Question

The beam is subjected to a moment of 15 kip*ft. Determine the resultant force the bending stress produces on the top flange A and the bottom flange B. Also compute the maximum bending stress developed in the beam.

Answer 1

Concepts and reason

Centroid is the mean position of all points in a geometric figure. The summation of the displacements of every point in the geometric figure from the centroid is zero. Centroid of an area is nothing but the centroid of mass. It is the point whose coordinates are the mean of all points in the geometric area.

The bending stress distribution in the cross-section of a beam subjected to bending moment varies linearly from zero at neutral axis to a maximum at a distance farthest from the neutral axis when the beam material is assumed to behave in a linear elastic manner.

Fundamentals

The y-coordinate of the centroid of composite area can be determined using the formula,

Moment of inertia of a rectangular area about an axis passing through the centroid can be determined using the parallel axis theorem,

Here, width of the rectangle is , height of the rectangle is , and y-coordinate of the centroid of the rectangle is .

Normal bending stress at a fiber which is at a distance of from the neutral axis can be calculated using the formula,

Draw the diagram of the cross-section as shown in the figure,

Find the y-coordinate of the neutral axis using the formula,

Find the moment of inertia of the lower flange about the neutral axis using the formula,

Find the moment of inertia of the web about the neutral axis using the formula,

Find the moment of inertia of the upper flange about the neutral axis using the formula,

Find the moment of inertia of the entire cross-section about the neutral axis using the relation,

Find the bending stress in the bottom layer of the cross-section using the formula,

Find the bending stress in the top layer of the cross-section using the formula,

Since, . The maximum bending stress in the beam is .

Find the bending stress in the bottom fiber of the web section using the formula,

Find the bending stress in the top layer of the web section using the formula,

Draw the profile view of the bending stress distribution as shown in the figure,

Find the resultant force in the top flange A using the relation,

Find the resultant force in the bottom flange B using the relation,

Ans:

Therefore, the maximum bending stress developed in the beam is .

Therefore, the resultant force the bending stress produces on top flange A is .

Therefore, the resultant force the bending stress produces on bottom flange B is .