A novel beam finite element with singularities for the dynamic analysis of discontinuous frames
Tóm tắt
In this work, a model of the stepped Timoshenko beam in presence of deflection and rotation discontinuities along the span is presented. The proposed model relies on the adoption of Heaviside’s and Dirac’s delta distributions to model abrupt and concentrated, both flexural and shear, stiffness discontinuities of the beam that lead to exact closed-form solutions of the elastic response in presence of static loads. Based on the latter solutions, a novel beam element for the analysis of frame structures with an arbitrary distribution of singularities is here proposed. In particular, the presented closed-form solutions are exploited to formulate the displacement shape functions of the beam element and the relevant explicit form of the stiffness matrix. The proposed beam element is adopted for a finite element discretization of discontinuous framed structures. In particular, by means of the introduction of a mass matrix consistent with the adopted shape functions, the presented model allows also the dynamic analysis of framed structures in presence of deflection and rotation discontinuities and abrupt variations of the cross-section. The presented formulation can also be easily employed to conduct a dynamic analysis of damaged frame structures in which the distributed and concentrated damage distributions are modelled by means of equivalent discontinuities. As an example, a simple portal frame, under different damage scenarios, has been analysed and the results in terms of frequency and vibration modes have been compared with exact results to show the accuracy of the presented discontinuous beam element.
Tài liệu tham khảo
Yavari A., Sarkani S.: On applications of generalised functions to the analysis of Euler-Bernoulli Beam-columns with jump discontinuities. Int. J. Mech. Sci. 43, 1543–1562 (2001)
Yavari A., Sarkani S., Moyer E.T.: On applications of generalised functions to beam bending problems. Int. J. Solids Struct. 37, 5675–5705 (2000)
Yavari A., Sarkani S., Reddy J.N.: On nonuniform Euler-Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory. Int. J. Solids Struct. 38, 8389–8406 (2001)
Yavari A., Sarkani S., Reddy J.N.: Generalised solutions of beams with jump discontinuities on elastic foundations. Arch. Appl. Mech. 71(9), 625–639 (2001)
Khiem N.T., Lien T.V.: A simplified method for natural frequency analysis of a multiple cracked beam. J. Sound Vib. 245, 737–751 (2001)
Li Q.S.: Free vibration analysis of non-uniform beams with an arbitrary number of cracks and concentrated masses. J. Sound Vib. 252, 509–525 (2002)
Sorrentino S., Fasana A., Marchesiello S.: Analysis of non-homogeneous Timoshenko beams with generalized damping distributions. J. Sound Vib. 304(3-5), 779–792 (2007)
Shifrin E.I., Ruotolo R.: Natural frequencies of a beam with an arbitrary number of cracks. J. Sound Vib. 222, 409–423 (1999)
Anifantis N., Dimarogonas A.: Stability of columns with a single crack subjected to follower and vertical loads. Int. J. Solids Struct. 19(4), 281–291 (1983)
Wang Q.: A comprehensive stability analysis of a cracked beam subjected to follower compression. Int. J. Solids Struct. 41, 4875–4888 (2004)
Wang C.Y., Wang C.M., Aung T.M.: Buckling of a weakened column. J. Eng. Mech. ASCE 130, 1373–1376 (2004)
Li Q.S.: Buckling of multi-step cracked columns with shear deformation. Eng. Struct. 23, 356–364 (2001)
Gurel M.A.: Buckling of slender prismatic circular columns weakened by multiple edge crack. Acta Mech. 188, 1–19 (2007)
Takahashi I.: Vibration and stability of non-uniform cracked Timoshenko beam subjected to follower force. Comput. Struct. 71, 585–591 (1999)
Zheng D.Y., Fan S.C.: Natural frequency changes of a cracked Timoshenko beam by modified Fourier series. J. Sound Vib. 246, 297–317 (2001)
Fan S.C., Zheng D.Y.: Stability of a cracked Timoshenko beam column by modified fourier series. J. Sound Vib. 264, 475–484 (2003)
Irwin G.R.: Analysis of stresses and strainsnear the end of a crack traversing a plate. J. Appl. Mech. 24, 361–364 (1957)
Irwin, G.R.: Relation of Stresses Near a Crack to the Crack Extension Force. 9th Congr. Appl. Mech., Brussels (1957)
Freund L.B., Hermann G.: Dynamic fracture of a beam or plate in plane bending. J. Appl. Mech. 76, 112–116 (1976)
Gounaris G., Dimarogonas A.D.: A finite element of a cracked prismatic beam for structural analysis. Comput. Struct. 28, 309–313 (1988)
Rizos P.F., Aspragathos N., Dimarogonas A.D.: Identification of crack location and magnitude in a cantilever beam from the vibration modes. J. Sound Vibr. 138(3), 381–388 (1990)
Ostachowicz W.M., Krawczuk C.: Analysis of the effect of cracks on the natural frequencies of a cantilever beam. J. Sound Vibr. 150(2), 191–201 (1991)
Paipetis S.A., Dimarogonas A.D.: Analytical Methods in Rotor Dynamics. Elsevier Applied Science, London (1986)
Biondi B., Caddemi S.: Closed form solutions of Euler-Bernoulli beams with singularities. Int. J. Solids Struct. 42, 3027–3044 (2005)
Biondi B., Caddemi S.: Euler-Bernoulli beams with multiple singularities in the flexural stiffness. Eur. J. Mech. A/Solids 26(5), 789–809 (2007)
Caddemi S., Caliò I.: Exact solution of the multi-cracked Euler-Bernoulli column. Int. J. Solids Struct. 45(16), 1332–1351 (2008)
Caddemi S., Caliò I.: Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. J. Sound Vibr. 327(3–5), 473–489 (2009)
Caddemi S., Caliò I.: The influence of the axial force on the vibration of the Euler-Bernoulli beam with an arbitrary number of cracks. Arch. Appl. Mech. 82(6), 827–839 (2012)
Caddemi S., Caliò I., Marletta M.: The non-linear dynamic response of the Euler–Bernoulli beam with an arbitrary number of switching cracks. Int. J. Non-Linear Mech. 45, 714–726 (2010)
Caddemi S., Caliò I., Cannizzaro F.: Closed-form solutions for stepped Timoshenko beams with internal singularities and along-axis external supports. Arch. Appl. Mech. 83(4), 559–577 (2013)
Caddemi S., Caliò I.: The exact explicit dynamic stiffness matrix of multi-cracked Euler–Bernoulli beam and applications to damaged frame structures. J. Sound Vibr. 332(12), 3049–3063 (2013)
Caddemi S., Morassi A.: Multi-cracked Euler–Bernoulli beams: Mathematical modeling and exact solutions. Int. J. Solids Struct. 50(6), 944–956 (2013)
Bagarello F.: Multiplication of distribution in one dimension: possible approaches and applications to δ-function and its derivatives. J. Math. Anal. Appl. 196, 885–901 (1995)
Bagarello F.: Multiplication of distribution in one dimension and a first application to quantum field theory. J. Math. Anal. Appl. 266, 298–320 (2002)
Liebowitz H., Vanderveldt H., Harris D.W.: Carrying capacity of notched column. Int. J. Solids Struct. 3, 489–500 (1967)
Liebowitz H., Claus W.D.S. Jr.: Failure of notched columns. Eng. Fract. Mech. 1, 379–383 (1968)
Okamura H., Liu H.W., Chorng-Shin Chu, Liebowitz H.: A cracked column under compression. Eng. Fract. Mech. 1, 547–564 (1969)
Bilello, C.: Theoretical and experimental investigation on damaged beams under moving systems. Ph.D. Thesis, University of Palermo, Italy (2001)
Caddemi, S., Caliò, I.: The influence of concentrated cracks on framed structures by means the dynamic stiffness method. Fifth European Workshop on Structural Health Monitoring Sorrento, Naples, Italy (2010)
Williams F.W., Wittrick W.H.: An automatic computational procedure for calculating natural frequencies of skeletal structures. Int. J. Mech. Sci. 12(9), 781–791 (1970)
Caddemi, S., Caliò, I.: The exact stability stiffness matrix for the analysis of multi-cracked frame structures. Comput. Struct. (in press). doi:10.1016/j.compstruc.2013.05.003