Dynamics of rocking elements with horizontal restraints

The seismic vulnerability of historic masonry structures is a quite known issue for the preservation of the architectural heritage. Historic buildings or churches are properly analyzed with local approaches if no effective connections between structural elements are present. Structurally independent parts of the building, called macro-elements, can be regarded as rigid rocking blocks oscillating under the seismic action. Their motion may be described by means of the Housner's formulation [1], which describes the response of a rectangular free-standing block subjected to several types of ground motions. The significant geometric parameters of the block are the slenderness ratio and the length of the semi-diagonal R. The damping effects are evaluated by means of the restitution coefficient, which decreases the velocity after each impact [2]. The general equation of motion, obtained in [1], is here modified to account for different boundary conditions in terms of horizontal restraints. These restraints might recreate vaults, transverse walls or retrofitting techniques such as steel tie-rods. The horizontal restraint that reproduces vaults or tie-rods may be described by concentrated spring, while continuous transverse walls can be regarded as Winkler spring bed. The equations of motions are obtained in both cases by applying the principle of virtual work and the Euler-Lagrange equation. The modified equation of motion is then implemented in a specifically written MATLAB code [3] suitable to assess the survival or the collapse of the equivalent block. The macro-element seismic vulnerability can be therefore determined by choosing proper seismic records and dissipation properties. A masonry façade of a church struck by L'Aquila earthquake (Italy, 2009) is modeled as equivalent rectangular block with different boundary conditions: free-standing block, block connected to the inner vault and block connected to the inner vaults and steel tie-rods. A comparison with kinematic analysis suggested by the Italian code [4] is also performed. The kinematic approach is shown to be over-conservative with respect to the rocking analysis, as already found in literature [5].