2.2 PREVIOUS STUDIES RELATED TO STATIC LOADING:
2.2.1 Arching around Box Culverts
The redistribution of ‘Free Field’ stresses as the result of the presence of buried structure will result in a decrease in loading over the deflecting or yielding areas of the structure, and an increase over adjoining rigid or stationary parts. This transfer of load is termed arching (Terzaghi, 1943). Shear resistance tends to keep the yielding mass in its original position resulting in a change of the pressure for both the yielding surface and the adjoining area of soil. If the yielding part moves downward, the shear resistance will act upward and reduce the stress at the base of the yielding mass as shown in Figure 2.1. On contrary, if the yielding part moves upward, the shear resistance will act downward to cause increase in the stress at the support of the yielding part (Bjerrum et al., 1972).
The experimental study by Terzaghi (1943) is the most famous one for studying arching in sand using a yielding horizontal trap door for plane strain conditions. The vertical stress on the trap door up to the surface resulting from his experiment is shown in Figure 2.2. The sand cover (H) is more than four times the width of the trap door (b) with the distance (h) measured upward from the trap door. Reduction in the vertical stress ('v) was observed starting from h/b = 2.5 down to the trap door. At h = H, the vertical stress (i.e. immediately above the trap door) is 10% of the vertical stress if there is no arching ('vh).
Figure 2.1: Stress distribution in the soil above yielding base (After Bjerrum et al., 1972)
Figure 2.2: Terzaghi’s trapdoor experiments (note: z measured upwards from trapdoor). (After Bulson, 1985).
Janssen (1895) assumed that the vertical pressure on the yielding element of width b and at depth z as shown in Figure 2.3 is equal to the difference between the soil pressure due to the weight of the soil prism ABCD above the element and the frictional resistance along the sides of the element. The shear resistance can be determined by:
c tan (2.1)
where, c′ is the effective cohesion of the soil, ′ is the effective angle of internal friction of the soil and ′ is the normal effective stress on the plane of shearing. The ratio between the horizontal and vertical stress is an empirical constant K. Further experiments by Terzaghi (1943) suggested that K increased from unity immediately over the yielding surface (of width b) to 1.5 at an elevation of b above the centerline of the surface. At elevations greater than 2.5b, the movement of the trapdoor did not alter the state of stress.
q z dz b D A B C K′v c’+K′vtan ’ ′v ′v+d′v Figure 2.3: Janssen’s analysis (After Bulson, 1985).
By resolving the vertical equilibrium on a unit length of the section, the following relation can be obtained:
dzb d b 2cdz2K dztan
b v v v v (2.2)
where, is the unit weight of the soil. With a uniform surcharge q at the surface, this equation can be solved for ′v
b z K q b z K K b c b v 2 tan exp 2 tan exp 1 tan 2 2 (2.3)The negative exponential indices in Eqution 2.3 imply ‘active arching’, which occurs due to the downward movement of the trapdoor. In the case of ‘passive arching’, the door will move upwards and the indices will be positive (Bulson, 1985).
Arching can be either active or passive depending on the relative stiffness between the soil and the structure. Active arching occurs when the structure is more compressible than the surrounding soil as illustrated in Figure 2.4a. When the system subjected to loads, the resulting stress distribution on plane AA and BB is similar as shown in Figure 2.4b, where the stresses on the structure are less than the soil. If the structure deforms uniformly on plane AA and BB, the stresses on it tend to be lower at the edges and that is because of the mobilized shear stresses in the soil (Evans, 1984).
Passive arching occurs when the soil is more compressible than the structure as shown in Figure 2.5a. In this case, large deformations and mobilized shear stresses will happen in the soil and causes the total pressure on the structure to increase while the pressure decreases in the adjacent soil as illustrated in Figure 2.5b. If the structure deforms uniformly, the stresses will be high at the edges and low at the center (Evans, 1984).
Figure 2.5: Passive arching (After Evans, 1984)
Buried structures usually do not have uniform deformations and that cause a more complex stress distributions than active and passive stresses shown above. As an example, a structure that has more flexible spans towards the center can result in a deformation pattern as presented in Figure 2.6. The horizontal and vertical stress distributions suggest that the faces of the structure experience active and passive arching at the same time.
Figure 2.6: Typical deformation and stress distributions around rectangular structure with flexible sides (After Evans, 1984)
Allen and Russ (1978) analyzed loads on box culverts under high embankments and presented the active and passive arching in the diagrams illustrated in Figure 2.7. The culvert dimensions were 1.90 m × 1.90 m and the height of fill was 21.3 m. The height of equal settlement He was equal to 2.5BC. In the active case, the failure lines were drawn tangent to the line of (45o + '/2) from the horizontal line, located at the top of the culvert, up to the plane of the equal settlement and vertically thereafter. For a high embankment they determined that the case was a passive arching condition because the failure lines were tangent to the line of (45o - '/2) from the horizontal to the elevation of the plane of equal settlement and vertical thereafter. Allen and Russ (1978) show the result of a photoelastic model that was constructed to simulate the positive projection box culvert under high fill using gelatin material around it as shown in Figure 2.8. The lines indicate that the arching occurs in such elastic material, and the pressure above the culvert is greater than the weight of the material above it. The darkest areas in the photograph are areas of zero strain, and from this observation, it is apparent that two large strain bulbs formed at the corners of the culvert and reached the surface.
Figure 2.7: Active and Passive arching conditions over a box culvert (after Allen and Russ, 1978)
Figure 2.8: Photoelastic modeling of arching and stress bulb above the box culvert (after Allen and Russ, 1978)
To examine the mechanism of arching, Stone and Newson (2002) performed centrifuge tests on the rectangular culvert shown in Figure 2.9. Two tests were performed under a soil cover of 200 mm; Test A: stiff sides and flexible top and base, and Test B: flexible sides and stiff top and base. The results were collected at three g levels: 5g, 10g, and 20g.
For Test A, the results of the loads are consistent with active arching due to the inward deflection of the top slab. The loads attracted to the top slab were about 45% of the theoretical soil load. Since the ratio of the measured to prism loads were almost constant over all the g levels, it can be deduced that the same soil volume is involved in the loading of the top slab and the attracted loads to the top slab is independent of its deflection. For Test B, the load ratio increased with g level. At the 5g level, the load ratio was the same as in Test A, but after that started increasing until it exceeded 100% at 20g. As shown in Figure 2.9, it is assumed that if sufficient inward movement of the side wall occurs, then active Rankine zones will form right beside them. As a result of that, an active soil wedge (a-b) extending toward the free surface will develop. This wedge will extend to (a-b-c), which results in increasing the load attracted to the culvert. This indicates that the volume of soil involved in the loading on the top slab increases as the deflection of the side wall increases and explains the reason behind the increase in the load ratio as the g level increases.
Figure 2.9: Schematic diagram illustrating the arching mechanism for Test A and Test B (after Stone and Newson, 2002)
Newson et al. (2006b, 2007) revisited the tests shown above, but with a 100 mm soil cover. Similar conclusions were observed for Test A, and the expected positive arching occurred above the culvert. The culvert top slab attracted fewer loads than expected based on the calculation of the overburden due to the rectangular prism of soil above. On the top slab of the culvert, the shaded area on the left-hand side shows the volume of soil applying loads to the top slab in the absence of arching, while the right-hand side shows the reduced volume of soil involved in loading when arching occurs. This occurs due to the transfer of load to the stationary mass of soil as shear stresses are mobilized on the vertical shear plane.
For Test B, the assumed soil arching mechanism was as presented in Figure 2.10. Above the top slab of the culvert, the self weight of the prism (iklm) is partially transferred to the next prism (fmih) and to the corners of the culvert causing positive arching over the top slab as it deflects under the loading. When the sidewall deflects laterally, the soil in (acfd) will move in the same deflection direction but the loads are supported partially by the shear stresses on the surfaces (ac) and (df). This will increase the loads experienced by the upper parts of the culvert sidewalls.
2.2.2 Investigation of Soil Structure Interaction for Box Culverts