# How can I understand shear flow in a beam cross section?

I am trying to understand shear flow and I am just not getting it. For example, one of the problems in Beer has an I-beam and there is both a horizontal and vertical force acting on it. So it has shear in x and y. I understand that, but in the solution the instructor shows this as the shear flow due to the right-facing horizontal force:

And this one due to the downward-facing vertical force:

My question is, how come the shear flow due to the vertical force has arrows flowing towards (and away from) the center but the shear flow due to the horizontal force doesn't have something similar - all the arrows are pointing the same direction? If the vertical force-induced shear flow has arrows pointing towards the center on the top pieces, I would expect the horizontal force-induced shear flow arrows to be pointing to the center on the vertical section of the beam.

How can I understand shear flow in general so that these images make sense? Beer is not very clear about it. The arrows just appear without much explanation. What is a good source for a physical explanation of this phenomenon?

• The shear flow arrows are used to visualize a rough approximation of the distribution of shear stresses in the beam. A good explanation is given in "Mechanics of Materials" by Bedford and Liechti. amazon.com/Mechanics-Materials-Anthony-Bedford/dp/3030220818 Apr 16, 2022 at 22:06

More likely the figure on top with shear flow universally going from left to right is due to direct horizontal shear, like the shear under a centered force or due to a support reaction.

The one with the shear flow on the flanges going to the right and left and on the web going down is the shear flow due to the moment stresses in the beam.

## Edit

In a beam with varying bending moment (not uniform bending moment) If we cut a small section of the beam, $$dx,$$ and investigate the stresses on a thin horizontal plane of this section the bending stresses on the two sides of the section are not equal and change by $$dm.$$

For static equilibrium to exist there has to be shear stress on this plane $$\tau_{xy}= dm$$ This stress varies over the cross-section of the beam and it is equal to

$$\tau_{xy}=\frac{VQ}{IB}$$

• V= section shear
• Q= first area moment of the area above or below the plane of consideration
• I= beam's area moment of inertia
• B= section width

here is a link to driving the equation. shear stress in beams

• Perhaps. Is there a reason why the one direction considers shear flow and the other doesn't? Also, do you know of a text that explains shear flow well? I still don't know why arrows point towards the center. Apr 16, 2022 at 2:36

Shear in a beam caused by a lateral force is generally highest near the center and usually zero at the peripheries considering the direction of the lateral force. So when you draw the shear flow, you should start in the center piece of the section, where the shear will be simply in the direction of the lateral force and as you go further from the center, the shear "flow" should smoothly connect to the center portion.

When applied to your pictures, it is very simple in the first one. In the second one, you just start at the centre and the flow there will "suck in" the top flange and "flow out" into the bottom flange.

Here is image of shear stress in an I beam calculated using Zhuravskii formula (from answer to another question), which does not consider the flow like in your images (here the flow is always only in direction of the lateral force), but it still shows the highest shear at the center and lowest shear at the top and the bottom.