Beer Johnston asked 12 months ago
The centric force P is applied to a short post as shown. Knowing that the stresses on plane a–a are σ = −15 ksi and σ = 5 ksi, determine (a) the angle β that plane a-a forms with the horizontal, (b) the maximum compressive stress in the post.
Mazurek Gravity answered 12 months ago
Step No: 1
Draw the free body diagram of the plane a-a.
Step No: 2
Consider the area of the section of the post on the plane DE be A.
Calculate the area of the cross section of the post on the plane DF by using the equation:
Here, 
is the angle that the plane a-a forms with the horizontal in the counter clockwise direction, 
is the area of the section of the post on the plane DE and 
is the area of the section of the post on the plane DF.
Substitute 
for 
.
Step No: 3
(a)
Calculate the normal force acting on the plane DF by using the equation:
Here, 
is the normal force acting on the plane DF and
is the normal stress on the plane.
Substitute 
for 
.
Calculate the shear force acting on the plane DF by using the equation:
Here, 
is the shear force acting on the plane DF and
is the shear stress on the plane.
Substitute 
for 
.
Step No: 4
Apply force equilibrium equation along the x-axis.
Substitute 
for 
and 
for 
.
Therefore, the angle 
that the plane a-a forms with the horizontal in the counter clockwise direction is 
.
Step No: 5
Consider 
is the maximum compressive stress in the post.
Apply force equilibrium equation along the x-axis.
Substitute 
for 
, 
for 
, 
for 
and 
for 
.
Therefore, the maximum compressive stress on the post is 