Experimental and finite element studies on buckling of skew plates under uniaxial compression

2.1 Experimental procedure

Several procedures have been used by different investigators to evaluate the critical buckling load of rectangular plates [42]. These are depicted in Figure 1. The procedures use applied load vs. deflection, applied load vs. end shortening, and applied load vs. strain plots. In the present study, five different methods are used, which are designated as Method I, Method II, etc. Method I employs a plot of applied load (P) vs. out-of-plane deflection (W) at midspan. Method II employs a plot of applied load (P) vs. end shortening in the direction of applied load (Δ). Method III employs a plot of applied load (P) vs. square of out-of-plane deflection (W²). Method IV uses a plot of applied load (P) versus average in-plane strain in the direction of load. Method V uses a plot of applied load (P) vs. surface strain difference at midspan in the direction of load. Method IV utilizes the fact that the surface strain on one side of the specimen becomes tensile when the specimen has buckled. In this method, the applied load is plotted against the algebraic mean strain, ε_A=(ε₁+ε₂)/2, where ε₁ and ε₂ are strains at the two surfaces of the specimen at midspan in the direction of loading. Method V employs a plot applied load vs. strain difference ε_D=(ε₁-ε₂).

Figure 1

Methods used to determine critical buckling load.

2.2 Finite element solution

A classical linear buckling analysis was made using MSC/NASTRAN software. CQUAD8 (eight-node isoparametric curved shell element with 6 degree of freedom per node) was employed in the present study. The CQUAD8 element has been preferred to CQUAD4 element in the present study as the former is more accurate as revealed by the investigation reported in paper [43].

3.1 Test specimens

For isotropic plate, the aluminum 7075-T6 material supplied by the Rio-Tinto Alcon, Canada was used. The composite plate specimens were made using unidirectional glass fiber, epoxy-556 resin, hardener (HY951), and polyvinyl alcohol (releasing agent) supplied by Ciba Geigy India Ltd. The specimens were fabricated by hand lay-up technique. The fiber weight percentage is 50:50. The appropriate ASTM procedures were followed while preparing the test specimens. At least three replicate specimens were tested, and the result was taken as the average of the tested specimens. The plate specimens were prepared carefully so as to avoid residual stress at the cutting edges. The tests were conducted at standard laboratory conditions of 27°C and 46% relative humidity. The material properties of the test plates are: for aluminum 7075-T6, E=71.7 GPa, μ=0.33 and for glass epoxy, E₁=38.07 GPa, E₂=8.1 GPa, G₁₂=3.05 GPa, and ν₁₂=0.22. The aspect ratio was varied from 1.0 to 2.5.

3.2 Experimental procedure

The fixture for holding the test specimen is shown in Figure 2. The test specimen was inserted between the end plates of the fixture, and the screws were tightened properly so that no slippage of the test specimen occurs. The tests were conducted in a 40-t computerized universal testing machine after positioning properly the test specimen using universal vice as shown in Figure 3. The measuring instrumentation consists of strain gauges and LVDTs. The strain gauges were placed at each surface of the test plate at midspan. Two LVDTs were fixed symmetrically at midspan along the width of the test specimen to measure the out-of-plane deflection, and one more was fixed to the moving jaw of the universal testing machine to measure the in-plane deflection. The testing was carried out with unloaded edges completely free, and one loaded edge restrained completely, and the other loaded edge restrained except translationally in the direction of loading.

Figure 2

Fixture for holding the specimen.

Figure 3

The experimental set-up.

4.1 Isotropic skew plates

Isotropic plates made of aluminum 7075-T6 were tested under uniaxial compression, varying the skew angle from 0° to 45° and aspect ratio from 1.0 to 2.5. The experimental values of the critical buckling load were determined in accordance with the Methods I to V. Classical linear buckling analysis was performed using MSC/NASTRAN and the finite element solution for the critical buckling load obtained. A typical plot of applied load (P) vs. out-of-plane deflection (W) for a skew angle of 15° and aspect ratio ranging from 1.0 to 2.5 is shown in Figure 4. Figure 5 shows a typical buckled shape of the test specimen. The values of the critical buckling load obtained are tabulated in Table 1 and presented in the form of a bar chart in Figure 6. The standard deviations are given within parentheses in Table 1 and are also indicated in Figure 6. The following are observed from Table 1 and Figure 6:

• Method IV yields the highest experimental value for critical buckling load, and Method III yields the lowest value. The experimental values are in good agreement with the finite element solution, the values given by Method IV being closest to the finite element solution. The percentage of discrepancy between the finite element solution and Method IV is very small and may be neglected for practical purposes.

• For a particular skew angle, the critical buckling load decreases as the aspect ratio increases. The rate of decrease is initially large and becomes smaller for higher values of aspect ratio.

• For a particular aspect ratio, the critical buckling load is observed to increase with the skew angle. The rate of increase is initially small and becomes larger for higher values of skew angle.

Figure 4

A typical plot of applied load (p) vs. out-of-plane deflection (w_center) for isotropic skew plate (α=15°).

Figure 5

A typical buckled shape of the tested specimen (α=15°, a/b=2.0).

Figure 6

Critical buckling load for isotropic skew plate (aluminum 7075-T6).

4.2 Laminated composite skew plates

Laminated composite skew plates were tested in uniaxial compression, varying the skew angle from 0° to 45°, aspect ratio from 1.0 to 2.5, and stacking sequence. The number of layers was kept constant at 20, and the total thickness of the laminate was 2.0 mm. Four laminate stacking sequences viz., antisymmetric angle-ply [+0°/-0°/…/-0°], antisymmetric angle-ply [+45°/-45°/…/-45°], antisymmetric angle-ply [+90°/-90°/…/-90°], and antisymmetric cross-ply [0°/90°/…/90°] were considered. The experimental values of the critical buckling load were determined according to Methods I to V. Classical linear buckling analysis was performed and the finite element solution for the critical buckling load determined. The values of the critical buckling load for various values of skew angle=0°, 15°, 30°, and 45° and aspect ratio are tabulated in Tables 2–5 and plotted in the form of bar chart in Figures 7–10 for antisymmetric angle-ply [+0°/-0°/…/-0°], antisymmetric angle-ply [+45°/-45°/…/-45°], antisymmetric angle-ply [+90°/-90°/…/-90°], and antisymmetric cross-ply [0°/90°/…/90°]. The standard deviations are presented in Tables 2–5 and indicated in Figures 7–10. The following are observed from Tables 2–5 and Figures 7–10.

Figure 7

Critical buckling load for laminated skew plate (α=0°).

Figure 8

Critical buckling load for laminated skew plate (α=15°).

Figure 9

Critical buckling load for laminated skew plate (α=30°).

Figure 10

Critical buckling load for laminated skew plate (α=45°).

• Method IV yields the highest experimental value for critical buckling load, and Method III yields the lowest experimental value. The experimental values are in good agreement with the finite element solution, the value given by Method IV being closest to the finite element solution. The percentage of discrepancy between the finite element solution and Method IV is about 10–15%.

• The critical buckling load is observed to decrease sharply as the aspect ratio increases from 1.0 to 2.5 for all the stacking sequences.

• The critical buckling load has the highest value for antisymmetric angle-ply [+0°/-0°/…/-0°] and the lowest for antisymmetric angle-ply [+90°/-90°/…/ -90°]. The critical buckling load depends on the stiffness of the cross-section of the laminate among other factors such as boundary conditions, etc. The stiffness of the cross-section depends upon the contribution made by extensional stiffness, coupling stiffness, and bending stiffness terms [44]. The stiffness is maximum when all the plies have 0° fiber orientation and minimum when all the plies have 90° fiber orientation with respect to the direction of loading. The critical buckling loads for the other stacking sequences lie between the previously mentioned maximum and minimum values.

• The critical buckling load for any stacking sequence increases as the skew angle increases and becomes maximum when the skew angle is 45°.