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• Rachmad Fadillah

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CIVL4760

Lecture 2: Mohr Circle Stress Analysis Prof. Gang Wang Hong Kong University of Science and Technology March 5, 2019

Stress State in 2D and 3D Matrix notation of the stress tensor

 xx   yx  zx 

Right-handed coordinate system

 xy  xz    yy  yz   zy  zz 

By convention, compression is positive (Geomechanics Sign Convention)

Principal Stresses 2

3

1

Three principal stresses (orthogonal)  xx   yx  zx 

 xy  xz    yy  yz   zy  zz 

0  1 0 0   0 2    0 0  3 

Stress Transformation A B

 y' y' Use Geomechanics Sign Convention

y'

 y'x'

 x' y'

 x'x'

A’ B’

 x' y'  y'x'

 x' y' 

 x'x'

 y' y'

x'

Mohr Circle – Sign Convention A

Positive Stresses

B

+



A

𝜎𝑦𝑥 Sign Convention (only for Mohr circle):

(1) Normal stresses are considered POSITIVE if they are COMPRESSIVE (2) A shear stress is considered POSITIVE if it is COUNTERCLOCKWISE (right hand rule!)

𝜎𝑥𝑥 (−) B

(+) 𝜎𝑦𝑦



Mohr Circle: representing the same stress at a point in different coordinates A’



A

A



B

2



2

B

 y' y'

B’ 𝜎𝑥𝑥 + 𝜎𝑦𝑦 2

y'

 y'x'



 x' y'

 x'x'

A’ B’

 y'x'

 y' y'

 x'x'  x' y' 

 x' y'



x'

Mohr-Circle: find the pole The point on the Mohr circle called the POLE or Origin of Planes has a special property: 𝜎1

𝜎2 𝜎2

𝜎1 The principal stresses

 1,2 

 xx   yy 2

    yy    xy2   xx  2  

2

Any straight line drawn through the POLE will intersect the Mohr circle at a point which represents the state of stress on a plan inclined at the same orientation in space as the line.

Example 1 • Known: stresses on an element ( 1 ,  3 ) • Required: stresses on inclined plane of   35o

Example 1 

A

40

B 30 20

• Plot stress on plane A and B; construct the Mohr circle through A and B.

10

B

0 0 -10

-20

10

20

30

40

50 A

60

n

Example 1

Hint: (1) If you draw a horizontal line through the pole, it will pass A (2) If you draw a vertical line through the pole, it will pass B.



A

Where is the Pooh? 40

B 30 20

10

• Find the “pole”.

B

0 0 -10

-20

10

Pole

20

30

40

50 A

60

n

Example 1  40 30 20

(39,18.6)

10

• Find the stress on any direction through the pole.

  35o

0 0 -10

-20

10

Pole

20

30

40

50

60

n

Example 1  40 30 20

(39,18.6)

10

• Interpret the stresses using sign convention.

  35o

0 0 -10

-20

10

Pole

20

30

40

50

60

n

Example 1  40 30 20

10

• Stress state on any arbitrary plane

0 0 -10

-20

10

Pole

20

30

40

50

60

n

Example 2 • Known: stresses on an element ( 1 ,  3 ) • Required: stresses on inclined plane of   35o

Example 2 

A B

40 30 20

• Plot stress on plane A and B; construct the Mohr circle through A and B.

10

B

0 0 -10

-20

10

20

30

40

50 A

60

n

Example 2 The SAME solution as Example 1



A B

40 30 20

• Blue line pass through point A and pole; • Orange line passes through point B and pole • Where is the pole?

(39,18.6) Pole

10

B

0 0 -10

-20

10

20

30

40

50 A

60

n

Example 3 Given: Stress shown on the element in the left figure

A C

B

Determine: • Stresses on incline plane C (𝛼 = 30°); • Principal stresses and direction • Maximum shear and direction

Example 3  A 4

B

A (6,2)

2

n • Plot stress on plane A and B; construct the Mohr circle through A and B.

-4

B (-4, -2)

-2

0 -2

-4

2

4

6

Example 3  A 4

B

A (6,2)

2

Pole

n -4

• Blue line pass through point A and pole; • Orange line passes through point B and pole • Where is the pole?

B (-4, -2)

-2

0 -2

-4

2

4

6

Example 3  C (1.8,5.3)

A C

4

B Pole (-4, 2)

n -4

• Stresses on inclined plane C;

A (6,2)

2

B (-4, -2)

-2

0 -2

-4

2

4

6

 1  6.4

Example 3

3  4.4

 A

11° 4

B Pole (-4, 2)

3

• Principal stresses and direction  1,2 

h v 2

  v     h   2 

 4.4)

-4

-2

0 -2

2

6  (4)  4  6    (2) 2    2  2   6.4 (or

1  n

B (-4, -2)

2

-4 2

A (6,2)

2

2

4

6

Example 3  shear stress is maximum

4

A (6,2)

2

Pole

n -4

• Maximum shear and direction

B (-4, -2)

-2

0 -2

-4

2

4

6

Example 4 Two planes, A and B, are separated by an unknown angle 𝜃. On plane A, 𝜎𝑎 =10 kPa, 𝜏𝑎 = +2 kPa. Plane A lies 15° from the horizontal, as shown in the figure. The stress on plane B are 𝜎𝑏 =9 kPa, 𝜏𝑏 = −3 kPa.

A

H

Plane B ?

Determine: • Major and minor principal stresses and their orientation • Stresses on horizontal plane H; • Find the angle between planes A and B.

Example 4  A Pole H

𝜃 = 46°

A (10, 2)

n Plane B ?

Determine: • Principal stresses and direction • Stresses on horizontal plane H; • Angle between planes A and B.

B (9, -3) Plane B

Example 4  A

H

n Plane B ?

Determine: • Principal stresses and direction • Stresses on horizontal plane H; • Angle between planes A and B.

Example 5

The stress on an element is known Determine: • The magnitude and direction of principal stresses

X (4, 2)

Where is Pole?

Y (8, -2)

Example 5

The stress on an element is known Determine: • Principal stresses and direction

Example 6 (Triaxial Compression) 1

3

τ

45o   / 2

 3

1

σ