Dynamic buckling of laminated composite stringer stiffened CFRP panels under axial compression

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1 MATEC Web o Conerences 149, (018) Dynaic buckling o lainated coposite stringer stiened CFRP panels under axial copression Mouhat Ouadia 1, Khalichi Abdellati, Hasnae Boubel 1, Ounia Elrabet 1, Mohaed Rougui 1, El Mehdi Echebba 1 and Ahed El Bouhidi 1 1 Mohaed V University, Structure LGCE l EST, Salé -CED de l EMI, Rabat, MOROCCO Abdelalek Essaadi University, Counications Systes and Detection Laboratory, Tetouan, MOROCCO Abstract. In this work, the dynaic buckling o stiened panels is evolved nuerically through a nonlinear increental expression through aking use o a speciic tie integration procedure via the inite eleent sotware progra. the buckling and post-buckling behaviours o hat-stringer-stiened coposite curved panel under axial copression load.dynaic buckling is extracted ro the curve abandoning the very last shortening as a characteristic o tie while the shape is subjected with the aid o a square copression pulse oveent carried out inside the axial direction. The duration o the heart beat and the aplitude o curvature o decreasing o the cloth inside the band torented by the warth, the dynaic buckling otion, are constant. The ethod approach was proposed to predict the dynaic buckling load o curved panel. Finite eleent analysis was used to investigate these tests and the FE odels were perored by ABAQUS.Approach to deterine the reliability o the stiened panel in dynaic buckling state. Keywords: Dynaic buckling, stiened panels, axial copression, inite eleent ethod, reliability. 1. Introduction Stiened coposite curved panel is a odel structure widely used in aeronautic industry and spacecrat in the last ew years, in the industry eiciency and perorance o the structure is the essential objective or the design which requires insistence on the need or strong and light aterials. So that the resistance ust be high or this in utilised the reinorced polyer ibres or this we ind the Stiened coposite curved panel is in several ields o application. it is necessary or the design to study the stability and buckling o the sti coposite panel o the any actors that can aect the buckling characteristics and the post-buckling, Many experiental tests and nuerical calculations have been realized to study the behaviour o static and dynaic buckling several researchers who are published in this axis on the static buckling but we are interested in this work by the dynaic buckling. During the last two decades any articles studied stiened curved panels under axial copression loading[1-3].recently, the project that has been the ocus o attention in previous years is the project POSICOES and COCO MAT [4],Both supported by the European Coission, to study the behavior o coposite panels stiened by all the buckling and post buckling ore and ore o attention is given to buckling and The aerospace industry has asked to reduce developent and operating costs by 0% and 50% in the short and long ter. There is a diiculty in preaching the critical load o buckling and the ultiate load by the classical ethods. Von Karan et al. [5] a proposed approach "eective width" or the optiization this has been odiied by several carters aong these loaders Wang Zhening [6]. There are several experiences o stiened curved panels under axial copression loading but ost are oring I-stiened panels [7]. And T-stiened panel [8]. Deterination o relative response or the boring load paraeters by equating the equations o the otion and trace the selected axial displaceent curve as a unction o tie. The speciic value o the charge ro this change The Authors, published by EDP Sciences. This is an open access article distributed under the ters o the Creative Coons Attribution License 4.0 (

2 MATEC Web o Conerences 149, (018) coincides with the critical value associated with dynaic buckling according to the Budiansky-Roth criterion [9] (Budiansky and Roth, 196). This criterion sees ore practical in the study o dynaic buckling o thin structures. In the aeronautical and aerospace industries the liespan o the aircrat is civil or ilitary, generally the aircrat that exceeds 0 years o service or not even, so the buckling perorance o stiened coposite panels ust also be a big concern or the airplane designer. Scientists now ocus on the dynaic buckling that was oten ignored coposite panels. This article discusses the nonlinear ethod and dynaic buckling o the hat-stringerstiened Coposite lat panel subjected to axial copression and ade a reliability analysis using the Matlab Monte Carlo ethod to calculate the probability o ailure. ollowing boundary conditions ux x z 0. The border z 0is supposed to be perectly wet uxuyuz x y z 0, as the uniorly distributed load P z is applied to the edge z awith rigid boundary conditions on the side ux x z 0. Fig. 1. The initial coniguration in o the stiener panel. Panel description The stiened coposite panel is coposed o three stieners. The noinal length and the noinal width o the panel are 95 and The gap between two stieners is 5.5. The two terinals o the panel were recessed ensuring a unior distributed load calculation. The echanical properties o the aterial are deined in Table 1. The layout o the lainate at each zone o the panel is reported in Table and speciied graphically in the stieners are L-shaped and have the constant height t h w thickness t w.95, 64, the thickness o the lange 4.3 and the height o the lange b 1. Speciied graphically in Fig. 1,Noinal ply thickness is In Fig.. Shows the boundary conditions the coniguration o the stiened panel. Fig.. Geoetry o panel and boundary condition Table 1. Mechanical properties Stiness strength E GPa X T 700MPa E 8.85 GPa X C 1600MPa G 1 =G GPa Y T 88MPa G 3 4. GPa Y C 71MPa 0.33 S 143MPa 1 By indicating u the displaceent coordinates and rotations, the boundary conditions exained in the sotware abaqus are as ollows. The lateral ends have the

3 MATEC Web o Conerences 149, (018) Table.Lay-up deinition Skin 45 / 45 / 90 / 90 / 0 / 90 / 0 t w /90 / 0 / 90 / 90 / 45 / 45 Flange t 45 / 45 / 0 / 90 / 90 / 0 / 45 / Buckling analysis o stiened panels 3.1 Coposite aterials Coposite eans in the ter coposite aterials either two or ore aterials are acerated on a acroscopic scale to or a third useul aterial. Dierent aterials can be ixing at the acroscopic scale as in the etal alloy, but the resulting aterials are, or all practical purposes, acroscopically hoogeneous, that is to say that the coposite cannot be distinguished with the naked eye and act essentially together. 3. Classiications o coposite aterials - A lainate consists o a stack o onolayers each a proper orientation with respect to a coon rae o reerence layers and designated as the lainate reerence rae. - Particulate Coposites consist o a atrix reinorced by a dispersed phase in or o particles. - Fibrous coposite aterials that consist o ibres in a atrix. - Cobination o soe or all o the irst three types. 3.3 Stiness and strength o a laina The orientation o the ibres can be randoly in the aterial; in reality the ibres in the preerred directions can be oriented in expectation o the highest stresses. Such a aterial is said to be anisotropic, which is dierent ro their physical properties (elasticity oduli, Poisson coeicients, theral conductivity, etc.) which have dierent values depending on the spatial orientation o the physical body. The icroscopic level, the properties o these coposites are ound by the orientation and distribution o the ibres, as well as by the properties o the ibre and atrix substances. Consider a region o unit diensions in a aterial type, which aterial contains a volue raction, V we chose all orientations in a single direction, with the volue raction o the atrix is written by this ollowing orula: V 1 V (1) This place can be ade in congloerate all the ibers together, disissal the atrix to occupy the reaining volue. I a stress l applied along the direction o the ibre, the phases o the ibre and the atrix proceed in siilar to support the load. In these parallel bonds the constraints in each period ust be the sae, so that the strain l in the direction o the ibre can be written in this way as: l () With: the indices L and denote the laina and the ibres and the atrix respectively). For the balance o the total charge in the coposite aterial are added the orces in each phase. Since the orces in each period are the period constraints ties the area (here theoretically equal to the volue raction), we have: V V E V E V (3) t l l The stiness in the direction o the ibres is ound by dividing the stress by the strain: (4) l E EV EV l With: E is the longitudinal Young's odulus. For the prediction o the ixtures o the global odule using this relation which was according to the ters o the odules o hashes which constitutes the volue ractions. 3

4 MATEC Web o Conerences 149, (018) Rule assessents o ixtures or orce low along lines siilar to those or rigidity. For exaple, consider a unidirectional reinorced coposite that is stretched to the value at which the ibre begins to break. I the atrix is ore ductile than the ibres, then the ultiate tensile strength o the laina in the equation (3) changed to: u u V 1 V (5) t When the power u shows an extree value, and is the stress o the atrix when the ibres break as in ig.3. It is visible that i the volue raction o the ibre is not strong, the attitude o the laina is exained by the atrix. Matheatically expressed by this ollowing relation: u l u 1 V (6) i it is assued that the laina is to be used in a practical application, the iniu raction o ibre volue is deduced this raction ust be added to the atrix, this orula obtained by solving the equations respectively (5) and (6). Matheatically expressed by this ollowing relation: V in u u u Fig.3.Stress-strain or iber and atrix 3.4 Nonlinear dynaic buckling analysis For the dynaic buckling test, the explicit dynaic instruction akes it possible to peror the whole sall (7) increent tie deterination to ind a working approxiation to the resolution o the proble. This procedure is based on the tie integration rule. The increent calculation ethod is a bit expensive copared to the iplicit orward dynaic integration dynaic analysis instruction does not involve any inversion o a linear syste. The explicit ederation dissiilarity anipulator satisies the dynaic equilibriu equations at the origin o the tie increent t tt/, the increents calculated at that tie are used to advance the tie-to-tie solution t t and the ontie ove solution.. Generously the principle o preliinary kineatics o an arbitrary tie step in acceleration and velocity copletions. Giving the kineatics state at the beginning o a given tie step in ters o the acceleration u n and the velocity un 1/, the equations o otion are integrated using the explicit central-dierence integration rule according to t t u u u u u t u n1 n n1/ n1/ n n1 n n1 n1/ With n reers to the step nuber. 4. Reliability analysis o coposite stringer stiened panels To coplete a reliability analysis, an analytical odel giving an explicit abstract o the proble is developed. This is copleted by selecting appropriate application points on the deep division o explored variables according to an experiental design scale (DOE). This table is generated by setting distinguished levels o interediate actors. Beginning with the objectives received ro the siulation generated or these particular points, a odel is originated in order to air the state unction in an explicit or over the whole doain used or the regression. A rectangle polynoial response surace abbreviation is provided. (8) 4

5 MATEC Web o Conerences 149, (018) The variability o the dynaic buckling load is assued to start ro the ollowing three constituent sources which include the agnitude o the geoetric iperection, the Young's odulus, and the square pulse period. All reerees geoetric and echanical paraeters. Keeping only these three active variables, the buckling load can be expressed explicitly as P P (w,e,t) (9) (6) cr cr 0 In this anticipation which akes it possible to ocus on the undaental eects governing the proble o dynaic buckling o stiened panels, the derivation o a predictive odel is considered under the adaptation o a rectangle polynoial response surace. 5. Results and discussion P. Table.3. gives the levels o the three paraeters selected to peror the reliability analysis. The colun T corresponds to the ollowing three values: T T T T T 0.5 0, 0.5 0, , 0with T0 cr 1.45s (b) Paraeter level w ( ) E( GPa ) T ( s ) 0 Low ediu High Table 3. Levels o actors considered in the reliability analysis Table. Is used to construct a coplete actorial table consisting o 7 cobinations involving the levels speciied. (a) (c) Fig.4. Three sequences o deoration o the stiened panel in the or o the iso-values o the transverse displaceent. In order to obtain a response surace that represents the dynaic buckling critical load in an explicit or with coeicients o the sae order o agnitude, the actors o Table.3 are diensioned by asking: w0 w0 / w0,ax, E E/ Eax et T TT / ax (10) The dynaic buckling critical stress and coeicients o this polynoial are obtained by the regstats coand o Matlab and the response surace polynoial o the coputed syste is: (MPa) w 4.51E cr 0 0.5T 1.84w E 1.4w T 0.961ET w 7.58E 3.65T 0 (11) 5

6 MATEC Web o Conerences 149, (018) The interpolation deined by equation (11) has an excellent coeicient o deterination R 98.13%. It should be noted that even a linear polynoial regression odel also gives a very good coeicient o deterination that is worth R 99.3%. 5.1Analysis o variance The analysis o variance perored on the results presented in Table.3 using the Matlab anovan coand we can notice that the actors and their interactions correctly explain the variability o the dynaic buckling load because the residual error does not exceed. I one works with a linear odel then the actors explain by theselves the results with an error liited to 3.4%. These results are very revealing o the phenoenology o the dynaic buckling which appears in this way as a phenoenon essentially governed by the duration o the ipulse loading. aplitude o curvatures, with dierent panel thicknesses these odels based on ABAQUS sotware. This allowed the derivation o a surace response representation o dynaic buckling load variations. Reerences 1. E. Gal,R. Levy,H. Abraovich, P.Pavsner. Buckling analysis o coposite panels.copos Struct J.E 73, (006).. C.Bisagni, C.Potito. An experiental investigation into the buckling and post-buckling o CFRP shells under cobined axial and torsion loading. Copos Struct J.E 60, (003). 3. R.Zierann, H.Klein, A.Kling. Buckling and postbuckling o stringer stiened ibre coposite curved panels tests and coputations. Copos Struct,J E 73, (006). 4. R.Zierann, R.Roles. Posicoss-iproved postbuckling siulation ordesign o iber coposite stiened uselage structures. Copos Structure, J.E 73,4-171,(006). 5. T.Von Karan, E.E.Sechler, L.H.Donnel. The strength o thin plates in copression. ASME Appl Mech Trans,J.E 54 (193). 6. X.M.Wang, W.Cao, C.H.Deng. Experiental and nuerical analysis or the post-buckling behavior o stiened coposite panels with ipact daage. Copos Struct,J.E 133 (015). 7. F.Caputo, R.Esposito, P.Perugini, D.Santoro. Nuerical-experiental investigation on post-buckled stiened coposite panels. Copos Struct,J.E 55, (00). 8. L.Lanzi, V.Giavotto. Post-buckling optiization o coposite stiened panels: coputations and experients. Copos Struct,J.E 73,08 0 (006). 9. B.Budiansky, J.W.Hutchinson. Dynaic buckling o iperection-sensitive structures Proc. 11th Int. Congress o Applied Mechanics (Berlin: Springer),J.E.13, , (1964). Fig 5. Relative inluence o the actors describing the variability o the dynaic buckling critical load; "Other" includes interactions and error The Fig.5 gives the relative percentage o the inluence o each actor on the variability o the dynaic buckling critical load. Conclusions This paper discusses dynaic buckling on coposite stiened panels when exposed to axial copression in the anner o a hat and axially reinorced. To obtain a variety o dynaic buckling, panels with dierent 6