Damage assessment and diagnosis of hydraulic concrete structures using optimization-based machine learning technology

Abstract

Concrete is widely used in various large construction projects owing to its high durability, compressive strength, and plasticity. However, the tensile strength of concrete is low, and concrete cracks easily. Changes in the concrete structure will result in changes in parameters such as the frequency mode and curvature mode, which allows one to effectively locate and evaluate structural damages. In this study, the characteristics of the curvature modes in concrete structures are analyzed and a method to obtain the curvature modes based on the strain and displacement modes is proposed. Subsequently, various indices for the damage diagnosis of concrete structures based on the curvature mode are introduced. A damage assessment method for concrete structures is established using an artificial bee colony backpropagation neural network algorithm. The proposed damage assessment method for dam concrete structures comprises various modal parameters, such as curvature and frequency. The feasibility and accuracy of the model are evaluated based on a case study of a concrete gravity dam. The results show that the damage assessment model can accurately evaluate the damage degree of concrete structures with a maximum error of less than 2%, which is within the required accuracy range of damage identification and assessment for most concrete structures.

Graphical abstract

Keywords

hydraulic structure / curvature mode / damage detection / artifical neural network / artificial bee colony

1 Introduction

Concrete is widely used in large construction projects owing to its high compressibility, plasticity, and durability. However, its low tensile strength is disadvatagous, as cracks can easily be generated and propagated in materials with low ductility, e.g., concrete [1–3]. Cracks in dams and bridges weaken the structure and shorten their lifetime [4]. In other words, the failure of these large engineering structures can reults in significant levels of casualties and economic losses [5–7]. Thus, appropriate techniques must be developed for detecting damage in concrete structures such that damages and potential dangers can be identified effectively and abrupt accidents can be avoided strategically.

Damage to large concrete structures occurs randomly and cryptically in a spatio–temporal manner, which further complicates damage detection and diagnosis [8–10]. Measurable modal parameters, such as frequency and curvature, can reflect the overall condition of the structure. Additionally, they can be used to identify the origin of damage as well as to monitor damage propagation [11,12]. For example, the curvature mode is susceptible to local structural damage. Therefore, advanced instruments are typically used to detect the response data (acceleration, strain, displacement, etc.) of the structure under complex conditions [13,14]. Subsequently, the method for identifying the modal parameter is used to extract the frequency, vibration, and curvature modes of the structure. Structural damage can be identified and evaluated by analyzing the changes and characteristics of the modal parameters of a structure [15].

Changes in the state of hydraulic concrete structures will change the structural parameters, such as the frequency mode and curvature mode, based on which the location of structural damage can be determined and the damage degree can be evaluated [16–19]. Curvature is a parameter associated with the second derivative of displacement; therefore, every vibration of the displacement mode presents its corresponding curvature distribution, and the inherent curvature state of the structure corresponding to the vibration is known as the curvature mode [20]. Local damage to the structure will deteriorate the local rigidity of the corresponding section and result in the local mutation of the structure’s curvature mode [11]. Therefore, the local mutation of the structural curvature mode can be used to diagnose structural damage. Compared to the frequency mode, the curvature mode is more sensitive to structural damage; however, the acquisition of the curvature mode requires more measuring points [21,22]. Thus, in this study, the damage diagnosis of concrete structures based on the curvature mode is regarded as a key issue, and the relationship between strain, displacement, and curvature mode is considered.

2 Methodology

2.1 Curvature mode characteristics

2.1.1 Differential equation of concrete structure vibration

We consider a concrete gravity dam as an example for structural stress analysis. As shown in Fig.1(a), we set y\left(x,t\right) as the transverse displacement of the gravity dam, which is a binary function of cross-section position x and time t. The moment of inertia of the cross-section to the kingbolt is I\left(x\right), the density of concrete is \rho, the cross-sectional area of the gravity dam is A\left(x\right), and distributed forces p\left(x,t\right) per unit length are exerted on the upstream surface of the gravity dam.

A microsegment \mathrm{d}x is extracted at elevation x, and its force diagram is shown in Fig.1(b), where F\left(x,t\right) is the surface pressure, Q\left(x,t\right) is the shear, and M\left(x,t\right) is the bending moment. Subsequently, a differential equation for the structural vibration is established based on the D’Alembert principle. An imaginary inertial force \rho A\mathrm{d}x{\displaystyle \frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}} is added to the microsegment such that the microsegment is in the imaginary equilibrium state. Subsequently, the motion equation of the microsegment along the y-axis can be established based on \sum F_y=0.

(1)\frac{\mathrm{\partial }Q}{\mathrm{\partial }x}\mathrm{d}x-\rho A\mathrm{d}x\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}+p\left(x,t\right)\mathrm{d}x=0.

After simplification, we obtain

(2)\frac{\mathrm{\partial }Q}{\mathrm{\partial }x}-\rho A\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}+p\left(x,t\right)=0.

As \sum m_o\left(F\right)=0, where m_o denotes the weight, the rotation equation of the microsegment can be obtained as follows:

(3)M+\frac{\mathrm{\partial }M}{\mathrm{\partial }x}\mathrm{d}x-M-Q\mathrm{d}x-\rho A\mathrm{d}x\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}\cdot \frac{1}{2}\mathrm{d}x+p\left(x,t\right)\mathrm{d}x\cdot \frac{1}{2}\mathrm{d}x=0.

The second-order trace is extremely small and negligible; therefore,

(4)\frac{\mathrm{\partial }M}{\mathrm{\partial }x}=Q.

Based on the relationship between bending moment and deflection in material mechanics,

(5)M=EI\left(x\right)\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}.

Substituting Eqs. (2) and (5) into Eq. (4) yields

(6)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\left[EI\left(x\right)\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}\right]+\rho A\left(x\right)\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}=p\left(x,t\right).

Equation (6) is a partial differential equation for the transverse vibration of a concrete gravity dam.

If the structure is a homogeneous uniform section, i.e., EI\left(x\right) and A\left(x\right) are constant, then Eq. (6) can be written as

(7)EI\frac{{\mathrm{\partial }}^{4}y}{\mathrm{\partial }{x}^4}+\rho A\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}=p\left(x,t\right).

If viscous damping c is considered, then Eq. (6) can be written as

(8)\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\left[EI\left(x\right)\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}\right]+\rho A\left(x\right)\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{t}^2}+c\frac{\mathrm{\partial }y}{\mathrm{\partial }t}=p\left(x,t\right).

Equation (8) is the partial differential equation of the structural transverse vibration with viscous damping c.

2.1.2 Superposition of curvature modes

According to the theory of modal analysis, the solution of Eq. (6) can be expressed as the superposition of the displacement modes of each order [23].

(9)y\left(x,t\right)=\sum _{i=1}^{\mathrm{\infty }}{q}_i\left(t\right){\varphi }_i\left(x\right),

where q_i\left(t\right) represents the generalized vibration mode coordinates, {\varphi }_i\left(x\right) is the function of the displacement modal, and i is the mode order.

Based on Eq. (9), any reasonable displacement mode of the structure can be represented by the superposition of the structure displacement modes that correspond to the amplitude.

Substituting Eq. (9) into Eq. (8) yields

(10)\begin{array}{rl}& \sum _{i=1}^{\mathrm{\infty }}\rho A\left(x\right){\varphi }_i\left(x\right){\ddot{q}}_i\left(t\right)+\sum _{i=1}^{\mathrm{\infty }}c{\varphi }_i\left(x\right){\dot{q}}_i\left(t\right)\\ & +\sum _{i=1}^{\mathrm{\infty }}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\left[EI\left(x\right)\frac{{\mathrm{d}}^2{\varphi }_i\left(x\right)}{\mathrm{d}{x}^2}\right]q_i\left(x\right)=p\left(x,t\right).\end{array}

Multiplying each of the terms in Eq. (10) by {\varphi }_n\left(x\right)\mathrm{d}x, the following expression is obtained when the integrations are performed from 0 to l for x.

(11)\begin{array}{rl}& \sum _{i=1}^{\mathrm{\infty }}{\int }_0^l\rho A\left(x\right){\varphi }_i\left(x\right){\varphi }_n\left(x\right)\mathrm{d}x{\ddot{q}}_i\left(t\right)\\ & +\sum _{i=1}^{\mathrm{\infty }}{\int }_0^{l}c{\varphi }_i\left(x\right){\varphi }_n\left(x\right)\mathrm{d}x{\dot{q}}_i\left(t\right)\\ & +\sum _{i=1}^{\mathrm{\infty }}{\int }_0^l\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\left[EI\left(x\right)\frac{{\mathrm{d}}^2{\varphi }_i\left(x\right)}{\mathrm{d}{x}^2}\right]{\varphi }_n\left(x\right)\mathrm{d}x{q}_i\left(t\right)\\ =& {\int }_0^{l}p\left(x,t\right){\varphi }_n\left(x\right)\mathrm{d}x.\end{array}

Using the orthogonality of Eq. (12), Eq. (11) can be simplified as

(12)\begin{array}{rl}& {\ddot{q}}_n\left(t\right)+\frac{{\displaystyle \sum _{i=1}^{\mathrm{\infty }}{\dot{q}}_i\left(t\right){\displaystyle {\int }_0^{l}c{\varphi }_i\left(x\right){\varphi }_n\left(x\right)\text{d}x}}}{{\displaystyle {\int }_0^l\rho A\left(x\right){\varphi }_n^2\left(x\right)\text{d}x}}+{\omega }_n^2{q}_n\left(t\right)=F_n\left(t\right), \\ & (n=1,2,3,...),\end{array}

where {\omega }_n is the frequency mode of the structure and F_n\left(t\right) is the generalized exciting force of the external exciting force p\left(x,t\right), which corresponds to the modal coordinate q_n\left(t\right).

Let \beta ={\displaystyle \frac{c}{2\rho A\left(x\right)}} be a constant that represents the attenuation coefficient of viscous damping. Based on the orthogonality of the displacement mode, the second term on the left side of Eq. (12) can be written as

(13)\begin{array}{rl}& \frac{{\displaystyle \sum _{i=1}^{\mathrm{\infty }}{\dot{q}}_i\left(t\right){\int }_0^{l}2\beta \rho A\left(x\right){\varphi }_i\left(x\right){\varphi }_n\left(x\right)\mathrm{d}x}}{{\int }_0^l\rho A\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x}\\ =& \frac{{\dot{q}}_n\left(t\right){\int }_0^{l}2\beta \rho A\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x}{{\int }_0^l\rho A\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x}=2\beta {\dot{q}}_n\left(t\right).\end{array}

Subsequently, Eq. (13) can be written as

(14){\ddot{q}}_n\left(t\right)+2\beta {\dot{q}}_n\left(t\right)+{\omega }_n^2{q}_n\left(t\right)=F_n\left(t\right).

Let {\zeta }_n={\displaystyle \frac{\beta }{{\omega }_n}}={\displaystyle \frac{c}{2\rho A\left(x\right){\omega }_n}} represent the damping ratio of the structure at the nth order. Therefore, Eq. (15) can be transformed into

(15){\ddot{q}}_n\left(t\right)+2{\zeta }_n{\omega }_n{\dot{q}}_n\left(t\right)+{\omega }_n^2{q}_n\left(t\right)=F_n\left(t\right), (n=1,2,3,...).

The solution for the damped single degree-of-freedom vibration differential equation \rho A\left(x\right){\ddot{q}}_n\left(t\right)+c{\dot{q}}_n\left(t\right)+kx=F_0\left(t\right) can be obtained as follows:

(16)\begin{array}{rl}{q}_n\left(t\right)=& {\mathrm{e}}^{-{\zeta }_n{\omega }_{n}t}\left(q_{n0}\cos {\omega }_{nd}t+\frac{{\dot{q}}_{n0}+{\zeta }_n{\omega }_n{q}_{n0}}{{\omega }_{nd}}\sin {\omega }_{nd}t\right)\\ & +\frac{1}{{\omega }_{nd}}{\int }_{}^{}{\mathrm{e}}^{-{\zeta }_n{\omega }_n\left(t-\tau \right)}{F}_n\tau \sin {\omega }_{nd}\left(t-\tau \right)\mathrm{d}\tau ,\end{array}

where {\omega }_{nd}=\sqrt{1-{\zeta }_n^2}{\omega }_n is the natural frequency of the damped structure; k is the weight coefficient; τ is the time-varying coefficient; and q_{n0} and {\dot{q}}_{n0} are the initial conditions of the beam in generalized coordinates, whose values can be calculated as

(17)\{\begin{array}{rl}& q_{n0}={\displaystyle \frac{{\int }_0^l\rho A\left(x\right){\varphi }_n\left(x\right)y_0\left(x\right)\mathrm{d}x}{{\int }_0^l\rho A\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x}},\\ & {\dot{q}}_{n0}={\displaystyle \frac{{\int }_0^l\rho A\left(x\right){\varphi }_n\left(x\right){\dot{y}}_0\left(x\right)\mathrm{d}x}{{\int }_0^l\rho A\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x}}.\end{array}

If the first N order displacement mode is used for superposition, then the corresponding expression of the beam under the action of the external exciting force p\left(x,t\right) is

(18)y\left(x,t\right)=\sum _{n=1}^N{\varphi }_n\left(x\right)q_n\left(t\right).

Based on the definition of curvature, the following relationship can be derived:

(19)C_1\left(x,t\right)=\frac{\mathrm{\partial }\theta }{\mathrm{\partial }x}=\frac{{\mathrm{\partial }}^{2}y}{\mathrm{\partial }{x}^2}=\sum _{n=1}^N{\varphi }_n^{{}^{″}}\left(x\right)q_n\left(t\right),

where C₁ is the structural curvature, {\varphi }_n^{{}^{″}} is the second derivative of the displacement mode, and q_n\left(t\right) is the generalized vibration coordinate. Equation (19) indicates that the curvature mode exhibits superposition.

2.1.3 Orthogonality of curvature modes

Equation (12) can be obtained by performing partial integration twice for the left side of Eq. (12), as follows:

(20)\begin{array}{rl}& {\displaystyle {\int }_0^l{\varphi }_k\left(x\right)\frac{{\text{d}}^2}{\text{d}{x}^2}\left[EI\left(x\right)\frac{{\text{d}}^2{\varphi }_n\left(x\right)}{\text{d}{x}^2}\right]\text{d}x}\\ =& {\varphi }_k\left(x\right)\frac{\text{d}}{\text{d}x}\left[EI\left(x\right)\frac{{\text{d}}^2{\varphi }_n\left(x\right)}{\text{d}{x}^2}\right]-{\varphi }_n^{{}^{\prime }}\left(x\right)EI\left(x\right)\frac{{\text{d}}^2{\varphi }_n\left(x\right)}{\text{d}{x}^2}\\ & +{\displaystyle {\int }_0^l{\varphi }_n^{{}^{″}}\left(x\right){\varphi }_k^{{}^{″}}\left(x\right)EI\left(x\right)\text{d}x}={\displaystyle {\int }_0^l{\varphi }_n^{{}^{″}}\left(x\right){\varphi }_k^{{}^{″}}\left(x\right)EI\left(x\right)\text{d}x}.\end{array}

Subsequently, Eq. (12) can be simplified to

(21){\displaystyle {\int }_0^l{\varphi }_n^{{}^{″}}\left(x\right){\varphi }_k^{{}^{″}}\left(x\right)EI\left(x\right)\text{d}x}=\{\begin{array}{ll}0, & n\ne k,\\ {\omega }_k^2{m}_k, & n=k.\end{array}

For a discrete system, i.e., where EI\left(x\right)=K, the equation above can be reduced to

(22){\left({\varphi }^{{}^{″}}\right)}^{\mathrm{T}}K{\varphi }^{{}^{″}}={\omega }^{2}m,

where K is the stiffness of the discrete system and {\omega }^{2}m=diag\left[{\omega }_1^2{m}_1,{\omega }_2^2{m}_2,...,{\omega }_n^2{m}_n\right] is the stiffness matrix of the system modal. Equation (22) indicates that the curvature mode exhibits orthogonality.

2.2 Curvature mode calculation

Curvature modes cannot be easily measured directly. Currently, the curvature modes of structures are obtained indirectly using one of the following two methods.

1) The strain mode method, which considers the relationship between strain modes.

2) The central difference method, which considers the relationship between displacement and curvature modes.

2.2.1 Calculating curvature modes from strain modes

A diagram showing the bending deformation analysis of the structure is presented in Fig.2. Based on the relationship between a static and a bent straight beam in material mechanics, the curvature can be obtained as follows:

(23)C_1=\frac{M}{EI},

where M is the moment, EI is the flexural rigidity, and C₁ is the curvature.

The curvature can be obtained from the approximate equation of bending deformation for a straight beam as follows:

(24)C_1=\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2},

where x is the coordinate along the length of the straight beam and \varphi is the displacement mode.

Equation (25) can be rewritten in a different form and substituted into Eq. (24) as follows:

(25)C_i=\frac{{\varphi }_{i+1}-2{\varphi }_i+{\varphi }_{i-1}}{l^2}=\frac{M_i}{E_i{I}_i},

where the subscript i represents the measuring point and l is the distance between two measuring points.

In addition, the bending deformation of the beam corresponds to the strain (\epsilon), which can be expressed as

(26)\epsilon =-hC=-h\frac{{\varphi }_{i+1}-2{\varphi }_i+{\varphi }_{i-1}}{l^2},

where h is the distance from the measuring point to the neutral layer.

Equation (27) shows the relationship among the strain, curvature, and displacement modes of the beam. The deduced curvature mode was obtained from bending deformation calculations of the neutral surface. Therefore, the strain mode can be measured on any plane parallel to the neutral plane, and the curvature mode of the structure can be calculated based on the relationship between the strain and curvature modes.

2.2.2 Calculation of curvature modes from displacement modes

Consider a one-dimensional linear structure, as shown in Fig.3, as an example. The structure is segmented into n units of equal length, namely n+1 nodes, and each unit is l, numbered 1 to n+1 from left to right. If \varphi \left(x\right) is the displacement modal curve of the structure, then \varphi \left(x\right) is a variate function of coordinate x, and \varphi \left(x\right) can be obtained via the Taylor expansion as follows:

(27)\begin{array}{rl}\varphi \left(x\right)=& \varphi \left(x_i\right)+{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i\left(x-x_i\right)+\frac{1}{2!}{\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i{\left(x-x_i\right)}^2\\ & +\frac{1}{3!}{\left(\frac{{\mathrm{d}}^3\varphi }{\mathrm{d}{x}^3}\right)}_i{\left(x-x_i\right)}^3+\frac{1}{4!}{\left(\frac{{\mathrm{d}}^4\varphi }{\mathrm{d}{x}^4}\right)}_i{\left(x-x_i\right)}^4+\cdots .\end{array}

The coordinate value at node i is x_i=\left(i-1\right)l; and the coordinate values at nodes i-1 and i+1 are x_{i-1}=\left(i-2\right)l and x_{i+1}=i\cdot l, respectively. Therefore, x_{i-1}-x_i=-l and x_{i+1}-x_i=l. Substituting these expressions into Eq. (28) yields

(28)\begin{array}{rl}\varphi \left(x_{i-1}\right)=& \varphi \left(x_i\right)-l{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i+\frac{l^2}{2!}{\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i\\ & -\frac{l^3}{3!}{\left(\frac{{\mathrm{d}}^3\varphi }{\mathrm{d}{x}^3}\right)}_i+\frac{l^4}{4!}{\left(\frac{{\mathrm{d}}^4\varphi }{\mathrm{d}{x}^4}\right)}_i-\cdots ,\end{array}

(29)\begin{array}{rl}\varphi \left(x_{i+1}\right)=& \varphi \left(x_i\right)+l{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i+\frac{l^2}{2!}{\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i\\ & +\frac{l^3}{3!}{\left(\frac{{\mathrm{d}}^3\varphi }{\mathrm{d}{x}^3}\right)}_i+\frac{l^4}{4!}{\left(\frac{{\mathrm{d}}^4\varphi }{\mathrm{d}{x}^4}\right)}_i+\cdots .\end{array}

If the structure is segmented finely, then l is sufficiently small. Subsequently, the terms of the third and higher powers in the Taylor expansion are negligible; thus, Eqs. (29) and (30) can be simplified as

(30)\varphi \left(x_{i-1}\right)=\varphi \left(x_i\right)-l{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i+\frac{l^2}{2!}{\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i,

(31)\varphi \left(x_{i+1}\right)=\varphi \left(x_i\right)+l{\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i+\frac{l^2}{2!}{\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i.

The differential equations of the first and second derivatives of the displacement modes can be obtained by solving the two equations above.

\begin{array}{}\text{(32a)}& {\left(\frac{\mathrm{d}\varphi }{\mathrm{d}x}\right)}_i=\frac{\varphi \left(x_{i+1}\right)-\varphi \left(x_{i-1}\right)}{2l},\end{array}

\begin{array}{}\text{(32b)}& {\left(\frac{{\mathrm{d}}^2\varphi }{\mathrm{d}{x}^2}\right)}_i=\frac{\varphi \left(x_{i+1}\right)-2\varphi \left(x_i\right)+\varphi \left(x_{i+1}\right)}{l^2}.\end{array}

Therefore, for general engineering structures, the center difference method can be used to approximate the parameters of the curvature mode.

(33)c={\varphi }_i^{{}^{″}}=\frac{{\varphi }_{i-1}-2{\varphi }_i+{\varphi }_{i+1}}{l^2},

where i is the number of measuring points, {\varphi }_i is the displacement mode, {\varphi }_i^{{}^{″}} is the curvature mode represented by the second derivative of the displacement mode, and l is the distance between two adjacent measurement points.

When the structure is not segmented into equal intervals, i.e., the distance between nodes is not always the same, then Eq. (34) should be modified as follows [24]:

(34){\varphi }_i^{{}^{″}}={\displaystyle \frac{{\displaystyle \frac{{\varphi }_{i+1}-{\varphi }_i}{l_i}}-{\displaystyle \frac{{\varphi }_i-{\varphi }_{i-1}}{l_{i-1}}}}{{\displaystyle \frac{l_i+l_{i-1}}{2}}}}={\displaystyle \frac{2}{l_i+l_{i-1}}}\left({\displaystyle \frac{{\varphi }_{i+1}-{\varphi }_i}{l_i}}-{\displaystyle \frac{{\varphi }_i-{\varphi }_{i-1}}{l_{i-1}}}\right),

where l_i is the distance between the ith and (i+1)th nodes of the structure.

3 Damage diagnosis index

According to material mechanics, the curvature of a structure can be expressed as

(35)c\left(x\right)=\frac{M}{EI\left(x\right)},

where M denotes the section moment, c\left(x\right) is the curvature, and EI\left(x\right) is the bending stiffness of the structure.

Substituting Eq. (36) into Eq. (20) yields

(36)\frac{M}{EI\left(x\right)}=\sum _{n=1}^N{\varphi }_n^{{}^{″}}\left(x\right)q_n\left(t\right).

The local damage of the structure will decrease the local stiffness EI\left(x\right) of the structure and, based on Eq. (24), the curvature c\left(x\right) at the damaged location will increase correspondingly. Based on Eq. (37), the reduction of EI\left(x\right) causes a mutation in the local curvature mode. Therefore, the location where {\varphi }_n^{{}^{″}}\left(x\right) is mutated is that of the structural damage, and the damage degree of the structure can be qualitatively determined based on the mutation amplitude of {\varphi }_n^{{}^{″}}\left(x\right).

In actual hydraulic concrete structures, the cross-sectional shape along the axial direction of the structure is typically not static. EI\left(x\right) will change with the structure section. This change in non-damaged stiffness may result in a misjudgment in structural damage identification; however, the modal difference value of the curvature modes can avoid such misjudgment.

The difference value of the curvature modes can be expressed as

(37)D{\varphi }^{{}^{″}}={\varphi }_{\mathrm{o}}^{{}^{″}}-{\varphi }_{\mathrm{d}}^{{}^{″}},

where D{\varphi }^{{}^{″}} is the difference value of the curvature modes, {\varphi }_{\mathrm{o}}^{{}^{″}} is the curvature mode before damage, and {\varphi }_{\mathrm{d}}^{{}^{″}} is the curvature mode after damage.

Damage to the structure will change the structural stiffness E, but the quality of the structure itself will remain unchanged, and the shape of the section I\left(x\right) at which damage occurs will not change. Thus, the damage factor D\left(x\right) can be expressed as

(38)D\left(x\right)=\frac{E_{\mathrm{d}}\left(x\right)}{E_{\mathrm{o}}\left(x\right)},

where E_{\mathrm{d}}\left(x\right) and E_{\mathrm{o}}\left(x\right) represent the elastic moduli after and before structural damage, respectively.

Using Eqs. (36) and (37), the curvature mode of the structure without damage can be obtained as follows:

(39){\varphi }_{\mathrm{o}}^{{}^{″}}=c_{\mathrm{o}}\left(x\right)=\frac{M}{E_{\mathrm{o}}I\left(x\right)}.

By substituting Eq. (39) into Eq. (40), the curvature mode after structural damage can be expressed as follows:

(40){\varphi }_{\mathrm{d}}^{{}^{″}}=c_{\mathrm{d}}\left(x\right)=\frac{M}{E_{\mathrm{d}}I\left(x\right)}=\frac{M}{D\left(x\right)E_{\mathrm{o}}I\left(x\right)}.

Substituting Eqs. (40) and (41) into Eq. (38) yields

(41)\begin{array}{rl}D{\varphi }^{{}^{″}}\left(x\right)=& {\varphi }_{\mathrm{o}}^{{}^{″}}-{\varphi }_{\mathrm{d}}^{{}^{″}}=\frac{M}{E_{\mathrm{o}}I\left(x\right)}-\frac{M}{D\left(x\right)E_{\mathrm{o}}I\left(x\right)}\\ =& \frac{M}{E_{\mathrm{o}}I\left(x\right)}\left(1-\frac{1}{D\left(x\right)}\right)={\varphi }_{\mathrm{o}}^{{}^{″}}\left(1-\frac{1}{D\left(x\right)}\right).\end{array}

The equation above shows that when no the structure is not damaged, D\left(x\right)=1 and D{\varphi }^{{}^{″}}\left(x\right)=0; meanwhile, when the structure is damaged, and D{\varphi }^{{}^{″}}\left(x\right)\ne 0. The difference in the curvature mode at the corresponding position of the damage is mutated. As the degree of damage increases, the damage factor D\left(x\right) decreases gradually. Furthermore, the mutation of the curvature mode difference value becomes increasingly evident.

The ratio of the curvature mode can be expressed as follows:

(42)\mathrm{\Delta }{\varphi }_{\mathrm{r}i}^{{}^{″}}=\frac{{\varphi }_{\mathrm{o}i}^{{}^{″}}-{\varphi }_{\mathrm{d}i}^{{}^{″}}}{{\varphi }_{\mathrm{d}i}^{{}^{″}}},

where \mathrm{\Delta }{\varphi }_{\mathrm{r}i}^{{}^{″}} is the curvature mode ratio at node i, {\varphi }_{\mathrm{o}i}^{{}^{″}} is the curvature mode at node i before damage, and {\varphi }_{\mathrm{d}i}^{{}^{″}} is the curvature mode at node i after damage.

The mean curvature mode difference value can be expressed as

(43)CDF=\frac{1}{N}\sum _{n=1}^N\left({\varphi }_{\mathrm{d}}^{{}^{″}}-{\varphi }_{\mathrm{o}}^{{}^{″}}\right),

where N is the order of the mode considered, {\varphi }_{\mathrm{d}}^{{}^{″}} is the curvature mode after structural damage, and {\varphi }_{\mathrm{o}}^{{}^{″}} is the curvature mode of the structure before structural damage. This index synthesizes the mean value of the curvature mode difference before structural damage and, to some extent, can eliminate error interference.

Based on a theoretical analysis of the curvature mode and the difference value of the curvature mode based on the diagnosis of structural damage, the difference value of the curvature mode, the curvature mode ratio, and the mean curvature mode can all be used as diagnostic indices of structural damage. Based on the location and amplitude of mutation obtained from these diagnostic indices, the damage location can be identified, and the degree of damage can be determined qualitatively.

4 Damage assessment model of concrete structure based on improved artificial bee colony backpropagation neural network

4.1 Backpropagation neural network-based multimodal parameter fusion damage assessment

After the concrete structure is damaged, its modal parameters (frequency and curvature) change. However, different modal parameters and modal parameters of different orders have different sensitivities toward structural damage, which renders it difficult to directly derive the functional relationship between modal parameters and damage degree. Therefore, the structural modal parameters or their derivative values can be used as damage assessment factors. Using a neural network and other intelligent methods, an implicit function model that maps between the damage degree and damage assessment factor was established; subsequently, the damage degree of the concrete structure was evaluated based on modal parameter changes [23–25].

The curvature mode c\left(x\right) of concrete structures is sensitive to local damage; therefore, it can be directly used as a damage assessment factor. For damage conditions at different positions and degrees, the variation in the structural frequency modes is insignificant; therefore, the frequency mode is less sensitive to structural damage, and the frequency is not suitable as a damage assessment factor. Therefore, the derivative values of the following frequency modes is selected as the damage assessment factor: the reciprocal of the variation in the frequency mode {\displaystyle \frac{1}{\mathrm{\Delta }\omega }} and the reciprocal of the percentage in the frequency mode variation {\displaystyle \frac{\omega }{\mathrm{\Delta }\omega }}.

When damage occurs a structure at a certain modal node, the modal parameters of that node do not change significantly (i.e., the modal parameters are not sensitive to the damage at the modal node). Hence, the first three curvature modal parameters of the structure were selected as damage assessment factors in this study. In total, nine damage assessment factors were selected from the following two categories to evaluate the construction of the model: three curvature mode factors (c_1, c_2, and c_3) and six frequency mode factors ({\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_1}}, {\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_2}}, {\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_3}}, {\displaystyle \frac{{\omega }_1}{\mathrm{\Delta }{\omega }_1}}, {\displaystyle \frac{{\omega }_2}{\mathrm{\Delta }{\omega }_2}}, and {\displaystyle \frac{{\omega }_3}{\mathrm{\Delta }{\omega }_3}}).

After the damage assessment factors are selected, the implicit function relationship between the damage degree of the structure and the modal parameters can be expressed as follows:

(44)S=\sum _{i=1}^{3}f\left(c_i\right)+\sum _{i=1}^{3}t\left(\frac{1}{\mathrm{\Delta }{\omega }_i}\right)+\sum _{i=1}^{3}u\left(\frac{{\omega }_i}{\mathrm{\Delta }{\omega }_i}\right),

where i is the order of the mode, S is the degree of structural damage; and f\left(x\right), t\left(x\right), and u\left(x\right) represent unknown functions.

Subsequently, the mapping relationship between the modal parameters of concrete structures and the damage degree, as shown in Eq. (45), was fitted by exploiting the high learning ability of the backpropagation (BP) neural network. A damage assessment model of concrete structures was established, and the damage degree was evaluated based on the variation in the modal parameters.

The BP neural network is a type of multilayer feedforward neural network that is composed of three layers: an input layer, a hidden layer, and an output layer, as shown in Fig.4. Neurons between the layers are connected to the thresholds and weights. Its main characteristic is the forward transmission of data with backward propagation of error. During data transmission, the input data are processed from the input layer toward the output layer via the hidden layer. If the error between the output results and expected output is significant, then the error propagates backward. The connection weight and threshold between each neuron in the network are adjusted such that the output value of the BP neural network continues to approach the expected output until the error becomes sufficiently small, of which the latter signifies that learning is completed [26].

We set Z_1,Z_2,...,Z_n as the input vectors of the neural network, which represent the damage assessment factors of the structure under different damage conditions. Y_1,Y_2,...,Y_m represent the output vectors of the neural network, i.e., the damage degree of the concrete structure element vector. For the sequence values \left(Z,Y\right) of the input and output, the input value Z of each set of damage assessment factor have a corresponding damage degree output value, Y. The mapping model between the structural modal parameters and damage degree can be obtained from the training process. The specific modeling process is as follows.

1) Based on the input and output sequence \left(Z,Y\right) of the system, namely the damage assessment factor converted by the modal parameters, the number of nodes in the input layer n, number of nodes in the output layer m, and number of nodes in the hidden layer p are specified. Initialize the connection weights ({\omega }_{ij} and {\omega }_{jk}) and thresholds (a and b) between the neuron layers to set the learning rate and the activation function of the neural cells.

2) Calculate the output of the hidden layer H from the input Z, connection weight {\omega }_{ij}, and threshold value a between the input and hidden layers.

(45)H_j=f\left({\displaystyle \sum _{i=1}^n{\mathit{\text{z}}}_i{\omega }_{ij}-a_j}\right), j=1,2,...,p,

where {\mathit{\text{z}}}_i is the input data; and f is the activation function, which is expressed as

(46)f\left(t\right)=\frac{1}{1+{\mathrm{e}}^{-t}}.

3) The output result O of the hidden layer can be calculated using the connection weight {\omega }_{jk} and threshold value b, both of which are between the hidden and output layers.

(47)O_k={\displaystyle \sum _{j=1}^p{H}_j{\omega }_{jk}-b_k}, k=1,2,...,m.

4) The error of the damage assessment e can be calculated by comparing the actual output of the damage assessment O with the expected output Y.

(48)e=Y_k-O_k.

5) The network weights and thresholds are adjusted to reduce the BP error e.

(49)\begin{array}{c}{\omega }_{ij}={\omega }_{ij}+\eta H_j\left(1-H_j\right){\mathit{\text{z}}}_i{\displaystyle \sum _{k=1}^m{\omega }_{jk}{e}_k},\\ i=1,2,...,n;j=1,2,...,p,\\ {\omega }_{jk}={\omega }_{jk}+\eta H_j{e}_k,j=1,2,...,p;k=1,2,...,m,\end{array}

(50)\begin{array}{c}{a}_j=a_j+\eta H_j\left(1-H_j\right){\mathit{\text{z}}}_i{\displaystyle \sum _{k=1}^m{\omega }_{jk}{e}_k},j=1,2,...,p,\\ b_k=b_k+e_k,k=1,2,...,m,\end{array}

where \eta is the learning efficiency; a_j denotes the weight coefficient after updating; z_i denotes the correction factor.

6) The iterations are terminated when the error is smaller than the set error, and the assessment result of the damage degree is output; otherwise, the flow is repeated from Step 2).

4.2 Artifical bee colony-based damage assessment model for hydraulic concrete structure

Despite its high learning ability, the BP neural network presents some disadvantages: Owing to its time-consuming training process and slow convergence, it is easily trapped by local optima. Moreover, using a BP netural network renders its difficult to determine the initial weight, threshold, and structure of a network; in fact, these parameters can only be determined by experts and via trial calculations. To optimize the BP neural network, an improved artificial bee colony (ABC) algorithm was introduced. The improved ABC algorithm in the BP neural network facilitates the development of a multimodal parameter fusion assessment model of concrete structure damage, thus increasing the search efficiency of the BP neural network.

4.2.1 Standard artificial bee colony algorithm

The ABC algorithm is a new group optimization method proposed by Karaboga, a Turkish scholar, based on the intelligent behavior of bees in nature. Karaboga established this new method by simulating the communication, transformation, and cooperation behaviors among three types of ABCs: lead, follow, and scout bees. In the ABC algorithm, each nectar source represents a feasible solution to an optimization problem and corresponds to a lead bee. The nectar content of the source represents the fitness of a feasible solution. The higher the nectar content, the better is the fitness. The optimal solution to this problem is obtained by identifying the nectar source with the highest nectar content. The colony circularly seeks the source with the highest amount of nectar, which is the best solution for the abovementioned problem. For a D-dimension optimization problem, the colony initializes the feasible solution of the D-dimension and continuously seeks new food sources near the current food source via iterations. Subsequently, it is replaced to improve the solution quality if better fitness can be achieved. The specific search process is as follows.

1) The lead bee updates the position of the nectar source that is adjacent to the nectar source in its memory based on Eq. (51). The fitness value of the corresponding nectar source is calculated. If the fitness is higher, then the original nectar source is replaced with the calculated value; otherwise, the nectar source remains fixed.

(51)v_{ij}=x_{ij}+r_{ij}\left(x_{ij}-x_{kj}\right),

where v_{ij} represents the location of the adjacent nectar source, where i,k\in \left\{1,2,...,SN\right\} and i\ne k; j\in \left\{1,2,...,D\right\}; and r_{ij} is a random number in the interval \left[-1,1\right].

2) Based on the location and fitness of the nectar source provided by the leading bee, the follow bee selects a lead bee to follow with selection probability P_i, as expressed in Eq. (52). Subsequently, the follow bee seeks the nectar source with better fitness in the adjacent nectar source, similar to the lead bee.

(52)P_i={\displaystyle \frac{f_i}{{\displaystyle \sum _{i=1}^{SN}{f}_i}}},

where f_i represents the fitness of the nectar source.

3) If the location of the nectar source where the lead bees forage has not been replaced after the maximum number of cycles is reached, i.e., the fitness has not improved during the iterations, then the lead bee becomes a scout bee and replaces the original nectar source with

(53)x_i=x_{i,min}+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\left(0,1\right)\left(x_{i,max}-x_{i,min}\right).

4.2.2 Improvement to artificial bee colony algorithm

1) Improvement to the probability of processing bee selecting nectar source

In the standard ABC algorithm, greedy selection is performed by a follow bee based on the probability expressed in Eq. (52). However, this selection method decreases the population diversity and degrades local optimization. Therefore, an improvement is performed such that the selection probability changes in a timely manner based on the different search stages.

(54)P_i=\frac{{\left(f_i\right)}^{\lambda }}{{\displaystyle \sum _{i=1}^{SN}{\left(f_i\right)}^{\lambda }}},\lambda ={\displaystyle \frac{k}{{\displaystyle \frac{f_{max}}{\overline{f}}}-1}},

where f_{max}=max\left\{f_i;i=1,2,...,SN\right\}; and \overline{f}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{SN}{f}_i}}{SN}}.

Based on an analysis of the equation above, the following conclusions can be inferred.

a) When {\displaystyle \frac{\overline{f}}{f_{max}}}\to 0, then \lambda \to 0, {\left(f_i\right)}^{\lambda }\to 1, P_i={\displaystyle \frac{1}{SN}}, and P_i is small, which implies that the poor fitness nectar source still contain living spaces that are sufficiently large to ensure the population diversity.

b) When {\displaystyle \frac{\overline{f}}{f_{max}}}\to 1, then \lambda \to \mathrm{\infty }. If , then P_i=0. If f_i=f_{\mathrm{m}\mathrm{a}\mathrm{x}}, then P_i={\displaystyle \frac{1}{M}} (M represents the number of individuals with the highest fitness), which implies no living space for the poor fitness nectar source, and the selection is only performed in the nectar source with high fitness.

The modification above enables the following.

a) In the early stage of selection, owing to the random generation of the initial nectar sources, the difference among the nectar sources is significant; therefore,the average and optimal fitness of the nectar sources differ significantly. When \lambda \to 0, the algorithm presents a higher wide-ranging searching ability and ensures the diversity of the nectar sources; b) in the middle stage of selection, \lambda changes with the characteristics of the nectar source to balance itself between seeking generality and precision; c) in the later stage of selection, when the average fitness of the nectar source is similar to the fitness of the optimal nectar source, \lambda \to \mathrm{\infty }, and the search space is reduced significantly, which increases the convergence rate.

2) Improvement based on neighborhood selection of bee

In the standard ABC algorithm, the method adopted by a lead bee and a follow bee to select the neighborhood is consistent, as expressed in Eq. (54). However, in a the actual nectar-gathering process, the lead bees share information regarding the nectar source with the follow bees in the form of a waggle dance. Processing bees select a nectar source based on the yield rate and select the nectar source in the neighborhood of the food source. In fact, the two groups of bees select their neighborhood differently. Therefore, the neighborhood search equation of the follow bees must be modified to introduce the average Euclidean distance, as follows [25,26]:

(55)m{d}_m=\frac{{\displaystyle \sum _{j=1}^{SN}d\left(m,j\right)}}{SN-1},

where d(m,j) represents the Euclidean distance between nectars x_m and x_j.

When d(m,j)⩽m{d}_m, x_j is regarded as a neighborhood nectar source of x_m.

4.2.3 Flowchart of damage assessment

The improved ABC algorithm is combined with a BP neural network to construct a damage assessment model for concrete structures. The process is shown in Fig.5, and the specific implementation process is as follows.

1) By calculating and analyzing different damage conditions, the frequency and curvature mode factors of different damage conditions are obtained to determine the topology of the BP neural network.

2) The main parameters of the improved ABC algorithm are set as follows: Number of artificial colonies, 2SN; number of lead and follow bees, SN, separately; limit time of nectar source updates, limit; maximum number of iterations, MCN; dimension of nectar source, D. Here, D=N_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}\ast N_{\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}}+N_{\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}}+N_{\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}}\ast N_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}}+N_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}}, where N_{\mathrm{i}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}}, N_{\mathrm{h}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}}, and N_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}} are the number of neurons in the input, hidden, and output layers of the BP neural network, respectively.

3) The initial nectar source X_i (i=1,2,...,SN) is generated using the small section method, and the fitness value of each nectar source is calculated as f_i={\displaystyle \frac{1}{E+1}}, where E is the mean square error of the BP neural network for the ith solution.

4) The lead bees identify a new nectar source based on Eq. (52). If the fitness of the new nectar source is higher than that of the old nectar source, then the old nectar source is replaced; otherwise, 1 is added to the update times for the old honey source.

5) The selection probability P_i is calculated based on the adaptive scale expressed in Eq. (55) to select the nectar source. Subsequently, the neighborhood is determined from the average Euclidean distance, and a new nectar source is identified in the neighborhood.

6) If the update time of the nectar source exceeds the preset number, then the lead bees become scout bees. Subsequenetly, a new nectar source is generated based on Eq. (54) to replace it, and the nectar source with the best fitness is selected.

7) If the time of the ABC algorithm iterations reaches MCN, then the optimal solution is assigned to the weight and threshold value of the BP neural network; otherwise, return to Step 4).

8) The frequency and curvature mode factors are input, and the BP neural network is trained. Subsequently, the output damage assessment results are obtained.

5 Case study

In this section, a concrete gravity dam is used as an example to establish a damage assessment model; subsequently, the damage degree is analyzed.

5.1 Project description

The finite element model developed and its measuring point arrangement for a concrete gravity dam are shown in Fig.6. The concrete dam featured a height of 100 m, bottom width of 80 m, and crest width of 10 m. The elevation of the foundation plane was 0 m and that of the turning point of the backward dam port was 90 m. The upstream face of the dam was set with 11 measuring points, numbered from 1 to 11 from the bottom to the crest. The elevation of the measuring points was 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 m, respectively. In the finite element model, the heights in the upstream, downstream, and depth directions of the dam foundation were combined into one dam height. The elastic modulus, Poisson’s ratio, and density of the dam concrete were 28 GPa, 0.17, and 2400 kg/m³, respectively, whereas the deformation modulus, Poisson’s ratio, and density of the bedrock were 30 GPa, 0.2, and 2700 kg/m³, respectively. The finite element model featured 1883 nodes and 1840 elements; in particular, the elements were four-node two-dimensional flat elements.

Based on the analysis presented in Subsection 4.1, nine damage assessment factors were selected to evaluate the construction of the model, i.e., three curvature mode factors (c_1, c_2, and c_3) and six frequency mode factors ({\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_1}}, {\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_2}}, {\displaystyle \frac{1}{\mathrm{\Delta }{\omega }_3}}, {\displaystyle \frac{{\omega }_1}{\mathrm{\Delta }{\omega }_1}}, {\displaystyle \frac{{\omega }_2}{\mathrm{\Delta }{\omega }_2}}, and {\displaystyle \frac{{\omega }_3}{\mathrm{\Delta }{\omega }_3}}).

5.2 Single-point damage assessment

A gravity dam with unit damage at 40 m upstream, as shown in Fig.7, was selected as the research object for damage assessment. The damage degree of the unit was established under 11 conditions for structural calculation and analysis, i.e., 5%, 10%, 12%, 15%, 20%, 22%, 25%, 30%, 32%, 35%, and 40%. Among them, damage assessment factors in eight conditions (5%, 10%, 15%, 20%, 25%, 30%, 35%, and 40%) were selected as training samples, whereas those in the other three operating conditions (12%, 22%, and 32%) were selected as test samples. When the damage degree of the gravity dam damage unit is 12%, the structural curvature and frequency mode factors are as shown in Tab.1 and Tab.2, respectively. Owing to space limitations, only the damage evaluation factors for the abovementioned damage condition are listed.

The data of the curvature modes, frequency modes, and other damage assessment factors under different damage conditions were used to train the improved artificial bee colony-backpropagation (ABC-BP) neural network. The established damage assessment model was used to evaluate damage on the test samples. The damage assessment results are listed in Tab.3. As shown in Tab.3, the structural damage fusion evaluation model based on multimodal parameters can be used to evaluate the damage to concrete gravity dams. The damage identification accuracy is high, with a maximum error of 1.48%, which satisfies the accuracy requirement for the damage identification and evaluation of most hydraulic structures.

5.3 Multipoint damage assessment

A gravity dam with unit damage at 10 and 40 m upstream, as shown in Fig.8, was selected as the research object for damage assessment. The damage degrees of the unit were established under 11 conditions for the structural calculation and analysis: 5%, 10%, 12%, 15%, 20%, 22%, 25%, 30%, 32%, 35%, and 40%. The damage assessment factors in eight conditions (5%, 10%, 15%, 20%, 25%, 30%, 35%, and 40%) were selected as training samples, whereas those in the other three operating conditions (12%, 22%, and 32%) were selected as test samples. When the damage degree of the gravity dam damage unit is 12%, the structural curvature frequency mode factors are as shown in Tab.4 and Tab.5, respectively.

The data of the curvature modes, frequency modes, and other damage assessment factors under different damage conditions were used to train the improved ABC-BP neural network. The established damage assessment model was used to evaluate damage on the test samples. As shown by the results presented in Tab.6, the structural damage fusion evaluation model based on multimodal parameters can be used to evaluate damage to concrete gravity dams. The accuracy of damage identification is relatively high, with a maximum error of 1.96%, which satisfies the accuracy requirement for the damage identification and evaluation of most hydraulic structures.

Tab.7 presents an evaluation and comparison of the damage identification results. Compared with other optimization algorithms, the ABC algorithim offers significantly better optimization results via the metaheuristic effect. The relative error of the ABC algorithm is significantly lower than those of the other methods.

6 Conclusions

Herein, the construction principle and realization process associated with the damage assessment model of a concrete structure were presented. The accuracy of damage assessment was evaluated via numerical experiments based on the modal information of a concrete structure, such as frequency and curvature, as well as via comprehensive applications of neural networks, ABC algorithms, and other intelligent learning methods.

1) Damage evaluation factors based on the frequency and curvature of concrete structures were selected to identify damage to concrete structures. Using a neural network, the complex mapping relationship between the structural damage assessment factors above and the damage degree was characterized. A neural network model for concrete structure damage assessment using multimodal parameters was proposed.

2) The optimal weight and threshold value of the neural network were determined based on an improved ABC algorithm. By training the neural network, a multimodal parameter fusion evaluation model for concrete structure damage was constructed.

3) Based on a concrete gravity dam as an example, various damage conditions with different degrees were proposed to accurately identify the structural damage location. The damage assessment model efficiently evaluated the damage degree of concrete structures, with a maximum error of less than 2%, which satisfies the requirements of most damage identification and assessment accuracy of concrete structures.

Notably, the artifical neural network model was designed with a shallow architecture, which may result in issues such as slow convergence or stagnancy in local minima. Nonetheless, researchers are current adopting deep learning in deep neural networks, which significantly benefits regression and classification. Even when only low amounts of data are available, deep learning via the additiona of regularization terms performs well for recognition and inference. Therefore, methods to expand data richness and improve effectiveness should be identified in future research.

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