Application of Dynamic Relaxation in Thermo-Elastic Structural Analysis of Highway Pavement Structures

Transcription

1 9 th Iteratioal LS-DYNA Users Coferece Simulatio Techology () Applicatio of Dyamic Relaxatio i Thermo-Elastic Structural Aalysis of Highway Pavemet Structures Samir N. Shoukry, Gergis W. William, Mourad Riad West Virgiia Uiversity Morgatow, WV , USA Abstract This paper describes the applicatio of the dyamic relaxatio techique implemeted i LS-DYNA i aalyzig large trasportatio structures as dowel joited cocrete pavemets uder the effect of temperature variatios. The mai feature of the pavemet model is the detailed modelig of dowel bars ad their iterfaces with the surroudig cocrete usig extremely fie mesh of solid elemets, while i the bridge structure, it is the detailed modelig of the girder-deck iterface as well as the bracig members betwee the girders. The 3DFE results were foud to be i a good agreemet with experimetally measured data obtaied from istrumeted pavemets sectios costructed i West Virgiia. Thus, such a techique provides a good tool for aalyzig the respose of large structures to static loads i a fractio of the time required by traditioal implicit fiite elemet methods. Itroductio Modelig cocrete structures is very difficult because of the material ad geometrical oliearity ivolved i such structures. For example, whe modelig a cocrete slab of a highway sectio, the iterface betwee cosecutive slabs must be modeled. If the adjacet slabs are joied usig steel dowel bars embedded withi the slab, this will create oliearity withi the cocrete slab. The same is true for a cocrete bridge deck icludig steel rebar reiforcemets or the iterfaces with the steel girders. I geeral, whe modelig cocrete structures, their oliearity creates a situatio that must be accouted for whe choosig a modelig techique. A type of modelig focused o i this study ivolves the aalysis of the iterface betwee two coected cocrete slabs joied by steel dowel bars, which has bee modeled previously by various other researchers [1, ]. Previous studies have made simplificatios or assumptios by idealizig dowel bars as sprig or beam elemets. This approach igores the dowel-cocrete cotact that leads to triaxial stresses ear the edges of the slab [3]. Also, the cotact iterface itroduces the axial dowel forces that will resist the cocretes expasio ad cotractio due to temperature chage that cause stresses to develop at the mid-slab [4]. I order to accurately model the stresses caused by the chage i temperature, the cotact iterface betwee the cocrete ad the dowel must be accurately modeled. Accurately modelig the cocrete-dowel iterface requires a fie mesh aroud the dowel bar to accurately simulate the circular shape of the dowel. I the model demostrated i this paper, a mesh of approximately 73,000 elemets is used. Solvig this problem usig traditioal implicit techiques is ot practical because the calculatio ad recalculatio of the stiffess matrix aloe would be too expesive i terms of computig time. However, whe lookig at the literature ad 8-53

2 Simulatio Techology () 9 th Iteratioal LS-DYNA Users Coferece studyig other possible techiques, dyamic relaxatio was determied to be the best techique to use for this study [5]. Dyamic relaxatio (DR) is a explicit iterative process that ca be used to obtai static solutios of structural mechaics problems. DR ca calculate static solutios because it is based o the fact that a damped system beig excited by a costat force udergoes vibratios, however; the system will ultimately come to a fixed displacemet at the positio of the systems static equilibrium [6]. Dyamic relaxatio becomes a attractive techique because of its low storage requiremets as well as its faster computig time compared with other implicit techiques. Also, aother characteristic of the dyamic relaxatio method is that it is very useful i solvig problems that exhibit oliear geometric ad material behavior. Actually, the mai egieerig use of dyamic relaxatio is for problems with oliearity. I fact, Newto s methods with iterative techiques are more useful i obtaiig solutios of liear structural mechaics problems. This paper explores the use of the dyamic relaxatio techique to ivestigate the effect of temperature variatios o cocrete structures. Two examples of 3DFE models built for ewly costructed cocrete pavemet ad Reiforced Cocrete Bridge are preseted. The results obtaied from the 3DFE models are validated versus field measured data obtaied from the istrumetatio istalled i such structures. The success of the dyamic relaxatio techique is maifested by the good agreemet betwee the 3DFE-calculated results ad the field measuremets. Dyamic Relaxatio Formulatio i LS-DYNA As stated before, DR is a techique that uses a dyamic aalysis to produce a static solutio. Sice DR is a explicit dyamic aalysis techique based o viewig the steady-state solutio of the damped system as the static solutio, the equatio of motio goverig dyamic respose of a system ca be the startig poit i fidig a solutio: M x + Cx + P( x ) = f ( t ) (1) where M is the mass matrix, C is the dampig matrix, f is the vector of exteral forces, t is the time, is the th time icremet, ad the dots represet derivatives with respect to time. I this equatio, P( x ) is substituted for Kx where K is the stiffess matrix, P is the vector of iteral forces, ad x is the vector of depedet discrete variables (the solutio vector) as stated by Uderwood [18]. Usig z as the fixed time icremet ad employig the cetral differece method, the followig expressios ca be stated: ad the equality for x = ( x x ) / z () x = ( x x ) / z (3) x ca be obtaied usig the average value x = ( x + x ) (4) By substitutig Eqs. ()-(4) ito Eq. (1), the future values for displacemet ad velocity ca be foud:

3 9 th Iteratioal LS-DYNA Users Coferece Simulatio Techology () ( f ( t ) P( x )) + 1 M z 1 C 1 x = x + M 1 C (5) z + ( M z + 1 C) x = x + zx (6) I order to maitai the explicit form of the cetral differece itegrator, M ad C must be diagoal matrices. I the dyamic relaxatio techique, C ca be writte as: C = cm (7) where c is a scalar quatity. Substitutig this value for C ito Eqs. (5)-(6) = + 1 cz 1 f P x x + zm 1 (8) + cz + cz x = x + zx (9) where f = f ( t ) ad P = P( x ) for simplicity. M -1 is the iverse of M. Sice matrix M is a diagoal matrix, we ca write Eqs. (8)-(9) i terms of each of the idividual compoets of the vectors or matrices. + 1 cz 1 f i Pi x = x + zm i i ii (10) + cz + cz x = x + zx (11) i Where the subscript i idicates the i th compoet i the vector ad m ii correspods to the i th diagoal elemet of M -1. The itegratio caot be started by usig Eqs. (8) ad (10)-(11) because they require the velocities to be kow at t -1/, which they caot be. However, the velocity is kow at t 0. Sice 0 x must be zero for a static solutio, the startig coditio of the DR procedure ca be assumed to be: x 0 = 0; x 0 = 0 (1) ad by usig Equatio (4) ad the secod equatio i Eq. (1), aother equality ca be deduced: x 1 = x 1 By combiig Eqs. (10)-(11) ad (13), a expressio ca is obtaied for the velocity after the first time icremet: (13) x = zm ( f P ) / (14) Combiig Eqs. (8)-(9) ad (14) the followig set of iterative equatios is developed that ca be used to obtai coverged values for the velocity ad the displacemet of a system: = + 1 cz 1 f P x x + zm 1 (15) + cz + cz x = x + zx (16) x = zm ( f P ) / (17) 8-55

4 Simulatio Techology () 9 th Iteratioal LS-DYNA Users Coferece With iterative processes, there are parameters that must be chose that allow the solutio to either coverge slowly or quickly. These parameters are either ot kow, ad a estimated value is chose ad verified with covergece, or they ca be varied for sake of speedig up covergece. For DR, the dampig coefficiet c, the time icremet z, ad the mass matrix M are all values that could be varied i the aalysis. However, for a specific fiite elemet mesh, the mass matrix is kow ad the time icremet is specified, so the dampig coefficiet is the variable that must be optimized to obtai covergece i miimal time. This procedure ca be followed i Hallquist [7]. Figure 1 illustrates the effect of the dampig ratio o the covergece of the dyamic solutio compared to the static solutio for a sigle degree of freedom system. The plots i Figure 1 idicate that the DR techique acts as a critically damped system ζ= 0.05 ζ= 0.5 δ/ δstatic ζ= 0.50 ζ= ω t Figure 1. Displacemet-Time Histories for Differet Dampig Ratios. Respose of Cocrete Pavemet to Temperature Variatios The 3D fiite elemet model developed to simulate this pavemet sectio cosists of oe full legth slab havig a half slab before ad after it i the directio of traffic supported by both a base ad a subgrade as show i Figure. To allow for cotractio ad expasio of the slabs ad to esure that the dowel bars are the oly meas of load trasfer betwee the slabs, a 10 mm gap was assumed betwee the two slabs. I order to avoid problems that may arise with boudary coditios, the full width of the cocrete slab was modeled as 3.66 m. Also, the base ad subgrade were wideed 0.61 m o each side of the slabs. The cotact betwee the base ad the slab is maitaied by the self-weight of the slabs ad o other exteral costraits are applied. The subgrade depth is chose as 1.9 m ad o-reflective boudaries are applied to the bottom of the model alog with the sides of the base ad subgrade layers. The o-reflective boudaries simulate the semi-ifiite extet of the subgrade by allowig the propagatig stress waves to pass through the edges. A mesh of 8-ode solid brick elemets is used throughout the model. However, a fier mesh is used whe modelig the dowel bars as well as the cocrete aroud the dowel bars ad the subgrade layers i close proximity to the dowel joit. This fie mesh allows modelig of the circular cross sectio of the dowel bar which is itegral i order to accout for dowel cotact as well as the associated states of stress that develop surroudig the dowels due to traffic ad/or thermal loadig. The cocrete elastic modulus was estimated based o the measured 8-day compressive stregth f c usig the ACI relatio (E c = 57,000 f c 0.5 ). The elastic moduli of the base ad subgrade layers were backcalculated from several Fallig Weight 8-56

5 9 th Iteratioal LS-DYNA Users Coferece Simulatio Techology () Deflectometer tests coducted moths after costructio. The model cosists of 35,018 odes ad 7,99 elemets with 1,637,95 degrees of freedom. The loadig placed upo this model is a temperature gradiet profile that is applied through the Figure Fiite elemet model of the cocrete pavemet thickess of the cocrete slab. The gravity was activated i the model by specifyig it as base acceleratio actig i the vertical directio. A data file cotaiig the distorted model ad the built up stresses becomes the startig file to which ay magitudes ad cofiguratios of traffic load ad/or temperature gradiet profile are applied prior to its reprocessig i a static or dyamic mode. This model is a excellet example of whe it is beeficial to apply the dyamic relaxatio techique to obtai a static solutio for the followig reasos: The model cosists of four differet materials ad a major area of cocer for this study was the state of triaxial stress at the iterface betwee the dowel bar ad cocrete. This clearly shows that the problem this study was cocered with is oliear i ature. The model cosists of a high umber of odes ad a high umber of elemets. Furthermore, because of the eed for very accurate modelig at the dowel-cocrete iterface, there are a large umber of very small elemets preset i the model. It ca clearly be see that the dyamic relaxatio techique is more efficiet to use tha the iterative techiques to solve this problem because the storage requiremets ad computig time required for the iterative techiques would be much higher tha that required for the dyamic relaxatio techique. From the istrumeted highway sectio, temperature variatio through the cocrete slab thickess ad the chage i this variatio ca be measured. These values are used to calculate temperature gradiets ad temperature chage profiles. The chage i the measured temperature gradiet profile through the slab thickess over a short period of time amely 6-10 hours was calculated. The 3DFE model was processed for differet temperature gradiets ragig from 8 C to +8 C; each is accompaied by a correspodig uiform temperature variatio calculated 8-57

6 Simulatio Techology () 9 th Iteratioal LS-DYNA Users Coferece usig Equatio 18. This temperature chage is applied to the odal poits of the cocrete slab, ad the model is processed usig LS-DYNA equatio solver. The 3DFE-computed strais are foud for all poits i the model that correspod to the locatio of strai gages i the istrumeted slab. The measured ad 3DFE-computed logitudial strais at slab top ad bottom are show i Figure 3. The compariso idicates a excellet agreemet betwee the measured ad 3DFE-calcuted strais. The differeces observed for the bottom strais ca be attributed to the built i costructio curlig iduced i the slab at the early age ad ot accouted for i the 3DFE model, which iflueced the cotact area betwee the slab ad the base layer. I the 3DFE model, the slab was assumed to be iitially flat ad i full cotact with the uderlyig base layer. Figure 3. 3DFE-Calculated versus Measured Logitudial Strais. Coclusios The study preseted i this paper presets the LS-DYNA dyamic relaxatio as a effective method for computig static solutios for large structures. Not oly is this techique acceptable, but it is also efficiet i solvig highly oliear problems. To exhibit the superiority of DR i solvig highly oliear structural problems, the logitudial ad trasverse strais throughout a cocrete slab due to temperature loadig were compared from DR results obtaied through FE aalysis ad experimetally obtaied values from a istrumeted highway sectio. The results show the effectiveess ad versatility of the dyamic relaxatio techique i modelig problems with itricate FE meshes alog with a high degree of oliearity. 8-58

7 9 th Iteratioal LS-DYNA Users Coferece Simulatio Techology () Ackowledgmets The work preseted i this paper is supported by research grats from the West Virgiia Divisio of Highways (WVDOH). The authors wish to ackowledge the support of the WVDOH staff from Materials Divisio ad Districts 4 ad 8 where the istrumeted sectio was costructed. Refereces [1] C. Chaakeshava, F. Barzegar, ad G.Z. Voyiajis, Noliear FE Aalysis of Plai Cocrete Pavemets with Doweled Joits, Joural of Trasportatio Egieerig, ASCE, vol. 119, o. 5, pp , [] W.G. Davids, 3D Fiite Elemet Study o Load Trasfer at Doweled Joits i Flat ad Curled Rigid Pavemets, The Iteratioal Joural of Geomechaics, vol. 1, o., pp , 001. [3] S.N. Shoukry, G.W. William, ad M. Riad, Characteristics of Cocrete Cotact Stresses i Doweled Trasverse Joits, The Iteratioal Joural of Pavemet Egieerig, vol. 3, o., pp , 00. [4] G.W. William ad S.N. Shoukry, 3D Fiite Elemet Aalysis of Temperature Iduced Stresses i Dowel Joited Cocrete Pavemets, The Iteratioal Joural of Geomechaics, ASCE, vol. 1, o. 3, pp , 001. [5] T. Belytschko, W.K. Liu, ad B. Mora, Noliear Fiite Elemet for Cotiua ad structures, Joh Wiley ad Sos, LTD, West Sussex, Eglad, 000. [6] M. Papadrakakis, A Method for the Automatic Evaluatio of the Dyamic Relaxatio Parameters, Computer Methods i Applied Mechaics ad Egieerig, vol. 5, pp , [7] J. Hallquist, LS-DYNA Theoretical Maual, Livermore Software Techology Corporatio, Livermore, Califoria,

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