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 when all lateral strain is prevented, we can use the formula for the longitudinal strain caused by compressive stress.
The formula for longitudinal strain caused by compressive stress is given by:
ε = σ / E
Where:
ε = Longitudinal strain
σ = Compressive stress
E = Modulus of elasticity
In this case, the material is in compression under a load of 100 kN, which can be converted to Newtons (1 kN = 1000 N). So, the compressive stress (σ) is given by:
σ = F / A
Where:
F = Load or force applied (in Newtons)
A = Cross-sectional area of the material (in square meters)
The cross-sectional area of the material is given by the length of one side of the square. So, the area (A) is:
A = side^2
In this case, the length of one side of the square is 2.5 cm, which can be converted to meters (1 cm = 0.01 m). So, the length of one side of the square (side) is:
side = 2.5 cm * 0.01 m/cm = 0.025 m
Now, we can substitute the values into the formulas. First, let's calculate the compressive stress (σ):
σ = (100 kN * 1000 N/kN) / (0.025 m)^2 = 160,000,000 N/m^2
Next, let's calculate the longitudinal strain (ε):
ε = σ / E = 160,000,000 N/m^2 / 105 GPa = 160,000,000 N/m^2 / (105 * 10^9 N/m^2) = 1.5238 * 10^-3
Finally, to find the alteration in length, we can use Poisson's ratio. Poisson's ratio (ν) is the ratio of lateral strain to longitudinal strain.
ν = - lateral strain / longitudinal strain
Since we are applying uniform lateral external pressure to prevent any lateral strain, the lateral strain is zero.
Therefore, we can rearrange the equation to solve for the alteration in length (ΔL):
ΔL = -(ν * ε) * length
In this case, the length is given as 15 cm, which can be converted to meters (1 cm = 0.01 m). So, the length (length) is:
length = 15 cm * 0.01 m/cm = 0.15 m
Now, let's substitute the values into the formula:
ΔL = -(0.25 * 1.5238 * 10^-3) * 0.15 m = -0.00569 m (rounded to five decimal places)
Therefore, the alteration in length, when all lateral strain is prevented by the application of uniform lateral external pressure, is approximately -0.00569 meters. Note that the negative sign indicates a decrease in length due to compression.