Abstract
A general energy formulation to predict the thermal post buckling behavior of uniform isotropic beams is presented in this paper. The hinged ends of the beam contain elastic rotational restraints to represent the actual practical support situation. The large amplitude vibration behavior of beams is deduced from the post buckling results. The classical hinged and clamped conditions can be obtained as the limiting cases of the rotational spring stiffness. The numerical results, in the form of the ratios of the post buckling to buckling loads for various maximum deflection ratios, are presented in the digital form. An alternate independent formulation, based on the nonlinear finite element formulation, is also used in this paper to validate the numerical results of the present work. Further, the results for the large amplitude vibrations, deduced from the thermal post buckling results are also presented and these results compare very well with the finite element results, available in the literature, for the large amplitude vibration problem. These comparisons show an excellent agreement not only for the present work on the proposed thermal post buckling formulation but also on the deduced results for the large amplitude vibration of beams with the ends elastically restrained against rotation (spring–hinged beams). The numerical results presented confirm the efficacy of the proposed methodology used for predicting the post buckling behavior and deducing the large amplitude vibration behavior of the spring–hinged beams.
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Abbreviations
- A :
-
Area of cross-section (m2)
- a, b, c :
-
Constants in Eq. (4); ‘a’ also represents the maximum lateral deflection (m)
- C, D :
-
Constants in Eq. (6) (–)
- E :
-
Young’s modulus (N/m2)
- \(\boldsymbol{g}_{g_{e}}\) :
-
Element generalized geometric stiffness matrix (–)
- g e :
-
Element geometric stiffness matrix (N/m)
- G 1, G 2 :
- G :
-
Assembled geometric stiffness matrix (N/m)
- I :
-
Area moment of inertia (m4)
- \(\boldsymbol{k}_{g_{e}}\) :
-
Element generalized stiffness (–)
- k e :
-
Element stiffness matrix (N/m)
- K :
-
Rotational spring stiffness (Nm/radian)
- K :
-
Assembled stiffness matrix (N/m)
- L :
-
Length of the beam (m)
- r :
-
Radius of gyration (m)
- P :
-
Thermal compressive load on the beam (N)
- P ∗ :
-
Axial load develop in the beam due to large lateral deflections (N)
- T o :
-
Stress free temperature (∘C)
- ΔT :
-
Rise in temperature from T o (∘C)
- T, T 1 :
-
Transformation matrices (–)
- u :
-
Axial displacement (m)
- U :
-
Maximum strain energy (Nm)
- w :
-
Lateral deflection (m)
- W :
-
Work done due to the large lateral deflections by the thermal load P (Nm)
- x :
-
Axial coordinate of the column (m)
- α :
-
Coefficient of linear thermal expansion (1/∘C)
- α 1 to α 8 :
-
Generalized coordinates
- β :
-
Coefficient in Eq. (18) (–)
- {δ}:
-
Eigenvector (–)
- γ :
-
Rotational spring stiffness parameter (=KL/EI) (N/m)
- ε :
-
Axial strain (–)
- λ :
-
Non-dimensional thermal buckling load parameter (=PL 2/EI)
- L :
-
Denotes linear
- NL :
-
Denotes nonlinear
References
Dym CL (1974) Stability theory and its applications to structural mechanics. Noordhoff International, Leyden
Thompson JMT, Hunt GW (1973) A general theory of elastic stability. Wiley, London
Rao GV, Rao KK (1984) Thermal postbuckling of columns. AIAA J 22:850–851
Raju KK, Rao GV (1984) Finite element analysis of thermal postbuckling of tapered columns. Comput Struct 19:617–620
Rao GV, Raju KK (1983) A reinvestigation of post-buckling behaviour of elastic circular plates using a simple finite element formulation. Comput Struct 17:233–236
Ziegler F, Rammerstorfer FG (1989) Thermo elastic stability. In: Hetnarski RB (ed) Thermal stresses III. Elsevier, Amsterdam, pp 107–189
Librescu L, Lin W, Nemeth MP, Starnes JH (1995) Thermo-mechanical postbuckling of geometrically imperfect flat and curved panels taking into account tangential edge constraints. J Therm Stresses 18:465–482
Elishakoff I (2001) Apparently first closed-form solution for frequency of beam with rotational spring. AIAA J 39:183–186
Emam SA, Nayfegh AH (2009) Postbuckling and free vibrations of composite beams. Compos Struct 88:636–642
Li S, Ch C (2000) Analysis of thermal post buckling of heated elastic rods. J Appl Math Mech 21:133–140
Rao GV, Varma RR (2009) Heuristic thermal post buckling and large amplitude vibration formulations of beams. AIAA J 47:1977–1980
Rao GV, Raju KK (2002) Thermal postbuckling of uniform columns: a simple intuitive method. AIAA J 40:2138–2140
Rao GV, Meera KS, Ranga GJ (2008) Simple formula to study the large amplitude free vibrations of beams and plates. J Appl Mech 75:14505
Rao GV, Raju KK (2002) Large amplitude vibrations of spring hinged beams. AIAA J 40:1912–1915
Azrar L, Benamar R, White RG (1999) A semi analytical approach to the nonlinear dynamic response problem of S-S and C-C beams at large amplitudes. Part 1: general theory and application to the single mode approach to free and forced vibration analysis. J Sound Vib 224:183–207
Acknowledgements
The authors thank the managements of their respective Institutes for their encouragement during the course of this work. The first author is grateful to the Indian National Academy of Engineering for the award of the Distinguished Professorship.
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Rao, G.V., Reddy, G.K., Jagadish Babu, G. et al. Prediction of thermal post buckling and deduction of large amplitude vibration behavior of spring–hinged beams. Forsch Ingenieurwes 76, 51–58 (2012). https://doi.org/10.1007/s10010-012-0150-2
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DOI: https://doi.org/10.1007/s10010-012-0150-2