## MOMENT-INDUCED SHEAR STRESS

#### by J. Vazquez

###### January 31, 2020

While working on a project a few months ago, I was reviewing a shear stress contour plot for a large beam, and I was surprised to see that the peak values did not occur in the proximity of the Neutral Axis, as one would expect for a beam made up of web(s) and flanges. Instead, the peak shear stress was near the top of the web. This prompted a thorough review of the model and the creation of a simple model to investigate further.

As all structural engineers know, shear stress on a beam is given by

τ = ~VQ/Ib, where V is the shear load,

Q is the 1st moment of area above (or below) the elevation of interest,

I is the 2nd moment of area of the entire cross-section, about its neutral axis, and

B is the thickness of the web(s)

The above formula clearly shows that shear stress is not a function of Moment, only shear load, and therefore the peak shear stress can be expected to be where Q is maximum or where b is minimum (i.e., at the Neutral Axis, or at the thinnest section of the web. But that was not what our analyses were showing.

The simple models we created in order to either validate or refute the findings from the original model immediately showed that for cases where the beam does not have a uniform cross-section, the shear stress is indeed dependent on the bending moment, not just the shear load.

To illustrate the point, I present the results from a simple test-run on two beams. Beam 1 is a typical “I-shape” beam. Beam 2 has top and bottom doubler plates that do not extend the full length of the beam. In order to ensure a valid stress comparison, the two beams have the exact same mesh. The two beams were subjected to pure bending.

The figure below shows the shear stresses for both of these beams, as determined from an ANSYS model using solid elements. As expected, the beam with uniform cross-section throughout its length has zero shear stress, but that’s not the case for the beam that has changes in cross-section due to the addition of the doubler plates.

Shear Stresses on Beams in Pure Bending, with and without doubler plates

In my search for information on this topic, I came across an article by Lechner, Taras and Greiner, from the Institute of Steel Structures and Shell Structures, Graz University of Technology, in Austria (see full reference below).

Lechner, A. Taras and R. Greiner. “FLANGE THICKNESS TRANSITIONS OF BRIDGE GIRDERS – BUCKLING BEHAVIOR IN GLOBAL BENDING” SDDS’ Rio 2010 Stability and Ductility of Steel Structures. Rio de Janeiro, Brazil, September 8-10, 2010.

As indicated in the title, their paper focuses on the effects that transitions in flange thickness have on the beam stresses inducing buckling. Along the way, they present the below figure (Figure 3 in their paper), indicating that a beam subject to pure bending has shear stress that peaks in the vicinity of the thickness transition. (In their paper, they only focus on the shear at the web lower edge).

I looked for more information on this topic, and not finding any, I decided to derive the relationship between shear stress and bending moment for a beam with a discontinuity in its cross-section. This work is in progress, and I intend to write a paper about it. For now, I would invite you to let me know if this is something you’ve come across in the past, and whether you are aware of these effects being accounted for in the various design codes or Class rules. I look forward to your comments on this. If after hearing from you, and if my efforts to find more information on the subject continue to be in vain, I will go ahead and publish the paper I’ve been working to address this topic.

Figure from the above-referenced article by Lechner et al.