draw 3d mohr's circle

Mohr's Circle for two-D Stress Assay

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

You can know almost the theory of  Mohr's circles from any text books of Mechanics of Materials. The post-obit two are good references, for examples.

     1.  Ferdinand P. Beer and Due east. Russell Johnson, Jr, "Mechanics of Materials", 2d Edition, McGraw-Hill, Inc, 1992.
     2 . James K. Gere and Stephen P. Timoshenko, "Mechanics of Materials", 3rd Edition, PWS-KENT Publishing Company, Boston, 1990.

The 2-D stresses, then chosen plane stress problem, are normally given by the three stress components s _x , southward _y , and t _xy ,  which consist in a two-by-two symmetric matrix (stress tensor):

(one)

What people usually are interested in more are the two prinicipal stresses s ₁ and southward ₂ , which are the two eigenvalues of the two-by-2 symmetric matrix of Eqn (1), and  the maximum shear stress t _max , which tin exist calculated from s ₁ and s _two . Now, meet the Fig. one below, which represents that a land of plane stress exists at point O and that information technology is defined by the stress components south _ten , s _y , and t _xy associated with the left element in the Fig. 1. We  suggest to determine the stress components s _{x q} , due south _{y q} , and t _{xy q} associated with the right element after it has been rotated through an angle q about the z axis. Fig. one  Plane stresses in different orientations

Then, we have the post-obit relationship:

s _{x q} = s ₁₀ cos ² q + s _y sin ² q + 2 t _xy sin q cos q

(2)

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

(iii)

Equivalently, the above two equations can be rewritten as follows: s _{x q} = (southward _x + due south _y )/2 + (s _x - s _y )/2 cos twoq + t _xy sin twoq

(4)

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

(5)

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

(6)

From the  relations for s _{10 q} and s _{y q} , one obtains the circumvolve equation: (south _{x q} - south _ave ) ⁱⁱ + t ^two _{xy q = R} ² _m

(7)

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

(viii)

This circle is with radius _R ² _m and centered at C = (south _{ave  ,} 0) if  let s = s _{x q} and t = - t _{xy q} equally shown in  Fig. 2 below - that is correct the Mohr's Circle for aeroplane stress problem  or two-D stress problem! Fig. 2  Mohr'due south circumvolve for airplane (2-D) stress In fact, Eqns. 4 and 5 are the parametric equations for the Mohr's circle!  In  Fig. 2, 1 reads   that  the betoken
X = (s _x , - t _xy )

(nine)

which corresponds to the indicate 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 ! Annotation that tan 2 q _{p =} 2t _xy /(south _x - due south _y )

(xi)

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

(12)

which corresponds to the signal at which q = xc ^o and the point B = (southward ₂ , 0 )

(13)

which corresponds to the point at which q = q _p + ninety ^o that gives the primary stress s ₂ ! To this end, one can selection the maxium normal stressess as south _{max =} max(south ₁ , s _ii ), due south _{min =} min(s _i , south ₂ )

(fourteen)

As well, finally ane can also read the maxium shear stress equally t _{max = R 1000 =  [} (south _x - s _y ) ⁱⁱ / four + t ⁱⁱ _{xy ]} ^1/2

(15)

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

Mohr's Circles for 3-D Stress Analysis

The 3-D stresses, so called spatial stress problem,  are usually given past the six stress components due south ₁₀ , s _y , s _z , t _xy , t _yz , and t _zx , (see Fig. 3) which consist in a three-by-3 symmetric matrix (stress tensor):

(xvi)

What people unremarkably are interested in more are the iii prinicipal stresses southward _i , s _two , and southward _iii , which are eigenvalues of the  three-past-three symmetric matrix of Eqn (16) , and the three maximum shear stresses t _max1 , t _max2 , and t _max3 , which can exist calculated from s ₁ , s ₂ , and s ₃ . Fig. 3  3-D stress state represented past axes parallel to X-Y-Z

Imagine that there is a plane cutting through the cube in Fig. 3 , and the unit normal vector n of  the cut plane has the direction cosines five _x , five _y , and 5 _z , that is

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

(17)

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

(eighteen)

There be three sets of direction cosines, n _one , n _ii , and due north ₃ - the three main axes, which brand southward _northward achieve farthermost values south _i , s ₂ , and southward ₃ - the three principal stresses, and on the corresponding cutting planes, the shear stresses vanish!  The trouble of finding the principal stresses and their associated axes is equivalent to finding the eigenvalues and eigenvectors of the post-obit problem: (sI ₃ - T ₃ )n = 0

(19)

The iii eigenvalues of Eqn (19) are the roots of  the following characteristic polynomial equation: det(southI ₃ - T ₃ ) = s ^three - Adue south ⁱⁱ + Bdue south - C = 0

(20)

where A = s ₁₀ + southward _y + s _z

(21)

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

(22)

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

(23)

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

(24)

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

(25)

s ₁ s _two s _three = C

(26)

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

(27)

due south ₃ = min( s _{1 ,} s _{2 ,} south ₃ )

(28)

s ₂ = (A - s ₁ - s ₂ )

(29)

Now, substituting s ₁ , s _two , or s ₃ into Eqn (xix), one can obtains the respective principal axes northward ₁ , n ₂ , or n ₃ , respectively.

Like to Fig. 3,  one can imagine a cube with their faces normal to n ₁ , n _ii , or n ₃ . For example, one tin can do so in Fig. three by replacing the axes X,Y, and Z with north ₁ , n ₂ , and n _iii , respectively,  replacing  the normal stresses south _x , s _y , and s _z with the primary stresses s _ane , 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 northward ₁ , northward ₂ , and n _three . Let the cube be rotated near the centrality n ₃ , then the corresponding transformation of stress may be analyzed by means of Mohr's circle as if it were a transformation of plane stress. Indeed, the shear stresses excerted on the faces normal to the due north _iii centrality remain equal to nothing, and the normal stress southward ₃ is perpendicular to the airplane spanned by n ₁ and north ₂ in which the transformation takes place and thus, does not touch on this transformation. I may therefore utilise the circle of diameter AB to determine the normal and shear stresses exerted on the faces of the cube as it is rotated well-nigh the n ₃ axis (see Fig. four). Similarly, the circles of diameter BC and CA may exist used to decide the stresses on the cube as it is rotated virtually the n ₁ and n _ii axes, respectively.

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

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

Note the notations to a higher place (which may be different from other references), one obtains that

southward _max =  south ₁

(30)

s _min =  southward _three

(31)

t _max = (s ₁ - s _three )/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 _i , northward ₂ , and n ₃ , respectively. (The end.)

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

Mohr's circumvolve(s) tin be used for strain analysis and for moments and products of inertia  and other quantities as long every bit they can exist represented by two-by-2 or iii-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