Damage identification in connections of moment frames using time domain responses and an optimization method

Abstract

Damage is defined as changes to the material and/or geometric properties of a structural system, comprising changes to the boundary conditions and system connectivity, adversely affecting the system’s performance. Inspecting the elements of structures, particularly critical components, is vital to evaluate the structural lifespan and safety. In this study, an optimization-based method for joint damage identification of moment frames using the time-domain responses is introduced. The beam-to-column connection in a metallic moment frame structure is modeled by a zero-length rotational spring at both ends of the beam element. For each connection, an end-fixity factor is specified, which changes between 0 and 1. Then, the problem of joint damage identification is converted to a standard optimization problem. An objective function is defined using the nodal point accelerations extracted from the damaged structure and an analytical model of the structure in which the nodal accelerations are obtained using the Newmark procedure. The optimization problem is solved by an improved differential evolution algorithm (IDEA) for identifying the location and severity of the damage. To assess the capability of the proposed method, two numerical examples via different damage scenarios are considered. Then, a comparison between the proposed method and the existing damage identification method is provided. The outcomes reveal the high efficiency of the proposed method for finding the severity and location of joint damage considering noise effects.

Graphical abstract

Keywords

damage identification / beam-to-column connection / time-domain response / optimization

1 Introduction

Damage occurrence in all structural systems and their critical components is inevitable during the lifetime. If the damage is identifiable, damaged elements can be repaired or replaced, which leads to preventing the overall failure or sudden collapse of the structure [ 1]. Most of the inverse problems of identifying damage have focused on detecting damage of structural members and do not consider detecting damage of connections, mostly seen [ 2]. Beam-to-column connections of moment frames play an important role in the performance of moment-resisting frame structures, as the structure’s performance mainly depends on the stiffness of connections. The joint damage will reduce the stiffness, mass, and damping properties of the structure leading to altering dynamic responses such as displacement and acceleration. This principle can be used as a way to detect damage in connections.

In recent years, fewer studies have been carried out aiming at identifying joint damage. Yun et al. [ 3] estimated structural joint damage using a neural network technique. Joint damage was simulated by reducing the connection rotational stiffness. All rigid joints converted to semi-rigid connections via zero-length rotational springs. Results showed that the damage can be correctly estimated by a few measured data. Ritdumrongkul and Fujino [ 4] identified the location and severity of damage using piezoceramic actuator-sensors and spectral element method (SEM). Damage was simulated by loosening some bolts. The outcomes revealed that the damage location could be identified, but identifying the exact damage location depended on the place and number of piezoceramic sensors. Dilena and Morassi [ 5] obtained damage location and its severity in shear connections. They utilized vibration methods and frequency measurements using the Euler-Bernoulli model of a composite beam with the aim of limiting damage in connections. Damage was simulated by a crack of increasing depth fashioned on one end connector. The results demonstrated that the resolution had been improved by recommending a Timoshenko-type model for the composite beam. In addition, it has been shown that the frequency variations contain good information, which can detect the severity and location of the damage. Xu et al. [ 6] could identify the damage of shear connectors for composite structures using the acceleration time series. They used a nonparametric method based on the neural network without needing structural parameters and its finite element model. Damage was simulated by loosening or removing some connections. The difference between the output and the measurement provided an index for joint damage identification. The outcomes showed that the method introduced is effective for detecting the local damage in composite structures. Labuz et al. [ 7] identified local damage in beam-to-column connections using an accelerometer sensor via dense clustered sensor networks. Damage was simulated by reducing the stiffness which was in a part of the beam near the connection. The results showed that damage could be identified by the proposed method. He and Zhu [ 8] detected diverse kinds of damage utilizing the vibration-based method and changes in natural frequencies. They used the finite element modeling method and a stout iterative algorithm utilizing a trust-region technique, called the Levenberg-Marquardt method. Damage was modeled by loosening some bolted joints. Results showed that the method could detect the damage location and its severity. Furthermore, the convergence speed of the iterative algorithm increased. Hegenderfer et al. [ 9] utilized a model analysis to identify joint damage of steel frames. Damage was modeled by removing some bolts at the base connection of one column. They’ve used modal properties to detect damage location. Yang et al. [ 10] have presented a joint damage detection method based on a combination of reduced-order finite element model and the adaptive quadratic sum-square error with unknown inputs (AQSSE-UI). This approach aimed to reduce the number of degrees of freedom of the finite element model using static condensation. Damage in connections was simulated by loosening some bolted joints in a steel frame. Results demonstrated that the proposed method could remove response quantities, which are difficult to measure, like the rotational acceleration. The mentioned technique could detect loosened bolts in the steel frame joints. Dackermann et al. [ 11] utilized the frequency response functions (FRFs) and artificial neural networks (ANNs) to identify damage in the steel frame. Damage was simulated by modifying the member connectivity conditions or changing the mass of the structure. Principal component analysis (PCA) techniques were pursued to decrease the size of FRF data and to separate out the noise. The obtained outcomes illustrated that the offered method is highly accurate and trustworthy to assess and detect joint damage and to filter the noise. It was also mentioned that the FRF data should be taken from numerous locations. Ho et al. [ 12] evaluated damage in steel column connections using a method based on impedance and piezoelectric sensors. Numerical and experimental results were compared from the impedance response. Results showed that the method could correctly identify loosened bolt joints. Moreover, the change in impedance responses was more sensitive to the position of damage that was closer to the piezoelectric sensor. Nanda et al. [ 13] used a unified particle swarm optimization (PSO) and modal parameters (mode shapes and natural frequencies) to identify the damaged joints in frame structures. Damage was simulated by a reduction in the joint fixity factor ratio. They have found that the mentioned method can evaluate the quantity of joint damage with reasonable accuracy. Yin et al. [ 14] identified the location of bolted connection damage employing a probabilistic method based on a finite element model reduction and utilizing the Bayesian inference technique. Damage was assumed by loosening a small part of beam-to-column connections. In the mentioned method, only noisy defective modal parameters with a moderate number of sensors were utilized.

As can be observed in the previous studies, different structural responses with some advantages and drawbacks have been used to identify damage [ 15– 17]. The natural frequencies are severely susceptible to environmental changes such as temperature and humidity. Moreover, in large symmetric structures, the method based on frequencies cannot correctly identify damage location and severity. On the other hand, methods based on mode shapes are often susceptible to the incompleteness of the measured modal data and hence require measurements from a large number of sensors to ensure the effectiveness of outcomes. Additionally, the FRF data are more popular dynamic features for vibration-based damage detection. However, they cannot be directly measured and need more data post-processing. Furthermore, time-domain data such as acceleration responses can be directly measured with a lower cost and require much less post-processing than modal parameters and FRF, which reduce the sensibility in contaminating the essential information.

This study aims to identify the location and severity of joint damage in metallic moment frames using a minimum number of dynamic data. At first, the simulation of beam-to-column connection is carried out using zero-length rotational springs, and the Newmark method is used to extract the time-domain responses. Then, the problem of joint damage detection is defined as a standard optimization problem introducing an objective function based on the nodal point accelerations. The location and severity of single and multiple damage cases in connections are determined by IDEA with high accuracy. The performance of the method is validated by numerical examples considering noise effects. Moreover, a comparative study between the proposed method and an existing method is carried out to assess the effectiveness of the method.

2 Modeling semi-rigid joints

To identify damage in joints, the beam-to-column connections are simulated as semi-rigid joints. Several techniques exist to model the behavior of connections, which is divided into two main groups, including mathematical and mechanical models that the present study focuses on the second one. Mechanical models sometimes are called spring model-based connection simulations, which are modeled using springs to convert the rigid joints into semi-rigid ones. Figure 1 illustrates the scheme of a beam element with semi-rigid connections at both ends [ 13].

First, beam-to-column connections are modeled by zero-length rotational springs. Then an end-fixity factor ( p) is defined in each member connection, which is associated with the member stiffness ( EI/ L) and rotational spring stiffness ( k) as given in Eq. (1) [ 13, 18]:

(1) p_i=\frac{1}{1+{\displaystyle \frac{3EI}{L{k}_i}}}, i=1,2,

where i represents a member with two ends. This parameter has a theoretical value that varies from 0 to 1 that zero is for pinned joint and one is for a quite rigid joint. Hence, a damaged connection has a p value less than one. It can be said that a semi-rigid member has an end-fixity factor between 0 and 1, which the joint damage can be identified from this conception in rigid joints of moment frames. The rotational stiffness is commonly illustrated in relevance to member stiffness, which attached at the semi-rigid joint as represented by Eq. (2) [ 18]:

(2) k_i={\gamma }_i\frac{EI}{L},

where k _i is the rotational stiffness of semi-rigid joints and E, I, L are Young’s modulus, the second moment of area of the cross-section, and the element length, respectively. The parameter γ _i is a multiplication factor named the stiffness index. The end-fixity factor pertained to the γ factor can be defined as below:

(3) p_i=\frac{{\gamma }_i}{3+{\gamma }_i}.

By considering the p _i for both two ends of a beam (‘e’ refers to end and ‘f’ refers to first), the mass matrix for semi-rigidly connected frame element at the local coordinate system can be given by Eq. (4) [ 18]:

(4) {\mathit{M}}^{\mathrm{e}} =\frac{\overline{m_{\mathrm{l}}}L}{420d^2}\left[\begin{array}{cccccc}140d^2& 0& 0& 70d^2& 0& 0\\ 0& 4f_1(p_{\mathrm{e}},p_{\mathrm{f}})& 2L{f}_2(p_{\mathrm{e}},p_{\mathrm{f}})& 0& 2f_3(p_{\mathrm{e}},p_{\mathrm{f}})& -L{f}_4(p_{\mathrm{e}},p_{\mathrm{f}})\\ 0& 2L{f}_2(p_{\mathrm{e}},p_{\mathrm{f}})& 4L^2{f}_5(p_{\mathrm{e}},p_{\mathrm{f}})& 0& L{f}_4(p_{\mathrm{e}},p_{\mathrm{f}})& -L^2{f}_6(p_{\mathrm{e}},p_{\mathrm{f}})\\ 70d^2& 0& 0& 140d^2& 0& 0\\ 0& 2f_3(p_{\mathrm{e}},p_{\mathrm{f}})& L{f}_4(p_{\mathrm{e}},p_{\mathrm{f}})& 0& 4f_1(p_{\mathrm{e}},p_{\mathrm{f}})& -2L{f}_2(p_{\mathrm{e}},p_{\mathrm{f}})\\ 0& -L{f}_4(p_{\mathrm{e}},p_{\mathrm{f}})& -L^2{f}_6(p_{\mathrm{e}},p_{\mathrm{f}})& 0& -2L{f}_2(p_{\mathrm{e}},p_{\mathrm{f}})& 4L^2{f}_5(p_{\mathrm{e}},p_{\mathrm{f}})\end{array}\right],

where the mass per unit length is m_{\mathrm{l}}=\rho A , where A and ρ are the cross-sectional area and mass density of the element, respectively. The parameter d=4-p_{\mathrm{e}} p_{\mathrm{f}} and the functions f _i are given in Refs. [ 13, 18].

The stiffness matrix for a semi-rigidly connected frame element at the local coordinate system is given by Eq. (5) as [ 18, 19]:

(5) {\mathit{K}}^{\mathrm{e}} =\frac{EI}{L}\left[\begin{array}{cccccc}{\displaystyle \frac{A}{I}}& 0& 0& -{\displaystyle \frac{A}{I}}& 0& 0\\ 0& {\displaystyle \frac{4(b_{11}+b_{12}+b_{22})}{L^2}}& {\displaystyle \frac{2(2b_{11}+b_{12})}{L}}& 0& -{\displaystyle \frac{4(b_{11}+b_{12}+b_{22})}{L^2}}& {\displaystyle \frac{2(2b_{12}+b_{22})}{L}}\\ 0& {\displaystyle \frac{2(2b_{11}+b_{12})}{L}}& 4b_{11}& 0& -{\displaystyle \frac{2(2b_{11}+b_{12})}{L}}& 2b_{12}\\ -{\displaystyle \frac{A}{I}}& 0& 0& {\displaystyle \frac{A}{I}}& 0& 0\\ 0& -{\displaystyle \frac{4(b_{11}+b_{12}+b_{22})}{L^2}}& -{\displaystyle \frac{2(2b_{11}+b_{12})}{L}}& 0& {\displaystyle \frac{4(b_{11}+b_{12}+b_{22})}{L^2}}& -{\displaystyle \frac{2(2b_{12}+b_{22})}{L}}\\ 0& {\displaystyle \frac{2(2b_{12}+b_{22})}{L}}& 2b_{12}& 0& -{\displaystyle \frac{2(2b_{12}+b_{22})}{L}}& 4b_{22}\end{array}\right],

where the coefficients b_{11},b_{12},b_{22} are given in Refs. [ 13, 18].

The element stiffness matrix ( {\mathit{K}}_{\mathrm{g}}^{\mathrm{e}} ) and mass matrix ( {\mathit{M}}_{\mathrm{g}}^{\mathrm{e}} ) in the global coordinate system are calculated by Eqs. (6) and (7), respectively:

(6) {\mathit{K}}_{\mathrm{g}}^{\mathrm{e}}= {\mathit{T}}^{\mathrm{T}} {\mathit{K}}^{\mathrm{e}} \mathit{T},

(7) {\mathit{M}}_{\mathrm{g}}^{\mathrm{e}}= {\mathit{T}}^{\mathrm{T}} {\mathit{M}}^{\mathrm{e}} \mathit{T},

where T is a transformation matrix.

Damping property is significant in determining the responses in the dynamic analysis. Rayleigh-type damping is used here to calculate the damping matrix due to the effectiveness of the method to find the damping of a structure which can be obtained by Eq. (8) as [ 20]:

(8) \mathit{C}={\alpha }_{\mathrm{m}}\mathit{M}+{\alpha }_{\mathrm{s}}\mathit{K},

where C , M , K are the global damping matrix, global mass matrix and global stiffness matrix of the structure, respectively, and factors α _m, α _s are the mass-proportional and stiffness-proportional Rayleigh damping coefficients, correspondingly, which are given by Eqs. (9) and (10), respectively:

(9) {\alpha }_{\mathrm{m}}=\frac{2\xi {\omega }_1{\omega }_2}{{\omega }_1+{\omega }_2},

(10) {\alpha }_{\mathrm{s}}=\frac{2\xi }{{\omega }_1+{\omega }_2} ,

where the angular frequencies ( ω ₁, ω ₂) of the first two modes each with 5% damping ratio ( ξ) is used to define α _m and α _S. The selection of ω ₁ and ω ₂ is because the first few modes of vibration, called significant modes, are the most important modes in dynamic analysis. The first two orders are generally obtained as reference frequencies in traditional methods, and they often indicate the dynamic characteristics well [ 21– 23]. On the other hand, considering more or other frequencies does not influence the effectiveness of the proposed method. Also, according to the Iranian Code of Practice for Seismic Resistant Design of Buildings (Standard No. 2800) [ 22], the damping ratio is considered 5% for all kinds of structures for dynamic analysis.

3 Dynamic analysis of structure via semi-rigid connections

As damage occurrence in the structure can lead to changes in acceleration responses with high sensitivity compared with other data, the nodal point accelerations are used here as appropriate responses for damage identification. Moreover, as mentioned, nodal accelerations can be practically obtained with lower costs than other dynamic responses. The equation of motion for a structural system of n degrees of freedom (DOFs) is given by Eq. (11) as [ 20]:

(11) \mathit{M}\ddot{\mathit{X}} (t)+\mathit{C}\dot{\mathit{X}} (t)+\mathit{K}\mathit{X}(t)=\mathit{F}(t),

where \mathit{F}(t) is the dynamic force vector; \ddot{\mathit{X}}(t) , \dot{\mathit{X}} (t) , and \mathit{X}(t) are the acceleration, velocity, and displacement vectors, respectively. Several techniques have been presented to solve the dynamic equation and calculate the displacement, velocity, and acceleration components. Newmark procedure has been commonly used to solve the equation owing to its advantages compared to other methods. Hence, the nodal displacement can be calculated from Eq. (12) as:

(12) X_{n+1}={\mathit{K}}_{\mathrm{e}}^{-1}{\mathit{F}}_{\mathrm{e}},

where K _e and F _e are the equivalent stiffness and force of the structure given by Eqs. (13) and (14), respectively.

(13) {\mathit{K}}_{\mathrm{e}} ={\alpha }_0 \mathit{M}+{\alpha }_1\mathit{C}+\mathit{K},

(14) {\mathit{F}}_{\mathrm{e}} =\mathit{F}+\mathit{M}({\alpha }_0{X}_n- {\alpha }_2{\dot{X}}_n -{\alpha }_3{\ddot{X}}_n )+\mathit{C}({\alpha }_1{X}_n- {\alpha }_4{\dot{X}}_n -{\alpha }_5{\ddot{X}}_n ).

For attaining the velocity and the acceleration vector, Eqs. (15) and (16) should be solved as below:

(15) {\dot{X}}_{n+1} ={\dot{X}}_n -{\alpha }_6{\ddot{X}}_n-{\alpha }_7 {\ddot{X}}_{n+1},

(16) {\ddot{X}}_{n+1}={\alpha }_0(X_{n+1}- X_n)-{\alpha }_2 {\dot{X}}_n-{\alpha }_3 {\ddot{X}}_n,

where Newmark coefficients ( {\alpha }_i ,i=0,1,...,7 ) can be obtained from Ref. [ 20].

4 Joint damage identification using an optimization method

This research aims to identify the joint damage in the moment frames by an optimization method. In this case, the damage detection problem becomes a nonlinear optimization problem which may have many local solutions with one global solution. The general form of the optimization-based damage detection problem in a classic term can be defined by Eq. (17):

(17)

where \mathit{X}\in R^n is the damage variables’ vector, including the location and severity of n unknown damages. Moreover, the lower and upper bounds of damage vector are shown by {\mathit{X}}^{\mathrm{l}}\in R^n and {\mathit{X}}^{\mathrm{l}}\in R^n , respectively, and w\in R is the objective function that should be minimized.

4.1 Damage variables

In this study, the joint damage severity at endpoint i of a frame element defined as reducing the end-fixity factor of damaged state to the end-fixity factor of the intact state is considered the damage variable. The mathematical form of the ith variable is expressed as Eq. (18):

(18) x_i=1-\frac{p_{\mathrm{x}i}}{p_{\mathrm{h}i}},

where p _{x i} and p _{h i} represent the damaged and undamaged end-fixity factors, respectively. The amount of x _i changes from 0 to 1 and with assuming p _{h i} = 1, the zero value and one value corresponding to the undamaged and completely damaged state, respectively.

4.2 The objective function

The objective function is a criterion that specifies the convergence of the algorithm implementation. Hence, the objective function plays an important role in the optimization algorithm. In this study, the objective function is defined as Eq. (19):

(19) w(\mathit{X})=-\frac{1}{2}\left(\frac{{{\mathit{a}}_{\mathrm{d}}^{\mathrm{T}}\cdot \mathit{a}(\mathit{X})}^2}{\left({\mathit{a}}_{\mathrm{d}}^{\mathrm{T}}\cdot {\mathit{a}}_{\mathrm{d}}\right)\left(\mathit{a}{(\mathit{X})}^{\mathrm{T}}\cdot \mathit{a}(\mathit{X})\right)}+\frac{1}{nm}\sum _{i=1}^{nm}\frac{min(a_{\mathrm{x}i},a_{\mathrm{d}i})}{max(a_{\mathrm{x}i},a_{\mathrm{d}i})}\right),

where a _d and a ( X ) are the acceleration response vectors of the damaged structure and analytical model, respectively, and nm is the number of acceleration vector components. Likewise, a _{x i} and a _{d i} are the ith component of a ( X ) and a _d, respectively. For more measuring points (sensors), the nodal acceleration vectors will be considered as a single columnar vector. Additionally, w( X ) varies from a minimum value −1 to a maximum value 0. It will be minimal whenever the vector of analytical acceleration responses becomes equal to the acceleration vector of the damaged structure.

The damage detection problem transformed into an optimization problem needs to be solved by an appropriate algorithm to obtain the location and severity of joint damage. It should be noted that the problem cannot be solved by an analytical method due to the complexity, and it needs to be solved by a numerical technique. Moreover, during the last years, the effectiveness of using the heuristic methods has been proven from two aspects. They enhance the probability of achieving the globally optimal solution and independence to evaluating the derivatives of functions [ 24– 26]. Since the optimization-based damage identification problems may have numerous local solutions, selecting an efficient algorithm for solving the problem is essential. One of the most important aspects of an algorithm is how to achieve the global solution using fewer structural analyses. For this purpose, an improved differential evolution algorithm (IDEA) is used here. The performance of the DEA for identifying structural damage has been assessed in Refs. [ 24– 26]. and results indicated the accuracy of the algorithm. However, for better performance, the DEA is modified here.

4.3 Differential evolution algorithm (DEA)

DEA is a probable strategy based on population, which is introduced by Storn and Price in 1995 [ 27]. The optimization process begins with some offered solutions, and the final solution is achieved during sequential repetitions. Three control parameters are needed to start the algorithm, including population size ( np), mutation factor ( mf), and the crossover rate ( cr) [ 28]. The general procedure of DEA is shown in Fig. 2.

According to the procedure, the lower bound {\mathit{X}}^{\mathrm{l}} and the upper bound {\mathit{X}}^{\mathrm{u}} are introduced for the variable vector at the start, then the initial population is generated randomly within the range \left[{\mathit{X}}^{\mathrm{l}},{\mathit{X}}^{\mathrm{u}}\right] . For a given parameter vector X_{i,G} ( i = 1,2,3,…, np) from Gth generation, the mutated vector (donor vector) is produced by a combination of three vectors, X_{r1,G}, X_{r2,G} ,X_{r3,G} which randomly are selected where the three different indices r ₁, r ₂, and r ₃ \in \left\{1,2,3,...,np\right\} are randomly chosen to be different from index I [ 24, 29]. Afterward, the crossover and selection processes are carried out to produce the new generation. This procedure is continued until the convergence is achieved.

4.4 Improved differential evolution

There are several strategies to produce a new generation. A different mutated vector ( V_{i,G+1} ) is used here to produce the novel generation to enhance the efficiency of the algorithm, which is defined as Eq. (20) [ 30]:

(20) V_{i,G+1}=\{\begin{array}{rl}& X_{r1,G}+mf(X_{r2,G}-X_{r3,G}),\mathrm{i}\mathrm{f}w(X_{r1,G})⩽w(X_{r2,G}),\\ & X_{r2,G}+mf(X_{r1,G}-X_{r3,G}),\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\end{array}

Moreover, one of the control parameters of DEA directly affecting the diversity of the algorithm is the crossover ratio cr [ 31]. The cr is chosen here via a formula which leads to a better solution for the algorithm as Eq. (21):

(21) cr=0.2{\exp }^{\left(\frac{-k}{k_{max}}\right)} ,

where k is the number of iterations of the algorithm that changes from 1 to the maximum number of iteration k _max.

5 Numerical studies

To assess the competence of the proposed method for identifying damage, two numerical examples are considered. The examples are a two-story, one-bay frame and a four-story, three-bay frame, and some parametric studies are conducted. Both frames are subjected to a trapezoidal impact load within 5 s, as shown in Fig. 3. Finally, the first example results are compared with those of an available method selected from Ref. [ 13].

The flowchart of the damage identification method using responses and IDEA is shown in Fig. 4.

5.1 Two-story frame with 30 elements

The finite element model of a 30-element planar frame shown in Fig. 5 is selected to assess the performance of the proposed method [ 13]. By eliminating the constrained degrees of freedom at supports, the structure contains 84 active degrees of freedom. The excitation load is applied to node 22 in the horizontal degree of freedom, which P₀ = 1 N, t₁ = 3 s and t₂ = 5 s. The structural properties and two damage scenarios are given in Tables 1 and 2, respectively. The optimization continues as long as the total number of iterations reaches k_max = 10000 or the objective function value does not change considerably after 400 successive iterations. Also, the population size is considered as np = 30. The sensor placement, the number of sensors, and mutation factor ( mf) are specified after parametric perusals which are provided in the next section. It should be noted that all sensors are installed in horizontal degrees of freedom.

5.1.1 Checking the effects of sensor number and placement

Some factors, such as the number and place of sensors, may affect the efficiency of the proposed method. Despite some existing methods for optimal sensor placement, a trial and error method is used to obtain the best case [ 16]. The place and the number of sensors are given in Table 3. Damage identification diagrams are shown in Fig. 6 for the damage case G1 considering seven various sensor placements without noise and with 3% noise.

It is observed that although changing the sensor placement does not have a significant effect on detecting the joint damage for the damage case G1, however, increasing the number of sensors leads to detection improvement and better accuracy. The same process is repeated for the damage case G2 shown in Fig. 7, considering seven various sensor placements.

According to damage identification diagrams, it can be seen that the joint damage detection is almost independent of the sensor location because of the small size of the frame and few connections. Consequently, damage location and severity can be detected by a lower number of sensors.

5.1.2 Checking the mutation factor ( mf)

Another parameter that may affect the performance of the proposed method is mf or mutation factor, which varies from zero to one. To attain the best mf value, a trial and error method is made here. The procedure is carried out on two damage cases (G1, G2) considering the case 5 sensor place, and the program is performed 10 times for each case. The result of the damage case G1 is shown in Fig. 8, for instance. As revealed in the figure, by considering mf = 0.6, the joint damage can be precisely identified in every 10 runs, which is considered the best value.

5.1.3 Joint damage identification in two-story frame

Using parametric perusal, one of the sensor placement (case 5) and the mutation factor mf = 0.6 are considered the best value. The identification and the convergence diagram of IDEA without considering noise for the damage case G1 and G2 are shown in Fig. 9.

The figures show that the proposed method can identify the damage location and severity accurately without considering the noise effects, and the outcomes demonstrate the high performance of the method. Furthermore, according to the convergence diagrams, it can be seen that the algorithm is repeated until 800 iterations, and in almost 400 last iterations, no significant change has been made, and the algorithm is converged to the optimal solution after 400 iterations.

5.2 Four-story frame with 28 elements

A frame model with 28 elements shown in Fig. 10 is considered to verify the efficiency of the method. By omitting the constrained degrees of freedom at supports, the structure contains 48 active degrees of freedom. The excitation load is applied to node 5 in the horizontal degree of freedom with considering P₀ = 10 ton, t₁ = 3 s and t₂ = 5 s. Also, the effects of dead masses are considered in which the mass per unit length is defined as equal to \overline{m} = 2000 kg/m. The structural properties and three damage scenarios are given in Tables 4 and 5, respectively. The population size is considered as np = 50. To demonstrate the effectiveness of the proposed method for joint damage detection, the column connections to a node are considered in addition to the beam-column connections. The maximum number of iteration is limited to 10000 for the example. The sensors’ placement, the number of sensors, and mutation factor ( mf) are specified after parametric perusals shown in the next section. (All sensors are installed in the horizontal DOFs).

5.2.1 Checking the effects of sensor number and placement

As earlier mentioned, the number and place of sensors may affect the outcomes of the proposed method, and a suitable placement needs to be chosen by the trial and error method. In this case, the damage case G1 is considered, for instance. The number and place of sensors are given in Table 6. The damage identification diagrams are shown in Fig. 11 for ten different sensor places.

Figure 11 shows that the proximity of sensors to the impact load and damage locality provides a better result. Also, the sensors must be located in some positions, which receive the responses well. In addition, increasing the number of sensors enhanced the accuracy of the damage identification procedure. If the sensors are located in an inappropriate position, the method may not identify damage easily. Theoretically, it is better to place sensors where damage has more effects on sensor data to efficiently identify damage; however, the damage location is unknown, and the idea may be inefficient. Thus one can say that in larger structures, the damage detection results are dependent on the sensor location. One way for placing the sensors is to employ the trial and error procedure which is time-consuming for real-world structures. It can be advantageous to employ an optimization for the sensor placement.

5.2.2 Checking the mutation factor ( mf)

As mentioned, an effective parameter on damage detection procedure is mf or mutation factor. For the present example, by considering case 9 as the sensor place, the best value of mf is obtained by a parametric perusal for both damage scenarios. The result of damage case G1 is shown in Fig. 12, for instance.

As one can see in the figure, the joint damage can be accurately identified in every 10 runs, when the mutation factor is equal to mf = 0.7.

5.2.3 Joint damage identification in 4-story planar frame

For damage identification, case 9 is selected as the finest sensor placement and the mutation factor mf = 0.7 is considered the best value. The identification and the convergence diagram of IDEA for the cases G1, G2, and G3 are shown in Fig. 13.

The diagrams confirm that the proposed method can identify damage location and severity precisely regardless of the measurement noise. According to the convergence diagrams, the algorithm is repeated until 1600 iterations, and approximately during 400 last iterations, the objective function has not changed significantly.

5.3 The results contaminating standard noise

To investigate the effects of measurement noise on the performance of the method, the results are assessed when responses are perturbed by noise using the standard error of 1%, 2%, and 3% for each example. The effect of measurement noise is considered here by contaminating the measured acceleration of damaged structure, which can be expressed as Eq. (22):

(22) a_{\mathrm{d}}^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{y}}= a_{\mathrm{d}} \left[1+(2(\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m})-1)noise\right],

where a_{\mathrm{d}}^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{y}} and a_{\mathrm{d}} are the noisy nodal acceleration vector and the nodal acceleration vector of the damaged structure, respectively. The symbol random is the production function of accidental values, which generates a uniformly distributed random number on the interval [0 1]. It has the same size as the nodal acceleration vector of the damaged structure and the noise is the level of noise considered.

5.3.1 Two-story frame with 30 elements

Damage identification and convergence diagram for the case G1 and G2 of the two-story frame are shown in Fig. 14, considering 1%, 2%, and 3% standard noise. It should be noted that the convergence diagrams are related to considering the maximum level of noise (3%).

As shown in identification diagrams, though the noise effect, damage location and severity are accurately detected. As shown in convergence diagrams, the algorithm is repeated until almost 800 iterations, and approximately during 400 last iterations, the objective function has not changed significantly, and the algorithm has converged to a correct solution.

5.3.2 Four-story frame with 28 elements

The identification and convergence diagrams of the four-story frame for the damage case G1, G2, and G3 are shown in Fig. 15 in which the responses are perturbed by the standard noise 1%, 2%, and 3%. It should be noted that the convergence diagrams are of noise 3%.

Although the noise affected finding the extent of damage severity accurately, however, damage location is correctly detected. As shown in the convergence diagrams, the algorithm is repeated until 1600 iterations, and approximately during 400 last iterations, the objective function has not changed considerably.

5.4 Comparison of results with an existing method

To demonstrate the high accuracy of the proposed method, the results of the method and those of an existing damage identification method in Ref. [ 13] are compared. The outcomes of joint damage identification obtained by the proposed method and the referenced work for both damage cases G1 and G2 of the 30-element planar frame are given in Table 7. Five runs are conducted for each damage case, and the mean of different runs considering 5% and 10% noise levels are provided.

The results demonstrate that the efficiency of the proposed method for damage localization and quantification compared to the existing method is better. Without considering noise, the damage values obtained by the proposed method are the same as the actual values; while, some small errors in damage values obtained by the existent method can be observed. As shown in the table, in the proposed method for the two damage cases G1 and G2, the CE has zero values. In contrast, the existing method produced some errors. The CE values of the proposed method to both the noise levels 5% and 10% are less than 10%, however, these values are more than 10% for the existing method. The error that occurred in damage values for the proposed method is much lower than the existing method. It is demonstrated that the proposed method has a better performance than the existing method for identifying damage.

6 Conclusions

An efficient optimization-based method was proposed to detect the location and severity of damage in the moment frame connections. First, the beam-to-column connections in a moment frame structure were modeled by a zero-length rotational spring at the end of the beam element. An end-fixity factor was specified for each connection changing between 0 and 1. The damage severity at any connection was defined as the reduction of the end-fixity factor. Then, the problem of joint damage detection was transformed into a standard optimization problem. An objective function was defined by the nodal acceleration vectors of the damaged structure and an analytical model based on the Newmark method. The optimization problem was solved by IDEA. Two numerical examples through different scenarios were considered where the location and number of sensors and the mutation factor of the optimization algorithm were specified by a parametric perusal. Finally, a comparison between the outcomes obtained by the proposed method and an existing method was implemented. The results show the efficiency of the proposed method for identifying the location and severity of joint damage considering a limited number of measuring points regardless of the noise effects. It can be observed that the effect of sensor places on the damage identification results is negligible in smaller structures. For a larger structure, although increasing the number of sensors could enhance the accuracy of the proposed method, however, the damage detection has mainly pertained to the placement of the sensors. Finding the suitable places for sensors was achieved by a trial and error method, which may practically be a time-consuming process for a large structure. In addition, even though considering the measurement noise and dead masses have caused some inaccuracy to find the damage severity, however, the exact damage locations were detected.

References

[1] SeyedpoorS M, YazdanpanahO. An efficient indicator for structural damage localization using the change of strain energy based on static noisy data. Applied Mathematical Modelling, 2014, 38( 9−10): 2661– 2672

[2] NanthakumarS S, LahmerT, ZhuangX, ZiG, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24( 1): 153– 176

[3] YunC B, YiJ H, BahngE Y. Joint damage assessment of framed structures using a neural networks technique. Engineering Structures, 2001, 23( 5): 425– 435

[4] RitdumrongkulS, FujinoY. Identification of the location and level of damage in multiple-bolted-joint structures by PZT actuator-sensors. Journal of Structural Engineering, 2006, 132( 2): 304– 311

[5] DilenaM, MorassiA. Vibrations of steel-concrete composite beams with partially degraded connection and applications to damage detection. Journal of Sound and Vibration, 2009, 320( 1−2): 101– 124

[6] Xu B, Tan T, Hao H. Local shear connectors damage detection for a concrete-steel composite beam using acceleration time series. In: Proceedings of the 12th Biennial International Conference on Engineering, Construction, and Operations in Challenging Environments. Honolulu: ASCE, 2010

[7] Labuz E L, Chang M, Pakzad S N. Local damage detection in beam-column connections using a dense sensor network. In: Proceedings of Structures Congress 2010. Orlando: ASCE, 2010

[8] He K, Zhu W D. Structural damage detection using changes in natural frequencies: Theory and applications. In: Proceedings of the 9th International Conference on Damage Assessment of Structures (DAMAS 2011). Oxford: IOP Publishing Ltd, 2011

[9] Hegenderfer J, Atamturktur S, Gillen A. Damage detection in steel structures using bayesian calibration techniques. In: Conference Proceedings of the Society for Expeimental Mechanics Series. New York: Springer, 2012

[10] YangJ N, XiaY, Loh C H. Damage identification of bolt connections in a steel frame. Journal of Structural Engineering, 2013, 140 : 1– 9

[11] DackermannU, LiJ, Samali B. Identification of member connectivity and mass changes on a two-storey framed structure using frequency response functions and artificial neural networks. Journal of Sound and Vibration, 2013, 332( 16): 3636– 3653

[12] HoD D, NgoT M, KimJ T. Impedance-based damage monitoring of steel column connection: Numerical simulation. Structural Monitoring and Maintenance, 2014, 1( 3): 339– 356

[13] NandaB, MaityD, MaitiD K. Modal parameter based inverse approach for structural joint damage assessment using unified particle swarm optimization. Applied Mathematics and Computation, 2014, 242 : 407– 422

[14] YinT, Jiang Q H, YuenK V. Vibration-based damage detection for structural connections using incomplete modal data by Bayesian approach and model reduction technique. Engineering Structures, 2017, 132 : 260– 277

[15] Dinh-CongD, Ho-HuuV, Vo-DuyT, Ngo-ThiH Q, Nguyen-ThoiT. Efficiency of Jaya algorithm for solving the optimization-based structural damage identification problem based on a hybrid objective function. Engineering Optimization, 2018, 50( 8): 1233– 1251

[16] Dinh-CongD, Dang-TrungH, Nguyen-ThoiT. An efficient approach for optimal sensor placement and damage identification in laminated composite structures. Advances in Engineering Software, 2018, 119 : 48– 59

[17] Dinh-CongD, Nguyen-ThoiT, NguyenD T. A FE model updating technique based on SAP2000-OAPI and enhanced SOS algorithm for damage assessment of full-scale structures. Applied Soft Computing, 2020, 89 : 106100–

[18] KatkhudaH N, DwairiH M, ShataratN. System identification of steel framed structures with semi-rigid connections. Structural Engineering and Mechanics, 2010, 34( 3): 351– 366

[19] FilhoM S, GuimaraesM J R, SahlitC L, BritoJ L V. Wind pressures in framed structures with semi-rigid connections. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2004, 26 : 180– 189

[20] Clough R W, Penzien J. Dynamics of Structures. New York: McGraw-Hill, 1993

[21] SongZ, SuC. Computation of Rayleigh damping coefficient for the seismic analysis of a hydro-powerhouse. Shock and Vibration, 2017, 2 : 1– 11

[22] Standard No. 2800. Iranian Code of Practice for Seismic Resistant Design of Buildings. Tehran: Road, Housing and Urban Development Research Center, 2015

[23] Manie J, Kikstra W P. DIANA Finite Element Analysis User’s Manual Release 10.0, 2015

[24] SeyedpoorS M, ShahbandehS, YazdanpanahO. An efficient method for structural damage detection using a differential evolution algorithm-based optimization approach. Civil Engineering and Environmental Systems, 2015, 32( 3): 230– 250

[25] SeyedpoorS M, MontazerM. A damage identification method for truss structures using a flexibility-based damage probability index and differential evolution algorithm. Inverse Problems in Science and Engineering, 2016, 24( 8): 1303– 1322

[26] SeyedpoorS M, MontazerM. A two stage damage detection method for truss structures using a modal residual vector based indicator and differential evolution algorithm. Smart Structures and Systems, 2016, 17( 2): 347– 361

[27] StornR, PriceK. Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization, 1997, 11( 4): 341– 359

[28] StutzL T, TenenbaumR A, CorreaR A P. The differential evolution method applied to continuum damage identification via flexibility matrix. Journal of Sound and Vibration, 2015, 345 : 86– 102

[29] NobahariM, SeyedpoorS M. Structural damage detection using an efficient correlation-based index and a modified genetic algorithm. Mathematical and Computer Modelling, 2011, 53( 9−10): 1798– 1809

[30] DasS, Mullick S S, SuganthanP N. Recent advances in differential evolution-an updated survey. Swarm and Evolutionary Computation, 2016, 27 : 1– 30

[31] ErbaoC, MingyongL. A hybrid differential evolution algorithm to vehicle routing problem with fuzzy demands. Journal of Computational and Applied Mathematics, 2009, 231( 1): 302– 310

RIGHTS & PERMISSIONS

Higher Education Press 2021.