Modeling the Total Surface Area and Volume of Potholes on a Length of Road

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1 Modeling the Total Sufae Aea and Volue of Potholes on a Length of oad S.A. Oke, M.S. * and O. Ajayi 1 Depatent of Mehanial Engineeing, Uniesity of Lagos, Nigeia. * E-ail: sa_oke@yahoo.o ASTACT Potholes ae essentials in ino oad epais. Infoation about thei total sufae aea and olue ae useful in pediting the tie spent on oad epais, whih inole light aintenane paties. In addition, ost infoation ay be oputed using these data. This pape pesents a atheatial odel fo ealuating the total sufae aea and olue of potholes on a length of oad. The poble of potholes is appoahed fo diffeent dietions. One pespetie elates to the wea aused by the atiities of as on the sufae of oads wheeby the ties of the as ae assued to be ubbing the sufae of the oad as it oes along the oad. The othe dietion is the effets of eosion by wate of the bituen in aeas whee aks and depessions ou. A thid dietion is in fo of pobabilities of ouene on the oad with aeage (assue) adius fo the potholes. Case studies ae deeloped whih illustates the patial signifiane f the odel foulated. It is enisaged that the wok would benefit the stakeholdes in oad anageent. (Keywods: aea, olue, total aea, sufae of a -D objet, thee diensional geoety, oad engineeing, aintenane, epai) INTODUCTION Tanspotation odes ae of thee ajo ategoies: land, ai, and wate. Howee, oad tanspot, whih is a signifiant pat of land tanspotation, is of inteest to us in this pape. The oad on whih ehiles tael is of piay inteest to oad stakeholdes, patiulaly the goenent. Many oads hae potholes, equiing light aintenane paties to estoe the to an aeptable ondition. Unfotunately, sientifi estiation of the total sufae aea and olue of potholes on a length of a oad is a ajo poble. This has depied oad anages of adequate planning sine they annot asetain the aount of ateials and ost of the epais fo suh potholes. This pape attepts to odel the total olue, sufae aea and ost of efilling of potholes along oads in the Nigeian enionent onsideing fatos suh as aount of ainfall, ehiula speed, ass of ehiles (Mabsout et al., 4). It is assued that wate is the only ajo natual soue of pothole popagation on the oad. The otion of ehiles on the oads espeially the ubbing of the a ties on the sufae of the oad auses wea and the weight of ehiles ause depessions on the oad espeially if the speed of tael of suh ehiles ae not that uh and if suh ehiles ae heay ehiles. The depessions and aks whih ae aused by suh ehiles then allow fo the gatheing of wate in the ausing dissolution and weaing away of the bituen eating a pothole (Singh, 7). The pape attepts to inopoate ost of these fatos into this odel. Fo the liteatue eseah by Mabsout (4) and Singh (7) poides a stong suppot fo the study. Mabsout et al. (4) pesents a paaeti study of the effet of potholes on load distibution in einfoed onete slab bidges using finite-eleent analysis (FEA). Consideation was gien to the pothole loation. It was epoted that the pesene of one o two potholes in the fist two lanes eated at the idspan of a bidge ineased the iu longitudinal bending oent by about 5% when opaed with FEA esults of slabs with no potholes, and by as uh as 7% when opaed with the AASHTO standad speifiations. Singh (7) olleted data of 4 ases of pot-hole subsidene fo Son-Mahanadi Maste Coal asin and analyzed fo stand point of thei ausatie fatos, pedition, ehanis, and suggested itigation easues. The esults show that the isk of pothole subsidene is high when the pothole potential atings of ausatie fatos ae between 8 and 1. The stutue of the pape is as The Paifi Jounal of Siene and Tehnology 4 Volue 8. Nube. Noebe 7 (Fall)

2 follows. The intodution poides a stong otiation fo the study and its justifiation. Subsequent setions will pesent the ethodology of the study; disuss ase study appliation of the poble in eal life, and also pesent a disussion of esults. Conluding eaks ae gien in the final setion of ou pape. METHODOLOGY The poble of potholes will be appoahed fo diffeent dietions one will be by wea aused by the atiities of as on its sufae wheeby the ties of the as ae assued to be ubbing the sufae of the oad as it oes along the oad. And the othe will be the effets of eosion by wate of the bituen in aeas whee aks and depessions ou. Anothe way will be in the fo of pobabilities of ouene on the oad with aeage (assue) adius fo the potholes. f t n t W W f n V t fequeny of tuks nube f tuks (total) suation ean weight of as weight of as fequeny of as total nube of as eloity of tuks Assuing that as a a oes along the oad, as in Figue 1: Definition of Tes V eloity of as M ass of a a aeleation of a F b the bonding foe of bituen F foe fo the as ties.e bonding enegy, K.E kineti enegy, ½M V C speed of light p ass of one patile n p nube of patiles n p sufae aea of patile ass of tie adius of tie ω angula eloity of tie total ass of patiles displaed itial eloity at whih the patile of bituen ae detahed beadth of the tie adius of the tie X distane taeled by the as Sine total sufae won SA in 1 e. nube of eolutions No. of e. total distane π adius F foe A aea l length δ etension E odulus of elastiity/igidity ε longitudinal stain ν Poisson atio a 1 b 1 1 diensions of a epesentatie ube W t weight of tuks The Paifi Jounal of Siene and Tehnology 41 Figue 1: Ca Moeent Along the oad. The ties not only otate but also tanslates due only to a gain in oentu, as in Figue : Figue : Inteation of the Wheel and Tie with the Sufae of the oad. As the a oes, the tie not only uses the fition between it and the oad to oe but also soe of the fition is oeoe and thee is a oesponding sliding of the tie on the oad. F μ N : Sliding ous when M a > F Assuing also that fo otion of the a thee has to be fition between the tie and the oad sufae. Volue 8. Nube. Noebe 7 (Fall)

3 As the tie otates we assue that soe patiles of bituen ae displaed o upooted due to the otion of the tie (Figue ). S A 4 b as i 1 πb 1 + πb + πb + + πb n when is onstant, S A 4 as i 1 whee b is onstant. π 1 b + π b + π b + + π n b, Figue : Inteation of the Tie/Wheel with the oad while in Motion. That is a fo of elation between the otation of the tie and the displaeent of the patiles. To deelop the fist, taking a iew at the patiulate leel (Figue 4): F Figue 4: Foe Analysis at the oad-tie/wheel Inteation. The tiny laye of bituen will detah itself when F > F b o oe, appopiately when K.E >.E, i.e. ½M V Δ, whee M, V and C ae onstant, Δ 1 MV C. Fo the seond assuption that soe patiles ae displaed as the tie otates, we assue onseation of enegy, i.e. ½ Iω ½ V, whee I, then n. p p The total sufae aea though whih the enegy is tansitted to the oad is A π 4. Fo as with diffeent tie adii and beadth, the total sufae aea won by these as will oe into play: This gies a double integal when eged beause of the fat that thee ae two aiables S A 4 ties. a in in πb d db in one otation of the Multiplying the esult of the integal by X X ties. d. Theefoe total adius d of the Total nube of eolutions in in in in πb d db π in in d d b d db d π d in d in d [( - in )( - in )] ( ) ( ) 4( in ) in Consideing the effets of the weight of a ehile on the bituen on the oad, we eaine the illustation of Figue 5. The Paifi Jounal of Siene and Tehnology 4 Volue 8. Nube. Noebe 7 (Fall)

4 Sine stess stain E F l E A δ beause all solid bodies hae a fo of elastiity befoe fatue. If the stess >>> E, thee will be a elatiely lage stain whih will ause a hange in olue of the bituen suh that: V o a 1 b in M g/ M g/ Figue 5: Foe Analysis at of the Vehile-oad Inteation. Fo Figue 6, it an be seen that the bituen epeienes opessie stess in the fo of Foe (weight)/aea: V f a 1 b 1 1 (1 + ε) (1 - νε) (1 - νε) a 1 b 1 1 (1 - νε + ε - νε ) (1 - νε) a 1 b 1 1 (1 - νε - νε + ν ε + ε - νε - νε + ν ε ) disegading all ultiples with powes: a 1 b 1 1 (1 - νε - νε + ε) a 1 b 1 1 (1 + ε - νε). When the elasti liit of the bituen has been eeeded aks and faults eege on the sufae of the oad. Assuing an eponential elationship between the weight of the ehile and the speed of the ehile, i.e. W W e -/1 suh that when, W W and when V V, W W M g/ M g/ 1 M g/ bituen onet soil Figue 6: Futhe Analysis of the Vehile-oad Inteation. The Paifi Jounal of Siene and Tehnology 4 Volue 8. Nube. Noebe 7 (Fall)

5 Fo this elationship, it an be seen that when fully loaded tuks play a oad at ey slow speeds thee is the tendeny that that oad will get bad fast if they ae not einfoed. Let us assue that a wide ange of ehiles ply a etain oad with a wide ange of weights and at aying eloities with diffeent fequenies on a ontinuous basis. The aeage o ean weight of the tuks Wtf Wtf t, while the ean weight of as, n t ft Wf Wf W, and the ean eloity n f Vtf t Vtf t of tuks, V t. Also, the ean n t f t Vf Vf eloity of as, V. n f Getting these alues and using the with the assued elationship between eloity and effetie weight of both the as and the tuks the tendeny of the oad to deelop potholes an be desibed (Nelkon and Pake, 1964). Using Fik s law of diffusion, the poess of disintegation of bituen an be eplained: dt d k(a ), C() Co whee C o onentation of the liquid initially C o < a a onentation of the diffusing body so that a (t) (a ())e -kt. Fo the dissoled patiles of bituen in oing wate, to desibe thei elatie oeent fo the enionent, ontinuity equations fo thee-diensional flow using Catesian oodinates will be ideal (i.e. assuing a ontol olue ACDEFGH) (Stoud, 1; Stoud and ooth, ). Figue 7 is taken in the fo of a etangula pis of sides δ, δy, δz in the, y, z dietions espetiely. While the aeage alues of the eloities in these dietions as V, Vy, Vz. Mass inflow though ACD in unit tie ρv. Taking a geneal ase whee ass density ρ and eloity V will hange in the dietion, the following equations apply: Y G A C H δy X D E δz δ Figue 7: Contol Volue. The Paifi Jounal of Siene and Tehnology 44 Volue 8. Nube. Noebe 7 (Fall)

6 Mass outflow though EFGH in unit tie ρv + ( ρv) δ Thus, net outflow in unit tie in dietion ( ρv) δ Siilaly, net outflow in unit tie in y dietion y ( ρvy) δ Net outflow in unit tie in z dietion z ( ρvz) δ Theefoe, total net outflow in unit tie y z ( ρv) + ( ρvy) + ( ρvz) δ also, sine unit tie. p t is the hange in ass density pe Change of ass in ontol olue in unit tie - p t δ (the negatie sign indiates that a net outflow has be assued). Then, total net outflow in unit tie hange of ass in ontol olue in unit tie. p - t y z ( ρv) + ( ρvy) + ( ρvz) δ δ y z ( ρv) + ( ρvy) + ( ρvz) Fo inopessible fluids like wate: ( V) ( Vy) ( Vz) 1995) + y + z - p t (Douglas et al., In this pape we hae assued potholes to be of a iula paaboloid shape with the foula z + y. Sine the ethod of getting the sufae aea of a funtion is: S z z y z z ds 1+ + s y fo a paaboloid ( y) Theefoe δ ( ) 4y and ddy ddy Coneting the oodinates to pola oodinates, whee: osθ, y sinθ + y, and ddy ddθ. + Theefoe δ 1 4 ddθ Taking the bounday onditions of the paaboloid fo to and θ to θ π, π δ 1 4 ddθ + π / 1 8/ 4 sufae aea of a paaboloid π 6 ( 1+ 4 ) dθ ( 1+ 4 ) The appoiate olue of a solid is found by haing a ultiple intenal of: V 1 1 Howee, V y y z z 1 dz dy d 1 1 fo pola oodinates. θ θ z1 dz z fo Catesian oodinates. d dθ Fo a paaboloid z y +, whee: The Paifi Jounal of Siene and Tehnology 45 Volue 8. Nube. Noebe 7 (Fall)

7 y sinθ osθ ddy d dθ z sin θ + os θ (sin θ + os θ) (1) Theefoe olue of a paaboloid (iula) π V π π 4 π 4 dz d dθ d dθ d dθ dθ 4 π olue of a paaboloid (iula) Due to the any paaetes o aiables affeting the foation of potholes on a patiula length of oad, we would use pobability to desibe the ouene of potholes on a oad suh that: The epeted nube of potholes on the entie length of a oad total length of the oad pobability of a pothole ouing on 1k of oad, i.e. E NX P(A), whee E is the epetation, N is the total pobability spae, while P(A) is the pobability saple. Using an aeage adius fo eah pothole the total sufae aea and total olue fo the potholes on the oad an be dedued. CASE STUDY AND DISCUSSION OF ESULTS Case Study 1: On the Lagos-Ibadan epessway, it was notied that fo eey 1k taele thee seeed to be between 1 to potholes with an aeage adius of.6. To find the total sufae aea and total olue fist of all, we find the epetation of potholes on the oad. Taking the length of the oad as 1k a pobability of a pothole as between.1 and.. Using.1 the epeted nube of potholes 1 potholes on the entie length of the oad and 6 potholes when using., all with an aeage adius of.6. Using this alue we get a total sufae aea of π [ 1+ 4(.6) ].57 fo one pothole, and π olue fo one pothole. Fo these figues and using the pobability of.1, thee will be a total pothole sufae aea of and a total olue of Case Study : On a oad of 1.5k an annual ainfall of about 1, lites of wate falls on it and the ean speed of flow of the wate off the oad is 11.11s -1. On this oad also is assued that about 1, as ply it annually with ties with a ean adius of.8 and a ass of 8kg (inluding the i). The iniu tie beadth and iu tie beadth ae.9 and.1 espetiely. The as that ply this oad ae assued to go at 1k/h. The oad has the following popeties: Its onentation is 1kg/ and a eady 9kg/ dissoles annually when wate aies. The density of one patile of bituen is about kg/, adius of.5 and diffusion onstant of 1-6 s -1. Citial eloity of bituen patile 1 8. Solution: Gien that δ 1.5k, V 1 lites 1, 11.11s -1, n 1. + in ,.8, 8kg, 1k/h, w (of ties) 99.ad/s. a 1kg/, C o 9kg/, g kg/, p.5, k 1-6 s -1, and 1 8. Sufae aea (total) 4 (.5) ( 1 ) + 4(.1) The Paifi Jounal of Siene and Tehnology 46 Volue 8. Nube. Noebe 7 (Fall)

8 Volue ( ) 8 ( 1 ) ( 1-9) Case Study : Total sufae aea won sufae aea won by the otion of ties + sufae of patiles won by eosion. 4 Sufae aea of patiles n p π p (using the sufae aea of a sphee as standad). Theefoe p 4 πp, Fo ½ Iω ½, Iω and taking I as, ω. Sufae aea won by ties 4( in ). Theefoe Total Sufae Aea + 4 ( in ) Volue of bituen won e ω 4 π p total ass won density of bituen total ass due to wea ing + ol.of ain diffusion ate density of bituen Iω Volue ω ω + ol.of wate - a - Whee: n p nube of patiles, p adius of patiles, total ass of patiles, g ( a - C ) o e -k δ p ass of one patile, ass of tie, adius of tie, eloity (itial) fo the eoal of a patile, iu beadth of ties, in iniu beadth of ties, a onentation of diffusing bodies, C o onentation of diffusing body in liquid, K diffusing oeffiient, δ distane taeled, and eloity of fluid (ean). Case Study 4: The fat that potholes do not ou ontinuously but andoly on a steth of oad akes the poble of odeling the total sufae aea of potholes on a steth of oad pobabilisti one. The potion on whih potholes ae found an be assued to be haateized by a weak soil unde laye with high seeping of goundwate into the onete foundation of the oad dissoling it. Fo this stateent, it an be seen that the nube of potholes that will ou on a oad, the depth and the size in tes of adius ae pobles of both statistis and pobability. Unde this peise theefoe, the odeling that will be deeloped will be on the dynais of the foation of a pothole whose soil undelaye is being o has been washed away fo unde the oads foundation. The only ausal fato fo the foation of potholes that is assued is that of wate. Assuing that the piniple o steps that poeed befoe the appeaane of a pothole on a oad ae these: (i) infiltation, (ii) dissolution, (iii) ass oeent of soil, (i) defletion of oad unde its own weight, () gatheing of wate on the sufae of the oad, (i) dissolution and washing away of the bituen. Infiltation: Assuing that the poess of infiltation is poweed by both apillay ation and soil wate table pessue. The total height h eahed by the wate o head of the wate is H H + H w, H apillay head, H w wate table head, Whee H 4σ osθ/ρgd. Note that: σ sufae tension, θ angle of ontat, d diaete of apillay as θ and d δd, dh i 4σ/ρgδd, H i 4nσ/ρgd The Paifi Jounal of Siene and Tehnology 47 Volue 8. Nube. Noebe 7 (Fall)

9 While Hw p/ρg; h 4nσ/ρgd + p/ρg 1 nσ + p ρg d The dissolution ate of any solute in a solent is k(t) ((o - ov) - (t) ), whee V is the olue of V the solute. Using the ontinuity equations, the oeent of soils an be odeled. Assuing a ontol olue ACDEFGH is taken in the fo of a etangula pis of sides δ, δy, δz in the, y, z dietions espetiely. While the aeage alues of the eloities in these dietions ae V, Vy, Vz.. Mass inflow though ACD in unit tie ρv. Taking a geneal ase whee ass density ρ and eloity V will hange in the α-dietion. Mass outflow though EFGH in unit tie ρv + ( ρv ) δ. Thus, net inflow in unit tie in -dietion ( ρv ) δ. Siilaly, net outflow in unit tie in y-dietion ( ρvy ) δ. Net outflow y in unit tie in z-dietion ( ρvz ) δ. z Theefoe, total net outflow in unit tie ( ρv ) ( ρvy ) ( ρvz ) δ y z + +. Also, ρ sine is the hange in ass density pe unit t tie. Change of ass in ontol olue in unit tie ρ δ (the negatie sign indiates that a t net outflow has been assued). Then, total net outflow in unit tie Change ass in ontol olue in unit tie y ρ δ t z ( ρv ) + ( ρv ) + ( ρv ) δ y Theefoe ( ρv ) + ( ρvy ) + ( ρvz ) y z ρ. Now fo the oeent of ass in wate we t assue a one-diensional flow i.e. ( ) ρv ρ. t z Y G A F δy C H δz X D E Z δ Figue 8: Futhe Analysis Contol Volue. The Paifi Jounal of Siene and Tehnology 48 Volue 8. Nube. Noebe 7 (Fall)

10 ρ n Now assuing that ate of dissolution: t t no - n(t) Kn(t) - o as δ V ( V ) d( V ) d, theefoe no - n(t) Kn(t) - o. V d( V ) d Whee: n ass pe unit tie of solute o iu onentation of ass pe unit tie of solute V olue of solent K oeffiient of dissolution (o assuing diffusion) ρ V o1 V Kn(t) (integating fo to ) ( n - n V) - n(t) o Assuing that eah ateial has its own ate of dissolution, theefoe Kn(t) ( no - nov) - n(t) an be epesented by p (Tioshenko, 195). p(t) Theefoe ass, whee length o adius V of aea, and t tie. Now due to the loss of the oad unde laye at a potion, that potion of oad an be assued to deflet unde its own weight. Assuing the oad to be a siply suppoted bea of span aying a unifoly distibuted load w pe unit un oe the whole span. Theefoe, the defletion at id-span whih is the 5wl 4 iu defletion aailable y, whee E 84EI the odulus of igidity, and I the oent of inetia of the bea/oads oss-setion. This y that has been got an be used as a alue of head if the effet of pessue of a pool of wate wee to be onsideed. ut sine the effet of the defletion being onsideed is that of the aking of the sufae, it an be negleted. o Sine only the effet of wate in dissoling and washing away the bituen is onsideed we an use the sae foula deied peiously fo the pt sufae eoal of the bituen i.e.. To get the olue obtained fo the pothole,, while fo the total sufae aea. ρ Assuing that all the patiles ae sphees, theefoe the sufae aea adius of patile. Theefoe total sufae aea n p p ass of one patile. 4 V π p, whee p n p 4 π p. ut, whee n p nube of patiles, and p Assuing that diffeent substanes in whatee fo hae a onstant diffusion ate, theefoe assuing fo bituen p kg/se and the adius of a patile is taken as 1 while the ass of one patile is also taken as.5g with a bituen of density of,kg -s assued. Theefoe the foulae beoes: (t), V and V,. The sufae aea π.1 Case Study 5: If a pothole wee to ou on a potion of a oad, it is desied to know the sufae aea and olue of bituen that will be washed away in yeas if the adius of the pothole would be about.5. The eloity of wate that noally flow on the sufae of that oad due to the egion it -1 is in, is s. Gien that t yeas ,.5, V 11.11s -1. The Paifi Jounal of Siene and Tehnology 49 Volue 8. Nube. Noebe 7 (Fall)

11 The ass that will be washed away is () kg kg. -4 Now the olue is V , while the sufae aea is about With the ase study just onluded, the total sufae aea and the total olue of potholes on a oad an be effetiely found, out afte onsideing and analyzing the data fo the diffeent paaetes that affet its foation. CONCLUSION With the ineasing effots by goenents to epai oads, the issue of pope planning and budgeting has been a ajo hallenge. Suh a poble has liited the popt epais of potholes on oads, whih equie light aintenane paties. The otiation to sole this ipotant poble has podued the uent study. Consequently, a odel is deeloped to analyze the total sufae aea and olue of potholes on oads. The odel deied within this pape an effetiely gie an epeted aount of potholes on any length of oad. 7. Tioshenko, S.P Stength of Mateials. MGaw Hill Publishes: New Yok, NY. AOUT THE AUTHOS Sunday A. Oke, M.S. gaduated in Industial Engineeing fo the Uniesity of Ibadan, Nigeia with a ahelo and Maste's degees in 1989 and 199, espetiely. He woked fo the IDM Seies Liited as a onsultant. M. Oke letues in the Depatent of Mehanial Engineeing, Uniesity of Lagos. He has eiewed papes fo seeal intenational jounals. Oluwadailae Ajayi, is an undegaduate student of the Depatent of Mehanial Engineeing, Uniesity of Lagos. SUGGESTED CITATION Oke, S.A. and O. Ajayi. 7. Modeling the Total Sufa e Aea and Volue of Potholes on a Length of oad. Paifi Jounal of Siene and Tehnology. 8():4-5. Paifi Jounal of Siene and Tehnology EFEENCES 1. Douglas, J.F., Gasioek, J.M., and Swaffield, J.A., Fluid Mehanis. Longan Goup: London, UK. ISN Mabsout M., Tahini K., Awwad E., and Fedeik G. 4. Effet of Potholes on Load Distibution in Conete Slab idges. Pat. Peiodial on Stut. Des. and Const.. 9(): Nelkon, M. and Pake, P. 1964, Adaned Leel Physis. Heineann Eduational Publishes: London, UK. 4. Singh, K.. 7. Pot-Hole Subsidene in Son- Mahanadi Maste Coal asin. Engineeing Geology, 89(1-): Stoud, K.A. and ooth D.J.. Adaned Engineeing Matheatis (4th edition). Palgae. ISN Stoud, K.A. 1. Engineeing Matheatis, 5th Edition. Palgae. ISN The Paifi Jounal of Siene and Tehnology 5 Volue 8. Nube. Noebe 7 (Fall)