On the Correspondence between 2D Force Networks and Polyhedral Stress Functions

Carpentieri, Gerardo

doi:10.1260/0266-3511.29.3.145

Cited by 22 publications

(25 citation statements)

References 50 publications

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“…The open web in Figure 7(c) supports these three forces, and the associated Airy function (up to addition of an affine function, see also [9]) is…”

Section: Stuck Loadingsmentioning

confidence: 76%

“…This extension field is differentiable a.e. and, at every point x of differentiability, inequality (9) implies that e( u) ≥ −κI. In order to apply Green's formula, we introduce a regularized field u η := u * ρ η , ρ η being a smooth convolution mollifier (i.e., non negative, supported in the ball of radius η, and such that ρ η = 1).…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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On the forces that cable webs under tension can support and how to design cable webs to channel stresses

et al. 2019

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In many applications of Structural Engineering the following question arises: given a set of forces f1, f2, . . . , fN applied at prescribed points x1, x2, . . . , xN , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x1, x2, . . . , xN in the two-and three-dimensional case. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two-dimensions we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f1, f2, . . . , fN applied at points strictly within the convex hull of x1, x2, . . . , xN . In three-dimensions we show that, by slightly perturbing f1, f2, . . . , fN , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for distributing stress in desired ways.1 By "admissible stress state" we mean an equilibrium state in which all bars are either in tension, or carrying no load. Thus, by "supporting", we mean supporting with all bars either in tension, or carrying no load.2 Notice that we solve this problem within the context of infinitesimal elasticity: examples of applications of the finite deformation theory to describe the geometric nonlinearity are given, for instance, by [6,18,17].

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“…The open web in Figure 7(c) supports these three forces, and the associated Airy function (up to addition of an affine function, see also [9]) is…”

Section: Stuck Loadingsmentioning

confidence: 76%

Section: Proof Of Theorem 11mentioning

confidence: 99%

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

et al. 2019

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“…Given that the web supports, under tension, the forces t i at the points x i it will also support the forces t i at the pointsx i , wherẽ 4) and one extends the web, as shown in Figure 2(b), by attaching n short wires of length > 0 between x i andx i , i = 1, 2, . .…”

Section: Forces At the Vertices Of A Convex Polygonmentioning

confidence: 99%

“…or equivalently 4) where the quantity on the right is negative when λ > 0 and j + n > i > j + 1 because the truss structure that supports the forces t i = λs i at the points x i has all its truss elements under compression, and thus must satisfy the reverse inequality to (2.17) when the t i are replaced by the forces t i . The parameter λ > 0 must be chosen so this inequality holds for all i and j with i > j, i.e., λ > max…”

Section: Forces At An Arbitrary Collection Of Points In the Planementioning

confidence: 99%

The set of forces that ideal trusses, or wire webs, under tension can support

Milton

2017

International Journal of Solids and Structures

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The problem of determining those multiplets of forces, or sets of force multiplets, acting at a set of points, such that there exists a truss structure, or wire web, that can support these force multiplets with all the elements of the truss or wire web being under tension, is considered. The two-dimensional problem where the points are at the vertices of a convex polygon is essentially solved: each multiplet of forces must be such that the net anticlockwise torque around any vertex of the forces summed over any number of consecutive points clockwise past the vertex must be non-negative; and one can find a truss structure that supports under tension, and only supports, those force multiplets in a convex polyhedron of force multiplets that is generated by a finite number of force multiplets each satisfying the torque condition. Progress is also made on the problem where only a subset of the points are at the vertices of a convex polygon, and the other points are inside. In particular, in the case where only one point is inside, an explicit procedure is described for constructing a suitable truss, if one exists. An alternative recipe to that provided by Milton, and Onofrei [8], based on earlier work of Camar Eddine and Seppecher [2], is given for constructing a truss structure, with elements under either compression or tension, that supports an arbitrary collection of balanced forces at the vertices of a convex polygon. Finally some constraints are given on the forces that a three-dimension truss, or wire web, under tension must satisfy.

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“…The modern Discrete Element Modeling (DEM) of masonry structures includes computerassisted, funicular-network procedures [11], Lumped Stress Models [12,13,14,15], and Thrust Network Approaches (TNA) [16,17,18]. A recent study [19] has presented a tensegrity approach to the 'minimal-mass' FRP-/FRCM reinforcement of masonry vaults and domes.…”

Section: Introductionmentioning

confidence: 99%

Minimal Mass Design of Strengthening Techniques for Planar and Curved Masonry Structures

Carpentieri¹,

Fabbrocino²,

Piano³

et al. 2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

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Abstract. We present a discrete element model of a masonry structure strengthened through the application of reinforcing elements designed to work in tension. We describe the reinforced masonry structure as a tensegrity network of masonry rods, mainly working in compression, and tension elements corresponding to fiber-reinforced composite reinforcements, which are assumed to behave as elastic-perfectly-plastic members. We optimize a background structure connecting each node of the discrete model of the structure with all the neighbors lying inside a sphere of prescribed radius, in order to determine a minimal mass resisting structure under the given loading conditions and prescribed yielding constraints. Fiber-reinforced composite reinforcements can be naturally replaced by any other reinforcements that are strong in tension (e.g., timber or steel beams/ties). Some numerical examples illustrate the potential of the proposed strategy in designing tensile reinforcements of a three-dimensional structure composed of a masonry vault and supporting walls.

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On the Correspondence between 2D Force Networks and Polyhedral Stress Functions

Cited by 22 publications

References 50 publications

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

On the forces that cable webs under tension can support and how to design cable webs to channel stresses

The set of forces that ideal trusses, or wire webs, under tension can support

Minimal Mass Design of Strengthening Techniques for Planar and Curved Masonry Structures

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