Stability Analysis of Composite Columns under Eccentric Load
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Abstract
Short thinwalled CFRP columns with channel section were investigated in this study. The tested columns were subjected to compressive load, both axial and eccentric. Eccentric load was applied in two mutually perpendicular directions. The scope of the study included examining the effect of load eccentricity on the critical load of the column. The study involved both experimental tests performed on physical models of columns created by the autoclave technique and a numerical analysis by the finite element method. The numerical analysis of critical state involved solving an eigen problem by the minimum potential energy criterion in order to determine the buckling mode and a corresponding critical load. For physical models, the critical load was determined based on tensometric measurements by Koiter’s approximation method. The numerical results were compared with experimental findings. The study determined the effect of eccentric compressive load on the critical load of the tested column and that of composite layup on the structure’s stability.
Keywords
Buckling Critical load Eccentric load Thinwalled structures Laminates Finite element method1 Introduction
Thinwalled columns of complex crosssectional shapes are structural members with high strength and stiffness yet relatively low specific weight. They are widely used in aviation and automotive design to reinforce thin coverings by acting as a specific loadcarrying framework of the structure. The primary assumption in the design of thinwalled columns is that they will carry axial loads, both tensile and compressive. Columns under compressive loads can either lose their stability or operate in a postcritical range, which greatly reduces their loadcarrying capacity [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In light of the above, it is of vital importance to know the value of critical load under which the structure loses stability. It turns out that when it comes to real structures, the methods for determining critical loads are not wellestablished, which makes the design of such systems difficult [12, 13, 14]. An alternative solution is to perform FEM numerical analyses to determine critical loads [15, 16, 17, 18, 19]. In numerical analysis, the critical load is determined by solving a linear eigen problem in compliance with the minimum potential energy criterion. Moreover, to ensure the equilibrium path stability, the other variant of minimum potential energy must be positive.
Another important aspect of stability analysis is that it must take account of real operating conditions of the thinwalled structure. This is due to the occurrence of various inaccuracies in real structures, including geometric imperfections, production inaccuracy, nonuniform boundary conditions and the eccentricity of compressive load [20, 21, 22]. In particular, under eccentric load, the structure operates in a complex loading state (compression and bending), which may cause the buckling to occur earlier than expected. These problems are investigated by the authors of the present study, which examines the effect of eccentric load on the operation of a thinwalled channelsection column made of carbonepoxy composite. The scope of the study included experimental and numerical investigation of the critical state of composite columns under axial and eccentric load. Obtained results enabled a qualitative and quantitative analysis of the stability of thinwalled composite structures with respect to their sensitivity to eccentricity of the applied load.
2 Object of the Study
Figure 1 shows a schematic design of the tested channelsection column and the axes e_{1} and e_{2} describing the directions of applied eccentric load. Eccentric load was defined as a displacement of the point of application of the compressive load in the end sections of the column relative to its centreline, in two directions: e_{1}: 0 < e_{1} < 10 mm and e_{2}: −10 < e_{2} < 10 mm. A single ply of the laminate was described the following mechanical properties: Young’s modulus in fibre direction – E_{1} = 130,710 MPa; modulus of elasticity perpendicular to fibre direction – E_{2} = 6360 MPa; Poisson’s ratio in ply plane –ν_{12} = 0.32; shear modulus – G_{12} = 4180 MPa.
3 Methods

Direction 1: e_{1} = 0 mm, e_{1} = 10 mm,

Direction 2: e_{2} = 0 mm, e_{2} = −5 mm and e_{2} = 10 mm.
In parallel with experimental tests, we performed a numerical analysis to develop FEM models suitable for determining the effect of eccentric load on the structure’s operation in a critical state. In numerical analysis, the eccentric load was applied in the range from 0 < e_{1} < 10 mm to −10 < e_{2} < 10 mm, every 1 mm. The discrete model of the column was created using 8node layered shell finite elements, S8R, with three translational and three rotational degrees of freedom in every node. S8Rs have quadratic shape function and reduced integration. This type of finite element enables the definition of the laminate’s structure in compliance with a vector normal to the surface of the element. The numerical models were created using an orthotropic material in onedimensional state of stress, based on experimentally obtained mechanical properties of the composite material.
The boundary conditions of the numerical model reflect pinjointed support of the column’s ends – Fig. 2b. Reference points were marked in the centres of gravity of the ends, and their degrees of freedom were linked with the degrees of freedom of the nodes located on the ends of the channelsection column. Three translational degrees of freedom were constrained for the point located in the lower end of the column (u_{x} = u_{y} = u_{z} = 0). The point located in the upper end of the column was defined by the condition of uniform displacement of all ends in the centreline of the column, i.e., u_{z} = const., while its other translational degrees of freedom were constrained (u_{x} = u_{y} = 0). The eccentricity e was described by changing the position of both reference points in the load eccentricity direction. The loading of the structure was realized by the application of load concentrated at the reference point located in the upper end of the column.
4 Results
Critical loads
e  P_{cr} [0/45/−45/90]s  P_{cr} [45/−45/90/0]s  

FEM e_{1}  EXP e_{1}  MES e_{2}  EXP e_{2}  FEM e_{1}  EXP e_{1}  MES e_{2}  EXP e_{2}  
[mm]  [N]  [N]  [N]  [N]  [N]  [N]  [N]  [N] 
−10  3641.8  4507.4  
−9  3774.3  4652.8  
−8  3915  4801.6  
−7  4021.2  4950.4  
−6  4022.9  5092.1  
−5  3974.3  3732.6  5213.4  5354.7  
−4  3863.1  5288.7  
−3  3688.7  5274.1  
−2  3467.6  5119.1  
−1  3225.9  4822.2  
0  2986.4  3108.2  2986.4  3108.2  4453.8  4582.3  4453.8  4582.3 
1  2975.2  2762.3  4403.9  4084.9  
2  2944.1  2559  4300.5  3748.1  
3  2898.4  2377.3  4184.1  3451.1  
4  2843.6  2215.8  4066.7  3191.8  
5  2783.8  2072.5  3952.1  2965.5  
6  2721.9  1945  3841.7  2767.4  
7  2659.6  1831.2  3736  2592.9  
8  2597.9  1729.2  3635.1  2438.4  
9  2537.5  1637.5  3538.8  2300.9  
10  2478.7  2684  1554.7  1380.9  3447.1  3677.3  2177.6  2325.3 
The application of eccentric compressive load in Direction e_{1} reveals that the load eccentricity has a slight effect on the critical load value. Here, the highest stiffness of the column was obtained for the axial compression case – for ply orientation [0/45/−45/90]s – P_{cr} = 2986,4 N and for ply orientation [45/−45/90/0]s – P_{cr} = 4453,8 N; the stiffness slightly decreases with increasing the load eccentricity to the highest value of e_{1} = 10 mm: for ply orientation [0/45/−45/90]s – P_{cr} = 2478.7 N and for ply orientation [45/−45/90/0]s – P_{cr} = 3477.1 N, respectively. Given the two cases, the critical load decreased by 17% and 22%, respectively.
A completely different behaviour of the structure was observed when the eccentric load was applied in Direction e_{2}. With increasing the load eccentricity in the direction from the web of the column, the critical load decreases; for the eccentricity value of e_{2} = 10 mm, it takes the following values: for ply orientation [0/45/−45/90]s – P_{cr} = 1554.7 N and for ply orientation [45/−45/90/0]s – P_{cr} = 2177.6 N, which means that the critical force decreased by 48% and 52%, respectively. These values are much higher than those obtained when the eccentric load was applied in Direction 1. A significant change in the structure’s operation was observed when the eccentric load was applied toward the wall web of the column. With increasing the eccentricity of compressive load, the critical force also increases when compared to the eccentricity value of e_{2} = −6 mm (ply orientation [0/45/−45/90]s) and e_{2} = −5 mm (ply orientation [45/−45/90/0]s). The critical load increased in the following way: for ply orientation [0/45/−45/90]s – P_{cr} is 4022.9 N and for ply orientation [45/−45/90/0]s – P_{cr} is 5213.4 N, which is equal to an increase in the critical load by 26% and 15%, respectively. A further increase in load eccentricity leads to a decrease in the critical load; however, for the limit value of e_{2} = −10 mm, the critical loads were still higher than those observed for axial compression (Table 1). This means that the displacement of the compressive force axis towards the web of the tested channelsection column causes a significant increase in the stiffness of the structure, thus increasing its resistance to stability loss.
The study also enabled the determination of the effect of laminate ply orientation on the critical load of the structure. As shown in the diagrams (Figs. 6 and 7), the change in orientation of the 0^{0} plies from external (ply orientation [0/45/−45/90]s) to internal (ply orientation [45/−45/90/0]s) leads to increased stuffiness of the structure, which results in an approx. 33% increase in the critical load in the axial compression case. This trend does not change when load eccentricity is increased in the two tested directions.
5 Conclusions
This paper presented the results of a study investigating the buckling of thinwalled channelsection composite columns under compression. The study examined the effect of eccentric load on the critical load of the columns. The results confirm that eccentric load has a significant effect on the buckling of columns under compression depending on the direction of load eccentricity relative to the centre of gravity of cross section of the column. When the eccentric load is applied in Direction 1 (Fig. 1), an increase in the load eccentricity leads to a decrease in the critical load; however, compared to the axial compression case, this decrease does not exceed 22%. A much more considerable effect was observed when the eccentric load was applied in Direction 2, where the load eccentricity was increased by moving the direction of action of the compressive force away from the web plane of the channel section column. In this case, the compressive load decreased by 50%. This was the most dangerous loading case observed in this study, as the structure was significantly loaded by the bending moment resulting from the eccentricity of the compressive force. A different behaviour of the structure was observed when the eccentric load was applied in Direction 2; nonetheless, the force shifted toward the web wall of the column. In this case, an increase in load eccentricity led to an increase in the stiffness of the entire structure, as demonstrated by a 15–26% increase in the critical load depending on the laminate ply orientation. The results also demonstrate that ply orientation of the laminate has a significant effect on the critical load value – in the axial compression case, this difference amounted to 33%.
The experimental findings show quantitative and qualitative agreement with the numerical results, and the highest discrepancy between the results does not exceed 10%. This confirms that the developed numerical models of the tested thinwalled structures were correct.
Notes
Acknowledgements
The research has been conducted under the project No UMO2015/19/B/ST8/02800 financed by the National Science Centre Poland.
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