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

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bENdINg
Notice that the bending stress in Material 1 is calculated from the flexure formula. Recall
that the actual area of Material 1 was not modified in developing the transformed section.
Substituting Equation (8.16) into the second equation of Equation (8.13) gives the
bending stress at those locations corresponding to Material 2 in the actual cross section:
σ
x2
E2
M E
y
2 My
=− =− ⎛ ⎞
Ey
n My
2
ρ ⎝ ⎜ EI ⎠
⎟ =− =− (8.18)
E I I
1 t 1 t t
When the transformed-section method is used to calculate bending stresses at locations
corresponding to Material 2 (i.e., the transformed material) in the actual cross section, the
flexure formula must be multiplied by the modular ratio n.
For a cross section consisting of two materials (Figure 8.12a), the strains caused by a
bending moment are distributed linearly over the depth of the cross section (Figure 8.12b),
just as they are for a homogeneous beam. The corresponding normal stresses are also
distributed linearly; however, there is a discontinuity at the intersection of the two materials
(Figure 8.12c)—a consequence of the differing elastic moduli of the materials. To account
for this discontinuity, in the transformed-section method the normal stresses for the material
that was transformed ( Material 2 in this instance) are calculated by multiplying the flexure
formula by the modular ratio n.
To recapitulate, the procedure for calculating bending stresses by the transformed-section
method depends upon whether or not the area of the material was transformed:
• If the area was not transformed, then simply calculate the associated bending stresses
from the flexure formula.
• If the area was transformed, then multiply the flexure formula by n when calculating
the associated bending stresses.
In this discussion, the actual cross section of the beam was transformed into an equivalent
cross section consisting entirely of Material 1. It is also permissible to transform the
cross section to one consisting entirely of Material 2. In that case, the modular ratio is
defined as n = E 1 /E 2 . The bending stresses in Material 2 of the actual cross section will
then be the same as the bending stresses in the corresponding portion of the transformed
cross section. The bending stresses at those locations corresponding to Material 1 in the
actual cross section will be obtained by multiplying the flexure formula by n = E 1 /E 2 .
y
y
h 2
(2)
ε x
M y
σ x2 = – n
It
σ x
h 1
(1)
ε x = – y ρ
M y
σ x1 = –
It
(a) Original cross section
b
(b) Distribution of
normal strains
FIGURE 8.12 Beam with two materials: strain and stress distributions.
(c) Distribution of normal stresses