The discussion centers on calculating maximum shear stress using Mohr's Circle, specifically addressing the confusion surrounding the sign convention and the relationship between shear and normal stresses. Participants emphasize the importance of correctly applying the equations for normal and shear stresses on a plane of arbitrary orientation, as detailed in the Wikipedia article on Mohr's Circle. The key equations include the transformation of stress components and the determination of maximum shear stress, which is derived from the principal stresses. The consensus is that understanding the sign convention is crucial for accurate calculations.
PREREQUISITESMechanical engineers, civil engineers, and students studying mechanics of materials who need to understand shear stress analysis and the application of Mohr's Circle in real-world scenarios.
Hope you can help me with the other thread . My lecturer asked me to do self -study . He is on leave . I have been thinking of this for the whole week . I have read a lots of online notes and also books, but stlil don't understandChestermiller said:Well, there's something wrong with the solution in the pdf file you presented. The diagram clearly shows that tauxy is negative, and yet they treat it as positive. So there's now way I can answer this question.
I have some other stuff to do today, so I'll get back to that other thread you cited tomorrow.
Are you familiar with matrix operations, such as multiplying a matrix by a column vector?fonseh said:Hope you can help me with the other thread . My lecturer asked me to do self -study . He is on leave . I have been thinking of this for the whole week . I have read a lots of online notes and also books, but stlil don't understand
Chestermiller said:What I am going to do is solve this problem using the method that I am familiar with and that I know works. I will solve it for our two cases, one with ##\tau_{xy}=+25## and the other with ##\tau_{xy}=-25##.
Case of ##\tau_{xy}=+25##
First calculate ##\tan{2\theta_p}##:
$$\tan{2\theta_p}=\frac{2\tau_{xy}}{(\sigma_x-\sigma_y)}=\frac{(2)(25)}{(-80-50}=-0.38462$$So, $$2\theta_p=-21.04\ degrees$$
Next calculate the average of ##\sigma_x## and ##\sigma_y##:
$$\frac{\sigma_x+\sigma_y}{2}=\frac{-80+50}{2}=-15$$
Next calculate radius of Mohr's circle:$$R=\tau_{max}=\sqrt{\left(\frac{(\sigma_x-\sigma_y)}{2}\right)^2+\tau_{xy}^2}=\sqrt{(-65)^2+(25)^2}=69.6$$
Next, use the following equations to calculate the normal and shear stress components on a plane normal to a unit vector pointing in the direction of a the angle ##\theta## measured counterclockwise relative to the x axis:
$$\sigma_n=\left(\frac{\sigma_x+\sigma_y}{2}\right)+\frac{(\sigma_x-\sigma_y)}{2}\cos{2\theta}+\tau_{xy}\sin(2\theta)$$
$$\sigma_t=-\frac{(\sigma_x-\sigma_y)}{2}\sin{2\theta}+\tau_{xy}\cos(2\theta)$$
The minimum principal stress occurs at ##2\theta=2\theta_p=## = -21.04 degrees. The value is -84.6.
The maximum principal stress occurs at ##2\theta=2\theta_p+180## =158.96 degrees. The value is +54.63.
The following is a table of values for ##2\theta## (degrees), ##\sigma_n##, and ##\sigma_t##:
View attachment 110530
The sequence of values is in the clockwise direction.
I changed my mind about doing tau=-25. This has gotten too time-consuming. I leave that case for you.
No , i learned it , but i didnt master it well. so , i am having difficuties of understanding itChestermiller said:Are you familiar with matrix operations, such as multiplying a matrix by a column vector?
Is there any other way than the matrix method to check my sign convention ??Chestermiller said:Are you familiar with matrix operations, such as multiplying a matrix by a column vector?
I'm trying to think of another way, but have not been successful yet? Are you saying that you could not multiply a 2x2 matrix by a 2X1 column vector? By any chance, are you familiar with dyadic tensor notation?fonseh said:Is there any other way than the matrix method to check my sign convention ??
i haven't learn this beforeChestermiller said:By any chance, are you familiar with dyadic tensor notation?
perhaps you can try yo help me on this thread first , https://www.physicsforums.com/threads/angle-of-principal-stress-vs-maximum-shear-stress.897353/Chestermiller said:I'm trying to think of another way, but have not been successful yet? Are you saying that you could not multiply a 2x2 matrix by a 2X1 column vector? By any chance, are you familiar with dyadic tensor notation?
Not having the most appropriate mathematical tools to analyze this stuff makes it much more difficult to learn. It's like trying to do mechanics without knowing calculus.