A nonlinear composite beam theory

Abstract

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 yon 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 yon Karman type nonlinear theory.

Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium

“…Our heuristic homogenization procedure is applied first for a class of nonlinear onedimensional continua (beams), focusing on modelling phenomena in which both extensional and bending deformations are of relevance, and subsequently for the class of two-dimensional continua studied in [6]): in both cases, we limit our attention to planar motions. In fact, there are relatively few results in the literature of nonlinear beam theory: we recall here the very first classical results by Euler and Bernoulli [41,42] and the research stemming from von Kármán [43,44] for moderately large rotations but small strains. Moreover, very often in the literature, the simultaneous extension and bending deformation for nonlinear beams are not considered: however, when considering two-dimensional continua embedding families of fibres (e.g.…”

Highly Flexible Structures: Modeling, Computation, and Experimentation

“…77 shows the use of two rectangular frames, each with one enhancement ring, to construct a structural cell. The rings AB of both frames need to be bound or interlocked at the beginning of erection, and points C, D, E, F, G, 1, 2, 3, 4, 5, 6, and 7, 72 Packing of a circular frame into a) three rings (perspective views) and b) nine rings (see MOVIE572.mpg, MOVIE572.avi, MOVIE572_1.mpg).…”

Exact mathematical solution for nonlinear free transverse vibration of beams

“…Now, consider the Lagrangian , , x y z and Eulerian * * * , , x y z coordinates systems with a common origin before deformation ( Figure 2). Equations of motion are derived in Lagrangian coordinates 6 system because it has a fixed origin. In the linear case, on each face of the infinitesimal element there is only one component of stress which is in the direction of the coordinate axes.…”

“…In this case bending and stretching stiffness have interaction with each other [5]. Pai and Nayfeh [6] showed that Von-Karman strains cannot be used to derive fully nonlinear beam model. To fully account geometry nonlinearity, the second Piola-Kirchhoff stresses should be used instead of Cauchy stresses.…”