*2.2. Mathematical Expression of the Ageing Displacement of the Concrete Dam Considering Viscoelastic Deformation*

Dam structural properties primarily consist of three states: elasticity, viscoelasticity, and unstable failure [33]. The dam monitoring displacement consists of two components: an elastic component and a nonlinear component that fluctuates with load and time (commonly known as the time effect). The elastic displacement is influenced and calculated by the instantaneous elastic modulus. The actual ageing displacement is a process of increasing and decreasing around a certain base value, including irrecoverable and recoverable terms [34]. The factors that generally affect the ageing displacement include the material properties of the dam concrete, such as autogenous volume displacement, wet and dry displacement, plastic displacement, creep of concrete, the material properties of the dam base rock, and the creep of the dam structure. Creep deformation reflects an inherent property of the dam concrete and the rock. It consists of an irreversible viscoplastic component and a reversible viscoelastic (also known as hysteretic elastic) component. During a long period of creep displacement, the reversible viscoelastic component accounts for a significant portion of the total elastic displacement [35], which is influenced by the delayed elastic modulus separated by the Burgers model shown in Figure 1. On a macroscopic scale, the displacement is characterised by a high degree of variation at the beginning of the storage period, followed by a gradual stabilisation over time. However, laboratory tests on concrete and rock have shown that a portion of the creep displacement of concrete and rock recovers after the unloading lag. Part of the recovery is influenced by the size of the unloading and the time of unloading, as well as the age of the concrete, as shown in Figure 2.

**Figure 1.** The Burgers model for dam concrete.

**Figure 2.** The radial displacement and the reservoir water level of the NO.9 dam section.

In the case of intermittent loading, the creep is a combination of increasing and decreasing curves. Because the water level in a reservoir is constantly changing, its effect on the dam can be seen as intermittent loading. As the water level is lowered, the dam and rock body thus undergo some non-linear recovery of creep with time, which can be linked to the delayed elastic moduli of the dam concrete. Therefore, it would not be accurate enough to describe the ageing displacement simply as a monotonically increasing function. A new model needs to be investigated to make the fit more accurate. As mentioned above, the ageing displacement is caused by the creep of the concrete and rock foundation and the plastic deformation caused by the compression of fractures and joints in the concrete and rock base. Their effects are discussed separately as follows.

#### 2.2.1. Creeping Law of Concrete

The total strain *ε*(*t*) of concrete at age *τ*1, under the action of the external force *σ*(*τ*1) can be expressed by the following equation [36]:

$$
\varepsilon(t) = \frac{\sigma(\mathbf{r}\_1)}{E(\mathbf{r}\_1)} + \sigma(\mathbf{r}\_1)\mathbf{C}(t, \mathbf{r}\_1),
\tag{4}
$$

where *C*(*t*, *τ*1) is the creep degree of the concrete; *E*(*τ*1) is the modulus of elasticity of concrete at age *τ*1.

In general, dams are old enough when they are built to hold water. According to the theory of elastic creep, the creep at load changes conforms to the principle of superposition, i.e., [37]:

$$
\varepsilon\_c(t) = \sigma(\tau\_1)\mathbb{C}(t, \tau\_1) + \sum \Delta \sigma\_i \mathbb{C}(t, \tau\_i), \tag{5}
$$

where Δ*σ<sup>i</sup>* is the stress increment at the moment *τi*.

The variation pattern of Equation (5) can be seen in Figure 2, which shows the effect of load history on strain.

2.2.2. Derivation of the Equation for the Creep Displacement of a Dam on a Rigid Foundation

Assume that the modulus of elasticity of the foundation *E*r→∞. The dam is built on a rigid foundation, and the water pressure can be approximated as proportional loading; according to the theory of creep mechanics, the total displacement *u* at the top of the dam can be expressed as [38]:

$$
u = \mathfrak{u}^{\varepsilon} f(t),$$

where *ue* is the elastic displacement under external load.

*f*(*t*) can be expressed as follows [39]:

$$f(t) = 1 + E(t) \int\_{\tau\_1}^{t} \varphi(t, \tau) \xi(t, \tau, \tau\_1) d\tau. \tag{7}$$

For a dam, its physical force is a constant and can be taken as *ϕ*(*t*,*τ*) = 1. So, the above equation becomes

$$f(t) = 1 + E(t) \int\_{\tau\_1}^{t} \xi(t, \tau, \tau\_1) d\tau. \tag{8}$$

After deducting the elastic displacement from Equation (6), we can obtain

$$
\delta\_\theta = \mu - \mu^\varepsilon = \mu^\varepsilon [f(t) - 1]. \tag{9}
$$

Substitute Equation (8) into Equation (9) and give

$$\delta\_{\emptyset} = u^{\varepsilon} E(t) \int\_{\tau\_1}^{t} \xi(t, \tau, \tau\_1) d\tau. \tag{10}$$

*ξ*(*t*,*τ*,*τ*1) in Equation (10) is the kernel of integration and using the conclusions of elastic creep theory, can be expressed as [40]:

$$\zeta^{\tau}(t,\tau,\tau\_{1}) = \sum\_{i=1}^{m} \mathbb{C}\_{i} r\_{i} e^{-r\_{i}(t-\tau\_{1})} + \sum\_{j=m+1}^{p} \mathbb{C}\_{i} r\_{j} e^{-r\_{i}(t-\tau\_{1})}.\tag{11}$$

Substitute *ξ*(*t*,*τ*,*τ*1) into Equation (10) and give

$$\delta\_{\theta} = u^{\varepsilon} E(t) \sum\_{i=1}^{p} \mathbb{C}\_{j} [1 - e^{-r\_{i}(\tau\_{1} - t)}].\tag{12}$$

When taking *p* = 1, it can be expressed as

$$\delta\_{\theta} = u^{\varepsilon} E(t) \mathbb{C}\_{1}[1 - e^{-r\_{i}(\tau\_{1} - t)}].\tag{13}$$

It should be noted that in Equation (13), *δθ* is the creep displacement at time *t* when the water level is *H* (the water level at the age of the dam concrete at *τ*1). In fact, the water level in the reservoir is constantly changing from time *τ*<sup>1</sup> to time *t*. Therefore, considering the water level change, the displacement expression at the moment *t* is

$$\delta\_{\theta1} = u^{\varepsilon} E(t) \mathbb{C}\_1[1 - e^{r\_1(\tau\_1 - t)}] + E(t) \int\_{\tau\_1}^{t} [u^{\varepsilon}(H, t) - u^{\varepsilon}(H, \tau)] \cdot \mathbb{C}\_2[1 - e^{r\_2(\tau - t)}] d\tau,\tag{14}$$

where *u<sup>e</sup>* (*H*,*t*) and *ue* (*H*,*τ*) are the elastic displacements of the dam crest due to the reservoir water pressure at moments *t* and *τ*.

The age of the dam concrete is generally large due to reservoir storage. Therefore, it can be assumed that *E*(*t*) is a constant. Thus, *u*e, *E*(*t*), and *C*<sup>1</sup> are combined as *C* <sup>1</sup> and *E*(*t*) and *C*<sup>2</sup> are combined as *C* <sup>2</sup>. *τ*<sup>1</sup> is the age of the concrete at the start of the monitored displacement. Therefore, *τ* − *t* can be replaced by −*t* . *t* indicates that the first day of the period starts at zero. In this way, Equation (14) can be rewritten as [38]

$$\delta\_{\theta1} = \mathbb{C}\_1'[1 - e^{-r\_1 t \tau}] + \mathbb{C}\_2' \int\_{\tau\_1}^t [u^\varepsilon(H, t) - u^\varepsilon(H, \tau)][1 - e^{r\_2(\tau - t)}] d\tau. \tag{15}$$

2.2.3. Creep Displacement of Intact Rock Masses under External Loading

According to the Poynting–Thomso rheological model, the intrinsic structure of the rock is related to

$$\frac{d\sigma}{dt} + \frac{E\_1 \sigma}{\eta\_1} = (E\_1 + E\_2)\frac{d\varepsilon}{dt} + \frac{E\_1 E\_2}{\eta\_1} \varepsilon. \tag{16}$$

When *σ* = *σ*<sup>0</sup> = const and *ε t*=<sup>0</sup> <sup>=</sup> *<sup>σ</sup>*<sup>0</sup> *<sup>E</sup>*1+*E*<sup>2</sup> . The solution to the above equation is

$$\varepsilon = \frac{\sigma}{E\_2} + \sigma (\frac{1}{E\_1 + E\_2} - \frac{1}{E\_2}) e^{-\frac{E\_1 E\_2}{E\_1 + E\_2} \cdot \frac{t}{\eta\_1}}.\tag{17}$$

The first term on the right-hand side of the above equation represents the instantaneous deformation and the second term is the creeping deformation. Therefore, the above equation can be rewritten as

$$
\varepsilon = \frac{\sigma}{E\_0} + \frac{\sigma}{E'} (1 - e^{-\frac{E'}{\eta} \cdot t}),
\tag{18}
$$

where *E*<sup>0</sup> is the instantaneous elastic modulus; *E* is the delayed elastic modulus; *η* is the corresponding viscosity coefficients.

From the above equation, the creep deformation can be expressed as follows:

$$
\varepsilon\_p(t) = \frac{\sigma}{E'} (1 - e^{-\frac{E'}{\eta}t}).\tag{19}
$$

The above equation indicates that when the stress is constant, the creep deformation varies with time as an exponential function, and after considering the integrated constants, the above equation is rewritten as

$$
\varepsilon\_p(t) = \mathbb{C}\_3(1 - e^{-r\_3t}).\tag{20}
$$

However, the water level in the reservoir is constantly changing and there is a certain amount of creep recovery in the rock mass as it is unloaded. Therefore, the total expression for the deformation of the foundation in the case of variable water level is

$$\varepsilon\_p(t) = \mathbb{C}\_3(1 - e^{-r\_3t}) + \mathbb{C}\_4 \int\_{\tau\_1}^t [\mathbb{G}(H, t) - \mathbb{G}(H, \tau)][1 - e^{r\_4(\tau - t)}] d\tau,\tag{21}$$

where *G*(*H*,*τ*) is the effect of water level *H* on the function of creep deformation. It is generally difficult to obtain its expression, thus, it can be used instead of the water pressure displacement component.

2.2.4. The Effect of Fractures and Joints in a Rock Body under Water Pressure on the Ageing Displacement

Rock bodies where fissures and joints are present will gradually close under the weight of water, and weak inclusions will deform plastically. In general, this part of the deformation does not recover from unloading, it is a monotonically increasing function of time, and it is difficult to deduce physically an expression for this part of the deformation with time. However, based on its characteristics of rapid change in the early stages of water storage and gradual stabilisation, the solution of the first-order decay differential equation can be used to describe the whole process of this change.

Let *C*<sup>5</sup> be the final stable value of the displacement, displacement *δθ*<sup>3</sup> with time *t*, and the rate of gradual decay and deformation residual *C*<sup>5</sup> − *δθ*<sup>3</sup> is proportional to

$$\frac{d\delta\_{\ell 3}}{dt} = r\_5(\mathbb{C}\_5 - \delta\_{\ell 3}).\tag{22}$$

Its solution can be expressed as

$$
\delta\_{\theta3} = \mathbb{C}\_5 (1 - e^{-r\_5 t}). \tag{23}
$$

Combining Equations Equations (14), (21) and (23), it can be seen that the ageing displacement of the dam under the action of water pressure can be expressed as

$$\delta\_{\theta} = \mathbb{C}(1 - e^{-rt}) + \mathbb{C}' \int\_{\tau\_1}^{t} [u^e(H, t) - u^e(H, \tau)] [1 - e^{r'(\tau - t)}] d\tau. \tag{24}$$

When the reservoir level varies with the seasons, the elastic displacement *ue* (*H*,*t*) of the dam caused by the cyclic load, the water pressure, also varies in an approximate annual cycle. As mentioned above, the hysteretic elastic hydraulic deformation caused by the viscoelastic creep feature is divided into the ageing component. Therefore, a set of basic functions (cos <sup>2</sup>*π<sup>t</sup>* <sup>365</sup> , sin2*π<sup>t</sup>* <sup>365</sup> , cos <sup>4</sup>*π<sup>t</sup>* <sup>365</sup> , sin4*π<sup>t</sup>* <sup>365</sup> , ··· , cos <sup>2</sup>*πnt* <sup>365</sup> , sin2*πnt* <sup>365</sup> ) is taken and their linear combination is used to express the term of the product function in Equation (24).

The decay term 1 <sup>−</sup> *<sup>e</sup><sup>r</sup>* (*τ*−*t*) in Equation (24) does not affect the periodicity of the product function. Clearly,

$$u^c(H, t) \leftrightarrow \sum\_{i=1}^{m} \left[ A\_i \sin \frac{2\pi it}{365} + B\_i \cos \frac{2\pi it}{365} \right]. \tag{25}$$

After replacing the original integral equation with the right-hand end of Equation (25), the resulting function is still a periodic function based on the trigonometric system, i.e.,

$$\int\_{\tau\_1}^{t} \sum\_{i=1}^{m} \left[ A\_i \sin \frac{2\pi it}{365} + B\_i \cos \frac{2\pi it}{365} \right] dt = \sum\_{i=1}^{m} \left[ C\_i \sin \frac{2\pi it}{365} + K\_i \cos \frac{2\pi it}{365} \right],\tag{26}$$

where *Ci* = <sup>365</sup>*Ai* <sup>2</sup>*π<sup>i</sup>* , *Ki* <sup>=</sup> <sup>365</sup>*Bi* <sup>2</sup>*π<sup>i</sup>* .

The second term on the right side of the Equation (24) is replaced by a periodic function term that varies with time. At this point, the practical expression for the ageing displacement can be expressed as

$$\delta\_{\theta}(t) = \mathbb{C}(1 - e^{-rt}) + \sum\_{i=1}^{2} \left(\mathbb{C}\_{i} \sin \frac{2\pi it}{365} + K\_{i} \cos \frac{2\pi it}{365}\right). \tag{27}$$

Using Equation (27) in combination with the measured data of the ageing displacement, the multiple linear regression method can be used to estimate the coefficient *C*, *r*, *Ci*, and *Ki*.

The Equation (27) contains the creep recovery part of the water level component, which makes the inversion obtained in the process of calculating the water level component and the elastic modulus shall be the instantaneous elastic modulus. In the HST model, the modulus of elasticity of the dam concrete obtained by the elastic inversion method is a composite reflection of the instantaneous elastic deformation and the deformation after viscoelasticity. Therefore, the viscoelastic inversion method must be used to separate the instantaneous elastic modulus from the viscoelastic modulus.

#### *2.3. Temperature Kernel Principal Components Analysis*

As a result, the temperature component is assumed to follow an annual cycle and is formulated in a predefined periodic harmonic factor, which may not accurately represent the thermal displacement effect of concrete dams [41]. For dams with buried thermometers, kernel principal components analysis has the advantage of better performance when extracting the feature and reducing the dimensionality of the thermometer measurement data [42]. Based on the principal component analysis (PCA) method, the KPCA method maps the input space into a high-dimensional feature space through nonlinear mapping, which makes the PCA able to perform in Hilbert space. The nonlinear problem can then be solved in high-dimensional space. Consider a set of *n* thermometer measurement data *X* = [*x*1, *x*2,... , *xn*], and then standardize it as follows:

$$\widehat{\mathbf{x}\_{i}} = \frac{\mathbf{x}\_{i} - A\_{i}}{S\_{i}},\\ A\_{i} = \frac{1}{n} \sum\_{i=1}^{n} \mathbf{x}\_{i\prime}\\ S\_{i} = \sqrt{\frac{1}{n-1} \sum\_{i=1}^{n} \left(\mathbf{x}\_{i} - A\_{i}\right)^{2}},\tag{28}$$

where *Ai* is the mean value of the thermometer measurement data, and *Si* is the standard deviation of the thermometer measurement data.

After the standardization, use the mapping function *φ*(·) to map *xi* (*i =* 1, 2, ... , *n*) to the high-dimensional feature space *<sup>ψ</sup>* (*φ*: *m*→*ψ*). As a result, the dot set corresponding to the original thermometer measurement data can be written as Φ ={*φ*(*xi*)} (*i =* 1,2, ... ,*n*). In the high-dimensional feature space, the initial input data of the thermometers satisfies the equation shown as follows:

$$\sum\_{i=1}^{n} \phi(x\_i) = 0.\tag{29}$$

Furthermore, the feature space covariance matrix **C** is computed as follows:

$$\mathbf{C} = \frac{1}{n} \sum\_{i=1}^{n} \boldsymbol{\phi}(\mathbf{x}\_{i})^{T} \boldsymbol{\phi}(\mathbf{x}\_{i}) = \frac{1}{n} \boldsymbol{\Phi}^{T} \boldsymbol{\Phi}.\tag{30}$$

The characteristic equation of the covariance matrix can thus be expressed as follows:

*λξ<sup>i</sup>* = **C***ξi*, (31)

where *λ* denotes the eigenvalue of the covariance matrix **C**; *ξ<sup>i</sup>* is the eigenvector *λ*, corresponding to the eigenvalue.

The nonlinear mapping function *φ*(·) is implicit, it is difficult to calculate the covariance matrix **C** directly, so a kernel matrix **K** is introduced. As shown below, the kernel matrix **K** should meet the Mercer condition:

$$\mathbf{K} = \boldsymbol{\phi}(\boldsymbol{x}\_{i}) \cdot \boldsymbol{\phi}(\boldsymbol{x}\_{i})^{T}. \tag{32}$$

The characteristic equation of the kernel matrix **K** is expressed as follows:

$$
\lambda\_i \alpha\_i = \mathbf{K} \alpha\_i \tag{33}
$$

where *λ<sup>i</sup>* = *nλ<sup>i</sup>* has the meaning of the eigenvalue of the kernel matrix **K**.

To normalize the eigenvectors *αi*, the following condition must be met [43]:

$$\mathbf{1} = \sum\_{i=1}^{n} \boldsymbol{\alpha}\_{i} \boldsymbol{\alpha}\_{j} \mathbf{K} = \lambda\_{k} (\boldsymbol{\alpha}^{k} \cdot \boldsymbol{\alpha}^{k}). \tag{34}$$

Following that, the covariance matrix **C** and the kernel matrix **K** are entered into the characteristic equation of the kernel matrix **K**. In addition, the eigenvector *ξ<sup>i</sup>* of the covariance matrix **C** can be introduced by the nonlinear function *φ*(*xi*), which can be obtained as follows:

$$\zeta\_i = \sum\_{i=1}^n \alpha\_i^k \phi(x\_i),\tag{35}$$

where *α<sup>k</sup> <sup>i</sup>* is the *i*-th coefficient associated with *ξi*.

The eigenvalues *λ<sup>i</sup>* of the kernel matrix **K** are ordered in descending order as follows:

$$
\overline{\lambda}\_1 \ge \overline{\lambda}\_2 \ge \cdots \ge \overline{\lambda}\_s \ge \cdots \ge \overline{\lambda}\_n > 0. \tag{36}
$$

The proportion of the information contained in the *i*th and the first *k* principal components in the total amount of the information (contribution rate *li* and cumulative contribution rate *Q*) of these eigenvalues are calculated in the following equations:

$$\left\{ \begin{array}{c} l\_i = \frac{\overline{\lambda\_i}}{\sum\overline{\lambda\_i}} \\ \stackrel{\text{\tiny \overline{\lambda\_i}}}{\sum\overline{\lambda\_i}} \\ Q = \frac{\overline{\frac{n}{n-1}}}{\sum\overline{\lambda\_i}} \end{array} \right\}.\tag{37}$$

When the cumulative contribution rate *Q* of the cumulative *s*th eigenvalue exceeds 85%, the information corresponding to these eigenvalues is thought to be adequate to convey the information of the original input thermometer measurement data.

Finally, the kernel PCs *Pi*(*x*) of the mapped input thermometer measurement data *φ*(*xi*) in the feature space employed to the eigenvalue *ξ<sup>i</sup>* is the *i*th principal component, as presented in Equation (38):

$$P\_i(\mathbf{x}) = \xi\_i \boldsymbol{\phi}(\mathbf{x}\_i) = \sum\_{i=1}^n a\_i^k \boldsymbol{\phi}(\mathbf{x}\_i) \boldsymbol{\phi}(\mathbf{x}\_i)^T = \sum\_{i=1}^n a\_i^k \mathbf{K}. \tag{38}$$

The matrix **P** = {*P*1(*x*), *P*2(*x*), ... , *Ps*(*x*)} is made up of the kernel PCs, which are the temperature principal component matrices obtained after reducing the dimensionality of the original input thermometer measurement data set. It retains enough information from the original data. After the KPCA process, the redundant information in the original thermometer measurement data is removed, resulting in a robust temperature database for developing a data-driven model.

#### **3. Mathematical Expression of the New Hybrid Model**

The water pressure, temperature, and ageing components are used as independent variables to perform an MLR with the measured displacement components. As mentioned above, the elastic hydraulic component and the hysteretic hydraulic component are separated from the original hydraulic component in the HST model by the Burgers model. In this paper, the hysteretic hydraulic component was subsumed into the ageing component. As for the temperature component, it can be determined by the KPCA method through the thermometers embedded in the dam and the foundation. The mathematical expression of the new hybrid model can be expressed as follows:

$$\delta = \delta\_{H\varepsilon} + \delta\_T + \delta\_\theta = X\delta\_{H\varepsilon}' + \sum\_{i=1}^n b\_i T\_{P\gets i} + \mathbb{C}(1 - e^{-rt}) + \sum\_{j=1}^2 \left(\mathbb{C}\_j \sin\frac{2\pi jt}{365} + K\_j \cos\frac{2\pi jt}{365}\right) \tag{39}$$

where *X* is the adjustment coefficient of the elastic hydraulic component and the FEMcalculated elastic hydraulic component; *n* is the total number of the principal components; *TPCi* is the principal components extracted from the thermometers by the KPCA method; *bi*, *C*, *Cj*, and *Kj* are all coefficients.
