Design Loading on Deeply Buried Box Culverts

Design Loading on Deeply Buried Box CulvertsKyungsik Kim1and Chai H.Yoo,F.ASCE2Abstract:The current American Association of State Highway and Transportation Officials(AASHTO)standard specifications for highway bridges and AASHTO LRFD bridge design specifications stipulate the computation of the design load on the box culvert primarily based on research by Marston and Spangler.Although this procedure may be applied conservatively for most ordinary culverts, an opportunity exists to evaluate a more realistic soil-structure interaction behavior based on modernfinite element analyses of deeply buried concrete box culverts.The Duncan soil model,represented by hyperbolic stress–strain curves,has been used for properties of backfill and in situ soil.The backfill heights are varied from15.2to61.0m͑50–200ft͒for the embankment condition and15.2–45.7m ͑50–150ft͒for the trench condition.An optimum combination of parameters has been identified for use in the imperfect trench instal-lation method.The data from several hundred hypothetical models with various parameters under three typical installation methods,i.e., embankment,trench,and imperfect trench installation are characterized and quantified using regression analysis.DOI:10.1061/(ASCE)1090-0241(2005)131:1(20)CE Database subject headings:Backfills;Culverts;Embankments;Finite element method;Installation;Soil-structure interaction; Buried pipes.IntroductionCast-in-place or precast reinforced concrete box culverts are widely used throughout the world to provide safe and relatively economical structures for the conveyance of water,vehicles,utili-ties,or pedestrians.Although single cell or multicell box culverts are rather simple structures,the loadings applied to these struc-tures during their construction and subsequent service life can be complex.A culvert as an underground structure causes a redistri-bution of stresses of surround soil layers and the nature of this redistribution influences the load that reaches the structure.The load that reaches the structure is governed by the characteristics of the soil,and the geometry and stiffness of the structure itself. Marston pioneered research on the behavior of underground con-duits analytically and experimentally in the early years of the20th century.One of the outstanding contributions of the Marston theory of loads on buried conduits is its demonstration,by the principles of mechanics,that the load on a structure is affected by installation conditions in addition to the height offill over the structure(Marston and Anderson1913;Marston1930).Succeed-ing Marston’s study,Spangler(1950a,b)has shown that the pri-mary factors influencing the load are associated with the installa-tion conditions that control the magnitude and direction of settling of the soil prism over the structure relative to settling of the exterior soil prisms immediately adjacent to this central soil prism.These relative settling generate friction forces or shearing stresses that are added to or subtracted from the dead weight of the central prism and affecting the resultant load on the structure, as shown in Fig.1.When the relative settlement of the soil prism directly above the structure is less than that of the adjacent soil prisms,as usually found in embankment installations[Fig.1(a)], the layers of soil in the central prism are subjected to an arch shape deformation and the earth pressure on the structure is in-creased,which is referred to as negative arching(Selig1972; Vaslestad et al.1993).Likewise,when the relative settlement of the soil prism directly above the structure is greater than that of the adjacent soil prisms,as depicted in the trench installations in Fig.1(b),the layers of soil in the central prism are subjected to a reverse arch shape deformation and consequently the earth pres-sure on the structure is reduced by the amount of the shearing forces exerted on the central soil prism,which is referred to as positive arching.Up until the twelfth edition of the American Association of State and Highway Transportation Officials(AASHTO)standard specifications for highway bridges(AASHTO1977),the stipu-lated vertical loading was essentially70%of the weight of the earth prism above the top slab.The current AASHTO LRFD bridge design specifications(AASHTO1998)and AASHTO stan-dard specifications for highway bridges(AASHTO2002)require-ments for the design loadings on cast-in-place or precast concrete box culverts are based on the Marston–Spangler theory of the soil–structure interaction.These criteria were adopted by AASHTO in the13th edition of the standard specifications in (AASHTO1983).At the same time,the0.7vertical soil pressure reduction factor,was eliminated.The current AASHTO(2002) soil–structure interaction factors are given byF e1=1+0.20HB c͑1͒for embankment installations1PhD Candidate,Dept.of Civil Engineering,Auburn Univ.,Auburn,AL36849-5337.2Professor,Dept.of Civil Engineering,Auburn Univ.,Auburn,AL36849-5337(corresponding author).E-mail:chyoo@Note.Discussion open until June1,2005.Separate discussions mustbe submitted for individual papers.To extend the closing date by onemonth,a written request must befiled with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on November12,2003;approved on May10,2004.Thispaper is part of the Journal of Geotechnical and GeoenvironmentalEngineering,V ol.131,No.1,January1,2005.©ASCE,ISSN1090-0241/2005/1-20–27/$25.00.20/JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING©ASCE/JANUARY2005F e 2=C d B d2HB c͑2͒for trench installations where F e 1,F e 2=soil–structure interaction factor;H =backfill height;B c =width of the structure;C d =load coefficient for trench installation;and B d =horizontal width of trench.BackgroundIf the result of a soil–structure interaction analysis is to be real-istic and meaningful,it is important that the stress–strain charac-teristics of the soil be represented in a reasonable way.There are two general categories of stiffness and/or compressibility models for soils:(1)models based on constant confining pressure triaxial tests,and (2)models based on uniaxial strain confined compres-sion tests.Soil models in the first category usually incorporate a failure condition,whereas those in the second category do not,because the failure condition is not defined in the uniaxial tests.Examples of the first category are the Hardin model (Hardin 1971),the Duncan model (Duncan and Chang 1970),and the bilinear model (McVay and Selig 1981),while the overburden-dependent model (Katona et al.1976)falls into the second cat-egory.Since the Duncan model is capable of obtaining its com-pressibility parameters from the uniaxial compression test,it is the most general of the models in the first category.Duncan and Chang (1970)expanded the basic hyperbolic stress–strain rela-tionship for soil suggested by Kondner (1963)to derive the tan-gent modulus of elasticity and Poisson’s ratio as follows:E t =ͫ1−R f ͑1−sin ␾͒͑␴1−␴3͒2c cos ␾+2␴3sin ␾ͬ2Kp aͩ␴3p aͪn͑3͒␯t =G −F logͩ␴3p aͪͭ1−Kp aͩ␴3p aͪnͫd ͑␴1−␴3͒͑1−sin ␾͒2c cos ␾+2␴3sin ␾ͬͮ2͑4͒where R f =failure ratio;␴1,␴3=maximum and minimum principal stresses;K =modulus number;n =modulus exponent;c =cohesion intercept;␾=friction angle;G,F,d =Poisson’s ratio parameters;and p a =atmospheric pressure.An attempt to reduce the load on the structure led to the de-velopment of the imperfect trench method (sometimes called the induced trench method )of construction,where larger relative ver-tical displacements of the soil prism above the structure are in-duced by replacing some part of the fill with lightweight material.Baled straw,leaves,compressive soil,or expanded polystyrene blocks are examples of the types of lightweight material that can be used.It was intuitively expected that the result of this light-weight backfill was to introduce a reverse arch shape deforma-tion,causing a radical alteration of the culvert pressure distribu-tion due to the fill subsequently placed above the soft zone.Brown (1967)quantified the pressure reduction effect of the hay layers based on the finite element method of plane elasticity.Vaslestad et al.(1993)revisited this imperfect trench installation method to examine the long-term behavior of the load reduction characteristics on rigid culverts beneath high fills.Linear and nonlinear stress analyses in geotechnical engineer-ing for static and dynamic loading were introduced in the late 1960s and early 1970s.Kulhawy et al.(1969)developed LSBUILD ,a finite element program for stresses and movements in embankments during construction.In ISBILD ,an updated ver-sion of LSBUILD ,Ozawa and Duncan (1973)incorporated a non-linear incremental finite element procedure employing a hyper-bolic stress–strain relationship.Techniques for determining values of the hyperbolic parameter were later presented and updated (Wong and Duncan 1974;Duncan et al.1980).Allen and Meade (1984)used the program ISBILD to predict loads on and settle-ments of concrete culverts that had been actually installed in Ken-tucky.Their results indicated that the finite element method pre-dicted the pressure on box culverts more accurately than other analytical methods available at the time.Katona et al.(1976)developed CANDE ,which is a special-purpose finite element pro-gram primarily intended for the design and analysis of buried culverts.The CANDE program has been widely used for soil–structure analysis and evaluation of buried box culvert designs by a number of researchers,including Chang et al.(1980),Katona and Vittes (1982),and Tadros et al.(1989).CANDE has been upgraded several times,with the latest version being CANDE-89(Musser 1989).Most recently,Kim and Yoo (2002a,b )investi-gated load distributions on deeply buried concrete box culverts using the programs ISBILD and CANDE-89.Finite Element ModelIn the present study,ABAQUS (1998)and ISBILD were used pri-marily for the analysis and CANDE-89for verification and com-parison.The ISBILD code incorporates a nonlinear incremental finite element method employing the hyperbolic stress–strain re-lationship and incremental analysis procedures based on plane strain elements.As required ABAQUS inputs for the material properties of soils,the tangent modulus of elasticity and Poisson’s ratio,are evaluated for each construction layer using Eqs.(3)and (4).In order to quantify the maximum and minimum principal stresses that are needed in Eqs.(3)and (4),the following equa-tions are derived based on the assumption that soil layers are subjected to normal stresses without any induced shear stress:␴1͑i ͒=␥i ͑h i /2͒+͚j =i +1n␥j h j͑5͒Fig.1.Pressure transfer within soil-structure system:(a )embank-ment installation and (b )trench installationJOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ©ASCE /JANUARY 2005/21␴3͑i ͒=R ␴1͑i ͒͑6͒where h i ,␥i =depth and density of the i th soil layer (numberingfrom the bottom to the top of the backfill,or original ground )and R ,the coefficient of lateral pressure,is taken to be 0.5in this study.Taking advantage of the symmetry,only one half of the struc-ture and the surrounding soil were modeled.Exactly the same geometric mesh,boundary conditions,and numbers of construc-tion layers were considered in both ABAQUS and ISBILD runs.It is expected that the width of the surrounding soil taken in the finite element model will affect the analysis results of soil-structure models.Soil–structure interaction factors or the effective densities (equivalent hydrostatic pressure )of the overburden soil increase as the ratio of the width of the soil layer ͑W s ͒to the width of the box culvert ͑B C ͒increases.Kim and Yoo (2002b )reported that the value of the soil–structure interaction factor in-crease gradually until W s /B c reaches approximately 12–14,after which no significant increase was observed.Therefore,W s /B c was taken to be 14in all subsequent analyses of soil–structure models.The width of the surrounding soil was taken to be six times the culvert width in the trench installations and the depth of the foundation below the bottom slab was taken to be four times the culvert height,as models based on a greater amount of in situ soil either vertically or horizontally did not affect the results.Typical soil–structure models with incremental sequences consid-ered for the embankment and trench installations are illustrated in Fig.2.Heights of backfill varied from 15.2m ͑50ft ͒to 61.0m ͑200ft ͒for the embankment installations,and from 15.2m ͑50ft ͒to 45.7m ͑150ft ͒for the trench installations.The constant exterior dimensions of the culvert were 2.4m ͑8ft ͒by 2.4m ͑8ft ͒,with 305mm ͑12in.͒thick walls and slabs.In cases where the height of the backfill exceeds 47.5m ͑150ft ͒,the thickness of walls and slabs may have to be increased to resist the induced shearing force near the wall face.It was found from a series of numerical analyses that the thickness of the culvert did not sig-nificantly alter the total load at the top of the culvert,having an effect only in the order of 1%.The modulus of elasticity of con-crete box culverts was computed to be 25,181MPa ͑3,605ksi ͒,assuming the 28day strength of concrete f c Јis 27.58MPa ͑4,000psi ͒and Poisson’s ratio ␯is 0.20.Hyperbolic parameters for the Duncan soil model were taken from Allen and Meade (1984).Effects of Soil–Structure Interface ModelingAn interface phenomenon that is frequently discussed is slip be-tween the soil and sidewalls of a culvert.Research has shown that there are basically two modeling techniques for the soil–structure interface examination,one of which is a shear element interface and the other a spring element (Katona et al.1976).Fig.3illus-trates the basic concept of these two modeling techniques.Table 1summarizes soil-structure interaction factors evaluated based on models with and without the effect of soil–structure interface.The hypothetical models examined in Table 1were assumed buried at a depth of 30.5m ͑100ft ͒.The dimensions of the box culvert are the same as those used for the comparative study presented here and other details such as the spring constant can be found in Kim and Yoo (2002b ).As can be seen from Table 1,the effect of a potential slip of the soil along the exterior culvert wall is insig-nificant.Therefore,the effect of interface action was not consid-ered in subsequent analyses.Comparison of Finite Element ModelingIn order to assess the validity of the soil modeling technique adopted in this study,an example used by Katona and Vittes (1982)was reanalyzed.The example is a single cell box culvert with interior dimensions of 1.22m ϫ1.22m and 254mm thick walls (4ft ϫ4ft and 10in.thick )tested in Kentucky in 1975.Pressure gage readings were taken at the backfill height of 6.6m ͑21.6ft ͒and 23.5m ͑77ft ͒.Fig.4shows the soil pressure around the box culvert predicted analytically by CANDE and ABAQUS .As can be seen from Fig.4,the analytically predicted values from ABAQUS are fairly close to those from CANDE .The analytically predicted soil pressure values are also fairly close to those deter-mined experimentally.The soil pressure directly above the side-Table parison of Soil–Structure Interaction Factors (SSIFs )SSIFDifferences a (%)Without interface 1.264—Spring model 1.2760.9Shear modelE S /E I bϭ101.277 1.0E S /E I bϭ201.288 1.9E S /E I bϭ501.310 3.6E S /E I bϭ1001.333 5.5E S /E I bϭ5001.3859.6aDifferences were computed based on the case without soil–structure interface.bE S /E I =ratio of the modulus of elasticity of soil to the fictitious shearelement.Fig.2.Incremental sequences:(a )embankment installation and (b )trenchinstallationFig.3.Interface modeling of culvert wall and soil elements:(a )shear element model and (b )spring element model22/JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ©ASCE /JANUARY 2005wall is substantially higher than the soil pressure at the center of the top slab,where the largest relative vertical deflection is ex-pected to occur.Embankment InstallationsThe pressure values on the topflange of the box culverts were averaged and conservatively converted to the effective densities or soil-structure interaction factors F e1and F e2,given by Eqs.(1) and(2).The effective density or the soil-structure interaction fac-tor is higher for an unyielding foundation than for a yielding foundation.Examination of the soil settlement adjacent to the culvert reveals that soil layers on an unyielding foundation[Fig. 5(b)]settle more at the level of the topflange than those on a yielding foundation[Fig.5(a)].As explained earlier and shown in Fig.1,the magnitude of relative settlements of soil prisms adja-cent to the central soil prism affects the effective soil density.Fig.6shows the effective density versus backfill height for culverts on yielding foundations.Effective densities increase as the backfill heights increase.The relative settlement of soil layers on a yielding foundation,illustrated in Fig.5,increases slightly more than linearly.Effective densities from ABAQUS are slightly higher than those from ISBILD.The predicted values from both ABAQUS and ISBILD lie between the values given for the com-pacted and the uncompacted side-fill by AASHTO(2002).Termi-nologies for compaction requirements in AASHTO such as“un-compacted”or“compacted,”used without quantitative references,are difficult to interpret in modern construction con-tracts.The compactness offill along the sides of the box section does not appear to significantly affect the effective density for deeply buried box culverts.It is clear that the effective density is most sensitive to the foundation characteristics.However,concrete box culverts are most likely to be installed on yielding foundations unless a solid rock layer is encountered immediately under the concrete box culverts.The AASHTO stipulates that a special analysis is re-quired for a culvert on an unyielding foundation.Fig.7shows the effective density versus backfill height for box culverts installed on unyielding foundations.The value of the effective density from ABAQUS showed somewhat higher values than those ob-tained from ISBILD.The following equations are proposed for the effective densities for yielding and unyielding foundations,re-spectively:D E=1.047H0.055;yielding foundation͑7͒D E=1.200H0.059;unyielding foundation͑8͒where DE and H stand for effective density andfilling height, respectively.While there are only small differences in the effec-tive densities determined from ABAQUS and ISBILD runs for yielding foundations,as shown in Fig.6,there are fairly large differences in the effective densities determined from ABAQUS and ISBILD for unyielding foundations,as shown in Fig.7.Eqs.Fig.5.Deformed shapes of soil layers adjacent to box culvert:(a) yielding foundation and(b)unyieldingfoundation Fig.6.Effective density for embankment installations on yieldingfoundations parison of soil pressure around box culvertTrench InstallationsThe pressure values on the top flange of the box culverts were averaged and conservatively converted to effective densities.The effect of sloping trench walls and vertical sidewalls was investi-gated.The effective density asymptotically approaches that for embankment installation as the ratio of the horizontal width of the trench to the box width ͑B d /B c ͒increases,as expected.The ef-fective density for vertical sidewalls is plotted as a function of the ratio of backfill height to horizontal width ͑H /B d ͒on yielding and unyielding foundations in Figs.8and 9,respectively.As in the case of embankment installation,the effective densities deter-mined from ABAQUS are slightly higher than those computed from ISBILD .Most values of the soil–structure interaction factor from AASHTO (2002)are 1.4,the upper limit value.It is clear from Fig.8that the effective density given by the current AASHTO is conservative compared to the analytically predicted values from this study.The curves shown are fitted from a regres-sion analysis on the data from ABAQUS and ISBILD as functions of H /B d .It is noted that different effective densities can be found for the same value of H /B d as the magnitude of the backfill height H may have a nonlinear effect on the effective density,as men-tioned earlier.It should also be noted that there are no effective densities plotted in Fig.9,as there are no procedures presented in the current AASHTO other than stipulating a special analysis re-quirement.Predictor equations for effective densities formulated from ABAQUS runs are proposed as follows:D E =exp ͓0.012͑H /B d ͒2−0.288͑H /B d ͒+0.375͔;yielding foundation͑9͒D E =exp ͓0.011͑H /B d ͒2−0.273͑H /B d ͒+0.465͔;unyielding foundation͑10͒In most theoretical treatments,the trench walls are generally taken to be vertical.However,for deep trenches,it is not practical to maintain the trench walls to be vertical.Sloping trench side-walls are expected to have a significant effect on the soil pressure on culverts.For cases where the trench is constructed with slop-ing walls,it is usual that the width of the trench B d is taken as the horizontal distance between the sloping walls at the top of the box.In order to investigate the effect of the sloping sidewalls on the effective density,a number of hypothetical concrete culverts installed in a trench with sloping sidewalls were analyzed.Fig.10shows the effective densities for trench installations with sloping sidewalls.It is believed that such quantified information on the effect of the sloping sidewall has not previously been available.The ranges of the parameters used to generate Fig.10were:ratio of the trench width to the concrete box culvert width B d /B c =2,3,4;and fill height H =15.2m ͑50ft ͒,30.5m ͑100ft ͒,45.7m ͑150ft ͒.␪represents the angle between the vertical line and the sloping sidewall of the trench.Effective densities increase to ap-proximately 90%of those for embankment installations when ␪approaches 45°.Although Fig.8shows large differences between the values of the effective density computed from AASHTO pro-visions and finite element analyses where the trench walls are assumed to be vertical,the actual differences are likely to be smaller when the sloping trench walls are taken into account.Imperfect Trench InstallationsThe loads on concrete box culverts in embankment installations are greater than the weight of the soil directly above the structure,as stated previously.An attempt to reduce the load on the struc-ture led to the development of the imperfect trench method of construction,as shown in Fig.11.As the embankment is con-structed,the soft zone compresses more than the surrounding fill and thus induces a reverse arch deformation above the culvert.Concrete box culverts buried with a lightweight material zone were analyzed for several different geometric configurations and backfill material properties.The modulus of elasticity of the light-Fig.7.Effective density for embankment installations on unyieldingfoundationsFig.8.Effective density versus H /B d for trench installations on yieldingfoundationsFig.9.Effective density versus H /B d for trench installations on unyielding foundations4.79kPa ͑10ksf ͒and the weight was taken to be 1.57kN/m 3͑10lb/ft 3͒.A typical average value of modulus of elasticity of soil E is about 10–20kPa at ground level and 200kPa at a depth of 30m.The width of the lightweight backfill or the loose mate-rial zone w was varied from 1to 2.5times the width of the culvertB c .The exterior dimensions of the box culvert were 2.44m ͑8ft ͒high and 2.44m ͑8ft ͒wide,and the thickness of the slab and wall were both 305mm ͑12in.͒.Table 2shows the load reduction rate of the imperfect trench method in terms of percentage for a concrete box culvert buried at a depth of 30.5m ͑100ft ͒with a lightweight material zone above 0.6m ͑2ft ͒from the top of the culvert.The height of the lightweight material zone was set at 2.44m ͑8ft ͒.Table 2clearly demonstrates that the width of the lightweight backfill zone need not be greater than 1.5times the width of the culvert,as no significant load reduction is realized beyond that.This number agrees well with the recommendation by Vaslestad et al.(1993).The effects of the height of the lightweight material zone h and the distance between the culvert and the lightweight material zone h Јwere also examined.h and h Јwere varied from 0.75m ͑2.5ft ͒to 4.5m ͑15ft ͒,and from 0to 2.44m ͑8ft ͒,respectively.The load reduction rates as functions of the ratio of the height of the soft zone to that of the culvert h /H c are plotted with different values of w /B c in Fig.12and with different moduli in Fig.13.It can be seen from Figs.12and 13that the ratio h /H c need not be greater than 1.5;after which no significant load reduction is real-ized.The variations of load reduction rates are shown in Fig.14as a function of the ratio of the distance between the culvert and the lightweight material zone to the height of the culvert h Ј/H c .Note that the maximum load reduction rate is realized if the light-weight backfill or the soft zone is placed immediately above the box culvert.However,it may be necessary to place a nominal layer of backfill over the box in order to facilitate the construction of the lightweight material zone.Typical effective density distri-butions on the box culverts are shown in Fig.15for different values of w /B c ,E ,and h Ј/H c .It is evident from Figs.12–15that the imperfect trench method can significantly reduce the effective densities or the soil–structure interaction factors.The notion of inducing artificial reverse arch shape deformations in the backfill above buried conduits,thereby reducing the vertical soil pressure,Fig.12.Effects of w /B c and h /H c in imperfect trench installationsFig.10.Effect of sloping sidewall in trench installationsFig.11.Concept and notation in imperfect trench installationsTable 2.Load Reduction Rates (%)Due to Imperfect Trench Installation Modulus of lightweight materialWidth of back fill layer,w /B c 1.0 1.5 2.0 2.547.9kPa ͑100ksf ͒32.841.139.136.723.9kPa ͑50ksf ͒43.658.657.054.34.79kPa ͑10ksf ͒56.983.785.384.8JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ©ASCE /JANUARY 2005/25has been known and tried intermittently (Brown 1967).It is be-lieved,however,that numerical values of the parameters for the optimum size of the lightweight material zone and its desired location have not been available elsewhere before this study.However,the imperfect trench method is an emerging technology that so far lacks a great deal of construction experience.Vaslestad et al.(1993)recommend a judicious selection of the lightweight backfill material and its careful installation,as many unsatisfac-tory experiences have been reported.Summary and Concluding RemarksLinear and nonlinear finite element analyses have been conducted to investigate the effective density or soil–structure interaction factor for deeply buried concrete box culverts.The effective den-sities are sensitively affected by many parameters,including in-stallation methods and foundation characteristics.Although con-servative classical methods will continue to be applied to the design and construction of buried conduits,an opportunity exists to utilize the modern numerical techniques made possible by ad-vances in finite element methods.The findings from this study presented in the form of proposed regression equations and charts should find their applicability.Highlights of the study are as fol-lows:1.Soil–structure interaction factors for deeply buried box cul-verts are more sensitively affected by the foundation charac-teristics.Predictor equations for soil–structure interaction factors were derived based on numerical data for culverts on both yielding and unyielding foundations constructed by em-bankment,trench,and imperfect trench installation methods.2.The effect of possible slips of the sidefill materials along the exterior culvert wall on the vertical soil pressure on the box culvert was found to be negligibly small.3.The current AASHTO provisions and most other references stipulate the effective densities for the trench installations under the assumption that the trench walls are excavated ver-tically.However,it was found that the effect of sloping trench walls on the soil–structure interface factors is very high.Fig.10presents variations of the soil–structure interac-tion factors as functions of major parameters,including the fill height,the degree of slope,and other trench geometry.4.Effective densities or the soil–structure interaction factors may be significantly reduced by properly implementing the imperfect trench method.Load reduction rates afforded by inducing artificial reverse arch shape deformations can be as high as 85%,depending on the dimensions of the lightweight material zone and the modulus of elasticity of the lightweight material.The height and the width of the lightweight mate-rial zone need not be greater than 1.5times those of the box culvert,beyond which no significant load reduction is real-ized.The greatest effect on load reduction is also obtained if the lightweight material zone can be placed immediately above the box culvert,although it may be necessary to place a nominal backfill layer immediately above the box to facili-tate the construction of the lightweight material zone.It is believed that the optimum values defining the lightweight material zone and its location have not been available else-where before this study.AcknowledgmentFunding for this research project was provided by the Highway Research Center,Auburn University.This financial support is gratefullyacknowledged.Fig.13.Effects of Young’s modulus and h /H c in imperfect trenchinstallationsFig.15.Effective density affected by imperfect trenchinstallationsFig.14.Effects of w /B c and h Ј/H c in imperfect trench installations26/JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ©ASCE /JANUARY 2005。