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Concept# Bending moment

Résumé

In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. The diagram shows a beam which is simply supported (free to rotate and therefore lacking bending moments) at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed (known as encastre beam); therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end (neither simple or fixed). In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.
The internal reaction loads in a cross-section of the structural element can be resolved into a resultant forc

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MICRO-200: Mechanism Design I

Ce cours introduit les bases de la mécanique des structures : calcul des contraintes et déformations provoquées par les forces extérieures et calcul des déformations. Ces enseignements théoriques sont appliqués à la conception des éléments importants des mécanismes de précision.

CIVIL-238: Structural mechanics (for GC)

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CIVIL-430: Concrete bridges

Ce cours traite les principaux aspects de la conception et du dimensionnement des ponts en béton armé et précontraint. L'accent est mis sur les ponts poutres. Etude des aspects suivants : optimisation du comportement structural, dimensionnement, dynamique des structures et technologie des matériaux.

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.

Glass fiber-reinforced polymer (GFRP) pultruded decks and sandwich panels currently represent two of the most extensive applications of FRP materials for load-bearing structural components in the bridge and building domains. Based on the state of the art, the global structural behavior of both systems has been fairly well investigated. Nonetheless, local effects governing in most cases the global behavior have been barely addressed. Selected local structural effects relevant to the global structural performance of pultruded GFRP bridge decks and GFRP-foam web-core sandwich structures are therefore investigated in this research. The effect of the core geometry of pultruded GFRP decks on the systemâs behavior in its transverse-to-pultrusion direction was experimentally investigated. The experimental work conducted on two deck designs with trapezoidal- and triangular-cell cross sections showed that the transverse structural performance depends on the cell geometry. Furthermore, the systemsâ transverse bending and in-plane shear stiffness were evaluated and the results indicated that a triangular core causes a more pronounced bi-directional behavior of the deck when it is subjected to concentrated loads. The local behavior of the web-flange junctions (WFJs) of the pultruded deck with trapezoidal cells was experimentally investigated regarding energy dissipation capacity and recovery subsequent to unloading. The experimental responses reported for two junction types with similar geometry and fiber architecture but different initial imperfections demonstrated that dissimilar imperfections could significantly affect WFJ behavior and change it from brittle to ductile. The time-dependent recovery and energy dissipation mechanisms of the WFJs exhibiting a ductile response were evaluated; the viscoelastic effects were found to be small in both cases. The rotational behavior of all WFJ types present in the trapezoidal-core deck was characterized. An experimental procedure based on three-point bending and cantilever experiments conducted on the web elements was developed and used for this purpose. The rotational stiffness, strength and failure modes of the WFJs differed depending on the web type, location of the WFJ within the deck profile, existing initial imperfections and direction of the applied bending moment. Numerical simulations of the full-scale deck were performed to demonstrate the validity of the experimental moment-rotation (M-Ï) relationships and simplified M-Ï curves provided. The effects of creep on the load-bearing behavior of GFRP-foam web-core sandwich structures were investigated. A study of the creep behavior of polyurethane (PUR) foams was conducted and showed that in order to assess the long-term structural performance of the sandwich system, the foam anisotropy, density and loading type should be considered. The creep behavior of web-core sandwich panels, and specifically the structural aspects affected by the web-core interaction, were analyzed using the GFRP-PUR sandwich roof of the Novartis Campus Main Gate Building as case study and currently available design guidelines. The resulting sandwich designs depended on the applied design recommendations. Finally, provisions for the cross-sectional design of the hybrid web-core were proposed.

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The web's local buckling is a major failure type of pultruded glass fiber-reinforced polymer (GFRP) bridge deck subjected to concentrated wheel loads. Three kinds of external actions-that is, in-plane shear force, local compressive force and bending moment-are coupled at web. Besides, the couple ratio of those three external actions is changed due to varied wheel's loading position, resulting in a more complicated web buckling mechanism. Hence, a study of the web buckling behavior was firstly conducted through two experiments with different concentrated loading cases, namely mid-span loading and quarter-span loading. After then, parametric analysis of FE model with detailed simulated web-flange junctions (WFJ) were performed to reveal a more comprehensive understanding. The experimental results showed that both specimens went through buckling failure of middle web, post-buckling strengthening owing to structural redundancy, and final failure caused by crack propagation on the top flange. The parametric analysis showed that the compression buckling and shear buckling coupled together, and that the compression buckling is dominant. In addition, as the shear-span ratio decreased, the principal compressive stress and the shear buckling became more pronounced, and consequently the GFRP bridge deck's bearing capacity reduced.