The Inversion Analysis and Material Parameter Optimization of a High Earth-Rockfill Dam during Construction Periods

Abstract

Inversion analysis is usually an efficient solution to process the monitoring data of earth-rockfill dams. However, it is still difficult to obtain calculation results that are consistent with monitoring data due to different construction statuses. To deal with this situation and to introduce a new solution to improve calculation accuracy, the general method of inversion analysis based on back-propagation neural networks and the original step-by-step inversion method assuming that the parameters of the constitutive model vary with construction periods are introduced and verified in this work. Then, both methods are applied in the inversion analysis of a high gravelly soil core rock-fill dam during construction periods. Moreover, the relationship between the inversed material parameters and the stress values of the core wall is discussed. The material parameters are further optimized to obtain more accurate displacement values. The results show that the step-by-step inversion method has a higher accuracy in vertical compression values compared with the conventional inversion method, the trend of material parameter K is more significant than other parameters, and the proposed variable parameter constitutive model has an accuracy between the step-by-step and conventional inversion methods. Conclusions can be drawn that the original step-by-step inversion method has more advantages than the conventional method and the variable parameter constitutive model proposed in this paper might be more suitable for the analysis of a high earth-rockfill dam during construction periods.

1. Introduction

Numerous monitoring data can be obtained from earth-rockfill dams [1,2] due to the development of monitoring technology. How to deal with massive amounts of monitoring data has recently become a hot research issue of high rock-fill dams. Generally, the monitoring data can be processed and analyzed for displacement predictions using data fitting methods [3,4]. X. Li et al. [3] proposed a monitoring model of dam safety with the random forest intelligent algorithm. V. Khoroshilov et al. [4] presented the most successfully designed predictive mathematical models and discussed the advantages of using a mathematical model with a separate introduction of the main effective factors into the model. Although monitoring data can be simulated well with data fitting methods, mathematical models without the theoretical basis of mechanics may still be unconvincing. On the other hand, inversion analysis [5,6] can be carried using processed monitoring data in order to get more accurate calculation parameters. Traditional inversion methods usually consist of optimization methods and forward analysis. Moreover, inversion analysis has been improved with the introduction of artificial intelligence technology. There are many kinds of artificial intelligence methods, such as the bat algorithm [7], the genetic algorithm (GA) [8], the particle swarm optimization (PSO) algorithm [9], and the neural network algorithm [10,11]. The neural network algorithm [12,13] has become a relatively popular algorithm with the advantages of separating forward and inversion analysis steps using a training strategy and obtaining multiple parameters quickly after training. The back propagation neural network [14] (BP neural network), as a member of the neural network methods, has been widely applied to deal with inversion analysis problems [15,16] and can also be adopted in this paper. Material parameters closer to engineering practice can be obtained using traditional inversion methods, but researchers might find that it is still difficult to obtain calculated results (displayed, stress and so on) close enough to the monitoring data with the inversed material parameters due to the complexity of practical engineering, especially in projects during construction periods.

As we all know, the finite element method based on the nonlinear elastic or elastic-plastic constitutive model is often applied in the forward analysis [17] of earth-rockfill dams. The Duncan–Chang constitutive model [18], cam-clay model [19], modified cam-clay model [19], PZ model [20], and so on are the universally accepted nonlinear elastic or elastic-plastic constitutive models. The Duncan–Chang constitutive model has been widely used in engineering practices [21,22,23], as it can reflect the main deformation laws of rock and soil, and the model parameters can be simply obtained through a conventional triaxial shear test. Numerous works in the literature can be found on increasing the simulation ability of the Duncan–Chang model in soil materials [21,24,25]. Among all these achievements, few researchers have studied the feasibility of optimizing the constitutive model by inversion analysis.

To sum up, monitoring data prediction has the advantage in processing monitoring data, but the data fitting method is not convincing enough. Inversion analysis has the advantage in obtaining material parameters relevant to the monitoring data, but it is still difficult to obtain calculation results that are highly consistent with the monitoring data by inversed material parameters. The Duncan–Chang model can be modified to obtain results closer to engineering practice, but few researchers have studied the feasibility of optimizing the constitutive model by inversion analysis. Thus, increasing the inversion accuracy of earth-rockfill dams during construction periods and improving the simulation ability of the Duncan–Chang constitutive model by studying the characteristic of inversed material parameters will be the main purpose of this article. Firstly, the inversion analysis based on neural networks and the basic formulations of the Duncan–Chang constitutive model are introduced in this work. Furthermore, a new idea using a step-by-step inversion analysis method is introduced to study the variations of the Duncan–Chang constitutive model parameters in high earth-rockfill dams. Finally, the analysis results of a high earth-rockfill dam are shown to make a comparison between the conventional inversion and step-by-step inversion.

2. Briefings on the Inversion Method for Earth-Rockfill Dams during Construction Periods

The constitutive model, inversion parameters selection, and conventional inversion method based on the BP neural network are briefly introduced in this section.

2.1. Constitutive Model and Inversion Parameters Selection

The Duncan–Chang EB constitutive model, as one member of the Duncan–Chang models, is applied in the forward analysis. The Young’s modulus and the Poisson’s ratio in the Duncan–Chang EB model [18] can be calculated as follows:

(1)$\mathrm{Tangent} \mathrm{modulus}: E_t={\left[1-R_f\frac{\left(1-\sin \phi \right)\left({\sigma }_1-{\sigma }_3\right)}{2c·\cos \phi +2{\sigma }_3·\sin \phi }\right]}^2·K·p_a {\left(\frac{{\sigma }_3}{p_a}\right)}^n ,$

(2)$\mathrm{Rebound} \mathrm{modulus}: E_{ur}=K_{ur} p_a{\left(\frac{{\sigma }_3}{p_a}\right)}^n ,$

(3)$\mathrm{Tan}\mathrm{gent} \mathrm{bulk} \mathrm{modulus}: B_t=K_b p_a {\left(\frac{{\sigma }_3}{p_a}\right)}^m ,$

(4)${\mathrm{Poisson}}^{’}\mathrm{s} \mathrm{ratio}: \upsilon =\frac{1}{2}-\frac{E}{6B_t} ,$

where $E_t$ and $E_{ur}$ are the Young’s modulus in loading and unloading conditions; $c$ and $\phi$ are the cohesion and internal friction angle; ${\sigma }_1$ and ${\sigma }_3$ are the first and third principal stress; $K$, $K_{ur}$, and $K_b$ are modulus numbers; $m$ and $n$ are modulus indexes; and $R_f$ is the break ratio defined as

(5)$R_f=\frac{{\left({\sigma }_1-{\sigma }_3\right)}_f}{{\left({\sigma }_1-{\sigma }_3\right)}_u} ,$

where ${( )}_f$ and ${( )}_u$ mean the condition when the soil is in destruction and the condition corresponding to the asymptote of hyperbolic line.

The internal friction angle can be determined by

(6)$\phi ={\phi }_0-\Delta \phi \lg \left(\frac{{\sigma }_3}{p_a}\right) ,$

where $\Delta \phi$ is called the internal friction angle correction parameter and ${\phi }_0$ is the internal friction angle when ${\sigma }_3$ equals to $p_a$.

There are nine material parameters in a Duncan-Chang EB constitutive model: $c$, ${\phi }_0$, $\Delta \phi$, $K$, $R_f$, $K_{ur}$, $K_b$, $m$, $n$. In this paper, $K$, $n$, $R_f$ are selected as the inversion parameters for $c$, ${\phi }_0$, $\Delta \phi$ can be easily determined by trial and $K_{ur}$, $K_b$, $m$ have only a limited impact during construction periods.

2.2. Conventional Inversion Method Based on the BP Neural Network

2.2.1. Back Propagation Neural Network

The back propagation (BP) neural network is applied as the optimization method of inversion analysis in this paper. The sketch map of the BP neural network is shown in Figure 1. The BP neural networks consist of several layers. The input of the network is obtained in the first layer, called the input layer. Each subsequent layer is connected to the previous layer. The network’s output is obtained from the final layer. In this figure, x is the input value; w, v, and o are the weighting factors; and f is the excitation function. Since the theory of the BP neural network is quite mature, its main theory can be drawn from many other works, such as [14].

The input value x and output value y of node M in the ith layer can be calculated as

(7)$$\begin{gathered} x_{Mi}={\displaystyle \sum }{w}_{j\left(i-1\right)}{y}_{j\left(i-1\right)}, \\ {y}_{Mi}={\displaystyle \sum }f\left(x_{Mi}\right), \end{gathered}$$

where j is the node sequence in the (i − 1)th layer, $w$ is the weighting factor, $y$ is the output value, and $f(\ast )$ is an excitation function which generally uses a sigmoid function:

(8)$f\left(x\right)=\frac{1}{1+e^{-x}} ,$

2.2.2. Conventional Inversion Process

The steps of inversion analysis for a high earth-rockfill dam during construction periods based on the BP neural networks are introduced as follows:

Step 1: Building the earth-rockfill dam construction model and inputting the material parameters.

Step 2: Analyzing the monitoring data and matching the monitoring points.

Step 3: Calculating using FEM with different combinations of material parameters.

Step 4: Training the BP neural network with combinations of material parameters and displacement data.

Step 5: Obtaining the output material parameters with monitoring data by the trained network.

3. Step-by-Step Inversion Method

A step-by-step inversion method to get the material parameters in each construction period is introduced in this section. The flowchart is displayed in Figure 2.

Step 1: Building the earth-rockfill dam construction model and inputting the material parameters.

Step 2: Analyzing the monitoring data and matching the monitoring points.

Step 3: Designing combinations of sample material parameters for each construction step.

Step 4: Calculating the displacement of the select points of the ith construction step with FEM and the calculated material parameters before the ith construction step.

Step 5: Training the BP neural network with combinations of the material parameters and displacement data of the ith construction step.

Step 6: Obtaining the output material parameters of the ith construction step with monitoring data by the ith trained network.

Step 7: Going back to Step 4 until all material parameters are obtained.

4. Numerical Examples

4.1. Benchmark Problem: Simple Inversion Analysis Based on the Results of Forward Analysis

4.1.1. Problem Description

The validity of the inversion solutions applied in this manuscript is checked by the inversion analysis of a simple model in this section. A two-dimensional construction problem is investigated in this section. The outline and discrete grids of the model can be seen in Figure 3. The bottom of the model is fixed in all directions. The materials of the three construction layers are set as the Duncan–Chang EB model with different material parameters. Material information can be found in Table 1.

4.1.2. Forward Analysis and Assumed Monitoring Data

Forward analysis is carried out to obtain assumed monitoring data for inversion analysis. Point 2 (P2) in Figure 3 is selected as a monitoring point and the calculated vertical displacement results are listed in Figure 4.

4.1.3. Validation Checks of Conventional and Step-by-Step Inversion Analysis

Conventional and step-by-step inversion analysis are carried out in this section with the combinations of material parameters in Table 2.

The material parameters calculated by conventional and step-by-step inversion analysis are listed in Table 3. Calculated vertical displacement and errors are listed in Table 4.

It can be seen from Table 3 and Table 4 that the calculated results obtained by both inversion methods are all close to the target values. From Table 4, it can be seen that the maximum error is less than 1.5%, which means both conventional and step-by-step methods are effective for inversion analysis during construction periods.

4.2. Inversion Analysis of a High Earth-Rockfill Dam Using the Step-by-Step Method

4.2.1. Problem Description

In this section, inversion analysis using the step-by-step method is carried out and the material parameters of the bottom of a high earth-rockfill dam during construction periods are obtained. Calculated values are compared with those by conventional method.

The model of a 240-meter high earth-rockfill dam is displayed in Figure 5. The material parameters for design are listed in Table 5. Fifty construction steps are considered in this discrete model and the monitoring arrangement of the core wall can be seen in Figure 6. The monitoring compression values of the bottom of the core wall (marked by red lines in Figure 6) are displayed in Figure 7. The combinations of material parameters for inversion analysis are listed in Table 6.

4.2.2. Step-by-Step Inversion Analysis

The step-by-step inversion analysis method is applied to get the material parameters of the bottom of the core wall (marked by red lines in Figure 6) during construction periods. Calculated material parameters can be found in Table 7. It can be seen from Table 7 that the trend of change of the material parameter K is quite obvious; this trend will be further studied in the following section. The compression values of the bottom of the core wall by different methods are all close to the original values; detailed values are listed in Figure 8.

In order to compare the accuracy of different inversion methods, the relative errors between the monitoring data and calculated data are calculated by Equation (9):

(9)$err=\frac{{{\displaystyle \sum }}_{i=1}^m\left|\frac{q_i-p_i}{q_i}\right|}{m}\times 100\% ,$

where err is the relative error, m is the number of monitoring data, and $q_i$ and $p_i$ are the monitoring data and calculated data in the ith construction step. The relative errors of step-by-step inversion and conventional inversion are 4.6% and 14.4%, respectively. Thus, results closer to monitoring data can be obtained by the step-by-step inversion analysis from the perspective of relative error.

4.2.3. Further Study on Material Parameter Optimization

From the variation range of each parameter, it can be seen that the material parameter K has the most obvious trend of change. Figure 9 is the trend diagram of K and the third principal stress with construction periods. It can be seen from Figure 9 that K is unchanged when the third principal stress is small, K begins to increase significantly and keeps a high value when the third principal stress keeps increasing, and K starts to decrease at the end of construction periods. Thus, Equation (10) can be proposed to represent this phenomenon:

(10)$\{\begin{array}{ll}{K}_1=ample1\times K , & S3\le 500 \mathrm{kPa} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{Stage}1\\ K_2=ample2\times K_1-\left(ample2-1\right)\times K_1\times {\left(\frac{S3}{p_a}-4\right)}^{-2} , & 500 \mathrm{kPa}<S3\le 2000 \mathrm{kPa} \mathrm{Stage}2\\ K_3=K_{2last}\times {\left(\frac{S3}{p_a}-19\right)}^{-2}+ample3\times K\times \left(1-{\left(\frac{S3}{p_a}-19\right)}^{-2}\right) , & S3>2000 \mathrm{kPa} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Stage}3\end{array}$

where K is the material parameter used for designing; ample1, ample2, and ample3 are scale factors; S3 is the third principal stress; and $K_{2last}$ is the material parameter obtained at the end of Stage2.

A new constitutive model (called the variable parameter constitutive model in this paper) can then be obtained by adding Equation (10) in the Duncan–Chang constitutive model. The vertical compression values in the bottom of the dam can be found in Figure 10. The relative error of the compression values using the variable parameter constitutive model is 9.74%, which is smaller than that of the conventional inversion (14.4%) and larger than that of the step-by-step inversion (4.6%). Considering that more time is needed when using step-by step inversion and enough calculation accuracy cannot be reached using conventional inversion, the variable parameter constitutive model proposed in this article might be more suitable for the calculation of earth-rockfill dams during construction periods.

5. Discussion and Conclusions

5.1. Discussion

The inversion analysis and material parameter optimization of a high earth-rockfill dam during construction periods was carried out in this paper. Firstly, the Duncan–Chang EB constitutive model and conventional inversion analysis method were introduced and the step-by-step inversion analysis method, a new inversion analysis solution for construction periods, was proposed in this article. Secondly, the validation of the step-by-step inversion analysis solution was checked by a simple inversion analysis based on the results of forward analysis. Thirdly, the inversion analysis for the material parameters of the bottom of a high earth-rockfill dam was carried out using the verified step-by-step analysis method and conventional analysis method. The material parameters were obtained and the relevant vertical compression values were compared. Finally, the material parameters were optimized by studying the trend of material parameter K. The results calculated by the optimized material parameters were displayed and compared with other results.

This work focused on the improvement method for the inversion analysis of earth-rockfill dam during construction periods. The original step-by-step inversion analysis method and the variable parameter constitutive model were put forward and verified in this study. The step-by-step inversion method assumes that material parameters vary with construction, which is significantly different from conventional inversion methods assuming that parameters remain unchanged during construction in articles such as [5,6]. The variable parameter constitutive model comes from the study of the material parameter K’s change trend. The idea of studying the trend of material parameters from monitoring data distinguishes the variable parameter constitutive model from other modified models in papers such as [21].

5.2. Conclusions

Conclusions can be drawn as follows. Firstly, the step-by-step inversion method proposed in this article is as effective as the conventional inversion method in the simple inversion analysis based on the results of forward analysis. The maximum relative error between material parameters calculated by both methods and actual material parameters was less than 1.5%. Secondly, the step-by-step inversion method applied in the inversion analysis during construction periods showed a higher accuracy in vertical compression values compared with the conventional inversion method. The relative error between compression values calculated by step-by-step inversion method and monitoring compression values was 4.6%, which is smaller than that (14.4%) between compression values calculated by conventional inversion method and monitoring compression values. Thirdly, the trend of material parameter K was studied in this article and the variable parameter constitutive model was then proposed. The relative error of compression values using the variable parameter constitutive model was 9.74%, which is smaller than that of the conventional inversion (14.4%) and larger than that of the step-by-step inversion (4.6%). Considering that more time is needed using step-by step inversion and adequate calculation accuracy cannot be reached using conventional inversion, the variable parameter constitutive model proposed in this paper might be more suitable for the analysis of a high earth-rockfill dam during construction periods.

Meanwhile, there are still some limitations in this study. The proposed step-by-step inversion method may have the disadvantage of being time consuming as the material parameters must be inversed at each construction steps. Also, the applicability of the variable parameter constitutive model in three-dimensional conditions or other earth-rockfill dams cannot be ensured due to the lack of data.

In further study, the feasibility of 3D inversion problems using step-by-step inversion method and the applicability of the variable parameter constitutive model in other earth-rockfill dam projects can still be probed.

Author Contributions

Conceptualization, S.P.; Investigation, H.Z.; Methodology, S.P.; Supervision, T.L.; Validation, S.P.; Writing—original draft, S.P. and H.Z.; Writing—review & editing, S.P., G.S., Z.C. and L.Y. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The study did not report any data.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Tables

Figure 1. Sketch map of the BP neural network.

Figure 2. Flowchart of step-by-step inversion method.

Figure 3. Outline and discrete grids of the benchmark model.

Figure 4. Vertical displacement values of monitoring point.

Figure 5. Construction layered model of a high rockfill dam.

Figure 6. Monitoring arrangement of core wall.

Figure 7. Monitoring compression values of the bottom of the core wall.

Figure 8. Comparison of compression values of the bottom of the core wall.

Figure 9. The trend diagram of K and the third principal stress with construction periods.

Figure 10. Comparison of compression values of the bottom of the core wall using variable parameter constitutive model.

Table 1

Material parameters.

Layer	Density(kg/m3)	c(kPa)	${\mathit{\phi }}_{0 }(°)$	${\mathit{R}}_{\mathit{f}}$	$\mathit{K}$	$\mathit{n}$	${\mathit{K}}_{\mathit{b}}$	$\mathit{m}$	${\mathit{K}}_{\mathit{u}\mathit{r}}$	$\mathbf{\Delta }\mathit{\phi } (°)$	${\mathit{p}}_{\mathit{a}}$(kPa)
1	2000	40	20	0.85	400	0.45	150	0.25	460	0	101
2	2000	40	20	0.85	300	0.45	150	0.25	360	0	101
3	2000	40	20	0.85	350	0.45	150	0.25	360	0	101

Table 2

Combinations of material parameters.

Sequence	$\mathit{K}$	$\mathit{n}$	$\mathit{R}\mathit{f}$	Sequence	$\mathit{K}$	$\mathit{n}$	$\mathit{R}\mathit{f}$
1	360	0.44	0.83	42	360	0.43	0.85
2	370	0.41	0.87	43	370	0.41	0.85
3	380	0.45	0.77	44	380	0.44	0.77
4	390	0.46	0.87	45	390	0.48	0.77
5	400	0.42	0.77	46	400	0.41	0.77
6	410	0.48	0.83	47	410	0.44	0.75
7	420	0.42	0.83	48	420	0.4	0.81
8	430	0.42	0.9	49	430	0.42	0.87
9	360	0.47	0.87	50	360	0.48	0.75
10	370	0.43	0.77	51	370	0.4	0.9
11	380	0.44	0.87	52	380	0.41	0.75
12	390	0.42	0.79	53	390	0.42	0.89
13	400	0.42	0.81	54	400	0.4	0.89
14	410	0.45	0.89	55	410	0.48	0.79
15	420	0.45	0.75	56	420	0.47	0.83
16	430	0.43	0.79	57	430	0.4	0.79
17	360	0.47	0.75	58	360	0.4	0.83
18	370	0.45	0.79	59	370	0.4	0.77
19	380	0.48	0.81	60	380	0.42	0.75
20	390	0.47	0.77	61	390	0.43	0.87
21	400	0.44	0.9	62	400	0.48	0.9
22	410	0.46	0.89	63	410	0.47	0.89
23	420	0.47	0.81	64	420	0.46	0.75
24	430	0.47	0.79	65	430	0.47	0.85
25	360	0.41	0.9	66	360	0.44	0.85
26	370	0.46	0.83	67	370	0.4	0.87
27	380	0.47	0.9	68	380	0.41	0.89
28	390	0.48	0.87	69	390	0.46	0.9
29	400	0.46	0.81	70	400	0.48	0.89
30	410	0.44	0.81	71	410	0.4	0.85
31	420	0.45	0.85	72	420	0.46	0.77
32	430	0.41	0.79	73	430	0.46	0.79
33	360	0.41	0.83	74	360	0.41	0.81
34	370	0.45	0.81	75	370	0.42	0.85
35	380	0.45	0.9	76	380	0.45	0.83
36	390	0.43	0.81	77	390	0.44	0.89
37	400	0.45	0.87	78	400	0.43	0.75
38	410	0.46	0.85	79	410	0.48	0.85
39	420	0.4	0.75	80	420	0.43	0.89
40	430	0.43	0.9	81	430	0.44	0.79
41	360	0.43	0.83

Table 3

Calculated material parameters.

Material Parameters	Conventional Inversion Analysis	Step-by-Step Inversion Analysis
		Construction Period 1	Construction Period 2	Construction Period 3
K	400	396	399	395
n	0.44	0.44	0.44	0.44
Rf	0.83	0.83	0.83	0.83

Table 4

Calculated vertical displacement and errors.

Construction Steps	Vertical Displacement of Monitoring Point (m)			Errors betweenStep-by-Step Inversion Method and Original Data	Errors betweenConventional Inversion Method and Original Data
	Original Data	Step-by-Step Inversion Method	Conventional Inversion Method
1	−0.00277	−0.00274	−0.00277	−1.19%	−0.13%
2	−0.00824	−0.00822	−0.00819	−0.23%	−0.56%
3	−0.01858	−0.0184	−0.0183	−0.94%	−1.47%

Table 5

Material parameters for design.

Sections	Density (t/m3)	C (kPa)	${\mathit{\phi }}_0 (°)$	${\mathit{R}}_{\mathit{f}}$	$\mathit{K}$	$\mathit{n}$	${\mathit{K}}_{\mathit{b}}$	$\mathit{m}$	${\mathit{K}}_{\mathit{u}\mathit{r}}$	$\mathbf{\Delta }\mathit{\phi } (°)$
Overburden	2.145	98	38	0.83	438.263	0.39	269.26	0.39	1360	5.26
Cofferdam	2.145	98	38	0.83	438.263	0.39	269.26	0.39	1360	5.26
Rock-fill body and ballast	2.31	0	50	0.76	1238	0.36	1000	0.32	2200	8.5
Core wall	2.14	98	38	0.83	438.263	0.39	269.26	0.39	1360	5.26
Filter layer	2.25	0	50.6	0.8	1105	0.31	400	0.25	2105	8.4

Table 6

Combinations of material parameters.

Sequence	K	n	Rf	Sequence	K	n	Rf
1	1200	0.4	0.75	42	2000	0.35	0.8
2	800	0.25	0.85	43	400	0.25	0.8
3	1000	0.45	0.6	44	400	0.4	0.6
4	1600	0.5	0.85	45	1400	0.25	0.6
5	1200	0.3	0.6	46	600	0.25	0.6
6	1600	0.25	0.75	47	1400	0.4	0.55
7	2000	0.3	0.75	48	1200	0.2	0.7
8	1800	0.3	0.6	49	1400	0.3	0.85
9	400	0.2	0.85	50	1800	0.25	0.55
10	1600	0.35	0.6	51	600	0.2	0.6
11	600	0.4	0.85	52	1000	0.25	0.55
12	800	0.3	0.65	53	400	0.3	0.55
13	600	0.3	0.7	54	1000	0.2	0.55
14	800	0.45	0.55	55	400	0.25	0.65
15	2000	0.45	0.55	56	1000	0.2	0.75
16	1200	0.35	0.65	57	1400	0.2	0.65
17	1200	0.2	0.55	58	800	0.2	0.75
18	600	0.45	0.65	59	1800	0.2	0.6
19	800	0.25	0.7	60	1600	0.3	0.55
20	800	0.2	0.6	61	1800	0.35	0.85
21	1000	0.4	0.6	62	2000	0.25	0.6
22	1200	0.5	0.55	63	1800	0.2	0.55
23	2000	0.2	0.7	64	600	0.5	0.55
24	1600	0.2	0.65	65	600	0.2	0.8
25	1200	0.25	0.6	66	800	0.4	0.8
26	400	0.5	0.75	67	2000	0.2	0.85
27	1400	0.2	0.6	68	1600	0.25	0.55
28	1000	0.25	0.85	69	800	0.5	0.6
29	1400	0.5	0.7	70	600	0.25	0.55
30	1600	0.4	0.7	71	1600	0.2	0.8
31	1400	0.45	0.8	72	2000	0.5	0.6
32	2000	0.25	0.65	73	1000	0.5	0.65
33	1400	0.25	0.75	74	1800	0.25	0.7
34	400	0.45	0.7	75	1000	0.3	0.8
35	1600	0.45	0.6	76	1800	0.45	0.75
36	1000	0.35	0.7	77	2000	0.4	0.55
37	1200	0.45	0.85	78	800	0.35	0.55
38	1800	0.5	0.8	79	1200	0.25	0.8
39	400	0.2	0.55	80	1400	0.35	0.55
40	400	0.35	0.6	81	1800	0.4	0.65
41	600	0.35	0.75

Table 7

Calculated material parameters.

Construction Steps	K	n	Rf
27	1050.001	0.355	0.754
28	1050.000	0.355	0.751
29	1050.004	0.356	0.751
30	1050.000	0.355	0.752
31	1050.095	0.355	0.750
32	1557.785	0.371	0.750
33	1653.524	0.360	0.750
34	1690.780	0.356	0.750
35	1769.950	0.371	0.750
36	1796.199	0.452	0.750
37	1783.091	0.453	0.750
38	1757.106	0.390	0.750
39	1796.517	0.457	0.750
40	1793.926	0.431	0.750
41	1527.351	0.363	0.750
42	1799.598	0.460	0.750
43	1798.960	0.459	0.750
44	1794.462	0.458	0.750
45	1716.357	0.434	0.750
46	1789.575	0.437	0.750
47	1768.449	0.417	0.750
48	1797.258	0.367	0.750
49	1660.474	0.456	0.751
50	1050.000	0.355	0.898

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