(3.170) we obtain, on multiplying by u, integrating over the beam length and
rearranging, that:
1
2
qAl€ a þ
p
4 EI
2l 3 þ
p
2 EA
2l
g
a þ
p
4 EA
8l 3 a
3
¼ ÀQ 0 cosðXtÞ:
ð3:172Þ
This is a Duffing equation of the form (3.152), with parameters:
uðtÞ ¼ aðtÞ; x
2
0 ¼
EI
qA
p
l
4
1 þ g
l
p
2 A
I
!
; c ¼
1
4
E
q
p
l
4 ; q ¼
À2Q 0
qAl
:
ð3:173Þ
The nonlinearity appears to be hardening (c > 0). The linear stiffness coefficient
x 0
2 equals the squared lowest natural frequency EIp
4 /qAl
4 of an ordinary
hinged-hinged beam, though, with an additional term that accounts for a stiffening
or weakening effect when η 6 ¼ 0.
Note that the linear stiffness can be negative, since x 0
2 < 0 whenever:
g\ À g crit ; g crit ¼
p
l
2 I
A
¼
p
s
2 ;
ð3:174Þ
where s = l/r is the beams slenderness ratio, and r ¼
ffiffiffiffiffiffiffi ffi
I=A
p
the radius of gyration
of the beam cross section. This condition is met when an initial contraction of
supports implies a compressive load of the beam that exceeds the static buckling
load EIp
2 /l
2 . In that case, it appears from (3.172), there are three possible configurations of static equilibrium: The unstable, straight configuration a = 0, and two
stable, symmetrical states of buckling:
a ¼ Æ
ffiffiffiffiffiffiffiffiffi ffi
Àx 2
0
c
s
¼ Æ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
À
I
A
À
l
p
2
g
s
:
ð3:175Þ
It appears that a ! ∞ when c ! 0, so that the buckling amplitude is bounded
only in the presence of nonlinearity (c 6 ¼ 0). This is yet an illustration that even
though critical states may be determined by linearization, the prediction of
post-critical states requires consideration to nonlinear effects.
Nonlinear Elasticity or Midplane Stretching – Which is the Stronger? If
axial motion is unrestricted, then midplane stretching is usually not relevant;
nonlinear elasticity is obviously the dominating nonlinearity, of the two. But what if
boundary conditions are such that both could be at play? We can estimate the
relative effect by comparing the coefficients of the cubic nonlinearity for the two
cases. By (3.161) and (3.173) we have, with subscripts NE and MS denoting the
coefficients of nonlinear elasticity and midplane stretching, respectively:
152
3 Nonlinear Vibrations: Classical Local Theory
rearranging, that:
1
2
qAl€ a þ
p
4 EI
2l 3 þ
p
2 EA
2l
g
a þ
p
4 EA
8l 3 a
3
¼ ÀQ 0 cosðXtÞ:
ð3:172Þ
This is a Duffing equation of the form (3.152), with parameters:
uðtÞ ¼ aðtÞ; x
2
0 ¼
EI
qA
p
l
4
1 þ g
l
p
2 A
I
!
; c ¼
1
4
E
q
p
l
4 ; q ¼
À2Q 0
qAl
:
ð3:173Þ
The nonlinearity appears to be hardening (c > 0). The linear stiffness coefficient
x 0
2 equals the squared lowest natural frequency EIp
4 /qAl
4 of an ordinary
hinged-hinged beam, though, with an additional term that accounts for a stiffening
or weakening effect when η 6 ¼ 0.
Note that the linear stiffness can be negative, since x 0
2 < 0 whenever:
g\ À g crit ; g crit ¼
p
l
2 I
A
¼
p
s
2 ;
ð3:174Þ
where s = l/r is the beams slenderness ratio, and r ¼
ffiffiffiffiffiffiffi ffi
I=A
p
the radius of gyration
of the beam cross section. This condition is met when an initial contraction of
supports implies a compressive load of the beam that exceeds the static buckling
load EIp
2 /l
2 . In that case, it appears from (3.172), there are three possible configurations of static equilibrium: The unstable, straight configuration a = 0, and two
stable, symmetrical states of buckling:
a ¼ Æ
ffiffiffiffiffiffiffiffiffi ffi
Àx 2
0
c
s
¼ Æ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
À
I
A
À
l
p
2
g
s
:
ð3:175Þ
It appears that a ! ∞ when c ! 0, so that the buckling amplitude is bounded
only in the presence of nonlinearity (c 6 ¼ 0). This is yet an illustration that even
though critical states may be determined by linearization, the prediction of
post-critical states requires consideration to nonlinear effects.
Nonlinear Elasticity or Midplane Stretching – Which is the Stronger? If
axial motion is unrestricted, then midplane stretching is usually not relevant;
nonlinear elasticity is obviously the dominating nonlinearity, of the two. But what if
boundary conditions are such that both could be at play? We can estimate the
relative effect by comparing the coefficients of the cubic nonlinearity for the two
cases. By (3.161) and (3.173) we have, with subscripts NE and MS denoting the
coefficients of nonlinear elasticity and midplane stretching, respectively:
152
3 Nonlinear Vibrations: Classical Local Theory