In this article, an exact analytical approach is presented to analyze free vibration of a thin piezoelectric spherical shell, using thin shallow shell theory. The piezoelectric spherical shell is modeled as a sensor or an actuator. The piezoelectric material is polarized through the thickness of the shell. Using the separation of variables method as well as some new potential functions, the equations of motion and Maxwell’s equation are exactly solved, simultaneously. First, the equation of the transverse displacement of the shell is separately obtained and after extracting the transverse displacement, other unknowns such as the in-plane displacements and electrical potential function are obtained. Then, applying mechanical and electrical boundary conditions, the natural frequencies of the shell are obtained for the sensor and actuator cases. In order to validate the accuracy of the present method, the obtained results are compared to those obtained by a finite element analysis in ABAQUS software. Also, the effects of various parameters such as inner radius to radius of curvature of the shell ratio, thickness to inner radius ratio, and different boundary conditions on the natural frequencies are considered. Results show that piezoelectricity effect causes an increase in strain energy of the structure leading to increasing the natural frequencies for both sensor and actuator shells. Also, by changing the conditions from actuator state to sensor one, the structure experiences an increase in the natural frequencies.

Abstract
Surface fault rupture is very dangerous for critical buildings and infrastructures located in or near active faults and can cause irreparable damages. These structures must be designed by considering the undesirable effects of surface faulting. In this case, geotechnical measures, especially the construction of reinforced earth foundations are very effective in reducing the adverse effects of surface faults. The ASTM designation primarily recommends avoiding constructions to the adjacent of active faults probable of causing rupture at ground surface during an earthquake, although it is hard to determine the exact location of surface faulting. The increasing growth of population and the need to develop cities, particularly in metropolitan areas with economic limitations or land restrictions, have attracted the attention of the engineering community more than before to carry out feasibility studies on the construction of buildings in active fault zones. Such a consideration does not negate that the primary recommendation to avoid construction of buildings over active fault zones is the most convenient solution; it rather aims at examining and making engineering arrangements for the construction of buildings in zones with surface faulting potential governable by engineering methods. In addition to buildings, linear projects such as roads, highways, and tunnels must cross regions probable of surface faulting. Therefore, geotechnical measures, particularly designing reinforced soil foundations, contribute significantly to the reduction of undesirable effects of surface faulting. This research is conducted based on a series of tests on foundations reinforced with geogrid, geocell, and a combination of both, subject to normal faulting, to reduce surface faulting ruptures. The tests simulate the behavior of a 1.5 m wide strip foundation, placed over 6 m thick alluvium, subjected to a displacement of 60 cm. Seven tests were performed by different types and numbers of reinforcement, which were scaled to 10. The image analysis was carried out to examine the ground settlement profile, angular distortion, and fault propagation path. The results showed that the geotextiles used in the reinforced soil foundation could effectively reduce the angular distortion, cause uniform settlement, and divert fault propagation path, all protecting the structure against faulting. In a foundation reinforced with one layer of geogrid, a uniform settlement occurred at fault-induced displacement. In particular, the geogrid largely affected fault distribution, angular distortion reduction, and uniform ground settlement. Also, the settlement occurred at a wider zone and reduced the angular distortion by 60%. It means that the geocell affected the reduction of angular distortion and creation of uniform settlement by about 30%; however, it did not affect faulting diversion. The results indicate that the foundation reinforced with a combination of geocell and geogrid reduces angular distortion by 70% acted almost the same as the foundation reinforced with one layer of geogrid. Due to the increased stiffness and compressive strength of geocell, the shear band was more diverted toward the left side, compared to the foundation reinforced with one layer of geogrid. The right boundary of the shear band was also moved to the left corner of the structure. Likewise, the foundation reinforced with a combination of two layers of geogrid and three layers of geogrid reduces angular distortion by 80%. The results also reveal that adding more than two layers of geogrid had no effect on angular distortion reduction and the fault propagation path was more diverted as the number of geogrid layers increased.