The In uence of Composite Lay-Up on the Stability of a Structure with Closed Section

This paper presents a study on the s tability of a composite structure with a closed section. The research was carried out solely based on numerical simulations using the fi nite element method. ABAQUS software was used to conduct numerical simulations, which allowed to determine the values of critical loads, as well as the corresponding forms of buckling. As part of the research, the infl uence of composite material arrangement of layers on the stability of the structure was analyzed. The study was conducted on a composite material, which is a carbon-epoxy composite (CFRP). The obtained research results will be verifi ed in subsequent studies by the results of experimental tests.


INTRODUCTION
The stability of thin-walled composite profi les, is an ongoing research issue, especially from an engineering perspective. Stability, as the ability of a mechanical system to independently return to the equilibrium state, from which it was knocked out, e.g. as a result of external loads, is a very desirable property of construction [1][2][3][4][5][6]. This type of characteristic are characterized by thin-walled structures, in recent times, especially made of composite materials, such as carbon-epoxy composite. Due to the fact that thin-walled structures are used as load-carrying profi les, such structures often undergo a phenomenon commonly referred to as loss of stability (which is commonly known as buckling) [7][8][9][10]. This phenomenon occurs due to axial or non-axial compression of thin-walled structures, which results in the aforementioned buckling. In the initial stage of axial compression, a thin-walled structure is subjected to a state in which its walls are purely in compression (membrane state). In the next stage, there is a state in which there is a loss of stability, where the form of deformation of the structure changes -the critical state. The last post-buckling state is a state in which the increase in load is accompanied by a simultaneous increase in defl ection of the structure [11][12][13][14][15][16].
In this paper, special consideration will be given mainly to the phenomenon of loss of stability of composite structure with closed cross-section. In the literature, there are already many scientifi c studies on the issue of loss of stability of thinwalled composite structures, however, the vast majority of studies concern structures with open cross-sections [17][18][19][20][21]. Profi les with open crosssections are more prone to loss of stability, mainly due to lower structural stiff ness than would be the case with closed profi les. A number of already conducted research on the stability of composite structures with open sections provided important information on the discussed issue. Therefore, it is important to conduct research on composite profi les with closed sections, because this type of structure has not yet been thoroughly analyzed in terms of buckling phenomenon [22,23].
The fi gure presented below shows the theoretical assumption according to which the ideal structure (numerical simulations -FEM) and the actual structure (experimental testing -EXP) behaves. In case of numerical simulations, it is The In uence of Composite Lay-Up on the Stability of a Structure with Closed Section Patryk Różyło 1* , Hubert Dębski 1 possible to determine the bifurcation (critical) load (P cr ). In case of the actual construction, predefl ection (w o ) occurs, while the determination of the critical loads is based on approximation methods, which will be the subject of further study.
The reason for conducting this research is primarily due to the fact that the topic of stability, and load-carrying capacity of thin-walled structures, made of composite materials is still a current engineering issue, which is important especially from the point of view of behavior of the structure in the critical state.

THE SUBJECT OF STUDY
The subject of the study are thin-walled composite columns, made of carbon-epoxy composite (CFRP). The material (mechanical and strength) properties of the composite structure, are presented in Table 1 [12,13].
In this paper, a composite structure with a square cross-section, subjected to axial compression, is numerically analyzed. The geometrical parameters of the test specimen are shown as an example in Figure 2.
The structure made of composite material consisted of 8 layers (each layer had a thickness of 0.1 mm) with their symmetrical arrangement with respect to the central plane of the laminate. In this study, four diff erent confi gurations of composite material layers were considered, which were des-

NUMERICAL SIMULATIONS
The study was conducted using numerical simulations based on the fi nite element method (FEM). Numerical simulations were carried out on the basis of the solution of the linear stability problem of the structure, based on the solution of the linear eigenproblem, using the minimum potential energy criterion. The stability problem in mathematical terms could be determined by the following relation: K + λH = 0 (1) where: K -stiff ness matrix, H -geometric stiffness matrix, λ -critical variable (in analyzed case -bifurcation load).
In the conducted numerical research, a few assumptions were adopted, which allowed to determine the phenomenon of loss of stability (buckling) of the structure. As part of the research, a numerical model was prepared, consisting of both a thin-walled composite structure and two plates,   The boundary conditions were defi ned at special reference points (RP) which were coupled to the elements representing the supports. The boundary conditions are presented in Figure 3.
The discrete model was prepared based on the use of shell fi nite elements. For the composite structure, S4R type linear shape function elements with reduced integration were used, having 4 computational nodes per fi nite element, where each node contains 6 degrees of freedom (3 translational and 3 rotational). For non-deformable supports, shell fi nite elements of type R3D4 were used. For the composite profi le, a mesh with a global density of 4 mm was used. The overall number of fi nite elements (within the discrete model) was 7000 while the nodes 7242 (2000 elements of type S4R and 5000 of type R3D4).
The numerical simulations were mainly focused on the buckling behavior of the composite structure, with particular emphasis on the eff ect of the laminate layup confi guration on the stability of the structure. The numerical models prepared, diff ered only in the lay-ups of the composite material, which were detailed in Section 2. In this paper, only numerical studies are presented, while further research will be the experimental verifi cation of the prepared numerical models. The results obtained are presented using the fi rst form of buckling of the structure as an example. Based on the study, a particular behavior of thin-walled composite structures was observed under buckling analysis. Based on the obtained critical state results, it was observed that each confi guration of the composite material layers aff ects the diff erential behavior of the composite structure. The form of buckling obtained for the structure with designation P1, was characterized by the occurrence of 3 half-waves, on each wall of the composite profi le, in the longitudinal direction. The critical load, which occurred in this form of buckling was P cr = 7661.8 N. For the P2 structure, it was observed that there were 4 half-waves on the composite profi le walls in the longitudinal direction of the column. The critical load which corresponded to the buckling form obtained was P cr = 6535.5 N. For structure P3, 5 half-waves were observed, which corresponded to a critical load of P cr = 10315 N. Structure P4 was characterized by the occurrence of 7 half-waves, while the critical load in this case was P cr = 7364.8 N.

NUMERICAL SIMULATION RESULTS
In each of the analyzed cases, the occurrence of a local form of buckling was observed. It was observed that the maximum level of deformation, which is related to the highest recorded defl ection of the structure, occurs in the middle part of the analyzed profi les, regardless of the arrangement of the composite material layers.
Additionally, within the qualitative assessment, it was noted that the use of each subsequent arrangement of composite layers, from P1 to P4, resulted in a change in the form of buckling, where an increase in the number of half-waves was registered, on the walls of the composite profi le in the longitudinal direction of the structure. The increase in the number of half-waves from 3 -for the P1 structure, to 7 -for the P4 structure, did not have a direct impact on the increase of critical loads, and consequently on the stiff ness of the structure. Based on the buckling tests, it was estimated that the structure with the P2 lay-up had the lowest critical load of P cr = 6535.5 N, while the structure with the P3 lay-up showed a load with the highest value of P cr = 10315 N. Regarding the above, it was observed that structure P3 showed 1.58 times higher critical load than structure P2. As part of the critical state results, a very important conclusion can be derived. The highest value of the critical load occurring in the P3 structure is caused by the fact that the high stiff ness is determined on the basis that the outer layers of the composite begin at an angle of 45 after which -45 degrees. The P2 structure is characterized by the occurrence of several 90-degree layers, where the fi bers are arranged transversely to the compression direction, which reduces the stiff ness of the composite structure, thus resulting in a significantly lower critical load than in the other cases.
Moreover, a preliminary analysis of the eff ect of mesh density on buckling results was conducted (based on the P1 profi le). The base case, where the composite structure was described by a mesh density of 4 mm, was compared with a case with a lower composite profi le mesh density of 3 mm and a higher density of 5 mm.
It was observed that both decreasing the mesh density (from 4 to 3 mm) - Figure 5a, and increasing (from 4 to 5 mm) - Figure 5c, has no signifi cant eff ect on the stability of the structure. The buckling forms are almost identical, while a +/-1 mm change in mesh density showed diff erences in critical load from the base case ( Figure  5b), at a range of about 1 %.
These observations, signifi cantly contribute to the correct ability to design composite structures -laminates. Conducted numerical simulations will be the ground for further research in the fi eld of nonlinear stability of the structure taking into account the behavior in the post-buckling state. In further studies, comparative analyses of the critical state for structures with other closed sections will be carried out, as well as analyses related to the failure of the composite material, under the infl uence of axial compressive load. Research presented in this paper, is a preliminary study carried out under the project of the National Science Centre, Poland (project No. 2021/41/B/ST8/00148).

CONCLUSIONS
A critical state analysis, using the FEM, has been carried out to provide a preliminary assessment of the stability of the structure. This study is a preliminary research for further scientifi c studies in the fi eld of stability of thin-walled composite structures with closed sections. On the basis of the study, the following issues were observed: • it is possible to determine the critical state for composite structures with closed sections using the fi nite element method within the framework of the linear eigenproblem solution; • signifi cant infl uence on the stability of composite structures with closed sections has the arrangement of composite material layers; • the use of the arrangement of layers, where there is a predominance of layers at an angle of +/-45 degrees, especially in the outer layers, increases the critical load of the composite structure; • a slight change in the mesh density of the composite structure has little effect on the buckling results.