APPROXIMATE ANALYSIS OF RIGID PLATE LOADING ON ELASTIC MULTI-LAYERED SYSTEMS

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1 6th ICPT, Sapporo, Japan, July 008 APPROXIMATE ANALYSIS OF RIGID PLATE LOADING ON ELASTIC MULTI-LAYERED SYSTEMS James MAINA Prncpal Researcher, Transport and Infrastructure Engneerng, CSIR Bult Envronment P.O.Box 395, Pretora 000, South Afrca Yoshak OZAWA Department of Cvl and Envronmental Engneerng, Tokyo Denk Unversty Ishaka, Hatoyama Town, Hk-gun, Satama , Japan Kunhto MATSUI Department of Cvl and Envronmental Engneerng, Tokyo Denk Unversty Ishaka, Hatoyama Town, Hk-gun, Satama , Japan ABSTRACT GAMES software s well known for ts capablty to compute responses for unformly dstrbuted load actng on the surface of a mult-layered lnear elastc system. In ths study a method was developed to approxmate rgd plate loadng to be used n GAMES software. When a rgd plate load s actng on the surface of a multlayered system, the resultng nterface contact stress dstrbuton s not unform. Snce contact stress dstrbuton between rgd plate and sem-nfnte medum s well known, ths dstrbuton was approxmated usng unformly dstrbuted multple loads and analyss performed usng GAMES. Results have shown good agreement wth the theory for the case of a sem-nfnte medum. Furthermore, extenson of ths method to multlayered system has shown good results f the rato of the frst layer thckness over the loadng plate radus s greater than. KEY WORDS rgd plate loadng, elastc multlayered system, GAMES, ax-symmetrc analyss 43

2 w (cm) Approxmate Analyss of Rgd Plate Loadng on Elastc Mult-layered Systems INTRODUCTION Rgd plate loadng test s generally used to evaluate load bearng capacty of pavement layers lke subgrade, subbase and base layers. When structures lke pavements are to be analyed, most of the tme the structural system s consdered to be multlayered lnear elastc. Furthermore, unformly dstrbuted crcular load s assumed to act on the surface of the pavement and t s possble to use software lke ELSA, CHEVRON, BISAR ) as well as GAMES ). However, when a test lke rgd plate loadng s performed, the nterface contact stresses are not unformly dstrbuted and software such as the ones lsted above may not be used for analyss. Authors of ths paper were unable, through lterature search, to fnd any document descrbng the applcaton of multlayer lnear elastc theory to the rgd plate loadng. Because of the facts hghlghted above, ths research evaluates the method of approxmatng theoretcal solutons of rgd plate loadng by usng GAMES software. unformly dstrbuted loads Sneddon derved the solutons for a ( P 49kN, E 00 MPa, 0.35 ) sem-nfnte medum subected to crcular rgd body loadng as shown n Fgure 3). In ths case, surface deflecton under the base of the rgd body s constant and normal stresses at the surface but outsde the base become ero. Fgure was drawn to compare surface deflectons from the applcatons of unformly dstrbuted loadng and rgd plate loadng. Ths fgure clearly shows that n the vcnty of the load applcaton area, deflectons from the analyss of unformly dstrbuted loadng are dfferent from results of rgd body loadng. Ths phenomenon s supported by Sant - Venant prncple, whch states that there s hgh nfluence of load dstrbuton near the pont of applcaton and ths nfluence dmnshes as you move away from the pont of load applcaton. It s clear from Fgure that there s a very good match of deflecton results for r a, there s a good match of deflecton results. In ths research, dstrbuton of contact stresses derved by Sneddon was approxmated by multple unformly dstrbuted loads, and responses for sem-nfnte medum were analyed usng GAMES software. Accuracy of ths approach was evaluated by comparng results obtaned by Sneddon for a sem-nfnte layer to results from usng approxmate stress dstrbuton. Extenson of ths approach to a multlayered system was evaluated by checkng whether deflectons under the base of a rgd plate remaned constant when layer modul were vared for a two-layer system. w φ = a r a Fgure Deformaton due to rgd body loadng Rgd plate loadng Unform loads ρ = r /a Fgure Surface deflectons due to rgd plate and 44

3 Mana, Oawa and Matsu Contact stress dstrbuton under rgd plate Theoretcal stress dstrbuton Assume a rgd plate load actng on the surface of a sem-nfnte body to be consdered an ax-symmetrc problem, and then the boundary condtons can be represented as shown below: w (r,0) 0 ra (a) ( r,0) 0 a r (b) s the amount of compresson (surface deflecton) and a s radus of the loadng plate. Accordng to Sneddon, shear modulus G, Posson s rato, and dstrbuton of contact stresses on a sem-nfnte medum p (r) are related as shown n the followng equaton: G p( r) ( ) a r () From equaton (), contact stress at the center of the load s G ( ) a, whle at the load edge, contact stresses becomes. Equaton () can be ntegrated across the crcular loadng secton to obtan the total load P as shown below: P a 4aG p( r) rddr 0 0 (3) From equatons () and (3), the relaton between p (r) and P can be obtaned as follows: P p( r) (4) a a r Furthermore, Sneddon determned surface deflectons usng the followng equaton: 0 r a ( r,0) a sn r a r w (5) Approxmate contact stresses Contact stress p (r) as expressed by equaton (4), becomes P ( a ) at the load center, where r 0 and at the load edge where r a. Such knd of a load dstrbuton can not be handled n any multlayer lnear elastc software such as GAMES. In order for such a contract stress dstrbuton to be analyed, t s mportant to approxmate ts dstrbuton by multple unformly dstrbuted loads. The method that was used to approxmate ths dstrbuton s explaned below. The load was subdvded nto N dfferent crcular loads of dfferent rad wth same centers. The smallest radus s r and the largest radus s r a N. Assumng every radus of each rgd plate crcle to represent the total loadng of P, we can obtan N domans comprsed of the nner most crcle of radus r and rngs formed by dfference between adacent crcles 45

4 Approxmate Analyss of Rgd Plate Loadng on Elastc Mult-layered Systems r r (, ).The value of r s determned by equalty wth the total loadng. Ths means, total load n each doman s equal to P P N. Ths total load n each doman s assumed to be unformly dstrbuted and we can ntegrate equaton (4) from r to r to obtan total th load for the doman. Consderng that total load n each doman can be expressed as P P / N, the followng equaton s obtaned: P P r a r a (6) N Makng use of the relatonshp between r and r shown n Equaton (5), yelds the followng equaton: r a r a N (7) Replacng the contact stresses between r and r by unformly dstrbuted load obtan: p to p N ( r P r ) (8) Assume a load of magntude P s actng on a crcular rgd plate. In order for ths load to be analyed usng GAMES software, t wll be approxmated as multple concentrc crcular th loads whose centers are on the loadng plate. Furthermore, for the load of radus r, responses (.e. stresses and dsplacements) at pont can be represented by ( r ), and for th ( ) load of radus r, responses can be represented by ( r ). When a unformly dstrbuted load of magntude p s actng on rng of wdth ( r r r ), responses at pont can be obtaned from the followng equaton: r ) ( r ) (9) ( th represents responses at pont when a unformly dstrbuted unt load s actng wthn a rng. Supermposng responses on pont due to the effect of the loads actng on all the rngs, the total response S may be obtaned as follows: S P N N p ( r r ) (0) p s the unformly dstrbuted load on rng, whle P s ts total load. Results from multple unformly dstrbuted loadng When multple unformly dstrbuted loads were analyed, there was mprovement n accuracy for bgger number of multple loads but the down sde of t was the ncrease n computatonal tme. Three methods of load subdvsons were used where N 5,0, 5, and 46

5 Mana, Oawa and Matsu results from numercal analyses were compared wth those obtaned by Sneddon. -σ (MPa) Fgure 3 s a comparson of contact stress dstrbutons between theoretcal analyss and approxmated load dstrbuton usng the three dfferent numbers of subdvsons. There s a sgnfcant dfference between contact stresses near the load edge for the theoretcal and approxmate loads but as the number of multple loads ncreases the contact stresses approach theoretcal values. At the load edge, theoretcal contact stress s nfnte whle approxmate contact stress stll mantans a fnte value. Fgure 4 shows comparson of surface deflectons for theoretcal load as determned by equaton (5) and surface deflectons as computed by GAMES software for the three dfferent N load subdvsons. Ths fgure shows that as the number of load subdvsons N ncreases, results of surface deflectons approach the theoretcal values. However, t s not the same tendency for deflectons near the load edge, where there s very lttle mprovement as the number of load subdvsons ncreases. The reason may be attrbuted to the fact t s dffcult for approxmate loadng to smulate theoretcal loadng near the edge, whch approaches nfnte. Deflecton results by fewer load subdvsons, n ths example 5 subdvsons, show poor accuracy near the load center whle results from bgger number of subdvsons, that s N 0, 5 show very good agreement wth theoretcal results. It s -aσ/(gδ ) An approxmate value by 5 ponts An approxmate value by 0 ponts An approxmate value by 5 ponts Theoretcal responses r /a Fgure 3 Theoretcal and approxmate Fgure 4 Subdvsons and surface deflectons contact stresses ( P 49kN, E 00 MPa, 0.35 ) ( P 49kN, 00 MPa, 0.35 ) Theory 0.0 Theory 0. Theory 0.4 Theory 0.6 Theory 0.8 Theory.0 GAMES 0.0 GAMES 0. GAMES 0.4 GAMES 0.6 GAMES 0.8 GAMES r / a * Legend value s a Fgure 5 Comparson of responses between approxmate and theoretcal loads w (cm) An approxmate value by 5 ponts An approxmate value by 0 ponts An approxmate value by 5 ponts Theoretcal responses r /a E therefore recommended that at least 0 subdvsons are mportant to obtan reasonably good results. Fgure 5 shows comparson of normal stresses ( ) at dfferent depths from the surface ( a ) n order to confrm ablty of 0 load subdvsons to approxmate theoretcal analyss. In ths fgure, dmensonless parameters of normal stresses ( ) dvded by G a 47

6 Approxmate Analyss of Rgd Plate Loadng on Elastc Mult-layered Systems was plotted aganst r a. Numbers n the legend of Fgure 5 ndcate a ratos. In ths fgure, sold lnes represent theoretcal results whle approxmate results are represented by the marks as defned n the legend. The fgure shows for surface results a dfference of about 5% between theoretcal and approxmate loads and a very good agreement for ponts deeper than a 0.. EXTENSION TO A TWO-LAYER STRUCTURE Method for applcaton evaluaton Equaton (4) shows dstrbuton of contact stresses resultng from a rgd plate loadng actng on the surface of a sem-nfnte medum and ths dstrbuton s not the same as for a two-layer structure. However, Sant-Venant s prncple states that nfluence of load dstrbuton s n the vcnty of the load applcaton pont and t s assumed that f the thckness of the surface layer s bg enough, contact stress dstrbuton for sem-nfnte medum may be used n two-layer system wthout affectng computatonal accuracy. Fgure 6 shows a two-layer analytcal model. Thckness of the frst layer and the rato of the elastc modul between the frst and second layer were vared and analyses performed n order to evaluate the lmtaton of the approxmate method explaned above. The accuracy of the analyss was evaluated n terms of deflectons under the base of the rgd plate n order to see whether they remaned constant. The root mean square error shown n equaton () between mean deflecton under the rgd plate and each deflecton wthn the doman was used as evaluaton crtera. RMSE () w 5 5 w w where w s the mean surface deflecton and w s the deflecton wthn r 0~4cm. ( 0 r a ) at cm each ( r a ). Good agreement of dsplacements for all the ponts of nterest wll make RMSE equal to ero. Ths wll be an ndcaton that approxmate soluton s equal to the theoretcal soluton. It s clear from Fgure 4 that rrespectve of the number of load subdvsons, there s poor agreement between approxmate soluton and theoretcal soluton at the load edge, and hence results at the load edge poston were not consdered n equaton (). The scope of applcaton of the approxmate loadng was evaluated by usng a two-layer system shown n Fgure 6 by followng the procedure explaned below: ) Thckness of the frst layer h was vared from h a to 3 n 0. steps resultng n thckness varatons. ) For each value of h a, E E was vared between 0.9~0 resultng n varatons. 3) h a and E E varables resulted n combnatons, for whch analyses were performed and example of the results s shown n Fgure 7. 4) Usng equaton (), each combnaton was evaluated and analytcal errors RMSE are gven n Table. 48

7 Mana, Oawa and Matsu Vertcal Load 0 P a 5cm 49 kn 0.04 r / a h varable r Layer modul Posson's rato E E varable 0.35 E 00MPa 0.35 w (cm) Fgure 6 Two layer structure ( h a. 8, E E 3 ) Fgure 7 Surface deflectons E /E Table Least mean square error (LMSE) from Eq.() h / a Table shows errors ( RMSE) for each combnaton consdered. Wth regards to the ratos of layer modul and rrespectve of the h a value, rato 7 gves the largest RMSE value. Smaller ratos of layer modul, where the two-layer system s close to sem-nfnte medum, result n smaller errors n the surface deflectons. Practcal applcaton of the approxmate method should be for all combnatons wth h a and RMSE less than %. All the errors fallng outsde ths crteron have been grayed out n Table. Fgure 7 shows surface deflecton w for the acceptable regon of applcaton wth the hghest error ( h a. 8, E E 3). Surface deflecton at the center of the load s cm, whle at the load edge where ( r a ) the surface deflecton s 0.08 cm, showng consderably smaller error. Rgd plate versus unformly dstrbuted loads An example for the above case wth largest RMSE was assumed. Thckness of the frst layer was h 30cm and layer modulus was E 700 MPa. Elastc modulus of the second layer was E 00 MPa. Radus of the load was a 5cm and analyses for a 49 kn was performed consderng a rgd plate load and unformly dstrbuted load. Results were compled and presented n Cartesan coordnates where Fgure 8(a) shows contour plot of x due to rgd plate load and Fgure 8(b) s for unformly dstrbuted load. At the boundary between the frst and second layer where 30cm and ust drectly under the load, both 49

8 Approxmate Analyss of Rgd Plate Loadng on Elastc Mult-layered Systems (a) Rgd plate load (b) Unformly dstrbuted load Fgure 8 Contour plots for n X-Z plane loads result n tensle strans but results from the rgd plate loadng cover a relatvely smaller area. For the case of unformly dstrbuted load, there are hgh strans, whch are transmtted downwards whle for the rgd plate loadng strans are transmtted from the load edge downwards. CONCLUSIONS An approxmate method was developed whereby GAMES software, whch s capable of analyng effect of unformly dstrbuted load on a multlayered lnear elastc system, was utled to analyse a multlayered lnear elastc system under the acton of rgd plate loadng. Results obtaned are summared as follows: ) Wth an excepton of rgd plate edge, very accurate results were obtaned when load subdvson s greater or equal to 0. ) It was possble to accurately evaluate a multlayered lnear elastc system usng rgd approxmate contact stresses for a sem-nfnte medum. 3) There were very hgh strans developed at the layer nterface between the frst and second layers for the case of unformly dstrbuted load. However, because of better load dsperson from the rgd plate edge downward, relatvely low strans were developed at the layer nterface rght below the load center. 4) For h a, contact stresses for rgd plate loadng on a sem-nfnte medum may be used to analye a multlayered lnear elastc system rrespectve of the magntude of layer modul ratos. REFERENCES ) De Jong, D.E. Peut, M.G.F. and Korswagen, A.R.:COMPUTER PROGRAM BISAR, Layered Systems under Normal and Tangental Surface Loads, Konnklke/Shell -Laboratorum, Amsterdam, 979. ) Mana, J.W. and Matsu, K.: Development of Software for Elastc Analyss of Pavement Structures due to Vertcal and Horontal Surface Loadngs, Journal of Trasportaton Research Record No.896, TRB, pp.07-8, ) Sneddon, I. FOURIER TRANSFORMS, McGraw-Hll Book Company, pp , 95. x 50