Souza et al 2021b - Revision history

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 [10] Soltani M., Sistani A. Elastic stability of columns with variable flexural rigidity under arbitrary axial load using the finite difference method. Journals Numer. Method Civ. Eng., 1(4):23–31, 2017. [Online]. Available: [http://nmce.kntu.ac.ir/article-1-91-en.pdf%0Ahttp://nmce.kntu.ac.ir/browse.php?a_code=A-10-50-3&sid=1&slc_lang=en. http://nmce.kntu.ac.ir/article-1-91-en.pdf%0Ahttp://nmce.kntu.ac.ir/browse.php?a_code=A-10-50-3&sid=1&slc_lang=en.]
-[11] O’Rourke M., Zebrownski, T. Buckling load for nonuniform columns. Comput. Struct., 7(4):717–720, 1977.
+[11] O’Rourke M., Zebrownski T. Buckling load for nonuniform columns. Comput. Struct., 7(4):717–720, 1977.
 [12]  Krishnan A.,  George G., Malathi P. Use of finite difference method in the study of stepped beams. Int. J. Mech. Eng. Educ., 26(1):11-24, 1998.

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 [10] Soltani M., Sistani A. Elastic stability of columns with variable flexural rigidity under arbitrary axial load using the finite difference method. Journals Numer. Method Civ. Eng., 1(4):23–31, 2017. [Online]. Available: [http://nmce.kntu.ac.ir/article-1-91-en.pdf%0Ahttp://nmce.kntu.ac.ir/browse.php?a_code=A-10-50-3&sid=1&slc_lang=en. http://nmce.kntu.ac.ir/article-1-91-en.pdf%0Ahttp://nmce.kntu.ac.ir/browse.php?a_code=A-10-50-3&sid=1&slc_lang=en.]
-[11] O’Rourke M. Buckling load for nonuniform columns. 7(4):717–720, 1977.
+[11] O’Rourke M., Zebrownski, T. Buckling load for nonuniform columns. Comput. Struct., 7(4):717–720, 1977.
 [12]  Krishnan A.,  George G., Malathi P. Use of finite difference method in the study of stepped beams. Int. J. Mech. Eng. Educ., 26(1):11-24, 1998.

@@ Line 536: / Line 536: @@
 {| style="margin:auto;width: 100%;"
 |-
-| <math>\phi ={\phi }_{1}+{U}_{Ri}\left( {\phi }_{1}\right) .</math>
+| <math>\phi ={\phi }_{1}+{U}_{Ri}( {\phi }_{1}) .</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref26791905'></span>(37)

@@ Line 422: / Line 422: @@
 {| style="margin:auto;width: 100%;"
 |-
-| <math>\epsilon \, \left( {\phi }_{1}\right) =\Phi -{\phi }_{1}.</math>
+| <math>\epsilon \, ( {\phi }_{1}) =\Phi -{\phi }_{1}.</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='eq28'></span><span id='_Ref26791192'></span>(28)

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 A reliable numerical solution should have a null error or very close to zero. In order to reduce the numerical error caused by discretization problems, the Richardson Error Estimator and Convergent Error Estimator were used to verify the convergence for FDM approaches. To perform the convergence methods analysis, one necessary to define 3 different meshes: coarse <math display="inline">({\phi }_{3})</math>, medium <math display="inline">({\phi }_{2})</math> and fine <math display="inline">{(\phi }_{1})</math>, with grades <math display="inline">\Delta {h}_{3}</math>, <math display="inline">\Delta {h}_{2}</math> e <math display="inline">\Delta {h}_{1}</math>, respectively.
-According to Marchi and Silva [4,5] the error estimation could be performed following the steps:
+According to Marchi and Silva [4,5] the error estimation could be performed following the steps.
 ===4.1 Richardson error estimator===

← Older revision		Revision as of 15:10, 10 November 2021
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−	In the stepped columns illustrated in <span id='cite-_Ref45029125'></span>[[#_Ref45029125\|~~Figure~~ 1]]a and <span id='cite-_Ref45029125'></span>[[#_Ref45029125\|~~Figure~~ 1]]b, the bending rigidity <math display="inline">(EI)</math> changes suddenly; then, it is necessary to analyze each section separately. The governing differential equation in a clamped-free stepped column (<span id='cite-_Ref45029125'></span>[[#_Ref45029125\|Figure 1]]a), for section 1 of length <math display="inline">{L}_{1}</math>, can be written as	+	In the stepped columns illustrated in <span id='cite-_Ref45029125'></span>[[#_Ref45029125\|Figures 1]]a and <span id='cite-_Ref45029125'></span>[[#_Ref45029125\|1]]b, the bending rigidity <math display="inline">(EI)</math> changes suddenly; then, it is necessary to analyze each section separately. The governing differential equation in a clamped-free stepped column (<span id='cite-_Ref45029125'></span>[[#_Ref45029125\|Figure 1]]a), for section 1 of length <math display="inline">{L}_{1}</math>, can be written as

	{\| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"		{\| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"

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 Finally, the use of error estimators proved to be adequate to verify the reliability of the discretization errors for the proposed problem, in addition to being possible to define the regions with asymptotic or unstable behavior of the numerical solutions. Moreover, applied extrapolations technics in FDM can be possible reaching minor errors than the model by FEM using commercial software.
-==8. Acknowledgements==
+==Acknowledgements==
-This study was financed in part by the Coordination for the Improvement of Higher Education Personnel – Brazil (CAPES), Brazilian Council for Scientific and Technological Development (CNPq) and Research Support Foundation of Federal District (FAPDF). The authors are grateful to Group of Experimental and Computational Mechanics (UnB-FGA/GMEC) for provide computational resources and become possible the work development. The second author acknowledges Laboratory of Vibration and Dynamic Metrology and Group of Dynamics of Systems (UnB-FT/EnM/GDS) for its support to make possible the present experiments<span style="text-align: center; font-size: 75%;">.</span>
+This study was financed in part by the Coordination for the Improvement of Higher Education Personnel – Brazil (CAPES), Brazilian Council for Scientific and Technological Development (CNPq) and Research Support Foundation of Federal District (FAPDF). The authors are grateful to Group of Experimental and Computational Mechanics (UnB-FGA/GMEC) for provide computational resources and become possible the work development. The second author acknowledges Laboratory of Vibration and Dynamic Metrology and Group of Dynamics of Systems (UnB-FT/EnM/GDS) for its support to make possible the present experiments.
 ==9. References==

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 {| style="margin:auto;width: 100%;"
 |-
-| <math>{U}_{Ri}\left( {\phi }_{1}\right) =\, sg\left( {\phi }_{1}-{\phi }_{2}\right) \mathrm{max}\,\left\{ \left| {U}_{Ri}\left( {\phi }_{1},PL\right) \right| ;{\vert U}_{Ri}\left( {\phi }_{1},PU\right) \vert \right\} ,</math>
+| <math>{U}_{Ri}\left( {\phi }_{1}\right) =\, sg\left( {\phi }_{1}-{\phi }_{2}\right) \mathrm{max}\,\left\{ \left| {U}_{Ri}\left( {\phi }_{1},PL\right) \right| ; \left|{U}_{Ri}\left( {\phi }_{1},PU\right) \right| \right\} ,</math>
 |}
 |  style="width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref26790570'></span>(33)