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

272
bENdINg
Similarly, the bending stress in the wooden beam is
(140,000 Nmm)(20mm)
⋅
σ wood =
= 5MPa
4
560,000 mm
Thus, the bending stress in the wood is one-seventh of the stress in the aluminum; therefore,
equivalent beams do not necessarily have the same bending stresses, only the same ρ and ε.
In this example, the elastic moduli, the beam widths, and the bending stresses all
differ by a factor of 7. Compare the moment–curvature relationships for the two beams:
1 M
ρ = E I
= E
alum alum
M
I
wood wood
Expressing the moments of inertia in terms of the respective beam widths b alum and b wood
and the common beam height h gives
which can be simplified to
M
M
=
⎛ b h ⎞ ⎛ bwoodh
E
3 ⎞
⎝
⎜
12 ⎠
⎟
⎝
⎜
12 ⎠
⎟
E alum
alum 3 wood
b
b
wood
alum
E
=
E
The ratio of the elastic moduli will be termed the modular ratio and denoted by the symbol
n. For the two materials considered here, the modular ratio has the value
n =
E
E
alum
wood
alum
wood
70 GPa
= = 7
10 GPa
Hence, the factor of 7 that appears throughout this example stems from the modular ratio
for the two materials. The required width of the wooden beam can be expressed in terms of
the modular ratio n as
b
b
wood
alum
Ealum
= = n ∴ bwood
= nbalum
= 7(15 mm) = 105 mm
E
wood
As mentioned, the bending stresses from the two beams also differ by a factor of 7. Consequently,
since the aluminum and wooden beams are equivalent, the bending strains are the
same for the two beams:
( ε ) = ( ε )
x
alum
Stress is related to strain by Hooke’s law; therefore, the bending strains can be expressed as
x
wood
σ
( ε x) alum = ⎛ ⎞
and ( ε x)
⎝ E⎠
alum
wood
σ
= ⎛ ⎞
⎝ E⎠
The relationship between the bending stresses of the two materials can now be expressed in
terms of the modular ratio n:
σalum
σwood
σ alum Ealum
= or = = n
E E
σ E
alum
wood
Once again, the ratio of the bending stresses differs by an amount equal to the modular ratio n.
To summarize, a beam made of one material is transformed into an equivalent beam
of a different material by modifying the beam width (and only the beam width). The ratio
between the elastic moduli of the two materials (termed the modular ratio) dictates the
change in width required for equivalence. Bending stresses are not equal for equivalent
beams; rather, they, too, differ by a factor equal to the modular ratio.
wood
wood
wood