7.1 Summary and Conclusions

A complete summary of the results obtained through the present research is provided in the following paragraphs.

In Chapter 3, a brief introduction of the theory of the experimental modal analysis is presented. The dynamic parameters of the two tile domes are experimentally obtained in terms of natural frequencies and the mode shapes identifiable by means of the experimental data. Although the axisymmetric mode shapes are not strongly observed in experimental data, the natural frequencies of the first two axisymmetric mode estimates of the FE model are confirmed by weak peaks in the FRF plots. Recommendations for the future applications of the techniques of experimental modal analysis to large-scale masonry domed structures are provided.

Chapter 3 also introduces another nondestructive experimental technique, impact-echo, which is useful to obtain the thickness of the dome webbing. In this study, the underlying theory of impact-echo is applied with alternative equipment, and satisfactory results are obtained.

Chapter 4 presents theories of thin elastic shells and their application to the structures under consideration. Since it is necessary to establish several assumptions through the application of the thin elastic shell theory to masonry structures, the results are not anticipated to represent the actual behavior of the tile domes. Instead, an approximate solution to compare the relevancy of EMA and FE is intended.

The thin elastic shell theory and experimental modal analysis results are not fully comparable as the theory of thin elastic shell only presents axisymmetric modes, and they were not strongly observed with EMA in this study. Therefore, FEA is used as an intermediate tool to examine the correlation between the two formerly discussed methods.

In Chapter 5, based on comparisons of the two methods discussed in Chapter 3 and Chapter 4, the boundary condition inputs to the FE models are calibrated. Through this calibration process, the material properties obtained from the laboratory tests are tuned. Based on the material tests conducted on the samples obtained from the SEB and the results obtained from the tuning process, a basis for Young’s modulus, Poisson’s ratio and density are presented for future studies on the Guastavino tile and mortar assembly. It is seen in the results of the Fairbanks tests that manufacturing consistent tile and mortar was difficult in the late nineteenth – early twentieth century. Additionally remembering the continuous experimentation process the company went through, the mechanical properties of the tile and mortar assembly provided in this study present reasonable starting values for preliminary analysis of future studies.

The validated FE models are then subjected to sensitivity analysis in order to identify the input entry to which the model is most responsive. The sensitivity analysis reveals that a 1% change in the material properties –Young’s modulus or density– results
in 0.46% change in natural frequencies. If the desired level of accuracy in the FE solutions is very high, it may be necessary to conduct material tests on samples obtained from the structure. In this case, Young’s modulus, Poisson’s ratio and density need to be established for both tile and mortar. Once the material properties of each component are know, the effective values for the assembly can be achieved through homogenization equations.

The mesh sensitivity analysis revealed that using elements of proper size has importance while validating the FE model based on the modal analysis solutions. Too coarse a mesh imposes unnecessary restraints and overestimates the natural frequencies. For future studies, a mesh size of approximately 20 cm is found to be of the same order of accuracy as the material property estimates. For future studies on Guastavino tile masonry structures at the scale of 5-10 m radius, this mesh size is appropriate, as it does not promote spatial aliasing in the first few modes. FE model results deliver remarkable accuracy for the modal behavior of the domes. The pendentive domes analyzed in the present study do not exhibit the third and higher order axisymmetric mode shapes. This aspect is possibly due to the geometric deviation from a spherical segmental dome to a pendentive one, which disturbs the axisymmetric nature of a spherical segmental dome and imposes unsymmetric boundary conditions.

The analysis continues on the static behavior of the two non-bearing tile domes. The contour plots of principal stresses as well as the normal stresses are obtained through the static analysis solutions of the FE model. The results are compared to the visual condition investigations on the existing domes. The FE model of the CCB reveals critical stress concentrations around the buttress locations where the cracks occur in the actual structure. The FE model of the SEB results in a very limited stress level.

The present study illustrates that the domes built in the Guastavino system behave linearly and elastically provided that the maximum allowable tensile stress capacity of the tile and mortar assembly is not exceeded. Where significant cracks are present in the structure –as in the CCB– the accuracy of the validated FE model is degraded. For these cases, nonlinear analysis may be useful for better accuracy.

For both structures, validated FE models exhibit significant horizontal thrust acting toward the adjacent members around the domes. The amplitude of the horizontal reaction forces are found to be of the same order as the vertical load acting on one pier.

The final chapter introduces Rafael Guastavino to the reader and provides brief background information on the cohesive construction system. Based on the results obtained from the completed research program, the claims established in public about erecting this type of domes are investigated. The existence of a horizontal thrust is verified, and Guastavino’s attitude towards the subject is discussed. A limited discussion on the contribution of Guastavino to the architectural design of the buildings in this study is also presented.

7.2 Recommendations for Future Research

The findings of this thesis yield several possible future research topics:

• A further study can be done to experimentally acquire the axisymmetric mode shapes of large-scale masonry pendentive domes. This enables the comparisons between the three methods utilized in this study, the thin elastic shell theory, EMA and FEA.
• The investigation of the thickness variation from abutments toward the apex by the techniques of impact-echo can provide valuable information for the accurate modeling of the masonry domes. The technique also has the potential to identify the thickness of the each tile and mortar layer, as well as to detect the existence of a possible delaminating between tiles. Furthermore, for cases where the thickness of the shell is known, the techniques of IE can deliver the E/ρ ratio for the masonry assembly.
• Since in the CCB the massive brick arches constrain the rotation and translation of the dome webbing, it is necessary to force the fixed end boundary condition to the solutions of the governing differential equations. The application of the thin elastic shell theory to the CCB is limited to a hinge end condition. Therefore, a successful representation of the actual structural behavior is not achieved in this study. Developing a computer code, which is easily adjustable to any dimension and geometry of dome as well as any boundary condition and material property, can serve as a reference to future researchers to validate the FE models when experimental data are not available.
• The Guastavino material properties are difficult to assess unless material samples are available. Solid information is not available in literature other than the present study and 2001 paper of Saliklis. A thorough investigation, providing chronological data on the material properties used by the Guastavino Company throughout their active career can add an important knowledge to the literature.
• The linearly elastic assumption gives reasonably accurate results for domes with minor cracks, while utilizing a nonlinear FE analysis yields better accuracy for cases when the domes have severe cracks. Defining a level for crack severity, which invokes nonlinear behavior, can be a valuable reference for future researchers.
• A complete study on the available documentation–for instance the letters of Guastavino to all the construction engineers and architects he worked with–can appraise the role of Guastavino as a consultant in the architectural design process