A composite structure as shown in figure is overhanged from roof of a chamber.The chamber temperature has been increased and kept to 90℃ from ambient temperature 20℃.Find out:(1)net elongation in structure (2)stresses induced in constituting material.

To analyze the composite structure that is overhanging from the roof of a chamber, we need to consider the effects of temperature change on the materials involved. When the temperature increases, materials expand, and this expansion can lead to elongation and stress within the structure. Let's break down the problem step by step to find the net elongation and the stresses induced in the materials.

Understanding Thermal Expansion

Thermal expansion is the tendency of materials to change their dimensions in response to temperature changes. The amount of expansion can be calculated using the formula:

ΔL = α × L0 × ΔT

• ΔL = change in length (elongation)
• α = coefficient of linear expansion (specific to each material)
• L0 = original length of the material
• ΔT = change in temperature (final temperature - initial temperature)

Step 1: Calculate Net Elongation

Given the temperature change from 20℃ to 90℃, we can find ΔT:

ΔT = 90℃ - 20℃ = 70℃

Now, we need the original lengths and coefficients of linear expansion for each material in the composite structure. Let's assume we have two materials, Material A and Material B, with the following properties:

• Material A: αA = 12 × 10^-6 /℃, L0A = 2 m
• Material B: αB = 15 × 10^-6 /℃, L0B = 3 m

Now we can calculate the elongation for each material:

• For Material A: ΔLA = αA × L0A × ΔT = 12 × 10^-6 × 2 × 70 = 1.68 × 10^-3 m
• For Material B: ΔLB = αB × L0B × ΔT = 15 × 10^-6 × 3 × 70 = 3.15 × 10^-3 m

The net elongation of the composite structure will depend on how these materials interact. If they are bonded together, the total elongation can be approximated as the sum of the individual elongations:

ΔLnet = ΔLA + ΔLB = 1.68 × 10^-3 + 3.15 × 10^-3 = 4.83 × 10^-3 m

Step 2: Determine Induced Stresses

Next, we need to find the stresses induced in each material due to the temperature change. The stress can be calculated using the formula:

σ = E × α × ΔT

• σ = stress (Pa)
• E = modulus of elasticity (specific to each material)

Assuming the following modulus of elasticity values:

• Material A: EA = 200 GPa
• Material B: EB = 150 GPa

Now we can calculate the stresses:

• For Material A: σA = EA × αA × ΔT = 200 × 10⁹ × 12 × 10^-6 × 70 = 168 × 10⁶ Pa
• For Material B: σB = EB × αB × ΔT = 150 × 10⁹ × 15 × 10^-6 × 70 = 157.5 × 10⁶ Pa

Summary of Results

In summary, we have calculated:

• Net Elongation: 4.83 × 10^-3 m
• Stresses Induced:
• Material A: 168 MPa
• Material B: 157.5 MPa

This analysis shows how temperature changes can significantly affect composite structures, leading to elongation and stress in the materials involved. Understanding these principles is crucial for designing structures that can withstand thermal variations without failing.