Mohr's circle for Plane stress conditions, its construction and the important results that can be deduced from it

Mohr’s circle is a geometric representation of the 2 dimensional transformation of stresses.

Mohr’s Circle is commonly used by engineers to graphically analyze the principal and maximum shear stresses, as well as to determine the inclination of these planes.

1. Mohr’s Circle for Plane Stress Dr. George Rapheal Dept. of Automation & Robotics

2. Mohr’s Circle for Plane Stress  Transformation equations for plane stress.  Construction of Mohr’s circle.  Stresses on an inclined element.  Principal stresses and maximum shear stresses.  Introduction to the stress tensor.

3. Rectangular Stress Components These stress components acting on a small cubic rectangular element in the Fig are: s xx , t xy , t xz on x plane s yy , t yx , t yz on y plane s zz , t zx , t zy on z plane For eg.,  xy is the stress component on the x plane in y direction. Similarly,  xz is the stress component on the x plane in z direction. By equality of cross-shear, t xy = t yx ; t yz = t zy ; t zx = t xz To distinguish between a normal stress and a shear stress , the normal stress is denoted by  and the shear stress by  . The first subscript indicates the plane on which the stresses are acting and the second subscript indicates the direction of the component.

4. Plane Stress Condition Also knows as bi-axial stress. This occurs when two parallel faces of the cubic element are free of any stress. If the parallel faces free of stress are perpendicular to the z axis, then σ z = τ zx = τ zy = 0 and the only remaining stress components are σ x , σ y , and τ xy . This situation occurs in thin plates as well as on the free surface of a structural element or machine component where any point of the surface is not subjected to an external force. Example of plane stress: Free surface of a structural component. Example of plane stress : Thin plate subjected to only in-plane loads.

5. Transformation of stress in 2D problems

6. Re-arrange the transformation equations and eliminate θ by squaring both sides of each equation and adding the two equations together. This is the equation of a circle of the form: ( h,k ) being the origin of the circle and R = radius Mohr’s circle The transformation equations for plane stress can be represented in graphical form by a plot known as Mohr’s circle . Origin: Radius:

7. Important deductions  1 and  2 are the Principal stresses (Max. and Min. normal stresses that occur within a material when subjected to complex loading conditions) No shear stresses exist together with either one of these principal stresses

8. Important deductions .....  The principal planes are at right angle to each other ( 2q =180°)  The largest shear stress  max is equal to the radius of the circle  A normal stress equal to acts on each of the planes of Max. shear stress.  If  1 =  2 Mohr’s circle degenerates into a point (  x =  y =  xy =0), and no shear stresses at all develop in the xy plane.  The maximum shear stress is equal to one half the difference of the principal stresses.

9. Important deductions .....  If  x +  y = 0, the center of Mohr’s circle coincides with the origin of the  - t coordinates, and the state of pure shear exists. The sum of the normal stresses on any two mutually perpendicular planes is invariant, i.e. Planes of maximum shear stress occur at 45° to the principal planes Characteristic cup and cone fracture of Ductile materials under Uniaxial tension

10. Thank You