Optimal shakedown analysis of plane reinforced concrete frames according to Eurocodes

We present an updated mathematical model of shakedown optimization for reinforced concrete plane frames subjected to variable and repeated uncertain loading within a known domain. In such structures, plastic redistribution of forces is known to occur, and various mechanisms of system collapse at shakedown have been identified, such as plastic yielding and sign-changing. We develop a general nonlinear mixed-integer optimization problem that reduces to a linear programming problem, and we demonstrate the duality of the linear programming problem for the static and kinematic formulations. We derive strength conditions according to Eurocode 2 and an iterative process of optimization, where stiffness properties of frame elements are allowed to vary. The frame cross-sections are rectangular and made from doubly reinforced concrete; the material is considered composite. We successfully demonstrate the numerical optimization procedure on a two-storey reinforced concrete plane frame. We present variations of interaction loci of each optimized section in graphical form.     

Backscattering from tapered resistive strips

Resistive strips have lower backscattering cross sections than perfectly conducting strips, and this is true in particular when the illumination is edge-on with the electric vector parallel to the edge. The scattering then consists of edge contributions and attention is confined to this case. Using the available expressions for the edge contributions of uniform resistive strips it is shown that the front edge scattering decreases with increasing resistivity, whereas the rear edge scattering increases. This suggests that the resistivity should be tapered from a maximum at the front to zero at the rear, and numerical results are presented for the particular case of quadratic (parabolic) tapers. Values for the front and rear edge contributions are extracted from the scattered field data, and for strips more than about a half-wavelength in width it is found that the front edge contribution is almost identical to that for a uniform resistive half-plane. An empirical expression for the rear edge contribution is also derived, and the implications of the results are examined.     

Effects of errors in the resistive taper on the scattering patternsof strips

The relationship between random errors in the resistive taper of a strip using physical optics and integral equation models is shown. Error and no-error terms are calculated for the resistivity, strip current density, and scattering patterns. It is shown that the integral equation and physical optics models produce very similar results. These models allow one to separate the error and no-error terms in the strip current density and the scattering patterns in order to study the effects of random and correlated errors in the resistive taper