Empirical estimates of the fundamental frequency of tall buildings vary inversely with their height, a dependency not exhibited by the various familiar models of beam behavior. This paper examines and explains this apparent discrepancy by analyzing the consequences of using two models to estimate such natural frequencies: A two-beam model that couples the bending of a classical cantilever to that of a shear beam by imposing a displacement constraint; and a Timoshenko beam in which the Euler-Bernoulli beam model is extended by adding a shear-displacement term to the classical bending deflection. A comparison of the two beam models suggests that the Timoshenko model is appropriate for describing the behavior of shear-wall buildings, while the coupled two-beam model is appropriate for shear-wall-frame ͑e.g., tube-and-core͒ buildings, and that the coupled-beam model comes much closer to replicating the parametric dependence of building frequency on height.

This paper presents analytical estimates of the behavior exhibited by curved, archlike structures under radially directed and gravitational line loads. The behavior is shown to range from elementary beam bending at one end to a state of pure compression at the other, and its behavior can be tracked by an arch rise parameter that is a function of the arch's semivertex angle, radius and thickness. The principal results are useful estimates of the dependence of the major displacements and stress resultants on the arch rise parameter. The results also offer some insight into the assumptions underlying Robert Maillart's arch designs.

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