Strength of Materials - Lecture Notes / Mehmet Zor
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23 Agust 2024
Plane Strain Transformations
6.3
(tvid- 6.3)
Strength of Materials - Lecture Notes / Mehmet Zor
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23 Agust 2024
k
m
Final length of vertical edge:
Final length of horizontal edge :
Strain :
Strain:
y
x
Figure 6.3.2.a
Figure 6.3.1
extension
in x direction
Figure 6.3.2.b
extension
in y direction
Figure 6.3.2.c
Shear deformation
Figure 6.3.2.d
(6.3.1)
(6.3.2)
Plane Strain Transformations
Strength of Materials - Lecture Notes / Mehmet Zor
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23 Agust 2024
Figure 6.3.3
Figure 6.3.4
In the ACB triangular plate as in Figure 6.3.4, for the plane elastic strain state, the changes in the side lengths are again the side length x (1+ strain in the side direction). Examine the figure on the side carefully.
Plane Strain Transformations
Strength of Materials - Lecture Notes / Mehmet Zor
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If we apply the cosine theorem of the triangle 𝐴′ 𝐵′ 𝐶′, which is the final deformed shape of the plate:
We will write the clearest expression of the last equation, where
Neglected terms: Strains generally have coefficients such as 10-4, 10-5. Therefore, the strains are both unitless and very small compared to other terms. For this reason, 2nd or 3rd order products (multiplications with each other or with themselves) are neglected. Neglected terms have been crossed out.
(6.3.3)
Figure 6.3.4
Equation 6.3.3 gives the strain of plane AB in the 𝑥′ direction in terms of knowns.
6.3.2 Calculation of strain components in an inclined plane:
Plane Strain Transformations
Strength of Materials - Lecture Notes / Mehmet Zor
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Figure 6.3.5
k
m
(6.3.3)
(6.3.4)
(6.3.6)
If we consider the axis set as 𝑥′−𝑦′ and apply equation 6.3.5 for the plane that makes an angle of 45o with 𝑥′, and therefore 𝜑+ 45o with the 𝑥 axis..>>
(6.3.7)
Plane Strain Transformations
If we substitute equations 6.3.9.c and 6.3.9.d on the next page into 6.3.10, we get the following equation in terms of angle 𝜑..>>
(6.3.8)
(6.3.5)
Figure 6.3.6
Strength of Materials - Lecture Notes / Mehmet Zor
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Using the following trigonometric transformation relations, we will express the equations in terms of 2𝜑.
Equations 6.3.6 and 6.3.7 will be especially useful in the next topic, experimental strain measurements.
(6.3.11)
(6.3.12)
If equations 6.3.11 and 6.3.12 are summed up:
(6.3.13)
From equ. 6.3.6
(6.3.15)
If we rewrite and rearrange equation 6.3.3:
Plane Strain Transformations
(6.3.14)
(6.3.10)
(a)
(b)
(c)
(d)
Strength of Materials - Lecture Notes / Mehmet Zor
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6.3.3 Mohr Circle for Plane Strain State
x
a
y
b
We can compare this last equation to the equation of a circle on the x-axis; (b=0)
and we call this circle the Mohr Circle. At a point there are an infinite number of planes with different angles 𝜑. Mohr's Circle is a geometric expression that shows strains in all planes.
Drawing and reading Mohr's Circle:..>>
We pass the first term on the right of equation 6.3.11 to the left. We write equation 6.3.15 below. We square both sides of the last equations and add these two equations.
(6.3.11)..>>
(6.3.15)..>>
Plane Strain Transformations
Figure 6.3.7
Strength of Materials - Lecture Notes / Mehmet Zor
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6.3.3.1 Drawing Order of Mohr Circle:
4- A circle with center C and passing through points H1 and H2 is drawn.
C
H1(εx , γxy/2)
H2(εy , - γxy /2)
εx
γxy/2
−γxy/2
εy
ε1
ε2= εmin
k
the point where we are
m
1- Point H1 (εx , γxy/2) is marked.
2- Point H2 (εy , - γxy /2 )is marked.
3- The H1-H2 line is drawn. The point C where this line intersects the x axis is determined.
= εmax
k
H1
> 0
m
Plane Strain Transformations
Figure 6.3.8
Figure 6.3.9
Principal Strains:
Maximum Principal Strain plane normal angle:
(6.3.17.a)
(6.3.17.b)
(6.3.18)
(6.3.16)
Radius from right triangleCH1εx
Strength of Materials - Lecture Notes / Mehmet Zor
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23 Agust 2024
Example 6.3.1
O
C
Principal strains and their planes:
30o
Solution a-)
If we follow the steps in topic 6.3.3.1 respectively:
3-) The H1-H2 line is drawn and point C is determined.
4-) A circle with center C and passing through H1 and H2 is drawn.
b-) Let's first find the radius ( R) of Mohr's circle:
Plane Strain Transformations
Figure 6.3.10
Figure 6.3.11
From right triangle CBH1:
Strength of Materials - Lecture Notes / Mehmet Zor
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1st way: From the Circle of Mohr
k
D
c-)
If the Mohr circle is drawn to scale, the coordinates of point k can also be found by measurement. It should not be overlooked that the coordinates are distances from the origin point O, not from the circle center C.
O
C
Our current point
30o
Plane Strain Transformations
Figure 6.3.12
Figure 6.3.13
Strength of Materials - Lecture Notes / Mehmet Zor
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2nd way: From equations
Since it is in the upward direction on the H1 surface, the 𝛾𝑥𝑦 value is taken positive in the equations. As a result, 𝛾𝑥𝑦 should be the same magnitude as the value we found from the Mohr circle, but with the opposite sign.
From equ. 6.3.3 :
(or the same result is found from equation 6.3.11)
From equ. 6.3.15
(or the same result is found from equation 6.3.8.)
d-)
Answers:
30o
The strain components in the m plane are asked. Carefully examine the steps we have made so far and try to find the answers below yourself, using similar steps for the m plane, in both ways.
Plane Strain Transformations
Figure 6.3.14