# Determine the range of τxz values for which the maximum tangential stress is less than or equal to 60 MPa

Consider the state of tension indicated in the Figure. Determine the range of τxz values for which the maximum tangential stress is less than or equal to 60 MPa.

I did σ(med)=(60+0)/2 = 30

To find the radius I did σ(max) = σ(I) = σ(med)+r <=> 100=30+r <=> r=70

But in the solution is

My question is: why is my radius wrong?

• Do not self-vandalize your questions, here or on other Stack Exchange sites like electronics.SE. This was a well-received question that was upvoted, with a good answer. The people answering have put in time and effort to write an answer -- These edits that remove the question contents are disrespectful to their work. – nanofarad Jan 14 at 21:53
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• @Buraian The answer saying the same thing is even stranger -- OP should seriously stop -- the question has been flagged for moderator intervention here and they have been notified on multiple sites about these edits (by mods and non-mod users alike) – nanofarad Jan 14 at 22:52

I'm not 100% sure, however one way to feel more confident on my reasoning, is to check whether my final value of $$\tau_{xz}$$ is the same with the solution. .

My understanding is the following :

$$\sigma_{xx}$$ and $$\sigma_yy$$ are principal stresses in the x-y plane. That means that $$\sigma_1 \leftarrow \sigma_{yy}$$, $$\sigma_2 \leftarrow \sigma_{xx}$$. The greatest tangential stress is going to be obtained by the $$\sigma_{I}$$ and $$\sigma_{III}$$.

In order to have $$\tau_max=60[MPa]$$, you'd need $$\sigma_{I}-\sigma_{III}=120[MPa]$$ Therefore, since, $$\sigma_1=100[MPa]$$, $$\sigma_{III} \rightarrow -20[MPa]$$.

Then on the x-z plane, you'd need to determine the $$\tau_{xz}$$, so that the lowest principal stress is -20[MPa]. Since, on that plane the $$\sigma_{med} = 30[MPa]$$, then the radius will be $$\tau_{xz,max} = 50[MPa]$$, which means that you need to have

$$50^2 \ge \tau_{xz}^2+ \left(\frac{\sigma_x - \sigma_z}{2}\right)^2$$ $$\tau_{xz}^2 \le 50^2 -\left(\frac{\sigma_x - \sigma_z}{2}\right)^2$$ $$\tau_{xz} \le \sqrt{50^2 -\left(\frac{60}{2}\right)^2}$$ $$\tau_{xz} \le \sqrt{50^2 - 30^2}$$

$$\tau_{xz} \le 40 [MPa]$$

Is this the right solution you are given?