Two vertical forces are applied to a beam of the cross section shown. Determine the maximum tensile and compressive stresses in portion BC of the beam.
Fig. P4.192
Find the maximum tensile and compressive stress in the portion BC of the beam.
Answer to Problem 192RP
The maximum tensile and compressive stress in the portion BC of the beam are
Explanation of Solution
Given information:
The load P acting on the beam is
Consider the radius of the semi-circular region is
Calculation:
Show the cross-section of the beam as shown in Figure 1.
Refer Figure 1.
The cross-section of the beam consist of a semi-circle 1 and a rectangle 2.
Calculate the area of the semi-circle 1 and a rectangle 2 as follows:
Consider the distance of the centroid of the region 1 and 2 from their bases are denoted by
Calculate the value of the distances
Substitute
Calculate the value of the distances
Substitute
Calculate the distance
Substitute
Calculate the total moment of inertia of the cross-section (I) using the rerlation:
Substitute
Show the forces acting on the beam as shown in Figure 2.
Calculate the value of moment M as follows:
Calculate the stress at the top fiber as follows:
Here,
Calculate the value of
Substitute
Calculate the stress at the top fiber as follows:
Here,
Calculate the value of
Substitute
Thus, the maximum tensile and compressive stress in the portion BC of the beam are
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