Research on damage identification in T-beam bridges based on strain influence lines

Abstract

Bridge damage induces immediate stiffness reductions in bridge structures, manifested as variations in static parameters such as strain and deflection, which pose significant safety risks. This paper presents a rapid damage diagnosis method for T-beam bridges using strain influence lines. The method derives the analytical strain influence line formula for T-beam bridges and examines its variation under transverse and longitudinal damage conditions. By analyzing peak values and envelope area shifts in the strain influence line, a diagnostic difference curve is generated, enabling the development of a qualitative damage assessment system. This system localizes damaged regions and provides transverse/longitudinal positioning and quantitative evaluation methods for T-beam bridge damage. Experimental validation using a scaled T-beam bridge model confirms the method’s applicability and feasibility for damaged-state structural assessment.

Introduction

Damage detection plays a critical role in safety assessment of in-service bridges and serves as an essential reference for formulating bridge maintenance strategies. The study of damage identification in aging bridge structures is of paramount importance for preventing safety incidents^1,2,3.

With advancements in modern technologies, materials, evaluation techniques, and construction methods over recent decades, considerable progress has been made in bridge maintenance. Accurately diagnosing bridge damage during service guarantees the safety not only of national infrastructure construction but also the lives and property of individuals. Therefore, since the 1950s, scholars have carried out considerable research and summary work on bridge damage diagnosis methods^4,5, and to date, the appearance investigation method^6,7 the expert empirical diagnosis method⁸, the design specification method⁹, the sensor diagnosis method for health monitoring system^10,11, and the load experiment method^12,13 have primarily been used. These diagnostic methods can be divided into two categories: dynamic information-based and static information-based damage diagnosis techniques.

Dynamic information-based methods typically compare measured dynamic parameters (e.g., frequency, mode shapes, modal curvature, and strain modes) against reference values. Static information-based approaches rely on detecting stiffness variations caused by structural damage, which manifest as changes in static measurements like strain and deflection. By comparing these measurements with theoretical values from undamaged states, structural anomalies can be identified.

For in-service bridges, the most prevalent diagnostic techniques remain visual inspections and load testing. While experienced technicians can assess bridge conditions through visual examination, this method suffers from subjectivity, depending heavily on inspector expertise, and often fails to detect hidden structural defects. Load testing, encompassing both static and dynamic components, provides more objective assessments through controlled loading scenarios. Static load tests examine structural responses (deflection, stress, crack propagation, etc.) under graduated loading, comparing measured values with theoretical calculations. Dynamic load tests analyze transient responses (dynamic deflection, strain, modal parameters^14,15) through various excitation methods. However, these methodologies require extended traffic disruptions, involve complex multi-stage processes, and offer limited diagnostic resolution with relative low accuracy. Damage detection via prestress loss monitoring has become a prominent research topic in recent years^16,17; however, this method necessitates periodic and costly on-site testing.

To address these limitations, researchers like Tang Guangwu and Liao Jingbo¹⁸ have pioneered bridge damage diagnosis systems based on influence line theory. This approach utilizes various influence line types (deflection^19,20, rotation^21,22, displacement^23,24, and strain influence lines^25,26) to detect structural damage through single-point measurement and multi-point excitation. Compared to conventional methods, influence line-based techniques demonstrate superior reliability, accuracy, and stability²⁷. Current methodologies primarily employ differential methods and model calibration approaches: The former quantifies damage by comparing measured and theoretical influence line differences, while the latter adjusts finite element model parameters until they converge with undamaged state characteristics.

This study integrates existing literature and begins with theoretical exploration, conducting research on bridge damage identification through a combined approach of analytical derivations, finite element simulation analyses, and experimental validation. Firstly, theoretical analyses were performed to investigate the global and local variations in strain influence lines under both T-beam damage and diaphragm damage scenarios. Secondly, the finite element software Midas Civil v1.1(http://www.MidasUser.com) was utilized to simulate transverse damage (e.g., weakened diaphragms) and longitudinal damage (e.g., weakened longitudinal main girders) in T-beam bridges by applying stiffness reduction techniques. Characteristic changes in strain influence lines under these two damage states were systematically summarized, and appropriate damage indicators (e.g., envelope area of strain influence lines and interval envelope area) were selected for damage diagnosis. Finally, scaled model testing was conducted using an acrylic glass T-beam bridge model to validate the strain influence line-based rapid damage diagnosis method proposed in this study in practical applications.

Theoretical basis of strain influence line for T-girder bridges

A simply supported single-beam structure is the simplest form of bridge structure, and its mechanical model is shown in Fig. 1. When a force of size F is applied to the beam at a distance x from the beam end, according to Hooke’s law, the cross-sectional strain expression at point B with any length l from the beam can be determined by using Eq. 1:

Fig. 1

Simply supported single-beam model.

Fig. 2

Strain influence line at point B under moving load.

$$\:\epsilon\:\left(x\right)=\left\{\begin{array}{cc}\frac{(L-l){F}_{xy}}{LEI}&\:(0<x\le\:l)\\\:\frac{(L-x){F}_{ly}}{LEI}&\:(l<x<L)\end{array}\right.$$

(1)

Where L is the span of the simply supported beam, EI is the stiffness of the measurement point section, and y is the distance from the lower edge to the section’s neutral axis.

Under a moving load F, the strain influence line at measurement point B manifests a triangular shape as shown in Fig. 2. The strain influence line exhibits a monotonic increasing trend on the left side of point B and a monotonic decreasing trend on the right side. The peak magnitude of this strain influence line is given by (L-l)F_ly/(LEI), located at the section of measurement point B.

In practical applications, T-beam bridges typically consist of multiple interconnected T-beams. When loads are applied to the bridge, the transverse connections between T-beams cause each beam to participate in load carrying to varying degrees. Therefore, the analytical formulation of strain influence lines for T-beam bridges must consider both the transverse distribution coefficient η and longitudinal distribution coefficient α. When a moving load F is applied to Beam i of a T-beam bridge (Fig. 2), the load carried by Beam i becomes F_i = αη_ii F, while the load carried by Beam k is $\:{F}_{k}=\alpha\:{\eta\:}_{ki}F$. This leads to the strain influence line expression shown in Eq. 2 for measurement point B_k on Beam k.

$$\:\epsilon\:\left(x\right)=\left\{\begin{array}{cc}\frac{\left(L-l\right){F}_{xy}\alpha\:{\eta\:}_{ki}}{LEI}&\:\left(0<x<l\right)\\\:\frac{\left(L-x\right){F}_{ly}\alpha\:{\eta\:}_{ki}}{LEI}&\:\left(l<x<L\right)\end{array}\right.$$

(2)

In the formula, the meaning of $\:{\alpha\:\eta\:}_{\text{k}\text{i}}$ is in the actual project, when the load is applied to beam i, the proportion of the load assigned to beam k. That is the transverse distribution coefficient of the beam k.

Transverse damage in T-beam bridges fundamentally alters load distribution mechanisms by weakening transverse connections between beams, thereby compromising transverse load transfer efficiency. This damage-induced connectivity reduction creates an asymmetrical load redistribution pattern: as the loaded beam (Beam i) assumes an increased load proportion, the non-loaded adjacent beam (Beam k) experiences corresponding load reduction. These opposing effects are quantitatively expressed through transverse distribution coefficient variations, where the loaded beam’s coefficient $\:{\alpha\:\eta\:}_{\text{i}\text{i}}$ demonstrates damage-dependent amplification at the affected location, while the non-loaded beam’s coefficient $\:{\alpha\:\eta\:}_{\text{k}\text{i}}$ exhibits proportional attenuation. To systematically characterize these phenomena, two dimensionless parameters are introduced - K_i (K_i >1) quantifying the transverse distribution coefficient enhancement rate for the loaded Beam i, and K_k (K_k <1) representing the attenuation rate for non-loaded Beam k.

The mid-span cross-section’s structural uniqueness renders it particularly sensitive to both load-bearing characteristics and damage patterns, establishing it as a critical analytical domain. As demonstrated in Figs. 3 and 4, transverse damage induces mathematically distinguishable strain influence line behaviors at mid-span sections: Eq. (3) governs the mechanical response at section B_i of the loaded Beam i, whereas Eq. (4) describes the modified behavior at section B_k of the adjacent non-loaded Beam k. This differential formulation enables precise quantification of damage-induced load redistribution through comparative analysis of strain influence line morphologies.

Fig. 3

Schematic view of the cross-section.

Fig. 4

The longitudinal mid-span measurement point.

$$\:\epsilon\:\left(x\right)=\left\{\begin{array}{cc}\frac{{{F}_{ly}K}_{i}\alpha\:{\eta\:}_{ii}}{2EI}&\:(0<x<\frac{l}{2})\\\:\frac{{(L-x)F}_{ly}{K}_{i}\alpha\:{\eta\:}_{ii}}{2EI}&\:(\frac{l}{2}<x<L)\end{array}\right.$$

(3)

$$\:\epsilon\:\left(x\right)=\left\{\begin{array}{cc}\frac{{F}_{xy}{K}_{k}\alpha\:{\eta\:}_{ki}}{2EI}&\:(0<x<\frac{l}{2})\\\:\frac{(L-x){F}_{ly}{K}_{k}\alpha\:{\eta\:}_{ki}}{2EI}&\:(\frac{l}{2}<x<L)\end{array}\right.$$

(4)

K_i (K_i>1) representing the amplification factor of the transverse distribution coefficient for the loaded beam (Beam i), and K_k (K_k<1) denoting the attenuation factor for the non-loaded adjacent beam (Beam k). The coefficient αη_ii defines the actual load proportion allocated to Beam i when subjected to direct loading, while αη_ki quantifies the load proportion transferred to Beam k under the same loading configuration.

Longitudinal damage in T-beam bridges induces localized flexural stiffness degradation within the damaged zone while preserving original stiffness EI in undamaged regions. This localized deterioration nevertheless propagates systemically, reducing the global load-bearing capacity to $\:\text{R}\alpha\:\eta\:\text{F}$ (R < 1), as empirically validated in Fig. 5. The mechanical model assumes a stiffness transition within a 2c-length zone adjacent to Section b at the left beam end, where the effective stiffness diminishes to $\:\text{E}{\text{I}}_{0}=\text{P}\text{E}\text{I}$ (ρ < 1), contrasting with the intact region’s maintained EI. The resultant strain influence line at mid-span measurement point BL/2 is analytically described by Eq. (5.), governing the structural response under compromised load-bearing conditions.

Fig. 5

Schematic diagram of the mechanical model of T beam under longitudinal damage.

$$\:\epsilon\:\left(x\right)=\left\{\begin{array}{c}\begin{array}{cc}\frac{{R}_{i}{F}_{xy}\alpha\:{\eta\:}_{ii}}{2EI}&\:(0<x<b)\\\:\frac{{R}_{i}{F}_{xy}\alpha\:{\eta\:}_{ii}}{2{P}_{i}EI}&\:(b<x<b+2c)\end{array}\\\:\begin{array}{cc}\frac{{R}_{i}{F}_{xy}\alpha\:{\eta\:}_{ii}}{2EI}&\:(b+2c<x<\frac{L}{2})\\\:\frac{(L-x){R}_{i}{F}_{y}\alpha\:{\eta\:}_{ii}}{2EI}&\:(\frac{L}{2}<x<L)\end{array}\end{array}\right.$$

(5)

Where P < 1 is the degree of stiffness reduction in the damaged area after longitudinal damage of the T-beam. R < 1 is the degree to which the load borne by the T beam decreases after longitudinal damage occurs.

As illustrated in Fig. 6, the strain influence line curves derived from Eqs. (3) and (4) exhibit critical transverse damage indicators in T-beam bridges. Transverse damage-induced degradation in load transfer efficiency leads to asymmetric load redistribution: the loaded beam assumes increased load proportions, resulting in elevated strain influence line peaks and expanded envelope areas compared to the undamaged state, while the non-loaded beam experiences reduced load shares accompanied by diminished peaks and contracted envelope areas. This inverse proportionality between loaded and non-loaded beam responses establishes the diagnostic validity of strain influence line peak variations and envelope area modulations for transverse damage assessment in T-beam bridges.

Fig. 6

Comparison of strain influence lines before and after transverse damage at the mid-span measurement point: (a) loading beam (I beam), (b). unloaded beam (k beam).

As demonstrated in Fig. 7, the strain influence line curves derived from Eq. (5) reveal distinct longitudinal damage signatures in T-beam bridges. Compared to the undamaged state, the measurement point on the damaged beam exhibits significant reductions in both strain influence line peak magnitude and envelope area, with these effects being particularly pronounced within the damaged zone. Conversely, the non-damaged beam’s measurement point displays a proportional increase in peak values and envelope areas. This diametrical response, characterized by localized parameter attenuation in the damaged region alongside compensatory amplification in adjacent intact areas, establishes a diagnostic framework for longitudinal damage assessment through comparative analysis of strain influence line peak variations and envelope area modulations.

Fig. 7

Comparison of strain influence lines before and after longitudinal damage at the mid-span measurement point: (a) loading beam (I beam), (b). unloaded beam (k beam).

Rapid diagnostic method for beam bridge damage

In terms of mechanical analysis, from Eq. 3 to Eq. 5, it can be seen that the main manifestations are the reduction of the bridge bending stiffness EI the reduction of the distributed bearing F_i, and then the change of strain influence line characteristics. Therefore, the change of strain-influenced line characteristics can better reflect the bridge structure’s damage condition and bearing performance.

Given the analytical derivation results above, a T-girder bridge is used as an example for numerical simulation verification. The bridge is a 5-piece simply supported T-beam bridge with a span diameter of 40 m, a bridge width of 11.25 m, and a T-beam height of 2.5 m. Material properties of T beam: elastic modulus E = 3.45 × 1010 Pa, torsional modulus G = 0.425E = 1.47 × 1010 Pa, bulk density = 2500Kg/m3, Poisson’s ratio = 0.2. A diaphragm is arranged at the fulcrum, 1/4 section, 2/4 section, and 3/4 section of the bridge, and the thickness of the diaphragm is 0.2 m and the height is 2.25 m. The number of T beams is from A to E, and the size of T beams is shown in Fig. 8.

Fig. 8

Schematic diagram of the cross-section of the T-beam bridge (unit: mm).

Diagnostic method for longitudinal damage of T-beam bridge

Damage case setting: The T beam is numbered from A to E, and measurement points A, E, and C are arranged in the span position of beams A, E, and C. In this section, the simulated damage stiffness reduction is 10% as a gradient with a maximum of 40%, which is realized by reducing the elastic modulus in the finite element software Midas Civil, the simulated damage area is shown in Fig. 9 and the damage case setting is shown in Table 1.

Fig. 9

Schematic diagram of longitudinal damaged area (unit: m).

Table 1 The setting of longitudinal damage conditions.

To enhance interpretability of strain influence line alterations between intact and damaged states, comparative visualization is achieved through total envelope area analysis. This methodological approach transforms abstract strain variations into quantifiable spatial damage profiles.

(1) The changes of the total envelope area of strain influence line of the measuring points A, C, and E under working conditions H1 to H4, compared to the baseline working condition 0, are shown in Fig. 10.

Fig. 10

Total envelope area of strain influence line of each measuring point under working conditions H1-H4.

As observed in the Fig. 10, when the H-area of beam A sustains progressive damage levels of 10%, 20%, 30%, and 40%, the total envelope area of the strain influence line at measurement point A decreases by 2.02%, 4.43%, 7.37%, and 11.02%, respectively, while the corresponding values at measurement point C increase by 0.87%, 1.91%, 3.17%, and 4.74%. In contrast, the total envelope area at measurement point E remains nearly unchanged across all damage levels. These results indicate that the mechanical performance degradation in the side beam (beam A) of the T-beam bridge is characterized by two complementary phenomena: a progressive reduction in the total envelope area at measurement point A on the damaged beam and a simultaneous increase at measurement point C on the undamaged beam. Furthermore, this contrasting behavior becomes increasingly pronounced as the damage severity in the side beam escalates, demonstrating a clear correlation between structural deterioration and strain redistribution within the bridge system.

(2) The changes of the total envelope area of strain influence line of the measurement points A, C, and E of working conditions J1 to J4 compared to the baseline working condition 0, are shown in Fig. 11.

Fig. 11

Total envelope area of strain influence line of each measuring point under working conditions J1-J4.

As shown in Fig. 11, when the I-area of the middle beam (C beam) in the T-beam bridge sustains progressive damage levels of 10%, 20%, 30%, and 40%, the total envelope area of the strain influence line at measurement point A increases by 0.68%, 1.46%, 2.38%, and 3.47% respectively compared to the undamaged state. Meanwhile, the corresponding values at measurement point C decrease significantly by 4.28%, 6.21%, 8.94%, and 10.67%. Measurement point E also shows increases of 0.68%, 1.46%, 2.38%, and 3.47%, mirroring the trend at point A. These results indicate that as the middle beam (C beam) sustains increasing levels of damage, its mechanical performance degrades. This degradation is evidenced by the progressive reduction in the total envelope area at measurement point C. Conversely, the strain influence lines at measurement points A and E on undamaged beams exhibit proportional increases corresponding to the severity of the damage. Notably, this contrasting behavior becomes more pronounced with greater damage levels in the middle beam. The phenomenon provides a direct quantitative basis for evaluating the mechanical performance of the middle beam (C beam).

(3) The changes of the total envelope area of the strain influence line of the measuring points A, C, and E under working conditions K1 to K4 compared to the baseline working condition 0, are shown in Fig. 12.

Fig. 12

Total envelope area of strain influence line of each measuring point under working conditions K1-K4.

As shown in Fig. 12, the total envelope area of the strain influence line for measurement points A, C, and E exhibits negligible changes when the K-area of the C beam sustains damage levels of 10%, 20%, 30%, and 40%. This observation confirms that if the measurement points on the T-beam bridge are positioned too far from the damaged area, the strain influence line parameters become insensitive to damage, creating a critical limitation for effective damage detection.

Based on the principles established in the preceding analysis, the total envelope area variations for measurement points A and E can be quantitatively evaluated using the methodology defined in Eq. (6):

$$\:{\Delta\:}{S}_{i}=\frac{{\int\:}_{0}^{L}{\epsilon\:}_{i}\left(x\right)dx-{\int\:}_{0}^{L}{\epsilon\:}_{0}\left(x\right)dx}{{\int\:}_{0}^{L}{\epsilon\:}_{0}\left(x\right)dx}\times\:100\%$$

(6)

Where L is the total span of the bridge; $\:{\epsilon\:}_{i}\left(x\right)$ is the expression of the measured strain influence line; $\:{\epsilon\:}_{0}\left(x\right)$ is the theoretical strain influences line expression; and $\:\varDelta\:{S}_{i}$ is the variation in the total envelope area at point i.

A histogram is established with the total change in the envelope area of the obtained measurement point A and measurement point E as the ordinates, which can locate and qualitatively diagnose the longitudinal damage status of the T-beam bridge. There are several types of histogram trends, as shown in Fig. 13:

Fig. 13

Qualitative diagnosis of the longitudinal damage of the T-beam bridge: (a) non-damaged; (b) The A-beam is damaged; (c) The B-beam is damaged; (d) The C-beam is damaged; (e) The D-beam is damaged. (f) The E-beam is damaged.

Figure 13 illustrates longitudinal damage-induced variations in the total envelope areas of measurement points A and E, positioned on opposing side beams of each T-girder. Field inspectors can leverage these data to construct diagnostic histograms (referencing Fig. 13) plotting the variation range of envelope areas at these measurement points, enabling structural integrity assessment across both individual girders and the entire bridge system.

For quantitative damage evaluation, the strain influence lines at measurement points are longitudinally partitioned into n equidistant intervals of length c along intact and damaged beams. Higher spatial resolution (achieved through smaller c values and correspondingly larger n) enhances diagnostic accuracy. These intervals are systematically labeled Interval 1 through Interval n.

Damage-induced deviations in strain influence lines are quantified through comparative analysis of total envelope areas between intact and damaged states. The envelope area difference (ΔA) for each interval is computed as:

$$\:\varDelta\:{S}_{i}={\int\:}_{i-c}^{i+c}{\epsilon\:}_{ii}\left(x\right)dx-{\int\:}_{i-c}^{i+c}{\epsilon\:}_{i0}\left(x\right)dx$$

(7)

where c is the divided length of the interval; $\:{{\upepsilon\:}}_{\text{i}\text{i}}\left(\text{x}\right)$ is the strain expression for the damage condition of interval i; $\:{{\upepsilon\:}}_{\text{i}0}\left(\text{x}\right)$ is the non-destructive condition strain expression for intervals i; and $\:\varDelta\:{\text{S}}_{\text{i}}$ is the difference between the envelope area of the damaged condition and the envelope area of the lossless condition in interval i.

Fig. 14

Longitudinal damage localization and quantitative diagnosis diagram system of the T-beam bridge damage: (a) Damage state of the edge beam; (b) Damage state of the middle beam.

Finite element simulation analysis demonstrates that longitudinal damage in T-beam bridges manifests through characteristic envelope area variations in strain influence lines. Specifically, when longitudinal damage occurs, the envelope area reduction at damaged intervals generally exceeds those at undamaged positions, with the most pronounced decrease consistently observed at the damaged interval itself. This pattern intensifies progressively with escalating damage severity. Quantitative relationships reveal that structural degradation severity correlates with envelope area reductions by a factor of twofold for loaded beams and fourfold for non-loaded beams. Based on these mechanical principles, practical damage diagnosis can be achieved by first plotting the envelope area difference curve between measurement points on the damaged beam, then identifying the peak interval corresponding to the damage location through comparison with the diagnostic criteria in Fig. 14. Finally, the longitudinal damage severity is quantified by analyzing the variation magnitude of the strain influence line’s envelope area within the identified damage interval.

Diagnostic method for transverse damage of T-beam bridge

Simulate multiple damaged areas as shown in Fig. 15, with various damage conditions set as indicated in Table 2.

Fig. 15

Schematic diagram of transverse damaged area (unit: m).

Table 2 The setting of transverse damage conditions.

To facilitate the interpretation of the change of the transverse damage state of the strain-influence line of each measurement point compared with the non-destructive state, the total envelope area of the strain-influence line is intuitively expressed by calculating and comparing the strain-influence line.

(1) As shown in Fig. 16, when the mid-span diaphragm (Diaphragm 3) experiences damage levels of 40%, 50%, 60%, and 70%, the total envelope area of the strain influence line at measurement point A decreases by 2.56%, 3.58%, 4.88%, and 6.62% respectively. In contrast, the corresponding values at measurement point C increase by 5.23%, 7.37%, 10.17%, and 14%. These changes indicate a compromised transverse load-transfer capacity of the mid-span diaphragm (Diaphragm 3) near the measurement cross-section. The increased load-bearing ratio of the loaded beam (Beam C) compared to the undamaged state results in an expansion of the strain influence line’s total envelope area at measurement point C. Conversely, the reduced load-bearing ratio of the unloaded beam (Beam A) relative to the intact condition causes a contraction of the total envelope area at measurement point A.

Fig. 16

The total envelope area of the strain influence line of each measuring point under working conditions L1-L4.

(2) As shown in Fig. 17, progressive damage of 40%, 50%, 60%, and 70% to the left diaphragm (diaphragm 2) leads to corresponding increases in the total envelope area of the strain influence line at measurement point A by 0.94%, 1.31%, 1.79%, and 2.42%. Meanwhile, measurement point C shows inversely proportional decreases of 1.68%, 2.37%, 3.26%, and 4.46% respectively. This suggests that damage to the unilateral diaphragm (diaphragm 2) located remotely from the measurement cross-section compromises the transverse load distribution capacity. This effect is opposite to that seen with mid-span transverse diaphragm damage. Specifically, the loading beam (C-beam) has reduced strain influence line envelope areas at measurement point C, while the non-loading beam (A-beam) shows increased values at measurement point A. This highlights the inverse relationship in load distribution between these two damage cases.

Fig. 17

The total envelope area of strain influence line of each measuring point under working conditions M1- M4.

Based on the above research and analysis results, it can be concluded that for the transverse damage of the T-beam bridge, the variation trend of the total envelope area of the strain influence line at the two measurement points of the side beam and the middle beam can be used as the diagnostic index to achieve the preliminary qualitative diagnosis.

Through Eq. 7, a histogram is established with the total change in the envelope area of the obtained measurement point A and measurement point C as the ordinates, which can be used to qualitatively diagnose the damage to the diaphragm of the T-beam bridge. As shown in Fig. 18, the histogram demonstrates the following situations:

Figure 18 shows the trends of the total envelope area of edge beam measurement point A and middle beam measurement point C under the damaged state of the diaphragm at different positions. According to the test results, the inspector can make a histogram of the change range of the total envelope area of measurement point A and measurement point C, and diagnose whether the transverse diaphragm of the T-beam bridge is damaged by comparing with Fig. 18, and can determine the approximate position of the damaged diaphragm.

The difference in the curve of the envelope area of each interval is obtained by fitting Eq. 7.

Fig. 18

Qualitative diagnosis of the transverse damage of the T-beam bridge.

Finite element simulation analysis revealed that for a T-beam bridge, whether it is damage to the mid-span diaphragm or damage to the unilateral diaphragm, the peak interval of the envelope area difference curve of the strain influence line interval of the measurement point is always consistent with the damage position interval of the transverse diaphragm. Therefore, according to the actual test results, the difference curve of the envelope area between the measurement point interval of the middle beam and the edge beam can be drawn, and the location and diagnosis of the damage location can be made by referring to the diagnostic diagram of Figs. 19 and 20. The difference curve of the envelope area between the strain influence line of the measuring point at the side beam and the middle beam can be used as the diagnostic index, the peak position can be used to locate and diagnose it, and the peak growth range can be used to make a rough quantitative diagnosis.

Fig. 19

The curve of the difference between the envelope area of each interval of the measurement point across the damaged state of the diaphragm in the mid-span: (a) The edge beam; and (b) The middle beam.

Fig. 20

The curve of the difference between the envelope area of each interval of the measurement point of the unilateral diaphragm damage state: (a) The edge beam; (b) The middle beam.

Diagnostic experiment of beam bridge damage based on the strain influence line

Model of a simple support T-beam bridge

The experimental investigation employs a scaled physical model based on a 40 m prototype T-girder bridge, fabricated at an 18:1 geometric scale ratio using acrylic (PMMA) material. The model configuration comprises five T-beams interconnected by five diaphragms positioned according to conventional bridge design principles: at support locations, quarter-span (1/4L), midspan (2/4L), and three-quarter-span (3/4L). Material properties under standard laboratory conditions (20 °C) include elastic modulus E = 2.93 GPa, Poisson’s ratio ν = 0.37, and unit weight γ = 11.9 kN/m³. Critical dimensional parameters feature a total span length of 225 cm, deck width of 62.5 cm, T-beam height of 14 cm, flange thickness of 1.8 cm, end/mid diaphragm thicknesses of 1.4 cm/1.0 cm, web thickness of 1.8 cm, and uniform diaphragm spacing of 45 cm. The elevation, cross-sectional geometry, and physical specimen are detailed in Fig. 21.

A custom two-axle vehicular loading system was implemented with mass calibration through fixed iron block counterweights on the upper structure, achieving a total operational mass (vehicle plus counterweights) of 20.38 kg during testing. The loading configuration, geometrically defined in Fig. 22, employed a quasi-static protocol with low-speed uniform traversal (a≪1), ensuring structural equivalence to static loading conditions through velocity control below critical speed v_cr. Traversal velocity was adaptively regulated by test personnel to meet operational requirements while maintaining analytical validity.

Fig. 21

Schematic diagram of the specific dimensions(unit: cm) of the test model: (a) Elevation view of the test model (unit: cm); (b) Cross-sectional view of the test model; and (c) Actual view of the test model.

Fig. 22

Schematic diagram of the test loading model vehicle (unit: cm): (a) Top-down view (b) Side-view.

The experimental program investigates four distinct working conditions: undamaged (baseline), midspan diaphragm damage, middle beam damage, and end beam damage, as detailed in Table 3 and visualized in Fig. 23. The simulation study approach adopted in this section involves reducing the haunch height. According to the stiffness calculation formula EI (where E is the elastic modulus of the section material and I is the section moment of inertia), E is constant for a specific cross-section, while I varies with changes in the haunch reduction height. Precise calculations using the section property calculator in the finite element analysis software Midas Civil revealed that, prior to the concrete cracking zone in the T-beam section reaching the level of the reinforcement bottom, for every 2 cm increase in the extent of cracking, the degree of section stiffness reduction increases by 5–6%, Within the same cracking range, diagonal concrete cracking induces an 8–10% greater stiffness reduction compared to vertical cracking. Therefore, transverse damage simulation involves introducing 14 mm-height cracks at diaphragm interfaces between the middle beam and adjacent secondary beams within the model span, achieving a 50% flexural stiffness reduction in the cracked diaphragm zones (Fig. 24). Longitudinal damage is replicated through 18 mm-height cracks at midspans of the middle and end beams, resulting in 50% stiffness degradation within the cracked T-beam regions (Fig. 25). All damage configurations employ bottom-flange cracking at critical sections to simulate realistic failure mechanisms while maintaining controlled stiffness reduction magnitudes for comparative analysis.

Table 3 Working conditions of the experiment.

Fig. 23

Schematic diagram of damaged area under working conditions (unit: cm): (a) working condition 1;(b) working condition 2, and (c) working condition 3.

Fig. 24

Schematic diagram of damage to the diaphragm.

Fig. 25

Schematic diagram of damage to the longitudinal beam.

The following tests are carried out on the T-beam model: a nondestructive test, a diaphragm damage test, a middle beam damage test, and an end beam damage test. Each test is conducted based on the model after the previous test. Since the elastic modulus of the material used in this model is sensitive to changes in temperature, it is necessary to maintain the indoor temperature at 20 °C during the test and repeat the test 5 times under the same working conditions.

Measurements taken with sensors are influenced by environmental factors and therefore require data processing. The existing methods mostly rely on signal filtering techniques²⁸, such as moving averages²⁹ or dynamic and static parts, to remove noise in the original signal through techniques such as empirical mode decomposition (EMD)³⁰. Through noise reduction of the data, the stress map lines can be smoothed, which can better reflect the static characteristics of the model.

Analysis of diagnostic results based on total envelope area

The total envelope area of the strain influence lines for measurement points A and C under different working conditions obtained through experiments is shown in Table 4.

Table 4 Statistical analysis of the total envelope area of the strain influence line (unit: µε•m).

(1) Verification of qualitative diagnosis for transverse damage.

The mid-span transverse partition damage is introduced only in working condition 1 compared to working condition 0. Therefore, the diagnostic effect of transverse damage can be verified by analyzing the change in the total envelope area of the strain influence line at the measuring point between these two conditions. Based on Table 4, the variation amplitude of the total envelope area for each measurement point under working condition 1 relative to working condition 0 can be obtained. This data is used to construct a diagnosis diagram for transverse damage in the T-beam model, as shown in Fig. 26(a).

In Fig. 26(a), when comparing condition 1 to condition 0, the total envelope area of the strain influence line at measurement point A on the edge beam decreases, while the total envelope area at measurement point C on the middle beam increases. This trend aligns with Fig. 18(b) in the qualitative diagnostic diagram for transverse damage presented in this paper. Consequently, the evaluation result indicates that, under the experimental model state in condition 1, damage has occurred in the mid-span diaphragm compared to the state in condition 0.

The transverse damage results of the experimental model, obtained using the diagnostic method proposed in this paper, are consistent with the preset test results. This confirms the rationality and feasibility of the diagnostic method.

(2) Verification of the qualitative diagnostic effect of longitudinal damage.

For the diagnosis of middle beam damage, the newly added damaged area in working condition 2, compared to working condition 1, is the damage in the middle beam area. Therefore, the change in the total envelope area of the strain influence line at the measurement point between working condition 2 and working condition 1 can be used to verify the diagnostic effect of middle beam damage. Based on Table 4, the variation amplitude of the total envelope area for each measurement point under working condition 2 relative to working condition 1 can be obtained. This data is used to construct a qualitative diagnosis diagram for middle beam damage in the T-beam model, as shown in Fig. 26(b).

In Fig. 26(b), relative to condition 1, the total envelope area of the strain influence line at measurement point A on the edge beam decreases under condition 2, while the total envelope area at measurement point C on the middle beam increases. This change trend corresponds to Fig. 13(d) in the qualitative diagnostic diagram for transverse damage presented in this paper. Consequently, the diagnostic result indicates that longitudinal damage of the middle beam occurs in condition 2 of the test model compared to condition 1.

For the diagnosis of edge beam damage, the newly added damaged area in working condition 3, compared to working condition 2, is the damage in the edge beam area. Thus, the qualitative diagnosis effect of edge beam damage in longitudinal damage can be verified by the change in the total envelope area of the strain influence line at the measuring point under working condition 3. According to Table 4, the variation amplitude of the total envelope area for each measurement point in working condition 3 relative to working condition 2 can be further obtained to create a qualitative evaluation chart of the longitudinal force performance of the edge beam of the test model, as shown in Fig. 26(c).

Figure 26(c) shows that, in condition 3 compared to condition 2, the total envelope area of the strain influence line at edge beam measurement point A significantly decreases, while the total envelope area at the opposite edge beam measurement point E remains unchanged. This trend corresponds to Fig. 13(b) in the qualitative diagnostic diagram for transverse damage presented in this paper. Therefore, the diagnostic result is that under the state of test model condition 3, longitudinal damage occurs in the edge beam compared to the state of condition 2.

The damage results of the middle beam and edge beam obtained using the diagnostic method proposed in this paper are consistent with the preset test results. This confirms the rationality and feasibility of this method for diagnosing longitudinal damage in both the middle beam and edge beam.

Fig. 26

Damage Qualitative Diagnosis Diagram: (a) transverse damage, (b) longitudinal damage of middle beam, (c) longitudinal damage of edge beam.

Verification of the localization and quantitative diagnosis effect of T-beam damage

The total span length of the test model is 2.25 m. The equal spacing of the strain influence line of the damaged beam body measurement point is divided into 8 intervals, as shown in Table 5. The change in the envelope area of each interval is calculated and analyzed statistically to determine the position and quantity of the force performance of the test model.

Table 5 Interval division of the strain influence line (unit: m).

(1) Verification of the localization and quantitative diagnosis effect of middle beam damage.

The newly added damaged area of working condition 2 relative to working condition 1 is the damaged area of the middle beam. To verify the damage location and quantitative diagnosis effect of the middle beam, it is necessary to use the working condition 1 model as the benchmark and use the change in the envelope area of each interval of measurement point C as the strain influence line of working condition 2 relative to working condition 1. The statistical information is shown in Table 6.

Table 6 Statistical table of the envelope area of each interval of the strain influence line at measurement point C (unit: µε•m).

According to Table 6, the variation amplitude and difference curve of the envelope area of each interval of measurement point C under working condition 2 relative to working condition 1 can be constructed further, as shown in Fig. 27.

Fig. 27

Statistical diagram of envelope area in each interval of measurement point C: (a) Variation in the envelope area in each interval; and (b) Curve of the difference between the envelope area of each interval.

Figure 27 shows that the peak position and change amplitude peak of the envelope area difference curve of each interval of the strain influence line of measurement point C in condition 2 and condition 1 all appear in interval 4; that is, the interval (0.84375 m, 1.125 m) and the envelope area change amplitude of the interval are − 26.22%. The conclusion drawn from the third section of this paper is that when the damaged beam is a loading beam, the damage degree of the beam is about 2 times the reduction of the envelope area in this interval. Therefore, the results can be described as follows: the damage to the middle beam in condition 2 of the test model is greater than that in condition 1, the damage location is within the interval of the middle beam (0.84375 m, 1.125 m), and the damage degree is 52.44%.

For the above diagnostic results, the localization diagnosis results for middle beam damage are consistent with the preset damage interval of the test, which proves the accuracy of the diagnostic effect and the feasibility of the method. The quantitative diagnosis result of 53.4% of the middle beam damage was greater than the preset damage degree of 50%, but its deviation remained within 5%. This basically proves the accuracy of the diagnostic effect and the feasibility of the method.

(2) Verification of the localization and quantitative diagnosis effect of end beam damage.

Compared with that in working condition 2, the damage to the end beam area is increased only under working condition 3. To verify the damage location and diagnosis effect of the end beam, it is necessary to use the model of working condition 2 as the benchmark. In addition, the change in the envelope area of each interval is used as measurement point A. Its statistics are shown in Table 7.

Table 7 Statistical table of the envelope area of each interval of the strain influence line of measurement point A (unit: µε•m).

According to Table 7, the variation amplitude chart and difference curve of the envelope area of each interval of measurement point A under condition 3 relative to condition 2 can be further constructed, as shown in Fig. 28.

Fig. 28

Statistical diagram of the change in envelope area in each interval of measurement point A: (a) Variation in the envelope area in each interval; and (b) Curve of the difference between the envelope area of each interval.

From Fig. 28, the peak position and the change amplitude of the peak of the envelope area difference curve for each interval of the strain-influence line of measurement point A all appear in interval 4, specifically within the range (0.84375 m, 1.125 m). Additionally, the envelope area change amplitude for this interval is calculated to be -12.17%. Based on the diagnostic method established above, the diagnostic results are as follows: the end beam is damaged under working condition 3 of this test model, with the damage located in the end beam within the range (0.84375 m, 1.125 m), and the damage degree is determined to be 48.68%.

According to the diagnostic results, the localization diagnosis for end beam damage aligns with the preset damage interval of the test. This demonstrates the accuracy of the diagnostic effect and the feasibility of the method. A key conclusion from this paper is that when the damaged beam is unloaded, the damage degree of the beam is approximately four times the reduction in the envelope area within this interval. The quantitative diagnosis result for end beam damage was calculated as 48.68%, which is slightly lower than the preset damage degree of 50%. However, the deviation remains within 5%, further proving the accuracy of the diagnostic effect and the feasibility of the method.

Conclusion

Based on influence line theory, this study systematically investigates the characteristic variations of strain influence lines in T-beam bridges under both undamaged and damaged conditions by integrating theoretical analysis, finite element simulation, and experimental verification. The proposed methodology commences with the derivation of analytical formulas for strain influence lines in T-beam bridges. By analyzing variations in peak values and envelope areas of these influence lines, differential influence line curves were constructed. Subsequently, positioning and quantification methods for damage in both transverse and longitudinal structural components were developed. The approach was validated through finite element modeling and analysis of a scaled T-beam bridge, supplemented by experimental verification using an acrylic scale model. The principal findings are summarized below:

(1) The influence line method adopted in this study overcomes limitations inherent in conventional load testing approaches, such as inaccurate diagnostics, traffic disruption, and cumbersome procedures. By deriving analytical expressions for strain influence line functions, we theoretically examine global and local variations in these lines following damage to T-beams and diaphragms, thereby paving the way for advanced investigations.

(2) Finite element simulations of transverse and longitudinal damage states revealed that regardless of damage orientation, envelope areas at damaged sections are significantly smaller than those at intact sections, with maximum reduction observed at primary damage locations. As damage severity increased, this trend became more pronounced. When the loaded beam sustains damage, the envelope area reduction coefficient approximates half the damage degree. For unloaded beam damage, this coefficient reduces to approximately one-fourth of the damage degree. Quantitative diagnosis was achieved by correlating damage severity with envelope area variation ranges.

(3) Experimental verification using a scaled acrylic bridge model demonstrated agreement between transverse damage diagnosis results and preset conditions. For mid-beam damage, localization results matched the preset damage interval, with quantitative assessment showing 53.4% damage versus the 50% preset value (deviation < 5%). End-beam damage localization aligned with test specifications, while quantitative evaluation yielded 48.68% versus 50% preset value (deviation < 5%), confirming method accuracy and feasibility.

(4) While this study focuses on simply supported T-beam bridges, and its findings are applicable exclusively to grillage bridges where a limited contribution of the deck slab is expected, extending this methodology to complex non-beam structures (e.g., arch bridges and suspension bridges) by analyzing strain influence line variations post-damage offers significant research potential.