How To Draw 3d Mohr's Circle

Mohr'south Circle for ii-D Stress Analysis

If you want to know the principal stresses and maximum shear stresses, you lot tin can just brand it through two-D or 3-D Mohr's cirlcles!

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

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

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

(1)

What people normally are interested in more are the two prinicipal stresses s ₁ and south ₂ , which are the 2 eigenvalues of the ii-by-2 symmetric matrix of Eqn (1), and the maximum shear stress t _max , which can be calculated from s _ane and s _ii . Now, see the Fig. 1 below, which represents that a state of aeroplane stress exists at signal O and that it is divers past the stress components south _ten, s _y, and t _xy associated with the left element in the Fig. 1. We propose to determine the stress components s _{10 q} , southward _{y q} , and t _{xy q} associated with the right chemical element after it has been rotated through an bending q about the z axis.

Fig. 1 Plane stresses in different orientations

Then, we have the post-obit human 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} = -(southward _ten - south _y ) cos ² q + t _xy (cos ⁱⁱ q - sin ² q)

(three)

Equivalently, the to a higher place two equations tin can exist rewritten as follows: s _{x q} = (s _x + s _y )/2 + (due south ₁₀ - s _y )/two cos 2q + t _xy sin twoq

(iv)

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

(five)

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

(half-dozen)

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

(vii)

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

(eight)

This circumvolve is with radius _R ² _m and centered at C = (s _{ave ,} 0) if allow southward = southward _{x q} and t = - t _{xy q} every bit shown in Fig. two below - that is right the Mohr's Circumvolve for plane stress problem or 2-D stress trouble!

Fig. 2 Mohr's circle for airplane (2-D) stress In fact, Eqns. four and 5 are the parametric equations for the Mohr'south circle! In Fig. 2, one reads that the signal
10 = (southward ₁₀ , - t _xy )

(9)

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

(10)

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

(11)

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

(12)

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

(13)

which corresponds to the bespeak at which q = q _p + 90 ^o that gives the principal stress s ₂ ! To this finish, one tin can pick the maxium normal stressess every bit due south _{max =} max(s ₁ , s _two ), s _{min =} min(s ₁ , s _two )

(14)

Likewise, finally i tin can also read the maxium shear stress as t _{max = R m = [} (s _ten - s _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 stop.)

Mohr's Circles for three-D Stress Analysis

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

(16)

What people unremarkably are interested in more are the iii prinicipal stresses s ₁ , s _two , and s ₃ , which are eigenvalues of the three-by-three symmetric matrix of Eqn (16) , and the 3 maximum shear stresses t _max1 , t _max2 , and t _max3 , which can be calculated from s _one , s ₂ , and s _iii .

Fig. 3 three-D stress state represented by axes parallel to X-Y-Z

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

north = (5 ₁₀ , five _y , 5 _z )

(17)

then the normal stress on this aeroplane tin be represented by s _north = s _x v ² _ten + s _y five ² _y + southward _z v ² _z + 2 t _xy five _x v _y + 2 t _yz v _y v _z + 2 t _xz v ₁₀ five _z

(18)

There exist three sets of direction cosines, n ₁ , n ₂ , and due north ₃ - the iii principal axes, which make s _northward achieve extreme values s ₁ , s ₂ , and s _three - the three principal stresses, and on the corresponding cut planes, the shear stresses vanish! The trouble of finding the master stresses and their associated axes is equivalent to finding the eigenvalues and eigenvectors of the following problem: (sI ₃ - T _three )n = 0

(19)

The 3 eigenvalues of Eqn (19) are the roots of the following characteristic polynomial equation: det(southI _three - T _iii ) = due south ^three - Adue south ^two + Bs - C = 0

(xx)

where A = south ₁₀ + s _y + southward _z

(21)

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

(22)

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

(23)

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

(24)

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

(25)

s _one south ₂ s _iii = C

(26)

Numerically, one can e'er discover one of the iii roots of Eqn (xx) , e.yard. s _one , using line search algorithm, e.k. bisection algorithm. And then combining Eqns (24)and (25), one obtains a simple quadratic equations and therefore obtains ii other roots of Eqn (20), e.1000. s ₂ and southward _{3 .} To this cease, one can re-social club the three roots and obtains the three master stresses, e.g. s ₁ = max( s _{one ,} s _{2 ,} southward ₃ )

(27)

s _three = min( due south _{1 ,} s _{two ,} s _iii )

(28)

south ₂ = (A - south _i - s _ii )

(29)

Now, substituting s _one , s ₂ , or s ₃ into Eqn (xix), one tin can obtains the corresponding principal axes n ₁ , n ₂ , or n _iii , respectively.

Similar to Fig. 3, i can imagine a cube with their faces normal to n _i , n ₂ , or n ₃ . For instance, ane can do so in Fig. 3 by replacing the axes 10,Y, and Z with n _ane , n ₂ , and n _three , respectively, replacing the normal stresses s ₁₀ , s _y , and s _z with the principal stresses south ₁ , southward _two , and s _three , respectively, and removing the shear stresses t _xy , t _yz , and t _zx .

Now, pay attention the new cube with axes n ₁ , n ₂ , and n ₃ . Let the cube be rotated about the axis due north _three , so the corresponding transformation of stress may be 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 n _three centrality remain equal to null, and the normal stress southward ₃ is perpendicular to the airplane spanned past n _one and n ₂ in which the transformation takes place and thus, does not bear upon this transformation. One may therefore utilize the circle of bore AB to determine the normal and shear stresses exerted on the faces of the cube as it is rotated about the n ₃ axis (meet Fig. iv). Similarly, the circles of diameter BC and CA may exist used to determine the stresses on the cube every bit information technology is rotated nigh the n ₁ and northward ₂ axes, respectively.

Fig. 4 Mohr'due south circles for space (3-D) stress What if the rotations are about 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 point located within the area which is bounded by 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 as shown in Fig. iv!

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

s _max = due south ₁

(30)

s _min = s ₃

(31)

t _max = (s _one - s ₃ )/ii = t _max2

(32)

Note that in Fig. 4, t _max1, t _max2, and t _max3are the maximum shear stresses obtained while the rotation is well-nigh n ₁ , n ₂ , and n ₃ , respectively. (The end.)

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 tin can be represented by two-by-2 or three-by-iii symmetric matrices (tensors). (The end.)