Flexible Pipes in Trenches with Stiff Clay Walls

Transcription

1 Flexible Pies in Trenches with Stiff Clay Walls D. A. Cameron University of South Australia, South Australia, Australia J. P. Carter University of Sydney, New South Wales, Australia Keywords: flexible buried ies, trench, state arameter, finite element analysis ABSTRACT: This aer describes numerical modelling of field tests involving traffic loading on flexible upvc ies buried in sand within trenches formed in stiff clay. The modelling was erformed using three-dimensional finite elements with the backfill soil reresented by a state arameter model. A major difficulty was identified with current simlified methods of estimating deflections of ies buried in trenches and subjected to construction loading; rediction of the stress distribution above a ie is difficult with a yielding backfill. A reliminary method of estimating stresses is roosed, based on results from the finite element analyses, which can be used with simle methods of calculating ie deflections. 1 Introduction Flexible ie design has traditionally treated the ie as a thin elastic ring and assumed that most of the ie suort is afforded by the lateral sidefill ressure, which is generated as the ie deflects downward and outward under vertical load. This lateral earth ressure resonse deends on the flexibility of the ie, the nature of the backfill and its level of comaction, as well as the stiffness of the sidewalls of the trench containing the buried ie. It is also a common assumtion that the lateral diametric increase of the ie is equal to the vertical diametric reduction, i.e., ie deformation is ellitical. Early models for design of flexible ies required a constant horizontal modulus of soil reaction, E', (or Sangler s modulus ) relating the lateral ie dislacement and the generated soil reaction ressure. The Sangler modulus is not a true soil roerty as it is simly the alied ressure divided by the horizontal diametric strain of the ie, which deends uon the ie stiffness and its dimensions. The Iowa or Sangler s equation for estimating the ie deflection may be exressed as:.1w ε y = ( ) (1) EI +.61E' 3 R where ε y = y/(r) = vertical diametric strain (deflection/diameter of ie), w = the ressure above the ie crown (assumed uniform) and (EI) = flexural stiffness of the ie. More rigorous theoretical analyses have been alied to the roblem of the deflection of flexible buried ies for dee burial and embankment loading (e.g., Gumbel and Wilson, 1981; Moore, 1993), which suggest shortening of the ie due to hoo thrust should be included in design. According to Dhar et al. (), AASHTO have incororated the following term for rediction of average deflection of the ie due to hoo force to be used in conjunction with Eqn 1:

2 FVAw ε H = () ( EA) +.57E R where (EA) is the cross-sectional stiffness of the ie; F VA is a vertical arching factor, given as: S H 1.17 F VA = (a) SH +.9 where S H, the hoo stiffness factor, = (E'R)/(EA). In this case the vertical diametric strain is the sum of -ε y and -ε H, while the horizontal diametric strain is the sum of ε y and -ε H. Dhar et al. () reorted good corresondence between three laboratory ie tests and the AASHTO aroach. Current study The influence of wheel loads on buried ies was investigated in a series of laboratory and field tests (Cameron, 4). Only the field tests are reviewed in this aer. Profiled ies of upvc were buried in trenches cut into stiff clay and dry, oorly graded sand was used as backfill. The backfill surface was loaded by a mm wide by 5 mm long rigid late, until either the design wheel ressure or the maximum diametric strain (5%) of the ie had been reached. The late was laced above the centre of the ie with the long side arallel to the longitudinal ie axis. Details of the field tests are rovided in Table 1. The density index of the sand was assessed by dynamic cone enetration testing. Pressure cells were laced within two of the installations. Test Nominal ie dia. (mm) Table 1. Details of the buried ie tests Height of backfill (mm) Trench width (mm) Estimated density index (%) Surround Backfill Pressure cells? F3/ no F375/ to 85 yes F45/ yes F45/ to 9 8 no 3 3D finite element analyses A 3D soil element was develoed for the AFENA finite element ackage (Carter and Balaam, 1995) to reresent the sand backfill. This element incororated a non-linear, state arameter constitutive model (Cameron and Carter, 5). Inut data included initial void ratio and stress state and the eight material constants needed to define the non-linear elastic re-yield behaviour. The four remaining constants defined the critical state condition and the relationshi between the current soil strength and state arameter. The natural soil in which the trench was formed was assumed to be linearly elastic and loaded in an undrained condition. An undrained Young s modulus between 3 and 9 MPa was assumed, based on the results of triaxial tests. The ie was modelled as an elastic continuum, using 8-noded, non-conforming, aralleleied elements with 8 Gauss oints (after Taylor et al., 1976). In order to balance the secified values of cross sectional stiffness, EA, and flexural stiffness, EI, for the rofiled ies, both the equivalent thickness of the ie, t, and the equivalent Young s modulus, E ie, were adjusted to rovide the correct axial and bending stiffness. The ie elements were used successfully in analysis of ie stiffness testing, in which a 3 mm length of ie was laid on a flat surface and vertically loaded

3 with a line load alied to its crown (refer ASTM D41-96a). The differences in deflections redicted numerically and by the theoretical solution for the deflections caused by vertical oint loading of a ring and numerical analyses ranged between 1 to +4.5%. 3.1 Modelling the buried ie tests The trench widths were nominally 6 mm and 75 mm for the 3 mm diameter and other size ies, resectively. The mesh was extended ast the trench wall into the native soil by an extra.54 trench widths, on average. An end view of a mesh at z = is rovided in Figure 1. Initial stresses in the soil were small comared to the alied surface ressures, and for convenience were assumed to be uniform. External loading through the rigid late was simulated with enforced, increments of vertical dislacement of the late nodes (. mm er increment)..375 m.775 m.6 m backfill 1.15 m.1 m.45 m.15 m ie Side Soil bedding Figure 1. Tyical mesh for simulation of field tests, 45 mm dia. ie with.6 m of backfill cover 4 Discussion of results Plots of redicted and observed diametric ie strain against alied ressure for the tests are rovided in Figures to 4. The test deflections are relative to the ie osition after installation of the backfill; initial construction deformations were not considered in the current study. The match between the exerimental and redicted behaviour was accetable, although the FEA redictions tended to overestimate the vertical diametric strain of the ie in the early stages of loading and could not match the rate of increase of deflection that was recorded subsequently. The redictions of the alied surface ressure for a 3% vertical diametric strain were within 15 to 5% of the observed values, with the 45 mm diameter ies being least well matched. Increasing the natural soil s Young s modulus to 9 MPa, increased the required ressure further, which was as exected; the stiffer the natural soil, the less the ie deflections. The ratio of the current cover height to the initial cover height lotted against the alied ressure has been included in Figures and 3 for tests, F3/3 and F375/7.The reduction in the cover height to the crown of the ie was adequately matched u to a surface ressure of 3 to 4 kpa. U to this oint, settlement was almost roortional to alied ressure. At greater ressures, dramatic

4 4 horizontal Test F3/3 1. Diametric Strain (%) D FEA, side soil, E = 3MPa AASHTO and Eqn 3 vertical.7 Cover height ratio Figure. Soil deflection data for the test on 3 mm diameter ie with 475 mm of cover horizontal Test F375/7 1. Diametric Strain (%) D FEA, side soil, E = 3MPa AASHTO vertical and Eqn 3.7 Cover height ratio Figure 3. Soil deflection data for tests F375/7 with 7 mm of cover F45/3 Test F45/4 Horizontal Horizontal Diametric Strain (%) - 3D FEA, side soil, E = 3 MP 3D FEA, side soil, E = 9 MP Diametric Strain (%) - -4 Vertical AASHTO and Eqn 3-4 Vertical Figure 4. Soil deflection data for the tests on 45 mm diameter ie

5 losses of cover height were observed as the rigid late enetrated into the soil cover above the ie. In contrast, the FEA redicted an almost linear increase of settlement over the full range of alied ressure. 4.1 Earth ressures Earth ressures in the backfill redicted by the FEA are shown against alied surface ressure in Figure 5, along with the exerimental observations. In the case of test F375/7, vertical ressure 15 mm above the crown of the ie at the central cross-section (z = ) was redicted reasonably well, u to an alied surface ressure of 65 kpa, and thereafter was underestimated. This observation is in reasonable agreement with the vertical deflection data; the vertical diametric strain was greater than the FEA estimate after 8 kpa or so. For test F45/3, the ressure measured 15 mm above the crown of the ie at the central cross-section rose sharly after an alied surface ressure of 35 kpa, due to acceleration of late settlement. The FEA significantly underestimated this increase of soil ressure. By the end of the test, the FEA rediction was just 65% of the observed ressure. For both tests, the vertical and horizontal ressures in the at the level of the sringline of the ie were also underestimated, articularly for test F375/7 (aroximately 5%). The FEA estimates of vertical soil ressures for test F45/3 were considerably closer (aroximately 7%), however the measured earth ressures in the sidefill were less than half those of test F375/7. 5 Test (V) 15 mm above ie 5 Soil Pressure (kpa) FEA (V), 15 mm above ie Test (V), FEA (V), Soil Pressure (kpa) Test (H), FEA (H), a) Test F375/7 (E = 3 MPa) b) Test F45/3 (E = 9 MPa) Figure 5. Earth ressure comarisons (H = horizontal ressure, V = vertical ressure) 4. Predicted deflections, hoo forces and moments Predicted crown and sringline deflections were greatest directly below the centre of the loaded area. Although an increase in diameter laterally was redicted directly below the loaded area, the ie was anticiated to contract slightly away from the end of the loading late. As exected, triling the Young s modulus of the side soil in the 45 mm ie FEA reduced the deflections. Hoo forces and moments at z = were determined from the FEA solutions for deflections at a late dislacement of 4 mm (test F3/3 did not reach this dislacement) and are shown in Figure 6. The maximum ositive moment was estimated to occur at an angle of 15 to from the ie crown, near a ossible inflection oint of the ie. The minimum negative moment was redicted almost midway between the crown and sringline. Tensile hoo force was a maximum at the ie

6 crown. The hoo force was comressive below the midoint between the crown and the sringline. The ie cross-section was considered under-stressed for the measured tensile caacity of the upvc ie of 33 MPa. The greatest redicted stress was 3%. of the tensile caacity. 5 Preliminary ie design model 5.1 Aroximation of soil ressure The FEA of the four ie tests were backed u with data from FEA of three fictitious tests, to extend the redictions for further combinations of installation geometry and ie diameter. The extra analyses are referred to as F37 (3 mm ie, with 7 mm of cover), F3745 (375 mm ie with 45 mm of cover) and F4545 (45 mm ie with 45 mm of cover). All extra analyses were erformed with all soil in the installation having an initial density index of 75%. Although the redicted vertical ressure 15 mm above the ie crown varied markedly with distance across the trench, the average ressure across the ie diameter at this height varied almost linearly with the alied surface ressure for all the analyses. The redicted average ressure above the ie has been lotted against alied surface ressure in Figure 7 for three tests. All ressures were normalized against atmosheric ressure, a (1 kpa assumed). From these examles, the average ressure varied linearly with the surface loading after a certain ressure, which aeared to deend uon installation geometry. The equation to the line can be exressed by a gradient and a constant. The constant is ossibly related to resistance to loading from the mobilized shear stress against the vertical trench walls. It was found that the constant was directly roortional to Z/B, where B is the width of the trench and Z is the height of backfill above the 15 mm level above the ie. As well, the constant was a function of the relative stiffness of the backfill above the ie and the natural soil, E 1 /E 4. In the search for an exression for the gradient, an equivalent radius of loaded area, r, was first adoted for the late. Other geometric factors were found to be imortant, such as the relative width of the loaded area, b, to the deth, Z, as well as the ratio of the trench width, B, to the ie diameter, D. The stiffness of the ie, S, and the Young s moduli of the soil seemed to be of little significance when comared with the geometric factors. However in these buried ie tests, S, was low and varied between.76 and 1. kn/m ; E 1 /E ranged between.7 and 1.5 (E is the Young s modulus of the sidefill) and E 1 /E 4 ranged between.1 and.35. The final exression relating the alied surface ressure, surface, to the average soil ressure 15 mm above the crown, av,15mm, is given in Eqn 3. In the early stages of loading, negative ressures generated by the equation are ignored. The dashed lines in Figure 7 are based on the roosed exression and comare well with the FEA data. av,15mm a b B.7.14 Z D Z 1 E = surface (3) B E 1 a 4 1 a ( r ) Z 5. Estimates of ie movements The findings of the FEA were investigated in an attemt to roduce simle guidelines for ie designers, without resort to finite element analysis. Subsequently the ie deflections redicted by

7 Moment (kn.m er m length).1 crown invert. sringline Angle measured from x-axis (degrees) F375/7 E = 3 MPa F45/3 E = 3 MPa F45/3 E = 9 MPa F45/4 E = 3 MPa F45/4 E = 9 MPa Hoo force (kn er m length) 1 invert crown -1 sringline Angle measured from x-axis (degrees) Figure 6. Distribution of moments and hoo forces about ie cross-section at z = Normalized Average Pressure above Pie, av 15 mm / a (kpa) Normalized Surface Pressure, surface/ a (kpa) F3/3 F4545 F45/4 Sangler's Modulus, E' (MPa) AS/NZS Trendline Density Index (% ) Figure 7. Average soil ressure 15 mm above crown with ressure, from FEA and from Eqn 3 Figure 8. Sangler s soil modulus calculated from ie tests the AASHTO equations (basically Eqns 1 and ) were comared with the exerimental data. Since Sangler s modulus is needed, the modulus was evaluated for each ie test in which horizontal earth ressure was measured close to the ie sringline. The modulus was estimated as the ratio of the horizontal earth ressure to the lateral diametric strain, after adjustment of the earth ressure for the relative distance of the ressure cell from the side of the ie. The exerimentally determined values of Sangler s modulus have been lotted against the estimated initial density of the sand in Figure 8. This Figure includes data from laboratory ie tests on similar ies. The recommendation of Standards Australia (1998) for the variation of the modulus of soil reaction with density index is also shown on this lot. It is evident that these values rovide a satisfactorily conservative estimate of Sangler s modulus for design. The emirical earth ressure aroximation in Eqn 3 and values of modulus of soil reaction from Standards Australia, 1998, were combined with the AASHTO Eqns to roduce estimates of ie movements in the four field tests. No correction was made to the soil modulus for the influence of the stiffness of the adjoining clay of the trench walls. The vertical arching factor was calculated to be 1.4 for all ies. Plots comaring the estimates of ie deflection with the actual deflections

8 have been included in Figures to 4 ( AASHTO and Eqn 3 ). It is evident that the lots reflect the measured trends well, although generally the estimates are conservative, resumably due to the use of conservative estimates of Sangler s modulus and the questionable alicability of an arching factor, devised for ies under dead weight loads. 6 Conclusions Three-dimensional finite element analysis of the field tests revealed that the redicted moments develoed in the articularly flexible ies investigated in this study are relatively insignificant. However, the hoo forces in the ie ring can be areciable, although remaining within the caacity of the ie. Simle methods of estimating ie movements require an estimate of the vertical ressure above the ie crown. Although this ressure is assumed to be uniform, the FEA clearly showed the soil ressure was concentrated closer to the ie crown. Estimates of ressure from the theory of elasticity under-redicted the vertical ressure below the centre of the loaded rectangular area. Further study of the FEA of the ie tests revealed that the redicted average vertical stress above the ie, was directly roortional to the average alied surface ressure. A relatively simle emirical exression has been derived subsequently between the two ressures. The exression requires the geometry of the installation and the ratio of the Young s modulus of the backfill soil to the Young s modulus of the natural soil forming the trench walls. Conservative design estimates of vertical ie deflection can be made with the AASHTO equations over the initial stages of external loading, using a modulus of soil reaction suggested by Standards Australia (1998), and an average vertical soil ressure determined from Eqn 3. Simle estimates of ie deflection arising from traffic assing over an unaved backfill are not ossible at this stage for the case of greater loadings, largely owing to the indeterminate nature of backfill settlement and the uncertainty of soil ressure develoment. Indeed, even the FEA as described in this aer was unable to yield good redictions at greater alied surface ressures and ie strains. 7 References American Society for Testing Materials (1996). Standard Test Method for Determination of External Loading Characteristics of Plastic Pie by Parallel Plate Loading. ASTM D41-96a, Annual book of ASTM Standards, V 8.1 Cameron, D.A. (4). Analysis of Buried Flexible Pies in Granular Backfill Subjected to Construction Traffic. PhD thesis in Civil Engineering, University of Sydney. Cameron, D.A. and Carter, J.P. (5). A Model for Sand with Limited Dilation. Proceedings of 11th International Conference of IACMAG, Turin, Italy, June. Carter, J.P. and Balaam, N.P. (1995) AFENA Users Manual. Centre for Geotechnical Research, The University of Sydney. Dhar, A.S., Moore, I.D. and McGrath, T.J. (). Evaluation of Simlified Design Methods for Buried Thermolastic Pie, Proc. of Pielines, Beneath our feet: Challenges and Solutions, American Society of Civil Engineers, Cleveland, OH, July, 11. Gumbel, J.E. and Wilson, J. (1981). Interactive Design of Buried Flexible Pies - a Fresh Aroach from Basic Princiles. Ground Engineering, May 1981, V14 No 4, Moore, I.D. (1993). Structural Design of Profiled Polyethylene Pie. Part 1 Dee Burial. Research Reort, University of Western Ontario, Geotechnical Research Centre, GEOT-8-93, March. Standards Australia/Standards New Zealand (1998). Buried Flexible Pies: Part 1: Structural Design. AS/NZS566.1:1998, Standards Australia, Homebush, Sydney, Australia.