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← Older revision		Revision as of 08:35, 19 October 2018
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−	Published in ''Comput. Methods Appl. Mech. Engrg.'' Vol. 195, pp. 5297–5315, 2006<br />	+	Published in ''Comput. Methods Appl. Mech. Engrg.'', Vol. 195, pp. 5297–5315, 2006<br />
	doi:10.1016/j.cma.2005.08.021		doi:10.1016/j.cma.2005.08.021

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 ==Abstract==
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 In this paper a finite element for the non-linear analysis of two dimensional beams and axisymmetric shells is presented. The element uses classical thin shell assumptions (no transverse shear strains). The main feature of the element is that it has no rotational degrees of freedom. Curvatures are computed using geometrical information from the patch of three elements formed by the main element and the two neighbour (adjacent) elements. Special attention is devoted to non-smooth geometries and branching shells. An elastic-plastic material law is considered. Large strains are treated using a logarithmic strain measure and a through-the-thickness numerical integration of the constitutive equations. Several examples are presented including linear problems to study convergence properties, and non-linear problems for both elastic and elastic-plastic materials and large strains.
-'''keywords''' finite elements, beams, axisymmetric shells, rotation-free element, large strains
+'''Keywords''': finite elements, beams, axisymmetric shells, rotation-free element, large strains
 ==1 Introduction==

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 ==2 Governing equations of beams and thin two-dimensional shells==
-In this section a summary of the most relevant equations governing the kinematic behavior of thin shells is presented. More details can be found in the wide literature dedicated to the subject<span id='citeF-1'></span>[[#cite-1|[1]]]. In order to introduce the problem, the kinematics of two dimensional beams are considered. These equations are latter extended to axisymmetric shells.
+In this section a summary of the most relevant equations governing the kinematic behavior of thin shells is presented. More details can be found in the wide literature dedicated to the subject <span id='citeF-1'></span>[[#cite-1|[1]]]. In order to introduce the problem, the kinematics of two dimensional beams are considered. These equations are latter extended to axisymmetric shells.
 <div id='img-1'></div>
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 ==6 Numerical examples==
-In this section several examples are presented to assess the element described. This will be denoted as B2 (2-node Beam). Initially three linear examples are considered, then three geometrically non-linear (elastic) examples are shown, and finally a couple of examples including large strain plasticity (in smooth surfaces) are studied. All units are in the International System. The present element performance is compared with two transverse shear deformable two-node elements with 3 nodal degrees of freedom (two displacements and one rotation) and one integration point. Namely element B21 present in program ABAQUS<span id='citeF-20'></span>[[#cite-20|[20]]], denoted here by Abq, and the element of Ref <span id='citeF-17'></span>[[#cite-17|[17]]] denoted here by Sh, that should give identical results at least in the small strain elastic range. Note in the comparisons that using the shear deformable elements implies 50% more DOFs than when using element B2 for the same mesh.
+In this section several examples are presented to assess the element described. This will be denoted as B2 (2-node Beam). Initially three linear examples are considered, then three geometrically non-linear (elastic) examples are shown, and finally a couple of examples including large strain plasticity (in smooth surfaces) are studied. All units are in the International System. The present element performance is compared with two transverse shear deformable two-node elements with 3 nodal degrees of freedom (two displacements and one rotation) and one integration point. Namely element B21 present in program ABAQUS <span id='citeF-20'></span>[[#cite-20|[20]]], denoted here by Abq, and the element of Ref <span id='citeF-17'></span>[[#cite-17|[17]]] denoted here by Sh, that should give identical results at least in the small strain elastic range. Note in the comparisons that using the shear deformable elements implies 50% more DOFs than when using element B2 for the same mesh.
 ===6.1 Linear examples===
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 ====6.2.3 Elastic large deflection response of a Z-shaped cantilever====
-This third example was extracted from a set of benchmarks for geometric non-linear behavior of structures <span id='citeF-19'></span>[[#cite-19|[19]]]. This benchmark is intended to assess membrane and bending actions, tension stiffening and change in sign of bending moments under large rotations and large displacements. Figure [[#img-10|10]].a shows the cross section properties, the material parameters and the initial geometry of a Z-shaped cantilever under a conservative load at the free end. Figure [[#img-10|10]].b shows the deformed configurations for three different values of the point load. These load values are defined as target values in Ref. <span id='citeF-19'></span>[[#cite-19|[19]]] to assess the element formulation.
+This third example was extracted from a set of benchmarks for geometric non-linear behavior of structures  <span id='citeF-19'></span>[[#cite-19|[19]]]. This benchmark is intended to assess membrane and bending actions, tension stiffening and change in sign of bending moments under large rotations and large displacements. Figure [[#img-10|10]].a shows the cross section properties, the material parameters and the initial geometry of a Z-shaped cantilever under a conservative load at the free end. Figure [[#img-10|10]].b shows the deformed configurations for three different values of the point load. These load values are defined as target values in Ref. <span id='citeF-19'></span>[[#cite-19|[19]]] to assess the element formulation.
 In Figure [[#img-10|10]].c and d the results for meshes with 3, 6 and 12 elements per zone are shown along all the process. In Figure [[#img-10|10]].c the load versus the tip displacement has been plotted while the bending moment at A versus the load is shown in Figure [[#img-10|10]].d. The bending moment plotted is obtained as the average of the bending moments computed at the integration points adjacent to point A. The results obtained with the B2 element show a very good agreement with the three expected values for the mesh with 6 elements per zone.
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-To assess convergence as the mesh is refined in the non-linear range Figures [[#img-10|10]].e and f plots for <math display="inline">P=104.5</math> and <math display="inline">P=1263</math> the normalized tip displacement and normalized moment at point A as a function of the number of DOFs in the mesh. The target values from Ref <span id='citeF-19'></span>[[#cite-19|[19]]] where used for normalization. It can be seen that the element has good convergence properties for both displacements and moments, while the element B21 of program ABAQUS <span id='citeF-20'></span>[[#cite-20|[20]]] shows excellent convergence properties for the displacement but a very slow convergence for the moments.
+To assess convergence as the mesh is refined in the non-linear range Figures [[#img-10|10]].e and f plots for <math display="inline">P=104.5</math> and <math display="inline">P=1263</math> the normalized tip displacement and normalized moment at point A as a function of the number of DOFs in the mesh. The target values from Ref. <span id='citeF-19'></span>[[#cite-19|[19]]] where used for normalization. It can be seen that the element has good convergence properties for both displacements and moments, while the element B21 of program ABAQUS <span id='citeF-20'></span>[[#cite-20|[20]]] shows excellent convergence properties for the displacement but a very slow convergence for the moments.
 ===6.3 Examples including large elastic-plastic strains===
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 ====6.3.1 Stretching of a circular sheet with an hemispherical punch====
-First a well studied benchmark is considered<span id='citeF-22'></span>[[#cite-22|[22]]]. Figure [[#img-11|11]].a shows the initial configuration of the stretching of circular plate with an hemispherical punch. The material of the blank is a mild steel with <math display="inline">E = 6.9 \times{10}^{10}</math>, <math display="inline">\nu = 0.3</math> and tensile strength <math display="inline">\sigma _{y} = 5.89 \times{10}^{8} (0.0001+e^{p})^{0.216}</math>.
+First a well studied benchmark is considered <span id='citeF-22'></span>[[#cite-22|[22]]]. Figure [[#img-11|11]].a shows the initial configuration of the stretching of circular plate with an hemispherical punch. The material of the blank is a mild steel with <math display="inline">E = 6.9 \times{10}^{10}</math>, <math display="inline">\nu = 0.3</math> and tensile strength <math display="inline">\sigma _{y} = 5.89 \times{10}^{8} (0.0001+e^{p})^{0.216}</math>.
-The tools are assumed to be rigid and a frictionless interface was used. Contact between the tools and the sheet was considered via a penalty formulation. Two different meshes of 28 and 56 elements where used in the simulations with no practical differences in the results. Punch force versus punch travel is plotted in Figure [[#img-11|11]].b. Results obtained with the B2 element have been compared with those obtained using two solid element: a non-conforming triangle<span id='citeF-23'></span>[[#cite-23|[23]]] (TR) and a standard quadrilateral with constant pressure<span id='citeF-24'></span>[[#cite-24|[24]]] (QUAD). The results show a excellent agreement.
+The tools are assumed to be rigid and a frictionless interface was used. Contact between the tools and the sheet was considered via a penalty formulation. Two different meshes of 28 and 56 elements where used in the simulations with no practical differences in the results. Punch force versus punch travel is plotted in Figure [[#img-11|11]].b. Results obtained with the B2 element have been compared with those obtained using two solid element: a non-conforming triangle <span id='citeF-23'></span>[[#cite-23|[23]]] (TR) and a standard quadrilateral with constant pressure <span id='citeF-24'></span>[[#cite-24|[24]]] (QUAD). The results show a excellent agreement.
 Figures [[#img-11|11]].c and [[#img-11|11]].d show the profiles of thickness ratio and equivalent plastic strain along the original radius for different stages of the simulation. These values compare quite well with the results obtained with the solid elements (not plotted for clarity).
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 ====6.3.2 Deep drawing and springback of a flat strip====
-The last example is a benchmark of NUMISHEET'93 <span id='citeF-25'></span>[[#cite-25|[25]]]. The aim of this example is to assess the forming process (deep drawing) of a thin sheet and the effects of the elastic springback after the tools are removed. Figure [[#img-12|12]].a shows the setting of the experiment. The geometrical parameters to be computed and compared with experimental results are shown in Figure [[#img-12|12]].b. Namely the angles <math display="inline">\theta _{1}</math> and <math display="inline">\theta _{2}\,</math> and the radius of curvature <math display="inline">\rho </math> of the lateral wall, obtained through the circle defined by points A, B and C. Due to the friction with the tools the shell is assumed to work in plane strain conditions (direction '''e'''<math display="inline">_{3}</math>) during the forming process. This assumption is kept for the springback step although the restriction of the tools is removed. Two different materials have been used for the simulations: aluminum and a mild steel. A complete definition of these materials can be found in <span id='citeF-25'></span>[[#cite-25|[25]]]. The material is assumed elastic-plastic. An isotropic behavior is used for the elastic component and an orthotropic yield function <span id='citeF-26'></span>[[#cite-26|[26]]] with exponential isotropic hardening is considered for the plastic component. For the simulations a low blank holder force (<math display="inline">2.45</math> kN) has been used.
+The last example is a benchmark of NUMISHEET'93 <span id='citeF-25'></span>[[#cite-25|[25]]]. The aim of this example is to assess the forming process (deep drawing) of a thin sheet and the effects of the elastic springback after the tools are removed. Figure [[#img-12|12]].a shows the setting of the experiment. The geometrical parameters to be computed and compared with experimental results are shown in Figure [[#img-12|12]].b. Namely the angles <math display="inline">\theta _{1}</math> and <math display="inline">\theta _{2}\,</math> and the radius of curvature <math display="inline">\rho </math> of the lateral wall, obtained through the circle defined by points A, B and C. Due to the friction with the tools the shell is assumed to work in plane strain conditions (direction '''e'''<math display="inline">_{3}</math>) during the forming process. This assumption is kept for the springback step although the restriction of the tools is removed. Two different materials have been used for the simulations: aluminum and a mild steel. A complete definition of these materials can be found in <span  id='citeF-25'></span>[[#cite-25|[25]]]. The material is assumed elastic-plastic. An isotropic behavior is used for the elastic component and an orthotropic yield function <span id='citeF-26'></span>[[#cite-26|[26]]] with exponential isotropic hardening is considered for the plastic component. For the simulations a low blank holder force (<math display="inline">2.45</math> kN) has been used.
 The deformed shape in Figure [[#img-12|12]].b corresponds to the simulation with aluminum. Figures [[#img-12|12]].c and d (obtained for aluminum and mild steel, respectively) show the geometrical parameters defined above obtained by different experimental groups (each abscissa corresponds to a different group). The average of the experimental values and those obtained with the B2 element are also shown. The numerical simulation agrees reasonably well between the rather scattered experimental values.

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 ==Acknowledgments==
-The first author is a member of the scientific staff of the Science Research Council of Argentina (CONICET). The support provided by a grant of CONICET is gratefully acknowledged. Financial support from CIMNE and support from Quantech ATZ (Barcelona-Spain) for providing Stampack, are also acknowledged
+The first author is a member of the scientific staff of the Science Research Council of Argentina (CONICET). The support provided by a grant of CONICET is gratefully acknowledged. Financial support from CIMNE and support from Quantech ATZ (Barcelona-Spain) for providing Stampack, are also acknowledged.
 ==References==

@@ Line 1,362: / Line 1,362: @@
 ====6.3.2 Deep drawing and springback of a flat strip====
-The last example is a benchmark of NUMISHEET'93 <span id='citeF-25'></span>[[#cite-25|[25]]]. The aim of this example is to assess the forming process (deep drawing) of a thin sheet and the effects of the elastic springback after the tools are removed. Figure [[#img-12|12]].a shows the setting of the experiment. The geometrical parameters to be computed and compared with experimental results are shown in Figure [[#img-12|12]].b. Namely the angles <math display="inline">\theta _{1}</math> and <math display="inline">\theta _{2}\,</math> and the radius of curvature <math display="inline">\rho </math> of the lateral wall, obtained through the circle defined by points A, B and C. Due to the friction with the tools the shell is assumed to work in plane strain conditions (direction '''e'''<math display="inline">_{3}</math>) during the forming process. This assumption is kept for the springback step although the restriction of the tools is removed. Two different materials have been used for the simulations: aluminum and a mild steel. A complete definition of these materials can be found in <span id='citeF-25'></span>[[#cite-25|[25]]]. The material is assumed elastic-plastic. An isotropic behavior is used for the elastic component and an orthotropic yield function (Hill<span id='citeF-26'></span>[[#cite-26|[26]]]) with exponential isotropic hardening is considered for the plastic component. For the simulations a low blank holder force (<math display="inline">2.45</math> kN) has been used.
+The last example is a benchmark of NUMISHEET'93 <span id='citeF-25'></span>[[#cite-25|[25]]]. The aim of this example is to assess the forming process (deep drawing) of a thin sheet and the effects of the elastic springback after the tools are removed. Figure [[#img-12|12]].a shows the setting of the experiment. The geometrical parameters to be computed and compared with experimental results are shown in Figure [[#img-12|12]].b. Namely the angles <math display="inline">\theta _{1}</math> and <math display="inline">\theta _{2}\,</math> and the radius of curvature <math display="inline">\rho </math> of the lateral wall, obtained through the circle defined by points A, B and C. Due to the friction with the tools the shell is assumed to work in plane strain conditions (direction '''e'''<math display="inline">_{3}</math>) during the forming process. This assumption is kept for the springback step although the restriction of the tools is removed. Two different materials have been used for the simulations: aluminum and a mild steel. A complete definition of these materials can be found in <span id='citeF-25'></span>[[#cite-25|[25]]]. The material is assumed elastic-plastic. An isotropic behavior is used for the elastic component and an orthotropic yield function <span id='citeF-26'></span>[[#cite-26|[26]]] with exponential isotropic hardening is considered for the plastic component. For the simulations a low blank holder force (<math display="inline">2.45</math> kN) has been used.
 The deformed shape in Figure [[#img-12|12]].b corresponds to the simulation with aluminum. Figures [[#img-12|12]].c and d (obtained for aluminum and mild steel, respectively) show the geometrical parameters defined above obtained by different experimental groups (each abscissa corresponds to a different group). The average of the experimental values and those obtained with the B2 element are also shown. The numerical simulation agrees reasonably well between the rather scattered experimental values.
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 | colspan="2" | '''Figure 12:''' 2-D Draw bending. (a) Geometry. (b) Geometrical parameters to measure the springback. (c) Comparison with experimental values (aluminum); (d) Comparison with experimental values (mild steel).
 |}
 ==7 Conclusions==

← Older revision	Revision as of 08:35, 19 October 2018
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 |[[Image:Draft_Samper_148409403-fig9a.png|228px|]]
 |[[Image:Draft_Samper_148409403-fig9b.png|372px|]]
 |-
 | colspan="2"|[[Image:Draft_Samper_148409403-fig9c.png|330px|Krauss' branching shell, (a) geometry R=20, L₁=20, L₂=10, h₁=0.3, h₂=0.4, and h₃=0.5 (b) normal displacement along the cylinder. (c) Normal displacements at points 1 and 2 versus the internal pressure.]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 9:''' Krauss' branching shell, (a) geometry <math>R=20</math>, <math>L_{1}=20</math>, <math>L_{2}=10</math>, <math>h_{1}=0.3</math>, <math>h_{2}=0.4</math>, and <math>h_{3}=0.5</math> (b) normal displacement along the cylinder. (c) Normal displacements at points 1 and 2 versus the internal pressure.
 |}
 ===6.2 Examples including geometric non-linearity===

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 |[[Image:Draft_Samper_148409403-fig10b.png|300px|]]
 |-
-|[[Image:Draft_Samper_148409403-fig10cd.png|600px|]]
+|colspan="2"|[[Image:Draft_Samper_148409403-fig10cd.png|600px|]]
-|[[Image:Draft_Samper_148409403-fig10e.png|300px|]]
 |-
-| colspan="2"|[[Image:Draft_Samper_148409403-fig10f.png|300px|Z-shaped cantilever. (a) Geometry and material; (b) Normalized displacements and moments as the mesh is refined; (c) Load versus tip displacement; (d) Bending moment at A versus load (e) and (f) convergence of the results in the non-linear range as the mesh is refined for P = 104.5 and P = 1263 respectively.]]
+|[[Image:Draft_Samper_148409403-fig10e.png|300px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 10:''' Z-shaped cantilever. (a) Geometry and material; (b) Normalized displacements and moments as the mesh is refined; (c) Load versus tip displacement; (d) Bending moment at A versus load (e) and (f) convergence of the results in the non-linear range as the mesh is refined for P = 104.5 and <math>P = 1263</math> respectively.

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 |[[Image:Draft_Samper_148409403-fig8b.png|300px|]]
 |-
-| colspan="2"|[[Image:Draft_Samper_148409403-fig8c.png|330px|Simple frame under point load. (a) geometry (b) horizontal displacements on the vertical member. (c) Deformed configurations in the non-linear range.]]
+|(a)
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 8:''' Simple frame under point load. (a) geometry (b) horizontal displacements on the vertical member. (c) Deformed configurations in the non-linear range.