# Moment and reaction of shaft - Statically Indeterminate

• Tekneek
In summary: You'll also need to study the elastic flexure formula for a beam under a transverse load. That will be necessary to figure the transverse loading for your continuous beam. Once you have the transverse loading figured, you can determine the reactions using the method of consistent deformations. You can find that method explained in any reasonable strength of materials textbook.
Tekneek

A shaft of diameter 10 inch is supported by 3 ball bearings. An external force (F, given) from pulley (diameter not given) also acts on the shaft. The supports are subjected to bending load (moment and transverse shear) and torsion load due to pulley. Yield Stress at Tension given. Assume no axial load. I need to calculate the reactions and moment equation for the shaft.

My approach:
• Statically Indeterminate
(1) ∑Fy = 0 = Ra + Rb + Rc -F
(2) ∑Mc (CCW +) = 0 = 5F-6Ra-3Rb
Solve (2) for Rb and Plug it in (1)

To write a moment equation, I make a cut between Rb and Rc
Summing the moment at the Cut (o)
∑Mo = 0 = -Ra(x) -Rb (x-2) + F(x-1) + M = 0
(3) M = Ra(x) + Rb(x-2) - F(x-1)

After this I can calculate the reactions using deflection method (singularity equation).

However, I am not sure that my moment equation is correct. Since the problem states that the shaft is subjected to bending load and torsion load at the ball bearings, do I have to account for these in my moment equation? If so, how?

Tekneek said:

A shaft of diameter 10 inch is supported by 3 ball bearings. An external force (F, given) from pulley (diameter not given) also acts on the shaft. The supports are subjected to bending load (moment and transverse shear) and torsion load due to pulley. Yield Stress at Tension given. Assume no axial load. I need to calculate the reactions and moment equation for the shaft.

My approach:
• Statically Indeterminate
(1) ∑Fy = 0 = Ra + Rb + Rc -F
(2) ∑Mc (CCW +) = 0 = 5F-6Ra-3Rb
Solve (2) for Rb and Plug it in (1)

To write a moment equation, I make a cut between Rb and Rc
Summing the moment at the Cut (o)
∑Mo = 0 = -Ra(x) -Rb (x-2) + F(x-1) + M = 0
(3) M = Ra(x) + Rb(x-2) - F(x-1)

After this I can calculate the reactions using deflection method (singularity equation).

However, I am not sure that my moment equation is correct. Since the problem states that the shaft is subjected to bending load and torsion load at the ball bearings, do I have to account for these in my moment equation? If so, how?

Assuming the shaft is free to rotate in the ball bearings, the torsion load should not affect the bearing reactions.

It's not clear to me how you have derived your moment equations. Are there some lengths of shaft segments which are not included in this thread?

In any event, the bearing reactions are produced only by the transverse loading of the shaft. There is a concentrated load due to the tension on the pulley (F), and I think you want to include the weight of the shaft as a distributed load, since the shaft is 10 inches in diameter. Once these transverse loads are figured, there are several different techniques which can be used to calculate the bearing reactions, including singularity functions.

SteamKing said:
Assuming the shaft is free to rotate in the ball bearings, the torsion load should not affect the bearing reactions.

It's not clear to me how you have derived your moment equations. Are there some lengths of shaft segments which are not included in this thread?

In any event, the bearing reactions are produced only by the transverse loading of the shaft. There is a concentrated load due to the tension on the pulley (F), and I think you want to include the weight of the shaft as a distributed load, since the shaft is 10 inches in diameter. Once these transverse loads are figured, there are several different techniques which can be used to calculate the bearing reactions, including singularity functions.

Sorry, that was my fault.
∑Mo = 0 = -Ra(x) -Rb (x-3) + F(x-1) + M = 0
(3) M = Ra(x) + Rb(x-3) - F(x-1)

Using singularity function

dy/dx = 1/EI [ (Ra(x-0)^2 )/2 + (Rb(x-3)^2)/2 - (F(x-1)^2)/2 + C1 ]
y(x) = 1/EI [ (Ra(x-0)^3 )/6 + (Rb(x-3)^3)/6 - (F(x-1)^3)/6 + C1x + C2 ]

Applying Boundary Condition

@x=0, dy/dx = 0 --> C1 = 0

@x=3, y(0) = 0 --> 0 = 4.5Ra - 267 + C2
@x=3, dy/dx = 0 --> 0 = 13.5Ra - 400 --> Ra = 29.62

Then using Ra I can find C2, then using that information I can find Rb and finally Rc. Am I correct so far?

Tekneek said:
Sorry, that was my fault.
∑Mo = 0 = -Ra(x) -Rb (x-3) + F(x-1) + M = 0
(3) M = Ra(x) + Rb(x-3) - F(x-1)

Using singularity function

dy/dx = 1/EI [ (Ra(x-0)^2 )/2 + (Rb(x-3)^2)/2 - (F(x-1)^2)/2 + C1 ]
y(x) = 1/EI [ (Ra(x-0)^3 )/6 + (Rb(x-3)^3)/6 - (F(x-1)^3)/6 + C1x + C2 ]

Applying Boundary Condition

@x=0, dy/dx = 0 --> C1 = 0

@x=3, y(0) = 0 --> 0 = 4.5Ra - 267 + C2
@x=3, dy/dx = 0 --> 0 = 13.5Ra - 400 --> Ra = 29.62

Then using Ra I can find C2, then using that information I can find Rb and finally Rc. Am I correct so far?
No, that's not how this works.

Your shaft is a continuous beam, since it rests on more than two supports. As a continuous beam, the equations of static equilibrium are insufficient to determine the reactions in the supports. Without figuring out the reactions first, you don't have enough equations to determine the integration constants for the slope and deflection of the shaft.

Have you studied anything about continuous beams and how to calculate the reactions at the supports?

SteamKing said:
No, that's not how this works.

Your shaft is a continuous beam, since it rests on more than two supports. As a continuous beam, the equations of static equilibrium are insufficient to determine the reactions in the supports. Without figuring out the reactions first, you don't have enough equations to determine the integration constants for the slope and deflection of the shaft.

Have you studied anything about continuous beams and how to calculate the reactions at the supports?
Not exactly sure. If you could link me to something I could look up that would be helpful.

Tekneek said:
Not exactly sure. If you could link me to something I could look up that would be helpful.
Google "continuous beam" and you'll get enough hits to keep you busy into next week.

## 1. What is the definition of a moment and reaction of a shaft?

A moment is a force that causes a body to rotate around a point, also known as the moment of force. A reaction of a shaft is the force exerted by the shaft on its supports or bearings in response to external loads.

## 2. How does the concept of static indeterminacy apply to moments and reactions of a shaft?

Static indeterminacy refers to a situation where the number of unknown forces or reactions is greater than the number of available equations. In the case of moments and reactions of a shaft, this means that the number of external loads acting on the shaft is greater than the number of unknown reactions at its supports, making the calculation of moments and reactions more complex.

## 3. What methods are used to solve for moments and reactions in statically indeterminate shafts?

There are several methods used to solve for moments and reactions in statically indeterminate shafts, including the slope-deflection method, the moment distribution method, and the virtual work method. These methods involve using equations of equilibrium and compatibility to determine the unknown reactions.

## 4. How does the material and geometry of a shaft affect its moment and reaction?

The material and geometry of a shaft can greatly influence its moment and reaction. For example, a longer and thinner shaft will have a higher moment of inertia and will be more prone to bending under external loads, resulting in higher moments and reactions at its supports. Additionally, the material properties, such as strength and stiffness, will also affect the moments and reactions in a shaft.

## 5. What are some real-world applications of understanding moments and reactions in shafts?

Understanding moments and reactions in shafts is crucial in engineering and design, particularly in structures that experience high levels of stress and loading, such as bridges, cranes, and buildings. It is also important in the design and analysis of mechanical systems, such as rotating machinery and power transmission systems, where the proper distribution of moments and reactions is necessary for safe and efficient operation.

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