Sigmund A. Freeman proposed the capacity spectrum method in 1987, which simplifies down the responses to a single-degree-of-freedom system (referred to as the capacity curve, hereafter) and compare it with the relationship between the response acceleration and displacement spectra (referred to as the demand curve, hereafter). Recently, some researches have been conducted in developing a method to calculate the capacity curve with measured accelerations during an earthquake to evaluate the building damage. Authors had proposed a method to predict the maximum response for the possible maximum aftershock to predict the building damage during the aftershock. In order to evaluate the damage due to an aftershock, safety limit deformation on the performance curve needs to be predicted. Although the safety limit deformation of each story can be estimated from a structural calculation or seismic evaluation, the safety limit deformation cannot be predicted with only them because it depends on the equivalent vibration mode shape at/to the safety limit state.
The vibration mode shape may change according to the damage. Therefore, a new method to evaluate the safety limit deformation with considering the vibration mode shape after yielding is proposed. First, a method to model the performance curve to tri-linear model with a simple error function is proposed. Second, a method to predict the safety limit deformation with an assumption that the incremental deformation mode shape from the second corner point to the maximum response point of the tri-linear model is constant. Mode adaptive pushover analysis results with 4-story, 6-story (soft-first-story), and 12-story buildings conducted by Dr. Matsumoto et. Al., which considered the change of vibration mode shape due to damage, and E-Defense shaking table test with 3-story R/C frame structure with pile foundation were used to verify the validity of the proposed modeling and prediction methods. The results were discussed with the relationship between error functions according to the assumed displacement at the second corner point and accuracy of the predicted safety limit deformation. With the pushover analysis results, the accuracy of the predicted safety limit deformation with pushover results that were intentionally stopped before the safety limit state was also discussed.
As the results, the following findings were obtained;
1) A modeling method for the performance curve to tri-linear model is modified and proposed.
2) It is confirmed by the pushover analysis results and shaking table test results that the proposed method can find an appropriate second corner point with a proposed error function.
3) A prediction method of the safety limit deformation on the performance curve with the tri-linear model from measured responses is proposed.
4) It was found that the safety limit deformation can be predicted within 10% error when the maximum response is around the second corner point of the tri-linear model, and within the few percent error when the maximum response exceeds the second corner point.
5) The error of the prediction converges faster for a story failure mechanism than for a total yielding mechanism.
