# The state of a Stress element isolated from a beam

Consider a beam subjected to an arbitrary loading, where the loads are only present in the plane of bending. The x-y plane is the plane of bending.

If I take a stress element in the beam, there will be normal stresses and transverse and longitudinal shear stresses acting on it as shown.

I'm interested in knowing whether shear stresses appear on the shaded face and on the face opposite to it. The books that I've been following say that there will be no stress on this face, and the state of stress will be a plane stress one.

However, I feel like there will be a tendency of vertical layers slipping past each other (as shown below) when the loads will be applied and that should result in a shear stress on the shaded face.

Why the stresses on the shaded face and the face opposite to it ignored?

• Given the loading, the shearing will not be as shown in your last figure. Commented Feb 24, 2022 at 15:38
• @BobD I don't understand why. Consider for example, the distributed load in the first figure, this distributed load is in the plane x-y, so an element taken just below the load will have a tendency to slide past the nearby elements. So that must give rise to a shear force on the shaded face. Isn't it? Commented Feb 24, 2022 at 16:09
• The distributed load is generally considered uniform into the page, such as the weight of the beam Commented Feb 24, 2022 at 16:39
• What about point loads? I could take an element in the vicinity of the point laod, and that element will have a tendency to slip past nearby elements. Commented Feb 24, 2022 at 16:45
• The distribution of the shear stress do to a concentrated load is also generally assumed to be uniform across the width of the beam. See wp.optics.arizona.edu/optomech/wp-content/uploads/sites/53/2016/… Commented Feb 24, 2022 at 17:30

Regarding your first diagram, the convention upon viewing is to assume that the loads are applied to the entire top width (in and out of the page). This, along with a few other assumptions, allows straightforward application of Euler-Bernoulli beam theory to obtain closed-form expressions for the stress, deflection, slope, and curvature, for example.

In this sense, "point load" is a misnomer because those loads appears as a point only in the side view. (Alternatively, we might idealize the beam as a 2D object, and then certain loads would appear as points.)

In real life, true point and line loads do not exist, but Saint-Venant's principle tells us that a distributed load can look like an idealized point or line load as we move farther away.

If you choose to specify that the loads are true 3D point loads that are applied only in the plane you show in your second image, then a shear stress would indeed arise on the plane you shaded. Solving this problem is much more complex, which may be why you haven't found an associated discussion in the references you're consulting.

• I couldn't thank you enough, I've been struggling on this from long, the second paragraph you wrote made everything clear. I always thought point loads were POINT loads. Knowing that in 3D indeed shear stresses would appear on the plane I showed, gives me a satisfaction now. A quick question, say my load is actually a point load, I could think of this point load to be applied via a plate, as in, a plate kept on the beam and then on the plate I apply the point load, that would too result in a point load which is spread along the width. Am I right on this? Commented Feb 24, 2022 at 18:09
• If the plate is sufficiently thick, Saint-Venant's principle tells us that a point load on one side will be distributed nearly evenly over the other side, if that's your question. Commented Feb 24, 2022 at 18:16
• Understood, thanks a lot Commented Feb 24, 2022 at 18:18
• No, that’s not necessarily true. The shear stress would be infinity directly under the load on either side. This is the problem with looking too close at point loads. Commented Feb 24, 2022 at 20:06
• Find some finite element analysis software and explore! Commented Feb 26, 2022 at 16:12