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.

The equation of the N.A. is obtained by setting

equation to zero,

(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.

The distribution of stress across the section canbe given by

• 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

Eccentric loading on two axes

contd…

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,

This is a linear expression in h and k producing

the line SS in the figure below.

• 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.

Therefore for zero tensile stress in the

presence of an eccentric compressive load

Thus the limiting region for application of the

load is the shaded circular area of diameter

d/4 which is termed the middle quarter.

References

1. Hearn E.J, (2000), Mechanics of Materials, ISBN 0 7506 3265 8