Lesson 6 Shear and Bending Stresses

Added: 2026-05-31

[00:00:00]now that we have formulas defining our internal loads let's talk about the stresses occurring inside the member and we already discussed bending stresses a bit in the foam demonstration so let's start with Shear stresses as you can see here the actual stress profile of a rectangular member is shaped like a parabola with a maximum amount of stress occurring at the center of the member or the neutral axis and a more complicated shape like say a wide flange beam will have a somewhat different stress profile but it will be similar to this Parabola shape so let's just stick with rectangular shapes for now the important thing to note here is that the maximum stress occurs at the neutral axis and the formula to Define that maximum shear stress is given by this formula here which looks a bit complicated but for a rectangular section it can be simplified to three times the maximum Shear force over two times the cross-sectional area for a wide flange beam however this formula would get more complicated but don't worry we're not going to be using this complex formula for this course instead recall our stress strain curve which looked something like this linear until we reached yield and then we have our plastic region of the Curve until we reach rupture last week we talked about how we typically design structural members to approach but not exceed this yield point where we go from elastic deformation to plastic but in the case of Steel design where we Define that yield stress is actually slightly offset from where yielding truly starts to occur so our design yield stress will be here somewhere slightly beyond the actual yield stress and that offset is defined by a plastic deformation of 0.2 percent and so we are actually going to allow just the slightest amount 0.2 percent of plastic deformation in steel design when designing for ultimate or factored loading that slight bit of plastic deformation allows us to simplify our calculation of our demand stresses if we return to our stress diagram here as the member starts to exhibit plastic deformation the stresses are going to redistribute in other words the areas of peak stress will spread toward the areas of lower stress and so we end up with a more uniform average stress across the section and so it's this average stress we will design for and similar to how we calculated stress for axial forces the average shear stress is going to be equal to the shear Force divided by the cross-sectional area similarly with bending we have a formula for calculating the maximum tension and compression stresses due to bending that occur in the member this formula is actually quite simple it's the maximum moment divided by the section modulus of the member section modulus is a geometric property and is defined by the moment of inertia divided by this value C and C is the distance from the centroid of the member to the outermost extreme fiber or in the case of a symmetric shape like this rectangle it will be equal to half the height and so if we solve this formula by plugging in our equation for moment of inertia which we know is equal to the width times the height cubed over 12.

[00:04:44]and our c is equal to H over 2 we get that the elastic section modulus is equal to B H squared over six and so to find our elastic stresses due to bending we would take the maximum moment and divide by our elastic section modulus s however as we had with Shear stresses due to that slight bit of plastic deformation that we will allow to occur under our maximum loading we will have some redistribution of stresses and so the plastic stress profile due to bending will look like this where we have sort of these parabolic curves and note that the peak values are lesser than they were for the elastic stresses and so we need a different geometric property to Define our plastic stresses due to bending and that geometric property is the plastic section modulus and note that the plastic section modulus will always be greater than the elastic suction modulus and therefore our plastic stresses will always be lesser than our elastic stresses and the way that we quantify this plastic section modulus is by taking this shape and dividing it into two equal areas where we draw this line is called the plastic neutral axis in the case of a symmetric shape like this rectangle the elastic neutral axis and plastic neutral axis will be the same and for all the shapes that we analyze in this course they will be the same now we simply calculate the areas on each side of our axis and multiply them by the distance from their centroid to our neutral axis we add those up and we get our plastic section modulus so in the case of this rectangle we have this area on this side of the neutral axis which has an area equal to B times H over 2 and its distance from its centroid to the neutral axis is H over 4.

[00:07:37]and since the area on the other side has the same area and the same centroid we can simply take this and multiply it by two and so we get a formula for our plastic section modulus which is BH squared over four and I'm not really going to spend too much time with defining the derivation for some of these geometric properties for some of you it may be easier just to Simply memorize these equations but I also recognize that some of you may actually like to see the derivation and it may help you with memorizing these formulas and understanding where they come from so post formulas are given here also for those curious I have derived a few formulas for how to calculate the elastic and plastic section moduli for an eye shaped beam note that a true wide flange beam has curved edges around where the web meets the flanges and so the actual formulas for calculating these properties for a true wide flange are even more complex but again I'm not going to require you to memorize these formulas though you may need them for the are for this course if I ask you what the stresses are in a wide flange beam with a bending moment I'm going to provide you with the plastic section modulus which is also given in tables from aisc and you can actually download this spreadsheet yourself so for example for a w16 by 26 beam it has an area of 7.68 inches squared and if we continue moving to the right here eventually we will find Z values the plastic section modulus and a w16 by 26 beam has a plastic section modulus of 724 inches cubed so again I'm not going to require you to know these two formulas so you though you may need them for the are I may ask you to use these two formulas however so it's important to know these ones also if you are really curious about the difference between an elastic neutral axis and a plastic neutral axis and why said that they are the same for asymmetric shape but not the same for an asymmetric shape you can see I've provided an example of calculating each of these two axes for this asymmetric T shape as I said before I'm not going to require you to know this for this course but it could be helpful for the are or at some point in your career Maybe or if you're just curious because I know of course that all of you are just as excited about structures and geometric properties as I am the truly important concept to know here from this discussion is that elastic Shear stresses in a beam follow a parabolic curve with a maximum stress at the center of the beam also elastic bending stresses are linear from one edge of the beam from the top of the beam to the bottom with the maximum stresses at the outermost edges of the member so if we scroll if I scroll down we have a summary here so maximum bending stresses will occur at the top and the bottom maximum shear stress occurs in the middle for ultimate load design however we allow for a slight amount of plastic deformation and thus redistribution of Shear stresses and bending stresses into a more uniform average shear stress and a more uniform plastic bending stress profile but that's only for ultimate load design a member may only see its ultimate load once very briefly in its entire lifespan or possibly even never at all for the vast majority of the time structural members will remain completely in the elastic region and therefore the actual stress profiles in the member at any given time will be elastic so if we look at the cross section of a wide flange beam we now know that the middle of the beam sees the highest shear stress and so it's the web that controls shear we also know that the outermost regions of the profile will see the maximum bending stress and so it's the flanges that primarily control moment capacity and if we look at our shear and moment diagram for a simply supported Beam with a uniformly distributed load recall that we calculated maximum Shear at the support and zero internal Shear at the mid span conversely we calculated a maximum internal bending Force at mid-span and zero bending Force at the supports therefore if we have zero bending Force at the supports we don't really need the flanges anymore as the flanges contribute primarily to moment capacity and therefore we can safely reduce the suction here without drastically impacting the capacity of the beam and so that answers the question that we had from our previous discussion last week so understanding where maximum stresses occur in the profile and where maximum internal forces occur along the length of a beam is important when you want to decide where to penetrate the beam or to reduce its section now in the next video we will continue our discussion of beam capacities for shear and bending moment as well as introduce the formulas we use for calculating deflection of a beam