Mechanics of Materials: Stress Concentration

Mechanics of Materials: Stress Concentration

Stress concentration or often called stress risers are locations on a body or an object due to its geometric changes throughout the object or the body. For example, this bar does not have any geometric changes throughout its length, so when I pull or push or bend or torque it, it will not have, stress is concentrated throughout its body However, this bar has a hole in it. Where stress flows throughout the body, will cause stress to concentrate around the hole and increase the stresses where the body could fail. Or, another bar this bar has two different geometric changes along its length Where stress is going to concentrate here as well as here. These concentrations depending on the geometric changes, geometric changes of the of the fillet or the whole or basically in a torque situation to fillet that is existing in the bar In order to calculate stresses due to stress concentrations, we will look at two stresses, one stress is called Sigma max, which is shown in this equation and the other one is called Sigma nominal where Sigma nominal is calculated from the geometric part of the body where the minimum area is considered when calculating that stress so For example, nominal stress in this part where the geometric changes are not present. The nominal area would be the area of the cross section where it is basically “b” multiplied by “h” In this bar however, the nominal area is the area minimum area going across this hole between this part and this part. I have shown that in this picture here where the nominal area considers these two areas added up in order to calculate the nominal stress or the nominal area in order to consider in the stress concentration, for this case which there is a fille t would be this area where is the smallest area in the body. Let’s look at an example the example that we have for you is shown on the figure over here as well as on the board there are two stress concentrations that will happen in this bar as we put 8 kN of force pulling this bar apart, one of them is due to fillet where the fillet size is 10 millimeter as well as the sides of the smallest area which we were consider for the calculation of the nominal stress is 20 mm by 5 mm as its shown in the figure below this is five millimeter thickness bar, in order to calculate the maximum stress due to the stress concentration factor which is called k. We will consider two calculations, one due to the fillet, where the fillet size is 10 millimeter and as seen from the figure that is present over here shows that the different curve for calculation of the factor k. we will have to find the ratio of “w” where is shown in this picture in this drawing, divided by “h”, which is the reduced size of the bar it comes out to be 40 mm divided by 20 millimeter, that number comes out to be 2. So, in the figure here we will pick w/h as 2 and then we would have to also calculate r, which is the finished size which is 10 mm divided by the h which is 20 mm and we calculate to be 0.5 Now, combining these two and taking the graph here where the horizontal axis represent w/h and the different curves that correspond to r/h values. If you cross these two values, if you go up and hit the right curve which represent r/h, we will find our stress concentration factor to be 1.4 Then we would have to go back to this equation to find our maximum stress by taking 1.4 to be the stress concentration factor 8 times 10 to the power of three, kN which is pulling the bar apart divide by the nominal area. As you see the nominal area i have shown here is 20 millimeter by five millimeters where it’s also shown over here, this is the nominal area for the part or the fillet. Considering the fillet for the stress consentration that number comes out to be 112 MPa Then we would have to calculate another stress concentration where existing this part and that’s due to the hole, the hole as you see is 10 mm in radius and it is cut through in a 40 millimeter width of the bar and by looking at this figure which shows the curve for hole stress concentration factor we would have to calculate r/w which “r” is 10 and “w” is 40 and that factor from this graph approximately comes out to be 2.375, then we would proceed to calculate Sigma max. Where? From this equation is inputting the stress concentration factor which has been found to be 2.375 multiply by the nominal stress and if you look at this nominal stress, I have calculated teh nominal area in the hole to be this area, where the hole is cut in the part. So, this is the nominal area we would have to consider for calculating this area and so by multiplying these numbers maximum stress in the part becomes 190 MPa However, as we look at the whole picture together, our answer to consider for design or for any other purpose of presenting this bar for part of a design or machinery we cannot exceed 112 MPa in the part due to the fact that the fillet maximum stress is 112MPa. So, basically the maximum absolute maximum with in these two numbers, that is going to be 112 for us to consider now if one of these numbers exceed the maximum strength of the material the material of course with fail, but also looking at these stress concentration factors, we look at that the fillet is 1.4 basically rises the stress by 1.4 but the whole rises by 2.375 now it’s interesting to look at the graph for the hole where you see the stress concentration can cause the stress to rise all the way up to three times the nominal stress and if you look at it carefully for that to happen the hole size would be smallest possible rather than a big hole is small drilled through the part will make that stress concentration to be three times as much. So, if you ignore just concentration you could fail the part without reaching your maximum stress due to stress concentration that you do not include in your calculation. [Music]

One thought on “Mechanics of Materials: Stress Concentration”

  1. If the hole size in the body was changed , would that change or impact the stress concentration factor ?

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