СРАВНИТЕЛЕН АНАЛИЗ НА МОДЕЛИ НА ГРЕДИ НА ЕЛАСТИЧНА ОСНОВА Милко Стоянов Милошев 1, Константин Савков Казаков 2 Висше Строително Училище Л. Каравелов - София COMPARATIVE ANALYSIS OF ELASTIC FOUNDATION MODELS FOR BEAMS Miko Stoyanov Mioshev, Konstantin Savkov Kazakov Higher Schoo of Civi Engineering L. Karaveov - Sofia Abstract: The Winker mode is the most popuar eastic foundation mode used in practice. Its ony parameter is the stiffness coefficient of the inear eastic springs that support the structura eement. The deficiency of this mode is the fact that it does not account for the interaction between individua springs. In attempt to improve this mode different scientists such as Pasternak, Kerr and others have proposed modes with two and three parameters in attempt to introduce an interaction between the individua springs. Such modes are caed mechanica eastic foundation modes. Another approach to the probem for beam on eastic foundation are the so caed simpified continuum modes where the foundation is treated as semi-infinite media. The constitutive reations are derived by introducing simpifying assumptions. Such a mode was proposed by Vasov and Leontiev. In this paper a comparison is made between beam on eastic foundation with one and two parameters derived by the Vasov's approach and the resuts are compared with a 2D FEM eement mode of the beam-foundation system. Key words: eastic foundation, foundation parameters 1. Introduction The most widey used method for cacuation of beams and pates ying on eastic foundation was the one proposed by Winker [8]. It assumes that the structura eement is supported by cosey spaced independent springs with stiffness k. We known drawback of this approach is the fact that no interaction between the springs is assumed, negecting in this way the shear stresses that occur in soi. Because of this discontinuity of the deformations between the oaded and the unoaded part of the soi appears (Fig. 1 a)). Different authors have tried to extend Winker's method in attempt to introduce interaction between the springs. Fionenko Borodich proposed a mode in which the springs are connected by an eastic membrane under constant tension [2]. Aternativey in [6] Pasternak proposed to connect the springs with a shear ayer of stiffness G (Fig.1 b)). Kerr went even further and in addition to the Pasternak's shear ayer added 1 Miko Stoyanov Mioshev, PhD candidate, Dip eng, VSU L. Karaveov, Sofia, miko_mi@abv.bg 2 Konstantin Savkov Kazakov, Prof. Dr. Eng, Structura mechanics dept, VSU L. Karaveov, Sofia, kazakov@vsu.bg I - 54
In the foowing sections of this artice a brief theoretica presentation of Vasov's approach is given. Furthermore, the stiffness matrix of a beam eement ying on one and two parameter eastic foundation is derived. These matrices are impemented in a numerica exampe and the resuts are compared to FEM mode of the beam-founadtion system which serves as reference. 2. Theoretica background Vasov considers a beam ying on soi ayer with thickness H, moduus of easticity E s and Poission ratio s. The function of the vertica distribution of the defection is chosen such that (2.1) 0 =1 and H =0 The dispacement fied is then expressed as the product of functions of a singe variabe I - 55
(2.2) v x, z =w x z and no horizonta dispacements are considered (2.3) u x, z =0 Under these assumptions the differentia equation of the beam-foundation system is derived as (2.4) E b I b d 4 w dx 4 k 1 w k 2 d 2 w dx 2 =q x where E b and I b are the moduus of easticity and the inertia moment of the beam. The the function of the vertica distribution of the dispacements is chosen as sinh 1 z /H (2.5) z = sinh The two foundation parameters can be computed by the foowing expressions (2.6) k 1 = 1 s E s b 1 s 1 2 s H sinh cosh 2 sinh 2 (2.7) k 2 = E b H s sinh cosh 2 1 s 2sinh 2 In the expressions above b is the beam width and accounts for the rate of decay of the dispacements aong the axis z. 3. Finite eement formuation A beam finite eement on two parameter eastic foundation is considered. If the nodes of the eement are denoted as i and j, the vector of the unknown noda dispacements is (3.1) {d e } T ={w i i w j j } Using the we-known Hermite cubic interpoation functions N i, the dispacement fied over the eement is expressed as foows (3.2) w e x =N 1 x w i N 2 x i N 3 x w j N 4 x j =[ N i ] T {d e } The stiffness matrix of an eement on two parameter eastic foundation can be written as the sum of stiffness matrices of the beam itsef and stiffness matrices reated to the two parameters. The expressions are shown beow. Can be seen that if the second shear parameter is set to zero the stiffness matrix of an eement on Winker foundation is obtained. ' (3.3) [k e ]=EI 0 [ N ' i ] T ' [ N ' j ]dx= E b I b [ 12 3 6 12 6 6 4 2 6 2 2 12 6 12 6 6 2 2 6 4 2] I - 56
22 54 13 (3.4) [k e,1 ]=k 1 0 [ N i ] T [N j ]dx= k 1 22 4 420[156 ] 2 13 3 2 54 13 156 22 13 3 2 22 4 2 36 3 36 3 3 4 2 3 2 36 3 36 3 [ (3.5) [k e,2 ]=k 2 0 [ N ' i ] T [ N ' j ]dx= k 2 30 3 2 3 4 2] 4. Numerica exampe A beam of ength 10000 mm is ying on a soi ayer with depth 5000 mm. The moduus of easticity and the Poisson ratio of the soi are E s = 20 MPaand s =0,25. The moduus of easticity and the Poisson ratio of the beam are E b = 27000 MPa and b =0,20. The beam width is b=500 mm and the beam height is varied from 400 mm to 1000 mm. This is done in order to investigate the contribution of the second subgarde parameter k 2 with respect to the bending stiffness of the beam. Two oad cases are considered: A) a concentrated force F=350 kn in the midde of the beam and B) two concentrated forces F=175 kn at the ends. The beam is discretized with eements having one and two parameters and the resuts are compared with a 2D FEM mode of the beam-foundation system. For this purpose the software package ANSYS is Tabe 2 contains the resuts for maxima bending moments taken with their absoute vaue. I - 57
Beam section Tabe 2. Maxima bending moment Max bending moment oad case A), [knm] Max bending moment oad case B), [knm] b/h, [m] 2 parameters 1 parameter FEM 2 parameters 1 parameter FEM 0,5/1 421.39 424.84 481.92 397.5 416.94 328.20 0,5/0,8 392.84 413.71 456.63 367.6 398.89 301.16 0,5/0,6 349.79 386.53 402.19 292.33 354.88 243.31 0,5/0,4 260.85 314.02 292.69 156.71 238.42 130.01 The resuts in Tabe 1 show that for both oad cases the difference between the dispacements of the one and two parameter mode is negigeabe if the beam stiffness is very arge. This is iustrated aso by the graph in Fig.4 where this difference denoted as Deta is potted over the beam stiffness E b I b. 0.0090000 0.0080000 0.0070000 0.0060000 Deta, [m] 0.0050000 0.0040000 0.0030000 0.0020000 0.0010000 0.0000000 1125000 576000 243000 72000 Stiffness EI, [knm^2] LC A) LC B) Fig. 4. Dispacement difference over the beam stiffness 5. Concusions The foowing concusions can be drawn form this investigation: For arge beam stiffness negigeabe difference between the dispacement resuts for one and two parameter foundation mode is observed for both oad cases. For oad case A) (i.e. concentrated force in the midde) adding a second foundation parameter yieds resuts that are coser to the FEM mode On the other hand, for oad case B) (i.e. two forces at the ends of the beam) the mechanica modes produce sighty conservative resuts. Furthermore, for oad case B), adding a second subgrade parameter yieds bending moments which get coser to the FEM soution as the beam stiffness decreases. REFERENCES [1] Bankov B., Pavova J.: The Finite Eement Method in structura mechanics, UACEG, Sofia 1999 (In Bugarian) [2] Fionenko-Borodich M., Some approximate theories for eastic foundation, Moscow 1940 (in Russian) I - 58