Effects of Elliptical Hole on the Correlation of Natural Frequency with Buckling Load of Basalt Laminates Composite Plates

2.1 Materials

In this study, basalt plain woven fabrics was provided by Basalt Fiber Tech company where situated in Australia. The filament diameters of basalt plain woven fabrics are about 13-20μm. Epoxy resin was supplied by Kukdo Chemical company in South Korea. Its viscosity (cps at 25^∘C) range between 11500-13500 and its mixing ratio (resin: hardener) is of 100:33. The material properties of basalt fiber as well as epoxy resin were listed in Table 1. To obtain the mechanical properties of epoxy resin, ASTM D2344/D2344M-03 [23] standard was conducted.

2.2 Testing program

In this work, five plies of BFRP were made of basalt plain woven fabrics and epoxy resin by means of a vacuum infusion process as shown in Figure 1(a). after completing the vacuum process 24 hours, the BFRP plate was kept in the temperature controlled room for two hours at the temperature of 80^∘C. Then the manufactured BFRP plate was cut to the required dimension for tensile and shear coupon test specimens by a water jet cutter. The dimension of tensile and shear test was conducted accordingly to ASTM D 3039/D 3039M-00 [24] and D 4255/D 4255M – 01 [25], respectively. Two aluminum plates of 25 mm length, 15 mm width and 1.5 mm thickness were attached on each side of the tensile specimens to ensure uniform load transfer and avoid stress concentration. To obtain the stain of specimens, strain gauges were glued at the center of the specimens for tensile and shear testing as shown in Figure 1(b) and 1(c), correspondingly.

Figure 1

(a) vacuum infusion process, (b) tensile testing and (c) shear testing

To conduct the experiment of coupon test, Universal testing machine (model AST-MHA) in Gangneung-Wonju National University with capacity of 100kN was used as shown in Figure 1(b) and 1(c). Monotonic testing was performed with the strain rate of 0.01/mm. The five specimens were conducted with the same condition of strain rate. Finally, the average results of tensile as well as shear testing were obtained from the each of five specimens.

2.3 Experimental results

The average stress-strain curve results of tensile and shear testing were plotted as illustrated in Figure 2(a) and 2(b), respectively. It was observed that the stress of the tensile specimen was increased linearly with proportional to increasing the tensile strain. This can be noted elastic region, where the deformation of the specimen returned to the original region when removing loading. Based on this tensile curve, it can be concluded that basalt fiber reinforced polymer (BFRP) is the elastic material. Elasticity module can be defined from the slop of proportional limit as described in (ASTM D 3039/D 3039M-00 [24]). Figure 2(b) was demonstrated that the stress-strain curve of the shear specimen was increasing non-linearly. It can be observed that this almost followed the bilinear curve. Firstly, it was increased rapidly and then changed to increas dramatically until failing. To obtain the shear module, it was determined in accordance with ASTMD2435/D2435M-01 [25]. To end, the elastic and shear module can be found and listed in Table 2. The mechanical properties of BFRP were used as input in numerical modeling to determine the natural frequencies as well as the correlation of natural frequencies with buckling load of laminate composite plates.

Figure 2

Stress-strain curve of: (a) tensile and (b) shear

3.1 Numerical modeling

The commercial finite element program ABAQUS was used to simulate the numerical investigation of the natural frequency of laminate composite plate as illustrated in Figure 3a. Appropriate elements were designated from the ABAQUS software library to suitably simulate the characteristic of the component of the free vibration analysis of laminate composite plate. A quadrilateral type eight-node shell elements (S8R) was sued to present the BFRP laminate composite that shell element is appropriate for modeling thin to adequately thick shell structure with membrane option. The shell element has six degrees of freedom at each node: 3 translations and other 3 rotations. 19874 quadrilateral S8R elements were used with a mesh size refine in the area of the elliptical hole as shown in Figure 3a. The analysis was carried out in plane stress. The plate was established of eight plies, each ply having a thickness of 0.125 mm.

Figure 3

(a) Meshing of the plate with elliptical hole and (b) geometric modeling

In this study, a thin rectangular plate of length (L) and width (H) with an elliptical hole was considered. The ratio between the length L = 200 mm and the width W = 100 mm of the plate is L/W = 2 and thickness t = 1 mm. The material properties of BFRP were listed in Table 2 and laid up in 8 plies. In application of laminate composite plates, various forms of the cut-out can be implemented for design purpose. The form of the cut-out was supposed to be an elliptical cut-out in the center of the plate. Thus, the effect of different elliptical hole inclination on the natural frequency was highlighted. The elliptical hole inclination was conducted with angle of 0^∘ 15^∘, 30^∘, 45^∘, 60^∘, 75^∘ and 90^∘. When the angle of the elliptical hole (ϕ = 0^∘), it means that major axis is parallel to the xx axis and when the angle (ϕ = 90^∘), the major axis is parallel to the yy axis. The major and minor diameter of elliptical hole was represented by ‘b’ and ‘a’, respectively as demonstrated in Figure 3b. Furthermore, the effect of different elliptical hole aspect ratios (a/b) were considered. The major axis of the elliptical hole (b) is equal to 20 mm and the minor axis (a) is varied 4, 8, 12, 16, and 20 mm.

3.2 Validation of numerical modeling

In this validation, the dimension of symmetric laminate composite plates was 76 mm × 76 mm × 1.04 mm (plate aspect ratio 1) and 152 mm × 76 mm × 1.04 mm (plate aspect ratio 2) with 8 layers. Each ply thickness was taken as 0.13 mm. The plates with each aspect ratio were orientated of [0/0/30/−30]_s, [0/45/−45/90]_s and [45/−45/−45/45]_s. The present finite element analysis was used the same material properties to those used by Crawley [26]. The results of this present study were validated for plates having aspect ratios 1 and 2 as illustrated in Table 3. It can be seen that the present finite element and observed results was initiated to be reasonable, but it has some differences. This inconsistency was due to reduction values of transverse shear stiffness in the finite element model. However, a reduction of 20% in transverse shear modulus lowered frequencies less than 1%. Furthermore, the in-plane stiffness used in the finite element analysis is too high, and that the dynamic flexural modulus is lower than the static extensional modulus as explained by Pingulkar and Suresha [27].

4.1 Effect of stacking sequence on the natural frequency

To illustrate the effect of the fiber orientation angle on the natural frequency of symmetrical laminate composite plates with regards to the elliptical hole as shown in Figure 4, the hole geometric ratio (a/b = 0.5) with clamp supports for all edges was considered. It can be seen that the largest values of natural frequency were obtained when the major axis of elliptical hole equals to ϕ = 90^∘ regardless ply angle. And the composite plates with elliptical hole ϕ = 0^∘ offer the smallest resistance to free vibration. Furthermore, it can be noted that the composite plate with fiber orientation angle (θ = 45^∘) provided the smallest natural frequency, while the greatest values of natural frequency were achieved for the plate with ply angle θ = 0^∘ and 90^∘. Thus, it can be noted that, to avoid reduction of the resistance to free vibration, the laminate composite plate should not be oriented at θ = 45^∘.

Figure 4

Natural frequency fiber orientation angle versus elliptical hole inclination (a/b = 0.5; BC = CCCC)

4.2 Effect of hole geometric ratio on the natural frequency

The effect of the hole geometric ratio (a/b) was also studied for different stacking sequences of laminate composite plate with elliptical hole inclination ϕ = 45^∘. The major axis of elliptical hole was maintained constant (b = 20 mm) and minor axis (a) varied for each case. Clamp support for all edges were performed. As illustrated in Figure 5, it can be seen that the composite plate with stacking sequence [90/0/90/0]_s provided the greatest natural frequency, while the lowest natural frequency values were obtained with stacking sequence [45/−30/60/−45]_s. It can also be observed that the natural frequency increases linearly with the increasing of hole aspect ratio for all stacking sequences. Although cut-out can reduce stiffness of composite plate, the reduced mass of the plate was more prominent which resulted in the increment of natural frequency. Thus, it can be concluded that the bigger the composite plate cut-out, the greater the resistance of composite plate to the free vibration.

Figure 5

Effect of a/b ratio on the variation of natural frequency (ϕ = 45; BC = CCCC)

The influence of the elliptical hole inclination on the variation of natural frequency was illustrated in Figure 6 for several geometric ratios (a/b). The laminate composite plate studied in this case was symmetrical of type [90/0/90/0]_s. Figure 6 shows an exponential increase of natural frequency for the increment of elliptical hole inclination of the laminate composite plate for the geometric ratios (a/b) smaller than 1.0. For cases where ϕ < 60^∘, it can be noted that the maximum values of natural frequency were obtained for the composite plate with geometric ratio (a/b = 1.0). While, the composite plate with geometric ratio (a/b = 0.2) yields the minimum values of natural frequencies. In contrast, for cases where ϕ ≥ 60^∘, it can be seen that the composite plate with the greatest resistance to free vibration were obtained, when a/b = 0.2. While the composite plate with geometric ratio (a/b = 1) provided the smallest values of natural frequencies. Additionally, according to Figure 6, it can be concluded that the natural frequency was less affected by the hole inclination when the geometric ratio getting bigger.

Figure 6

Effect of elliptical hole inclination on the variation of natural frequency (BC = CCCC; θ = [90/0/90/0]s)

4.3 Effect of the hole position

Figure 7 shows the natural frequency curve when the position of the center of the elliptical hole changes with regards to the free vibration. The effect of fiber orientation angle with the geometric ratio (a/b = 0.5) was highlighted. The presence of elliptical hole inclination ϕ ≤ 45^∘ was considered. Clamp support on one edge and three free edge (CFFF) was studied. Figure 7 shows that the maximum values of natural frequency were obtained when the composite plate had a stacking sequence [90/0/90/0]_s. While the plate with ply angle [45/−30/60/−45]_s provided the minimum values of natural frequencies. It can also be seen that the natural frequency increases linearly when the center of elliptical hole position farther away from the support. This was due to the decrement of the composite plate stiffness when transferring vibration to the support when cut-out was closer to the support. Therefore, it can be concluded that, to avoid obtaining lower resistance to free vibration, the laminate composite plate shouldn’t be cut-out near the support.

Figure 7

Effect of hole position on the natural frequency (a/b = 0.5; ϕ = 45; BC = CFFF)

5.1 Mode shapes under free vibration and buckling load of laminate composite plates

Figure 8 shows the mode shapes of free vibration and buckling load (bi-axial load). The symmetric laminate composite plate with stacking sequence [0/0/0/0]_s was studied. Also, the geometric ratio (a/b = 0.6) and clamp support for all edges was considered. Five mode shapes of free vibration and buckling load were investigated. Based on Figure 8, it is interesting to observed that mode shapes of composite plate under free vibration were similar to those of buckling load, for mode shapes I, II and III. However, those mode shapes were different for the mode shapes IV and V.

Figure 8

Mode shapes of free vibration and buckling load

5.2 Effect of elliptical hole inclination

The correlation of free vibration and buckling load of the composite plate with clamp support on all edges were studied. The effects of elliptical hole inclination and fiber orientation angle were considered. The correlation curves were determined based on the plotted of data points of various fiber angles. With the presence of hole cut-out, four cases (a/b = 0.2, 0.4, 0.6 and 0.8) were investigated. As shown in Figure 9 and 10, the correlation coefficient (r) of natural frequency and buckling load had a negative value. This negative correlation coefficient showed the inverse proportion between natural frequency and buckling load, i.e. natural frequency decrease as the buckling load increased. Moreover, it was illustrated that the composite plate with elliptical hole inclination ϕ = 0^∘ had the strongest correlation between natural frequency and buckling load, while the lowest was obtained from the plate with elliptical hole inclination ϕ = 90^∘ regardless geometric ratio. Thus, it can be concluded that elliptical hole inclination as well as fiber orientation angle induced the inverse proportion between natural frequency and buckling load.

Figure 9

Correlation of natural frequency and buckling load: (a) a/b = 0.2 and (b) a/b = 0.4

According to Figure 9(a) and 9(b), it can also be seen that each correlation curve was widely separated from each other. This was due to the significant effect of elliptical hole inclination on the laminate composite plate that caused a large gap of buckling load. In contrast, the small gap between each curve was obtained when the composite plates had the geometric ratio greater than 0.6 as illustrated in Figure 10(a) and 10(b). Thus, it can be noted that when the geometric ratio a/b ≥ 0.6, the curves were close to each other. From these curves, it was also observed that the plate with elliptical hole inclination parallel to the width of the plate provided the maximum values for both natural frequency and buckling load, while the minimum values were obtained for the plates with hole inclination perpendicular to width of the plate.

Figure 10

Correlation of natural frequency with buckling load: (a) a/b = 0.6 and (b) = 0.8