Abstract
The stability of clamped stepped and stiffened rectangular plate subjected to in-plane forces is examined. The plate is divided into 900 rectangular meshes and the partial derivatives are approximated using central difference formula. Altogether 841 equations of equilibrium and 248 equations representing boundary conditions are formed, finally leading to the solution of eigenvalue problem. The buckling coefficients are calculated for various types of stepped plates and the results are presented in tables for ready use by designers. The results are compared with the published results and they are in close agreement.
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- a :
-
length of the plate
- a 1 :
-
step length
- b :
-
width of the plate
- a/b :
-
panel aspect ratio
- D :
-
flexural rigidity of the plate
- E :
-
Young’s modulus
- K :
-
flexural stiffness
- \({\overline{K}_{G}}\) :
-
geometric stiffness
- k,k e :
-
buckling coefficients
- l :
-
mesh length
- m :
-
mesh breadth
- M :
-
number of divisions in y direction
- M :
-
mass matrix
- N :
-
number of divisions in x direction
- nj :
-
number of joints
- N x ,N y ,N xy :
-
in-plane forces
- Pcr :
-
critical load
- t :
-
thickness of the plate
- t i :
-
step thickness
- w :
-
lateral deflection
- x,y :
-
Cartesian coordinates
- α :
-
ratio between the breadth and length of a mesh
- β,γ,δ :
-
tracers corresponding to N x ,N y ,N xy
- λ :
-
eigenvalue
- ν :
-
Poison’s ratio
- τ :
-
shear stress
- σ x ,σ y ,σ :
-
compressive stresses
References
Timoshenko SP, Gere JM (1961) Theory of elastic stability. McGraw-Hill, New York
Allen HB, Bulson PS (1980) Background to buckling. McGraw-Hill, London
Szilard R (2004) Theory and applications of plate analysis—classical and numerical and engineering methods. Wiley, New York
Azhari M (1993) Local and post-local buckling of plates and plate assemblies using finite strip method. Ph.D. thesis, The University of New South Wales Kensington
Wittrick WH, Ellen CH (1962) Buckling of tapered rectangular plates in compression. Aeronaut Q 13:308–326
Navaneethakrishnan PV (1968) Buckling of nonuniform plates: spline method. J Eng Mech 114(5):893–898
Chung MS, Cheung YK (1971) Natural vibration of thin flat walled structures with different boundary conditions. J Sound Vib 18(3):325–337
Przemieniecki JS (1973) Finite element Analysis of local instability. AIAA J 11:33–39
Chehil DS, Dua SS (1973) Buckling of rectangular plates with general variation in thickness. J Appl Mech, Trans ASME 40:745–751
Hwang SS (1973) Stability of plats with piecewise varying thickness. J Appl Mech, Trans ASME 40(4):1127–1128
Hancock GJ (1978) Local distortional and lateral buckling of I beams. J Struct Div 104(ST11):1787–1798
Chen WF, Lui EM (1987) Structural stability theory and implementation. Elsevier, New York
Singh JP, Dey SS (1990) Variational finite difference approach to buckling of plates of variable thickness. Comput Struct 36:39–45
Harik IE, Liu X, Ekambaram R (1991) Elastic stability of plats with varying rigidities. Comput Struct 38:161–168
Subramanian K, Elangovan A, Rajkumar R (1993) Elastic stability of varying thickness plates using the finite element method. Comput Struct 48(4):733–738
Bradford MA, Azhari M (1995) Buckling of plates with different end conditions using the finite strip method. Comput Struct 56(1):75–83
Nerantzaki MS, Katsikadalils JT (1996) Buckling of plates with variable thickness and analog equation solution. Eng Anal Bound Elem 18:149–154
Bradford MA, Azhari M (1997) The use of bubble functions for the stability of plates with different end conditions. Eng Struct 19(2):151–161
Yuan S, Yin Y (1998) Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method. Comput Struct 66(6):861–867
Xiang Y, Wang CM (2002) Exact buckling and vibration solutions for stepped rectangular plates. J Sound Vib 250(3):503–517
Eisenberger M, Alexandrov A (2003) Buckling loads of variable thickness thin isotropic plates. Thin-Walled Struct 41:871–889
Xiang Y, Wei GW (2004) Exact solutions for buckling and vibration of stepped rectangular Mindlin plates. Int J Solids Struct 41:279–294
John Wilson A, Rajasekaran S (2012) Elastic stability of all edges simply supported, stepped and stiffened rectangular plate under uni-axial loading. Appl Math Model 36:5758–5772
Elishakoff I (2005) Essay on the contributors to the elastic stability theory. Meccanica 40:75–110
Malekzadeh P, Golbahaar Haghigh MR, Atashi MM (2011) Free vibration of elastically supported functionally graded annular plates subjected to thermal environment. Meccanica 46:893–913
Eftekhari SA, Jafari AA (2012) A simple and accurate method FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions. Meccanica. doi:10.1007/s111012-012-9657-8
Malekzadeh P, Golbahaar Haghigh MR, Alibeygi Beni A (2012) Buckling analysis of functionally graded arbitrary straight sided quadrilateral plates on elastic foundation. Meccanica 47:321–333
Ravari MRK, Shahidi AR (2013) Axi-symmetric buckling of the circular annular nanoplate using finite difference method. Meccanica. doi:10.1007/s11012-012-9589-1012-012-9493
Gambir ML (2004) Stability analysis and design of structures. Springer, Berlin
Acknowledgements
The first author wishes to express his thanks to Dr. T. Venkatachlam department of physics of Coimbatore Institute of Technology for his help in preparing the diagrams and charts. The second author thanks the management and Principal of P.S.G College of Technology for giving necessary facilities to carry out the research work reported in this paper. The authors would like to express their gratitude to the anonymous reviewers for their constructive comments and thoughtful suggestions towards enhancing the quality of the paper.
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A. John Wilson was retired from Dept. of Mathematics, Coimbatore Institute of Technology.
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John Wilson, A., Rajasekaran, S. Elastic stability of all edges clamped stepped and stiffened rectangular plate under uni-axial, bi-axial and shearing forces. Meccanica 48, 2325–2337 (2013). https://doi.org/10.1007/s11012-013-9751-6
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DOI: https://doi.org/10.1007/s11012-013-9751-6