Opposing Moments on Levered Beam

[nickbarker77]

Homework Statement

Calculate the force F1 to bend the beam AB to a radius of 7.5m. Assume the beam is a steel tube 200mm OD with wall thickness 20mm. The arm's AC and BD can be assumed rigid.

The force F1 acts on pivots at C and D and is always in the x direction.

Homework Equations

I = pi (do4 - di4) / 64

The Attempt at a Solution

My first thought was to assume the beam was fixed at one end and then work out the deflection required to achieve a a radius of 7.5m. From this I substituted into the deflection formula to work out a point load need on the end of the cantilevered beam to produce the deflection.

I think i should be calculating the moment at point C and D but I'm struggling a bit, any help would be greatly appreciated.

 

[rock.freak667]

Why not just use

M/I = σ/y = E/R

E is given for the material and you know R. Just get M.

 

[nickbarker77]

Thanks rock.freak667.

i have since found this formula and came out with an answer of 1.25 x 10^6 N/m for the moment required.

this is using a value of 4.637 x 10^-5 for the second moment of area and a young's modulus of 200GPa for the steel.

As the arms at the end of the beam are 1.25m long then the required force at points C and D would be 9.9 x 10^5 N.

This seem a little high?

 

[HyperSniper]

When I calculated the second moment of area I got 2.701*10^-5 m^4 based on this:
$I=\frac{\pi*\left[\left(200mm\right)^{4}-\left(200mm-20mm\right)^{4}\right]}{64}=2.701\times10^{-5}m^{4}$

This would change your results a little bit. Solving out the ratio before of M/I = E/R gives:

$\frac{M}{I}=\frac{E}{R}$

$M=\frac{EI}{R}$

$M=1.441\times10^{5}N$

I would definitely double check what I did. 32,394 pounds of force seems reasonable to me though.

 

[DeeNos]

As a scientist, the first step in solving this problem would be to clarify any uncertainties or missing information. It is important to know the material properties of the steel tube, such as its modulus of elasticity and yield strength, in order to accurately calculate the required force. Additionally, the location of the pivots at C and D should be specified in order to determine the lever arm and moment arm.

Assuming these values are known, the next step would be to calculate the moment required to bend the beam to a radius of 7.5m. This can be done by using the formula for the moment of inertia of a hollow cylinder (I = pi (do4 - di4) / 64) and substituting in the values for the outer and inner diameters of the steel tube. This moment can then be equated to the force F1 multiplied by the lever arm, which is the distance from the pivot to the point of force application.

Finally, the force F1 can be calculated by dividing the calculated moment by the lever arm. It is important to note that the force F1 will vary depending on the location of the pivots at C and D. If the pivots are closer to A, the required force will be larger, and if they are closer to B, the required force will be smaller.

In conclusion, to accurately calculate the force F1 required to bend the beam to a radius of 7.5m, we need to know the material properties of the steel tube, the location of the pivots, and the lever arm. These values can then be used to calculate the moment required and subsequently the force F1.