Is non-linear Tension Force possible?

[karen_lorr]

I seem to remember from my school days that Tension Force can only be linear.

Is this true?

In 1 (in the graphic) the tension will follow the line of the rope

In 2 there is an unbendable, unbreakable, steel cable formed into an arch.
a ] In 3 which direction will the Tension be?
b ] In 3 if there was a gauge on this arch would it still read 1000kn, the same as on the rope, or would the shape change this.

Thank you

 

[PeroK]

First, conceptually, what is the difference between the rigid cable and the chassis of a car? The point about a rope or a string is that it can only apply a force along its length. That's not true of a rigid body, like a car or your cable.

In your case, you could look at it as one rigid body (made up or two cars plus the cable) with equal and opposite forces acting it. There is no concept of tension, per se. Just one extended rigid body.

Instead of tension, we generally talk about stresses and strains for a deformable material in a case like this. The pattern of internal forces in the cable would be more complicated, associated with some deformation of the cable. See, for example:

https://www.nde-ed.org/EducationResources/CommunityCollege/Materials/Mechanical/StressStrain.htm

 

[sophiecentaur]

Summary:: Is tension altered by the shape of the cable?

I seem to remember from my school days that Tension Force can only be linear.

It may be better to re-state the question - which could actually answer it at the same time. Tension is a Force that can sometimes be considered as the only force acting on an object. A massless (totally flexible) string, under tension will experience equal and opposite forces, on each of its ends, acting along the line of the string; it will go straight. If you pass the string over 'perfect' pulleys and rollers, the same thing can be said to apply to the sections of the string. Your question is taking us away from the ideal to the practical so we can't necessarily categorise forces by using a word like Tension.
Connecting two things with a rigid structure can sometimes be treated as two forces pulling the connecting points together without considering any other forces. Your arch of rigid steel could be 'replaced' by a rope and simple Tension but only if there is no mass involved (for both, of course).
Try to avoid worrying too much about when a term like Tension should or should not be used. Concentrate, rather, on the magnitudes and directions of forces involved and on which of those forces can be ignored in any particular situation. Don't get involved with the 'Word Police' and aim at understanding.

 

[Lnewqban]

Summary:: Is tension altered by the shape of the cable?

I seem to remember from my school days that Tension Force can only be linear.

Is this true?

In 1 (in the graphic) the tension will follow the line of the rope

In 2 there is an unbendable, unbreakable, steel cable formed into an arch.
a ] In 3 which direction will the Tension be?
b ] In 3 if there was a gauge on this arch would it still read 1000kn, the same as on the rope, or would the shape change this.

Thank you

I may be wrong, but this is how I see it:
Non-linear tension force is not possible.
Tension, as well as compression force, is considered a vector, which represents a physical quantity that has both magnitude and direction.

I believe that you are asking about whether or not the internal forces of the arc follow a curvilinear path.
For the external forces at the points of connection car-arc-car, we can consider those hinges or articulations, upon which only the reactive horizontal forces opposing the car's thrust and the reactive vertical forces opposing the weight of the arc (all linear forces) act.

If you disconnect that arc from the cars, still imaginarily adding those four linear forces to both ends or connecting points, you could cut that arc section by section and calculate the internal forces acting on each little section.
Then, you will see that a combination of new shearing and bending forces appear, which are a result of the action of those four external forces on the geometry of the arc (which introduces strong moments).

Even example #1 is ideal, since a horizontal rope has some weight, which makes it take the shape of a catenary (curve close to the shape of your cable, but inverted).
Because of that, each infinitesimal section of that rope will "feel" tension forces in alignment with the longitudinal axis of that section plus a force that pulls it sideways, which induces shear stress.
There are no moments in that case, since the rope is flexible, unlike your rigid arc.

 

[Vanadium 50]

The material doesn't matter: rope or steel. The tension force is in the direction of the "rope" at the attachment points to the cars.

 

[sophiecentaur]

. . . . . . if the rope is totally flexible.

 

[zoki85]

I seem to remember from my school days that Tension Force can only be linear.

Linear or constant? Constant is a special case of linear. Think about it-it's not the same.

b ] In 3 if there was a gauge on this arch would it still read 1000kn, the same as on the rope, or would the shape change this.

If the arc is somehow supported in the mid point-the apex of the arc (say by a vertical pole), then the gauge at that point will read 1000 kN. Otherwise no. There will be variation of stress along cross section of the curved geometry in a rigid body.

 

[Vanadium 50]

. . . . . . if the rope is totally flexible.

Not really. A rigid object can apply a force in a direction other than its length, but it's not a tension.

 

[sophiecentaur]

Not really. A rigid object can apply a force in a direction other than its length, but it's not a tension.

Fair enough. We are only discussing the use of a particular word and a Force Diagram wouldn't need to commit one way or the other. That seems to be a needless worry in the thread.

 

[Chestermiller]

In the case of the curved rigid cable, at any cross section along the cable, the tensile stress within the cable varies over the cross section. The tension in the cable at a given location is equal to the integral of the stress distribution over the area of the cross section. But in addition to the tension, as a result of the stress distribution, there is also a bending moment acting on each cross section. This is equal to the integral of the stress times the distance from the center of the cross section, integrated over the area of the cross section. The stress distribution is such that the tensile stress on the inside of the bend in your figure is higher than the tensile stress at the outside of the bend. This is what gives rise to the bending moment. In the system you have described, both the tension and the bending moment varies with axial position along the cable (unlike the case of a straight cable where the bending moment is zero everywhere and the tension is constant).

 

[PeroK]

In the case of the curved rigid cable, at any cross section along the cable, the tensile stress within the cable varies over the cross section. The tension in the cable at a given location is equal to the integral of the stress distribution over the area of the cross section. But in addition to the tension, as a result of the stress distribution, there is also a bending moment acting on each cross section. This is equal to the integral of the stress times the distance from the center of the cross section, integrated over the area of the cross section. The stress distribution is such that the tensile stress on the inside of the bend in your figure is higher than the tensile stress at the outside of the bend. This is what gives rise to the bending moment. In the system you have described, both the tension and the bending moment varies with axial position along the cable (unlike the case of a straight cable where the bending moment is zero everywhere and the tension is constant).

That's the answer we've all been waiting for!

 

[some bloke]

With regard to the title question rather than the specific example, which has already been covered, would the fact that any cable with weight under tension in gravity would have a curve, dipping toward the center, give any answers? By my understanding, the tension in such a cable remains constant and the direction is aligned to the cable. Or does tension not really cover a cable with additional forces at play (E.G. a suspension bridge cable with a constant side-wind)?

 

[sophiecentaur]

By my understanding, the tension in such a cable remains constant

No. The tension at each end has to support the weight of the cable so it cannot be constant over the length. A special case of this would be a cable, suspended at one end. There would be zero tension in the bottom mm of the cable.