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Concept# Bending

Summary

In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two. When the length is considerably longer than the width and the thickness, the element is called a beam. For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending.
In the absence of a qualifier, the term bending is ambiguous becaus

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The student will acquire the basis for the analysis of static structures and deformation of simple structural elements. The focus is given to problem-solving skills in the context of engineering design.

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The primary cilium, a hair-like projection from the cellular membrane, is involved in fluid flow sensing. Bending the primary cilium is known to trigger several signaling pathways. In this study, the primary cilium is modeled as a thin beam undergoing large deflection due to fluid drag forces. For the first time, the bending response is analyzed with a model combining large angle rotations with the assumption of a linear drag force along the ciliary length. In addition, the initial curvature and the angle between the cell membrane and the cilium are integrated into the model. The model is applied on three dimensional confocal images of fluorescent cilia in living cells that are exposed to laminar shear flow. The coordinates of the cilia are computed by a simple and effective image processing algorithm. By fitting the equations of the model to the coordinates of the bent cilia, the flexural rigidity and the angle between the cilium and the cell membrane are estimated. The flexural rigidity of the primary cilium is approximately ${10}^{-23}$ Nm, but the values vary for different cilia, which suggests that the mechanical properties of the primary cilia are heterogeneous. Interestingly, the base of the cilium doesn't deflect under fluid flow which means that the primary cilium is firmly anchored. Using immunocytochemistry, the connections of the microtubule network to the base of the cilium could be resolved. The microtubules may provide the mechanical stability of the cilium. Incorporating a linear drag force and allowing a basal tilt is an important step towards a more realistic mechanical model of the primary cilium and towards more accurate values of its flexural rigidity. The model together with the imaging technique is a useful tool to study the mechanical behavior of the primary cilium and will eventually lead to new insight in the poorly understood mechanisms of mechanotransduction.

2010The present research represents a theoretical and experimental contribution to the understanding of the structural behaviour of elements made of ultra-high performance fibre-reinforced concrete (UHPFRC). UHPFRC is investigated as an advanced cementitious material offering particular potential in innovative bridge design. The optimised material composition results in high compressive strength and non-negligible tensile strength and ductility, provided by multi-microcracking. This allows significant tensile forces to be sustained by elements in bending even without the use of ordinary reinforcement. Thanks also to the material's resistance to environmental degradation, very thin structural elements can be constructed. This research focuses primarily on the bending behaviour and design of thin UHPFRC beams and slabs. Punching-shear is also investigated as a possible failure mode of thin UHPFRC slabs. One of the main differences between other concretes and UHPFRC is that the latter requires mechanical models capable of taking tensile behaviour into account for rational structural application. Analytical and numerical models are developed in this study to simulate the non-linear bending response of UHPFRC beams and slabs. This permits the assessment of element behaviour at service states and prediction of failure loads. Theoretical research on both bending and punching-shear failure is supported by experimental research on beams and slabs made of BSI® UHPFRC with 2.5 % volume of 20-mm long steel fibres. The analytical model for beams in bending takes both material multi-microcracking and macrocrack propagation with tensile softening into consideration. Multi-microcracking is modelled as a pseudo-plastic tensile behaviour, while the macrocrack is simulated based on the assumptions of the fictitious crack model. The results are in good agreement with experimental data and simulations obtained from a developed finite element model. Using theoretical results and experimental data it is demonstrated that pre-peak behaviour and bending strength are mainly governed by multi-microcracking. The propagation of the macrocrack provides only a minor additional contribution to bending strength, but, in the case of thin beams, plays an important role in providing ductility in bending. Theoretical results demonstrate that, due to the presence of pronounced pseudo-plastic deformations, size effect on bending strength is much less significant for UHPFRC than for other quasi-brittle materials, which corresponds to experimental observations. It is however shown that, even if the pseudo-plastic phase is less pronounced, thin elements develop behaviour similar to that of elements with high pseudo-plastic tensile deformations, owing to the low stress decrease in tensile softening. Nonetheless, in the absence of pseudo-plastic tensile deformations, the behaviour of thick elements approaches that of typical quasi-brittle materials, with a more pronounced size effect. In the case of thin statically indeterminate beams and slabs, it is shown that a high level of tensile ductility can allow sufficient internal force redistribution to occur, leading to a significant increase in load-bearing capacity. Moreover, high rotations can be sustained after cracking while almost constant bending strength is maintained, resulting in a plastic-like behaviour. It is demonstrated that the theory of plasticity can thus be applied: a formulation is proposed to predict the resistant plastic moment, enabling easy estimation of the bending failure load for thin elements. The analysis results show good agreement with test results for slabs of different thicknesses. However, due to the remarkable size effect on ductility in bending, the rotation capacity of UHPFRC elements thicker than approximately 100 mm is limited, and the theory of plasticity does not apply. Experimental and theoretical research on the punching-shear failure of thin UHPFRC slabs demonstrates the influence of structural parameters on achieved shear resistance. A proposal is made for considering fibre contribution in shear resistance as a structure-dependent parameter, relating the critical shear crack opening to slab rotation. This approach results in more accurate predictions for thin elements with larger deformations as compared to current code predictions that overestimate resistances for such elements. With a view to the structural application of UHPFRC in bridge design, the concept of ribbed deck slab is studied. Based on the theoretical and experimental results it is demonstrated that thin UHPFRC slabs (40-60 mm) without ordinary reinforcement can be effectively used in this concept: sufficient bending and punching-shear resistances to locally applied traffic loads can be ensured. With prestressed ribs, UHPFRC ribbed slabs attain high load-bearing capacity, while structural dead weight is significantly decreased. This concept could open up new vistas in the design of new structures and offers effective possibilities for the structural repair or widening of existing bridges.

Plane reinforced concrete (RC) elements are used in a large variety of structures. Their principal function is to carry forces that act in the plane of the element, but external actions and connections to other structural elements generally introduce additional out-of-plane forces. In practice, the design of such elements is often performed in a simplified manner, neglecting the interaction between these different internal forces. However, especially for existing structures the need for more precise and kinematically consistent analysis tools arises. This thesis provides novel tools based on the elastic-plastic stress field (EPSF) method to investigate the interaction between in-plane and out-of-plane forces in plane RC elements in general and the effect of transverse bending on the longitudinal shear resistance of beams in particular. A multi-layered (ML) EPSF approach is developed. Applied to a unitary web segment, in-plane shear-transverse bending interaction diagrams are established and compared to existing rigid-plastic (RP) interaction models. In general it is found that the influence on the shear resistance is less pronounced, especially in case of small transverse moments. The shear transfer actions admitted in RP models that consist in a shift of the compression field to the bending compression side and a rearrangement of the stirrup forces are confirmed. However, it is shown that the stress field is highly non-linear in the transverse directions (stress/strain distribution and inclination) and strongly depends on the intensity of the applied transverse moment. The concrete strength reduction factor Î·Îµ is generally higher and high shear reinforcement ratios or asymmetric layouts allow equilibrating small moments without disturbing the stress field in the concrete. This increases the predicted shear resistance. The longitudinal deformation is shown to have a non-negligible effect on the overall interaction and ultimate resistance. A simplified verification method for beams in practice is proposed. Based on the EPSF finite element method (FEM), it considers the influence of the transverse moment by means of a reduced web width and an effective shear reinforcement ratio. Validation with tests from the literature gave safe but not overly conservative results and consistent predictions of the failure modes. The method provides enhanced lower-bound solutions. Plane EPSF analyses of experimental tests suggest that the influence of the transverse bending moment in beams is less pronounced than predicted by interaction models, especially if ductile failure modes occur. But more experimental data is required to validate this observation. A non-linear FEM based on the ML-EPSF is developed. It aims to extend the field of application of the EPSF FEM by accounting for in-plane (normal and shear) and out-of-plane (bending and shear) actions in plane RC elements. Concrete is modelled by ML in-plane elements that are combined with out-of-plane shear elements. Reinforcing steel is modelled separately by bar elements. Benchmark tests and validation with experimental data show that the proposed FEM is a promising tool for the design and assessment of plane reinforced concrete elements primarily subjected to combinations of in-plane forces and out-of-plane bending moments.