A piece of material 15 cm long by 2.5 cm square is in compression under a load 100kN. If the modulus of elasticity of the material is 105 GPa and Possion's ratio is 0.25, find the alternation in length if all lateral strain is prevented by the application of uniform lateral external pressure of suitable intensity
A Piece of material 15 cm long by 2.5 cm square is in compression under a load 100 KN. If the modulus of elasticity of the material s 105 GPa and Poisson’s ratio is 0.25, find the alteration in length if all lateral strain is prevented by the application of uniform lateral external pressure of suitable intensity.
To find the alteration in length of the material under the given conditions, we need to use the concept of axial stress and strain along with the formula for lateral strain.
First, let's calculate the axial stress on the material:
Stress (σ) = Force / Area
Given that the load on the material is 100 kN and the cross-sectional area is (2.5 cm)^2, we can convert the load to Newtons:
Load = 100 kN = 100,000 N
Area = (2.5 cm)^2 = 6.25 cm^2 = 6.25 x 10^-4 m^2
Now, we can calculate the stress:
Stress (σ) = Load / Area = 100,000 N / (6.25 x 10^-4 m^2) = 1.6 x 10^8 N/m^2
Next, we need to calculate the longitudinal strain using Hooke's law:
Longitudinal strain (ε) = Stress / Modulus of Elasticity
Given that the modulus of elasticity is 105 GPa = 105 x 10^9 N/m^2, we need to convert it to N/m^2:
Modulus of Elasticity = 105 x 10^9 N/m^2
Now, we can calculate the longitudinal strain:
Longitudinal strain (ε) = Stress / Modulus of Elasticity = (1.6 x 10^8 N/m^2) / (105 x 10^9 N/m^2) = 1.52 x 10^-3
Finally, we can calculate the alteration in length using Poisson's ratio and the formula for lateral strain:
Lateral strain = -Poisson's ratio * Longitudinal strain
Given that Poisson's ratio is 0.25, we can calculate the alteration in length:
Lateral strain = -0.25 * 1.52 x 10^-3 = -3.8 x 10^-4
The negative sign indicates that the length of the material has decreased. To find the alteration in length, we multiply the lateral strain by the original length:
Alteration in length = Lateral strain * Original length
Given that the original length of the material is 15 cm = 0.15 m, we can calculate the alteration in length:
Alteration in length = -3.8 x 10^-4 * 0.15 m = -5.7 x 10^-5 m
So, the alteration in length is -5.7 x 10^-5 meters (or approximately -0.057 mm).