- Email: [email protected]
Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation”
European Journal of Mechanics A/Solids xxx (2014) 1e2
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Discussion
Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation” Ravi Kumar Kovvali a, *, Dewey H. Hodges a, Wenbin Yu b, Heung Soo Kim c, Jun-Sik Kim d a
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-2045, USA Department of Mechanical, Robotics and Energy Engineering, Dongguk University-Seoul, Seoul 100-715, Republic of Korea d Department of Intelligent Mechanical Engineering, Kumoh National Institute of Technology, Gumi, Gyeongbuk 730-701, Republic of Korea b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2014 Accepted 18 February 2014 Available online xxx
In their paper (Kim and Kim, 2013), Kim and Kim presented predictions that exhibited significant differences in bending-shear coupling for the CUS box beam from their own analysis vis-à-vis from VABS (Yu et al., 2002, 2012). The actual differences between results obtained from VABS (Yu et al., 2002, 2012) and the RTV theory (Kim and Kim, 2013), when executed properly, are shown herein to be very minor. Indeed, both stiffness models yield very acceptable results for a static analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions, in contrast to results presented in Kim and Kim (2013). Published by Elsevier Masson SAS.
Keywords: Anisotropic beams Mixed variational theorem VABS
1. Observations In their paper (Kim and Kim, 2013), Kim and Kim develop a refined shear deformable beam theory (referred to as a RankineTimoshenko-Vlasov (RTV) beam theory), which is based on the Formal Asymptotic Method-based Beam Analysis (FAMBA) (Kim et al., 2008), to analyze and design anisotropic beams via the mixed variational theorem (MVT). After a detailed mathematical treatment of MVT to systematically blend an asymptotic stress field and an intuitive displacement field to derive a beam model, they consider various examples of thin-walled composite beams with closed and open cross sections for verification of the developed RTV theory, and the numerical results are compared with those from 3D FEM and other methods, such as the Variational-Asymptotic Beam Section analysis (VABS) (Yu et al., 2002, 2012). In particular, Section 3.1 of their paper describes a 15 deg CUS1 thin-walled composite box beam (whose ply material properties
DOI of original article: http://dx.doi.org/10.1016/j.euromechsol.2013.01.004. * Corresponding author. E-mail addresses: [email protected], [email protected] (R.K. Kovvali).
are listed in Table 1 of (Kim and Kim, 2013)) analyzed with VABS by taking the stiffness model given in Yu et al. (2002) and applying it to a typical Timoshenko beam model. The induced center deflection under a horizontal unit shear force computed by 3D FEM, FAMBA, VABS, etc., is plotted against the beam slenderness ratio in Fig. 4 and Table 3 of Kim and Kim (2013). While it is clear that FAMBA2nd predictions are reasonably close to those of 3D FEM, the authors (Kim and Kim) also show, and hence claim, that results from VABS deviate significantly (in excess of 50%) from those of 3D FEM for the bending-shear coupling behavior. These claims call for the following comments: 1. The stiffness model given in Yu et al. (2002, 2012) (as calculated by versions of VABS till v.3.1), when used in tandem with a 1D beam analysis code based on a geometrically exact intrinsic beam theory (GEBT), provides results that are both qualitatively and quantitatively in close agreement to 3D FEM unlike the results reported in Kim and Kim (2013). This can be seen in Fig. 1 and Table 1 which have now been recalculated by the authors (Kim and Kim). We can safely assume that the fault lies neither with the Rankine-Timoshenko model that the authors used nor the geometrically-exact beam theory (since the
http://dx.doi.org/10.1016/j.euromechsol.2014.02.012 0997-7538/Published by Elsevier Masson SAS.
Please cite this article in press as: Kovvali, R.K., et al., Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation”, European Journal of Mechanics A/Solids (2014), http://dx.doi.org/10.1016/j.euromechsol.2014.02.012
2
R.K. Kovvali et al. / European Journal of Mechanics A/Solids xxx (2014) 1e2
Table 1 Induced deflection values (103) along the axial locations presented in Fig. 1 (Table 3 in (Kim and Kim, 2013)). x1 (in.)
3D FEM
FAMBAa
NABSA etcb
Present
3 6 9 12 15 18 21 24 27
1.1055 2.0289 2.6784 3.0542 3.1563 2.9846 2.5393 1.8202 0.8274
1.2319 2.1882 2.8688 3.2737 3.403 3.2566 2.8346 2.137 1.1636
1.2589 2.2348 2.9274 3.3370 3.4634 3.3066 2.8668 2.1437 1.1376
1.1414 2.0148 2.6203 2.9578 3.0273 2.8288 2.3623 1.6279 0.6254
a b
FAMBA 2nd. NABSA, Jung et al. (2002) and VABS, Yu et al. (2002, 2012). −3
x 10
former seemed to have worked with other models and the latter is designed to capture all the geometrical nonlinearities obtainable by a beam model). Therefore, it seems logical to conclude that the source of discrepancy was an error in applying the given stiffness model correctly to the macroscopic analysis instead of an error in the calculation of the 6 6 stiffness matrix by VABS. 2. The discrepancy has been examined by the authors (Kim and Kim). It is found that the sign convention used in VABS (Yu et al., 2002, 2012) was not consistently applied to the calculation of VABS results in Kim and Kim (2013). The static induced deflection results from VABS and others have been corrected and re-plotted as shown in Fig. 1 and Table 1. The results indicate that all the results including VABS match closely with the 3D FEM results for the bending-shear coupling behavior.
2. Conclusion In conclusion, we find that although there are minor differences in the prediction of the bending-shear coupling for the CUS box beam between VABS (Yu et al., 2002, 2012) and the RTV theory (Kim and Kim, 2013), both stiffness models yield very acceptable results for a static analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions, unlike shown in Fig. 4 and Table 3 of Kim and Kim (2013).
3
Induced deflection (u ), inches
0
−4
References FAMBA 0th (Kim et al., 2008) FAMBA 2nd (Kim et al., 2008) VABS(2012) and NABSA, Jung et al. (2002) VABS (Yu et al., 2002) 3D FEM (Kim et al., 2008) Present RTV −8 0
5
10
15
20
25
30
Axial coordinate, inches Fig. 1. Induced deflection of the box beam, l ¼ 30 in (Fig. 4 in (Kim and Kim, 2013)).
Jung, S.N., Nagaraj, V.T., Chopra, I., 2002. Refined structural model for thin- and thick-walled composite rotor blades. AIAA J. 40, 105e116. Kim, H.S., Kim, J.-S., 2013. A Rankine-Timoshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation. Eur. J. Mech. A/ Solids 40, 131e138. Kim, J.-S., Cho, M., Smith, E.C., 2008. An asymptotic analysis of composite beams with kinematically corrected end effects. Int. J. Solids Struct. 45, 1954e1977. Yu, W., Hodges, D.H., Volovoi, V.V., Cesnik, C.E.S., 2002. On Timoshenko-like modeling of initially curved and twisted composite beams. Int. J. Solids Struct. 39 (19), 5101e5121. Yu, W., Hodges, D.H., Ho, J.C., 2012. Variational asymptotic beam sectional analysis e an updated version. Int. J. Eng. Sci. 59, 40e64.
Please cite this article in press as: Kovvali, R.K., et al., Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation”, European Journal of Mechanics A/Solids (2014), http://dx.doi.org/10.1016/j.euromechsol.2014.02.012
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Discussion
Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation” Ravi Kumar Kovvali a, *, Dewey H. Hodges a, Wenbin Yu b, Heung Soo Kim c, Jun-Sik Kim d a
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-2045, USA Department of Mechanical, Robotics and Energy Engineering, Dongguk University-Seoul, Seoul 100-715, Republic of Korea d Department of Intelligent Mechanical Engineering, Kumoh National Institute of Technology, Gumi, Gyeongbuk 730-701, Republic of Korea b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 February 2014 Accepted 18 February 2014 Available online xxx
In their paper (Kim and Kim, 2013), Kim and Kim presented predictions that exhibited significant differences in bending-shear coupling for the CUS box beam from their own analysis vis-à-vis from VABS (Yu et al., 2002, 2012). The actual differences between results obtained from VABS (Yu et al., 2002, 2012) and the RTV theory (Kim and Kim, 2013), when executed properly, are shown herein to be very minor. Indeed, both stiffness models yield very acceptable results for a static analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions, in contrast to results presented in Kim and Kim (2013). Published by Elsevier Masson SAS.
Keywords: Anisotropic beams Mixed variational theorem VABS
1. Observations In their paper (Kim and Kim, 2013), Kim and Kim develop a refined shear deformable beam theory (referred to as a RankineTimoshenko-Vlasov (RTV) beam theory), which is based on the Formal Asymptotic Method-based Beam Analysis (FAMBA) (Kim et al., 2008), to analyze and design anisotropic beams via the mixed variational theorem (MVT). After a detailed mathematical treatment of MVT to systematically blend an asymptotic stress field and an intuitive displacement field to derive a beam model, they consider various examples of thin-walled composite beams with closed and open cross sections for verification of the developed RTV theory, and the numerical results are compared with those from 3D FEM and other methods, such as the Variational-Asymptotic Beam Section analysis (VABS) (Yu et al., 2002, 2012). In particular, Section 3.1 of their paper describes a 15 deg CUS1 thin-walled composite box beam (whose ply material properties
DOI of original article: http://dx.doi.org/10.1016/j.euromechsol.2013.01.004. * Corresponding author. E-mail addresses: [email protected], [email protected] (R.K. Kovvali).
are listed in Table 1 of (Kim and Kim, 2013)) analyzed with VABS by taking the stiffness model given in Yu et al. (2002) and applying it to a typical Timoshenko beam model. The induced center deflection under a horizontal unit shear force computed by 3D FEM, FAMBA, VABS, etc., is plotted against the beam slenderness ratio in Fig. 4 and Table 3 of Kim and Kim (2013). While it is clear that FAMBA2nd predictions are reasonably close to those of 3D FEM, the authors (Kim and Kim) also show, and hence claim, that results from VABS deviate significantly (in excess of 50%) from those of 3D FEM for the bending-shear coupling behavior. These claims call for the following comments: 1. The stiffness model given in Yu et al. (2002, 2012) (as calculated by versions of VABS till v.3.1), when used in tandem with a 1D beam analysis code based on a geometrically exact intrinsic beam theory (GEBT), provides results that are both qualitatively and quantitatively in close agreement to 3D FEM unlike the results reported in Kim and Kim (2013). This can be seen in Fig. 1 and Table 1 which have now been recalculated by the authors (Kim and Kim). We can safely assume that the fault lies neither with the Rankine-Timoshenko model that the authors used nor the geometrically-exact beam theory (since the
http://dx.doi.org/10.1016/j.euromechsol.2014.02.012 0997-7538/Published by Elsevier Masson SAS.
Please cite this article in press as: Kovvali, R.K., et al., Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation”, European Journal of Mechanics A/Solids (2014), http://dx.doi.org/10.1016/j.euromechsol.2014.02.012
2
R.K. Kovvali et al. / European Journal of Mechanics A/Solids xxx (2014) 1e2
Table 1 Induced deflection values (103) along the axial locations presented in Fig. 1 (Table 3 in (Kim and Kim, 2013)). x1 (in.)
3D FEM
FAMBAa
NABSA etcb
Present
3 6 9 12 15 18 21 24 27
1.1055 2.0289 2.6784 3.0542 3.1563 2.9846 2.5393 1.8202 0.8274
1.2319 2.1882 2.8688 3.2737 3.403 3.2566 2.8346 2.137 1.1636
1.2589 2.2348 2.9274 3.3370 3.4634 3.3066 2.8668 2.1437 1.1376
1.1414 2.0148 2.6203 2.9578 3.0273 2.8288 2.3623 1.6279 0.6254
a b
FAMBA 2nd. NABSA, Jung et al. (2002) and VABS, Yu et al. (2002, 2012). −3
x 10
former seemed to have worked with other models and the latter is designed to capture all the geometrical nonlinearities obtainable by a beam model). Therefore, it seems logical to conclude that the source of discrepancy was an error in applying the given stiffness model correctly to the macroscopic analysis instead of an error in the calculation of the 6 6 stiffness matrix by VABS. 2. The discrepancy has been examined by the authors (Kim and Kim). It is found that the sign convention used in VABS (Yu et al., 2002, 2012) was not consistently applied to the calculation of VABS results in Kim and Kim (2013). The static induced deflection results from VABS and others have been corrected and re-plotted as shown in Fig. 1 and Table 1. The results indicate that all the results including VABS match closely with the 3D FEM results for the bending-shear coupling behavior.
2. Conclusion In conclusion, we find that although there are minor differences in the prediction of the bending-shear coupling for the CUS box beam between VABS (Yu et al., 2002, 2012) and the RTV theory (Kim and Kim, 2013), both stiffness models yield very acceptable results for a static analysis. Specifically, VABS does not significantly deviate, qualitatively or quantitatively, from 3D FEM predictions, unlike shown in Fig. 4 and Table 3 of Kim and Kim (2013).
3
Induced deflection (u ), inches
0
−4
References FAMBA 0th (Kim et al., 2008) FAMBA 2nd (Kim et al., 2008) VABS(2012) and NABSA, Jung et al. (2002) VABS (Yu et al., 2002) 3D FEM (Kim et al., 2008) Present RTV −8 0
5
10
15
20
25
30
Axial coordinate, inches Fig. 1. Induced deflection of the box beam, l ¼ 30 in (Fig. 4 in (Kim and Kim, 2013)).
Jung, S.N., Nagaraj, V.T., Chopra, I., 2002. Refined structural model for thin- and thick-walled composite rotor blades. AIAA J. 40, 105e116. Kim, H.S., Kim, J.-S., 2013. A Rankine-Timoshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation. Eur. J. Mech. A/ Solids 40, 131e138. Kim, J.-S., Cho, M., Smith, E.C., 2008. An asymptotic analysis of composite beams with kinematically corrected end effects. Int. J. Solids Struct. 45, 1954e1977. Yu, W., Hodges, D.H., Volovoi, V.V., Cesnik, C.E.S., 2002. On Timoshenko-like modeling of initially curved and twisted composite beams. Int. J. Solids Struct. 39 (19), 5101e5121. Yu, W., Hodges, D.H., Ho, J.C., 2012. Variational asymptotic beam sectional analysis e an updated version. Int. J. Eng. Sci. 59, 40e64.
Please cite this article in press as: Kovvali, R.K., et al., Comments on “A Rankine-Timonshenko-Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation”, European Journal of Mechanics A/Solids (2014), http://dx.doi.org/10.1016/j.euromechsol.2014.02.012