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Content

- Stress Transformation
- AMini Quiz
- Strain Transformation

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Approximate Duration: 20 minutes

x

Plane Stress Loading~ where all elements of the body are subjected to normal and shear stresses acting along a plane (x-y); none perpendicular to the plane (z-direction)

z = 0; xz = 0; zy = 0

y

xy

x

A

A

y

x

Plane Stress LoadingTherefore, the state of stress at a point can be defined by the three independent stresses:

x; y; and xy

Transformation

Solving equilibrium equations for the wedge…

gives two values (p1 and p2)

Principal Planes & Principal StressesPrincipal Planes

~ are the two planes where the normal stress () is the maximum or minimum

~ there are no shear stresses on principal planes

~ these two planes are mutually perpendicular

~ the orientations of the planes (p) are given by:

Principal Planes & Principal Stresses

Principal Stresses

~ are the normal stresses () acting on the principal planes

gives two values (s1 and s2)

Maximum Shear (max)~ maximum shear stress occurs on two mutually perpendicular planes

~ orientations of the two planes (s) are given by:

max = R

Equation of a circle, with variables being x’ and xy’

Mohr CirclesFrom the stress-transformation equations (slide 7),

Mohr Circles

- A point on the Mohr circle represents the x’ and xy’values on a specific plane.
- is measured counterclockwise from the original x-axis.
- Same sign convention for stresses as before. i.e., on positive planes, pointing positive directions positive, and ….

= 0

x’

= 90

xy’

Mohr CirclesWhen we rotate the plane by 180°, we go a full round (i.e., 360°, on the Mohr circle. Therefore….

From the three Musketeers

Mohr circle represents the state of stress at a point; thus different Mohr circles for different points in the body

Mohr circle is a simple but powerful technique

Get the sign convention right

Quit

Continue

200 kPa

60 kPa

A

40 kPa

A Mohr Circle Problem

The stresses at a point A are shown on right.

Find the following:

- major and minor principal stresses,
- orientations of principal planes,
- maximum shear stress, and
- orientations of maximum shear stress planes.

200 kPa

60 kPa

A

40 kPa

R = 100

60

120

40

(kPa)

60

(kPa)

Positions of x & y Planeson Mohr Circle

tan = 60/80

= 36.87°

200 kPa

60 kPa

A

40 kPa

71.6°

(kPa)

36.9°

major principal plane

18.4°

(kPa)

Orientations of Principal Planes

minor principal plane

200 kPa

26.6°

60 kPa

A

40 kPa

53.1°

(kPa)

36.9°

(kPa)

116.6°

Orientations of Max. Shear Stress Planes

90 kPa

40 kPa

A

30 kPa

The state of stress at a point A is shown.

What would be the maximum shear stress at this point?

Answer 1: 50 kPa

Press RETURN for the answer

Press RETURN to continue

90 kPa

40 kPa

A

30 kPa

At A, what would be the principal stresses?

Answer 2:

10 kPa, 110 kPa

Press RETURN to continue

Press RETURN for the answer

90 kPa

40 kPa

A

30 kPa

At A, will there be any compressive stresses?

Answer 3:

No. The minimum normal stress is 10 kPa (tensile).

Press RETURN to continue

Press RETURN for the answer

90 kPa

0 kPa

B

90 kPa

The state of stress at a point B is shown.

What would be the maximum shear stress at this point?

Answer 4:

0

This is hydrostatic state of stress (same in all directions). No shear stresses.

Press RETURN to continue

Press RETURN for the answer

x

Plane Strain Loading~ where all elements of the body are subjected to normal and shear strains acting along a plane (x-y); none perpendicular to the plane (z-direction)

z = 0; xz = 0; zy = 0

y

x

x

before

after

Plane Strain TransformationSign Convention:

Normal strains (x andy): extension positive

Shear strain ( ): decreasing angle positive

e.g.,

x positive

y negative

positive

Plane Strain Transformation

Same format as the stress transformation equations

Principal Strains

~ maximum (1) and minimum (2) principal strains

~ occur along two mutually perpendicular directions, given by:

Gives two values (p1 and p2)

electrical resistance strain gauge

Strain Gauge~ measures normal strain (), from the change in electrical resistance during deformation

90

measured

45

45°

0

45°

x

Strain Rosettes~ measure normal strain () in three directions; use these to find x, y, and xy

e.g., 45° Strain Rosette

x = 0

y = 90

xy = 2 45– (0+ 90)

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