Stability of trusses by graphic statics

This paper presents a graphical method for determining the linearized stiffness and stability of prestressed trusses consisting of rigid bars connected at pinned joints and which possess kinematic freedoms. Key to the construction are the rectangular areas which combine the reciprocal form and force diagrams in the unified Maxwell–Minkowski diagram. The area of each such rectangle is the product of the bar tension and the bar length, and this corresponds to the rotational stiffness of the bar that arises due to the axial force that it carries. The prestress stability of any kinematic freedom may then be assessed using a weighted sum of these areas. The method is generalized to describe the out-of-plane stability of two-dimensional trusses, and to describe three-dimensional trusses in general. The paper also gives a graphical representation of the ‘product forces’ that were introduced by Pellegrino and Calladine to describe the prestress stability of trusses.

The paper is exceptionally well written, and the figures present difficult graphical concepts in a clear way. The following comments are all minor and are mostly suggestions to help further improve the clarity of the paper. • Title: Suggest to remove "prestress" from the title. The method is not only applicable to prestressed structures, but simply stressed (or loaded) structures as well. However, the rigidity assumption is a big one, and this should be added to the title. Particularly in light of the fact that other recently submitted papers seem to focus on non-rigid structures, so the unique contribution of the paper is the rigid assumption. • Intro: Suggest to clarify what is meant by "prestress" and clarify the structural determinacy for which the method is applicable. • Page 3, line 58: State more specifically (here or later) what is meant by rotated 90 degrees. Also clarify whether this paragraph only refers to 2D, or 3D as well?
• Section 3a -Beautifully and very briefly demonstrates the concept. • Figure 3 (& following): Suggest to simply add the bar "p" and make the right support a roller. The current presentation was at first confusing, since it wasn't clear if p was there or not, or what the dashed line of p meant, and then "P" shows up in the force diagram. Alternatively, explicitly state that the solution which follows assumes that the dashed bar p exists, and the support forces are ignored. • Figure 3 (right): Suggest to explain in the text that line P would overlap with lines Q and R, but these lines are drawn offset for clarity, and this convention is used throughout the paper. • Page 7, line 56: maybe revise to "triangle 4 of the force diagram (in Fig. 3)" • Figure 10 caption: suggest to change to "…when viewing the buckled shape…". • End of section 4: Suggested edit: " Fig. 13 shows a potentially more economical solution to the design problem". • Page 13, line 56: should "term" be "terms". • Page 14, line 24: should "follows" be "follow" • Page 14, bottom: Suggest to add "between the two tension rings" to make the following: "consisting of a flying column (between the two tension rings) and two radial tension "spokes" that connect it to the surrounding compression ring". • Section 5: Tottenham roof: Agreed that the analysis nicely demonstrates stability, though the graphic statics contribution in this case is not particularly obvious. Does the graphic statics methodology shed any new light on this stability? Couldn't the same thing be said using section 2a? Or the simple pendulum analogy? • Figure 15 (right) = FYI. One of the gray arrows is still on the figure. The other is deleted. • Figure 20 (right): The forces in this diagram should all be multiplied by W/H, correct? • Figure 21: I see value in the proposed method to graphically display the effective rotational stiffness. This might be beneficial in design. However, what additional information does Figure 21 hold compared to 17? The benefit (to the structural designer, or otherwise) of the graphical product force method versus matrix methods or the displacement diagram in Figure 17 could be stated more explicitly at the end of this section. • Final paragraph = is reference [15] one of the companion papers mentioned? If so, add reference to text.

Decision letter (RSOS-201970.R0)
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Dear Mr McRobie
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-201970 "Prestress Stability by Graphic Statics" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
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Best regards, Lianne Parkhouse Editorial Coordinator Royal Society Open Science openscience@royalsociety.org on behalf of Professor R. Kerry Rowe (Subject Editor) openscience@royalsociety.org Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) The authors are to be commended for a series of elegant graphical solutions to long-standing problems in structural mechanics. The problems have been solved previously elsewhere, but the authors confirm the results with new methodologies. This reviewer does not know of other similar work verifying stability through graphical solutions.
Major comments: 1) Given that the new methods are confirming previously known results, it would be helpful to better articulate exactly how the graphical methods are superior to matrix methods. Shorter computational time? More robust solutions? Easier to generalize? This reviewer would like to hear a clearer argument as to why graphical solutions are preferable to matrix solutions.
2) The authors mention companion papers and they cite several recent papers, particularly references [11, 13, 15, and 16], which they have published or submitted. It would be useful if they could clarify how the current paper differs from those previous papers. The paper provides a novel theoretical contribution by providing a new graphical solution to the stability of truss structures. More specifically, it provides a framework that enables a graphical depiction of the effective stiffness, and therefore stability. The paper focuses only on structures with "rigid elements", which allows the formulations to provide conceptual understanding of stability in an accessible manner. The authors appear to have extended this solution to non-rigid elements in another paper. The paper is exceptionally well written, and the figures present difficult graphical concepts in a clear way. The following comments are all minor and are mostly suggestions to help further improve the clarity of the paper. • Title: Suggest to remove "prestress" from the title. The method is not only applicable to prestressed structures, but simply stressed (or loaded) structures as well. However, the rigidity assumption is a big one, and this should be added to the title. Particularly in light of the fact that other recently submitted papers seem to focus on non-rigid structures, so the unique contribution of the paper is the rigid assumption. • Intro: Suggest to clarify what is meant by "prestress" and clarify the structural determinacy for which the method is applicable. • Page 3, line 58: State more specifically (here or later) what is meant by rotated 90 degrees. Also clarify whether this paragraph only refers to 2D, or 3D as well?
• Section 3a -Beautifully and very briefly demonstrates the concept. • Figure 3 (& following): Suggest to simply add the bar "p" and make the right support a roller. The current presentation was at first confusing, since it wasn't clear if p was there or not, or what the dashed line of p meant, and then "P" shows up in the force diagram. Alternatively, explicitly state that the solution which follows assumes that the dashed bar p exists, and the support forces are ignored. • Figure 3 (right): Suggest to explain in the text that line P would overlap with lines Q and R, but these lines are drawn offset for clarity, and this convention is used throughout the paper. • Page 7, line 56: maybe revise to "triangle 4 of the force diagram (in Fig. 3)" • Figure 10 caption: suggest to change to "…when viewing the buckled shape…". • End of section 4: Suggested edit: " Fig. 13 shows a potentially more economical solution to the design problem". • Page 13, line 56: should "term" be "terms". • Page 14, line 24: should "follows" be "follow" • Page 14, bottom: Suggest to add "between the two tension rings" to make the following: "consisting of a flying column (between the two tension rings) and two radial tension "spokes" that connect it to the surrounding compression ring". • Section 5: Tottenham roof: Agreed that the analysis nicely demonstrates stability, though the graphic statics contribution in this case is not particularly obvious. Does the graphic statics methodology shed any new light on this stability? Couldn't the same thing be said using section 2a? Or the simple pendulum analogy? • Figure 15 Figure  21 hold compared to 17? The benefit (to the structural designer, or otherwise) of the graphical product force method versus matrix methods or the displacement diagram in Figure 17 could be stated more explicitly at the end of this section. • Final paragraph = is reference [15] one of the companion papers mentioned? If so, add reference to text.

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Author's Response to Decision Letter for (RSOS-201970.R0) See Appendix A.

Decision letter (RSOS-201970.R1)
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Dear Mr McRobie,
It is a pleasure to accept your manuscript entitled "Stability of Trusses by Graphic Statics" in its current form for publication in Royal Society Open Science. The comments of the reviewer(s) who reviewed your manuscript are included at the foot of this letter.
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