By Angelo Luongo
Nonlinear types of elastic and visco-elastic onedimensional non-stop constructions (beams and cables) are formulated by way of the authors of this identify. numerous types of accelerating complexity are provided: straight/curved, planar/non-planar, extensible/inextensible, shearable/unshearable, warpingunsensitive/ delicate, prestressed/unprestressed beams, either in statics and dynamics. usual engineering difficulties are solved through perturbation and/or numerical methods, comparable to bifurcation and balance lower than strength and/or tangential a lot, parametric excitation, nonlinear dynamics and aeroelasticity.
Contents
1. A One-Dimensional Beam Metamodel.
2. immediately Beams.
3. Curved Beams.
4. Internally limited Beams.
5. versatile Cables.
6. Stiff Cables.
7. Locally-Deformable Thin-Walled Beams.
8. Distortion-Constrained Thin-Walled Beams.
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Extra resources for Mathematical Models of Beams and Cables
Sample text
E. 30] General linear constraints The constraints εc = 0, so far considered, are probably so simple that they hide some interesting aspects of the problem. 31] where B is an Mc × M constant matrix. Of course, if B = [0, I], the previous case is recovered. Later on in the book (Chapter 8), constraints like this will be addressed. 32] H=A S S B ˙ T pds + w ˙ (σ Ta ε˙ + λT B ε)ds = where we denoted by σa the active stresses and by λ the Lagrangian multipliers. Note that, differently from the particular case examined previously, we did not introduce the constraint in the active part of the internal power.
We consider the internal action that two parts of the beam mutually exchange at T the abscissa s and time t, and denote by f = (fi (s, t)) , i = 1, . . e. forces and couples) acting on one of the two parts, conventionally assumed as positive. Note that the internal force components are in the same number of the degrees of freedom (translations and rotations) of the “rigid” point P . We then consider an infinitesimal element of length ds, loaded by external T forces per unit length p := (pi (s, t)) , at whose ends, internal forces f (s, t) and f (s + ds, t) = f (s, t) + f (s, t)ds act.
However, if wi is needed, for example to evaluate inertia forces proportional to w ¨i , we should integrate, thus obtaining wi = f wj ds, and also using geometric boundary conditions. In these circumstances, elimination of the variables could be inconvenient, and the mixed formulation would be preferable. Hybrid procedures, in 16. A similar circumstance occurs in Lagrangian mechanics, when non-holonomic constraints exist, which involve time-differentiated displacements. A One-Dimensional Beam Metamodel 19 which only a sub-set of the variables is eliminated, are also possible, and relevant examples will be illustrated further on in the book (Chapter 4).