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Closure to discussion by O. Bedair of “Semi-analytical buckling strength analysis of plates with arbitrary stiffener arrangements”
Journal of Constructional Steel Research 63 (2007) 1618–1620 www.elsevier.com/locate/jcsr
Comment
Closure to discussion by O. Bedair of “Semi-analytical buckling strength analysis of plates with arbitrary stiffener arrangements” Lars Brubak a,∗ , Jostein Hellesland a , Eivind Steen b a Mechanics Division, Department of Mathematics, University of Oslo, 0316 Oslo, Norway b Section for Hydrodynamics, Structures and Stability, DNV Maritime, Det Norske Veritas, 1322 Høvik, Norway
Received 31 January 2007; accepted 19 May 2007
The authors would like to thank the discusser for his comments. They are limited to that aspect of the paper that is related to the computation of elastic buckling stress limits (abbreviated as ESL in the paper). The elastic buckling load is used as a parameter in the approximate strength analysis, labelled buckling strength limit (BSL) analysis, in the paper. It is the presentation and discussion of the BSL analysis that has been the main objective of the paper. The discusser has misunderstood when he states that another main objective has been to compare the computational efficiency of the method with the finite element method. If it had been so, we would certainly have devoted more than the present few lines to the topic. The discusser presents a number of comments and questions on details of the ESL analysis. We will try to do justice to the items raised by the discusser. A point-to-point reply is listed below. (1) The discusser must have misunderstood when he writes that the authors have mentioned “that all the semi-analytical methods are restricted to plates with regular stiffeners parallel to the boundaries” (Introduction, last paragraph). The word “all” has been added by the discusser. As can be seen in the paper, we made specific reference to the studies mentioned in the two previous paragraphs of the paper, and for those the statement is correct. The discusser proceeds to provide a total of one reference [1 (in the discusser’s ref. list)] to previous literature on plates ”with arbitrary oriented stiffeners”. That paper is on vibration analysis. Moreover, a stiffener is required to pass through the centre point of the plate.
DOI of original article: 10.1016/j.jcsr.2007.05.009. ∗ Corresponding author.
E-mail address: [email protected] (L. Brubak). c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.05.012
It cannot be placed and oriented arbitrarily at different locations, which limits the generality of the method. However, we appreciate this information as vibration and buckling analysis have several aspects in common. (2) The discusser states that the authors have overlooked the literature on the subject. The authors can appreciate that there are many papers on semi-analytical methods. Most of them are on elastic buckling (eigenvalue) analysis as such, and some of them are on such analysis for plates with various stiffening configurations parallel to the plate boundaries. Work by the discusser, for instance [1,2], are examples of such studies. Studies of plates with inclined stiffeners are very limited (see also item (1) above). The main objective of the paper is strength rather than elastic buckling as such. Therefore, the studies reviewed by the authors are primarily on semi-analytical strength rather than on buckling analysis. We concede that the review of papers on buckling possibly could have been somewhat more extensive. However, the buckling load computation is only a part of our strength analysis, and we decided to adopt a rather conventional buckling analysis approach that we considered to be suitable in the wider context of the paper. Our primary contribution with regard to the buckling analysis, that includes some simplifications that increase the computational efficiency, has been to verify and document the applicability of it to a wide variety of parameters. Although, we consider that to be an important contribution, we do not pretend to contribute any significant novel aspects to the approach. Therefore, we feel that an exhaustive literature review of elastic buckling literature would not have been altogether appropriate in the context of our strength analysis study. The discusser feels that comparisons of the computational efficiency with other semi-analytical methods should have been performed. Considering that the elastic buckling
L. Brubak et al. / Journal of Constructional Steel Research 63 (2007) 1618–1620
(eigenvalue) computations are not the most time consuming part in our strength predictions, comparisons with methods that do not consider strength predictions are not totally relevant. Nevertheless, this is not to say that separate comparisons of the computational efficiency of the buckling analysis part with more rigorous and therefore, most likely, less computationally efficient methods, are not of interest. Such comparisons, such as for instance with a buckling analysis method by the discusser [1,2], poses a practical problem since sufficient details in conjunction with the computational time are normally missing in the papers. Therefore, a comparative study of the type suggested by the discusser, would probably require a paper with that as one of the major objectives. (3) In the considered applications of the study, and in the cases studied, the stiffeners are torsionally rather weak and the neglect of the torsional stiffness is acceptable. The authors agree that it could have been mentioned that this neglect will be conservative for stiffeners with strong torsional stiffness. As the discusser knows, it is a trivial matter to include the torsional stiffness, and in another work by the authors [3], it has been included. (4) Since the axial stiffener stiffness is neglected in the paper, the stiffeners are only connected to the out-ofplane displacements of the plate and not to the in-plane displacements. See also item (5) below. The discusser must have misunderstood when he states that we have used two coordinate systems. The stiffener curvature w,ss in the stiffener direction is related to one single, global Cartesian coordinate system (xi, yj). Possibly, the paper could have given more details on this aspect. The stiffener curvature was taken according to w,ss = ∇(∇w · s) · s
(1)
where ∇ = i∂/∂ x + j∂/∂ y and s = (L x i + L y j)/L s is the unit direction vector in the direction of the stiffener. (5) As explained in the paper, but overlooked by the discusser, eccentric stiffeners are reflected in a simplified manner. An eccentric stiffener is included in the same manner as a concentric stiffener, but with its effective second moment of area Ie (Eq. (16) in the paper). For instance, for the T-stiffener illustrated in Fig. 1, Ie is calculated about an estimated axis of bending (z = z c ) for the cross-section consisting of the stiffener and an assumed effective plate width be (the second moment of area of the effective plate width about its own axis is not included as it is included in the energy formulation itself). The required assumption of be represents an uncertainty, but results are found not to be too sensitive to reasonable variations in this value. This is clearly a simplified approach, but numerous comparison with finite element analysis, including those given in the paper, have shown that it is acceptable for the plates, boundary conditions and loadings considered in the paper. (6) According to the discusser, the neglect of the axial stiffener stiffness will have the most pronounced effect in cases with inclined stiffeners. This is exactly the kind of cases studied
1619
Fig. 1. Stiffener profile of a stiffener with an effective plate width be .
in the paper. Comparison with finite element analysis for a wide range of stiffener and plate dimensions, have shown that this neglect had only minor effects on the computed buckling loads of the considered (free-to-move in-plane) plates. The same conclusion can be drawn for the selected ESL results shown in the paper. (7) It is quite correct that we have not given any guidelines on how to choose the rotational spring coefficient kr , or the translational (out-of-plane) spring coefficient kt for that matter. We did not consider that it was necessary to provide further details, but left it to the user to choose appropriate values. Engineers are normally quite familiar with such coefficients. The rotational springs along the edges are included in the formulations in order to be able, for instance, to do parametric studies of the effect of different boundary conditions, varying from simply supported to rotationally fully clamped. For clamped plates, the rotational springs must be chosen large enough so as to prevent edge rotations. To model the rotational stiffness provided by an edge girder, kr must be chosen so as to reflect the torsional stiffness of the edge girder. Alternatively, in the same manner as in Brubak and Hellesland [3], the torsional energy of the edge girder can be included in the total potential energy formulation. (8) In the paper we give some information on computational speed of the presented method compared to nonlinear FEM analyses. The discusser states that the given numbers are for a plate with “an intermediate stiffener” (dividing the plate into two halves). This is not correct. The given relative numbers (Section 6, Validation) are provided as general information, typical for the plates analysed, most of which had irregular stiffener arrangements. The comments in the discusser’s item 8(a) is based on the same misunderstanding mentioned above. For irregularly stiffened plates, it is not possible to take advantage of symmetry. In the paper we state that the relative computational speed numbers are given “as an indication of the relative computational efficiency” (Section 6, Validation). We further state that “these numbers are just given as an indication of the relative computational efficiency, and do not reflect the results of an in-depth study of the different factors affecting computer time consumption”. In a more
1620
L. Brubak et al. / Journal of Constructional Steel Research 63 (2007) 1618–1620
in-depth discussion, the points made by the discusser in his item 8(b) and (c) naturally belong. In such a discussion, it is also necessary to study the effect of optimising the efficiency of computer routines in the present method. The present research version of the program can be significantly improved in that respect. For comparisons of such a version with a more efficient finite element analysis, the relative difference in computational times (CPU) will still be very significant. This is an important advantage of the method in engineering applications and weight optimisation and reliability studies. (9) The discusser concludes from inspection of some of the figures in the paper that the boundaries “were treated as fully restrained against in-plane motion”. This conclusion is not correct. Edges are required to remain straight and free in-plane movements are allowed. This is mentioned for
the finite element model, but unfortunately not explicitly for the present model. Even so, it should be reasonably clear in view of Fig. 1 in the paper that in-plane movements must be allowed, since external loads can be specified on all the four edges. Nevertheless, we agree that the latter condition should have been stated clearly. References [1] Bedair O. The application of mathematical programming techniques to the stability of plate/stiffener assemblies. Computer Methods in Applied Mechanics and Engineering 1997;148(3–4):353–65. [2] Bedair O. The elastic behaviour of multi-stiffened plates under uniform compression. Thin-Walled Structures 1997;27(4):311–35. [3] Brubak L, Hellesland J. Approximate buckling strength analysis of arbitrarily stiffened, stepped plates. Engineering Structures 2007;13. doi:10.1016/j.engstruct.2006.12.002.
Comment
Closure to discussion by O. Bedair of “Semi-analytical buckling strength analysis of plates with arbitrary stiffener arrangements” Lars Brubak a,∗ , Jostein Hellesland a , Eivind Steen b a Mechanics Division, Department of Mathematics, University of Oslo, 0316 Oslo, Norway b Section for Hydrodynamics, Structures and Stability, DNV Maritime, Det Norske Veritas, 1322 Høvik, Norway
Received 31 January 2007; accepted 19 May 2007
The authors would like to thank the discusser for his comments. They are limited to that aspect of the paper that is related to the computation of elastic buckling stress limits (abbreviated as ESL in the paper). The elastic buckling load is used as a parameter in the approximate strength analysis, labelled buckling strength limit (BSL) analysis, in the paper. It is the presentation and discussion of the BSL analysis that has been the main objective of the paper. The discusser has misunderstood when he states that another main objective has been to compare the computational efficiency of the method with the finite element method. If it had been so, we would certainly have devoted more than the present few lines to the topic. The discusser presents a number of comments and questions on details of the ESL analysis. We will try to do justice to the items raised by the discusser. A point-to-point reply is listed below. (1) The discusser must have misunderstood when he writes that the authors have mentioned “that all the semi-analytical methods are restricted to plates with regular stiffeners parallel to the boundaries” (Introduction, last paragraph). The word “all” has been added by the discusser. As can be seen in the paper, we made specific reference to the studies mentioned in the two previous paragraphs of the paper, and for those the statement is correct. The discusser proceeds to provide a total of one reference [1 (in the discusser’s ref. list)] to previous literature on plates ”with arbitrary oriented stiffeners”. That paper is on vibration analysis. Moreover, a stiffener is required to pass through the centre point of the plate.
DOI of original article: 10.1016/j.jcsr.2007.05.009. ∗ Corresponding author.
E-mail address: [email protected] (L. Brubak). c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.05.012
It cannot be placed and oriented arbitrarily at different locations, which limits the generality of the method. However, we appreciate this information as vibration and buckling analysis have several aspects in common. (2) The discusser states that the authors have overlooked the literature on the subject. The authors can appreciate that there are many papers on semi-analytical methods. Most of them are on elastic buckling (eigenvalue) analysis as such, and some of them are on such analysis for plates with various stiffening configurations parallel to the plate boundaries. Work by the discusser, for instance [1,2], are examples of such studies. Studies of plates with inclined stiffeners are very limited (see also item (1) above). The main objective of the paper is strength rather than elastic buckling as such. Therefore, the studies reviewed by the authors are primarily on semi-analytical strength rather than on buckling analysis. We concede that the review of papers on buckling possibly could have been somewhat more extensive. However, the buckling load computation is only a part of our strength analysis, and we decided to adopt a rather conventional buckling analysis approach that we considered to be suitable in the wider context of the paper. Our primary contribution with regard to the buckling analysis, that includes some simplifications that increase the computational efficiency, has been to verify and document the applicability of it to a wide variety of parameters. Although, we consider that to be an important contribution, we do not pretend to contribute any significant novel aspects to the approach. Therefore, we feel that an exhaustive literature review of elastic buckling literature would not have been altogether appropriate in the context of our strength analysis study. The discusser feels that comparisons of the computational efficiency with other semi-analytical methods should have been performed. Considering that the elastic buckling
L. Brubak et al. / Journal of Constructional Steel Research 63 (2007) 1618–1620
(eigenvalue) computations are not the most time consuming part in our strength predictions, comparisons with methods that do not consider strength predictions are not totally relevant. Nevertheless, this is not to say that separate comparisons of the computational efficiency of the buckling analysis part with more rigorous and therefore, most likely, less computationally efficient methods, are not of interest. Such comparisons, such as for instance with a buckling analysis method by the discusser [1,2], poses a practical problem since sufficient details in conjunction with the computational time are normally missing in the papers. Therefore, a comparative study of the type suggested by the discusser, would probably require a paper with that as one of the major objectives. (3) In the considered applications of the study, and in the cases studied, the stiffeners are torsionally rather weak and the neglect of the torsional stiffness is acceptable. The authors agree that it could have been mentioned that this neglect will be conservative for stiffeners with strong torsional stiffness. As the discusser knows, it is a trivial matter to include the torsional stiffness, and in another work by the authors [3], it has been included. (4) Since the axial stiffener stiffness is neglected in the paper, the stiffeners are only connected to the out-ofplane displacements of the plate and not to the in-plane displacements. See also item (5) below. The discusser must have misunderstood when he states that we have used two coordinate systems. The stiffener curvature w,ss in the stiffener direction is related to one single, global Cartesian coordinate system (xi, yj). Possibly, the paper could have given more details on this aspect. The stiffener curvature was taken according to w,ss = ∇(∇w · s) · s
(1)
where ∇ = i∂/∂ x + j∂/∂ y and s = (L x i + L y j)/L s is the unit direction vector in the direction of the stiffener. (5) As explained in the paper, but overlooked by the discusser, eccentric stiffeners are reflected in a simplified manner. An eccentric stiffener is included in the same manner as a concentric stiffener, but with its effective second moment of area Ie (Eq. (16) in the paper). For instance, for the T-stiffener illustrated in Fig. 1, Ie is calculated about an estimated axis of bending (z = z c ) for the cross-section consisting of the stiffener and an assumed effective plate width be (the second moment of area of the effective plate width about its own axis is not included as it is included in the energy formulation itself). The required assumption of be represents an uncertainty, but results are found not to be too sensitive to reasonable variations in this value. This is clearly a simplified approach, but numerous comparison with finite element analysis, including those given in the paper, have shown that it is acceptable for the plates, boundary conditions and loadings considered in the paper. (6) According to the discusser, the neglect of the axial stiffener stiffness will have the most pronounced effect in cases with inclined stiffeners. This is exactly the kind of cases studied
1619
Fig. 1. Stiffener profile of a stiffener with an effective plate width be .
in the paper. Comparison with finite element analysis for a wide range of stiffener and plate dimensions, have shown that this neglect had only minor effects on the computed buckling loads of the considered (free-to-move in-plane) plates. The same conclusion can be drawn for the selected ESL results shown in the paper. (7) It is quite correct that we have not given any guidelines on how to choose the rotational spring coefficient kr , or the translational (out-of-plane) spring coefficient kt for that matter. We did not consider that it was necessary to provide further details, but left it to the user to choose appropriate values. Engineers are normally quite familiar with such coefficients. The rotational springs along the edges are included in the formulations in order to be able, for instance, to do parametric studies of the effect of different boundary conditions, varying from simply supported to rotationally fully clamped. For clamped plates, the rotational springs must be chosen large enough so as to prevent edge rotations. To model the rotational stiffness provided by an edge girder, kr must be chosen so as to reflect the torsional stiffness of the edge girder. Alternatively, in the same manner as in Brubak and Hellesland [3], the torsional energy of the edge girder can be included in the total potential energy formulation. (8) In the paper we give some information on computational speed of the presented method compared to nonlinear FEM analyses. The discusser states that the given numbers are for a plate with “an intermediate stiffener” (dividing the plate into two halves). This is not correct. The given relative numbers (Section 6, Validation) are provided as general information, typical for the plates analysed, most of which had irregular stiffener arrangements. The comments in the discusser’s item 8(a) is based on the same misunderstanding mentioned above. For irregularly stiffened plates, it is not possible to take advantage of symmetry. In the paper we state that the relative computational speed numbers are given “as an indication of the relative computational efficiency” (Section 6, Validation). We further state that “these numbers are just given as an indication of the relative computational efficiency, and do not reflect the results of an in-depth study of the different factors affecting computer time consumption”. In a more
1620
L. Brubak et al. / Journal of Constructional Steel Research 63 (2007) 1618–1620
in-depth discussion, the points made by the discusser in his item 8(b) and (c) naturally belong. In such a discussion, it is also necessary to study the effect of optimising the efficiency of computer routines in the present method. The present research version of the program can be significantly improved in that respect. For comparisons of such a version with a more efficient finite element analysis, the relative difference in computational times (CPU) will still be very significant. This is an important advantage of the method in engineering applications and weight optimisation and reliability studies. (9) The discusser concludes from inspection of some of the figures in the paper that the boundaries “were treated as fully restrained against in-plane motion”. This conclusion is not correct. Edges are required to remain straight and free in-plane movements are allowed. This is mentioned for
the finite element model, but unfortunately not explicitly for the present model. Even so, it should be reasonably clear in view of Fig. 1 in the paper that in-plane movements must be allowed, since external loads can be specified on all the four edges. Nevertheless, we agree that the latter condition should have been stated clearly. References [1] Bedair O. The application of mathematical programming techniques to the stability of plate/stiffener assemblies. Computer Methods in Applied Mechanics and Engineering 1997;148(3–4):353–65. [2] Bedair O. The elastic behaviour of multi-stiffened plates under uniform compression. Thin-Walled Structures 1997;27(4):311–35. [3] Brubak L, Hellesland J. Approximate buckling strength analysis of arbitrarily stiffened, stepped plates. Engineering Structures 2007;13. doi:10.1016/j.engstruct.2006.12.002.