‘reanoportation Recearch Record 1043 It Se clear from both of these figures that a sub- Stantial Increase in" longitudinal cracking can be ‘Obtained by reducing the standard deviation (see the examples in Figures ¢ and § shown by a dotted Line for a depth ratio = 0.75). This indicates that re sonable control of the quality of this construction material (concrete), pavenant thickness, and say-cut ‘epth, all conbined, can contribute to a reasonable ev-cut depths conctusrons The reaulte of this study can be cumarized as fol lows: 1. The development of longitudinal cracks in a savecut groove can be explained by a model using the Concepts of variebility in concrete strength and thickness of pavenent sections. 2." The model developed for this study is sensi- tive to the construction quality of pavenent. An in Provenent in construction quality can result In Feduction of sawcut depthe, The rellability of Lon ituainal eracks (being confined to saw-cut groove) ig also improved. This can save construction costs fas well as future maintenance and repair costa, 3. Figures 2-5 show that it is poseible to in~ Gace’ any” desired anount of longitedinal cracking along the sav-cut groove if an appropriate say-cut Gepth is provided. ‘4. The aggregates used in concrete affect the evelopment ‘of longitudinal cracks along a saw-cut groove. This finding 1e based on a study of two ag- Grogates (river gravel and Linestone) AcKNOMLBDGNENTS he authors ace pleased to acknowledge the combined efforts and support of the Center for Transportation Research at the University of Texas at austin and the Texas State Department of Highways and Public ‘Feangportation, in cooperation with the PKA, U.S. Department of Transportation. REPERENCE 4J.R. Benjamin and C.A, Cornel. Probability sta~ tiatics and Decision for Civil’ engineers. MeGraw-il1 Book Company, New York, 1970. she contents of this poper reflect the views of the authors, who are responsible for the facts and the accuracy Of the data presented herein. ‘The contents 0 not necessarily reflect the official. views ot policies of the Federal Highey Adninistration. ‘This Paper does not constitute s standard, epecifica tien, or regulation. Publication of thie paper sponsored by Committee on Rigid Pavements. Westergaard Solutions Reconsidered A. M. JOANNIDES, M. R. THOMPSON, and E. J. BARENBERG ‘mhe pioneering analytical work of Harold Maloon Westergaard (1 31-1950) has been at the heart of slab-on-grade pavenent design since the 1920s. Every code Of practice published since then Rokes reference to the thon: rd. cols ‘These solutions are only available for thee particular leading condi~ Eons (interior, edge, and corner) and acsune 2 sieb of infinite or ent-inti- nite dinensions. Since their first appearance, beginning in the early 1920s, Westers: publications. r4 equations have often been misquoted or misapplied in subsequent To remedy thie situation, a reexamination of these solution? Using the finite elenent method ie described in thie paper. A nusber of inter esting resulte ace. presented: (a) Several equations the Literature are erroneous, usually as a result of a jeribed to Westergserd in ies of typographical etrore or misapplicatione, or both, ‘The correct form of these equations and their Linitations have now been conclusively established. (@) Westergoara® original equation for edge strese 1c incorrect. The long-ignored equation given {in his 1948 paper should be used instead. corner loading responses have been developed. the development of Westergaard r (c) “Improves expressions for maxinun (@) Slab eize cequirenente for poner have also been established.u me pioneering analytical work of Harald Malcom Westergaara (1880-1950) -has "been at the heart of Slab-on-grade pavenent design aince the 1920s. Every code of ‘practice published since then makes cefer~ ‘nce to the *Westergaara solutions." these solutions fare only available for three particular loading con @itions (interior, edge, and corner) and assume a Slab of ‘infinite’ or semi-infinite dimensiones In Practice, the slab size required for the development Of Westersaard recponces ie detersined empirically. Several investigatora, however, have noted repeat= edly that although the Westergaard solution agreed fairly well with their observations for the interior Loading ‘condition, it failed to give even a close ‘estinate of the response in the cases of ease end Corner loading. ‘he time-honored Westergaard, coli tions deserve a thorough “reexanination using the tool of Finite elenent analysis now available. ‘The highlights of an effort to reevaluate the Westergaara solutions (1) ‘are presented. ‘the forms theoretical background, Linitationa, and applicebia~ ity of these equations’ have been exanineds and what are considered to be the most accurate formulae are Presented herein. Several empiricel adjustments to the Westergaard solutions are also considered, and slab size” requirenonts, for the Govelopnent of Westergaard responses are established. ‘Te basic tool for this study 1¢ the TLLI-stas Einite element computer program developed and exton= sively used at the University of tlinois. (2). The TLLT-SLAB wodel {= based on classical wediun-ehick plate theory, and employs the 4-noded, 12-degree-of— Ereedon plate-bending elenent, known in finite ele= iment Titerature as ACH or RIB? (3). ‘the Winkler= type subgrade assuned by Westergaaré {e nodeled a @ uniforn, distributes subgrade thecugh an equivalent wage formlation (4) » As defined by Westergaard, this is the case of « wheel load at 2 “considerable distance from the edges," with pressure "assumed to be uniformly dis tributed over the area of a small circle vith radius a" (G). After an oxtensive Literature survey and Comparisons with finite elenent resulte (6), the following interior loading equations are considered to be in their most general form. Maximum bending stress, o; Ordinary theory E3PC + wh /aah™f [en (28/0) +05-91 + 888207 ) speidtheory BIST f[(L + 1/24} (ea (2ND) 05-1 rBs1087 aw Foc se sts {OR +s] 2h [in 2tle') 405-9] 851250 9) Sepplemeniany og ESI2OT {[BRC +N /6AN (Ha) (eeary tea) Suplementars on BSIRST= {38C. +] /6%} (C0) cc) (gecil teory} Supplementary, og BSI25Q= {130 +N]/64"7} (MF 1 orsqoxe) Maximum deflection, 6; ree DERIC= ret) {1 +172) (a2) +7514) (Ff ae) ‘Transportation Research Record 1043, P = total applied esd, E © slab Young's modulus; » Slab Poteson's ratio; hf slab thickness: K © modulus of subgrade reaction; a = radius of clvesiar load; © = side length of equare loads w= (hi ~ uA)el) woh de cadive of reta- tive stiffness) be (a6 + m2) ~ o.675n if ac a.704h a deg 2 rainy ett teeter one y= Huler's constant’ (= 0,577 215 664 90). ‘ese equations have been incorporated into WESTER, ‘2 computerized compendium of closed-form eolutions for slabs on grade, developed in the course of thie research (1) Equation la follova from Equation 50 given in 1939, with the term [tn (21/2) ~ y]_ replacing the erm {un (21/ya)] used by Weatergnara (2)- Note that the symbol yes used in bquation 1a is the euler ‘constant, whereas Westergaerd uses this sysbol to @enote the antilog of the Puler constant. Bguation 1a also includes supplenentary stress, eg, Eiret cerived by Westergaard in 1939. This is calculated according to Equation 1a, which ie the sane as Equation 6 in "Stresses in Concrete Runways Of Airports" (2). qhis adéitional term vas intro- Guced to account ‘for the effect of the finite size of the loaded area and is "outistactorily appiicebie When a does not exceed 7 (7). Its contribution is Seuatly email, but it ie includes because of ite Figorous ‘analytical nature. the effect of the size of the loaded area will be discvased further here> Pgiation 1b employs Westergaard's “special theory,” fist proposed in 1926, in vhich radius Eeplaces the true radius, a, of the loaded aren. This was Introduced to account for the effect of shear stresses in the vicinity of the load, which 1e neglected in the “ordinary theory" of nediun-thick Plates. As Westergaard stated, "the effect of the Enlernese of the slab ie equivalent to a. rounding off of the peak in the diajrans of mosente* (5) 10 Setermine the relation anong hy ay and by Wester~ Seard’ (5) performed "nunericel computations s+ in Secordence “with an analysia vbich iz duo’ to” Ay Nadal." Results were fitted vith a hyperbola, the equation of which may be weitten in the form pre~ Sented earlier, "which ig custable for numerics caleulations™ (5). The valisity of Westergsard: Semienpirical adjustaent and of the resulting “pe cial theory” has been debated by various investiga tors [see, for example, Scott (8)], bute full di cussion of this tecue would be beyond the scope of this paper. The authors reconmond, however, ueing ordinary theory" when comparisons with finite ele= ent results are nade. ‘To obtain the interior stress in the cere of @ square Loaded area, radius a 1s replaced in Bauation Ane by & constant, ¢", related to the length of the aide of the square, cy as follows: i o's rI1/2%) en 0573904,..¢ he resulting expression is not stated explicitly by Westergaard, but follows directly fron his theory (S,10). ‘TMmoshenko “and Woinoweky-Keieger (11) Bro~ vide a theoretical justification for thie substitu ‘tion by shoving that, loeded by the same totel losd Py a square cide c and a circle radios a give the ‘sane mainum interior etreas.Toannides ot al. In 1948 Westergaaré presented an equation for the stress under an elliptical loaded area. (equation 2 in "Now Formulas for Stresses in Concrete Pavenonte Of Airfields" (o)}. setting both axes of the el- Lipse toa, this equation can be compromised with Equation 1a provided that’ the following sesumption ie made, mere rye lem 20-4) For 4 = 0.15, this aseunption gives asis916~o.61ss374 ‘This indicates that the term 0.6189 in Equation 9 in “Stresses in Concrete Pavenenta Computed by Theoret ical Anelysis™ (S) is a truncated form of the term involving) Euler"s constant, not lsh Poisson's ratio, y+ Equation la ie, therefore, more general than the 1940 equation. Equation 1g follows from Equation 51 in "Stresses in Concrete funways of Airports" (7), described pre viously, with the introdsetion of Buler's constants ‘his fore Le more general than the one obtained from Equation 5 in “Mow Formulas for Stresses in Concrete Pavenents of Airfields" (Ud), which makes the ap= roxization noted in the previous paragraph: Westergiard alco presented an equation for up- plementary stress, yy to account for’ "the ef fects of a plausible redistribution" of subgrade reactions (12,7). This vas a semiempirical aéjuct- hnent to redsce ‘calculated stresses 20. that they ‘agreed better with the 1932 Aelington teste (3) ~ Bergstrom et al. (14) note that "it appears sdvie— able to neglect "93 in design," because it is aifti- Gult to evaluate and causes’ considerable reduction fn strecs, Further discussions of this term are pre sented by Bradbury (13) and Kelley (26). Slab Size Requienents for the Development of Interior Loading westergaaré Responses As mentioned earlier, the closed-form Westergeard ‘solutions assume a slab of infinite dimensions, al~ as though in practice empirical guidelines have been Geveloped for the Least slab dimension, Ly required to achieve the Westergaard “infinite elab™ condt- tion. in this section, analyses will be presented to establish similar guidelines using the finite ele ent method. A slab with a radiue of relative stiff ness, t, of 2316 in. wae used with a mesh fine hess’ ratio (2a/a) of 1.8. an earlier study (6) Indicated that this ratio of elenent size, 2a, 4i~ vided by elab thicknese, hy must be about 0.6 for 98 percent accuracy. TILI-SIAB results fron this investigation are shown ‘in Pigure 1. Both maximun deflection and bend- ing stress converge to large ole values. ‘The con- vergence of geflection is from above, indicating that a smaller slab settles more than a bigger one ina "panch-like" fashion, Bending atcess converges fron velow, ae expected. ‘the rate of convergencey defined a6 the slab size at which the solution is essentially that for an infinite slab, se different Tor deflection (1/2 = 8.0) than for bending stress (/.'= 3.5). Surprisingly, deflection appears to be’ much “more ‘sensitive to slab size changes for (1/1) values of less than 3, because of the previ~ ously Wentioned punch-like effect. ‘the Limit value approached by maximm deflection ie the Westergeard Solution (Byuation 1g). ‘The value to which bending stress converges when. elab size is expanded {= slightly lover than Westergaara's (Equation 1a) due to the coarseness of the mesh used. EEfect of Size of Loaded Aces In his attenpt to develop equations for a loaded area of finite size, Westergasrd used an approsch in Which a solution for @ point lose is first derived. Ten, ‘the loaded arcs ie split into. a nunber of tml aubaren, and each subarea is replaced bys statically equivalent point load acting at its cen~ ter. A cuanation is perfornea over these subaress. In the Limit of refinenent, this eumnation tends to fan exact integration (27). Westergaaré suggested that his equations were valid for any size of loaded y | of | { \ \ yee % Westergoors Slob Size, L(f) —— Es 1 sas co 3 L ¥ FIGURE 1. Effet of dla sive on maximum interior lading responses,a area and that hie ‘New Formulas" (10) assume that “the average width and length of the footprint. of tthe tire de greater than the thickness of the elab Loeverg (8) showed that the stress and deflec tion equations presented by Westergaard are only the Hirst one or two terns of repidiy converging. inti nite series. Westergaard'e supplenentary stcens, entioned. previously, for example, is an additional term of this series. The rete of convergence can be expected to vary depending on, among other things, the size of the loadee area. ‘Timoshenko and Woinowsky-Releger (11) state that tthe ‘equations “apply only when the radius of the Joade area is "enall in comparison with 1." Scott (B) attributes enie restriction to the fact that "in the derivation of the equation a term of approxinate valve 0.1 3!/1" was omitted." thie cannot be the real cause of the restriction {aposed by ‘Timoshenko and Woinowaky-Reieger, because in most cases (even. when the Fadius of the losdea area, a, ie not "emell. in cosparison with i") thie term is, indeed, negli~ gible. Th’ this study the effect of the size of the loaded area vas investigated using the finite ele ment method. ‘To eliminate slab size, mesh fineness, land elenont asyect ratio ‘effects, ‘a large. (L/t 9-33) and fine (2a/h = 0.6) mesh, which consicted of sglare elenents (aspect ratio = 1.0}, wae used. he Fesulte are plotted in Figure 2. Tt is observed that Kestergaara strece values (equation 1a) agree with finite elenent results for a Toaded area whose Gide length, © (if square), 32 about 0.2 tines the radius of relative stiffness, ty Lf the load is circular, {te racive, ay mast bo about 0.1L. he (6/1) oF (a/t) increase,” Einite ele~ ment stresses become progressively higher than Westerganra's. Therefore the consequences of Wester gaara's truncation, mentioned previously, must be Borne in mind when’ attempting such couparicones The results in Figure 2 also sugyest an effect related to the internal finite elenent discretization of the applied load. this ie discussed in more detail else where ()+ Westergaard defines edge loading at the cace in Which "the wheel losd is at the edge, but at a con ‘Transportation Research Record 1043, siderable distance fron any corner." The pressure is Aeounea to be “dieteibuted uniformly over the eres Of a enall semivcircle with the center at the edge" (6). Bqvations. for # circular load at the edge were Eirst presented in 1948 (10). the most general forms of the edge loading formlas follow. Maxim bending stress, ¢ eda eoy —_BSEWOT=0529(1 +0544) A?) Nao (EN (emi #4) 020) es Syst tency BSENST=0529(1 +0544) Hogue? (Gmc smx)-071} eo New" formula BSEIC= [1 Pn 0 tn) ic $1.86 4403 + (4-02 +1180+200} @ New" frm SEIS 51+ Pe +] (CE Oc) (onic) 4384-4313 050 +20) (4/0) 0D Simpl “ew"_BSELS= (6) (1 +051) [0489 (I) formals Gems) = 0091-0027 /6) ° Sip “new™ —_BSELC™ CRN?) +054) (48906) formal (ee) 0012-006) en Cra formula DEFEW = (1/6%)(14 0459 0%) eo wien fora DEFEIC=(0(2 +1259} (68°91) (ice) 710276 +0419) en New" formule DEFEIS=(P(@2 + 1.24)"]}/ (4°R)"1) £1 (ome Som +01 I) @ Spite “new” DEFELS= (464). +040) €RE) foul emi) 70323 (14054) 4/91 © Simple “oow" —DEFELC= 1/8) (1 +044) PEE (1 fem) =1.760(1+054) 9) eo % Westergoord FIGURE2 Effect of tize of loaded ares on maximum interior loading sponees |Toannides et al- were ag = cadius of semicircle, Bh = ((.6ey + 472) ~ 6.6755 fe ay < 2.7200 say $6 ag > 1.724h, and other dymbols ave as defined for Equations 2. Equation 28 is identical to Byuation 3 in *Ana- lytical Tools for Juéging Resulte of structural ‘Tests of Concrete Pavements” (12). Equation 2b en~ loys the “special theory,” which ie also used for Interior loading. in his 1948 paper, Westergaard (G0) presented generalized solutions for, maximum sg ané deflection produced by elliptical and Semleitiptical loaded areas placed at a ela edge. Setting the lengths of both the major and minor Semlaxce of the ellipse to a or az lends tothe corresponding solutions for a circle fadiue, a, of a senicirele radius, a2, given by Equations Ze and Tosherg (18) presented simplified versions of these solutions by introdscing “simplifications of ‘the sane type as Westergaara (19) himself introduced An hie original formila for the case of edge losé~ ing” to eliminate the "complicated functional rela~ tionship" in which appears in these equations. Losberg (18) stated ‘that his simplifies equations Tare vell applicable, for the spall yevalues here concerned." There are fquations ze and 2¢. Conpar= isons. nade curing this study shov thet Losberg's ‘simplified equations lead to recults that are tyPi- Cally about 1 percent greater than those obtained by the general Equations 20 and 22. equation 23 is Westergsara's original equation for edge deflection (5), and Equations 2h and 24 can be obtained from his 1948 paper (10), as indicated previously. The corresponding Locberg formlas (28) fre given by Squations 25 and 2k, Setting the radius OF the Loaded ares to sero, these formulas reduce to Byyation 29. Alternative Hestergsard solution Xt was pointed out eariier, vestigators (14,18), that in the cage of interior Geflection and stront, az well as edge deflection, when the “new" formulas are specialized for a circu- well as by other in- lar (or a semicircular) loaded area, they become identical ‘to the corresponding original (5,12) equa tions. Results ‘from thie ‘study chow, however, that edge stresses calculated fron the tnev" formula are considerably dicterent than those computes using the Original formula. A munber of alternative Wester- Geard solutions are considered in this section, in Order to determine which one, if any, sgreet best with finite elenent results ‘TABLE 1 Altomative Westergard Solutions v In Table 1, five different Westergaard solutions fare conpered. "In all of these, total applied load, Py and applied pressure, p, are matched in the Westergaard and finite clenent analyses. A previous study (6) confireed thet this Le an appropriate cep resentation of the square loaded area used. in the Limite elenent solution. The solutions given in Table 1 are code naned WESI through WESV and were obtained using WESTER, vhich incorporates Equations The range of results in Table 1 is extcamely wide. Therefore careful use of Kestergaard'e. theory Cannot be overemphasized. he most obvious eftact te that the "new" formulas typically lead to stresseo 55 percent higher and deflections € percent Lower than the values obtained using the original formlae (compare WESIV with WESTI2) - 2 comparison of WESIV and WESV indicates that the seaicircular load ie nore severe than the circular Toad (ives, leads to higher etresces and deflec= tions), ‘as expected. Tf both the circular and the Semicitcular loads ‘are reduced to an equivalent point load acting at the respective center of gfav- ty, this expectation is shown to be justified be- ‘cause the center of gravity of the circle is further foward the interior of the elab then ie that of the semicircle. ‘This argunent elso leads to the conclusion thet the difference in response from a cizcular and a semicircular load should be fairly spall an propor Uonal to the difference in the distance between the respective centers of gravity and the slab edge. The Gitference between WESIV and WESV stresses is about 2 percent, and deflection difference is about 5 per~ cent. These differences are much nore compatible With’ expected values than is the stress aifference Sotained using the original equation (compare WEST fea WEStr2)- Table 2 gives a comparison of WESY, TLLI-stAB, ‘and B51 results, the Latter ie a computerised ver Sion of the Pickett and Ray (20) chart for edge Loading (21). streases exhibit almost perfect agree pent even at lov. (i/t) values. Deflections are more sensitive to tlab size effects, as shown in Figure 3. Teis graph shows thet an (1/1) value of about 5.0 is required for the development of Wester~ eard stresses and about 8.0 is required for Wester gaard deflections. The trends shovn in Pigure 3 are Similar to those observed for the interior condi= tion. ote, hovever, that the requirement for the development’ of matimn edge trent (L/1 = 5.0) te higher: than for maximum Interfor stress (l/t = 3-5) ‘The excellent agreement between ILLI-SIAB and R-Si results and the "ney" formsla confirns Losberg's ob- Servation that tthe original formule for edge load= ing according to Westergaara (5) is, at least fron a theoretical viewpoint, completely erroneous." ‘eteston 8 (nis) Beoting Sues 25 Ge) on No, Gein) tin) _WESI WES _WESITWESIV WESY WEN WEST WEST WES WESV 7 naa ig 997 ~=Ce eS 2 mo Sea sos 30 esl S9 TB 3 oS S13 ma 3) ea 4 so 6232 cmeaswest, = 80 ots 8 fon wtsar 5 moe ea Bo Sik fl Me ae eS é So is Ba ios ok S73) ee kek 5 som ae Be 62 3 mom te a med He 830m bee 5 So ea i243 eo ees te Ce ec Se Ne ea ec a Seyae ‘Teangportation Research Record 1043, ‘TABLE 2 Comparison with “New” Edge Loading Formulas Detection 5, Bending Sts ey x » WEsY ILLISLAB WELISLAD WESVWLLISUAB Ws) TELESLAB un No.Goalte) Gn) ye) mis) (is) WES pate dW r er 70970 e097 351098 a ao eS“ joe mo foe O99 i So 1 me 388 motos mn 4 soe ms Ne wo 3) 3 mo 16 mi tar Sy Ss oe é Sos zor toe a3 ase a 5 so} a 1S ays 6 a 2 oB Be tn 39 se xe 5 jo ite ms a Has as ents ea me iuui-suaB Wesv po 1 L 7 FIGURE 3. Effect of ala size on maximum edge loading responses. The Pickett and Ray charts (20) for the ease oading ‘condition on a dense Liguld ‘subgrade ‘are based on a pait of integral equations identical to those presented by Wertergaaré (U0). the results from these charts, therefore, agree with, the "new formulas, ab indicates by Sl results in Table 2, Te is interesting to note thet, although in several design codes reference 1s made to the original equa tion, the fact that multiple wheel leads are often coneiacred inplics, that design charts in these coSes Nave been obtained using the Pickett end Ray cherts (Gsesy the ew" foemilas) > The question of the source of the discrepancy be- tween Westergaard'e original and ney. formlaa for ending resains wnanawored. I ie too early to 28 the original fornilaa aa altogether false and uselees. Bergatzon et al. (14) reported that Values celouiated using these equations are in Fel= Stively close agreement with test results.” They furthermore suggested that” there are "no resvons to use the new formla for edge Loading.* On the other hand, soot (6) suggests that vexperimental indica tons are that ‘the edge stresses experienced in practice are higher than the Westergaara. (original) Equation "indicates." Laboratory model ‘esto by Gerlton and Behcnann (23) produced edge stresses 10 to 12 percent, tover than the new formula. predicts, Felnforcing the expectation that in eit values Probably Ite between the two Westergasrd equations, ‘The theoretical backscound for. the oFipinal edge stcere equation is also open to debates the derive lon of thie equation is not presented in any of Wostecgaard's papers. Attenpts by Losberg” da) to rive at this forma through Integration ‘of the expressions for the concentrated oad (19) id not Produce agceenent with Westergoard'e result. If the Probability of a gross theoretical blunder by euch possible explanation for this discrepancy is sone assumption regarding subgrade support at the slab edge, which 1s (aaliost inthe original formula bot i ‘never explicitly conmER LOADING Of the eheee fundamental cases of loading investi= gated by Westergaaré, cornet loading 42 undoubtedly: he most obscure and debatable, ‘the theoretical beckground for maxinm corner deflection and stress equations is particularly vesk. ‘Their semiempirical land approxinate nature has lea to numerous ‘ions and modifications in the yeare since their original publication, in an attempt. to reconcile observed slab behavicr with theory. These ave die cussed by Kelley (16) and Pickett (23) and sre sum Sarized ae follows: Deslection 82 PRE) [11-088 4,09] Weerard (5) ee eu"(@PM) Goldbeck (25) Oder(26) oe ee" PMA) (= (i094) Westrgand (5) 69 ehToannides et al. ¢= (3P/B?) [1 = (a/-] ‘o¢= (P/M?) {1 ~ (ay 8)"]_— Kelley (16), Teller vad Sutherland/ 3) (3e) a." 2PM?) [1 =(0/0)] Spangler (28) eo ocean) (1= {1609110925 +022 }]) Bradbury (15) os) Picket (23) Ge) Distance to point of maximm stress along corner angle bisector Xy=21@0%] Wester (5) oe» were 2 = radius of circular Load tangent to both edges at comer and ‘ay ~ distance to point of action of resultant ‘long corner’ angle bisector = (ZA, A Gee Bguations 1 for other symbols In the early 19202 4 ehort and simple piece of analytical work vas heralded as "the most important Bingle ttep in the Lnvestigation of the mechanics of Fond slabs" (24). This vas the flea attempt to Solve the problen of the “corner break" by two. proo- nent engineers of the day working independently, AuB. Goldbeck of the Bureau of Public Rosde and Ciittord clder of the Tlinoke Highway Department. By ascuning that in the corner region the slab acts as a cantilever of uniform strength (1.0. that in Enis region the subgrade reaction is negligible com pared to the applied losd), Equation 3b was pro- pored for the maximin stress, Gg; due £0 # concen Erated load, Py acting et the comer of @ slab, of thickness b”(25,26) . ‘A few yoare Later, Westergaard (5) took up the problem ajain, trying to account for the effect of a Toed distributed over some area, the ceaultant of which could be represented by # point losd P acting at a Gmall distance a from the corner, along the Bisector of the corner angle. Using a “simple ap- proximate process" involving the use ‘of the prin~ Ciple of minim potential eneray (27) he hoped to achieve an. "improved approximation" of corner Berens. ‘Thus he first arrived at Equation 3a. for corner deflection. He considered this equation *ap~ proxinately applicable for plausible ranges of ay ‘nd t" (presumably (ay/1) is not much greater than O11). From thie, he obtained bending monents by Ln kegration snd concluded that the maxinun stress Swould be represented with satisfactory accurscy" by Bquation 2c. Furthernore, the distance to the point Of maxinun stress along the corner angle bisector ae found to be given "roughly" by Bquation 3h. Equations for the Corner Loading Conéition Based on he Finite Element Method In this section, ILLI-SIAB is used to establish et of equations that vould accurately predict the Fesponse of a slab, in full contact with a Winkler foundation, to a single load distributed over a small area at its corner. Equations 3, propored by Previous investigators, suggest that, from a theo Eetical viewpoint, the parameters involved in the Geternination of slab response cen be lumped into ‘hee nondinensional ratios to. be investigated, namely, (éckt!/2) (ach*/2) and (2/1) or (ay/t) for & chroalar, of (c/t) for a equare load. ‘the results obtained fron eeveral ILLI-SLAB anal~ yees are plotted in a nondineneional fashion in Pig- res 4 end 5. Other available closed-form solutions ‘are also show in these figures for comparison. ‘The Inteer were obtained using a circular load of the same area as the square one in TLLT-SIAB but are » Sets oop) aeeskefucor of} oe, a o os “ FIGURE Compuson of ILLISLAB and Westead comer detections plotted with the corresponding (¢/#) values along the ordinate. Though not mathenatically rigorou! this facilitates direct comparison of the results. ILLI-SIAB stresses are values of the ainor (tensile) Principal stress occurring at the top fiber of the Slab. the maximin normal strese obtained fron finite lenent analysis (P. Frey, "Develoment of a Finite Elenent Based Expression to Predict. Maximum Corner Loading stresaee’ in a” Uniformly Supported Rigid Pavement on an Elastic Subgrade,* University of 11 Linois, Urbana, 1983) significantly underestimates ‘the critical stress. ‘Curves were fitted to TLLIMSIAB dats with a spe- cial effort made to keep the general form of the ‘equations the sane as that of the Mestergaaré formi~ las. Shue a straight Line may be used to deceribe corner deflections, fg, obtained using the finite ‘element method (Figure 4). This Line hax the follov= ling equation: 55= ha) [1.205 069.201 o where © o the side Length of eguare loséed area. ‘The similarity to Westergaard's equation indi~ cates that Westerganrd's approxination war foirly 008. the finite elenent results obtained are tYpi- cally about 10 percent higher than those predicted by Westergaard. & emall part of this discrepancy is Que to the lack of a theoretical solution for 2 square Loaded area, as vell as limitations of the finite elenent solution with respect to mesh fine hese and slab size. In the case of ILLI-SLAR maxinun corner stresses, cer curve Fitting suggested the following equation: se GPM?) [1.0-(1H°7} o ‘The Golabeck-older equation Le obviously a rough ‘approximation of the theoretical solution, mich nore 80 at high values of (c/t). This is expected, in View of the Coldbeck-older secumption of a concen~ trated toad acting on a cantilever. Assuming that Finite elenont results give a fairly accurate pic~‘Transportation Research Record 1043 F L L L oa) FI ture of the theoretical solution, the Westergaaré equation represents a considerable improvement over the Coldbeck-Older one. ‘The finite elenent method Gives results that fali between those predicted by estergaard (5) and thove predicted ‘by Bradbury (Us). Note that the empirical modifications to the Westergaaré formula proposed in the last 60 yeare-= with the exception of Brecbury's--nave tended to. in- Crease the discrepancy between calculated and theo- Fetical stresses. These modified expressions are such closer to the Goldbeck-older equation than to the theoretical solution. scott (8) pointe out that Sexporinental indications are that the corner stres~ ses experienced in practice are higher than. the Westergaaré cquation indicates." ote, however, that, in the model teste mentioned previously, neal ‘sured maximin comer stresses "were only 65 to 75 Percent a great as. those determined fron the Kestergaaré equation" (22). the very significant Limitations of the Winkler subgrade {deat ization for corner loading are reflected in Figure 5. At = re~ Sult, discrepancies between measured responses and theory may be expected. In the absence of nore. con elusive field data, it i prudent to design for a higher corner eteess than indicated by Westergoaré!e formula (Equation 3c). Location of Maximum ste ‘The ceaulte from 2 selected nusber of TLLI-SLAR runs fare given in Teble 3, whore Xz a2 obteined from TABLE 3. Location of Maximum Corner Stross IRE 5 Nondimensional maximum hending stress for corner loading. Equation 3h is compared with the location of the minor (Coneile) principal strese given by TuLY-SIAD, ‘This shows wat TLLI-SLAB usually gives somewhat greater distance than westergeard. On the other hand, model tests suggest values shout €5 percent of Westergeard's (22), Curve fitting through the values of 1 obtained by extrapolation from TLLI-StAB, Fesulted in the following equation: y= ns0ee9 gute © ‘This best-£1t equation indicates that the infiuence of the radiue of relative stiffness, ty is moch Greater than that of the size of the loaded area Westergaard'e equation suggente that these “tho. foo Eaneters contribute equally to the determination of Slab Size Requirenente for Corner tosding Westergaara Responses The pertinent results from this study are shown in Figure €, in which ILLI-SLAB deflections and stres~ fse8 axe Shown as percentages of the values given by the best-fit equations. The validity of any conclu~ sions drawn from cuch a comparison is not considered to be greatly affected by the numerical accuracy of the proposed formulas. The patterns observed in Fig~ lure 6 are the sane as those observed for the other Loading conditions. Once again, stresses converge faster, requiring tinimin (L/t) value of bose Toston of oe kr in) fc Bauer Bun No, ny tin) Gn) Equation 3h foo or oa Geos ho 3S 435 So go ae coor so in cont so da coe) woo 856 TLESLAB [ArNode By Extapolaton Popoted Equation BB is a Saat 3600 260s so ike ae ist iia Pa Ep aos Pes nae HiseYoannides et a1. a Bok q | ® ‘ene \ 1 ! ! 1 | os + os 5 ts os FIGURES et of abi on maxima core odin pons 4.0, ant are tose sensitive te changes in (t/H) Bllact of sizeof toned Axes Si, Ae Se haa afta ay ER sitive even to snail changes for (L/i), smaller Vigor than about 3.0. Infinite slab deflection requires en spans 7. shows finite element corner leading re- aes function of (6/1), wnere cis the | W/t) ratio of at least 5.0, Slab size require- sige length of the applieé square load. Rosponsec rente establiched ducing thie study are sunmarized are normalized with respect fo the values obtained in Table ¢ Using proposed Equations 4 and 5. ‘TABLE 4 Slab Size Requirements for Westergaard Responses Based on the Finite Blement Method (Eye values for inter eo u Comer 58 40 per (28-086 é thierreerzs Eo . n L L n ar cH as 3 cy e FIGURE fet of sie of onded arson maximum comer reponse2 Deflections are not very sensitive to changes in (c/t), but stresses diverge from the “theoretical” values as (c/t) exceeds about 0.2, ‘The trend. ex- hibited by corner stresses ia the reverse of that for interior loading. The effect of (e/t) on cor ner stresses is lose proncunced than on interior Stresses. Coments made earlier with respect to this effect are also generally applicable to corner Load ing. ‘concuustoN Since their first appearance, beginning in the carly 1920s, Westergaaré equations have often been mis= ‘quoted of misapplied in subsequent publications. To Fenedy this situation, a reexamination of these so- Lotions using the finite element method hae been Presented. This exercise yielded = nunber of inter” esting results: 1, Several equations ascribed to westergaard in the Literature are erroneous, usually as a result of a series of typogeaphical errors or misapplications, for both. ‘the correct form of these equations and their Limitations have now been conclusively eatal- Lished (Equations 1-3) 2. Westergaard's original equation for edge stress (5) 18 incorrect. The leng-ignored equation Given in his 1948 paper (10) should be used instead. 3. Inproved expressions for maxinum corner losd~ ing fesponses have been developed (equations 4-6). ‘4, Slab size requirenents for the development. of Westergaard responses have also been” established (oable 4) AcraoLzDcRT ‘The investigations for this paper were conducted under a research project (grant AFOSR-02-0143) spon sored by the Air Force Office of Scientific Research (E0SR), Ar Force Syctene Command, Bolling Air Force Base, District of columbia, Lt. Cols LD. Hokanson vas the Program Manager. 1. AM, Toannides. Analysis of Slabe-on-crade for 3" Variety of Loading and support Conditions: Ph.b. dissertation. University of TLlinois, Ure bana, 1984. Aull,” Tabstabaie, £.3. Barenberg, and _R.E. Salth. “Longitadinal Soint systens in siip= Forued Rigid Pavenents, Vol. II: Analysis of Load Transfer Systens for Concrete Pavenents. Report FAA“HD-79-4,, II, U.S. Department of ‘Transportation, Nov. 1979+ 0.c. Zlenkievice. ‘The Finite Element Method. Bra ed. MeGeaw-iil Book Company, New York; wn. uJ." Dawe A Finite Blonont Approach to Plate Vibration Problens. Journal of Mechanical Engi= neering Science, Vol. 7, Ho: 1, 1965. Him. Westergaaré. Stresses in Concrete Pave~ ments Computed by Theoretical Analysis. Public Roads, Vol: 7, No. 2, April 1926. Also Proc., Sth Annual Meeting.” HRB, National Research Councit, Washington, D.C., 1926, a= Conpatation of Stresses in Conorete fssds. H.R. ‘Thompson, EoJ. Barenberg, A.M. Tosnnides, and’ J.A. Fischer. Development of » Streas De~ Pendent Finite Element Slab Model. Report TR S3-1061. U.8. air Force Office of Solentific Research, Air Force syatens Command, Bolling ‘Air Force Base, Washington, D.C.» May 1983, HM. Westergaard. Stresses in Concrete Ramways of Airports. Proc., 19th Annual Heeting, ED, ao. ae az, aa. ae an ae as. 20. ae 2. 23. 2s. 26. an. ‘Transportation Research Record 1043 National Research Council, Kashington, D.C. 1939. Also in Stresses in Concrete. Runways Of Airports. Portland Cement Astociation, Chicago, Tite, Dec. 1941, RP. Scott. Foundation Analysis. Prentice-tait, Ine. Englewood Cliffs, W.3-, 1902. un.” Westergaard. Stress" Concentrations in Plates Loaded over Small Areas. ASCE ‘Trancac~ tions, Vol. No. 108, 1943. Hun. Wostergaard. slew Pormlas for steesses in Concrete Pavenents of Mrfielés, ASCE Trensac” tions, Vol. 113, 1948. 5. ‘Tnoshenko 2nd 5.” Wolnowsky-Refeger. Theory of Plates and shells. 2nd ed-, McGraw-Hill Book Company, New York, 1959. Buk. Westergaard. Analytical Tools for sudging Resulte of structural Tests of Concrete Fave~ ments. Public Roads, Vol. 14, No. 10, Dec. 1933. Lim, Teller and #.C. Sutherland. the Stroctural Design of Concrete Pavenents, Part $+ An Exper imental stuay of the Westergaara Analysis of Strece Condition in Concrete Paverent Slabs of Uniform Thickness. Puble Beads, Vol. 22, lov &y Apcil-dune 1943. SG. Bergstrom, £. Fronen, and 8. Linderhoim. Tavestigation of eel load Streseee in. Con crete Pavenents. Proceedings 13. swedich Cesent ‘and Concrete Research Institute, Royal Insti= ute of Technology, Stockholm, 1949. R.D. Bradbury. Reinforced Concrete Pavenents. Wire Reinforcenent Institute, Washington, D.C. 1938. E.B. Relley. Application of the Results of Re- search to. the Structural Design of Concrete Pavenant. Public Roads, Vol. 20, Ko. 5) July 41935; Vol. 20, No, 6, Aug. 2939, G. Pickett, Miz. Raville, W.c. Janes, ané oJ. Mccormick.” Detiections,. Yonents and Reactive Pressures for Concrete Pavements. Bulletin 65. Engineering Experiment Station, Kansar State Coltege, Pittsburg, Oot, 1951. A. Losberg. Structurally Reinforced concrete Pavenents. Doktorsavhandlingar Vie Chalmers Tekniska Hogckole, Cotesborg, Eweden, 1960. EAM. Westergaacd. Om Beregning af Plader pan elastik Underlag med saerlight™ Henbilk pas Sporgseaalet om Spaendinger { Botonvese (On the Design of Slabs on Elastic Foundation with Spe- cial Reference to stresses in Concrete Pave- Rents). Ingeniocen (Copenhagen), Vol- 32, 1923- Gs Pickett and G.K. Ray. Influence Charts for Concrete Paverents, ASCE ‘Traneactions, Vols ine, 1951. W.C. Kroger. Computerized Aircraft Ground Plo~ tation Analysis--zage Loaded Rigid Pavenent- Research Report HRA-RWHS72.Coneca2 Dynamics Corps, Fort Korth, Tex., Jan. 1967. P,P. Carlton and RM. Sohemann. A Model stoay of ‘Rigid Pavement Behavior Under Corner and Page Loadings. Proc., 35th Annual Meeting, HRD, National Research Council, Washington, D.C. sss. G. Pickett. concrete Pavenent Design, Appendix Tits A Stady of Stresses in the Corner Raglon ‘of Concrete. Pavonent Slabs Under Large Corner Yoads. Portland Conent Association, Skokie, Ti1., 1946, repeint 1981. Hai Weatergeard. Theory of stresses in Road Slabs. Proc., 4th Annual Mecting, HRB, National Reseach Council, Washington, D.C, 1925. A.T. Goldbeck. ‘thickness “Of Concrete Slabs. Public Roads, Vol. 1, Now 12, april 1919. ©. Older. Wighvay Research’ in T11inots. ‘ansactions, Vol. 87, 192 H.M. Westergaard. What Le frown of stresses. Mugineering News Record, Jan. 1937.‘Teansportation Research Racoré 1043 28, M.G. Spangler. stresses in the Corner Region of ‘concrete Pavements. Bulletin 157. Engineering Experinent Station, Tova State College, Anes, wa. accuracy Of the data presented herein. the contents 0 not necessarily reflect the official views or policies of the U.8. Alr Force. ‘this paper does not Constitute a standard, specification, or regulation: The contents of this paper reflect the views of the Publication of this paper sponsored by Committee on authors who are responsible for the facto and th Rigte Pavenents, ‘ Establishing Load Transfer in Existing Jointed Concrete Pavements WOUTER GULDEN and DANNY BROWN In this paper are described the results of @ research project that had the ob- jective of developing construction procedures for restoring load tranefer in ‘existing jointed concrete pavenents and of evaluating the effectiveness of the restoration nethods. A total of 28 test sections with various load transfer Ge~ vioes were placed. ‘the devices include split pipe, figure eight, vee, double vee, and dovel bars. Patching materials used on the project included the types of fast-setting grouts, three brands of polymer concrete, and plain port Land cenent concrete. The nunber and spacing of the devices and dowel bare were ‘also variables. in the project. Dovel bare and double vee devices were used on tthe major portion of the project. Performance evaluations vere based on deflee~ tion tests: conducted with a 20,000-Ib azle losd. Horizontal joint movenent. mea= surements and visual observations were also made. the ahort-term performance Gata indicate ‘good results with the dowel bar installations regardless, of Patching materials. the sections with split pipe, figure eight, and vee devices Eaiiea in bond ducing the first winter cycle. The resulte with the double vee Sections indicate the importance of the patching material to. the eucces: oF failure of the load transfer system: sone sections are performing well and other sections are performing poorly with double vee devices, lorizontal joint ovenent messurenonts indicate that neither the dowel bare nor the double vee Gevices are restricting Joint novenent. Many niles of Interstate pavenent have been con structed using plain jointed concrete pavenents of various thicknesses and joint spacings. ‘The presence The variability of the amount of lose transfer available fron aggregate "interlock created. by Changes in joint openings pointe out the need to Of a joint is a discontinuity that ceuses higher Stresees and deflections in the pavenent eapectally in the outelde corner area, Hany designs of Jointed concrete pavenent relied on aggregate interlock to Provide for the transfer of the load across the Joint, thereby reducing stress concentration and de- ‘fections under oad. Laboratory studies conducted by the Portland Cenent Rescetation (BCA) found that the effectiveness of “load transfer fron aggregate interlock depended on load magnitude, number of rep etitions, ‘slab thickness, Joint opening, subgrage value, and aggregate angularity (1). Tt vas also found’ that the effectiveness decreased with cum lative load applications. Provide for a nore positive means of load transfe Tn Georgia, and in many other states, dovel bere are placed in ‘newly constructed pavementa. Many older concrete pavonants do not have the dowel bare and this absence of @ pocitive means of load tranefer ie a factor that contributes to the deterioration of ‘these pavenent sections. Faulting measurenente made in Georgia in 1972 on projects that contained both eoveled and ‘nondoveled Joints indloated that the Presence of dovels reduced the rate of faulting (2) ‘The distrece found in plain Jointed concrete pavenents in Georgia generally has been caused bY the presence of an erogible base or subgrade, infll~ tration of surface water into the pavement system,