In many structural components such as beams, columns or plates, failure develops not only from excessive stresses but also from buckling. The buckling behavior of plates subjected to in-plane loads is an important aspect in the preliminary design of any structural component. The knowledge of critical buckling loads, mode shapes is vital for reliable structural design. There is a necessity for providing the holes of various shapes in the plates for various purposes. In this paper, the buckling behavior of rectangular plates with holes of various shapes is investigated. The finite element modeling and buckling analysis of the plates has been carried out using ANSYS 14.5. Plates with three varieties of shapes of holes i.e., Triangular, Square and Rhombus have been studied. The plates are investigated for different positions of the holes (i.e. Center, Top) and for various a/b ratios ranging from 2 to 12 at an increment of 2 and for various b/t ratios ranging from 20 to 100 at an increment of 20. Finally, buckling loads of the plates with holes is compared wrt. shape of the hole, location of the hole and some useful conclusions are drawn.
The buckling characteristics of structures are often considered with less significance than the static and dynamic response. When structural engineers carry out finite element analysis of plates in most cases, they are primarily interested in stress at various points and displacements rather than buckling. However the buckling analysis of plate structures is of a sufficient practical important class of problems in literature. The knowledge of the buckling behavior of plate structures is vital for a sound design.
Buckling loads are the critical loads, where the structures become unstable. Each load has an associated Buckled mode shape. Methods of performing buckling analysis are primarily are of two types. (i.e. Eigen buckling analysis and Nonlinear buckling analysis). Eigen buckling analysis computes eigenvalues for a given system of loading and constraints. A nonlinear buckling analysis is carried out using the standard geometric nonlinear solver. In this method load is applied incrementally, from zero up towards the maximum.
Hsuan-The Hu, and Jiing-Sen Yang (1999) studied the buckling resistance of fiber-reinforced laminated cylindrical panels with a given material system and subjected to uniaxial compressive force is maximized with respect to fiber orientations by using a sequential linear programming method.
Abdulkareem Al Humdany and Emad Q. Hussein (2012) studied the buckling behavior of antisymmetrically angle ply laminated composite plate under uniaxial compression using eigen value buckling analysis. It is found that, in design of an antisymmetric laminate; the thickness is not only the element which must be determined, to satisfy the constraints of the problem, but in addition, the number of layers in the laminate and their relative fiber orientation must also be considered.
Adnan Naji Jameel, and Kawther Khalid Younus, (2012) investigated the buckling behavior of composite laminated plates subjected to mechanical loads. Levy method of classical laminated plate theory and Finite element coded by ANSYS 13.0 is used to formulate the theoretical model. It was noted that inserting cutout doesn't always decrease the buckling loads.
T. Subramani, and Athulya Sugathan, (2012) found out the buckling strength of cylindrical and elliptical members. Both the members are created using ANSYS top down approach. The result shows reduction of nonlinear buckling loads compared to the elastic buckling loads.
Nagendra Singh Gaira, Nagendra Kumar Maurya, and Rakesh Kumar Yadav (2012) determined the buckling load factors for different aspect ratios, d/b ratio & d/D ratio. It is noted that the presence of cutout lowers the buckling load.
T. Susmitha, V. Rama Krishna Rao and S Mahesh Babu (2012) carried out the buckling analysis in FRP (fibre Reinforced Plastic) thin cylinder, which is subjected to uniaxial compression using 2-D finite element analysis.
M.M. Jadhav, and P.V. Gunjavate (2012) investigated about the optimum laminate which can sustain maximum critical buckling load. The homogenization method can be used to find optimum laminate.
Joshi .A, P. Ravinder Reddy, V.N. Krishnar Reddy, and V. Sushma (2013) determined the buckling load per unit length in rectangular plate with circular cut-outs under biaxial compression using 2D finite element analysis. It was noted that, the buckling load/unit length decreases with increases of aspect ratio. As the b/t ratio increases, the buckling load decreases.
Sa'el Saleh Al-Jameel, and Rafi, K. Albazzaz (2014) carried out buckling analysis of multilayer composite plates consisting of Aluminum, Brass, and Steel with Unsaturated Polyester Resin (UPE) as a core material. Result shows that, the buckling load highly dependent on the type of composite material, and the buckling load increased by increasing elliptical hole orientation angle.
Rajappan, Magesh, and Gurunathan (2015) conducted the buckling experiments on uni-directional laminated composite plate specimens and the, length to thickness ratio, orientation and aspect ratio are examined and determined experimentally.
1) Go to Preferences=>select structural.
2) Preprocessor=>ElementType=>Add/Edit=> select (Shell =>3D 4 node 181 from the list of the elements.
3) Material properties=>Material Models=> select Structural =>linear =>elastic=> orthotropic=> values of EX, EY, EZ, PRXY, PRYZ, PRXZ,GXY,GYZ and GXZ were given.
4) Sections=> Shell=> Lay-up=>Add/Edit=> Value of thickness is given.
5) Modeling=>Create=>Areas=>rectangle=>By dimensions=> Enter Coordinates=> o.k
6) Meshing =>Mesh Tool (smart size and the shape of the mesh were chosen and then the structure was meshed).
7) Go to File=> Change Job name => Buckling1.
8) Go to File=> Change Title => Buckling1.
9) Go to Solution =>Analysis Type => Static Analysis.
10) Solution => Analysis Type=> Analysis options => Pressure effects ON
11) Solution=> Define Loads => Apply=> Structural=> Displacement=> on Lines
12) Solution=> Define Loads =>Apply=> Structural=> Pressure=> on Lines.
13) Solve =>Current LS
14) Solution => Analysis Type=> New Analysis => Eigen buckling=> o.k
15) Solution=> Analysis Type=>Analysis options => select Mode Extraction Method as Block Lanczos=> Give no. of Modes to extract as 5=> o.k
16) Solution=> Load step opts.=>Expansion Pass=> Single expand => expand Modes=> Give no. of Modes to Expand as 5=> o.k
17) Solve =>Current LS General Post processor=> Results Summary General Post processor=> Read Results=> Last set
18) General Post processor=> Plot Results=> Deformed Shape=> select Deformed +Undeformed => o.k
In this paper, Buckling analysis has been carried out for a rectangular plate, varying a/b ratio, b/t ratio and boundary conditions. Finite element modelling of the plates is carried out using the layered shell element. The buckling load factors are evaluated for a rectangular plate subjected to uniaxial compression are found using ANSYS 14.5. In this study, shell 3D 4 noded 181 elements are used for modeling. The plate has a length 'a', width 'b' & thickness 't'. The width of plate is taken as constant i.e. b = 1 m. The properties of the plate shown in Table 1.
Table 1. Material Properties of the Plate
Length of the Plate, a (in m) : 2,4,6,8,10,12
Width of the Plate, b (in m) : 1(constant)
a/b ratio : 2,4,6,8,10,12
b/t ratio : 20,40,60,80,100
Position of Holes : Centre, Top
Shapes of Holes : Triangle, Square & Rhombus
Side of the Square hole : 0.5m
Side of the Rhombus hole : 0.5m
Side of the Triangular hole : 0.5m
Load : 1 N/m2
Type of analysis : Eigen buckling
Element chosen : 3D 4 Noded Shell element in ANSYS
Rectangular plates with three varieties of shapes of holes i.e. Triangular, Square and Rhombus have been investigated. i.e., plate with triangular hole, plate with square hole, and plate with rhombus hole. Each sub case has been investigated for two position of holes. (i.e. center, top) and for various a/b ratios from 2 to 12 at an increment of 2 and for various b/t ratios from 20 to 100 at an increment of 20.
From Figure 13, it is observed that for a plate with a triangular hole at the centre (T-C) with a constant a/b ratio (i.e., 2), the buckling load decreased by 99.1%, as the b/t ratio increases from 20 to 100. Finite Element Model of Plate with Triangular Hole at the Centre, Fundamental Mode Shape of Plate with Triangular Hole at Centre, Buckling loads of Plates with Triangular Hole at the Centre, are shown in Figures 1, 2 & 13 respectively.
Figure 1. Finite Element Model of Plate with Triangular Hole at Centre
Figure 2. Fundamental Mode Shape of Plate with Triangular hole at Centre
It is observed that for a plate with a triangular hole at centre (T-C) with constant b/t ratio (i.e., 20), the buckling load decreased by 98.96%, as a/b ratio increases from 2 to 12.
From Figure 14, it is observed that for a plate with a triangular hole at the top (T-T) with constant a/b ratio (i.e., 2), the buckling load decreased by 99.1%, as b/t ratio increases from 20 to 100. Finite Element Model of Plate with Triangular Hole at Top, Fundamental Mode Shape of Plate with Triangular hole at Top, Buckling loads of Plates with Triangular Hole at Top are shown in Figures 3, 4 & 14 respectively.
Figure 3. Finite Element Model of Plate with Triangular Hole at Top
Figure 4. Fundamental Mode Shape of Plate with Triangular Hole at Top
It is observed that for a plate with a triangular hole at the top (T-T) with a constant b/t ratio (i.e., 20), the buckling load decreased by 98.79%, as a/b ratio increases from 2 to 12.
From Figure 15, It is observed that for a plate with Square hole at centre (S-C) with constant a/b ratio (i.e., 2), the buckling load decreased by 99.1%, as the b/t ratio increases from 20 to 100. Finite Element Model of Plate with Square Hole at Centre, Fundamental Mode Shape of Plate with Square hole at Centre, Buckling loads of Plates with Square Hole at Centre are shown in Figures 5, 6 & 15 respectively.
Figure 5. Finite Element Model of Plate with Square Hole at Centre
Figure 6. Fundamental mode shape of Plate with Square Hole at Centre
It is observed that for a plate with Square hole at the centre (S-C) with a constant b/t ratio (i.e., 20), the buckling load decreased by 98.89%, as a/b ratio increases from 2 to 12.
From Figure 16, It is observed that for a plate with Square hole at the top (S-T) with a constant a/b ratio (i.e., 2), the buckling load decreased by 99.1%, as b/t ratio increases from 20 to 100. Finite Element Model of Plate with Square Hole at Top, Fundamental Mode Shape of Plate with Square Hole at Top, Buckling loads of Plates with Square Hole at Top are shown in Figures 7,8 & 16 respectively.
Figure 7. Finite Element Model of Plate with Square Hole at Top
Figure 8. Fundamental Mode Shape of Plate with Square Hole at Top
It is observed that for a plate with Square hole at the top (S-T) with a constant b/t ratio (i.e., 20), the buckling load decreased by 98.7%, as a/b ratio increases from 2 to 12.
From Figure 17, It is observed that for a plate with Rhombus holes at the centre (R-C) with a constant a/b ratio (i.e., 2), the buckling load decreased by 99.14%, as the b/t ratio increases from 20 to 100. Finite Element Model of Plate with Rhombus Hole at the Centre, Fundamental Mode Shape of Plate with Rhombus hole at Centre, Buckling loads of Plates with Rhombus Hole at Centre are shown in Figures 9,10 & 17 respectively.
Figure 9. Finite Element Model of Plate with Rhombus Hole at Centre
Figure 10. Fundamental Mode Shape of Plate with Rhombus Hole at Centre
It is observed that for a plate with Rhombus holes at centre (R-C) with a constant b/t ratio (i.e., 20), the buckling load decreased by 98.93%, as a/b ratio increases from 2 to 12.
From Figure 18, It is observed that for a plate with Rhombus hole at the top (R-T) with a constant a/b ratio (i.e., 2), the buckling load decreased by 99.12%, as b/t ratio increases from 20 to 100. Finite Element Model of Plate with Rhombus Hole at Top, Fundamental Mode Shape of Plate with Rhombus hole at Top, Buckling loads of Plates with Rhombus Hole at Top, are shown in Figures 11,12 &18 respectively.
Figure 11. Finite Element Model of Plate with Rhombus Hole at Top
Figure 12. Fundamental Mode Shape of Plate with Rhombus Hole at Top
Figure 13. Buckling loads of Plates with Triangular Hole at Centre
Figure 14. Buckling loads of Plates with Triangular Hole at Top
Figure 15. Buckling loads of Plates with Square Hole at Centre
Figure 16. Buckling loads of Plates with Square Hole at Top
Figure 17. Buckling loads of Plates with Rhombus Hole at Centre
Figure 18. Buckling loads of Plates with Rhombus Hole at Top
It is observed that for a plate with Rhombus hole at the top (R-T) with a constant b/t ratio (i.e., 20), the buckling load decreased by 98.83%, as a/b ratio increases from 2 to 12.
Buckling loads of Plates of various Shapes with Central hole are shown in Figure 19, Buckling loads of Plates of various Shapes with Top hole are shown in Figure 20, buckling loads of Plates with Triangular Hole at different positions is shown in Figure 21, buckling loads of Plates with Square Hole at different positions is shown in Figure 22, buckling loads of Plates with Rhombus Hole at different positions is shown in Figure 23.
Figure 19. Buckling loads of Plates of various Shapes with Central Hole
Figure 20. Buckling loads of Plates of various Shapes with Top Hole
Figure 21. Buckling loads of Plates with Triangular Hole at different positions
Figure 22. Buckling loads of Plates with Square Hole at different positions
Figure 23. Buckling loads of Plates with Rhombus Hole at different positions
The present research work can be further extended as given below.
E11 - Longitudinal Elasticity Modulus
E22 - Transverse Elasticity Modulus
E33 - Longitudinal Elasticity Modulus
ʋ12 - In-plane Poisson's ratio
ʋ23 - Transverse Poisson's ratio
ʋ31 - In-plane Poisson's ratio
G12 - In-plane Shear Modulus
G23 - Transverse Shear Modulus
G31 - In-plane Shear Modulus
T-C - Triangular hole at Centre
T-T - Triangular hole at Top
S-C - Square hole at Centre
S-T - Square hole at Top
R-C - Rhombus hole at Centre
R-T - Rhombus hole at Top
