7.11
Câble soumis à des charges concentrées
Les câbles flexibles sont couramment utilisés dans diverses applications pour le support et la transmission de charges. Considérez un câble fixé à deu…
Les câbles flexibles sont utilisés pour le support et la transmission de charge dans diverses applications.
Considérons un câble AB fixé en deux points, subissant plusieurs charges verticales concentrées.
Déterminez la forme du câble et la tension dans chaque partie du câble, en connaissant les distances horizontales entre les charges et le support.
Pour l’analyse, on suppose que le câble est flexible, inextensible et a un poids négligeable.
Le câble se compose de plusieurs segments en ligne droite, chacun soumis à une force de traction constante.
Dessinez un schéma du corps libre du câble pour déterminer les forces de réaction sur les supports. Ici, le nombre de composants de réaction inconnus dépasse les équations d’équilibre.
Considérons le point D sur le câble à une distance connue et dessinez un schéma du segment AD. En utilisant l’équation d’équilibre du moment au point D, on obtient une équation supplémentaire.
En rappelant les équations d’équilibre et en utilisant les forces de réaction, on obtient la distance verticale entre le support A et chaque charge concentrée.
De même, la tension dans chaque segment peut être obtenue, qui est maximale lorsque le segment a le plus grand angle d’inclinaison.
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Q1: What assumptions are made when analyzing a cable subjected to concentrated loads?
Cables are assumed to be flexible, allowing them to change shape under applied loads, and inextensible, meaning their length does not change under tension. Additionally, the cable's weight is assumed negligible, so it does not significantly impact the cable's behavior during analysis of concentrated loads.
Q2: Why is a free-body diagram of a cable segment necessary for solving concentrated load problems?
A free-body diagram of the entire cable often yields more unknown reaction components than available equilibrium equations. By isolating a segment at a known distance and applying the moment equilibrium equation at that point, an additional equation is obtained, allowing the system to be solved for reaction forces and tensions.
Q3: How does the shape of a cable change when subjected to multiple concentrated loads?
A cable subjected to concentrated loads forms several straight-line segments between load points and supports. Each segment maintains constant tensile force and connects at the load application points, creating a polygonal shape that reflects the distribution and magnitude of the applied vertical loads.
Q4: What is the relationship between cable segment inclination and tension magnitude?
Tension in a cable segment is maximum when the segment has the largest inclination angle relative to the horizontal. Segments with steeper angles experience greater tensile forces because they must support larger vertical load components while maintaining static equilibrium across the cable system.
Q5: How are vertical distances from support to load points determined in cable analysis?
Once reaction forces are known from equilibrium equations, vertical distances from support A to each concentrated load are determined by recalling the equilibrium equation that states the sum of vertical forces acting on the cable must equal zero. This allows calculation of the cable's vertical geometry.
Q6: What information is needed to determine tension in each cable segment?
To calculate tension in each cable segment, you need the reaction forces at supports, the horizontal distances between loads, and the vertical distances from support to each load point. With these values, the tension in each segment can be computed, with maximum tension occurring in segments with the largest inclination angle.
Q7: How does the moment equilibrium equation help solve indeterminate cable systems?
When the number of unknown reaction components exceeds available equilibrium equations, applying the moment equilibrium equation at a point on the cable segment provides an additional independent equation. This transforms the indeterminate system into a determinate one, enabling complete solution for all reaction forces and segment tensions.