![Two different structural systems and the degree of static and kinematic indeterminacy shown in column one. The kinematic mechanism of the first structure drawn in grey. The respective self-stress states shown in the second column (red and blue indicating compression and tension; magnitude of force scaled with line thickness) and the redundancy distribution shown in colorscheme in the third column. The example is adapted from Pellegrino [16].](https://www.researchgate.net/profile/David-Forster-3/publication/383562923/figure/fig1/AS:11431281274742276@1725039711632/Two-different-structural-systems-and-the-degree-of-static-and-kinematic-indeterminacy_Q320.jpg)
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Proceedings of the IASS 2024 Symposium
Redefining the Art of Structural Design
August 26-30, 2024, Zurich, Switzerland
Philippe Block, Giulia Boller, Catherine DeWolf,
Jacqueline Pauli, Walter Kaufmann (eds.)
Structural Analysis Using the Redundancy Matrix and
Graph Theory
David FORSTERa,*, William F. BAKERb, Manfred BISCHOFFa
aUniversity of Stuttgart, Institute for Structural Mechanics
Pfaffenwaldring 7, 70569 Stuttgart
∗forster@ibb.uni-stuttgart.de
bSkidmore, Owings & Merrill, LLP
224 S. Michigan Avenue, Chicago, IL 60604, USA
Abstract
Structural engineers often want to have a redundant structure where the loss of a member would not lead
to structural collapse. For a truss, adding a bar beyond that required for static determinacy renders the
structure redundant, but what is the spatial distribution of the static indeterminacy within the individual
elements of a framework? Can an additional bar be redundant with several existing bars? Are there
truss topologies and geometries that enhance redundancy? The assessment of structures based on such
load-independent quantitative measures can be useful in early design stages to achieve an integrative
planning process for designers and engineers. The degree of static indeterminacy and in particular its
spatial distribution, quantified with the redundancy matrix can be used for assessing structural integrity
of a framework. Focusing on structural properties independent of the individual member stiffness, such
as geometry and topology, graph theory offers yet another tool to assess structural performance. This
paper explores the integration of the Maxwell-Calladine count with the redundancy matrix from theo-
retical structural mechanics and with contributions of graph theory to explore a deeper understanding of
structural redundancy.
Keywords: structural redundancy, static indeterminacy, redundancy matrix, graph theory, structural analysis
1. Introduction
A fundamental question for structural engineers, especially in early design stages, is whether a struc-
tural assembly is rigid or not. Here, the term “rigid” refers to being free from kinematic modes with zero
stiffness, rather than “infinitely stiff”. One answer lies in the degree of static indeterminacy, which quan-
tifies the amount of redundant load-transfer mechanisms as an integer number, introduced by Maxwell
in 1864 [1]. This counting rule, however, is not sufficient to assess whether a structure has states of
self-stress and kinematic mechanisms at the same time. Therefore, Calladine completed the counting
theory with recognition that the Maxwell count equals the number of mechanisms less the number of
states of self-stress [2].
One drawback of the counting rules is that they do not provide any information about the spatial distribu-
tion of static indeterminacy over the individual elements of a framework. In the group of Klaus Linkwitz
at the University of Stuttgart, the so-called redundancy matrix was derived, which quantifies this distri-
bution of the degree of static indeterminacy in a structure [3, 4, 5]. A detailed derivation for truss and
beam structures can be reviewed in von Scheven et al. [6]. The redundancy matrix proves feasible as a
performance indicator for the design and optimization of structures. Examples with regards to robustness
Copyright ©2024 by David FORSTER, William F. BAKER, Manfred BISCHOFF.
Published in the Proceedings of the IASS Annual Symposium 2024 with permission.
Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
and structural on-site assembly are shown in Forster et al. [7]. In the context of integrative computational
design and construction processes, as described in Knippers et al. [8], the redundancy matrix can be used
to quantify the individual element’s importance in coreless filament-wound structures [9].
Besides these well-established concepts for structural engineers, the rigidity and redundancy of frame-
works can also be analyzed with the help of graph theory, a branch of mathematics, elaborating on
the connectivity of frameworks [10]. Bolker and Crapo present a special way to solve the problem
of placing braces in a rectangular framework by encoding the diagonal truss elements with a bipartite
graph [11, 12]. However, the resulting graph only includes information about the spatial distribution of
the braces in the framework and lacks information about the redundancy of the individual elements. The
contribution of Achi and Tibert [13] proposes to use graph theory and the redundancy distribution for
interdisciplinary exchange to assess structural behavior by providing an overview of existing literature
in the two fields.
This contribution presents the analysis of frameworks combining the Maxwell-Calladine count, the re-
dundancy matrix and graph theory. The representation of frameworks using bipartite graphs is enriched
with information about the redundancy distribution. In Section 2, we introduce the fundamentals of
matrix structural analysis necessary to calculate the redundancy matrix and to understand the counting
rules as well as the basics of graph theory. In Section 3, the interaction of the above-mentioned structural
assessment methods is showcased with truss examples. Section 4 summarizes the work and points out
open research questions.
2. Matrix structural analysis and graph theory
This section describes the fundamentals of matrix structural analysis, the redundancy matrix, the count-
ing rules for the determination of static and kinematic indeterminacy, and graph theory. We follow the
notation in von Scheven et al. [6]. For a more in-depth study of matrix structural analysis and details
on the fundamental subspaces of the equilibrium matrix, the reader is referred to Livesley [14] as well
as Pellegrino and Calladine [15]. We consider a discrete truss structure with neelements, nnnodes, nc
kinematic constraints and ndunconstrained nodal displacements. The kinematic equations, describing
the relation between nodal displacements d∈Rndand element elongations ∆l∈Rnevia the compati-
bility matrix A∈Rne×ndcan be written as
Ad = ∆l.(1)
The equations of equilibrium, relating the external loads f∈Rndvia the equilibrium matrix AT∈Rnd×ne
to the internal forces N∈Rneread
ATN=f.(2)
To complete the governing equations of linear elasto-statics, the material matrix C∈Rne×ne, containing
the individual element’s member stiffness EA/l on its main diagonal, is introduced. Thus, the elastic
material law can be written as
N=C∆lel.(3)
If a structural system is statically determinate, the rank of the equilibrium matrix equals the number of
elements. In this case, the internal forces can be calculated by solely using the equilibrium equation (2).
The definition of the degree of static indeterminacy nsdates back to Maxwell [1] and the counting rule
is therein defined as
ns=ne+nc−2nn(4)
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Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
for plane trusses. If a bar is added to an initially statically determinate system, equation (4) gives ns= 1.
The system becomes statically indeterminate and contains a possible state of self-stress due to the ad-
ditional bar, meaning that in this potential stress state the internal forces are in equilibrium without any
external load. All states of self-stress in a system can be found in the nullspace of AT[15]. On the other
hand, if one bar is removed from an initially statically determinate structure, a kinematic mechanism
will be present and the Maxwell count results in ns=−1. The nullspace of Aspans the subspace of
kinematic mechanisms [15]. Thus, the potential states of self-stress and the kinematic mechanisms are
properties of the structure only using information about topology and geometry, independent of cross
sectional stiffness.
Due to the interplay of self-stress states and kinematic mechanisms, which can both be present in a
structure at the same time, the Maxwell count is extended by Calladine [2] to
ns−nm=ne+nc−2nn(5)
for plane trusses, nmbeing the degree of kinematic indeterminacy. This extension completes the as-
sessment of structures in terms of the degree of static and kinematic indeterminacy. Figure 1 shows two
different truss examples and the respective results for the analysis with the counting rules. The lower
truss is statically indeterminate by degree two and contains two states of self-stress, which are both il-
lustrated in the second column using red and blue colors for tension and compression, respectively. The
thickness of the lines represents the magnitude of the member force. In contrast to this, the upper truss
has a kinematic mechanism, shown in grey in the first column, and a degree of static indeterminacy of
one. In this case, equation (4) would fail to recognize the kinematic mechanism of this special geometry.
The self-stress state for the upper truss, which forms only in the two vertical members on the left, is also
shown in the second column.
Equations (4) and (5) both result in a single integer number, which provides little insight into the load-
bearing behavior of the structure, especially in structures with a large number of elements. Therefore,
an open question remains when applying the above-mentioned counting rules: How is the static inde-
terminacy distributed within the structure? This question can be answered using the redundancy matrix,
derived in the group of Klaus Linkwitz [3, 4, 5]. The redundancy matrix R∈Rne×neis described as a
mapping of initial, prescribed elongations ∆l0to negative elastic elongations ∆lel as
∆lel =−(1−AK−1ATC)∆l0=−R∆l0,(6)
K=ATCA being the elastic stiffness matrix of the system. As opposed to the self-stress states and
kinematic mechanisms, the redundancy distribution is calculated using information about the topology
and geometry as well as the elastic properties of the structure, namely the cross-sectional stiffness in the
case of trusses, as it quantifies the constraint of the structure on the individual elements. If the structure
has a kinematic mechanism (nm>0), the stiffness matrix is singular and thus not invertible. Therefore,
the generalized inverse must be used in equation (6), as presented by Chen et al. [17]. The main diagonal
entries of the redundancy matrix quantify the spatial distribution of the static indeterminacy. Thus, the
trace sums up to the degree of static indeterminacy [6]:
tr(R) = ns.(7)
The maximum redundancy for a single truss element is one, meaning that this element can be removed
without altering the load-bearing-behavior of the structure. The minimum redundancy value of zero
means that the non-redundant truss element is indispensable for structural integrity and a removal of
such an element would result in a (partial) collapse of the structure. The image of the transpose of the
3
Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
ns= 1, nm= 1
0.0
1.0
Redundancy
ns= 2, nm= 0
Structural system RedundancySelf-stress states
kinematic
distribution
mechanism
Figure 1: Two different structural systems and the degree of static and kinematic indeterminacy shown
in column one. The kinematic mechanism of the first structure drawn in grey. The respective self-stress
states shown in the second column (red and blue indicating compression and tension; magnitude of force
scaled with line thickness) and the redundancy distribution shown in colorscheme in the third column.
The example is adapted from Pellegrino [16].
redundancy matrix, im(RT), is equal to the nullspace of the equilibrium matrix [18]. This means, that
the states of self-stress can also be identified with the redundancy matrix. The redundancy distribution
of the two truss examples in Figure 1 are shown in colorscheme on the right. One can see in dark
blue color in the upper truss, that the elements that are not stressed in the self-stress state have zero
redundancy. In other words, they are statically determinate. In the lower example, each element has a
certain redundancy, meaning that failure of any one element would not lead to structural collapse in this
case. The redundancy of an individual element increases with the number of available alternative load
paths and the stiffness associated with these.
Another way to assess structures comes with graph theory, a branch of mathematics that can in general
be used to model many kinds of relationships. In the context of structural mechanics, the so-called
structure graph describes how a framework is connected and thus the relationship between the nodes.
For a detailed and extensive introduction to this field, we refer to the textbooks of Graver [10] and
Wilson [19]. In order to apply graph theory in the context of structural assessment, some necessary
4
Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
AB
C D
1
2
3
4
5
S1
S2
Figure 2: A connected graph with vertices Ato Dand edges 1 to 5 is shown on the left. A bipartite
graph with the two vertex sets S1and S2is shown on the right, indicating the partitioning of the vertices.
definitions will be recapitulated. A graph (V, E )is defined as a pair of vertices Vand edges E. Therein,
the edges describe the collection of pairs of vertices. A graph (V, E )is defined as connected, if no
partition into two nonempty sets Aand Bexists (V=A∪B, A ∩B=∅, A =∅, B =∅). Figure 2
shows on the left a graph with four vertices (Ato D) and five edges (1 to 5). It is a connected graph, since
no partition of the vertices into two nonempty sets exists. In other words, starting at a random vertex,
one can reach every other vertex by using the available edges. Furthermore, the notion of a bipartite
graph is introduced, which can be used for encoding bracing elements in a truss grid, as shown in Bolker
and Crapo [12]. A graph (V, E)is called bipartite, if its vertex set can be partitioned into two sets S1
and S2of disjoint vertices. This means that each edge Ehas an endpoint in S1and an endpoint in S2.
Such an exemplary bipartite graph is shown in Figure 2 on the right.
3. Structural assessment beyond stress and strain
In this section, the structural assessment of trusses going beyond the typical performance indicators, like
stresses, strains or displacements, is presented by using an interplay of the Maxwell-Calladine count, the
redundancy distribution and graph theory. For this purpose, a special grid bracing problem is shown, in
which the bracing of the structure can be abstracted with the help of graph theory, as first presented by
Bolker and Crapo [11, 12] and described in detail in Graver [10]. We consider a r×crectangular grid
of truss members, where rand crefer to the number of rows and columns, respectively. The structure is
constrained by a fixed support and a roller support in a statically determinate manner.
An eigenvalue decomposition of the stiffness matrix of the grid leads to (r+c−1) zero eigenvalues,
meaning that at least (r+c−1) diagonal bracings are necessary to eliminate the kinematic mechanisms.
Bolker and Crapo describe, how the bracing elements can be encoded with a bipartite graph by number-
ing the rows and columns of the grid. We will refer to the numbering as rows riand columns ck. With
the help of the associated bipartite graph, the bracing can be identified to be rigid if and only if this graph
is connected [10]. This means, that a path connecting all vertices must exist as described in Section 2.
With this, one can assess the structure in such a way that the existence of kinematic mechanisms and
redundant bracing elements are easy to locate. Nevertheless, little insight into the load-bearing behavior
of the structure and the quantitative impact of different bracing options are given. Therefore, the infor-
mation about the redundancy of the bracings, as described in Section 2, is added to the bipartite graph in
colorscheme. This enriches the visual encoding of the bracings via a bipartite graph and quantifies the
structural importance of the individual bracings.
Figure 3 shows three different solutions of the bracing problem. On the left, the minimal number of
bracings is used to render the structure rigid, as the associated bipartite graph is connected. As it can be
seen in colorscheme of the graph below the structure, the bracings are all non-redundant. Removing a
bracing would lead to a kinematic mechanism. In the center of Figure 3, an additional brace is added,
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Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
r1
r2
r3
c1c2c3
r1r2r3
c1c2c3
0.00 0.21
Redundancy
r1
r2
r3
c1c2c3
r1r2r3
c1c2c3
r1
r2
r3
c1c2c3
r1r2r3
c1c2c3
ns= 0, nm= 0 ns= 1, nm= 0 ns= 2, nm= 0
Figure 3: Three different options of bracing the 3×3grid with the associated bipartite graph below. The
edges represent the bracing elements of the associated column and row of the grid. The colorscheme
indicates the redundancies of the diagonal elements only.
represented in the bipartite graph by connecting r2and c2. This additional bracing is redundant with
every other member, meaning that any one of the diagonals can be removed without the structure col-
lapsing subsequently. The redundancies are distributed homogeneously throughout the diagonals, which
indicates a robust design, as described by Forster et al. [20]. On the right of Figure 3, another bracing
is added, which increases the degree of static indeterminacy by one and thus, the redundancies of the
diagonals are distributed from a minimum of 0.11 to a maximum of 0.21. The lastly added diagonal on
the top right (r1, c3) is the most redundant one, meaning that its removal would have the least impact
regarding structural performance.
Figure 4 shows a bracing that also uses five diagonals, like the statically determinate version above.
However, their specific placement results in a structure that is kinematic and has one state of self-stress.
According to equation (5) it is ns=nm= 1. On the left, the bracing and the associated bipartite graph
are shown, indicating that the four outer diagonals are equally redundant. Since the bipartite graph is not
connected, the structure has a kinematic mechanism, which can be seen on the top right. On the bottom
right, the state of self-stress is shown. The diagonal bracing in the center of the grid is in an unstressed
state, which relates to the fact that this member is statically determinate, as can be seen in dark blue
color on the bipartite graph, and thus it cannot be prestressed.
The investigation of the trusses has shown that by integrating redundancy analysis and graph theory,
information about the structural importance of individual elements can be seen right away in the asso-
ciated graph. Information about the distribution of constraint within a structure is added to the purely
topological representation, adding insight into the load-bearing behavior of the structure to the graphical
encoding. Moreover, the involvement of the individual elements in states of self-stress can be evaluated
through the graphical representation.
6
Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
r1
r2
r3
c1c2c3
r1r2r3
c1c2c3
0.00 0.12
Redundancy
ns= 1, nm= 1
Figure 4: A special option of bracing the 3×3grid with the associated bipartite graph below. The edges
represent the bracing elements of the associated column and row of the grid. The colorscheme indicates
the redundancies of the diagonal elements only. Kinematic mechanism and state of self-stress shown on
the right.
4. Conclusion and outlook
The Maxwell counting rule lacks information about the distribution of redundancy in statically indeter-
minate structures. By adding this information with the redundancy matrix, insight into the load-bearing
behavior of structures can be improved. The redundancy matrix for load bearing structures, quantifying
the distributed static indeterminacy, was first derived by the group of the Geodesist Klaus Linkwitz in
Stuttgart. The present contribution shows an integration of the Calladine-Maxwell counting rule, the
redundancy distribution and graph theory to assess structural performance. By enriching the graphical
encoding of a grid bracing in a bipartite graph with the redundancies of the individual elements, the
structural importance of the bracing elements can now be evaluated on a quantitative basis. This leads to
a representation of the structural configuration beyond only the topology by means of the connectivity
of the graph. As an extension of the present work, the application of a graphical representation of beam
elements and the relation between the Airy stress function and the redundancy distribution is current
work of the authors.
Acknowledgments
This research was supported by the Deutsche Forschungsgemeinschaft (DFG; German Research Foun-
dation) under Germany’s Excellence Strategy – EXC 2120/1 – 390831618.
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Proceedings of the IASS Annual Symposium 2024
Redefining the Art of Structural Design
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