A Brief Explanation of the Issue
A common problem type for vector calculus students or introductory physics/engineering students is tension problems. For instance, if a mass of $10$ kg is suspended from two taut cables of length $3$ m and $5$ m at angles of $52^\circ$ and $40^\circ$, respectively, then the corresponding tensions in the cables are $75.12$ N and $60.37$ N.
To make this concrete to students (in terms of application), one might talk about the strength of the cable material, e.g., would this arrangement break either of the cables. Looking up material strengths, these are most often measured in kPa. So from my understanding, the 'failure' in this situation is a measure of resistance to pressure in terms of tensile strength.
How would one determine if catastrophic failure of the cables would occur? Naturally, this criterion is expressed as a pressure, but what is the area I would be using? Would this be the area of a slice of the cable attaching the cable to the object and to where the cable is 'grounded'? Would it be the area of the cross section where the cable is thinnest? Is it an average size of a cross section of the cable? While I assume the tension is constant in the cable as a whole, the pressure from the object is not constant throughout the cable, correct? Or given that the cables are at an angle, would some analysis of the forces from a cable's tendency to rotate until it is vertical due to the weight weight be more relevant, i.e., some type of twisting force? If anyone could clarify the situation in this case, or more generally, this would be wonderful.
[Note: I am a mathematician, with only an undergraduate background in physics, if this helps with the type of explanation that would be most understandable to me or my students.]
I teach college students, typically freshmen/sophomore engineering students who have taken their calculus sequence up to multivariable/vector calculus but have had only an introductory physics course—if any at all.