Abstract
The complex structural behavior of shallow arches can be remarkably affected by many parameters. In this paper, the structural responses of a halfsine pinended shallow arch under sinusoidal and step loadings are accurately calculated. Additionally, the effects of environmental temperature changes are considered. Three types of sinusoidal loadings are separately investigated. Displacements, loadbearing capacity, the magnitude of the axial force and the locus of critical points (including limit and bifurcation points) are directly obtained without tracing the corresponding equilibrium path. Furthermore, the boundaries identifying the number of critical points are investigated. All mentioned structural responses are formulized based on the rise of the arch and the environmental temperature change, which are introduced in a dimensionless form. The proposed formulation is also developed for generalized sinusoidal loadings. Additionally, the structural behavior of the shallow arch under two types of step loadings is investigated. Finally, the accuracy of the suggested approach is examined by a nonlinear finite element method.
Keywords
Halfsine shallow arch; equilibrium path; critical point; bifurcation; stability analysis
1 INTRODUCTION
Shallow arches are widely used in structural, mechanical and aerospace engineering, and the investigation of structural stability has always been of the researchers’ interest. The failure of such structures is in the form of material failures, structural instability or a combination of them.
The tendency of structure to return to the static state, after creating a perturbation in the degrees of freedom, is called stability (Thompson and Hunt, 1973Thompson, J.M.T. and Hunt, G.W. (1973). A General Theory of Elastic Stability, J. Wiley.; Khalil, 2002Khalil, H.K. (2002). Nonlinear Systems, Prentice Hall.). In the analysis of structural stability, since the structure experiences the sudden deformations, the investigation of critical points (such as limit and bifurcation point) is crucial. Such deformations cause severe changes in strains and stresses. The geometry of the arch is an influential parameter on its loadbearing capacity (Cai et al., 2012Cai, J., Xu, Y., Feng, J. and Zhang, J. (2012). Inplane elastic buckling of shallow parabolic arches under an external load and temperature changes, Journal of structural engineering 138(11): 13001309.; Bateni and Eslami, 2015Bateni, M. and Eslami, M.R. (2015). Nonlinear inplane stability analysis of FG circular shallow arches under uniform radial pressure, ThinWalled Structures 94: 302313.; Bradford et al., 2015Bradford, M.A., Pi, Y.L., Yang, G. and Fan, X.C. (2015). Effects of approximations on nonlinear inplane elastic buckling and postbuckling analyses of shallow parabolic arches, Engineering Structures 101: 5867.; RezaieePajand and RajabzadehSafaei, 2016RezaieePajand, M. and RajabzadehSafaei, N. (2016). An explicit stiffness matrix for parabolic beam element, Latin American Journal of Solids and Structures 13: 17821801.). In addition, various loadings (e.g., the sinusoidal (Plaut and Johnson, 1981Plaut, R.H. and Johnson, E.R. (1981). The effects of initial thrust and elastic foundation on the vibration frequencies of a shallow arch, Journal of Sound and Vibration 78(4): 565571.), concentrated (Pi et al., 2008Pi, Y.L., Bradford, M.A. and TinLoi, F. (2008). Nonlinear inplane buckling of rotationally restrained shallow arches under a central concentrated load, International Journal of NonLinear Mechanics 43(1): 117.; Chandra et al., 2012Chandra, Y., Stanciulescu, I., Eason, T. and Spottswood, M. (2012). Numerical pathologies in snapthrough simulations, Engineering Structures 34: 495504.; Tsiatas and Babouskos, 2017Tsiatas, G.C. and Babouskos, N.G. (2017). Linear and geometrically nonlinear analysis of nonuniform shallow arches under a central concentrated force, International Journal of NonLinear Mechanics 92: 92101.), distributed (Moghaddasie and Stanciulescu, 2013bMoghaddasie, B. and Stanciulescu, I. (2013b). Equilibria and stability boundaries of shallow arches under static loading in a thermal environment, International Journal of NonLinear Mechanics 51: 132144.) and end moment loads (Chen and Liao, 2005Chen, J.S. and Liao, C.Y. (2005). Experiment and analysis on the free dynamics of a shallow arch after an impact load at the end, ASME Journal of Applied Mechanics 72(1): 5461.; Chen and Lin, 2005Chen, J.S. and Lin, J.S. (2005). Exact critical loads for a pinned halfsine arch under end couples, ASME Journal of Applied Mechanics 72(1): 147148.)), geometric imperfections (Virgin et al., 2014Virgin, L.N., Wiebe, R., Spottswood, S.M. and Eason, T.G. (2014). Sensitivity in the structural behavior of shallow arches, International Journal of NonLinear Mechanics 58: 212221.; Zhou et al., 2015aZhou, Y., Chang, W. and Stanciulescu, I. (2015a). Nonlinear stability and remote unconnected equilibria of shallow arches with asymmetric geometric imperfections, International Journal of NonLinear Mechanics 77: 111.), and boundary conditions (Pi and Bradford, 2012Pi, Y.L. and Bradford, M.A. (2012). Nonlinear buckling and postbuckling analysis of arches with unequal rotational end restraints under a central concentrated load, International Journal of Solids and Structures 49(26): 37623773.; Pi and Bradford, 2013Pi, Y.L. and Bradford, M.A. (2013). Nonlinear elastic analysis and buckling of pinnedfixed arches, International Journal of Mechanical Sciences 68: 212223.; Han et al., 2016Han, Q., Cheng, Y., Lu, Y., Li, T. and Lu, P. (2016). Nonlinear buckling analysis of shallow arches with elastic horizontal supports, ThinWalled Structures 109: 88102.) are other important factors in the structural design.
In most cases, shallow arches become elastically unstable when the lateral load reaches a critical value (Chen and Li, 2006Chen, J.S. and Li, Y.T. (2006). Effects of elastic foundation on the snapthrough buckling of a shallow arch under a moving point load, International Journal of Solids and Structures 43(14): 42204237.). This means that a large deformation could be observed while the material remains elastic. Practical experiences also confirm this issue (Chen and Liao, 2005Chen, J.S. and Liao, C.Y. (2005). Experiment and analysis on the free dynamics of a shallow arch after an impact load at the end, ASME Journal of Applied Mechanics 72(1): 5461.; Chen and Yang, 2007aChen, J.S. and Yang, C.H. (2007a). Experiment and theory on the nonlinear vibration of a shallow arch under harmonic excitation at the end, ASME Journal of Applied Mechanics 74(1): 10611070.; Chen and Ro, 2009Chen, J.S. and Ro, W.C. (2009). Dynamic response of a shallow arch under end moments, Journal of Sound and Vibration 326(1): 321331.). Consequently, the behavior of shallow arches can be explained by the nonlinear theory of elastic stability. In some analyses, it is assumed that the displacements of the arch are limited to avoid a material failure (Pippard, 1990Pippard, A.B. (1990). The elastic arch and its modes of instability, European Journal of Physics 11: 359365.; Xu et al., 2002Xu, J.X., Huang, H., Zhang, P.Z. and Zhou, J.Q. (2002). Dynamic stability of shallow arch with elastic supportsapplication in the dynamic stability analysis of inner winding of transformer during short circuit, International Journal of NonLinear Mechanics 37(4): 909920.; Chen and Hung, 2012Chen, J.S. and Hung, S.Y. (2012). Exact snapping loads of a buckled beam under a midpoint force, Applied Mathematical Modelling 36: 17761782.). In addition, the variation in the environment temperature can be influential on the stability of structures (Matsunaga, 1996Matsunaga, H. (1996). Inplane vibration and stability of shallow circular arches subjected to axial forces, International Journal of Solids and Structures 33(4): 469482.; Hung and Chen, 2012Hung, S.Y. and Chen, J.S. (2012). Snapping of a buckled beam on elastic foundation under a midpoint force, European Journal of MechanicsA/Solids 31(1): 90100.; Stanciulescu et al., 2012Stanciulescu, I., Mitchell, T., Chandra, Y., Eason, T. and Spottswood, M. (2012). A lower bound on snapthrough instability of curved beams under thermomechanical loads, International Journal of NonLinear Mechanics 47(5): 561575.; Kiani and Eslami, 2013Kiani, Y. and Eslami, M.R. (2013). Thermomechanical buckling of temperaturedependent FGM beams, Latin American Journal of Solids and Structures 10: 223246.).
Several approaches can be applied to investigate the structural behavior of shallow arches. Previously, both analytical and numerical methods are discussed in the literature (Plaut and Johnson, 1981Plaut, R.H. and Johnson, E.R. (1981). The effects of initial thrust and elastic foundation on the vibration frequencies of a shallow arch, Journal of Sound and Vibration 78(4): 565571.; Reddy and Volpi, 1992Reddy, B.D. and Volpi, M.B. (1992). Mixed finite element methods for the circular arch problem, Computer methods in applied mechanics and engineering 97(1): 125145.; Pi et al., 2002Pi, Y.L., Bradford, M.A. and Uy, B. (2002). Inplane stability of arches, International Journal of Solids and Structures 39(1): 105125.; Xenidis et al., 2013Xenidis, H., Morfidis, K. and Papadopoulos, P.G. (2013). Nonlinear analysis of thin shallow arches subject to snapthrough using truss models, Structural Engineering and Mechanics 45(4): 521542.). In some analytical techniques, the displacement field is replaced by a set of orthogonal functions to derive the nonlinear equilibrium and buckling equations (Xu et al., 2002Xu, J.X., Huang, H., Zhang, P.Z. and Zhou, J.Q. (2002). Dynamic stability of shallow arch with elastic supportsapplication in the dynamic stability analysis of inner winding of transformer during short circuit, International Journal of NonLinear Mechanics 37(4): 909920.; Chen et al., 2009Chen, J.S., Ro, W.C. and Lin, J.S. (2009). Exact static and dynamic critical loads of a sinusoidal arch under a point force at the midpoint, International Journal of NonLinear Mechanics 44(1): 6670.; Chen and Hung, 2012Chen, J.S. and Hung, S.Y. (2012). Exact snapping loads of a buckled beam under a midpoint force, Applied Mathematical Modelling 36: 17761782.; Moghaddasie and Stanciulescu, 2013bMoghaddasie, B. and Stanciulescu, I. (2013b). Equilibria and stability boundaries of shallow arches under static loading in a thermal environment, International Journal of NonLinear Mechanics 51: 132144.; Zhou et al., 2015aZhou, Y., Chang, W. and Stanciulescu, I. (2015a). Nonlinear stability and remote unconnected equilibria of shallow arches with asymmetric geometric imperfections, International Journal of NonLinear Mechanics 77: 111.). Using the principle of stationary potential energy is another robust analytical approach to investigate the equilibrium and stability of shallow arches (Moon et al., 2007Moon, J., Yoon, K.Y., Lee, T.H. and Lee, H.E. (2007). Inplane elastic buckling of pinended shallow parabolic arches, Engineering Structures 29(10): 26112617.; Pi et al., 2007Pi, Y.L., Bradford, M.A. and TinLoi, F. (2007). Nonlinear analysis and buckling of elastically supported circular shallow arches, International Journal of Solids and Structures 44(78): 24012425.; Pi et al., 2008Pi, Y.L., Bradford, M.A. and TinLoi, F. (2008). Nonlinear inplane buckling of rotationally restrained shallow arches under a central concentrated load, International Journal of NonLinear Mechanics 43(1): 117.; Pi et al., 2010Pi, Y.L., Bradford, M.A. and Qu, W. (2010). Energy approach for dynamic buckling of shallow fixed arches under step loading with infinite duration, Structural Engineering and Mechanics 35(5): 555570.). On the other hand, the nonlinear finite element method has been widely applied by researchers to trace the equilibrium path (Chandra et al., 2012Chandra, Y., Stanciulescu, I., Eason, T. and Spottswood, M. (2012). Numerical pathologies in snapthrough simulations, Engineering Structures 34: 495504.; Saffari et al., 2012Saffari, H., Mirzai, N.M. and Mansouri, I. (2012). An accelerated incremental algorithm to trace the nonlinear equilibrium path of structures, Latin American Journal of Solids and Structures 9: 425442.; Stanciulescu et al., 2012Stanciulescu, I., Mitchell, T., Chandra, Y., Eason, T. and Spottswood, M. (2012). A lower bound on snapthrough instability of curved beams under thermomechanical loads, International Journal of NonLinear Mechanics 47(5): 561575.; Zhou et al., 2015bZhou, Y., Stanciulescu, I., Eason, T. and Spottswood, M. (2015b). Nonlinear elastic buckling and postbuckling analysis of cylindrical panels, Finite Elements in Analysis and Design 96: 4150.). Identifying the corresponding critical point(s) and finding the relationship between imperfections and loadbearing capacity are the capability of this numerical technique (Eriksson et al., 1999Eriksson, A., Pacoste, C. and Zdunek, A. (1999). Numerical analysis of complex instability behaviour using incrementaliterative strategies, Computer methods in applied mechanics and engineering 179(34): 265305.; Moghaddasie and Stanciulescu, 2013b; RezaieePajand and Moghaddasie, 2014RezaieePajand, M. and Moghaddasie, B. (2014). Stability boundaries of twoparameter nonlinear elastic structures, International Journal of Solids and Structures 51(5): 10891102.).
This paper provides an analytical method to find the exact response of the halfsine shallow arch under the sinusoidal and step loads. Furthermore, the effect of temperature change on the equilibrium paths is investigated. For this purpose, the displacements of the structure are rewritten in the form of the Fourier series. By the substitution of displacements into the governing equations of the arch, the initial and bifurcated equilibrium path are obtained. On the other hand, the critical (limit and bifurcation) points on the static paths are achieved when the stiffness matrix is singular. In this paper, the behavior of the shallow arch under five types of distributed loads are separately investigated by the suggested approach.
The advantages of the proposed method are: (1) obtaining the exact solution of displacement field, equilibrium paths and the locus of critical points, (2) performing one parametric analysis instead of multiple analyses with specified values, and (3) finding the critical points without tracing the equilibrium paths. On the other hand, some limitations of the supposed method can be listed as (1) the changes in environment temperature are gradual, (2) the theory of plane stress is applied, (3) the height of the arch is limited, and (4) the material remains elastic during the analysis.
In the following section, the governing equations of the halfsine shallow arch under an arbitrary load are provided and the relative equilibrium paths are obtained. Then, the way of finding the locus of critical (limit and bifurcation) points is proposed (Section 3). The behavior of the halfsine arch under a number of distributed loadings is investigated by using the suggested method in Section 4. Finally, concluding remarks are given.
2 THE GOVERNING EQUATIONS OF THE SHALLOW ARCH
In this section, the governing equations of a halfsine shallow arch under an arbitrary loading
A halfsine pinned shallow arch under (a) arbitrary, (b) halfsine, (c) onesine, (d) one and halfsine, (e) symmetric and (f) asymmetric loadings
The assumptions used for the analysis are as follows: (1) The axial force is constant over the span (Xu et al., 2002Xu, J.X., Huang, H., Zhang, P.Z. and Zhou, J.Q. (2002). Dynamic stability of shallow arch with elastic supportsapplication in the dynamic stability analysis of inner winding of transformer during short circuit, International Journal of NonLinear Mechanics 37(4): 909920.; Plaut, 2009Plaut, R.H. (2009). Snapthrough of shallow elastic arches under end moments, ASME Journal of Applied Mechanics 76: 014504.; Chen and Hung, 2012Hung, S.Y. and Chen, J.S. (2012). Snapping of a buckled beam on elastic foundation under a midpoint force, European Journal of MechanicsA/Solids 31(1): 90100.); (2) The material is elastic (Chen and Li, 2006Chen, J.S. and Li, Y.T. (2006). Effects of elastic foundation on the snapthrough buckling of a shallow arch under a moving point load, International Journal of Solids and Structures 43(14): 42204237.); (3) outofplane deflections are neglected (Chen and Yang, 2007aChen, J.S. and Yang, C.H. (2007a). Experiment and theory on the nonlinear vibration of a shallow arch under harmonic excitation at the end, ASME Journal of Applied Mechanics 74(1): 10611070.); and (4) The range of displacements and curvatures of the arch is small in comparison with the length of the span
where,
Here,
Since the supports are fixed at the ends, the displacement of the arch in the direction
where,
Here,
By considering the boundary conditions, the initial and deformed shapes of the arch can be rewritten in the Fourier series form:
In Eq. (7),
By substituting Eqs. (7)(9) into (3), a set of equations will be obtained:
Here,
At equilibrium state,
where,
3 THE CRITICAL POINTS
In this section, the critical load of the shallow arch is addressed. If the external load is a function of an independent parameter
In some equilibrium states, sudden changes can be observed in the behavior of the structure. These such points, which are part of the equilibrium path, are called the critical points. The critical points are categorized into limit and bifurcation points. At limit points, the slope of the equilibrium path is zero (Point A in Figure 2), while bifurcation points are located at the intersection of equilibrium paths (Point B in Figure 2).
One way to obtain critical points is equating the determinant of tangent stiffness matrix to zero. The (modal) tangent stiffness matrix is calculated by the derivation of the unbalanced force. This can be done by substituting the magnitude of
Here,
The magnitude of the determinant
By using algebraic operations, Eq. (17) becomes the determinant of an upper triangle matrix:
Consequently, Eq. (15) is rewritten as
or
4 RESULTS FOR LOADING PATTERNS
In this section, the behavior of shallow arch under a number of distributed loads are separately investigated. The patterns of loadings, respectively, are halfsine
4.1 Halfsine loading
By considering the type of loading shown in Figure 1(b), the values of
For
By substituting
From this equation,
By substituting (22) into Eq. (8), the displacement field of the shallow arch is obtained:
This equation is compatible with the results previously presented in the literature for a pinended shallow arch under a halfsine distributed loading (Plaut and Johnson, 1981Plaut, R.H. and Johnson, E.R. (1981). The effects of initial thrust and elastic foundation on the vibration frequencies of a shallow arch, Journal of Sound and Vibration 78(4): 565571.). Eqs. (24) and (25) reveals the equilibrium path in the space
Figure 3 shows the equilibrium path for four different values of
The comparison between the formation of curves and the result of finite element method displays the performance of the proposed strategy. It is noteworthy that the nonlinear FEM procedure obtains a number of discrete equilibrium points, while a continuous equilibrium curve is given by the suggested method.
As it is mentioned previously, there is a nonzero term in bifurcation paths (
Similar to the initial equilibrium path, the displacement field relative to the
This equation shows a linear relationship between
The gray curves in Figure 4 represent the calculated bifurcation paths given by Eq. (29).
As it can be seen, for greater values of
To investigate the structural behavior, a number of static states are displayed in Figure 5. These states are related to points ag specified in Figure 4(b). In this Figure, the points ac and dg are, respectively, corresponding to the initial and bifurcation paths.
A particular case which can be of interest is the relationship between the external load parameter and the axial force along the equilibrium path. Figure 6 draws this relationship by considering Eq. (23). In this figure, four cases corresponding to the values of
Relationship between load parameter and axial force for four different values of
The solid black and gray curves are relative to the initial and bifurcated paths, respectively. Note that, negative values for the axial force
In order to obtain limit points, Eq. (19) can be rewritten in a simpler form:
By substituting the obtained values
In this equation,
Figure 7 shows the relationship between the magnitude of critical load and the values of
The projection of the surfaces displayed in Figure 7 on the plane of
It can be proven that the magnitude of the axial force on the boundary (
As previously mentioned, the bifurcation points have the following characteristics:
Since bifurcation points are located on both initial and bifurcation paths, these points include all properties of both paths (especially, the condition
The equation
In a similar way, the boundaries which are identifying the number of bifurcation points on the equilibrium path (
Figure 10 shows the boundaries
It is noteworthy that, all initial equilibrium paths corresponding to the states between two specific curves include the same number of bifurcation points.
4.2 Onesine loading
Figure 1(c) shows the second loading type
If the procedure, which is previously described in the Subsection 4.1, is applied, the values of
Additionally, the dimensionless displacement field for the initial equilibrium path is obtained from Eq. (8):
Consequently, the displacement of the midpoint is as follows:
Figure 11 displays the equilibrium paths for four different values of
Similar to the Subsection 4.1.,
Eqs. (42) and (43), respectively, show the displacement field and displacement of the midpoint for the
In Figure 12, bifurcation equilibrium paths are shown by gray curves. Although the initial equilibrium path does not include any critical point, the second path has limit and bifurcation points.
In addition, to have a better analogy, the relationship between the load parameter and the axial force is given in Figure 13.
Relationship between load parameter and axial force for four different values of
Figure 14 shows the equilibrium states relative to the points ag in Figure 12(a).
In order to obtain the locus of limit points, the values of
Eqs. (38) and (44) reveal the location of limit points as a set of surfaces in the space of
On the other hand, the equation (45) should be satisfied for the bifurcation points on the
By considering
Figure 16 shows surfaces
4.3 One and halfsine loading
If the loading pattern is assumed to be
Furthermore, the calculated parameters
Figure 17 shows the equilibrium paths for four different values of
In a similar way, the displacement field and displacement of midpoint for the
By considering Eq. (12),
In Figure 18, the bifurcation paths are denoted by gray curves.
Figure 19 shows the structural states corresponding to the points af in Figure 18(b). Here, Figure 19(a)(e) are relative to the initial and secondary equilibrium paths, while the last figure is related to the bifurcation path.
Figure 18 shows that the initial equilibrium path does not include any critical point, and all critical points (limit and bifurcation) are located on the second equilibrium path. By substituting the values of
The locus of limit points in the interval
On the other hand, the bifurcation points should satisfy the following conditions:
By substituting the value
The locus of bifurcation points for the interval
4.4 ksine loading
In this subsection, the effect of general sinusoidal loading pattern (
By considering Eqs. (13), (58) and (59), the coefficients
The States I and II are corresponding to
Subsequently, the dimensionless displacement field for the initial equilibrium path is achieved by considering Eq.(8) for the States I and III:
On the other hand, the States II and IV, are corresponding to the bifurcation equilibrium path. Similarly, the displacement field for the
The displacement of the midpoint is obtained when
In order to find limit points on the initial equilibrium path, Eq. (30) is applied:
Furthermore, the condition (74) should be satisfied for the
By substituting Eq. (74) into Eqs. (64) and (65), the locus of bifurcation points is computed:
It is noteworthy that bifurcation points are located on both initial and bifurcated paths.
4.5 Symmetric step loading
A type of symmetric step load is shown in Figure 1(e). This loading pattern can be defined in the following form:
By using the Fourier series, the values of
Note that for even values of
where, the function
Figure 22 displays the equilibrium paths for four different values of
In the case of bifurcation path, there is a nonzero
Eqs. (84) and (85), respectively, show the displacement field and displacement of the midpoint for the
In Figure 23, the bifurcation paths are given.
By substituting the values of
The locus of limit points in the interval
On the other hand, by substituting the value
The locus of bifurcation points for the interval
4.6 Asymmetric step loading
An asymmetric step load is shown in Figure 1(f). This loading pattern is defined as
The values of
It is noteworthy that the magnitude of
where, the function
The initial equilibrium paths for different values of
As it is observed, the procedure of FEM becomes divergent in Figure 26(d).
In the case of asymmetric loading, there can be a nonzero
Accordingly, the displacement field and displacement of the midpoint for the
In the asymmetric step loading, there is only one bifurcation path which can be seen in the case of
Similar to the previous subsection, the locus of limit points can be determined. In this way, the critical constraint (30) is rewritten in the following form by considering Eq. (90):
The locus of limit points in the interval
In order to find the location of bifurcation points in the space of
The locus of bifurcation points for the interval
5 CONCLUSIONS
The stability behavior of shallow arches is always being of the researchers’ interest. In this paper, an analytical method to find the exact solution of a halfsinusoidal elastic shallow arch in the thermal environment under sinusoidal and step loads is proposed. For this purpose, the structural displacement is rewritten in a form of Fourier series, and subsequently, both initial and bifurcated equilibrium paths are obtained by substituting the transformed displacements into the governing equations of the arch. In addition, the critical points (such as limit and bifurcation points) are calculated by equating the determinant of stiffness matrix to zero. Furthermore, a new generalized formulation for various types of sinusoidal loadings is proposed.
In this research, the stability behavior of a halfsine shallow arch under three types of sinusoidal and two types of step function loads is separately investigated. Simultaneously, a nonlinear finite element method is applied to show the accuracy and robustness of the suggested approach. In some cases, FEM becomes divergent during the path following procedure, while the proposed method is able to obtain the equilibrium path(s) comprehensively. Moreover, finding the critical points without tracing the equilibrium path is the superiority of the suggested technique.
Acknowledgements
The authors gratefully acknowledge the helpful suggestions received from the anonymous reviewers. The quality of this article has benefited substantially from their comments.
References
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Available online: June 06, 2018
APPENDIX
Here, the functions
Publication Dates

Publication in this collection
2018
History

Received
22 Oct 2017 
Reviewed
21 Jan 2018 
Accepted
28 Feb 2018