# Shear flow in thin-walled members -- Hibbeler confusion

• arestes
In summary, Hibbeler's Engineering Mechanics, Mechanics of Materials discusses thin-walled members under shear force and how the moment of inertia is computed using different methods. The first calculation is an approximation, while the second calculation is exact.
arestes
TL;DR Summary
Hibbeler computes moment of inertia of thin-walled member with exact dimensions. In more recent edition, it uses "centerline" dimensions. Results are close to each other but which one is "more" right?
Hello:
I was reading about thin-walled members under shear force, specifically example 7-7 of Hibbeler's Engineering Mechanics, Mechanics of Materials.
First, the fourth Edition:

As you can see above. He starts by computing the moment of inertia on the first equation by subtracting a rectangle to another rectangle (the hole). This is clear.

However, the most recent edition I have (Ninth edition) computes the moment of inertia (again, same problem, same data) on the first equation:

Here Hibbeler uses "centerline" dimensions (which, I guess it's just an approximation). I understand that both computations are close. I could argue that one is just a convenient approximation. However, I read some parts of the theory about shear flow. For example, shear stress due to torsion needs centerline area (see the chapter on torsion on this same book (thin-walled tubes having closed cross sections) and the concept of "mean area" appears. This makes me believe there's something going on with mean or centerline dimensions. After all, isn't all this based on approximations of continuum mechanics theories?

If the only difference in taking centerline dimensions and exact moment of inertia is just a matter of convenience, I'm puzzled as to why Hibbeler decided to change his first exact calculation to an approximate one, given that the exact computation was fairly straightforward.

Lnewqban
arestes said:
I'm puzzled as to why Hibbeler decided to change his first exact calculation to an approximate one, given that the exact computation was fairly straightforward.
The first calculation is also an approximation from continuum mechanics. One of the larger assumptions being that the shear stress is constant over the width of a member. So, the "regular" shear stress formula is an average value assuming uniform stress distribution. This assumption quickly falls apart as the width/height ratio of a member increases, e.g. in thin-walled members.

The thin-walled approach explicitly ignores thickness terms above 2nd order in it's derivation/simplification. It also ignores shear flow perpendicular to the thin-walled members.

It would be interesting to compare both approaches to the theory of elasticity, but I don't have much insight into the magnitudes of deviations from deeper theory.. Maybe a continuum mechanics expert could chime in.

Lnewqban

## 1. What is shear flow in thin-walled members?

Shear flow in thin-walled members refers to the distribution of shear stress along the cross-section of a structural member, such as a beam or column, that has a thin wall compared to its overall length. It is an important concept in structural engineering, as it affects the strength and stability of the member.

## 2. How is shear flow calculated?

Shear flow is calculated by dividing the applied shear force by the moment of inertia of the cross-section. This results in a shear stress value that is distributed along the cross-section, with the highest values occurring at the edges and decreasing towards the center.

## 3. What is the difference between shear flow and shear stress?

Shear flow and shear stress are related concepts, but they are not the same thing. Shear stress is a measure of the force per unit area that is acting on a material, while shear flow is a measure of the distribution of that shear stress along a cross-section.

## 4. How does shear flow affect the design of thin-walled members?

Shear flow is an important consideration in the design of thin-walled members, as it can significantly affect the strength and stability of the member. Engineers must carefully calculate and account for shear flow when designing these types of structures to ensure they can withstand the expected loads.

## 5. Why is there confusion surrounding shear flow in thin-walled members?

The concept of shear flow can be confusing for some because it involves both shear stress and moment of inertia, which are separate concepts. Additionally, the distribution of shear stress along a cross-section can be difficult to visualize and understand. It is important for engineers to have a strong understanding of shear flow in order to properly design and analyze thin-walled members.

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