# Shear stress in an annular plate

• Hoser415
In summary, Roark's equations for shear stress calculations for an annular plate with a line load at a radius r seem to give conflicting results as to the direction of the shear stress. The results seem to depend on the radius of the line load and the radius of the inner and outer edges of the plate.
Hoser415
Hi Everyone,
I am having a problem finding the shear stresses in an annular plate. The problem is as follows. I have an annular plate with outer radius a, inner radius b. Outer and inner edges are fixed. I also have an annular line load at radius r. I am trying to calculate the shear stresses at a and b. Using Roarks equations to find the shear line forces Qa and Qb I get one of the forces as a positive number and one as a negative number...this makes no sense to me as both forces should be in the same direction as the applied force, at least that's what it seems to me. So am I calculating them correctly and just don't fully understand what is going on or is there some mistake in my math...I've checked the equations about ten times and can't find anything wrong. Any hlep would be great. Thanks.

normal to the plate

According to Roark's: Case 1h- Outer edge fixed, inner edge fixed

Shear at the inner radius support:
$$Q_{b}=w\frac{C_{2}L_{6}-C_{5}L_{3}}{C_{2}C_{6}-C_{3}C_{5}}$$

Shear at the outer radius support:
$$Q_{a}=Q_{b}\frac{b}{a}-\frac{wr_{o}}{a}$$

Where:

$$C_{2}=\frac{1}{4}[1-(\frac{b}{a})^2(1+2ln\frac{a}{b})]$$

$$C_{3}=\frac{b}{4a}[[(\frac{b}{a})^2+1]ln\frac{a}{b}+(\frac{b}{a})^2-1]$$

$$C_{5}=\frac{1}{2}[1-(\frac{b}{a})^2]$$

$$L_{3}=\frac{r_{o}}{4a}[[(\frac{r_{o}}{a})^2+1]ln\frac{a}{r_{o}}+(\frac{r_{o}}{a})^2-1]$$

$$L_{6}=\frac{r_{o}}{4a}[(\frac{r_{o}}{a})^2-1+2ln\frac{a}{r_{o}}]$$

Whew! Hope I got all the latex right...

The only way i can see one being opposite in sign to the other is if you accidentally messed up the radius the load is acting at...

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That is exactly what I used, However here is my problem. w is a positive number, and for constants come out to a number that is greater than one. Therefore Qb is greater than w, which makes sense as it is on a smaller radius than w. However, this means that Qb*(b/a)-w*(ro/a) makes Qa a negative number at some point when ro approaches b. My question is how can that be...It doesn't seem like there should be shear in the opposite direction of the applied force on the outer edge. If I am wrong could someone please explain this to me, I am pretty sure my constants are right becuase I've checked them a bunch of times. Just in case anyone wants to check them I have
a=19.75 in
b=6in
ro=10.976
v=.33

I get c2=0.171948169
c3=0.029897023
c5=0.453853549
c6=0.112031464
L3=0.010799721
L6=0.067209158

Sorry I mean Qa gets negative at some point as r0 approaches a.

Also my w= 6162 lbs/in

giving me a Qb of ~7213 and a Qa of ~ -1233

Your numbers look good, I think the problem is that Qa and Qb are not the reactions, they are the calculated shear force at that location... In a shear force diagram for your system to be in equilibrium it must have a shear force that passes through 0 and into negative values. So the shear force at the inner edge is positive, and the shear force at the outer edge is negative.

However, like you said the reactions will both be in the opposite direction of the load, so just take the absolute value of the calculated values of Qa and Qb to get Ra and Rb.

I noticed that deflections could be pretty large depending on the thickness of your plate. Be sure to check that your maximum deflection is less than half of the thickness of the plate or the equtions being used are not valid.

Last edited:
Thanks so much, that makes sense.

In order to calculate the maximum deflection I'm assuming you use the Ky values in the table for this particular case and use the formula y=Ky*(w*a^3)/D, but for this case I'm in between the k values given, do you know of another place that has a table of more k values or where I might find a formula to calculate a specific value (that is if they are calculated and not experimentally determined).

I would do what Roark's recommends to do for interpolation in Table 23. That is to adjust your problem to match the values listed in the table and then use those values to interpolate in between to get your specific condition. You could simply interpolate between the values given in the table, but they say that that could lead to a lesser degree of accuracy.

## 1. What is shear stress in an annular plate?

Shear stress in an annular plate is a type of stress that occurs when two forces act parallel to each other but in opposite directions on a plate with a circular inner hole and a circular outer boundary. It is a result of the shear force acting on the plate, which causes the plate to deform and experience stress.

## 2. How is shear stress in an annular plate calculated?

The shear stress in an annular plate can be calculated using the formula τ=Q/It, where τ is the shear stress, Q is the shear force acting on the plate, I is the moment of inertia, and t is the thickness of the plate. This formula is derived from the shear stress formula for a rectangular plate and is applicable to annular plates as well.

## 3. What factors affect the shear stress in an annular plate?

The shear stress in an annular plate is affected by several factors, including the magnitude of the shear force, the thickness and material properties of the plate, and the radius of the inner and outer boundaries. The shape and location of any cutouts or holes in the plate also affect the distribution of shear stress.

## 4. How does shear stress in an annular plate impact the structural integrity of a component?

Shear stress in an annular plate can cause the plate to deform and potentially lead to failure if the stress exceeds the material's yield strength. This can compromise the structural integrity of the component and may result in buckling, cracking, or permanent deformation. It is important to consider and properly manage shear stress in the design and analysis of annular plates to ensure the safety and reliability of the component.

## 5. What are some real-world applications of shear stress in annular plates?

Shear stress in annular plates is a crucial consideration in the design and analysis of various engineering structures, such as pressure vessels, tanks, and pipes. It is also relevant in the aerospace industry for components such as engine discs and turbine blades. Additionally, shear stress in annular plates is important in the design of bearings and other mechanical components that experience rotational forces.

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