressure distribution that can be used in conjunction with the Heger distribution devel- oped for earth loads. The indirect procedures require a bed- ding factor that relates the reinforcing requirements to resist live loads in ground to the reinforcing requirements to pass the three-edge bearing test. 2.3.1.3 Metal Culverts Metal culverts fall into the categories of pipe (including ellipses and closed bottom arches) and long-spans. Metal pipe is designed solely for compressive thrust as are long-span culverts. NCHRP Report 473 (McGrath et al., 2002) recom- mended that long-span metal culverts be designed for moment as well as thrust. The research team analyzed the moment data and did not develop an improved recommendation relative to NCHRP Report 473, thus no equation is proposed for moment. 2.3.1.4 Thermoplastic Culverts Although only one category of culvert is made of thermo- plastic materials, these culverts have the most detailed design procedures of the types included in current AASHTO Specifi- cations. Thermoplastic culverts are evaluated for thrust, flex- ure, and deflection. Current AASHTO Specifications for ther- moplastic pipe include polyethylene (PE) and PVC. 2.3.2 Evaluation of Computer Study Data Proposed simplified design methods were developed from the numerical modeling reported in the preceding section. The approach taken was to consider concrete box culverts first, because box sections have flat top slabs which afford an opportunity to evaluate not only the design forces but the ver- tical soil stresses on the top slab. Normal pressures on round or elliptical culverts are more difficult to interpret. The find- ings from box sections were then used as a basis to investigate the other culverts. 2.3.2.1 Concrete Box Sections Vertical Soil Pressure on the Top Slab. Figure 2-17 pres- ents the vertical soil pressure resulting from live load alone on the cross section of 48-in.-span by 48-in.-rise and 96-in.-span by 96-in.-rise box sections at 24 in. depth of fill directly under the load center. The x-axis is in degrees from the center of the box sections so that the edges of the top slab and bottom slabs are at +45 and −45 degrees, respectively. The various curves represent variations in soil type. Figure 2-17 shows that the peak vertical pressures are approximately the same and that soil type does not have any significant affect on the distribution. In the figure, the live load distribution on the 48-in.-span box sections has spread to the edge of the boxes, while it is still clearly contained on the sur- face of the 96-in.-span sections. Distributions for other depths of fill are similar, except at shallower depths of fill the load is within the edges of the top slab for both box dimensions and for deeper fills the load has spread past the edges of the top slab for both box sizes. The mean load was calculated for all spans and depths of fill as the total force applied to the culvert for a 1-inch length directly under the load. The computed values were compared to the applied soil pressure using the current AASHTO distri- bution, calculated using Equation 5 (and shown in Table 2-19): where LLpres is the live load pressure at depth H, psi LL is the total live load applied at surface, lb LL LL w LLDF H l LLDF H pres t t = +( ) +( )• • • ( )2 23 Note: Soil types not differentiated in the figure as there were no substantial differences between the various gradations and compaction levels. Figure 2-17. Vertical live load soil pressure on box sections, 24 in. cover.

wt is the width of tire (or axle length plus tire width at depths where wheels interact), in. lt is the length of tire parallel to span of culvert, in. LLDF is the rate of increase of load spread with increasing depth of fill, taken as 1.15 H is the depth of fill, in. Table 2-19 indicates the following: • Within any size and depth of culvert, the coefficient of vari- ation is small confirming that soil type has very little effect on the load distribution. • The variation between the model load and the calculated live load is about 10% at a depth of 24 inches and increases with depth for the 48-in. culverts and decreases with depth for the 96-in. culverts. • When the total load is considered by adding the weight of soil, the variation between the model live load and the cal- culated live load is a maximum at a depth of 24 inches and decreases to less than 5 percent for all depths 4 feet and greater. The same analysis completed for an LLDF of 1.0 shows a better fit to the data, with the live load delta less than 5 percent except for the deeper fills and the total load delta less than 3 percent at all depths. However, this analysis only evaluates the contact pressure. The final distribution width must also consider the further distribution of load within box sections due to the structural stiffness of the slabs. This is addressed in the following section. Bending Moment in Box Sections. To analyze the bend- ing moments from the computer study, values were taken at midspan, top slab for the maximum positive moments, and at the tip of the top haunch in the sidewall which is the typical lo- cation of the design negative moment. Table 2-20 presents the bending moments averaged for all soil types, and, as for the live load normal pressures, indicates that soil type is not a fac- 24 Depth Model live load Coeff. of variation Calc. live load Live load delta (1) Total load delta (2) 48 in. Span Box Sections in. lbs/in. % lbs/in. % % 12 576 4.4% 520 -9.8% -9.0% 24 382 4.6% 318 -16.7% -13.0% 48 153 3.9% 135 -11.7% -4.8% 72 87 3.6% 74 -14.3% -3.0% 96 57 3.4% 46 -20.1% -2.3% 144 34 3.0% 25 -28.6% -1.4% 96 in. Span Box Sections in. lbs/in. % lbs/in. % % 12 582 4.3% 520 -10.7% -9.0% 24 381 3.7% 318 -16.4% -10.4% 48 190 1.2% 175 -7.7% -2.3% 96 84 3.7% 90 7.1% 0.6% Notes: 1. Live load delta is the variation between the calculated live load and the model live load. 2. Total load delta is the variation between the calculated and model total load, where the total load is the live load plus the weight of soil directly over the box section, assuming the soil prism load and a soil density of 120 pcf. Table 2-19. Comparison of model and calculated load on culvert. Top slab, midspan Sidewall, tip of tophaunch Depth (ft) Bending moment (in.- lb/in.) Coeff. of variation (percent) Bending moment (in.- lb/in.) Coeff. of variation (percent) 48 in. Span Box Sections 12 2414 3.5% -1208 2.8% 24 1383 5.1% -849 5.8% 48 584 4.7% -424 7.6% 72 341 3.5% -250 3.9% 96 227 2.5% -166 4.4% 144 135 2.5% -99 2.8% 96 in. Span Box Sections 12 3251 3.0% -1003 1.9% 24 2060 3.6% -884 3.7% 48 997 1.6% -600 1.6% 96 422 4.2% -301 4.1% Table 2-20. Computer model bending moments.

tor in live load distribution onto box sections. The bending moments versus depth of fill are shown in Figure 2-18. To evaluate the model bending moments relative to current practice, the research team compared the values with the bend- ing moments from the 2D frame analysis program BOXCAR, which was used to generate reinforcing designs for current ASTM and AASHTO box section standards. The BOXCAR analyses were completed for the same load as the computer models (i.e., LRFD design truck [32,000-lb axle load], LRFD impact, and 1.2 multiple presence factor). The LLDF in the BOXCAR analyses was taken as 1.15. Table 2-21 presents the ratios of the BOXCAR bending moments to the computer model bending moments and suggests that current practice is very conservative, especially for the 96 in. span culverts. The comparison is only made for depths of fill 24 inches and greater because AASHTO uses a strip width approach to live load distribution for box sections under depths of fill less than 24 inches which was outside the scope of this project. Table 2-21 indicates that the conservatism increases with the span, which is reasonable as longer span structures are somewhat more flexible and should distribute loads longitu- dinally to a greater extent. There are several issues to be con- sidered in increasing load distribution widths as much as the ratios suggested in Table 2-21. • The analysis assumes equal stiffness in both the span and longitudinal directions of the culvert slabs. Although this is typical, cracking parallel to the culvert span can decrease the distribution length slightly. • Box culverts typically are designed for single lane loadings. Under current distribution widths, the use of a multiple presence factor of 1.2 ensures conservative designs for multi- lane loadings. Increasing the distribution width would result in overlapping lane loads such that multiple lane loadings would control design, thus, the reduction in design load would not be realized to the extent suggested in Table 2-21. • The computer models show the entire vertical live load being carried through vertical shear stresses at the sides of the box sections. There is no vertical reaction on the bottom slabs. We remain uncertain if this would be the case for multiple live load cycles. A distributed load on the bottom slab would cause some increase in the top slab moments. • Precast box sections are rarely longer than 8 feet. Thus, if the load spreading results from internal forces within the culvert, then the spread will be limited. For the time being, the research team is reluctant to increase the distribution for live loads to box sections significantly greater than allowed by the current LRFD specifications. Thus, the research team proposes the following live load distribution equation for box sections: where LLpres is the live load pressure at depth H, psi LL is the total live load applied at surface, lb wt is the width of tire (or axle length plus tire width at depths where wheels interact), in. lt is the length of tire parallel to span of culvert, in. LLDF is the rate of load spread with increasing depth of fill, taken as 1.15 for box culverts in all soil types H is the depth of fill, in. Di is the inside span of the culvert, in. The term 0.06  Di allows for a modest increase in the dis- tribution width. The longitudinal distribution width at 2 ft of fill essentially matches the longitudinal distribution for slabs of box sections given in the current LRFD Specifications. Equation 6 increases the longitudinal distribution length and sets the distribution perpendicular to the span slightly longer than in current practice. LL LL w LLDF H D l LLDF H pres t i t = + +( ) +( )• • • •. ( )0 06 3 25 Figure 2-18. Box culvert bending moments vs. depth of fill. BOXCAR/Table 20 bending moments Depth Top slab, midspan Sidewall, tip of top haunch 48 in. Span Culverts 24 185% 196% 48 171% 151% 96 149% 130% 96 in. Span Culverts 24 328% 424% 48 361% 362% 96 313% 268% Table 2-21. Comparison of 2D and computer model bending moments.

Comparison with Past Practice. The proposed design method for box culverts can be evaluated by calculating the applied vertical load on the sections and comparing it with the applied load using the current AASHTO Standard and LRFD specifications. The comparison is made for service loads; how- ever, to make the design service loads equivalent, the loads com- puted according to the Standard Specifications were increased 20 percent to account for the multiple presence factor included in the LRFD Specifications and was modified for the difference in impact factor between the two specifications. Table 2-22 presents the results of the calculations for an 8-ft-span culvert; results are similar for all sizes. Table 2-22 shows the following: • For depths less than 2 feet of fill, there is no change in the load calculation. The strip width for box section with less than 2 feet of fill was recently addressed by AASHTO. • There is no significant jump in the proposed load at 2 feet where the calculation method changes from the strip load to the distributed load. • At a depth of 2 feet, the proposed load calculation is about 10 percent less than current LRFD because of the inclusion of the term “0.06 Span.” The drop from the current Stan- dard Specification is more significant because the Standard Specification conservatively applies the surface load as a point load. • At depths greater than 2 feet, the proposed calculation method is reduced slightly from the current LRFD. • At depths greater than about 2 feet, the proposed calculation is approximately equal to the Standard Specification for depths from 3 to 4 feet and then increases to about 1.5 times the Standard Specification load at greater depths. A comparison calculation shows that the load on a single lane with multiple presence factor of 1.2 is larger than in a mul- tiple lane loading for all depths at which live loads are signifi- cant. Given that the more liberal distribution of the Standard Specification would result in the two-lane condition control- ling, the research team recommends the narrower distribution in part to keep the single-lane condition controlling the design. 2.3.2.2 Concrete Arch Structures An analysis of concrete arches indicates that the same distri- bution width proposed for box sections can be used for analyz- ing reinforced concrete arch sections. A similar finding was made in NCHRP Report 473 (McGrath et al., 2002) on long- span culverts. 2.3.2.3 Concrete Pipe Thrust. In evaluating live load distribution onto con- crete pipe, the research team first looked at the peak and springline thrusts. Figures 2-19 through 2-21 compare the model values of springline and peak thrusts with the thrusts based on Equation 6. The curve “Calc’d-Crown” was calcu- lated from Equation 6 using a depth of fill to the pipe crown and the curve “Calc’d-Springline” was computed using the depth of fill to the springline. 26 Current Ratios Depth* (ft) Proposed (lb/ft) LRFD (lb/ft) Modified Stnd** (lb/ft) Proposed /LRFD Proposed / Modified Stnd 1 5523 5523 5523 1.00 1.00 1.001 5523 5523 5523 1.00 1.00 1.999 5347 5347 5347 1.00 1.00 2 5387 6038 6844 0.89 0.79 2.001 5385 6036 6840 0.89 0.79 2.999 4139 4528 4012 0.91 1.03 3 4138 4526 4412 0.91 0.94 4 3510 3647 3442 0.96 1.02 5 3105 3216 2675 0.97 1.16 6 2763 2854 1920 0.97 1.44 7 2223 2291 1431 0.97 1.55 8 1765 1815 1097 0.97 1.61 9 1485 1525 949 0.97 1.56 10 1268 1300 830 0.98 1.53 11 1096 1121 732 0.98 1.50 12 978 1000 650 0.98 1.50 *Incremental depths included to show steps in load due to stepwise function for Standard Specification impact **Modified to negate differences in multiple presence and impact (= actual Standard * 1.2* (LRFD impact/Standard impact) Table 2-22. Comparison of proposed and current live loads on 8-ft-span box culvert.

Figures 2-19 through 2-21 show that peak thrust is reason- ably predicted for all spans and depths of fill by using Equa- tion 6 with the depth of fill taken to the top of the pipe. The springline thrust is reasonably predicted for the 12-in. and 24-in.-span pipe using Equation 6, but is somewhat conser- vative for the 48-in. and 96-in.-span pipe under shallow fill. These figures also show that the soil type has very little influ- ence on the load distribution for concrete pipe, as was the case with box sections. The research team investigated the effect of improving the quality of prediction for peak thrust by modifying the LLDF based on diameter and developed the following equations: where LLDFcp is the live load distribution factor for concrete pipe Di is the inside span of the culvert, in. These equations increase the LLDF for pipe diameters larger than 24 inches, giving a value of 1.35 for 48-in.-diameter pipe For D in LLDFi cp> =96 1 75 6. ( ) For <24 96 0 00833 0 95 5in D in LLDF Di cp i≤ = +. . ( )• For D in LLDFi cp≤ =24 1 15 4. ( ) 27 12 in. Span 24 in. Span 96 in. Span48 in. Span Figure 2-19. Comparison of concrete pipe model thrust with eqn (6)-based values. 96 in. Span48 in. Span Figure 2-20. Comparison of proposed SDE with model peak thrust in concrete pipe.

28 Figure 2-21. Ratios of calculated to computer model thrusts. Figure 2-22. Bedding factors computed from computer study. and 1.75 for pipe with diameters of 96 inches and larger. The upper limitation on LLDF is imposed because there is prior successful experience with using a distribution factor of 1.75 for design. Figure 2-20 compares the model data with the modified form of Equation 6 for 48-in.-diameter concrete pipe and Figure 2-21 does the same for 96-in.-diameter concrete pipe. The figure shows a better match—ratios of calculated thrust from the modified equation to model thrust average 1.00 with a standard deviation of 0.22. This compares with an average of 1.14 and a standard deviation of 0.29 for thrusts calculated with equations above. The ratios between the proposed model-predicted peak thrusts and the computer model val- ues are presented in Figure 2-21. Bedding Factor. Once the live load was determined using the equation above, a bedding factor was required to calculate the required pipe capacity in the indirect design method. The philosophy of indirect design is to size reinforcement to pass the three-edge bearing test under a load that produces the same peak positive bending moment as imposed by the field conditions. The bedding factor is the ratio of the field load to the three-edge bearing load that produces the same bending moment. The bedding factor can thus be calculated as the ratio of the three-edge bearing moment due to the field load to the actual field moment. For earth loads, the bedding fac- tors developed for the current AASHTO Specifications com- puted a reduced effective field moment to account for the presence of lateral thrust. The approach is presented in ACPA’s Concrete Pipe Design Manual (1998). In this study, the research team did not consider modifications due to thrust be- cause the location of peak thrust was not necessarily at the crown or invert where the peak moment would occur, and the live load also generated very little thrust as a result of lateral soil pressure. Thus, the research team computed the bedding factor using the simple equations: where Bf is the bedding factor MTEB is the bending moment in the three-edge bearing test, in.-lb/in. MFLD is the field bending moment after reduction for com- pressive thrust, in.-lb/in. NFLD is the peak thrust in the field, lb/in. Di is the inside pipe diameter, in. t is the pipe wall thickness, in. Using this approach, the research team computed the bed- ding factors shown in Figure 2-22 using the peak thrusts and peak positive bending moments from the computer study. In completing these calculations, the research team did not con- sider installation type as a factor because the peak moments occur in the crown area which is not significantly affected by installation condition. This is consistent with current AASHTO practice. Figure 2-22 shows a trend of bedding fac- tors increasing as the depth of fill goes from 12 to 24 inches and then decreasing with further increases in depth of fill. The general trend of decreasing bedding factor with in- creasing depth of fill is the reverse of the trend in the current LRFD live load bedding factors for concrete pipe, which were developed based on 2D analyses. However, both the computer model and the current AASHTO LRFD trend toward a bed- ding factor of 2.2 for deeply buried pipe. The reason for this change in trend is simply in the method of calculation. In this study, the peak compressive thrust was used as the basis for computing the bedding factor, because there was less variabil- ity in predicting the peak thrust than the springline thrust. M N D tTEB FLD i= +( )0 318 8. ( )• B M M f TEB FLD = ( )7

The current AASHTO live load bedding factors were com- puted using the springline thrusts. Given that the AASHTO analyses were based on 2D modeling, the springline thrusts would be higher than the 3D values, thus very different bed- ding factors from current practice would be expected, regard- less of the method of calculation. The research team proposes a constant bedding factor of 2.2 for all installation conditions, diameters, and depths of fill. This is a lower bound for all conditions and, as seen in the next section, still results in a significantly reduced load on concrete pipe due to the change in load calculation. Comparison with Past and Current Practice. The vari- ation in required D-load to carry live load is quite variable between the proposed method and current practice. For con- crete pipe the research team includes a method from the ACPA Handbook to complete the comparison. Variations in the inputs for the procedure are summarized in Table 2-23. The results of the calculations are presented in Table 2-24 and Table 2-25 for 4-ft-diameter and 12-ft-diameter pipe, respectively. Table 2-24 and Table 2-25 show a wide discrepancy in cur- rent practice. The Standard Specification is generally the most conservative at shallow depths due to the concentrated surface load but is less conservative with depth due to the larger LLDF of 1.75. The ACPA method is the least conser- vative, but represents a procedure that has been used for many years. 2.3.2.4 Corrugated Metal Pipe Thrust. Initial calculations for evaluating peak thrust in corrugated metal pipe were completed using the live load pressure computed from Equation 6, multiplied by one-half the smaller of the load spread parallel to the span direction or the pipe diameter. Peak thrust is calculated as where Npeak is the estimated peak thrust, lb/in. LLpres is the nominal live load pressure, psi lt is the length of tire parallel to span of culvert, in. LLDF is the rate of load spread with increasing depth of fill, 1.15 N LL D l LLDF Hpeak pres i t= +( ){ }• • •. min ; ( )0 5 9 29 Calculation Method Parameter Proposed LRFD Standard ACPA Footprint (in x in) 10 x 20 10 x 20 0 x 0 10 x 20 LLDF 1.15 to 1.75 1.15 (granular soils) 1.75 1.75 Elevation of calculation top of pipe top of pipe top of pipe 0.75 iD below top of pipe Bedding factor 2.2 1.1 to 2.2 1.1 to 2.2 1.7 (user option) Table 2-23. Input variables for calculating live load on pipe. Current Practice Ratios Depth (ft) Proposed ( fB = 2.2) (lb/ft/ft) LRFD (lb/ft/ft) Modified Stnd* (lb/ft/ft) Modified ACPA Hndbk* ( fB = 1.7) (lb/ft/ft) Proposed/ LRFD Proposed/ Modified Standard Proposed/ Modified ACPA Hndbk 1 863 1464 2357 537 0.59 0.37 1.61 2 591 755 856 492 0.78 0.69 1.20 3 437 514 462 400 0.85 0.95 1.09 4 296 369 270 266 0.80 1.10 1.11 5 213 268 183 186 0.79 1.16 1.14 6 160 203 131 136 0.79 1.22 1.17 7 123 157 99 102 0.78 1.25 1.20 8 97 125 76 80 0.78 1.28 1.21 9 81 105 65 71 0.77 1.25 1.14 10 70 89 56 64 0.79 1.24 1.10 11 62 77 50 58 0.81 1.23 1.08 12 56 69 44 42 0.81 1.26 1.09 *Standard and ACPA modified to match impact and multiple presence of LRFD Table 2-24. Comparison of proposed live load D-load on 4-ft-diameter concrete pipe with past practice.

H is the depth of fill, in. Di is the inside span of the culvert, in. Figures 2-23 and 2-24 show reasonable agreement between the computer model thrusts and the Equation 6 predicted thrusts, except for the large-diameter pipe under shallow con- ditions. These pipes, with low bending stiffness, probably develop higher thrusts to carry the load as a membrane force rather than in flexure. Also, analysis of the live load spread versus depth [lt + LLDF  H] indicates that for these pipes the live load has not spread to the width of the crown, thus, there is probably a concentration effect. The following factor was introduced to account for this effect and a slight modification for the 12-in.-diameter pipes. 30 2.27 1.62 1.27 1.23 1.19 1.18 1.17 1.19 1.14 1.10 1.08 227 154 129 110 95 83 73 64 56 49 44 39 666 387 269 203 149 119 96 86 81 76 72 67 1080 422 217 164 120 97 79 73 64 55 49 43 100 95 101 89 80 70 62 54 49 44 41 37 0.34 0.40 0.48 0.54 0.64 0.70 0.76 0.74 0.69 0.64 0.61 0.58 0.21 0.36 0.59 0.67 0.79 0.85 0.92 0.87 0.88 0.89 0.89 0.90 1.05 Current Practice Ratios Depth (ft) Proposed ( fB = 2.2) (lb/ft/ft) LRFD (lb/ft/ft) Modified Stnd* (lb/ft/ft) Modified ACPA Hndbk* ( fB = 1.7) (lb/ft/ft) Proposed/ LRFD Proposed/ Modified Standard Proposed/ Modified ACPA Hndbk 1 2 3 4 5 6 7 8 9 10 11 12 *Standard and ACPA modified to match impact and multiple presence of LRFD Table 2-25. Comparison of proposed live load D-load on 12-ft diameter concrete pipe with past practice. 12 in. 24 in. 96 in.48 in. Figure 2-23. Comparison of thrusts between LRFD and computer model for metal pipe.

31 Figure 2-24. Peak thrust ratios for metal pipe— proposed design equation. where LLpres  {0.5  min(Di;lt + LLDF  H)} is the peak thrust prior to correction, from Eq. 12 lt is the length of tire parallel to span of culvert, in. LLDF is the rate of load spread with increasing depth of fill, taken as 1.15 H is the depth of fill, in. Di is the inside span of the culvert, in. F1 is the correction factor F1−Lim is the limit value for F1 With this modification, the ratios of the calculated thrust to the computer model thrust are shown in Figure 2-24. The ratios have a mean value of 1.15 and a standard deviation F D FLim t Lim1 1 15 1 0 12 − − = >subject to: . ( ) F D l LLDF H Fi t Lim1 10 75 11= + ≥ − . ( )• • N F LL D l LLDF Hpeak pres i t= +( ){ }1 0 5 10• • • •. min ; ( ) of 0.21. The only values notably less than 1.0 are the small- diameter pipe under deep fills, where the live load is a trivial component of the total load. Comparison with Current Practice. Thrust forces com- puted with the proposed equation are compared with thrust forces computed using the procedures of the LRFD and Stan- dard Specifications in Table 2-26 and Table 2-27. Table 2-26 and Table 2-27 indicate that metal culverts develop a larger thrust force under shallow fills than predicted by current practice. This becomes more pronounced as the pipe diameter increases. For 4-ft-diameter pipe, this thrust increase exists only at the shallowest depth of fill, 1 ft. For 12-ft- diameter pipe, however, the increase runs to a depth of 6 ft. This finding is consistent with NCHRP Report 473 (McGrath et al., 2002) results for long-span metal culverts. 2.3.2.5 Corrugated Metal Arches Predicted thrusts computed using the live load pressure of Equation 6 are compared with the model thrusts in Figure 2-25. Similar to metal pipe, the thrusts show a high maximum thrust at shallow depths relative to that predicted by Equation 6. Thrusts in corrugated metal arches were studied in NCHRP Report 473—a modifying coefficient using the same form was developed: where Fm.arch is the thrust modifier for long-span metal arches Span is the culvert span, in. LLDF is the live load distribution factor, 1.15 H is the depth of fill, in. wt is the width of tire (or axle length plus tire width at depths where wheels interact), in. F Span w LLDF H Span m arch t . . . ( ) • • • = + + 0 54 0 03 13 6122 2847 2019 1317 960 726 564 447 376 321 277 247 4392 3019 2113 1342 977 738 573 454 381 325 280 250 7070 3422 1681 983 669 479 357 275 238 208 182 162 1.39 0.94 0.96 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.87 0.83 1.20 1.34 1.44 1.51 1.58 1.63 1.58 1.55 1.52 1.52 Current Ratios Depth (ft) Thrust from Proposed SDE (lb/ft) LRFD (lb/ft) Modified Standard (lb/ft) Proposed/ LRFD Proposed/ Modified Standard 1 2 3 4 5 6 7 8 9 10 11 12 Table 2-26. Comparison of proposed and existing equations for live load thrust in 4-ft-diameter metal pipe.

The effect of soil type was expected because thermoplastic pipe has low hoop stiffness which increases the arching of load around a pipe in stiff soils. Backfill stiffness is a significant factor in earth load arching. Further analysis showed that the thrust in thermoplastic pipe could be predicted from Equa- tion 13 with the following correction for the soil effect: where Fth is the correction factor for effect of soil type on thrust SH is the hoop stiffness factor computed in accordance with AASHTO LRFD Equation 12.12.3.4-4, assuming that the vertical confining stress is 0.15 ksf. and using the short-term modulus of elasticity of the pipe material. (Note: the assumption of the vertical confining stress is somewhat arbitrary, but as the soil stresses due to live F S th H = + 0 95 1 0 6 16 . . ( ) N F F LL D l LLDF Hpeak th pres i t= +( ){ }• • • • •. min ;1 0 5 ( )15 32 Current Ratios Depth (ft) Thrust from Proposed SDE (lb/ft) LRFD (lb/ft) Modified Standard (lb/ft) Proposed/ LRFD Proposed/ Modified Standard 1 15874 4392 7070 3.61 2.25 2 7340 3019 3422 2.43 2.15 3 4169 2263 2206 1.84 1.89 4 2853 1823 1721 1.57 1.66 5 2087 1608 1463 1.30 1.43 6 1582 1427 1260 1.11 1.26 7 1232 1272 1073 0.97 1.15 8 1092 1138 823 0.96 1.33 9 1025 1066 712 0.96 1.44 10 939 975 623 0.96 1.51 11 812 841 548 0.97 1.48 12 725 750 487 0.97 1.49 Table 2-27. Comparison of proposed and existing equations for live load thrust in 12-ft-diameter metal pipe. Figure 2-25. Comparison of SDE and computer model thrusts for corrugated metal arches. Figure 2-26. Comparison of proposed and model thrusts for long-span metal culverts. The proposed design equation is The peak thrusts computed with Equation 17 were compared to model thrusts. The ratios of the computed values to the model values are shown in Figure 2-26. The mean value of the ratios is 1.10 with a standard deviation of 0.15. The two values that plot below 0.9 were culverts embedded in clay backfill which is not allowed by AASHTO for long-span culverts. 2.3.2.6 Thermoplastic Pipe (Profile Wall) Thrust. Analysis of thrust for thermoplastic pipe (profile wall), using Equation 13, is presented in Figures 2-27 through 2-30. The comparison of the calculated values with the com- puter model thrusts is similar to the other types of pipe ex- cept that there is a distinct effect of soil type as shown by the vertical scatter within each data set for depth and diameter. N F LL D l LLpeak m arch m arch pres i t. . . • • •. min ;= +0 5 DF H• ( )( ){ } 14

load are variable around the pipe, this provides an ade- quate approximation.) Applying this correction, the ratio of the predicted thrusts to the computer model thrusts is presented in Figure 2-28. The mean value of the ratio is 1.21 with a standard deviation of 0.25. Comparison of thrust due to combined earth plus live load is shown in Figure 2-29. In this comparison, long-term earth load was calculated using the current AASHTO procedures for ther- moplastic pipe. The long-term earth load is substantially less than the short-term earth load for pipe embedded in SW95 soils, thus the comparison is conservative. Figure 2-29 shows an average ratio of calculated to computer model total thrust of 1.14 with a standard deviation of 0.14. The figure also shows that the variation reduces significantly for depths of fill greater than 24 inches. Current AASHTO specifications require at least 2 feet of cover for thermoplastic pipe. Bending Moment. The live load bending moment is required to calculate strains required for thermoplastic pipe design. The research team computed non-dimensional bend- ing moment coefficients as where cm is a dimensionless moment coefficient M is the bending moment, in.-lb/in. c M T R m = ( )• ( )17 33 12 in. 24 in. 60 in.48 in. Figure 2-27. Comparison of SDE and computer model thrusts for thermoplastic pipe. Ratio of Calculated to Computer Model Thrust Ca lc ul at ed /M od el Figure 2-28. Ratio of calculated (Eqn (18)) to computer model peak thrust for thermoplastic pipe.

T is the peak thrust, lb/in. R is the mean pipe radius. in. The moment coefficients are plotted by diameter in Fig- ure 2-30 which shows some variation as a result of diameter (symbol type) and soil type (variations within symbols of the same type). The variation due to diameter is chiefly in the smaller diameters under shallow fill. The correction factors in Eq. 21 and Table 2-28 were developed for the two effects. Computing a revised moment coefficient as gives the data plot in Figure 2-31. The data is fairly consoli- dated and a simple upper bound curve was plotted with the equation c c F Fm m th th.mod . . ( )= 3 4 19 F D Fth i th3 3 48 1. . (= < in. constrained by: 0.65 < 18) The upper bound equation is exceeded for 48-in.- and 60-in.-diameter pipe under 12 inches of fill; however, this is far below the minimum allowable depth of fill for these products. Combining the upper bound curve with the coefficients gives the equation for computing bending moment: where Mth is the design bending moment, in.-lb/in. Npeak is peak thrust computed from Eq. 18, lb/in. R is the radius to centroid of pipe wall, in. H is the depth of fill to top of pipe, in. F3.th is the coefficient for diameter from Eq. 21 F4.th is the coefficient for diameter from Table 2-28 M N R H F F th peak th th = ( ) −[ ]• • • • . . ( . . 0 045 0 00032 3 4 21) UP Hcm = − ( )0 045 0 00032 20. . ( )• in. 34 Figure 2-30. Dimensionless moment coefficients from computer model. Ratio of Total Calculated to Computer Model Thrust Ca lc ul at ed /M od el Figure 2-29. Thermoplastic pipe—ratio of calculated to model thrust for earth load plus peak live load thrust.

Using Eq. 24 bending moments were calculated using the peak thrusts from Eq. 18. The ratios of these calculated mo- ments to the computer model moments are presented in Fig- ure 2-32. The data shows more scatter than the thrust predic- tions but indicates that the proposed equation is generally conservative. The average ratio is 1.35 with a standard devia- tion of 0.29. Comparison with Current Practice. Thrusts computed with the proposed equations are compared with current AASHTO practice in Table 2-29 for 24-in.-diameter pipe in SW95 soil (called Sn95 in the LRFD Specifications) and in Table 2-30 for 24-in.-diameter pipe in ML85 soil (called Si85 in the LRFD Specifications). The tables show a substantial live load reduction in high- quality backfill where the thermoplastic PE pipe with low hoop and flexural stiffness deforms and allows the soil to carry load around the pipe. In weaker soils, such as ML85, the load reduc- tion is modest relative to the LRFD Specifications and increases relative to the Standard Specifications. 2.3.2.7 PVC Pipe The PVC pipe modeled in the study was solid wall pipe that is not representative of PVC pipe that would be installed in typ- ical highway culvert installations (which are typically profile wall pipe). Pipe stiffnesses are in the range of 500 psi versus about 50 psi for culvert pipe. Analysis of the data shows that the 35 Soil Type 4.thF SW95 1.00 SW90, ML95 0.75 SW85, ML90, CL95 0.50 ML85, CL90 0.38 CL85 0.25 Table 2-28. Soil correction factor, F4.th, for bending moment in thermoplastic pipe. Figure 2-31. Corrected moment coefficient with upper bound curve, thermoplastic pipe. Figure 2-32. Calculated to model moment ratios vs. fill depth for thermoplastic pipe.

36 2457 1091 602 388 282 213 166 131 110 94 81 72 4392 1927 1057 671 489 369 286 227 191 162 140 125 7132 1880 696 422 298 222 172 137 119 103 91 82 0.56 0.57 0.57 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.34 0.58 0.86 0.92 0.95 0.96 0.97 0.96 0.93 0.91 0.89 0.88 Current Ratios Depth (ft) Thrust from Proposed SDE (lb/ft) LRFD (lb/ft) Modified Standard (lb/ft) Proposed/ LRFD Proposed/ 1.2 x Standard 1 2 3 4 5 6 7 8 9 10 11 12 Table 2-29. Comparison of proposed thermoplastic thrust with current AASHTO specifications for SW95 soil. 3645 1618 893 575 419 317 246 195 164 140 121 107 0.83 0.84 0.84 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.51 0.86 1.28 1.36 1.41 1.43 1.43 1.43 1.38 1.36 1.33 1.31 4392 1927 1057 671 489 369 286 227 191 162 140 125 7132 1880 696 422 298 222 172 137 119 103 91 82 Current Ratios Depth (ft) Thrust from Proposed SDE (lb/ft) LRFD (lb/ft) Modified Standard (lb/ft) Proposed/ LRFD Proposed/ 1.2 x Standard 1 2 3 4 5 6 7 8 9 10 11 12 Table 2-30. Comparison of proposed thermoplastic thrust with current AASHTO specification for ML85 soil. pipe can be analyzed for load using Eq. 6. However, this may not be appropriate for culvert pipe, which can be analyzed with the same procedures as proposed for profile wall pipe in the previous section. Figure 2-33 shows the comparison of the calculated to model thrusts for the PVC pipe. The model ratios average 1.09 with a standard deviation of 0.23, which is high but tolerable for a solid wall pipe with very low hoop stresses. 2.4 Effect of SDEs on Culvert Forces The research team calculated and compared the critical structural responses for the following culvert types and depth- span combinations: • Concrete box—6 combinations • Concrete pipe—100 combinations • Corrugated metal pipe—42 combinations • Thermoplastic (profile wall)—80 combinations • Metal arch—6 combinations • Concrete arch—8 combinations The research team provides direct comparison of the struc- tural responses generated under the AASHTO Standard Spec- ification, AASHTO LRFD Specification, and the proposed SDEs. (Appendix D.1 contains MathCAD templates illustrat- ing Standard, LRFD, and Proposed calculations for all struc- ture types; Appendix D.2 lists parametric study results.) 2.4.1 Live Load Equations from AASHTO Standard and LRFD Codes This section summarizes the live load equations from the AASHTO Standard and LRFD codes. These equations were used to compute the live loads for all culvert types in the comparisons described below. The proposed live load

equations for each culvert type are presented in the appro- priate section. 2.4.1.1 AASHTO Standard The live load equations for the AASHTO Standard Speci- fication are LHS20 is the wheel load from the HS20 load case, 16,000 lb: where LL is the live load force, lb Determine the wheel interaction depth: where Hint is the wheel interaction depth, ft sw is the wheel spacing, 6 ft LLDFs is the live load distribution factor, 1.75 Determine the live load area and pressure where H is the culvert depth, ft ALL is the live load area, sf WLL is the live load pressure, psf Determine the impact fraction W LL ALL LL= 2 27i ( ) For H H A s LLDF H LLDF HLL w s s≥ = +( ) ( )int • • • ( )26 W LL ALL LL= ( )25 For H H A LLDF H LLDF HLL s s< = ( ) ( )int • • • ( )24 H s LLDF w s int ( )= 23 LL LHS= 20 22( ) where I is the impact fraction (maximum 30 percent) The service live load is computed from where Di is the inside diameter or span of the culvert, inches 2.4.1.2 AASHTO LRFD The live load equations for the AASHTO LRFD Specifica- tion are where LHS20 is the wheel load from the HS20 load case, 16,000 lb LL is the live load force, lb Determine the wheel interaction depth where Hint is the wheel interaction depth, ft sw is the wheel spacing, 6 ft wt is the tire patch width, 20 in. LLDFl is the live load distribution factor, 1.15 Determine the live load area and pressure where H is the culvert depth, ft lt is the tire patch length, 10 in. ALL is the live load area, sf WLL is the live load pressure, psf W LL ALL LL= 2 35i ( ) For H H A w s LLDF H l LL LL t w l t ≥ = + +⎛⎝⎜ ⎞⎠⎟ + int • • 12 12 DF Hl • ( ) ⎛⎝⎜ ⎞⎠⎟ 34 W LL ALL LL= ( )33 For H H A w LLDF H l LLDFLL t l t l< = + ⎛⎝⎜ ⎞⎠⎟ +int • •12 12 • ( )H ⎛⎝⎜ ⎞⎠⎟ 32 H s w LLDF w t l int ( )= − 12 31 LL LHS= 20 30( ) W I W D LLDF HL LL i s= +( ) ( )1 12 29• • •min , ( ) For 0 1 0 3 28 1 2 0 2 2 3 0 1 3 0 ≤ ≤ = ≤ ≤ = ≤ ≤ = ≤ = H I H I H I H I . ( ) . . 37 Figure 2-33. Ratios of model thrust to predicted thrust in PVC pipe using Equation 6.

38 AASHTO LRFD AASHTO Standard Simplified Design EquationCulvert Type Dead Load Live Load Dead Load Live Load Dead Load Live Load Concrete Pipe 1.3 1.75 1.3 2.17 1.3 1.75 Concrete Box 1.35 1.75 1.35 2.17 1.35 1.75 Corrugated Metal Pipe 1.95 1.75 1.95 2.17 1.95 1.75 Thermoplastic (Profile Wall) 1.95 1.75 1.95 2.17 1.95 1.75 Metal Arch 1.95 1.75 1.95 2.17 1.95 1.75 Concrete Arch 1.3 1.75 1.3 2.17 1.3 1.75 Table 2-31. Dead load and live load factors, d and l. Determine the governing load length where Lt.gov is the governing load length, ft Determine the dynamic load allowance where IM is the dynamic load allowance Determine the service live load where MPF is the multiple presence factor, 1.2 Di is the inside diameter or span of the culvert, in. 2.4.2 Proposed SDEs 2.4.2.1 Live Loads The proposed live load equations differ for each culvert type, so they are presented in sections 2.4.3.1 through 2.4.8.1. 2.4.2.2 Dead Loads Dead loads vary according to the culvert type, so they are presented in sections 2.4.3.2 through 2.4.8.2. 2.4.2.3 Service and Factored Loads Service and factored loads used in the design comparisons were TL DL LLs = + ( )41 W MPF IM W D LL LL i t gov= +( ) ( )• • • min , ( ).1 12 40 For H ft IM≥ =8 0 39( ) For H ft IM H < = −⎛⎝⎜ ⎞⎠⎟8 33 1 8 100 38• ( ) For H L l LLDF Ht gov t l≥ = +0 833 12 37. ( ). • For H L lt gov t< =0 833 12 36. ( ). where DL is the total dead load LL is the total live load TLS is the total service load TLF,STD is the factored load for the AASHTO Standard Specification TLF,LRFD is the factored load for the AASHTO LRFD Speci- fication TLF,SDE is the factored load for the proposed specification γd is the dead load factor, from Table 2-31 γl is the live load factor, from Table 2-31 Table 2-31 contains the dead load and live load factors referred to above. 2.4.3 Concrete Box Comparison Concrete box design calculations were done using the direct design method, using the software BOXCAR to calculate struc- tural responses. BOXCAR (BOXCAR, 2000) is a four member frame program with the stiffness matrix modified to account for the haunch stiffness. For live load input to BOXCAR, the same total live load was used for each case, but with live load distribution areas. These distribution areas were determined from AASHTO Standard, AASHTO LRFD, and SDEs. 2.4.3.1 Live Load Equations The proposed live load equations used for concrete box design calculations are as presented in Section 2.4.1.2, except the interaction depth, live load area, and pressure are as follows. Determine the wheel interaction depth H s w D LLDF w t i l int . ( )= − − 12 0 06 12 45 TL DL LLF SDE d l, • • ( )= +γ γ 44 TL DL LLF LRFD d l, • • ( )= +γ γ 43 TL DL LLF STD d l, • • ( )= +γ γ 42

where Hint is the wheel interaction depth, ft sw is the wheel spacing, 6 ft wt is the tire patch width, 20 in. LLDFl is the live load distribution factor, 1.15 Di is the inside span of the culvert, in. where H is the culvert depth, ft wt is the tire patch width, 20 in. lt is the tire patch length, 10 in. LLDFl is the AASHTO LRFD live load distribution fac- tor, 1.15 Figure 2-34 compares the variation of live load with depth for concrete boxes, for the Standard, LRFD, and proposed For H H A w s LLDF H DLL t w l i≥ = + + + ⎛⎝⎜ ⎞int • •.12 0 06 12⎠⎟ + ⎛⎝⎜ ⎞⎠⎟• • ( ) l LLDF Ht l 12 47 For H H A w LLDF H DLL t l i< = + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 12 l LLDF Ht l 12 46+ ⎛⎝⎜ ⎞⎠⎟• ( ) 39 Figure 2-34. Live load variation with depth for concrete box culverts. Parameter Value Soil density 120 pcf Minimum lateral pressure coefficient 0.25 Maximum lateral pressure coefficient 0.5 Installation type Embankment/ Compacted Soil-structure interaction factor 1.083 Fluid density 62.5 pcf Table 2-32. BOXCAR dead load parameters for box culvert parametric study. SDE. The SDE distribution starts out wider than LRFD, but increases width with depth at the same rate. In BOXCAR, the details of live load input were 1. Use live load option “Other” in Boxcar-Design (page 4) and specify 1 wheel 2. Set the LLDF to 0.00001 (to avoid dividing by zero) 3. Set the tire footprint to the area defined by the denomina- tor of the appropriate live load equation above 2.4.3.2 Dead Load Equations The dead loads were computed by BOXCAR from the parameters listed in Table 2-32. 2.4.3.3 Geometry and Material Properties Table 2-33 lists the concrete box geometry and material properties used in the design comparison.

2.4.3.4 Comparison of Standard, LRFD, and Proposed Figures 2-35 through 2-38 compare the structural responses for reinforced concrete box culverts. These four figures, and comparison figures for all culvert types, have a common for- mat. Each figure contains two comparisons: AASHTO Stan- dard versus proposed SDEs, and AASHTO LRFD versus pro- posed SDEs. The figures are constructed with the AASHTO value as the abscissa and the SDE as the ordinate. Each figure has a thick line for abscissa values equal to ordinate values. The region below this line, where SDE values are less than the AASHTO values, is shaded one gray tone, to distinguish from the region above this line, where SDE values are greater than AASHTO values, shaded in a different gray tone. All culvert depths are plotted in one figure. In some graphs, the data pairs created by plotting one data point for the (Stnd,SDE) combi- nation and one data point for the (LRFD,SDE) combination are easy to discern (for example Figure 2-35). Comparing the location of these data pairs provides insight into the differ- ences between the Standard and LRFD values. Top-slab middle moments range from about 30 kip-in/ft to about 180 kip-in/ft, while bottom-slab middle moments range from about 30 kip-in/ft to about 140 kip-in/ft. About two- thirds of the comparisons have SDE values slightly greater than the AASHTO values. Top-slab and bottom-slab maximum shear comparisons are similar. For both moment and shear comparisons, the SDEs, on average, produce slightly higher moments and shears. The research team expects similar results for larger-span concrete box culverts. 40 Figure 2-35. Top middle moment comparison for concrete boxes. Span (ft) Rise (ft) Thickness (in) Haunch (in) Cover (ft) Reinforcing Yield Stress (psi) Concrete Strength (psi) Concrete Unit Weight (pcf) 4 4 5 5 2 65,000 5,000 150 4 4 5 5 4 65,000 5,000 150 4 4 5 5 8 65,000 5,000 150 8 8 9 9 2 65,000 5,000 150 8 8 9 9 4 65,000 5,000 150 8 8 9 9 8 65,000 5,000 150 Table 2-33. Concrete box culvert properties for parametric study.

41 Figure 2-36. Bottom middle moment for concrete boxes. Figure 2-37. Top slab shear comparison for concrete boxes.

42 Figure 2-38. Bottom slab shear comparison for concrete boxes. 2.4.4 Concrete Pipe Comparison Concrete pipe design comparisons were done using the direct design method. The determination of earth loads and live load pressure distributions on the structure were selected by the bedding and installation conditions. The thrust, moment and shear analysis for each installation type was performed using the SIDD methods (ASCE, 2000). 2.4.4.1 Live Load Equations The proposed live load equations used for concrete pipe design calculations are similar to the AASHTO LRFD equa- tions presented in Section 2.4.1.2, except as follows: where LLDFcp is the live load distribution factor for concrete pipe Determine the wheel interaction depth Determine the live load area and pressure H s w D LLDF w t i cp int . ( )= − − 12 0 06 12 51 For D in LLDFi cp> =96 1 75 50. ( ) For 24 96 0 00833 0 95 49in D in LLDF Di cp i< ≤ = +. . ( )• For D in LLDFi cp≥ =24 1 15 48. ( ) Determine the governing load length where H is the culvert depth, ft wt is the tire patch width, 20 in lt is the tire patch length, 10 in. Di is the inside span of the culvert, in. Lt.gov is the governing load length Figure 2-39 compares the variation of live load with depth for concrete pipe, for the Standard, LRFD and proposed SDE. The SDE distribution starts out wider than LRFD, and increases in width with depth at a faster rate. For H L l LLDF Ht gov t cp≥ = +0 833 12 55. ( ). • For H L lt gov t< =0 833 12 54. ( ). For H H A w s LLDF H DLL t w cp i≥ = + + + ⎛⎝⎜int • •.12 0 06 12 ⎞⎠⎟ + ⎛⎝⎜ ⎞⎠⎟• • ( ) l LLDF Ht cp 12 53 For H H A w LLDF H DLL t cp i< = + + ⎛⎝⎜ ⎞⎠⎟int • •.12 0 06 12 • • ( ) l LLDF Ht cp 12 52+ ⎛⎝⎜ ⎞⎠⎟

2.4.4.2 Dead Load Equations Concrete pipe dead loads were computed from where H is the culvert crown depth, in. Di is the culvert inside diameter, in. Do is the culvert outside diameter, in. Wp is the pipe dead load, lb/ft We is the earth dead load, lb/ft Wf is the fluid dead load, lb/ft wc, we, and wf are the concrete unit weight, earth unit weight, and fluid unit weight, respectively. Fe is the soil-structure interaction factor, 1.35 The total service dead load is 2.4.4.3 Moment, Thrust and Shear Calculations To find the design moments, thrusts and shear, the SIDD nondimensional coefficients Cmi, Cni, and Cvi were used for determining the moment thrust and shear, respectively, at governing locations at the crown, invert, springline and at the critical locations for shear in the invert and crown re- DL W W WP E F= + + ( )59 W w D f f i = ⎛⎝⎜ ⎞⎠⎟• • ( ) π 4 12 58 2 W F w H D De e e o o= +( )• • • •. ( )0 0089 12 57 W w D Dp c o i= −( )• • ( )π 4 144 562 2 gions. Calculations were done for the four SIDD installa- tion types. where Wi takes the values WP, WE, WF and WL. Additional details may be found in ASCE 2000. 2.4.4.4 Comparison of Standard, LRFD, and Proposed SDEs Figures 2-40 through 2-43 compare the structural responses for reinforced concrete pipe. These figures, and comparison figures for all culvert types, have a common format, described in Section 2.4.3.4. All four figures for RCP are similar. For low values, corresponding to small-diameter culverts, the SDE val- ues are similar to or slightly greater than LRFD or Standard values. For higher values, corresponding to larger diameter culverts, the SDE values are similar to or less than (sometime significantly less than) AASHTO values. In all four figures, one group of four data points has SDE values substantially below the AASHTO values. This data is for a 48-in.-diameter RCP at 1-ft depth of burial and com- pares the AASHTO standard to proposed SDE. The mo- ment or shear values plot so far off the 1:1 because the Stan- dard Specifications treat the live load as a point load for burial depths of 1 foot. V C Wi vi i= ∑ • ( )62 N C Wi mi i= ∑ • ( )61 M D C Wi m mi i= ∑ 2 60• ( ) 43 Figure 2-39. Live load variation with depth for concrete pipe culverts.

44 Figure 2-40. Crown moment comparison for RCP. Figure 2-41. Invert moment comparison for RCP.

45 Figure 2-42. Crown shear comparison for RCP. Figure 2-43. Invert shear comparison for RCP.

2.4.5 Corrugated Metal Pipe Comparison The corrugated metal pipe culvert comparison was made using the peak factored thrust per unit length of wall. 2.4.5.1 Live Load Equations The proposed live load equations used for corrugated metal pipe are as presented in Section 2.4.1.2, except the live load area is as follows. Determine the wheel interaction depth where H is the culvert depth, ft wt is the tire patch width, 20 in. lt is the tire patch length, 10 in. LLDFl is the AASHTO LRFD live load distribution fac- tor, 1.15 For H H A w s LLDF H D LL t w l i≥ = + + +⎛⎝⎜ ⎞int • •.12 0 06 12 ⎠⎟ + ⎛⎝⎜ ⎞⎠⎟• • ( ) l LLDF Ht l 12 65 For H H A w LLDF H D LL t l i< = + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 12 l LLDF Ht l 12 64+ ⎛⎝⎜ ⎞⎠⎟• ( ) H s w D LLDF w t i l int . ( )= − − 12 0 06 12 63 Di is the inside span of the culvert, in. sw is the wheel spacing, 6 ft Figure 2-44 compares the variation of live load with depth for corrugated metal pipe, for the Standard, LRFD and pro- posed SDE. The SDE distribution starts out wider than LRFD, but increases in width with depth at the same rate as the LRFD. 2.4.5.2 Dead Load Equations Corrugated metal pipe dead loads were computed from where We is the earth dead load, lb/ft we is the earth unit weight, lb/cubic ft Fe is the soil-structure interaction factor, 1.0 The total dead load is 2.4.5.3 Thrust Calculations For AASHTO Standard Specifications, the total factored thrust is where Tt is the factored thrust per unit length (lb/ft) T TL t F STD = , ( ) 2 68 DL We= ( )67 W F w H D De e e o o= +( )• • • •. ( )0 0089 12 66 46 Figure 2-44. Live load variation with depth for corrugated metal pipe.

For AASHTO LRFD Specifications For the proposed live load equations, the following live load adjustment is required: 2.4.5.4 Comparison between Standard, LRFD and Proposed Figure 2-45 compares the peak thrust for corrugated metal pipe. This figure, and comparison figures for all culvert types, have a common format, described in Section 2.4.3.4. The figure illustrates that, for most cases, the design peak thrust determined from the Standard Method, LRFD Method, and the SDEs are similar. In most cases, the peak thrust from the SDEs are slightly greater than from the Standard Method and about the same or slightly less than the LRFD Method. Outliers occur for large-diameter pipes under shallow burial, T DL LL F t d t = + ( )γ γ• • • ( ) 1 2 72 F F D L LLDF H i t l 1 1 0 75 12 12 = + ⎛ ⎝⎜ ⎞ ⎠maximum ,lim , . • • ⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ ( )71 F Dt1 15 1 70,lim , ( )= ( )maximum T TL t F LRFD = , ( ) 2 69 typically diameters above 5 feet and depths of 2 to 3 feet and less. The high peak thrusts required by the SDEs are the result of the computer model results from this study and are also con- sistent with NCHRP Report 473 (McGrath et al., 2002) results for long-span metal culverts. Additional details may be found in Section 2.3.2.4. 2.4.6 Thermoplastic Pipe (Profile Wall) This section compares profile wall thermoplastic pipes design on the basis of the peak factored thrust. The vertical confining stress is 0.15 ksf (the assumption of the vertical confining stress is somewhat arbitrary, but as the soil stresses due to live load are variable around the pipe, this provides an adequate approximation) using the short term modulus of elasticity of the pipe material. The comparison was done for four soil types (i.e., SW95, SW85, ML85, and CL85). 2.4.6.1 Live Load Equations The proposed live load equations used for profile wall ther- moplastic pipe are as presented in Section 2.4.1.2, except the live load area is as follows: Determine the wheel interaction depth H s w D LLDF w t i l int . ( )= − − 12 0 06 12 73 47 Figure 2-45. Peak thrust comparison for corrugated metal pipe.

where H is the culvert depth, ft wt is the tire patch width, 20 in. lt is the tire patch length, 10 in. LLDFl is the AASHTO LRFD live load distribution fac- tor, 1.15 Di is the inside span of the culvert, in. sw is the wheel spacing, 6 ft Figure 2-46 compares the variation of live load with depth for thermoplastic pipe, for Standard, LRFD, and the proposed SDE. The SDE distribution starts out wider than LRFD, but increases in width with depth at the same rate as the LRFD. 2.4.6.2 Dead Load Equations Profile wall pipe dead loads were computed from S M R E A h s s p = φ • • • ( )77 PL w H D De o o= +( )• • •. ( )0 0089 12 76 For H H A w s LLDF H D LL t w l i≥ = + + +⎛⎝⎜ ⎞int • •.12 0 06 12 ⎠⎟ + ⎛⎝⎜ ⎞⎠⎟• • ( ) l LLDF Ht l 12 75 For H H A w LLDF H D LL t l i< = + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 12 l LLDF Ht l 12 74+ ⎛⎝⎜ ⎞⎠⎟• ( ) where we is the earth unit weight, pcf We is the earth dead load, lb/ft H is the culvert depth, ft Do is the outside span of the culvert, in. φs is the resistance factor for soil stiffness, 0.9 Ms is the constrained soil modulus at 150 psf, per the tables in Figure 2-47, excerpted from the AASHTO code R is the radius to the centroid of the culvert wall, in. E is the pipe material modulus of elasticity, as specified in Table 12.12.3.3-1 of the AASHTO code, 110,000 psi Ap is the pipe unit area, in2/in Values of Do, R and Ap used in the calculations are provided in Table 2-34. The total dead load is 2.4.6.3 Thrust Calculations For AASHTO Standard Specifications, the total factored thrust is where Tt is the factored thrust per unit length (lb/ft) T TL t F STD = , ( ) 2 81 DL We= ( )80 W VAF PLe = • ( )79 VAF S S h h = − − + ⎛⎝⎜ ⎞⎠⎟0 76 0 71 1 17 2 92 78. . . . ( )• 48 Figure 2-46. Live load variation with depth for thermoplastic pipe culverts.

49 Nominal Diameter (in) Outside Diameter Do (in) Mean Radius R (in) Pipe Unit Area Ap (in2/in) 24 28.32 13.08 0.344 48 54.24 25.56 0.48 60 67.02 31.75 0.541 Table 2-34. Outside diameter, mean radius and pipe unit area. Figure 2-47. AASHTO tables with modulus coefficients for thermoplastic pipe (pro- file wall).

For AASHTO LRFD Specifications For the proposed live load equations, the following live load adjustments are required: 2.4.6.4 Comparison between Standard, LRFD and Proposed Figure 2-48 provides comparisons of the peak thrust for thermoplastic pipe. This figure and comparison figures for all culvert types have a common format, described in Section 2.4.3.4. T DL LL F F t d l = +γ γ• • • • ( )1 2 2 86 F Sh 2 0 95 1 0 6 85= + . . ( ) • F F D L LLDF H i t l 1 1 0 75 12 12 = + ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝⎜max , .,lim • • ⎞ ⎠⎟ ( )84 F Di1 15 1 83,lim max , ( )= ( ) T TL t F LRFD = , ( ) 2 82 The figure shows that the peak thrust for most cases are very similar. In general, the peak thrust from the SDEs is slightly greater than the peak thrust from the AASHTO Standard Method. Similarly, the peak thrust from the SDEs is slightly less than the peak thrust from the AASHTO LRFD Method. Several data clusters are noteworthy. Four Standard Method data points at about 13.5 kips/ft correspond to four LRFD Method data points at about 8.5 kips/ft. These eight values are for 48-in.-diameter pipe buried 1 foot. The difference between the values is the different live load factors between the Standard and LRFD Methods, and special treatment required by the Standard Method for burial depths of less than 2 feet. The data points that plot below the 1:1 line between 6.0 and 7.5 kips/ft are for 2 feet of burial. For these cases, the SDEs are consistently less than either the Standard or LRFD values. 2.4.7 Corrugated Metal Arches This section compares peak factored thrust for corrugated metal arches. In flexible large-span culverts, the design and per- formance depend on the interaction of the structure and the surrounding soil. Properties of the backfill envelope as well as in situ material have a major effect on the performance of these structures. AASHTO does not currently specify a factor for long-span metal arch to handle the interaction of the struc- ture and the surrounding soil. A thrust modifier for long-span metal arch will be used in the SDE. 50 Figure 2-48. Peak thrust comparison for thermoplastic pipe (profile wall).

2.4.7.1 Live Load Equations The proposed live load equations used for corrugated metal arches are as presented in Section 2.4.1.2, except the live load area is as follows. Determine the wheel interaction depth where H is the culvert depth, ft wt is the tire patch width, 20 in. lt is the tire patch length, 10 in. S is the culvert span, feet LLDFl is the AASHTO LRFD live load distribution fac- tor, 1.15 sw is the wheel spacing, 6 ft The service live load is determined from For H H A w s LLDF H SLL t w l≥ = + + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 l LLDF Ht l 12 89+ ⎛⎝⎜ ⎞⎠⎟• ( ) For H H A w LLDF H S l LL t l t < = + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 12 88+ ⎛⎝⎜ ⎞⎠⎟LLDF Hl • ( ) H s w S LLDF w t l int . ( )= − − 12 0 06 87 Where S is the culvert span, ft, and the other factors are as defined in previous sections. Figure 2-49 compares the variation of live load with depth for corrugated metal arches, for the Standard, LRFD and pro- posed SDE. The SDE distribution starts out wider than LRFD, but increases in width with depth at the same rate as the LRFD. 2.4.7.2 Dead Load Equations Corrugated metal arch dead loads are calculated from where we is the earth unit weight, pcf We is the earth dead load, lb/ft H is the culvert depth, ft S is the outside span of the culvert, ft The total dead load is DL We= ( )94 W w H Se e= • • ( )93 For proposed SDEs W MPF IM W S L L LL t = +( )• • • min , , 1 gov( ) ( )92 For AASHTO Standard W I W S LLDF L LL s = +( )1 • • •min , H( ) ( )91 For AASHTO LRFD W MPF IM W S L L LL t go = +( )• • • min , , 1 v( ) ( )90 51 Figure 2-49. Live load variation with depth for corrugated metal arches.

2.4.7.3 Thrust Calculations For AASHTO Standard Specifications, the total factored thrust is where Tt is the factored thrust per unit length (lb/ft) For AASHTO LRFD Specifications For the proposed live load equations, the following live load adjustments are required: 2.4.7.4 Comparison between Standard, LRFD and Proposed Figure 2-50 compares the peak thrust for corrugated metal arches. This figure and comparison figures for all culvert types have a common format, described in Section 2.4.3.4. T DL LL F t d l m arch = +γ γ• • • . ( ) 2 98 F S w LLDF H S m arch t l . . . ( ) • • • = + + 0 54 12 0 03 97 T TL t F LRFD = , ( ) 2 96 T TL t F STD = , ( ) 2 95 Design calculations were done for two arch sizes (20′-1″ × 9′-1″ and 30′-1″ × 18′) and four depths of burial. The 20-ft- span arch was designed for 1, 4, and 8 feet of burial, and the 30-ft-span arch was designed for 1, 2, and 8 feet of burial. Results for depths of 4 and 8 feet, where the dead loads are significant, show that the SDE results are essentially identical to the results from the Standard and LRFD methods (these are the data pairs at 12, 21, and 30 kip/ft). For burial depths of 1 foot and 2 feet, where live loads are greater, the SDEs pro- duce peak thrusts 1.3 to 2.1 times greater than the correspond- ing values from Standard and LRFD Methods. The principal cause for the difference is the factor Fm,arch. 2.4.8 Concrete Arches This section describes design calculations for concrete arches for AASHTO LRFD, AASHTO Standard, and proposed SDEs. Comparisons were made for peak thrust, peak shear, peak pos- itive moment, and peak negative moment. Structural responses to the applied dead and live loads were computed using the 2D structural analysis program (SAP, 2000). 2.4.8.1 Live Load Equations The proposed live load equations used for concrete arches are as presented in Section 2.4.1.2, except the live load area is as follows. Determine the wheel interaction depth H s w S LLDF w t l int . ( )= − − 12 0 06 99 52 Figure 2-50. Peak thrust comparison for corrugated metal arches.

where H is the culvert depth, ft wt is the tire patch width, 20 in. lt is the tire patch length, 10 in. S is the culvert span, ft LLDFl is the AASHTO LRFD live load distribution fac- tor, 1.15 sw is the wheel spacing, 6 ft The service live loads were determined from For AASHTO Standard W I W S LLDF L LL s = +( )1 • • •min , H( ) ( )103 For AASHTO LRFD W MPF IM W S L L LL t go = +( )• • • min , , 1 v( ) ( )102 For H H A w s LLDF H SLL t w l≥ = + + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 l LLDF Ht l 12 101+ ⎛⎝⎜ ⎞⎠⎟• ( ) For H H A w LLDF H S l LL t l t < = + + ⎛⎝⎜ ⎞⎠⎟int • • • . 12 0 06 12 100+ ⎛⎝⎜ ⎞⎠⎟LLDF Hl • ( ) Where S is the culvert span, ft, and the other factors are as defined in previous sections. Live loads were applied to the SAP2000 model in the loca- tions illustrated in Figure 2-51, which also compares the variation of live load with depth for the Standard, LRFD, and proposed SDE. The SDE distribution starts out wider than LRFD, but increases in width with depth at the same rate as the LRFD. 2.4.8.2 Dead Load Equations Typical earth-pressure distributions were used to evaluate thrust, shear, and moment in large-span arch culverts. Soil loads were calculated as follows: where σye is the vertical pressure at the culvert edge, psf σyc is the vertical pressure at the culvert centerline, psf σxx is the horizontal pressure along the side of the culvert, psf σxx h eK z w= • • ( )107 σ yc c eK z w= • • ( )106 σ ye e eK z w= • • ( )105 For proposed W MPF IM W S L L LL t gov = +( ) ( ) • • • min , , 1 ( )104 53 Figure 2-51. Live load variation with depth for concrete arches.

we is the earth unit weight, 120 pcf z is the vertical distance from the surface to the point of interest, ft Ke is the edge pressure coefficient, 1.2 Kc is the center pressure coefficient, 1.0 Kh is the horizontal pressure coefficient, 0.4 Concrete loads were calculated as follows: where Warch is the weight of the concrete arch, lb/ft Aa is the area of the concrete arch, square ft wc is the unit weight of concrete, 150 pcf The dead loads applied to the SAP2000 model are illus- trated in Figure 2-52. 2.4.8.3 Geometry and Material Properties The geometry of the concrete arches is listed in Table 2-35. The concrete unit weight was 150 pcf. 2.4.8.4 Moment, Shear and Thrust Calculations The 2D structural analysis program, SAP2000, was used to calculate the structural response to the loads described in the previous sections. Two analyses were performed for each W A warch a c= i arch-depth combination: one analysis was done with no lat- eral footing movement, and one analysis was done with lateral footing movement equal to 0.001 times the mean span, based on the recommendations of NCHRP Report 473 (McGrath et al., 2002). 2.4.8.5 Comparison between Standard, LRFD, and Proposed SDEs Figures 2-53 through 2-56 compare the peak thrust, peak shear, peak positive moment, and peak negative moment for concrete arches, respectively. These figures and comparison figures for all culvert types have a common format, described in Section 2.4.3.4. Figure 2-53 illustrates that the peak thrusts calculated using the proposed SDEs are very similar to those calculated using the Standard and LRFD Methods. The peak thrusts are not significantly influenced by the footing boundary conditions. Figure 2-54 illustrates the peak shear values calculated using the three methods. This graph shows that the footing movement condition has a much greater influence on peak shear values than peak thrust values. Peak shear values calcu- lated using the proposed SDEs are very similar to those from the LRFD Method. There is no clear pattern to the SDE- Standard comparison—some are significantly more, some significantly less. Figure 2-55 compares positive moments for the three live loads. The footing movement condition has a profound effect on positive moment—all data points with SDE-calculated val- ues less than 10 kip-in/ft are for no footing movement, while all those greater than 10 kip-in/ft are for footing movement. In general, the positive moments determined using SDE live loads are the same or less than those determined from using Standard or LRFD loads, sometimes significantly less. Figure 2-56 illustrates the peak negative moments from the three live load cases. SDE-based values on average are about 54 Figure 2-52. Concrete arch dead loads. Nominal Span Actual Span Rise Inside Radius 25'-4" 25' - 3-9/16" 10'-0" 13'-0" 43'-11" 43' - 11-3/8" 13'-8" 24'-6" Table 2-35. Concrete arch geometry.

55 Figure 2-53. Peak thrust values for concrete arches. Figure 2-54. Peak shear values for concrete arches.

56 Figure 2-55. Peak positive moment values for concrete arches. Figure 2-56. Peak negative moment values for concrete arches. the same as from the Standard and LRFD loads, although there is a lot of scatter. 2.5 Guidelines for Use of Refined Analysis Methods This project has conducted extensive 3D modeling of the transfer of surface live loads to buried culverts. From the re- sults, the research team has proposed SDEs that permit culvert design without modeling. However, many design situations have conditions not covered by the SDEs. In these situations, 2D and 3D modeling may be used for design. The research team developed guidelines for conducting 2D and 3D model- ing, based on the work reported in previous sections and on additional 2D and 3D modeling. The guidelines are presented in detail in Appendix E. The 2D guidelines provide a means for selecting the surface load inten- sity to be applied to a 2D model. The 3D guidelines address

software, live load application, representations of the pave- ment and the soil, model dimensions, element size, symmetry and boundary conditions, representations of culvert structures, and the soil-culvert interface. 2.5.1 Guideline for 2D Analysis 2.5.1.1 Longitudinal and Transverse Subsurface Spreading 2D computer models have an inherent limitation when computing the effect of surface live loads. Because the mod- els are 2D, the load spreading that occurs in the longitudinal direction, parallel to the axis of the culvert, cannot be cor- rectly computed. The model represents a single, vertical slice through the real-world geometry. Figure 2-8 illustrated the location and intensity of the live load. The vehicle centerline is in the left-right plane of sym- metry and the culvert centerline is in the up-down plane of symmetry. This section refers to the left-right direction as the transverse direction and the up-down direction as the longi- tudinal direction. The fundamental equation in all live load spread calcula- tions is that the total force at depth H is equal to the total force at the surface: The surface pressure is For 3D spreading, the live load pressure at depth is the force divided by the area: where PH is the vertical pressure at depth H LL is the live load force at the surface, 16,000 lb unfactored wt is the transverse dimension of the tire patch, typically 10 in. lt is the longitudinal dimension of the tire patch, typically 20 in. LLDF is the live load distribution factor, 1.75 for Standard, for LRFD: 1.15 for granular fill, 1.0 for other fills In the case of 2D modeling, models correctly determine the load spread in the transverse direction (in the plane of the model). In the longitudinal direction, 2D models do not com- pute load spreading. Hence, the live load must be factored (or “spread”) to achieve the spreading that cannot be modeled. P LL w LLDF H l LLDF H H t t = +( ) +( )• • ( )110 P LL w l S t t = • ( )109 F FS H= ( )108 Assuming that the transverse live load spread will be com- puted by the model, the vertical pressure at depth is and at the surface Hence, the ratio of the live load pressure at depth to the surface live load pressure is 2.5.1.2 2D and 3D Modeling Preceding sections report the results of extensive 3D mod- eling of a range of culvert types, soils, and depths. All model- ing was done with service live loads. Selected 3D models, with Mohr-Coulomb soil behavior were rerun with 16,000 lb live load, for comparison with analogous 2D modeling. The cul- vert types, sizes, depths, and soils were as follows: 1. Materials: Concrete Pipe (RCP), Corrugated Metal Pipe (CMP), Profile Wall Pipe (PW) 2. Size: a. RCP using 24-, 48-. and 96-in. dia. b. CMP using 12-, 24-, 48-, and 96-in. dia. c. PW using 12-, 24-, 48-, and 60-in. dia. 3. Soil Type: SW85 4. Cover Depth: 12, 24, 48, and 96 in. The 2D models used FLAC3D (with a 2D geometry), un- factored loads, and elastic soil behavior. Elastic soil behavior was chosen because elastic models are most common. Peak thrust and crown moment were compared by com- puting the following ratios of 3D structural response to 2D structural response: where TDRR i T is the Two-Dimensional Response Ratio for peak thrust, and TDRRi MC is the Two-Dimensional Response Ratio for crown moment. TDRR M M i MC c d c d = , , ( )3 2 115 TDRR T T i T p d p d = , , ( ) 3 2 114 P P l l LLDF H H D S D t t 2 2 113= +( )• ( ) P LL l S D t 2 112= ( ) P LL l LLDF H H D t 2 111= +( )• ( ) 57

Figures 2-57 through 2-59 illustrate the peak thrust 2D response ratio. The peak thrust generally does not occur at the crown of the culvert. Each figure includes the curve result- ing from Eq. 113, with f equal to 1.15. The peak thrust figures show that the TDRR is strongly influenced by culvert type. Figure 2-57, illustrating RCP results, shows that peak thrust response is very close to the longitudi- nal spread equation (Eq. 113). In comparison, Figure 2-58 shows that the profile wall response is about 1.5 times greater than Eq. 113. Figure 2-59 shows that the CMP response is similar to Eq. 113 at depths greater than 48 inches, but signif- icantly higher at shallower depths. Figures 2-60 through 2-62 illustrate the crown moment 2D response ratio. Each figure includes the curve resulting from 58 Figure 2-57. Peak thrust TDRR for concrete pipe (AASHTO refers to Eqn (113)). Figure 2-58. Peak thrust TDRR for profile wall (AASHTO refers to Eqn (113)).

Equation 113, with LLDF equal to 1.15. Like the results for peak thrust, the crown moment results are strongly influenced by culvert type. For RCP (Figure 2-60), the model response is less at shallow depths and greater at 96 inches. The profile wall data (Figure 2-61) shows significant variation due to culvert diam- eters and is also significantly greater than Equation 113 for all depths. CMP results (Figure 2-62) are about the same as Equa- tion 113 at 12 inches, but increase with increasing depth. 2.5.1.3 2D Guideline To characterize the variations illustrated in the figures of the previous section, nonlinear curve fitting was used to select parameters for a variant of Equation 116, with one additional parameter: P P TDRR a l l b H H D S D t t 2 2 116= = +( ) • • ( ) 59 Figure 2-59. Peak thrust TDRR for corrugated metal pipe (AASHTO refers to Eqn (113)). Figure 2-60. Crown moment TDRR for concrete pipe (AASHTO refers to Eqn (113)).

where a is the additional parameter and LLDF is replaced by b. These parameters are to be selected from curve fitting. Microsoft Excel’s Solver function was used to select values for these parameters, but minimizing the sum of the square dif- ferences between the function in Equation 116 and the data points. The nonlinear fit curves plotted on each figure illus- trate that the curves fit the data relatively well. Table 2-36 illustrates the resulting parameter values for Equation 116; and Figures 2-63 and 2-64 illustrate the com- posite graphs (all data and curves) for peak thrust and crown moment, respectively. 60 Figure 2-61. Crown moment TDRR for profile wall (AASHTO refers to Eqn (113)). Figure 2-62. Crown moment TDRR for corrugated metal pipe (AASHTO refers to Equation 113).

Figure 2-63 illustrates that nearly all data shows a TDRR greater than the longitudinal load spread from Eq. 113, and that there is a significant variation in the data depending on culvert type. Hence, the fitted curves also have significant variations. Figure 2-64 illustrates similar results for crown moment, except that the RCP data is less than Equation 116 for depths of 48 inches or less. The resulting guideline for the surface pressure to be used for conducting 2D analyses is where PS 2D is the 2D surface pressure, PS 3D is the 3D surface pressure, a and b are parameters from Table 2-36. P P a l l b H S D S D t t 2 3 117= +( ) • • • ( ) The parameters for peak thrust and crown moment are sufficiently different that separate analyses should be con- ducted for each. 2.5.2 Guideline for 3D Analysis 2.5.2.1 General Software Guidelines Following are general software guidelines for conducting 3D analyses of live loads on culverts: 1. 3D elements, geometry, boundary conditions, etc. 2. Shell structural elements: a. Isotropic shell elements for isotropic culvert materials b. Orthotropic shell elements for orthotropic culvert materials 3. Ability to model live loads placed on the soil surface 4. At least the following constitutive models a. Elastic (for pavement) b. Mohr-Coulomb (for soil) 5. Soil-culvert interface logic that permits arbitrary interface strength and stiffness 2.5.2.2 Live Load Magnitude, Contact Area, Location Analyses were conducted for dead load (soil loading only) and combined dead plus live load. The dead load response was subtracted from the combined response to determine the 61 Figure 2-63. Composite graph for peak thrust two-dimensional response ratio (AASHTO refers to Equation 113). Culvert Type Structural Response Constant a Constant b Peak Thrust 1.387 1.595 Concrete Pipe Crown Moment 0.509 0.411 Peak Thrust 1.303 0.757 Profile Wall Crown Moment 1.195 0.787 Peak Thrust 2.132 2.379 Corrugated Metal Pipe Crown Moment 0.794 0.511 Table 2-36. Two-dimensional response ratio equation parameters.

live only response. Dead loads, that is soil loads, were not fac- tored. Live loads were applied and factored as follows: where mmpf is the multiple presence factor (1.2) P is the wheel load magnitude (16,000 lb) IM is the dynamic load allowance H ≤ 8 H is the depth of cover from road surface to top of culvert, in. In Figure 2-8, two surface load patches are included in the models, either explicitly or via symmetry, while the other load patches at the front and rear of the vehicle are not included. For some conditions, it may be necessary to include the other load patches. 2.5.2.3 Factored versus Unfactored Live Loads The culvert community is divided on the issue of model- ing using factored versus unfactored live loads. As a result, the study included a comparison of structural responses to un- factored and factored live loads. If the structure and surrounding soil have linear-elastic material properties, structural responses to the factored live 33 1 0 125 12 − ⎡ ⎣⎢ ⎤ ⎦⎥ . , • H LL m IM Pmpf= + ⎡ ⎣⎢ ⎤ ⎦⎥1 100 118( ) loads will differ from those to the unfactored live loads by a load factor. However, backfill surrounding the structure is nonlinear, and the ratio of structural response to the factored load to the response to the unfactored live load will not be exactly equal to the load factor. To examine the effect of soil nonlinearity, soil-structure interaction analyses were per- formed and compared for culverts subjected to factored and unfactored live loads. The analyses were for various 2D and 3D conditions, structure types, soil behavior, and software. Based on the cases we examined, structural responses to the factored live load can be estimated by scaling unfactored live load responses by the load factor. The exceptions are thrusts for shallow burial. 2.5.2.4 Representing the Pavement During the study, the number of models with and without concrete pavement was approximately equal. In models with concrete pavement, the pavement was represented by a single layer of zones with the elastic behavior and properties suitable for concrete. In models without pavement, live loads produced excessive localized bearing failure of the soil. As a result, the surface layer of zones was modeled using the same properties as the underlying zones, but with elastic rather than Mohr- Coulomb behavior. Results showed that pavement spreads the load and shields the culvert. Because of the significant affect of this load spread- ing and shielding and given that live loads are possible prior to 62 Figure 2-64. Composite graph for crown moment two-dimensional response ratio (AASHTO refers to Equation 113).

paving or during roadway rehabilitation, the research team concluded that unpaved is the controlling case. The influence of pavement is greater for shallow culvert cover depth and flexible culverts and is smaller for stiffer culverts and deeper burial. For example, the ratios of unpaved to paved response were 1.0 to 3.3, for a 48-inch RCP culvert, with 2 feet of cover and SW85 soil. In contrast, the ratios of unpaved to paved response were 0.85 to more than 30, for a 48-inch profile wall culvert with 2 feet of cover and SW85 soil. Ratios were com- puted for crown and invert moment, crown and springline thrust, and crown and invert displacement. 2.5.2.5 Soil Constitutive Models In this study, the research team found that a linearly elas- tic, perfectly plastic model with a Mohr-Coulomb failure cri- terion was appropriate. This selection offers the best mix of capturing the important aspects of soil behavior in transmit- ting live loads to structures. The Mohr-Coulomb constitutive model does not offer all of the benefits of the Duncan-Selig/ hardening-soil models in capturing stress-dependent stiffness behavior of soil, but for a live load study, the Mohr-Coulomb model appears to provide sufficient accuracy. 2.5.2.6 Soil Properties All models used one of four soil materials: well-graded or gravelly sand at 85-percent standard compaction (SW85), well- graded or gravelly sand at 95-percent standard compaction (SW95), inorganic silts and fine sands at 85-percent standard compaction (ML85), and inorganic clays at 85-percent stan- dard compaction (CL85). The research team recommends that angles of friction at a reference confinement of 14.7 psi from Selig’s parameters (1988) be used at any depth in the 3D analysis instead of variable angles of friction calculated from the stress state before the live load application. (Mohr-Coulomb soil parameters for SW95, SW85, ML85, and CL85 are pre- sented in Chapter 3.4.2 of Appendix E. If site-specific soil properties are known, these values should be used instead of the values presented in the appendix.) Near-surface soil modulus measurements using the Hum- boldt GeoGauge, lightweight deflectometer, and dynamic cone penetrometer (DCP) produce near-surface values significantly higher than the values presented in the preceding paragraphs. The modulus values for SW85 and SW95 are lower bounds for DCP data from one site. Many DCP values are 2 to 5 times greater. Because the focus of NCHRP Project 15-29 was live load effects, inhomogeneous culvert bedding was not modeled, so the research team cannot offer any guidelines for modeling culvert bedding. 2.5.2.7 Model Dimensions and Element Size In general, model dimensions were larger for increasing cover depth and increasing culvert diameter (span). In some instances, the research team initially used smaller model widths, observed results indicating that the models were too narrow, and reran the models using greater width. (Chapter 3.5 of Appendix E summarizes the model dimensions.) The size of continuum elements used in the study varied depending on the size of the culvert and the location of the element in the model. In general, smaller elements were used for smaller culvert diameters and larger elements were used for larger culvert diameters. Element size also in- creased with distance away from the live load—the largest elements were typically at the bottom corner of the model at the end farthest from the live load. Table 2-37 lists the continuum and structural element sizes for nine selected models. Given that the continuum element sizes near the culvert are wedge-shaped, the inner and outer width are listed. During the study, the research team found the element sizes were sufficiently small to produce good results. 63 Case Minimum Continuum Element Size (inches) Minimum Structural Element Size(inches) Maximum Structural Element Size(inches) Maximum Continuum Element Size(inches) 12-inch round 1.6-3.8 x 6 x 6 6 x 6 x 11 1.6 x 6 1.6 x 11 24-inch round 2.4-4.1 x 6 x 6 6 x 10 x 11 2.4 x 6 2.4 x 11 48-inch round 2.4-3.2 x 6 x 6 9 x 11 x 11 2.4 x 6 2.4 x 11 96-inch round 3.8-4.5 x 6 x 6 12 x 12 x 11 3.8 x 6 3.8 x 11 120-inch round 3.4-3.8 x 3.9 x 6 23 x 24 x 11 3.4 x 6 3.4 x 11 25.4 ft x 10 ft conc. arch 12 x 6 x 6 12 x 12 x 18 6 x 6 6 x 18 43.1 ft x 13.8 ft conc. arch 3 x 7 x 6 18 x 18 x 18 7 x 6 7 x 18 20.1 ft x 9.1 ft metal arch 3 x 6 x 6 17 x 12 x 18 6 x 6 6 x 18 30.1 ft x 18 ft metal arch 3 x 8 x 6 17 x 18 x 18 8 x 6 8 x 18 Table 2-37. Continuum and structural element sizes for selected models.

2.5.2.8 Symmetry and Boundary Conditions Planes of symmetry may be used to reduce the size of and simplify models. All culvert structures modeled were sym- metric about a vertical plane of symmetry through the culvert axis. The culvert structures were also of uniform cross sec- tion, so planes of symmetry could be used to reduce the length of the models. The live load magnitudes and geometry illustrated in Fig- ure 2-8 are symmetric about the centerline of the vehicle (which is perpendicular to the centerline of the culvert). The live load is not symmetric about the centerline of the culvert, if all three axles are included. However, for a single, relatively shallow culvert, the front and rear axles do not significantly affect culvert loads. As a result, the research team ignored the live loads from the front and rear axles. The result is a live load distribution that is symmetric about the culvert axis. Nearly all models in this study employed two planes of symmetry to reduce model size. A few analyses did not use the plane of sym- metry through the culvert axis to check the analysis results. Boundary conditions for the continuum (soil) parts of the models were straightforward—live loads were applied as pressures on the model top surface and all other surfaces had free or fixed displacements. The conditions were as follows: 1. Model top—The top was free, with a 10- by 20-inch patch of live load applied as shown in Figure 2-8. 2. Model bottom—The top was fixed in the vertical direction and free otherwise. 3. Model ends—The model ends, where the ends of the cul- vert were exposed, were fixed in the direction parallel to the culvert axis and free otherwise. 4. Model sides—The model side that contained the culvert centerline was fixed in the horizontal direction perpendi- cular to the culvert axis and free otherwise. The model side opposite the culvert centerline had the same boundary condition, meaning that it was also a plane of symmetry. Boundary conditions for the culvert were similar: 1. Culvert ends—The culvert ends, where the end of the cul- vert was exposed, were fixed in translation in the direction of the culvert axis and were fixed in rotation about the ver- tical and transverse direction. All other degrees of freedom were free. 2. Culvert crown and invert—The culvert crown and invert, where cut by the plane of symmetry, were fixed in transla- tion perpendicular to the culvert axis and were fixed in ro- tation about the longitudinal and vertical direction. All other degrees of freedom were free. During the study, the research team became concerned that translational fixity of the culvert ends was increasing the stiffness of the overall culvert structure. A few analyses were rerun with no translation fixity of the culvert ends. The results were only slightly different, confirming that this boundary condition was not affecting the results. 2.5.2.9 Culvert Structure Representation Culvert structures may be represented as 1. Continuum elements, where the structure is built up as a series of continuum elements across the thickness of the structure. This method was used in 2D analyses to model box culvert haunch behavior, as a basis for selecting struc- tural element properties. 2. Multiple structural elements, where the culvert is built up of structural elements. This method was used to model the complex interior structure of profile wall pipe for compar- ison with orthotropic structural elements. 3. Single structural elements, where a single element (of zero thickness) is used to represent a segment of the culvert. This method was used for most analyses. In all cases, the structures were linear elastic. The three methods have advantages and disadvantages, but in general, the built-up methods (either continuum elements or structural elements) were only used in special cases where a single, zero-thickness structural element was not adequate. In this study, single structural elements were used to repre- sent the culverts in all production analyses. In 3D, structural elements for representing culverts must accommodate both bending action and membrane action. As a result, shell elements are necessary and were used for all production analyses. The formulation of shell elements does not permit the calculation of transverse shear forces. 2.5.2.10 Requirements for Iso- and Orthotropic Structural Elements Culverts composed of solid material and regular geometry may be represented by isotropic structural elements (i.e., bending and membrane properties are the same in all direc- tions). This category of culverts includes concrete boxes and concrete pipe, smooth steel pipe, and smooth thermoplastic pipe. Both plastic and metal culvert products use cross-sectional shapes that are orthotropic, meaning the structural proper- ties vary by direction. These culvert shapes typically have much higher circumferential bending stiffness than longitudinal bending stiffness. In addition, the circumferential membrane stiffness is much higher than the longitudinal membrane stiffness. In plane shear stiffness is reduced from that of flat plate of the same thickness. In order to accurately model 64

buried pipes with such properties, accurate and well-behaved 3D orthotropic structural elements that permit specification of different stiffnesses for bending and membrane behavior are needed. Before a discussion of modeling with 3D shells, the analo- gous issue in two dimensions will be described. A 2D beam formulation permits specification of the following properties: 1. Material Young’s modulus, E 2. Member area, A, used to calculate the axial stiffness EA of the member 3. Member moment of inertia, I, used to calculate the bend- ing stiffness EI of the member If the beam were of solid cross section, A and I could be calculated from the beam width and thickness. However, commonly used beams are not of solid cross section and hence the area, A, and moment of inertia, I, must be speci- fied separately. Modeling of a 3D shell presents a similar challenge. The current formulation of shell elements in many structural analysis programs is based on a “solid” representation of the shell. For profile wall pipes, which are not solid, and for cor- rugated metal culverts, which are not “solid” due to the cor- rugations, two bending stiffnesses and two membrane stiff- nesses must be specified to capture the structural behavior. Typical software permits only three of the four stiffness pairs (i.e., EAtransverse, EAlongitudinal, EItransverse, and EIlon- gitudinal) to be specified independently. At the start of the study, FLAC3D also had this restriction, but Itasca Consult- ing Group modified the software to permit the four stiffness pairs to be input independently. Results were confirmed by several culvert and non-culvert test cases. Two guidelines resulted. When modeling orthotropic culverts 1. Use orthotropic shell elements for all culvert types that are orthotropic, and 2. Confirm via simple demonstration analyses that the 3D analysis software correctly models orthotropic materials. (The research team found that a model of a plywood plate with 2×4 stiffeners in one direction was effective in con- firming model behavior.) 2.5.2.11 Culvert Joints In distributing live loads through fill onto buried struc- tures, practice has been to ignore the presence of joints in a pipe. This results in two potential issues: 1. The discontinuity created by a pipe joint will prevent load spreading though the pipe, resulting in an overstress. 2. A joint loaded on one side, but not the other, will undergo differential deflection, resulting in a joint leak. Parameters that could affect this condition include 1. Pipe bell and spigot joints are often heavier and stronger than the barrel, providing more strength to resist the live load. Bells are typically thicker than pipe barrels, and the spigots, which may not be thicker than the barrels, are contained within the bell which provides additional con- finement. 2. Pipe joints completed by wrap around couplings provide a mechanical connector to two adjacent lengths of pipe that likely provides shear transfer. 3. Unlike box culvert slabs, buried pipes are not assumed to have any inherent load distribution capability (i.e., in a box culvert under less than 2 feet of fill, a live load is dis- tributed over a width about 4 feet wider than the actual loaded width, while in pipe the loaded length is typically assumed to carry the entire load.) 4. Most thermoplastic pipes are required to pass joint shear tests that require imposition of an unbalanced load with- out causing leakage. Concrete pipes have a joint shear test, but drainage pipes are not typically subjected to it. Metal pipes do not currently have a joint shear test. 5. Most pipes have excess structural capacity at the mini- mum depths of fill allowed by specifications. Minimum depths of fill are set to control road surface performance and are virtually always, if not always, set at depths where the pipe has extra capacity to carry unanticipated loads. The research team is not aware of any definitive studies on the above issues and thus cannot state with certainty that the presence of a joint can be ignored when distributing live loads through earth fills; however, the lack of any problems associated with this matter is compelling. It is well known that pipes can be subjected to severe abuse during installa- tion and are often installed in backfill conditions that do not comply with specifications or are subjected to large con- struction loads that exceed design loads. Despite this, prob- lems are limited and often only result from extreme loading conditions. Each culvert type has one or more different joint types, each with different behavior. The research team’s full-scale models use shell elements to represent the culverts. In order to provide the basis for incorporating joints in full-scale mod- els, large, complex models using continuum elements would need to be developed, tested, and analyzed for each joint type. The macro structural properties of each joint could then be incorporated in a structural element model. Consideration of gasket pressures would also be required. 65

The research team believes that the technical points out- lined above are sufficiently compelling that, when combined with the practical considerations, support a decision to address jointed culverts in the commentary. 2.5.2.12 Soil-Culvert Interface The soil-culvert interface connects the continuum ele- ments representing the soil to the structural elements repre- senting the culvert. Historically, it was common to model soil-culvert interaction with no interface—the soil and cul- vert structure were bonded and, in fact, had common nodes. Now, in typical formulations, the interface has stiffness and strength properties, which vary depending on compres- sion or tension loading. If modeled in this manner, nonlinear behavior may occur in the interface between the culvert and the soil, or in the soil. The influence of the soil-culvert interface stiffness on cul- vert response was not investigated, so the research team of- fers no guidelines. For preliminary 2D analyses, the interface strength was 50% of the soil shear strength. To examine the effect of inter- face strength on structural response, the research team analyzed the concrete and thermoplastic pipe with backfill modeled by the Mohr-Coulomb constitutive model with the interface strength equal to 100% of the soil shear strength. Structural responses to live loads did not change significantly when the interface strength was changed from 50% of the soil shear strength to 100%, although the cases with the 100% strength showed slightly larger peak responses than those with the 50% strength, except for moments of the thermoplastic pipe with 2-ft cover. A change in the interface strength affected thrusts more than moments. Structural responses of the thermoplas- tic pipe were affected more by a change of interface strength than those of the concrete pipe. Structural responses of the 6-ft cover cases were affected more by a change of interface strength than those of the 2-ft cover cases; however, responses of the 6-ft cover cases were much smaller than those for the 2-ft cover cases. For 3D analyses, the influence of soil-culvert interface strength was investigated, for a few culvert types, sizes, and depths, by varying the interface strength. Four interface strengths were considered: 1. Fully bonded—No relative deformation was permitted between the soil and the culvert. 2. 100% soil strength—Interface strength was 100% of the soil friction angle and 100% of the soil cohesion. 3. 50% soil strength—Interface strength was 50% of the soil friction angle and 50% of the soil cohesion. 4. Unbonded—Interface friction and cohesion were zero. Interfaces had a Mohr-Coulomb failure criterion. Most production analyses were conducted for interfaces with strength properties of 100% of the soil strength. As with the 2D results, the 3D results show that the reasonable inter- face strengths do not have significant influence on the struc- tural response. 2.5.2.13 Modeling Sequence Three states of the model were analyzed and saved for each analysis conducted: • State 1 is the soil mass in equilibrium, with no culvert or live load. State 1 was achieved by creating the model grid, applying material properties to the soil materials, and plac- ing stresses in the grid. • State 2 (dead load) is the soil mass plus the culvert, in equi- librium. This state was achieved by excavating the soil (with no cycling of the model), installing the culvert in the soil, and then cycling to equilibrium. • State 3 (dead load plus live load) is State 2 plus application of the live load defined above. While saving and reviewing States 1 and 2 is not necessary in order to find State 3, the research team recommends con- ducting analyses in this manner. 66