A beam with an unsymmetrical cross-section has a width of 120 mm, a depth of 200 mm, and a thickness of 10 mm on one side and 20 mm on the other side. The centroid of the cross-section is located at a distance of 50 mm from the larger side. The beam is subjected to a bending moment of 50 kN-m about the y-axis. Determine
the maximum tensile and compressive stresses developed in the beam
To determine the maximum tensile and compressive stresses developed in the beam, we need to calculate the moment of inertia of the cross-section with respect to the y-axis, and then use the bending stress formula:
σ = M*y/I
where:
σ = bending stress
M = bending moment (50 kN-m)
y = distance from the centroid to the furthest edge of the beam (100 mm)
I = moment of inertia
The moment of inertia of an unsymmetrical cross-section can be calculated by summing the moments of inertia of the individual rectangles that make up the cross-section. The moment of inertia of a rectangle about its centroidal axis is given by:
I_rect = (1/12)*b*h^3
For the larger rectangle (20 mm x 200 mm):
I_larger = (1/12)*20*200^3 = 10666667 mm^4
For the smaller rectangle (10 mm x 200 mm):
I_smaller = (1/12)*10*200^3 = 2666667 mm^4
The total moment of inertia of the cross-section about the y-axis is:
I_total = I_larger + (200-50)^2*20 + I_smaller = 10666667 + 150^2*20 + 2666667 = 12116667 mm^4
Now, we can calculate the maximum tensile and compressive stresses:
Maximum tensile stress:
σ_max = M*y_max/I_total = 50*100/12116667 = 0.041 kN/mm^2 or 41 MPa
Maximum compressive stress:
σ_min = -M*y_min/I_total = -50*50/12116667 = -0.021 kN/mm^2 or -21 MPa
Therefore, the maximum tensile stress developed in the beam is 41 MPa, and the maximum compressive stress developed in the beam is 21 MPa.