Forschung im Ingenieurwesen

, Volume 76, Issue 1–2, pp 51–58 | Cite as

Prediction of thermal post buckling and deduction of large amplitude vibration behavior of spring–hinged beams

  • G. V. Rao
  • G. K. Reddy
  • G. Jagadish Babu
  • V. V. S. Rao


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.


Admissible Function Lateral Deflection Closed Form Analytical Solution Large Amplitude Vibration Classical Boundary Condition 
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List of symbols


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) (–)


Young’s modulus (N/m2)


Element generalized geometric stiffness matrix (–)


Element geometric stiffness matrix (N/m)

G1, G2

Constants in Eq. (14) and Eq. (15)


Assembled geometric stiffness matrix (N/m)


Area moment of inertia (m4)


Element generalized stiffness (–)


Element stiffness matrix (N/m)


Rotational spring stiffness (Nm/radian)


Assembled stiffness matrix (N/m)


Length of the beam (m)


Radius of gyration (m)


Thermal compressive load on the beam (N)


Axial load develop in the beam due to large lateral deflections (N)


Stress free temperature (C)


Rise in temperature from T o (C)

T, T1

Transformation matrices (–)


Axial displacement (m)


Maximum strain energy (Nm)


Lateral deflection (m)


Work done due to the large lateral deflections by the thermal load P (Nm)


Axial coordinate of the column (m)


Coefficient of linear thermal expansion (1/C)

α1 to α8

Generalized coordinates


Coefficient in Eq. (18) (–)


Eigenvector (–)


Rotational spring stiffness parameter (=KL/EI) (N/m)


Axial strain (–)


Non-dimensional thermal buckling load parameter (=PL 2/EI)



Denotes linear


Denotes nonlinear



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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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • G. V. Rao
    • 1
  • G. K. Reddy
    • 2
  • G. Jagadish Babu
    • 3
  • V. V. S. Rao
    • 4
  1. 1.Department of Mechanical EngineeringVardhaman College of EngineeringHyderabadIndia
  2. 2.Department of Mechanical EngineeringSreenidhi Institute of Science and TechnologyHyderabadIndia
  3. 3.Advanced Systems LaboratoryHyderabadIndia
  4. 4.Department of Mechanical EngineeringJNTU College of EngineeringKakinadaIndia

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