Answer:
See explanation
Explanation:
See the document for the complete FBD and the introductory part of the solution.
Static Balance ( Sum of Forces = 0 ) in all three directions
∑
∑
∑
Where, ( ) are internal forces at section ( G ) along the defined coordinate axes.
Static Balance ( Sum of Moments about G = 0 ) in all three directions
Where,
r_OG: The vector from point O to point G
F_OG: The force vector at point O
- The vector ( r_OG ) and ( F_OG ) can be written as follows:
- Then perform the cross product of the two vectors ( r_OG ) and ( F_OG ):
- The internal torque ( T ) and shear force ( V ) that act on slice ( G ) are due to pressure force ( P ) as follows:
- For the state of stress at point "C" we need to determine the the normal stress along x direction ( σ_x ) and planar stress ( τ_xy ) as follows:-
σ_x =
Where,
A: The area of pipe cross section
z*: The distance of point "C" along z-direction from central axis ( x )
I_YY: The second area moment of pipe along and about "y" axis:
y*: The distance of point "C" along y-direction from central axis ( x )
- The normal stress ( σ_x ) becomes:
σ_x =
- The planar stress is ( τ_xy ) is a contribution of torsion ( T ) and shear force ( V ):
τ_xy =
Where,
c: The radial distance from central axis ( x ) and point "C".
J: The polar moment of inertia of the annular cross section of pipe:
Q: The first moment of area for point "C" = semi-circle
I: The second area moment of pipe along and about "y" axis:
t: The effective thickness of thin walled pipe:
- The planar stress is ( τ_xy ) becomes:
τ_xy =
- The principal stresses at point "C" can be determined from the following formula:-
σ_x = 15.55 ksi, σ_y = 0 ksi , τ_xy = 52.4 ksi
σ_1 =
σ_2 =
σ_1 =
σ_2 =
- The angle of maximum plane stress ( θ ):
θ =
Note: The plane stresses at point D are evaluated using the following procedure given above. Due to 5,000 character limit at Brainly, i'm unable to post here.