Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 1, pp. 227-242, Warsaw 2011 NUMERICAL-EXPERIMENTAL ANALYSIS OF THE POST-BUCKLING STATE OF A MULTI-SEGMENT AND MULTI-MEMBER THIN-WALLED STRUCTURE SUBJECTED TO TORSION Tomasz Kopecki Rzeszów University of Technology, Department of Aircraft and Aircraft Engines, Rzeszów, Poland e-mail: tkopecki@prz.edu.pl A three-segment ten-member thin-shell structurewith flatwallsmade of a material with an instantaneous characteristic approximated bymeans of an ideally elastic-plastic material model is considered. The structure material (polycarbonate) demonstrates the temporary double refraction effect in polarized light. The system is subject to twisting resulting in the state of local post-critical deformation of skin segments within the structure area. As a result of non-linear numerical analysis in the course of which conformance of equilibrium paths obtained numerically and by means of the experiment is assured, the stress field is determined taking into account the flexural andmembrane state of the structure. Key words: shell, nonlinear analysis, finite elements, constrained torsion, buckling 1. Introduction Thin-shell load-bearing systems of aviation structures are characterised by admissibility of certain local loss of stability of skin elements in conditions of operational loads (Arborcz, 1985; Lynch, 2000). That results from the fact that the commonly adopted staticmodel of a structure composed of a framing and a skin represents a semi-monocoque structure in which it is assumed that the function of the skin consists in transfer of shear interactions only. The framing, composed of transversally situated ribs (frames) characterised with large stiffness in their planes andmembers demonstrating large stiffness for normal forces and relatively small stiffness for bending, is a mechanism. Joinedwith the skin, the framing creates a structure able to transfer any loads resulting from any potentially arising admissible flight phases. 228 T. Kopecki As for the transfer of loads by individual structure elements, the cases in which twisting is the dominant form of load generating the pure she- ar state in isolated skin elements between neighboring frames and members turn out to be the dimensioning ones for the skin elements. These elements, when subject to shearing, loose quickly their stability at relatively low cri- tical compressive stress values. An important stage in the design work on an aircraft load-bearing structure with significant effect on relation betwe- en its mass, stiffness and load capacity limit, consists in choice of the num- ber of frames and members resulting in certain level of internal load exer- ted on the skin elements in post-critical deformation conditions (Kopecki and Dębski, 2007). This paperpresents a concept concerning analysis of amulti-segment semi- monocoque structure subjected to dominant twisting, resulting in a local post- critical deformation state in skin elements, basedonan example of a 3-segment 10-member structure. The considered structure is a simplified part of a torsion box applied in constructional solutions of wings, fuselages and tail planes of aircraft. The mentioned structures often contain various discontinuities, such as cut-outs, usually circular or rectangular with rounded corners, the presence of which is justified for exploatational reasons. They are, for example, inspection ope- nings or cut-outs allowing quick replacements of equipment parts. In case of emergency events, there is a possibility of apperance of irregurar shapes holes, which can be a consequence of collisions with birds or any other obstacles. Then the best temporary method to minimize crash effets is a correction of shape of the damaged zone to the circular opening with possible small ra- dius. In spite of reinforcements of edges, cut-outs, especially localised in the zones including longerons, significantly reduce stiffnes of the structure cau- sing local stress redistributions, which can be a reason of fatique degradation. Aiming at the determination of the extremal values of the stress field, in order to determine fatique life, the considered structure was subject to nonlinear numerical analysis in the finite element approach verified by means of expe- rimental work. The results of experimental research created the possibility to perform corrections of the numerical model in such a way that, at any stage of the structure deformation advancement, the conformance of equili- brium paths and deformation form was assured. The conformance of the re- sults constituted a base for acknowledgement of the obtained structure stress patterns. Numerical-experimental analysis of the post-buckling state... 229 2. Subject and scope of research The subject of research is a three-segment thin-shell structure with ten strin- gers, the general view of which is presented in Fig.1. Joints between the structure elements were realised by means of densely distributed bolts (pitch t=15mm). Fig. 1. Schematic view of the structure (dimensions in millimeters) Fig. 2. Schematic view of the structure mounting and load application The experimental researchwas carried out bymounting the structure on a specially designed test stand (Fig.4) making it possible to introduce a load in the form of dominant twisting with negligible bending effect and transversal force. One of the structure boundary frames was fixed, while the other was connected by means of a stiff rib closing the cross-section, with a lever, by 230 T. Kopecki means of which the load was applied gravitationally. A schematic view of the mounting and the introduction of load to the structure is presented in Fig.2. Thestructurewasmadeofpolycarbonate amaterialwithan instantaneous characteristic presented in Fig.3. Fig. 3. Structure material (polycarbonate) tensile graph Fig. 4. View of the test stand The permanent deformations range, resulting from changes of the polymer molecules position and shape, corresponds in its nature to the plastic ran- ge of an elastic-plastic material. That enabled one to approximate the actual characteristic with an ideally elastic-plasticmaterialmodel in the course of nu- merical analysis. Moreover, polycarbonate demonstrates a temporary double refraction effect. Observation of optical effects in circularly polarized light cre- Numerical-experimental analysis of the post-buckling state... 231 ates a possibility to obtain qualitative information about the existence and location of strong stress concentration zones (Kopecki, 1991; Laerman, 1982). In order to enable observation of the above-mentioned effects, the inner surfa- ce of skin elements were coated with a reflexive layer. The observations were carried out using the reflected light method. In the course of experiment, the load was increased gradually, with very small increment values and theangle of torsionbeingmeasuredaccordingly.As a result, the dependence between the twistingmoment and the structure total angle of torsion was obtained, i.e. the parameters determining representative equilibrium path of the system (Fig.9). Fig. 5. Advanced phase of deformation of the structure Fig. 6. Optical effect patterns 232 T. Kopecki Alreadyat relatively small values of the twistingmoment, all skin segments reached the post-critical deformation state. After full release of the load, the structure returned to its initial form. Therefore, despite significant deforma- tions in the advanced phase, a permanent set did not occur. In Fig.5, an advanced phase of deformation of the structure is presented. Figure 6 shows corresponding patterns of optical effects. 3. Nonlinear numerical analysis In nonlinear analysis of load-bearing structures, relations between a set of static parameters and the corresponding set of geometric parameters can be presented in the form of matrix equation g=K−1(g)f (3.1) where g is a set of geometric parameters describing the system deformation state causedby the load, f is a set of static parameters,while K is the stiffness matrix depending on the set of geometric parameters determining the current deformation state and a nonlinear constitutive relation. Inviewof theoccurrenceof permanentdeformations observed in the course of experiments, thephysical characteristic of the structurematerial determined through uniaxial tensile stress tests (Fig.3) was approximated by an ideally elastic-plastic bodymodel (Fig.7). Fig. 7. Constitutive model of the material In the constitutive equation, in the description related to the linear-elastic range σ=Dve (3.2) Numerical-experimental analysis of the post-buckling state... 233 an assumption about invariance of normal segment (vez = 0) is kept in force. Therefore, the plate stress state is represented by the vector σ = = {σx,σy,τxy,τyz,τzx} ⊤, while D= E 1−ν2            1 ν 0 0 0 ν 1 0 0 0 0 0 1−ν 2 0 0 0 0 0 1−ν 2k 0 0 0 0 0 1−ν 2k            (3.3) is the material constants matrix in which the effect of non-dilatational strain on the plate elastic energy was accounted for by introducing the correction coefficient k=1.2 ve=              vex vey γxy γyz γzx              =                                    ∂u ∂x + 1 2 [(∂u ∂x )2 + (∂v ∂x )2 + (∂w ∂x )2] ∂v ∂y + 1 2 [(∂u ∂y )2 + (∂v ∂y )2 + (∂w ∂y )2] ∂u ∂y + ∂v ∂x + (∂u ∂x )(∂u ∂y ) + (∂v ∂x )(∂v ∂y ) + (∂w ∂x )(∂w ∂y ) ∂u ∂z + ∂w ∂y + (∂u ∂y )(∂u ∂z ) + (∂v ∂y )(∂v ∂z ) + (∂w ∂y )(∂w ∂z ) ∂u ∂z + ∂w ∂x + (∂u ∂x )(∂u ∂z ) + (∂v ∂x )(∂v ∂z ) + (∂w ∂x )(∂w ∂z )                                    (3.4) is a vector containing deformation state components corresponding to the Green-Saint-Venant deformation tensor (Marcinowski, 1999); and u, v,w are displacement vector components in local system of coordinates x,y,z. Numerical representations of the system nonlinear deformations are based on the assumption that at any solution stage with the corresponding load, the deformed system always retains a static equilibrium state. Thus, for a defined discrete system it is possible to formulate a system of equilibrium equations that, with respect to nonlinear structural analysis in its displacement-based representation, can be expressed in the form of matrix residual force equation r(u,Λ)=0 (3.5) where u is the state vector containing displacement components of the struc- ture nodes corresponding to current geometrical configuration, Λ is a matrix composed of control parameters corresponding to the current load state, while 234 T. Kopecki r is the residual vector containing uncompensated components of forces re- lated to the current system deformation state (Felippa, 1976; Felippa et al., 1994). In numerical algorithms, the components of matrix Λ are expressed as functions of the parameter λ described as the state control parameter. It is a measure of the increase of load indirectly or directly related with the pseudo- time parameter t. Thus, the system of equilibrium equations (3.5) can also be presented in the form r(u,λ)=0 (3.6) The above equation is known as the monoparametric residual force equation. Its solution includes a finite number of consecutive structure deformation sta- tes, where each state corresponds to a combination of varying control parame- ters related to the system load, expressedbyasingle state control parameter λ. Transition from the current state to the consecutive one, representing the in- crement step, is initialized by a change of the control parameter to which a new geometrical form is related determined by a new state vector. Development of numerical methods reflected in contemporary algorithms implemented in professional commercial codes resulted in constitution of the- ir two fundamental types. The first one includes purely incremental methods known also as prediction methods, while the other type encompasses correc- tionmethods, called alsoprediction-correction or increment-iterationmethods. The first of thementioned groups is characterised with limited, often unsatis- factory accuracy of obtained results.Moreover, they do not provide possibility to continue calculations after crossing critical points on the equilibrium path. Introduction of the iteration phase is therefore aimed at reduction of the so- lution error and possibility of determination of critical points. That enables analysis of a structure in advanced deformation states. A feature common for both types of themethod consists in the presence of the incremental phase.With respect to arbitrary increment, at the transition from n-th to (n+1)-th state, the undetermined quantities are ∆un =un+1−un ∆λn =λn+1−λn (3.7) In order to determine them, an additional increment control equation is for- mulated, known as the equation of constraints, expressed in the form of a condition c(∆un,∆λn)= 0 (3.8) The fundamental component of the increment phase consists in its pre- diction step, determining a point in the state hyperspace corresponding to Numerical-experimental analysis of the post-buckling state... 235 the consecutive state configuration defined by determination of the increment ∆u for the assumed ∆λ, with equation (3.8) satisfied at the same time. The solution error at each increment step depends on the increment control equ- ation and the adopted extrapolation formula. In each consecutive increment step, the value of total error may increase, resulting in the occurrence of the so-called drift error. Its minimization is ensured by the iteration phase. The fundamental method used in the solution of structural mechanics nonlinear problems is the Newton-Raphson method with numerous program realisations and variations constituting a family of methods (Crisfied, 1997; Felippa, 1976; Felippa et al., 1994; Kopecki and Dębski, 2007). The core idea of these methods consists in expansion of the residual forces equation, r=0, and the increment control equation, c=0, into Fourier series. Assuming that, as a result of the k-th correction iteration step, values uk and λk are obtained, the equations will take the following forms r k+1 = rk+ ∂r ∂u d+ ∂r ∂λ η+H =0 (3.9) ck+1 = ck+ ∂c ∂u d+ ∂c ∂λ η+H =0 where d=uk+1−uk η=λk+1−λk (3.10) Terms H in both equations include the neglected residual values of higher orders. In the iteration process, consecutive values of d and η are determi- ned with respect to which the solution convergence condition is checked at the assumed tolerance. The set obtained that way, constituting a solution to the nonlinear algebraic equations with respect to the unknown nodal di- splacements, creates a base for determination of the equilibrium path. The path, representing the relation between static parameters corresponding to the structure and geometrical parameters related to displacements of its in- dividual points creates a hyper-surface in a multidimensional space with the number of dimensions corresponding to the number of degrees of freedom of the system taken into account. In practice, representative relations between the two parameters are usually developed. Thenumerical analysiswasperformedbymeansofMSCMARC7software. In the modeling of the skin of the examined structure, one used a bilinear thin-shell element. This is a four-node thin element with global displacements and rotations as the degrees of freedom. The bilinear interpolation is used for the coordinates, displacements and rotations. Themembrane strains were obtained from the displacement field and the curvatures from the rotation 236 T. Kopecki field. In the modeling of frames and loading system four-node, bilinear thick- shell elementswereused.As in thecase of thin-shell element,membrane strains were obtained fromthedisplacementfield and the curvatures fromthe rotation field. The transverse shear strains were calculated at the middle of the edges and interpolated to the integration points. The stringers were represented bymeans of beam-type elements based on the Euler-Bernoulli method. All the above-listed elements had six degrees of freedom in the node. The total of 25300 nodes was obtained. The nonlinear analysis was based on the Newton-Raphson prediction me- thod and the Crisfield hyperspherical correction (Bathe, 1996; Doyle, 2001; Crisfied, 1997; Felipa, 1976; Felippa et al., 1994; Rakowski and Kacprzyk, 1993). Reliability of the obtained results was assessed by comparing both equ- ilibrium path shapes and deformation geometries. The two elements created a base for repeated corrections of the numerical model. A series of tests was performed leading to a development of numerical models in which the nature of deformation fully qualitatively corresponded to deformations obtained in the course of experiment. For all model versions, likewise in the course of experimental research, a dependence was determined between the overall angle of torsion and torque, constituting representative equilibrium paths. Figure 8presents thegeometricalmodel of the structureobtainedbymeans ofMSCPATRANsoftware and the finite element grid adopted for calculations with theMSCMARC program. Fig. 8. Geometrical model (left) and the finite element grid (right) Numerical-experimental analysis of the post-buckling state... 237 The comparison of representative equilibrium paths is presented in Fig.9. Fig. 9. Comparison of representative equilibrium paths In Fig.10, the reduced stress distribution according to H-M-H hypothesis is presented corresponding to the advanced deformation phase. Fig. 10. Calculation results obtained withMSCMARC 7 software: the numerical model deformation picture and the effective stress distribution according to H-M-H hypothesis The comparison of equilibrium paths and the deformation type enables one to acknowledge the results of nonlinear numerical analysis as satisfactory. The refinement of numericalmodels was possible in view of systematical com- parison of results of calculations with those obtained in the experiment. Even minimal corrections to stiffness of the structure made significant variance in the equilibrium paths course, and even the lack of the solution convergence. 238 T. Kopecki On the grounds of the solution uniqueness rule, providing that a specific deformation form can correspond only to one stress distribution, the effective stress distributionsdeterminednumerically canbeacknowledged as correspon- ding to the actual ones. The qualitative form of verification of the calculation results consists in comparison of the obtained effective stress distribution pat- terns with pictures of optical effects observed in the course of experiments. Nonlinear analysis was carried out with the use of identical iteration pa- rameters. Using the test stand described above, an experiment was carried out with the use of a model provided with circular opening. Figure 11 presents an advanceddeformation state of the structure corresponding to twistingmoment equal to 350Nm. Fig. 11. Advanced phase of post-critical deformations in the structure with a cut-out – result of experiment Figure12presents acomparisonofpicturesof the structurepost-critical de- formation obtained from the experiment on one hand and nonlinear numerical analysis on the other, while Fig.13 shows a comparison of the corresponding representative equilibrium paths. The presented results confirm full relevance of the deformation form and the representative equilibrium paths. The effective stress pattern presented in Fig.14 can be considered as a reliable one, so it can constitute the base for fatique life calculations. Numerical-experimental analysis of the post-buckling state... 239 Fig. 12. Advanced phase of post-critical deformations in the structure with a circular cut-out: result of the experiment (left) and nonlinear numerical analysis Fig. 13. Comparison of representative equilibrium paths – the model with circular opening 4. Conclusions In the final conclusion, it is important to emhasise the difficultties in repro- ducing the stiffnes of the considered structure model, identified during the experiment. In fact, this problem comes down to the method of modeling the connection between the skin, longerons and frames. Making attempts to re- alise discrete connections, as have been practiced in the structure subjected to experimental tests, in the case of numerical modeling it led to strong local stress concentrations, disabling nonlinear numerical procedures. In that ca- 240 T. Kopecki Fig. 14. Effective stress distribution according to H-M-H hypothesis for the model with circular opening se, one decided to model the mentioned connections in a continuous form. It turned out to be an effective way to provide the conformity of deformation forms and equilibriumpaths. In the succesfull realisation of such research, the problem with modeling of the object assigned to experimental tests seemed to be very significant. These problems do not confine to the compatibility of measurements of structure parts, but also to the maintaining of the technolo- gical limits during the assembly process. They significantly influence the state of initial deformations, as they led to incompatibility between the results of nonlinear numerical calculations and experimental tests. It seems to be important that the presented methodology allows one to make modifications to the structure in the virtual environment. It enables refinement of the structure before very expensive and laborious prototype making. References 1. Arborcz J., 1985,Post-buckling behavior of structures, Numerical techniques for more complicated structures, Lecture Notes in Physics, 228 2. Bathe K.J., 1996,Finite Element Procedures, Prentice Hall 3. Doyle J.F., 2001, Nonlinear Analysis of Thin-Walled Structures, Springer- Verlag, NewYork Berlin Heidelberg Numerical-experimental analysis of the post-buckling state... 241 4. CrisfieldM.A., 1997,Non-LinearFiniteElementAnalysis of Solid andStruc- tures, J.Wiley & Sons, NewYork 5. Felippa C.A., 1976, Procedures for Computer Analysis of Large Nonlinear Structural System in Large Engineering Systems, A.Wexler (Edit.), Pergamon Press, London 6. Felippa C.A., Crivelli L.A., Haugen B., 1994, A survey of the core- congruential formulation for nonlinear finite element, Arch. of Comput. Meth. in Enging., 1 7. Kopecki H., 1991, Problemy analizy stanów naprężenia ustrojów nośnych, w świetle badań eksperymentalnych metodami mechaniki modelowej [Problems in analysis of load-bearing structures stress states in the light of experimen- tal methods using the mode mechanics methods, in Polish], Zeszyty Naukowe Politechniki Rzeszowskiej, 78 8. Kopecki T., Dębski H., 2007, Buckling and post-buckling study of open sec- tion cylindrical shells subjected to constrained torsion,Arch. ofMech. Enging., LIV, 4 9. Laerman K.H., 1982, The principle of integrated photo-elasticity applied to experimental analysis of plates with nonlinear deformation,Proc. 7th Conf. on Experimental Stress Analysis, Haifa 10. LynchC., 2000,AFinite Element Study of the Post Buckling State Behaviour of a Typical Aircraft Fuselage Panel, PhD. Thesis, Queen’s University Belfast 11. Marcinowski J., 1999,Nieliniowa stateczność powłok sprężystych [Nonlinear stability of elastic shells, inPolish],OficynaWydawniczaPolitechnikiWrocław- skiej,Wrocław 12. Rakowski G., Kacprzyk Z., 1993, Metoda elementów skończonych w me- chanice konstrukcji [The finite elements method in mechanics of structures, in Polish], OficynaWydawnicza PolitechnikiWarszawskiej,Warszawa Numeryczno-eksperymentalna analiza stanu deformacji zakrytycznej wielosegmentowej, wielopodłużnicowej konstrukcji cienkościennej poddanej skręcaniu Streszczenie Rozważano trzysegmentową, dziesięciopodłużnicową strukturę cienkościenną o ściankach płaskich, wykonaną zmateriału o charakterystyce natychmiastowej przy- bliżanej modelem materiału idealnie sprężysto-plastycznego. Materiał konstrukcji (poliwęglan) wykazuje efekt dwójłomności wymuszonej w świetle spolaryzowanym. 242 T. Kopecki Konstrukcję poddawano skręcaniu, wskutek czego w obszarze struktury pojawiał się stan lokalnej deformacji zakrytycznej segmentówpokrycia.Wwynikunieliniowej ana- lizy numerycznej, w trakcie której zachowywano zgodność ścieżek równowagi otrzy- manych na drodze numerycznej oraz badań eksperymentalnych, wyznaczano pole na- prężeń, uwzględniające stan giętny i błonowy ustroju. Manuscript received February 5, 2010; accepted for print October 28, 2010