Calculating the Radius of Curvature for a Deflected Beam

In summary, the conversation discusses the problem of comparing the curve of a beam to its horizontal deflection for an experiment. The attachment shows a flexible beam being deflected by a force and the displacement x is known, but the vertical displacement y cannot be measured. It is assumed that the beam bends in a circular shape with radius R. The question is how to calculate R with a L0 of 120mm and an X-displacement of 55mm. The conversation also mentions the difference between small and large deflection theories and the use of elastica theory for flexible beams.
  • #1
Jerry1989
4
0
Hey Guys,

I'm stuck with this problem, we want to compare the curve of a beam to it's horziontal deflection, it's for an experiment we're performing on a very elastic beam.

In the attachment you see a flexibel beam getting deflected by a force, this causes displacement x. We don't know (or can't measure) displacement in vertical y-direction.
However, we can assume the beam bends in a way that it's curvature resembles a circle with radius R.

Now, if we have an L0 of 120mm and an X-displacement of 55mm. What would that radius R be?

So far, I've tried an arc-length menthod, and I tried looking at the circles, but I just can't figure it out. Can you guys help me out?
 

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  • #2
Most beam theories which are used in structural analysis are what are known as 'small deflection' theories. Because of the mathematics involved with relating the bending moment in the beam to its curvature, a great simplification in calculating the deflection of the beam is obtained when the slope of the beam is very small. For very flexible beams, where the slopes can no longer be considered small, the 'small deflection' theories can no longer be applied, and more complicated 'large deflection' theories must be used for analysis.

What you are looking for is called 'elastica theory':

http://en.wikipedia.org/wiki/Elastica_theory

For obvious reasons, most structural engineers don't deal with such theories, but some engineers, like those involved in laying submarine pipelines, may be familiar with the necessary mathematics.
 
  • #3
thanks, I'm looking into it.
So far, it's looking t be more of a trigonometry problem. But a tricky one..
 

What is the difference between a curved beam and a straight beam?

A curved beam is a structural element that has a curved shape along its length, while a straight beam has a linear shape. This means that a curved beam can withstand different types of loads and forces compared to a straight beam.

What are the main factors that affect the geometry of a curved beam?

The main factors that affect the geometry of a curved beam are the radius of curvature, the cross-section shape, and the material properties. These factors determine the curvature, thickness, and overall shape of the beam.

How does the geometry of a curved beam affect its structural strength?

The geometry of a curved beam can greatly affect its structural strength. A beam with a larger radius of curvature and thicker cross-section will have a higher strength compared to a beam with a smaller radius and thinner cross-section. The shape of the beam, such as a parabolic or circular curve, can also affect its strength.

What are some common applications of curved beams?

Curved beams are commonly used in architecture, such as in the construction of domes and arches. They are also used in the design of bridges, cranes, and other structures that require a strong and flexible support. In addition, curved beams are used in the aerospace industry for the design of aircraft wings and fuselages.

What are the advantages of using curved beams in construction?

Using curved beams in construction offers several advantages. They can distribute loads more evenly, allowing for a more efficient use of materials. Curved beams also have a higher resistance to bending and torsion forces, making them suitable for a variety of structural designs. In addition, their unique shape can add aesthetic appeal to a building or structure.

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