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A new damage index, called strain change based on flexibility index (SCBFI), is introduced to locate damaged elements of truss systems. The principle of SCBFI is based on considering strain changes in structural elements, between undamaged and damaged states. The strain of an element is evaluated using the columnar coefficients of the flexibility matrix estimated via modal analysis information. Two illustrative test examples are considered to assess the performance of the proposed method. Numerical results indicate that the method can provide a reliable tool to accurately identify the multiple-structural damage for truss structures.

Structural damage detection has a great importance in civil engineering. Neglecting the local damage may cause the reduction of the functional age of a structural system or even an overall failure of the structure. Therefore, damage detection is an important issue in structural engineering. The basis of many damage identification procedures is observing the changes in structural responses. Damage reduces structure’s stiffness and mass, which leads to a change in the static and dynamic responses of the structure. Therefore, the damage detection techniques are generally classified into two main categories. They include the dynamic and static identification methods requiring the dynamic and static test data, respectively. Because of the global nature of the dynamic responses of a structure, techniques for detecting damage based on vibration characteristics of structures have been gaining importance.

Presence of a crack or localized damage in a structure reduces its stiffness leading to the decrease of the natural frequencies and the change of vibration modes of the structure [

In this paper, a new index for structural damage detection is proposed. The principle of the index is based on comparison of structural element’s strains obtained from two sets related to intact and damaged structure. The calculation of the strain is based on a flexibility matrix estimated from modal analysis information. Taking advantage of highly converged flexibility matrix using only few vibration modes related to low frequencies in the first phase and then determining the elemental strain using the flexibility coefficients develop a robust tool for damage localization of truss structures.

In recent years, many damage indices have been proposed to identify structural damage. In this paper, a number of widely used indices are first described and then the new proposed damage index is introduced.

Based on the basic modal parameters of structures such as natural frequencies, damping ratios, and mode shapes, some coefficients derived from these parameters can be useful for damage detection. The MAC and COMAC factors [

Let

The COMAC factors are generally used to identify where the mode shapes of the structure from two sets of measurements do not correlate. If the modal displacements in a coordinate

It has been proved that the presence of damage in a structure increases its flexibility. So, any change observed in the flexibility matrix can be interpreted as a damage indication in the structure and allows one to identify damage [

From (

The principle of flexibility method is based on a comparison of the flexibility matrices from two sets of mode shapes. If

The methods based on modal strain energy of a structure have been commonly used in damage detection [

A normalized form of MSE considering ^{e}. So, by determining the parameter

In this study, a new damage detection index based on considering strain changes in a structural element, due to damage, is developed. The new index

As the first step for constructing the proposed damage index, a modal analysis is required to be performed. The modal analysis is a tool to determine the natural frequencies and mode shapes of a structure [

At the second step, the flexibility matrix of healthy and damaged structure (

It can be observed that all components of the mode shapes are required to be measured and it is not a realistic assumption for operational damage detection. However, for actual use of the suggested method, it is not needed to measure the full set of mode shapes. The mode shapes of the damaged structure in partial degrees of freedom are first measured, and then the incomplete mode shapes are expanded to match all degrees of freedom of the structure by a common technique such as a dynamic condensation method [

Since each column of the flexibility matrix represents the displacement pattern of the structure, associated with a unit force applied at the corresponding DOF of that column, therefore they can be used as nodal displacements to calculate the strains of structural elements. The strain of each element of a 2-D truss structure can be expressed as [

So, at the next step, the strain of

Now, the strain change matrix SCM can be defined as the difference between the strain matrix of damaged structure

Theoretically, damage occurrence leads to increasing the SCM and consequently the index

In order to obtain a more accurate damage extent for an element, the damage indicator of (

The process of constructing the SCBFI index can also be briefly shown in Figure

The process for constructing the SCBFI index.

In order to show the capabilities of the proposed method for identifying the multiple-structural damage, two illustrative test examples are considered. The first example is a 31-bar planar truss and the second one is a 47-bar planar truss. The effect of measurement noise on the performance of the method is considered in the first example.

The 31-bar planar truss shown in Figure ^{3} and 70 GPa, respectively. Damage in the structure is simulated as a relative reduction in the elasticity modulus of individual bars. Five different damage cases given in Table

Five different damage cases induced in 31-bar planar truss.

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | |||||
---|---|---|---|---|---|---|---|---|---|

Element number | Damage ratio | Element number | Damage ratio | Element number | Damage ratio | Element number | Damage ratio | Element number | Damage ratio |

11 | 0.25 | 16 | 0.30 | 1 | 0.30 | 13 | 0.25 | 6 | 0.20 |

25 | 0.15 | — | — | 2 | 0.20 | 30 | 0.2 | 15 | 0.25 |

— | — | — | — | — | — | — | — | 26 | 0.30 |

Planar truss having 31 elements.

Damage identification for 31-bar truss for case 1 considering (a) 1 mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

Damage identification for 31-bar truss for case 2 considering (a) 1 mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

Damage identification for 31-bar truss for case 3 considering (a) 1 mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

Damage identification for 31-bar truss for case 4 considering (a) 1 mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

Damage identification for 31-bar truss for case 5 considering (a) 1 mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

It can be observed that the new index achieves the actual site of damage truthfully in most cases. It is also revealed that the general configuration of identification charts does not change after considering more than two modes. It means that the elements detected as damaged elements will be constant via increasing the measured mode shapes. Thus, requiring only two mode shapes for damage localization is one of the most important advantages of the proposed index.

The 47-bar planar power line tower, shown in Figure ^{3} and 30,000 ksi, respectively. Damage in the structure is also simulated as a relative reduction in the elasticity modulus of individual bars. Four different damage cases given in Table

Four different damage cases induced in 47-bar planar power line tower.

Case 1 | Case 2 | Case 3 | Case 4 | ||||
---|---|---|---|---|---|---|---|

Element number | Damage ratio | Element number | Damage ratio | Element number | Damage ratio | Element number | Damage ratio |

27 | 0.30 | 7 | 0.30 | 3 | 0.30 | 25 | 0.30 |

— | — | 19 | 0.20 | 30 | 0.25 | 38 | 0.25 |

— | — | — | — | 47 | 0.20 | 43 | 0.20 |

Forty-seven-bar planar power line tower.

Comparison of damage index

Comparison of damage index

Comparison of damage index

Comparison of damage index

It is observed that the new proposed index can find the actual site of damage truthfully. Moreover, the general configuration of identification charts dose not change when more than 3 structural modes are considered. It means that the elements identified as the damaged elements will be constant via increasing the mode shapes. Thus, requiring only three vibration modes for damage localization is one of the most important advantages of the proposed index that is due to high convergence of the flexibility matrix. Moreover, the comparison of the SCBFI with MSEBI index in Figures

In order to investigate the noise effect on the performance of the proposed method, the measurement noise is considered here by an error applied to the mode shapes [

The SCBFI for 31-bar truss considering 1% noise for (a) case 1 and (b) case 2.

The SCBFI for 31-bar truss considering 2% noise for (a) case 1 and (b) case 2.

The SCBFI for 31-bar truss considering 3% noise for (a) case 1 and (b) case 2.

It can be seen from the Figures

An efficient damage indicator called here as strain change based on flexibility index (SCBFI) has been proposed for locating multiple damage cases of truss systems. The SCBFI is based on the change of elemental strain computed from the flexibility matrix of a structure between the undamaged structure and damaged structure. Since the flexibility matrix used in the calculation of elemental strains converges rapidly with lower frequencies and mode shapes, it will be useful in decreasing the computational cost. In order to assess the performance of the proposed method for structural damage detection, two illustrative test examples selected from the literature have been considered. The numerical results considering the measurement noise demonstrate that the method can provide an efficient tool for properly locating the multiple damage in the truss systems while needing just three vibration modes. In addition, according to the numerical results, the independence of SCBFI with respect to the mode numbers is the main advantage of the method for damage identification without needing the higher frequencies and mode shapes which are practically difficult and experimentally limited to measure.

The authors declare that there is no conflict of interests regarding the publication of this paper.