# How to calculate Shear Flow distribution through an Annulus

• spggodd
In summary, the conversation discusses the problem of calculating the shear flow distribution in an open cross-section with a uniform thickness of 2mm and a 100 kN downwards force applied. The equations and terms needed for the calculation are also provided. The attempt at a solution involves calculating the second moment of area and shear flows up to a certain point, but the solution becomes more complex when considering the annular section. The conversation ends with a request for reference materials and the expert suggesting breaking the annulus into differential elements and developing an expression for Q to solve the problem.
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) = {(IxxVx(z) - IyxVx(z)) / (IyyIxx - Ixy2)} ∫s0tx*ds - {(IyyVy(z) - IyxVx(z)) / (IyyIxx - Iyx2)} ∫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)-(Vy(z)/Ixx)∫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..

Steve

#### Attachments

• Section.JPG
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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

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

Ok I will have a crack at it tomorrow.

Could you offer a link to an reference material?

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!

## What is shear flow?

Shear flow refers to the distribution of shear stress along a cross-section of a structural member. It is caused by external forces acting on the member and is an important factor in determining the strength and stability of the structure.

## Why is calculating shear flow important?

Calculating shear flow is important in designing and analyzing structures, as it helps determine the distribution of stress and the potential for failure. It is also crucial in selecting appropriate materials and dimensions for structural members.

## What is an annulus?

An annulus is a geometric shape that consists of two concentric circles, where the area between the circles is filled in. It is commonly used in engineering and mathematics to represent circular objects with a hollow center, such as pipes or rings.

## How do you calculate shear flow through an annulus?

The formula for calculating shear flow through an annulus is: Q = 2πrt, where Q is the shear flow, r is the distance from the center of the annulus to the point of interest, and t is the thickness of the annulus. This formula assumes that the shear stress is constant throughout the thickness of the annulus.

## What are some practical applications of calculating shear flow through an annulus?

Calculating shear flow through an annulus is commonly used in the design and analysis of circular structures, such as pipes, cylinders, and rings. It is also used in the analysis of shear and bending stresses in aircraft wings and other curved structures. Additionally, understanding shear flow can help engineers optimize the design and minimize material usage for cost-effective construction.

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