Issue 59

A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03

Residual forces In this section, we explore structural damage identification based on the residual forces, details about residual forces can be found in Ref. [22]. In such approaches, the structure is divided into small elements via the finite element method, and the damage index of each element is expressed as the change of the rigidity i.e.:                      e e e e u d u i i i i i K K K K (14) i K is the difference in stiffness, and  is a value between 0 and 1, which indicates a loss of rigidity in each element. In other words, for undamaged elements,  is equal to 0, and to 1 in fully damaged elements. In this study, we assume that the damage does not affect the mass matrix of a damaged structure, therefore the rigidity matrix of the damaged element is expressed as follows:                       0 1, , e e u j j j M K K j m (15) and the modal residual force vector is expressed by this equation: where   u e i K and   d e i K are the elementary matrices of the undamaged and damaged structures, respectively. In the above equation, Eqn. (14),    e

  1 2                      m

     

                         e e d R K f f f f

     F

e

(16)

i

i

i

m

1

2

i

As consequence, Eqn. (16) can be represented in a matrix form by the following expression:

        F R

(17)

The coefficient matrix   F is:

      e u e d

 

F K

(18)

ij

j

ij

Here,   ij F is the force vector in the actual node

th i on the th j element, written in the global coordinates.

The vector of residual force modal is expressed in this manner:



    d

  R K M     d    i

(19)

i

i

and Eqn. (17) is rewritten in this manner:                 11 12 1 m F F F F F F

    R R                                 1 2   1 2

    

m

21

22

2

(20)





 

    F F

  F

    n R

     m



n

n

nm

1

2

40