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A timbre beam 100mm wide and 200mm deep is to be reinforced by bolting on two steel flitches each 150mm by 12.5mm in section. Calculate the moment of resistance in the following cases: (a) flitches attached symmetrically at the top and bottom (b) flitches attached symmetrically at the sides. Allowable stress in timbre is 6N/mm2. What is the maximum stress in the steel in each case? Take Es=200000N/mm2 and Et=10000N/mm2.

User Mthurlin
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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

User Intuited
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