The steel pipe AB has a 102-mm outer diameter and a 6-mm wall thickness. Knowing that arm CD is rigidly attached to the pipe, determine the principal stresses and die maximum shearing stress at point K.
Step No: 1
Use the following relation to calculate the inner diameter of the Steel pipe:
Here, the outer diameter is and thickness of the pipe is t.
Substitute 102 mm for and 6 mm for
.
Use the following relation to calculate the polar moment of inertia of the Steel pipe:
Substitute 102 mm for and
for
.
Step No: 2
Calculate the moment of inertia of the Steel pipe.
Substitute for:
Step No: 3
Calculate the moment about y-direction.
Here, is the force in the x-direction and
is the perpendicular distance.
Substitute 10000 N for and 200 mm for
.
Calculate the moment about z-direction.
Here, is the force in the x-direction and
is the perpendicular distance.
Substitute 10000 N for and 150 mm for
.
Here, the negative sign in the moment about z-direction is due to the moment acts in the clockwise direction.
The force creates the moment about z axis and the force
creates the torsion in y direction. Therefore, the moments
and
cause for the bending and twisting moments respectively and the respective stresses are bending stress and torsional shear stress.
Step No: 4
Calculate the bending stress at K as follows:
Here, distance of the point K from the axis is c.
Substitute for
,
for
and 0.051 m for y.
Here, the negative sign indicates that the stress at the point K is compressive in nature.
Step No: 5
Calculate the shear stress at K as follows:
Substitute for
,
for
and 0.051 m for r.
Step No: 6
Calculate the average stress as follows:
Substitute for
and 0 for
Calculate the maximum shear stress as follows.
Substitute for
,
for
and 0 for
Therefore, the maximum shear stress is .
Step No: 7
Calculate the maximum normal stress at K as follows.
Substitute for
and
for
.
Therefore, the maximum normal stress at K is .
Step No: 8
Calculate the minimum normal stress at K as follows.
Substitute for
and
for
.
Therefore, the minimum normal stress at K is .