7.14
Cable: Resolución de problemas
Cuando se trata de un cable que está fijo a dos soportes y se somete a una carga uniforme, es crucial determinar la máxima tensión en el cable. Este p…
Considere un cable fijado a dos soportes sometidos a una carga uniforme. Determine la tensión máxima en el cable.
Para el análisis, considere el origen en el centro del cable debido a su simetría.
Recuerde la ecuación de forma del cable para la carga distribuida y sustituya el valor de carga constante conocido.
A continuación, se integra la ecuación, y aplicando las condiciones de contorno en el origen, se determina la constante C2.
Tomando la primera derivada de la ecuación de la forma del cable, se puede determinar la pendiente. Aplicando las condiciones de contorno para el talud en el origen, se obtiene C1.
Sustituyendo las constantes de integración y las coordenadas de posición del soporte y reorganizando la ecuación, se obtiene la fuerza de tracción horizontal.
Recuerde la ecuación de la pendiente y sustituya el valor de la coordenada x en el soporte donde el ángulo es máximo.
La tensión de Cabel cambia con el ángulo, que es el máximo cerca del soporte.
Aplicando la relación trigonométrica y reordenando los términos, se obtiene la ecuación de máxima tensión.
Finalmente, sustituyendo la ecuación de tensión horizontal y los valores conocidos, se obtiene la tensión máxima en el cable.
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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.