ABSTRACT
When box culverts are located under engineering structures, the overlying foundation loading may induce considerable soil pressure on the culverts causing excessive culvert deformation. To minimize possible adverse effect of foundation loading on the culvert performance, the soil pressures induced by the foundation loading need to be properly considered. In this study, the finite element method of analysis developed based on the elasto-plasticity theory was used to analyze soil pressures on square reinforced concrete box culverts induced by overlying strip footings. Variables considered in the analysis were culvert size, location, and wall thickness. Based on the results of analysis, the distributions of vertical soil pressure on the vertical planes through culvert center as well as culvert edge, and on the horizontal plane through top of culvert were obtained. Also obtained was the normal soil pressure distribution along culvert periphery. It is concluded that the soil pressure distribution around box culverts induced by the overlying footings is strongly dependent on soil-culvert interactions. Thus, an optimal structural design of box culverts may require iteration procedures. The results of study have provided an insight into soil-culvert interaction mechanism. More data are needed, however, not only for better understanding of the interaction mechanism but also for the development of a rational method for structural design of square box culverts overlain by strip footings.
Keywords: Box culverts, strip footings, soil pressure, elasto-plastic analysis, finite element method.
INTRODUCTION
There are numerous types of culvert in terms of material and geometric shape. Of the different culvert types, reinforced concrete box culverts are popularly used. Culverts are often installed under engineering structures such as roadways and buildings. When culverts are near the foundations of structures, the overlying foundation loading may induce considerable soil pressure on the culverts. As a result, the culverts may undergo excessive deformation or failure rendering the culverts unserviceable. Thus, the design of culverts located below foundations requires full understanding of the soil pressure induced by the overlying foundation loading.
Earth pressure distribution around concrete box culverts has been the subject of a few studies. However, nearly all of the studies deal only with embankment loading that crosses over culverts. Examples include the studies of Young et al. (1997, 1999), Vaslestad et al. (1994), Russ (1975), James et al. (1986), and Awwad et al. (2000). Embankment loading normally spans over the entire culvert width, while footing loading may cover only part of the width. The difference in loading size compared with culvert width may result in considerable difference in the loading-induced soil pressure around box culverts.
Because plastic yielding within the surrounding soil may occur due to considerable culvert deformation under the foundation loading, a reasonable approach to analyze soil pressures on the culvert requires a full consideration of both elastic and plastic behaviors of the soil. Accordingly, in this study, the analysis was made using the finite element method based on the elasto-plasticity theory. The conditions analyzed included reinforced concrete strip footings centered above parallel reinforced concrete square box culverts for different culvert size, location, and wall thickness. The soil pressures analyzed were vertical pressure distributions on the vertical planes through culvert center and culvert edge, and on the horizontal plane through top of culvert. Also, the normal soil pressure distribution along culvert periphery was determined. The method and results of analysis together with other pertinent information are presented and discussed below.
FINITE ELEMENT ANALYSIS
The analysis was made using a two-dimensional plane strain finite element computer program, which was developed, based on the elasto-plasticity theory (Jao, 1995). In the analysis, both the concrete culvert and the surrounding soil were idealized as nonlinear elastic-perfectly plastic materials. Within the elastic range, the stress-strain relation of concrete obeyed Hooke’s law, while the stress-strain behavior of soil followed the hyperbolic function of Duncan and Chang (1970). Beyond elastic range, the Drucker-Prager yield criterion (Drucker and Prager, 1952) was adopted to model the plastic behavior. Important features and applications of the computer program are available elsewhere (Jao and Wang, 2000).
The footing analyzed was a reinforced concrete strip footing, which had a width (B) of 3 ft (0.9 m) and was embedded to a depth (D) of 3 ft (0.9 m). The square reinforced concrete culvert was centered below the footing. The variables investigated included three levels of culvert wall thickness (t), and four levels each of culvert width (W) and depth to culvert (D); they were t = 1 in. (25.4 mm), 3 in. (76.2 mm), and 6 in. (152.4 mm); W = 3 ft (0.9 m), 6 ft (1.8 m), 9 ft (2.7 m), and 12 ft (3.7 m); and D = 3 ft (0.9 m), 6 ft (1.8 m), 9 ft (2.7 m), and 12 ft (3.7 m). A typical finite element mesh together with the various symbols used in the analysis is shown in Figure 1, which shows only one-half of the footing-culvert-soil system analyzed.
Figure 1. Finite Element Mesh
FOUNDATION SOIL
The foundation soil analyzed is compacted kaolin, which is commercially available under the name Edgar Plastic Kaolin (EPK). The EPK has a median grain size of D50 = 0.001 mm with a clay size fraction (D< 0.002 mm) of approximately 78%. Its liquid limit, plasticity index, and specific gravity are 58%, 22, and 2.60, respectively; the soil classification is MH in the Unified System and A-7-5 in the AASHTO classification system.
The compacted soil used in the analysis had a water content of 23% with a dry unit weight of 86.0 pcf (13.5 kN/m2). The soil properties together with the concrete properties analyzed are summarized in Table 1. In the analysis, the modulus of deformation in compression of the soil was computed using the following equation of the hyperbolic stress-strain model, which was developed by Duncan and Chang (1970):
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(1) |
in which Ec = the tangential modulus of deformation in compression, Ei = the initial compressive modulus, and Rf = a model parameter that has been determined to be 0.77 for the soil used this study from triaxial compression test results.
Table 1. Material Properties of Soil and Concrete Culvert/Footing.
![]() | Parameters | Soil | Concrete |
![]() | Angle of internal friction, in degrees (*) | 8 | 39.6 |
Cohesion, in psi (kN/m2) (*) | 23 (158.7) | 810 (5581) |
Initial modulus in compression, in psi (kN/m2) | 2880 (19,872) | 3.3*106 (22.8*106) |
Initial modulus in tension, psi (kN/m2) | 7,000 (48,300) | 3.3*106 (22.8*106) |
Tensile strength, psi (kN/m2) | 7 (48.3) | 300 (2070) |
Poissons ratio | 0.39 | 0.30 |
Density (wet density for soil), in pci (kN/m2) | .06 (16.28) | 0.09 (23.60) |
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(*) i.e., the shear strength parameters (undrained / total stress) |
Vertical Soil Pressure Distribution on Vertical Planes
The distribution of vertical soil pressure around the culvert is affected by many factors. The effect of culvert size on the vertical pressure distribution with depth below footing center is shown in Figure 2 for a footing pressure of 5,000 psf (239.4 kN/m) with D = B = 3 ft (0.9 m) and t = 1 in (25.4 mm). Also included in the figure for comparison is the distribution of vertical pressure computed from Boussinesq equation plus the geostatic vertical pressure for no-culvert condition. It is seen that, for the larger culverts with sizes of 6 ft (1.8 m), 9 ft (2.7 m), and 12 ft (3.7 m), the vertical pressure distributions with depth between footing base and culvert top differ from the shape of Boussinesq distribution. This is primarily due to the effect of culvert roof flexibility. The roofs of larger culverts are more flexible causing greater deflections under the same footing pressure. As the culvert roof deflects, the overlying soil mass follows downward losing its ability to carry the pressure induced by the footing. As a consequence, Boussinesq solution becomes no longer valid. In the meantime, the vertical pressures at both footing base and culvert roof decrease with increasing culvert size. The figure also shows a little tensile stress at the roof center of larger culverts. The development of tensile stress can be attributed to the sagging effect resulted from the greater culvert roof deflection. At the bottom of the sagged soil mass, tension develops in both horizontal and vertical directions.
Below culvert bottom, the vertical pressure distribution of larger culverts is greatly influenced by geostatic stress, which increases linearly with increasing depth. For smaller culverts, however, the vertical soil pressure is still under a considerable influence of footing pressure. As a consequence, the shape of distribution curve for smaller culverts differs from that of larger culverts. Meanwhile, the magnitude of vertical pressure decreases with increasing culvert size, primarily because the influence of footing pressure diminishes with depth. It is also seen that for larger culverts, a tensile vertical stress occurs. A possible explanation for the development of tensile stress is that the soil at the bottom is squeezed upward from both sides due to the upward deformation of culvert bottom.
Figure 2. Vertical Soil Pressure Distribution below Footing Center for different Culvert Widths
(1 psf = 0.048 kN/m, 1 ft = 0.3 m, 1 in. = 25.4 mm)
For a given culvert size of 12 ft (3.7 m), the vertical soil pressure below the footing center decreases with decreasing culvert wall thickness as shown in Figure 3. This is as would be expected, because thinner culverts are more flexible so that the culvert roofs undergo more deflection resulting in a lower vertical soil pressure. When compare the culvert at 3 ft (0.9 m) with that at 12 ft (3.7 m) depths both having a 3 in. (76.2 mm) thick wall, the vertical pressure near the footing base is much higher for the 12 ft (3.7 m) deep culvert primarily due to the thicker soil layer between the footing and culvert for the 12 ft (3.7 m) deep culvert. The thicker soil layer is able to support a greater load. Furthermore, the vertical pressure above the culvert roof is much smaller for the 12 ft (3.7 m) deep culvert than that of the 3 ft (0.9 m) deep culvert, simply because of the diminishing effect of footing pressure with depth. Below the culvert bottom, the vertical soil pressure is influenced more by bottom deflection than by footing pressure. As before, the negative vertical pressure immediately below the center of culvert bottom can be attributed to the upward soil displacement.
Figure 3. Vertical Soil Pressure Distribution below Footing Center for different Culvert Depth and Thicknesses
(1 psf = 0.048 kN/m2, 1 ft = 0.3 m, 1 in. = 25.4 mm)
On the vertical plane located at the culvert edge, the vertical pressure distributions for three different culvert sizes of 6 ft (1.8 m), 9 ft (2.7 m), and 12 ft (3.7 m) with two culvert wall thicknesses of 1 in. (25.4 mm) and 6 in. (152.4 mm) are presented in Figure 4. Note that all four culverts have a depth of 3 ft (0.9 m) and the footing base is at 3 ft (0.9 m) below the ground surface. It is interesting to see that along the vertical plane, the vertical pressure increases with depth to a maximum near the upper culvert corner. Between the upper and bottom corners, the vertical pressure first decreases then increases to another maximum value exceeding that at the upper corner. Below the bottom corner, the vertical pressure also decreases first then increases with depth. As the depth increases further, the four curves tend to merge together approaching the geostatic stress distribution.
Between the top and bottom corners, the considerably greater pressures near the corners than near the mid point reflect the effect of stress transfer from mid portion to corners in response to the deformed culvert wall. A significantly different shape of pressure distribution curves between the 6 in. (152.4 mm) and 1 in. (25.4 mm) walls is seen. It is primarily due to the difference in culvert wall deformation behavior between the 1 in. (25.4 mm) and 6 in. (152.4 mm) walls. It appears that the greater stiffness of 6 in. (152.4 mm) wall undergoes much less lateral deformation so that the pressure increases somewhat linearly with depth.
Figure 4. Vertical Soil Pressure Distribution along Culvert Edge
(1 psf = 0.048 kN/m2, 1 ft = 0.3 m, 1 in. = 25.4 mm)
Vertical Soil Pressure Distribution on Horizontal Planes
Under a footing pressure of 5000 psf (239.4 kN/m2), the vertical soil pressure on the horizontal plane at culvert roofs is presented in Figure 5 for three culverts, each having a depth of 3 ft (0.9 m), a width of 12 ft (3.7 m) and a wall thickness of 1 in. (25.4 mm), 3 in. (76.2 mm), or 6 in. (152.4 mm). Also shown in the figure is the vertical pressure distribution on the same horizontal plane computed from Boussinesq solution for no-culvert condition. As shown, for no-culvert condition, the Boussinesq distribution resembles a bell shape having its peak under the footing center. On the top of culverts having 1 in. (25.4 mm) and 3 in. (76.2 mm) wall thicknesses, the vertical pressures increase with horizontal distance from a minimum at the mid section to a maximum at the culvert edge then decrease abruptly to a constant value equal to the geostatic vertical stress. The stress concentration at the edge is primarily due to stress transfer from the mid section to the edge of culvert caused by excessive culvert roof deflection. For the culvert having 6 in. (152.4 mm) wall, the roof is stiffer and deflects less compared with the culverts with 1 in. (25.4 mm) and 3 in. (76.2 mm) wall. As a result, the pressure distribution at the mid section bears some resemblance to the Boussinesq distribution.
Figure 5. Vertical Soil Pressure Distribution on Horizontal Plane at Top of Culvert with D3W12.
(1 psf = 0.048 kN/m2, 1 ft = 0.3 m, 1 in. = 25.4 mm)
A comparison of vertical pressure distributions on the horizontal plane at the top of culvert between 12 ft (3.7 m) and 3 ft (0.9 m) deep culverts both having a width of 12 ft (3.7 m) with a wall thickness of 3 in. (76.2 mm) is shown in Figure 6. As illustrated in the figure, the shape of the two distribution curves is very similar. The similarity in the shape of distribution seems to indicate that the degree of stress transfer caused by culvert top deflection is approximately the same between the two culverts. However, except near the center of the culvert roof, the vertical pressure on the horizontal plane at the top of culvert is greater for the 12 ft (3.7 m) than 3 ft (0.9 m) deep culverts. Such a difference can be attributed to the greater geostatic stress for the 12 ft (3.7 m) deep culvert. The smaller stress near the center of the culvert roof for the 12 ft (3.7 m) deep culvert can be a result of the slightly greater deflection than the 3 ft (0.9 m) culvert.
Figure 6. Vertical Soil Pressure Distribution on Horizontal Plane at Top of Culvert with W12t3.
(1 psf = 0.048 kN/m2, 1 ft = 0.3 m, 1 in. = 25.4 mm)
Normal Soil Pressure Distribution along Culvert Periphery
The distribution of soil pressure normal to the culvert periphery is plotted against the central angle in Figure 7. With the origin at culvert center, the central angle is measured clockwise from the vertical center line. The data are for 12 ft (3.7 m) wide square culverts with 1 in. (25.4 mm), 3 in. (76.2 mm), and 6 in. (152.4 mm) wall thicknesses and located at 3 ft (0.9 m) and 12 ft (3.7 m) below the footing base. Note that the figure presents only the distribution along the right half of the culvert. As shown, the normal soil pressure distribution is not quite symmetrical about the 90° angle, which is the horizontal line through the mid wall height. This is as would be expected because the normal soil pressure in the upper half of culvert is influenced more by footing pressure, while in the lower half the dominant effect is geostatic stress.
It is interesting to see from Figure 7 that, because of the inward deflection of culvert side wall, the normal soil pressure distribution along the culvert side also reveals stress transfer from the mid height to the corners. Meanwhile, for the 3 ft (0.9 m) deep culvert with 1 in. (25.4 mm) wall, the normal soil pressures near the mid sections of top, side, and bottom are almost equal to zero. At these locations, the pressure becomes larger as the wall thickness increases. For the two culverts with 6 in. (152.4 mm) wall thickness, the normal soil pressure is significantly greater for the 12 ft (3.7 m) than 3 ft (0.9 m) deep culverts, except near the centers of culvert roof and bottom.
Figure 7. Normal Soil Pressure Distribution along Box Culvert Periphery.
(1 psf = 0.048 kN/m2, 1 ft = 0.3 m, 1 in. = 25.4 mm)
DISCUSSIONS
The results of analysis presented above indicate that soil pressure distribution strongly depends on culvert size and wall thickness. The dependency reflects the effect of culvert wall /roof stiffness on soil pressure distribution. As the stiffness decreases, either due to the decreased wall thickness or increased wall span, the wall deflection increases and causes stress transfer from mid section to the supports at both ends. Thus, the determination of soil pressure distribution requires a consideration of soil-culvert interaction.
An essential criterion of structural design of box culverts is the tensile stress developed in the culvert wall. The design must be such that the soil pressure-induced tensile stress in the wall is within the limit of tensile strength of culvert material. Since the tensile stress varies with wall deflection and the deflection is coupled with the soil pressure, an iteration process is needed in the rational structural design of box culverts.
The results of study have demonstrated that the footing-induced soil pressure distributions around box culverts are strongly influenced by interactions between the culvert and the surrounding soil. While the results provide insight into the nature of soil-culvert interaction, additional data are needed to establish a broader database not only for better understanding of the interaction mechanism but also for the development of a rational method for structural design of square box culverts overlain by strip footings.
SUMMARY AND CONCLUSIONS
The behavior of soil pressure distribution around square reinforced concrete box culverts induced by overlying strip reinforced concrete footings was investigated. The study was made using a 2-D finite-element computer program developed based on the theory of elasto-plasticity. Conditions analyzed included a 3-ft (0.9 m) strip footing centered above and parallel with square box culverts having four different sizes together with three levels of wall thickness and four levels of depth to culvert. Variables analyzed were the vertical soil pressure distributions along the vertical planes through culvert center as well as culvert edge, and along the horizontal plane through top of culvert. Also analyzed was the normal soil pressure distribution along culvert periphery.
Results of analysis show, among other things, that the vertical soil pressure distribution varies considerably with culvert size, wall thickness, and depth to culvert. On the vertical plane through culvert center, the vertical pressure distribution between footing base and culvert top bears some resemblance to the Boussinesq vertical stress distribution for smaller culverts. For larger culverts, however, the vertical soil pressure decreases with depth, and the magnitude of pressure decreases with increasing culvert size. Below the culvert bottom, the vertical soil pressure increases with depth. Along the vertical plane at culvert edges, the vertical soil pressure increases from zero at the ground surface to a maximum at the culvert shoulder then decreases to a minimum at mid height of the culvert side and increases again to another maximum value at the bottom. From the bottom, the vertical pressure first decreases abruptly then increases gradually with depth. On the horizontal plane at top of culvert, the vertical soil pressure reaches a maximum at both shoulders. The minimum value of vertical pressure occurs at the mid point of the roof for culverts with thinner walls. For culverts with thicker walls, however, the pressure at mid point is not the smallest. Along the culvert periphery, the normal soil pressure distribution on top and bottom of culvert is the same as the distribution on the horizontal plane mentioned before. On the culvert sides, the normal pressure distribution also shows stress transfer from mid section to both corners.
It is concluded that the soil pressure distribution around box culverts induced by the overlying footings is strongly dependent on culvert-soil interactions. Specifically, the soil pressure causes culvert wall/roof deflections and the deflection results in a readjustment of soil pressure distribution. Thus, iteration procedures are needed to obtain an optimal structural design of box culverts. The results of study have provided an insight into soil-box culvert interaction mechanism. More data are needed, however, to establish a database for the development of a rational method for design of square box culverts overlain by strip footings.
references
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