- Email: [email protected]
Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams
ARTICLE IN PRESS
Thin-Walled Structures 45 (2007) 845–848 www.elsevier.com/locate/tws
Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams F. Guarracino Dipartimento di Ingegneria Strutturale, Universita` degli Studi di Napoli ‘‘Federico II’’, Via Claudio 21, 80125 Napoli, Italy Available online 27 September 2007
Abstract In this work, an exceedingly simple examination of the von Ka´rma´n theory for the buckling analysis of thin plates is carried out. It is shown that the von Ka´rma´n assumption, based on the interaction between axial and flexural deformation, is intrinsically less ‘‘robust’’ in capturing the target phenomenology than the usual treatment of Euler inextensible strut. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thin plates buckling; von Ka´rma´n theory; Axial and flexural deformation
1. Introduction It is common experience that when a simply compressed thin plate of ductile material, supported to be in-plane along its edges, buckles, it develops out-of-plane ripples along its length. The plate as a whole sustains increases in load after buckling, but the axial stiffness reduces and the maximum stress grows at an increased rate. This behaviour is similar to the standard Euler strut problem. However, while the bearing of beams can be analysed by means of a highly idealised model of an inextensible, shear undeformable rod, the analysis of buckling and initial post-buckling behaviour of compressed thin plates usually takes into account the membrane strains and deformations. This is generally done by means of the von Ka´rma´n theory [1,2], essentially based on an a priori assumption on the non-linear strain–displacement relations that neglect the quadratic terms in the derivatives of the in-plane displacements. von Ka´rma´n model provides a relatively easy and effective way of analysing the buckling of thin plates; but, to what extent is it accurate? This model has been systematically employed both by means of a system of two simultaneous non-linear differential equations and Rayleigh–Ritz energy procedures (the field is so wide Tel.: +39 081 768 3733; fax: +39 081 768 3332.
E-mail address: [email protected] 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.08.004
ranging that it is worth making reference only to survey works as, for example, Refs. [3,4]). On the other hand, it is well known that from a mathematical standpoint the derivation of von Ka´rma´n equations seems to present a decided lack of rigour, therefore several works have been dedicated to provide rigorous justifications of the von Ka´rma´n theory, showing, for example, that these equations can be seen arising either as the leading terms in an asymptotic expansion with respect to the thickness parameter in the framework of three-dimensional elasticity [5] or constituting the reduction, through an explicit approximation procedure, of a two-dimensional, quasi-linear system of equations [6]. It is unquestionable from an engineering standpoint that the von Ka´rma´n theory is very effective in capturing the phenomenology of the plate post-buckling problem and that for over a century it has constituted a valuable tool in this extremely important topic in structural mechanics since plates are possibly unique in their extensive use as load-carrying structural components up to and into the post-buckling range. This theory has been also largely employed to explain the ability of thin plates loaded in edge compression to sustain loads well above the classical critical load in presence of abrupt changes in wave-form after initial buckling on account of non-linear coupling of buckling modes at simultaneous or near-simultaneous critical loads and these findings have been confirmed by extensive experiments.
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
846
Since the non-linear coupling does not affect the postbuckling strength of the plate, plates under in-plane loading are considered to be well-behaved structural components such as the classic Euler strut, while it is not possible to say automatically the same is true for structures composed of assemblages of plates. In fact, such structures may behave as thin shells and display unstable and potentially dangerous post-buckling behaviour. In the last decades of the past century, finite strip methods have been quite popular among non-linear analysts for studying the buckling of assemblage of plates [7], until the computing expense of finite element programs has become quite negligible and therefore ubiquitously used. Nevertheless, all these tools rely, more or less, on the original von Ka´rma´n theory with some sort of modification, aimed to the effective treatment of the overall systems of equations. Yet, when dealing with approximate methods in nonlinear mechanics there is always the possibility of a misleading result, therefore a clear understanding of the physics underlying the mathematical formulations employed is necessary. In his 2002 Timoshenko Medal Lecture, J.W. Hutchinson said: ‘‘Buckling problems of all kinds arise continually in many areas of technology. Sometimes I wonder where the expertise on buckling will reside when all of us aging bucklers cross the bar. ABAQUSs can solve buckling problems, but it can not pose or understand them.’’ For this reason, in the present work, an exceedingly simple examination of the implications of the intuition at the basis of the von Ka´rma´n theory is carried out. The limit case of a strip of such a limited depth so that its stress state can be approximated to that of an axially loaded strut is taken into consideration. It is shown that the von Ka´rma´n assumption, based on the interaction between axial and flexural deformation, requires that all the terms of fourth order in the derivatives of the out-of-plane displacement are taken into consideration in order to obtain the correct initial postbuckling behaviour of the problem, differently from the usual treatment of the Euler strut, which, in a certain sense, results as intrinsically more ‘‘robust’’ in capturing the target phenomenology. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind. 2. Buckling and initial post-buckling analysis of a slender strip Let us take into consideration a simply supported rectangular thin plate uniformly compressed in one direction, as the one shown in Fig. 1. The von Ka´rma´n non-linear strain–displacement relations are x ¼ u;x þ 12ðw;x Þ2 ; y ¼ v;y þ 12ðw;y Þ2 ; gxy ¼ u;y þ v;x þ w;x w;y ;
(1)
b q
x y
a
w Fig. 1. Simply supported plate under uniform compression in one direction.
where the comma denotes partial differentiation, i.e. (),x ¼ q()/qx, etc.; u, v and w are the components of displacement in the x, y and z direction, respectively, and the angles are positive if measured in a clockwise direction. It is evident that Eq. (1) is derived from the classical Green–Lagrange strain tensor by neglecting the quadratic terms in the derivatives of the in-plane displacements u and v. By considering bba, we can assume that the resulting slender strip behaves approximately like a Bernouilli beam, so that its deformation is independent of y and, further, the displacement v is equal to zero [2]. Therefore, for a linearly elastic isotropic material and in the framework of the Kirchhoff–Love kinematics, it is u ¼ uðx; zÞ
u0 ðxÞ ¼ uðx; 0Þ,
w ¼ wðxÞ, u ¼ u0 zw;x , x ¼ u0;x zw;xx þ 12ðw;x Þ2 , Z Z b 1 F¼U L¼ sx x dV qa 0x dx 2 V 0 Z 1 E2 dV qa½u0 ð0Þ u0 ðbÞ, ¼ 2 V x
ð2Þ
ð3Þ
where F is the total potential energy of the structure, E is Young’s modulus and q is the uniform compressive load per unit length at the edges. u0 and e0x are referred to the middle plane of the plate. After simple calculations, we have ! Z b w4;x 1 2 2 F ¼ EA u0;x þ u0;x w;x þ dx 2 4 0 Z b 1 þ EI w2;xx dx qa½u0 ð0Þ u0 ðbÞ, ð4Þ 2 0 where A ¼ at and I ¼ at3/12, t being the plate thickness.
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
According to a straightforward Ritz procedure, we take as trial functions [8] npx C2x ; uo ¼ (5) w ¼ C 1 sin b b and, imposing F to be stationary with respect to the parameters C1 and C2 [9], we obtain the following relationship between the load qa and the displacement parameter C1: qa ¼
n2 p2 EI n2 p2 EAC 21 þ . b2 8b2
(6)
From Eq. (6) it is clear that the value of the critical loads is yielded by the bending terms, while the post-buckling behaviour is entirely governed by the membrane terms. In fact, for C1 ¼ 0, we are simply left with the critical loads of the Euler strut. The initial post-buckling behaviour is correctly depicted as stable symmetric. Now, if we arbitrarily neglect the fourth power of w,x in the Eq. (4) and repeat the Ritz procedure, we get qa ¼
n2 p2 EI n2 p2 EAC 21 . b2 4b2
(7)
It is worth pointing out that this truncation leaves us with a load–deflection relationship which still furnishes the correct values of the critical loads, but results as totally unreliable in predicting the initial post-buckling behaviour, which now mistakenly has the unstable-symmetric type of behaviour. Of course, the omission of what might seem a ‘‘small’’ term cannot be justified with respect to the original von Ka´rma´n model, yet a great number of technical theories and procedures can be found in literature for the analysis of assemblage of plates which are based upon the neglect of certain kinematical terms, vaguely regarded as ‘‘small’’. In order to check the robustness of the von Ka´rma´n theory against the usual treatment of the highly idealised model of the inextensible Euler strut, let us now write the total potential energy associated with the latter. In terms of bending moment and curvature in the framework of the Bernouilli kinematics we have, for the slender strip under analysis, M y ¼ EIfy;x , w;xx wy ¼ fy;x ¼ ½arcsinðw;x Þ;x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 w2;x F¼U L¼
1 2
Z
Z
b
b
M y fy;x dx 0
1 ¼ EI 2
Z 0
qað1 cos fy Þ dx 0
b
ð8Þ
w2;xx dx 1 w2;x
Z 0
b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qa 1 1 w2;x dx,
ð9Þ where My, wy and fy are the bending moment, the curvature and the angle of rotation about the y axis, respectively. It is worth stressing that under the above-
847
mentioned assumptions the expression of the total potential energy Eq. (9) is exact. By expanding the following terms in power series about the point 0 1 ¼ 1 þ w2;x þ w4;x þ , 1 w2;x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ð10Þ 1 w2;x ¼ 1 w2;x w4;x þ 2 8 and taking the terms up to the second power, Eq. (9) can be written as Z b Z 1 1 b F ¼ EI ðw2;xx þ w2;xx w2;x Þ dx qaw2;x dx. (11) 2 2 0 0 Repeating once again the Ritz procedure with w as in Eq. (5), it is n2 p2 EI C 21 n4 p4 EI þ , (12) b2 2b4 which yields the correct values of the critical loads and predicts the correct type of the initial post-buckling behaviour in terms of the flexural stiffness only. If we want to take into consideration all the possible fourth-order powers in the displacement gradients when expanding Eq. (9), we have Z b Z 1 1 b 1 F ¼ EI ðw2;xx þ w2;xx w2;x Þ dx qa w2;x þ w4;x dx 2 2 0 4 0 (13) qa ¼
and the Ritz approach gives n2 p2 EI 8b2 þ 4C 21 n2 p2 qa ¼ b2 8b2 þ 3C 21 n2 p2
! (14)
that again furnishes the correct values of the critical loads and the correct type of the initial post-buckling behaviour. Without discussing the precise mathematical meaning of these approximations (Eq. (13) is simply obtained from the arbitrary assumption that the terms w2;xx w2;x and w4;x are of the same order of magnitude), it is evident that the von Ka´rma´n theory results as less ‘‘robust’’ against omission of what might seem ‘‘small’’ terms than the usual approach for Euler strut. More importantly, as a result of the truncation the von Ka´rma´n model does not lose the capacity to determine the initial post-buckling behaviour of the structure, but gives a misleading value. In order to enlighten further such a circumstance, in the next section, the case of a hinged cantilever will be taken into account. 3. Buckling and initial post-buckling analysis of a hinged cantilever Let us consider a hinged cantilever comprising a linear extensional link of stiffness kv and length L, pinned to a rigid foundation and supported by a linear rotational spring of stiffness kf, as shown in Fig. 2(a). Apart from the axial deformability of the link this is unquestionably the
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
848
(a)
(b) P P L(1-cos ) + Lcos P
L
kv
exact
approximate
k
Fig. 2. (a) Simply supported plate under uniform compression in one direction. (b) Exact vs. approximate load–deflection plots.
simplest example to discuss the fundamental and deflected equilibrium paths and therefore constitutes an introductory example in the vast majority of stability textbooks. The exact expression of the total potential energy of the system is F ¼ 12kf f2 þ 12kv DL2 P½Lð1 cos fÞ þ DL cos f
(15)
and the equilibrium conditions F,f ¼ F,DL ¼ 0 give the expression P¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kv L sin f kv sin fðkv L2 sin f 4kf f cos fÞ sin 2f
(16) that represents the two post-critical equilibrium paths, which can be easily verified to be of the stable symmetrical type. However, if we expand cos f in power series about the point 0 and take the terms up to the second power, Eq. (15) becomes F ¼ 12kf f2 þ 12kv DL2 P DL 12Pf2 ðL DLÞ
Euler inextensible strut. Since load-shortening relations arise in the vast majority of plate buckling modelling and in the scientific literature have been treated in rather general terms [10], it was felt that exceedingly simple cases, as the ones stated hitherto, could be useful in stressing once more that in the non-linear analysis of structures any approximation, even in the rare case of closed-form solution, has to be taken with a clear understanding of the physics underlying the problem at hand. Two of the simple examples cited here have the great advantage in that an exact solution is available and can be used to ‘‘calibrate’’ the use of formulations in which apparently negligibly small terms have been eliminated. The results highlight the great dangers of using numerical modelling without either having a calibration model or physical test available and without a sufficiently deep understanding of the effects of eliminating small terms in the numerical treatment. References
(17)
and the equilibrium conditions F,f ¼ F,DL ¼ 0 yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kv L k2v L2 þ 2kv kf ðf2 2Þ P¼ . (18) 2 f2 This result shows that on account of the interaction between the extensional and the rotational stiffnesses as it happens in the von Ka´rma´n model, what seems a simple approximation may produce an unexpected outcome: that is for the lowest critical load, a post-buckling equilibrium path of the unstable-symmetric type, as shown for example in Fig. 2(b). 4. Conclusions Starting from the von Ka´rma´n theory for the buckling analysis of thin plates and by means of very simple examples, it has been shown that the interaction between axial and flexural deformation is intrinsically less ‘‘robust’’ in capturing the required phenomenology than the usual treatment of the
[1] von Ka´rma´n T. Festigkeitsprobleme in Maschinenbau. In: Klein F, Mu¨ller C, editors. Encyclopa¨die der Matematisches Wissenschaften. Stuttgart: Teubner; 1910. [2] Novozhilov V. Foundations of the nonlinear theory of elasticity. New York: Graylock Press; 1953. [3] Walker AC. A brief review of plate buckling research. In: Rhodes J, Spence J, editors. Behaviour of thin-walled structures. London: Elsevier; 1984. [4] Rhodes J. Buckling of thin plates and members—and early work on rectangular tubes. Thin-Walled Struct 2002;40:87–108. [5] Ciarlet PG. A justification of the von Ka´rma´n equations. Arch Rational Mech Anal 1980;73:349–89. [6] Podio-Guidugli P. A new quasilinear model for plate buckling. J Elasticity 2003;71:157–82. [7] Hancock GJ. Finite strip buckling and nonlinear analyses and distorsional buckling analysis of thin-walled structural members. In: Coupled instabilities in metal strucutres: theoretical and design aspects. Udine: CISM; 1996. [8] Timoshenko SP, Gere JM. Theory of elastic stability. New York: McGraw-Hill; 1961. [9] Guarracino F, Walker AC. Energy methods in structural mechanics. London: Telford; 1999. [10] Budiansky B. Theory of buckling and post-buclkling behaviour of elastic structures. Adv Appl Mech 1974;14:2–65.
Thin-Walled Structures 45 (2007) 845–848 www.elsevier.com/locate/tws
Considerations on the numerical analysis of initial post-buckling behaviour in plates and beams F. Guarracino Dipartimento di Ingegneria Strutturale, Universita` degli Studi di Napoli ‘‘Federico II’’, Via Claudio 21, 80125 Napoli, Italy Available online 27 September 2007
Abstract In this work, an exceedingly simple examination of the von Ka´rma´n theory for the buckling analysis of thin plates is carried out. It is shown that the von Ka´rma´n assumption, based on the interaction between axial and flexural deformation, is intrinsically less ‘‘robust’’ in capturing the target phenomenology than the usual treatment of Euler inextensible strut. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thin plates buckling; von Ka´rma´n theory; Axial and flexural deformation
1. Introduction It is common experience that when a simply compressed thin plate of ductile material, supported to be in-plane along its edges, buckles, it develops out-of-plane ripples along its length. The plate as a whole sustains increases in load after buckling, but the axial stiffness reduces and the maximum stress grows at an increased rate. This behaviour is similar to the standard Euler strut problem. However, while the bearing of beams can be analysed by means of a highly idealised model of an inextensible, shear undeformable rod, the analysis of buckling and initial post-buckling behaviour of compressed thin plates usually takes into account the membrane strains and deformations. This is generally done by means of the von Ka´rma´n theory [1,2], essentially based on an a priori assumption on the non-linear strain–displacement relations that neglect the quadratic terms in the derivatives of the in-plane displacements. von Ka´rma´n model provides a relatively easy and effective way of analysing the buckling of thin plates; but, to what extent is it accurate? This model has been systematically employed both by means of a system of two simultaneous non-linear differential equations and Rayleigh–Ritz energy procedures (the field is so wide Tel.: +39 081 768 3733; fax: +39 081 768 3332.
E-mail address: [email protected] 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.08.004
ranging that it is worth making reference only to survey works as, for example, Refs. [3,4]). On the other hand, it is well known that from a mathematical standpoint the derivation of von Ka´rma´n equations seems to present a decided lack of rigour, therefore several works have been dedicated to provide rigorous justifications of the von Ka´rma´n theory, showing, for example, that these equations can be seen arising either as the leading terms in an asymptotic expansion with respect to the thickness parameter in the framework of three-dimensional elasticity [5] or constituting the reduction, through an explicit approximation procedure, of a two-dimensional, quasi-linear system of equations [6]. It is unquestionable from an engineering standpoint that the von Ka´rma´n theory is very effective in capturing the phenomenology of the plate post-buckling problem and that for over a century it has constituted a valuable tool in this extremely important topic in structural mechanics since plates are possibly unique in their extensive use as load-carrying structural components up to and into the post-buckling range. This theory has been also largely employed to explain the ability of thin plates loaded in edge compression to sustain loads well above the classical critical load in presence of abrupt changes in wave-form after initial buckling on account of non-linear coupling of buckling modes at simultaneous or near-simultaneous critical loads and these findings have been confirmed by extensive experiments.
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
846
Since the non-linear coupling does not affect the postbuckling strength of the plate, plates under in-plane loading are considered to be well-behaved structural components such as the classic Euler strut, while it is not possible to say automatically the same is true for structures composed of assemblages of plates. In fact, such structures may behave as thin shells and display unstable and potentially dangerous post-buckling behaviour. In the last decades of the past century, finite strip methods have been quite popular among non-linear analysts for studying the buckling of assemblage of plates [7], until the computing expense of finite element programs has become quite negligible and therefore ubiquitously used. Nevertheless, all these tools rely, more or less, on the original von Ka´rma´n theory with some sort of modification, aimed to the effective treatment of the overall systems of equations. Yet, when dealing with approximate methods in nonlinear mechanics there is always the possibility of a misleading result, therefore a clear understanding of the physics underlying the mathematical formulations employed is necessary. In his 2002 Timoshenko Medal Lecture, J.W. Hutchinson said: ‘‘Buckling problems of all kinds arise continually in many areas of technology. Sometimes I wonder where the expertise on buckling will reside when all of us aging bucklers cross the bar. ABAQUSs can solve buckling problems, but it can not pose or understand them.’’ For this reason, in the present work, an exceedingly simple examination of the implications of the intuition at the basis of the von Ka´rma´n theory is carried out. The limit case of a strip of such a limited depth so that its stress state can be approximated to that of an axially loaded strut is taken into consideration. It is shown that the von Ka´rma´n assumption, based on the interaction between axial and flexural deformation, requires that all the terms of fourth order in the derivatives of the out-of-plane displacement are taken into consideration in order to obtain the correct initial postbuckling behaviour of the problem, differently from the usual treatment of the Euler strut, which, in a certain sense, results as intrinsically more ‘‘robust’’ in capturing the target phenomenology. Finally, the same observation is carried out in the case of a hinged cantilever, which constitutes the simplest structural example of a similar kind. 2. Buckling and initial post-buckling analysis of a slender strip Let us take into consideration a simply supported rectangular thin plate uniformly compressed in one direction, as the one shown in Fig. 1. The von Ka´rma´n non-linear strain–displacement relations are x ¼ u;x þ 12ðw;x Þ2 ; y ¼ v;y þ 12ðw;y Þ2 ; gxy ¼ u;y þ v;x þ w;x w;y ;
(1)
b q
x y
a
w Fig. 1. Simply supported plate under uniform compression in one direction.
where the comma denotes partial differentiation, i.e. (),x ¼ q()/qx, etc.; u, v and w are the components of displacement in the x, y and z direction, respectively, and the angles are positive if measured in a clockwise direction. It is evident that Eq. (1) is derived from the classical Green–Lagrange strain tensor by neglecting the quadratic terms in the derivatives of the in-plane displacements u and v. By considering bba, we can assume that the resulting slender strip behaves approximately like a Bernouilli beam, so that its deformation is independent of y and, further, the displacement v is equal to zero [2]. Therefore, for a linearly elastic isotropic material and in the framework of the Kirchhoff–Love kinematics, it is u ¼ uðx; zÞ
u0 ðxÞ ¼ uðx; 0Þ,
w ¼ wðxÞ, u ¼ u0 zw;x , x ¼ u0;x zw;xx þ 12ðw;x Þ2 , Z Z b 1 F¼U L¼ sx x dV qa 0x dx 2 V 0 Z 1 E2 dV qa½u0 ð0Þ u0 ðbÞ, ¼ 2 V x
ð2Þ
ð3Þ
where F is the total potential energy of the structure, E is Young’s modulus and q is the uniform compressive load per unit length at the edges. u0 and e0x are referred to the middle plane of the plate. After simple calculations, we have ! Z b w4;x 1 2 2 F ¼ EA u0;x þ u0;x w;x þ dx 2 4 0 Z b 1 þ EI w2;xx dx qa½u0 ð0Þ u0 ðbÞ, ð4Þ 2 0 where A ¼ at and I ¼ at3/12, t being the plate thickness.
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
According to a straightforward Ritz procedure, we take as trial functions [8] npx C2x ; uo ¼ (5) w ¼ C 1 sin b b and, imposing F to be stationary with respect to the parameters C1 and C2 [9], we obtain the following relationship between the load qa and the displacement parameter C1: qa ¼
n2 p2 EI n2 p2 EAC 21 þ . b2 8b2
(6)
From Eq. (6) it is clear that the value of the critical loads is yielded by the bending terms, while the post-buckling behaviour is entirely governed by the membrane terms. In fact, for C1 ¼ 0, we are simply left with the critical loads of the Euler strut. The initial post-buckling behaviour is correctly depicted as stable symmetric. Now, if we arbitrarily neglect the fourth power of w,x in the Eq. (4) and repeat the Ritz procedure, we get qa ¼
n2 p2 EI n2 p2 EAC 21 . b2 4b2
(7)
It is worth pointing out that this truncation leaves us with a load–deflection relationship which still furnishes the correct values of the critical loads, but results as totally unreliable in predicting the initial post-buckling behaviour, which now mistakenly has the unstable-symmetric type of behaviour. Of course, the omission of what might seem a ‘‘small’’ term cannot be justified with respect to the original von Ka´rma´n model, yet a great number of technical theories and procedures can be found in literature for the analysis of assemblage of plates which are based upon the neglect of certain kinematical terms, vaguely regarded as ‘‘small’’. In order to check the robustness of the von Ka´rma´n theory against the usual treatment of the highly idealised model of the inextensible Euler strut, let us now write the total potential energy associated with the latter. In terms of bending moment and curvature in the framework of the Bernouilli kinematics we have, for the slender strip under analysis, M y ¼ EIfy;x , w;xx wy ¼ fy;x ¼ ½arcsinðw;x Þ;x ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , 1 w2;x F¼U L¼
1 2
Z
Z
b
b
M y fy;x dx 0
1 ¼ EI 2
Z 0
qað1 cos fy Þ dx 0
b
ð8Þ
w2;xx dx 1 w2;x
Z 0
b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qa 1 1 w2;x dx,
ð9Þ where My, wy and fy are the bending moment, the curvature and the angle of rotation about the y axis, respectively. It is worth stressing that under the above-
847
mentioned assumptions the expression of the total potential energy Eq. (9) is exact. By expanding the following terms in power series about the point 0 1 ¼ 1 þ w2;x þ w4;x þ , 1 w2;x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ð10Þ 1 w2;x ¼ 1 w2;x w4;x þ 2 8 and taking the terms up to the second power, Eq. (9) can be written as Z b Z 1 1 b F ¼ EI ðw2;xx þ w2;xx w2;x Þ dx qaw2;x dx. (11) 2 2 0 0 Repeating once again the Ritz procedure with w as in Eq. (5), it is n2 p2 EI C 21 n4 p4 EI þ , (12) b2 2b4 which yields the correct values of the critical loads and predicts the correct type of the initial post-buckling behaviour in terms of the flexural stiffness only. If we want to take into consideration all the possible fourth-order powers in the displacement gradients when expanding Eq. (9), we have Z b Z 1 1 b 1 F ¼ EI ðw2;xx þ w2;xx w2;x Þ dx qa w2;x þ w4;x dx 2 2 0 4 0 (13) qa ¼
and the Ritz approach gives n2 p2 EI 8b2 þ 4C 21 n2 p2 qa ¼ b2 8b2 þ 3C 21 n2 p2
! (14)
that again furnishes the correct values of the critical loads and the correct type of the initial post-buckling behaviour. Without discussing the precise mathematical meaning of these approximations (Eq. (13) is simply obtained from the arbitrary assumption that the terms w2;xx w2;x and w4;x are of the same order of magnitude), it is evident that the von Ka´rma´n theory results as less ‘‘robust’’ against omission of what might seem ‘‘small’’ terms than the usual approach for Euler strut. More importantly, as a result of the truncation the von Ka´rma´n model does not lose the capacity to determine the initial post-buckling behaviour of the structure, but gives a misleading value. In order to enlighten further such a circumstance, in the next section, the case of a hinged cantilever will be taken into account. 3. Buckling and initial post-buckling analysis of a hinged cantilever Let us consider a hinged cantilever comprising a linear extensional link of stiffness kv and length L, pinned to a rigid foundation and supported by a linear rotational spring of stiffness kf, as shown in Fig. 2(a). Apart from the axial deformability of the link this is unquestionably the
ARTICLE IN PRESS F. Guarracino / Thin-Walled Structures 45 (2007) 845–848
848
(a)
(b) P P L(1-cos ) + Lcos P
L
kv
exact
approximate
k
Fig. 2. (a) Simply supported plate under uniform compression in one direction. (b) Exact vs. approximate load–deflection plots.
simplest example to discuss the fundamental and deflected equilibrium paths and therefore constitutes an introductory example in the vast majority of stability textbooks. The exact expression of the total potential energy of the system is F ¼ 12kf f2 þ 12kv DL2 P½Lð1 cos fÞ þ DL cos f
(15)
and the equilibrium conditions F,f ¼ F,DL ¼ 0 give the expression P¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 kv L sin f kv sin fðkv L2 sin f 4kf f cos fÞ sin 2f
(16) that represents the two post-critical equilibrium paths, which can be easily verified to be of the stable symmetrical type. However, if we expand cos f in power series about the point 0 and take the terms up to the second power, Eq. (15) becomes F ¼ 12kf f2 þ 12kv DL2 P DL 12Pf2 ðL DLÞ
Euler inextensible strut. Since load-shortening relations arise in the vast majority of plate buckling modelling and in the scientific literature have been treated in rather general terms [10], it was felt that exceedingly simple cases, as the ones stated hitherto, could be useful in stressing once more that in the non-linear analysis of structures any approximation, even in the rare case of closed-form solution, has to be taken with a clear understanding of the physics underlying the problem at hand. Two of the simple examples cited here have the great advantage in that an exact solution is available and can be used to ‘‘calibrate’’ the use of formulations in which apparently negligibly small terms have been eliminated. The results highlight the great dangers of using numerical modelling without either having a calibration model or physical test available and without a sufficiently deep understanding of the effects of eliminating small terms in the numerical treatment. References
(17)
and the equilibrium conditions F,f ¼ F,DL ¼ 0 yield qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kv L k2v L2 þ 2kv kf ðf2 2Þ P¼ . (18) 2 f2 This result shows that on account of the interaction between the extensional and the rotational stiffnesses as it happens in the von Ka´rma´n model, what seems a simple approximation may produce an unexpected outcome: that is for the lowest critical load, a post-buckling equilibrium path of the unstable-symmetric type, as shown for example in Fig. 2(b). 4. Conclusions Starting from the von Ka´rma´n theory for the buckling analysis of thin plates and by means of very simple examples, it has been shown that the interaction between axial and flexural deformation is intrinsically less ‘‘robust’’ in capturing the required phenomenology than the usual treatment of the
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