Structural Design Using Matrix-based Performance Indicators
On 12 May 2025, David Forster presented the outcome of his doctoral research titled "Structural Design Using Matrix-based Performance Indicators" in front of the doctoral committee. The doctoral committee consisted of chair Prof. Wolfgang Nowak (IWS), the supervisor Prof. Manfred Bischoff (IBB), and the second examiners Prof. Steffen Freitag (KIT), and Prof. William F. Baker (University of Cambridge).
Many congratulations to David Forster on this outstanding great achievement!
Abstract Doctoral Research
This thesis deals with the matrix-based assessment of structures to enrich the existing performance evaluation at the verge of design and engineering. Load-independent performance indicators are used to quantify structural properties. The main focus lies on the interplay between architectural design and structural mechanics to make a contribution towards a truly integrative design process of load-bearing structures. The thesis seeks to generate a deeper understanding of structural assessment measures, like for example robustness and assemblability, based on fundamental performance indicators like the degree of static indeterminacy. Special emphasis on the use of these structural assessment strategies lies in early planning stages and design exploration stages. The degree of static indeterminacy is probably the most widely known and a loadindependent measure for evaluating structural performance. The fundamental question of how many structural elements are dispensable for the load-transfer can be answered by determining the degree of static indeterminacy. In other words: How many elements are redundant? This assessment strategy dates back to the 19th century, when James C. Maxwell introduced what is today known as the Maxwell counting rule. The degree of static indeterminacy is one of the first assessment strategies that civil engineering students are taught in mechanics classes. Nevertheless, the counting rule leaves one important aspect of the degree of static indeterminacy unanswered. Which element is redundant with which other element, and how are structural redundancies distributed within the structure? This question can be answered using the so-called redundancy matrix. This matrix was derived by German geodesists from the group of Klaus Linkwitz at the University of Stuttgart by transferring adjustment calculus for geodesic networks to structural mechanics problems. This redundancy matrix captures the distributed degree of static indeterminacy on its main diagonal and by this quantifies the individual element’s contribution to the overall degree of static indeterminacy. It is worth mentioning that the equivalence of redundancy and distributed static indeterminacy only holds true for the case of linear static analysis. Detailed investigations on influences of the redundancy distribution regarding structural topology, cross-sectional stiffnesses, and the shape of the structure are presented in this thesis. The concept is shown for truss and beam structures, and a newly developed visualization of the redundancy distribution for spatial beam structures is introduced. Similar to the degree of static indeterminacy, the extension of the Maxwell counting rule by Calladine accounts for the degree of kinematic indeterminacy. The degree of kinematic indeterminacy quantifies the number of independent mechanisms of a structure. As the degree of static indeterminacy is distributed amongst the elements, the degree of kinematic indeterminacy can be distributed amongst the unconstrained nodes. A matrix, introduced within this thesis as distributed kinematic matrix, is derived for truss structures and discussed in detail. Investigating structures based on the static and kinematic indeterminacy offers a load-independent structural assessment and does not rely on explicit knowledge about load cases.
Robustness as a structural property is closely linked to the notion of structural safety and structural reliability. The formal description of robustness, as it is written in building codes, says that failure of any individual part of the structure may not lead to an overall collapse. Thus, the damage and its cause must not be disproportionate. In this work, the redundancy distribution is used to assess structural robustness. This is done by studying the effect of the removal of individual elements and the resulting change in element elongation as a local criterion and nodal displacements as a global criterion. Beyond single element failure, a removal of multiple elements, or multiple load-carrying modes in case of beam structures, is showcased. Within an optimization scheme, an investigated objective is to homogenize the redundancy distribution and by this achieve a robust structural design. This strategy is presented with two different examples using shape optimization and cross-sectional optimization. Regarding imperfection sensitivity, an application of the redundancy matrix for the assessment of structural assembly is presented. How does an element imperfection influence the initial stress and strain distribution in a structure? Element imperfections are considered as imperfectly manufactured elements with a deviation in length in case of truss elements. The redundancy matrix, more precisely a column-wise interpretation of the redundancy matrix, is used to derive a compact, matrix-based calculation scheme for assessing the effect of element imperfections and for assessing assembly sequences. Combining redundancy analysis with graph-theoretical structural assessment offers yet another possibility to evaluate structural performance. Graph theory, a branch of mathematics, can be used to assess structures. For the special problem of bracing a rectangular grid, the topological information from the structure can be transferred to a graphical visualization by means of a bipartite graph. To account for the individual element’s importance regarding the load transfer, adding information about the redundancy distribution to the representing graph is newly introduced within this work. This leads to an accessible and easy-to-interpret method of analyzing rectangular grids. Besides, information about self-stress states and kinematic mechanisms is encoded in the bipartite graph and discussed in detail. Aside from redundancy-based and graph-theoretical assessment strategies, flexibility ellipses are used for structural analysis within this work. These ellipses offer a simple visualization of the flexibility of individual nodes and are easy to interpret. The larger the half-axis of the ellipse, the more flexibility is offered by the structure in the direction of the axis. Within an optimization scheme, the formulation of an objective function using the axes of the flexibility ellipse is shown. Throughout all the examples that are presented, matrix-based performance indicators are used to generate a deeper understanding of structural behavior as the central aspect of this thesis. The optimization examples do not claim to produce perfect structures, but rather to understand the interplay between performance indicators and structural properties like static indeterminacy, redundancy, flexibility, robustness, assemblability, and the relation to shape, topology, and material properties.
Relation of redundancy, states of self-stress and graph theory. © David Forster
David Forster's doctoral thesis "Structural design using matrix-based performance indicators" is a monograph. It is available via OPUS: http://dx.doi.org/10.18419/opus-16484.