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Bilateral bounds for the shear and torsion factors: comments on elementary derivations
No full textThis paper proposes a new simple derivation of bilateral bounds for the strain energy-based shear and torsion factors, χi, of an elastic beam together with some comments about the coherence of the current formulations. A rearrangement of the definition as a mean over the cross-section and an original decomposition of the shear stress in two parts — τeqv that is equivalent to the external force and a residual Δτ — allow (i) to interpret (χ-1) as the mean quadratic deviation of the shear field with respect to the distribution τeqv and (ii) to easily evaluate an upper bound, using minimal information about the stress field. In this formulation the lower bound becomes trivial. Several numerical examples illustrate the accuracy and suitability of the results obtained
Exact deflation in the complex modal analysis of low-rank nonclassically damped structures
No full textThis paper presents the complex modal analysis for a proportionally damped structure
equipped with linear non-proportionally damped viscous elements (substructures or discrete real
devices) giving a low-rank contribution (r) to the non-proportional part of the damping matrix. Using
the classical undamped modes and a special low-rank matrix update formulation of the original
problem, the original Quadratic Eigenproblem (QEP) is hugely deflated, without approximations, to an
equivalent Rational Eigenproblem (REP) of dimension r << n (Theorem 2), as an alternative to the
linearized Standard Eigenproblem of order 2n over the complex field. The existence of classical modes
in non-classically damped structures is also discussed. The REP is solved by the homotopy method: a
robust predictor-corrector continuation algorithm is designed in order to determine the required
eigenpairs. Some application to simple models of both traditional and base-isolated structures,
together with an outline of future work, end the paper
Spectral Acceleration and Pseudo-spectral acceleration proximity
No full textThis paper deals with the deterministic relationship between spectral and pseudo-spectral acceleration, SA(csi,omega0) and PSA(csi,omega0) respectively, and gives the reason for their proximity on theoretical basis. The key step is to estimate the velocity at the acceleration extrema using the exact integral representation of the solution. When the product csi T0 < 1, each maximum displacement occurs csiT0/ Pi after each acceleration maximum.
In this case the maximum displacement is evaluated in closed form as a simple function of the maximum acceleration: the well-known proximity between spectral and pseudo-spectral acceleration is thus quantified for the first time. As an alternative, a very simple procedure is presented using a “first-order” version of the motion equation; despite its coarseness the result agrees with the more complex analysis discussed previously.
On the other hand, in Seismic Base Isolation systems (with large modal periods and damping ratios) a discussion of the relationship between PSA(csi,omega0) – giving the forces on the structure – and SA(csi,omega0) – governing the forces on the foundation – becomes crucial. Comparison of the obtained results with spectral quantities derived from a number of real accelerograms, along with a description of a typical counterexample in which displacement and acceleration extrema are not correlated, end the paper
Rigid folding representation by the Stereographic Projection
Full text linkThe Stereographic Projection machinery is applied to the qualitative and quantitative description
of the folding problem of an origami-like structure: the facets of a creased rigid sheet undergo
successive rotation about several creases arbitrarily located in the three-dimensional space. The
theoretical apparatus is revisited and some basic problems are depicted to emphasize both the
simplicity and the great efficiency of the representation. Several original results, useful to simplify
the trajectory tracing of the facet, are derived. The Representation of the Pole Path around a corner
is finally presented as an alternative synthetic description of the folded configuration. An
application to a multi-folded carton erection demonstrates the effectiveness of the proposed
formulation
Exact deflation in the complex modal analysis of low-rank non-proportionally damped structures
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