How to Draw 3d Mohrs Circle

Mohr's Circle for 2-D Stress Assay

If you lot want to know the chief stresses and maximum shear stresses, you can simply make it through 2-D or 3-D Mohr's cirlcles!

You can know about the theory of  Mohr's circles from whatever text books of Mechanics of Materials. The following 2 are good references, for examples.

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

The 2-D stresses, so called plane stress problem, are commonly given by the three stress components southward _ten , s _y , and t _xy ,  which consist in a 2-past-ii symmetric matrix (stress tensor):

(1)

What people usually are interested in more are the two prinicipal stresses s _i and south _ii , which are the two eigenvalues of the ii-by-2 symmetric matrix of Eqn (1), and  the maximum shear stress t _max , which can be calculated from due south ₁ and s ₂ . At present, see the Fig. one below, which represents that a state of aeroplane stress exists at point O and that it is divers by the stress components southward _x , s _y , and t _xy associated with the left element in the Fig. ane. We  suggest to determine the stress components southward _{x q} , s _{y q} , and t _{xy q} associated with the right element after information technology has been rotated through an angle q about the z axis. Fig. 1  Aeroplane stresses in different orientations

So, nosotros have the following human relationship:

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

(2)

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

(3)

Equivalently, the above 2 equations can exist rewritten as follows: s _{x q} = (southward _ten + s _y )/2 + (s _x - s _y )/2 cos iiq + t _xy sin iiq

(4)

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

(5)

The expression for the normal stress s _{y q} may  be obtained past replacing the q in the relation for south _{x q} in Eqn. 3 past q + 90 ^o ,  information technology turns out to be s _{y q} = (s _x + s _y )/two - (s _x - south _y )/two cos 2q - t _xy sin 2q

(6)

From the  relations for due south _{x q} and south _{y q} , 1 obtains the circle equation: (s _{x q} - s _ave ) ² + t ² _{xy q = R} ² ₁₀₀₀

(vii)

where s _ave = (s _x + southward _y )/two  = (s _{ten q} + s _{y q} )/2 ; _{R m =  [} (s ₁₀ - southward _y ) ² / 4 + t ^two _{xy ]} ^1/2

(eight)

This circumvolve is with radius _R ² _thousand and centered at C = (s _{ave  ,} 0) if  let due south = southward _{x q} and t = - t _{xy q} as shown in  Fig. 2 beneath - that is right the Mohr's Circumvolve for plane stress problem  or ii-D stress problem! Fig. 2  Mohr's circumvolve for plane (2-D) stress In fact, Eqns. 4 and five are the parametric equations for the Mohr's circle!  In  Fig. two, one reads   that  the betoken
X = (s _x , - t _xy )

(9)

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

(ten)

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

(11)

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

(12)

which corresponds to the bespeak at which q = xc ^o and the indicate B = (s _ii , 0 )

(13)

which corresponds to the point at which q = q _p + 90 ^o that gives the primary stress south ₂ ! To this terminate, one can pick the maxium normal stressess as southward _{max =} max(southward ₁ , s ₂ ), s _{min =} min(s ₁ , southward ₂ )

(14)

Besides, finally one tin also read the maxium shear stress every bit t _{max = R m =  [} (s _x - due south _y ) ² / four + t ² _{xy ]} ^1/2

(15)

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

Mohr's Circles for three-D Stress Analysis

The three-D stresses, so called spatial stress trouble,  are usually given by the half-dozen stress components due south _x , s _y , south _z , t _xy , t _yz , and t _zx , (meet Fig. 3) which consist in a three-by-iii symmetric matrix (stress tensor):

(16)

What people normally are interested in more than are the iii prinicipal stresses due south ₁ , south ₂ , and s ₃ , which are eigenvalues of the  three-past-three symmetric matrix of Eqn (sixteen) , and the iii maximum shear stresses t _max1 , t _max2 , and t _max3 , which can be calculated from due south ₁ , southward _ii , and s ₃ . Fig. 3  3-D stress state represented by axes parallel to X-Y-Z

Imagine that at that place is a airplane cut through the cube in Fig. 3 , and the unit normal vector north of  the cut plane has the direction cosines five _x , 5 _y , and 5 _z , that is

northward = (v _x , v _y , v _z )

(17)

and so the normal stress on this aeroplane tin can be represented past s _n = southward _x 5 ² _x + s _y 5 ² _y + s _z v ² _z + 2 t _xy five _x v _y + ii t _yz v _y v _z + two t _xz v _ten v _z

(18)

At that place exist three sets of direction cosines, north ₁ , n ₂ , and n ₃ - the three principal axes, which brand s _n achieve extreme values s ₁ , s ₂ , and due south ₃ - the three principal stresses, and on the respective cut 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: (due southI ₃ - T ₃ )northward = 0

(xix)

The three eigenvalues of Eqn (nineteen) are the roots of  the following characteristic polynomial equation: det(sI ₃ - T _iii ) = s ⁱⁱⁱ - As ² + Bs - C = 0

(twenty)

where A = due south _x + south _y + s _z

(21)

B = s ₁₀ s _y + southward _y s _z + s _x s _z - t ⁱⁱ _xy - t ² _yz - t ² _xz

(22)

C = s _x s _y southward _z + 2 t _xy t _yz t _xz - southward ₁₀ t ^two _yz - southward _y t ² _xz - s _z t ² _xy

(23)

In fact,  the coefficients A, B, and C in Eqn (20) are invariants equally long every bit the stress state is prescribed(encounter e.g. Ref. 2) . Therefore, if the three roots of Eqn (20) are s _i , s ₂ , and s _three , one has the post-obit equations: s _one + south ₂ + s ₃ = A

(24)

s ₁ southward ₂ + s _ii due south ₃ + due south ₁ s _three = B

(25)

s ₁ s ₂ southward ₃ = C

(26)

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

(27)

south ₃ = min( due south _{one ,} s _{ii ,} due south ₃ )

(28)

s _ii = (A - south _ane - due south ₂ )

(29)

Now, substituting s ₁ , due south _ii , or southward ₃ into Eqn (19), one tin can obtains the corresponding principal axes due north ₁ , n ₂ , or north ₃ , respectively.

Similar to Fig. iii,  i can imagine a cube with their faces normal to n _ane , n _two , or n _three . For example, 1 can do so in Fig. iii by replacing the axes X,Y, and Z with north _ane , n _ii , and n ₃ , respectively,  replacing  the normal stresses due south _x , s _y , and due south _z with the principal stresses s ₁ , s ₂ , and s ₃ , respectively, and removing the shear stresses t _xy , t _yz , and t _zx .

Now,  pay attention the new cube with axes n ₁ , north ₂ , and due north _iii . Let the cube be rotated almost the axis n ₃ , and so the respective transformation of stress may exist analyzed by means of Mohr's circumvolve as if it were a transformation of plane stress. Indeed, the shear stresses excerted on the faces normal to the northward ₃ centrality remain equal to zero, and the normal stress s _iii is perpendicular to the airplane spanned past n _one and due north ₂ in which the transformation takes place and thus, does non affect this transformation. One may therefore use the circle of bore AB to determine the normal and shear stresses exerted on the faces of the cube as it is rotated nearly the n _three axis (see Fig. four). Similarly, the circles of diameter BC and CA may be used to determine the stresses on the cube as it is rotated about the n ₁ and n ₂ axes, respectively.

Fig. 4  Mohr's circles for space (3-D) stress What if the rotations are about the axes rather than chief axes? It can be shown that whatsoever other transformation of axes would lead to stresses represented in Fig. 4 by a point located inside the area which is bounded by the bigest circumvolve with the other ii circles removed!

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

Note the notations above (which may exist different from other references), one obtains that

s _max =  s ₁

(xxx)

s _min =  due south ₃

(31)

t _max = (south ₁ - s ₃ )/2 = t _max2

(32)

Note that in Fig. 4, t _max1 , t _max2 , and t _max3 are the maximum shear stresses obtained while the rotation is about northward ₁ , n ₂ , and n ₃ , respectively. (The stop.)

Mohr's Circles for Strain and for Moments and Products of Inertia

Mohr's circumvolve(south) can be used for strain analysis and for moments and products of inertia  and other quantities as long every bit they can be represented past ii-by-two or three-by-three symmetric matrices (tensors). (The end.)

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

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