Mohrs circle and stress transformation equations

Mohrd circle

1. 2D - Principal stresses One exam based question And three methods to solve it Use the slide show! Etchi Regoli Gioia Material Science and Engineering student

2. A material is subjected to the plane stress state represented in the figure. Determine : The principal stresses and The maximal shear stresses and The principal directions and The angle of maximum shear stress   Example Before determining the variables with appropriate equations , the figure has to be interpreted ; Its three values refer to the normal stresses and and the shear stress .         2D - Principal Stresses - Etchi Regoli Gioia 2

3. Normal s tresses can be horizontal or vertical forces . Respect to the chosen coordinates system : horizontal forces have a - axis direction horizontal forces are called   Respect to the chosen coordinates system : vertical forces have a - axis direction vertical forces are called           For planes subjected to a stress state: chosen coordinates system     How to determine what and are and their signs (1)   2D - Principal Stresses - Etchi Regoli Gioia 3

4. Horizontal and vertical forces seem to converge on the same point this leads to a compression the forces are negative Horizontal and vertical forces seem to diverge from the same point this leads to a tension the forces are positive How to determine what and are and their signs (2)   2D - Principal Stresses - Etchi Regoli Gioia 4

5. How to determine what is and its sign   When the forces seem to converge to the same point: by convention the force is positive When the forces seem to diverge from the same point: by convention the force is negative * HINT To not get confused always look at the paired vectors at top-right corner of your plane ! Shear stresses are represented by ‘’ paired ’’ vectors at a corner of a plane is the symbol for the shear stress           2D - Principal Stresses - Etchi Regoli Gioia 5

6. ( horizontal , compressive force) ( vertical , tensile force) (‘’corner’’, shear force)   … going back to the example Once we have learned how to interprete the figure given , we can name its values in respect to the chosen coordinates system and attribute the correct sign . chosen coordinates system           Therefore the stresses from the figure given are:   =   =   2D - Principal Stresses - Etchi Regoli Gioia 6

7. 3 Different methods Now that we have understood the figure given , we can answer the four initial questions of the problem . To do that , 3 different methods can be used : MOHR’S CIRCLE MATRICES AND DETERMINANTS FORMULAE Despite their diversity , they are all mathematical methods and they all have to lead to the same answer . 2D - Principal Stresses - Etchi Regoli Gioia 7

8. Mohr’s circle Mohr’s circle gives a graphical solution to the principal stresses . This method is for t hose who prefer a visual approach . The set up of Mohr’s circle will be explained over the next three slides. The set up is really a demonstration of the derived equations; once the demonstration is clear, the setting up is not necessary anymore. In the other lecture notes we are still going to deal Mohr’s circle, but the use of the equations will be enough for the building of the circle. 2D - Principal Stresses - Etchi Regoli Gioia 8

9. Mohr’s circle : setting up (1) τ         Mohr’s circle is set up: The - axis is called (sigma ); all the principle stresses lie on the - axis ; The - axis is called τ (tau ) and it points downwards ; all the shear stresses lie on the points of the circle ; T he center always lies on the - axis ; We find on the - axis and consider it to be the distance between (zero, the origin of the axis ) and We could find find on the - axis but it is not useful for the following steps .   2D - Principal Stresses - Etchi Regoli Gioia 9

10. Mohr’s circle : setting up (2)             Proceeding with the setting up: We find on the - axis and consider it to be the distance between (zero, the origin of the axis ) and The distance between the center and (zero, the origin of the axis ) is called . The center is at . The point is found . T he distance between the center and the point is the radius . Knowing the radius the circle can be plotted       2D - Principal Stresses - Etchi Regoli Gioia 10 τ

11. Mohr’s circle : setting up (2) Let’s now focus on the four initial questions of the problem . In Mohr’s circle : The values of the principal stresses and are determined by the interceptions of the circle with the - axis ; the values of shear stresses and are determined by the intersection of the circle with the diameter parallel to the - axis ; The value of the principal direction is given by the angle between the - axis and the radius of the circle . that we call for convenience .                           2D - Principal Stresses - Etchi Regoli Gioia 11 τ

12. Determining: The principal stresses and The maximal shear stresses and The principal directions and The angle of maximum shear stress With MOHR’S CIRCLE   2D - Principal Stresses - Etchi Regoli Gioia 12

13. To find the values of principal stresses and , the distances between (zero, the origin of the axis ) and the points and have to be determined . As the radius is the distance between the center and any point of the circumference , i t can be seen that :                       Determining the principle stresses and with Mohr’s circle (1)   2D - Principal Stresses - Etchi Regoli Gioia 13 τ    

14.                 Determining the principal stresses and with Mohr’s circle (2)   Now to find the radius we apply Pitagoras theorem * to the highlited triangle . The sides of this right triangle are: The segment The radius ; The segment   *HINT: Pitagora’s theorem           2D - Principal Stresses - Etchi Regoli Gioia 14 τ    

15. a nd remebering that : We find that :                 The segment is simply the difference between the distance and the the distance .         Determining the principal stresses and with Mohr’s circle (2)     By applying Pitagoras theorem , w e find that: 2D - Principal Stresses - Etchi Regoli Gioia 15 τ    

16.   and again remebering that : we find that :               Now that we have found the value of the radius , it’s possible to compute the values of principal stresses and As we have previously seen :       Determining the principal stresses and with Mohr’s circle (3)               2D - Principal Stresses - Etchi Regoli Gioia 16 τ

17. τ τ τ τ           To calculate the maximal shear stresses and , the distances pointed in the figure have to be found . As previously stated the values of any shear force is given by a point of the circumference of the circle . And the values of maximal shear stresses and are equal to the radius being distances from the center of the circle and the circumference . Therefore : But because we need to stick with the chosen coordinates system , the result is :           2. Determining The maximal shear stresses and with Mohr’s circle         ;   2D - Principal Stresses - Etchi Regoli Gioia 17 τ    

18. 3. Determining t he principal directions and with Mohr’s circle (1)               Now to find the principal direction we apply the trigonometric ratio to the triangle previously examined . The sides of this right triangle are: The segment The radius ; The segment Therefore to find the following equation is performed *         *HINT To get the value of , use the calculator and type :   And we find that     2D - Principal Stresses - Etchi Regoli Gioia 18 τ  

19. 3. Determining the principle directions and with Mohr’s circle (2)       To find the value of the principal direction the equation below is used .       This is a close -up of what we are dealing with now . The figure on the left shows the right triangle we took into consideration ; the two principal directions and act as the angles pointed in the figure, in respect to the chosen coordinates system . *   Direction 1 Direction 2 We find that :   τ     2D - Principal Stresses - Etchi Regoli Gioia 19 *NOTE THAT In Mohr’s circle the angles should be represented as twice their value .

20.     Direction 1   τ   Direction max The angle of maximum shear stress is easily calculated with the equation below .       We find that   4. Determining t he angle of maximum shear stress with Mohr’s circle   2D - Principal Stresses - Etchi Regoli Gioia 20 *NOTE THAT In Mohr’s circle the angles should be represented as twice their value .

21. Some clarifications before the next method As principal stresses can also be solved with Matrices and Determinants , we will name the data interpreted from the figure given according to the matrices and determinants theory .   We build a 2x2 matrix which determinant is   a nd we put in our variables As this is a 2x2 matrix , a table with two rows and two colums is built : is in the first row and first column ; is in the first row and second column ; is in the second row and first column ; is in the second row and second column .     2D - Principal Stresses - Etchi Regoli Gioia 21

22. Determining: The principal stresses and The maximal shear stresses and The principal directions and The angle of maximum shear stress With MATRICES AND DETERMINANTS   2D - Principal Stresses - Etchi Regoli Gioia 22

23.   1. Determining the principal stresses and with matrices and determinants (1)         *NOTE THAT and refer to the same variable . That is This is a property of symmetric matrices .     The identity matrix is multiplied by λ   The difference between the two matrices is performed The equation is set up: is the identity matrix λ is the constant to find   T he multiplication between the two brackets is performed The rule is applied                             2D - Principal Stresses - Etchi Regoli Gioia 23

24. 1. Determining the principal stresses and with matrices and determinants (2)     Rearranging we find that this is a quadratic equation in     By further rearranging we get the equation in the form         We substitute our values in the equation according to the equalities below       And performing some calculations we get :   Obtaining: 2D - Principal Stresses - Etchi Regoli Gioia 24

25. 1. Determining the principle stresses and with matrices and determinants (3)     After equating our equation to zero, the quadratic rule can be applied         These are the values of the principal stresses and   The unknown values are found   Finding that the coefficients and are:       2D - Principal Stresses - Etchi Regoli Gioia 25

26. To determine t he maximal shear stresses and , the principal directions and and the angle of maximum shear stress , the next method can be used ( FORMULAE )   2D - Principal Stresses - Etchi Regoli Gioia 26

27. Determining: The principal stresses and The maximal shear stresses and The principal directions and The angle of maximum shear stress With FORMULAE   2D - Principal Stresses - Etchi Regoli Gioia 27

28. Taking into account the values found from the given figure, the following equation is used : By substituting the values of principal stresses (on the right) into the equation , the two principal stresses are found .       Determining the principal stresses and with Formulae   2D - Principal Stresses - Etchi Regoli Gioia 28

29. Once we have found the principal stresses , this equation allows us to find the maximal shear stresses . By substituting the values of principal stresses (on the right) into the equation , the two maximal shear stresses are calculated .       2 . Determining t he maximal shear stresses and with Formulae   2D - Principal Stresses - Etchi Regoli Gioia 29

30.   To calculate the principal direction the following equation is used   By substituting the values below : the principal stress is found : *     *HINT To get the value of y, use the calculator and type :     3. Determining the principlal directions and with Formulae   To compute the principal stress the equation below is used .       And w e find that : 2D - Principal Stresses - Etchi Regoli Gioia 30

31. The angle of maximum shear stress is easily calculated with the equation below .       The angle of maximum shear angle refers to the maximum shear stress . The maximum shear stress happens to be maximum or minimum every . Therefore the angle of maximum shear stress can also be found in the following way:     We find that : 4. Determining t he angle of maximum shear stress with Formulae   2D - Principal Stresses - Etchi Regoli Gioia 31

32. 2D - Principal Stresses - Etchi Regoli Gioia 32 REFERENCES: Websites: http ://web.mst.edu/~mecmovie/ Textbooks: Hibbeler , R. C., 2014.  Statics and mechanics of materials . 4th ed. Cape Town, Singapore: Pearson Education South Asia Lecture notes : DEN4102-12-Multi-dimensional Stress and Strain by Professor Emiliano Bilotti