Statically Indeterminate Beam to the Sixth Degree

• 6Stang7
In summary, the static indeterminate beam with support at points B and C may not be the correct solution because the equations used to solve for the angle and deflection do not assume a linear differential equation. Additional boundary conditions must be specified in order to determine all the reacting forces and moments in the beam.
6Stang7
As the title says, I have a statically indeterminate beam to the sixth degree and I'm attempting to use the superposition method (aka force method) to solve for the reactions. My additional equations will be the angle at points A, B, C, and D as well as the deflection at points B and C.

Is this the correct method to solve this, or is this the wrong approach? My thought is that this might not be right because the equations which are used to solve for the angle and deflection assume a linear differential equation, which I _think_ is not the case here.

The knife edge supports at B and C can allow a vertical reaction to form, but the beam is free to rotate at these points. Consequently, the boundary conditions for this beam are that the deflection = 0 at A, B, C, and D and the slope = 0 at A and D. For small deflections, the differential equation governing the beam's behaviour can be linearized. This configuration is known as a continuous beam, and there are several techniques which can be used to solve for the unknown reaction forces and moments.

The knife edge supports at B and C can allow a vertical reaction to form, but the beam is free to rotate at these points.

Ah, then I used the wrong support representation; the support method used at points B and C in the physical item will not allow for rotation, so a moment will be produced there.

If a moment reaction can develop at B and C but the loading is only applied to segment BC, then it would appear that segments AB and CD are not affected by the load F. Whatever the case, additional boundary conditions for B and C are required in order to determine all the reacting forces and moments in the beam. As long as deflections are assumed sufficiently small, the linearized Euler-Bernoulli equation will still apply.

1. What is a statically indeterminate beam to the sixth degree?

A statically indeterminate beam to the sixth degree is a structural element that cannot be analyzed using traditional methods due to the complexity of its geometry and loading conditions. It requires advanced mathematical techniques, such as the slope-deflection method or the moment distribution method, to solve for the unknown forces and reactions.

2. How is a statically indeterminate beam to the sixth degree different from other beams?

A statically indeterminate beam to the sixth degree has six unknown reactions, while a typical beam has only three. This means that there are more forces and moments acting on the beam, making it more complex to analyze and design.

3. What are the applications of a statically indeterminate beam to the sixth degree?

Statically indeterminate beams to the sixth degree are commonly found in structures with complex geometries, such as bridges, cranes, and high-rise buildings. They are also used in advanced engineering problems to develop new techniques and solutions.

4. How do you solve for the unknown forces and reactions in a statically indeterminate beam to the sixth degree?

To solve for the unknown forces and reactions, you can use advanced structural analysis methods such as the slope-deflection method or the moment distribution method. These methods involve setting up a system of equations based on the beam's geometry and loading conditions and solving for the unknowns using iterative techniques.

5. What are the limitations of using a statically indeterminate beam to the sixth degree?

The main limitation of using a statically indeterminate beam to the sixth degree is the complexity of its analysis. It requires advanced mathematical skills and can be time-consuming. Additionally, the assumptions made in the analysis may not accurately reflect the real-world behavior of the beam, leading to potential errors in the design.

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