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!

Feynstein 100Post authorAwesome video. Wish it were longer like the rest tho ?

Alek DincoffPost authorThe animation is absolutely great.

Ramvel SelvarajanPost authorThe video is marvelous

And pls do for stress strain curve

Sunchit AnandPost authorGreat, fresh content. Also amazing animations! Where did you learn animation? Can you direct me to the resource?

Prashant SinhaPost authorGreat work

Shivam RaiPost authorAmazing

dinesh cutePost authorsuper. please make more videos

Bart KoningPost authorHi. Thanks doe the geest video. Do you have any sources of this subject? I need it for a scool project, thanks!

MOHA KHALAFPost authorThanks (:

Bradley PringPost authorGreat video, loved it!

I think you meant 3×3 matrix at 2:47.

Ajay rajendranPost authorShort and understandable ❤️❤️ Thank you

Louis LiuPost authorWhy can’t uni professors teach like this?