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Effect of load eccentricity on the buckling and post-buckling states of short laminated Z-columns
Composite Structures 210 (2019) 134–144
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Effect of load eccentricity on the buckling and post-buckling states of short laminated Z-columns
T
H. Debskia, , A. Teterb ⁎
a b
Department of Machine Design and Mechatronics, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland Department of Applied Mechanics, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20–618 Lublin, Poland
ARTICLE INFO
ABSTRACT
Keywords: Buckling Critical load Load misalignment Thin-walled structures Laminates Finite element method
This paper was dealt with buckling and post-buckling behavior of short thin-walled Z-columns made of a carbonepoxy laminate, subjected to eccentric compressive load. The buckling mode and the buckling load of real structures versus load eccentricity were discussed. The study involved both experimental tests were carried out on real laminated structures and numerical simulations by the finite element method. The buckling load of real structures was determined by approximation methods on the basis of experimental and numerical post-buckling equilibrium paths of the structure. Additionally, in numerical simulations, the bifurcation load value was determined by solving an eigen problem. In all cases, experimental findings and numerical results show high agreement. The study determined the quantitative influence of the direction and value of compressive load eccentricity on the buckling load and rigidity of the structure in the post-buckling state.
1. Introduction Thin-walled elements with an open cross-section are used in various load-carrying structures as structural reinforcement, hence their rigidity and strength often determine the strength of the entire structure. Among most popular structures with complex cross-sectional shapes, one can distinguish C-columns, angle sections and Z-columns. Under compression, columns of this type – due to their thin-walled construction and high compressive strength – are exposed to destruction by the loss of stability. The use of thin-walled plate columns shows that in many cases they are capable of carrying loads in post-buckling states, because their post-buckling equilibrium paths are stable [1,2]. This makes designers and researches thoroughly examine strength and rigidity parameters of structures under full load conditions. These problems have not yet been fully investigated, especially with reference to thin-walled structures made of composite materials such as laminates [3–6]. Due to their very good strength properties in relation to weight, laminates are more and more widely used in technologically advanced load-carrying structures e.g. in the aircraft or automotive industry. An additional advantage of this material is the possibility to shape its mechanical properties, through the application of appropriate configuration of laminate plies already at the stage of their design [7–10]. Studies on compressed composite structures reported in the literature of the subject are in most cases on analytical and numerical
⁎
considerations, and are usually carried out on structures operated in ideal conditions [11–13]. Only to a small extent are these studies are verified by experimental tests conducted on real structural components [14–17]. Real composite structures are usually operated in complex load states resulting from the specific nature of the structure’s work as well as from various types of manufacturing inaccuracy or assembly. This produces additional undesired effects that significantly increase the level of material effort [9,18,19]. In particular, the eccentricity of load can lead to a fundamental change in the distribution of stresses and strains when compared to the structure subjected to uniform compression or simple bending. This state leads to a significant change in the way the structure is loaded, which can be a direct cause of decreased buckling load and operation of the structure in a post-buckling state in terms of permissible operating loads. A premature buckling state of a thin-walled structure may result in a significant reduction of its strength properties and lead to faster failure of load-carrying elements of the structure. This problem was the subject of study by the authors of this work conducted on real thin-walled Z-columns made of a carbon-epoxy laminate, subjected to eccentric compression. The studies involved the determination of qualitative and quantitative impact of the eccentricity of compressive load on the buckling load and post-buckling equilibrium paths of the structure. Experimental results served as a basis for verifying developed numerical models by the finite element method (FEM).
Corresponding author. E-mail addresses: [email protected] (H. Debski), [email protected] (A. Teter).
https://doi.org/10.1016/j.compstruct.2018.11.044 Received 23 August 2018; Received in revised form 29 October 2018; Accepted 15 November 2018 Available online 16 November 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved.
Composite Structures 210 (2019) 134–144
H. Debski, A. Teter
Fig. 1. Geometric parameters of the investigated composite column.
2. Methods 2.1. Object and scope of the study The scope of the study included experimental tests of real laminated Z-columns and a numerical analysis of the buckling and post-buckling states of the structure by the finite element method. The effect of the eccentricity of compressive load on the buckling load and the mode of stability loss was investigated, together with the structure’s work in the post-buckling state assuming a small overload. The experiments were carried out on short thin-walled columns made of a carbon-epoxy laminate. The columns were produced by the autoclave method to ensure high mechanical properties of the composite material and repeatability of the manufacturing process thanks to fully automatic control of this process. The structures produced in this way are also characterized by a very low degree of porosity of below 1%. The tests were carried out on short columns with ply orientation [0/-45/45/90]s in a symmetrical arrangement with respect to the mid-plane, the geometry of which is shown in Fig. 1. Mechanical properties of the produced laminate were determined in experimental tests in compliance with relevant ISO standards [3]. Average values are listed in Table 1. Experimentally determined mechanical properties of the composite material were used to develop a model of orthotropic material for numerical analysis. This additionally allowed to verify the repeatability of the production process for laminated Z-columns.
specimen
2.2. Experimental Fig. 2. Experimental test stand with a mounted specimen.
Experimental tests was carried out under laboratory conditions at a room temperature of about 22 °C using a universal testing machine. The tests included axial and eccentric compression of laminated Z-columns under load ranging up to about 150% of the bifurcation load
determined by solving an eigen problem. The test stand with a mounted test specimen is shown in Fig. 2. Eccentric load was applied on the testing machine with the use of a specially designed ball joint heads and a table enabling accurate control of the value of eccentricity of specimen mounting relative to the machine axis every 0.1 mm – Fig. 3. The eccentric load was applied in two mutually perpendicular directions, the eccentricity of load was changed every 1 mm in a range from 0 to 10 mm (direction e1) and in a range from 0 to 5 mm (direction e2) – Fig. 1. Additionally, thin pads made of
Table 1 Average values of mechanical properties of carbon-epoxy laminate. Tensile modulus [GPa]
Poisson’s ratio [–]
Shear modulus [GPa]
E1 (0°) 145.528
ν12 0.36
G12 3.845
E2 (90°) 5.826
135
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movable table with micrometric screw
electronic sensor head with ball joint Fig. 3. Application of eccentric compressive load.
soft plastic were put in the cross-section of the column support to eliminate any inaccuracies and strong boundary effects as well as the concentration of stress that could interfere with test results and lead to failure of the specimen. In the tests, strains of the column web were measured using resistance strain gauges that were attached on both sides of the web at 1/ 4 of the height of the column (half-wave centre). The measurement was made and recorded with the Hottinger MGCplus measurement system at a frequency of 1 Hz. The measurement was quasi-static. In the tests, the compression was applied at a constant velocity of the cross beam set equal to 1 mm/min. In addition, deflection perpendicular to the column flange was measured with the use of a laser sensor – Fig. 4. The experimental post-buckling equilibrium paths for load-strain and load-deflection relationships served as a basis for determining the experimental value of buckling load by approximation methods [20]. Four independent methods were employed: the vertical tangent method, the curve intersection method P-ɛm, the P-w2 method and the inflection point method. In the case of the vertical tangent and curve intersection methods, the buckling load was determined on the basis of the post-buckling dependence between load and mean strain obtained by tensometric measurements [21]. In the P-w2 and inflection point methods, the buckling load was determined on the basis of the loaddeflection relationship directly obtained from laser sensor measurements. In the case of structures with small inaccuracies, the P-w2 method enables determination of the lower estimate for buckling load. In the case of a real structure, the post-buckling equilibrium path shows no bifurcation point corresponding to the buckling load. Therefore, the
gauges
buckling load is measured at the point of change in rigidity of the postbuckling equilibrium path. Thereby determined buckling load is always lower than the buckling load measured at the bifurcation point for a geometrically perfect structure. Importantly, the error increases with an increase in the overload factor. In the tests, the specimen was applied loaded with a force equal to approx. 1.5 of the expected bifurcation load for the perfect structure [22]. The buckling load was determined by the P-w2 method on the basis of a load versus square deflection relationship, approximating the relationship with a linear function in the form [20]:
P=
1
(1)
+ Pcr
where: α1 is a constant function coefficient, P is the compressive load, Pcr is the value of an unknown buckling load, while = w 2 is the square of deflection increase measured perpendicular to the profile wall. In the case of the P-w2 method, the buckling load can be defined as a point of intersection between the approximation function (1) and the Y-axis, i.e. it is equal to the free term in Eq. (1). In the case of the inflection point method, the post-buckling equilibrium path obtained under the full load conditions was approximated by cubic polynomial [21]. Using the perturbative Koiter’s method for conservative systems [23,24], the post-buckling equilibrium path for a column with geometric deflection can be described by the dependence [25–27]:
1
w P w + a111 P1 t t
2
+ a1111
w t
3
=
wo P t P1
(2)
where: P is the compressive load, P1 is the bifurcation load of a perfect structure, a111, a1111 are the constant post-buckling coefficients of firstand second-order approximation [28], t is a thickness of the column walls, w is the maximum increase in deflection, and wo is the amplitude of initial deflection corresponding to the mode of buckling. For a geometrically perfect column, this equation can be written as:
laser sensor
1
P w w + a111 + a1111 P1 t t
2
=0
For the Z-column, the coefficient a111
1 Fig. 4. Measurement of strain and deflection in the column web and flange.
or 136
P w + a1111 P1 t
2
=0
(3)
0 , i.e.: (4)
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P = a1111 P1
w t
2
+ P1
(5)
Eq. (5) describing the post-buckling equilibrium path of a geometrically perfect column is a quadratic polynomial. The post-buckling coefficient of the Z-column is a1111 > 0 , which means that the path is stable. At the point w = 0, the buckling load is equal to the bifurcation load, so P1≈Pcr (see Eqs. (1) and (5)), and the function described with Eq. (5) reaches its minimal value in this point. To determine the buckling load for a real structure, the inflection point method assumed that the bifurcation point corresponded to the point of inflection on the post-buckling equilibrium path of a real structure. The inflection point method is used to calculate bifurcation load for columns with small initial deflection, producing its upper estimate. It must be emphasized that with initial deflections that are higher than the column wall thickness, the P-w2 method can significantly underrate the value of bifurcation load, so a much more accurate value of this load can be obtained with the inflection point method. During the approximation process, the correction coefficient R2 describing the degree of fitting of the approximation function to the experimental post-buckling equilibrium path of the structure was constantly verified. In the experiments, the coefficient of correlation was set to the minimum value of R2 ≥ 0.9. 2.3. Numerical analysis Numerical analysis of the stability and post-buckling behaviour of compressed Z-columns was performed by the finite element method (FEM). The first stage of the analysis involved solving an eigen problem, in which the extreme potential energy conditions are used – the system’s state of equilibrium corresponds to the minimum potential energy [29]. This means that for static systems, the second variation of potential energy must be positively determined. In this case, the solution of a stability problem consists in solving the equation:
K+
H= 0
Fig. 5. Discrete model of the Z-column.
to the axis crossing the centre of gravity of the column’s end sections that connects the reference points of support. The compressive load was applied as a concentrated force to the reference point associated with the upper ball joint of column mounting – Fig. 5.
(6)
3. Results and discussion
where K is the matrix of system rigidity, H is the matrix of geometric rigidity of a system, and λ is the critical variable, e.g. bifurcation load. The solution of Eq. (6) makes it possible to determine the bifurcation load corresponding to the buckling load of a geometrically perfect column and its corresponding eigen mode. The second stage of the analysis involved determination of the post-buckling state with implemented initial geometric imperfections corresponding to the lowest eigen mode, their amplitude being set to 0.05 mm. The non-linear stability problem was solved by the Newton-Raphson method. Discretization of the structure was performed using layered shell finite elements with a linear shape function and reduced integration. The finite elements were four-node elements with dimensions of 4x4mm, having 6 degrees of freedom in each node. The shell finite elements enabled defining the laminated structure with respect to element thickness. A general view of the numerical model is shown in Fig. 5. The boundary conditions assigned to the numerical model reflected pin support of the composite column under compression. Reference points corresponded to the centre of gravity of the testing machine mounting heads that were linked by coupling all translational and rotational degrees of freedom to the edges of rigid plates. Three translational degrees of freedom and the ability to rotate relative to the longitudinal axis of the structure were restrained in the reference point of the lower ball joint (i.e. Ux = Uy = Uz = 0 and URz = 0, where: Ux, Uy, Uz – displacements parallel to the respective axes x, y, z of the local Cartesian system of co-ordinates and URz – rotation relative to the axis z. Details can be found in Fig. 5), while in the reference point of the upper ball joint two translational degrees of freedom and the ability to rotate relative to the longitudinal axis of the column were restrained (i.e. Ux = Uy = 0 and URz = 0). The load eccentricity (denoted as e1/2) was determined by changing the position of the column model relative
Numerical analysis and experimental tests of thin-walled Z-columns under eccentric compression provided information enabling the assessment of strains in a real structure as a function of compressive load. Obtained results make it possible to perform a qualitative and quantitative assessment of the buckling and post-buckling states of the compressed thin-walled structure. The numerical solution of the eigen problem (denoted as Case 0) led to identification of the buckling state of the investigated column, reflecting the lowest buckling mode and its corresponding bifurcation load. The obtained value and eigen mode describe the buckling load and the buckling mode for a geometrically perfect structure. Since geometric imperfections are unavoidable in the case of real structures, the buckling load was determined by two approximation methods: P-w2 (denoted as Case 1 (FEM)) and the inflection point method (denoted as Case 2 (FEM)), providing numerical postbuckling paths. Similarly, in the experimental tests, the P-w2 method (denoted as Case 1 (Exp)) and the inflection point method (denoted as Case 2 (Exp)) were used to determine the experimental buckling load. Additionally, based on the relationships obtained from the strain measurements, the solutions were verified using the vertical tangent method (denoted as Case 3) and the curve intersection method, P- ɛm (denoted as Case 4). Non-destructive tests were performed assuming a small overload factor for the investigated columns. All the discussed methods were employed to investigate the effect of compressive load eccentricity on the buckling mode and the buckling load value. Finally, the post-buckling behaviour of the structure was analysed by examining obtained equilibrium paths. The loss of stability of axially compressed short Z-columns with ply orientation of [0/-45/45/90]s reflected a local buckling mode. The numerical and experimental results demonstrate that, under axial 137
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H. Debski, A. Teter
(a)
(b)
Fig. 6. Lowest buckling mode of the Z-column under compression: (a) experimental, (b) FEM numerical analysis.
compression, two buckling semi-waves occur over the web and flange of the Z-column (see Fig. 6). To describe the behaviour of the Z-columns under compression, numerical and experimental post-buckling equilibrium paths were determined for assumed load eccentricity values. Fig. 7 shows a comparison of obtained equilibrium paths for selected values of the eccentricity e1. In the numerical simulations and experimental tests the load eccentricity e1 was changed from 0 to 10 mm. Obtained results show high agreement in all cases. The maximum compressive load values were determined by solving
a numerical eigen problem. It was assumed that the maximum compressive load cannot be greater than 150% of the bifurcation load. Analysing the eigen problem solutions for the load eccentricity e1 ranging from 0 to 10 mm, it was found that the bifurcation load practically does not change (see Table 2, Case 0) and is approximately 2100 N. The application of the maximum compressive load is of vital importance when determining the buckling load of real columns with geometric imperfections. The application of a too high maximum load value significantly reduces the buckling loads. On the other hand, if the maximum load value is too low, some of the above methods may prove
Fig. 7. Comparison of post-buckling equilibrium paths obtained in experiments and numerical analysis (FEM) for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm. 138
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line in the P-w2 plot. The assumptions of both methods are identical. The upper buckling load was estimated by the inflection point method. With this method, the equilibrium path is approximated with the 3rd degree polynomial. This is in line with Koiter's theory, because in a single mode buckling case Eq. (2) can be written for any value of the amplitude of geometric imperfections. Fig. 9 shows examples of equilibrium paths described by 3-degree polynomials. After that, the point of inflection was determined for the approximation curves, which enables determination of the approximation value of the buckling load. In the case of small geometric imperfections, this method significantly overestimates the buckling load. In the case of initial deflections that are higher than the column wall thickness, this method is much better for calculating the buckling load. All the discussed methods were used to determine buckling loads for the analysed Z-column – the results are listed in Table 2 and in Fig. 10a. The numerical solution of the eigen problem indicates that the structure is not sensitive to the eccentricity in direction e1. The bifurcation load (Case 0) differs by 3.5% (i.e. 76 N) when the load eccentricity e1 is changed from 0 to 10 mm. In experimental tests, this difference is unmeasurable. Naturally, one can observe the trend that for higher values of the load eccentricity e1 the buckling load decreases. With the P-w2 method, the underestimation of the bifurcation load ranges from −7.5% to −12% (Fig. 10b) for the FEM solution (Case 1 (FEM)) and from −9% to −20% (Fig. 10b) for the experimental result (Case 1 (Exp)). In both cases, the underestimation increases with increasing the load eccentricity e1. It is worth emphasizing that experimental and numerical buckling loads obtained by the P-w2 method are very similar. The experimental buckling load (Case 1 (Exp)) is underestimated in relation to the numerical result (Case 1 (FEM)) by several percent. For the inflection point method, the overestimation of the bifurcation load is ranges from 16% to 20% (Fig. 10b) for the FEM solutions (Case 2 (FEM)) and from 4% to 16% (Fig. 10b) for the experimental results (Case 2 (Exp)). In both cases, the overestimation decreases with increasing the load eccentricity e1. The experimental and numerical buckling loads determined with the inflection point method are also very similar. The experimental buckling load (Case 2 (Exp)) is underestimated in relation to the FEM result (Case 2 (FEM)) by several percent. Fig. 11 compares the numerical equilibrium paths of the Z-column for selected values of the load eccentricity e1. The application of
Table 2 Buckling loads obtained by P-w2 method (Case1), inflexion point method (Case2) and FEM (Case 0) for load eccentricity e1. Eccentricity e1 in mm
Buckling load in N Case 1 (FEM)
Case 1 (Exp)
Case 2 (FEM)
Case 2 (Exp)
Case 0
0 1 2 3 4 6 7 8 9 10
2010 1968 1969 1963 1951 1916 1901 1883 1861 1846
1972 1906 1966 1920 1932 1855 1755 1736 1710 1670
2600 2560 2568 2563 2550 2517 2494 2474 2450 2434
2520 2440 2577 2472 2600 2632 2348 2330 2316 2179
2174 2173 2171 2166 2160 2144 2134 2123 2111 2098
ineffective. The methods marked as Cases 1–3 worked perfectly well for the analysed Z-column, but in Case 4 the overload was too low, and hence it was not possible to determine the buckling load for all applied eccentricity values (see Table 4). The buckling load of a real Z-column was first determined by the Pw2 method. It was assumed that the buckling equilibrium path for the load range: P1 < P < Pmax where: P1 – bifurcation load obtained from the eigen problem, and Pmax = 1.5P1 – the maximum compressive load. Since deflections in the pre-buckling state for small geometric imperfections can be assumed to be small, their square tends to 0, so the intersection of the determined straight line with the Y-axis in the P-w2 plot represents the approximate buckling load. Selected solutions are shown in Fig. 8 for different values of the load eccentricity e1. In all cases, the determined buckling load is the lower estimate of the bifurcation load. It should be mentioned that an increase in the maximum load always leads to a high underestimation of the buckling load. What is more, the error increases with an increase in the amplitude of geometric imperfections. It is therefore recommended using this method for structures with very small geometric imperfections that do not exceed half of the wall thickness. Identical solutions are obtained by Koiter’s method. The difference lies in the fact that in the P-w system the post-buckling equilibrium path has the form of a symmetrical parabola described with Eq. (5), and this corresponds to the straight
Fig. 8. Comparison of the approximated post-buckling paths obtained by P-w2 method for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm. 139
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Fig. 9. Comparison of approximate post-buckling equilibrium paths obtained by inflection point method for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm.
Fig. 10. Values (a) and error (b) of buckling loads obtained by P-w2 method (Case 1), inflexion point method (Case 2) and FEM (Case 0) for load eccentricity e1.
deflection, and thus failure of the structure. The next stage of the study involved investigation of the effect of load eccentricity in the e2 direction. Fig. 12 compares the equilibrium paths obtained in the numerical analysis and experimental tests for selected values of the eccentricity e2. The results show high qualitative and quantitative agreement. Similarly to the above case, approximation functions were determined for post-buckling paths. Fig. 13 shows the approximation lines obtained with P-w2, while Fig. 14 shows the cubic polynomials obtained with the inflexion point method. All buckling loads obtained with the discussed methods are listed in Table 3 and Fig. 15a. The numerical solution of the eigen problem indicates that the structure is sensitive to the load eccentricity in direction e2. The bifurcation load (Case 0) differs by approx. 50% (i.e. 763 N) when the load eccentricity e2 changed from 0 to 5 mm. One can observe that the buckling load drops rapidly for a higher eccentricity e2. As for the P-w2 method, the underestimation of the bifurcation load amounts to several percent (Fig. 15b) for the FEM solution (Case 1 (FEM)) and to approx. 10% (Fig. 15b) for the experimental result (Case 1 (Exp)). In both cases, the underestimation remains practically unchanged with increasing the load eccentricity e2. It is worth emphasizing that in the case of the P-w2 method, the buckling loads obtained from the FEM analysis and experimental tests are very similar. The experimental
Fig. 11. Effect of load eccentricity e1 on post-buckling equilibrium paths (FEM).
eccentricity leads to reduced rigidity of the post-buckling equilibrium path. In all cases, the equilibrium paths are stable. The experimental results demonstrate that the behaviour of the Z-column is identical in qualitative and quantitative terms. The reduced rigidity can be explained by the fact that the presence of load eccentricity induces deflection from the onset of structure loading. This phenomenon is the same as in the case of structure with geometric imperfections. This accelerates the loss of stability and leads to a rapid increase in 140
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Fig. 12. Comparison of post-buckling equilibrium paths obtained in experiments and numerical analysis (FEM) when load eccentricity e2 is: (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively.
buckling load (Case 1 (Exp)) is underestimated in relation to the FEM result (Case 1 (FEM)) by several percent. As for the inflection point method, the overestimation of the bifurcation load ranges from 1% to 20% (Fig. 15b) for the FEM solution (Case 2 (FEM)) and from 9% to 16% (Fig. 10b) for the experimental result (Case 2 (Exp)). In both cases, the overestimation decreases with increasing the load eccentricity e2. The experimental and numerical buckling loads obtained by the inflection point method are very similar. The experimental buckling load (Case 2 (Exp)) is underestimated in relation to the FEM numerical result (Case 2 (FEM)) by several percent. Fig. 16 shows the numerical equilibrium paths of the Z-column for tested values of the load eccentricity e2. The application of eccentricity reduces rigidity of the post-buckling equilibrium path. In all cases, the equilibrium paths are stable. Experimental results demonstrate that the
behaviour of the Z-column is identical in qualitative and quantitative terms. The behaviour of the Z-column is identical to the corresponding case of the load eccentricity e1. The use of strain gauges in experimental tests enables an independent determination of the buckling load. Fig. 17 shows the results obtained with two strain gauges denoted as T1 and T2 for selected values of the load eccentricity e1. In addition, the figure shows the mean strains (denoted as Tm = εm) obtained with the gauges where:
Tm =
m
= 0.5(T 1 + T 2)
(6)
For detailed calculations, the vertical tangent method (Case 3) and the P-εm intersection method (Case 4) were used. With the first method, experimental buckling load is measured at the point where average strains (i.e. Tm = εm) are the lowest. With the other, the pre-buckling and post-buckling states for average strains are approximated by
Fig. 13. Comparison of approximate post-buckling equilibrium paths obtained with P-w2 method when load eccentricity e2 is (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively. 141
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Fig. 14. Approximated post-buckling equilibrium paths obtained with inflexion point method when load eccentricity e2 is (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively. Table 3 Buckling loads in [N] obtained by P-w2 method (Case1), inflexion point method (Case 2) and FEM (Case0) for load eccentricity e2. Eccentricity e2 in mm
Buckling load in N Case 1 (FEM)
Case 1 (Exp)
Case 2 (FEM)
Case 2 (Exp)
Case 0
0 1 2 3 4 5
2010 1925 1771 1700 1450 1302
1972 1852 1778 1540 1422 1299
2600 2471 2259 2009 1676 1419
2520 2471 2295 1902 1760 1539
2174 2075 1890 1708 1547 1411
Fig. 16. Effect of load eccentricity e2 on post-buckling equilibrium paths (FEM).
straight lines. The point of their intersection determines the experimental buckling load. Obtained buckling loads are listed in Table 4 and in Fig. 18 for the load eccentricity e1, and in Fig. 19 for the load eccentricity e2. Comparing the buckling loads obtained by both methods based on strain measurement, one can observe that they are similar. The difference in the buckling load values does not exceed 5%. Due to a low overload, the P-εm intersection method could not be applied to all cases. The obtained post-buckling state was very short and transient, so obtained solutions were rejected. Analysing the buckling loads obtained
by the vertical tangent method (Case 3), it was found that the determined values are approx. 1.5 higher than their corresponding bifurcation load (Case 0). Hence, they are not acceptable in quantitative terms. In contrast, there is a qualitative agreement between the bifurcation loads (Case 0) and buckling loads determined by the vertical tangent method (Case 3) – one can observe a lack of sensitivity to the load eccentricity in e1 direction (Fig. 18) and high sensitivity to the load eccentricity in direction e2 (Fig. 19).
Fig. 15. Values (a) and errors (b) of buckling loads obtained by P-w2 method (Case 1), inflexion point method (Case 2) and FEM (Case 0) for load eccentricity e2. 142
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Fig. 17. Comparison of the results obtained with strain gauges T1 and T2 and approximation lines of inflexion point method, P-εm, for load eccentricity e1: (a) 0 mm, (b) 5 mm, (c) 10 mm. Table 4 Buckling loads obtained by horizontal tangent method (Case 3), inflexion point P- ɛm method (Case 4) and FEM (Case 0) for load eccentricities e1 and e2. Eccentricity e1 in mm
Buckling load in N Case 3
Case 4
Case 0
0 1 2 3 4 5 6 7 8 10
3054 3275 3196 3268 2806 3221 3281 3187 3142 2910
3027 – – – 2949 – – – – 2912
2174 2173 2171 2166 2160 2153 2144 2134 2123 2098
Eccentricity e2 in mm 1 2 3 4 5
Case 3 2742 2762 2518 2328 1871
Case 4 – – – – –
Case 0 2075 1890 1708 1547 1411
Fig. 18. Buckling loads obtained by vertical tangent method (Case 3) and FEM (Case 0) for load eccentricity e1.
4. Summary and conclusions The presence of load eccentricity means that the structure works in the deflection state from the onset of load application. The behaviour of a structure under eccentric load is identical to that of a geometrically imperfect structure under axial compression. The post-buckling equilibrium paths obtained for the tested laminated Z-column were always stable, and their rigidity decreased with increasing the eccentricity of load. A comparison of the post-buckling equilibrium paths determined by numerical and experimental methods demonstrates that they show high agreement in both quantitative and qualitative terms. The numerical results demonstrate that the effect of load eccentricity in two
Fig. 19. Buckling loads obtained by horizontal tangent method (Case 3) and FEM (Case 0) for load eccentricity e2.
143
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mutually perpendicular directions, e1 and e2, is varied. As for eccentricity applied in the e1 direction, the decrease in the bifurcation load with increasing the eccentricity value from 0 mm to 10 mm was insignificant and amounted to 3.5%, while with the application of eccentricity in the e2 direction in the range from 0 mm to 5 mm the bifurcation load decreased by over 50%. The approximation methods P-w2 and the inflection point method enable the calculation of lower and upper estimates of the buckling load. The first method works well for very small values of eccentricity, while the other is suitable for high values of eccentricity. It should be remembered that the P-w2 method always underestimates the buckling load and this underestimation increases rapidly with increasing the compressive load eccentricity. On the other hand, the inflection point method overestimates the buckling load, and this overestimation decreases with increasing the value of load eccentricity. The application of strain measurement did not work in this case due to a low overloading of the specimens in the experiments. The vertical tangent method and the P-εm intersection method produce close results, but both methods significantly overestimate the value of bifurcation load. Therefore, there is no quantitative agreement, however obtained results show a qualitative agreement, because in all cases the increase in load eccentricity caused a decrease in the buckling load determined by both methods.
Mater 1971:58–80. [9] Turvey GJ, Zhang YA. computational and experimental analysis of the buckling, postbuckling and initial failure of pultruded GRP columns. Comput Struct 2006;84:1527–37. [10] Wong PMH, Wang YC. An experimental study of pultruded glass fibre reinforced plastics channel columns at elevated temperatures. Compos Struct 2007;81:84–95. [11] Bloom F, Coffin D. Handbook of thin plate buckling and postbuckling. London, New York, Washington, D.C.: CHAPMAN & HALL/CRC Boca Raton; 2001. [12] Chou SM, Rhodes J. Review and compilation of experimental results on thin-walled structures. Comput Struct 1997;65(1):47–67. [13] Ascione F, Lamberti M, Razaqpur G. Modifications of standard GFRP sections shape and proportions for improved stiffness and lateral-torsional stability. Compos Struct 2015;132:265–89. [14] Rozylo P, Teter A, Debski H, Wysmulski P, Falkowicz K. Experimental and numerical study of the buckling of composite profiles with open cross section under axial compression. Appl Compos Mater 2017;24:251–1264. [15] Debski H, Jonak J. Failure analysis of thin-walled composite channel section columns. Compos Struct 2015;132:567–74. [16] Debski H. Experimental investigation of post-buckling behavior of composite column with top-hat cross-section. Eksploatacja i Niezawodność – Maint Reliab 2013;15(2):106–10. [17] Debski H, Teter A. Numerical and experimental studies on the limit state of fibrereinforced composite columns with a lipped channel section under quasi-static compression. Compos Struct 2015;133:1–7. [18] Kim SC, Kim JS, Yoon HJ. Experimental and numerical investigations of mode i delamination behaviors of woven fabric composites with carbon, kevlar and their hybrid fibers. Int J Precis Eng Manuf 2011;12(2):321–9. [19] Orifici AC, Thomson RS, Degenhardt R, Kling A, Rohwer K, Bayandor J. Degradation investigation in a postbuckling composite stiffened fuselage panel. Compos Struct 2008;82:217–24. [20] Paszkiewicz M, Kubiak T. Selected problems concerning determination of the buckling load of channel section beams and columns. Thin-Walled Struct 2015;93:112–21. [21] Coan JM. Large-deflection theory for plates with small initial curvature loaded in edge compression. ASME, J Appl Mech 1951;18:143–51. [22] Venkataramaiah KR, Roorda J. Analysis of local plate buckling experimental data. 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, 45–74, 1982. [23] Koiter WT. Elastic stability and post-buckling behaviour. Proceedings of the Symposium on Nonlinear Problems. Wisconsin: Univ. of Wisconsin Press; 1963. [24] van der Heijden AMA, editor. W.T. Koiter’s Elastic Stability of Solids and Structures. Cambridge University Press; 2009. [25] Kubiak T. Postbuckling behaviour of thin-walled girders with orthotropy varying widthwise. Int. J. Solids Struct. 2001;38:4839–55. [26] Teter A. Static and dynamic interactive buckling of isotropic thin-walled closed columns with variable thickness. Thin-Walled Struct 2007;45:936–40. [27] Teter A, Kolakowski Z. Lower bound estimation of load-carrying capacity of thinwalled structures with intermediate stiffeners. Thin-Walled Struct 2001;39(8):649–69. [28] Kolakowski Z, Teter A. Coupled static and dynamic buckling modelling of thinwalled structures in elastic range review of selected problems. Acta Mechanica et Automatica 2016;10(2):141–9. [29] Teter A, Kolakowski Z. Buckling of thin-walled composite structures with intermediate stiffeners. Compos Struct 2005;69(4):421–8.
Acknowledgment The research has been conducted under the project No. UMO2015/ 19/B/ST8/02800 financed by the National Science Centre Poland. References: [1] Bazant ZP, Cedolin L. Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories. UK: Oxford University Press; 2010. [2] Singer J, Arbocz J, Weller T. Buckling experiments. Experimental methods in buckling of thin-walled structure. Basic concepts, columns, beams, and plates, vol. 1 (vol. 2: 2002). New York: John Wiley & Sons Inc.; 1998. [3] Adams DF, Carlsson LA, Pipes RB. Experimental characterization of advanced composite materials. CRC Press LLC; 2003. [4] Reddy JSN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton: CRC Press; 2004. [5] Kolakowski Z. Static and dynamic interactive buckling of composite columns. J Theor Appl Mech 2009;47:177–92. [6] Parlapalli MR, Soh KC, Shu DW, Ma G. Experimental investigation of delamination buckling of stitched composite laminates. Compos A 2007;38:2024–33. [7] Taheri F, Nagaraj M, Khosravi P. Buckling response of glue-laminated columns reinforced with fiber-reinforced plastic sheets. Compos Struct 2009;88:481–90. [8] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos
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Effect of load eccentricity on the buckling and post-buckling states of short laminated Z-columns
T
H. Debskia, , A. Teterb ⁎
a b
Department of Machine Design and Mechatronics, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland Department of Applied Mechanics, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20–618 Lublin, Poland
ARTICLE INFO
ABSTRACT
Keywords: Buckling Critical load Load misalignment Thin-walled structures Laminates Finite element method
This paper was dealt with buckling and post-buckling behavior of short thin-walled Z-columns made of a carbonepoxy laminate, subjected to eccentric compressive load. The buckling mode and the buckling load of real structures versus load eccentricity were discussed. The study involved both experimental tests were carried out on real laminated structures and numerical simulations by the finite element method. The buckling load of real structures was determined by approximation methods on the basis of experimental and numerical post-buckling equilibrium paths of the structure. Additionally, in numerical simulations, the bifurcation load value was determined by solving an eigen problem. In all cases, experimental findings and numerical results show high agreement. The study determined the quantitative influence of the direction and value of compressive load eccentricity on the buckling load and rigidity of the structure in the post-buckling state.
1. Introduction Thin-walled elements with an open cross-section are used in various load-carrying structures as structural reinforcement, hence their rigidity and strength often determine the strength of the entire structure. Among most popular structures with complex cross-sectional shapes, one can distinguish C-columns, angle sections and Z-columns. Under compression, columns of this type – due to their thin-walled construction and high compressive strength – are exposed to destruction by the loss of stability. The use of thin-walled plate columns shows that in many cases they are capable of carrying loads in post-buckling states, because their post-buckling equilibrium paths are stable [1,2]. This makes designers and researches thoroughly examine strength and rigidity parameters of structures under full load conditions. These problems have not yet been fully investigated, especially with reference to thin-walled structures made of composite materials such as laminates [3–6]. Due to their very good strength properties in relation to weight, laminates are more and more widely used in technologically advanced load-carrying structures e.g. in the aircraft or automotive industry. An additional advantage of this material is the possibility to shape its mechanical properties, through the application of appropriate configuration of laminate plies already at the stage of their design [7–10]. Studies on compressed composite structures reported in the literature of the subject are in most cases on analytical and numerical
⁎
considerations, and are usually carried out on structures operated in ideal conditions [11–13]. Only to a small extent are these studies are verified by experimental tests conducted on real structural components [14–17]. Real composite structures are usually operated in complex load states resulting from the specific nature of the structure’s work as well as from various types of manufacturing inaccuracy or assembly. This produces additional undesired effects that significantly increase the level of material effort [9,18,19]. In particular, the eccentricity of load can lead to a fundamental change in the distribution of stresses and strains when compared to the structure subjected to uniform compression or simple bending. This state leads to a significant change in the way the structure is loaded, which can be a direct cause of decreased buckling load and operation of the structure in a post-buckling state in terms of permissible operating loads. A premature buckling state of a thin-walled structure may result in a significant reduction of its strength properties and lead to faster failure of load-carrying elements of the structure. This problem was the subject of study by the authors of this work conducted on real thin-walled Z-columns made of a carbon-epoxy laminate, subjected to eccentric compression. The studies involved the determination of qualitative and quantitative impact of the eccentricity of compressive load on the buckling load and post-buckling equilibrium paths of the structure. Experimental results served as a basis for verifying developed numerical models by the finite element method (FEM).
Corresponding author. E-mail addresses: [email protected] (H. Debski), [email protected] (A. Teter).
https://doi.org/10.1016/j.compstruct.2018.11.044 Received 23 August 2018; Received in revised form 29 October 2018; Accepted 15 November 2018 Available online 16 November 2018 0263-8223/ © 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Geometric parameters of the investigated composite column.
2. Methods 2.1. Object and scope of the study The scope of the study included experimental tests of real laminated Z-columns and a numerical analysis of the buckling and post-buckling states of the structure by the finite element method. The effect of the eccentricity of compressive load on the buckling load and the mode of stability loss was investigated, together with the structure’s work in the post-buckling state assuming a small overload. The experiments were carried out on short thin-walled columns made of a carbon-epoxy laminate. The columns were produced by the autoclave method to ensure high mechanical properties of the composite material and repeatability of the manufacturing process thanks to fully automatic control of this process. The structures produced in this way are also characterized by a very low degree of porosity of below 1%. The tests were carried out on short columns with ply orientation [0/-45/45/90]s in a symmetrical arrangement with respect to the mid-plane, the geometry of which is shown in Fig. 1. Mechanical properties of the produced laminate were determined in experimental tests in compliance with relevant ISO standards [3]. Average values are listed in Table 1. Experimentally determined mechanical properties of the composite material were used to develop a model of orthotropic material for numerical analysis. This additionally allowed to verify the repeatability of the production process for laminated Z-columns.
specimen
2.2. Experimental Fig. 2. Experimental test stand with a mounted specimen.
Experimental tests was carried out under laboratory conditions at a room temperature of about 22 °C using a universal testing machine. The tests included axial and eccentric compression of laminated Z-columns under load ranging up to about 150% of the bifurcation load
determined by solving an eigen problem. The test stand with a mounted test specimen is shown in Fig. 2. Eccentric load was applied on the testing machine with the use of a specially designed ball joint heads and a table enabling accurate control of the value of eccentricity of specimen mounting relative to the machine axis every 0.1 mm – Fig. 3. The eccentric load was applied in two mutually perpendicular directions, the eccentricity of load was changed every 1 mm in a range from 0 to 10 mm (direction e1) and in a range from 0 to 5 mm (direction e2) – Fig. 1. Additionally, thin pads made of
Table 1 Average values of mechanical properties of carbon-epoxy laminate. Tensile modulus [GPa]
Poisson’s ratio [–]
Shear modulus [GPa]
E1 (0°) 145.528
ν12 0.36
G12 3.845
E2 (90°) 5.826
135
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movable table with micrometric screw
electronic sensor head with ball joint Fig. 3. Application of eccentric compressive load.
soft plastic were put in the cross-section of the column support to eliminate any inaccuracies and strong boundary effects as well as the concentration of stress that could interfere with test results and lead to failure of the specimen. In the tests, strains of the column web were measured using resistance strain gauges that were attached on both sides of the web at 1/ 4 of the height of the column (half-wave centre). The measurement was made and recorded with the Hottinger MGCplus measurement system at a frequency of 1 Hz. The measurement was quasi-static. In the tests, the compression was applied at a constant velocity of the cross beam set equal to 1 mm/min. In addition, deflection perpendicular to the column flange was measured with the use of a laser sensor – Fig. 4. The experimental post-buckling equilibrium paths for load-strain and load-deflection relationships served as a basis for determining the experimental value of buckling load by approximation methods [20]. Four independent methods were employed: the vertical tangent method, the curve intersection method P-ɛm, the P-w2 method and the inflection point method. In the case of the vertical tangent and curve intersection methods, the buckling load was determined on the basis of the post-buckling dependence between load and mean strain obtained by tensometric measurements [21]. In the P-w2 and inflection point methods, the buckling load was determined on the basis of the loaddeflection relationship directly obtained from laser sensor measurements. In the case of structures with small inaccuracies, the P-w2 method enables determination of the lower estimate for buckling load. In the case of a real structure, the post-buckling equilibrium path shows no bifurcation point corresponding to the buckling load. Therefore, the
gauges
buckling load is measured at the point of change in rigidity of the postbuckling equilibrium path. Thereby determined buckling load is always lower than the buckling load measured at the bifurcation point for a geometrically perfect structure. Importantly, the error increases with an increase in the overload factor. In the tests, the specimen was applied loaded with a force equal to approx. 1.5 of the expected bifurcation load for the perfect structure [22]. The buckling load was determined by the P-w2 method on the basis of a load versus square deflection relationship, approximating the relationship with a linear function in the form [20]:
P=
1
(1)
+ Pcr
where: α1 is a constant function coefficient, P is the compressive load, Pcr is the value of an unknown buckling load, while = w 2 is the square of deflection increase measured perpendicular to the profile wall. In the case of the P-w2 method, the buckling load can be defined as a point of intersection between the approximation function (1) and the Y-axis, i.e. it is equal to the free term in Eq. (1). In the case of the inflection point method, the post-buckling equilibrium path obtained under the full load conditions was approximated by cubic polynomial [21]. Using the perturbative Koiter’s method for conservative systems [23,24], the post-buckling equilibrium path for a column with geometric deflection can be described by the dependence [25–27]:
1
w P w + a111 P1 t t
2
+ a1111
w t
3
=
wo P t P1
(2)
where: P is the compressive load, P1 is the bifurcation load of a perfect structure, a111, a1111 are the constant post-buckling coefficients of firstand second-order approximation [28], t is a thickness of the column walls, w is the maximum increase in deflection, and wo is the amplitude of initial deflection corresponding to the mode of buckling. For a geometrically perfect column, this equation can be written as:
laser sensor
1
P w w + a111 + a1111 P1 t t
2
=0
For the Z-column, the coefficient a111
1 Fig. 4. Measurement of strain and deflection in the column web and flange.
or 136
P w + a1111 P1 t
2
=0
(3)
0 , i.e.: (4)
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P = a1111 P1
w t
2
+ P1
(5)
Eq. (5) describing the post-buckling equilibrium path of a geometrically perfect column is a quadratic polynomial. The post-buckling coefficient of the Z-column is a1111 > 0 , which means that the path is stable. At the point w = 0, the buckling load is equal to the bifurcation load, so P1≈Pcr (see Eqs. (1) and (5)), and the function described with Eq. (5) reaches its minimal value in this point. To determine the buckling load for a real structure, the inflection point method assumed that the bifurcation point corresponded to the point of inflection on the post-buckling equilibrium path of a real structure. The inflection point method is used to calculate bifurcation load for columns with small initial deflection, producing its upper estimate. It must be emphasized that with initial deflections that are higher than the column wall thickness, the P-w2 method can significantly underrate the value of bifurcation load, so a much more accurate value of this load can be obtained with the inflection point method. During the approximation process, the correction coefficient R2 describing the degree of fitting of the approximation function to the experimental post-buckling equilibrium path of the structure was constantly verified. In the experiments, the coefficient of correlation was set to the minimum value of R2 ≥ 0.9. 2.3. Numerical analysis Numerical analysis of the stability and post-buckling behaviour of compressed Z-columns was performed by the finite element method (FEM). The first stage of the analysis involved solving an eigen problem, in which the extreme potential energy conditions are used – the system’s state of equilibrium corresponds to the minimum potential energy [29]. This means that for static systems, the second variation of potential energy must be positively determined. In this case, the solution of a stability problem consists in solving the equation:
K+
H= 0
Fig. 5. Discrete model of the Z-column.
to the axis crossing the centre of gravity of the column’s end sections that connects the reference points of support. The compressive load was applied as a concentrated force to the reference point associated with the upper ball joint of column mounting – Fig. 5.
(6)
3. Results and discussion
where K is the matrix of system rigidity, H is the matrix of geometric rigidity of a system, and λ is the critical variable, e.g. bifurcation load. The solution of Eq. (6) makes it possible to determine the bifurcation load corresponding to the buckling load of a geometrically perfect column and its corresponding eigen mode. The second stage of the analysis involved determination of the post-buckling state with implemented initial geometric imperfections corresponding to the lowest eigen mode, their amplitude being set to 0.05 mm. The non-linear stability problem was solved by the Newton-Raphson method. Discretization of the structure was performed using layered shell finite elements with a linear shape function and reduced integration. The finite elements were four-node elements with dimensions of 4x4mm, having 6 degrees of freedom in each node. The shell finite elements enabled defining the laminated structure with respect to element thickness. A general view of the numerical model is shown in Fig. 5. The boundary conditions assigned to the numerical model reflected pin support of the composite column under compression. Reference points corresponded to the centre of gravity of the testing machine mounting heads that were linked by coupling all translational and rotational degrees of freedom to the edges of rigid plates. Three translational degrees of freedom and the ability to rotate relative to the longitudinal axis of the structure were restrained in the reference point of the lower ball joint (i.e. Ux = Uy = Uz = 0 and URz = 0, where: Ux, Uy, Uz – displacements parallel to the respective axes x, y, z of the local Cartesian system of co-ordinates and URz – rotation relative to the axis z. Details can be found in Fig. 5), while in the reference point of the upper ball joint two translational degrees of freedom and the ability to rotate relative to the longitudinal axis of the column were restrained (i.e. Ux = Uy = 0 and URz = 0). The load eccentricity (denoted as e1/2) was determined by changing the position of the column model relative
Numerical analysis and experimental tests of thin-walled Z-columns under eccentric compression provided information enabling the assessment of strains in a real structure as a function of compressive load. Obtained results make it possible to perform a qualitative and quantitative assessment of the buckling and post-buckling states of the compressed thin-walled structure. The numerical solution of the eigen problem (denoted as Case 0) led to identification of the buckling state of the investigated column, reflecting the lowest buckling mode and its corresponding bifurcation load. The obtained value and eigen mode describe the buckling load and the buckling mode for a geometrically perfect structure. Since geometric imperfections are unavoidable in the case of real structures, the buckling load was determined by two approximation methods: P-w2 (denoted as Case 1 (FEM)) and the inflection point method (denoted as Case 2 (FEM)), providing numerical postbuckling paths. Similarly, in the experimental tests, the P-w2 method (denoted as Case 1 (Exp)) and the inflection point method (denoted as Case 2 (Exp)) were used to determine the experimental buckling load. Additionally, based on the relationships obtained from the strain measurements, the solutions were verified using the vertical tangent method (denoted as Case 3) and the curve intersection method, P- ɛm (denoted as Case 4). Non-destructive tests were performed assuming a small overload factor for the investigated columns. All the discussed methods were employed to investigate the effect of compressive load eccentricity on the buckling mode and the buckling load value. Finally, the post-buckling behaviour of the structure was analysed by examining obtained equilibrium paths. The loss of stability of axially compressed short Z-columns with ply orientation of [0/-45/45/90]s reflected a local buckling mode. The numerical and experimental results demonstrate that, under axial 137
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H. Debski, A. Teter
(a)
(b)
Fig. 6. Lowest buckling mode of the Z-column under compression: (a) experimental, (b) FEM numerical analysis.
compression, two buckling semi-waves occur over the web and flange of the Z-column (see Fig. 6). To describe the behaviour of the Z-columns under compression, numerical and experimental post-buckling equilibrium paths were determined for assumed load eccentricity values. Fig. 7 shows a comparison of obtained equilibrium paths for selected values of the eccentricity e1. In the numerical simulations and experimental tests the load eccentricity e1 was changed from 0 to 10 mm. Obtained results show high agreement in all cases. The maximum compressive load values were determined by solving
a numerical eigen problem. It was assumed that the maximum compressive load cannot be greater than 150% of the bifurcation load. Analysing the eigen problem solutions for the load eccentricity e1 ranging from 0 to 10 mm, it was found that the bifurcation load practically does not change (see Table 2, Case 0) and is approximately 2100 N. The application of the maximum compressive load is of vital importance when determining the buckling load of real columns with geometric imperfections. The application of a too high maximum load value significantly reduces the buckling loads. On the other hand, if the maximum load value is too low, some of the above methods may prove
Fig. 7. Comparison of post-buckling equilibrium paths obtained in experiments and numerical analysis (FEM) for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm. 138
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line in the P-w2 plot. The assumptions of both methods are identical. The upper buckling load was estimated by the inflection point method. With this method, the equilibrium path is approximated with the 3rd degree polynomial. This is in line with Koiter's theory, because in a single mode buckling case Eq. (2) can be written for any value of the amplitude of geometric imperfections. Fig. 9 shows examples of equilibrium paths described by 3-degree polynomials. After that, the point of inflection was determined for the approximation curves, which enables determination of the approximation value of the buckling load. In the case of small geometric imperfections, this method significantly overestimates the buckling load. In the case of initial deflections that are higher than the column wall thickness, this method is much better for calculating the buckling load. All the discussed methods were used to determine buckling loads for the analysed Z-column – the results are listed in Table 2 and in Fig. 10a. The numerical solution of the eigen problem indicates that the structure is not sensitive to the eccentricity in direction e1. The bifurcation load (Case 0) differs by 3.5% (i.e. 76 N) when the load eccentricity e1 is changed from 0 to 10 mm. In experimental tests, this difference is unmeasurable. Naturally, one can observe the trend that for higher values of the load eccentricity e1 the buckling load decreases. With the P-w2 method, the underestimation of the bifurcation load ranges from −7.5% to −12% (Fig. 10b) for the FEM solution (Case 1 (FEM)) and from −9% to −20% (Fig. 10b) for the experimental result (Case 1 (Exp)). In both cases, the underestimation increases with increasing the load eccentricity e1. It is worth emphasizing that experimental and numerical buckling loads obtained by the P-w2 method are very similar. The experimental buckling load (Case 1 (Exp)) is underestimated in relation to the numerical result (Case 1 (FEM)) by several percent. For the inflection point method, the overestimation of the bifurcation load is ranges from 16% to 20% (Fig. 10b) for the FEM solutions (Case 2 (FEM)) and from 4% to 16% (Fig. 10b) for the experimental results (Case 2 (Exp)). In both cases, the overestimation decreases with increasing the load eccentricity e1. The experimental and numerical buckling loads determined with the inflection point method are also very similar. The experimental buckling load (Case 2 (Exp)) is underestimated in relation to the FEM result (Case 2 (FEM)) by several percent. Fig. 11 compares the numerical equilibrium paths of the Z-column for selected values of the load eccentricity e1. The application of
Table 2 Buckling loads obtained by P-w2 method (Case1), inflexion point method (Case2) and FEM (Case 0) for load eccentricity e1. Eccentricity e1 in mm
Buckling load in N Case 1 (FEM)
Case 1 (Exp)
Case 2 (FEM)
Case 2 (Exp)
Case 0
0 1 2 3 4 6 7 8 9 10
2010 1968 1969 1963 1951 1916 1901 1883 1861 1846
1972 1906 1966 1920 1932 1855 1755 1736 1710 1670
2600 2560 2568 2563 2550 2517 2494 2474 2450 2434
2520 2440 2577 2472 2600 2632 2348 2330 2316 2179
2174 2173 2171 2166 2160 2144 2134 2123 2111 2098
ineffective. The methods marked as Cases 1–3 worked perfectly well for the analysed Z-column, but in Case 4 the overload was too low, and hence it was not possible to determine the buckling load for all applied eccentricity values (see Table 4). The buckling load of a real Z-column was first determined by the Pw2 method. It was assumed that the buckling equilibrium path for the load range: P1 < P < Pmax where: P1 – bifurcation load obtained from the eigen problem, and Pmax = 1.5P1 – the maximum compressive load. Since deflections in the pre-buckling state for small geometric imperfections can be assumed to be small, their square tends to 0, so the intersection of the determined straight line with the Y-axis in the P-w2 plot represents the approximate buckling load. Selected solutions are shown in Fig. 8 for different values of the load eccentricity e1. In all cases, the determined buckling load is the lower estimate of the bifurcation load. It should be mentioned that an increase in the maximum load always leads to a high underestimation of the buckling load. What is more, the error increases with an increase in the amplitude of geometric imperfections. It is therefore recommended using this method for structures with very small geometric imperfections that do not exceed half of the wall thickness. Identical solutions are obtained by Koiter’s method. The difference lies in the fact that in the P-w system the post-buckling equilibrium path has the form of a symmetrical parabola described with Eq. (5), and this corresponds to the straight
Fig. 8. Comparison of the approximated post-buckling paths obtained by P-w2 method for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm. 139
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Fig. 9. Comparison of approximate post-buckling equilibrium paths obtained by inflection point method for load eccentricity e1: (a) 0 mm, (b) 3 mm, (c) 7 mm and (d) 10 mm.
Fig. 10. Values (a) and error (b) of buckling loads obtained by P-w2 method (Case 1), inflexion point method (Case 2) and FEM (Case 0) for load eccentricity e1.
deflection, and thus failure of the structure. The next stage of the study involved investigation of the effect of load eccentricity in the e2 direction. Fig. 12 compares the equilibrium paths obtained in the numerical analysis and experimental tests for selected values of the eccentricity e2. The results show high qualitative and quantitative agreement. Similarly to the above case, approximation functions were determined for post-buckling paths. Fig. 13 shows the approximation lines obtained with P-w2, while Fig. 14 shows the cubic polynomials obtained with the inflexion point method. All buckling loads obtained with the discussed methods are listed in Table 3 and Fig. 15a. The numerical solution of the eigen problem indicates that the structure is sensitive to the load eccentricity in direction e2. The bifurcation load (Case 0) differs by approx. 50% (i.e. 763 N) when the load eccentricity e2 changed from 0 to 5 mm. One can observe that the buckling load drops rapidly for a higher eccentricity e2. As for the P-w2 method, the underestimation of the bifurcation load amounts to several percent (Fig. 15b) for the FEM solution (Case 1 (FEM)) and to approx. 10% (Fig. 15b) for the experimental result (Case 1 (Exp)). In both cases, the underestimation remains practically unchanged with increasing the load eccentricity e2. It is worth emphasizing that in the case of the P-w2 method, the buckling loads obtained from the FEM analysis and experimental tests are very similar. The experimental
Fig. 11. Effect of load eccentricity e1 on post-buckling equilibrium paths (FEM).
eccentricity leads to reduced rigidity of the post-buckling equilibrium path. In all cases, the equilibrium paths are stable. The experimental results demonstrate that the behaviour of the Z-column is identical in qualitative and quantitative terms. The reduced rigidity can be explained by the fact that the presence of load eccentricity induces deflection from the onset of structure loading. This phenomenon is the same as in the case of structure with geometric imperfections. This accelerates the loss of stability and leads to a rapid increase in 140
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Fig. 12. Comparison of post-buckling equilibrium paths obtained in experiments and numerical analysis (FEM) when load eccentricity e2 is: (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively.
buckling load (Case 1 (Exp)) is underestimated in relation to the FEM result (Case 1 (FEM)) by several percent. As for the inflection point method, the overestimation of the bifurcation load ranges from 1% to 20% (Fig. 15b) for the FEM solution (Case 2 (FEM)) and from 9% to 16% (Fig. 10b) for the experimental result (Case 2 (Exp)). In both cases, the overestimation decreases with increasing the load eccentricity e2. The experimental and numerical buckling loads obtained by the inflection point method are very similar. The experimental buckling load (Case 2 (Exp)) is underestimated in relation to the FEM numerical result (Case 2 (FEM)) by several percent. Fig. 16 shows the numerical equilibrium paths of the Z-column for tested values of the load eccentricity e2. The application of eccentricity reduces rigidity of the post-buckling equilibrium path. In all cases, the equilibrium paths are stable. Experimental results demonstrate that the
behaviour of the Z-column is identical in qualitative and quantitative terms. The behaviour of the Z-column is identical to the corresponding case of the load eccentricity e1. The use of strain gauges in experimental tests enables an independent determination of the buckling load. Fig. 17 shows the results obtained with two strain gauges denoted as T1 and T2 for selected values of the load eccentricity e1. In addition, the figure shows the mean strains (denoted as Tm = εm) obtained with the gauges where:
Tm =
m
= 0.5(T 1 + T 2)
(6)
For detailed calculations, the vertical tangent method (Case 3) and the P-εm intersection method (Case 4) were used. With the first method, experimental buckling load is measured at the point where average strains (i.e. Tm = εm) are the lowest. With the other, the pre-buckling and post-buckling states for average strains are approximated by
Fig. 13. Comparison of approximate post-buckling equilibrium paths obtained with P-w2 method when load eccentricity e2 is (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively. 141
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Fig. 14. Approximated post-buckling equilibrium paths obtained with inflexion point method when load eccentricity e2 is (a) 0 mm, (b) 1 mm, (c) 3 mm and (d) 5 mm, respectively. Table 3 Buckling loads in [N] obtained by P-w2 method (Case1), inflexion point method (Case 2) and FEM (Case0) for load eccentricity e2. Eccentricity e2 in mm
Buckling load in N Case 1 (FEM)
Case 1 (Exp)
Case 2 (FEM)
Case 2 (Exp)
Case 0
0 1 2 3 4 5
2010 1925 1771 1700 1450 1302
1972 1852 1778 1540 1422 1299
2600 2471 2259 2009 1676 1419
2520 2471 2295 1902 1760 1539
2174 2075 1890 1708 1547 1411
Fig. 16. Effect of load eccentricity e2 on post-buckling equilibrium paths (FEM).
straight lines. The point of their intersection determines the experimental buckling load. Obtained buckling loads are listed in Table 4 and in Fig. 18 for the load eccentricity e1, and in Fig. 19 for the load eccentricity e2. Comparing the buckling loads obtained by both methods based on strain measurement, one can observe that they are similar. The difference in the buckling load values does not exceed 5%. Due to a low overload, the P-εm intersection method could not be applied to all cases. The obtained post-buckling state was very short and transient, so obtained solutions were rejected. Analysing the buckling loads obtained
by the vertical tangent method (Case 3), it was found that the determined values are approx. 1.5 higher than their corresponding bifurcation load (Case 0). Hence, they are not acceptable in quantitative terms. In contrast, there is a qualitative agreement between the bifurcation loads (Case 0) and buckling loads determined by the vertical tangent method (Case 3) – one can observe a lack of sensitivity to the load eccentricity in e1 direction (Fig. 18) and high sensitivity to the load eccentricity in direction e2 (Fig. 19).
Fig. 15. Values (a) and errors (b) of buckling loads obtained by P-w2 method (Case 1), inflexion point method (Case 2) and FEM (Case 0) for load eccentricity e2. 142
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Fig. 17. Comparison of the results obtained with strain gauges T1 and T2 and approximation lines of inflexion point method, P-εm, for load eccentricity e1: (a) 0 mm, (b) 5 mm, (c) 10 mm. Table 4 Buckling loads obtained by horizontal tangent method (Case 3), inflexion point P- ɛm method (Case 4) and FEM (Case 0) for load eccentricities e1 and e2. Eccentricity e1 in mm
Buckling load in N Case 3
Case 4
Case 0
0 1 2 3 4 5 6 7 8 10
3054 3275 3196 3268 2806 3221 3281 3187 3142 2910
3027 – – – 2949 – – – – 2912
2174 2173 2171 2166 2160 2153 2144 2134 2123 2098
Eccentricity e2 in mm 1 2 3 4 5
Case 3 2742 2762 2518 2328 1871
Case 4 – – – – –
Case 0 2075 1890 1708 1547 1411
Fig. 18. Buckling loads obtained by vertical tangent method (Case 3) and FEM (Case 0) for load eccentricity e1.
4. Summary and conclusions The presence of load eccentricity means that the structure works in the deflection state from the onset of load application. The behaviour of a structure under eccentric load is identical to that of a geometrically imperfect structure under axial compression. The post-buckling equilibrium paths obtained for the tested laminated Z-column were always stable, and their rigidity decreased with increasing the eccentricity of load. A comparison of the post-buckling equilibrium paths determined by numerical and experimental methods demonstrates that they show high agreement in both quantitative and qualitative terms. The numerical results demonstrate that the effect of load eccentricity in two
Fig. 19. Buckling loads obtained by horizontal tangent method (Case 3) and FEM (Case 0) for load eccentricity e2.
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mutually perpendicular directions, e1 and e2, is varied. As for eccentricity applied in the e1 direction, the decrease in the bifurcation load with increasing the eccentricity value from 0 mm to 10 mm was insignificant and amounted to 3.5%, while with the application of eccentricity in the e2 direction in the range from 0 mm to 5 mm the bifurcation load decreased by over 50%. The approximation methods P-w2 and the inflection point method enable the calculation of lower and upper estimates of the buckling load. The first method works well for very small values of eccentricity, while the other is suitable for high values of eccentricity. It should be remembered that the P-w2 method always underestimates the buckling load and this underestimation increases rapidly with increasing the compressive load eccentricity. On the other hand, the inflection point method overestimates the buckling load, and this overestimation decreases with increasing the value of load eccentricity. The application of strain measurement did not work in this case due to a low overloading of the specimens in the experiments. The vertical tangent method and the P-εm intersection method produce close results, but both methods significantly overestimate the value of bifurcation load. Therefore, there is no quantitative agreement, however obtained results show a qualitative agreement, because in all cases the increase in load eccentricity caused a decrease in the buckling load determined by both methods.
Mater 1971:58–80. [9] Turvey GJ, Zhang YA. computational and experimental analysis of the buckling, postbuckling and initial failure of pultruded GRP columns. Comput Struct 2006;84:1527–37. [10] Wong PMH, Wang YC. An experimental study of pultruded glass fibre reinforced plastics channel columns at elevated temperatures. Compos Struct 2007;81:84–95. [11] Bloom F, Coffin D. Handbook of thin plate buckling and postbuckling. London, New York, Washington, D.C.: CHAPMAN & HALL/CRC Boca Raton; 2001. [12] Chou SM, Rhodes J. Review and compilation of experimental results on thin-walled structures. Comput Struct 1997;65(1):47–67. [13] Ascione F, Lamberti M, Razaqpur G. Modifications of standard GFRP sections shape and proportions for improved stiffness and lateral-torsional stability. Compos Struct 2015;132:265–89. [14] Rozylo P, Teter A, Debski H, Wysmulski P, Falkowicz K. Experimental and numerical study of the buckling of composite profiles with open cross section under axial compression. Appl Compos Mater 2017;24:251–1264. [15] Debski H, Jonak J. Failure analysis of thin-walled composite channel section columns. Compos Struct 2015;132:567–74. [16] Debski H. Experimental investigation of post-buckling behavior of composite column with top-hat cross-section. Eksploatacja i Niezawodność – Maint Reliab 2013;15(2):106–10. [17] Debski H, Teter A. Numerical and experimental studies on the limit state of fibrereinforced composite columns with a lipped channel section under quasi-static compression. Compos Struct 2015;133:1–7. [18] Kim SC, Kim JS, Yoon HJ. Experimental and numerical investigations of mode i delamination behaviors of woven fabric composites with carbon, kevlar and their hybrid fibers. Int J Precis Eng Manuf 2011;12(2):321–9. [19] Orifici AC, Thomson RS, Degenhardt R, Kling A, Rohwer K, Bayandor J. Degradation investigation in a postbuckling composite stiffened fuselage panel. Compos Struct 2008;82:217–24. [20] Paszkiewicz M, Kubiak T. Selected problems concerning determination of the buckling load of channel section beams and columns. Thin-Walled Struct 2015;93:112–21. [21] Coan JM. Large-deflection theory for plates with small initial curvature loaded in edge compression. ASME, J Appl Mech 1951;18:143–51. [22] Venkataramaiah KR, Roorda J. Analysis of local plate buckling experimental data. 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, 45–74, 1982. [23] Koiter WT. Elastic stability and post-buckling behaviour. Proceedings of the Symposium on Nonlinear Problems. Wisconsin: Univ. of Wisconsin Press; 1963. [24] van der Heijden AMA, editor. W.T. Koiter’s Elastic Stability of Solids and Structures. Cambridge University Press; 2009. [25] Kubiak T. Postbuckling behaviour of thin-walled girders with orthotropy varying widthwise. Int. J. Solids Struct. 2001;38:4839–55. [26] Teter A. Static and dynamic interactive buckling of isotropic thin-walled closed columns with variable thickness. Thin-Walled Struct 2007;45:936–40. [27] Teter A, Kolakowski Z. Lower bound estimation of load-carrying capacity of thinwalled structures with intermediate stiffeners. Thin-Walled Struct 2001;39(8):649–69. [28] Kolakowski Z, Teter A. Coupled static and dynamic buckling modelling of thinwalled structures in elastic range review of selected problems. Acta Mechanica et Automatica 2016;10(2):141–9. [29] Teter A, Kolakowski Z. Buckling of thin-walled composite structures with intermediate stiffeners. Compos Struct 2005;69(4):421–8.
Acknowledgment The research has been conducted under the project No. UMO2015/ 19/B/ST8/02800 financed by the National Science Centre Poland. References: [1] Bazant ZP, Cedolin L. Stability of Structures. Elastic, Inelastic, Fracture and Damage Theories. UK: Oxford University Press; 2010. [2] Singer J, Arbocz J, Weller T. Buckling experiments. Experimental methods in buckling of thin-walled structure. Basic concepts, columns, beams, and plates, vol. 1 (vol. 2: 2002). New York: John Wiley & Sons Inc.; 1998. [3] Adams DF, Carlsson LA, Pipes RB. Experimental characterization of advanced composite materials. CRC Press LLC; 2003. [4] Reddy JSN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton: CRC Press; 2004. [5] Kolakowski Z. Static and dynamic interactive buckling of composite columns. J Theor Appl Mech 2009;47:177–92. [6] Parlapalli MR, Soh KC, Shu DW, Ma G. Experimental investigation of delamination buckling of stitched composite laminates. Compos A 2007;38:2024–33. [7] Taheri F, Nagaraj M, Khosravi P. Buckling response of glue-laminated columns reinforced with fiber-reinforced plastic sheets. Compos Struct 2009;88:481–90. [8] Tsai SW, Wu EM. A general theory of strength for anisotropic materials. J Compos
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