An analytical exact, locking free element formulation for thin-walled composite Timoshenko beams - Jäger, Michael; Albertsen, Jacqueline; Wartzack, Sandro (Computer Methods in Applied Mechanics and Engineering, 2025)
Spatial truss structures represent a robust, cost-effective, and efficient lightweight design, especially when isotropic materials are substituted with lightweight materials such as composites. During early design phases, truss structures are often subject to optimisations. In order to achieve this in an efficient manner, it is essential to employ a precise yet cost-effective computational model. The most common methodology for the analysis of spatial truss structures employs hinged joints in conjunction with struts that are only subject to tension or compression. However, this approach does not account for the bending and coupling effects inherent to struts manufactured from composite materials. In particular, when employing asymmetric laminates, these effects can no longer be ignored. In order to incorporate these effects, it is common practice to use Finite Element Analysis tools. Particularly for large spatial truss structures comprising struts with slender and thin-walled cross-sections, a large number of solid or shell elements is required, which results in time-consuming simulations. This contribution presents a fully analytical thin-walled composite beam element, applicable to an arbitrarily shaped, closed cross-section. The beam model incorporates two distinct composite material models, namely the Classical Laminate Plate Theory and the First Order Shear Deformation Theory. Moreover, it is capable of simulating asymmetric laminates and modelling the coupling effects within these laminates. Utilising the exact third-order solution of a composite Timoshenko-Ehrenfest beam enables the locking-free representation of an individual strut with a single beam element. In comparison to the conventional shell / solid Finite Element Analysis, this approach results in a substantial reduction in the number of degrees of freedom, by a factor of several orders of magnitude. As a result, the required computational time is significantly reduced. In the case of a single strut, the computational time is reduced by a factor between 160 and 430. For an exemplary truss structure comprising 64 struts, a reduction in computational time of approximately 100 000 times is reached. The numerical comparisons presented in this contribution demonstrate that the model is highly accurate, particularly for tubular and elliptical cross-sections including symmetric and asymmetric laminate
Keywords: Lightweight design; Spatial truss structures; Structural optimisation; Thin-walled composite beams; Timoshenko beam

A New Design Method for Generating Surrogate Kinematic Truss Orthoses to support Pathological Gait Patterns in Human Model Simulations - Steck, Patrick; Scherb, David; Mayer, Johannes; Jäger, Michael; Miehling, Jörg; Völkl, Harald; Wartzack, Sandro (Proceedings of the Design Society, 2023)
With increasing life expectancy, the risk of diseases of the central nervous system, such as cancer,strokes, etc., also increases. Strokes often result in injury to the sciatic nerve, which is responsible forcontrolling the calf muscles (plantar and dorsal flexors). A so-called ankle joint orthosis (AFO) helps tosupport the pathological gait and to avoid foot drop during gait. Passive orthoses are of particularimportance for research, as they do not require additional incoming energy from outside to the orthoticsystem. However, current passive orthoses are often not personalized. On the one hand, because theyusually have only a temporary muscle-building function and, on the other hand, because the individualdesign process is computationally time consuming and thus expensive. This paper presents a possibilityto pre-dimension and pre-design passive orthoses fast and cost-efficiently by reducing the complexityof the model based on volume-optimized truss elements. Therefor a traditional high calculation intensivedesign procedure is compared with the complexity reduced model to show its effectiviness and thesimilarity of the results. Witl incoming energy from outside to the orthoticsystem. However, current passive orthoses are often not personalized. On the one hand, because theyusually have only a temporary muscle-building function and, on the other hand, because the individualdesign process is computationally time consuming and thus expensive. This paper presents a possibilityto pre-dimension and pre-design passive orthoses fast and cost-efficiently by reducing the complexityof the model based on volume-optimized truss elements. Therefor a traditional high calculation intensivedesign procedure is compared with the complexity reduced model to show its effectiviness and thesimilarity of the results. <br
Keywords: Orthoses; User centred design; Lightweight design; Truss optimization; Design methods

Efficient Computation of Spatial Truss Structures for Design Optimization Approaches Using Tube-Shaped Thin-Walled Composite Beams - Jäger, Michael; Wartzack, Sandro (Proceedings of the Munich Symposium on Lightweight Design 2021, 2022)
Spatial truss structures are a stiff, economical, and effective lightweight design method, especially when using composites instead of isotropic materials for the struts. An efficient computation of these structures is crucial for optimization approaches during the product design process. The most common method for computing spatial truss structures relies on hinged connections with tension/compression-only struts, which ignores the bending and coupling effects of composite beams. However, especially when using asymmetric laminates, these effects are no longer neglectable. Within commercial finite element tools, the computation of large truss structures - which include these effects - is a very time-consuming process. Particularly for slender, thin-walled beams a large number of solid/shell elements is required. In this paper, an analytical solution of the stiffness matrix for a tube-shaped thin-walled composite beam is provided. It is based on the classical laminate plate theory and Timoshenko’s exact solutin including shear deformation and coupling effects. By using three-dimensional exact Timoshenko beam elements, the number of degrees of freedom can be reduced significantly while coupling effects are maintained. This results in a remarkably lower computation time especially needed for topology optimization. The results are compared to a commercial finite element tool using both solid and shell elements.<br
Keywords: Lightweight design; Spatial truss structures; Structural optimization; Thin-walled composite beams; Timoshenko beam

Konzept einer CAE-Methode zur systematischen Auslegung beanspruchungsgerechter, kurzfaserverstärkter AM-Fachwerksknoten für hochoptimierte Fachwerke - Jäger, Michael; Völkl, Harald; Wartzack, Sandro (Proceedings of the 31st Symposium Design for X (DFX2020), 2020)
Trusses represent an economical, rigid and effective lightweight construction method. However, the connecting parts (nodes) between the struts are a design challenge, especially in highly optimized trusses. Several struts with different geometries (diameter, cross-sections) must be connected at any spatial angle in one point in a manner that is suitable for the load. The use of fibre-reinforced materials for the struts is common practice, for the nodes, metals are usually used. However, these do not exploit the full lightweight construction potential of the truss. The proposed method allows the design of truss nodes using fibre-reinforced materials and additive manufacturing, while taking into account the geometric boundary conditions of the connection points for the strut
Keywords: Structural optimization; CFRP truss; lightweight design; short fibre reinforced plastics

Nodal force distribution of moments on arbitrarily shaped geometries without multi-point constraints - Jäger, Michael; Wartzack, Sandro (Engineering With Computers, 2025)
This paper presents a new approach for converting external moments into nodal forces for the Solid Finite Element Analysis. The proposed approach accurately distributes loads on arbitrarily shaped geometries by utilising the vertices of the Finite Element mesh as a point cloud. This eliminates the need for additional elements and Multi-Point Constraints, overcoming the challenges, complexity and computational cost introduced by these constraints. The proposed approach offers a precise and lightweight solution that enhances the performance of design optimisation scenarios with multi-axial moments on complex, non-convex surfaces. The paper presents comprehensive results that validate the effectiveness and accuracy of the proposed approach, particularly in engineering design optimisation scenarios that require a large number of simulations, such as the topology optimisation of truss nodes. Overall, this method offers a practical and efficient approach for handling moments in Solid Finite Element Analysis. In the presented example, computational cost and complexity were reduced by approximately 14 % compared to commonly used methods for solving Multi-Point Constraint
Keywords: Solid finite element analysis; Nodal forces; Multi-point constraints; Moment load; Point cloud; Optimisation

Strength Analysis of Structurally Optimized Aluminium-Composite Tubular Lap Joints - Jäger, Michael; Wartzack, Sandro (Proceedings of the Design Society, 2022)
Truss structures are a stiff, economical, and efficient lightweight design, the limiting factor of these structures are usually the load transfer elements. This paper presents an analytical design method for optimized adhesive tubular lap joints between CFRP tubes and aluminium inserts. The analytically optimized design agrees very well with the numerical simulations and experimental results. Although the experiments show a highly non-linear behaviour, where a linear elastic correlation is expected, the total load capacity is only reached when the adhesive is fully plasticised.<br
Keywords: lightweight design; design optimisation; tubular lap joint; experimentation; composites

Symbolic Parametric Representation of the Area and the Second Moments of Area of Periodic B-Spline Cross-Sections - Denk, Martin; Jäger, Michael; Wartzack, Sandro (Applied Mechanics, 2023)
The calculation of moments of area is one of the most fundamental aspects of engineering mechanics for calculating the properties of beams or for the determination of invariants in different kind of geometries. While a variety of shapes, such as circles, rectangles, ellipses, or their combinations, can be described symbolically, such symbolic expressions are missing for freeform cross-sections. In particular, periodic B-spline cross-sections are suitable for an alternative beam cross-section, e.g., for the representation of topology optimization results. In this work, therefore, a symbolic description of the moments of area of various parametric representations of such B-splines is computed. The expressions found are then compared with alternative computational methods and checked for validity. The resulting equations show a simple method that can be used for the fast conceptual computation of such moments of area of periodic B-splines.
Keywords: B-spline; cross-section; beam; moments of area; parametric; symbolic; moments; area