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This paper investigates the behavior of a typical Chilean reinforced concrete wall building under sequential earthquake and tsunami actions using a double pushover analysis approach. A real concrete wall structure that was damaged after the M w = 8.8 2010 Maule earthquake is considered as a case study building and a simplified nonlinear finite element model of a fictitious slice of the building is subjected to earthquake and tsunami loading in sequence. The analysis of the building under these loadings consists of three stages: the structure is first subjected to seismic loading by means of a pushover analysis until a specific damage state is reached; then a pushover with the same load pattern but in the opposite direction is applied until the shear at the base is zero; and finally, the building is subjected to tsunami loading by means of a variable depth pushover analysis until the maximum capacity is reached. Different tsunami load cases are considered in this study, varying the Froude number and the direction of the tsunami loading. The results show that the tsunami response of the building is mainly dependent on the Froude number of the tsunami flow; however, when the seismic damage is severe, the tsunami capacity of the building is found to be reduced. This is more likely to occur when the effect of the tsunami increases the damage previously induced by the earthquake in the same direction.

The problem of enforcing two-variable nonlinear interaction code provisions within a spectral superposition approach to design, is addressed. An analytical procedure based on the linearization of the interaction curve through a set of tangents or secants is developed. Safety with respect to the interaction curve is approximated by requiring safety with respect to that set of straight lines. For calculating the required estimators, the cross-estimator based formula for estimators of the linear combination of variables, derived in a companion paper, is used. The analytical method developed was shown to be equivalent to a graphical method proposed by Gupta, based on inscribing an estimator ellipse within the interaction curve. The analytical method is more straightforward and handles the directional maximum in single-component excitation, which the graphical method does not. The analytical method is much easier to apply. However, if a study calls for actually drawing the estimator ellipse, straightforward construction methods are presented, making unnecessary a cumbersome equivalent modal response approximation that had been suggested. The inherent antisymmetry of the spectral superposition formula, and its implications, are discussed. The effect of static loading and of symmetry of the interaction curve is also analyzed. Within the analysis of an example, the work required for the application of the procedure for nonlinear interaction for design purposes is discussed. It is found that its implementation can be achieved through a very simple function written for any standard numerical computation software package. The example also makes quite apparent the advantages of the analytical over the graphical method. The application example, which considers the design of a concrete column in a simple 10-storey building structure, shows the overconservativeness of a design based only in the standard estimators. The design criterion used in the example is that of the most unfavorable direction of a single-component earthquake.

Modal spectral superposition for single-direction excitation is reviewed, with special attention to the intervening cross-directional estimator. A method for calculating this cross-estimator in terms of three standard estimators, the x-, y-, and 45°-direction excitation estimators, is presented. Addressing the question of the signs of cross-estimators, it is shown that the problem dwells in double-sum superposition formulas rather than in the cross-estimators themselves. In that respect, it is shown that in a double superposition formula, the modal coordinates of strongly coupled modes must enter with the same sign, so that there is no loss of information in the suppression of signs in spectral ordinates. This discussion is in itself a deterministic justification of double-sum spectral superposition formulas, showing that for closely spaced modal frequencies, the formula is an approximation rather than estimation. A formula is derived for the estimator of a linear combination of two variables, equivalent to calculating the estimator by directly applying the spectral superposition formula to the linear combination of the modal components of the two basic variables. It only requires the cross-variable estimator, in addition to the standard estimators of the two basic variables. It is stressed that contrary to common belief, statical or kinematical linear combinations can be checked or otherwise used, if the appropriate cross-estimators are retrieved from analysis. Cross-estimators are interpreted as the product of the standard estimator of one variable times the pseudo-synchronous value of the other variable. The appropriateness of the synchronous value concept is verified, and it is used to propose a member single or double curvature criterion.