A solid, round simply supported steel shaft has a diameter of 25mm and span length of 500mm. The shaft supports a concentrated load of 3kN at midspan. Calculate the maximum deflection of the shaft.
diameter, d = 25mm = 0.025m
I=%pi;d^4/64
Modulus of elasticity of steel, E = 200 GPa
Span, L = 500 mm = 0.5m
Point load, P = 3kN
Maximum deflection occurs at the centre of the span (by symmetry) and is equal to δ = PL³/(48EI)
To calculate the maximum deflection of the steel shaft, we'll be using the formula for deflection in a simply supported beam due to a concentrated load at midspan:
δ = (F * L^3) / (48 * E * I)
Where:
δ = deflection
F = load applied (3kN)
L = span length (500mm)
E = Young's modulus of elasticity of steel
I = moment of inertia of the shaft
First, let's convert the units to be consistent. The load applied, 3kN, can be converted to newtons by multiplying by 1000 (1kN = 1000N). Therefore, F = 3000N.
The length L is already given in millimeters, which can be kept as is: L = 500mm.
Next, we need to determine the Young's modulus of elasticity of steel and the moment of inertia of the shaft.
The Young's modulus of elasticity for steel is typically around 200 GPa (Gigapascals), which is equivalent to 200 x 10^9 N/m^2.
The moment of inertia, I, can be calculated using the formula for a solid circular shaft:
I = (π * D^4) / 64
Where:
D = diameter of the shaft (25mm)
Converting the diameter D to meters: D = 0.025m.
Now we can substitute these values into the deflection formula:
δ = (3000 * 0.5^3) / (48 * 200 x 10^9 * (π * 0.025^4) / 64)
Simplifying the equation:
δ = (3000 * 0.125) / (48 * 200 x 10^9 * (π * 0.0000390625) / 64)
δ = 0.125 / (48 * 200 x 10^9 * 0.00002454375 / 64)
δ = 0.125 / (48 * 0.001226875)
δ = 0.125 / 0.058875
δ ≈ 2.12 x 10^-3 meters or 2.12 mm
Therefore, the maximum deflection of the steel shaft is approximately 2.12 mm.
To calculate the maximum deflection of the shaft, we can use the formula for deflection of a simply supported beam under a concentrated load:
δ = (F * L^3) / (48 * E * I)
Where:
- δ is the maximum deflection of the beam
- F is the applied load
- L is the span length
- E is the Young's modulus (a material property)
- I is the moment of inertia of the shaft cross-section
To calculate the moment of inertia (I), we can use the formula for the moment of inertia of a solid round shaft:
I = (π * D^4) / 64
Where:
- π is the mathematical constant π (approximately 3.14159)
- D is the diameter of the shaft
Now, let's substitute the given values into the formulas to find the maximum deflection:
D = 25 mm (convert to meters: 25 / 1000 = 0.025 m)
L = 500 mm (convert to meters: 500 / 1000 = 0.5 m)
F = 3 kN (convert to newtons: 3 * 1000 = 3000 N)
E = Young's modulus of steel (typically around 200 GPa or 200,000 MPa, but may vary depending on the type of steel)
First, calculate the moment of inertia (I):
I = (π * D^4) / 64
= (π * (0.025^4)) / 64
≈ 1.54 x 10^-8 m^4
Next, substitute the values into the deflection formula:
δ = (F * L^3) / (48 * E * I)
= (3000 * (0.5^3)) / (48 * E * (1.54 x 10^-8))
≈ 0.0646 m
Therefore, the maximum deflection of the steel shaft under the given conditions is approximately 0.0646 meters.