Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Transformed-Section method
The concepts introduced in the preceding example can be used to develop a method for
analyzing beams made up of two materials. The basic idea is to transform a cross section that
consists of two different materials into an equivalent cross section of only one material. Once
this transformation is completed, techniques developed previously for flexure of homogeneous
beams can be used to determine the bending stresses.
Consider a beam cross section that is made up of two linear elastic materials (designated
Material 1 and Material 2) perfectly bonded together (Figure 8.11a). This composite
beam will bend as described in Section 8.2. If a bending moment is applied to such a beam,
then, as with a homogeneous beam, the total cross-sectional area will remain plane after
bending. It then follows that the normal strains will vary linearly with the y coordinate
measured from the neutral surface and that Equation (8.1) is valid:
273
FLExuRAL STRESSES IN bEAMS
OF TwO MATERIALS
ε x
1
=− y
ρ
In this situation, however, the neutral surface cannot be assumed to pass through the centroid
of the composite area.
We next wish to transform Material 2 into an equivalent amount of Material 1 and, in so
doing, define a new cross section made entirely of Material 1. In order for this transformed cross
section to be valid for calculation purposes, it must be equivalent to the actual cross section
(which consists of Material 1 and Material 2), meaning that the strains and curvature of the
transformed section must be the same as the strains and curvature of the actual cross section.
How much area of Material 1 is equivalent to an area dA of Material 2? Consider a
cross section consisting of two materials such that Material 2 is stiffer than Material 1; in
other words, E 2 > E 1 (Figure 8.11b). We will investigate the force transmitted by an area
element dA 2 of Material 2. Element dA has width dz and height dy. The force dF transmitted
by this element of area is given by dF = σ x dz dy. From Hooke’s law, the stress σ x can be
expressed as the product of the elastic modulus and the strain; therefore,
dF = ( E2ε)
dzdy
Since Material 2 is stiffer than Material 1, more area of Material 1 will be required to transmit
a force equal to dF. Now, the distribution of strain in the transformed section must be
the same as that in the actual cross section. For that reason, the y dimensions (i.e., the
dimensions perpendicular to the neutral axis) in the transformed section must be the same
In this procedure, Material 1 can
be thought of as a “common
currency” for the transformation.
All areas are converted to their
equivalents in the common
currency.
Suppose that Material 2 was a
“hard” material like steel and
Material 1 was a “soft” material
like rubber. If the strains in both
the rubber and the steel were the
same, then a much greater area
of rubber would be required to
transmit the same force that
could be transmitted by a small
area of steel.
nb
dA
dz
y
dA′
n dz
y
(2)
h 2
(1)
dy
(2)
dy
(2) Transformed
z
z
h 1
(1)
(1)
E 2 > E 1
E 2
(a) Original cross section
b
(b) Orginal cross section
with dA = dy dz
b
FIGURE 8.11 Beam with two materials: basic geometry and transformed geometry of the cross section.
b
n = E1
(c) Material 2 transformed by use of the
modular ratio n