A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.

[1]  V. V. Bolotin,et al.  Dynamic Stability of Elastic Systems , 1965 .

[2]  C. Sun,et al.  Vibration analysis of laminated composite thin-walled beams using finite elements , 1991 .

[3]  I. Etsion.  Literature Review : A Review of Mechanical Face Seal Dynamics , 1982 .

[4]  J. Whitney.  Structural Analysis of Laminated Anisotropic Plates , 1987 .

[5]  P. Cunniff,et al.  The Vibration of Cantilever Beams of Fiber Reinforced Material , 1972 .

[6]  J. Reddy,et al.  A higher order beam finite element for bending and vibration problems , 1988 .

[7]  M. Sathyamoorthy,et al.  Literature Review : Nonlinear Analysis of Beams Part I: A Survey of Recent Advances , 1982 .

[8]  M. Borri,et al.  Anisotropic beam theory and applications , 1983 .

[9]  Aviv Rosen,et al.  Nonlinear analysis of pretwisted rods using 'principal curvature transformation'. II - Numerical results , 1987 .

[10]  Rakesh K. Kapania,et al.  Nonlinear vibrations of unsymmetrically laminated beams , 1989 .

[11]  Alan D. Stemple,et al.  Finite-Element Model for Composite Beams with Arbitrary Cross-Sectional Warping , 1988 .

[12]  Earl H. Dowell,et al.  Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades , 1974 .

[13]  Rakesh K. Kapania,et al.  Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave Propagation , 1989 .

[14]  Peretz P. Friedmann,et al.  Recent developments in rotary-wing aeroelasticity , 1977 .

[15]  Aviv Rosen,et al.  Nonlinear analysis of pretwisted rods using 'principal curvature transformation'. I - Theoretical derivation , 1987 .

[16]  A. Krishn,et al.  Vibrations of short beams , 1970 .

[17]  K. Alkire.  An analysis of rotor blade twist variables associated with different Euler sequences and pretwist treatments , 1984 .

[18]  David A. Peters,et al.  On the lateral buckling of uniform slender cantilever beams , 1975 .

[19]  A Linear Theory for Pretwisted Elastic Beams , 1983 .

[20]  C. Chree.  The Equations of an Isotropic Elastic Solid in Polar and Cylindrical Co-ordinates their Solution and Application , 1889 .

[21]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[22]  G. V. Rao,et al.  Nonlinear vibrations of beams considering shear deformation and rotary inertia , 1976 .

[23]  M. R. Silva,et al.  Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion , 1978 .

[24]  T. R. Kane,et al.  Dynamics of a cantilever beam attached to a moving base , 1987 .

[25]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[26]  Robert D. Adams,et al.  Measurement of the flexural damping capacity and dynamic Young's modulus of metals and reinforced plastics , 1973 .

[27]  V. I. Weingarten,et al.  Non-linear response of a beam to periodic loading , 1972 .

[28]  Ali H. Nayfeh,et al.  Three-dimensional nonlinear vibrations of composite beams — I. Equations of motion , 1990 .

[29]  John Dugundji,et al.  Experiments and analysis for composite blades under large deflections. II - Dynamic behavior , 1990 .

[30]  Izhak Sheinman,et al.  The Effect of Shear Deformation on Post-Buckling Behavior of Laminated Beams , 1987 .

[31]  Olivier A. Bauchau,et al.  Large displacement analysis of naturally curved and twisted composite beams , 1987 .

[32]  Rakesh K. Kapania,et al.  Recent advances in analysis of laminated beams and plates. Part I - Sheareffects and buckling. , 1989 .

[33]  G. Cowper,et al.  On the Accuracy of Timoshenko's Beam Theory , 1968 .

[34]  M. Sathyamoorthy,et al.  Literature Review : Nonlinear Analysis of Beams Part Ii: Finite Element Methods , 1982 .

[35]  Olivier A. Bauchau,et al.  Nonlinear Composite Beam Theory , 1988 .

[36]  Clive L. Dym,et al.  Energy and Finite Element Methods In Structural Mechanics : SI Units , 2017 .

[37]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[38]  Earl H. Dowell,et al.  An experimental-theoretical correlation study of non-linear bending and torsion deformations of a cantilever beam☆ , 1977 .

[39]  Richard A. Scott,et al.  Non-planar, non-linear oscillations of a beam—I. Forced motions , 1975 .

[40]  S. Timoshenko,et al.  X. On the transverse vibrations of bars of uniform cross-section , 1922 .