On the dynamic behaviour of masonry beam–columns: An analytical approach

Highlights

• The paper presents an analytical approach for the study of the transverse oscillations of masonry columns.

• The approach is first applied to free dumped oscillations.

• Then applications to forced oscillations in primary resonance on the column's first mode are shown.

• Some examples are presented, comparing the analytical results with those obtained via the NOSA-ITACA code.

• In the Appendix some remarks are included on the use of the averaged Lagrangian method.

Abstract

The paper presents an analytical approach to the study of the transverse vibrations of masonry beam–columns. Starting with the constitutive equation for beams made of a masonry–like material and the averaged Lagrangian of the system, some explicit approximate solutions are found to the problem of free damped periodic oscillations and forced oscillations in the case of primary resonance on the beam's first mode. In particular, a set of equations is obtained that gives the modulation over time of the system's energy and of the fundamental frequency of the beam's response. The analytical results are compared to those obtained via the finite element code NOSA–ITACA, developed at ISTI–CNR.

Introduction

A constitutive model is proposed in De Falco and Lucchesi, 2002, Zani, 2004 for masonry–like materials with zero tensile strength and infinite compressive strength, where the constitutive equation for masonry–like materials (Del Piero, 1989), (Di Pasquale, 1992), (Lucchesi et al., 2008), is specialized for masonry beams. The nonlinear elastic equation provided in De Falco and Lucchesi, 2002, Zani, 2004, which expresses the internal forces, normal force and bending moment, as functions of the generalized strains, stretching and change of curvature of the beam axis, has proven to be simple enough to enable some explicit calculations (De Falco and Lucchesi, 2002), (Girardi, 2009), (Girardi and Lucchesi, 2010), (Zani, 2004). At the same time, its numerical implementation in the MADY code (Girardi et al., 2012), (Lucchesi et al.,), (Lucchesi and Pintucchi, 2007), represents a quick and effective way to asses the effects of the load's eccentricity on the static and dynamic behaviour of masonry columns, arches and towers.

For cyclic actions, this approach can furnish reasonable results for slender structures, for which the influence of shear forces on the dynamic equilibrium tends to decrease and the nonlinear behaviour is due essentially to the opening of cracks. In fact, accurate modelling of the dynamic behaviour of masonry still represents an open problem. It is influenced by many parameters, involving the mechanical characteristics of the constituents materials, the construction techniques, the geometric characteristics, the kind of loading and the soil characteristics. Such complexity is extremely difficult to capture with a single model.

Techniques for modelling masonry structures range from very complex micro–mechanical approaches (Oliveira and Lourenco, 2004), (Luciano and Sacco, 1998), to rigid block modelling for limit analysis (Chetouane et al., 2005), (Heyman, 1995), to homogenisation techniques (Briccoli Bati et al., 1999), (Oliveira and Lourenco, 2004), (Sacco, 2009), (Zucchini and Lourenco, 2002) and continuum models (Berto et al., 2002), (Betti et al., 2011), (Gambarotta and Lagomarsino, 1997), (Ivorra and Pallarés, 2006).

With regard to the seismic vulnerability of masonry structures, the main instruments for engineers are based on vulnerability analyses derived from a statistic classification of the earthquake damage (Lagomarsino, 2006), (Min, 2008) and on the use of numerical codes developed by defining macro–elements with a small number of degrees of freedom. Very few examples can be found of analytical approaches to the dynamic problem of masonry structures.

In Girardi and Lucchesi (2010) the authors present an analytical study of the transverse vibrations of masonry beam–columns based on the constitutive equation described in De Falco and Lucchesi (2002), (Zani, 2004). They limit themselves to considering free vibrations and obtain an explicit relation between the fundamental frequency of the beam and amplitude of the displacement. In the present paper the study is generalized in order to include damped and forced oscillations. In order to simplify computations, use is made of the averaged Lagrangian method proposed by G.B. Whitham to study the modulation of nonlinear dispersive waves (Debnath, 1997), (Whitham, 1970), (Whitham, 1974). This method reduces the problem to the study of a set of nonlinear differential equations – the so–called modulation equations – for some parameters of the problem, specifically energy and frequencies, which, if the nonconservative terms are small, can be considered slowly varying over time. The averaged Lagrangian method, whose use in the present context is justified in the Appendix, allows obtaining the modulation equations without the manipulations typical of other conventional methods based on series development, such as the multiple scales method (Nayfeh and Mook, 1995), (Nayfeh, 2000).

All results presented here have been obtained by assuming a unimodal expression for the beam's displacements and considering transverse vibrations only. With regard to the unimodal assumption, mainly depending on the frequency content of the excitation, some damage related to the presence of higher modes has been observed in many masonry towers and also modelled by numerical codes (Callieri et al., 2010), (Casolo and Peña, 2007). However, the use of one single mode to describe the motion of slender masonry structures has generally proven to be able to capture some global damage patterns, such as the maximum displacements and the maximum compressive stresses along the structure. This hypothesis is also accepted by Italian regulations (Min, 2008), provided the structure's geometry is regular. The interaction between longitudinal and transverse vibrations is also recognized to cause additional damage in slender masonry structures, especially for high values of the compressive stresses and in presence of vertical components of the dynamic excitation.

To face the complex calculations involved in the analysis, we limit ourselves to considering opportune sets of initial conditions and harmonic transverse loads in primary resonance of the first mode, for which, provided that internal resonance phenomena do not occur (Nayfeh and Mook, 1995), a unimodal solution is expected (Lacarbonara and Yabuno, 2006), (Nayfeh, 2000). Under these assumptions, the influence of the longitudinal vibrations on the transverse response of the beam can be neglected as well (Lucchesi and Pintucchi, 2002) and, provided that the normal force acting along the beam is known, the problems of the transverse and longitudinal vibrations of the beam–column can be dealt with separately.

The paper is divided into three parts. In the first, the averaged Lagrangian method is presented and the modulation equations obtained for a broad class of nonlinear elastic materials. In the second, the method is applied to masonry–like materials, in the case of free damped and forced damped oscillations. Finally, the third part presents a parametric study, by varying on the one hand the slenderness and modal damping coefficient of the structure and, on the other, the forcing amplitude and frequency. All results are compared with those obtained numerically via the finite element code NOSA–ITACA (Lucchesi et al., 2008), http://www.nosaitaca.it/, developed at ISTI–CNR for static and dynamic equilibrium problems of masonry structures and constructions.