## 1.INTRODUCTION

Thin steel plates exist in many structures, such as deck and bottom of ship structures, plate and box girders of bridges, platforms of offshore structures, and structures used in aerospace industries. Sometimes depending on requirements, it is needed to make holes in plates. The presence of holes in plates will change the strength and stiffness of the plate, so the amounts of stress and their distribution which induce the change in the load bearing capacity of the plate, which might be different from plates without holes (Choi *et al*. 2012; Cristopher *et al*. 2009).

When a plate element is subjected to direct compression, the plate may buckle locally before the member as a whole becomes unstable or before the yield stress of the material is reached. When a plate has a hole, the compressive stress arises near the hole, and the stress may cause local buckling of the plate (Gerard *et al*. 1957). Local buckling behavior is characterized by distortion of the cross-section of the member (Hong *et al*. 1993). The theoretical or elastic critical local bucking load is not a satisfactory basis for design purpose of plates. Ultimate strength of plates may be less than the critical local buckling load due to yielding, or may exceed the critical local buckling loads(Jodoin *et al*. 2012).

As an alternative several studies investigated the buckling of plates with or without holes from various aspects. Husam Al Qablan *et al*. (2009) investigated the buckling behavior of composite plates with hole subjected to in-plane shear. The results showed that, in the case of shear loading compared to uniaxial and biaxial compression, there is significant reduction in the buckling load by increasing the size of holes. For relatively small size cutouts, a better performance has been achieved if the cutout was kept close to the edge of the plate, however, for a relatively large size of cutout, a higher buckling load was achieved if the cutout was kept in the middle of the plate.

Khaled M. El-sawy and Aly S.Nazmy (2001), through their study, assessed the effects of aspect ratio (ratio of length ‘a’ to width ‘b’) on the plate buckling load. Two shapes of perforation, circular and rectangular with curved corners, with various location of center of the hole are considered to evaluate the buckling load. Results showed that the use of a rectangular hole, with curved corners, with its short dimension positioned along the longitudinal direction of the plate is a better option than using a circular hole, from the plate stability point of view.

This study deals with numerical analyses of buckling of plates having two holes from the viewpoint of plate thickness, to find the buckling coefficients. Finite element method using ABAQUS is applied to perform analysis and to investigate the plate buckling. The need for approximate methods arises from the fact that exact solutions can be found only for a limited number of buckling problems of practical importance (Cristopher *et al*. 2009).

To analyze the effects of thickness of the plate buckling, the thickness variation from 0.5mm to 10.0mm is considered while the holes having the diameter of 20mm exist in the plate. Besides, the buckling of plate without holes is also investigated to provide a means of comparison between two states of plates with and without holes.

## 2.BUCKLING COEFFICIENT

The general buckling stress equation is defined as follows(Tom 1993):

Where is the buckling coefficient, the elastic Poisson’s ratio, t the thickness of plate, ‘a’ the length of the loaded edge in uniaxial compression, and E is the modulus of elasticity of plate material.

## 3.MODELING OF PLATES

This study investigates the effects of thickness on buckling of plate with through-thickness holes to find buckling coefficient, Kc, for those kinds of plates. Because of multi-functional application of these plates, two holes of diameter of 20mm are considered in the plates to be simulated using ABAQUS from buckling point of view. The plate materials are considered to be of steel with Poisson’s ratio of 0.3, modulus of elasticity of 209GPa and yield stress of 400MPa.

### 3.1.Plate with Different Thickness

In order to investigate the effects of thickness on buckling of plates having through-thickness holes, a range of various thicknesses from 0.5mm to 10mm is considered. For thin plates, the values of thicknesses in the range between 0.5mm and 2.5mm vary by the value of 0.25 consecutively. For plates having thicknesses from 3.0mm to 10.0mm, thicknesses vary by 1.0mm. Left edge of the plate is constrained for displacement in all directions while for other edges only the displacement in direction 3, out of plane direction, is constrained.

The centers of left and right holes of diameter of 20.0mm are located at coordinates of x=-50.0mm, y=0.0mm and x=50.0mm, y=0.0mm, respectively. A distributed load of magnitude of 1.0N/m is applied in the negative x-direction on the right edge of the plate (Fig. 1). In order to mesh the plate, quadratic 8-node second order plate finite element is employed. This element uses the reduced integration to improve the solution and to eliminate the locking phenomenon. Approximate global size of seeds, determining the density of mesh, is considered to have the value of 0.02 (Fig. 2).

In order to provide a means of comparing the buckling stresses of plates having holes with those of plate without holes as well as making comparison between buckling coefficients of plate with and without holes, this study also analyzed the plates without holes having same dimension and properties with those having holes. By having results of these simulations, it is possible to have a better insight into effects of perforating holes on the plate buckling.

## 4.NUMERICAL INVESTIGATIONS

The buckling analysis of plate with through-thickness holes are performed to find the critical load of such plates for specific condition from which criteria for designing the size of hole which are implemented in plates as well as the dimensions of corresponding plate, for desired applying load, can be determined. A dimensionless parameter, a kind of aspect ratio, is presented here to generalize the analysis results. The ratio of diameter of the hole to the width of plate, α =d/a, is to be called hole aspect ratio for which the critical buckling loads for different situations will be obtained based on finite element analyses.

### 4.1.Effect of Plate Thickness and Corresponding Buckling Coefficients

Analyzing the effects of thickness on buckling of plate having through-thickness holes was undertaken to find the range of permissible applying load in terms of the plate thickness. Critical loads and critical stresses of plates with holes, with respect to different values of thickness, as well as those for plates without holes are provided in Table 1.

As given in Table 1, as the thickness of the plate increases, the buckling load increases as it may already expected. However, the trend is nonlinear because the buckling load is proportional to t^{3} approximately while the critical stress is a function of t^{2}. If we look into the results, we can find a logical relationship between obtained critical loads for each case and the amount of increase in thickness. It can be written as Eq. (2):

The relationship between critical stresses and thicknesses of the plates can be written as Eq. (3):

By having buckling coefficient for a plate of thickness of , it is possible to approximately calculate the critical load of any desired plate having the thickness of. It can provide designers a quick view about the desired plate, i.e., they can quickly find out the load carrying capacity of a desired plate.

Buckling loads of plates with and without holes as well as the ratio of critical loads of plate with holes to the plate without holes are provided in Fig. 3. The results denoted with triangle symbols shows the changes in ratio of buckling loads for plate with holes to that of plate without holes. As can be seen for thin plates, both plates provide buckling loads very close to each other but by increasing the thickness the differences become more considerable as can be seen by the line with triangle symbols. The minimum amount occurred for the plates having the thicknesses in the range from 5mm to 8mm for which the maximum decrease in buckling load due to the existence of hole is occurred i.e., the maximum difference between buckling loads of plates with and without holes happened.

The results for critical stresses are provided in Fig. 4. The stresses for both plates with and without holes having the thicknesses from 0.5mm to 5mm are close to each other but the differences in critical stresses are getting increased for plates having thicknesses from 5mm to 10mm. The results for critical stress ratios state that the existence of holes in plate has more effect on plates having the thicknesses in the range from 5mm to 8mm than the other plates.

By considering the results given in Table 1, obtained from finite element analyses, the buckling coefficients, Kc, can be calculated as given in Table 2. Buckling coefficients, K_{c}, given in Table 2 correspond to *α*=0.2. The given results for buckling coefficients show that as the thickness increases the buckling coefficients decreases for both plates with and without holes. Plates with holes provide smaller buckling coefficients having ratio of approximately 0.91.

## 5.DEMONSTRATION OF THE GENERALITY OF BUCKLING COEFFICIENTS

In order to investigate the generality of buckling coefficients proposed, we analyze plates in different dimensions having width of 200mm and length of 400mm and having *α* of 0.2. Two thicknesses of 1mm and 2mm corresponding to thin plates and other two of 7mm and 10mm corresponding to thick plates are considered. By using ABAQUS, the critical stresses are calculated. Then the theoretical critical stresses are calculated by using the presented buckling coefficients in Table 2 and Eq. (1). Thereafter the results obtained from ABAQUS models and those calculated by buckling coefficients are compared and the errors are calculated. The results are provided in Table. 3.

By taking look into the results it can be found that the buckling coefficients obtained in this study can be applied to plates having different dimensions but having the same ratio (*α*). The last column of Table 3 shows the differences between buckling stresses calculated using Kc and those obtained from FEM analyses. As can be seen in the table errors are small, and good agreement is observed for the results obtained from both FEM and proposed method.

## 6.CONCLUSIONS

This study dealt with analyzing buckling of plate with through-thickness holes under various conditions to find buckling coefficients, by which the buckling stresses of plates are calculated. To clarify the trend of buckling of plate several results are presented: the trends of buckling load and critical stress with respect to the thickness of plate are provided. They show that by increasing the thickness of plate both buckling load and critical stress increase. The results for plates without holes are also provided as well as the ratio of buckling load and critical stress of plates with holes to those of plate without holes. Results showed that buckling loads of plates with and without through-thickness holes corresponding to thin plates are close to each other and by increasing in thickness the differences are getting increased. The generality of buckling coefficients is demonstrated with plates in different dimensions.