Solution:

Given data:

Width of the wooden joist = 150 mm

Depth of the wooden joist = 300 mm

Thickness of steel plates = 10 mm

Depth of steel plates = 300 mm

Modulus of elasticity of steel = 20 times that of wood

To find: Width of the equivalent wooden section

Assumption:

The flitched beam is under the assumption of the modular ratio, i.e., the strain in the steel and wood is the same.

Steps involved:

1. Calculation of the equivalent section

The equivalent section is the one that has the same stiffness as that of the flitched beam. The stiffness of the flitched beam can be calculated using the formula,

Stiffness of the flitched beam = Modulus of elasticity of material × moment of inertia of the section

The moment of inertia of the section can be calculated by adding the moment of inertia of the wooden section and that of the steel plates.

Moment of inertia of the wooden section = (b × d^3)/12

where b is the width of the wooden joist and d is the depth of the wooden joist.

Moment of inertia of one steel plate = (b × d^3)/12

Total moment of inertia of the steel plates = 2 × (b × d^3)/12

Total moment of inertia of the flitched beam = [(b × d^3)/12] + [2 × (b × d^3)/12]

Stiffness of the flitched beam = [(Modulus of elasticity of wood × (b × d^3)/12)] + [(Modulus of elasticity of steel × 2 × (b × d^3)/12)]

The equivalent section can be calculated by equating the stiffness of the flitched beam with that of the wooden section.

Modulus of elasticity of steel/ Modulus of elasticity of wood = 20

So, the formula can be written as:

Stiffness of the equivalent wooden section = [(Modulus of elasticity of wood × b1 × d^3)/12]

Equating the stiffness of the flitched beam with that of the equivalent wooden section, we get:

[(Modulus of elasticity of wood × b1 × d^3)/12] = [(Modulus of elasticity of wood × (b × d^3)/12)] + [(Modulus of elasticity of steel × 2 × (b × d^3)/12)]

Simplifying the above equation, we get:

b1 = b + (2 × t × Modulus of elasticity of steel/Modulus of elasticity of wood)

where t is the thickness of the steel plates.

Putting the given values in the above equation, we get:

b1 = 150 + (2 × 10 × 20)

b1 = 550 mm

Therefore, the width of the equivalent wooden section is 550 mm.

2. Checking the assumption

The assumption of the modular ratio can be checked by calculating the strain in both materials.

Strain in wood = Stress in wood/ Modulus of elasticity of wood

Stress in wood = Stress in steel

Strain in steel = Stress in steel/ Modulus of elasticity of steel

The stress in steel can be calculated using the formula,

Stress in steel = (Modulus of elasticity of wood/ Modulus of elasticity of steel) × Stress in wood

Putting the given values in the above equation,