Resolving line tension using vectors

[Potat]

I was wondering how you would go about solving the diagram I have drawn below. It is a simple representation of a body in equilibrium due to the tension of the 4 lines represented as vectors. The two forces are along the respective x and y axis.

With the forces shown one of the lines is useless however but this is necessary as the forces can change direction.

Looking at this instant simultaneous equations don't working without a lot of assumptions, I can solve part the problem with matrices or get an answer using computer software but I am interested in the process to resolve the tensions.

Any ideas?

 

[Unrest]

It's a statically indeterminate problem so it can't be solved without knowing the stiffnesses of the lines.

What do you mean by "a lot of assumptions"? Like the lines being able to support compression as well as tension?

What's unsatisfactory about the ways you've already solved it? I'm not sure what you're asking for.

 

[Potat]

I'm looking for a general way to solve this type of problem (that doesn't involve modeling it in software) so I can write a series of functions etc. in excel and from the data entered it can automatically solve a series of similar problems. If its possible to do?

The matrix functions in excel are limited and the Reduced row-echelon form function I used on a calculator uses a different process every time so duplicating in excel has numerous issues.

By the stiffness do you mean the young's modulus of the material, or modulus x cross section, or another?

'a lot of assumptions' meaning as the force is the -x direction there is no tension along x-axis on T3 & T4 therefore you can negate it, and visa versa in y-direction, obviously this is not the case and what I am after.

 

[Unrest]

By the stiffness do you mean the young's modulus of the material, or modulus x cross section, or another?

Young's modulus x cross sectional area.

'a lot of assumptions' meaning as the force is the -x direction there is no tension along x-axis on T3 & T4 therefore you can negate it, and visa versa in y-direction, obviously this is not the case and what I am after.

You mean you had to oversimplify it and possibly allow wrong solutions?

Have you been able to write a system of equations to describe it?

I would think it certainly requires an iterative method to detect compression and set those tensions to zero. Unless you can make more assumptions. Like are the wires prestressed before the forces are applied? Do you know in advance if any of them are slack? The other way is to write a complete FEA solver in Excel VBA. Are you willing to include 3rd party libraries, say for equation solving? If you want I can send you a simple step-by-step tutorial (5 pages) for writing a 2D truss FEA solver. It would do exactly what you want except for diallowing compressive force. Then you could iterate through this solver to zero the stiffnesses of compressed members.

 

[CellCoree]

I would approach this problem by first identifying the variables and parameters involved. In this case, the variables would be the tensions of the four lines and the parameters would be the direction and magnitude of each tension vector.

Next, I would use vector addition to determine the resultant force acting on the body. This can be done by breaking down each tension vector into its x and y components and then adding them together to get the resultant force vector.

To resolve the tensions, I would use the principle of equilibrium, which states that the sum of all forces acting on a body must equal zero for the body to remain in equilibrium. This means that the resultant force vector calculated earlier must be equal in magnitude and opposite in direction to the force of gravity acting on the body.

Using this principle, I would set up a system of equations with the tensions as unknowns and solve for their values. This can be done using simultaneous equations, matrices, or computer software, as mentioned in the question.

However, it is important to note that this approach assumes that the body is in a state of static equilibrium, meaning that it is not moving. If the body is in motion, additional factors such as acceleration and velocity would need to be considered.

Overall, the process of resolving the tensions in this diagram would involve identifying variables and parameters, using vector addition and the principle of equilibrium, and solving for the unknown tensions. This approach can be applied to more complex diagrams and can provide valuable insights into the forces acting on a body in equilibrium.