Mechanics of Materials: Angle – Normal Stress

Moments of Area

In order to calculate stress (and therefore, strain) caused by bending, we need to understand where the neutral centrality of the axle is, and how to calculate the second moment of area for a given cross section.

Allow's kickoff by imagining an arbitrary cross section – something not round, not rectangular, etc.

In the above epitome, the capricious shape has an area denoted byA. We can expect at a small, differential areadA that exists some distance x and y from the origin. We tin look at the first moment of area in each direction from the post-obit formulas:

The first moment of area is the integral of a length over an area – that means information technology will accept the units of length cubed [50three]. It is important because it helps u.s. locate the centroid of an object. The centroid is defined as the "averagex(ory) position of the area". Mathematically, this statement looks like this:

The far correct side of the to a higher place equations will be very useful in this course – it allows us to break upwards a complex shape into simple shapes with known areas and known centroid locations. In most applied science structures in that location is at least ane centrality of symmetry – and this allows us to greatly simplify finding the centroid. The centroid has to be located on the centrality of symmetry. For instance:

For the cross department on the left, nosotros know the centroid has to prevarication on the centrality of symmetry, so we only need to find the centroid along they-axis. The cross section on the right is even easier – since the centroid has to line on the axes of symmetry, it has to be at the heart of the object.

Now that we know how to locate the centroid, nosotros can plow our attending to the second moment of area. Every bit you might remember from the previous section on torsion, this is divers equally:

And, finally sometimes we volition demand to determine the second moment of area about an arbitraryten ory axis – 1 that does not correspond to the centroid. In this instance, nosotros tin use the parallel axis theorem to summate it. In this case, we employ the second moment of area with respect to the centroid, plus a term that includes the distances betwixt the 2 axes.

This equation is referred to equally the Parallel Axis Theorem. It will exist very useful throughout this course. As described in the introductory video to this department, it tin can be straightforward to summate the second moment of area for a simple shape. For more complex shapes, nosotros'll demand to calculateI by computing the individual I'due south for each simple shape and combining them together using the parallel axis theorem.

Shear and Moment Diagrams

Transverse loading refers to forces that are perpendicular to a structure's long axis. These transverse loads will cause a angle moment M that induces a normal stress, and a shear strengthFive that induces a shear stress. These forces can and will vary forth the length of the axle, and we will employ shear & moment diagrams (V-G Diagram) to extract the most relevant values. Constructing these diagrams should exist familiar to y'all from statics, but we will review them here. There are two important considerations when examining a transversely loaded axle:

  1. How is the beam loaded?
  • betoken load, distributed load (uniform or varying), a combination of loads…
  1. How is the beam supported?
  • simply supported, cantilevered, overhanging, statically indeterminate…

Knowing about the loads and supports volition enable you to sketch a qualitative 5-One thousand diagram, and then a statics analysis of the free trunk will help you determine the quantitative description of the curves. Let'due south commencement by recalling our sign conventions.

These sign conventions should exist familiar. If the shear causes a counterclockwise rotation, it is positive. If the moment bends the beam in a fashion that makes the axle bend into a "grin" or a U-shape, it is positive. The best way to recollect these diagrams is to work through an example. Begin with this cantilevered beam – from here you can progress through more complicated loadings.

Normal Stress in Bending

In many ways, bending and torsion are pretty similar. Bending results from a couple, or a bending momentM, that is applied. Just like torsion, in pure bending there is an axis within the material where the stress and strain are zippo. This is referred to as the neutral axis. And, just like torsion, the stress is no longer uniform over the cross department of the construction – it varies. Let'south start by looking at how a moment about thez-axis bends a structure. In this example, we won't limit ourselves to circular cantankerous sections – in the effigy below, we'll consider a prismatic cross department.

Earlier we delve into the mathematics backside bending, let's try to get a feel for information technology conceptually. Peradventure the be way to see what's happening is to overlay the bent beam on peak of the original, straight axle.

What you tin observe now is that the bottom surface of the beam got longer in length, while the to surface of the beam got shorter in length. As well, along the center of the beam, the length didn't change at all – corresponding to the neutral axis. To restate this is the language of this class, we tin can say that the bottom surface is under tension, while the meridian surface is under compression. Something that is a little more subtle, but can still be observed from the above overlaid prototype, is that the displacement of the beam varies linearly from the pinnacle to the bottom – passing through zilch at the neutral axis. Remember, this is exactly what nosotros saw with torsion as well – the stress varied linearly from the center to the center.  We tin look at this stress distribution through the beam'southward cross section a bit more explicitly:

Now nosotros can await for a mathematical relation between the practical moment and the stress inside the beam. We already mentioned that beam deforms linearly from one edge to the other – this means the strain in thex-direction increases linearly with the distance along they-axis (or, along the thickness of the beam). Then, the strain will be at a maximum in tension at y = -c (since y=0 is at the neutral axis, in this case, the center of the beam), and will be at a maximum in pinch at y=c. We can write that out mathematically like this:

Now, this tells us something about the strain, what tin can we say almost the maximum values of the stress? Well, let's start by multiplying both sides of the equation pastE, Immature'due south rubberband modulus. At present our equation looks like:

Using Hooke's law, nosotros can relate those quantities with braces under them to the stress in thex-direction and the maximum stress. Which gives united states of america this equation for the stress in the10-management:

Our final step in this process is to empathize how the bending moment relates to the stress. To do that, we recall that a moment is a force times a distance. If we tin imagine simply looking at a very small chemical element inside the beam, a differential element, then we can write that out mathematically as:

Since we have differentials in our equation, we can determine the momentChiliad acting over the cross sectional expanse of the beam by integrating both sides of the equation. And, if we recall our definition of stress as being force per surface area, nosotros can write:

The final term in the last equation – the integral over y squared – represents the second moment of area about thez-axis (because of how we have defined our coordinates). In Cartesian coordinates, this 2d moment of expanse is denoted byI(in cylindrical coordinates, remember, it was denoted byJ). Now nosotros can finally write out our equation for the maximum stress, and therefore the stress at whatsoever point along they-axis, as:

It's important to note that the subscripts in this equation and direction along the cross section (hither, information technology is measured alongy) all will alter depending on the nature of the problem, i.e. the direction of the moment – which axis is the axle angle virtually? We based our note on the bent beam bear witness in the first image of this lesson.

Remember at the beginning of the section when I mentioned that angle and torsion were actually quite similar? Nosotros actually see this very explicitly in the terminal equation. In both cases, the stress (normal for angle, and shear for torsion) is equal to a couple/moment (M for bending, andT for torsion) times the location along the cross section, because the stress isn't uniform along the cross department (with Cartesian coordinates for bending, and cylindrical coordinates for torsion), all divided by the 2d moment of area of the cross department.

Summary

We learned aboutmoments of surface area andshear-moment diagrams in this lesson. From the first moment of expanse of a cross section nosotros tin can calculate thecentroid. We learned how to calculate the 2d moment of area in Cartesian and polar coordinates, and nosotros learned how the parallel axis theorem allows usa to the 2nd moment of area relative to an object's centroid – this is useful for splitting a complex cantankerous section into multiple simple shapes and combining them together. We reexamined the concept of shear and moment diagrams from statics. These diagrams volition exist essential for determining the maximum shear force and bending moment along a complexly loaded beam, which in turn volition be needed to calculate stresses and predict failure. Finally, we learned about normal stress from angle a beam. Both the stress and strain vary forth the cross section of the axle, with i surface in tension and the other in pinch. A plane running through the centroid forms the neutral axis – there is no stress or strain along the neutral axis. The stress is a function of the applied moment and second moment of area relative to the axis the moment is about.

This material is based upon work supported past the National Scientific discipline Foundation under Grant No. 1454153. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the writer(s) and do non necessarily reflect the views of the National Science Foundation.