7.14
Cable: Problem Solving
When dealing with a cable that is fixed to two supports and subjected to uniform loading, it is crucial to determine the maximum tension in the cable.…
Consider a cable fixed at two supports subjected to uniform loading. Determine the maximum tension in the cable.
For analysis, consider the origin at the cable's center due to its symmetry.
Recall the cable's shape equation for distributed load and substitute the known constant loading value.
Next, integrate the equation, and by applying the boundary conditions at the origin, the constant C2 is determined.
By taking the first derivative of the cable's shape equation, the slope can be determined. Applying the boundary conditions for the slope at the origin, C1 is obtained.
By substituting the integration constants and position coordinates of the support and rearranging the equation, the horizontal tensile force is obtained.
Recall the slope equation, and substitute the x-coordinate value at the support where the angle is maximum.
Cabel's tension changes with the angle, which is the maximum near support.
Applying the trigonometric relation and rearranging the terms, the maximum tension equation is obtained.
Finally, by substituting the horizontal tension equation and known values, the maximum tension in the cable is obtained.
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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.