Question

The wood beam has an allowable shear stress of 7 MPa. Determine the maximum shear force V that can be applied to the cross section. It is a 4 rectangles that make one rectangle with the left and right sides h=200mm b=50mm and the top and bottom are in line with the sides and inside each side and are h=50mm and b=100mm and V is in the center of it

Answer 1

Concepts and reason

First moment of inertia:

It is the summation of all products of area and its centroid distance from the reference axis. It is denoted by Q and its unit is .

Moment of inertia:

The product of area with the square of the distance from the reference axis is known as moment of inertia. It is also known as the second moment of area. It is denoted by and its unit is .

Shear stress:

Shear stress is the resistance offered by the material due to the load which is applied in parallel to the cross sectional area. It is denoted by and its unit is .

Fundamentals

The moment of inertia for a rectangle about centroidal axis is given as follows:

$=1 $

Here, width of the rectangle is and height of the rectangle is .

The equation for first moment of inertia is given as follows:

$Q=ΣΥΑ, $

Here, distance to the centroid of area element from the reference axis of area element is and area of the element is

The relation for shear stress is given as follows:

Here, shear force at cross section is and width of beam at cross section is .

The schematic of the wooden beam representing its geometry is shown in Figure (1).

Here, shear force acting at the section is .

Find the moment of inertia of the wooden beam.

Here, outer length of the wooden beam is B, outer breadth of the wooden beam is D, inner length of the wooden beam is b, and inner breadth of the wooden beam is d.

Substitute for , for , for , and for .

Find the first moment of inertia of the wooden beam.

Here, area of the various sections (1), (2) and (3) above the neutral axis are , and respectively and distance to the centroid of area from the neutral axis at various sections (1), (2) and (3) are , and respectively.

Substitute for , for , for , for , for and for .

Find the maximum shear force using the shear stress relation.

Substitute for , for , for and for .

Ans:

The maximum shear force that can be applied to the cross section is .