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in stress concentrations, say in a hole made in a beam loaded axially, we need the value of K ( the stress concentration value) which equals max stress\ avg stress for the given body.

the avg stress is given by the apllied force over the net area, but im not sure what's the max stress that satisfies this relation.

sorry if the question is more of a defenition.

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  • $\begingroup$ between the words "maximum" and "stress" which don't you understand? $\endgroup$
    – agentp
    Dec 1, 2017 at 1:51
  • $\begingroup$ well, i apologize, but what is it ? $\endgroup$
    – sarah
    Dec 1, 2017 at 2:17
  • $\begingroup$ its really not clear what you are asking. Stress is a quantity that varies from location to location, "maximum" means the value where it it largest. $\endgroup$
    – agentp
    Dec 1, 2017 at 2:28

3 Answers 3

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In your example (a hole in a bar loaded axially), the hole stretches. This causes the stress around the hole to vary. The locations where the hole changes curvature the most is where the stress is a maximum. From this, the stress concentration can be calculated: k=max stress/average stress.

Here is an image from a finite element analysis (FEA) showing the elongation and stress around the hole. Red is high stress, dark blue is low stress. Note that the minimum stress is lower than the average stress.

enter image description here

Edit: The hole in the original bar is circular, and the above image greatly exaggerates the displacement. Also, you need to check the reference for the stress concentration factor K to see if the average stress = force/w/t (the total area) or average stress = force/(w-d)/t (the net area), where w is the width of the bar, d is the diameter of the hole, and t is the thickness of the bar.

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  • $\begingroup$ Seems elementary but to check : min stress is the value of force over the total area (without cutting out the hole area ) right ? Your answer cleared a lot for me! Thanks $\endgroup$
    – sarah
    Dec 3, 2017 at 0:51
  • $\begingroup$ No. The minimum stress is not force/w/t nor force/(w-d)/t. Just like the stress concentration causes the maximum stress to rise above the average stress, whatever the opposite is of a stress concentration causes the minimum stress to drop below the average stress. (It is related to the " line of stress" from photoelastic images.) I doubt that there are any formulas for the minimum stress since those are usually not a problem in the design. $\endgroup$
    – JohnHoltz
    Dec 3, 2017 at 5:03
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In manual calculations you will pick or calculate a value of K from a table or graph based on the geometry or calculate from a formula. Depending on the material there may also be additional factors to take into account relating to surface roughness etc. There are also various online calculators which can work it out for you.

Essentially K is the ratio between average stress and max stress so you need to know at least two of the three values to calculate the third.

So in practice the average stress is easily calculable, multiply this by the appropriate K value and you get the max stress (assuming that the assumptions used to calculate K are valid).

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When talking about stress concentration, the goal is to calculate max stress, or you can measure it optically by measuring displacement like Aramis and calculating stresses.
In the literature about Mechanics of solids i.e. stress concentration factors eg.Roark you can see all relevant formulas and definition. Best is to use your course literature because different authors and different lecturers often use different notation.

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  • $\begingroup$ It'd be nice to show where these links go, or better yet include the key points in your answer, to remove the need for OP to trawl through pages which they may not fully understand, and to guard against Link Rot (see en.wikipedia.org/wiki/Link_rot) $\endgroup$
    – Jonathan R Swift
    Dec 1, 2017 at 10:33

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