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Numerical and experimental studies of compressed composite columns with complex open cross-sections
Composite Structures 118 (2014) 28–36
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Numerical and experimental studies of compressed composite columns with complex open cross-sections Hubert Debski a, Andrzej Teter b, Tomasz Kubiak c,⇑ a
Department of Machine Design, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland c Department of Strength of Materials, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Łódz´, Poland b
a r t i c l e
i n f o
Article history: Available online 29 July 2014 Keywords: Compressed column Buckling Post-buckling behavior Finite element analysis Carbon–epoxy laminate Experimental test
a b s t r a c t The article presents results of experimental investigations of thin-walled beams made with carbon fiber composite. Experimental studies were conducted to confirm results obtained from numerical calculations, which was performed using two different software based on finite element method and analytical–numerical method based on Koiter’s asymptotic stability theory of conservative systems modified by Byskov and Hutchinson. The studies consisted of axially compressed thin-walled columns with channel and top-hat cross section. The specimen for the tests were made of unidirectional composite – epoxy resin matrix with carbon fibers reinforcement. The specimen was manufactured using autoclave technique. Thin-walled channel columns with dimensions of 80 40 1.048 mm and length of 300 mm and with four different symmetrical ply arrangements were prepared. The resulting axial force, longitudinal shortening and lateral displacement were recorded during experimental tests. Additionally in chosen points the strain were registered using strain gauges measurement technique. The experimental critical loads using the different method were determined. The experimental results was compared with these obtained from finite element method (ANSYS and ABAQUS) and analytical– numerical method. The postbuckling equilibrium paths obtained from experimental tests and numerical calculation for all considered columns were compared. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled load-carrying structures are characterized by high rigidity and strength-to-weight ratio. Owing to these properties, thin-walled elements are applied in light weight constructions, such as aircraft structures, which must meet rigorous requirements with regard to operation under complex load. High strength parameters of modern structures can be ensured due to the use of modern constructional materials, including fibrous polymeric composite materials – the so-called laminates. An important aspect of the analysis of compressed thin-walled composite structures pertains to ways of increasing critical load and structure rigidity in post-buckling states, while, at the same time, keeping weight of the entire system to the minimum. One of the methods for preventing stability loss in thin-walled ⇑ Corresponding author. E-mail addresses: [email protected] (H. Debski), [email protected] (A. Teter), [email protected] (T. Kubiak). http://dx.doi.org/10.1016/j.compstruct.2014.07.033 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
structures is to use profiles with complex cross-sections and various types of reinforcements. The problem of nonlinear stability of thin-walled structures, including thin-walled profiles, has been extensively discussed. A detailed overview of relevant literature can be found, among others, in the monographs [1–6] and in the reviews [7–10]. These works present comprehensive analysis results of investigating behavior of isotropic structures subjected to compression. The problem has been investigated theoretically [1,2,11–16] and experimentally [5,7,8,17]. The literature offers, however, few studies on experimental investigation on thin-walled structures made of composite materials [9,18–27]. The overwhelming majority of available studies present only results of theoretical investigations. What is lacking are experimental studies investigating light elements of complex-shaped structures made of composite materials. In an attempt to fill in this gap, the authors of the present study have conducted experimental studies into short compressed composite columns [23–25]. This study deals with both experimental and numerical investigation of thin-walled composite structures
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with complex open cross-sections subjected to compression. The main objective of the study was to determine the effect of a given cross sectional profile of columns on their critical load and postbuckling behavior. The study also compared the effect of lay-up of individual plies in the composite on behavior of the structure being compressed; the comparison was made by examining critical load values and post-buckling equilibrium path. Other behavioraffecting factors, such as boundary conditions, material properties and dimensions of the investigates structure element, were identical in all examined cases. The obtained results were then used to validate FEM numerical models. In addition, numerical computations were made by the analytical–numerical method (ANM) based on the asymptotic Koiter theory [6,14,15,26–29].
Strength properties of the autoclave-produced laminates were determined in compliance with relevant ISO standards, which involved computing Young’s modulus E1 and E2, Kirchhoff’s modulus G and Poisson’s ratio m12. Also, basic boundary properties of the laminate were determined using destructive static tensile tests and compression tests for 0° and 90° directions, as well as using shear tests (tensile tests of samples with fiber orientation of ± 45°), the values of which are compared in Table 1. The determined mechanical properties of the laminate applied were then used to numerically define a material model to be used. The application of real mechanical properties of the produced composite material in numerical modeling allows a comparison of obtained numerical and experimental results.
2. Object of the study
3. Experimental tests
The studies were undertaken to investigate thin-walled columns with two types of cross section: a channel section and a top hat section, both produced by the autoclave method in the Department of Materials Engineering at the Lublin University of Technology. The columns were made of Hexcel’s unidirectional carbon–epoxy laminate prepreg tape with a nominal fiber volume of ca. 60%. In each examined case, the laminate texture was composed of 8 plies in a symmetrical lay-up each ply having a thickness of 0.131 mm. The columns had the following dimensions: channel section columns – 80 40 1.048, cross sectional area – 167.68 mm2; top hat section columns – 80 40 20 1.048, cross sectional area – 209.6 mm2 and column length – 300 mm (Fig. 1). Top hat section columns were produced by adding a wall to reinforce the channel section arm. The tests were conducted using four types of symmetrical ply lay-up: [0/45/45/90]s, [90/45/45/0]s, [(0/90)2]s, [45/45/90/0]s. The application of the autoclave technique to produce composite materials ensures that the manufacturing process is repeatable, which is essential if high quality structures are to be produced. Additionally, the technique produces parts with high strength properties, minimal porosity (<1%), and high surface quality with fewer geometrical imperfections of profile walls. All the investigated composite columns were subjected to structure quality control, which helped eliminate samples with internal defects, such as porosity or delamination at corners.
Experimental tests of compressed thin-walled structures were performed using Zwick Z100/SN3A, a universal class 1 accuracy testing machine with a 100 kN load range. The machine was operated by a computer system fitted with testXpertÒ II. The machine was equipped with custom-designed and made grips to ensure that the column was loaded axially (Fig. 2). The grips were set coaxially by mounting them in the loading pins of the machine. The balland-socket joint enabled free rotations of the grips. In order to determine buckling and post-buckling behaviors of the structures, it was necessary to measure load as well as true strains and displacements at selected points of the structure. To this aim, two Vishay strain gauges were fastened to the surface of the samples at the point of highest deflection of the web, in 0° direction parallel to the axis of the column. The two CEA-06-125UW-350 strain gauges had a constant k of 2.135 ± 0.5%, while resistance was set to 350 X ± 0.3%. Deflection was also measured by the optoNCDT 1605 laser sensor; these measurements, however, were made at some other point of highest deflection of the web or plate. All these devices were connected to Hottinger’s MGCplus measurement system loaded with the Catman software and equipped with strain bridges and analog inputs; the measurements were registered with a frequency of 1 Hz. The tests were performed under standard conditions at a temperature of 23 °C and steady cross-beam velocity set to 1 mm/min. The buckling and post-buckling behavior was investigated experimentally by subjecting composite columns to a load of ca. 150% of the expected critical load. The test results do not reveal construction failure initiation. Detailed tests were performed for four series of channel section samples (marked as C1 C4) and top hat section samples (marked as O1 O4), with three independently made samples per each series. In accordance with the devised schedule of experimental tests, each sample was measured 3 times. As a result, 9 measurements were obtained per each ply lay-up; the results were then used to determine buckling and post-buckling behavior of the columns. In order to determine critical loads based on experimental results, the following methods were employed [23,24,30]: the mean strains method (denoted as K1), the method of straight lines intersection in the plot of mean strains (denoted as K2), the P–w2 method (denoted as K3), the inflection-point method (denoted as K4), the Tereszkowski method [31] (denoted as K5), the Koiter method (denoted as K6).
Fig. 1. Dimensions of the examined thin-walled structure.
Local buckling behavior was put to both qualitative and quantitative analysis. The buckling behavior of the examined structures was identified based both on the obtained buckling modes equal
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Table 1 Mechanical properties of the carbon–epoxy laminate applied. Tensile strength FTU [MPa]
Tensile modulus ET [GPa]
Poisson’s ratio m12
Shear strength FSU [MPa]
Shear modulus G [GPa]
Compression strength FCU [MPa]
0° 1867.2
E1 (0°) 130.71
0° 0.32
±45° 100.15
±45° 4.18
0° 1531
90° 25.97
E2 (90°) 6.36
90° 214
Fig. 2. Test stand with a sample mounted in the grips.
to the lowest stability loss and on the corresponding values of critical load determined by the above mentioned methods. The stability loss in both channel section columns and top hat section columns involved buckling of particular column walls, which meant that a specific number m [] of longitudinal half waves was generated depending on the ply lay-up in the composite (Table 2). The results demonstrate that profiles with 0° ply lay-up (fibers oriented along the axis of the column), such as C1, C3 and O1, O3, exhibit fewer longitudinal half-waves than other ply lay-ups in the composite. Similar relationships between the number of half waves and ply lay-up can also be observed in both channel section and top hat section columns. Such behavior of structures demonstrates that the lay-up of plies has a direct impact on buckling mode in the examined structures, while a change in the rigidity of a particular section (channel section, top hat section) does not disturb the ratios obtained, leading only to a higher number of half-waves in the more rigid section, i.e. the top hat section. The experimental results of investigating buckling behavior of compressed thin-walled composite columns provide a great deal of information that may be useful as for the manner and methodology of the experiments. The scatter of measurement results demonstrates that experimental tests must be performed for several series of samples. Also, if possible, all the samples should be examined by means of several independent methods. The performed
Table 2 Mode of stability loss – number m [] of longitudinal half-waves in the column. Ply lay-up
Column
m []
Column
m []
[0/45/45/90]S [90/45/45/0]S [(0/90)2]S [45/45/90/0]S
C1 C2 C3 C4
2 4 2 3
O1 O2 O3 O4
3 7 4 5
analysis allowed to compare the results obtained by different methods, examples of these results for the first series of C1 and O3 column samples are given in Fig. 3. The high agreement between P–w and P–er equilibrium paths, as can be seen from the charts, proves that the methodology applied in the tests is suitable for registering parameters of test samples. Experimental results in all the cases of subjecting composite columns to compression load are repeatable. 4. Numerical computations Parallel to the experimental tests, stability of the compressed composite columns was investigated by numerical methods. To this aim, two independent methods were employed: the analytical–numerical method (ANM) and the finite element method (FEM). The applied analytical–numerical method for solving equations with Koiter’s asymptotic stability theory of conservative systems is used to determine critical loads, frequency of free vibrations, and coefficients describing post-buckling equilibrium paths [6,29]. To examine composite columns by the analytical–numerical method, a plate model was used in the computations. To every plate a full tensor of central surface of the plate was applied along with linear dependences to describe curvature growth [6,14,15,26,27]. Here, post-buckling equilibrium path for a noncoupled case (the so-called single mode case) of buckling in real structures [6,14,15,26,29] can be described as:
P P n þ a111 n2 þ a1111 n3 ¼ n 1 Pcr Pcr
ð1Þ
where Pcr denotes the value of critical load, a111, a1111 denote the post-critical coefficients, while:
n¼
ðDwÞmax h1
ð2Þ
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Fig. 3. Equilibrium path of a column: (a) with channel section and (b) with top hat section. The curves were determined by a laser sensor (Curve 1) and extensometer (Curve 2).
denotes the maximum dimensionless inflection increase relative to the thickness of the first plate, while
n ¼
ðDw0 Þmax h1
ð3Þ
is the dimensionless amplitude of initial inflection corresponding to a buckling mode. Moreover, the dimensionless amplitude of total inflection is expressed as
nt ¼ n þ n ¼
wmax h1
ð4Þ
The numerical computations were performed by the finite element method using two commercial software suites: ABAQUS [32] and ANSYS [33]. The investigation of buckling state involved solving the problem as formulated by the present authors. The post-buckling range required that a solution be found for the problem of nonlinear stability of structures with geometrical imperfections corresponding to the lowest local buckling mode. The value of amplitude of initial imperfections was set to 1/10 of profile wall thickness. The ABAQUS software was used to do numerical computations by the Newton–Raphson method, whereas the computations made by the ANSYS program were based on the Arc-length method. The discretization of the thinwalled column model was conducted using shell elements of the second order (square shape function), each element having 6 degrees of freedom at each node; the finite elements used were Ò respectively: 8-node elements (S8R) in ABAQUS and 8-node eleÒ ments (SHELL281) in ANSYS . The applied multilayer shell elements allow to define independently properties of individual plies in the composite, including thickness, material properties, or directions of main axes of orthotropic material. The composite material was assigned the properties of orthotropic material in plane stress state based on experimentally determined properties of the composite for the main orthotropic directions connected with fiber orientation (Table 1). An example of the simulated numerical models of a top hat section column is shown in Fig. 4. Boundary conditions were defined such to correspond to articulated support of the columns. These boundary conditions were defined either by fixing translational and rotational degrees of freedom at specified nodes of the model or by the coupling of degrees of freedom via defining generalized constant-value displacements for a selected group of nodes. To this aim, zero-displacements were applied to the nodes located at the edges of both lower and upper sections of columns, perpendicular to plane of each wall (displacements ux = 0 and uy = 0). In addition, the nodes belonging to the bottom column end were restrained from vertical displacements
Fig. 4. Discrete model of a top hat section column together with its the boundary conditions.
(uz = 0), while the nodes from the upper end of the column were assigned constant displacement uz = const by the coupling of displacements relative to the axis of the column. The numerical model was loaded such that the load was applied to the edge of the upper end section of the column, ensuring that the column was under uniform axial compression. 5. Results comparison: a discussion At the first stage of the tests on the buckling of thin-walled columns, critical load and corresponding stability loss modes were determined. Fig. 5 shows both experimentally and numerically determined lowest buckling modes of columns C2 and O2. The qualitative agreement between buckling of the real structure and its numerical model was observed in all the examined channel section and top hat section columns. The above observation is also confirmed by the results obtained with the analytical–numerical method (ANM), where the number of sinusoidal half waves, longitudinal to the column, obtained in each examined case was identical with that obtained in the experiments and numerical computations. The experimentally determined values of critical load of the compressed channel section and top hat section column profiles are compared with the numerical and analytical–numerical results
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Fig. 5. Buckling modes in top hat section columns (a) experimental, (b) FEM-determined for column O1, (c) column O2, (d) column O3 and (e) column O4.
Fig. 6. Critical stresses: a comparison of experimental and numerical results, K1 – mean strains method, K2 – method of straight lines intersection in the plot of mean strains, K3 – P–w2 method, K4 – inflection-point method, K5 – Tereszkowski method, K6 – Koiter method, K7 – ABAQUS, K8 – ANSYS, K9 – ANM.
(Fig. 6). The charts given in the figure confirm that critical loads determined by different methods show agreement. The maximum differences between these values do not exceed 15%. In order to compare critical loads of the examined columns, instead of comparing critical loads Pcr, the authors compared mean critical
stresses determined as a quotient of critical load and cross sectional area of a given column. As a result of the performed quantitative analysis of critical loads obtained, it is possible to evaluate the effect of ply lay-up and profile rigidity on stability loss in the examined composite
H. Debski et al. / Composite Structures 118 (2014) 28–36 Table 3 Critical load Pkr [N]: results comparison. Column
C1 C2 C3 C4
MES ABAQUS
ANSYS
2977.2 2995.2 2282.3 4402.4
2946.3 2969.3 2272.5 4359.6
ANM
Column
2848.3 2846.4 2274.8 4369.7
O1 O2 O3 O4
MES
ANM
ABAQUS
ANSYS
6994.6 6968.0 6655.5 9186.0
6898.2 6906.7 6565.0 9106.6
6681.6 6642.1 6629.1 9046.9
Table 4 Mean critical stresses rcr [MPa]: results comparison. Column
C1 C2 C3 C4
FEM
ANM
ABAQUS
ANSYS
17.76 17.86 13.61 26.25
17.57 17.71 13.55 26.00
16.99 16.98 13.57 26.06
Column
O1 O2 O3 O4
FEM
ANM
ABAQUS
ANSYS
33.37 33.24 31.75 43.83
32.91 32.95 31.32 43.45
31.88 31.69 31.63 43.16
structures. It was observed that the placing of ± 450 plies on the outside of the lay-up led to a substantial increase in critical load – in the case of channel section columns the critical load increased by 48%, while in the case of top hat channel profiles it increased by 27% compared to the configuration with the lowest critical load. Another step was to evaluate the effect of edge reinforcement in channel section profiles on critical load (mean critical stresses) in the columns being analyzed. As can be seen from the results given in Fig. 6, top hat cross section columns exhibit much higher mean critical stresses. Tables 3 and 4 list the ANM and FEM results with regard to critical loads (Table 3) and mean critical stresses (Table 4), which allows to compare them with a structure that has a similar ply lay-up in the composite. The results comparison demonstrates that critical load of the structure is significantly affected if walls of a channel section profile are made more rigid. Compared to columns C1 and C2, critical load in columns O1 and O2 increased by 135%, in C3 and O3 – it increased by 192%, while in C4 and O4 – it increased by 133%. By eliminating the effect of cross section change (see the comparison of mean critical stresses – Table 4), it is also possible to demonstrate that top hat cross sections with a ply lay-up analogous to that of channel cross sections are more advantageous due to their higher values of mean critical stresses. Compared to columns C1 and C2, the critical stresses of columns
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O1 and O2 are higher by ca. 86%; for C3 and O3 the increase amounts to 133%, while for O4 and C4 – it is 67%. This means that the applied modification to the open profile, which causes an increase in weight by ca. 25%, leads both to a more than double increase in critical load and to an increase in mean critical stresses by over 67%. The above observations are of vital importance for the practice of producing modern composite structures endangered with stability loss, including, among others, plate structures for thin-walled profiles of various shapes. The experimental and numerical tests of compressed composite columns after stability loss were carried out up using loads equal to ca. 150% of critical load. It was observed that buckling state of particular walls of the columns was increasing according to the initiated buckling mode that corresponded to the lowest critical values. The authors analyzed equilibrium paths describing the relationships between load applied to structure P in the function of deflection of the profile wall w for the nodes attaining the maximum displacement amplitudes. The results of post-buckling equilibrium paths obtained by ANM and FEM were then compared with the experimental results. Fig. 7 shows examples of post-buckling paths for columns C2 and O2. However, given the almost identical FEM results, the figure shows only the curve obtained in the ABAQUS program. The experimental results show satisfactory agreement with the FEM numerical and ANM analytical–numerical results in all the examined cases involving channel section and top hat section columns. The rigidity of post-buckling equilibrium paths determined by the analytical–numerical method is slightly lower compared to that obtained by the numerical method; this notwithstanding, it is to be observed that the FEM results generally agree with the experimental results, and the scatter of results is within the tolerable range. Fig. 8 presents post-buckling equilibrium paths obtained with all the investigation methods employed to examine channels section and top hat section columns, respectively; all these methods take into account the effect of ply lay-up in the composite on post-buckling rigidity of the columns. Comparing the results obtained for an analogous lay-up of plies in channel section and top hat columns, it can be observed that the results agreement is characteristic of structures with the highest rigidity, i.e. columns C2–O2 and C4–O4. Columns with lower post-buckling rigidity, i.e. C1, C3 as well as O1 and O3, reveal an ‘‘exchange’’ in the sequence of curves obtained; their characteristics, however, remain the same. The charts given in Fig. 8 show that the highest rigidity in the post-buckling range is exhibited by the columns in
Fig. 7. Post-buckling equilibrium paths: (a) channel section column C2 and (b) top hat section column O2 – ABAQUS and ANM numerical results along with experimental results.
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Fig. 8. Post-buckling equilibrium paths of channel section columns (left) and of top hat section columns (right): (a), (b) experimental results, (c), (d) FEM results, (e), (f) ANM results.
which the plies (±45°) are located outside the composite system, namely columns C4 and O4. These columns are also characterized by the highest values of critical load. Furthermore, it can be observed that the location of plies in the 0° direction, near the symmetry plane of the composite leads to higher structure rigidity in post-buckling state (columns C2, O2 and C4, O4). The fact that post-buckling paths in columns C1, O1 and C3, O3 are plane means that the applied ply lay-up can affect energy absorbing properties. The regularities observed can therefore provide significant insight into the process and its applications, specifically when it comes to the possibility of configuring mechanical properties of a composite material and its load carrying capacity.
Fig. 9 shows a comparison of equilibrium paths of channel section and top hat section columns that have analogous ply lay-ups. The comparison (Fig. 9) demonstrates that the equilibrium paths for ply lay-ups [0/45/45/90]S and [(0/90)2]S are the same as long as the load is below1.25 Pcr. In both cases, however, if the load exceeds 1.25 Pcr, the channel section columns become more rigid (rigid columns are more vertical). As a result of displacing a ply with longitudinal fiber orientation (0 ply) toward the plane of the central wall of the column (lay-ups [90/-45/45/0]S and [45/-45/90/0]S), the channel section columns with reinforced edges are more rigid in the post-buckling range (Figs. 9b and d). A clear increase in rigidity can be observed for the ply lay-up
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Fig. 9. Experimentally-determined dimensionless post-buckling paths (P/Pcr [] versus a/mcr []) for: (a) C1–O1, (b) C2–O2, (c) C3–O3 and (d) C4–O4.
[45/-45/90/0]S, in which ± 45° plies are located outside, while 0 ply is located in the vicinity of central plane. Similar conclusions concerning ply lay-up can be found, among others, in the work [34]. The above analysis of the results shows that although reinforcing the edges of the channel section leads to a higher critical load of such column, the same cannot be said of all types of ply lay-up in channel section columns. 6. Conclusions The study related to the investigation of both buckling and post-buckling behavior of compressed thin-walled composite columns with complex cross sections. The methodology for investigating buckling behavior and nonlinear stability of structures operating in the post-buckling range was presented. The examined flat-walled columns were modeled as plate structures. To solve the problem, the investigation involved both experiments and simulations, i.e. ANM analytical–numerical and FEM numerical computations. The conducted investigation demonstrated a considerable difference between the rigidity of channel section columns and that of columns with top hat cross section. The reinforcing of the edge of a channel section to obtain a top hat section leads to a considerable (double) increase in critical load, a 25% increase in weight, while the material of the column retains its properties. What exerts a significant effect on critical load and structure rigidity in the post-buckling range is the lay-up of plies in the composite material, particularly the location of ± 45° and 0° plies. If ± 45° plies are outside the symmetric system, this results in a
considerable increase in critical load (by 48% in channel section columns and by 27% in top hat section columns). The location of 0 ply is also important, as locating plies with longitudinal fiber orientation in the vicinity of central plane of column walls results in a higher rigidity of the column in the post-buckling range. If 0 ply is located at the surface of the laminate, the effect of this is that the edge is practically the same. The results, particularly the experimental ones, provide valuable information with regard to, first, better understanding and practical applications of the process for producing structures made of composite load-carrying elements and, second, optimization of such structures to prevent stability loss. What is more, these results can be useful and of vital importance for designers of task-specific constructional elements. As the present study demonstrates, to achieve the desired aim, designers can control not only cross section shapes, as is the case with columns made of conventional materials (e.g. steel), but also the number and ply lay-up in composites, for example when designing absorbers (both resistant and flexible structures) or safety critical elements (rigid and resistant structures). The agreement between the numerical and analytical–numerical results as well as the experimental results demonstrates that the designed simulation models are suitable for investigating the problem of stability of thin-walled composite structures. Acknowledgements The paper written as ministerial research project no. N N507 241440 financed by the Polish National Science Centre.
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[19] Wong PMH, Wang YC. An experimental study of pultruded glass fibre reinforced plastics channel columns at elevated temperatures. Compos Struct 2007;81:84–95. [20] Parlapalli MR, Soh KC, Shu DW, Ma G. Experimental investigation of delamination buckling of stitched composite laminates. Compos: Part A 2007;38:2024–33. [21] Turvey GJ, Zhang Y. A computational and experimental analysis of the buckling, postbuckling and initial failure of pultruded GRP columns. Comput Struct 2006;84:1527–37. [22] Teter A, Kolakowski Z. Buckling of thin-walled composite structures with intermediate stiffeners. Compos struct 2005;69(4):421–8. [23] Debski H, Kubiak T, Teter A. Buckling and postbuckling behaviour of thinwalled composite channel section column. Compos Struct 2013;100: 195–204. [24] Debski H, Kubiak T, Teter A. Experimental investigation of channel-section composite profiles behavior with various sequences of plies subjected to static compression. Thinn-Walled Struct 2013;71:147–54. [25] Debski H. Experimental investigation post-buckling behaviour of composite column with top-hat cross section. Maintenance Reliab 2013;2:105–9. [26] Kolakowski Z. Static and dynamic interactive buckling of composite columns. J Theor Appl Mech 2009;47:177–92. [27] Kolakowski Z, Kubiak T. Load-carrying capacity of thin-walled composite structures. Compos struct 2005;67(4):417–26. [28] Thompson JMT, Hunt GW. General theory of elastic stability. New York: Wiley; 1973. [29] van der Heijden AMA, editor. W.T. Koiter’s elastic stability of solids and structures. Cambridge University Press; 2009. [30] Venkataramaiah KR, Roorda J. Analysis of local plate buckling experimental data. In: Sixth international specialty conference on cold-formed steel structures (1982: November 16–17; St. Louis, Missouri), Missouri S&T (formerly the University of Missouri—Rolla); 1982. p. 45–74. [31] Tereszkowski Z. An experimental method for determining critical loads of plates. Arch Mech Eng 1970;3:485–93. [32] ABAQUS HTML Documentation. [33] User’s Guide ANSYS 14.5, ANSYS Inc, Houston, USA. [34] Kolakowski Z, Mania RJ. Semi-analytical method versus the FEM for analyzing of the local post-buckling of thin-walled composite structures. Compos struct 2013;97:99–106.
Contents lists available at ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Numerical and experimental studies of compressed composite columns with complex open cross-sections Hubert Debski a, Andrzej Teter b, Tomasz Kubiak c,⇑ a
Department of Machine Design, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland Department of Applied Mechanics, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland c Department of Strength of Materials, Lodz University of Technology, Stefanowskiego 1/15, 90-924 Łódz´, Poland b
a r t i c l e
i n f o
Article history: Available online 29 July 2014 Keywords: Compressed column Buckling Post-buckling behavior Finite element analysis Carbon–epoxy laminate Experimental test
a b s t r a c t The article presents results of experimental investigations of thin-walled beams made with carbon fiber composite. Experimental studies were conducted to confirm results obtained from numerical calculations, which was performed using two different software based on finite element method and analytical–numerical method based on Koiter’s asymptotic stability theory of conservative systems modified by Byskov and Hutchinson. The studies consisted of axially compressed thin-walled columns with channel and top-hat cross section. The specimen for the tests were made of unidirectional composite – epoxy resin matrix with carbon fibers reinforcement. The specimen was manufactured using autoclave technique. Thin-walled channel columns with dimensions of 80 40 1.048 mm and length of 300 mm and with four different symmetrical ply arrangements were prepared. The resulting axial force, longitudinal shortening and lateral displacement were recorded during experimental tests. Additionally in chosen points the strain were registered using strain gauges measurement technique. The experimental critical loads using the different method were determined. The experimental results was compared with these obtained from finite element method (ANSYS and ABAQUS) and analytical– numerical method. The postbuckling equilibrium paths obtained from experimental tests and numerical calculation for all considered columns were compared. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Thin-walled load-carrying structures are characterized by high rigidity and strength-to-weight ratio. Owing to these properties, thin-walled elements are applied in light weight constructions, such as aircraft structures, which must meet rigorous requirements with regard to operation under complex load. High strength parameters of modern structures can be ensured due to the use of modern constructional materials, including fibrous polymeric composite materials – the so-called laminates. An important aspect of the analysis of compressed thin-walled composite structures pertains to ways of increasing critical load and structure rigidity in post-buckling states, while, at the same time, keeping weight of the entire system to the minimum. One of the methods for preventing stability loss in thin-walled ⇑ Corresponding author. E-mail addresses: [email protected] (H. Debski), [email protected] (A. Teter), [email protected] (T. Kubiak). http://dx.doi.org/10.1016/j.compstruct.2014.07.033 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.
structures is to use profiles with complex cross-sections and various types of reinforcements. The problem of nonlinear stability of thin-walled structures, including thin-walled profiles, has been extensively discussed. A detailed overview of relevant literature can be found, among others, in the monographs [1–6] and in the reviews [7–10]. These works present comprehensive analysis results of investigating behavior of isotropic structures subjected to compression. The problem has been investigated theoretically [1,2,11–16] and experimentally [5,7,8,17]. The literature offers, however, few studies on experimental investigation on thin-walled structures made of composite materials [9,18–27]. The overwhelming majority of available studies present only results of theoretical investigations. What is lacking are experimental studies investigating light elements of complex-shaped structures made of composite materials. In an attempt to fill in this gap, the authors of the present study have conducted experimental studies into short compressed composite columns [23–25]. This study deals with both experimental and numerical investigation of thin-walled composite structures
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with complex open cross-sections subjected to compression. The main objective of the study was to determine the effect of a given cross sectional profile of columns on their critical load and postbuckling behavior. The study also compared the effect of lay-up of individual plies in the composite on behavior of the structure being compressed; the comparison was made by examining critical load values and post-buckling equilibrium path. Other behavioraffecting factors, such as boundary conditions, material properties and dimensions of the investigates structure element, were identical in all examined cases. The obtained results were then used to validate FEM numerical models. In addition, numerical computations were made by the analytical–numerical method (ANM) based on the asymptotic Koiter theory [6,14,15,26–29].
Strength properties of the autoclave-produced laminates were determined in compliance with relevant ISO standards, which involved computing Young’s modulus E1 and E2, Kirchhoff’s modulus G and Poisson’s ratio m12. Also, basic boundary properties of the laminate were determined using destructive static tensile tests and compression tests for 0° and 90° directions, as well as using shear tests (tensile tests of samples with fiber orientation of ± 45°), the values of which are compared in Table 1. The determined mechanical properties of the laminate applied were then used to numerically define a material model to be used. The application of real mechanical properties of the produced composite material in numerical modeling allows a comparison of obtained numerical and experimental results.
2. Object of the study
3. Experimental tests
The studies were undertaken to investigate thin-walled columns with two types of cross section: a channel section and a top hat section, both produced by the autoclave method in the Department of Materials Engineering at the Lublin University of Technology. The columns were made of Hexcel’s unidirectional carbon–epoxy laminate prepreg tape with a nominal fiber volume of ca. 60%. In each examined case, the laminate texture was composed of 8 plies in a symmetrical lay-up each ply having a thickness of 0.131 mm. The columns had the following dimensions: channel section columns – 80 40 1.048, cross sectional area – 167.68 mm2; top hat section columns – 80 40 20 1.048, cross sectional area – 209.6 mm2 and column length – 300 mm (Fig. 1). Top hat section columns were produced by adding a wall to reinforce the channel section arm. The tests were conducted using four types of symmetrical ply lay-up: [0/45/45/90]s, [90/45/45/0]s, [(0/90)2]s, [45/45/90/0]s. The application of the autoclave technique to produce composite materials ensures that the manufacturing process is repeatable, which is essential if high quality structures are to be produced. Additionally, the technique produces parts with high strength properties, minimal porosity (<1%), and high surface quality with fewer geometrical imperfections of profile walls. All the investigated composite columns were subjected to structure quality control, which helped eliminate samples with internal defects, such as porosity or delamination at corners.
Experimental tests of compressed thin-walled structures were performed using Zwick Z100/SN3A, a universal class 1 accuracy testing machine with a 100 kN load range. The machine was operated by a computer system fitted with testXpertÒ II. The machine was equipped with custom-designed and made grips to ensure that the column was loaded axially (Fig. 2). The grips were set coaxially by mounting them in the loading pins of the machine. The balland-socket joint enabled free rotations of the grips. In order to determine buckling and post-buckling behaviors of the structures, it was necessary to measure load as well as true strains and displacements at selected points of the structure. To this aim, two Vishay strain gauges were fastened to the surface of the samples at the point of highest deflection of the web, in 0° direction parallel to the axis of the column. The two CEA-06-125UW-350 strain gauges had a constant k of 2.135 ± 0.5%, while resistance was set to 350 X ± 0.3%. Deflection was also measured by the optoNCDT 1605 laser sensor; these measurements, however, were made at some other point of highest deflection of the web or plate. All these devices were connected to Hottinger’s MGCplus measurement system loaded with the Catman software and equipped with strain bridges and analog inputs; the measurements were registered with a frequency of 1 Hz. The tests were performed under standard conditions at a temperature of 23 °C and steady cross-beam velocity set to 1 mm/min. The buckling and post-buckling behavior was investigated experimentally by subjecting composite columns to a load of ca. 150% of the expected critical load. The test results do not reveal construction failure initiation. Detailed tests were performed for four series of channel section samples (marked as C1 C4) and top hat section samples (marked as O1 O4), with three independently made samples per each series. In accordance with the devised schedule of experimental tests, each sample was measured 3 times. As a result, 9 measurements were obtained per each ply lay-up; the results were then used to determine buckling and post-buckling behavior of the columns. In order to determine critical loads based on experimental results, the following methods were employed [23,24,30]: the mean strains method (denoted as K1), the method of straight lines intersection in the plot of mean strains (denoted as K2), the P–w2 method (denoted as K3), the inflection-point method (denoted as K4), the Tereszkowski method [31] (denoted as K5), the Koiter method (denoted as K6).
Fig. 1. Dimensions of the examined thin-walled structure.
Local buckling behavior was put to both qualitative and quantitative analysis. The buckling behavior of the examined structures was identified based both on the obtained buckling modes equal
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Table 1 Mechanical properties of the carbon–epoxy laminate applied. Tensile strength FTU [MPa]
Tensile modulus ET [GPa]
Poisson’s ratio m12
Shear strength FSU [MPa]
Shear modulus G [GPa]
Compression strength FCU [MPa]
0° 1867.2
E1 (0°) 130.71
0° 0.32
±45° 100.15
±45° 4.18
0° 1531
90° 25.97
E2 (90°) 6.36
90° 214
Fig. 2. Test stand with a sample mounted in the grips.
to the lowest stability loss and on the corresponding values of critical load determined by the above mentioned methods. The stability loss in both channel section columns and top hat section columns involved buckling of particular column walls, which meant that a specific number m [] of longitudinal half waves was generated depending on the ply lay-up in the composite (Table 2). The results demonstrate that profiles with 0° ply lay-up (fibers oriented along the axis of the column), such as C1, C3 and O1, O3, exhibit fewer longitudinal half-waves than other ply lay-ups in the composite. Similar relationships between the number of half waves and ply lay-up can also be observed in both channel section and top hat section columns. Such behavior of structures demonstrates that the lay-up of plies has a direct impact on buckling mode in the examined structures, while a change in the rigidity of a particular section (channel section, top hat section) does not disturb the ratios obtained, leading only to a higher number of half-waves in the more rigid section, i.e. the top hat section. The experimental results of investigating buckling behavior of compressed thin-walled composite columns provide a great deal of information that may be useful as for the manner and methodology of the experiments. The scatter of measurement results demonstrates that experimental tests must be performed for several series of samples. Also, if possible, all the samples should be examined by means of several independent methods. The performed
Table 2 Mode of stability loss – number m [] of longitudinal half-waves in the column. Ply lay-up
Column
m []
Column
m []
[0/45/45/90]S [90/45/45/0]S [(0/90)2]S [45/45/90/0]S
C1 C2 C3 C4
2 4 2 3
O1 O2 O3 O4
3 7 4 5
analysis allowed to compare the results obtained by different methods, examples of these results for the first series of C1 and O3 column samples are given in Fig. 3. The high agreement between P–w and P–er equilibrium paths, as can be seen from the charts, proves that the methodology applied in the tests is suitable for registering parameters of test samples. Experimental results in all the cases of subjecting composite columns to compression load are repeatable. 4. Numerical computations Parallel to the experimental tests, stability of the compressed composite columns was investigated by numerical methods. To this aim, two independent methods were employed: the analytical–numerical method (ANM) and the finite element method (FEM). The applied analytical–numerical method for solving equations with Koiter’s asymptotic stability theory of conservative systems is used to determine critical loads, frequency of free vibrations, and coefficients describing post-buckling equilibrium paths [6,29]. To examine composite columns by the analytical–numerical method, a plate model was used in the computations. To every plate a full tensor of central surface of the plate was applied along with linear dependences to describe curvature growth [6,14,15,26,27]. Here, post-buckling equilibrium path for a noncoupled case (the so-called single mode case) of buckling in real structures [6,14,15,26,29] can be described as:
P P n þ a111 n2 þ a1111 n3 ¼ n 1 Pcr Pcr
ð1Þ
where Pcr denotes the value of critical load, a111, a1111 denote the post-critical coefficients, while:
n¼
ðDwÞmax h1
ð2Þ
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Fig. 3. Equilibrium path of a column: (a) with channel section and (b) with top hat section. The curves were determined by a laser sensor (Curve 1) and extensometer (Curve 2).
denotes the maximum dimensionless inflection increase relative to the thickness of the first plate, while
n ¼
ðDw0 Þmax h1
ð3Þ
is the dimensionless amplitude of initial inflection corresponding to a buckling mode. Moreover, the dimensionless amplitude of total inflection is expressed as
nt ¼ n þ n ¼
wmax h1
ð4Þ
The numerical computations were performed by the finite element method using two commercial software suites: ABAQUS [32] and ANSYS [33]. The investigation of buckling state involved solving the problem as formulated by the present authors. The post-buckling range required that a solution be found for the problem of nonlinear stability of structures with geometrical imperfections corresponding to the lowest local buckling mode. The value of amplitude of initial imperfections was set to 1/10 of profile wall thickness. The ABAQUS software was used to do numerical computations by the Newton–Raphson method, whereas the computations made by the ANSYS program were based on the Arc-length method. The discretization of the thinwalled column model was conducted using shell elements of the second order (square shape function), each element having 6 degrees of freedom at each node; the finite elements used were Ò respectively: 8-node elements (S8R) in ABAQUS and 8-node eleÒ ments (SHELL281) in ANSYS . The applied multilayer shell elements allow to define independently properties of individual plies in the composite, including thickness, material properties, or directions of main axes of orthotropic material. The composite material was assigned the properties of orthotropic material in plane stress state based on experimentally determined properties of the composite for the main orthotropic directions connected with fiber orientation (Table 1). An example of the simulated numerical models of a top hat section column is shown in Fig. 4. Boundary conditions were defined such to correspond to articulated support of the columns. These boundary conditions were defined either by fixing translational and rotational degrees of freedom at specified nodes of the model or by the coupling of degrees of freedom via defining generalized constant-value displacements for a selected group of nodes. To this aim, zero-displacements were applied to the nodes located at the edges of both lower and upper sections of columns, perpendicular to plane of each wall (displacements ux = 0 and uy = 0). In addition, the nodes belonging to the bottom column end were restrained from vertical displacements
Fig. 4. Discrete model of a top hat section column together with its the boundary conditions.
(uz = 0), while the nodes from the upper end of the column were assigned constant displacement uz = const by the coupling of displacements relative to the axis of the column. The numerical model was loaded such that the load was applied to the edge of the upper end section of the column, ensuring that the column was under uniform axial compression. 5. Results comparison: a discussion At the first stage of the tests on the buckling of thin-walled columns, critical load and corresponding stability loss modes were determined. Fig. 5 shows both experimentally and numerically determined lowest buckling modes of columns C2 and O2. The qualitative agreement between buckling of the real structure and its numerical model was observed in all the examined channel section and top hat section columns. The above observation is also confirmed by the results obtained with the analytical–numerical method (ANM), where the number of sinusoidal half waves, longitudinal to the column, obtained in each examined case was identical with that obtained in the experiments and numerical computations. The experimentally determined values of critical load of the compressed channel section and top hat section column profiles are compared with the numerical and analytical–numerical results
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Fig. 5. Buckling modes in top hat section columns (a) experimental, (b) FEM-determined for column O1, (c) column O2, (d) column O3 and (e) column O4.
Fig. 6. Critical stresses: a comparison of experimental and numerical results, K1 – mean strains method, K2 – method of straight lines intersection in the plot of mean strains, K3 – P–w2 method, K4 – inflection-point method, K5 – Tereszkowski method, K6 – Koiter method, K7 – ABAQUS, K8 – ANSYS, K9 – ANM.
(Fig. 6). The charts given in the figure confirm that critical loads determined by different methods show agreement. The maximum differences between these values do not exceed 15%. In order to compare critical loads of the examined columns, instead of comparing critical loads Pcr, the authors compared mean critical
stresses determined as a quotient of critical load and cross sectional area of a given column. As a result of the performed quantitative analysis of critical loads obtained, it is possible to evaluate the effect of ply lay-up and profile rigidity on stability loss in the examined composite
H. Debski et al. / Composite Structures 118 (2014) 28–36 Table 3 Critical load Pkr [N]: results comparison. Column
C1 C2 C3 C4
MES ABAQUS
ANSYS
2977.2 2995.2 2282.3 4402.4
2946.3 2969.3 2272.5 4359.6
ANM
Column
2848.3 2846.4 2274.8 4369.7
O1 O2 O3 O4
MES
ANM
ABAQUS
ANSYS
6994.6 6968.0 6655.5 9186.0
6898.2 6906.7 6565.0 9106.6
6681.6 6642.1 6629.1 9046.9
Table 4 Mean critical stresses rcr [MPa]: results comparison. Column
C1 C2 C3 C4
FEM
ANM
ABAQUS
ANSYS
17.76 17.86 13.61 26.25
17.57 17.71 13.55 26.00
16.99 16.98 13.57 26.06
Column
O1 O2 O3 O4
FEM
ANM
ABAQUS
ANSYS
33.37 33.24 31.75 43.83
32.91 32.95 31.32 43.45
31.88 31.69 31.63 43.16
structures. It was observed that the placing of ± 450 plies on the outside of the lay-up led to a substantial increase in critical load – in the case of channel section columns the critical load increased by 48%, while in the case of top hat channel profiles it increased by 27% compared to the configuration with the lowest critical load. Another step was to evaluate the effect of edge reinforcement in channel section profiles on critical load (mean critical stresses) in the columns being analyzed. As can be seen from the results given in Fig. 6, top hat cross section columns exhibit much higher mean critical stresses. Tables 3 and 4 list the ANM and FEM results with regard to critical loads (Table 3) and mean critical stresses (Table 4), which allows to compare them with a structure that has a similar ply lay-up in the composite. The results comparison demonstrates that critical load of the structure is significantly affected if walls of a channel section profile are made more rigid. Compared to columns C1 and C2, critical load in columns O1 and O2 increased by 135%, in C3 and O3 – it increased by 192%, while in C4 and O4 – it increased by 133%. By eliminating the effect of cross section change (see the comparison of mean critical stresses – Table 4), it is also possible to demonstrate that top hat cross sections with a ply lay-up analogous to that of channel cross sections are more advantageous due to their higher values of mean critical stresses. Compared to columns C1 and C2, the critical stresses of columns
33
O1 and O2 are higher by ca. 86%; for C3 and O3 the increase amounts to 133%, while for O4 and C4 – it is 67%. This means that the applied modification to the open profile, which causes an increase in weight by ca. 25%, leads both to a more than double increase in critical load and to an increase in mean critical stresses by over 67%. The above observations are of vital importance for the practice of producing modern composite structures endangered with stability loss, including, among others, plate structures for thin-walled profiles of various shapes. The experimental and numerical tests of compressed composite columns after stability loss were carried out up using loads equal to ca. 150% of critical load. It was observed that buckling state of particular walls of the columns was increasing according to the initiated buckling mode that corresponded to the lowest critical values. The authors analyzed equilibrium paths describing the relationships between load applied to structure P in the function of deflection of the profile wall w for the nodes attaining the maximum displacement amplitudes. The results of post-buckling equilibrium paths obtained by ANM and FEM were then compared with the experimental results. Fig. 7 shows examples of post-buckling paths for columns C2 and O2. However, given the almost identical FEM results, the figure shows only the curve obtained in the ABAQUS program. The experimental results show satisfactory agreement with the FEM numerical and ANM analytical–numerical results in all the examined cases involving channel section and top hat section columns. The rigidity of post-buckling equilibrium paths determined by the analytical–numerical method is slightly lower compared to that obtained by the numerical method; this notwithstanding, it is to be observed that the FEM results generally agree with the experimental results, and the scatter of results is within the tolerable range. Fig. 8 presents post-buckling equilibrium paths obtained with all the investigation methods employed to examine channels section and top hat section columns, respectively; all these methods take into account the effect of ply lay-up in the composite on post-buckling rigidity of the columns. Comparing the results obtained for an analogous lay-up of plies in channel section and top hat columns, it can be observed that the results agreement is characteristic of structures with the highest rigidity, i.e. columns C2–O2 and C4–O4. Columns with lower post-buckling rigidity, i.e. C1, C3 as well as O1 and O3, reveal an ‘‘exchange’’ in the sequence of curves obtained; their characteristics, however, remain the same. The charts given in Fig. 8 show that the highest rigidity in the post-buckling range is exhibited by the columns in
Fig. 7. Post-buckling equilibrium paths: (a) channel section column C2 and (b) top hat section column O2 – ABAQUS and ANM numerical results along with experimental results.
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Fig. 8. Post-buckling equilibrium paths of channel section columns (left) and of top hat section columns (right): (a), (b) experimental results, (c), (d) FEM results, (e), (f) ANM results.
which the plies (±45°) are located outside the composite system, namely columns C4 and O4. These columns are also characterized by the highest values of critical load. Furthermore, it can be observed that the location of plies in the 0° direction, near the symmetry plane of the composite leads to higher structure rigidity in post-buckling state (columns C2, O2 and C4, O4). The fact that post-buckling paths in columns C1, O1 and C3, O3 are plane means that the applied ply lay-up can affect energy absorbing properties. The regularities observed can therefore provide significant insight into the process and its applications, specifically when it comes to the possibility of configuring mechanical properties of a composite material and its load carrying capacity.
Fig. 9 shows a comparison of equilibrium paths of channel section and top hat section columns that have analogous ply lay-ups. The comparison (Fig. 9) demonstrates that the equilibrium paths for ply lay-ups [0/45/45/90]S and [(0/90)2]S are the same as long as the load is below1.25 Pcr. In both cases, however, if the load exceeds 1.25 Pcr, the channel section columns become more rigid (rigid columns are more vertical). As a result of displacing a ply with longitudinal fiber orientation (0 ply) toward the plane of the central wall of the column (lay-ups [90/-45/45/0]S and [45/-45/90/0]S), the channel section columns with reinforced edges are more rigid in the post-buckling range (Figs. 9b and d). A clear increase in rigidity can be observed for the ply lay-up
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Fig. 9. Experimentally-determined dimensionless post-buckling paths (P/Pcr [] versus a/mcr []) for: (a) C1–O1, (b) C2–O2, (c) C3–O3 and (d) C4–O4.
[45/-45/90/0]S, in which ± 45° plies are located outside, while 0 ply is located in the vicinity of central plane. Similar conclusions concerning ply lay-up can be found, among others, in the work [34]. The above analysis of the results shows that although reinforcing the edges of the channel section leads to a higher critical load of such column, the same cannot be said of all types of ply lay-up in channel section columns. 6. Conclusions The study related to the investigation of both buckling and post-buckling behavior of compressed thin-walled composite columns with complex cross sections. The methodology for investigating buckling behavior and nonlinear stability of structures operating in the post-buckling range was presented. The examined flat-walled columns were modeled as plate structures. To solve the problem, the investigation involved both experiments and simulations, i.e. ANM analytical–numerical and FEM numerical computations. The conducted investigation demonstrated a considerable difference between the rigidity of channel section columns and that of columns with top hat cross section. The reinforcing of the edge of a channel section to obtain a top hat section leads to a considerable (double) increase in critical load, a 25% increase in weight, while the material of the column retains its properties. What exerts a significant effect on critical load and structure rigidity in the post-buckling range is the lay-up of plies in the composite material, particularly the location of ± 45° and 0° plies. If ± 45° plies are outside the symmetric system, this results in a
considerable increase in critical load (by 48% in channel section columns and by 27% in top hat section columns). The location of 0 ply is also important, as locating plies with longitudinal fiber orientation in the vicinity of central plane of column walls results in a higher rigidity of the column in the post-buckling range. If 0 ply is located at the surface of the laminate, the effect of this is that the edge is practically the same. The results, particularly the experimental ones, provide valuable information with regard to, first, better understanding and practical applications of the process for producing structures made of composite load-carrying elements and, second, optimization of such structures to prevent stability loss. What is more, these results can be useful and of vital importance for designers of task-specific constructional elements. As the present study demonstrates, to achieve the desired aim, designers can control not only cross section shapes, as is the case with columns made of conventional materials (e.g. steel), but also the number and ply lay-up in composites, for example when designing absorbers (both resistant and flexible structures) or safety critical elements (rigid and resistant structures). The agreement between the numerical and analytical–numerical results as well as the experimental results demonstrates that the designed simulation models are suitable for investigating the problem of stability of thin-walled composite structures. Acknowledgements The paper written as ministerial research project no. N N507 241440 financed by the Polish National Science Centre.
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