## Modal Shape Curvature

by Ardalan Sabamehr, Ph.D.

The mode shape curvature is an efficient technique in identifying damage by using the mode shapes of the structure. The curvature is related to the flexural stiffness of the beam’s cross section. If a crack or other damage is introduced in a structure, it reduces its flexural stiffness (EI) at its cracked section. The changes in curvature are local in nature and, hence, can be used to detect and locate a crack or damage in the structure. In order to calculate the curvature from a mode shape, a central difference approximation can be used as follows:

νi" = (νi+1 - 2νi + νi-1)/ h2

where h is the length of the elements (Pandey, A.K., Biswas, M. and Samman, M.M., 1991. Damage detection from changes in curvature mode shapes. Journal of sound and vibration, 145(2), pp.321-332).

Another technique used to detect damage locations is to use the Mode Shape Index (MSI). MSI has the following equation:

MSI = (Damage MSC) 2 – (Intact MSC) 2

In this blog, we will demonstrate how to detect damages and changes in a structure. The first case study is a multi-storey building. The schematic model of the building is shown in the following figure. An ambient vibration test using Sensequake Larzé sensors was performed on this building in only 3 hours.

After analyzing the measured vibration data with Sensequake 3D-SAM software, several vibration modes were derived. Floor 5 and 9 had gone through major retrofit and the sensing test was repeated after the upgrade. As shown in the following figure mode shapes, mode shape curvature and MSI were compared before and after the retrofit (only the 1st mode is shown). For this case study all three features (the mode shape, its curvature and its index) were able to detect the location of the change in the building.

The curvature of the mode shapes is more sensitive than the mode shapes for detecting damages. This can be better observed in cases with smaller damage. In the second case, the following bridge was modelled in a finite element software and damage was simulated by applying stiffness reduction factor to the following elements: elements 166, 167 (span 4) and 566, 567 (span 12) by 20% and elements 366, 367 (span 8) by 30%.

The MAC (modal assurance criteria) values between the damage and undamaged scenarios of the first six mode shapes were compared and are shown in the following figure. As seen, the MAC values are around 1 which indicate no change between the undamaged and damaged mode shapes. Therefore, comparing the mode shapes has been unsuccessful to detect the damages. However, the comparison of the intact and damaged curvatures for mode shape 1 and 3 show spikes at the location of the applied damage and reduced stiffness (shown by blue circles on the plots).

Therefore, mode shape curvature monitoring is the most sensitive feature for damage detection as compare to the frequency and mode shape monitoring.

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