This post is based on some old (2009) research of mine into catenary cables.
I had a situation where a cable was spanning between 2 points of different heights. At the time there was plenty of examples for the simple case of a catenary cable supported at points of equal height, but only scattered information and incomplete solutions for the case where the cable was supported at different heights. Ultimately I decided to go back to first principles to derive the catenary equations and I have included the full working to that solution in this post, as well as a copy of a spreadsheet which implements the solution.
Full working for deriving catenary cable equations
Consider the case below, where a 23m long cable is spanning 20m horizontally with a 5m change in height between supports.
The cable profile, max sag and lowest point are calculated from the spreadsheet. Only the 6 variables shown in grey below need to be input. You will see the spreadsheet also solves for the tension within the cable.
I typically size a cable with a breaking load ~5 times the calculated maximum tension when designing temporary cables for supporting electrical services during brown field structural remediation works.
The calculations are based on an inelastic cable with no bending stiffness. For critical works I suggest verifying with another method. Internal verification has shown good correlation between the spreadsheet and a linear analysis with Strand7 FEA software. Verification of the example above yields a maximum tension of 1884N in Strand7 c.f. 1887N calculated using the attached spreadsheet.
Screenshot of Strand7 verification:
I hope this helps other engineers when designing simple cable systems. I have referred back to this spreadsheet for designing simple cable systems such as temporarily supporting services for in cable trays or estimating tension loading on cable support shade structures. While modern structural design packages can easily model cables with the finite element method, in the context of a calculating forces in a single cable an exact solution, this is overkill (in my opinion) when an exact solution exists.