2.2. Typical problem employed for the recognition of engineering constants

A typical problem of backfilled region after a retaining wall is considered (Figure 1). The length and height of backfilled region is L = 20 m, H = 5 m, respectively. For simplicity, the

Figure 1. Schematic of a typical CLSM-backfilled region for the analysis of parameter recognition.

backfilled materials are considered to be linearly elastic and are within small deformation under external loadings; therefore, linear elastic analysis can be performed using only two engineering constants, that is, the Young's modulus, E (with the unit GPa), and the Poisson's ratio, ν. Furthermore, considering the width of backfilled zone is infinitely long, and a twodimensional analysis can be used. In order to evaluate the engineering constants through displacements and stresses, the backfilled region is assumed to be subjected to a concentrated vertical loading (surcharge) Q<sup>0</sup> = 72.5 kN acting at the point behind the retaining walla = 0.5 H.

In order to provide training data and testing data in parameter recognition using neural networks, the following four quantities are defined:

x1 = Uy1: vertical displacement (settlement) at upper surface (x = L/4), (positive downward, m). x2 = Uy2: vertical displacement (settlement) at upper surface (x = L/2), (positive downward, m). x3 = Sx1: horizontal normal stress (lateral earth pressure) on wall (y = 3H/4), (tensile positive, Pa). x4 = Sx2: horizontal normal stress (lateral earth pressure) on wall (y = H/2), (tensile positive, Pa). These sampling points are illustrated in Figure 1.

#### 2.3. FEM analysis using ANSYS

Finite element equation for elastic analysis of displacement and stress of a plane deformation problem can be expressed in matrix form [33]:

$$\{K\}\{X\} = \{F\}\tag{1}$$

where [K], {X}, {F} denotes global stiffness matrix, global degrees of freedom vector and global load vector respectively, defined as follows:

$$\begin{aligned} [K] = \sum\_{\epsilon=1}^{NE} [k]^{\epsilon} = \sum\_{\epsilon=1}^{NE} t \iint\_{A^{\epsilon}} [B]^{T} [D] [B] \, \, d\mathbf{x} \, dy \\ \{F\} = \sum\_{\epsilon=1}^{NE} \{f\}^{\epsilon} = \sum\_{\epsilon=1}^{NE} t \oint\_{S^{\epsilon}} \{N\}^{T} \, \{q\} \, \, dS \end{aligned} \tag{2}$$

If a plane 4-node isoparametric element is used, the shape functions in Eq. (2) can be expressed as

$$\begin{aligned} N\_1(\xi, \eta) &= (1 - \xi)(1 - \eta)/4 \\ N\_2(\xi, \eta) &= (1 + \xi)(1 - \eta)/4 \\ N\_3(\xi, \eta) &= (1 + \xi)(1 + \eta)/4 \\ N\_4(\xi, \eta) &= (1 - \xi)(1 + \eta)/4 \end{aligned} \tag{3}$$

where ξ, η are local coordinates. A typical PLANE42 element in ANSYS is shown in Figure 2.

Parameter Recognition of Engineering Constants of CLSMs in Civil Engineering Using Artificial Neural Networks http://dx.doi.org/10.5772/intechopen.71538 101

Figure 2. Definition of PLANE42 element of ANSYS [34].

backfilled materials are considered to be linearly elastic and are within small deformation under external loadings; therefore, linear elastic analysis can be performed using only two engineering constants, that is, the Young's modulus, E (with the unit GPa), and the Poisson's ratio, ν. Furthermore, considering the width of backfilled zone is infinitely long, and a twodimensional analysis can be used. In order to evaluate the engineering constants through displacements and stresses, the backfilled region is assumed to be subjected to a concentrated vertical loading (surcharge) Q<sup>0</sup> = 72.5 kN acting at the point behind the retaining walla = 0.5 H. In order to provide training data and testing data in parameter recognition using neural

x1 = Uy1: vertical displacement (settlement) at upper surface (x = L/4), (positive downward, m). x2 = Uy2: vertical displacement (settlement) at upper surface (x = L/2), (positive downward, m). x3 = Sx1: horizontal normal stress (lateral earth pressure) on wall (y = 3H/4), (tensile positive, Pa). x4 = Sx2: horizontal normal stress (lateral earth pressure) on wall (y = H/2), (tensile positive, Pa).

Finite element equation for elastic analysis of displacement and stress of a plane deformation

where [K], {X}, {F} denotes global stiffness matrix, global degrees of freedom vector and global

½ � K f g X ¼ f gF (1)

(2)

(3)

½ � <sup>B</sup> <sup>T</sup>½ � <sup>D</sup> ½ � <sup>B</sup> dx dy

<sup>t</sup> <sup>∮</sup> <sup>S</sup><sup>e</sup> <sup>½</sup>Ng<sup>T</sup> f g<sup>q</sup> dS

networks, the following four quantities are defined:

100 Advanced Applications for Artificial Neural Networks

These sampling points are illustrated in Figure 1.

problem can be expressed in matrix form [33]:

load vector respectively, defined as follows:

½ �¼ <sup>K</sup> <sup>X</sup> NE

f g<sup>F</sup> <sup>¼</sup> <sup>X</sup> NE

e¼1

e¼1

½ � <sup>k</sup> <sup>e</sup> <sup>¼</sup> <sup>X</sup> NE

e¼1 t ðð Ae

e¼1

If a plane 4-node isoparametric element is used, the shape functions in Eq. (2) can be expressed as

N1ð Þ¼ ξ; η ð Þ 1 � ξ ð Þ 1 � η =4 N2ð Þ¼ ξ; η ð Þ 1 þ ξ ð Þ 1 � η =4 N3ð Þ¼ ξ; η ð Þ 1 þ ξ ð Þ 1 þ η =4 N4ð Þ¼ ξ; η ð Þ 1 � ξ ð Þ 1 þ η =4

where ξ, η are local coordinates. A typical PLANE42 element in ANSYS is shown in Figure 2.

f g<sup>f</sup> <sup>e</sup> <sup>¼</sup> <sup>X</sup> NE

2.3. FEM analysis using ANSYS

#### 2.4. Numerical experiments for different combination of engineering constants

A finite element mesh with 40 10 = 400 elements, 41 11 = 451 nodes, 451 2 = 902 degrees of freedom (displacements) is employed for a numerical analysis of backfilled zone (Figure 3). The left-hand side and right-hand side can only move freely in vertical direction, while the bottom of backfilled zone is considered fixed in both directions. In order to provide enough sampling data for later training and testing process using supervised neural networks, the Young's modulus (E) ranges from 0.02 to 3 GPa (covering the general values of soil and

Figure 3. FEM mesh of the typical example of CLSM-backfilled region using ANSYS of PLANE42 elements.

CLSMs), while the Poisson's ratio (ν) is selected to be 0.1, 0.2, 0.25, 0.3, and 0.4. A total of 270 data samples with different combinations of E and ν are used.

### 2.5. Data preparation

#### 2.5.1. Data collection from FEM analysis using ANSYS

Table 2 shows part of the 270 numerical results of computed displacements and stresses for different Young's moduli and Poisson's ratio using the ANSYS PLANE42 model.

## 2.5.2. Normalization of data

Since the output range of the sigmoid function is within [0, 1], and in order to have a better performance of training of neural network, the input data should be normalized to have a uniformly ranged data.

The relationship between normalized and original data can be expressed as follows:

$$\begin{aligned} \check{\mathbf{x}} &= \mathbf{a}\_{\mathbf{x}} \mathbf{x} + \mathbf{b}\_{\mathbf{x}} \\ \check{\mathbf{y}} &= \mathbf{a}\_{\mathbf{y}} \mathbf{y} + \mathbf{b}\_{\mathbf{y}} \end{aligned} \tag{4}$$

where ax, bx are selected such that ~x ∈ ½ � �1; 1 , ay, by are chosen to let ~y ∈½ � 0 1 . In this case, the relations are

$$\begin{aligned} \ddot{\mathbf{x}}\_1 &= \mathbf{x}\_1 / 0.001\\ \ddot{\mathbf{x}}\_2 &= \mathbf{x}\_2 / 0.001\\ \ddot{\mathbf{x}}\_3 &= \mathbf{x}\_3 / 9000\\ \ddot{\mathbf{x}}\_4 &= \mathbf{x}\_4 / 9000\\ \ddot{\mathbf{y}}\_1 &= \mathbf{3}^\* y\_1 / 10^{10}\\ \ddot{\mathbf{y}}\_2 &= y\_2 \end{aligned} \tag{5}$$


Table 2. Computed displacements and stresses for different Young's moduli and Poisson's ratio using the ANSYS PLANE42 model.

Parameter Recognition of Engineering Constants of CLSMs in Civil Engineering Using Artificial Neural Networks http://dx.doi.org/10.5772/intechopen.71538 103


Table 3. Typical normalized FEM computed displacements and stress at sampling points.

Part of typical normalized FEM computed displacements and stress at sampling points are shown in Table 3. Among these normalized data, 216 set (216/270 ≒ 80%) are selected randomly for training and 54 set (54/270 ≒ 20%) for testing in the later neural work analyses.

#### 2.5.3. Data for verification

(4)

(5)

CLSMs), while the Poisson's ratio (ν) is selected to be 0.1, 0.2, 0.25, 0.3, and 0.4. A total of 270

Table 2 shows part of the 270 numerical results of computed displacements and stresses for

Since the output range of the sigmoid function is within [0, 1], and in order to have a better performance of training of neural network, the input data should be normalized to have a

> ~x ¼ axx þ bx ~y ¼ ayy þ by

where ax, bx are selected such that ~x ∈ ½ � �1; 1 , ay, by are chosen to let ~y ∈½ � 0 1 . In this case, the

x~<sup>1</sup> ¼ x1=0:001 x~<sup>2</sup> ¼ x2=0:001 x~<sup>3</sup> ¼ x3=9000 x~<sup>4</sup> ¼ x4=9000 <sup>y</sup>~<sup>1</sup> <sup>¼</sup> <sup>3</sup><sup>∗</sup>y1=1010

y~<sup>2</sup> ¼ y<sup>2</sup>

Data x1,Uy1 (m) x2,Uy2 (m) x3, Sx1 (Pa) x4, Sx2 (Pa) y1,E(GPa) y2,ν �0.00098 1.9447E-05 �4452.6 �4029.1 0.02 0.1 �4.91E-04 9.72E-06 �4452.6 �4029.1 0.04 0.1 �0.0008607 2.6951E-05 �5434.05 �5196 0.02 0.2 �0.0004303 1.3476E-05 �5434.05 �5196 0.04 0.2 �7.61E-04 4.36E-05 �5979.35 �5862.9 0.02 0.25 �3.80E-04 2.18E-05 �5979.35 �5862.9 0.04 0.25 �0.62826E-03 0.72850E-04 �6567.6 �6599 0.02 0.3 �3.14E-04 3.64E-05 �6567.6 �6599 0.04 0.3 �1.88E-06 1.50E-06 �7896.55 �8337.1 2.5 0.4 �1.56E-06 1.25E-06 �7896.55 �8337.1 3 0.4

Table 2. Computed displacements and stresses for different Young's moduli and Poisson's ratio using the ANSYS

different Young's moduli and Poisson's ratio using the ANSYS PLANE42 model.

The relationship between normalized and original data can be expressed as follows:

data samples with different combinations of E and ν are used.

2.5.1. Data collection from FEM analysis using ANSYS

2.5. Data preparation

102 Advanced Applications for Artificial Neural Networks

2.5.2. Normalization of data

uniformly ranged data.

relations are

PLANE42 model.

In order to verify the trained and tested neural networks, three data set of backfilled materials are used: (1) the first is commonly used soil (E = 0.1 GPa, ν = 0.30); (2) CLSM-B80/30% (E = 0.27 GPa, ν = 0.25); and (3) CLSM-B130/30% (E = 0.87 GPa, ν = 0.25). The CLSM-B80/30% and CLSM-B130/30% denote unit weight of binder of the CLSM is 80 and 130 kg/m<sup>3</sup> with 30% of replacement of cement by fly ash. Selection of materials for the CLSM mixture in this study consisted of fine aggregate, type I Portland cement, stainless steel reducing slag (SSRS), and water. Fine aggregate for CLSM was formed by well blending between river sand and residual soil with a given proportion (e.g., 6:4, by volume) for grading improvement. The soil was obtained from a construction site. The experimental work was conducted on two binder content levels in mixtures (i.e., 80- and 130 kg/m3 ). The water-to-binder ratio was selected via few trial mixes until the acceptable flowability for CLSM of 150300 mm was achieved. The detailed information can be seen in [8].

In Table 4, computed displacements and stresses for verified soil and two CLSMs using the ANSYS PLANE42 model are summarized, while the normalized data for verification are tabulated in Table 5.


Table 4. Computed displacements and stresses for verified soil and two CLSMs using the ANSYS PLANE42 model.


Table 5. Normalized FEM computed displacements and stress for verified soil and two CLSMs using the ANSYS PLANE42model.

### 3. Parameter recognition using BPANN

#### 3.1. Application of BPANN for parameter recognition

Figure 4 demonstrates a typical BPANN for the identification of engineering constants of CLSM. The BPANN shown in Figure 4 contains a single output, that is, the Young's modulus (y1 = E, y2 = ν), L input neurons (xi, i = 1, 2, ⋯, L), and M hidden neurons (hj, j = 1, 2, ⋯, M). The predicted outputs can be expressed as [35–37]:

a. from input layer (IL) to hidden layer (HL):

$$h\_{\vec{\jmath}} = f\left(n\_{\vec{\jmath}}\right) = f\left(\sum\_{i=1}^{L} w\_{\vec{\jmath}i} \mathbf{x}\_i - b\_{\vec{\jmath}}\right) \tag{6a}$$

b. from hidden layer (HL) to output layer (OL):

$$y\_k = f(n\_k) = f\left(\sum\_{j=1}^{M} w\_{kj} h\_j - b\_k\right) \tag{6b}$$

Figure 4. Schematic of a BPANN topology for single parameter recognition.

Where the activating function (the sigmoid function) and its derivative can be expressed as.

$$\begin{aligned} f(n) &= \frac{1}{1 + e^{-n}}\\ f'(n) &= f(n)[1 - f(n)]. \end{aligned} \tag{7}$$

#### 3.2. Learning algorithms of BPANN for parameter recognition

There exist many approaches for the determination of the network parameters in BPANN (wkj, bk, wji, bj). The most basic and popular generalized delta rule based on the method of the steepest descent along with two learning parameters (i.e., the learning rate η and the momentum factor μ)is employed, which can be expressed as

$$\begin{aligned} w\_{\not k}(p+1) &= w\_{\not k}(p) - \eta \quad \frac{\partial E}{\partial w\_{\not k}}(p) + \mu w\_{\not k}(p) \\ b\_k(p+1) &= b\_k(p) - \eta \quad \frac{\partial E}{\partial b\_k}(p) + \mu \, b\_k(p) \\ w\_{\not k}(p+1) &= w\_{\not k}(p) - \eta \, \frac{\partial E}{\partial w\_{\not k}}(p) + \mu \, w\_{\not k}(p) \\ b\_{\not}(p+1) &= b\_{\not\ell}(p) - \eta \, \frac{\partial E}{\partial b\_{\not\ell}}(p) + \mu \, b\_{\not\ell}(p) \end{aligned} \tag{8}$$

where

(6a)

3. Parameter recognition using BPANN

104 Advanced Applications for Artificial Neural Networks

PLANE42model.

predicted outputs can be expressed as [35–37]: a. from input layer (IL) to hidden layer (HL):

b. from hidden layer (HL) to output layer (OL):

Figure 4. Schematic of a BPANN topology for single parameter recognition.

3.1. Application of BPANN for parameter recognition

hj ¼ f nj

Figure 4 demonstrates a typical BPANN for the identification of engineering constants of CLSM. The BPANN shown in Figure 4 contains a single output, that is, the Young's modulus (y1 = E, y2 = ν), L input neurons (xi, i = 1, 2, ⋯, L), and M hidden neurons (hj, j = 1, 2, ⋯, M). The

Data x1,Uy1 (m) x2,Uy2 (m) x3, Sx1 (Pa) x4, Sx2 (Pa) y1,E(GPa) y2,ν Soil �0.126 0.0146 �0.72973 �0.73322 0.003 0.3 CLSM-B80/30% �0.0564 0.00323 �0.66437 �0.65143 0.0081 0.25 CLSM-B130/30% �0.0175 0.001 �0.66437 �0.65143 0.0261 0.25

Table 5. Normalized FEM computed displacements and stress for verified soil and two CLSMs using the ANSYS

� � <sup>¼</sup> <sup>f</sup> <sup>X</sup>

yk <sup>¼</sup> f nð Þ¼ <sup>k</sup> <sup>f</sup> <sup>X</sup>

L

i¼1

M

0 @

j¼1

wji xi � bj !

wkj hj � bk

1

A (6b)

$$E(p) = \frac{1}{2} \sum\_{k=1}^{N} e\_k^2(p) = \sum\_{k=1}^{N} \frac{1}{2} \left[ d\_k(p) - y\_k(p) \right]^2 \tag{9}$$

denotes the error between targets and trained output results.

$$\begin{aligned} \frac{\partial E}{\partial w\_{kj}}(p) &= -e\_k(p)f'(n\_k) \ \boldsymbol{h}\_j \\ \frac{\partial E}{\partial \boldsymbol{b}\_k}(p) &= -e\_k(p)f'(n\_k)^2 \\ \frac{\partial E}{\partial w\_{ji}}(p) &= -\sum\_{k=1}^N \left[ e\_k(p)f'(n\_k) \ \boldsymbol{w}\_{kj} \right] f'(n\_j) \ \boldsymbol{x}\_i \\ \frac{\partial E}{\partial \boldsymbol{b}\_j}(p) &= -\sum\_{k=1}^N \left[ e\_k(p)f'(n\_k) \ \boldsymbol{w}\_{kj} \right] f'(n\_j) \end{aligned} \tag{10}$$

The MATLAB toolbox nntool provides a lot of training methods along with BPANN [38], among which four kinds of training algorithms were tested in the current research as follows:

a. Generalized steepest decent (GD):

The learning rule can be written as Eq. (8) with η 6¼ 0, μ = 0.

b. Generalized steepest decent including momentum (GDM):

The learning rule can be written as Eq. (8) with η 6¼ 0, μ 6¼ 0.

c. Generalized steepest decent with adjustable learning rate (GDA):

In this algorithm, the basic learning rule is the same as Eq. (6) but adding a conditional judgment. When stable learning can be kept under a learning rate, then the learning rate is increased, otherwise it is decreased. The learning rate increment and decrement can be denoted as ζinc and ζdec [35].

d. Levenberg-Marquardt (LM):

The learning rule can be written as

$$
\Delta w\_{\vec{\eta}}(t+1) = w\_{\vec{\eta}}(t+1) - w\_{\vec{\eta}}(t) = -\left[ [J]^T [J] + \lambda [I] \right]^{-1} \left[ \mathfrak{f} \right]^T \{ \mathfrak{e} \} \tag{11}
$$

where λ denotes a constant to assure the inversion of matrix, and the learning rule becomes Gauss-Newton algorithm when λ = 0, while it approaches GD with small learning rate with large λ [35].
