A 6m long I beam of the cross section shown below is simply supported at each en

Question

A 6m long I beam of the cross section shown below is simply supported at each end and carries two loads of 3 kN each at its 1/3 points. What are the maximum tensile and compressive stresses of the beam if it is bent about (a) the x axis, (b) the y axis? What are the maximum tensile and compressive strains for each case (a) and (b), if the beam is made of steel? (Es=200GPa) What maximum loads for each case (a) and (b), could the beam take if "? * allowable stress is 150 MPa in tension or compression?

Explanation / Answer

The reaction at the supports are 1 kN and 2 kN, and the maximum bending moment occurs at the point where the force is acting, i.e., M = 2*1 = 4 kN.m = 4000 N.m

Ix = (0.1)(0.123)/12 - (0.09)(0.13)/12 = 6.9 x 10-6 m4

Iy = 2*(0.01)(0.13)/12 + (0.1)(0.013)/12 = 1.675 x 10-6 m4

(A) Due to symmetry, the maximum tensile and compressive stresses are the same

(a) max_xx = Mc/Ixx = 4000(0.06)/[6.9 x 10-6] = 34.78 x 106 Pa =34.78 MPa

(b) max_yy = Mc/Iyy = 4000(0.05)/[1.675x 10-6] = 119.40 x 106 Pa =119.40 MPa

(B) Due to symmetry, the maximum tensile and compressive strains are the same

(a) max_xx = max_xx /E = 34.78 x 106 Pa/(200 x 109 Pa) = 1.739x10-4 = 173.9

(b) max_yy = max_yy /E = 119.40 x 106 Pa/(200 x 109 Pa) = 5.97x10-4 = 5.97

(C) Due to symmetry, the maximum tensile and compressive stresses are the same

(a) Pmax = (allow/max_xx ) (3kN) = (150/34.78 ) (3kN) = 12.94 kN

(b) Pmax = (allow/max_yy ) (3kN) = (150/119.4 ) (3kN) = 3.77 kN

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