Derive the expression for shape factor due to section modulus

 

The shape factor due to section modulus is a measure of the efficiency of a structural member in resisting bending stresses. It is defined as the ratio of the cross-sectional area of a member to its section modulus. The section modulus is a geometric property that describes the resistance of a member to bending stresses and is given by:

 

Z = I / c

 

where Z is the section modulus, I is the moment of inertia of the cross-section about the neutral axis, and c is the distance from the neutral axis to the extreme fiber.

 

The shape factor due to section modulus can be expressed as:

 

k = A / Z

 

where k is the shape factor, A is the cross-sectional area of the member, and Z is the section modulus.

 

This expression can be derived as follows:

 

Consider a beam subjected to bending stresses. Let the cross-section of the beam have an area A and a section modulus Z. Let the distance from the neutral axis to the extreme fiber be c.

 

The maximum bending stress on the beam is given by:

 

σ_max = M * c / Z

 

where M is the bending moment on the beam.

 

The total bending moment on the beam can be expressed as:

 

M = σ_avg * A * d

 

where σ_avg is the average bending stress on the beam, and d is the distance from the neutral axis to the centroid of the cross-section.

 

Substituting the expression for M into the equation for σ_max, we get:

 

σ_max = σ_avg * A * c / Z

 

Dividing both sides by σ_avg, we get:

 

σ_max / σ_avg = A * c / Z

 

Since the ratio σ_max / σ_avg is a constant for a given beam and loading, we can define it as the shape factor k. Thus:

 

k = σ_max / σ_avg = A * c / Z

 

Dividing both sides by A, we get:

 

k / A = c / Z

 

Solving for k, we get:

 

k = A / Z

 

which is the expression for the shape factor due to section modulus.