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Steel Designers' Manual - 6th Edition (2003)
290 Introduction to manual and computer analysis
y
ey
y
Fig. 9.2 Compressive force applied eccentrically with reference to the centroidal axis
The total stress at any section can be obtained as the algebraic sum of the stresses
due to P, Mx and My.
9.3.2 Elastic analysis of line elements in pure bending
For a section having at least one axis of symmetry and acted upon by a bending
moment in the plane of symmetry, the Bernoulli equation of bending may be used
as the basis to determine both stresses and deflections within the elastic range. The
assumptions which form the basis of the theory are:
• The beam is subjected to a pure moment (i.e. shear is absent). (Generally the
deflections due to shear are small compared with those due to flexure; this is not
true of deep beams.)
• Plane sections before bending remain plane after bending.
• The material has a constant value of modulus of elasticity (E) and is linearly
elastic.
The following equation results (see Fig. 9.3).
M
I
f E
= =
y R
(9.3)
where M is the applied moment; I is the second moment of area about the neutral
axis; f is the longitudinal direct stress at any point within the cross section; y is the
distance of the point from the neutral axis; E is the modulus of elasticity; R is the
radius of curvature of the beam at the neutral axis.