Damage identification of bridge structure considering temperature variations based on particle swarm optimization

Abstract

Structures are always exposed to environmental conditions such as varying temperatures and noises; as a consequence, the dynamic features of structures are changed accordingly. But the model-based methods, used to detect damage using optimization algorithms to get global optimal solution, are highly sensitive to environmental conditions, experimental noises, or numerical errors. While the mechanisms of optimization algorithms are limited by local optimal solution, their convergences are not always assured. In the study, a model-based damage-identification method considering temperature variations, comprised of particle swarm optimization and cuckoo search, is implemented to detect structural damage. First, to eliminate the influence of environmental temperature, temperature change is considered as a parameter of structural material elastic modulus. A function relationship is established between environmental temperature and the material elastic modulus, and an objective function composed of natural frequency, mode shape and modal strain energy with different weight coefficients is constructed. Second, the hybrid optimization algorithm, a combination of particle swarm optimization and cuckoo search, is proposed. Third, to solve the problem of optimization algorithm convergence, the optimization performance of the hybrid optimization algorithm is validated by utilizing four benchmark functions, and it is found that the performance of the hybrid optimization algorithm is the best. In order to test the performance of the three algorithms in damage identification, a numerical simply supported beam is adopted. The results show that the hybrid optimization algorithm can identify the damage location and severity under four different damage cases without considering temperature variations and two cases considering temperature variations. Finally, the hybrid optimization algorithm is introduced to test the damage-identification performance of I-40 Bridge, an actual steel–concrete composite bridge under temperature variations, whose results show that the hybrid optimization algorithm can preferably distinguish between real damages and temperature effects (temperature gradient included); its good robustness and engineering applicability are validated.

Keywords

damage identification the hybrid optimization algorithm steel–concrete composite bridge temperature gradient temperature variations

Introduction

In structural damage identification, it is difficult to assess the damage by visual inspection at early stages through vibration measurements, structural dynamic parameters (such as natural frequencies and mode shapes), which are often used as identification indicators for determining whether a structure is damaged. However, the environmental variations, such as temperature change and experimental noise, cannot be ignored; the change of ambient temperature significantly generates changes in the elastic modulus of the material, which in turn affects the overall stiffness of the structure, ultimately resulting in the change of structural dynamic properties (Pham, 2009).

Farrar et al. (1997) conducted a full-day monitoring, once every 2 h, at the Alamos Grand Canyon Bridge and found that the first-order modal frequency of the structure changed more than 5% with the temperature-gradient changes along the concrete slab. Then, the in-depth study was carried out, whose results showed that the first three-order natural frequencies of the bridge changed by 4.7%, 6.6%, and 5.0%, respectively, within a 24-h monitoring. Actually, both research and practical experience shows that the temperature changes (Peeters and De Roeck, 1999), freeze–thaw effect (Alampalli, 1998), and the average seasonal variation (Askegaard and Mossing, 1988) have caused a change in natural frequencies.

Hence, it is reducing or even eliminating the effects of temperature change on structural modal parameters that plays a significant role in ensuring the accuracy of damage identification (Hao and Xia, 2002; Titurus et al., 2003). In recent years, many methods have been proposed to eliminate the influence of environmental conditions in damage identification. Some techniques aim at removing environmental influence on testing data (Cornwell et al., 2010; Dervilis et al., 2015; Gillich et al., 2019; Liang et al., 2018; Sohn et al., 2015; Spiridonakos et al., 2016; Vanlanduit et al., 2005; Yan et al., 2005). Other types of methods focus on trying to remove the variability due to the environment without measuring the environmental factors (Hu et al., 2009; Jiang et al., 2011; Kullaa, 2003; Lam and Ng, 2008; Li et al., 2011; Surace and Worden, 2010). Those methods mentioned above can identify the presence of damage; however, the location and extent of damage cannot be determined. In addition, the temperature effects are handled by data-processing means rather than combination of temperature effects and finite-element model. According to Friswell (2007), the most promising strategy is to integrate temperature effects into the numerical model. Thus, it is necessary to propose a model-based damage-identification method with a good capability to handle the variations of environment factors by numerical model and considering temperature effects on modal parameters.

In model-based damage-identification method, different modal information can be used to locate and even quantify damage, for example, frequencies (Carden and Fanning, 2004; Modak et al., 2000; Salawu, 1997), modal assurance criterion (MAC; Allemang and Brown, 1982; Deraemaeker and Preumont, 2006), coordinate modal assurance criterion (COMAC; Lieven and Edwins, 1988), Spatial MAC (SMAC) (Heylen and Janter, 1990), substructuring method (Weng et al., 2011), modal strain energy (Dinh-Cong et al., 2018; Jaishi and Ren, 2007), modal flexibility (Jaishi and Ren, 2006; Weng et al., 2013), and residual forces (Liu, 1995). Through the comparison between the undamaged and damaged state, correlating a numerical model with experimental data from the structure, the damage is modeled as a reduction of stiffness. An advantage of these methods is that they are reasonable and feasible; they can combine physical damage with numerical analysis. However, these methods depend on the accuracy of the numerical model; the structural modal data obtained are often affected by environmental factors such as temperature changes and numerical errors, so they are difficult to distinguish actual changes due to damage from other factors.

To identify and quantify structural damage accurately, some researchers have established finite-element model methods to ensure that the temperature effects were appropriately considered, but other researchers begin to study how to incorporate the effects of temperature in model-based damage-assessment algorithms, assuming that the elasticity modulus of the materials is influenced by temperature (Meruane and Heylen, 2011; Pham, 2009; Xia et al., 2012). In the model-based damage-identification method, a robust optimization algorithm with an objective function is significant; the global optimal solution and the convergence is the goal that most researchers focus on. Conventional optimization techniques used in damage identification or model updating employing sensitivity-based searching mechanisms are often limited by the disadvantage of local optimal solution and the convergence and so on. Many researchers tried to find and improve a robust optimization algorithm that achieves the global minima using an objective function. For example, genetic algorithm (GA; Guo and Li, 2012; Na et al., 2011; Shabbir et al., 2017), bee colony algorithm (BCA; Ding et al., 2018), particle swarm optimization (PSO; Seyedpoor, 2012; Shirazi et al., 2014), global artificial fish swarm algorithm (GAFSA; Yu and Li, 2014), and cuckoo search (CS; Yang and Deb, 2009).

There are some disadvantages in the former algorithms, such as local optimum and slow convergence rate, so a novel strategy with hybrid mechanism is often adopted to ensure global optimum and fast convergence. A method based on the Nelder–Mead algorithm (Simplex method) was presented (Begambre and Laier, 2009), which aimed to control parameters of the PSO; the stability and confidence of the algorithm are enhanced. Due to its good ability in local searching, Nelder–Mead algorithm was incorporated into firefly algorithm (FA) to improve the local searching ability (Pan et al., 2016), and combined with PSO, an improved Nelder–Mead method (INM) was proposed to detect multi-damage case (Chen and Yu, 2017). A new Structural damage detection (SDD) strategy of hybrid particle swarm optimization (HPSO) proposed by Chen and Yu (2018) was verified to be helpful to enhance the PSO global searching ability numerically and experimentally. For better detection accuracy and robustness, a novel optimization algorithm with better convergence and accuracy than pure PSO and GA, the hybrid of PSO and GA, is implemented by Sandesh and Shankar (2010). Chen et al. (2019) combined a weighted strategy and the trace Lasso into the objection function, and a novel ant lion optimizer algorithm was utilized to accurately locate structural damages and quantify the damage severities of structures.

In this article, a model-based damage-identification method considering temperature variations, comprised of PSO and CS, is implemented to detect structural damage. In the method, the objective function is composed of natural frequency, mode shape and modal strain energy, and temperature changes, defined as a parameter, affects structural material properties to avoid false damage identification. Four benchmark functions are used to validate the optimization performance of PSO-CS, and a numerical simply supported beam is adopted to test the performance of the three algorithms for damage identification. The method is verified with the example of I-40 Bridge, whose results showed that PSO-CS can preferably distinguish the real damage under temperature effects (temperature gradient included).

Damage-identification model

Temperature–elastic modulus relationship model

The main reason for the change of the modal parameters of the structure is that the elastic modulus of the structural material changes with the variations of environmental temperature, and the modal frequency changes of the concrete structure under temperature variations are much more than the steel structure (Xia et al., 2006). As shown in Figure 1, the assumed relationship between the elastic modulus and temperature was stated by Yan et al. (2005) and verified by many researchers (Huang et al., 2018; Meruane, 2010). Under temperature variations, the elastic modulus of concrete decreases with increasing temperature above 0°C, and for every 1°C increase in temperature, the elastic modulus of C50 concrete decreased by 0.4% on average, and C30 concrete decreased by 0.61% through a large number of experiments (Li, 2016).

Figure 1.

Assumed elastic modulus versus temperature: (a) concrete and (b) steel (Yan et al., 2005 and Meruane, 2010).

Structural elastic modulus varying with temperature change can be defined as

E (T) = E_{0} \cdot (1 - β_{E}) \cdot (T - T_{0})

(1)

where $E (T)$ is the elastic modulus when temperature is T; $E_{0}$ is the elastic modulus of the structure when temperature is $T_{0}$ ; and $β_{E}$ is the change coefficient of elastic modulus.

Structural damage can be expressed by the reduction of structural stiffness. In order to consider the influence of temperature on engineering structure, the element stiffness matrix of a structure can be expressed as

k_{T, D} = (1 - θ) \cdot k_{T, U}

(2)

where $k_{T, D}$ and $k_{T, U}$ are the stiffness matrices of the damaged and undamaged element when temperature is T; $θ$ is the stiffness reduction factor, $0 \leq θ \leq 1$ , where 0 means no damage and 1 means complete damage.

Correspondingly, the global stiffness matrix $K_{T, D}$ of a structure can be expressed as

K_{T, D} = \sum_{i}^{n} (1 - θ_{i}) \cdot k_{T, U}^{i}

(3)

where $K_{T, D}$ is the global stiffness matrix of the structure when temperature is T, $k_{T, U}^{i}$ is the elemental stiffness matrix of element i.

Construction of objective function

Dynamic parameters (such as natural frequencies and mode shapes) are common damage-identification indicators. Natural frequencies are sensitive to global damage, and mode shapes are more sensitive to local damage. Even damage identification of large structures, mode shapes containing the amount of local damage information is insufficient to meet the identification of small local damage. It is necessary to introduce new damage indicators, which are more sensitive to local damage of the structure, into the objective function to ensure the sensitivity of damage identification to local damage. The damage indicator of modal strain energy has higher sensitivity than mode shape for local damage in the identification of large structures. In the study, therefore, the objective function is determined as a function composed of three damage indicators with different weight coefficients, natural frequency, mode shape, and modal strain energy as follows

f (x) = f_{1} (x) + f_{2} (x) + f_{3} (x)

(4)

f_{1} (x) = C_{1} \cdot \sum_{j = 1}^{n mod} {(\frac{ω_{aj} - ω_{ej}}{ω_{ej}})}^{2}

(5)

f_{2} (x) = C_{2} \cdot \sum_{j = 1}^{n mod} {(\frac{1 - \sqrt{MA C_{j}}}{MA C_{j}})}^{2}

(6)

f_{3} (x) = C_{3} \cdot \sum_{j = 1}^{n mod} {(\frac{{MSE}_{aj}^{i}}{{MSE}_{ej}^{i}} - 1)}^{2}

(7)

where $f_{1} (x)$ is the natural frequency indicator; $f_{2} (x)$ is the mode shape indicator; $f_{3} (x)$ is the modal strain energy indicator (Dinh-Cong et al., 2018); $ω_{aj}$ and $ω_{ej}$ are the jth theoretical and experimental frequencies, respectively; $MA C_{j}$ is the jth MAC, $MA C_{j} = (φ_{aj}^{T} φ_{ej})^{2} / ((φ_{aj}^{T} φ_{aj}) (φ_{ej}^{T} φ_{ej}))$ , $φ_{aj}$ and $φ_{ej}$ are the jth theoretical and experimental mode shapes; ${MSE}_{aj}^{i} = \sum_{i = 1}^{nele} 1 / 2 {(Φ_{aj}^{i})}^{T} k^{i} Φ_{aj}^{i}$ , ${MSE}_{e j}^{i} = \sum_{i = 1}^{nele} 1 / 2 {(Φ_{e j}^{i})}^{T} k^{i} Φ_{ej}^{i}$ , $k^{i}$ is the stiffness matrix of the ith element, nmod is the number of considered modes, and nele is the number of elements, $Φ_{aj}^{i}$ and $Φ_{ej}^{i}$ are the vectors of nodal displacements of ith element corresponding to the jth theoretical and experimental mode shapes. In order to improve the effect of damage identification, the weight coefficients of different indicators in the objective function should be considered, which is based on the proper results (Huang et al., 2009), where different indicators are assigned different weight coefficients due to their confidences, in the paper, C₁ = 10, C₂ = 1, and C₃ = 0.1.

PSO-CS hybrid algorithm

PSO

PSO is a group intelligent optimization algorithm proposed by Kennedy and Eberhart (1995). PSO is suitable in dynamic multi-objective optimization condition for its advantages, such as faster calculation speed and better global search ability, which is not limited to the size of the population and insensitive to population size; the optimal position of the individual will be retained in the calculation. But its effectiveness is not good for ill-conditioned problems and multi-modal functions (Kennedy, 2011).

In PSO, a set of particles is randomly generated, all of which fly toward the global optimal particle of the current search, and update the speed and position. Suppose that in a N-dimensional optimization problem, the particle population size is M, the position of the ith particle is $P_{i}^{t} = (p_{i 1}, p_{i 2}, \dots, p_{iN})$ at tth iteration, and the velocity of the particle is $V_{i}^{t} = (v_{i 1}, v_{i 2}, \dots, v_{iN})$ . The position and velocity of the particle at (t + 1)-th iteration are updated in accordance

V_{i}^{t + 1} = ω V_{i}^{t} + c_{1} α_{1} \cdot (gbest - P_{i}^{t}) + c_{2} α_{2} \cdot (pbest - P_{i}^{t})

(8)

P_{i}^{t + 1} = P_{i}^{t} + V_{i}^{t + 1}

(9)

where t is the number of current iteration; $c_{1}$ and $c_{2}$ are learning factors, usually $c_{1}$ = $c_{2}$ = 2; $α_{1}$ and $α_{2}$ are random numbers between 0 and 1; gbest is the global optimal position of the current particle; and pbest is the optimal position of the individual.

The linear inertia weight $ω$ is related to the number of current iteration and the total number of iterations

ω = ω_{\max} - \frac{t \cdot (ω_{max} - ω_{min})}{N_{iteration}}

(10)

where $ω_{max}$ and $ω_{min}$ are the maximum and minimum linear inertia weights, respectively; $N_{iteration}$ is the total number of iterations (Shi and Eberhart, 1988).

CS

CS, the cuckoo looking for the best nest in its search space through Levy flight, is an emerging heuristic intelligent algorithm proposed by professors Yang and Deb in 2009, the main features of which are few parameters, simple operation, easy implementation, excellent random search path, and strong search ability. But concerns have been raised to overcome the disadvantages of traditional one, such as bad accuracy, low convergence rate, and easy to fall into local optimal value. To simplify the process, the principles can be summarized into three hypotheses (Yang and Deb, 2009, 2010, 2013);

Each cuckoo randomly selects a nest to lay eggs and produces only one egg at a time;

In the process of selecting the nest, the best nests with high-quality eggs will be carried over to the next generations;

The total number of available nests is fixed, and the egg laid by a cuckoo is discovered by the host bird with a probability $P_{a} \in (0, 1)$ . In this case, the host bird can either get rid of the egg, or simply abandon the nest and build a completely new nest.

Based on the above three assumptions, the way the cuckoo algorithm searches for the nest position and the path updates is updated as follows

x_{i}^{(t + 1)} = x_{i}^{(t)} + α \oplus L (s, β), i = 1, 2, \dots, n

(11)

where $x_{i}^{(t)}$ is the ith bird’s nest at the tth generation of the bird’s nest position; ⊕ is the point-to-point multiplication; $a (a > 0)$ is the step-size scaling factor used to control the search range of step size, which obeys the normal distribution; s is the step size; $1 < β < 3$ , usually $β = 1.5$

L (s, β) = \frac{λ Γ (β) \sin (\frac{π β}{2})}{π} \cdot \frac{1}{s^{1 + β}}, (0 < s_{0} \leq s)

(12)

where $L (s, β)$ is a random search path; $Γ$ is a standard Gamma function; $s_{0}$ is the minimum step size.

PSO-CS hybrid algorithm

The update mechanism of PSO is combined with CS to reduce the blindness of search while maintaining the high randomness of the algorithm. At the same time, retaining the random elimination mechanism in CS also makes the algorithm quickly leave the local extremum and avoid local optimization. The basic flow of the hybrid algorithm is as follows:

1. To confirm all the algorithm parameters including population number N, iterations number $N_{iteration}$ , system tolerance Tol, discovering probability $P_{a}$ , maximum and minimum value of inertia weight $ω_{max}$ and $ω_{min}$ , learning factors $c_{1}$ and $c_{2}$ , upper and lower bounds of searching range $U_{b}$ and $L_{b}$ . When the current number of iteration $t = 0$ , a set of initial population position matrix $P_{0}^{t = 0}$ and velocity matrix $V_{0}^{t = 0}$ are randomly generated, and the concept of particle velocity is introduced into the nest. The optimal individual in $P_{0}^{t}$ is defined as $P_{0}$ , and the fitness value of $P_{0}$ is assigned to the global best individual pbest in the initial population.

2. To take the fitness value of the ith bird’s nest in the nest for this iteration as $f_{i}^{t = 0}$ , the maximum, average, and minimum values of the overall fitness are $f_{max}^{t = 0}$ , $f_{ave}^{t = 0}$ , and $f_{\min}^{t = 0}$ , respectively; linear parameter is $d = 0.5$ ; the inertia weight $ω_{i}^{t = 0}$ of the nest i of this iteration can be as follows

ω_{i}^{t = 0} = ω_{min} + \frac{(f_{i}^{t = 0} - f_{\min}^{t = 0} + 0.1) \cdot (N_{iteration} - t) \cdot d}{N_{iteration} \cdot (f_{\max}^{t = 0} - f_{\min}^{t = 0} + 0.1)}

(13)

The updated formula for speed and nest position is given as

V_{i}^{t + 1} = ω V_{i}^{t} + c_{1} α_{1} \cdot (gbest - P_{i}^{t}) + c_{2} α_{2} \cdot (pbest - P_{i}^{t})

(14)

P_{i}^{t + 1} = P_{i}^{t} + V_{i}^{t + 1}

(15)

where $α_{1}$ and $α_{2}$ are random numbers ranging from 0 to 1.

3. To calculate the best fitness of $P_{0}^{t}$ and $P_{1}^{t}$ , generate $P_{2}^{t}$ based on the selected nests with the best fitness value according to the nests number, then give a random elimination probability $e_{i}$ to each individual in $P_{2}^{t}$ , and $P_{3}^{t}$ is calculated on the basis of elimination probability matrix $E = (e_{1}, e_{2}, \dots, e_{n})$

P_{3, i}^{t} = {\begin{matrix} P_{2, i}^{t}, if e_{i} \leq P_{a} \\ L_{b} + (U_{b} - L_{b}) \cdot randn (m, 1), else \end{matrix}

(16)

4. To calculate the fitness value of $P_{2}^{t}$ and $P_{3}^{t}$ , select the excellent partial composition matrix $P_{4}^{t}$ among $P_{2}^{t}$ and $P_{3}^{t}$ , select the best nest in $P_{4}^{t}$ as $T_{best}$ , and determine if the maximum number of iterations is reached. If the condition is met, the iteration is terminated, the target value is calculated, and the optimal solution is output. If the condition is not satisfied, $P_{4}^{t} = P_{0}^{t}$ , and the individual optimal and global optimal positions are updated according to formulas (14) and (15) to proceed to the next iteration.

Performance validation of PSO-CS

Benchmark functions

Four classical benchmark functions: Sphere function, Rosenbrock function, Rastrigin function, and Schaffer function (Figure 2), are used to evaluate the optimization performance of PSO-CS, which are shown as follows

Sphere

f_{Sphere} (x) = \sum_{i = 1}^{n} x_{i}^{2}, - 5.12 \leq x \leq 5.12

(17)

Rosenbrock

\begin{matrix} f_{Rosenbrock} (x) = \sum_{i = 1}^{10} (100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}), \\ - 5 \leq x \leq 5 \end{matrix}

(18)

Rastrigin

\begin{matrix} f_{Rastrigin} (x_{i}) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10], \\ - 5.12 \leq x_{i} \leq 5.12 \end{matrix}

(19)

Schaffer

\begin{matrix} f_{Schaffer} (x_{1}, x_{2}) = 0.5 + \frac{(\sin \sqrt{{x_{1}}^{2} + {x_{2}}^{2}}) - 0.5}{{[1 + 0.01 (x_{1}^{2} + x_{2}^{2})]}^{2}}, \\ - 10.0 \leq x_{1}, x_{2} \leq 10.0 \end{matrix}

(20)

Figure 2.

Graphs of benchmark functions: (a) Sphere, (b) Rosenbrock, (c) Rastrigin, and (d) Schaffer.

To evaluate the performance of PSO-CS, the optimization is performed, the algorithm parameters are shown as Table 1, and the iteration terminates when iteration time reaches 100. The iteration process of the three algorithms is shown in Figure 3. From Figure 3 and Table 1, it can be seen that the performance of PSO-CS is the best because PSO-CS combines the Levy flight of CS, the mechanism of inertia weight and particle updating, and it can reduce search blindness, quickly leave the local extremum, and avoid local optimum.

Table 1.

Optimization performance of benchmark function based on the three algorithms.

Benchmark function	Algorithm	Parameters	Best value	Worst value
Sphere	PSO	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	1.1791e–26	0.1280
	CS	$P_{a} = 0.25$	4.8719e–15	0.5879
	PSO-CS	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	2.6947e–27	0.0065
Rosenbrock	PSO	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	0	0.1503
	CS	$P_{a} = 0.25$	2.6752e–06	0.0731
	PSO-CS	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	0	0.0014
Rastrigin	PSO	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	3.7780e–06	0.1144
	CS	$P_{a} = 0.25$	6.6892e–10	0.1149
	PSO-CS	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	9.6581e–12	4.6558e–04
Schaffer	PSO	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	0	0.0019
	CS	$P_{a} = 0.25$	7.4349e–05	0.0071
	PSO-CS	$c_{1} = c_{2} = 2$ $ω_{max} = 0.9$ $ω_{min} = 0.4$	0	4.7886e–04

Figure 3.

Iteration process of benchmark functions: (a) Sphere, (b) Rosenbrock, (c)Rastrigin, and (d) Schaffer.

Damage identification of a simply supported beam

To validate the performance of the three algorithms for damage identification, numerical simulation of a simply supported concrete beam with material properties tabulated in Table 2 and four different predetermined cases without considering temperature variations are carried out, as demonstrated in Figure 4(a); in this stage of study, no temperature is considered.

Table 2.

Structural model parameters.

Parameters	Value	Unit
Mass density $(ρ)$	2500	kg/m³
Poisson’s ratio $(ν)$	0.166667	/
Elasticity modulus $(E)$	3.45 × 10¹⁰	Pa
Span	16 × 0.5	m
Width $(b)$	0.3	m
Height $(h)$	0.1	m

Figure 4.

(a) Model of the simply supported beam and predetermined damage cases, (b) Case 1, (c) Case 2, (d) Case 3, and (e) Case 4.

There are 16 elements in the analytical model and the damage is assumed in elements #2, #8, #15, and all 16 elements by 10%, 5%, 10%, and 10%, respectively. For four damage cases, damage is simulated by reducing the elemental stiffness of the members as introduced (Huang et al., 2018). It is assumed experimental modal responses of the beam before and after damage are generated through finite-element analysis based on program developed with MATLAB. Instead of experimental measurements, these numerically generated measurements are used to locate and assess damage. The four damage cases without considering temperature variations are defined as follows (Figure 4).

1. Case 1: 10% damage in element #2.

2. Case 2: 10% and 5% damage in element #2 and #8.

3. Case 3: 10%, 5%, and 10% damage in element #2, #8, and #15.

4. Case 4: 10% damage of all elements.

To verify the performance of PSO-CS under temperature variations, two more cases are adopted based on Case 3 and Case 4.

5. Case 5: 10%, 5%, and 10% damage in elements #2, #8, and #15, and temperature increases by 20°C (reference temperature is 0°C).

6. Case 6: 10% damage of all elements and temperature increases by 20°C.

The parameters of the three algorithms are same as before in the optimization of benchmark functions, and identification results of the six cases are shown as Figure 5.

Figure 5.

Damage-identification results of the three algorithms: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, and (f) Case 6.

In Case 1, the damage location and severity can be identified accurately by the three algorithms, but for CS, it is identified a damage of 0.02% in element #15, which is very small.

In Case 2, the damage location and severity can be identified accurately by PSO and PSO-CS, but for CS, it is identified a damage of 0.6%, 8.77%, 2.7%, and 4.8% in elements #1, #2, #3, and #8, but the predetermined damage is 10% and 5% in elements #2 and #8, that is to say, there is a false identification by CS.

In Case 3, the false identification as Case 2 also occurs when CS is used, it is identified a damage of 2.3%, 8.0%, 0.6%, 4.3%, and 9.0% in elements #1, #2, #7, #8, and #15, but the predetermined damage is 10%, 5%, and 10% in elements #2, #8, and #15.

In Case 4, the damage of all elements can be identified accurately by PSO-CS, whose maximum error is less than 1%, for PSO and CS, both of which are 1.28% and 2.60%, respectively, so the identification performance is not good.

In Case 5, the damage can be identified accurately by PSO-CS and PSO, but the performance is not good when CS are used; for CS, a damage of 0.13%, 0.19%, 0.09%, 0.40%, 0.01%, 0.25%, 0.01%, 0.18%, and 0.25% is identified in elements #1, #3, #4, #9, #10, #11, #12, #15, and #16, but the predetermined damage is 10%, 5%, and 10% in element #2, #8, and #14.

In Case 6, under temperature variations, from the damage-identification results of the three algorithms, it is shown that the maximum error for PSO, CS, and PSO-CS is 3.11%, 2.8%, and 0.33%, the performance of PSO-CS is the best, and large deviations occur when PSO and CS are used.

From the above discussion, it can be seen that the identification performance is the best when PSO-CS is used even when temperature variations are considered; PSO-CS can be applied in the later damage identification of actual bridge.

Damage-identification example

I-40 Bridge

I-40 Bridge over Rio Grande, located in Albuquerque, New Mexico, USA, a highway bridge connecting Mexico to the United States and razed in the summer of 1993, was supported by two welded-steel plate girder and three steel stringers and consisted of twin spans (made by concrete deck, there were separate bridges for each traffic direction). A vibration test experiment conducted by Farrar et al. (1994) is one of the most classic tests in bridge health monitoring research, whose results have been applied to the verification of various damage-identification methods. An elevation view of the portion of the tested bridge is showed in Figure 6(a).

Figure 6.

I-40 Bridge: (a) overall layout (unit: m) and (b) finite-element model.

Finite-element simulation

Based on MATLAB, with the finite-element calculation program of I-40 Bridge established, four-node shell elements are used to model the concrete bridge deck and the web of the plate girders, two-node beam elements are used to model the stringers, floor beam, and flanges of the plate girder, the diagonal bracing, and the pier are beam element. The bridge supports are modeled as spring elements, the individual spring constants are $k_{x} = 3.17 \times 10^{6} N / m$ , $k_{y} = 1.26 \times 10^{6} N / m$ , and $k_{z} = 4.29 \times 10^{7} N / m$ . The finite-element model is shown in Figure 6(b), and the generic material properties of the bridge are shown in Table 3.

Table 3.

Structural material properties in the finite-element model.

Material	Properties	Value	Unit
Concrete	Density	2323	kg/m³
	Elastic modulus	2.48 × 10¹⁰	Pa
	Poisson’s ratio	0.2	/
	Shear modulus	8.95 × 10⁹	Pa
Steel	Density	7850	kg/m³
	Elastic modulus	2.1 × 10¹¹	Pa
	Poisson’s ratio	0.3	/
	Shear modulus	8.04 × 10¹⁰	Pa

Vibration tests

On March 30 and 31, 1993, preliminary vibration measurements were made on I-40 Bridge, the dynamic features of the structure, such as resonant frequencies, modal damping, and the corresponding modes shapes, were identified by these tests (Farrar et al., 1994). The bridge was tested for the first time under non-destructive conditions, and the first six natural frequencies are shown in Table 4. On the basis of non-destructive structure, four different degrees of damage D1–D4 were introduced at different times. The damage location was located on the north side of the web and the bottom plate. Several vibration tests were carried out, respectively. The arrangement of the measuring points is shown in Figure 7(a). The introduced damage was intended to simulate fatigue cracking that has been observed in plate girder bridges. The various levels of damage are shown in Figure 7(b). More test details are shown in the technical report (Farrar et al., 1994).

Table 4.

Experimental frequency, theoretical frequency, MAC, and modal strain energy before and after temperature correction.

Mode	Experimental frequency (Hz)	After temperature correction				Before temperature correction
Mode	Experimental frequency (Hz)	Theoretical frequency ( Hz)	Frequency error %	MAC	Modal strain energy	Theoretical frequency (Hz)	Frequency error %	MAC	Modal strain energy
1	2.483	2.558	3.03	0.990	0.0037	2.483	0.00	0.996	0.0008
2	2.959	3.084	4.20	0.984	0.0673	2.959	0.00	0.991	0.0012
3	3.499	3.604	2.99	0.990	0.1156	3.495	0.12	0.990	0.0013
4	4.079	4.085	0.14	0.971	0.1207	4.073	0.16	0.977	0.0040
5	4.167	4.197	0.73	0.978	0.2388	4.161	0.15	0.990	0.0142
6	4.631	4.778	3.18	0.956	0.2429	4.605	0.57	0.968	0.0264

Figure 7.

I-40 Bridge: (a) experimental setup and (b) D1–D4 damage case.

Benchmark finite-element model

According to the supporting data of I-40 Bridge vibration test report and the record of river and climatological observations of U.S. Department of Commerce, the temperature of the bridge is 88°F (31°C) under vibration test in the non-destructive state. The first six modal parameters obtained by the MATLAB finite-element calculation program without considering the influence of temperature on the structural material are shown in Table 4. It is noticed that the finite-element model is built without considering the influence of temperature on the structure; the modal parameter data obtained contain during the vibration tests. The comparison error between the two is shown in Table 4, the minimum error is 0.14% and the maximum is 4.20%.

There is a certain error between the analytical values from temperature-free model and the measured ones, coming from the influence of temperature, model and test errors; the existing model cannot be used as a benchmark finite-element model for subsequent damage identification, which needs to be corrected. According to research by Xia et al. (2012), for every 1°C increase in temperature, the elastic modulus of concrete will decrease by 0.045%, and the elastic modulus of steel will decrease by 0.0036%, the elastic modulus of the material in the model is adjusted to the corresponding value of 88°F (31°C), the finite-element model considering the temperature influence is obtained. The frequencies, MAC and modal strain energy of the modified model are shown in Table 4, after temperature effects are considered, the maximum and minimum error are reduced to 0.57% and 0, respectively. At this time, the error between the finite-element model and the bridge of the undamaged state is small. The error of each frequency is below 0.6%, and the MAC value is above 0.97, the modified model can be used as a benchmark model for the subsequent damage identification.

Damage identification

The damage data of D1–D4 damage is introduced into the benchmark finite-element model, respectively, and combined with the modal data calculated by the benchmark finite-element model to form the objective function; the D1–D4 damage has a temperature change with the reference condition during the vibration test, the change is shown in Table 5.

Table 5.

Changes in temperature relative to the baseline model at each stage of damage.

Cases	Ambient temperature (°F)	Ambient temperature (°C)	Temperature change relative to the reference model (°C)
D1	60	15.5	−14.5
D2	84	28.9	−1.1
D3	79	26.1	−3.9
D4	68	20.0	−10.0

The unit stiffness variation parameters of the 48 elements on both sides (elements #1–#24 on the north side and elements #25–#48 on the south side) and the temperature change in the relative benchmark model (converted to varying elastic modulus) are both used as identification parameters for damage identification, and these 49 parameters are identified by PSO-CS. The actual damage is applied to the north side web and the corresponding unit stiffness of the D1–D4 damage is reduced to 5%, 10%, 32%, and 92% of element #12 of the north web (Meruane and Heylen, 2011). The damage-identification results and the identification temperature changes of the web unit in the damage conditions D1–D4 are shown in Table 6. The nonexistent damage of the south-side web is identified, such as #25, #36, and #37.

Table 6.

Damage-identification results of the north and south side web.

Cases	Damaged element	Degree of damage	Identified damaged element	Identified damage	Identified temperature changes (unit: °C)
D1	#12	5%	#1 #11 #12 #13 #24 #36 #37	1.6% 7.5% 9.2% 1.0% 0.9% 2.1% 1.2%	−21.3
D2	#12	10%	#1 #12 #13 #24 #36 #37	1.9% 12.99% 6.6% 1.7% 2.3% 0.9%	−6.6
D3	#12	32%	#1 #11 #12 #13 #24 #36	1.70% 1.70% 28.78% 5.73% 2.80% 6%	−8.57
D4	#12	92%	#1 #10 #11 #12 #13 #14 #24 #25 #36	3.0% 2.6% 2.6% 82.1% 11.6% 4.2% 3.10% 6.5% 7.1%	−13.11

The results of damage identification show that the temperature-variation trend of each damage cases, based on the benchmark finite-element model, can be better reflected; however, the actual unit damage identification is not accurate enough. The main reason is that the distribution of temperature in the structure is more inclined to the gradient distribution, and the discrete type of the structural material is more obvious, and the temperature distribution shows a distinct uneven trend. To improve the accuracy, the damage-detection algorithm needs temperature measurements at different points of the structure, it can be considered to convert the uniform temperature distribution into the temperature-gradient distribution in the general engineering practice to be closer to the real working state of the structure.

Temperature gradient is defined by American Standard Specification for Highway Bridges (AASHTO) as Figure 8. According to the gradient distribution and the solar radiation zone in I-40 Bridge, the temperature of upper concrete deck is 30°C $(T_{1})$ , the temperature of bottom deck is 7.3°C $(T_{2})$ . The temperature of steel girder is 4.36°C; therefore, the average temperature of concrete deck is 12.84°C, and there is a large gap between the average temperature and the initial temperature when bridge is undamaged, meaning the temperature gradient can be considered as damage to test whether PSO-CS can distinguish the temperature effects.

Figure 8.

Temperature gradient in I-40 Bridge based on AASHTO standard.

From Table 7, it is clearly demonstrated that the temperature change is identified, and the error between practical temperature change and identified temperature change is reasonably limited. But elements #12 and #13 are identified damaged in a certain degree; the expectation is no damage identified. As the assumed thermal-elasticity is relatively simple, it cannot simulate the accurate structure temperature gradient. Moreover, when the temperature difference is relatively large, the modal parameters change by temperature may be identified as structure damage in middle span for its importance in the global structure stiffness.

Table 7.

Identified damages and changes of temperature.

Structure component	Damage	Temperature change (°C)	Identified element	Identified damage %	Identified temperature change (°C)
Concrete deck	No damage	−18.16	#12	3.36	−19.58
Steel web	No damage	−26.64	#13	2.67	−29.56

Conclusion

This article proposes a model-based damage-identification method combined with PSO-CS, which can handle gradient temperature. The numerical model of the structure assumes that the elastic modulus of the materials is temperature dependent. Four benchmark functions and a simulated data of a simply supported beam validated the algorithm, and experimental example of I-40 Bridge is used to validate the performance of PSO-CS in actual structure.

In the construction of the objective function, natural frequency, mode shape, and modal strain energy indicators with different weight coefficients are used, which meets the needs of global damage identification and also considers the sensitivity of local damage. A hybrid algorithm based on PSO and CS is proposed; the results of four benchmark functions demonstrate that PSO-CS has better optimization performance than CS and PSO. The effectiveness and feasibility of the damage identification are verified by a numerical simply supported beam.

In the example of I-40 Bridge, temperature is considered in the finite-element model, the baseline finite-element model is established based on tests on the undamaged structure. The algorithm updates the temperature and damage parameters together. The algorithm can distinguish between damage and temperature variations in the four damaged cases, which successfully locates and quantifies the damage, even with gradient temperature variations.

In the four damaged cases, the damage detected using the algorithms reflects a good consistency with the experimental damage. Despite the presence of error, such as experimental noise, modeling errors, and varying temperature conditions, the damage of the middle of the north plate girder is detected correctly by the algorithm, the magnitude of the detected damage increases with the increment of the experimental damage, whose results show that the algorithm has a good recognition result for large structures such as I-40 Bridge, providing a new idea for large-scale structural damage identification. However, large-scale bridge structures with temperature distribution inhomogeneities and time variations, it is necessary to consider the temperature distribution more accurately and precise gradient temperature measurement may be needed.

Footnotes

Author Note

Shaoxi Cheng is also affiliated to China Shipbuilding Industry Corporation, Beijing, China.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Minshui Huang

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Damage identification of bridge structure considering temperature variations based on particle swarm optimization - cuckoo search algorithm

Abstract

Keywords

Introduction

Damage-identification model

Temperature–elastic modulus relationship model

Construction of objective function

PSO-CS hybrid algorithm

PSO

CS

PSO-CS hybrid algorithm

Performance validation of PSO-CS

Benchmark functions

Damage identification of a simply supported beam

Damage-identification example

I-40 Bridge

Finite-element simulation

Vibration tests

Benchmark finite-element model

Damage identification

Conclusion

Footnotes

Author Note

Declaration of Conflicting Interests

Funding

ORCID iD

References