Mohr’s Stress circle

 Mohr’s Stress circle 

GRAPHICAL METHOD: MOHR’S STRESS CIRCLE METHOD

 We know from the analytical method that

 σθ = (σx + σy)/2  + ((σx — σy)/2) (Cos2θ) + τ Sin2θ

τθ = (( σx — σy)/2) (Sin2θ) — τ Cos2θ

Square and add, we get the equation of a circle as given below.

θ – (σx — σy)/2)]22  = [(1/2) (σx — σy) + 4τ2 ]1/2

Thus it was concluded that the complex system can be analysed graphically by a circle.

A point on this circle will represent the state of stress on a certain plane in the stressed element.

MOST IMPORTANT THREE THINGS TO REMEMBER

(i) All measurement of stresses are made from the pole.

(ii) Every radius (radial line) of the circle is a plane in the stressed element.

(iii) θ on stressed element is 2θ on Mohr’s circle or vice versa.

HOW TO FIND THE VARIOUS QUANTITIES FROM MOHR’S CIRCLE

 (i) To find principal stresses σ1 and σ2—–points of zero shear stress

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Mark two extreme ends on the horizontal diameter as S and R. These are the points of zero shear stress. Therefore measure            OS = σ1 

                               OR = σ2

 (ii) To find principal planes θpp1 and θpp2

Since every radial line is a plane.

Point S joined to centre P becomes a radial line.Similarly radial lines PN,PQ, PS, PN, PT and PV are radial lines and hence are planes.

Hence Radial line PS is the principal plane because of zero shear stress on it.

But the given reference plane is AB on stressed element and PN on Mohr’s circle.

Measure angle NPS.

Angle NPS = 2θpp1

Since angle on Mohr’s circle is twice of angle on the stressed element, angle θpp1 can be found.

Angle NPR will be 2θpp2.

NOTE: Both the angles 2θpp1 and2θpp2 must be measured clockwise or anticlockwise. 

Angle NPR = 2θpp2= 1800 + 2θpp1

θpp2= 900 + θpp1

Hence Angle between principal planes will be 900

OR Principal planes at right angles to each other.

 (iii) Determination of planes of maximum shear and maximum shear stress

The point of maximum shear stress will be exactly above as well as exactly below the center of the circle.

The two ends of the vertical diameter will be points of maximum shear.

Let the point on top be T and at the bottom be V.

Measure PT = + Maximum shear stress

Measure PV= — maximum shear stress

Since TP is a radius, becomes the plane of maximum shear

Measure angle TPN = 2θms1

Another plane of maximum shear will be radius PV.

Measure angle NPV = 2θms2

ms2 = 2θms1 + 1800

θms2 = θms1 + 900

 Thus planes of maximum shear will be at right angles.

  (iv) To Find angle between a principal plane and a plane of  maximum shear i.e. between θpp1 and θms1

Angle between θpp1 and θms1 on Mohr’s circle will be 900.

But actually angle between θpp1 and θms1 will be 450because angle on Mohr’s circle is twice of angle on stressed element.

 HENCE ANGLE BETWEEN A PRINCIPAL PLANE AND A PLANE OF MAXIMUM SHEAR WILL BE  450