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Mohr's Circle for 2-D Stress Analysis

If you lot desire to know the principal stresses and maximum shear stresses, y'all can just make it through 2-D or iii-D Mohr's cirlcles!

You tin can know about the theory of  Mohr's circles from any text books of Mechanics of Materials. The following two are good references, for examples.

     1.  Ferdinand P. Beer and East. Russell Johnson, Jr, "Mechanics of Materials", Second Edition, McGraw-Loma, Inc, 1992.
     2 . James M. Gere and Stephen P. Timoshenko, "Mechanics of Materials", Third Edition, PWS-KENT Publishing Company, Boston, 1990.

The two-D stresses, so called plane stress problem, are usually given by the three stress components s ₁₀ , south _y , and t _xy ,  which consist in a two-by-ii symmetric matrix (stress tensor):

(1)

What people usually are interested in more are the two prinicipal stresses due south ₁ and due south _two , which are the two eigenvalues of the two-past-two symmetric matrix of Eqn (1), and  the maximum shear stress t _max , which can be calculated from due south _ane and southward ₂ . Now, encounter the Fig. 1 beneath, which represents that a state of plane stress exists at point O and that it is divers past the stress components s _x , s _y , and t _xy associated with the left element in the Fig. 1. We  propose to determine the stress components southward _{10 q} , southward _{y q} , and t _{xy q} associated with the right chemical element after information technology has been rotated through an angle q about the z axis. Fig. 1  Aeroplane stresses in different orientations

Then, we take the post-obit relationship:

s _{x q} = due south _x cos ² q + south _y sin ² q + 2 t _xy sin q cos q

(2)

and t _{xy q} = -(southward _x - s _y ) cos ² q +  t _xy (cos ^two q - sin ⁱⁱ q)

(3)

Equivalently, the above two equations tin can be rewritten as follows: s _{ten q} = (southward _x + s _y )/2 + (s ₁₀ - s _y )/2 cos 2q + t _xy sin iiq

(4)

and t _{xy q} = -(s _x - s _y )/2 sin 2q + t _xy cos 2q

(v)

The expression for the normal stress s _{y q} may  be obtained by replacing the q in the relation for s _{x q} in Eqn. 3 by q + 90 ^o ,  it turns out to be s _{y q} = (southward _ten + s _y )/2 - (s _x - s _y )/ii cos 2q - t _xy sin iiq

(6)

From the  relations for s _{x q} and due south _{y q} , i obtains the circle equation: (s _{x q} - s _ave ) ⁱⁱ + t ² _{xy q = R} ² _m

(7)

where s _ave = (s _x + s _y )/2  = (due south _{ten q} + s _{y q} )/2 ; _{R m =  [} (s _x - s _y ) ² / four + t ² _{xy ]} ^1/2

(8)

This circle is with radius _R ² ₁₀₀₀ and centered at C = (s _{ave  ,} 0) if  let south = southward _{10 q} and t = - t _{xy q} every bit shown in  Fig. 2 below - that is correct the Mohr'southward Circle for plane stress problem  or 2-D stress trouble! Fig. two  Mohr's circumvolve for airplane (2-D) stress In fact, Eqns. 4 and 5 are the parametric equations for the Mohr's circle!  In  Fig. 2, ane reads   that  the point
X = (s ₁₀ , - t _xy )

(nine)

which corresponds to the point at which q = 0 and the point A = (s ₁ , 0 )

(10)

which corresponds to the point at which q = q _p that gives the principal stress s ₁ ! Note that tan 2 q _{p =} 2t _xy /(s _x - s _y )

(11)

and the point Y = (s _y , t _xy )

(12)

which corresponds to the bespeak at which q = xc ^o and the point B = (due south ₂ , 0 )

(13)

which corresponds to the point at which q = q _p + 90 ^o that gives the primary stress s _two ! To this finish, one tin can pick the maxium normal stressess as s _{max =} max(s ₁ , s _ii ), s _{min =} min(southward _one , s _ii )

(14)

Too, finally one tin too read the maxium shear stress as t _{max = R m =  [} (s ₁₀ - s _y ) ⁱⁱ / iv + t ^two _{xy ]} ^i/2

(15)

which corresponds to the apex of the Mohr's circle at which q = q _p + 45 ^o ! (The cease.)

Mohr's Circles for 3-D Stress Analysis

The 3-D stresses, then called spatial stress problem,  are unremarkably given by the six stress components s _ten , s _y , s _z , t _xy , t _yz , and t _zx , (see Fig. three) which consist in a three-by-three symmetric matrix (stress tensor):

(16)

What people ordinarily are interested in more are the three prinicipal stresses south _one , southward ₂ , and s _three , which are eigenvalues of the  three-by-three symmetric matrix of Eqn (16) , and the three maximum shear stresses t _max1 , t _max2 , and t _max3 , which can be calculated from southward ₁ , s ₂ , and s ₃ . Fig. 3  3-D stress land represented by axes parallel to X-Y-Z

Imagine that there is a plane cut through the cube in Fig. 3 , and the unit normal vector north of  the cut plane has the management cosines five _x , five _y , and v _z , that is

n = (5 _x , v _y , v _z )

(17)

and then the normal stress on this plane tin can be represented by southward _northward = s ₁₀ five ² _x + south _y v ² _y + due south _z v ^two _z + 2 t _xy 5 _x five _y + 2 t _yz five _y v _z + ii t _xz 5 ₁₀ v _z

(18)

There be three sets of management cosines, n _i , due north _two , and north ₃ - the three principal axes, which make s _n achieve farthermost values due south _i , s ₂ , and southward ₃ - the three principal stresses, and on the corresponding cutting planes, the shear stresses vanish!  The problem of finding the principal stresses and their associated axes is equivalent to finding the eigenvalues and eigenvectors of the following problem: (sI ₃ - T _iii )north = 0

(19)

The three eigenvalues of Eqn (19) are the roots of  the post-obit characteristic polynomial equation: det(sI ₃ - T _iii ) = s ^three - Asouthward ² + Bdue south - C = 0

(xx)

where A = south _x + southward _y + southward _z

(21)

B = s _x south _y + s _y s _z + s _x s _z - t ² _xy - t ⁱⁱ _yz - t ² _xz

(22)

C = south _x south _y s _z + two t _xy t _yz t _xz - s _ten t ² _yz - s _y t ² _xz - s _z t ⁱⁱ _xy

(23)

In fact,  the coefficients A, B, and C in Eqn (20) are invariants as long equally the stress country is prescribed(see e.g. Ref. 2) . Therefore, if the three roots of Eqn (twenty) are s ₁ , s ₂ , and s ₃ , one has the following equations: s ₁ + s _ii + s _iii = A

(24)

southward _ane s _two + due south ₂ due south ₃ + southward ₁ s ₃ = B

(25)

due south ₁ southward ₂ s _three = C

(26)

Numerically, one tin can always find i of the three roots of Eqn (20) , e.m. south ₁ , using line search algorithm, eastward.thou. bisection  algorithm. Then combining Eqns (24)and (25),  one obtains a simple quadratic equations and therefore obtains two other roots of Eqn (20),  e.thou. s ₂ and s _{iii .} To this stop, one can re-society the three roots and obtains the 3 principal stresses, due east.g. south ₁ = max( due south _{1 ,} s _{2 ,} southward ₃ )

(27)

s ₃ = min( southward _{1 ,} s _{ii ,} due south _three )

(28)

s ₂ = (A - s _one - s _ii )

(29)

At present, substituting s _one , due south _ii , or s ₃ into Eqn (nineteen), i can obtains the respective principal axes north _ane , north ₂ , or n ₃ , respectively.

Similar to Fig. three,  one tin imagine a cube with their faces normal to n ₁ , due north _two , or n ₃ . For case, one tin practise so in Fig. three by replacing the axes X,Y, and Z with north ₁ , northward _two , and n ₃ , respectively,  replacing  the normal stresses due south _x , s _y , and s _z with the main stresses s _one , s ₂ , and due south ₃ , respectively, and removing the shear stresses t _xy , t _yz , and t _zx .

Now,  pay attention the new cube with axes northward _ane , n _ii , and n ₃ . Let the cube exist rotated well-nigh the axis n _iii , then the respective transformation of stress may be analyzed by means of Mohr's circle as if information technology were a transformation of plane stress. Indeed, the shear stresses excerted on the faces normal to the n ₃ axis remain equal to zero, and the normal stress s ₃ is perpendicular to the plane spanned by due north ₁ and n ₂ in which the transformation takes identify and thus, does non affect this transformation. Ane may therefore use the circle of diameter AB to determine the normal and shear stresses exerted on the faces of the cube every bit information technology is rotated about the due north _three axis (see Fig. iv). Similarly, the circles of diameter BC and CA may be used to make up one's mind the stresses on the cube equally it is rotated most the northward ₁ and northward ₂ axes, respectively.

Fig. 4  Mohr'due south circles for infinite (3-D) stress What if the rotations are virtually the axes rather than principal axes? It can be shown that any other transformation of axes would lead to stresses represented in Fig. 4 past a betoken located within the area which is divisional past the bigest circle with the other two circles removed!

Therefore,  i can obtain the maxium/minimum normal and shear stresses from Mohr's circles for 3-D stress as shown in  Fig. 4!

Annotation the notations above (which may be different from other references), ane obtains that

s _max =  s _i

(thirty)

s _min =  s ₃

(31)

t _max = (southward ₁ - s ₃ )/ii = t _max2

(32)

Annotation that in Fig. 4, t _max1 , t _max2 , and t _max3 are the maximum shear stresses obtained while the rotation is about n _one , n ₂ , and n _three , respectively. (The end.)

Mohr'due south Circles for Strain and for Moments and Products of Inertia

Mohr'south circle(south) can be used for strain assay and for moments and products of inertia  and other quantities as long as they can be represented past two-by-2 or 3-by-three symmetric matrices (tensors). (The stop.)

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Source: https://www.engapplets.vt.edu/Mohr/java/nsfapplets/MohrCircles2-3D/Theory/theory.htm

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