A 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 shear force of 30 kN, determine the location of the shear centre
The shear centre of a steel channel section can be determined by calculating the shear flow in the two flanges and web of the section. The shear flow in each component is equal to the applied shear force divided by the total area of that component.
Given:
Depth (d) = 200 mm
Flange width (b) = 75 mm
Thickness (t) = 10 mm
Shear force (V) = 30 kN = 30,000 N
Area of web = t * d = 10 mm * 200 mm = 2000 mm^2 = 0.002 m^2
Area of each flange = b * t = 75 mm * 10 mm = 750 mm^2 = 0.00075 m^2
Therefore, the area of the two flanges is 2 * 0.00075 m^2 = 0.0015 m^2.
The shear flow in the web is:
q_web = V / A_web = 30,000 N / 0.002 m^2 = 15,000 N/m^2
The shear flow in each flange is:
q_flange = V / (2 * A_flange) = 30,000 N / (2 * 0.0015 m^2) = 10,000 N/m^2
The shear centre is located at a distance 'd' from the centroid of the section, where d can be calculated using the formula:
d = (q_web * y_web + q_flange * y_flange) / (q_web + q_flange)
where y_web is the distance from the neutral axis to the centroid of the web, and y_flange is the distance from the neutral axis to the centroid of the flanges.
For a symmetrical section like the steel channel, the shear centre is located at the centroid of the section. Therefore, y_web = d/2 = 100 mm and y_flange = (d + t/2) = (200 + 10/2) = 105 mm.
Substitute these values into the formula to find the location of the shear centre:
d = (15,000 N/m^2 * 100 mm + 10,000 N/m^2 * 105 mm) / (15,000 N/m^2 + 10,000 N/m^2)
d = (1,500,000 N/m + 1,050,000 N/m) / 25,000 N/m^2
d = 60 mm
Therefore, the shear centre of the steel channel section is located 60 mm away from the centroid of the section.