How to Draw a Mohr's Circle
Draw two perpendicular lines., Label the intersection of the coordinates, i.e., Mark off both sigma "x" and sigma "y" to some scale on the Normal Stress axis on your drawing sheet., Divide the tangential force acting on the body by the area to...
Step-by-Step Guide
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Step 1: Draw two perpendicular lines.
They're like axes in the 'x' and "y" coordinate systems but here, you call them stress coordinates.
The horizontal line will represent the Normal Stress coordinate and the vertical line will represent the Shear Stress coordinate. -
Step 2: Label the intersection of the coordinates
the origin, as "O".
Assuming that the external force in one of the directions ( say "x" direction) is F1, convert it to a stress by dividing the force by the cross section area normal to it.
Usually the dimensions will be available for you to calculate the stress.
Similarly calculate the stress in the mutually perpendicular direction also (i.e. the "y" direction).
You can call these stresses sigma "x" and sigma "y". , Follow the convention that all tensile stresses are in the positive direction i.e. to the right of the origin and compressive stresses are in the negative direction, i.e. to the left of the origin.
Call these points "J" and "K".
OJ then represents sigma "x" and OK represents sigma "y". , Remember that, for a body subjected to a shear stress, there must be an accompanying shear stress in the opposite direction on the opposite face.
These stresses will form a couple.
For the body to remain in equilibrium, an opposite couple will automatically develop.
This is called the complimentary shear i.e. every clockwise shear couple will be accompanied by an anti-clockwise shear couple when a body is subjected to simple shear.
You will also follow a convention that clockwise shears are positive and anti-clockwise shears are negative.
Correspondingly, the area above the horizontal Normal Stress axis, is positive and the area below it is negative. , Call this point "D".
Similarly,draw a vertical line (downwards) from "K" and mark on it the same value of shear stress.
This is downwards because the complimentary shear stress will be opposite to the main shear stress couple.
Call this point "E".
Note that the value of shear stress and the complimentary shear stress will be the same because it is a complimentary couple. , This will cut the horizontal axis, OJ, at some point.
Call it "C".
Draw a circle with "C" as the centre and CJ as the radius.
This is called the Mohr's Circle.
Mark the points where the Mohr's Circle cuts the horizontal axis as G and H respectively (G being the farthest point on the axis and H the nearer or negative cutting point). , Drop a perpendicular from "P" to the horizontal axis at "Q".
Join OP.OQ represents sigma N i.e. the normal stress at the plane, OG the maximum stress in the body, OH the minimum stress in the body, and PQ represents the shear stress on the plane.OP represents the resultant stress on the plane.
Similarly the stresses on any plane can be measured by choosing point P appropriately. -
Step 3: Mark off both sigma "x" and sigma "y" to some scale on the Normal Stress axis on your drawing sheet.
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Step 4: Divide the tangential force acting on the body by the area to calculate the shear stress on the body.
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Step 5: Draw a vertical line from "J" towards the positive (upwards) side and mark on it the value of the calculated shear stress.
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Step 6: Join points "D" and "E".
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Step 7: To find the values of the stress on a plane at any angle from the axis of the force
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Step 8: draw a radius CP at twice that angle
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Step 9: the angle being measured from the radius CE.
Detailed Guide
They're like axes in the 'x' and "y" coordinate systems but here, you call them stress coordinates.
The horizontal line will represent the Normal Stress coordinate and the vertical line will represent the Shear Stress coordinate.
the origin, as "O".
Assuming that the external force in one of the directions ( say "x" direction) is F1, convert it to a stress by dividing the force by the cross section area normal to it.
Usually the dimensions will be available for you to calculate the stress.
Similarly calculate the stress in the mutually perpendicular direction also (i.e. the "y" direction).
You can call these stresses sigma "x" and sigma "y". , Follow the convention that all tensile stresses are in the positive direction i.e. to the right of the origin and compressive stresses are in the negative direction, i.e. to the left of the origin.
Call these points "J" and "K".
OJ then represents sigma "x" and OK represents sigma "y". , Remember that, for a body subjected to a shear stress, there must be an accompanying shear stress in the opposite direction on the opposite face.
These stresses will form a couple.
For the body to remain in equilibrium, an opposite couple will automatically develop.
This is called the complimentary shear i.e. every clockwise shear couple will be accompanied by an anti-clockwise shear couple when a body is subjected to simple shear.
You will also follow a convention that clockwise shears are positive and anti-clockwise shears are negative.
Correspondingly, the area above the horizontal Normal Stress axis, is positive and the area below it is negative. , Call this point "D".
Similarly,draw a vertical line (downwards) from "K" and mark on it the same value of shear stress.
This is downwards because the complimentary shear stress will be opposite to the main shear stress couple.
Call this point "E".
Note that the value of shear stress and the complimentary shear stress will be the same because it is a complimentary couple. , This will cut the horizontal axis, OJ, at some point.
Call it "C".
Draw a circle with "C" as the centre and CJ as the radius.
This is called the Mohr's Circle.
Mark the points where the Mohr's Circle cuts the horizontal axis as G and H respectively (G being the farthest point on the axis and H the nearer or negative cutting point). , Drop a perpendicular from "P" to the horizontal axis at "Q".
Join OP.OQ represents sigma N i.e. the normal stress at the plane, OG the maximum stress in the body, OH the minimum stress in the body, and PQ represents the shear stress on the plane.OP represents the resultant stress on the plane.
Similarly the stresses on any plane can be measured by choosing point P appropriately.
About the Author
David White
Creates helpful guides on creative arts to inspire and educate readers.
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