Structure Design -I Lecture 12 General theory of Bending

1. STRUCTURAL DESIGN I
STRUCTURAL MECHANICS
GENERAL THEORY OF BENDING

2. Beams with non rectangular sections
• The formula M= fbd²/6 can not be use for beam sections that
are not rectangular as the properties of a rectangle were
taken to understand this relationship.
• To understand the moment of resistance of sections other
than rectangles we have to understand the general theory of
bending of beams.

3. GENERAL THEORY OF BENDING
The figure shows a portion of a
beam in the unbent and bent
conditions. AB and CD are two
vertical cross sections , assumed
so close to each other that that
the portion between them may
be regarded as bending to the
arc of a circle. EF is the part of
the neutral layer intercepted . GH
represents a typical layer of the
material at a distance y from the
neutral axis R is the radius of
curvature of the portion of the
neutral layer , in the bent beam.

4. The following are the steps in the development of the
theory :
1. Determination of the strain in the layer G’ H’ by
principles of geometry
2. Evaluation of the stress in this layer by means of
Young’s Modulus
3. Determination of the load carried by the little strip
of cross section at distance y from N.A.
4. Calculation of the moment this load has about the
N.A. and by summation the total moment of all
such strip loads

5. Step 1
Extension in layer G’H’ = G’H’ – GH
Strain in layer G’H’ = Extension /original length
= (G’H’ – GH) / GH
But GH = EF and EF = E’F’ (being on the neutral axis)
Strain in layer G’H’ = (G’H’ – E’F’) / E’F’
Expressing these distances in terms of R and θ(the angle in
radians contained by B’A’ and D’C’)
Strain in layer G’H’ = ((R+y) θ - R θ)/R θ = y θ/R θ = y/R
Step 2
Stress in G’H’/Strain in G’H’ = E
or Stress in G’H’ = E x Strain in G’H’
= E x (y/R) = Ey/R
If f = stress then f = Ey/R
Step 3
If a = area of cross sectional strip,
load carried by strip = stress x area = Ey x a or E x ay
R R

6. Step 4
Moment of the load on strip about NA
= Load x Distance
= E x ay x y = E x ay²
R R
The total ‘moment of resistance’ of the beam section is made up
of all such moments as this.
Total Moment of Resistance = ∑E x ay² = E x ∑ay²
R R
∑ay² is a geometric property of the beam section with reference
to the Neutral axis. It is termed as the moment of Inertia and is
denoted by the letter I
M = (E x I) / R but f = Ey/R or f/y = E/R
M = I x (f/y) or M/I = f/y = E/R
This is the important formula for finding the moment of resistance
for a beam section
In using this expression f will generally be the extreme stress so y
will be the distance of the extreme fiber from the NA

7. SECTION MODULUS
M/I = f/y = E/R
In this equation I and y are physical properties of a beam section
In structures it is sometimes convenient to amalgamate these in to a single
property. This relationship is called the Section Modulus. This is denoted by
the letter Z.
Z (Section Modulus) = INA / y
= Moment of inertia about NA/ Distance of extreme fiber from the NA
In most cases (i.e. in all cases where the NA of the beam section is an axis of
symmetry) y = d/2. In such cases M = fZ.
The symbols M and f may have a variety of special meanings according to the
beam problems in which they are involved. Usually M will stand for the ‘maximum
bending moment’ of the beam (e.g. Wl/8 or Wl/4 etc.) and f will represent will be
the maximum permissible stress.
So Z = M/f or Necessary Section modulus = Maximum bending moment/ Working
stress in bending

8. Values of Section Modulus
The values of section modulus for certain sections were shown.
These have been derived by the formula Z = I/y
So, for rectangular section b x d
Z = (bd³/12) / (d/2) = bd²/6
Units of Section Modulus
I is measured in in⁴/in = in³