Analyzing solid mechanics problems in three dimensions can be really hard work, and it can get very complicated very fast. Fortunately in a lot of cases there are some simplifications we can use to reduce a three-dimensional problem to a two dimensional, one making it much easier to solve.
The two main simplifications which are frequently used in solid mechanics are the plane stress and plane strain conditions. In this video we’re going to take a look at plane stress. So what does plane stress mean? A component is said to be in a condition of plane stress when all the stress is acting on it are in the same plane. A surprising number of common engineering problems can be approximated to plane stress conditions. It is most relevant for the analysis of thin components.
Let’s take a look at an example. To determine whether we could model this perforated plate using plane stress assumptions, we need to see whether it is reasonable to assume that all stresses are acting in the same plane. All the loads are applied in the same plane, the X-Y plane, so that’s a good start. But having all loads acting in the same plane is not enough for the plane stress condition to be met, as we could still have stresses in the Z direction. This is where the thickness of the plate comes into it. We know that normal and shear stresses at a free surface are always zero. This means that the stresses on the top and bottom faces of the plate must be zero.
And because this plate is very thin there can’t be much variation in stress through the plate’s thickness, meaning that the stresses in the Z direction will be close to zero all the way through the plate. Because the only non-zero stresses are acting in the X-Y plane, a condition of plane stress applies. Of course in reality the stresses in the Z direction are unlikely to be exactly zero. Deciding whether a plane stress condition is applicable will always require a degree of engineering judgment. Why is the plane stress assumption useful? We can answer by taking a look at a stress element in our perforated plate.
The stresses at a single point are defined by six different stress components, three normal stresses and three shear stresses. For plane stress conditions, sigma-Z, tau-XZ and tau-YZ are equal to zero This means that the six components defining the stress at a point are reduced to just three components – sigma-X, sigma-Y and tau-XY. This is a two-dimensional problem which will be much easier to solve. The stress tensor for a three-dimensional case is a 3×3 matrix. But if we consider plane stress conditions, it is reduced to a much more manageable 2×2 matrix. Let’s look at two more examples of situations where it might be appropriate to assume plane stress conditions.
Pressure vessels can sometimes be modeled using plane stress assumptions. The pressure load generates hoop stresses which are oriented around the circumference of the vessel, and axial stresses. If the vessel wall is thin compared to its diameter, radial stresses will be close to zero, and plane stress conditions will be applicable. The teeth of a spur gear can also sometimes be modeled using plane stress conditions, if the width of the gear is narrow enough. So, to summarize, plane stress is a simplification which can be used to turn a three-dimensional solid mechanics problem into a simpler two-dimensional one, by assuming that the stresses in one direction are equal to zero. It is normally applicable for thin structures which are loaded in a single plane. Thanks for watching, and stay tuned for more engineering videos!