If a rod of 1025 steel 0.5 m (19.7 in) long is heated from 20 to 80 deg C (68 to 176 deg F) while its ends are maintained rigid, determine the type and magnitude of stress that develops. Assume that at 20 deg C the rod is stress free.
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If a rod of 1025 steel 0.5 m long is heated from 20 to 80°C (293 to 353 K) while its ends are maintained rigid, determine the type and magnitude of stress that develops.
To determine the type and magnitude of stress that develops in the rod when it is heated, we can use the equation:
ΔL = α * L * ΔT
where:
ΔL is the change in length,
α is the coefficient of linear expansion,
L is the original length of the rod,
and ΔT is the change in temperature.
First, let's find the change in temperature and length of the rod:
Change in temperature:
ΔT = final temperature - initial temperature
= 80 - 20
= 60 °C (or 108 °F)
Change in length:
ΔL = α * L * ΔT
The coefficient of linear expansion for steel is typically around 12 × 10^-6 per degree Celsius (or 6.7 × 10^-6 per degree Fahrenheit). Let's assume α = 12 × 10^-6 per °C for this calculation.
ΔL = (12 × 10^-6) * (0.5) * (60)
= 0.036 m (or 1.417 in)
As the ends of the rod are rigid, the rod will experience a thermal stress due to the restriction of movement. The type of stress that develops is called "thermal stress." The magnitude of the thermal stress can be calculated using the formula:
Stress = E * α * ΔT
where:
E is the modulus of elasticity (Young's modulus) of the material.
The modulus of elasticity for steel is typically around 200 GPa.
Stress = (200 × 10^9 Pa) * (12 × 10^-6 per °C) * (60 °C)
= 144,000,000 Pa (or 144 MPa)
Therefore, a thermal stress of approximately 144 MPa develops in the rod when it is heated from 20 to 80 °C while its ends are maintained rigid.
To determine the type and magnitude of stress that develops in the rod, we can use the formula for thermal stress. Thermal stress occurs when there is a difference in temperature across a material, resulting in expansion or contraction.
The formula for obtaining thermal stress is:
Stress = (modulus of elasticity) * (coefficient of thermal expansion) * (change in temperature)
Let's break down the problem and gather the necessary information:
Given:
- The rod is made of steel.
- Length of the rod = 0.5 m (19.7 in)
- Initial temperature = 20°C
- Final temperature = 80°C
To calculate the stress, we need to find the modulus of elasticity and the coefficient of thermal expansion for the steel.
1. Modulus of Elasticity (E):
The modulus of elasticity is a property of the material. For steel, the modulus of elasticity is typically around 200 GPa (Gigapascals) or 200,000 MPa (Megapascals).
2. Coefficient of Thermal Expansion (α):
The coefficient of thermal expansion is also a material property. For steel, the average coefficient of thermal expansion is approximately 11.7 x 10^(-6) per °C or 11.7 x 10^(-6) per °F.
Now, we have all the required information to calculate stress:
Change in temperature, ΔT = Final temperature - Initial temperature
= 80°C - 20°C
= 60°C
Coefficient of thermal expansion (α) = 11.7 x 10^(-6) per °C
Stress = (Modulus of Elasticity) * (Coefficient of Thermal Expansion) * (Change in Temperature)
= 200,000,000 Pa * 11.7 x 10^(-6) / °C * 60°C
= 140,400 Pa (or 140.4 kPa)
Therefore, the stress that develops in the steel rod is approximately 140,400 Pa (or 140.4 kilopascals). The stress is compressive, as the rod contracts when heated.