During the last decades, great efforts have been done in developing numerical formulations and their implementation in computer codes for the simulation of the nonlinear dynamic response of RC structures, for example a recent state of the art review for the case of concrete structures can be found in [295].
The engineering community agrees with the fact that the use of general fully 3D numer- ical technics, such as finite elements with appropriated constitutive laws, constitute the most precise tools for the simulation of the the behavior of RC buildings subjected to earthquakes [156, 296] to other kind of loads [172]. However, usually the computing time required when using full models of real structures became their application unpractical. Several approaches have been developed to overcome this difficulty; some authors propose the use of the so called macro–elements, which provide simplified solutions for the anal- ysis of large scale problems [90, 91, 101]. Considering that most of the elements in RC buildings are columns or beams, one–dimensional formulations for structural elements, obtained trough the reduction of spatial dimensions by means of kinematic assumptions [4, 67], appear as a solution combining both numerical precision and reasonable computa- tional costs [203]. Experimental evidence [37] shows that nonlinearity in beam elements can be formulated in terms of cross sectional forces and/or moments and displacements
and/or curvatures, which is frequently quoted in literature as plastic hinges models [78] (see §2.2). Some formulations of this type have been extended to take into account geomet- ric nonlinearities [241, 293, 317, 318] allowing to simulate the P –∆ effect, which occurs due to the changes of configuration of the structure during the earthquake [65, 118, 293, 309]. Several limitations have been reported to this kind of models, specially for the modeling of RC structures with softening behavior in the dynamic range [299] (this aspect is covered in §4.1). A discussion about topics such as step-by-step methods, path bifurcation, overall stability, limit and deformation analysis in the context of the plastic hinges formulation for beam structures can be consulted in [77].
An additional refinement is obtained considering inhomogeneous distributions of materi- als on arbitrarily shaped beam cross sections [226]. Therefore, using this approach the mechanical behavior of beams constituted by complex combinations of materials, such it is the case of RC beams, can be simulated [121, 122, 30]. In general, the engineering community agree with the fact that although this models are more expensive, in terms of computational cost, than the plastic hinges ones, they allow to estimate more precisely the response nonlinear response of RC and other kind of structures [23, 89, 299]. Formulations of this type, considering both constitutive and geometric nonlinearity are rather scarce [94]; moreover, most of the geometrically nonlinear models for beams are limited to the elastic range of materials, as it can be consulted for example in Refs. [138, 201, 277] and the treatment of constitutive nonlinear behavior has been mainly restricted to plasticity [48, 117]. In reference [106] a theory for the stress analysis of composite beams is pre- sented, however the formulation is only valid for moderated rotations and the behavior of the materials remain in the elastic range. Recently, Mata et al. [203, 205] has extended the geometrically exact formulation for beams due to Reissner–Simo [257, 256, 277, 278, 280] for considering and arbitrary distribution of composite materials con the cross sections for the static and dynamic cases.
Geometrically exact formulation for
rods
This chapter is devoted to the presentation of a geometrically exact formulation for rods capable of considering large displacements and rotations. The present formulation is based on that originally proposed by Simo [277] and extended by Vu-Quoc [278, 280], which gen- eralize to the full three-dimensional dynamic case the formulation originally developed by Reissner [257, 256] for the plane static problem. These works are based on a convenient parametrization of the three-dimensional extension of the classical Kirchhoff–Love1 [182]
model. The approach can be classified as a director type’s one according to Antman [4, 6], which allows to consider finite shearing, extension, flexure and torsion. In the present case, an initially curved and unstressed rod is considered as the reference configuration in an analogous approach as Ibrahimbegovi´c et al. [138, 142].
First, a detailed description of the kinematic assumptions of the rod model is carried out in the framework of the configurational description of the mechanics. Due to its impor- tance in the development of time–stepping schemes in next chapters, special attention is paid to the formal definition of the nonlinear differentiable manifolds that constitute the configuration, placement and their tangent spaces. After defining translational and rotational strain vectors and calculating the deformation gradient tensor, a set of strain measures at material point level on the cross section are described following the develop- ments of Kapania and Li [167, 168]. However, the developments are not limited to the static case and explicit expressions for the material, spatial and co–rotational versions of the strain rate vectors as functions of the spin variables are also provided. At material point level, the conjugated stress measures are deduced from the principles of continuum mechanics and using the power balance condition for deducing the stress measure ener- getically conjugated to the cross sectional strain measures.
The equations of the motion of the rod are deduced starting from the local form of the lin- ear and angular balance conditions and integrating over the rod’s volume. A form (weak) appropriated for the numerical implementation is deduced for the nonlinear functional corresponding to virtual work principle, considering the noncommutative nature of a part of the admissible variation of the displacement field.
1The Kirchhoff–Love formulation can be seen as the finite strain counterpart of the Euler formulation
for beams frequently employed in structural engineering [24, 89].
3.1. Kinematics 24 Finally, a discussion about the deduction of reduced constitutive relations considering hyperelastic materials is presented, leaving the detailed treatment of the rate dependent and independent constitutive nonlinearity for the next chapter.
3.1
Kinematics
For an appropriated description of the three-dimensional motion of rods and shells in finite deformation (and in the rigid body dynamics [4, 47, 86, 148]) it is necessary to deal with the (finite) rotation of a unit triad and therefore, the results of Appendix A will be used repeatedly here to describe the Reissner–Simo geometrically exact formulation for rods.
First, it is necessary to define the orthogonal frame { ˆEi} which corresponds to the material reference frame of the configurational description of the mechanic, and it is defined to be coincident with the fixed spatial frame {ˆei} by convenience. The concept of spatially fixed means that the corresponding spatially fixed objects are fixed in an arbitrarily chosen orthogonal frame2 {ˆe
i} that has no acceleration nor rotation in the 3D inertial physical
space [280].