Timoshenko Beam Theory for the Flexural Analysis of Moderately Thick Beams – Variational Formulation, and Closed Form Solution

Timoshenko Beam Theory for the Flexural Analysis of Moderately Thick Beams – Variational Formulation, and Closed Form Solution

Charles Chinwuba Ike

Department of Civil Engineering, Enugu State University of Science & Technology, Enugu State, Nigeria

Corresponding Author Email: 
ikecc2007@yahoo.com
Page: 
34-45
|
DOI: 
https://doi.org/10.18280/ti-ijes.630105
Received: 
22 January 2019
|
Accepted: 
19 March 2019
|
Published: 
31 March 2019
| Citation

OPEN ACCESS

Abstract: 

In this study, the Timoshenko first order shear deformation beam theory for the flexural behaviour of moderately thick beams of rectangular cross-section is formulated from vartiational principles, and applied to obtain closed form solutions to the flexural problem of moderately thick rectangular beams. The total potential energy functional for the moderately thick beam flexure problem was formulated by considering the contribution of shearing deformation to the strain energy. Euler-Lagrange conditions were then applied to obtain the system of two coupled ordinary differential equations of equilibrium. The problem of moderately thick beam with simply supported ends subject to uniformly distributed transverse load on the entire span was solved in closed form to obtain the transverse deflection as the sum of the flexural and shear components. Another problem of moderately thick cantilever beam under point load at the free end was solved in closed form to illustrate the solution of the governing equations. The transverse deflection was similarly obtained as the sum of shear and bending components. The bending component of deflection was found to be identical with the Euler-Bernoulli results while the shear component was found to be dependent on the square of the ratio of the beam thickness, t, to the span, l. It was found that as t/l<0.02, the contribution of shear to the overall deflection is insignificant; but becomes significant for t/l>0.10. The findings are in excellent agreement with the technical literature.

Keywords: 

Timoshenko beam theory, moderately thick beams, total potential energy functional, Euler-Lagrange differential equations, differential equations of equilibrium, shear deformation

1. Introduction
2. Theoretical Framework
3. Results
4. Discussion
5. Conclusions
  References

[1]    Ike CC, Ikwueze EU. (2018). Ritz method for the analysis of statically indeterminate Euler-Bernoulli beams. Saudi Journal of Engineering and Technology (SJEAT) 3(3): 133-140. https://doi.org/10.21276/sjeat20183.3.3

[2]    Ike CC, Ikwueze EU. (2018). Fifth degree Hermittian polynomial shape functions for the finite element analysis of clamped simply supported Euler-Bernoulli beam. American Journal of Engineering Research (AJER) 7(4): 97-105.

[3]    Ghugal YM, Dahake AG. (2013). Flexure of simply supported thick beams using refined shear deformation theory. World Academy of Science, Engineering and Technology, International Journal of Civil, Architectural, Structural and Construction Engineering 7(1): 82-91. https://doi.org/10.1999/1307-6892/9996869

[4]    Nimbalkar VN, Dahake AG. (2015). Displacement and stresses for thick beam using new hyperbolic shear deformation theory. International Journal of Pure and Applied Research in Engineering and Technology 3(9): 120-130.

[5]    Chavan VB, Dahake AG. (2015). A refined shear deformation theory for flexure of thick beam. International Journal of Pure and Applied Research in Engineering and Technology 3(9): 109-119.

[6]    Pankade PM, Tupe DH, Salve SB. (2016). Static flexural analysis of thick isotropic beam using hyperbolic shear deformation theory. International Journal of Engineering Research 5(3): 565-571.

[7]    Sayyad A. (2012). Static flexure and free vibration analysis of thick order shear deformation theories. International Journal of Applied Mathematics and Mechanics 8(14): 71-87.

[8]    Ghugal Y. (2006). A two dimensional exact elasticity solution of thick beam. Departmental Report 1. Department of Applied Mechanics, Government Engineering College, Aurangabad India, pp 1-96.

[9]    Timoshenko SP. (1921). On the correction for shear of differential equation for transverse vibrations of prismatic bars. Philosophical Magazine, Series 6 41: 742-746.

[10]    Cowper GR. (1966). The shear coefficients in Timoshenko beam theory. ASME Journal of Applied Mechanics 33: 335-340.

[11]    Ghugal YM, Sharma R. (2009). Hyperbolic shear deformation theory for flexure and vibration of thick isotropic beams. International Journal of Computational Methods 6(4): 585-604.

[12]    Ghugal YM, Shimpi RP. (2002). A review of refined shear deformation theories for isotropic and anisotropic laminated beams. Journal of Reinforced Plastics and Composites 21: 775-813.

[13]    Dahake AG, Ghugal YM. (2012). Flexure of thick simply supported beam using trigonometric shear deformation theory. International Journal of Scientific and Research Publications 2(11): 1-7. 10.29322

[14]    Ghugal YM, Dahake AG. (2012). Flexure of thick beams using refined shear deformation theory. International Journal of Civil and Structural Engineering 3(2): 321-335.

[15]    Levinson M. (1981). A new rectangular beam theory. Journal of Sound and Vibration 74: 81-87. https://doi.org/10.1016/0022-460x(81)90493-4.

[16]    Krishna Murty AV. (1984). Toward a consistent beam theory. AIAA Journal 22: 811-816.

[17]    Baluch MH, Azad AK, Khidir MA. (1984). Technical theory of beams with normal strain. Journal of the Engineering Mechanics, Proceedings of the ASCE 110: 1233-1237.

[18]    Bhimaraddi A, Chandreshekhara K. (1993). Observations on higher-order beam theory. Journal of Aerospace Engineering, Proceedings of ASCE, Technical Note 6: 403-413.

[19]    Brickford WB. (1982). A consistent higher order beam theory. Development in Theoretical Applied Mechanics SECTAM 11: 137-150.

[20]    Touratier M. (1991). An efficient standard plate theory. International Journal of Engineering Science 29(8): 901-916.

[21]    Vlasov VZ, Leontiev UN. (1996). Beams, plates and shells on elastic foundation. Chapter 1. pp. 1-8 (Translated from Russian by Barouch A. and Plez T.). Israel program for scientific translation Ltd, Jerusalem.

[22]    Stein M. (1989). Vibration of beams and plate strips with three dimensional flexibility. Transaction ASME Journal of Applied Mechanics 56(1): 228-231.

[23]    Kapdis P, Kalwane U, Salunkhe U, Dahake A. (2018). Flexural analysis of deep aluminium beam. Journal of Soft Computing in Civil Engineering 2(1): 71-84. https://doi.org/10.22115/SCCE.2018.49679

[24]    Ghugal YM. (2006). A simple higher order theory for beam with transverse shear and transverse normal effect. Departmental Report No 4, Applied Mechanics Department, Government College of Engineering, Aurangabad, India, pp. 1-96.