Mohr Circle Procedure PPT file - Copy.pptx

Mohrs circle for transformation of stress, strain and moment of inertia

1.                         O B A F   J H G E Mohr’s Circle for Stress Measure OA and OB equal to and respectively to some scale along the x-axis. At A and B, draw perpendicular AG and BF on the x–axis and equal to . AG is taken downward on plane which is anticlockwise, and BF is taken as upward as the direction on the plane is clockwise. Join F and G, cutting the x–axis in C, which is the center of the stress circle (i.e., Mohr’s Circle) With C as the center and radius equal to CG or CF, draw a circle. At C, make CD at an angle of with CG in the anticlockwise direction. From D, draw perpendicular DE on x–axis. Then, DE represents , OE represents and OD represent . The absicca of H gives the maximum principal stress and absicca of J gives the minimum principal stress Angle 2θp(anticlockwise) is used to describe principal plane where θ p is the principal plane Angle 2θs(anticlockwise) is used to describe plane of max shear where θ s is the required plane and is the maximum shear stress      

2. Verification of Mohr’s Circle   Ƴɛ θ

3. For the given state of stress of an element a)determine the stress on an inclined plane incline at 30°with the vertical b) principal planes and principal stresses c) maximum shear and plane of maximum shear

4. Plot points G( σ x, - τ xy ) and F ( σ y, + τ xy ) . Here in question τ xy tries to rotate σ x in anticlockwise direction and is taken as negative while tries to rotate σ y in clockwise direction and is taken as positive Joing CF find point C and draw circle using radius =GF=EF For plane inclined at 30° in real which is equivalent to rotating CG by 60° in Mohr’s circle, OE gives normal stress ( σ n )=54.4Mpa ED gives the shear stress ( τ θ ) = 93.5Mpa which is negative means tries to rotate σ n =54.4Mpa in anticlockwise direction stress on an inclined plane incline at 30°

5. Principal Stresses and Principal Planes Rotate CG by angle 134° to reach CH gives 2 θ p i.e., θ p=67° at which OH gives principal stress σ 1=124.7Mpa OJ obtained by rotation of CG by (134°+180°) degree in clockwise or -46° meaning anticlockwise in Mohr's circle meaning 23°in anticlockwise in real or (67°+90°) in clockwise in real. OJ gives other principal stress σ 2=69.7Mpa CN obtained by rotating CG by 44°inmohrs circle ( i.e 22°in real) gives τ max which is negative in this plane meaning tries to rotate the horizontal stress in anticlockwise direction and vertical stress in clockwise direction)

6. Mohr’s Circle for Strain Analysis The element(with black outline) is subjected to normal strain ɛ x and ɛ y and shear strain Ƴ xy such that the normal strains are tensile in nature and shear strain is such angle at bottom left corner is decreased and at bottom right corner is increased The elements just attains shape denoted by red dotted line as shown We need to evaluate principal strains, principal planes, maximum shear strain and plane of maximum shear strain. We also need to evaluate the strains corresponding to X’Y’ axes obtained by rotating XY axes by angle θ as shown

7.                         O B A F   J H G E Mohr’s Circle for strain Measure OA and OB equal to and respectively to some scale along the x-axis. At A and B, draw perpendicular AG and BF on the x–axis and equal to . AG is taken downward on plane means the angle is increased in bottom right corner, and BF is taken as upward the plane as the angle decreases in bottom right corner Join F and G, cutting the x–axis in C, which is the center of the strain (i.e., Mohr’s Circle) With C as the center and radius equal to CG or CF, draw a circle. At C, make CD at an angle of with CG in the anticlockwise direction. From D, draw perpendicular DE on x–axis. Then, DE represents , OE represents and OD represent . The absicca of H gives the maximum principal strain and absicca of J gives the minimum principal strain Angle 2θp(anticlockwise) is used to describe principal plane where θ p is the principal plane Angle 2θs(anticlockwise) is used to describe plane of max shear strain where θ s is the required plane and is the maximum shear strain      

8. For the given state of strain of an element, determine the strains corresponding to X’Y’ axis inclined at angle of 60 degrees with XY-axis. Also determine the principal strains and maximum shear strain with its orientation Ɛ x =340X10 -6 Ɛ y = 110X10 -6 Ƴ xy =180X10 -6

9. Measure OA and OB equal to =340x10 -6 and = 110x10 -6 respectively to some scale along the x-axis. At A and B, draw perpendicular AG and BF on the x–axis and equal to /2= 90x10 -6 . AG is taken downward on plane means the strain tries to rotate the corresponding axis in anticlockwise direction, and BF is taken as upward plane along tries to rotate clockwise Join F and G, cutting the x–axis in C, which is the center of the strain (i.e., Mohr’s Circle) With C as the center and radius equal to CG or CF, draw a circle. At C, make CD at an angle of =60° with CG in the anticlockwise direction. From D, draw perpendicular DE on x–axis. Then, DE represents /2= 55x10 -6 , OE represents =340x10 -6 , OL represents =340x10 -6 and OD represent =364.17x10 -6 . At 60° the shear stress /2 is observed positive in the circle which means the plane along tries to rotate in clockwise and that along tries to rotate anticlockwise  

10. Then, CM represents /2= 146x10 -6 , OH represents =370x10 -6 , OJ represents =340x10 -6 At angle ACG=38° , 3. The maximum shear stress /2 is observed positive in the circle at angle MCG=128° which means that due to maxm shear strain the plane along tries to rotate in clockwise direction and that along tries to rotate in anticlockwise direction  

11.                 O B A F J H G Mohr’s Circle for Principal MOI Measure OA and OB equal to and respectively to some scale along the x-axis. At A and B, draw perpendicular AG and BF on the x–axis and equal to . If is positive plot it upwards with and if negative plot it upwards with Join F and G, cutting the x–axis in C, which is the center of the strain (i.e., Mohr’s Circle) With C as the center and radius equal to CG or CF, draw a circle. The absicca of H gives the maximum principal MOI and absicca of J gives the minimum principal MOI Angle 2θp(anticlockwise + ve )) is used to describe principal plane where θ p is the principal plane In this figure CG is to be rotated clockwise to reach point H i.e Imax which means the horizontal x-axis is to be rotated by θp in clockwise to obtain the plane of maximum MOI in real    

12. Find the principal planes and principal moment of inertia about the centroidal axes for the given built up section using Mohr’s Circle

13.       =3.16*10         O B A F J H G Mohr’s Circle for Principal MOI Measure OA and OB equal to and respectively to some scale along the x-axis. At A and B, draw perpendicular AG and BF on the x–axis and equal to . If is positive plot it upwards with and if negative plot it upwards with Join F and G, cutting the x–axis in C, which is the center of the strain (i.e., Mohr’s Circle) With C as the center and radius equal to CG or CF, draw a circle. The absicca of H gives the maximum principal MOI and absicca of J gives the minimum principal MOI Angle 2θp(anticlockwise + ve )) is used to describe principal plane where θ p is the principal plane In this figure CG is to be rotated clockwise to reach point H i.e Imax which means the horizontal x-axis is to be rotated by θp in clockwise to obtain the plane of maximum MOI in real From the plot, 2θp=-57.7° which means θp =-28.85° (clockwise)   =3.16*10 6