BS02-10

Section 2-10 Beams with Unsymmetrical Sections

The relations developed in section [2-09] can also be applied to beams having unsymmetrical sections provided the plane of bending coincides with one of the two principal axes of the section. We have found that the stress at a distance y from the neutral axis is


     sigma = - E * y / rho              [a]

Therefore the force on the element of area dA in Figure [2-15] is


     sigma * dA = - E * y / rho * dA

Taking moments of this force about the y axis and integrating across the section gives


     M_y = INTEGRAL(sigma * dA) =
          - E / rho * INTEGRAL(y*z*dA)      [b]

We recognize the second integral in equation [b] as the product of inertia I_yz. If the bending moment is in the plane of one of the principal axes, then


     I_yz = INTEGRAL(y*z*dA) = 0            [c]

and with this restriction the relations hold for any cross-sectional shape. In design, of course, this does not come about unless the designer designs the parts which connect to, and transfer the load into, the beam such that the plane of theses bending loads is the same as one of the principal planes of the beam.

Mechanical Engineering Design Section 2-10 Beams with Unsymmetrical Sections