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

This equation reduces to
∫
A
∫
ydA + yndA = 0
(8.14)
A
1 2
275
FLExuRAL STRESSES IN bEAMS
OF TwO MATERIALS
The area of the transformed cross section can be expressed as
∫
A
∫
dA + ndA = dA
A
1 2
so Equation (8.14) can be rewritten simply as
∫
A
t
t
∫ ydA t = 0
(8.15)
A t
Therefore, the neutral axis passes through the centroid of the transformed section, just as
it passes through the centroid of a homogeneous beam.
Does the transformed section have the same moment–curvature relationship as the
actual cross section? From the relationships of Equation (8.13), the moment–curvature
relationship for a beam of two materials is
∫
M =− yσ
dA
∫
A
=− yσ
dA − yσ
dA
A
1
1
= ⎡
∫ Ey dA +
ρ ⎣⎢ A1
x
x
∫
A
x
Ey dA⎤
⎦⎥
1 2 2 2
A
By the modular ratio, the elastic modulus of Material 2 can be expressed as E 2 = nE 1 ,
reducing the preceding equation to
M = E 1 ⎡ 2 2
y dA y ndA
ρ
∫ + ⎤
⎣⎢ A ∫
1 A2
⎦⎥
The term in brackets is just the moment of inertia, I t , of the transformed section about its
neutral axis (which was previously shown to pass through the centroid). Therefore, the
moment–curvature relationship can be written as
1 t
2
2
∫
2
∫
M = EI where It
= y dAt
(8.16)
ρ
At
In other words, the moment–curvature relationship of the transformed cross section is
equal to that of the actual cross section.
How are bending stresses calculated for each of the two materials, according to the
transformed-section method? Equation (8.16) can be expressed as
1
ρ =
M
EI
1 t
and substituted into the stress relationships of Equation (8.13). Substituting into the first
equation gives the bending stress at those locations corresponding to Material 1 in the
actual cross section:
σ
x1
E1
M My
=− y =− ⎛ ⎞
Ey 1
ρ ⎝ ⎜ EI ⎠
⎟ =− (8.17)
I
1 t
t