Bending a patterned beam (FOSDT)

In summary, according to First order shear deformation theory, the flexed pattern on a wooden beam will look like the middle picture. However, if the material has a non-zero Poisson's ratio, then the beam will deform into an S-shaped curve with different amounts of curvature depending on the location and magnitude of the shear forces.
  • #1
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Say I have a wooden beam with an angled pattern in it if the beam is flexed (and assuming First order shear deformation theory holds: parallel lines remain parallel after flexing) what would the flexed pattern look like?

I think it would look like the pattern in the middle picture since the strain is linear but I'm being told by somebody else that it will look something like the bottom picture when flexed.

Can somebody with some knowledge on deformation confirm one or the other.

http://img858.imageshack.us/i/beamflexed.png/ [Broken]
 
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  • #2
You can answer the question yourself by drawing a grid of vertical and horizontal lines on the beam and drawing your sloping lines through the intersections of those lines.

For simple beam theory, if the beam is in pure bending (i.e. there is a bending moment applied to each end, no shear forces, so the neutral axis of the beam bends into an arc of a circle), then the "vertical" lines will remain straight and at right angles to the neutral axis. So the grid will deform into a set of concentric circles and radial lines. You can plot the shape of the diagonal lines from that information. They will be slightly curved.

Actually it is more complcated than this if Poisson's ratio for the material is not zero. When the beam is bent, the neutral axis is not mid way between the top and bottom because of the strains caused by Poisson's ratio. Google "anticlastic curvature" for more on this if you are interested.

If there are shear forces on the beam, the "vertical" lines also bend into S-shaped curves, with different amounts of curvature as the shear varies along the length of the beam. A good book on Timoshenko beam theory should explain that in detail.
 
  • #3
Thanks Alephzero, are there any particular Timoshenko beam theory books that you recommend?
 
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  • #4
Why not go straight to the source and look at Timoshenko's own books? They were first published in about 1950, but still in print.
 
  • #5


I understand the importance of accurate and reliable information. In this case, the flexed pattern of a wooden beam with an angled pattern would depend on various factors such as the material properties of the beam, the magnitude and direction of the applied load, and the geometry of the pattern itself. Therefore, it is not possible to definitively confirm which of the two predicted patterns (middle or bottom picture) would result without further information and analysis.

However, based on the assumption that First Order Shear Deformation Theory (FOSDT) holds, it is likely that the flexed pattern would resemble the middle picture. This is because FOSDT assumes that parallel lines on the beam will remain parallel after flexing, which would result in a linear strain pattern. The bottom picture, on the other hand, shows a more complex and non-linear strain pattern, which may not be consistent with FOSDT.

It is important to note that FOSDT is a simplified theory and may not accurately predict the behavior of real-world structures. Therefore, it would be beneficial to conduct further analysis and experiments to determine the actual flexed pattern of the beam in question.
 

1. What is FOSDT?

FOSDT stands for "Flexural One-Step Digital Transformation" and is a process for bending and shaping patterned beams using digital control.

2. How does FOSDT work?

FOSDT involves using a digital control system to precisely manipulate the shape of a patterned beam by applying localized forces at specific points along its surface.

3. What are the practical applications of FOSDT?

FOSDT has many potential applications, including in the development of advanced electronic devices, in the creation of complex 3D structures, and in the production of microscale medical devices.

4. What are the advantages of using FOSDT compared to traditional bending methods?

FOSDT offers several advantages, including greater precision, faster production times, and the ability to create more complex and intricate shapes that would not be possible with traditional bending methods.

5. Are there any limitations or challenges to using FOSDT?

One of the main challenges of FOSDT is the need for specialized equipment and expertise, making it more difficult and expensive to implement compared to traditional bending methods. Additionally, the materials used for patterned beams must have certain properties in order to be successfully bent using FOSDT.

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