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Module : Strengths of Materials
Lecturer : Mr. Mihiran Galagedara
Batch : Batch 02
Copies : 17
Date : 27/01/2012
Session : 12
By: Mihiran GalagedaraB.Sc. Eng (Hons)UOM, AMIESL
Faculty of Engineering and Construction, ICBT Campus-MT,
Sri Lanka.
• Analysis of pure bending has been limited tomembers subjected to bending couplesacting in a plane of symmetry.
• Members remain symmetricand bend in the plane ofsymmetry.
• The neutral axis of thecross section coincideswith the axis of thecouple
• Columns are the vertical prismatic members
supporting axial loads.
• In practical applications, the loads applied on
the columns are not always axial or act
symmetrically to the cross section.
• Therefore theories need to be modified to
describe such situations– i.e. for skew loading (Bending of symmetrical sections
about axes other than the axes of symmetry)
• Consider the simple
rectangular-section
beam shown.
• It is subjected to a load
inclined to the axes of
symmetry.
• In such cases bending will take place about an
inclined axis
• In this kind of cases it is convenient to resolve the
load P, and hence the applied moment, into its
components parallel with the axes of symmetry and to
apply the simple bending theory to the resulting
bending about both axes.
• It is thus assumed that simple bending takesplace simultaneously about both axes ofsymmetry
• The total stress at any point (x, y) being givenby combining the results of the separatebending actions algebraically
• The normal conventions for the signs of thestress is used,
• i.e. tension-positive, compression-negative.
(a) Eccentric loading on one axis
• There are numerous examples in engineering
practice where tensile or compressive loads
on sections are not applied through the
centroid of the section
• Which thus will introduce not only tension or
compression as the case may be but also
considerable bending effects.
• In concrete applications, for example,
where the material is considerably
weaker in tension than in compression,
any bending and hence tensile stresses
which are introduced can often cause
severe problems.
• Consider the beam shown in the figure where
the load has been applied at an eccentricity e
from one axis of symmetry.
• The stress at any point is determined by
calculating the bending stress at the point on
the basis of the simple bending theory and
combining this with the direct stress
(load/area), taking due account of sign,
• The positive sign between the two terms of theexpression is used when both parts have thesame effect and the negative sign when oneproduces tension and the other compression.
• It should now be clear that any eccentric loadcan be treated as precisely equivalent to adirect load acting through the centroid plus anapplied moment about an axis through thecentroid.
• In some cases the applied load will not be
applied on either of the axes of symmetry so
that there will now be a direct stress effect plus
simultaneous bending about both axes.
• Thus, for the section shown with the load
applied at P with eccentricities of h and k, the
total stress at any point (x, y) is given by
Again the equation of the N.A. is obtained by
equating eqn. (4.26) to zero, when
This equation is a linear equation in x and y so
that the N.A. is a straight line such as SS which
may or may not cut the section.
• As cast iron and concrete are not strong in
tension, considerable problems may arise
when they are subjected to eccentric loads.
• It will be convenient provided that the load
is applied within certain defined areas, no
tension will be produced whatever the
magnitude of the applied compressive load.
• Middle Third Rule states that no tension is
developed in a wall or foundation if the
resultant force lies within the middle third of
the structure.
• Consider, the rectangular cross-section of
The stress at any point (x, y) is given by eqn
Thus, with a compressive load applied, the
most severe tension stresses are introduced
when the last two terms have their maximum
value and are tensile in effect,
• Thus, with a compressive load applied, the
most severe tension stresses are introduced
when the last two terms have their maximum
value and are tensile in effect,
• For no tension to result in the section, 0 must
be equated to zero,
• If we follow the same
procedure for other
three quadrants, we’ll
get the shaded area
with diagonals of b/3
and d/3
• Hence termed the
middle third rule.
• whatever the positionof application of P, anaxis of symmetry willpass through thisposition
• So that the problemreduces to one ofeccentricity about asingle axis of symmetry.
Thus the limiting region for application of the
load is the shaded circular area of diameter
d/4 which is termed the middle quarter.