UNSYMMETRICAL BENDING:

PRINCIPAL AXIS

The principal axis can also be related the members in pure bending, but also applies to material under many different types of stress. Essentially it is the orientation of a material under which the normal stresses are maximum and shear stresses are zero. Instead of looking at the material as a whole you are looking at a differential piece of the material.

Neutral Axis:

The Neutral axis is the axis passing through the section of a member undergoing pure bending where the stress is zero. Basically it is the point where the member goes from tension to compression.

Direct Stress Distribution due to Bending Moment:

Consider a beam having the arbitrary cross-section shown in Fig. 9.4(a). The beam supports bending moments Mx and My and bends about some axis in its cross-section which is therefore an axis of zero stress or a neutral axis (NA). Let us suppose that the origin of axes coincides with the centroid C of the cross-section and that the neutral axis is a distance p from C. The direct stress σ on an element of area dA at a point (x, y) and a distance ζ from the neutral axis is, then

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If the beam is bent to a radius of curvature p about the neutral axis at this particular section then, since plane sections are assumed to remain plane after bending, and by a comparison with symmetrical bending theory

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Substituting for ez in Eq. we have

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The beam supports pure bending moments so that the resultant normal load on any section must be zero. Hence

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Therefore, replacing az in this equation from Eq. (9.2) and cancelling the constant E/r gives

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i.e. the first moment of area of the cross-section of the beam about the neutral axis is zero. It follows that the neutral axis passes through the centroid of the cross-section as shown in Fig. 9.4(b).

Suppose that the inclination of the neutral axis to Cx is a (measured clockwise from Cx), then

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The moment resultants of the internal direct stress distribution have the same sense as the applied moments Mx and My. Thus

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Substituting for a= from Eq. (9.4) in Eqs. (9.5) and defining the second moments of area of the section about the axes Cx, Cy as

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In the case where the beam cross-section has either (or both) Cx or Cy as an axis of symmetry the product second moment of area Ixy is zero and Cxy are principal axes. Equation (9.7) then reduces to

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Further, if either Mx or My is zero then

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Equations (9.8) and (9.9) are those derived for the bending of beams having at least a singly symmetrical cross-section. It may also be noted that in Eqs. (9.9) σz = 0 when, for the first equation, y = 0 and for the second equation when x = 0. Therefore, in symmetrical bending theory the x axis becomes the neutral axis when My = 0 and the y axis becomes the neutral axis when Mx = 0. Thus we see that the position of the neutral axis depends on the form of the applied loading as well as the geometrical properties of the cross-section.

There exists, in any unsymmetrical cross-section, a centroidal set of axes for which the product second moment of area is zero. These axes are then principal axes and the direct stress distribution referred to these axes takes the simplified form of Eqs. (9.8) or (9.9). It would therefore appear that the amount of computation can be reduced if these axes are used. This is not the case, however, unless the principal axes are obvious from inspection since the calculation of the position of the principal axes, the principal sectional properties and the coordinates of points at which the stresses are to be determined consumes a greater amount of time than direct use of Eqs. (9.6) or (9.7) for an arbitrary, but convenient set of centroidal axes.

Position of the neutral axis

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