In his work on stress functions Maxwell noted that given a planar truss the internal force distribution may be described by a piecewise linear, \(C^0\) continuous version of the Airy stress function. Later Williams and McRobie proposed that one can consider planar moment-bearing frames, where the stress function need not be even \(C^0\) continuous. The two authors also proposed a discontinuous stress function for the analysis of space-frames, which however suffers from incompleteness. This paper provides a discontinuous stress function for n-dimensional space frames that is complete and minimal, along with its derivation from an n-dimensional continuous stress function. The continuous stress function generalizes both Günther’s and Maxwell’s stress functions.
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1 Introduction
In his work on stress functions Maxwell [1] noted that given a planar truss the internal force distribution may be described by a piecewise linear, \(C^0\) continuous (polyhedral) version of the Airy stress function [2]. The bar forces correspond to the change of slope of the polyhedral Airy stress function, as a discrete version of the second derivatives. The same paper also deals with graphic representation of force distributions. It relates the polyhedral Airy stress function with the reciprocal diagram (Cremona force-plan [3] rotated with 90 degrees) of the planar truss. It also gives a scalar valued stress function for spatial problems the discrete analogue of which is compatible with force-diagrams of spatial trusses according to the representational idea of Rankine [4]; although Maxwell himself shows this stress function is not a complete description of static equilibrium. He then gives a vector valued stress function for spatial problems that is a superposition of three orthogonal copies of Airy’s planar description. This vector valued stress function was later shown to be complete [5] for a large amount of engineering purposes.
The marriage of graphic representation with stress functions was also heavily emphasised in some recent works in graphic statics: Williams and McRobie [6] proposed that one can consider planar moment-bearing frames, where the stress function need not be even \(C^0\) continuous. The three dimensional version of this discontinuous function, intended for space-frames was also introduced [7, 8] relying on the scalar valued stress function of Maxwell; where the authors themselves have shown their stress function to be an incomplete description of static equilibrium. It was later shown [9] that although in general incomplete, this description is complete for single layer gridshells loaded unidirectionally and may be derived from Pucher’s equation and Airy’s stress function.
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This paper provides a dimension-independent discontinuous stress function for space-frames that is complete and minimal. On the application-side this allows for the efficient computation of space-frames without the method prescribing spatial or combinatorical constraints on the structure. On the theoretical-side we believe the derivation of the stress function has additional explaining power and as such we present it here. During the derivation process we were guided by three requirements we set observing past works.
1.
The discontinuous stress function has to be derived from a sufficiently complete continuous one.
2.
The continuous stress function has to be derived from the static equilibrium equations.
3.
The description should be dimension independent.
In the first requirement, the completeness of a stress-function is intimately connected to the topology of the problem [10]. Both the Beltrami Stress Function and Günther’s Stress Function for Cosserat continua are only complete in their original forms for stress fields that are totally self-equilibrated [11, 12] and have been generalized to include a broader set of solutions. If the solid considered has a single closed boundary (hyper-) surface then static equilibrium and the single closed boundary surface together imply that the stress field is totally self-equilibrated. We will only consider solids in our continuous description that are homeomorphic to a n-dimensional cube in \(\mathbb {R}^n\) (we have based our description on the Poincaré Lemma), as such it will only describe totally self-equilibrated stress fields. The reason for this is that we will capture the topology of the problem in the discrete part of the description, using tools from graph theory. As such we will only give the connection with the non generalized stress functions of Beltrami and Günther.
The second requirement should be self explanatory. The third is motivated by the desire to solve the problem entirely, to give a description that does not loose its explaining power as the dimension of the space grows. When describing spatial rotations Weyl [13] phrased this rather eloquently: "The above treatment of the problem of rotation may, in contradistinction to the usual method, be transposed, word for word, from three-dimensional space to multi-dimensional spaces. This is, indeed, irrelevant in practice. On the other hand, the fact that we have freed ourselves from the limitation to a definite dimensional number and that we have formulated physical laws in such a way that the dimensional number appears accidental in them, gives us an assurance that we have succeeded fully in grasping them mathematically.”
1.1 Notation, preliminaries
We will use k-vector valued differential forms. One way of looking at it is that one has the Grassmann algebra in \(\mathbb {R}^n\) where the scalar coordinates are replaced with differential forms. (Scalar valued forms in this interpretation, and the degree of them has to be the same in all coordinates). Real-number multiplication of the coordinates is replaced with the wedge product of differential forms. We will omit the wedge-product sign if possible and denote the "directions" of the Grassmann algebra with \(\varvec{x}_i,\varvec{x}_i \varvec{x}_j, \dots\) (for 1-vectors, 2-vectors, \(\dots\)) while the components of the differential forms with \(\varvec{d}_i, \varvec{d}_i \varvec{d}_j\) and so on. The coordinates (functions) will be labelled by upper indices corresponding to the index-sets, for instance a 3-form coordinate of a 2-vector will have components like \(\alpha ^{12,123}\varvec{d}_1\varvec{d}_2\varvec{d}_3 \ \varvec{x}_1\varvec{x}_2\). We will mostly follow the convention to list \(\varvec{x}_i\) and \(\varvec{d}_i\) in lexicographic order.
Remark 1
Admittedly, this is far from standard notation. The point of choosing it is to keep track of what object has what mechanical meaning. See for instance forces and moments below.
We will need the Hodge-duals [14] of both k-vectors and k-forms (we assume the usual Euclidean metric). We denote the Hodge-star on k-vectors with \(\star _x\), mapping a k-vector to an \((n-k)\)-vector. This map may be given on the base vectors as
where \((1 \dots n )\) is the disjoint union of \(\mathcal {I}\) and \(\mathcal {J}\) and \(\tau\) is the number of permutations required to bring \((\mathcal {I},\mathcal {J})\) to \((1 \dots n)\). Similarly we have the one on scalar valued differential forms as
where \((1 \dots n )\) is the disjoint union of \(\mathcal {I}\) and \(\mathcal {J}\) and \(\tau\) is the number of permutations required to bring \((\mathcal {I},\mathcal {J})\) to \((1 \dots n)\). Under the Hodge-dual of a k-vector valued differential form we will mean the form achieved by applying composition of the two stars, i.e: \(\star =\star _x\circ \star _d=\star _d \circ \star _x\). This way \(\varvec{\alpha } \wedge \star \varvec{\alpha }\) gives an n-vector valued n-form.
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We will need the scalar product of k-vector valued m-forms \(\varvec{\alpha }\) and \(\varvec{\beta }\), defined as
Forces will correspond to 1-vectors while moments to 2-vectors. Force \(\varvec{F}=\sum F^i \varvec{x}_i\) acting at point \(\varvec{x}=\sum x^i \varvec{x}_i\) has moment \(\varvec{x} \wedge \varvec{F}\) with respect to the origin. In \(\mathbb {R}^n\) this is a 2-vector with \({n \atopwithdelims ()2}\) coordinates, as there exists a moment with respect to the ortho-complement of each plane. (A moment is a rotating effect of a force and rotations can happen in each plane of the space.) The moment introduced this way differs from the engineering moment in \(\mathbb {R}^3\) having the sign of the second component flipped. This is captured in the relations \(\varvec{a}\times \varvec{b} =\star _x(\varvec{a}\wedge \varvec{b})\), \(\star _x(\varvec{a}\times \varvec{b}) =\varvec{a}\wedge \varvec{b}\) (where \(\varvec{a},\varvec{b}\in \mathbb {R}^3\)).
Stresses (force-like) in general are \((n-1)\)-forms, that need to be integrated along \(n-1\) dimensional hyper-surfaces. As a consequence they will be measured in \([N/mm^{d-1}]\). As an example, we express the \(\varvec{x}_1\) directional stresses in \(\mathbb {R}^3\) as
where the minus sign comes from the lexicographic ordering (\(1 \varvec{d}_1\varvec{d}_3=-1 \varvec{d}_3\varvec{d}_1\)).
Couple-stresses are also \((n-1)\)-forms, we will denote them with \(\varvec{\mu }\).
Strains may be represented with a 1-vector valued 1-form \(\varvec{\epsilon }\). Given some volume V of the material the work of the stresses on the strains inside V may be expressed by integrating the volume form
which is an n-vector, thus isomorphic to a scalar. Here \(\star _x \varvec{\epsilon }\) is an \((n-1)\)-vector valued 1-form. The spaces of 1-vectors and \((n-1)\) vectors play the role of the usual vector-space and the dual space of linear functionals. Since in case of \(\mathbb {R}^n\) they are isomorphic to each other, no attempt is made here to assign the role of primal-space and dual-space.
Body forces (self-weight/volume) correspond to vector valued n-forms, we will denote them with \(\varvec{\rho }\). Body-moments (per volume) are two-vector valued n-forms, we will denote them with \(\varvec{\xi }\).
The exterior derivative is taken coordinate-wise on the k-vectors, we will denote it with \(\text {d}( \ )\), omitting the parentheses if no confusion arises. The exterrior derivative satisfies \(\text {d}^2\varvec{\alpha }=\text {d}(\text {d}\varvec{\alpha })=\varvec{0}\) for all twice differentiable forms. It follows from the computational rules that given k-vector valued forms \(\varvec{\alpha }\) and \(\varvec{\beta }\) the Leibniz Rule
holds, where \(\text {deg}(\varvec{\alpha })\) denotes the degree of \(\varvec{\alpha }\). We will heavily rely on the following two results [15]:
Lemma 1
(Poincaré Lemma) If \(\text {d} \varvec{\alpha } =\varvec{0}\) throughout a simply connected region, then \(\exists \varvec{\beta }: \ \varvec{\alpha }=\text {d} \varvec{\beta }\).
Theorem 1
(Fundamental Theorem of Exterior Calculus / Generalized Stokes’s Theorem) Given a compact, oriented \((p+1)\) dimensional region R, its boundary \(\partial R\) and p-form \(\varvec{\alpha }\): \(\int _R \text {d}\varvec{\alpha } = \int _{\partial R} \varvec{\alpha }\).
2 The continuous stress function
We will be able to handle the case when both the body forces and moments have potential forms, that is there exist \(\varvec{\pi }\) and \(\varvec{\nu }\) such that \(\varvec{\rho }=\text {d}\varvec{\pi }\) and \(\varvec{\xi }=\text {d}\varvec{\nu }\), holds. In general potential forms are not unique, we will chose the ones having appropriate shapes later.
Let us cut out some volume V of the material. The equilibrium of the stresses acting on its boundary surface \(\partial V\) and of the body forces acting on V may be expressed as
implying the existence of \((n-2)\)-form \(\varvec{\psi }\) such that \(\varvec{\sigma }+\varvec{\pi }=\text {d}\varvec{\psi }\).
The moment of the force-like stresses acting on a small piece of hyper-surface at location \(\varvec{x}\in \mathbb {R}^n\) is calculated as \(\varvec{x} \wedge \varvec{\sigma }\) (with respect to the origin of the coordinate system). Similarly the moment of the body-forces may be expressed as \(\varvec{x} \wedge \varvec{\rho }\). Cutting out some volume V of the material, the equilibrium of moments may be expressed as
(here \(\text {d}\varvec{x}=\sum _{i=1\dots n} 1 \ \varvec{d}_i \varvec{x}_i\)). At this point we have to go back to the fact that \(\varvec{\pi }\) is not unique and prescribe
It is shown in the “Appendix” that this condition is without the loss of generality (through the example of \(\varvec{\psi }\)). This guarantees \(\text {d}\varvec{x}\wedge \varvec{\pi }=\varvec{0}\) and we may write:
This may be considered a dimension-independent generalization of Günther’s stress function [16]. By setting \(\varvec{\pi }=\varvec{0}=\varvec{\xi }\) this can be compared with Theorem 4.1 and 4.2 of Carlson’s paper on this stress function [12], where he denoted \(\varvec{\psi }\) with F and \(\varvec{\omega }\) with G (and used tensorial notation).
We may now set \(\varvec{\mu }=\varvec{0}=\varvec{\xi }\) and see how the Beltrami stress function is related to what we derived. The question becomes: Can we reconstruct \(\varvec{\psi }\) from \(\text {d}\varvec{\omega }\) and if so how? For an arbitrary differential form \(\varvec{\alpha }\) the map \(\varvec{\alpha } \mapsto \text {d}\varvec{x}\wedge \varvec{\alpha }\) contains information loss, but for certain forms it is actually reversible. The idea can be seen in \(\mathbb {R}^3\) with the cross product as:
For differential forms \(\varvec{\psi }\) satisfying the orthogonality condition \(\left[ \varvec{\psi }; \text {d}\varvec{x} \right] =0\) map (19) interchanges the indices as \(\overline{\psi }^{\mathcal {P},i}=(-1)^n \psi ^{i,\mathcal {P}}\) taking the vector valued \((n-2)\)-form to a \((n-2)\)-vector valued 1-form; furthermore map (20) is the inverse of map (19). Due to the non uniqueness of the potential-forms we may prescribe some constraints on \(\varvec{\phi }\) and \(\varvec{\omega }\) enforcing \(\left[ \varvec{\psi }; \text {d}\varvec{x} \right] =0\), and use the maps above.
For \(n=2\)\(\varvec{\omega }\) is a function and there is nothing to prescribe, the orthogonality condition automatically holds. In fact this function is exactly the Airy stress function. For \(n=3\)\(\varvec{\omega }\) is a 1-form and we can have its components correspond to the Beltrami stress function. Using the notation of Sadd [17] this would mean \(\omega ^{12,3}=\Phi ^{33}\), \(\omega ^{13,2}=-\Phi ^{22}\) and \(\omega ^{23,1}=\Phi ^{11}\) for Maxwell’s part and \(\omega ^{13,1}=-\omega ^{23,2}=-\Phi ^{12}\), \(-\omega ^{12,1}=-\omega ^{23,3}=-\Phi ^{13}\) and \(-\omega ^{12,2}=\omega ^{13,3}=-\Phi ^{23}\) for Morera’s part. This however is a redundant parametrization, we only need \({n \atopwithdelims ()2}\) parameters [18]. In the general case for arbitrary n we may set
where \(\mathcal {P}\) is an index-set of \(n-1\) elements. We show in the “Appendix” how to do this from a form \(\varvec{\alpha }\) satisfying \(\varvec{\sigma }+\varvec{\pi }=d\varvec{\alpha }\). This is sufficient (but not necessary) condition to satisfy \(\left[ \varvec{\psi }; \text {d}\varvec{x} \right] =0\), but we have to prescribe the corresponding constraint on \(\varvec{\omega }\) for this to be relied on. If \(\varvec{\alpha }=\text {d}\varvec{\omega }\) then
must hold. In other words for any \(\varvec{x}_i\varvec{x}_j\) moment component there is a single non-zero component \(\omega ^{ij,\mathcal {Q}}\), exactly the one where \((1 \dots n )\) is the disjoint union of (i, j) and \(\mathcal {Q}\).
Thus in the absence of couple stresses and loads, given \(\varvec{\rho }\) one has to find the potential function \(\varvec{\pi }\) satisfying Eq. (13). Then one may choose any \((n-2)\)-form \(\varvec{\omega }\) satisfying Eq. (23) and the boundary conditions corresponding to the problem. The stresses may be determined as
Rewriting Maxwell’s less-continuous interpretation of the Airy stress function in this language given a planar truss the internal force distribution may be described by a piecewise linear, \(C^0\) continuous stress function \(\varvec{\omega }\), implying \(\text {d}\varvec{\psi }=\varvec{\sigma }=\varvec{0}\) where there is no structure (between the rods) and \(\text {d}\varvec{\psi }\) not being defined where there is structure. (As we are dealing with a pin-jointed truss, \(\varvec{\rho }=\varvec{\xi }=\varvec{\mu }=\varvec{0}\) has to hold everywhere.) The planar case is somewhat degenerate as although \(\text {d}\varvec{\psi }=\varvec{0}\) holds \(\varvec{\psi }\) cannot be the exterior derivative of anything since it is a 0-form. Regardless, it is not hard to see that \(\text {d}\varvec{\psi }=\varvec{\sigma }=\varvec{0}\) also implies the piecewise linearity of \(\varvec{\omega }\). Williams and McRobie proposed [6] that one can consider planar moment-bearing frames, where the stress function need not be even \(C^0\) continuous and have a well defined value at the axes of the rods. The method they proposed works in case of structures where the rod axes correspond to planar graphs, or equivalently the rod axes are edges of a planar mosaic. We will be able to actually strengthen this description to include structures with non planar-graphs, but for now we will assume the rod axes of the framework to correspond to edges (1-faces) of a convex, polyhedral n-dimensional mosaic.
If the case of moment bearing frames in \(\mathbb {R}^n: n\ge 3\) we will keep the \(\varvec{\rho }=\varvec{0}=\varvec{\xi }\) condition, as Maxwell’s idea turned the loads into self-stresses of the structure. The non-existence of force-like stresses outside the rod axis imply that in these points there exists \(n-3\) form \(\varvec{\Psi }\) such that \(\text {d}\varvec{\Psi }=\varvec{\psi }\). This, together with the non-existence of moment-like stresses outside the rod axes imply that in these points we have
implying the existence of \(n-3\)-form \(\varvec{\Omega }\) such that \(\varvec{\gamma }=\text {d}\varvec{\Omega }\).
If in the continuous case one cuts the solid in half resulting in cut hyper-surface S, the moment 2-vector resultant of the stresses (with respect to the origin) may be expressed as
When transferring this to the less continuous case \(\partial S\) will fall in multiple cells of the mosaic and will fall apart into smaller domains, such that \(\partial S = \cup _i \partial S_i\). These smaller domains will have their boundary of their own and may contain different stress function pieces. Thus we may continue with the potential forms of the potential forms
The use of this equation is that it tells us the shape of the stress function pieces on the continuous parts, the actual equilibrium will depend on what happens at the \(C^1\) or even \(C^0\) discontinuities.
Based on the above, when taking the discontinuous analogue of the continuous function we prescribe the following rules (not stricter than what has been done before) that will determine the discontinuous stress function:
1.
The function need not be defined at the rod axes.
2.
To every point outside the rod axes a single stress function piece has to correspond, satisfying \(\text {d}\varvec{\Psi }_i=\varvec{\psi }_i\) and \(\text {d}\varvec{\psi }_i=\varvec{0}\) on the entirety of \(\mathbb {R}^n\).
We will look at the \(n=3\) case below, in a way that will generalize.
Consider a tetrahedral piece of material, vertices of the tetrahedron denoted by \(\varvec{p}_1,\varvec{p}_2,\varvec{p}_3,\varvec{p}_4\), point \(\varvec{p}\) being inside the tetrahedron (Fig. 1). We want to replace this with 4 pieces of rods running from \(\varvec{p}_i\) to \(\varvec{p}\) connected in a force and moment-bearing way. Choosing \(\varvec{q}_1,\varvec{q}_2,\varvec{q}_3\) to be points close to \(\varvec{p}_1\) on the edges of the tetrahedron, the force resultant of the stresses (in the continuous case) acting on the area enclosed by lines \(\varvec{q}_i,\varvec{q}_j\) may be calculated via line-integrals of 1-forms \(\varvec{\psi }_{ij}\) on the respective lines as
Here the three 1-forms are the same, we labelled them according to the curve-segments to introduce the logic of building up the resultant from parts.
Fig. 1
Tetrahedral piece of material and the discontinuous version of a line integral on it. The negative signs are due to the orientation of the boundary (endpoints) of the curve-segments
×
Distorting the geometry of the tetrahedron into the discontinuous case, we naturally get triangles \(\varvec{p},\varvec{p}_i,\varvec{p}_j\) spanning 2-faces of the mosaic where the continuity of the line-integral may break. To express the condition \(\varvec{\sigma }=\varvec{0}\) we have to define the stress function (it has to be single-valued) in every point that is not on a 1-face (rod axis). We define a separate stress function piece corresponding to the inside of each 3-face (cell) of the mosaic and to the relative inside to every 2-face. (Relative inside of a k-face: points of the k-face that are not elements of a j-face such that \(j<k\).) Thus the discrete version of Eq. (28) will be
where \(\varvec{\Psi }_{ij}\) are the potential functions as introduced above, while \(\varvec{\Psi }_i\) are the discrete, orientation sensitive jumps corresponding to the line-integral passing through the 2-faces of the mosaic. (Here every unique label may denote a unique form. We avoid defining \(\varvec{\Psi }_i\) through the Dirac-delta function since we believe this to be a technicality.) We remark that the positive signs are with respect to the positive orientation of the path integral, we will return to this later. Observing that the path-integral runs in a stress-free region since all the stresses are concentrated to the rod axis we can freely perturb the path without changing the resultant. This includes shrinking the path of integration to an arbitrarily small circle around the rod axis. Forms \(\varvec{\psi }_{ij}\) have an exterior derivative and thus they are finite (on the whole of \(\mathbb {R}^n\)) while their support will tend to a zero measure set, implying their effect is zero in the limit. We may also observe, that around the limit each \(\varvec{\Psi }_i\) has to be constant, since we may perturb the path integral such that the only effected, non-vanishing term is \(\varvec{\Psi }_i(\varvec{x})\). Thus we have to be able to describe the force distribution with constant functions, corresponding only to the two-faces of the mosaic. We extend this description around the limit to apply to any path of integration, effectively setting \(\varvec{\Psi }_{ij}=\varvec{0}\) on the basis that we wish to avoid superfluous parameters.
This argument carries over to \(n>3\) as follows: The domain of integration \(\partial S\) will intersect all the k-faces of the mosaic where \(k>1\). The intersections with the 2-faces will be a zero-measure set that supports the stress function pieces corresponding to each two face-of the mosaic. The intersections with faces \(k>2\) will have a non-zero measure and will support finite functions. As we shrink \(\partial S\) the measure of the support of these components will tend to zero and the effect of the corresponding functions will vanish.
With this established we may similarly write up the moment of the force system with respect to the origin as
The moment of the force system at \(\varvec{p}_1\) with respect to \(\varvec{p}_1\) is \(\varvec{M}_1-\varvec{p}_1\wedge \varvec{F}_1=\sum _{i=1\ldots n} \varvec{\Omega }_i(\varvec{p}_1)\). Thus to each 2-face corresponds a force system, and the stress function \(\varvec{\Omega }_i\) is the moment of the force system with respect to the points in the 2-face. As a consequence
holds, which is the discrete analogue of Eq. (15) in the absence of stresses and body-moments. The force components stored in \(\varvec{\Psi }_i\) may be restored from the derivative, for instance in the way introduced in the continuous case.
At this point we have to treat the signs in the expressions, preferably in way independent of local domains of integration, that can also handle structures not having convex polyhedral geometry. The question is already answered in Graph Theory. Observing Eqs. (29) and (30), we may look at force-and moment components coordinate-wise, each coordinate corresponding to what is often called a flow or circulation [19]. (At each node incoming flow may be identified with the positive direction of the corresponding coordinate). This requires a choice of orientation on each edge of the graph, but the following are valid regardless of orientation: The flow space of a graph is a vector space, spanned by the signed characteristic vectors of the cycles of the graph. A natural way of finding a basis in this vector space is by finding a spanning tree (forest in case of multiple components). Each edge not present in the spanning tree will give a cycle when added to the spanning tree and the corresponding characteristic vector will be a good base-vector.
We will thus orient each edge of the graph of the structure and attach a stress function piece \(\varvec{\Omega }_c(\varvec{x})=\varvec{M}_c-\varvec{x}\wedge \varvec{F}_c\) to each cycle found in the above described way. When determining the force and moment components of a rod we sum over each of the cycles the graph edge corresponding to the rod is present in, with positive sign if the cycle traverses the edge according to its orientation. Defined this way there is a stress function piece corresponding to each generator element of the Fundamental Group of the graph of structure; see section 1.2 in the Algebraic Topology book of Hatcher [20].
The method given here can incorporate planar problems if they are embedded in at least \(\mathbb {R}^3\), with potential functions \(\varvec{\Omega }\) and \(\varvec{\Psi }\) having some constant 0 components. Furthermore, this description strengthens the existing planar one as non-planar graphs can also be computed this way. This will be seen in the proofs below. Before we present these proofs we take a look at the usage of what we derived.
The original idea of Maxwell turned loads and support reactions into internal forces by adding fictitious bars and at least one joint representing the "world" outside the structure, where the loads and support reactions meet. This way the equilibrium of the entire structure is expressed by the equilibrium of the added joint(s). We will refer to the larger structure created this way as the extended structure, to the non-extended one as the original structure. Since loads and support forces of the original structure will have to be sums of stress function pieces, this imposes conditions on the stress function pieces which can be expressed as a system of linear equations. In case the original structure is statically determinate there is a single solution to this system of linear equations. If the original structure is statically indeterminate we have to find the actual solution the structure chooses, depending on its geometry and material properties. If linear elastic materials and small displacements are assumed one possible solution strategy may be quadratic programming, to which we give a (3 dimensional) example below.
3.1 Example
Consider a tetrahedral frame, with moment bearing joints at points \(\varvec{p}_1, \varvec{p}_2, \varvec{p}_3, \varvec{p}_4\)! Let us load it with a concentrated force \(\varvec{L}\in \mathbb {R}^3\) at point \(\varvec{p}_1\)! Support it with pinned supports at \(\varvec{p}_2,\varvec{p}_3\) and a roller at \(\varvec{p}_4\) allowing only \(\varvec{x}_3\) directional force. The structure is drawn in the top of Fig. 2. We will argue using the graph of the structure, joints at \(\varvec{p}_i\) will correspond to graph vertices \(v_i\) while a bar between \(\varvec{p}_i\) and \(\varvec{p}_j\) will correspond to edge \(\{v_i,v_j\}\). To account for the loads and supports, we extend the graph by adding vertex \(v_0\) and additional graph edges \(\{v_0, v_i\}\) representing the load and the supporting forces. This is drawn in the lower part of Fig. 2.
A spanning tree of the graph is given by all the edges containing vertex \(v_0\). Thus our cycles are
To each cycle corresponds a stress function piece \(\varvec{\Psi }_c=(\varvec{F}_c,\varvec{M}_c)\), \(c=1\ldots 6\), the coordinates of which will be our unknowns. We will adopt the notation that whenever referring to a force system present in a bar, we express the coordinates of the force system that acts on the joint with the smaller index. A stress function piece contributing to a bar force will appear with positive sign if the corresponding cycle traverses the vertices of the edge corresponding to the bar in ascending order. As the load corresponds to edge \(\{v_0, v_1\}\), the prescribed load turns into a condition on the stress function pieces as
where the last two equations represent directional constraint of the roller. Equations (32)–(37) may be collected as system of linear equations in the shape of
where \(\varvec{y}\) contains the unknowns \(\varvec{F}_c\) and \(\varvec{M}_c\), \(\varvec{C}\) is the coefficient matrix and \(\varvec{b}\) contains the effect of the load.
The elastic deformational energy stored in the bar between \(\varvec{p}_j\) and \(\varvec{p}_k\) may expressed from the \((\varvec{F}_c,\varvec{M}_c)\)-shaped dynames with the help of a 6-by 6 matrix [21], which may also be used to express this energy as a function of stress function coordinates. Rearranging these equations we may write up the total elastic deformational energy in the form \(E=\sum _{j,k} E_{j,k}=\frac{1}{2}\varvec{y}^T \varvec{Q} \varvec{y}\) (\(k>j\)) where \(\varvec{Q}=\varvec{Q}^T\) and arrive at a quadratic programming problem in the form
by relying on the Principle of the Minimum of Complementary Potential Energy.
Fig. 2
An example to handle loads and supports as fictitious internal bars. The world outside the structure is represented by vertex \(v_0\), the equilibrium of the loads and supports correspond to the equilibrium of vertex \({v_0}\)
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3.2 Formal proofs
Unsurprisingly, since we derived the continuous stress function from the equilibrium equations and the discontinuous stress function from the continuous one using only the definition of the discrete structure, the discontinuous stress function is equivalent with the static equilibrium of the extended structure. We will show this in two steps. First we show that each internal stress distribution the stress function gives is automatically in equilibrium, then we show that any solution of static equations can be represented this way by a stress function that is unique (up to the choice of the global coordinate system).
3.2.1 Automatic equilibrium
Theorem 2
The proposed discontinuous stress function gives internal force systems that are in static equilibrium.
Proof
Consider a joint of the structure where j rods meet, and consider the corresponding vertex of the graph of the structure. Each cycle of the graph that travels through the vertex enters on one graph-edge and exits on another. Thus when summing up the force resultants at the ends of the rods acting on the joint, each stress function component will be present twice, with opposite signs. For the equilibrium of forces we have (after some rearrangement)
where k runs in the indices of cycles passing through the vertex. Similarly, we can write up the sum of the moments with respect to \(\varvec{p}\), which after some rearrangement will take the shape of
with the same indices, completing the proof. \(\square\)
3.2.2 Completeness and minimality
We will argue using the extended structure. Here we don’t prescribe boundary conditions as they would only exclude certain loads and we are interested in parametrizing the general case.
Theorem 3
The proposed discontinuous stress function is a complete and minimal parametrization of the internal forces of the extended structure.
Proof
We have to show that whatever internal force distribution that is in equilibrium is given in the extended structure, there is a unique corresponding stress function built from the appropriate components. Recall how the force components can be calculated through component-wise summation, where the topology of the structure determines the summations. We will manipulate a graph that will act as a topological aid to write up the correct equations. The starting shape of this graph will be the graph of the extended structure. We will calculate the stress function coordinates component-wise, row-by row. Let \(g_i\) denote the \(\varvec{x}_1\) directional force component of each stress function piece, where i runs on all the generator-cycles of the cycle space of the graph of the extended structure. Let \(f_k\) denote the \(\varvec{x}_1\) directional component of the bar-force in bar k (k runs on all the bars). For each i in ascending order we may do the following:
Find the cycle in the graph corresponding to \(\varvec{\Psi }_i\) (Fig. 3, left). Choose any bar k in the cycle and express \(f_k\) as
where \(\mathcal {J}_i\) is some index set. After this equation is written up contract the cycle in the graph, unifying the involved vertices to a single one. By contracting the cycle we make sure not to use component \(g_i\) again. This way the index set \(\mathcal {J}_i\) will satisfy: \(j\in \mathcal {J}_i \implies j>i\).
After we do this for all i we get a system of linear equations \(\varvec{A}\varvec{g}=\varvec{f}\), whose coefficient matrix \(\varvec{A}\) is square, upper triangular and each element on the main diagonal is \(\pm 1\). The determinant of this matrix is \(\pm 1\) (the product of the elements on the main diagonal), thus it is invertible and we may solve for \(\varvec{g}\). We may also repeat the whole procedure for the other components of \((\varvec{F}_c,\varvec{M}_c)\), in total \(n+{n \atopwithdelims ()2}\) times.
As such we may find a suitable stress function distribution to any force system in the structure, that is in equilibrium. We may also note that the number of cycles equals \(n+{n \atopwithdelims ()2}\) the degree of static indeterminacy of the extended structure, implying that in the general case of a frame we can not get away with less stress function parameters.
We still have to see, that choosing different bars in each cycle does not give a different stress-distribution, or in other words \(\varvec{f}\) contains force components that parametrize the self-stresses of the structure. This can be seen by doing the cycle-contraction procedure backwards. At each backwards-step \(f_k\) may be used as a parameter and the rest of the unknowns in the step may be calculated from the static equilibrium equations (see Fig. 4). At each backwards-step the number of added nodes equals the number of unknown components and there is an independent equilibrium equation corresponding to each joint of the structure (corresponding to each vertex of the graph). The mechanical interpretation of the equilibrium equation of a fictitious vertex (one that contains a cycle contracted into it) is the sum of all the equilibrium equations of the joints that were present in the cycle. As the procedure restores the internal force distribution of the structure, the proof is concluded. \(\square\)
Fig. 3
Contracting cycle i corresponding to equation i in the process of determining the stress function corresponding to a given force-distribution
Fig. 4
Doing the cycle-contraction procedure backwards to determine non-parameter rod forces
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4 Conclusion
Motivated by the previously open problem of finding a complete three dimensional discontinuous stress function we investigated the subject of stress functions in a systematic way. We based our approach on the differential-form nature of stresses, one of the cornerstones of the connection between elasticity and geometry. We rewrote the static equilibrium equations into a differential-form shaped dimension-independent continuous stress function, that in simply connected domains is equivalent with static equilibrium. Then, we took the defining property of idealized space-frames (stresses are zero everywhere except at rod axes) and applied it to our continuous function, thereby deriving a dimension-independent discontinuous stress function that is equivalent with static equilibrium of space-frames. This approach allowed us not only to solve the previously open problem, but we also improved on the planar construction of Maxwell by being able to treat planar mechanical problems with non planar graphs. Apart from this efficiency we could also see how and why the stress function description works:
We saw that the stress functions are a parametrization of the self stresses of the structure and they should correspond to whatever is causing the static indeterminacy. In the continuous case the base-problem of elasticity is statically indeterminate, in the discontinuous case the roots of the indeterminacy are the cycles in the extended structure. These cycles correspond to the generating elements of the First Fundamental Group of the extended structure, showing that the number of function-pieces required in the discontinuous stress function is determined by the topology and not the metric properties of the structure. Sticking with the idealized line-model of the structure and not taking material properties into account, these metric properties become important if one prescribes constraints in the internal force distribution, like introducing ball-joints enforcing truss-like behaviour. This will tie the discrete stress functions to line-geometry, as for special cases the dynames (\(\varvec{F}_c,\varvec{M}_c\)) turn into projective line-coordinates. We hope to continue this work by investigating space-trusses this way. We would not mind arriving at some graphic representation of the internal force distribution of space-trusses if possible, but we do wish to derive it from geometry instead of relying on a representation scheme rooted only in tradition.
Furthermore, although variational methods at first might seem far from geometry, the use of the discontinuous geometric stress function being equivalent with static equilibrium can be seen when using the Principle of the Minimum of Complementary Potential Energy. This principle requires one to take variations enforcing static equilibrium, which may be cumbersome if tried from a direct description of internal forces. Using the discontinuous stress function provided here one only has to take variations in the space of the stress functions, making the use of this principle trivial.
Declarations
Conflict of interest
The author declares that he has no conflict of interest.
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Assume we have found \(\varvec{\alpha }\) such that \(\varvec{\sigma }+\varvec{\pi }=d\varvec{\alpha }\). We may create \((n-3)\)-form \(\lambda\) as
$$\begin{aligned}&\lambda ^{i,\mathcal {Q}}= 0 \Leftarrow i \in \mathcal {Q} \end{aligned}$$
(42)
$$\begin{aligned}&\lambda ^{i,\mathcal {Q}}= (-1)^\tau \int \alpha ^{i,(i, \mathcal {Q} )} d x_i \Leftarrow i \notin \mathcal {Q} \end{aligned}$$
(43)
where \(\mathcal {Q}\) is an index-set of \(n-2\) elements, \(\tau\) is the number of permutations required to bring the index-set \((i, \mathcal {Q} )\) to lexicographic order and \(dx_i\) denotes integration with respect to the \(\varvec{x}_i\) direction. We may now have \(\varvec{\psi }=\varvec{\alpha }-\text {d}\varvec{\lambda }\), satisfying Eq. (21) and \(\varvec{\sigma }+\varvec{\pi }=\text {d}\varvec{\psi }\) (since \(\text {d}^2\varvec{\lambda }=\varvec{0}\)). If \(\varvec{\sigma } \in C^1\) this is always doable, as shown below.
For \(n=2\) both \(\mathcal {P}\) and \(\mathcal {Q}\) is empty and any 0-form \(\varvec{\psi }\) is good.
For \(n=3\) only \(\mathcal {Q}\) is empty. As an example the \(\varvec{x}_2\) direction of \(\varvec{\alpha }\) looks like
the undesirable part being \(\alpha ^{2,2}\). By integrating it we get function \(\lambda ^2=\int \alpha ^{2,2} dx_2\). The stress and potential components pointing in the \(\varvec{x}_2\) direction will be calculated as
Equations (45) and (46) are the same if \(\frac{\partial ^2 \lambda ^2}{\partial x_i \partial x_j}=\frac{\partial ^2\lambda ^2}{\partial x_j \partial x_i}\) and \(\frac{\partial \alpha ^{2,2}}{\partial x_j}=\frac{\partial ^2 \lambda ^{2} }{\partial x_j \partial x_2}\). Both are satisfied since we have \(\varvec{\alpha } \in C^2\) due to \(\varvec{\sigma }\in C^1\), for Eq. (9) to make sense. We can see from the definition that the integration is always with respect to a single variable and is always possible without having to solve a system of differential equations.
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