A Numerical Method for Critical Buckling Load for a Beam Supported on Elastic Foundation

Transcription

1 A Numerical Method for Critical Buckling Load for a Beam Suorted on Elastic Foundation Guo-ing Xia Institute of Bridge Engineering, Dalian University of Technology, Dalian, Liaoning Province, P. R. China guoguoia@hotmail.com Zhe Zhang Institute of Bridge Engineering, Dalian University of Technology, Dalian, Liaoning Province, P. R. China ABSTRACT The differential equation of the beam on elastic foundation subjected to the comressive aial force was derived based on the Winkler theory. Accordingly, we obtained the formulations for the critical buckling load of such structures. And the numerical eamles are resented using finite element rogram ANSYS, by which the elastic foundation model also adoted the Winkler model, which regards the foundation as a series of searated srings without couling effects between each other. The results obtained by ANSYS are quite close to the numerical solutions. And the study shows a suitable sacing of the srings must be met when using ANSYS to calculate the critical load of the beam on elastic foundation, while the numerical method roosed in this aer is free from this restriction. The study also shows that the finite element method would become invalid when the elastic stiffness eceeds an ultimate stiffness. So the numerical method introduced in this aer is a convenient method with highly accurate solution comared with the finite element method. KEYWORDS: Beam suorted on elastic foundation, Critical buckling load, Numerical method INTRODUCTION Beams suorted on elastic foundation have wide alications in modern engineering, such as stri foundation, mat foundation of the buildings as well as such structure of underass bridges. To describe the interactions of the beam and foundation as more aroriate as ossible, scientists have roosed various kinds of foundation models. The most common model for the elastic foundation is the Winkler model, which regards the foundation as a series of searated srings without couling effects between each other. Static and dynamical analysis as well as the foundation settlement of such structures based on the Winkler theory has been etensively

2 Vol. 14, Bund. E covered by many investigators. For the buckling analysis of the beam on elastic foundation, only a few studies are available in the literature. This aer aims to rovide a new formulation for critical buckling load of the beam suorted on elastic foundation according to the Winkler hyothesis. And the resented formulation is eected to serve as a tool for future analyses of the in-lane stability for main beams of the cable-stayed bridge and the susension bridge, which has been suggested to be regarded as the beams on elastic foundation. GOVERNING EQUATIONS The Winkler hyothesis According to Winkler theory, it is assumed that the foundation settlement that at an arbitrary oint of the surface of the beam on elastic foundations is roortional to the force er unit length along the beam suffered at the same oint holding the relation as follows. q = ky Where q is the force er unit length along the beam; y is the foundation settlement; k is a coefficient assumed to be a roerty of the elastic medium; hysically seaking, it reresents the reaction force er unit area due to a unit foundation settlement. Derivation of basic differential equations A beam suorted on the elastic foundation suffers a uniformly distributed load reresented as () as shown in Fig.1-a, where y() is the dislacement of the beam and the foundation and q() is the force er length between the beam and the foundation. The force of the beam and the force of the foundation are showed in Fig.1-b and Fig.1-c resectively. In Fig.(a), a beam, simly suorted at both ends, is suorted on elastic foundation, with elasticity modulus of E, sectional moment of inertia of I and the elastic medium coefficient of k. The deformed configuration of the beam are shown in Fig.(b). The equation can be obtained by the force balance of the beam as showed in Fig.(c). (a) () (a) (b) y () (b) (c) y y q() q() M M+dM/d Figure 1 Figure 1. The equilibrium equation (c) y Q d q() Q+dQ/d dy

3 Vol. 14, Bund. E 3 Equation due to the balance of vertical loads is eressed as: dq + + = d ( ) Q Q d q d dq q( ) ky d = = (1-1) Similarly, equation due to the balance of moment can be obtained as: dm 1 M ( )( ) + dy + Qd M + d + q d = d (1-) When the quadratic term is neglected, Eq.(1-) becomes: Q dm d = dy d () So dq d M d y = = ky, and since d d d d M d d y =, hence d 4 4 Ⅳ y + y '' + ky = (3) Eq.(3) is the differential equation of the beam on elastic foundation subjected to the comressive aial force.. Solve the equation If setting / ( ) k α =, β = 4, Eq.(3) can be simlified as: 4 4 y α y β y Ⅳ + '' + 4 = (4) Introduced a differential oerator denoted by a symbol D to indicate the differentiation with resect to, i.e. d n dy d y. So we can write as Dy, n d d d as n D y, and Eq.(4) becomes: 4 4 ( D + α D + 4 β ) y = (5) 4 4 Writing LD ( ) = D + α D + 4β to eress the n-order olynomial of the differential oerator D, Eq.(5) can be written as : LDy= ( ),

4 Vol. 14, Bund. E 4 y r Setting y = e, and De r = e into Eq.(5) results in r r = re, D e = r e, r r D e r r = r e, so LDe ( ) = Lre ( ). Substituting 4 r 4 r r Lre () =, Then a 4-order algebraic equation is obtained as follows. Lr () =, i.e. r + α r + 4β = 4 4 (6) r Where r is the characteristic root of the Eq.(6). So y = e is a secial solution of Eq.(5). We call Eq.(6) an eigen-function of Eq.(5). Solving Eq.(6), the result is: 4 4 α ± α 16β r =±, 4 4 α α 16β i.e. r1, =± i, r comle root of Eq.(6). Letting 3,4 4 4 α + α 16β =± i. They are two airs of single A = 4 4 α α 16β and B = 4 4 α + α 16β the general solution of Eq.(5) is obtained as follows: y = C1cos A + Csin A + C3cos B + C4sin B (7-1) 3. Solve the equation according to the boundary conditions of the beam suorted on elastic foundation The first order derivative of Eq.(7-1) can be obtained as follows: y ' = AC sina + AC cosa BC sinb + BC cosb And the second order derivative of Eq.(7-1) can be obtained as:

5 Vol. 14, Bund. E 5 y'' = A C cos A A C sin A B C cos B B C sin B (7-) For a beam simly suorted at both ends, the related boundary conditions for the above equations are obtained as: y() = y( l) = y''() = y''( l) = (8) Substituting Eq.(8) into Eq.(7) leads to the following equations, y() = C + C = 1 3 (9-1) y() l = C cosal+ C sinal+ C cosbl+ C sinbl = (9-) y''() = A C B C = (9-3) 1 3 y ''( l) = A C cos Al A C sin Al B C cos Bl B C sin Bl = (9-4) To solve the Equation grou (9), the relationshi must be met as follows: ( A B )sinal sinbl = Several ossibilities will be discussed as follows. 1. A B =, thenα 4 16β 4 so = k. =, i.e. ( ). sin Al =, then Al = nπ, n =,1,, Al ( ) k α = /, 4 4 β = n π kl = + l n π, it is obtained that: k / 16 =, α α 16β = nπ, so l 3. In the same way, when sin Bl =, we also can obtain that: n π kl = + l n π 4. The critical load of the beam suorted on elastic foundation = n π, since

6 Vol. 14, Bund. E 6 For the etreme case in which d dn is zero, we have l k n = 4 π, hence = k. But as the half-wave numbers, i.e. modal number, n must be integers. Two ossibilities of calculating critical load according to the different n can be summarized as follows: 1when n 1, n can only be equal to 1, hence cr π kl = +, moreover, modal number is 1. l π when n > 1, setting i as the integer art of n. There are two values can be obtained for n, n π kl ( n+ 1) π kl one is cr1 = +, and the other is cr = +. We take the l n π l ( n+ 1) π smaller one as the critical load when n > 1, i.e. cr = min( cr1, cr ). At the same time, the modal number has the alternative of n or n + 1, the one make be equal to the critical value. The critical buckling load and modal numbers on beam suorted on elastic foundation can be found by above formulations, and the solutions are called the numerical results which will be comared to that obtained by the finite element method in the following tet. It is found that when n > 1, the result of critical load obtained by above formulations is quite close to the value π of k, so that we can consider that, when k >, the critical load is only related to k as l well as the material roerties and is insensitive to the san of the beam. NUMERICAL EXAMPLES In order to validate the resent formulations, numerical results for the critical buckling load on beam suorted on elastic foundation are comared to that obtained by the numerical eamles using the commercial finite element rogram ANSYS which has been used for many analyses of structures in recent years. For uroses of comarison, beams with four different sans are considered, and for each san of the beams, different elastic stiffness values of the foundation are used to calculate the critical load. The sans adoted in the calculations are m, 6m, 1m, and 3m, and the material and section roerties in all the calculations are showed in Table 1. Table 1: Material roerties adoted in all the calculations San(m) E(Kn/ m 4 ) 3.5E+1 3.5E+1 3.5E+1 3.5E+1 I(m 4 ) The model for the elastic foundation adoted in ANSYS is the Winkler model, which regards the foundation as a series of searated srings without couling effects between each other. Moreover, elastic stiffness of the srings is equivalent to the Winkler elastic medium coefficient according to the following form: 4

7 Vol. 14, Bund. E 7 k = ak where k is the equivalent elastic stiffness of the sring; and a is the sacing between two adjacent srings. The model adoted to calculate is showed in Fig. 3. k k Figure 3 The accuracy of the numerical solution The critical loads calculated by ANSYS are called FEM solutions. There are four tyes of beam to be considered with the sans of m, 6m, 1m and 3m by ANSYS. The FEM solutions are comared with the numerical ones when a= 1m. In order to verify the accuracy of the numerical results, the relative error is also evaluated based on the FEM solutions and are shown in Table. Here, relative error is defined as (Numerical solution- FEM solution)/fem solution. From this figure ecellent agreement between both results can be observed at the same value of the elastic stiffness for any san of the beam. It is obvious that the numerical method is accurate enough to obtain the Table : Comarisons of critical loads between FEM solution and numerical solution when a=1m m 6m k(n/m) FEM solution (Pa) Numerical solution (Pa) Relative error (%) FEM solution (Pa) Numerical solution (Pa) Relative error (%) 1.E E E E E+8. 1.E E E E E E E E E E E E E E E E E E E E E E E E E k(n/m) k k k k a a a 1m 3m FEM solution Numerical solution Relative error FEM solution Numerical solution Relative error

8 Vol. 14, Bund. E 8 (Pa) (Pa) (%) (Pa) (Pa) (%) 1.E E E E E E E E E E E E E E E E E E E E+1. 1.E E E E E E E E E E critical load with error less than 1%. So we can say the numerical eamles resented demonstrate that the resent method is both rational and reliable. Moreover, a same conclusion as the result of the numerical formulation mentioned above can be obtained that the critical load is only related to k as well as the material roerties and is insensitive to the san of the beam. Advantages of the numerical method The finite element method has been successfully emloyed for many engineering fields, but it has many disadvantages and subject to various conditions. For eamle, the sacing between the equivalent srings is an imortant arameter when using ANSYS to analyze the beam suorted on elastic foundation. It determines the calculation accuracy. When the sacing is too large the accuracy will not be satisfactory. Studies above all adot the sacing of a= 1m, while such small sacing could not be adot when the san of the beam is too large. Moreover, the effect of the elastic stiffness of the foundation can not be ignored, too. The effect of these factors on the solution obtained by ANSYS is investigated. Calculations using ANSYS are erformed to obtain the critical load of the beams with different sans and different sacing of the srings, and the results are showed in Table 3. Table 3: Comarisons of critical loads between FEM solution and numerical solution when the sacing of the srings taken different values san m sacing k(n/m) Eact FEM solution Relative error Solutions(Pa) (Pa) (%) L/a 1.E E E E E E E E E a=1m 1.E E E E E E E E E E E E E E E a=4m 1.E E E E E E E E E

9 Vol. 14, Bund. E 9 6m 1m 3m 5.863E E E E E E E+9.873E E E E E E E E E E E a=1m 1.E E E E E E E E E E E E E E E E E E E E E a=5m 1.E E E E E+9 6.4E E E E E E E E E E E E E E E E a=1m 1.E E E E E E E E E E E E E E E E E E E E E a=5m 1.E E E E+8 6.3E E E E E E E E E E E E E E E E E a=1m 1.E E E E E E E E E E E E a=5m 1.E E E

10 Vol. 14, Bund. E 1 1.E E E E E E E E E E+9 6.3E E E E E E E E The last column of Table 3 shows the values of L/ a, where L = 4 β = k. L is a general arameter related to the roerties of the beam and foundation, which is imortant to the behavior of the beam suorted on elastic foundation, so it is called the characteristic length. The shaded area in Table 3 list the results obtained using ANSYS when L/ a= The effective stiffness of elastic foundation It can be found that, when L/ a 1, FEM solutions are quite close to the eact solutions, and the ercentage error of FEM solution is all less than 1%. But when L/ a< 1, the ercentage error increases with decreasing the value of L/ a. So the elastic medium coefficient when L/ a= 1can be called effective stiffness k ' which identify a range of the stiffness lead to the eact solution. When k k', the results can be considered accurate. When k > k', the ercentage error increases with increasing the value of k. The value of effective stiffness k ' can be obtained by L/ a= 1, and be written as: 4 k ' = 4 a From this eression one can see that the sacing of the srings is smaller the value of k ' is bigger. That is to say, when the sacing of the srings decreases, the range of the eact solution is eanded. The same conclusion can be obtained from the figures in Table 3. So we can define a suitable scoe of the sacing of the equivalent srings according to the actual elastic medium coefficient of the foundation when we calculate the critical load of the beam suorted on elastic foundation using ANSYS. If a is less than the characteristic length L, the results can be considered accurate. Above all, a suitable sacing of the srings must be met when using ANSYS to calculate the critical load of the beam on elastic foundation, while the numerical method roosed in this aer is free from the restriction of the sacing. This is an advantage of the numerical method comared with the finite element method.. The ultimate stiffness of elastic foundation The last row of each section of Table 3 indicates that if k is more than the current stiffness value listed in this row, the results of critical load calculated by the finite element method don t increasing according to k. At that time it reaches an etreme case and the stiffness value are called the ultimate stiffness. Actually, elastic suorts become to rigid suorts when the elastic stiffness reaches the ultimate stiffness, while the beam suorted on elastic foundation becomes to a continuous beam with a serial of rigid suorted. This is a limitation of the finite element method, too.

11 Vol. 14, Bund. E 11 CONCLUSIONS This aer analyzes the in-lane stability for beams suorted on elastic foundations on the base of the Winkler hyothesis, and rooses a numerical method for critical buckling load of such structures. Eamles have also been resented for buckling analysis using ANSYS finite element software. Comarisons are made with results from numerical method as well as ANSYS, it is found that: (1) It can be concluded that this methodology as a convenient numerical tool with highly accurate solution is both rational and reliable. () The effect of the sacing of the srings on the accuracy of the critical loads of the beam on elastic foundation obtained by the finite element method is investigated. It is found that, for a bigger value of a, the ercentage error of FEM solution increases with increasing the sacing of the srings. (3) The concets of the effective stiffness and the ultimate stiffness of elastic foundation are roosed, with the conclusions that, using the finite element method, eact results are ossible if k less than the effective stiffness, and that, the finite element method could become invalid because the elastic suorts will become rigid suorts when the elastic stiffness eceeds the ultimate stiffness. (4) The numerical method resented in the aer has many advantages comared with the finite element method. ACKNOWLEDGEMENTS Author generated the data for this aer while working on the Project of West Traffic Construction Technology of Ministry of Transortation of the Peole's Reublic of China (No: ). REFERENCES 1. Long, Y. Q. (1981). The calculation for beam suorted on elastic foundation. Chinese Peole's education ress, Beijing, China, Xiang, H. F. (1). Higher bridge structure theory. China communications ress, Beijing, Lu, D. (6). The ractical and simlified calculation method of eloration for the main beam's stability in lane of the self-anchored susension bridge. Master's thesis, Univ. of Harbin Institute of Technology, Harbin, China. 4. Sheng, H. F., Lu, D. (6). The ractical and simlified method for analyzing selfanchored susension bridge girder's stability in lane. Chinese Technology and Economy in Areas of Communications, (5): 1- (in Chinese). 9 ejge