Modified model of a plate based on the Winkler foundation

Transcription

1 Computational Methods and Experimental Measurements XII 417 Modified model of a plate based on the Winkler foundation T. Karamanski & K. Kazakov University of Architecture, Civil Engineering and Geodesy, Bulgaria Abstract A modified model of a foundation plate based on Winkler's hypothesis is proposed. The "modified Winkler's model" takes into account the action of the soil massif beyond the contour of the foundation plate via the application of an additional stiffness along its contour. The formula for the evaluation of the Winkler's constant for this additional stiffness is derived. The comparison between the model "elastic media" and the "modified Winkler's model" is performed by the FEM using software system ANSYS. The verification of the proposed method is performed on the structure of a residential building founded on a foundation plate. The results for the vertical deflection and the bending moments in the foundation plate using the "modified Winkler's model" are close to the results when we used "elastic media" model. So we create an effective tool for engineers by using the simple Winkler's hypothesis which increases the accuracy of the results. 1 Introduction The equation describing the bending of a plate on elastic foundation is: D w( x, = q( x, р( x, (1) A great number of foundation models are known, i.e. of the connection between the vertical deflection w and the ground reaction q. The representation of the soil as a layer of solid finite elements is logical and plausible, however, it is complicated with respect to numerical modeling, and on the other hand not all software systems provide this possibility. Unlike this, the Winkler s model is much easier to apply. It is characterized by simple and visual representation of the process of modeling, as well as of the numerical results. 5 WIT Press

2 418 Computational Methods and Experimental Measurements XII However, this model has two significant shortcomings - it does not reflect the distribution potential of the soil and it does not take into account the action of the ground beyond the contour of the foundation plate. The main objective of this study is to create a modified model based on Winkler's hypothesis taking into account the action of the soil massif beyond the contour of the foundation plate via the application of an additional linear stiffness along its contour. Task formulation The comparison between the model elastic media and the modified Winkler's model shall be performed applying the Finite Element Method using software system ANSYS. The following elements are used: Shell 63, Solid 45, and Surf 154. The study will be performed on the structure of a real seven-floor residential building founded on a common foundation plate. The upper structure of both models includes the foundation plate, the vertical elements to level of the firstfloor slab, and the first-floor slab. With the first model the elastic medium is modeled as a layer of finite elements Solid 45, while the second one is modeled employing special option EFS (Elastic foundation stiffness) of Shell 63. In plan the models have the geometric characteristics given by the project. We shall compare the results of the vertical deflection w and the bending moments in the foundation plate of the two models along eight lines on the foundation plate. The characteristics of the real project are: For the concrete: Е = 5 GPa, ν =.; For the soil: E = 158 MPa, ν =.3; Winkler's constant: k s = kn/m 3. Constructing the elastic medium model the following options or specific features of the software system ANSYS are used. The nodes of the elements Solid 45 and Surf 154 coincide. The option ADJUSTMENT REGIONS (CEINTF command) is used for the coordination of the vertical deflection of the elements Solid 45 and Shell 63. This command is used to "glue" the two regions together, when the density of the finite element mesh is different. This is achieved by linking via constraint equations the degrees of freedom of the nodes of region A with the nodes of region B via the shape functions of the elements of the second region. For the purposes of this study this is performed only with respect to vertical deflection. The use of the shape functions of the plate elements Shell 63 proved to be relevant. 3 Computational results Figure 1 shows a comparison of the vertical deflection along line between the classic Winkler's model (the solid line) and the elastic medium model (the dashed line). Significant differences are observed. The diagram of w with elastic medium has lower values along the contour of the foundation plate due to the 5 WIT Press

3 Computational Methods and Experimental Measurements XII 419 action of the ground beyond the plate. The bending moments show also significant differences Line -1.5 UZ, mm Figure 1: Vertical deflection along line between "elasstic medium" model and classical Winkler s model. In order to approximate the results of the Winkler's model to the "elasstic medium" model, linear Winkler's constant [kn/m ] was also applied along the contiour of the plate in addition to area Winkler's constant [kn/m 3 ]. Numeric experiments were performed varying the value of this additional linear stiffness along the contour. Figure represents the comparison of the diagrams of the vertical deflection along line between the so obtained modified Winkler's model with linear stiffness along the edges of the plate k* = 5 kn/m and the elastic medium model. It can be noticed that these diagrames are considerable similar.,,,5,9 1,3 1,8,,7 3,1 3,6 3,9 4, 4,5 4,8 5, 5,6 6, 6,4 6,8 7, 7,5 7,8 8,3 8,7 9, 9,7 1, 1,7 11, 11,3 11,8 1,3 1,8 13,3 13,8 14,1 14,4 14,8 15,3 15,7 16, 16,5 16,8 17,3 17,7 18, 18,6 19, 19,4 19,8,3,7 1, 1,7,,5,8 3, 3,7 4,1 4,6 5, 5,5 5,9 -,5-1, Line UZ, mm -1,5 -, -,5-3, -3,5-4, Figure : Vertical deflection along line between "elasstic medium" model and modified Winkler's model. -6 Line MX, kn.m/m Figure 3: Bending moment Mx along line between "elasstic medium" model and modified Winkler's model. 5 WIT Press

4 4 Computational Methods and Experimental Measurements XII The same high degree of approximation is also observed if the diagrams of М х and М у are concerned. Figure 3 and figure4 show these diagrams along line. The diagrams of the vertical deflections and the bending moments along the rest of the lines of the modified Winkler's model almost coincide with the diagrams of the elastic medium model Line MY, kn.m/m Figure 4: Bending moment My along line between "elasstic medium" model and modified Winkler's model. 4 Theoretical investigation of the additional stiffness of the modified Winkler's model We shall use the theory of bending of rectangular plates on elastic foundation of Vlasov and Leontiev []. We shall give a brief account of certain elements of this theory. Vlasov and Leontiev propose the function of vertical deflection of the ground under the foundation plate to be represented as: w n ( x y, z) = W ( x, ( z), χ, () k = 1 where W k is the deflection of the surface of the ground; χ k is the function of distribution of this deflection in depth of the ground, which is represented as: or k k H z χ k ( z) =, (3) H ( H z) shγ χk () z =. (4) shγh Beyond the plate the function of the vertical deflection is represented as an exponential curve respectively in the direction of axes x and у and in the edge zone: ( x a) w a ( x, = w a ( е, ( y b) w b ( x, = w b ( x) е, (5) 5 WIT Press

5 Computational Methods and Experimental Measurements XII 41 ( x a) ( y b) w c ( х, = w c е e, where: w a, w b - vertical deflection at the edge of the plate paralel to axis y, respectively x; w c - vertical deflection at the edge point of the plate. This can be represented vissually as shown in figure 5. w b w c b a y x w a Figure 5: Example of plate on Vlasov and Leontiev s foundation. The differential equation of bending of the plate on a uni-layer foundation is: D w t w + kv w = p( x,, (6) where t and k are characteristics of the ground: H H E t = ( + ) E χ dz, kv = 4 1 ν χ' dz (7) (1 ν ) Taking into consideration the above expressions, we obtain the expressions of the so-called fictive forces Q ф and R ф (figure 6), with which the ground beyond the plate acts upon it. These fictive forces are of the following general type: ф w 1 w Q a = t α w +, a x a α y a ф w 1 w Qb = t α w +, (8) b y α x b b ф 3 R = tw c, k v where: α. (9) = t 5 WIT Press

6 4 Computational Methods and Experimental Measurements XII y x R ф Q ф a R ф Q ф b R ф Figure 6: Example of plate with so-called fictive forces. Now we shall try to transfrom the dependences (8) for the perposes of this analysis. If we ignore the members of the derivatives of w and take into consideration (9), the expression for Q ф will be: ф Q = tkv w (1) Hence, the required Winkler's constant along the edge will be obtained from the expression: * k = tk (11) If we asssume the dependence (3) to be valid, we obtain for t and k: E H E 1 t =, k 4( 1 +ν ) v = (1) 3 (1 ν ) H If we substitute (1) in (11), we obtain: E k * = (13) 6(1 +ν )(1 ν ) If we accept the formula of Tcitovich for the relation between the Winkler's constant and the ground elasticity module: 1 k s = E.87 b( 1 ν ) (14) and substitute Е from (14) in (13), we obtain: k* =.355ks b 1 ν (15) With plate width b = 1 m and ν =.3 we obtain: * k =. 97k (16) This result is relatively close to the experimentally obtained result: * k =. 5k (17) Recommending stiffness is given by (18): k c = k + k. (18) 5 WIT Press

7 5 Conclusions and recommendations It cannot be maintained that the expression (15) exhausts the problem of defining the additional stiffness along the boundaries of the plate as deriving it a number of assumptions have been made, some of which will be subjected to further clarification. In any case, however, it can be maintained that the modified Winkler's model simulates to a higher degree the elastic medium and it is easy to apply. References Computational Methods and Experimental Measurements XII 43 [1] Bozhinov, B. A., Etimov, T. G., and Gocheva-Raikova, D.S. "Analysis of structures on elastic foundation", Sofia, Technica, 198. [] Vlasov, V. Z. and Leontiev, N. N., Балки плиты и оболочки на упругом оснований - Москва, Физматгиз, 196. [3] Etimov, T. G., "Interaction of structures, foundations and ground", Sofia [4] Bathe K. J., Finite element procedures in engineering analyses New Jersey, Prentice-Hill, 198. [5] ANSYS, Structural Analysis Guide, Ansys Release WIT Press