Answer:
To calculate the moment of resistance in each case, we need to consider the properties and stresses of both the timber beam and the steel flitches.
Given data:
Width of timber beam (b): 100 mm
Depth of timber beam (d): 200 mm
Size of steel flitches (b1 x d1): 150 mm x 12.5 mm
Allowable stress in timber (σt): 6 N/mm²
Modulus of elasticity for steel (Es): 200,000 N/mm²
Modulus of elasticity for timber (Et): 10,000 N/mm²
Let's calculate the moment of resistance in each case:
(a) Flitches attached symmetrically at the top and bottom:
In this case, the steel flitches are attached to the top and bottom faces of the timber beam.
Calculation of moment of resistance:
The moment of resistance is given by the formula:
M = σt * Zt
where M is the moment of resistance and Zt is the section modulus of the timber beam.
The section modulus of a rectangular section is calculated as:
Zt = (b * d^2) / 6
Plugging in the values:
Zt = (100 * 200^2) / 6 = 6,666,667 mm^3
Now, let's calculate the maximum stress in the steel flitches:
Calculation of maximum stress in steel:
The stress in the steel flitches can be calculated using the formula:
σs = (M * Es) / Zs
where σs is the stress in the steel and Zs is the section modulus of the steel flitches.
The section modulus of a rectangular section is calculated as:
Zs = (b1 * d1^2) / 6
Plugging in the values:
Zs = (150 * 12.5^2) / 6 = 520.833 mm^3
Now, let's calculate the maximum stress in the steel:
σs = (M * Es) / Zs
= (σt * Zt * Es) / Zs
Plugging in the values:
σs = (6 * 6,666,667 * 200,000) / 520.833
= 45,454.5 N/mm²
Therefore, the maximum stress in the steel when flitches are attached symmetrically at the top and bottom is 45,454.5 N/mm².
(b) Flitches attached symmetrically at the sides:
In this case, the steel flitches are attached to the sides of the timber beam.
Calculation of moment of resistance:
The section modulus of the timber beam remains the same as in case (a).
Zt = 6,666,667 mm^3
Now, let's calculate the maximum stress in the steel flitches:
Calculation of maximum stress in steel:
The section modulus of the steel flitches changes as they are now attached differently. Since the flitches are attached symmetrically at the sides, their effective section modulus becomes:
Zs = (b1^2 * d1) / 6
Plugging in the values:
Zs = (150^2 * 12.5) / 6 = 46,875 mm^3
Now, let's calculate the maximum stress in the steel:
σs = (M * Es) / Zs
= (σt * Zt * Es) / Zs
Plugging in the values:
σs = (6 * 6,666,667 * 200,000) / 46,875
= 40,000 N/mm²
Therefore, the maximum stress in the steel when flitches are attached symmetrically at the sides is 40,000 N/mm².
To summarize:
(a) Flitches attached symmetrically at the top and bottom:
Moment of resistance: Calculated using σt and Zt.
Maximum stress in steel: 45,454.5 N/mm²
(b) Flitches attached symmetrically at the sides:
Moment of resistance: Calculated using σt and Zt.
Maximum stress in steel: 40,000 N/mm²