Hole effects on lateral buckling of laminated cantilever beams

Composites: Part B 40 (2009) 174–179

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Hole effects on lateral buckling of laminated cantilever beams Ersin Eryig˘it a, Mehmet Zor b, Yusuf Arman b,* a b

Turkish Air Force, NCO College Command, Gaziemir, Izmir, Turkey Dokuz Eylul University, Department of Mechanical Engineering, Bornova, Izmir, Turkey

a r t i c l e

i n f o

Article history: Received 7 September 2007 Received in revised form 29 July 2008 Accepted 31 July 2008 Available online 6 December 2008 Keywords: A. Layered structures B. Buckling C. Finite element analysis (FEA)

a b s t r a c t In this study, the effects of hole diameter and hole location on the lateral buckling behaviour of woven fabric laminated composite cantilever beams have been investigated. In the experimental studies, two different groups of samples were used; samples with a single circular hole and samples with no hole. The critical buckling load for each sample was then determined experimentally. For the numerical analyses, ANSYS 10.0 finite element program was utilized. It has been noted that there is a good agreement between experimental results and those of finite element analyses. On the basis of this harmony, the numerical analyses of some models having different dimensions and fiber orientations have been done by changing length and width of the beam, diameter and location of the hole. It has been concluded that the effects of the hole diameter and hole location on the lateral buckling behaviours is very important, especially for the short beams. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Composites have many desirable characteristics such as high ratio of stiffness and strength to weight, corrosion resistance and magnetic transparency. Composite beam members are widely used as a structural component in many types of structures within the fields of engineering. However, it is well known that thin composite beams are very susceptible to buckling in various modes, depending on the geometry of the cross-section, the material properties, and the boundary and loading conditions. When loaded in its plane of symmetry, the beam initially deflects. However, at a certain level of the applied load, the beam may buckle laterally while its cross-section rotates simultaneously about the beam’s axis. This phenomenon is called lateral buckling and the value of the load, at which buckling occurs, is the critical load. Therefore, the accurate prediction of the stability limit state is of fundamental importance in the design of these beams. Accordingly, many research activities have been conducted towards the development of theoretical and computational methods for the buckling analysis of thin beams. Most of the works concerning the lateral stability of these members focus on thin-walled composite beams. Several studies regarding lateral buckling of thin-walled beams have been done by using a linearized approach based on Vlasov’s theory. For example, buckling loads were determined for thin-walled beams made of composite [1,2] and metallic materials [3–6].

* Corresponding author. Tel.: +90 232 388 31 38; fax: +90 232 388 78 68. E-mail address: [email protected] (Y. Arman). 1359-8368/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2008.07.005

Bauld and Tzeng [7] presented a Vlasov-type theory for thinwalled fiber-reinforced beams with open cross-sections having midplane symmetry. They attempted to predict the lateral buckling capacity of beams including the influence of load forms, boundary conditions, and stacking sequences. Brooks and his collaborator conducted a series of lateral buckling tests on pultruded GRP I-section cantilever beams [8] and compared their results with simple theoretical ones [9]. Lee and Kim [10] presented an analytical model which accounts for flexural-torsional buckling of I-section composite beams. The model was capable of predicting accurate buckling loads and modes for various configurations. A general analytical model applicable to the lateral buckling of an I-section composite beam subjected to various types of loadings is developed by Lee et al. [2]. Kabir and Sherbourne [11] proposed an analytical solution to predict the lateral buckling capacity of composite channel-section beams. Lee [12] gave a closed-form expression for the location of center of gravity and shear center as a function of lamination stacking sequence as well as sectional properties. A few closed-form solutions have been obtained for critical loads considering the prebuckling deflections of the beam [13–16]. The effects of the prebuckling displacements as well as the shear deformation on the lateral buckling of bisymmetric thin-walled composite beams subjected to concentrated end moments, concentrated forces or uniformly distributed loads have been investigated numerically.[17]. They have considered simply supported and cantilever beams. Although composite beams having rectangular cross-sections have widespread application fields, there are not sufficient

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investigations about their lateral buckling behaviours. Experimental measurement and prediction of lateral-torsional buckling loads of composite woods having a rectangular cross-section has been investigated by Hindman et al. [18]. Turvey [19] presented the results of a series of tests on rectangular cross-section pultruded GRP cantilever beams, which are loaded in their stiffer plane of symmetry by means of a single point load acting through the centroid of the cross-section at the free end. Hodges and Peters [20] presented an analytical formula for the lateral-torsional buckling load of composite rectangular section and I-beams applying asymptotic (or perturbation) method and considering various refinements of previously published results, including the Vlasov effect, elastic coupling, the offset of the load from the centroid and the effect of the in-plane lateral deflection. Since one of the factors affecting dynamic stability zones is the critical buckling load, various solution methods have been developed for both analytical and finite element solutions of the mentioned problem [21–23]. In order to investigate static and dynamic stabilities of laminated composite cantilever beams having rectangular cross-sections, subjected to vertical end loading, Karaag˘aç et al. [24], first determined the critical buckling loads by using the finite elements method. They compared their results with those of the ANSYS, the-

P 30°

d

175

oretical [25] and analytical solutions [20]. It is shown that there are perfect harmonies among the results of these four studies. According to the application of the constructions, some holes may be made in the beams (cellular beams). It is clear that the effects of the sectional discontinuities, caused by the holes, on the beam strength depend on the dimensions, locations, shapes and numbers of the holes. There are some investigations about lateral buckling of the cellular beams made of isotropic materials in literature [23,26 and 27]. Lawson et al. [28] have examined lateral buckling behaviours of I-section composite cellular beams, subjected to uniform distributed loads, by using both finite element analyses and experimental studies. In the present study, lateral buckling behaviour of woven fabric laminated composite cantilever beams having a rectangular crosssection and including a single hole is examined. Firstly, test specimens with/without holes, for different stacking sequences, were prepared. Test specimens made of glass–epoxy were of eight plies. The detailed information regarding production of test samples can be found in another study of authors [29]. Single vertical load was acted at the free end of the cantilever beam specimens and critical buckling loads were then determined experimentally. After obtaining harmony between finite element results and experimental ones, further finite element analyses for other models having differences in terms of hole diameter and locations of holes were carried out. The results obtained are compared for each stacking sequence, i.e., [(0/90)4]s, [(30/ 60)4]s, [(±45)4]s, and some recommendations are made for each examined situation in the subsequent sections.

b

2. Materials and methods

-60°

a

t=1.6 mm L Fig. 1. The dimensions of the cantilever composite beam.

Table 1 Mechanical properties of composite beam [29]. E1 = E2 (MPa)

E3 = 0.6E1 (MPa)

m12

m13 = m23 = 0.6m12

G12 (MPa)

G13 = G23 = G12 (MPa)

27,000

16,200

0.15

0.09

7540

7540

The present study consists of two stages: experimental studies and finite element analyses. 2.1. Experimental study Critical lateral buckling loads of some woven composite beam specimens were obtained experimentally and then compared with those of finite element analyses. Test samples (Fig. 1) were prepared from the glass–epoxy laminated composite plates. The dimensions of the samples were 150  20  1.6 mm. Their mechanical properties are given in Table 1. For each fiber

Test Machine

Load Cell

Test Fixture

Specimen

Clamp

Fig. 2. The lateral buckling test apparatus for one-edge cantilevered boundary condition.

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E. Eryig˘it et al. / Composites: Part B 40 (2009) 174–179 Table 2 Comparison of the experimental and finite element results. Fiber orientations

Critical buckling loads (N) P cr (without hole)

[(0/90)4]s [(30/ 60)4]s [(±45)4]s

Pcr (with hole)

Experimental

FEM

Experimental

FEM

28.29 32.09 30.44

27.875 33.421 30.56

25.29 29.95 28.57

26.72 30.98 28.16

2.2. Finite element analyses

Fig. 3. A view from lateral buckling test process.

Firstly, two-dimensional models of the experiment samples were established and buckling analyses were then carried out by using ANSYS 10.0. The results obtained are shown in Table 2. Rectangular layered elements with eight nodes and six degrees of freedom were used in the models. Afterwards, the effect of beam dimensions (b, L), hole diameter (d) and hole location (a), see Fig. 1, on the lateral buckling behaviour was investigated by finite element analyses. The thickness (t) of all models was 1.6 mm. For the same dimensions of beam, Pcr stands for the critical buckling loads of beams with no hole while Pcr for the beams with a hole in the paper. For all cases examined, Pcr and Pcr values have been calculated and shown in Table 3. In Figs. 5–7, variation of the ratio Pcr/P cr versus diverse b/L ratios is presented for three different stacking sequences. 3. Results and discussions

Fig. 4. Determination of the experimental critical buckling point on the graphic.

orientation, i.e., [(0/90)4]s, [(30/ 60)4]s and [(±45)4]s, six test samples were prepared; half of them had a central circular hole (a = 75 mm) of 10 mm in diameter. In total, 18 specimens were used in tests. The buckling tests were done by using a universal tensile testing machine, Shimadzu AG-100 kNG. During the tests, an inhouse buckling test apparatus was used. The test apparatus (Fig. 2) was designed to ensure the accurate cantilever boundary condition for one edge of the beam. Single vertical load was acted at the free end of the beams. During the tests, all specimens were loaded laterally until the first buckling mode was reached, as shown in Fig. 3. Since the laminated composite beams become unstable as the first buckling mode takes place, other modes are not considered in this work. The load–displacement (P–d) graphics were constituted for each specimen. How critical lateral buckling loads were determined from P–d curves is illustrated in Fig. 4. The test results are compared with those of the FEA in Table 2.

As shown in Table 2, there seem negligible differences between the experimental and the finite elements (ANSYS) results. Therefore, it can be said that the solution strategy followed during finite element analyses were correct and results obtained can be confidently. The results given in Table 3 show that the critical lateral buckling load goings the highest values for beams having [(30/ 60)4]s fiber orientations, and as the buckling loads decreases significantly as the beam length (L) increases. Furthermore, the increase of the hole diameter also causes some decrease to some extent in the critical lateral buckling loads. Some important discussions drawn from Figs. 5–7 are summarized in below. When the location of the circular hole becomes closer to the clamped left edge of the beam, the ratio Pcr/Pcr decreases. Also the effect of the circular hole diameter on the lateral buckling load decreases with the decrease of hole diameter. Namely, when the hole location is unchanged, the effect of the holes near clamped edge on the critical buckling load decreases as the beam length increases and hole diameter decreases. In the figures, it is also seen that critical lateral buckling loads show important decreases from stacking sequence [(0/90)4]s, to [(30/ 60)4]s and from [(30/ 60)4]s to [(±45)4]. Pcr/Pcr values are close to 1 for all hole diameters and fiber orientations when the hole approaches to the right free end of the beam. It can be concluded that the effect of the hole close to the free end of beam on the buckling loads is negligible. Although the cantilever beams with [(30/ 60)4]s fiber orientation have the maximum buckling loads in all cases, they seems to be the most influenced beams by the existence of the holes. However, the existence of the hole has fewer effects on the shorter beams having [(0/90)4]s stacks. The less influenced fiber orientation is the [(±45)4]s for greater hole diameters and beam lengths. The existence of the holes with small diameter (d/b 6 1/4) increases the buckling loads at the locations near clamped edge for [(0/90)4]s orientations and near free edge for [(±45)4]s orientations.

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Table 3 Lateral buckling loads for all cases. Beam length (mm):

L = 100

L = 200

L = 300

Fiber orientation (°): P cr (N) (with no hole): Pcr (N) (with hole) d (mm) a/l 5 1/6 2/6 3/6 4/6 5/6

[(0/90)4]s 62.69

[(30/ 60)4]s 75.2

[(±45)4]s 68.74

[(0/90)4]s 15.846

[(30/ 60)4]s 18.723

[(±45)4]s 17.16

[(0/90)4]s 7.098

[(30/ 60)4]s 8.287

[(±45)4]s 7.61

63.90 61.71 60.93 61.84 62.51

69.58 69.75 70.95 74.64 74.56

64.36 66.21 68.15 69.39 68.53

16.14 15.64 15.56 15.30 15.81

17.59 18.05 18.56 18.47 18.86

16.48 16.91 17.14 17.66 17.32

7.115 6.973 6.9853 6.8219 7.0348

7.9655 8.0317 8.2062 8.1947 8.2718

7.319 7.5364 7.5795 7.8354 7.6063

10

1/6 2/6 3/6 4/6 5/6

58.65 56.93 57.55 59.65 61.46

62.33 62.62 65.50 70.21 72.66

57.54 58.75 61.43 65.21 67.60

15.24 15.165 14.96 15.356 15.727

16.672 16.908 17.801 18.405 18.765

15.584 15.797 16.506 16.856 17.209

6.7856 6.7081 6.7798 6.8983 7.001

7.5784 7.7389 7.9297 8.1703 8.201

7.1414 7.2516 7.4216 7.5278 7.5896

15

1/6 2/6 3/6 4/6 5/6

45.73 44.10 46.59 51.44 58.26

48.26 47.89 52.44 61.19 70.80

44.73 46.13 50.71 57.94 64.85

12.771 12.731 13.406 14.311 15.531

14.346 14.581 15.729 17.187 18.489

13.723 14.191 14.862 15.956 16.834

6.0592 5.9985 6.1616 6.4579 6.9315

6.8455 6.9717 7.198 7.9031 8.1144

6.552 6.6703 7.0668 7.4069 7.4566

Fig. 5. Variation of the Pcr/P cr versus a/L for b/L = 1/5.

4. Conclusions The conclusions drawn from the study can be summarized as: The locations of the hole have important effects on the buckling loads. The holes near the clamped edge (a/L 6 2/6) seems to have the highest influence. The hole effect decreases when its location approaches to the free end of the beam. For this reason, especially for the short beams, a hole if necessary should be drilled far away from the clamped edges as much as possible in terms of buckling safety. It is possible to say for all fiber orientations and hole diameters that the existence of the hole close to the free edge does not have important influences on the lateral buckling loads of these types of cantilever beams. If the hole diameter increases, the buckling loads decrease. Towards to the clamped edge, the influence of hole diameter increases. The small holes (d/b 6 1/4) have negligible effects on the buckling loads. Therefore, their locations are not so significant in terms of lateral buckling safety. Moreover, in some cases the small holes increase the buckling loads and constitute positive influences on them. However, the increase in the number of the small holes may have negative effects on the buckling behaviours.

The increase in the beam length (L) causes reduction in the values of buckling loads. In addition, the effects of the hole on the longer beams decrease. It is seen from Table 3 that the critical buckling load values of the beams having [(30/ 60)4]s fiber placements are the highest and consequently this orientation offers safer cases than others. The critical loads for all orientations becomes close to each other with the increase in hole diameter and beam length. It can be said especially for smaller beams (b/L 6 1/5) with small hole diameters (d/b 6 1/4) that [(30/ 60)4]s stacking sequences are more advantageous than the others. However, [(0/90)4]s and [(±45)4]s orientations show more stable situations and hole effects on the buckling loads for these orientations are lower than those of the [(30/ 60)4]s orientations. The number and geometry of the holes, thickness and crosssection conditions of the beam, load and boundary conditions, stacking sequences of the layers and material properties are the other important factors affecting the lateral buckling behaviour. Furthermore, the numbers, dimensions, shapes, locations of the delaminations, occurring for some reasons, among the layers of the composite beams may cause very different influences.

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Fig. 6. Variation of the Pcr/P cr versus a/L for b/L = 1/10.

Fig. 7. Variation of the Pcr/P cr versus a/L for b/L = 1/15.

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