Cao et al 2024a - Revision history

Rimni at 13:27, 18 April 2024

2024-04-18T13:27:33Z

Rimni at 13:27, 18 April 2024

2024-04-18T13:27:01Z

Rimni: /* Acknowledgments */

2024-04-18T13:26:05Z

‎Acknowledgments

Rimni at 13:22, 18 April 2024

2024-04-18T13:22:29Z

Rimni at 13:21, 18 April 2024

2024-04-18T13:21:25Z

Rimni: /* 5. Verification of the case */

2024-04-18T13:19:55Z

‎5. Verification of the case

Rimni at 13:17, 18 April 2024

2024-04-18T13:17:17Z

Rimni at 13:16, 18 April 2024

2024-04-18T13:16:37Z

Rimni at 13:15, 18 April 2024

2024-04-18T13:15:17Z

Rimni at 13:11, 18 April 2024

2024-04-18T13:11:33Z

@@ Line 398: / Line 398: @@
 When the velocity reaches <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m, [[#tab-1|Table 1]] presents both the measured and theoretical deflection values, while [[#img-4|Figure 4]] illustrates the fitting curve.
-<div class="center" style="font-size: 80%;">'''Table 1'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m)</div>
+<div class="center" style="font-size: 85%;">'''Table 1'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m)</div>
 <div id='tab-1'></div>
@@ Line 455: / Line 455: @@
 When the velocity reaches <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m, [[#tab-2|Table 2]] presents both the measured and theoretical deflection values, while [[#img-5|Figure 5]] illustrates the corresponding fitting curve.
-<div class="center" style="font-size: 80%;">'''Table 2'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m) </div>
+<div class="center" style="font-size: 85%;">'''Table 2'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m) </div>
 <div id='tab-2'></div>

@@ Line 398: / Line 398: @@
 When the velocity reaches <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m, [[#tab-1|Table 1]] presents both the measured and theoretical deflection values, while [[#img-4|Figure 4]] illustrates the fitting curve.
-<div class="center" style="font-size: 75%;">'''Table 1'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m)</div>
+<div class="center" style="font-size: 80%;">'''Table 1'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m)</div>
 <div id='tab-1'></div>
@@ Line 455: / Line 455: @@
 When the velocity reaches <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m, [[#tab-2|Table 2]] presents both the measured and theoretical deflection values, while [[#img-5|Figure 5]] illustrates the corresponding fitting curve.
-<div class="center" style="font-size: 75%;">'''Table 2'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m) </div>
+<div class="center" style="font-size: 80%;">'''Table 2'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m) </div>
 <div id='tab-2'></div>

@@ Line 520: / Line 520: @@
 ==Acknowledgments==
-The authors would like to thank The Youth Innovation Team of Shaanxi Universities(Key Technology Innovation Team for Urban Rail Transit Track Bed);Youth fund project at Xi’an Jiaotong Engineering Institute(2023KY-39) and The Scientific Research Program Funded by Education Department of Shaanxi Provincial Government (Program No.23JK0532).Hui-min Cao and Han-jun Zhao contributed equally to this work.
+The authors would like to thank The Youth Innovation Team of Shaanxi Universities (Key Technology Innovation Team for Urban Rail Transit Track Bed); Youth fund project at Xi’an Jiaotong Engineering Institute (2023KY-39) and The Scientific Research Program Funded by Education Department of Shaanxi Provincial Government (Program No.23JK0532). Hui-min Cao and Han-jun Zhao contributed equally to this work.
 ==References==

@@ Line 396: / Line 396: @@
 The model and inversion recognition method presented in this paper were utilized to compute the pavement surface of an airport located in Nanjing, with geometric dimensions of <math display="inline">a=6</math>m, <math display="inline">b=4</math>m, elastic modulus <math display="inline">E=3.5\times 1{0}^{4\, }</math>MPa, Poisson ratio <math display="inline">\mu =0.167</math>, density <math display="inline">\rho =2500\, \frac{{\rm kN}}{{\rm m}^{2}}</math>, thickness <math display="inline">h=15.75</math>mm. Additionally, considered were foundation shear modulus <math display="inline">G=35.17 \frac{{\rm N}}{{\rm cm}^{3}}</math>, rigidity coefficients <math display="inline">k1=1.5706\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math>, <math display="inline">k2=1.5830\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math> as well as loading equipment for aircraft (<math display="inline">{P}_{0} =215000</math>N).
-When the velocity reaches 2.59 m/s and <math display="inline">y=1.1</math>m, [[#tab-1|Table 1]] presents both the measured and theoretical deflection values, while [[#img-4|Figure 4]] illustrates the fitting curve.
+When the velocity reaches <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m, [[#tab-1|Table 1]] presents both the measured and theoretical deflection values, while [[#img-4|Figure 4]] illustrates the fitting curve.
 <div class="center" style="font-size: 75%;">'''Table 1'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.59</math> m/s and <math display="inline">y=1.1</math>m)</div>
@@ Line 453: / Line 453: @@
-When the velocity reaches 2.79 m/s and <math display="inline">y=0.68</math>m, [[#tab-2|Table 2]] presents both the measured and theoretical deflection values, while [[#img-5|Figure 5]] illustrates the corresponding fitting curve.
+When the velocity reaches <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m, [[#tab-2|Table 2]] presents both the measured and theoretical deflection values, while [[#img-5|Figure 5]] illustrates the corresponding fitting curve.
 <div class="center" style="font-size: 75%;">'''Table 2'''.  Inversion results of foundation modulus (velocity = <math display="inline">2.79</math> m/s and <math display="inline">y=0.68</math>m) </div>

@@ Line 496: / Line 496: @@
 |  style="text-align: center;"|0.011
 |-
-|  style="text-align: left;"|K(N/cm3)
+!  style="text-align: left;"|K(N/cm3)
 |  colspan='7'  style="text-align: center;"|58.40
 |}

@@ Line 394: / Line 394: @@
 ==5. Verification of the case==
-The model and inversion recognition method presented in this paper were utilized to compute the pavement surface of an airport located in Nanjing, with geometric dimensions of a=6m, b=4m, elastic modulus <math display="inline">E=3.5\times 1{0}^{4\, }</math>MPa, Poisson ratio <math display="inline">\mu =</math><math>0.167</math>, density <math display="inline">\rho =2500\, \frac{{\rm kN}}{{\rm m}^{2}}</math>, thickness <math display="inline">h=</math><math>15.75</math>mm. Additionally, considered were foundation shear modulus <math display="inline">G=35.17 \frac{{\rm N}}{{\rm cm}^{3}}</math>, rigidity coefficients <math display="inline">k1=1.5706\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math>, <math display="inline">k2=1.5830\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math> as well as loading equipment for aircraft (<math display="inline">{P}_{0} =215000</math>N).
+The model and inversion recognition method presented in this paper were utilized to compute the pavement surface of an airport located in Nanjing, with geometric dimensions of <math display="inline">a=6</math>m, <math display="inline">b=4</math>m, elastic modulus <math display="inline">E=3.5\times 1{0}^{4\, }</math>MPa, Poisson ratio <math display="inline">\mu =0.167</math>, density <math display="inline">\rho =2500\, \frac{{\rm kN}}{{\rm m}^{2}}</math>, thickness <math display="inline">h=15.75</math>mm. Additionally, considered were foundation shear modulus <math display="inline">G=35.17 \frac{{\rm N}}{{\rm cm}^{3}}</math>, rigidity coefficients <math display="inline">k1=1.5706\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math>, <math display="inline">k2=1.5830\times 1{0}^{4}\frac{{\rm kN}}{{\rm m}}</math> as well as loading equipment for aircraft (<math display="inline">{P}_{0} =215000</math>N).
 When the velocity reaches 2.59 m/s and <math display="inline">y=1.1</math>m, [[#tab-1|Table 1]] presents both the measured and theoretical deflection values, while [[#img-4|Figure 4]] illustrates the fitting curve.

@@ Line 373: / Line 373: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math> W\left( x,y,t\right) =\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{W\left( x,y\right) }{M{w}_{mn}\int_{0}^{a}\displaystyle\int_{0}^{b}{W}^{2}\left( x,y\right) dxdy}\displaystyle\int_{0}^{t}{P}_{0}W\left( {v}_{x}\tau ,\eta \right) \mathrm{sin}\,{w}_{mn}\left( t-\tau \right) d\tau  </math>
+| style="text-align: center;" | <math> W\left( x,y,t\right) =\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\frac{W\left( x,y\right) }{M{w}_{mn}\displaystyle\int_{0}^{a}\displaystyle\int_{0}^{b}{W}^{2}\left( x,y\right) dxdy}\int_{0}^{t}{P}_{0}W\left( {v}_{x}\tau ,\eta \right) \mathrm{sin}\,{w}_{mn}\left( t-\tau \right) d\tau  </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(19)

@@ Line 364: / Line 364: @@
 |}
-The coordinate factor after the aforementioned transformation not only satisfies the condition of zero boundary deflection but also fulfills equations (12) and (13). In this scenario, the actual system must adhere to boundary conditions (3). The substitution of <math display="inline">{B}_{n}</math>, <math display="inline">{C}_{m}</math>, <math display="inline">{D}_{m}</math>, <math display="inline">{E}_{n}</math>, <math display="inline">{F}_{n}</math>, <math display="inline">{G}_{m}</math> and <math display="inline">{H}_{m}</math> into Eq. (3) yields eight equations, the solutions of which provide the corresponding values. Substituting these values into Eq. (17) determines the coordinate factor  <math display="inline">W\left( x,y\right)</math>
+The coordinate factor after the aforementioned transformation not only satisfies the condition of zero boundary deflection but also fulfills equations (12) and (13). In this scenario, the actual system must adhere to boundary conditions (3). The substitution of <math display="inline">{B}_{n}</math>, <math display="inline">{C}_{m}</math>, <math display="inline">{D}_{m}</math>, <math display="inline">{E}_{n}</math>, <math display="inline">{F}_{n}</math>, <math display="inline">{G}_{m}</math> and <math display="inline">{H}_{m}</math> into Eq. (3) yields eight equations, the solutions of which provide the corresponding values. Substituting these values into Eq. (17) determines the coordinate factor  <math display="inline">W\left( x,y\right)</math>.
 By substituting the coordinate factor <math display="inline">W\left( x,y\right)</math>  into Eq. (10), one can derive the expression for  <math display="inline">W\left( x,y,t\right)</math>  and obtain an analytical formula for the deflection of the pavement panel

@@ Line 201: / Line 201: @@
 |}
-where, <math display="inline">{w}_{mn}</math> represents the natural frequency of undamped free vibration for a thin plate. The first term in Eq.<math display="inline">  </math>  (8) denotes the thin plate's free vibration, while the second term represents its forced vibration.
+where, <math display="inline">{w}_{mn}</math> represents the natural frequency of undamped free vibration for a thin plate. The first term in Eq. (8) denotes the thin plate's free vibration, while the second term represents its forced vibration.
-Among them, <math display="inline"> c1 </math>  and <math display="inline"> c2 </math> are constants that depend on the initial conditions of motion, while <math display="inline">B(t)</math>  is dependent on the load characteristics. In this study, the load is considered as a concentrated load <math display="inline"> P_0 </math>, neglecting its mass. Consequently, <math display="inline">B(t)</math> can be expressed as follows:
+Among them, <math display="inline"> c_1 </math>  and <math display="inline"> c_2 </math> are constants that depend on the initial conditions of motion, while <math display="inline">B(t)</math>  is dependent on the load characteristics. In this study, the load is considered as a concentrated load <math display="inline"> P_0 </math>, neglecting its mass. Consequently, <math display="inline">B(t)</math> can be expressed as follows:
 {| class="formulaSCP" style="width: 100%; text-align: left;"