Multiple damage detection in complex bridges based on strain energy extracted from single point measurement

Abstract

Strain Energy of the structure can be changed with the damage at the damage location. The accurate detection of the damage location using this index in a force system is dependent on the degree of accuracy in determining the structure deformation function before and after damage. The use of modal-based methods to identify damage in complex bridges is always associated with problems due to the need to consider the effects of higher modes and the adverse effect of operational conditions on the extraction of structural modal parameters. In this paper, the deformation of the structure was determined by the concept of influence line using the Betti-Maxwell theory. Then two damage detection indicators were developed based on strain energy variations. These indices were presented separately for bending and torsion changes. Finite element analysis of a five-span concrete curved bridge was done to validate the stated methods. Damage was simulated by decreasing stiffness at different sections of the deck. The response regarding displacement of a point on the deck was measured along each span by passing a moving load on the bridge at very low speeds. Indicators of the strain energy extracted from displacement influence line and the strain energy extracted from the rotational displacement influence line (SERIL) were calculated for the studied bridge. The results show that the proposed methods have well identified the location of the damage by significantly reducing the number of sensors required to record the response. Also, the location of symmetric damages is detected with high resolution using SERIL.

Keywords

damage detection / strain energy / influence line / complex bridges / rotation displacement

Introduction

The aging of road and rail infrastructures and the safety importance of these structures have prompted extensive research into structural health monitoring (SHM) in recent decades [¹–⁴]. A realistic prediction of the structural response is needed to study the behavior of complex geometry bridges, especially when subjected to dynamic loads [⁵]. Damage caused by traffic loads and extreme events such as earthquakes can affect the performance and behavior characteristics of the bridges. The existence of any structural damage can affect the service and final capacity of the structure, and subsequently, failure occurs. Despite significant scientific work in detecting the damage to bridges, still, the significant data collected from the bridges in-service is related to eye visits. Recently, the international research community has been standardizing the methods and definition in this regard to eliminate contradictions in the results of traditional and visual inspections [⁶,⁷].

Damage detection in the early stages will significantly reduce maintenance costs [⁸–¹⁰]. A useful option in maintenance planning is the use of known indicators in determining the status of the structure. Usually, these indicators used for inspection purposes are modal-based parameters. In recent years, many of these parameters (natural frequencies, mode shapes, modal curvature, modal flexibility, and damage index) have been proposed or developed, some of which are presented in Refs. [¹¹–²⁰].

Many studies have been conducted to compare the modal parameters mentioned. For example, Talebinejad et al. [²¹] used COMAC methods, damage index, modal curvature, and modal flexibility in a cable-stayed bridge finite element model. They found that only severe damages are detectable, and noise reduces the sensitivity to damage. By examining different methods for detecting damage in reinforced concrete beams, Ndambi et al. [²²] found that although the severity of the damage is not recognizable, the damage index and COMAC methods have been well- functioning in determining the damage location. Cruz and Salgado [²³] argued using the model of a composite bridge and the actual vibration data from a test that the clarity of the findings of the damage detection is related to higher modes and severe damages are also not recognized in low modes. Fan and Qiao [²⁴] also observed that the high sensitivity of the damage occurs in higher modes. Nanthakumar et al. [²⁵] proposed an algorithm to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure. They evaluated the algorithm with various noise levels. Other powerful and practical methods that can be used to solve inverse problems are investigated in the Refs. [²⁶–²⁸].

In the study by Meixedo et al. [²⁹], the accelerations of the deck under the influence of moving loads in the two-dimensional finite element model were tested. The results showed that a large number of vibration-based parameters have a high sensitivity to damage and has the potential to be used in determining the location and severity of the damage.

Limongelli [³⁰] stated by providing a deformation-based indicator for damage detection (IDDM) that the vibration amplitude mismatch can be used to determine the location of the damage. Dilena et al. [³¹] examined the performance of IDDM against modal curvature on a single-span concrete bridge under harmonic force and with increasing damage in different locations. The results showed that IDDM function is dependent on sensors distribution. Modal curvature for the first two modes has good sensitivity to damage, but accuracy decreases in higher modes. This reduction in accuracy is due to the need for a dense sensor network to determine the exact curvature of higher modes.

Generally, it can be said that natural frequencies can provide a simple and appropriate evaluation of structures with regular geometry, while the mode shapes and its derivatives give us useful spatial information if sufficient sensors and low levels of noise are provided. The use of strain energy according to its capabilities has been widely considered by researchers [³²–³⁵]. Damage index method is also provided based on modal strain energy and assuming pure bending. The use of this method in a structure with torsional and axial responses has an appropriate accuracy with increasing the number of sensors [³⁶].

As the studies show, the exact determination of the modal parameters of an in-service bridge as well as the complexity of the geometry of the structures reveals damage detection with numerous challenges and constraints. The effect of higher modes is significant in complex bridges, and using a denser sensor network is needed. On the other hand, the dynamic changes resulting from traffic are not constant and decreases with increasing load [³⁷]. The greater complexity of vibration-based health monitoring of in-service bridges is that modal parameters changes of the bridge may be due to the interaction of the bridge and the vehicle rather than the damage [³⁸]. Therefore, although the modal-based damage detection methods have the appropriate support theory, their application in real bridges has contradictory results. Alternative and practical methods should be provided to reduce the number of sensors, eliminate operational conditions, and reduce the weaknesses associated with the measurement of modal parameters, in order to improve the methods for detecting damage and their widespread use. In this paper, a new algorithm for damage detection was developed based on the strain energy of a deformed bridge under the influence of a concentrated vertical load. Betti’s theory [³⁹] was used on complex bridges instead of using a dense sensor network with an innovative idea, and a structural deformation vector was obtained from the response of only one sensor. First, the static displacement influence line of a deck point was determined between two supports by passing a concentrated load on the bridge. Then, the damage detection method was developed with the assumption of pure bending by linking the strain energy extracted from the influence line of the damaged and undamaged bridge. Subsequently, the modal strain-based damage detection method was proposed for pure torsion mode. The validity of the proposed methods was confirmed with several damages to the deck by finite element modeling of a concrete horizontally curved bridge.

An algorithm for detecting damage location

It is possible to derive from the results of previous research that strain energy is a sensitive indicator for detecting damage. In some cases, strain energy variations stored in a beam are calculated when it is deformed in a particular mode [⁴⁰]. It is worth noting, the effect of higher modes should be considered in complex structures using a large number of response measurement points to provide adequate accuracy in determining the location of the damage. In this section, the deformation of a Euler-Bernoulli beam was first obtained using the concept of the influence line of the displacement response resulting from the central load transfer P of the beam to reduce the number of sensors required. Then, a new method was developed to identify damage in the bridge’s decks, separately, using the strain energy theory extracted from bending and torsion.

Strain energy extracted from static displacement

It is necessary to establish a connection between responses and the structure deformation function to determine the location of the damage by using the response of a limited number of points. An accurate deformation function makes it possible to get the exact location of damage by calculating strain energy before and after damage. Now assume that a one-dimensional Euler-Bernoulli beam [⁴¹] with an element NE is deformed under the influence of a vertical point load (assuming that the beam behaves linearly). The value of the strain energy of the beam (U) is obtained from Eq. (1) [⁴²–⁴⁴].

(1)U={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}EI\left(x\right)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}y}{\partial x^{\mathrm{2}}}}\right]}}^{\mathrm{2}}\mathrm{d}x,

where y(x) is the bending deformation of the beam.

The vertical deformation vector of the beam can be defined according to Eq. (2) when a concentrated load P is located on the beam at the point i (y_{P_i}).

(2)\begin{array}{ll}{y}_{P_i}={\overrightarrow{\mathrm{\Delta }}}_{P_{i}j},\hfill & \mathrm{0}\le j\le NE.\hfill \end{array}

In this equation, {\overrightarrow{\mathrm{\Delta }}}_{P_{i}j} is the displacement value obtained from the system of force P_i at j point.

The influence line is also considered as a vertical displacement extracted from the passage of the moving load P from the bridge at very low speeds. The slow speed eliminates the dynamic effect of the moving load on the response. Given this definition, the vector for the displacement influence line is computed from Eq. (3).

(3)\begin{array}{ll}I{L}_i={\overrightarrow{\mathrm{\Delta }}}_{P_{i}j},\hfill & \mathrm{0}\le j\le NE\hfill \end{array}.

In this equation, {\overrightarrow{\mathrm{\Delta }}}_{P_{i}j} is the displacement value obtained from the system of force P_j at i point.

According to Betti’s law, if a beam at point i is subjected to a concentrated vertical force of P, then the value of displacement resulting from this force system at the point j ({\overrightarrow{\mathrm{\Delta }}}_{P_{i}j}) is equal to the displacement of the point i when the load P is at the point j ({\overrightarrow{\mathrm{\Delta }}}_{P_{j}i}). This principle is presented by Eq. (4).

(4){\overrightarrow{\mathrm{\Delta }}}_{P_{i}j}={\overrightarrow{\mathrm{\Delta }}}_{P_{j}i}.

Equation (5) is obtained concerning Eqs. (2) to (4). The concept of this equation is that the vector of displacement influence line for each point of the beam is precisely equal to the beam deformation vector when the concentrated load locates at the same point.

(5)y_{P_i}=I{L}_i.

The value of the contribution of the jth member of the beam strain energy (U_j) is obtained from Eq. (6) after determining the structure deformation by the displacement influence line using Eqs. (1) and (5). As previously mentioned, the total number of elements is NE.

(6)U_j={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \underset{j}{\overset{}{\int }}EI\left(x\right)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}}^{\mathrm{2}}\mathrm{d}x.

In this equation, IL(x) is the vector of influence line and EI(x) is the bending stiffness of the beam at an x distance from the beginning of the beam.

The contribution of the jth element from the total strain energy of the beam can be presented by Eq. (7).

(7)F_j={\displaystyle \frac{U_j}{U}}.

The parameters for Eqs. (1), (6), and (7) for the damaged structure are shown with the index “*”. For the first time, Eq. (8) will be established, with an approximation between two states of undamaged and damaged.

(8){{\displaystyle F}}_{\mathrm{1}}^{*}={\displaystyle \frac{{{\displaystyle U}}_{\mathrm{1}}^{*}}{U^{*}}}={{\displaystyle F}}_j\left(higher-orderterms\right).

Equations (7) and (8) show that the relationship between {{\displaystyle F}}_{ij} and {{\displaystyle F}}_{ij}^{*} terms is in the form of Eq. (9).

(9)\mathrm{\sum }_{j=\mathrm{1}}^{NE}{{\displaystyle F}}_{ij}=\mathrm{\sum }_{j=\mathrm{1}}^{NE}{{\displaystyle F}}_{ij}^{*}=\mathrm{1},{{\displaystyle F}}_{ij}\le \mathrm{1},{{\displaystyle F}}_{ij}^{*}\le \mathrm{1.}

Approximate Eq. (10) has been used to establish an appropriate relationship between an undamaged and damaged structure and concerning Eq. (9).

(10)\mathrm{1}+F_{ij}\cong \mathrm{1}+F_{ij}^{*}.

By inserting Eqs. (7) and (8) into Eq. (10), Eq. (11) is obtained as follows:

(11)\mathrm{1}={\displaystyle \frac{(U_j^{*}+U^{*})U}{(U_j+U)U^{*}}}.

Equation (11) is given in the form of Eq. (12) concerning Eq. (1) and the mean value theorem.

(12)\mathrm{1}={\displaystyle \frac{E{I}_j^{*}\left(x\right)}{E{I}_j\left(x\right)}}{\displaystyle \frac{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \frac{\mathrm{1}}{E{I}_j^{*}\left(x\right)}}{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}E{I}^{*}\left(x\right)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \frac{\mathrm{1}}{E{I}_j\left(x\right)}}{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}EI\left(x\right)\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}{\displaystyle \frac{EI\left(\widehat{x}\right){\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x}}{E{I}^{*}\left(\widehat{x}\right){\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x}}}.}

Equation (12) has been converted to Eq. (13) by assuming EI to remain constant along an undamaged and damaged state.

(13)SEDI{L}_j={\displaystyle \frac{E{I}_j\left(x\right)}{E{I}_j^{*}\left(x\right)}}={\displaystyle \frac{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}}{\displaystyle \frac{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x}}{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}^{\mathrm{2}}\mathrm{d}x}}}.

where SEDIL_j is the indicator based on strain energy extracted from displacement influence line (SEDIL) in element j. \left({\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right)and \left({\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right) are the curvature functions of influence line at the x distance along the beam and are related to the undamaged and damaged states, respectively. The SEDIL value can be calculated for each structural element of Eq. (14) if the distance between the deformation measurement points is the same.

(14)SEDI{L}_j={\displaystyle \frac{{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}_j^{\mathrm{2}}+{\displaystyle \sum _{k=\mathrm{1}}^m{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}_k^{\mathrm{2}}}}{{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}_j^{\mathrm{2}}+{\displaystyle \sum _{k=\mathrm{1}}^m{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}_k^{\mathrm{2}}}}}{\displaystyle \frac{{\displaystyle \sum _{k=\mathrm{1}}^m{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}IL}{\partial x^{\mathrm{2}}}}\right]}_k^{\mathrm{2}}}}{{\displaystyle \sum _{k=\mathrm{1}}^m{\left[{\displaystyle \frac{{\partial }^{\mathrm{2}}I{L}^{*}}{\partial x^{\mathrm{2}}}}\right]}_k^{\mathrm{2}}}}}\mathrm{.}

It should be noted that Betti’s law is valid in structures with linear elastic behavior without support settlement. Therefore, a point is needed in multi-span bridges to measure the response between the two supports. A unique index for each location j is determined by Eq. (15) when more than one sensor is used.

(15)SEDI{L}_j={\displaystyle \frac{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^{NP}NuSEDI{L}_{jk}}}{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^{NP}DeSEDI{L}_{jk}}}},

where NuSEDIL_jk is the numeric value of the numerator and DeSEDIL_jk is the denominator of the SEDIL index obtained from the kth sensor and at jth point on the bridge deck.

SEDIL_j index has two important characteristics, first, the amount of strain energy changes is determined using the influence line for the beam displacement before and after the damage; secondly, the right parameters of Eq. (13) are all measurable. The curvature value of the influence line is calculated using the central difference method by Eq. (16).

(16)I{L^{″}}_j={\displaystyle \frac{I{L}_{(j+\mathrm{1})}-\mathrm{2}I{L}_j+I{L}_{(j-\mathrm{1})}}{h^{\mathrm{2}}}},

where IL_j is displacement at j point and h is the mean distance between points in the vector of influence line.

Strain energy extracted from rotational displacement

The assumption of pure bending in the calculation of strain energy will not be correct in structures with torsional behavior. For this purpose, the amount of strain energy extracted from torsion is obtained from Eq. (17).

(17)U={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}GJ\left(x\right)\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}}^{\mathrm{2}}\mathrm{d}x,

where \psi \left(x\right) is the torsional deformation and GJ(x) is the torsional stiffness of the beam. The contribution of jth member of the torsion strain energy of the beam is according to Eq. (18).

(18)U_j={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{{\displaystyle \underset{j}{\overset{}{\int }}GJ\left(x\right)\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}}^{\mathrm{2}}\mathrm{d}x.

The SERIL_j damage detection index is obtained based on the strain energy extracted from rotational displacement influence line (SERIL) effect in element j from Eq. (19) by applying Eqs. (17) and (18) as well as Eqs. (7) to (11).

(19)SERI{L}_j={\displaystyle \frac{GJ\left(x\right)}{G{J}^{*}\left(x\right)}}={\displaystyle \frac{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}{\left({{\displaystyle \underset{j}{\overset{}{\int }}{\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}^{\mathrm{2}}\mathrm{d}x+{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}}}^{\mathrm{2}}\mathrm{d}x\right)}}{\displaystyle \frac{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}^{\mathrm{2}}\mathrm{d}x}}{{\displaystyle \underset{\mathrm{0}}{\overset{L}{\int }}{\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}^{\mathrm{2}}\mathrm{d}x}}}.

In this equation, {\displaystyle \frac{\partial {\psi }_x}{\partial x}} and {\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}are derivatives of rotational deformation at x distance from the beginning of the beam and are related to undamaged and damaged state, respectively.

The SERIL index is calculated from Eq. (20) for each structural element after determining the influence line vector.

(20)SERI{L}_j={\displaystyle \frac{{\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}_j^{\mathrm{2}}+{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^m{\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}_k^{\mathrm{2}}}}{{\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}_j^{\mathrm{2}}+{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^m{\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}_k^{\mathrm{2}}}}}{\displaystyle \frac{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^m{\left[{\displaystyle \frac{\partial {\psi }_x}{\partial x}}\right]}_k^{\mathrm{2}}}}{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^m{\left[{\displaystyle \frac{\partial {\psi }_x^{*}}{\partial x}}\right]}_k^{\mathrm{2}}}}}={\displaystyle \frac{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^{NP}NuSEDI{L}_{jk}}}{{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^{NP}DeSEDI{L}_{jk}}}}.

Given Eq. (5), the amount of rotational displacement of the bridge deck is obtained from Eq. (21) using the vertical displacement influence line of the points in two sides of the deck [I^{\prime }L\left(x\right) and IL(x)] which are placed at a distance w from each other in a cross section.

(21)\psi \left(x\right)={\displaystyle \frac{IL\left(x\right)+I^{\prime }L\left(x\right)}{w}}.

The finite difference approximation method used to calculate the derivative of the rotational displacement extracted from Eq. (21), and Eq. (22), is presented.

(22){\displaystyle \frac{\partial {\psi }_i}{\partial x}}={\displaystyle \frac{(I{L}_{i+\mathrm{1}}-I{L}_{i-\mathrm{1}})+(I^{\prime }{L}_{i+\mathrm{1}}-I^{\prime }{L}_{i-\mathrm{1}})}{\mathrm{2}wh}}.

Damage detection using finite element model

Finite element model of the bridge

In this research, a horizontally curved bridge with a subtended angle of 90 degrees was investigated. The bridge has five spans with a length of 15 m for each and a total length of 75 m, with a concrete system in place. The finite element model of the bridge was prepared. To model bridge members, the frame elements and shell elements were defined and used in the analytical model. The bridge deck was modeled using shell elements and intermediate piers with linear frame elements. The bridge model has 877 nodes, 36 frame elements, and 720 shell elements. A three-dimensional finite element model of the bridge can be found in Fig. 1. In addition, the geometric characteristics of the bridge’s central pier and deck are shown in Fig. 2.

The link elements were used to define the connectivity of the columns to cap beams and cap beams to the deck. The connections between these members in all degrees of freedom were considered to be rigid.

The type of supports used in the bridges examined is elastomeric supports, which deck is located on three neoprene. Only two expansion joints used in the bridges at the place of the abutments and there are no joints on the intermediate piers. The effect of expansion joints in the place of abutments and the stiffness of the neoprenes were considered as part of the effective stiffness of the abutments and as equivalent linear springs in the modeling. The equivalent stiffness in the longitudinal direction was calculated with the stiffness of neoprenes, and its value is 222 ton/m. Due to the use of concrete blocks along the transverse direction, the stiffness of abutments was considered to be rigid. With a combination of foundation transverse stiffness and abutments in series, the equivalent stiffness is equal to the foundation-abutment transverse stiffness. Also, the equivalent vertical stiffness will be equal to the vertical stiffness of the abutment foundation.

The foundation stiffness of the piers and abutments was calculated using the method presented in FEMA 357 [⁴⁵], and its values are presented in Table 1. In this table, k_x, k_y, and k_z are the transitive stiffness in the direction x, y, z, respectively, and k_xx, k_yy, and k_zz are rotational stiffness around the x-, y-, and z-axes respectively.

In this study, the materials used are elastic and homogeneous, and the structure was analyzed in the elastic region, assuming small deformations. Other details of modeling are provided in Ref. [¹⁹].

Damage simulation

It is emphasized that the presented indices do not depend on numerical modeling and expanded on the basis of the mathematical development of strain energy based on the deformed Euler-Bernoulli beam under the bending and torsion effects. However, for the evaluation of the indices, the damages were considered at different locations with different distances in the model. For this purpose, the thickness of the deck was reduced in four designated locations according to Table 2. These damages are selected according to the effective inertia of the cross section after the cracking of the concrete and the first yielding of the reinforcing bars, in accordance with recommendations of Caltrans [⁴⁶], and are presented in Table 3. It should be noted that the identification of low-intensity damages is always a challenge to damage detection methods. Therefore, for the evaluation of the sensitivity [a,b] of the indices, very low-intensity damage was also considered.

A concentrated load unit was moved along the deck about 25 cm length of steps to determine the influence line of the deck displacement. The amount of deck displacement in each loading step is recorded by static analysis of the undamaged and damaged bridge. The influence line is determined at the given point, by using the displacement resulting from the load passing on the deck. The static analysis in the finite element model was performed for 280 loading steps. The defined damage location and the deck response record points in the three-dimensional finite element model of the bridge are shown in Fig. 3. The calculation of the SEDIL index has been determined using the displacement influence line of points P₁ to P₅. These points are randomly assigned so that each point is located in a span. The influence line of two sides of the deck in addition to the points mentioned has also been used at points {{\displaystyle P\text{'}}}_{\mathrm{1}} to {{\displaystyle P\text{'}}}_{\mathrm{5}} to determine the SERIL index vector.

Damage detection results using SEDIL and SERIL indices are presented in Figs. 4 and 5, respectively. The variation of each index indicates the location of the damage. The location of all four damages is determined in both methods. Based on the calculated results, using the proposed indices, damages can be detected with only one measurement point. However, each of the damage is closer to the measurement location, regardless of their amount, detected more accurately. As shown in Fig. 4, damage 2 that is symmetric to damage 4 is less clearly identified. However, the use of the SERIL index covers this weakness, and the clarity of results is obvious.

The above points indicate the importance of the torsion effect in detecting the damage to curved bridges. Based on the results, using SEDIL index with just one measurement point, damages in the bridge deck and their location can be identified. However, on bridges where the participation of torsion modes is high or in cases where damage has been developed in part of the deck cross-section, SERIL index gives more accurate results.

Conclusions

Detection of deck damages in complex bridges, such as curved bridges is accompanied by double errors compared to straight bridges. The origin of these errors is the initial assumptions such as pure bending in the structure. It is necessary to use a dense sensor network in the application of modal-based methods, such as the Damage Index (DI) for bridges with torsional-respond, to extract the exact dynamic characteristics and determine the higher modes. Operational conditions are also limiting factors in determining the modal parameters. In this research, two SEDIL and SERIL indices were presented, which are based on the strain energy extracted from bending and torsion, using the concept of influence line. The location of the deck damage in each of these methods was obtained using the minimum number of sensors. Application of these indices in the finite element model of a damaged curved bridge indicates that these methods have a good performance. Also, determining the influence line using static analysis removes the effect of vehicle-bridge interaction. The results of this paper can be summarized in the following ways.

1) Symmetric damages are determined using the SERIL method.

2) In complex bridges, the simultaneous use of the SEDIL and SERIL indices has resulted in high clarity in determining damage.

3) The torsion is the determining factor in detecting the damage.

Based on the calculated results and characteristics and features of the proposed damage indices, they can be used for damage detection in complex bridges.

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