Modeling of the Beam Discontinuity with Two Analyses in Strong and Weak Forms using a Torsional Spring Model

Document Type : Persian

Authors

1 MSc Student, Department of Mechanical Engineering, Takestan Branch, Islamic Azad University, Takestan, Iran

2 Assistant Prof., Department of Mechanical Engineering, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali, Iran

3 Professor, Department of Mechanical Engineering, Bandar Anzali Branch, Islamic Azad University, Bandar Anzali, Iran

4 MSc Student, Department of Mechanical Engineering, University of Guilan, Rasht, Iran

Abstract

In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an analogy between the formulation of the Euler-Bernoulli and Timoshenko theories that indicates the influence of the shear deformation in discontinuous beams. The flexural stiffness of discontinuous beams is corrected by using the Dirac’s delta function.  In the weak form, the reduced stiffness matrix is derived from the strain energy equation established by the continuity, kinematics and constitutive equations. The linearity assumption of the geometry and material is considered to construct the kinematics and constitutive equations respectively. The continuity conditions mathematically connect two divided parts of the Euler-Bernoulli beam for which an improved Hermitian shape function is employed to interpolate displacement field. An application shows the comparison and validation of the results of the strong and weak forms, and also the static behavior of discontinuous beams

Keywords

References

[1] G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics, vol. 1, No. 24, 1957, pp. 361–364.
[2] A. D. Dimarogonas, C. A. Papadopulus, Vibration of cracked shafts in bending, Journal of Sound and Vibration, vol. 91, No. 4, 1983, pp. 583–593.
[3] H. Okamura, H. W. Liu, C. Chorng-Shin, A cracked column under compression, Engineering Fracture Mechanics, vol. 1, pp. 547–564, 1969.
[4] W. M. Ostachowicz, M. Krawczuk, Vibrational analysis of cracked beam, Composite Structures, vol. 36–22, 1990, pp. 245–250.
[5] M. Krawczuk, W. M. Ostachowicz, Influence of a crack on the dynamic stability of a column,Journal of Sound and Vibration, vol. 167, No. 3, 1993, pp. 541–555.
[6] M. Skrinar, T. Pliberšek, New linear spring stiffness definition for displacement analysis of cracked beam elements, Proceedings in Applied Mathematics and Mechanics, vol. 4, 2004, pp. 654–655.
[7] A. J. Dentsoras, A. D. Dimarogonas, Resonance controlled fatigue crack propagation in a beam under longitudinal vibrations, International Journal of Fracture, vol. 1, No. 23, 1983, pp. 15–22.
[8] T. G. Chondros, A. D. Dimarogonas, Identification of cracks in welded joints of complex structures, Journal of Sound and Vibration, vol. 4, No. 64, 1980, pp. 531–538.
[9] P. F. Rizos, N. Aspragathos, A. D. Dimarogonas, Identification of crack location and magnitude in a cantilever beam from the vibration modes, Journal of Sound and Vibration, vol. 3, No. 138, 1990, pp. 381–388.
[10] F. Bagarello, Multiplication of distribution in one dimension: possible approaches and applications to d-function and its derivatives, Journal of Mathematical Analysis and Applications, vol. 196, 1995, pp.  885–901.
[11] F. Bagarello, Multiplication of distribution in one dimension and a first application to quantum field theory, Journal of Mathematical Analysis and Applications, vol. 266, 2002, pp. 298–320.
[12] B. Biondi, S. Caddemi, Closed form solutions of Euler–Bernoulli beam with singularities, International Journal of Solids and Structures, vol. 42, 2005, pp. 3027–3044.
[13] B. Biondi, S. Caddemi, Euler–Bernoulli beams with multiple singularities in the flexural stiffness, European Journal of Mechanics A/Solids, vol. 26, 2007, pp. 789–809.
[14] A. Palmeri, A. Cicirello, Physically-based Dirac‘s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams, International Journal of Solids and Structures, vol. 48, 2011, pp.2184–2195.
[15] A. Cicirello, A. Palmeri, Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads, International Journal of Solids and Structures, vol. 51, 2014, pp. 1020-1029.
[16] M. Dona, A. Palmeri, M. Lombardo, Exact closed-form solutions for the static analyses of multi-cracked gradient-elastic beams in bending, International Journal of Solids and Structures, vol. 51, 2014, pp. 2744-2753.
[17] S. A. Eftekhari, A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads, Applied Mathematical Modelling, In Press, Corrected Proof, 2015.
[18] G. Gounaris, A. D. Dimarogonas, A finite element of a cracked prismatic beam for structural analysis, composite structures, vol. 3, No. 28, 1988, pp. 301–313.
[19] M. Skrinar, A. Umek, Plane beam finite element with crack, Journal of Gradbeni vestnik (in Slovenian), vol. 1, No. 2, 1996, pp. 2–7.
 
 
[20] M. Skrinar, On the application of a simple computational model for slender transversely cracked beams in buckling problems, Computational Materials Science, vol. 1, No. 39, 2007, pp. 242–249.
[21] J. N. Reddy, On locking-free shear deformable beam finite elements, Computer Methods in Applied Mechanics and Engineering, vol. 149, 1997, pp. 113-132.
[22] M. Abromowitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1965, pp. 773.
[23] M. Skrinar, Elastic beam finite element with an arbitrary number of transverse cracks, Finite Elements in Analysis and Design, vol. 45, 2009, pp. 181–189.
[24] T. R. Chandrupatla, A. D. Belegundu, Introduction to Finite Elements in Engineering, Second Edition, New Jersey: Prentice Hall, 1997, pp. 9-13.
[25] K. Rajagopalan, Finite Element Buckling Analysis of Stiffened Cylindrical Shells, A. A. Balkema: Rotterdam, 1993, pp. 13-25.
[26] J. N. Reddy, an Introduction to Nonlinear Finite Element Analysis, New York: Oxford 2005, pp. 87-126.
 
 
 
Journal of Simulation and Analysis of Novel Technologies in Mechanical Engineering
Volume 8, Issue 4 - Serial Number 4
Winter 2015
Pages 295-309

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