Statically Indeterminate Shaft

In summary, the problem is to determine the load on the three bearings at A, B, and C when the shaft is loaded with a force F at one end. The solution involves using beam deflection formulas to find the displacement at points B and C due to the force F, and then using the same formulas to find the displacement at these points due to the reaction forces at B and C. By setting the actual deflection at points B and C to zero, two equations with two unknowns can be solved to determine the load at each bearing. It is important to note that the supports at A, B, and C can only take vertical load and cannot carry moments.
  • #1
Spimon
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Homework Statement


Determine the load (vertical only) on the three bearings at A, B and C. The shaft is loaded at one end with force F (see attached diagram).

Homework Equations



Cantilever Beam Deflection
Deflection at End = PL^3/(3EI)
Deflection at Point X = Px^2/(6EI)*(3l-x)

The Attempt at a Solution



I'm having a few difficulties remembering the details of solving statically indeterminate problems. My attempt is as follows:

1. Remove reaction forces B and C. Applying beam deflection formulas I can find the displacement of points B and C due to force F. I called these δB1 and δC1.

2. Replace the reaction forces at B and C, but remove the force F. I can now find the displacement of points B and C due to the reaction forces, respectively. I called these δB2 and δC2.

3. Since the actual deflection at the fixed points B and C is zero, I can set:

δB1 + δB2 = 0

and similarly

δC1 = δC2

This leaves 2 equations and 2 unknowns which may be solved.Any hints, comments or corrections would be a world of help :smile:
 

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  • #2
All supports take vertical load only (simple supports) and cannot carry moment, so the support at A is not fixed and cannot act that way when supports B and C are removed. The beam is statically indeterminate to the first degree, so you need to initially remove just one support... like perhaps B.
 

1. What is a statically indeterminate shaft?

A statically indeterminate shaft is a type of structural element that cannot be fully analyzed using basic statics principles. This means that the forces and moments acting on the shaft cannot be determined solely by considering the external loads and the geometry of the shaft.

2. How is a statically indeterminate shaft different from a statically determinate shaft?

A statically determinate shaft is one in which all the forces and moments can be calculated using statics principles. In contrast, a statically indeterminate shaft requires more advanced analysis techniques, such as the method of superposition or the principle of virtual work, to determine all the forces and moments acting on it.

3. What are some common examples of statically indeterminate shafts?

Statically indeterminate shafts can be found in many structures, such as bridges, buildings, and machines. Some examples include truss bridges, cantilever beams with overhangs, and shafts with multiple supports or connections.

4. Why is it important to consider the statical indeterminacy of a shaft?

Ignoring the statical indeterminacy of a shaft can lead to inaccurate or unsafe designs. By fully understanding the forces and moments acting on a shaft, engineers can ensure that the structure is able to withstand the expected loads and conditions.

5. How can the statical indeterminacy of a shaft be solved?

There are several methods for solving the statical indeterminacy of a shaft, including the method of consistent deformations, the moment distribution method, and the slope-deflection method. These techniques involve creating a system of equations based on the known and unknown forces and moments, and solving for the unknowns using mathematical principles.

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