This paper discusses the mathematical
modeling for the mechanics of solid using the distribution theory of Schwartz
to the beam bending differential Equations. This problem is solved by the use
of generalized functions, among which is the well known Dirac delta function.
The governing differential Equation is Euler-Bernoulli beams with jump
discontinuities on displacements and rotations. Also, the governing
differential Equations of a Timoshenko beam with jump discontinuities in slope,
deflection, flexural stiffness, and shear stiffness are obtained in the space
of generalized functions. The operator of one of the governing differential
Equations changes so that for both Equations the Dirac Delta function and its
first distributional derivative appear in the new force terms as we present the
same in a Euler-Bernoulli beam. Examples are provided to illustrate the
abstract theory. This research is useful to Mechanical Engineering, Ocean
Engineering, Civil Engineering, and Aerospace Engineering.
References
[1]
Timoshenko, S. and Woinowsky-Kreiger, S. (1959) Theory of Plates and Shells. 2nd Edition, McGraw-Hill, New York.
[2]
Gorman, D.J. (2008) On the Use of the Dirac Delta Function in the Vibration Analysis of Elastic Structures. International Journal of Solids and Structures, 45, 4605-4614.
http://dx.doi.org/10.1016/j.ijsolstr.2008.03.009
[3]
Timoshenko, S.P. (1976) Strength of Materials: Part I. Krieger, Malabar.
[4]
Shames, I.H. (1989) Introduction to Solid Mechanics. Prentice-Hall, New York.
[5]
Kanwal, R.P. (1983) Generalized Functions Theory and Applications. Academic Press, New York.
[6]
Palmeri, A. and Cicirello, A. (2011) Physically-Based Dirac’s Delta Functions in the Static Analysis of Multi-Cracked Euler-Bernoulli and Timoshenko Beams. International Journal of Solids and Structures, 48, 2184-2195. http://dx.doi.org/10.1016/j.ijsolstr.2011.03.024
[7]
Yavari, A., Sarkani, S. and Moyer, E.T. (2000) On Applications of Generalised Functions to Beam Bending Problems. International Journal of Solids and Structures, 38, 8389-8406.
http://dx.doi.org/10.1016/S0020-7683(01)00095-6
[8]
Yavari, A. and Sarkani, S. (2001) On Applications of Generalised Functions to the Analysis of Euler-Bernoulli Beam-Columns with Jump Discontinuities. International Journal of Mechanical Sciences, 43, 1543-1562. http://dx.doi.org/10.1016/S0020-7403(00)00041-2
[9]
Bagarello, F. (1995) Multiplication of Distribution in One Dimension: Possible Approaches and Applications to Delta-Function and Its Derivatives. Journal of Mathematical Analysis and Applications, 196, 885-901. http://dx.doi.org/10.1006/jmaa.1995.1449
