Complex Static Stresses and Torsion

UAA School of Engineering

CE 334 - Properties of Materials

Lecture # 17

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Quiz 1: The State of Stress

ABCDE

Stresses at point A:

Stresses at point B:

Stresses at point C:

Stresses at point D:

Stresses at point E:

P

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Quiz 2: Stresses Acting on Other Directions

600 psi

250 psi

600 psi

250 psi

=?

=?

Welding Plan =37o

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Complex Static Stresses

• Complex Stress States:

and in three directions

• Typical complex stress states: Biaxial stresses are stresses applied

along two axes of the element and = 0.Triaxial stresses are stresses applied along

three axes of the element and = 0.

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Biaxial StressesConsider a small plane element with plane stresses. Cut an angle through the element and derive the equations for the stress normal to the new plane and the shear stress.

= 0

0

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Stress Normal to Plane

2cos22

sincos

sin)sin(cos)cos(

)sin()cos()(0

22

111

yxyxn

yxn

yxn

nynxnn dAdAdAF

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Shear Stress on Plane

F dA dA dAn n x n y n

n x y

n x y

n

x y

2 2 20

22

( ) ( cos ) ( sin )

( sin ) cos ( cos ) sin

( ) sin cos

sin

Note that the maximum shear stress occurs when the angle equals 45 degrees.

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Mohr’s Circle for Biaxial Stress

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Triaxial Stress• Normal stresses are in three dimensions.• Shear stress in each of these three dimensions is zero. • The following element is in triaxial stress state.

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Mohr’s Circle for Triaxial Stress

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Triaxial Strain• Stresses usually act in all three directions • It is necessary to modify Hooke’s Law regarding stress and strain

in the elastic range in order to include Poisson’s effects in each direction, caused by the strains in the other two directions.

• The generalized equations for Hooke’s Law are given below:

1 1 2 3

2 2 3 1

3 3 1 2

1

1

1

E

E

E 11

Purposes of Torsional Tests

The purposes of torsion tests usually parallel those for tension tests:

– Determine the proportional limit and the ultimate strength in shear

– Determine the modulus of rigidity (G)– Plot stress-strain curve for material

behavior in shear

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Torsion Procedure and Set up

• Torsion specimens: round bars

• Torsion apparatus: similar to tension

• The ends of a specimen should be securely gripped

• Apply torque

• Measure the torque and the angle of twist

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Torsional Specimens The Torsion test is a test which places the

specimen in pure shear.

R

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Torsional Quantities

Shear Strain:

Shear Stress:

L

r

J

Tr

R

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Elastic Shear Strength

Porportional Limit:

Yield Stress:

Modulus of Rigidity:

J

rTpl

pl

J

rTy

y

12 :also

EG

J

TL

r

L

J

TrG

(unit = radian/length)16

Torsional Modulus of Resilience

GR

r

rR

Lrr

L

r

JR

TR

R

pl

pl

plpl

pl

plpl

4

42

2

1

volume2

Treach tovolumework/unit

2

2

4

4

21

2

pl

(unit = radian/length)

)( radiusr

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Torsional Plastic Strength

• Yielding begins at outer surface.

• Stress variation is no longer linear.

• Yielding continues until the cross section is fully yielded.

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Torsional Plastic Shear Strength

Modulus of Rupture:

Ductility:

Modulus of Toughness:

J

RTuu

% L

100

T T

ALu pl f

2

u

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Torsional Fracture

• Ductile materials fracture in shear.

• Brittle materials fracture in tension.20

Review Mechanics

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