steel channel section has a depth of 200 mm, flange width of 75 mm, and thickness of 10 mm. If the section is subjected to a bending moment of 50 kN-m about the x-axis and 20 kN-m about the y-axis, calculate the maximum normal stress and the location of the neutral axis.
To calculate the maximum normal stress and the location of the neutral axis, we first need to determine the section properties of the steel channel.
Given:
Depth (d) = 200 mm
Flange width (b) = 75 mm
Thickness (t) = 10 mm
Bending moment about x-axis (Mx) = 50 kN-m
Bending moment about y-axis (My) = 20 kN-m
The centroid of the section can be calculated as follows:
Yc = (d - t) / 2 = (200 - 10) / 2 = 95 mm
The moment of inertia about the x-axis (Ix) can be calculated as follows:
Ix = (1/12) * b * d^3 + t * d * (Yc - t/2)^2 + b * t^3 = (1/12) * 75 * 200^3 + 10 * 200 * (95 - 5)^2 + 75 * 10^3
Ix ≈ 8875000 mm^4
The moment of inertia about the y-axis (Iy) can be calculated as follows:
Iy = (1/12) * d * b^3 + b * t * (b/2 - t/2)^2 + t * d^3 = (1/12) * 200 * 75^3 + 75 * 10 * (75/2 - 5/2)^2 + 10 * 200^3
Iy ≈ 5343750 mm^4
The maximum normal stress can be calculated using the bending stress formula:
σ = M * y / I
For bending about the x-axis:
σx = (Mx * (d/2 - Yc)) / Ix
σx = (50 * (200/2 - 95)) / 8875000
σx = 0.02831 N/mm^2
For bending about the y-axis:
σy = (My * (b/2)) / Iy
σy = (20 * (75/2)) / 5343750
σy = 0.01410 N/mm^2
The maximum normal stress is the higher of the two:
Maximum normal stress = σx = 0.02831 N/mm^2
The neutral axis is located along the centroid of the section, so the location of the neutral axis is at a distance of 95 mm from the bottom of the steel channel.