7.14
Câble : Résolution de problèmes
Lorsqu'on manipule un câble fixé à deux supports et soumis à une charge uniforme, il est crucial de déterminer la tension maximale dans le câble. Ce p…
Considérons un câble fixé à deux supports soumis à une charge uniforme. Déterminez la tension maximale dans le câble.
Pour l'analyse, considérons l'origine au centre du câble en raison de sa symétrie.
Rappelez-vous l'équation de forme du câble pour la charge répartie et remplacez-la par la valeur de charge constante connue.
Ensuite, intégrez l’équation, et en appliquant les conditions aux limites à l’origine, la constante C2 est déterminée.
En prenant la dérivée première de l'équation de forme du câble, la pente peut être déterminée. En appliquant les conditions aux limites de la pente à l’origine, on obtient C1.
En substituant les constantes d’intégration et les coordonnées de position du support et en réarrangeant l’équation, la force de traction horizontale est obtenue.
Rappelez-vous l’équation de pente et remplacez la valeur de la coordonnée x au niveau de l’appui où l’angle est maximal.
La tension de Cabel change avec l'angle, qui est le support maximal de près.
En appliquant la relation trigonométrique et en réarrangeant les termes, on obtient l’équation de tension maximale.
Enfin, en substituant l’équation de tension horizontale et les valeurs connues, on obtient la tension maximale dans le câble.
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Q1: How do you set up the coordinate system when analyzing a cable fixed at two supports?
Place the origin at the cable's center to leverage its symmetry. This positioning simplifies calculations by allowing you to apply boundary conditions at a single point. The symmetric setup reduces the complexity of the shape equation and integration constants, making the analysis more efficient and manageable for determining cable behavior under uniform loading.
Q2: What role does the cable's shape equation play in finding maximum tension?
The shape equation describes how the cable curves under uniform loading and relates the cable's geometry to applied forces. By integrating this equation and applying boundary conditions, you obtain integration constants that define the cable's exact profile. These constants are essential for calculating the horizontal tensile force and ultimately determining where maximum tension occurs along the cable.
Q3: Why is the slope of the cable important when calculating maximum tension?
The slope, found by taking the first derivative of the shape equation, determines the cable's angle at each point. Maximum tension occurs where this angle is greatest, typically near the supports. Using trigonometric relationships between the slope angle and horizontal tensile force allows you to express and calculate the maximum tension value accurately.
Q4: How do boundary conditions help determine integration constants?
Boundary conditions specify known values at particular locations, such as zero slope at the cable's center. Applying these conditions to the integrated shape equation and slope equation yields the integration constants C1 and C2. These constants are then substituted back into equations to find the horizontal tensile force and ultimately the maximum tension in the cable.
Q5: What is the relationship between horizontal tensile force and maximum tension?
Horizontal tensile force is the constant tension component acting along the cable's length. Maximum tension combines this horizontal force with the vertical component created by the cable's angle at the support. Using trigonometric relations, maximum tension equals the horizontal force divided by the cosine of the maximum angle, making it always greater than the horizontal component.
Q6: Where does maximum tension occur in a uniformly loaded cable, and why?
Maximum tension occurs near the cable's supports where the slope angle reaches its maximum value. At these locations, the cable must support both the accumulated load and resist the steepest angle, creating the highest combined tensile stress. The symmetric loading ensures maximum angles occur symmetrically at both supports.
Q7: How do you calculate the horizontal tensile force from the cable's support coordinates?
Substitute the support's position coordinates and integration constants into the rearranged shape equation to isolate the horizontal tensile force. This force remains constant throughout the cable under uniform loading and serves as the foundation for calculating maximum tension. Once determined, it combines with the angle at the support to yield the final maximum tension value.