stress transformtion problems

7/30/2019 Stress Transformtion Problems

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STRESS TRANSFORMATION PROBLEMS:-

For the following data, using the stress transformation equations, a) find the

principal stresses and show their sense on properly oriented elements; b)find the

maximum shear stresses with the associated normal stresses and show the results

on properly oriented elements; and c) check the invariance of the normal stresses

for solutions in (a) and (b).

Problem no 11-17:

Given: x=-30 ksi, y=+10 ksi and =-20 ksi

Solution:

Principal stresses:


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(tension)

Hence


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For maximum shear stress to occur

Problem no 11-18:

Given: x=0, y=+20 ksi and =+10 ksi


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Sol: Principal stresses:

( tension)


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Hence


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Problem no 11-19:

Given: x=-40MPa, y=+10MPa and =+20 MPa

Solution: Principal stresses:


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(tension)


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Hence

Problem no 11-20: MPa


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Given: x=20 MPa, y=-10MPa and =-20 MPa

Solution:

Principal stresses:

(tension)


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Hence



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Problem no 11-21: MPagiven: x=0 MPa, y=-40MPa and =-30 Mpa

Solution:

Principal stresses:


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(tension)

Hence


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For the following data, using Mohrs circles of stress and trigonometry,

Problem no 11-26

(a)Find the principal stresses and show their sense on properly orientedisolated elements

(b)Find the maximum shear stresses with the associated normal stressesand show the results on properly oriented elements in each case, check

the invariance of the normal stresses

Given: the principal and shear stress

Solution:

To determine the principal stresses trigonometrically is as follows

Therefore

Principal planes

We get Other plane is an angle 90 to this plane For shear stress


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Shear angle is

( )

The other angle is From the Mohrs circle method i.e. through Mohrs graphical method we get

Principal stresses


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And principal planes are and Maximum and minimum shear stresses are

Shear angles are


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Problem no 11-27





Solution:


Therefore

Principal planes We get Other plane is an angle 90 to this plane

For shear stress


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Shear angle is

( )

The other angle is From Mohrs circle we have


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From the Mohrscircle method i.e. through Mohrs graphical method we get

Principal stresses

And principal planes are

and

Maximum and minimum shear stresses are


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Shear angles are

Problem no 11-28





Solution:


Therefore

Principal planes



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Shear angle is

( )


Principal stresses


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Shear angles are


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(Actual values on Mohrs circle are twice the angles obtained numerically thus the

graphical value obtained is divide by 2 to get the appropriate value)

Problem no 11-29

(c)Find the principal stresses and show their sense on properly orientedisolated elements

(d)Find the maximum shear stresses with the associated normal stressesand show the results on properly oriented elements in each case, check



Solution:


Therefore

Principal planes We get

Other plane is an angle 90 to this plane For shear stress


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Shear angle is

( )


Principal stresses

Here shear stress is taken along y axis and normal stress along x axis


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and


Shear angles are

Problem no 11-30



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Given: the principal and shear stressSolution:


Therefore

Principal planes



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Shear angle is

( )

The other angle is From the Mohrs circle method i.e. through Mohrs graphical method we get Principal stresses


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Shear angles are


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Problem no 11-31





Solution:


Therefore

Principal planes We get Other plane is an angle 90 to this plane For shear stress


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Shear angle is

(

) The other angle is From the Mohrs circle method i.e. through Mohrs graphical method we get

Principal stresses


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Shear angles are

Problem no 11-32





Solution:


Therefore Principal planes


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Shear angle is ( )


Principal stresses


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and


Shear angles are

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