Draw Mohrs Circles Stress in X Direction Is 0

Mohr's Circumvolve for 2-D Stress Analysis

If you want to know the principal stresses and maximum shear stresses, y'all can just make information technology through 2-D or iii-D Mohr'due south cirlcles!

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

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

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

(one)

What people commonly are interested in more are the 2 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 can be calculated from s ₁ and s ₂ . Now, see the Fig. 1 below, which represents that a state of airplane stress exists at point O and that it is divers past the stress components southward ₁₀ , s _y , and t _xy associated with the left chemical element in the Fig. 1. Nosotros  propose to determine the stress components s _{x q} , s _{y q} , and t _{xy q} associated with the right element after it has been rotated through an angle q nigh the z axis. Fig. i  Plane stresses in different orientations

Then, we take the post-obit relationship:

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

(2)

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

(3)

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

(iv)

and t _{xy q} = -(s ₁₀ - s _y )/ii sin 2q + t _xy cos 2q

(five)

The expression for the normal stress s _{y q} may  be obtained past replacing the q in the relation for s _{10 q} in Eqn. 3 past q + 90 ^o ,  it turns out to exist s _{y q} = (s _x + s _y )/ii - (s _x - s _y )/2 cos 2q - t _xy sin 2q

(6)

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

(seven)

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

(8)

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

(ix)

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 master stress s _ane ! Notation that tan 2 q _{p =} 2t _xy /(s _ten - s _y )

(eleven)

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

(12)

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

(thirteen)

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

(14)

Besides, finally one can also read the maxium shear stress every bit t _{max = R thou =  [} (southward _x - s _y ) ² / 4 + t ^two _{xy ]} ^1/2

(xv)

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

Mohr's Circles for 3-D Stress Analysis

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

(xvi)

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

Imagine that at that place is a plane cutting through the cube in Fig. three , and the unit normal vector due north of  the cut plane has the direction cosines v ₁₀ , v _y , and v _z , that is

north = (v _ten , v _y , v _z )

(17)

so the normal stress on this airplane can exist represented by s _northward = s _x 5 ^two _x + s _y five ² _y + s _z 5 ² _z + 2 t _xy v _x v _y + 2 t _yz v _y v _z + 2 t _xz v _x v _z

(eighteen)

At that place be 3 sets of management cosines, n _i , north ₂ , and north ₃ - the three principal axes, which make southward _{due north} achieve extreme values s _one , south ₂ , and southward ₃ - the three principal stresses, and on the corresponding 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 trouble: (sI ₃ - T ₃ )n = 0

(19)

The iii eigenvalues of Eqn (19) are the roots of  the post-obit characteristic polynomial equation: det(sI ₃ - T ₃ ) = s ³ - As ⁱⁱ + Bs - C = 0

(twenty)

where A = s _x + s _y + s _z

(21)

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

(22)

C = s ₁₀ due south _y s _z + 2 t _xy t _yz t _xz - southward _x t ⁱⁱ _yz - south _y t ⁱⁱ _xz - s _z t ² _xy

(23)

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

(24)

s _one south ₂ + s _two south ₃ + s _i s _iii = B

(25)

s ₁ due south ₂ south ₃ = C

(26)

Numerically, one tin can e'er find one of the three roots of Eqn (xx) , east.chiliad. s _ane , using line search algorithm, e.thousand. bisection  algorithm. Then combining Eqns (24)and (25),  i obtains a simple quadratic equations and therefore obtains ii other roots of Eqn (xx),  e.g. due south _ii and s _{iii .} To this end, one can re-order the three roots and obtains the three primary stresses, e.g. s ₁ = max( south _{1 ,} due south _{2 ,} s _three )

(27)

s ₃ = min( s _{1 ,} s _{2 ,} s ₃ )

(28)

southward ₂ = (A - s ₁ - s ₂ )

(29)

Now, substituting s ₁ , s ₂ , or s ₃ into Eqn (19), one can obtains the corresponding primary axes n _ane , n _ii , or n ₃ , respectively.

Like to Fig. 3,  i can imagine a cube with their faces normal to n ₁ , n ₂ , or due north ₃ . For example, i can do so in Fig. 3 by replacing the axes X,Y, and Z with n _one , n ₂ , and n ₃ , respectively,  replacing  the normal stresses s _ten , south _y , and s _z with the chief 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 north ₁ , n ₂ , and due north _three . Let the cube be rotated near the centrality n ₃ , and so the corresponding transformation of stress may exist analyzed by ways of Mohr's circle equally if it were a transformation of plane stress. Indeed, the shear stresses excerted on the faces normal to the n _three axis remain equal to zippo, and the normal stress southward _iii is perpendicular to the aeroplane spanned by n ₁ and northward _ii in which the transformation takes place and thus, does not affect this transformation. One may therefore utilize the circle of diameter AB to determine the normal and shear stresses exerted on the faces of the cube every bit it is rotated about the n _iii 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 most the n _one and n _two axes, respectively.

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

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

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

southward _max =  s ₁

(thirty)

s _min =  s _iii

(31)

t _max = (s ₁ - southward 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 almost n ₁ , n ₂ , and due north _three , respectively. (The terminate.)

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 be represented by two-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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