Comparative Study of Using Displacement Influence Lines and Their Derivatives for Structural Damage Identification

Structural damage identification has recently become one of the most important topics for engineering structures due to its benefits in enhancing safety, reducing life-cycle cost, and providing guidance for system construction and maintenance. This research studies the accuracy of using displacement influence lines (DIL) and their derivatives (first and second derivative) for detecting structural damage characteristics (location and severity). The study includes analytical and numerical studies to investigate the sensitivity of displacement influence lines and their derivatives (first and second derivative) as a main parameter for damage identification. The results illustrate that the method can locate and quantify damage in both simply supported and continuous beams without the need for an optimization algorithm. Although, for simply supported beam, the optimal location of the displacement measurement point is at the middle of span. While, for continuous beam, one displacement sensor at the middle of each span enables locating the damages reliably. The advantages and disadvantages of using this index are also discussed.


Introduction
Civil engineering structures around the world are vulnerable to progressive deterioration and damage due to various reasons such as fatigue cracking, aggressive corrosions, extreme natural disasters, and man-made hazards [1].When the damage reaches a certain level, cracks, structural deformation, and even structural defects will appear and consequently will cause great harm to people's life safety and economic development [2].By detecting damage in its earlier stages, one may (1) extend the service life time of the structure by repairing and/or strengthening (2) prevent many disastrous failures, and (3) save a lot of money and time.
In the last few decades, many damage detection techniques (DDT) have appeared.Damage detection methods B Hazem O. Nady hazem0111@eng.aun.edu.egMohamed A.-B. Abdo abdo14@aun.edu.egFayez Kaiser fkaiser@aun.edu.eg 1 Civil Engineering Department, Assiut University, Asyût, Egypt can be categorized as either local or global damage detection techniques.The local damage detection methods (X-ray, ultrasonic,…etc.)focus non-destructive evaluation tools on specific structural components [3].The local techniques can be effective if the damage location is known a priori, which cannot be expected in most civil engineering structures [4].Therefore, the global DDT are, firstly, developed to detect the damage that occurs in the system and determine the position of the damage.
Global DDT can be categorized as dynamic or static.The dynamic-based DDT such as natural frequency [5,6], modal shape [7], modal assurance criterion (MAC) [8,9] co-ordinate modal assurance criterion (COMAC) [10], curvature of mode shape [11,12], high order mode shape [13,14], strain mode [15,16], modal flexibility [17,18] and modal damping [19,20] are more fully developed in comparison to static DDT.But, the static DDT have some advantages over the dynamic DDT, since the dynamic DDT require the use of stiffness, mass, and damping properties.Also, the influence of boundary conditions has a significant effect on measured frequencies and mode shapes [21].Higher modes are difficult to determine and measure and a large number of measurement points or measured frequencies are necessary to ensure the reliability of the damage assessment [22].Furthermore, structural dynamic characteristics are either insensitive to local structural damage or too sensitive to changes in the operational environment, such as temperature [23].On the other hand, the static DDT only require stiffness properties.Indeed, static DDT are usually cheaper and more accurate than vibration DDT [24], since the static equilibrium equation is solely related to the structural stiffness, and accurate static displacement and strain data can be obtained rapidly and cheaply.Also, static methods need less complicated calculation algorithms.
Damage index based on displacement influence lines (DILs) as a static damage detection technique has appeared recently.The proposed method only require the stiffness properties, and they can be obtained accurately and rapidly by inexpensive tools of measurement.Only one sensor is sufficient to obtain the influence line.When it is used in a bridge, applying load by moving a truck can get the DIL at a certain point in the structure.It can solve the problem of damage detection and can be used for bridge real-time detection [23].However, it has practical difficulties in its applications to real structures, since DIL is difficulty obtainable under operational conditions.Also, they are mostly applicable for periodic monitoring, but not readily suited for continuous monitoring.Grandic et al. [25] utilized DIL and its derivatives for undamaged and damaged beam structure to identify damage location.Displacement influence line can be obtained from measurements at only one point in the structure.The change in rotation of the displacement influence line shows the most reliable results in damage localization.
Zaurin and Necati [26] measured unit influence line of a four-span bridge model through the fusion of video imaging and sensing data.They also pointed out that an IL is a promising damage index.Huang et al. [23] proposed a method which uses the DIL to identify the damage in arch rib which can locate the structural damage positions accurately by observing the difference of curvature of the DIL.He et al. [27] presented a damage detection method for beam structures which were loaded by a quasi-static moving load.This deflection method is based on displacement measurements where the relationship between damage parameters and DIL is investigated.The results from the calculation and from the experiment showed peaks in the area of damages.However, the authors mentioned difficulties for damage detection due to measurement noise effects [28].Erdenebat et al. [29,30], and Erdenebat and Waldmann [31] introduced an efficient new method for damage detection called the deformation area difference (DAD) method.The method is based on a simple load-deflection experiment using a specific data processing.To validate the method on real structural elements, laboratory experiments on a gradually loaded reinforced concrete beam have been realized and several measurement techniques were applied and compared.The DAD-method has proven Fig. 1 A simply supported beam its potential for practical application through the successful localization of cracking in the concrete beam.
In this paper, a static DDT based on displacement influence lines and their derivatives (first derivative and second derivative) for identifying the location and severity of damage is presented.The study includes an analytical investigation of damage detection of multiple damages in a simply supported beam.Then, numerical studies have been conducted to investigate the effectiveness of the proposed method in different scenarios for simply supported beam, continuous beam with different damage characteristics.Furthermore, the most effective measuring point locations and number of measuring points for structural damage detection on simply supported and continuous beams are reported.Furthermore, noise intensity effects from the limited sensors precision, sensors faults and transmission errors are discussed in this paper.The advantages and disadvantages of using DILs for identifying damages are outlined.

Basic Theory of Displacement Influences Lines
Figure 1 presents a simply supported beam model based on the Euler-Bernoulli beam with constant bending stiffness.
According to the theory of virtual work, the beam displacement can be calculated by where Y(x) represents the displacement, EI represents the stiffness of the beam, M represents the bending moment.Hence, in principle, any change in a structure's stiffness, either locally or globally, should be evident in the displacement measurements of the structure.
According to Maxwell's theorem, the displacement at node

Derivation Method for Displacement Influence Lines (DILs) of Simply Supported Beam for Damage Detection
To verify the validity of the DILs method for damage detection, Euler-Bernoulli beam is used.Figure 3a shows a damaged simply supported beam with two damaged elements having different damage severities.The span length of the beam is L with bending stiffness EI. x is a variable used to represent the location of the concentrated load.Point S represents the measurement point (sensor).Y(x) represents the displacement at the measuring point, whereby the distance s corresponds to the measurement point S. The distance d corresponds to point D, which is the center point of a local damaged section.R is a random unknown variable used to represent the range of damage.E I is the bending stiffness of the damaged section.
The bending moment equations due to actual load F M 0 (x) The bending moment equations due to imaginary unit load M 1 (x) The used diagram-multiplication procedure to calculate the displacement diagram for Fig. 3 is as follows: 1.For x ∈ [0, s], the DIL can be computed from the following equation: the DIL can be computed from the following equation: the DIL can be computed from the following equation: 4. For x ∈ [d + R, L], the DIL can be computed from the following equation: 3 Derivation Method for Displacement Influence Lines for Simply Supported Beam with Multiple Damages

Damage Identification Based on the Difference in the Displacement Influence Lines (DDIL)
From Eqs. (4-7), it is clear that the DIL D of a damaged beam is different from the DIL for an undamaged beam so the occurrence of damage can be identified when the DDIL is non-zero as follows: The DDIL can be calculated as follows: Equations (9,11,13) show that the DDIL varies linearly with the location of the concentrated load F in an undamaged area.In contrast, Eqs. ( 10) and (12) show that the DDIL vary nonlinearly in the damaged areas.Figure 4 plots DDIL using Eqs.(9)(10)(11)(12)(13).It is obvious that the damage locations can be detected at the two abrupt changes between the three linear portions.

Damage Identification Based on the Difference
in the First Derivative of the Displacement Influence Lines (DFDDIL) 123 Fig. 6 Damage location concept of SS beam using DSDDIL  Equations (14,16,18) show that the DFDDIL is constant in an undamaged area.In contrast, Eqs. ( 15) and (17) show that the DFDDIL varies nonlinearly in the damaged areas.Figure 5 plots DFDDILs using Eqs.(14)(15)(16)(17)(18).It is evident that the damaged location is detected by a vertical jump in the function of DFDDIL.

For
Fig. 10 DDIL, DFDDIL and DSDDIL for scenario (6) Equation (19) shows that the DSDDIL equals zero value in an undamaged area.In contrast, Eqs. ( 20) and (21) show that the DSDDIL are non-zero in the damaged areas.Figure 6 plots DSDDIL using Eqs.(19)(20)(21).It is evident that damaged elements are located in regions where DSDDIL is not equal to zero.
From the previous analytical study, it can be seen that the difference of displacement influence lines and its derivatives (first derivative and second derivative) can identify damage location as shown in Figs. 4, 5 and 6.

Identification of Damage Severity
After the damage locations are determined in the region between (d − R, d + R), its severity information is important to enable higher levels of damage detection.Equations (10, 12, 15, 17, 13, 20, and 21) characterize the DDIL, DFD-DIL, DSDDIL of a simply supported beam with respect to a damage severity, but only DSDDIL in Eqs. ( 20) and ( 21) can be used to determine damage severity because it does not depend on the range of damage (R).At segment (d -R, d + R), one can calculate the damage severity coefficient (μ) as follows: The damage severity coefficient (μ) can be obtained easily by solving Eq. ( 23) and Eq. ( 24) without the need of an optimization algorithm, where Eq. ( 25) is used when the damage location before measuring point while Eq. ( 24) is used when damage location after measuring point.

Numerical Studies
Numerical studies have been conducted to investigate the effectiveness of the proposed method for damage identification of simply supported beam and two-span continuous beam with different damage characteristics.The displacement influence lines DILs are calculated for both the undamaged DIL U and the damaged state DIL D .The damage in this study is modeled by a reduction in Young's modulus (E) at the location of damage.The first and second derivatives of DIL can be calculated using the central difference method as follows: where y is the displacement and h is the length of the finite element.Then, the DDIL, DFDDIL, and DSDDIL can be calculated.

Parametric Study for Damage Detection of the Simply Supported Beam
The numerical analysis has been implemented for a simply supported beam with different damage characteristics.The displacement was obtained for an undamaged beam and a damaged beam using the software ABAQUS platform [32].
The span of the beam model is 30 m.The beam is divided into 60 segments and the length of each segment is h = 0.5 m.Young's modulus is defined as 210 GPa.The second moment of inertia is I = 0.084 m 4 .The applied force is F = 600 kN.
The DIL has been measured in node No 31 (the middle of the span).
Table 1 shows the damage characteristics of six damage cases.

Damage Location
Figures 7, 8 and 9 show that the DDIL, DFDDIL, and DSDDIL can identify the damage occurrence and damage location.The DDIL has a maximum value at the position of the damage.The damage location is detected by a vertical jump in the function of DFDDIL.However, the DSDDIL show peak at the damage location.From numerical analyses, it can be seen that the damaged element which is located near the measurement point will be detected with more accuracy than the damaged element which is located far from the measurement point.From Fig. 9, it can be seen that the difference in displacement influences lines and their derivatives increases as the damage increases.
Figure 10 shows similar results for multiple damage locations.Damage locations are displayed with a brake in the function of DDIL however, the DDIL are not a reliable indicator for multiple damages especially when damage locations are close to each other.The DFDDIL and DSDDIL are good Fig. 12 DFDDIL and DSDDIL for scenario (5) with 5% and 10% noise damage identification methods since all damage locations can be accurately located.

Damage Severity
After the damages are located, its severity is calculated directly from the damage severity coefficient (μ).By solving the equations, the damage severities are calculated, with minimal errors compared to the simulated severities.Figure 11 shows the relationship between the percentage reduction of stiffness μ, on vertical axis with the node number on horizontal axis for scenarios 1, 2, 3, and 6, respectively.The percentage errors are -0.3%, + 0.05% and -2.0% for scenarios 1, 2, and 3 respectively.Also, the percentage error does not exceed 1.2% for multiple damages as shown in Fig. 11d.Thus, the values of the calculated damage severity coefficient (μ) are in good agreement with the exact value in all positions of damages.

Measurement Noise
The noise is ubiquitous during the civil engineering structure test due to measurement errors.To investigate the noise effect on the performance of the proposed techniques, the measurement noise was introduced to the calculated influence lines of Scenario (5) and Scenario (6) by the following equation: where Y noise is the displacement influence line by the noise consideration, γ is the noise level (0% at 0, 100% at 1) and ρ is a random parameter obeying the truncated normal distribution in the range of [− 1, 1].It is assumed that the data of the damaged beam in Scenarios (5) (single damage) and ( 6) (multiple damages) are polluted by measurement noise with an error of ± 5%, 10%, as shown in Figs. 12 and 13.
The influence of measurement noise on damage identification using DFDDIL and DSDDIL has been investigated.Figure 13 shows that DFDDIL and DSDDIL accurately point out the locations of stiffness loss even with 10% noise and DFDDIL and DSDDIL are in good agreement with those obtained with error-free measurements.It can be concluded that measurement noise has a negligible effect on the damage localization approach and DFDDIL and DSDDIL are promising as good damage identification techniques.

Parametric Study for Damage Detection of Continuous Beam
A two-span continuous beam was established in ABAQUS platform [32].A continuous beam property has been simulated as those used for simple beam in Sect.4.1.The beam is divided into n = 120 finite elements.The length of elements for finite element analysis is h = 0.5 m.The DIL have been computed for point in the middle of the first span (Table 2).As it can be seen from Figs. 14, 15, 16, 17 and 18 that the DIL can be used for determining damage locations in indeterminate structures.However, the changes in the DDIL, are not always reliable for damage assessment especially, if the damage is located in one span and the measurement point is in the other span.The DFDDIL and DSDDIL provide reliable results in damage localization.Therefore, to determine the damage with high accuracy, at least one sensor should be placed in the middle of each span.
When the damage is in zero moment, it cannot be determined as shown in Figs.19 and 20.Although DFDDIL and DCDDIL can determine the damage location near the middle support (scenarios 8 and 11) where the bending moment is not equal to zero, they cannot determine the damage location in the end supports where the bending moments equal zero.The damage close to zero moment positions cannot be determined (shear failure).The closer the damage is to maximum moment, it can be easily determined (bending failure).

Conclusions
In this study, an analytical investigation of damage detection of multiple damages in a simply supported beam is presented.Then, numerical studies have been conducted to investigate the effectiveness of the proposed method for damage identification of simply supported beam (determinate structure), continuous beam (indeterminate structure) with different damage characteristics.The following conclusions can be drawn from this study: (a) due to a load acting at node (b) equals the displacement at node (b) due to the same load acting at node (a) (Y a = Y b ) as shown in Fig. 2. It means that only a single sensor is needed in the measurement of the DILs of a beam under a concentrated force.

Fig. 4
Fig. 4 Damage location concept of SS beam using DDIL

Fig. 5
Fig. 5 Damage location concept of SS beam using DFDDIL

Table 1
Damage scenarios of simply supported beam

Table 2
Damage scenarios of two-span continuous beam