Mohr Circle.pptx

Mohr's Circle. Mechanics of Solids.
construction of mohrs circle

1. Mohr ’ s Circle Mechanics of Solids CT-254 Department of Civil Technology Course In charge: Engr. Ameer Murad Khan. Mir Chakar Khan Rind University of Technology, DGK.

2. Mohr’s circle represents the graphical form of the transformation law. The mohr’s circle is the easiest method for finding the principal stresses and normal stresses at various planes. Each point on the circumference of mohr’s circle represents a certain plane which gives the value of normal stress and shear stress at that plane. Therefore the value of normal stress along with the shear stress indicates the particular plane of the object. The mohr’s circle can be drawn for the biaxial stresses and also for the triaxial stresses. What is Mohr’s circle?

3. The mohr’s circle method has the following uses and significances By drawing a single mohr’s circle, we can easily find normal and shear stresses at various inclined oblique planes which is difficult by using the analytical method. By using mohr’s circle, it is easy to calculate principal stresses, maximum shear stress, resultant stress, principal plane angle, etc. The method of mohr’s circle is a simple and faster method than the analytical method. Mohr’s circle uses and significances

4. For normal stress Tensile stress is considered as + ve and compressive stress is considered as - ve . Sign conventions for mohr’s circle

5. For shear stress If shear stresses are of clockwise nature then it is considered as positive. If shear stress is anticlockwise in nature then it is considered as negative. Sign conventions for mohr’s circle

6. For oblique angle If the angle is measured in a clockwise direction from the reference plane then it is considered as positive. If the angle is measured in an anticlockwise direction from the reference plane then it is considered as negative. Sign conventions for mohr’s circle

7. The plane which has maximum axial stress is considered as a reference plane. If  σx  >  σy then vertical plane is considered as reference plane. If  σy >  σx then horizontal plane is considered as reference plane. How to identify the reference plane?

8. Consider an example with stresses as shown below with  σx >  σy . The plane ‘A’ shown by red color is the reference plane (as  σx  >  σy ) at 0°. The plane B shown by green color is the horizontal plane at 90° clockwise. How to draw mohr’s circle?

9. Step 1]   Draw, the vertical and horizontal axis with normal stress ( σn ) as abscissa and shear stress ( τ ) as ordinate with the suitable scale. The steps to draw the Mohr’s circle from the normal and shear stresses are as follows

10. First locate  σx  on  σn  axis. If the  σx  is tensile then locate it on positive  σn  axis from origin. And if the  σx is compressive then locate it on negative  σn  axis from origin. From the endpoint of the  σx , draw  τxy  in the vertical direction. If the nature of  τxy  is clockwise then draw it in positive  τ  direction and if the nature of  τxy  is anticlockwise then draw it in negative  τ  direction. For the given example, plane A is located as A (+  σx , +  τxy ), as shown in the below figure. Step 2] Locate plane A on  σn  –  τ  plot.

11. Similarly, plot the plane B on  σn  –  τ plot. For the given example, plane B is located as B( σy , - τyx ). Step 3] Locate plane B on  σn  – τ plot.

12. Step 4] Join points A and B.

13. Step 5] Draw a circle with radius AC or BC with C as the center of the circle. In this figure, point A indicates plane A while point B indicates plane B.

14. The principal stresses are the normal stresses when the value of shear stress ( τ ) is zero. On the mohr’s circle, there are two points that are ‘F’ and ‘G’. Therefore the major principal stress is, σ 1  = Distance OF And the minor principal stress is, σ 2  = Distance OG What are the principal stresses on a mohr’s circle?

15. The magnitude of maximum shear stress ( τ max ) is equal to the radius of mohr’s circle in the verticle direction. To find maximum shear stress from mohr’s circle, follow the steps given below:- 1] Draw a perpendicular line to the  σn  axis from the center ‘C’ of mohr’s circle. 2] Find the value of shear stress at the point where the perpendicular line cuts the mohr’s circle. Or by using principal stresses, the maximum shear stress can be calculated as, τmax  = σ1−σ22 Maximum shear stress from mohr’s circle

16. In the mohr’s circle, the angle of the oblique plane is double of actual angle. In the stress diagram, if the angle of the oblique plane (θ) is measured anticlockwise from the reference plane then in mohr’s circle diagram the oblique plane should be drawn with an angle (2θ) measured anticlockwise from a reference plane. In the stress diagram, if the angle of an oblique plane (θ) is measured clockwise from the reference plane then in mohr’s diagram, the oblique plane should be drawn with angle (2θ) measured clockwise from a reference line. Stresses at the oblique plane by using mohr’s circle

17. The example of the oblique plane at a clockwise angle (θ) is shown. Stresses at the oblique plane by using mohr’s circle

18. Now after drawing the oblique plane on mohr’s circle, draw perpendicular lines on the  σn  and  τ  axis to get the values of normal stress and shear stress at this oblique angle. Here  σo and  τo are normal stress and shear stress for the oblique plane at angle θ. Stresses at the oblique plane by using mohr’s circle

19. CASE-I CASE-II CASE-III Mohr ’ s circle for different cases

20. For the object subjected to only shear stresses (pure shear), the stress diagram is drawn, For zero normal stresses, plane A and plane B has the following coordinates on Mohr’s circle A = (0,  τxy ) B = (0, - τyx ) The point A and B can be plotted on  σn  – 𝜏 graph, Case I: Mohr’s circle for pure shear stresses

21. The line AB intersects the  σn axis at point C. Thus draw a circle with a radius of ‘AC’ and ‘C’ as the center of the circle. In this case, the relation between principal stresses is given by, σ 1=− σ 2 Case I: Mohr’s circle for pure shear stresses

22. For the object subjected to only normal stresses and zero shear stresses ( τxy = τyx = 0), the stress diagram is drawn  For zero shear stresses, plane A and plane B has the following coordinates on Mohr’s circle A = ( σx , 0) B = ( σy ,0) The point A and B can be plotted on  σn – 𝜏 graph as follows, Case II: Mohr’s circle for only normal stresses

23. Divide the line ‘AB’ and find the center of Mohr’s circle. Draw a circle with ‘C’ as the center and ‘AC’ as the radius of Mohr’s circle. In this case, the maximum shear stress is given by, τ max = σ x− σ y2 Case II: Mohr’s circle for only normal stresses

24. If the normal stresses  σx  and  σy  are equal ( σx = σy =) and there is no shear force is acting on the object, ( τxy  =  τyx = 0) then in this situation both plane A and B lies on same location. Therefore the diameter of mohr’s circle becomes zero and in this situation, the mohr’s circle becomes a point. Case III: Mohr’s circle for zero shear stresses and same normal stresses

25. For the given stress condition, find the principal stresses by using mohr’s circle method. Mohr ’ s circle example σ x  = 100 MPa σ y  = 40 MPa τ xy  = 20 MPa (Clockwise) τ yx  = -20 MPa (Anticlockwise)

26. 1] Draw  σn and  τ  axis with suitable scale. 2] Plot plane ‘A’ with coordinates of A(100, 20). 3] Plot plane ‘B’ with coordinates of B(40, -20). 4] Join points ‘A’ and ‘B’. 5] The line AB intersects the  σn  axis at point ‘C’. 6] Draw a circle with ‘C’ as the origin and ‘AC’ as radius. 7] Measure distance ‘OG’ and ‘OF’ As,  σ 1 = OF and  σ 2 = OG Solution Therefore from above Mohr’s circle, principal stresses are, σ 1  = OF = 106.05 MPa σ 2  = OG = 33.95 MPa

27. Lets understand and learn together…. Any Question ?