# Damage Detection Problem

Objective function

In order to create an optimization routine that can locate and detect damage, an objective function must be found that achieves a minimum value at the damaged state of the structure. Without prior knowledge of the damage state it is difficult to create an objective function that reaches a minimum value at all possible damage states. In order to solve this challenge, it was hypothesized that the damage state of the structure would be that which produces the least damage compared to the healthy structure. Since damage has been defined as a reduction in stiffness of the structure, the objective function is written as to minimize the total reduction in stiffness. Although bending elements are used for each element, it was found that the reduction of the shear stiffness of each element produced the best results. As a result, the objective function is written as:

n At

Ф = . Eq(7)

i=1 Li

Ii = moment of inertia of the ith member Li = length of the ith member

Since each change in moment of inertia is divided by the length of each respective member, the function inherently weights the change in cross section properties according to the length of each member, thereby taking into account the affect each member has on the total stiffness of the structure.