Dynamic behavior of buildings with non-uniform stiffness along their height assessed through coupled flexural and shear beams

A novel model for assessing building behavior has been developed by coupling a Bernoulli beam with a quartic stiffness variation and a shear beam with a parabolic stiffness variation, trends that are expected in buildings designed for earthquake actions. Then the partial differential equation of motion governing the behavior of the model has been solved, obtaining analytic expressions (Closed form solutions) for mode shapes in terms of Legendre functions. These closed form solutions were validated with finite element model analyses and effects of non-uniformity of stiffness were assessed in a generalized manner. It was found that period lengthening is mild for the first mode, but for higher modes can be far more noticeable if shear stiffness at beam top is <20 % of its base value. Mode shapes also change notoriously for reductions beyond the same limit, potentially inducing large floor acceleration demands at unexpected locations. Also it was found that drift demands can be noticeably enhanced even if shear stiffness at top is 75 % of the base value, in what would be considered uniform buildings. This model has several applications for assessing the response or large stocks of buildings, calibrate complex models, assess damage on building contents, establishing in short time damage scenarios for large cities, and could be helpful for education, as emphasis is brought back on fundamental concepts.