Internal Equilibrium Section Properties

55 Just as external forces acting on a structure must be in equilibrium, the internal forces must also satisfy the equilibrium equations.

56 The internal forces are determined by slicing the beam. The internal forces on the "cut" section must be in equilibrium with the external forces.

57 The first equation we consider is the summation of axial forces.

58 Since there are no external axial forces (unlike a column or a beam-column), the internal axial forces must be in equilibrium.

where ax was given by Eq. 12.18, substituting we obtain

But since the curvature k and the modulus of elasticity E are constants, we conclude that

or the first moment of the cross section with respect to the z axis is zero. Hence we conclude that the neutral axis passes through the centroid of the cross section.

59 The second equation of internal equilibrium which must be satisfied is the summation of moments. However contrarily to the summation of axial forces, we now have an external moment to account for, the one from the moment diagram at that particular location where the beam was sliced, hence

Int.

where dA is an differential area a distance y from the neutral axis. 60 Substituting Eq. 12.18

6i We now pause and define the section moment of inertia with respect to the z axis as and section modulus as

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