An extensive parametric study is conducted to evaluate the effect of the different geometric parameters on the elastic buckling of strengthened and unstrengthened plates with circular openings in order to deduce a suitable formula for the buckling coefficient (K). Effects of the different considered parameters will be exhibited in the next sections for unstrengthened and strengthened plates separately.
Unstrengthened plate with circular opening
Influence of slenderness ratio (W/t
_{p})
Several runs were performed to study the influence of plate slenderness ratio on the buckling coefficient factor (K) as illustrated in Fig. 8, where symbols OP1, OP2, and OP3 present opening diametertowidth ratios of 0.1, 0.2, and 0.3, respectively. Numbers between brackets refer to the position of the opening center with respect to the X and Y axes. Figure 7 shows that the plate slenderness ratio has no effect on the buckling factor coefficient (K_{o}). If the buckling coefficient (K_{o}) for perforated plates is compared to the ones extracted from solid plate models, it can be observed that the coefficient (K_{o}) decreases by about 1–2.5% for plates with an opening near the corner. However, for models having small diameters, the decrease in buckling coefficient can be neglected. For plates having an aspect ratio equal to 1.5 and a central opening, the buckling coefficient of perforated plates slightly increased when compared to solid plates.
Influence of aspect ratio (W/D)
As indicated by previous researchers, the aspect ratio of plates has a great effect on the buckling coefficient factor (K). Figure 8 shows that the buckling coefficient factor (K_{o}) decreases linearly when the aspect ratio is less than 1 with a high slope. However, for aspect ratios ranging between 1 and 1.5, the buckling coefficient factor (K_{o}) increases slightly then decreases again as the aspect ratio reaches to 2. It is also clear from curves that the general relationship between the buckling coefficient factor (K_{o}) and plate aspect ratio is the same for all considered slenderness ratios.
Influence of opening position in the Xdirection (OPX/W)
For a small aspect ratio equal to 0.5 and a small opening diameter, the position of the hole affects the buckling coefficient factor significantly. As shown in Fig. 9, the local buckling factor (K) decreases as the opening moves towards the center of the plate. While for aspect ratios equal to 1 till 2, the buckling coefficient factor (K_{o}) slightly changes as the position of open changes.
For the different considered aspects, the buckling coefficient factor (K_{o}) decreases as the opening gets near to the center. The decrease ranges from 4 to 10.4% for the aspect ratio equal to 0.5. Meanwhile, the decrease in buckling coefficient factor (K_{o}) ranges from 2.8 to 4.1% for the aspect ratio equal to 1.0. For the aspect ratio equal to 2, the buckling coefficient factor (K_{o}) decreases as the opening gets near to the center except for the case of openings near the plate corner. In this case, a slight increase may occur and reaches 0.5%. While the decrease ranges from 1 to 4%. This shows that for aspect ratios larger than 1.0, the effect of the opening is less pronounced than other aspect ratios.
Influence of opening position in the Ydirection (OPY/W)
The position of the opening in the Ydirection is varied for different plate aspect ratios and the influence on the local buckling factor (K_{o}) is studied. It is generally observed that the buckling factor decreases as the opening approaches the center of the plate. Meanwhile, for an aspect ratio equal to 1 through 2, the buckling coefficient factor (K_{o}) slightly changes as the position of the open changes as shown in Fig. 10.
Figure 11 shows the effect of the opening diameter on the buckling coefficient factor (K_{o}) at the same aspect ratio which was taken to be 1. The results show that the buckling coefficient factors (K) decrease as the opening diameter increases.
Strengthened plate with longitudinal stiffener
Several runs are performed while considering longitudinal stiffeners around the introduced circular opening. Two cases are considered for the location of the circular opening: (1) a circular opening moving in the Xdirection and centered in the Ydirection and (2) a circular opening moving in the Ydirection and centered in the Xdirection. These cases are considered for circular openings having OPH/W ratios ranging between 0.1 and 0.5 while considering plate aspect ratios of 0.5, 1, 1.5, and 2. Accordingly, the buckling coefficient factor for stiffened plate (K_{s}) is compared to the buckling coefficient factor for unstiffened plates with a circular opening (K_{o}) and the buckling coefficient factor of the solid plate (K_{no}).
Results are exhibited for different plate aspect ratios while comparing normalized buckling factor with respect to the unstrengthened plate with opening (K_{s}/K_{o}) and normalized buckling factor with respect to the solid plate (K_{s}/K_{no}). Figure 12 shows the relation between the normalized buckling factor coefficient (K_{s}/K_{o}) with the position of the cutout with respect to the yaxis (OPX/W) and variation of B_{s}/OPH for plate aspect ratio equal to 0.5.
It can be deduced from the curves that:

For OPH/W = 0.1, as the location of the opening in Xdirection with respect to plate width (OPX/W) changes from 0.1 to 0.5, the buckling coefficient factor ratio (K_{s}/K_{o}) increases by a percentage ranging from 101 to 108.3%. Meanwhile, for OPH/W = 0.2, the buckling coefficient factor ratio (K_{s}/K_{o}) increases by a percentage ranging from 112.5 to 137.8%. This ratio increases between 122.5–200.7% and 178.9–329.6% for the OPH/W ratio equal to 0.3 and 0.4, respectively.

The results also show that the change of stiffener breadth (B_{s}) has no effect on the buckling coefficient factor (K_{s}).
For aspect ratio 0.5, Fig. 13 shows the relation between the normalized buckling factor coefficient (K_{s}/K_{o}) with the variation of B_{s}/OPH and the position of the cutout with respect to the xaxis (OPY/D). Examining the figure, it is clear from the curves that as the OPY/D ratio increases, the buckling coefficient factor ratio (K_{s}/K_{o}) increases. The percentage of increase ranges between 104.7 and 601.7%. The maximum increase is observed when OPH/W = 0.5 and OPY/D = 0.5.
Figure 14 shows the deformed shape for stiffened and unstiffened plates. It can be seen that the horizontal stiffener changes the elastic buckling mode with respect to the number of waves. This is due to the increase in plate stiffness upon strengthening.
Figure 15 shows a comparison between stiffened perforated plate and solid plate for an aspect ratio of 1.5. It is clear from the curves that the influence of adding stiffener for OPH/W = 0.1 is different than other ratios. At this ratio, the local buckling factor increases at OPX/W = 0.1–0.3 then it starts to decrease at OPX/W = 0.4 and 0.5. For the other ratios, the buckling coefficient increases as OPX/W increases by a percentage ranging between 108 and 214.3%. In addition, it can be observed that the stiffener breadth (B_{s}) value does not have a large influence on the local buckling coefficient (K_{s}) for the different values of OPH/W. Figure 16 shows the deformed shape of stiffened and unstiffened plates as the opening changes its location in the Xdirection. When the plate aspect ratio exceeds 1.0, the plate deforms in two half waves. Adding a circular opening near the edge or at the middle of these plates has relatively no effect on the plate deformed shape. For other locations of openings, it is observed that deformation of the part including the opening is larger than the one in the other half that does not include an opening. When adding stiffeners parallel to the load direction, a decrease in deformations is generally observed. Increasing the opening diameter gradually changes the deformed shape to a single wave which explains the decrease of the local buckling coefficient for stiffened plates.
Strengthened plate with transversal stiffener
Several runs are performed while considering transversal stiffeners around the introduced circular opening. Two cases are considered for the location of the circular opening: (1) a circular opening moving in the Xdirection and centered in the Ydirection and (2) a circular opening moving in the Ydirection and centered in the Xdirection. These cases are considered for circular openings having OPH/W ratios ranging between 0.1 and 0.5 while considering plate aspect ratios of 0.5, 1, 1.5, and 2. Accordingly, the buckling coefficient factor for stiffened plate (K_{s}) is compared to the buckling coefficient factor for unstiffened plates with a circular opening (K_{o}) and the buckling coefficient factor of the solid plate (K_{no}).
Strengthened plate using box stiffener
Combining longitudinal and transversal stiffeners, a box stiffener is introduced around the opening. Several runs were made on plates with an aspect ratio of 1.0 only to compare the effect of the box stiffener with each case separately. Figures 18 and 19 show results compared to unstiffened opening and opening stiffened with longitudinal and transversal stiffeners, respectively.
Examining Fig. 17, it is clear that the box stiffener strengthens the plate. When compared to longitudinal stiffener, using the box stiffener has a small effect for OPH/W = 0.1–0.3. However, its effect increases clearly for OPH/W = 0.4 and 0.5. For OPX/W = 0.3, strength increases by about 40%; meanwhile, it decreases by 20% when OPX/W = 0.5. This can be attributed to the fact that the box stiffener is totally inside the maximum deformed zone and has no effect on the deformed shape. When comparing box stiffener to transversal stiffener, the influence can be neglected when OPH/W = 0.1–0.3. However, strength increases by a percentage of 90% when OPH/W = 0.5 and OPX/W = 0.3. Figure 18 shows the comparison between the box stiffener and the transversal stiffener.
It can be seen that using the box stiffener strength doubles the elastic buckling strength when compared to a longitudinal stiffener for OPH/D = 0.5 and OPY/D = 0.3. However, for OPH/D = 0.1–0.3, the box stiffener strength provides the same strength of transversal stiffeners. Considering Figs. 19 and 20, it is clear how the deformed shape changes with the change of stiffener type and that explains the trend of curves above.