How to calculate Shear Flow distribution through an Annulus

[spggodd]

Homework Statement

My problem is how to calculate the Shear Flow Distribution through this open cross-section.
The section has a uniform thickness of 2mm and all other dimensions are on the attached picture.

There is also a 100 kN downwards force applied.

Homework Equations

q(s) = q(s₀) = {(I_xxV_x(z) - I_yxV_x(z)) / (I_yyI_xx - I_xy²)} ∫^s₀tx*ds - {(I_yyV_y(z) - I_yxV_x(z)) / (I_yyI_xx - I_yx²)} ∫^s₀ty*ds

Where the:
I terms are the second moment of areas.
V terms relate to the applied force.
t is the thickness
y is the distance to the overall centroid of the part from the centroid of the sub-section under consideration.
ds is the distance along the sub-section.

For the rectangular sections 1 - 4 I have simplified the equation to:

q(s)=q(s₀)-(V_y(z)/I_xx)∫^s₀ty*ds

The Attempt at a Solution

I have calculated the Second Moment of Area (500.172x10^-6) m⁴
I have calculated all the shears flows up until point 4 on the diagram.
To the left of point 4 is a small rectangular area which I have chosen to neglect, therefore I am concentrating on the Shear Flow around a the remaining quarter annulus which will bring me back to the x-axis.

At point 4 I believe the initial shear stress = 90.168 N/mm based on my workings through the rest of the cross-section up to this point.

I am now stuck at the annulus and I can't seem to find anything in books or my course notes etc..


Thank you for your time.
Steve

 

[SteamKing]

In the annulus, you'll have to squint at the diagram and pretend the thickness of the material is small w.r.t. the radius. You can split the annulus into differential elements where dA = t dθ. You'll have to develop an expression for Q as you go around the annulus, but the shear flow q = VQ/I

 

[spggodd]

Aw man, I had a feeling it was going to be something like that.

Ok I will have a crack at it tomorrow.

Thanks for your help!

 

[spggodd]

Could you offer a link to an reference material?

 

[rezeew]

Hello Steve,

Calculating the shear flow distribution through an annulus involves using the principles of mechanics of materials and the equations of equilibrium. The first step is to determine the shear flow at any given point on the cross-section, which can be done using the formula you have provided. This formula takes into account the second moment of area, the applied force, and the distance from the centroid of the sub-section.

To calculate the shear flow for the annulus, you will need to divide it into smaller sub-sections, similar to what you have done for the rectangular sections. The key here is to choose a suitable coordinate system and determine the distances and moments of inertia for each sub-section. Once you have all the necessary values, you can then use the formula to calculate the shear flow at each point along the annulus.

It is important to note that the shear flow will vary at different points along the annulus, so you will need to repeat this process for multiple points to get a complete distribution. Additionally, you will also need to consider the effect of the applied force on the shear flow, as it will change the distribution and magnitude of the shear flow.

I recommend consulting your course materials or a textbook on mechanics of materials for more detailed steps and examples on how to calculate shear flow for an annulus. It may also be helpful to seek guidance from your instructor or a tutor for further clarification. I hope this helps and good luck with your calculations!