Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_197

⚡ Mecademy AIENG정역학 · ch5  Problem Statement Determine the length of cable which will allow a sag-to-span ratio of 1/10 for the configuration shown. Problem 5/197 (a) Determination of the cable length ● Calculation Process 1. Present Final Formula: The configuration represents a cable hanging under its own weight, which takes the shape of a catenary. The governing equations for a catenary with its lowest point at the origin are: The sag is the difference between the elevation at the supports () and the lowest point (): The total length of the cable is given by: Alternatively, a useful identity relating length, sag, and the parameter is: 2. Substitute Values: From the diagram, the span is . The sag-to-span ratio is . Thus, the sag is: Substituting these into the sag equation to find the parameter : s (0,c) y= ccosh ( c x ) hx=±L/2 x=0 h= c cosh −1[( 2c L )] s s= 2csinh ( 2c L ) c s= 2 h+2ch 2 L=1200 m n= = L h 10 1 h= = 10 1200 m 120 m c 120= c cosh −1 ⟹ [( 2c 1200 )]120= c cosh −1[( c 600 )] 3. Partial Operations: Let , then . The equation becomes: Solving this transcendental equation numerically (e.g., using Newton-Raphson or trial and error): For : For : By interpolation, . Now, calculate the parameter : 4. Final Calculation: Using the length-sag identity: ● Final Conclusion: The length of the cable required is approximately . ✨ Final Answer Summary (a) Mecademy AI Solution · ENGProblem 5/197 u= c 600 c= u 600 120= (coshu− u 600 1)⟹0.2u=coshu−1⟹coshu−0.2u− u=0.39cosh(0.39)