Limit point buckling of a finite beam on a nonlinear foundation
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n Limit point buckling of a finite beam on a nonlinear foundation
R. Lagrange Massachusetts Institute of Technology,Department of Mathematics, Cambridge, MA 02139-4307, USA (Dated: January 29, 2021)
Abstract
In this paper, we consider an imperfect finite beam lying on a nonlinear foundation,whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases.Periodic equilibrium solutions are found analytically and are in good agreement witha numerical resolution, suggesting that localized buckling does not appear for a finitebeam. The equilibrium paths may exhibit a limit point whose existence is related tothe imperfection size and the stiffness parameter k through an explicit condition. Thelimit point decreases with the imperfection size while it increases with the stiffnessparameter. We show that the decay/growth rate is sensitive to the restoring forcemodel. The analytical results on the limit load may be of particular interest forengineers in structural mechanics. Keywords: Buckling, Imperfection, Finite beam, Nonlinear foundation, Limit point
I. INTRODUCTION
An elastic beam on a foundation is a model that can be found in a broad range ofapplications: railway tracks, buried pipelines, sandwich panels, coated solids in material,network beams, floating structures... The usual way to model the interaction between thebeam and the foundation is to replace the latter with a set of independent springs whoserestoring force is a linear [see e.g. 1–8] or a nonlinear [see e.g. 9–19] function of the localdeflection of the beam. In both cases, the nonlinear effects, from the beam’s deformationand/or from the restoring force, play a crucial role in the buckling and the post-bucklingbehaviors. In particular, for a softening nonlinear foundation, the equilibrium curves ofthe beam may exhibit a maximum load (i.e. limit point) at which the structure loses itsstability. Small imperfections, arising from various sources, usually have an appreciable effect W P − P W Fig. 1: Sketch of a beam on a nonlinear foundation. The beam has an imperfect shape W and itslateral displacement is W . The compressive force is P and the restoring force per unit length is − P . on this maximum load. The papers on deterministic imperfection sensitivity include thoseof [4, 5, 11, 15, 20–24] and extensive references for the stochastic imperfection sensitivityare compiled in [25]. As a general rule, the maximum load at which the beam becomesunstable diminishes with increasing imperfection size. Considering a finite beam on a bi-linear/exponential foundation, [24] has shown the existence of a critical imperfection size A c such that: if A < A c , then the maximum load diminishes with the imperfection size,from the critical buckling load predicted by the classical linear analysis [see 10], for A = 0,to the Euler load for A = A c . In this case, the decay rate is sensitive to the restoring forcemodel. For A > A c , the maximum load is the Euler load (i.e. buckling load of a beamwith no foundation).In the present paper we aim to extend these results to two restoring force models withmore general softening behaviors. We derive an analytical expression for A c and study theevolution of the maximum load with the imperfection size and the stiffness reduction. II. FORMULATION OF THE PROBLEM
We consider the effects of a compressive load P on a beam of length L , with bendingstiffness EI , lying on a foundation that provides a restoring force per unit length P (seeFig. 1). The beam and the foundation are assumed to be well bonded at their interface andremain bonded during deformation. Thus, interfacial slip or debonding is not considered.The mobilization of the foundation (also named the yield point) is noted ∆, its linear stiffness K and its nonlinear stiffness K .In its initial configuration, the beam has an imperfect shape W = A sin ( πX/L ), where w k p − k Fig. 2: Dimensionless restoring force p k . Red line: bi-linear model. Blue line: hyperbolic model.The stiffness ratio is k ≤ A is the imperfection size and X is the longitudinal coordinate.We introduce the characteristic length L c = ( EI/K ) / and the non-dimensional quan-tities l = LL c , x = XL c , w = W ∆ , w = W ∆ , a = A ∆ , λ = P L c EI , k = KK , p k = PK ∆ , (1)as the dimensionless beam length, longitudinal coordinate, lateral deflection (measured fromthe initial configuration), imperfection shape, imperfection size, compressive load, stiffnessratio and restoring force respectively.Two models for the restoring force p k are considered in this article (see Fig. 2). The firstone is p k ( w ) = − w − (1 − k ) (sgn ( w ) − w ) H ( | w | − , (2)where sgn denotes the sign function and H is the Heaviside function, defined as H ( | w | −
1) =0 if | w | < | w | >
1. This bi-linear restoring force refers to a foundation whosestiffness is instantaneously reduced from 1 to k ≤ w >
1. The particular case k = 1corresponds to a linear foundation. Reference [24] considered the particular case of k = 0.Here, we extend the study to k ≤
1, which leads to more general results.To reflect the experimental tests on railway tracks performed by [26], also reported in[27, 28], who showed that the lateral friction force acting on a track is a smooth function ofthe lateral displacement, we introduce a hyperbolic profile defined as p k ( w ) = − kw − (1 − k ) tanh( w ) . (3)This restoring force is a regularization of the bi-linear model as they share the same asymp-totic behaviors.We assume that λ and p k are conservative forces, that strains are small compared to unityand that the kinematics of the beam is given by the classical Euler-Bernoulli assumption.The imperfection is also assumed to be small so that terms with higher powers of w or itsderivatives are neglected in the expression of the potential energy. Under these assumptions,the potential energy V with low-order geometrically nonlinear terms is [see 10] V = l Z w ′′ − λ (cid:18) w ′ + w ′ w ′ (cid:19) − w Z p k ( t ) dt d x, (4)where a prime denotes differentiation with respect to x . The first term in the integral isthe elastic bending energy, the second is the work done by the load λ , the last term is theenergy stored in the elastic foundation.The equilibrium states are given by the critical values of V . Assuming a simply supportedbeam, the boundary conditions are w (0) = w ( l ) = 0. Variations of (4) for an arbitrary kine-matically admissible virtual displacement δw yields the weak formulation of the equilibriumproblem l Z h w ′′′′ + λ (cid:16) w ′′ + w ′′ (cid:17) − p k ( w ) i δw d x = 0 , (5)which is equivalent to the stationary Swift-Hohenberg equation w ′′′′ + λ (cid:16) w ′′ + w ′′ (cid:17) − p k ( w ) = 0 , (6)along with static boundary conditions w ′′ (0) = w ′′ ( l ) = 0.This boundary value problem is nonlinear because of the restoring force and its solutionsare highly sensitive to the length l , as shown in [4]. Therefore, it is unrealistic to describethe behavior of the system over a large range of variation for l . As done in [24], this studyis restricted to a finite length beam where l < √ π . For such values of l a classical linearanalysis [see 10] shows that the first buckling mode is the most unstable one and appearsfor λ c = λ e + λ − e , where λ e = ( π/l ) is the Euler load. III. SOLVING METHODS
To solve (5) we apply a Galerkin method with a trigonometric test function w of am-plitude y : w = y sin ( π x/l ), assuming that the deflection has the same shape as the firstbuckling mode and the initial imperfection. For more details about the principle of themethod, the reader is referred to [24], where the procedure has already been used. In thatpaper, this method has been shown to be reliable in the prediction of the equilibrium pathsof the system, for k = 0. We shall see in the present paper that it is actually reliable forany k ≤
1, thereby extending the results of [24].The insertion of δw = δy sin ( π x/l ) in (5) yields l Z sin (cid:16) πl x (cid:17) h w ′′′′ + λ (cid:16) w ′′ + w ′′ (cid:17) − p k ( w ) i d x = 0 . (7)Splitting the restoring force in a linear and a nonlinear term N ( w ) leads to p k ( w ) = − w − (1 − k ) N ( w ). With this decomposition and w = y sin ( π x/l ), (7) can be rewritten as λ k = 1 a + y (cid:20) λ c y + (1 − k ) Q ( y ) λ e (cid:21) , (8)where the subscript k denotes the dependance of λ on the parameter k . The function Q takes into account the nonlinear behavior of the restoring force and is given by Q ( y ) = 2 l Z l sin (cid:16) πl x (cid:17) N (cid:16) y sin (cid:16) πl x (cid:17)(cid:17) d x. (9)For the restoring force models (2) and (3), Q is negative and decreases monotonically tothe asymptote − y + 4 /π as y → + ∞ . Thus λ k is maximum for k = 1 (linear foundation)and has an horizontal asymptote λ ∞ k = λ e + kλ − e as y → + ∞ .Equilibrium paths predicted by (8) are traced out in the plane ( y = max ( w ) , λ ) bygradually incrementing y and evaluating λ k , k and a being fixed. Predictions are comparedwith a numerical solution of (6), using MATLAB’s boundary value solver bvp4c [this is afinite difference code that implements a collocation formula, details of which can be foundin 29]. IV. RESULTS
The equilibrium paths predicted by the Galerkin method and the numerical solution areshown in Fig. 3. A perfect agreement in the predictions is found for both restoring forcemodels (the relative error between the two methods being less than 0 . l < √ π , no localized buckling is observed for a beam on a bi-linear or hyperbolicfoundation. This behavior has also been reported by [4] for a linear foundation, showing atendency toward localization when increasing the beam length.As expected, the equilibrium paths traced out for the hyperbolic restoring force are belowthose traced out for the bi-linear force, the hyperbolic profile modeling a softer foundationthan the bi-linear one. However, the choice of the restoring force has little influence on theshape of the equilibrium paths.For small a , the equilibrium paths first increase to a maximum λ m that is smaller (orequals to in the case of a = 0) than the buckling load λ c . Then, the paths asymptoticallydecrease to λ ∞ k . In the case of a = 0, k = 1, they remain equal to λ c . For high a , theequilibrium paths increase monotonically to the asymptote λ ∞ k ≤ λ c . The asymptotic value λ ∞ k = λ c is reached for a = 0 and k = 1.Note that for k < − λ e , λ ∞ k is negative, so that equilibrium states with λ < k < − λ e the restoring force p k may becomenegative so that springs are compressed, pushing up the beam. In this situation, the restoringforce has a destabilizing effect on the beam. To counteract this effect, a tensile force λ < ( ) max y w = ( ) max y w = ( ) max y w = ( ) max y w =λ λλ λ ( ) a ( ) b ( ) c ( ) d λ ∞− c λ c λ c λ λ ∞ λ ∞ . a = . a = . a = a = Fig. 3: Equilibrium paths of a finite length beam on a nonlinear foundation. Circles: numericalpredictions. Lines: Galerkin solution. In red: bi-linear restoring force model. In blue: hyperbolicmodel. (a) k = −
2, (b) k = 0, (c) k = 0 .
75, (d) k = 1. On each subfigure, the equilibrium pathsare plotted (from top to bottom) for a = 0, a = 0 . a = 0 .
595 and a = 1 .
19, as shown in(d). The length of the beam is l = 3. a m λ k = − k = k = k = → → → → → → → λ ∞ c λ λ ∞ Fig. 4: Maximum load λ m that the beam may support versus the imperfection size a and thefoundation stiffness ratio k . In red: bi-linear restoring force model. In blue: hyperbolic model.Vertical dotted lines correspond to the critical imperfection sizes a c for k = 0 .
75 and k = 0. Thelength of the beam is l = 3. Limit point No limit point
Limit point whatever a k a ( ) max. load = k λ ∞ e k a k π λ −= + ( ) max. load k λ ∞ < ≤ c λ e λ− Fig. 5: Diagram of existence of a limit point for an imperfect finite beam on a bi-linear/hyperbolicfoundation. k is the stiffness ratio of the foundation and a the imperfection size. λ e = ( π/l ) isthe Euler load, λ c = λ e + λ − e and λ ∞ k = λ e + kλ − e . has to be applied.The evolution of λ m versus a is shown in Fig. 4. A gradual drop in the maximum loadadmissible by the structure from λ c to λ ∞ k is observed when increasing a (resp. decreasing k ). This gradual drop is highly sensitive to the restoring force model. A log scale appliedon Fig. 4 shows that, for small imperfection sizes, the decay rate does not depend on k : λ m − λ c scales as − a , for the bi-linear model and as − a / for the hyperbolic model.For a larger than a critical value a c , the equilibrium paths do not have a limit pointanymore. Actually, a path with no limit point may be seen as a path with a limit point at( ∞ , λ ∞ k ). Thereby, a c may be obtained from (8) by enforcing y → ∞ in dλ k /dy = 0. Bothrestoring force models leads to a c = 4 π − kλ e + k , (10)whose dimensional equivalent form is A c = 4 π ( K − K ) ∆ π EI/L + K . (11)The critical imperfection size predicted by [24] is therefore recovered in the particularcase K = 0, showing that A c only depends on the limiting plateau K ∆ of the restoringforce [as stated in 30].Finally, since a >
0, equation (10) shows that if k < − λ e then the equilibrium pathsalways have a limit point λ ∞ k < λ m < λ c . V. CONCLUSION
In this paper, we considered the buckling of an imperfect finite beam on a bi-linear/hyperbolic foundation. The imperfection has been introduced as an initial curvatureof size a and the foundation stiffness ratio as a parameter k ≤
1, extending the result of[24] derived for k = 0.Equilibrium paths of the beam have been predicted using a Galerkin method initiatedwith a single trigonometric function which has the same shape as the imperfection. Predic-tions compare well with a numerical solution and lead to the conclusion that only periodic0buckling can arise for a finite beam on a bi-linear/hyperbolic foundation, as also observedby [4] for an linear foundation.We have shown the existence of a critical imperfection size a c = 4 (1 − k ) [ π ( λ e + k )] − ,independent of the restoring force model, such that: • if a < a c , then the maximum load diminishes with increasing imperfection size, from λ c = λ e + λ − e , for a = 0, to λ e + kλ − e for a = a c , λ e = ( π/l ) being the Euler load.The decay rate has been shown to be sensitive to the restoring force model. In thelimit of small a , λ m − λ c ∼ − a for the bi-linear model and λ m − λ c ∼ − a / for thehyperbolic model. • If a > a c , then the maximum load simply corresponds to λ e + kλ − e .Finally, we have shown that if k < − λ e then an imperfect finite beam on a bi-linear/hyperbolic foundation can support a compressive load larger than λ ∞ k , and smallerthan λ c , whatever the imperfection size is. This feature is highly interesting for an engineersince a is usually hard to evaluate. The main results from this study are summarized inFig. 5.In the present paper, a bi-linear restoring force model for the foundation has been usedbut plasticity effects that would emerge from loading/unloading cycles have not been consid-ered. Future works will have to highlight the way those effects could modify the maximumload that the beam can support. A basic model would consist of considering a permanentdeflection as an imperfection whose size would grow up after each cycle. In that case, fromthe present study, it is expected a decrease of the maximum load after each cycle, at leastas long as the accumulated deflection remains smaller than a threshold equivalent to a c .The author acknowledges Dr. M. Brojan for introducing to him the hyperbolic restoringforce model and Dr. Alban Sauret and Dr. Jay Miller for their insightful comments on thispaper. [1] E. Winkler, H. Dominicus, Prague, Czechoslovakia. , 182 (1867).[2] J. G. Lekkerkerker, Proc. Royal Netherlands Acad. Sci. B65 , 190 (1962).[3] M. S. El Naschie, ZAMM. , 677 (1974). [4] S. H. Lee and A. M. Waas, International Journal of Non-linear Mechanics. , 313 (1996).[5] A. N. Kounadis, J. Mallis, and A. Sbarounis, Archive of Applied Mechanics. , 395 (2006).[6] W. T. Koiter, Koiter’s Elastic stability of solids and structures (Cambridge University Press.Herausgeber Arnold M. A. van der Heijden, 2009).[7] N. Challamel, Comptes Rendus M´ecanique. , 396 (2011).[8] E. Suhir, Journal of Applied Mechanics. , 011009 (2012).[9] H. Cox, J. Roy. Aeronaut. Soc. , 231 (1940).[10] M. Potier-Ferry, in Lecture notes in Physics , Buckling and Post-buckling, Vol. 288 (1987) pp.1–82.[11] G. W. Hunt, M. K. Wadee, and N. Shiacolas, Journal of Applied Mechanics. , 1033(1993).[12] G. W. Hunt and A. Blackmore, Journal of Applied Mechanics. , 234 (1996).[13] M. K. Wadee, G. W. Hunt, and A. I. M. Whiting, in
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