BEAMS ON ELASTIC FOUNDATION THE SIMPLIFIED CONTINUUM APPROACH

Transcription

1 BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Univeritatea Tehnică Gheorghe Aachi din Iaşi Tomul LV (LIX), Fac. 4, 009 Secţia CONSTRUCŢII. ĂRHITECTURĂ BEAMS ON ELASTIC FOUNDATION THE SIMPLIFIED CONTINUUM APPROACH BY IANCU-BOGDAN TEODORU Abtract. The key apect in the deign of flexible tructural element in contact with bearing oil i the way in which oil reaction, referred to qualitatively a oil reactive preure (p), i aumed or accounted for in analyi. A magnitude and ditribution of p might be preliminary aumed, or ome mathematical relationhip could be incorporated into the analyi itelf, o that p i calculated a part of the analyi. In order to eliminate the bearing oil reaction a a variable in the problem olution, the implified continuum approach i preented. Thi idealization provide much more information on the tre and deformation within oil ma compared to ordinary Winkler model, and it ha the important advantage of the elimination of the neceity to determine the value of the foundation parameter, arbitrarily, becaue thee value can be computed from the material propertie (deformation modulu, E, Poion ratio, ν and depth of influence zone, H, along the beam) for the oil. A numerical invetigation of the implified-continuum approach i alo preented. Key word: beam; elatic foundation; Winkler foundation; Vlaov foundation; two-parameter elatic foundation; EBBEFp computer code.. Introduction Generally, the analyi of bending of beam on an elatic foundation i developed on the aumption that the reaction force of the foundation are

2 38 Iancu-Bogdan Teodoru proportional at every point to the deflection of the beam at that point. The vertical deformation characteritic of the foundation are defined by mean of identical, independent, cloely paced, dicrete and linearly elatic pring. The contant of proportionality of thee pring i known a the modulu of ubgrade reaction, k. Thi imple and relatively crude mechanical repreentation of oil foundation wa firtly introduced by Winkler, in 867 [3], []. The Winkler model, which ha been originally developed for the analyi of railroad track, i very imple but doe not accurately repreent the characteritic of many practical foundation. One of the mot important deficiencie of the Winkler model i that a diplacement dicontinuity appear between the loaded and the unloaded part of the foundation urface. In reality, the oil urface doe not how any dicontinuity (Fig. ). a b Fig. Deflection of elatic foundation under uniform preure: a Winkler foundation; b practical oil foundation. Hitorically, the traditional way to overcome the deficiency of Winkler model i by introducing ome kind of interaction between the independent pring by viualiing variou type of interconnection uch a flexural element (beam in one-dimenion (-D), plate in -D), hear-only layer and deformed, pretenioned membrane [3]. The foundation model propoed by Filonenko and Borodich in 940 [3] acquire continuity between the individual pring element in the Winkler model by connecting them to a thin elatic membrane under a contant tenion. In the model propoed by Hetényi in 950 [3], interaction between the independent pring element i accomplihed by incorporating an elatic plate in 3-D problem, or an elatic beam in -D problem, that can deform only in bending. Another foundation model, propoed by Paternak in 954, acquire hear interaction between pring by connecting the end of the pring to a layer coniting of incompreible vertical element which deform only by tranvere hearing [3]. Thi cla of mathematical model have another contant parameter which characterize the interaction implied between pring and hence are called two-parameter model or, more imply, mechanical model (Fig. ). Another approach to develop, and alo to improve foundation model tart with the three complex et of differential equation with partial derivative (compatibility, contitutive, equilibrium) governing the behavior of the oil a a emi-infinite continuum, and then introduce implifying aumption with repect to diplacement or/and tree in order to render the

3 Bul. Int. Polit. Iaşi, t. LV (LIX), f. 4, remaining equation fairly eay to olve in an exact, cloed-form, manner. Thee are referred to a implified-continuum model. Fig. Beam reting on two-parameter elatic foundation. Reiner [4], [5] pioneered a traightforward application of the implified-continuum concept to produce what i referred to a the Reiner Simplified Continuum model. Auming a foundation layer in which all inplane tree are negligibly mall, () σ = σ = τ = 0, x y xy and the horizontal diplacement at the top and bottom urface of the foundation are zero, he obtain the analytical olution for elatic foundation given by c =, 4c () q( x,y) q( x,y) c w( x,y) c w( x,y) where q(x,y) i a ditributed load acting on the foundation urface, w(x,y) the diplacement of the foundation urface in the z-direction, and E HG (3) c= ; c =, H 3 were E, G and H are deformation modulu, hear modulu and depth of the foundation, repectively. A conequence of aumption () i that the hear tree in the zx- and zy-plane are independent of the z-coordinate and thu are contant through the depth of the foundation, for a given urface point, (x,y). Therefore, thi model may be applied only to tudy the repone near loading contact area and not to tudy tree inide the foundation [3]. Vlaov, in 960, adopted the implified-continuum approach baed on the variation principle and derived a two-parameter foundation model []. In hi method the foundation wa treated a an elatic layer and the contraint were

4 40 Iancu-Bogdan Teodoru impoed by retricting the deflection within the foundation to an appropriate mode hape, φ(z). The two-parameter Vlaov model (Fig. 3) account for the effect of the neglected hear train energy in the oil and hear force that come from urrounding oil by introducing an arbitrary parameter, γ, to characterize the vertical ditribution of the deformation in the uboil []; the author did not provide any mechanim for the calculation of γ. Jone and Xenophonto [] etablihed a relationhip between the parameter γ and the diplacement characteritic, but did not ugget any method for the calculation of it actual value. Following Jone and Xenophonto, Vallabhan and Da [7] determined the parameter γ a a function of the characteritic of the beam and the foundation, uing an iterative procedure. They named thi model a modified Vlaov model [7], [8]. Fig. 3 Beam reting on two-parameter Vlaov foundation.. Governing Differential Equation for Beam on Two-Parameter Elatic Foundation All foundation model hown foregoing lead to the ame differential equation. Baically, all thee model are mathematically equivalent and differ only in the definition of their parameter [9]. The variou two-parameter elatic foundation model define the reactive preure of the foundation, p(x), a [9] (4) wx d ( ) d wx ( ) px () = kbwx () kb = kwx () k, where: B i the width of the beam cro ection; w deflection of the centroidal line of the beam; k the econd foundation parameter with a different definition for each foundation model. A a pecial cae, if the econd parameter

5 Bul. Int. Polit. Iaşi, t. LV (LIX), f. 4, k, i neglected, the mechanical modelling of the foundation converge to the Winkler formulation. Uing the lat relation and the beam theory, one can generate the governing differential equation for the centroidal line of the deformed beam reting on two-parameter elatic foundation a [9] (5) 4 wx 4 d ( ) d wx ( ) EI + k( x) w( x) k ( x) = q( x), where: E i the modulu of elaticity for the contitutive material of the beam; I the moment of inertia for the cro ection of the beam; q(x) the ditributed load on the beam... Parameter Etimation It i difficult to interpret exactly what ubgrade material propertie or characteritic are reflected in the variou mechanical element (pring, hear layer, etc.), thu evaluation on a rational, theoretical bai i cumberome. The advantage of implified-continuum approach i the elimination of the neceity to determine the value of the foundation parameter, arbitrarily, becaue thee value can be computed from the material propertie (deformation modulu, E, Poion number, ν, and depth of influence zone, H, along the beam) for the oil. Thu there i inight into exactly what each model aume and implie in term of ubgrade behavior. Therefore, with the aumption of vertical diplacement, (6) v( x,z) = w( x) ϕ ( z), and horizontal diplacement, (7) u ( x,z ) = 0, uing variational calculu, Vlaov model parameter are expreed a [], [7], [8] (8) where k H E ( ν ) dϕ = dz, k ( + ν )( ν ) dz 0 z inhγ H (9) ϕ ( z) = inhγ H E = ϕ dz, ( + ν ) 0

6 4 Iancu-Bogdan Teodoru i a function defining the variation of the deflection, ν(x, z), in the z-direction, which atify the boundary condition hown in Fig. 3, and (0) γ H ν = ( ν ) + dw + w. Since γ i not known a priori, the olution technique for parameter evaluation i an iterative proce which depend upon the value of the parameter γ. Therefore, by auming an approximate value of γ initially, the value of k and k are evaluated uing eq. (8). From the olution of the deflection of the beam, the value of γ i computed uing eq. (0). The new γ value i again ued to compute new value of k and k. The proceeding i repeated until two ucceive value of γ are approximately equal [7], [8]. 3. Example Even though the continuum mechanic approach may look laboriou and difficult to ue for a cloed-form olution, the numerical model i quite imple and can be eaily implemented in application-pecific oftware. An example of uch oftware i EBBEFp [6] which i developed in MATLAB environment and can handle a wide range of tatic loading problem involving one-dimenional beam upported by one- or two-parameter elatic foundation, for any loading and boundary condition. Uing EBBEFp computer code, the reult for a ample beam reting on Vlaov Simplified Continuum (VSC) model are compared with olution obtained by two-dimenional finite element plane train analye (-D FEM) (Fig. 4). a b Fig. 4 Geometry of the conidered example: a VSC model; b -D FEM model.

7 Bul. Int. Polit. Iaşi, t. LV (LIX), f. 4, A beam of length L = 0 m, width b = 0.5 m and height h =.0 m, with modulu of elaticity E = 7,000 MPa, i conidered to be uported by foundation having depth H = 5 m, deformation modulu E = 0 MPa and Poion ratio, ν = 0.5 (Fig. 4). The beam carrie concentrated load at end, 50 kn each. A total of 905, 5-noded triangular element with a fourth order interpolation for diplacement and twelve Gau point for the numerical integration were ued to define the meh for the -D FEM model. In both VSC and -D FEM model, the beam i modelled with flexure beam element (Table ). Table Beam Modelling VSC model -D FEM model Element type linear with node linear with node Total number of node Total number of element 34 7 a b c Fig. 5 VSC v. -D FEM olution: a ettlement; b bending moment; c hearing force.

8 44 Iancu-Bogdan Teodoru The reult from both -D FEM and EBBEFp technique are hown for comparion in Fig. 5. It can be noted that both olution have almot the ame hape and they are in good agreement. However, a full comparion between thee two technique i not fair, becaue in the -D finite element olution, complete compatibility of diplacement at the beam oil interface i aumed, but only vertical diplacement compatibility exit in Vlaov model [8]. The reult of the final computed value of the oil parameter are preented in Table. Thi demontrate the veratility of the (Vlaov) implified continuum foundation model: olve beam on elatic foundation problem without having a need to etablih the value of foundation parameter. Table The Obtained Value of Vlaov Foundation Parameter k, [kn/m ] k, [kn] γ Number of iteration, , Concluion The implified continuum approach, by Vlaov model viewpoint, for tatic tructural analyi of foundation beam, i preented. To demontrate the veratility of the Vlaov Simplified Continuum (VSC) foundation model, the reult for a ample beam reting on VSC model are compared with olution obtained by two-dimenional finite element plane train analye (-D FEM). A a general obervation, the obtained VSM olution are reaonably cloe to thoe from more ophiticated finite element olution. Received, October 7, 009 Gheorghe Aachi Technical Univerity of Iaşi, Department of Tranportation Infratructure and Foundation bteodoru@ce.tuiai.ro R E F E R E N C E S. Hetényi M., Beam on Elatic Foundation: Theory with Application in the Field of Civil and Mechanical Engineering. Univ. of Michigan Pre, Ann Arbor, Michigan, Jone R., Xenophonto J., The Vlaov Foundation Model. Internat. J. of Mech. Science, 9, 6, (977). 3. Kerr A. D., Elatic and Vicoelatic Foundation Model. J. of Appl. Mech., 3, 3, (964). 4. Reiner E., A Note on Deflection of Plate on a Vicoelatic Foundation, J. of Appl. Mech., 5, (958). 5. Reiner E., Note on the Formulation of the Problem of the Plate on an Elatic Foundation, Acta Mechanica, 4,, 88 9 (967).

9 Bul. Int. Polit. Iaşi, t. LV (LIX), f. 4, Teodoru I.B., EBBEFp A Computer Code for Analying Beam on Elatic Foundation. Proc. of the 7 th Internat. Symp. Computational Civil Engineering, CCE, Gheorghe Aachi Techn. Univ., Faculty of Civil Engng., Jay, Vallabhan C.V.G., Da Y.C., A Parametric Study of Beam on Elatic Foundation. J. of Engng. Mech. Div., 4,, (988). 8. Vallabhan C.V.G., Da Y.C., Modified Vlaov Model for Beam on Elatic Foundation. J. of Geotechn. Engng., 7, 6, (99). 9. Zhaohua F., Cook D.R., Beam Element on Two-Parameter Elatic Foundation. J. of Engng. Mech., 09, 6, (983). GRINZI REZEMATE PE MEDIU ELASTIC Abordarea din perpectiva implificată a mecanicii continuului deformabil (Rezumat) Apectul cheie în proiectarea elementelor tructurale rezemate au în contact cu maivele de pământ e referă la modalitatea prin care intervine în analiză preiunea reactivă a terenului, p, ca o încărcare de intenitate şi ditribuţie compatibile cu potenţialul maivului de pământ de a prelua încărcări (cazul fundaţiilor izolate au metodelor implificate pentru calculul grinzilor de fundare), au ca o variabilă necunocută, prin încorporarea a într-o relaţie conceptuală de proporţionalitate cu taările fundaţiei (cazul maivelor de pământ idealizate printr-o erie de reorturi a căror contantă de rigiditate ete tocmai raportul în care e găec preiunile reactive şi taările modelul Winkler). Pentru a elimina preiunea reactivă a terenului, ca o variabilă necunocută în oluţia problemei, e introduce o abordare din perpectiva implificată a mecanicii continuului deformabil. Comparativ cu modelul Winkler au alte modele mecanice, idealizarea prezentată prezintă avantajul eliminării neceităţii determinării, în mod arbitrar, a parametrilor caracteritici modelului avut în vedere; aceştia unt calculaţi în funcţie de caracteriticile de material ale maivului de pământ (modulul de deformaţie liniară, E, coeficientul lui Poion, ν ) şi adâncimea zonei de influenţă, H. Performanţa şi acurateţea formulării, în decrierea răpunului anamblului grindă maiv de pământ, unt tetate prin compararea rezultatelor cu oluţiile obţinute pe modele numerice mai complexe, în ipoteza comportării liniar-deformabile a maivului de pământ.