Solution Methods for Beam and Frames on Elastic Foundation Using the Finite Element Method

Transcription

1 Solution Methods for Beam and Frames on Elastic Foundation Using the Finite Element Method Spôsoby riešenie nosníkov a rámov na pružnom podklade pomocou metódy konečných prvkov Roland JANČO 1 Abstract: This paper contains numerical methods for the solution of plane beams and frames on elastic foundation. In the first case, the solution is based on beam elements BEAM5 (ANSYS software) and the derivation of the stiffness matrix of the element is presented. The second approach uses a beam element with a contact element with the description of the derivation of the stiffness matrix for the frame on elastic foundation. Both solutions are compared with theoretical solutions. In conclusions, the influence of the number of divisions of the beam element on the accuracy of the solution is shown. Abstrakt: Príspevok obsahuje numerické spôsoby riešenia rovinných nosníkov a rámov na pružnom podklade. V prvom prípade na riešenie je použitý nosníkový prvok BEAM5 (program ANSYS) a odvodení matice tuhosti tohto prvku je uvedené. Druhý prístup je použitím nosníkového prvku s kontaktným prvkom aj s odvodením matice tuhosti. Oba spôsoby riešenia sú porovnané s teoretickým riešením. V závere je ukázaný aj vplyv delenia nosníka na prvku na presnosť riešenia. Keywords: elastic (Winkler's) foundation, Finite Element Method, beam element, mechanical contact Klíčová slova: pružný (Winklerov) podklad, metóda konečných prvkov, nosníkový prvok, mechanický kontakt 1. Introduction Solutions of the frames and beams on elastic foundation often occur in many practical cases. For example, solution of building frames and constructions, buried gas pipeline systems and in design of railway tracks for railway transport, etc. Solution of beam on elastic foundation is a statically indeterminate problem of mechanics. In this case, we have the beam with elastic foundation along the whole length and width or only over some part of the length or width. From the theoretical point of view, the exact solution of this type of problems exist only for some type of beams. Detailed explanation of theoretical solution can be found in [] and []. Not all problems can be solved by theoretical approach because theoretical solution is very complicated. In solution of this type of problems, it is possible to use the numerical methods. In this paper, the Finite Element Method approach for the solution of D beam and frame on elastic foundation is described. 1 M.Sc. Roland JANČO, Ph.D. ING-PAED IGIP, Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Applied Mechanics and Mechatronics, Nám. slobody 17, 81 1 Bratislava, Slovak republic, roland.janco@stuba.sk.

2 . Theoretical background for D beam on elastic foundation The Winkler's foundation model is easy to formulate using energy concepts. The Winkler's model postulates a foundation to resists only displacement normal to its surface, with resisting pressure p = Kv /Pa/, where K /Nm - / is the foundation modulus and v /m/ is lateral deflection at the foundation surface. This model deflects only where load is applied. The surrounding foundation is utterly unaffected (Fig. 1a). An area da of the foundation surface acts like a linear spring of stiffness C = p da / v = Kv da / v = K da /Nm -1 / Strain energy in a linear 1 spring is Cv. Now considering a structural element, perhaps a plate bending element or one face of a D solid element that has an area A /m / in contact with the foundation. ateral v, d contains D.O.F. of deflection of area A normal to the foundation, is [ N ]{ d } where { } = f f element nodes in contact with foundation. Strain energy U in foundation over area is U = Kv da = v Kv da = T T { f } [ f ]{ f } f d K d (1) in which the Winkler's foundation stiffness matrix for the element is T K = K N N da. () [ ] [ ] [ ] f f f (a) (b) Fig. 1 Deflection of elastic foundations under uniform pressure q applied directly to the foundation. (a) Winkler's foundation. (b) Elastic solid foundation [1].. Solution base on D beam element with elastic foundation properties (1. method of solution) When we consider the beam element with Winkler's foundation, than matrix [ N f ] is shape function matrix in the eqn. (), which is identical to the shape function matrix of the beam N are without elastic foundation. Components of matrix [ ] f x x N 1 = 1 +, x x N = x +, () x x N =, x x N = + and da = b dx, where b is the width of the beam face contact with the foundation.

3 We can input eqn. () into (), and get 1bK 11b K 9bK 1b K b K b K 1b K b K [ K f ] = () 9bK 1b K 1bK 11b K b K b K 11b K b K The stiffness matrix of beam without shear deformation can obtain the formal approach using equation T K B EI B dx, (5) [ b ] = d N where B is the strain-displacement matrix, which is defined for beam by B =. dx After mathematical solution of equation (5) using eqn. (), we obtain the stiffness matrix for beam considering only bending moment and transversal load at the nodes EI 6 6 [ K ] b =. (6) If we consider axial loading, stiffness matrix for the beam becomes AE AE EI 6EI 1EI 6EI 0 0 6EI EI 6EI EI 0 0 [ K b ] = AE AE. (7) EI 6EI 1EI 6EI 0 0 6EI EI 6EI EI 0 0 K (eqn. Resulting stiffness matrix for beam on elastic foundation is the sum of matrix [ ] f ) and [ K b ] (eqn. 7), which is following

4 [ K ] [ K ] [ K ] = + = bef f b AE AE EI 1bK 6EI 11b K 1EI 9bK 6EI 1b K EI 11b K EI b K 6EI 1b K EI b K = AE AE EI 9bK 6EI 1b K 1EI 1bK 6EI 11b K EI 1b K EI b K 6EI 11b K EI b K (8) Hence, in ANSYS software [], the element BEAM5 has a stiffness matrix derived from eqn. (8).. Solution base on D beam and contact elements (. method of solution) The second method of solving beam and frames is by using a beam and contact element. Stiffness matrix for beam without shearing deformation and axial loading is given in eqn. (6). Contact will be simulated by spring element between rigid ground and beam in the fig.. Stiffness matrix for spring element is as follows C C K spring = C C, (9) where C is stiffness of spring. Fig.. Beam and contact element. Global stiffness matrix for beam and spring element is given by combining eqn. (6) and eqn. (9), which is

5 where 1 + CM CM EI CM 6 0 C M [ K bc ] =, (10) CM CM CM 0 0 CM C is the modified stiffness of spring solved by the following equation M CM = C. (11) EI 5. Verification of numerical solution Consider the beam of elastic foundation in fig., where the length of one-half of beam is 1.8 m, this is the parameter. Beam is made from steel, which has the Young s modulus 5 E = x10 MPa with rectangular cross-section area by parameters b = 00 mm and h = 00 mm. Foundation modulus K is equal Nm - and applied force F is equal N Theoretical solution Fig.. Beam on elastic foundation. The theoretical solution of beam on elastic foundation in Fig.. is described in literature [1, chapter 9]. Solution from literature is in Tab. 1. d v EI kv 0 dx + = k ω = EI v( x ) = ω ω ω ω ωx -ωx e (A1cos x+asin x)+e (Acos x+asin x) ω ω ω A1 = B [1 + e +e cos( ) e sin( )] ω ω ω ω A = Be [ + e +cos(ω) + sin(ω)] ω ω A B [-1 e cos(ω) e sin(ω)] ω ω = + + A = Be [e - cos(ω) + sin(ω)] B = 8 EIω [ 1 e ω e ω + + sin( ω )] F d v ωx -ωx ω e [A cosωx A sinωx] e [A sinωx A cosωx] ( 1 ) M o = EI = EI dx d v ωx T = EI = EI ω (e [A (cosωx -sinωx) - A 1(cosωx + sinωx)] + dx + ω - ω - ω + ω -ωx e [A (cos x sin x) A (cos x sin x)]) Tab.1 Theoretical solution for beam on elastic foundation in fig.. x (0, )

6 From the given parameters using equations in Tab. 1 the maximum deflection of beam is m in x = 0, minimum deflection is m in x =, maximum bending moment is M o (x = 0) = 05.7 Nm and minimum bending moment is M o (x = = 1.8 m) = 0 Nm. Maximum transversal load is T (x = 0) = N. Minimum transversal load is T (x = = 1.8 m) = 0. Numerical solution using BEAM5 (1. method of solution) For numerical solution we use the program environment ANSYS, which includes a special D element. This element is BEAM5. Properties and the characteristics of crosssectional area are entered in the real constants. In Fig. the FEM model shows the numbering of nodes and elements. The theoretical solution of this model using the principle of FEM can be entered using the eqn. (6) and boundary conditions. Boundary conditions are as follows: the displacement of all nodes in the direction of x is equal to zero. At point number two the force F in the y direction is applied. Theoretical solution written in matrix form is as follows: Fig. FE model using BEAM5. K11, K1, K1, K1, 0 0 v1 0 K1 K K1 K 0 0,,,, θ 1 0 K1, K1, K11, 0 K1, K1, v F =, (1) K1, K, 0 K, K1, K, θ K1, K1, K11, K 1, v K1, K, K1, K, θ 0 1EI 1bK 6EI 11b K 1EI 9bK where K11, = +, K 1, = +, K 1, = +, EI 1b K EI b K EI b K k K1, =, K, = +, K, =, K = b In fig. 5 the deflection of beam is shown. The bending moment diagram is shown in Fig. 6 and transversal load diagram is shown in Fig. 7.

7 Fig. 5 Diagram of the deflection of beam in meters. Fig. 6 Diagram of the bending moment in Nm.

8 Fig. 7 Diagram of the transversal load in N. Numerical solution using BEAM5 (k=0) and CONTAC5 (. method of solution) The second method of solution is using the beam and the contact element. The model is shown in Fig. 8 with numbering of nodes and elements. Matrix form of the theoretical solutions using FEM approach, where the stiffness matrix for one element is given by eqn. (10) is as follows Fig. 8 FE model using BEAM and CONTACT element. EI 1 + CM v θ CM v F =, (1) θ CM 6 v θ 0

9 K where CM = C. Stiffness of spring in contact element is followingc =, where n contact EI n contact is number of contact element. Result of the solution for deflection is shown in Fig. 9. The bending moment diagram is shown in Fig. 10 and transverse load diagram is shown in Fig. 11. Fig. 9 Diagram of the deflection of beam in meters.

10 Fig. 10 Diagram of the bending moment in Nm. Fig. 11 Diagram of the transversal load in N.

11 Conclusions In this paper, the solution of plane beams and frames using the Finite Element Method is shown. Two approaches are shown for the numerical solution of beams. The first approach is using the BEAM5 element, where the element stiffness matrix is according to equation (8). This approach can be used when the foundation is without compression resistance. If we consider compression resistance, we have to applied the contact element approach (for example. CONTACT5), where compression resistance is input by a gap. In our example, the gap is equal to zero. Because the ANSYS software contains beam elements with shear deformation, only BEAM5 is without shear deformation. For verification an example of contact was considering using the element BEAM5, when the stiffness of elastic foundation is equal to zero. Of course the accuracy of the result for the numerical methods is influenced by the number of elements over the length of beam. The verification examples used only one element over the length of beam. Influence of the number of divisions in both the approaches is illustrated in the Fig. 1, where deflection of the beam is shown. In Fig. 1 a diagram of bending moment is shown and in Fig. 1 the transverse load is shown. 0,15 x 10-0,1 0,1 Deflection (m) 0,1 0,11 0,10 Max. deflection using Beam and Contact (. method) Min. deflection using Beam and Contact (. method) Max. deflection (theoretical solution) Min. deflection (theoretical solution) Max. deflection using Beam5 (1. method) Min. deflection using Beam5 (1. method) 0,09 0, Number of division (-) Fig. 1 The influence of number of element divisions along the length on the minimum and maximum deflection.

12 56000 Bending moment (Nm) Max. bending moment using Beam and Contact (. method) Max. bending moment (theoretical solution) Max. bending moment using BEAM5 (1. method) Number of division (-) Fig. 1 The influence of the number of elements division along the length on the bending moment Transverse force (N) Max. Transverse force using Beam and Contact (. method) Max. Transverse force (theoretical solution) Max. Transverse force using Beam5 (1. method) Number of division (-) Fig. 1 The influence of number of elements division along the length on maximum transverse force It is shown in Fig. 1 to 1, the difference between solutions using the element BEAM5 with given foundation stiffness and using the BEAM5, when the stiffness of foundation is zero

13 in the real constants, and elastic foundation is modelled by contact elements CONTAC5. Both solutions are compared with theoretical solution. Reasons of difference are shown in Fig. 1, where the solution without contact is shown in Fig. 1a and with the contact in Fig.1b. Based on the above information, it is possible to create the program code for the solution of plane beams on elastic foundations, for example in the program environment MATAB. Practical solutions of problem related to gas pipelines using the contact elements is explained in Chapter 16 of iterature []. Acknowledgements The paper was supported by grant from Grant Agency of VEGA no. 1/0100/08. References [1] Cook, R.D., Malkus D.S., Plesha, M.E., Witt, R.J.: Concepts and Applications of Finite Element Analysis, Fourth Edition, WIEY, ISBN , 00, pp.719 [] Frydrýšek, K.: Nosníky a rámy na pružném podkladu 1, Faculty of Mechanical Engineering, VŠB - Technical University of Ostrava, ISBN , Ostrava, Czech Republic, 006, pp.6. [] Frydrýšek, K. Jančo, R: Nosníky a rámy na pružném podkladu, Faculty of Mechanical Engineering, VŠB - Technical University of Ostrava, ISBN , Ostrava, Czech Republic, 008, pp.516. [] ANSYS Theoretical manual Reviewer: doc. Ing. Karel FRYDRÝŠEK, Ph.D., ING-PAED IGIP, VŠB - Technical University of Ostrava, Czech Republic