Seismic Behavior of Masonry Cloister Vaults

DOI: 10.22201/fa.2007252Xp.2018.18.67942 Abstract This paper presents the results of an investigation into earthquake resistance and the behavior of historical masonry cloister vaults. In a previous paper,1 the authors presented a simplified structural analysis approach, called the ‘strip method of analysis,’ to assess the lateral load behavior of a relatively shallow cloister vault. In this paper, two cloister vaults with the same span but with different crown heights are studied. The vault material is assumed to be elastic and a linear thin shell F.E.A. is used for analysis under gravity loads and seismic loads. Nodal reaction components and diaphragm shears are compared for the two cloister vaults, along with those obtained using the strip method of analysis. A compression surface is determined for a selected finite element strip based on in-plane compression forces and out-of-plane bending moments. This is to indicate where tensile stresses will develop due to bending and hence where the vault will develop cracking and go into a limit state. The results include contour plots of membrane stresses, shear stresses, bending stresses and deformations in the masonry cloister vaults under gravity and lateral loads, as well as a representation of the lateral load resistance mechanism of masonry cloister vaults when subjected to earthquakes.


Introduction
Historic masonry vaults have been an integral part of the rich architectural heritage of Europe, the U.K., U.S.A., Latin America and beyond. A significant portion of these low-strength masonry vaults are located in seismic zones. The widespread inventory of cloister vaults in historic structures around the world requires an examination of their performance and behavior during recent destructive earthquakes and the lessons that can be derived therefrom. The observed damage data on historic vaults and domes from past destructive earthquakes suggests that low-strength masonry vault structures provide improved performance during severe earthquakes. In general, the state of knowledge of the performance and behavior of low-strength masonry buildings has advanced a great deal during the past several decades. Despite these advances in the state of knowledge, there is a need for practical structural analysis approaches to understanding the seismic resis- • segunda época • año 9 • núm. 18 • México • unam • diciembre 2018 • 42-56 tance mechanism of historic masonry cloister vaults. A cloister vault is defined as a structure consisting of four singly curved surfaces intersecting along ridges that span diagonally across to opposite corners of a square plan, as shown in figure 1. The ridges meet at the crown of the vault (the high point).

Previous work
In a previous paper, 2 the authors used a simplified structural analysis approach, known as the strip method of analysis, to assess the gravity and lateral load behavior of a cloister vault. This vault was relatively shallow, that is, the crown-height-to-span ratio was low. The authors were interested in diaphragm action and out-of-plane bending in the vault. Diaphragm action consists of in-plane resistance of shear forces in the planes of the vault that are parallel to the applied lateral loads. Out-of-plane bending was assumed to be prevalent in planes perpendicular to the motion of the lateral loads. Tensions due to bending, combined with shear forces, would be an indication of the limit state of the cloister vault.   Nicola Augenti. 5 Furthermore, the observed damage of the cross vaults in the Cathedral of Modena during the 2012 Emilia, Italy earthquake and a detailed study of its seismic behavior and safety has been carried out and presented by Tomaso Trombetti, Simonetta Baracanni, Stefano Silvestri, Giada Gasparini and Michele Palermo. 6 A significant knowledge base of the observed damage to historic masonry vaults is further provided by the observed damage data from destructive earthquakes in Portugal, Turkey, Greece, Iran and India, among others, in recent decades.

Lateral (seismic) load analysis
The authors are interested in developing a better understanding of the mechanism of earthquake resistance and seismic behavior in lowstrength masonry cloister vaults. The cloister vaults selected for analysis have a square plan configuration and variable height, as shown in figures 2 and 3.    Seismic loading for seismic analysis is based on an approximate Peak Ground Acceleration (pgA) of 0.25g. Therefore, lateral (seismic) loading is applied as equivalent to 25% of the weight of the vault system.

Strip method of analysis
In this practical analysis method, the vault is assumed to be divided into a series of parallel strips, as shown in figure 4. Each of the strips is assumed to act as an arch spanning the edge supports and the diagonals. A free-body diagram of a typical arched strip is also presented in the lower part of figure 4. Based on the mass distribution along the finite arched strips, the seismic loads were assumed to be 25% of the strip weight and the resulting seismic load distribution for the strips was determined for 0.574m and 1.07m high cloister vaults, as presented in figure 5.

Results
The masonry cloister vaults of the two proportions investigated, i.e. heights of 0.574m and 1.07m with the material properties defined above, were analyzed using SAP 2000 under the action of gravity and lateral (seismic) loads. The typical deformed shape of the 1.07m high cloister vault obtained from the 3D F.E.A. model structural analysis is presented in figure 7.
The horizontal and vertical reactions to gravity loads are compared using the strip method and the F.E.A. model for the two vault heights in figures 8 and 9. The F.E.A. model shows lower maximum values at the centerline strip and distribute more across adjacent strips.  The distribution of seismic force in the walls parallel to the lateral force is presented in figure 10; diaphragm action can be seen in the vault planes connected to the walls. Diaphragm action appears to be greater in the 1.07m high vault due to its greater weight.
The distribution of seismic force in the walls perpendicular to the lateral force is presented in figure 11 for across the wall shears on top of the walls. It can be noted that the 1.07-meter-high vault has smaller reactions at or near the centerline of the vault. It appears that this vault is more 'flexible' when bending.   The thrustline diagram for the 1.07m high cloister vault with a center parallel to the line of action of the lateral (seismic) force is presented in figure  12. This figure indicates where the tensile and compressive stresses due to bending occur in the vault due to gravity and lateral (seismic) loads. The strip method indicates tensile stresses at the surface of the vault due to gravity and seismic loads. The F.E.A. method indicates the 'thrustline' to be within the 'middle third' of the vault section. Hence no or very low tensile stresses develop in the vault due to bending.
A comparison of the lateral (seismic) load transfer through diaphragm action and bending action was made for the shallow and the tall cloister vault based on the strip method of analysis and the F.E.A. model. These results are presented in tables 1 and 2.

Conclusions
• A comparison of the mechanism of lateral (seismic) load transfer for the shallow cloister vault (0.574m height) and the tall cloister vault (1.07 m height) shows that the strip method of analysis is conservative for the shallow cloister vault and about the same for the tall cloister vault when compared to results from the F.E.A. method, as shown in tables 1 and 2. • The Finite Element Analysis method indicates that both cloister vaults will be able to safely resist the lateral (seismic) loads of 25% of the vault weight and transfer them to the supporting walls. • The shear forces along the walls and across the walls are low for the assumed material. The shear stresses that develop, as presented in figures 17 and 18, are well within the material's limits. • A study of the 3D plots of maximum principal moments and principal stresses indicates the occurrence of higher stresses in the same general area as compared to those shown on the thrust line diagram shown in figure 12 and the 3D plots presented in figures 13-16. • The 'thrust-line' diagram from the F.E.A. model analysis results indicates that the 1.07m high vault will be able to withstand seismic forces through bending in the vault planes perpendicular to the line of action of the lateral (seismic) loads. • Tension stresses develop along the diagonals and at the corners, but are well within the material's limits, as shown in figures 14 and 16. The cloister vault will likely develop fracture lines across the vault where tension due to bending begins to develop across the vault, rather than in shear at the wall supports, for lateral forces greater than indicated in the paper (see figures 13 and 15). Ingeniera principal en Nabih Youssef & Associates con cuatro años de experiencia en la profesión de ingeniería estructural. Ha sido parte del Comité de Educación de la Asociación de Ingeniería Estructural del Sur de California (SeAoSC) en los últimos años. Tiene una licenciatura y maestría en ciencias en Ingeniería Arquitectónica por la Universidad Politécnica Estatal de California, en San Luis Obispo, California.