draw mohr circles for 3d stress states

Mohr'due south Circumvolve for 2-D Stress Analysis

If you desire to know the principal stresses and maximum shear stresses, you lot can only brand it through 2-D or iii-D Mohr'south cirlcles!

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

     one.  Ferdinand P. Beer and Eastward. 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 ii-D stresses, so called aeroplane stress trouble, are ordinarily given by the three stress components s _ten , s _y , and t _xy ,  which consist in a two-past-two symmetric matrix (stress tensor):

(1)

What people unremarkably are interested in more are the two prinicipal stresses s ₁ and s _ii , which are the ii eigenvalues of the 2-by-two symmetric matrix of Eqn (1), and  the maximum shear stress t _max , which can be calculated from s _one and due south _ii . At present, see the Fig. 1 beneath, which represents that a state of plane stress exists at point O and that it is defined by the stress components s ₁₀ , due south _y , and t _xy associated with the left element in the Fig. i. We  advise to determine the stress components s _{10 q} , due south _{y q} , and t _{xy q} associated with the right element after it has been rotated through an angle q well-nigh the z axis. Fig. i  Plane stresses in different orientations

Then, we take the post-obit relationship:

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

(two)

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

(3)

Equivalently, the to a higher place ii equations tin be rewritten as follows: due south _{x q} = (s _ten + s _y )/2 + (southward _x - s _y )/2 cos 2q + t _xy sin 2q

(iv)

and t _{xy q} = -(s _x - s _y )/2 sin iiq + 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. iii by q + ninety ^o ,  information technology turns out to exist s _{y q} = (s _ten + s _y )/2 - (south _x - south _y )/2 cos 2q - t _xy sin twoq

(half dozen)

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

(7)

where southward _ave = (southward ₁₀ + s _y )/2  = (s _{x q} + s _{y q} )/2 ; _{R m =  [} (s ₁₀ - s _y ) ² / four + t ⁱⁱ _{xy ]} ^ane/two

(eight)

This circle is with radius _R ² _yard and centered at C = (due south _{ave  ,} 0) if  let s = s _{ten q} and t = - t _{xy q} as shown in  Fig. 2 below - that is right the Mohr's Circle for aeroplane stress problem  or 2-D stress trouble! Fig. 2  Mohr'south circle for plane (2-D) stress In fact, Eqns. four and 5 are the parametric equations for the Mohr'southward circle!  In  Fig. ii, 1 reads   that  the indicate
10 = (s _x , - t _xy )

(9)

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

(x)

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

(eleven)

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

(12)

which corresponds to the point at which q = 90 ^o and the point B = (due south _two , 0 )

(13)

which corresponds to the betoken at which q = q _p + ninety ^o that gives the chief stress due south _ii ! To this end, ane tin selection the maxium normal stressess equally s _{max =} max(south ₁ , s _two ), s _{min =} min(south ₁ , s _two )

(xiv)

Besides, finally i can also read the maxium shear stress as t _{max = R m =  [} (due south _x - due south _y ) ² / 4 + t ⁱⁱ _{xy ]} ^1/2

(xv)

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

Mohr'southward Circles for three-D Stress Analysis

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

(16)

What people usually are interested in more than are the iii prinicipal stresses s ₁ , south ₂ , and south ₃ , which are eigenvalues of the  three-by-iii symmetric matrix of Eqn (16) , and the three maximum shear stresses t _max1 , t _max2 , and t _max3 , which tin can be calculated from s _one , s ₂ , and southward ₃ . Fig. three  3-D stress state represented by axes parallel to 10-Y-Z

Imagine that there is a aeroplane cutting through the cube in Fig. 3 , and the unit of measurement normal vector n of  the cut plane has the direction cosines 5 ₁₀ , 5 _y , and v _z , that is

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

(17)

then the normal stress on this plane tin can be represented past south _n = s _x five ² ₁₀ + due south _y five ² _y + s _z v ² _z + two t _xy 5 _x 5 _y + 2 t _yz five _y v _z + 2 t _xz v ₁₀ v _z

(eighteen)

There be three sets of direction cosines, northward ₁ , due north ₂ , and north ₃ - the three principal axes, which brand south _n achieve farthermost values south _one , s _two , and s _three - 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 following problem: (sI ₃ - T _three )north = 0

(xix)

The iii eigenvalues of Eqn (19) are the roots of  the following feature polynomial equation: det(sI _iii - T ₃ ) = s ^three - As ^two + Bs - C = 0

(20)

where A = s _x + s _y + s _z

(21)

B = s ₁₀ s _y + s _y due south _z + s _x s _z - t ² _xy - t ² _yz - t ² _xz

(22)

C = due south _ten s _y s _z + ii t _xy t _yz t _xz - s _x t ² _yz - s _y t ² _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 state is prescribed(run into east.g. Ref. 2) . Therefore, if the three roots of Eqn (20) are s ₁ , southward ₂ , and south ₃ , one has the post-obit equations: s ₁ + due south ₂ + southward _iii = A

(24)

s ₁ s _two + south ₂ s ₃ + s _i south ₃ = B

(25)

s ₁ s ₂ due south ₃ = C

(26)

Numerically, one can ever discover one of the three roots of Eqn (20) , east.g. southward ₁ , using line search algorithm, e.g. bisection  algorithm. Then combining Eqns (24)and (25),  1 obtains a simple quadratic equations and therefore obtains two other roots of Eqn (20),  e.grand. s _two and due south _{3 .} To this end, one can re-order the iii roots and obtains the iii primary stresses, e.g. s ₁ = max( s _{1 ,} s _{2 ,} s _three )

(27)

s ₃ = min( s _{1 ,} s _{ii ,} s ₃ )

(28)

s ₂ = (A - s ₁ - south _ii )

(29)

Now, substituting s _i , due south ₂ , or southward _iii into Eqn (nineteen), one can obtains the corresponding principal axes n ₁ , northward _two , or n ₃ , respectively.

Similar to Fig. 3,  one can imagine a cube with their faces normal to n ₁ , n ₂ , or n _iii . For instance, 1 can do then in Fig. 3 past replacing the axes X,Y, and Z with n _i , n _two , and northward _iii , respectively,  replacing  the normal stresses s _x , s _y , and s _z with the chief stresses s ₁ , due south ₂ , and southward _three , respectively, and removing the shear stresses t _xy , t _yz , and t _zx .

At present,  pay attending the new cube with axes n _ane , due north ₂ , and northward ₃ . Permit the cube be rotated about the centrality n ₃ , then the corresponding transformation of stress may be analyzed by ways of Mohr'southward circle every bit if it 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 south ₃ is perpendicular to the plane spanned by n ₁ and north _ii in which the transformation takes place and thus, does not affect this transformation. I may therefore use the circle of diameter AB to determine the normal and shear stresses exerted on the faces of the cube as it is rotated about the n ₃ axis (run across Fig. 4). Similarly, the circles of diameter BC and CA may be used to determine the stresses on the cube every bit information technology 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 master axes? It can be shown that any other transformation of axes would lead to stresses represented in Fig. iv by a bespeak located within the expanse which is bounded past the bigest circle 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 equally shown in  Fig. 4!

Note the notations higher up (which may exist different from other references), ane obtains that

south _max =  s _i

(30)

s _min =  s ₃

(31)

t _max = (s _one - due south ₃ )/2 = t _max2

(32)

Note that in Fig. iv, t _max1 , t _max2 , and t _max3 are the maximum shear stresses obtained while the rotation is about north ₁ , n _ii , and north _three , respectively. (The cease.)

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

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