Damage detection on structures using transfer matrix with lumped crack properties

A novel lumped crack transfer matrix (LCTM) and state vector based method is proposed in this paper for identification of damage in the beam like structures. Transfer matrix (TM) is a square matrix which contains all structural and crack parameters such as crack depth and its location. State vector at a node is the sum of internal and external contributions of displacements, forces and moment at that node on the structural element, when it is multiplied with TM the state vector at the adjacent point can be obtained. A cracked beam element is assumed as two intact beam elements are connected by a hinge or torsional spring. The crack is modelled as an element of zero length and mass but has elastic properties. Hence the LCTM for the cracked beam element is obtained by series multiplication of LCTMs for the intact beam elements and that for the crack element. State vector is formed at one node known as initial state vector from which state vectors at other nodes are predicted by multiplying with TM for the predicted values of crack parameters. Displacement responses are measured at a few nodes in the structure. The mean square error between measured and predicted responses is minimized using a heuristic optimization algorithm with crack depth and location as the optimization variables. Two numerical examples a cantilever and sub-structure of a frame with nine members are solved with multiple cracks in an element. The TM algorithm is also validated experimentally. The main advantage in this method is one or more cracks in the single element can be identified.