Wave propagation in a doubly tapered shear beam: Model and application to a pyramid‐shaped skyscraper

Abstract

One dimensional wave propagation is studied in a cantilever conical structure deforming in shear. The structure has rectangular cross-section both dimensions of which decrease linearly along the polar axis (hence, doubly tapered). The equation motion is derived and solved in terms of spherical Bessel functions, which can be expressed exactly as finite algebraic expressions in terms of powers and sine and cosine functions of the polar coordinate. The transfer-matrix for a truncated pyramid element is then derived and generalized to a chain of elements. The model is used to represent a 48-story pyramid-shaped steel-frame skyscraper, the Transamerica Tower in San Francisco, California, in which the Loma Prieta, 1989 earthquake (𝑀 w = 6.9, epicentral distance, 𝑅 ≈ 90 km) was recorded by an array of accelerometers. The building is modeled by homogeneous and layered truncated pyramids. Wave propagation through the building is studied by analysis of impulse response functions computed from the observed earthquake accelerations. In addition, its equivalent homogeneous beam shearwave velocity is identified solely from its geometry and observed fundamental frequency of vibration. Further, the variation of wave velocity along is height is identified by least squares fit of a layered pyramid model in the observed impulse response functions. The results reveal equivalent homogeneous pyramid wave velocity of about 150-160 m/s in both directions, which is similar to other steelframe structures. The identified wave velocity profiles are consistent with the structural design. The identified parameters can be used as reference in future structural health monitoring of this structure.