A cantilever beam with square cross-section of 6 mm side is subjected to a load of 2 kN normal to the top surface as shown in the figure. The Young’s modulus of elasticity of the material of the beam is 210 GPa,fl magnitude of slope (in radian) at Q (20 mm from the fixed end) is_____ :
Question
A cantilever beam with square cross-section of 6 mm side is subjected to a load of 2 kN normal to the top surface as shown in the figure. The Young’s modulus of elasticity of the material of the beam is 210 GPa,fl magnitude of slope (in radian) at Q (20 mm from the fixed end) is_____ :
Solution
The problem you're asking about is a classic problem in mechanics of materials, specifically beam bending. Here's how you can solve it:
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First, we need to calculate the moment of inertia (I) for the square cross-section of the beam. The formula for the moment of inertia for a square cross-section is I = (b*h^3)/12, where b is the base and h is the height. In this case, both b and h are 6 mm, so I = (6 mm * (6 mm)^3) / 12 = 648 mm^4.
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Next, we need to calculate the bending moment (M) at point Q. The bending moment for a cantilever beam subjected to a point load is M = P * L, where P is the load and L is the distance from the point of interest to the load. In this case, P = 2 kN and L = 20 mm, so M = 2 kN * 20 mm = 40 N*m.
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Now we can calculate the slope at point Q using the formula θ = ML / (2EI), where E is the Young's modulus of elasticity. In this case, E = 210 GPa = 210 * 10^9 Pa, so θ = (40 Nm * 20 mm) / (2 * 210 * 10^9 Pa * 648 mm^4) = 0.00000116 radians.
So, the magnitude of the slope at point Q is approximately 0.00000116 radians.
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