QIAO* et al 2021a - Revision history

Rimni at 08:40, 28 September 2021

2021-09-28T08:40:19Z

Rimni: /* References */

2021-09-28T08:40:05Z

‎References

Rimni at 08:39, 28 September 2021

2021-09-28T08:39:31Z

Rimni: /* 5. Sensitivity analysis for beam and stringer */

2021-09-28T08:27:20Z

‎5. Sensitivity analysis for beam and stringer

Rimni: /* 5. Sensitivity analysis for beam and stringer */

2021-09-28T08:26:25Z

‎5. Sensitivity analysis for beam and stringer

Rimni at 08:18, 28 September 2021

2021-09-28T08:18:36Z

Rimni: /* 5 Sensitivity analysis for beam and stringer */

2021-09-28T08:15:28Z

‎5 Sensitivity analysis for beam and stringer

Rimni at 08:08, 28 September 2021

2021-09-28T08:08:10Z

Rimni at 08:07, 28 September 2021

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Rimni at 08:07, 28 September 2021

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@@ Line 532: / Line 532: @@
 ==References==
 <div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
 [1] Guan D., Zhong K. Aeroelastic tailoring applied to composite wing. Acta Aeronautica et Astronautica Sinica, 10(5):221-226, 1989.

← Older revision		Revision as of 08:40, 28 September 2021
Line 531:		Line 531:

	==References==		==References==
		+	<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
		+

	[1] Guan D., Zhong K. Aeroelastic tailoring applied to composite wing. Acta Aeronautica et Astronautica Sinica, 10(5):221-226, 1989.		[1] Guan D., Zhong K. Aeroelastic tailoring applied to composite wing. Acta Aeronautica et Astronautica Sinica, 10(5):221-226, 1989.

@@ Line 516: / Line 516: @@
 We concluded that it has small impact by changing the beam and stringer thickness for position of stiffness center on full composite wing structure. However, it is large to the changing of the positions of front beam and rear beam. Comparing 4 different models with only different stringer spacing, it is found that the positions of wing cross sections stiffness center are all impacted largely for the 4 models.
-=6 Conclusion=
+==6. Conclusion==
 This paper studies the influence of the size and layout of the stringers and beams on the stiffness center of the wing cross section. The sensitivity formulas for stiffness center of wing cross sections were derived and expressed in terms of variations with dimension and position of beam and stringer. The important conclusions obtained through the research are as follows:
@@ Line 530: / Line 530: @@
 This work is funded by the Scientific Research Foundation for Doctors, the Natural Science Basic Research Plan in Shaanxi Province of China (No: 2019JQ-912) and the Natural Science Foundation of Xi′ an Aeronautical University (No: 2018KY1226).
-==Reference==
+==References==
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-[11]  Pandey P. C., Bakshi P. Analytical response sensitivity using hybrid finite elements [J]. Computers and Structures, 70: 525-534, 1999.
+[11]  Pandey P.C., Bakshi P. Analytical response sensitivity using hybrid finite elements. Computers and Structures, 70:525-534, 1999.
-[12]  Tafreshi A. Shape design sensitivity analysis with respect to the positioning of features in composite structures using the boundary element method [J]. Engineering Analysis with Boundary Elements, 30: 1-13, 2006.
+[12]  Tafreshi A. Shape design sensitivity analysis with respect to the positioning of features in composite structures using the boundary element method. Engineering Analysis with Boundary Elements, 30:1-13, 2006.
-[13]  Choi J. H. Shape design sensitivity analysis for stability of elastic continuum structures [J]. International Journal of Solids and Structures, 44: 1593-1607, 2007.
+[13]  Choi J.H. Shape design sensitivity analysis for stability of elastic continuum structures. International Journal of Solids and Structures, 44:1593-1607, 2007.
-[14]  Choi J. H. Shape design sensitivity analysis and optimization of general plane arch structures [J]. Finite Elements in Analysis and Design, 39: 119-136, 2002.
+[14]  Choi J.H. Shape design sensitivity analysis and optimization of general plane arch structures. Finite Elements in Analysis and Design, 39: 119-136, 2002.
-[15]  Antonio C. C., Hoffbauer L. N. Uncertainty analysis based on sensitivity applied to angle-ply composite structures [J]. Reliability Engineering and System Safety, 92: 1353-1362, 2007.
+[15]  Antonio C.C., Hoffbauer L.N. Uncertainty analysis based on sensitivity applied to angle-ply composite structures. Reliability Engineering and System Safety, 92:1353-1362, 2007.
-[16]  Hien T. D., Kleiber M. Stochastic structural design sensitivity of static response [J]. Computers and Structures, 38(5/6): 659-667, 1991.
+[16]  Hien T.D., Kleiber M. Stochastic structural design sensitivity of static response. Computers and Structures, 38(5/6):659-667, 1991.
-[17]  Sergeyey O., Mroz Z. Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints [J]. Computers and Structures, 76: 167-185, 2000.
+[17]  Sergeyey O., Mroz Z. Sensitivity analysis and optimal design of 3D frame structures for stress and frequency constraints. Computers and Structures, 76:167-185, 2000.
-[18]  Wang Z. D., Chen S. H. Sensitivity analysis of eigenmodes and dynamic responses for intelligent structures [J]. Finite Elements in Analysis and Design, 33: 71-81, 1999.
+[18]  Wang Z.D., Chen S.H. Sensitivity analysis of eigenmodes and dynamic responses for intelligent structures. Finite Elements in Analysis and Design, 33:71-81, 1999.
-[19]  Zhao Y, Wang F. Y. Sensitivity analysis for structures subjected to stationary random excitations [J]. Procedia Engineering, 14: 2475-2480, 2011.
+[19]  Zhao Y., Wang F.Y. Sensitivity analysis for structures subjected to stationary random excitations. Procedia Engineering, 14:2475-2480, 2011.
-[20]  Ohsaki M. Design sensitivity analysis and optimization for nonlinear buckling of finite-dimensional elastic conservative structures [J]. Comput. Methods Appl. Mech. Engrg., 194: 3331-3358, 2005.
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-[21]  Petrov E. P. Analysis of sensitivity and robustness of forced response for nonlinear dynamic structures [J]. Mechanical Systems and Signal Processing, 23: 68-86, 2009.
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-[22]  Etienne S., Pelletier D. A general approach to sensitivity analysis of fluid-structure interactions [J]. Journal of Fluids and Structures, 21: 169-186, 2005.
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-[23]  Kim M. G, Jang H. L., Cho S. Adjoint design sensitivity analysis of reduced atomic systems using generalized langevin equation for lattice structures [J]. Journal of computational physics, 240: 1-19, 2013.
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-[24]  Pajot J. M., Maute K. Analytical sensitivity analysis of geometrically nonlinear structures based on the co-rotational finite element method [J]. Finite Elements in Analysis and Design, 42: 900-913, 2006.
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-[25]  Mang H. A, Hofinger G., Jia X. On the interdependency of primary and initial secondary equilibrium paths in sensitivity analysis of elastic structures [J]. Comput. Methods Appl. Mech. Engrg., 200: 1558-1567, 2011.<span id="OLE_LINK1"></span>
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 ----
 Corresponding author.
 ''E-mail address:'' [mailto:qiaoshengjun123@163.com qiaoshengjun123@163.com]

@@ Line 474: / Line 474: @@
 |  0.0352
 |}
 When researching the impact of the front beam and rear beam layouts for stiffness center, it is necessary to keep the positions of other structure while only change the position of front beam or rear beam. According to the Eq. (10), the results of mean sensitivity of beam positions of stiffness center are obtained in [[#tab-7|Table 7]].
@@ Line 494: / Line 495: @@
 <div class="center" style="font-size: 75%;">'''Table 8'''. Mean sensitivity for layout of stringers on different models</div>
-<div id='tab-1'></div>
+<div id='tab-8'></div>
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
@@ Line 511: / Line 512: @@
 |  0.4174
 |}
 We concluded that it has small impact by changing the beam and stringer thickness for position of stiffness center on full composite wing structure. However, it is large to the changing of the positions of front beam and rear beam. Comparing 4 different models with only different stringer spacing, it is found that the positions of wing cross sections stiffness center are all impacted largely for the 4 models.

@@ Line 424: / Line 424: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align} \xi_{h_i}=\frac{\partial Y}{\partial h_i} & =\frac{1}{J_x}\int_{s}^{} \frac{\partial S_x}{\partial h_i}\rho ds-\frac{\partial \Omega}{\partial h_i}\frac{\oint \frac{S_x}{Gt}}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\oint \left(\frac{\partial S_x}{\partial h_i}\frac{1}{Gt}-\frac{S_x}{Gt^2}\frac{\partial t}{\partial h_i}\right)ds\cdot\oint \frac{ds}{Gt} \\ & = -\frac{\Omega}{J_x\left(\oint \frac{ds}{Gt}\right)^2}\cdot\oint \frac{S_x}{Gt}ds\cdot\oint \frac{1}{Gt^2}\frac{\partial t}{\partial h_i}ds-\frac{1}{J_x^2}\Bigl(\int_{s}^{} S_x \rho ds-\Omega\frac{\oint \frac{S_x}{Gt}ds}{\oint \frac{ds}{Gt}}\Bigr)\cdot\frac{\partial J_x}{\partial h_i} \\ \end{align}</math>
+| style="text-align: center;" | <math>\begin{align} \xi_{h_i}=\frac{\partial Y}{\partial h_i} & =\frac{1}{J_x}\int_{s}^{} \frac{\partial S_x}{\partial h_i}\rho ds-\frac{\partial \Omega}{\partial h_i}\frac{\oint \frac{S_x}{Gt}}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\oint \left(\frac{\partial S_x}{\partial h_i}\frac{1}{Gt}-\frac{S_x}{Gt^2}\frac{\partial t}{\partial h_i}\right)ds\cdot\oint \frac{ds}{Gt} \\ & = -\frac{\Omega}{J_x\left(\oint \frac{ds}{Gt}\right)^2}\cdot\oint \frac{S_x}{Gt}ds\cdot\oint \frac{1}{Gt^2}\frac{\partial t}{\partial h_i}ds-\frac{1}{J_x^2}\left(\int_{s}^{} S_x \rho ds-\Omega\frac{\oint \frac{S_x}{Gt}ds}{\oint \frac{ds}{Gt}}\right)\cdot\frac{\partial J_x}{\partial h_i} \\ \end{align}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(9)
@@ Line 435: / Line 435: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{array}{lcr} \frac{\partial J_y}{\partial h_i}=\oint z^2\varphi_i\frac{\partial t_i}{\partial h_i}ds+\sum_{i}\frac{\partial A_i}{\partial h_i}z_i^2 \\ \frac{\partial S_y}{\partial h_i}=\oint z\varphi_i\frac{\partial t_i}{\partial h_i}ds+\sum_{i}\varphi_i\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>
+| style="text-align: center;" | <math>\begin{array}{lcr} \displaystyle\frac{\partial J_y}{\partial h_i}=\oint z^2\varphi_i\displaystyle\frac{\partial t_i}{\partial h_i}ds+\sum_{i}\displaystyle\frac{\partial A_i}{\partial h_i}z_i^2 \\ \displaystyle\frac{\partial S_y}{\partial h_i}=\oint z\varphi_i\displaystyle\frac{\partial t_i}{\partial h_i}ds+\sum_{i}\varphi_i\displaystyle\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>
 |}
 |}
@@ Line 457: / Line 457: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math display="block">\begin{array}{lcr} \frac{\partial J_y}{\partial I_i}=\oint z^2\varphi_it_ids+\sum\frac{\partial A_i}{\partial h_i}z_i^2 \\ \frac{\partial S_y}{\partial I_i}=\oint \varphi_izt_ids+\sum\varphi_i\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>
+| style="text-align: center;" | <math display="block">\begin{array}{lcr} \displaystyle\frac{\partial J_y}{\partial I_i}=\oint z^2\varphi_it_ids+\sum\displaystyle\frac{\partial A_i}{\partial h_i}z_i^2 \\ \displaystyle\frac{\partial S_y}{\partial I_i}=\oint \varphi_izt_ids+\sum\varphi_i\displaystyle\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>
 |}
 |}
-<math display="block">\begin{array}{lcr} \frac{\partial J_y}{\partial I_i}=\oint z^2\varphi_it_ids+\sum\frac{\partial A_i}{\partial h_i}z_i^2 \\ \frac{\partial S_y}{\partial I_i}=\oint \varphi_izt_ids+\sum\varphi_i\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>Mean value sensitivity reflects the influencing degree of the average value of variable for structure. According to the formulation (9), when enlarging or reducing the thicknesses of beam and stringer at the same time, the results of mean sensitivity for the stiffness center of wing cross sections for the thickness of beam and stringer are obtained in Table 6.
+<div class="center" style="font-size: 75%;">'''Table 6'''. Mean value sensitivity for thickness of beams and stringers</div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div id='tab-6'></div>
-'''Table 6 Mean value sensitivity for thickness of beams and stringers'''</div>
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
+|-style="text-align:center"
-{| style="width: 100%;border-collapse: collapse;"
+! Variables !! Beams !! Stringers
-|-
+|-style="text-align:center"
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Variables
+|  Mean value sensitivity
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Beams
+|  1.691
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Stringers
+|  0.0352
-|-
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|Mean value sensitivity
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|1.691
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.0352
 |}
-When researching the impact of the front beam and rear beam layouts for stiffness center, it is necessary to keep the positions of other structure while only change the position of front beam or rear beam. According to the Eq (10), the results of mean sensitivity of beam positions of stiffness center are obtained in Table 7.
+When researching the impact of the front beam and rear beam layouts for stiffness center, it is necessary to keep the positions of other structure while only change the position of front beam or rear beam. According to the Eq. (10), the results of mean sensitivity of beam positions of stiffness center are obtained in [[#tab-7|Table 7]].
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="font-size: 75%;">'''Table 7'''. Mean value sensitivity for layout of beams</div>
-'''Table 7 Mean value sensitivity for layout of beams'''</div>
-{| style="width: 100%;border-collapse: collapse;"
+<div id='tab-7'></div>
-|-
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Variables
+|-style="text-align:center"
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Front beam
+! Variables !! Front beam !! Rear beam
-|  style="border-top: 2pt solid black;border-bottom: 1pt solid black;text-align: center;"|Rear beam
+|-style="text-align:center"
-|-
+|  Mean value sensitivity
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|Mean value sensitivity
+|  0.2906
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.2906
+|  0.288
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.288
 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+The distribution of the stringers for wing cross section was shown in [[#img-2|Figure 2]]. In order to research the impact of different stringer spacing for stiffness center, 4 models with different stringer spacing were obtained. Change the position of stringers at the same time when middle group stringer is the design variable of sensitivity, the results of mean value sensitivity for stringer positions about stiffness center for different models are obtained in [[#tab-8|Table 8]].
-'''Table 8 Mean sensitivity for layout of stringers on different models'''</div>
-{| style="width: 100%;border-collapse: collapse;"
+<div class="center" style="font-size: 75%;">'''Table 8'''. Mean sensitivity for layout of stringers on different models</div>
 |-
-|  style="border-top: 2pt solid black;text-align: center;"|Variables
+<div id='tab-1'></div>
-|  colspan='4'  style="border-top: 2pt solid black;text-align: center;"|Middle group stringer
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|-
+|-style="text-align:center"
-|  style="border-bottom: 1pt solid black;text-align: center;"|Models
+! Variables !!  colspan='4'  |Middle group stringer
-|  style="border-bottom: 1pt solid black;text-align: center;"|1
+|-style="text-align:center"
-|  style="border-bottom: 1pt solid black;text-align: center;"|2
+|  Models
-|  style="border-bottom: 1pt solid black;text-align: center;"|3
+|  1
-|  style="border-bottom: 1pt solid black;text-align: center;"|4
+|  2
-|-
+|  3
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|Mean value sensitivity
+|  4
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.4462
+|-style="text-align:center"
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.4413
+|  Mean value sensitivity
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.4328
+|  0.4462
-|  style="border-top: 1pt solid black;border-bottom: 2pt solid black;text-align: center;"|0.4174
+|  0.4413
 |}

@@ Line 415: / Line 415: @@
 |}
-==5 Sensitivity analysis for beam and stringer==
+==5. Sensitivity analysis for beam and stringer==
 Sensitivity is the derivative information, which reflects the impact of design variable on objective function or constraint function. The sensitivity formula for wing cross sections stiffness is derived according to the thickness <math display="inline"> h_i </math> of beam and stringer, which is written as
@@ Line 424: / Line 424: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align} \xi_{h_i}=\frac{\partial Y}{\partial h_i} & =\frac{1}{J_x}\int_{s}^{} \frac{\partial S_x}{\partial h_i}\rho ds-\frac{\partial \Omega}{\partial h_i}\frac{\oint \frac{S_x}{Gt}}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\oint \bigl(\frac{\partial S_x}{\partial h_i}\frac{1}{Gt}-\frac{S_x}{Gt^2}\frac{\partial t}{\partial h_i}\bigr)ds\cdot\oint \frac{ds}{Gt} \\ & = -\frac{\Omega}{J_x\bigl(\oint \frac{ds}{Gt}\bigr)^2}\cdot\oint \frac{S_x}{Gt}ds\cdot\oint \frac{1}{Gt^2}\frac{\partial t}{\partial h_i}ds-\frac{1}{J_x^2}\Bigl(\int_{s}^{} S_x \rho ds-\Omega\frac{\oint \frac{S_x}{Gt}ds}{\oint \frac{ds}{Gt}}\Bigr)\cdot\frac{\partial J_x}{\partial h_i} \\ \end{align}</math>
+| style="text-align: center;" | <math>\begin{align} \xi_{h_i}=\frac{\partial Y}{\partial h_i} & =\frac{1}{J_x}\int_{s}^{} \frac{\partial S_x}{\partial h_i}\rho ds-\frac{\partial \Omega}{\partial h_i}\frac{\oint \frac{S_x}{Gt}}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\oint \left(\frac{\partial S_x}{\partial h_i}\frac{1}{Gt}-\frac{S_x}{Gt^2}\frac{\partial t}{\partial h_i}\right)ds\cdot\oint \frac{ds}{Gt} \\ & = -\frac{\Omega}{J_x\left(\oint \frac{ds}{Gt}\right)^2}\cdot\oint \frac{S_x}{Gt}ds\cdot\oint \frac{1}{Gt^2}\frac{\partial t}{\partial h_i}ds-\frac{1}{J_x^2}\Bigl(\int_{s}^{} S_x \rho ds-\Omega\frac{\oint \frac{S_x}{Gt}ds}{\oint \frac{ds}{Gt}}\Bigr)\cdot\frac{\partial J_x}{\partial h_i} \\ \end{align}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(9)
@@ Line 439: / Line 439: @@
 |}
-As the distance <math display="inline"> l_i </math> [[Image:Draft_qiao_302098793-image34.png|12px]] between wing beam or middle group stringer and inertia axis of wing cross sections, the sensitivity formula for wing cross sections stiffness is derived as
+As the distance <math display="inline"> l_i </math>  between wing beam or middle group stringer and inertia axis of wing cross sections, the sensitivity formula for wing cross sections stiffness is derived as
 {| class="formulaSCP" style="width: 100%; text-align: left;"
@@ Line 446: / Line 446: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\begin{align} \varepsilon_{l,i}=\frac{\partial Y}{\partial l_i} & = \frac{1}{J_y}[\int\limits_{s}^{} \Bigl(\frac{\partial S_y}{\partial l_i}\rho+S_y\frac{\partial \rho}{\partial I_i}\Bigr)ds-\frac{\partial \Omega}{\partial l_i}\frac{\oint \frac{S_y}{Gt}ds}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\cdot\oint \frac{1}{Gt}\frac{\partial S_y}{\partial l_i}ds] \\ & -\frac{1}{J_y^2}\Bigl(\int\limits_{s}^{} S_y\rho ds-\Omega\frac{\oint \frac{S_y}{Gt}ds}{\oint \frac{ds}{Gt}}\Bigr)\cdot\frac{\partial J_y}{\partial l_i} \\ \end{align}</math>
+| style="text-align: center;" | <math>\begin{align} \varepsilon_{l,i}=\frac{\partial Y}{\partial l_i} & = \frac{1}{J_y}[\int\limits_{s}^{} \left(\frac{\partial S_y}{\partial l_i}\rho+S_y\frac{\partial \rho}{\partial I_i}\right)ds-\frac{\partial \Omega}{\partial l_i}\frac{\oint \frac{S_y}{Gt}ds}{\oint \frac{ds}{Gt}}-\frac{\Omega}{\oint \frac{ds}{Gt}}\cdot\oint \frac{1}{Gt}\frac{\partial S_y}{\partial l_i}ds] \\ & -\frac{1}{J_y^2}\left(\int\limits_{s}^{} S_y\rho ds-\Omega\frac{\oint \frac{S_y}{Gt}ds}{\oint \frac{ds}{Gt}}\right)\cdot\frac{\partial J_y}{\partial l_i} \\ \end{align}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" |(10)
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 with
 <math display="block">\begin{array}{lcr} \frac{\partial J_y}{\partial I_i}=\oint z^2\varphi_it_ids+\sum\frac{\partial A_i}{\partial h_i}z_i^2 \\ \frac{\partial S_y}{\partial I_i}=\oint \varphi_izt_ids+\sum\varphi_i\frac{\partial A_i}{\partial h_i}z_i^2 \end{array}</math>Mean value sensitivity reflects the influencing degree of the average value of variable for structure. According to the formulation (9), when enlarging or reducing the thicknesses of beam and stringer at the same time, the results of mean sensitivity for the stiffness center of wing cross sections for the thickness of beam and stringer are obtained in Table 6.

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