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3D Stress Transformations and 3D Mohr Circle
6.2
(tvid- 6.2.a)
(tvid- 6.2.b)
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3D Stress Transformations and 3D Mohr Circle
6.2.1 Reminder: How to Find Stress Transformation Equations and Mohr Circle in the Plane (2D)?
K
view from x-y plane
Our aim in this section is to calculate the stress components in different planes at the same point and draw the 3D Mohr circle while there is a three-dimensional stress state (6 stress components) at one point.
(It is the geometric expression of the stress components in all diagonal planes..)
(Chapter 6.1)
Mohr's Circle in the x-y Plane
σ
τ
C
D1
2θ
K
D2
For more detailed information, see:
Figure 6.2.1
Figure 6.2.2
Figure 6.2.3
Figure 6.2.4
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S
Now we will make similar calculations for the 3D Stress State with the same logic.
z
y
S
N
B
A
C
plane normal
The stress we want to find.
Our first goal is to find the S resultant stress that occurs on a 4th inclined plane.
Tip-6.2.1:
Known values: Cartesian stress components (σx , σy , σz , τyz , τxz , τyz), and the angles made by the normal of the inclined plane whose stress we want to calculate, with the axes (𝛼, 𝛽, 𝛾). In future calculations, we will always aim to find unknown values in terms of these known values.
Tip -6.2.2:
If the stress components with respect to a Cartesian axis system are known, the stress components with respect to all axis systems at the same point can be calculated.
3D Stress Transformations and 3D Mohr Circle
(Resultant stress)
We separate the Cubic Element from the inclined plane and examine the equilibrium of the left part.
Figure 6.2.5
Figure 6.2.6
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6.2.2- Calculation of Resultant Stress and Cartesian Components in an Inclined Plane (according to x, y, z reference coordinate system):
N
Wanted Values
Known Values
Sy
Sz
Sx
3D Stress Transformations and 3D Mohr Circle
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They are the cosines of the angles between the plane normal (N) and the positive Cartesian axes:
6.2.2.1 Direction Cosines
3D Stress Transformations and 3D Mohr Circle
The area of a triangle is equal to half the magnitude of the vector product of the position vectors of its two sides.
Accordingly, the areas of triangles OBC and ABC are:
(6.2.3a)
(6.2.3b)
(6.2.4)
Unit vector in normal direction:
Relationship Between Direction Cosines :
(6.2.2)
Relationship Between Direction Cosines and Area
Tip 6.2.3: We understand from equations (6.2.1a-c); The intensity of the component in any axis direction of a vector such as T parallel to the normal is equal to the product of "the direction cosine" and "the intensity of the resultant vector."
(6.2.1a)
(6.2.1b)
(6.2.1c)
(6.2.3c)
Figure 6.2.7
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Sy
Sz
Sx
ΣFz=0
ΣFx=0
ΣFy=0
intensity of resultant stress S :
(6.2.5a)
(6.2.5b)
(6.2.5c)
(6.2.6)
S stress vector :
(6.2.7)
From equ. (6.2.3a-c) :
6.2.2.2 Calculation of Cartesian Stress Components Based on x,y,z Reference Axis Set
(6.2.8)
If we write the equations (6.2.5a,b,c) in matrix format :
3D Stress Transformations and 3D Mohr Circle
Tip 6.2.4: To find the intensity of the components of a vector according to any axis set, we multiply that vector by scalar with the unit vectors of that axis set. According to this:
,
,
(6.2.9b)
(6.2.9a)
(6.2.9c)
Notice that if the vector S were in the direction of the plane normal, equations 6.2.9a-c would be the same as equations 6.2.1a-c.
(Similarly, we divide both sides of the bottom two balance equations by ABC)
area
Figure 6.2.8
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6.2.3 Calculation of Shear and Normal Components of Resultant Stress in Inclined Plane :
N
Wanted values
Known values
σN
τ
N
τ, σN :?
3D Stress Transformations and 3D Mohr Circle
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Sy
Sz
Sx
σN
τ
N (l,m,n)
intensity of the component σN :
(6.2.10a)
(6.2.10b)
Vector expression of σN components :
(6.2.11)
(6.2.12a)
intensity of the component τ :
(6.2.12b)
From equ. (6.2.7)
from equations (6.2.5):
or
6.2.3.1 Calculation of Shear and Normal Stress Components
3D Stress Transformations and 3D Mohr Circle
If we apply tip 6.2.4 to t-N axes
t
Figure 6.2.9
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Example -6.2.1
In the inclined plane with coordinates of the end points;
The stress components at point O of the body seen on the side, exposed to external loads and in equilibrium, are calculated as follows.
A
B
C
O
A (5,0,0) ,
B (0,10,0) ,
C (0,0,15)
a-) Calculate the resultant stress and Cartesian components, b-) Calculate the shear and normal stress components.
Solution:
3D Stress Transformations and 3D Mohr Circle
First, we will calculate the unit vector in the direction of the normal of the inclined plane.
A
B
C
O
Figure 6.2.10
Figure 6.2.11
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C (0,0,15)
Sy
Sz
Sx
σN
τ
N (l,m,n)
From eq.(6.2.6)
3D Stress Transformations and 3D Mohr Circle
t
From eq.(6.2.7) :
From Equation (6.2.10a), the normal component of stress S is:
From Equation (6.2.12a), the shear component of the resultant stress S is:
B (0,10,0),
A (5,0,0),
(Intensity of Resultant Force)
Figure 6.2.12
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Wanted values
Known Values
6.2.4 Stress Transformation Equations According to a Different Axis Set in Three Dimensions
3D Stress Transformations and 3D Mohr Circle
Our aim now is that while the stress components according to the x,y,z axis set are known, it is to find the stress components according to another axis set such as x',y'z'.
Direction Cosines
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N(l, m, n)
τ
Sy
Sx
Sz
6.2.4.1 Components of Resultant Stress in an Inclined Plane, According to a Different Axis Set
S
Note:
The shape of the inclined plane is not important. Triangular, rectangular and even circular can be examined. What is important is the direction of the normal of the inclined plane, that is, the value of the direction cosines.
(6.2.13)
(6.2.14)
3D Stress Transformations and 3D Mohr Circle
t
We also know that the intensity of the component of a vector in a direction is equal to the scalar product of that vector by the unit vector in that direction. According to this :
We calculated the component of the S resultant stress in the direction of the plane normal (N) from equation 6.2.10b as follows. According to this
We consider the normal N of the inclined plane to coincide with the 𝑦′ axis.
(6.2.10b)
Intensities of shear stress components of S:
The intensity of the normal component of S is:
Figure 6.2.13
Figure 6.2.14
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Knowns
6.2.4.2 Stress Transformation Equations According to a Different Axis Set in Three Dimensions:
Wanted values
We can find the normal stress components from equation 6.2.10b.
(6.2.15)
(6.2.16)
(6.2.17)
(6.2.19)
(6.2.18)
(6.2.20)
(6.2.21)
(6.2.22)
(6.2.23)
(6.2.24)
(6.2.25)
(6.2.26)
(6.2.28)
(6.2.27)
(6.2.29)
(6.2.30)
(6.2.31)
(6.2.32)
3D Stress Transformations and 3D Mohr Circle
From equ. 6.2.5
Shear Components From equ.
6.2.13 and 6.2.14
Figure 6.2.15
Figure 6.2.16
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Example6.2.2
The values of the stress components at a point of a machine element are given in the matrix on the right according to the x, y, z Cartesian axes. According to this;
Calculate the stress components according to the x′ - y′ -z' axis set obtained by rotating the cubic element counterclockwise θ = 37o around the z axis.
| x | y | z |
x' | | | |
y' | | | |
z' | | | |
SOLUTION
3D Stress Transformations and 3D Mohr Circle
The cosines of these angles, that is, their driection cosines, are given in the table below.
For θ=37o, Let's find the angles that Na, Nb and Nc normals make with the x, y, z axes:
Let's find the unit vectors:
Figure 6.2.17
Figure 6.2.18
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3D Stress Transformations and 3D Mohr Circle
Let's apply equation 6.2.6 for each surface and calculate the resultant stresses:
Figure 6.2.19
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Wanted values
Known values
6.2.5 Principal Stresses, Principal Planes and Principal Axes in Three Dimensions (1,2,3)
2
σ1
σ2
σ3
Three perpendicular planes where shear stresses are zero are called principal planes, and the stresses in these planes are called principal stresses. Our aim is to calculate these special stresses and their planes (Direction Cosines) in terms of known values.
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Let's consider the fourth (ABC) plane, whose normal is N.
Sx
Sz
Sy
S
If the intensity of a resultant vector in the N direction is multiplied by the cosine directions, Cartesian components are obtained. (from tip 6.2.3)
We also know from equations (6.2.5a-c) for Cartesian components that:
If we write these last equations in matrix format...>>
(6.2.34)
(6.2.35)
(6.2.36)
For this plane to be a principal plane, the stress vector S should not have a shear stress component. In this case, the S vector is in the direction of the N normal. (Can be in +N or –N direction)
3D Stress Transformations and 3D Mohr Circle
Figure 6.2.20
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If we remember equation (6.2.2):
However, according to equation (6.2.2), all direction cosines
cannot be zero at the same time.
In this case, the determinant of the first matrix to be multiplied must be zero. The problem becomes an eigenvalue problem.
The explicit form of the determinant is::
(6.2.37)
(6.2.38)
3D Stress Transformations and 3D Mohr Circle
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Calculation of Stress Invariants :
The three real roots of this last 6.2.39 equation, which is of the third degree, are the principal stresses (σ1, σ2, σ3).
(6.2.40a)
(6.2.40b)
(6.2.40c)
(6.2.39)
3D Stress Transformations and 3D Mohr Circle
Calculation from Principal Stresses
(6.2.41a)
(6.2.41b)
(6.2.41c)
Calculation from Cartesian Stress Components
If we write the equations of stress invariants more regularly in the last equation on the previous page:
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Since principal stresses and their axes are special values, they are symbolized with the notations 1, 2 and 3:
σ1
σ2
σ3
and they are always used as σ1 > σ2 > σ3.
The 1st, 2nd and 3rd axes, the principal axes, are perpendicular to each other and their directions, that is, their direction cosines, are not known initially. Each of them can be calculated in terms of known ones, that is, cartesian stress components.
Stress transformation relations can now be made according to this axis set. (Remember tip-6.2.1)
We accept that we found the principal stresses (σ1, σ2, σ3 ) from the roots of Equation 6.2.39.
3D Stress Transformations and 3D Mohr Circle
Figure 6.2.21
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6.2.5.1 Determination of Principal Axes (1,2,3)
The principal axis directions are also among the unknowns.
If the direction cosine of each is found, the Principal Axes and Planes are found.
Now we will learn how these are determined:
We can write Equation (6.2.1b), which was previously obtained for any inclined plane, and Equations (6.2.34-6.2.36), which we could write for a principal plane, for an 𝑖 plane as follows:
Equations (6.2.34-6.2.36):
Equation (6.2.2):
Similarly, the same calculation method is valid for 𝑖=2 and 3.
σ1
σ2
σ3
3D Stress Transformations and 3D Mohr Circle
Figure 6.2.22
Figure 6.2.23
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Sy
N
Sx
Sz
τ
S3
S2
S1
(6.2.43)
(6.2.44)
(6.2.2)
τ
(6.2.42d)
(6.2.42b)
(6.2.42a)
(6.2.42c)
3D Stress Transformations and 3D Mohr Circle
From Equation (6.2.10a-b), the normal component of the S stress is:
6.2.6-) 3D Mohr Circle:
Figure 6.2.24
Figure 6.2.25
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From equations (6.2.2), (6.2.43) ve (6.2.44)
(6.2.46a)
(6.2.46c)
If the explicit form of inequality (6.2.46a) is written:
If we rearrange the last equation…>>
Considering that σ1> σ2> σ3
(6.2.46b)
(6.2.45a)
(6.2.45b)
(6.2.45c)
3D Stress Transformations and 3D Mohr Circle
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When similar operations are performed for inequalities (6.2.46b) and (6.2.46c), the following equations are obtained:>>
3D MOHR Circle
(6.2.47)
(6.2.48)
(6.2.49)
The limit case of equation 6.2.47 is that the left term is equal to the right term, and this equation resembles the equation of a circle on the x-axis.
3D Stress Transformations and 3D Mohr Circle
Figure 6.2.26
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6.2.6.1 Finding the Stresses in the Fourth Plane by Drawing to Scale from the Three-Dimensional Mohr Circle
τP =?
b
a
c
P
N(l, m, n)
3D Stress Transformations and 3D Mohr Circle
Subsequent operations and drawings can be made optionally to check the accuracy of the scale drawing. One of the angles used may be g. By examining the figure, you can understand which alternative drawings can be used to obtain point P. A different 3D example should also provide all symmetrical points as in the figure on the side.
Even if the angles β and γ are taken in opposite directions, the same point P is found.
If α is taken in the opposite direction, the point where P is symmetrical with respect to the horizontal axis is found. The value of 𝜎𝑃 comes out the same. Only the sign of the τP value changes, which is not important in terms of strength.
d
d
e
e
Calculation Steps
Figure 6.2.27
Figure 6.2.28
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Example 6.2.3
The stress values at a point are given as shown in side:
According to this;
Solution:
First, let's find the Stress Inveriants from equations 6.2.40:
From equ.6.2.39:
The roots of this equation give us the principal stresses..>>
3D Stress Transformations and 3D Mohr Circle
Roots are found as:
Figure 6.2.29
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b-) To find the maximum principal stress (𝜎1) direction, we write Equation (6.2.2) and Equation (6.2.34-36) for i =1.
Equations (6.2.34-36):
Equ. (6.2.2):
(Two of these equations are independent.)
(It is an independent equation..)
There are a total of 3 independent equations and 3 unknowns. By using 2 of equations (2), (3), (4) together with equation (1):
Direction cosines of principal stress σ1 (l1, m1, n1) are obtained as follows:
(1)
(2)
(3)
(4)
For i =1
Note: By performing similar operations for i=2 and i=3, the direction cosines (l2, m2, n2) and (l3, m3, n3) of the principal stresses σ2 and σ3 are found.
3D Stress Transformations and 3D Mohr Circle
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c-) Let's draw the 3D Mohr Circle.
3D Stress Transformations and 3D Mohr Circle
Figure 6.2.30
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d-)
τP =?
P
N(l, m, n)
Direction Cosines
Finding the normal and shear stress components on the inclined plane P
From formulas:
With scale drawings from the Circle of Mohr:
3D Stress Transformations and 3D Mohr Circle
Components of the resultant stress S in the inclined plane P in the principal directions. From equations 6.2.42:
From equation 6.2.43, the component of the resultant stress S in the normal direction is:
Shear Component :
Figure 6.2.31
Figure 6.2.32
(6.2.2)
(Principal stresses calculated in option a)