Finite-element-based limit analysis for strength predictions of wood at different scales of observation and a novel algorithm for adaptive implementation of velocity discontinuities in upper bound approaches = Finite-Elemente-basierte Traglastanalyse für Festigkeitsvorhersagen von Holz auf unterschiedlichen Betrachtungsebenen und ein neuartiger Algorithmus zur adaptiven Implementierung von Diskontinuitäten in kinematischen Traglastansätzen

Abstract

Numerical limit analysis belongs to the field of computational plasticity and is mainly used to predict load bearing capacities of engineering structures. In contrast to conventional elastoplastic analysis methods, limit analysis exclusively focuses on the time instant of structural collapse, and thus the whole load history up to this point doesnt need to be examined. Especially, finite-element-based limit analysis formulations have evolved to powerful tools, able to solve complex large-scale problems efficiently and robustly. Within this thesis, existing finite-element-based limit analysis formulations are discussed in detail and, subsequently, systematically adapted and extended to be applicable to orthotropic materials. The performance of the proposed formulations is assessed by means of different engineering problems. The whole thesis addresses two main tasks and thus has been organised into two parts accordingly. The first part of the thesis is dedicated to the implementation of numerical limit analysis approaches for strength predictions of wood and wood-based products. Since wood is undergoing a revival and has recaptured market shares in recent years, a reliable and efficient strength prediction tool is urgently needed in timber engineering. However, due to the intrinsic hierarchical structure of wood, a sophisticated and realistic numerical description of its strength behaviour is only possible by means of multiscale considerations. For this reason, numerical limit analysis formulations are proposed for two different scales of observation, the annual ring scale and the clear wood scale. At each scale, effective failure surfaces and distinct failure modes at various stress states could be obtained, and a validation by means of biaxial tests at the clear wood scale has rendered this numerical approach as a powerful tool providing sufficient and reliable information to investigate failure mechanisms of wood at different scales. Furthermore, a comparison between the proposed numerical method with two other computational methods, the extended finite element method and an elastic limit approach in the framework of continuum micromechanics, was performed, showing their strengths and weaknesses on predicting wooden strengths. The numerical limit analysis approaches can, on the one hand, capture basic characteristics of failure modes and the overall strength behaviours correctly and, on the other hand, fulfil the requirement of simplicity and efficiency for being applicable in engineering practice. To assess the applicability in engineering problems, the proposed numerical limit analysis approaches are implemented to predict the load bearing capacity of wood-based products. According to a validation by means of experimental results, this numerical method can provide reliable predictions on bending capacities of cross-laminated timber plates and allows for stochastic studies, taking inhomogeneities and uncertainties of the material into account. In summary, numerical limit analysis can be expected to play an important role for fast strength predictions of wood and wood-based products in the future. The second part of the thesis is dedicated to the development of new finite-elementbased upper bound formulations, allowing for an efficient description of localised failure mechanisms in combine with the consideration of orthotropic strength behaviours. Adaptive mesh refinement is commonly used in numerical upper bound approaches to handle localised failure mechanisms, which normally leads to the use of very fine meshes in failure regions and thus requires high computational effort. Alternatively, by introducing velocity discontinuities (as additional degrees of freedom) in discretised structures and arranging them in a sensible way, localised failure mechanisms can be captured accurately and efficiently by velocity jumps across these discontinuities. To guarantee a consistent orthotropic strength behaviour within solid elements and across discontinuities, an algorithm is derived projecting the stress-based yield function into a traction-based yield function with respect to the plane of plastic flow localisation. Then, to automatically arrange velocity discontinuities in a sensible way, an adaptive strategy is developed to iteratively introduce new velocity discontinuities and adjust orientations of existing ones within the discretised structures. For selected examples, the adaptively-arranged velocity discontinuities can play dominant roles in the resulting upper bound failure modes and thus the plastic strain-rate within solid elements is reduced to a minimum, making adaptive mesh refinement for these examples obsolete.