A NEW APPROACH TO DYNAMIC BUCKLING LOAD ESTIMATION FOR PLATE STRUCTURES

Transcription

1 Stability o Structures XIII-th Syposiu Zakopane 202 A NW APPROACH TO DYNAMIC BUCKLIN LOAD STIMATION FOR PLAT STRUCTURS T. KUBIAK, K. KOWAL-MICHALSKA Departent o Strength o Materials, Lodz University o Technology Steanowskiego /5, Łódź, Poland In the paper the authors propose to estiate the dynaic buckling load relatively to static buckling load or iperect plates, thereore the dynaic load actor DLF * is introduced as a ratio o pulse load aplitude and the static buckling load or iperect plates. The calculations have been ade to check how the way o dynaic load actor deterination inluences the critical aplitude o pulse load leading to dynaic buckling. The results or coposite plate loaded by copressive pulse load o dierent durations or dierent iperection aplitude are presented in igures and discussed.. INTRODUCTION The proble o dynaic buckling o thin walled structures such as shells and plates subjected to in-plane pulse loading has been widely investigated starting ro sixties o previous century see e.g. works [2], [], [3]). These pulse loads ay be o various durations and shapes rectangular, sinusoidal, triangular, trapezoidal, etc.) being approxiations o real load courses. Depending on the so-called pulse intensity dierent phenoena ay occur ipact or pulses o high aplitudes and durations in range o icroseconds or quasi-static behaviour i aplitude is low and duration is twice o period o undaental natural vibrations. For pulses o interediate intensity aplitudes in range o static buckling load and durations close to one hal or one o period o undaental natural vibrations) the phenoenon o dynaic buckling occurs. It is known that at pulse loads o short duration in range o illiseconds) the dynaic structure carrying capacity is larger than static one. However it should be reebered that or plate structure, in contrary to the static behaviour, the biurcation dynaic load does not exist. The phenoenon o dynaic buckling takes place only or initially iperect structures. Initial iperections agnitude in connection with pulse shape and its duration are crucial paraeters in dynaic buckling load estiation [6], [7]. Dynaic buckling load is usually deterined on the basis o dynaic buckling that onesel sees to be probleatic. Coonly used Budiansky-Roth- Hutchinson [] was orulated or structures having liit point or unstable postbiurcation path. Its application to plate structure behaviour, with stable postbuckling path, is based rather on accepted practice. Thereore in subject literature one can ind a lot o stability criteria - ost o the are based on state o displaceents or state o stresses. The ost popular are Volir [3] and Budiansky-Hutchinson [3]. Soe degree o uncertainty o all entioned criteria brought the researches o new dynaic stability criteria basing on Jacobian atrix eigenvalues analysis see Kubiak [8]) or applying phase portraits see Teter [2]). 397

2 Stability o Structures XIII-th Syposiu Zakopane 202 In ost papers the results o calculations are presented as the relations between axial diensionless dynaic deection o a structure and ratio o pulse aplitude versus static buckling load deterined or perect structure i.e. biurcation load). In literature, ollowing Budiansky and Hutchinson, the quotient o pulse aplitude and static bucking load is tered as Dynaic Load Factor DLF). The critical value o DLF is deterined on the assued dynaic buckling. It can be easily seen that or very sall iperections the dynaic buckling load deterined on the basis o Budiansky- Hutchinson is greater than static one and in soe range o loads the dynaic delections are saller than static ones. For larger values o iperections the character o curves changes it becoes siilar to static course) and dynaic buckling load is less than static one or a perect plate. Thereore the question arises - should the buckling load o iperect structure be deterined relatively to the static critical load o a perect structure as it is coonly assued? Perhaps the inoration o dynaic load carrying capacity would be ore evident with regard to static buckling load deterined or iperect structure. In this paper the authors propose to estiate the dynaic buckling load relatively to static buckling load or iperect plates, thereore the dynaic load actor DLF * is introduced as a ratio o pulse load aplitude and the static buckling load or iperect plates. The calculations have been ade to check how the way o DLF deterination inluences the critical aplitude o pulse load leading to dynaic buckling. 2. MTHOD OF SOLUTION A non-linear stability proble has been solved by eans o the Koiter s asyptotic theory [5]. The displaceent ield U, and sectional orce ield N have been expanded into the power series with respect to the paraeter, - the buckling linear eigenvector aplitude noralised with the equality condition between the axiu delection and the thickness o the irst plate). U U N N 0 ) 0 ) U N i i i ) i ) U N i i 398 j j ij ) ij ) It was assued that the diensionless aplitude o the initial delections iperections) corresponding to the considered buckling ode or s-th buckling ode) is: * i ) U U. 2) s By substituting expansion into equations o equilibriu with neglected inertia ters static buckling proble) and boundary conditions, the boundary proble o the zero superscript 0) in quation and urther), irst superscript i) ) and second superscript ij) ) order has been obtained [8], [9]. The zero approxiation describes the prebuckling state, whereas the irst order approxiation allows or deterination o critical loads and the buckling odes corresponding to the, taking into account iniisation with respect to the nuber o hal-waves in the lengthwise direction. The second order approxiation is reduced to a linear syste o dierential heterogeneous )

3 Stability o Structures XIII-th Syposiu Zakopane 202 equations, which right-hand sides depend on the orce ield and the irst order displaceents only. Having ound the solutions to the irst and second order o the boundary proble, the coeicients a ijs, b ijks have been deterined [8]: a b ijs ijks σ i ) 2σ * L U i ) j ) * L U, U jk ), U ) 0. 5σ 0 ) sσ 0 ) sσ * L U 2 ) σ * L U ij ) * L U ) * L U where: s - is the critical load corresponding to the s-th ode, L is the bilinear operator, L 2 is the quadratic operator and σ i), σ ij) are the stress ield tensors in the irst and second order. The postbuckling static equilibriu paths or coupled buckling can be described by the equation: * s aijsi j bijksi jk s ; s =,, N), 4) s s which or the uncoupled proble have the or: 2 3 a b * 5) cr cr where cr is the critical load value. When the equilibriu path is syetrical a = 0) the eq. 5) have the or: 3 b * 6) cr cr In a special case, i.e. or the so-called ideal structure without initial iperections * =0) the postbuckling equilibriu path is deined by the equation: 2 b 7) cr In the dynaic analysis while inding the requency o natural vibrations [9]), the independent non-diensional displaceent and the load actor becoe a unction dependent on tie, and dynaic ters were added to equations describing postbuckling equilibriu path. Neglecting the orces associated with the inertia ters o prebuckling state and the second-order approxiations, and taking into account the orthogonality i ) conditions or the displaceent ield in the irst U and second-order approxiationu ij ) ) i ) k ), the Lagrange equations can be written as:.. s s aijsi j b 2 s s ijks s, U, U j ) ), ), 3) i jk s ; s=,2,, N) 8)

4 Stability o Structures XIII-th Syposiu Zakopane 202 where s is a natural requency with ode corresponding to buckling ode; a ijs and b ijks are the coeicients q. 4) describing the postbuckling behaviour o the structure independent o tie); however the paraeters o load and the displaceent are the unctions o tie t. For the uncoupled buckling, i.e. the single-ode buckling where index s = N = ), the equations o otion ay be written in the or: * a b ; 2 9) It is assued that in the initial oent o tie t = 0 the non-diensional displaceent, as well as the velocity o displaceent are equal to zero, i.e.: 3. DYNAMIC BUCKLIN CRITRIA. ξt=0) = 0 and t 0 ) 0. 0) The results o calculations presented in the paper allow discussing the eect o application o the ollowing criteria: the siplest, proposed by Volir [3] - the dynaic critical load corresponds to the aplitude o pulse orce o constant duration) at which the axial plate delection is equal to soe constant value k k=one hal or one plate thickness). Budiansky&Hutchinson [3] stability that states: dynaic stability loss occurs when the axial plate delection grows rapidly with the sall variation o the load aplitude. 4. SUBJCT OF CONSIDRATION A square siply supported coposite plate with length to thickness ratio equals 00 is considered. The plate subjects to unidirectional copression in or o a rectangular pulse load P o inite duration. The unloaded edges o plate reain straight and parallel during loading. The analysed plate is ade o epoxy-glass coposite with ibre volue raction equals 0.5. The aterial properties or epoxy resin and glass ibre are given in Table. Table. Assued aterial properties aterial type: [Pa] [kg/ 3 ] epoxy resin glass ibre For nuerical calculation the orthotropic odel was assued. The necessary values o Young s odules and Poison s ratios were calculated using equations based on ixture theory [4], which are as ollows: 400

5 Stability o Structures XIII-th Syposiu Zakopane ,,, yx y x ) where: and are the Young s odules o elasticity or atrix and ibre, respectively, and are the shear odules or atrix subscript ) and ibre subscript ), ν and ν are the Poisson s ratios or atrix and ibre and = V /V + V ) is the ibre volue raction. 5. RSULTS OF CALCULATIONS 5.. STATIC BUCKLIN LOAD AND POSTBUCKLIN BHAVIOUR a) b) Fig.. Diensionless load P/P cr vs. diensionless delection a) or square o diensionless delection b)

6 Stability o Structures XIII-th Syposiu Zakopane 202 For ideal plate structure the critical load can be deterined ro eigenvalue analysis but or structures with iperection the buckling load ay be deterined basing on preand post-buckling behaviour. The authors decided to eploy two well-known ethods or identiication o the critical load that are usually applied to the results o experiental investigations. The inlection point ethod P-w) which is very siilar to top o the knee ethod and alternative P-w 2 ) ethod was used [9]. The postbuckling equilibriu paths or plates with initial iperections corresponding to the buckling ode and dierent aplitudes w * =w 0 /h where: w 0 - aplitude o iperection, h - thickness o the plate) are presented in Fig.. Using entioned above ethods the buckling load P cr */P cr or a plate with geoetrical iperections has been ound and presented in Table 2 where P cr * - buckling copressive orce or iperect plate and P cr - biurcational load). Table 2. P cr * /P cr or dierent aplitude o initial iperection deterination ethod: P-w P-w 2 initial iperection aplitude w * P cr */P cr P cr */P cr It ollows ro the Table 2 that lower values o P cr * were obtained using P-w 2 ethod and the dierences between the results o both the ethods are growing with the increase o iperection aplitude value. In urther investigations the values o P cr * ound on the basis o the inlection point ethod P-w ethod) were taken into account ANALYSIS OF DYNAMIC RSPONS OF A PLAT In ost publications dealing with dynaic buckling proble the aplitude o initial iperection has been assued as w * = 0.0 or which the buckling load decrease is very sall ca %. The dynaic load actor DLF is deined as the ratio o an aplitude o pulse load to the critical static buckling load or ideal structures. Presented below calculations were ade in ai to check how the way o DLF estiation inluences the critical aplitude o pulse load leading to dynaic buckling. The authors propose to introduce a dynaic load actor DLF * =P/P * cr - a pulse load aplitude divided by the static buckling load or iperect structures. The calculations were perored or two values o pulse duration T 0 =T and T 0 =0.5T where T - period o natural undaent lexural vibration o a plate, or assued aterial properties and geoetry T=0.59 s). 402

7 Stability o Structures XIII-th Syposiu Zakopane 202 In Tables 3 and 4 the critical dynaic load actors DLF cr deterined in the conventional way) and DLF * cr deterined ro DLF * w/h) relations, where the aplitude o pulse loading is divided by buckling load or iperect structures) are presented. The critical value o dynaic load actor DLF * cr was calculated as the aplitude o pulse load divided by static buckling load or structure with initial iperection using inlection point ethod see Table 2, P-w colun). Table 3. DLF cr and DLF cr * or dierent aplitude o initial iperection, T 0 =T Assued : initial iperection aplitude w * Table 4. Assued : initial iperection aplitude w * Volir w/t) cr = DLF cr Budiansky- Hutchinson DLF cr Volir w/t) cr = DLF * cr Budiansky- Hutchinson DLF * cr DLF cr and DLF cr * or dierent aplitude o initial iperection, T 0 =0.5T Volir w/t) cr = DLF cr Budiansky- Hutchinson DLF cr Volir w/t) cr = DLF * cr Budiansky- Hutchinson DLF * cr The courses o DLFw/h) and DLF*w/h) or three values o iperection aplitudes and two pulse durations T 0 =T and T 0 =0.5T) are presented in Figs 2-4. In these igures the static postbuckling curves P/P cr or lat plate) and P/P cr * or iperect plate) are also drawn. It can be noticed that or relatively sall iperection aplitude w*=0.0 Fig. 2) the curves DLFw/h) and DLF*w/h) cover up or given pulse duration T 0. Also the static postbuckling curves overlap excluding the initial range o delections). In this case the character o dynaic responses strongly depends on the assued pulse duration - or shorter pulse the delections are sall and the dynaic buckling load is at least three ties greater see Table 3 and 4). When the aplitude o iperections grows up to the value w*=0. Fig. 3) the dierences between DLF and DLF* curves are clearly visible or both pulse durations and static postbuckling curves P/P cr or lat plate) and P/P cr * dier as well. The character

8 Stability o Structures XIII-th Syposiu Zakopane 202 o dynaic responses or pulse durations T 0 =T and T 0 =0.5T is siilar but the dynaic buckling load or shorter pulse is twice the one or T 0 =T. Fig. 2. Static and dynaic diensionless responses versus diensionless delection or w*=0.0. Fig. 3. Static and dynaic diensionless responses versus diensionless delection or w*=0.. For relatively large value o w* the iperection aplitude equals one hal o plate thickness) - Fig. 4, the results show that the dynaic responses o a plate do not depend on pulse load duration the relations DLFw/h) or T 0 =T and or T 0 =0.5T cover up and also DLF*w/h) or both assued pulses overlap) and oreover the courses o DLF*w/h) are alost identical as the static postbuckling curve P/P * cr. It should be underlined that the dierences between the courses o DLF estiated as the ratio o 404

9 Stability o Structures XIII-th Syposiu Zakopane 202 pulse load aplitude to static biurcational load) and DLF* estiated as the ratio o pulse load aplitude to static buckling load o iperect plate) are clearly visible. Fig. 4. Static and dynaic diensionless responses versus diensionless delection or w*= CONCLUSIONS On the basis o the results analysis o the conducted calculations ollowing conclusions can be drawn: For sall values o iperection aplitude in range o hundredth parts o plate thickness) the dierences between the dynaic response known as DLF w/h) ratio o pulse load aplitude to static biurcational load versus diensionless delection) and proposed relation DLF* w/h) ratio o pulse load aplitude to static buckling load or iperect plate versus diensionless delection) are less than % and the curves are practically identical. In this case the pulse load duration tie strongly aects the dynaic buckling load value and the character o dynaic response o considered plate. For rather large values o iperection aplitude in range o one hal and greater) the inluence o pulse duration on courses o DLFw/h) and DLF*w/h) has shown to be negligible but the dynaic responses DLFw/h) and DLF*w/h) dier signiicantly. It should be strongly underlined that the proposed relations DLF*w/h) calculated or two values o pulse load are alost identical as static postbuckling curve P/P * cr what allows us to conclude that or large iperection aplitude values the static and dynaic behaviour o a plate is practically the sae or considered pulse durations). This act can be only observed i proposed relation DLF*w/h) is applied in calculations. 405

10 Stability o Structures XIII-th Syposiu Zakopane 202 The authors are aware that the analysis conducted in this work is restricted to a single plate and or pulses o rectangular shape; thereore urther investigations should be perored to conir the above conclusions. ACKNOWLDMNTS This publication is a result o the research work carried out within the project subsidized over the years ro the state unds designated or scientiic research MNiSW - N N ). RFRNCS [] Budiansky B., Dynaic buckling o elastic structures: Criteria and stiates, Report SM- 7, NASA CR-66072, 965. [2] ryboś R., Stability o structures under ipact load, /in Polish/, PWN, Warsaw-Poznan, 980. [3] Hutchinson J.W., Budiansky B., Dynaic buckling estiates, AIAA Journal, 43), 966, pp [4] Kelly A. ed.), Concise ncyclopedia o Coposite Materials, Pergaon Press, 989. [5] Koiter W.T., eneral theory o ode interaction in stiened plate and shell structures, WTHD Report 590, Delt, 976. [6] Kowal-Michalska, K. ed.), Dynaic stability o coposite plated structures, in Polish), Warszawa Łódź, WNT, [7] Kowal-Michalska, K., About soe iportant paraeters in dynaic buckling analysis o plated structures subjected to pulse loading, Mechanics and Mechanical ngineering, Vol.4, 200, pp [8] Kubiak T., Interactive dynaic buckling o thin-walled coluns, /in Polish/, Scientiic Bulletin o Łódź Technical University, Łódź, [9] Kubiak T., stiation o dynaic buckling or coposite coluns with open crosssection, Coputers and Structures, 89, 20, pp [0] Rhodes J., Zaraś J., Deterination o critical loads by experiental ethods, in Statics, Dynaics and Stability o Structures, Vol.2., Statics, Dynaics and Stability o Structural leents and Systes, edited by Z. Kołakowski, K. Kowal-Michalska, Publishing House o Łódź University o Technology, 202. [] Siitses.J., Dynaic stability o suddenly loaded structures, Springer Verlag, New York,990. [2] Teter A., Multi-odal buckling o thin-walled stiened coluns loaded by copressive pulse, /in Polish/, Scientiic Bulletin o Łódź Technical University, Łódź, 200. [3] Volir S.A., Nonlinear dynaics o plates and shells, /in Russian/ Science, Moscow,