When are reactive forces enough to support this body?

[per persson]

Homework Statement

When are reactive forces enough to support the body such that no reactive moments need to be introduced for 2 or more supports

Relevant Equations

no equations

In the book engineering mechanics statics and dynamics Hibbeler says:" It should be noted that the single bearing supports in items (5) and (7), the single pin (8), and the single hinge (9) are shown to resist both force and couple-moment components. If, however, these supports are used in conjunction with other bearings, pins, or hinges to hold a rigid body in equilibrium and the supports are properly aligned when connected to the body, then the force reactions at these supports alone are adequate for supporting the body. In other words, the couple moments become redundant and are not shown on the free-body diagram. The reason for this should become clear after studying the examples which follow." And then he shows following examples

But I still don't understand when the reactive forces are enough to support the body such that no reactive moments need to be introduced. How can I determine from looking at the diagram that reactive moments are redundant?

 

[haruspex]

. How can I determine from looking at the diagram that reactive forces are redundant?

Suppose you have a support force acting along a certain line and another line that represents a possible axis of rotation. What is the geometric relationship between the two lines that determines whether the force exerts a torque about the axis?

 

[per persson]

Suppose you have a support force acting along a certain line and another line that represents a possible axis of rotation. What is the geometric relationship between the two lines that determines whether the force exerts a torque about the axis?

I'm thinking a disc centered at z=0 that lies on the xy-plane. The only forces that will not cause the disc to spin are those for which the action of line passes through the z-axis and those which are parallel to the z-axis. Correct?

Edit: Sorry it should say how can I determine from a diagram when the reactive moments are redundant not the reactive forces

 

[haruspex]

I'm thinking a disc centered at z=0 that lies on the xy-plane. The only forces that will not cause the disc to spin are those for which the action of line passes through the z-axis and those which are parallel to the z-axis. Correct?

Yes. More generally, since the magnitude of the torque a force exerts about an axis is proportional to the perpendicular distance between the line of action and the axis, if the line of action intersects the axis then there is no torque. (It is also proportional to the sine of the angle between the two lines as viewed along that perpendicular distance, so a parallel force exerts no torque either.)
If there is an axis about which the forces present are unable to control rotation, none of them is exerting a torque about it. So, given a set of forces, how would you look for such an axis?

Edit: Sorry it should say how can I determine from a diagram when the reactive moments are redundant not the reactive forces

Yes, I figured that out.

 

[per persson]

Yes. More generally, since the magnitude of the torque a force exerts about an axis is proportional to the perpendicular distance between the line of action and the axis, if the line of action intersects the axis then there is no torque. (It is also proportional to the sine of the angle between the two lines as viewed along that perpendicular distance, so a parallel force exerts no torque either.)
If there is an axis about which the forces present are unable to control rotation, none of them is exerting a torque about it. So, given a set of forces, how would you look for such an axis?

Are you saying given a set of forces how to find an axis where there is no torque about it? I would just check if the action of line of the forces pass through the axis or are parallel to the axis.

 

[Lnewqban]

But I still don't understand when the reactive forces are enough to support the body such that no reactive moments need to be introduced. How can I determine from looking at the diagram that reactive moments are redundant?

The diagrams in the book seem to be a little confusing because those are not showing the type of supports.
You can consider that any part has the potential of six degrees of freedom respect to certain selected 3-D axes.

When you see a part that should be in static balance, see if you can determine what force or moment may be missing, without which the part would be free to rotate or slide.

Please, see:
https://www.techtarget.com/whatis/definition/degrees-of-freedom

https://skyciv.com/docs/tutorials/beam-tutorials/types-of-supports-in-structural-analysis/

https://engineeringdiscoveries.com/types-of-supports-reactions-and-their-applications-in-structures/

 

[haruspex]

Are you saying given a set of forces how to find an axis where there is no torque about it? I would just check if the action of line of the forces pass through the axis or are parallel to the axis.

Yes, but I would express it this way: see if there is a line which either passes through or is parallel to each of the force lines.

In the first diagram (top left), the line would have to lie in the plane containing $A_y, A_z$ and that containing $B_x, B_z$. Those planes intersect in a vertical line. The A and B forces cannot prevent rotation about it, but the C forces both have a torque about that line.

Try applying this procedure to the other diagrams.

 

[per persson]

Yes, but I would express it this way: see if there is a line which either passes through or is parallel to each of the force lines.

In the first diagram (top left), the line would have to lie in the plane containing $A_y, A_z$ and that containing $B_x, B_z$. Those planes intersect in a vertical line. The A and B forces cannot prevent rotation about it, but the C forces both have a torque about that line.

Try applying this procedure to the other diagrams.

In the picture below consider the plane where T_z and T_y of T lie and the plane where A_x and A_y lie. They intersect in a line which is parallel to the x-axis. The force 300lb and A_z causes a torque around this axis but it does not prevent the rotation around it thus there is a reactive moment M_x. Correct?

So how am I supposed to know if the forces that causes torque about the axis are enough to prevent rotation?

 

[SammyS]

In the picture below consider the plane where T_z and T_y of T lie and the plane where A_x and A_y lie. They intersect in a line which is parallel to the x-axis. The force 300lb and A_z causes a torque around this axis but it does not prevent the rotation around it thus there is a reactive moment M_x. Correct?

So how am I supposed to know if the forces that causes torque about the axis are enough to prevent rotation?

Where is the picture ?

 

[per persson]

Where is the picture ?

 

[haruspex]

In the picture below consider the plane where T_z and T_y of T lie

Do you mean $T_z, T_x$? T does not appear to have a component in the y direction.

and the plane where A_x and A_y lie. They intersect in a line which is parallel to the x-axis.

Yes, if you meant $T_z, T_x$.

The force 300lb and A_z causes a torque around this axis

Looks to me that they certainly have a net torque about any axis parallel to the y axis. If they are unequal in magnitude then they reduce to a single force parallel to the z axis acting somewhere along the x axis. Exactly where depends on their relative magnitudes. They could produce a net torque about any axis that does not lie in the zx axes plane.

In the same way (calling the elbow point C) we can project AC out to point D where CD=8” then replace the 300lb force and the 200 lb ft torque by a 300 lb force at D.

What is T? Is it another restraint force or an arbitrary applied force?

So how am I supposed to know if the forces that causes torque about the axis are enough to prevent rotation?

Are you asking whether the forces have sufficient magnitude, given that no axis intersects all their lines of action? You can assume that restraint forces will adjust magnitude as necessary to prevent movement.

 

[Lnewqban]

Missing information, copied from:
https://www.pearsonhighered.com/assets/samplechapter/0/1/3/6/0136077900.pdf

 

[per persson]

Do you mean $T_z, T_x$? T does not appear to have a component in the y direction.

Yes, if you meant $T_z, T_x$.

Looks to me that they certainly have a net torque about any axis parallel to the y axis. If they are unequal in magnitude then they reduce to a single force parallel to the z axis acting somewhere along the x axis. Exactly where depends on their relative magnitudes. They could produce a net torque about any axis that does not lie in the zx axes plane.

In the same way (calling the elbow point C) we can project AC out to point D where CD=8” then replace the 300lb force and the 200 lb ft torque by a 300 lb force at D.

What is T? Is it another restraint force or an arbitrary applied force?

Are you asking whether the forces have sufficient magnitude, given that no axis intersects all their lines of action? You can assume that restraint forces will adjust magnitude as necessary to prevent movement.

Above is the pages of the problems. T is the force caused by a cable.
I'm still lost on how to determine if there is reactive moment or not when there are more than two supports.

If I understand your method right you find an axis where reactive forces of two supports do not cause a torque about it. Then u look at other forces which do cause a torque about this axis and say those forces prevent any rotation around the axis and thus there is no reactive moment of the supports around an axis parallel to this axis.

I don't understand this. How do we know that the forces that cause a torque around the axis are enough to prevent rotation, if the forces are not enough a reactive moment could be introduced to prevent rotation.

Also how do we conclude that the supports cause no reactive moments by looking at some random axis?

 

[haruspex]

Above is the pages of the problems. T is the force caused by a cable.
I'm still lost on how to determine if there is reactive moment or not when there are more than two supports.

The text you highlight doesn’t say there is no reactive moment, it says there need not be one. It is an example of static indeterminacy. A simple example is a table with four legs. If we take the idealised view that the legs are exactly equal in length and perfectly rigid, and the floor perfectly flat and rigid, there is no way to determine how much weight each leg is taking. In reality, tiny deviations from the idealisation lead to the actual forces.

In these torque examples, if there is a combination of forces at the joints which produces equilibrium then perhaps there are no torques at the joints, but if the joints are capable of producing torques then it could be that there are torques and the forces are in some other combination.
For example, a simple horizontal beam fixed by rigid joints at the ends. Either one of the joints can produce equilibrium by itself, meaning there need not be any force or torque from the other.

If I understand your method right you find an axis where reactive forces of two supports do not cause a torque about it.

Or more, if possible.

Then u look at other forces which do cause a torque about this axis and say those forces prevent any rotation around the axis

That they can prevent rotation. To be sure they will, they have to be constraint forces, i.e. forces that adjust as necessary to inhibit movement. Merely hanging a weight on one side, for example, might not do it.
Indeed, a non constraint force might be precisely what does cause rotation.

and thus there is no reactive moment of the supports around an axis parallel to this axis.

There need not be a reactive moment around such an axis.

I don't understand this. How do we know that the forces that cause a torque around the axis are enough to prevent rotation, if the forces are not enough a reactive moment could be introduced to prevent rotation.

Answered above. For example, an object weight W resting on the floor. The normal force from the floor is a constraint force, adjusting to be the minimum force to prevent the object from penetrating the floor.

Also how do we conclude that the supports cause no reactive moments by looking at some random axis?

The procedure as you describe it above only shows there need not be a reactive moment about the chosen axis. Consider a beam suspended by two parallel rods. Take a vertical axis along one rod. If we add a constraint holding the centre of the beam in position, but not restricting rotation in the horizontal plane, it can exert a torque about the axis, but if the rods do not then rotation about that axis possible.
Had we chosen a vertical axis through the centre of the beam we would have discovered that the added constraint does not prevent rotation.

 

[Lnewqban]

I'm still lost on how to determine if there is reactive moment or not when there are more than two supports.

I hope that you are still interested on understanding this subject.
Perhaps if we go back to the simpler 2-D problems, we could better deal with the 3-D ones.

The type of support determines "if there can be a reactive moment or not."
I also hope that you have studied the capabilities of each type of support shown in the link of post #6 and in post #12 above.

Only certain types of supports are able to naturally generate a punctual moment that resists the rotational movement of the link or member of one structure that is directly connected to it, keeping the structure in static equilibrium.

How do we know that the forces that cause a torque around the axis are enough to prevent rotation, if the forces are not enough a reactive moment could be introduced to prevent rotation.

You compute a summation of the torques caused by all the involved co-planar forces.

A reactive moment at the support can't be introduced: it would appear naturally and in the exact magnitude needed to keep the statis balance.
But that is true only if that type of support is able to resist rotation: a simple pin would never be able to do that, for example.

Also how do we conclude that the supports cause no reactive moments by looking at some random axis?

You perpendicularly look at each projection of the mechanism on each of the three single planes.
Then, you proceed with the summation of the torques caused by all the forces contained in each plane, one by one.