On beam model for buckling (and post-buckling)
analysis of multilayered rubber bearings
Antonio D. Lanzo
Dipartimento di Strutture, Geotecnica e Geologia Applicata
Università della Basilicata, Potenza
COFIN’07, 26 giugno 2009 / Catania
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
behavior and design of m.e. bearings are strongly affected by instability
phenomena in axial compression condition
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
simple and synthetic representation;
the specific design of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
simple and synthetic representation;
the specific design of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
simple and synthetic representation;
the specific design of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
a confused outline of models
several beam models have been suggested in literature;
several of these models come from the classic linear beam model, by
adding ad hoc nonlinear terms;
that caused confused discussions on higher or lower reliability of the
models;
example
the discussion about the buckling load evaluation using Haringx or Engesser
models
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
a confused outline of models
several beam models have been suggested in literature;
several of these models come from the classic linear beam model, by
adding ad hoc nonlinear terms;
that caused confused discussions on higher or lower reliability of the
models;
example
the discussion about the buckling load evaluation using Haringx or Engesser
models
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
a confused outline of models
several beam models have been suggested in literature;
several of these models come from the classic linear beam model, by
adding ad hoc nonlinear terms;
that caused confused discussions on higher or lower reliability of the
models;
example
the discussion about the buckling load evaluation using Haringx or Engesser
models
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
a confused outline of models
several beam models have been suggested in literature;
several of these models come from the classic linear beam model, by
adding ad hoc nonlinear terms;
that caused confused discussions on higher or lower reliability of the
models;
example
the discussion about the buckling load evaluation using Haringx or Engesser
models
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
framework and aims
beam models for buckling analysis of multilayered elastomeric bearings
evaluation of buckling load of m.e. bearings
general use of equivalent homogeneous beam models
a confused outline of models
aim of the work
to give a right framework to the problem and to revise some of the beam
models with their evaluation formula for the buckling of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
outlines
.
A nonlinear Cosserat beam model, geometrically exact and complete
of axial shear and flexural deformation, is presented
..
On the basis of this model, the buckling and post-buckling behavior of
m.e. bearings is evaluated
...
by the light of these results, the (only) buckling evaluation of Haringx
and Engesser beam models is discussed
....
at last, some considerations and investigations on a class of beam
models obtained as one-dimensional reduction of 3D solid of
hyperelastic nonlinear neo-hookean material, are carried out
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
outlines
.
A nonlinear Cosserat beam model, geometrically exact and complete
of axial shear and flexural deformation, is presented
..
On the basis of this model, the buckling and post-buckling behavior of
m.e. bearings is evaluated
...
by the light of these results, the (only) buckling evaluation of Haringx
and Engesser beam models is discussed
....
at last, some considerations and investigations on a class of beam
models obtained as one-dimensional reduction of 3D solid of
hyperelastic nonlinear neo-hookean material, are carried out
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
outlines
.
A nonlinear Cosserat beam model, geometrically exact and complete
of axial shear and flexural deformation, is presented
..
On the basis of this model, the buckling and post-buckling behavior of
m.e. bearings is evaluated
...
by the light of these results, the (only) buckling evaluation of Haringx
and Engesser beam models is discussed
....
at last, some considerations and investigations on a class of beam
models obtained as one-dimensional reduction of 3D solid of
hyperelastic nonlinear neo-hookean material, are carried out
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
outlines
.
A nonlinear Cosserat beam model, geometrically exact and complete
of axial shear and flexural deformation, is presented
..
On the basis of this model, the buckling and post-buckling behavior of
m.e. bearings is evaluated
...
by the light of these results, the (only) buckling evaluation of Haringx
and Engesser beam models is discussed
....
at last, some considerations and investigations on a class of beam
models obtained as one-dimensional reduction of 3D solid of
hyperelastic nonlinear neo-hookean material, are carried out
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
introduction
summary
1
Koiter’s strategy
the equilibrium bifurcation problem
critical and post-critical analysis
2
Cosserat’ beam model
general relations
buckling o m.e. bearings
post-buckling of m.e. bearings
some numerical results
3
neo-hookean constrained beam model
constrained solids models
the framework
reduced 1-D relations
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
Outline
1
Koiter’s strategy
the equilibrium bifurcation problem
critical and post-critical analysis
2
Cosserat’ beam model
general relations
buckling o m.e. bearings
post-buckling of m.e. bearings
some numerical results
3
neo-hookean constrained beam model
constrained solids models
the framework
reduced 1-D relations
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
the Koiter’s perturbation strategy
asymptotic reconstruction of a bifurcation problem
the equilibrium problem
stationary totale potential energy:
Π[u, λ] = Φ[u] − λP(u) = statu
Φ[u] strain energy, λP(u) load potentional
the fundamental equilibrium path u f [λ]
known (or extrapolated)
the branching equilibrium path
asymptotically reconstructed . . .
λd [ξ]
=
λb + λ̇b ξ + 12 λ̈b ξ 2
u d [ξ]
=
u f [λd [ξ]] + ξ v̇b + 12 ξ 2 v̈b
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
the Koiter’s perturbation strategy
asymptotic reconstruction of a bifurcation problem
the equilibrium problem
stationary totale potential energy:
Π[u, λ] = Φ[u] − λP(u) = statu
Φ[u] strain energy, λP(u) load potentional
the fundamental equilibrium path u f [λ]
known (or extrapolated)
the branching equilibrium path
asymptotically reconstructed . . .
λd [ξ]
=
λb + λ̇b ξ + 12 λ̈b ξ 2
u d [ξ]
=
u f [λd [ξ]] + ξ v̇b + 12 ξ 2 v̈b
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
the Koiter’s perturbation strategy
asymptotic reconstruction of a bifurcation problem
the equilibrium problem
stationary totale potential energy:
Π[u, λ] = Φ[u] − λP(u) = statu
Φ[u] strain energy, λP(u) load potentional
the fundamental equilibrium path u f [λ]
known (or extrapolated)
the branching equilibrium path
asymptotically reconstructed . . .
λd [ξ]
=
λb + λ̇b ξ + 12 λ̈b ξ 2
u d [ξ]
=
u f [λd [ξ]] + ξ v̇b + 12 ξ 2 v̈b
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
the Koiter’s perturbation strategy
bifurcation of equilibrium paths
fundamental and branching equilibrium paths define a bifurcation problem
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
critical and post-critical analysis
critical analysis
λb , v̇b bifurcation load and primary buckling mode, solutions of the critical
problem
Π00
b v̇b δu = 0 ∀δu , kv̇b k = 1
post-critical analysis
λ̇b initial post-critical slope of the branching path, evaluated by the ration of scalar
coefficients
Π000 v̇ 3
λ̇b = − 12 000b b 2 = 0
Πb ûb v̇b
(λ̇b = 0 in symmetric cases).
v̈b secondary buckling mode, solution of the constrained linear problem
000 2
Π00
b v̈b δu + Πb v̇b δu = 0 , ∀δu , v̇b ⊥ v̈b
λ̈b initial post-critical curvature, evaluated by the scalar coefficient
λ̈b = −
A. D. Lanzo (Università della Basilicata)
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
critical and post-critical analysis
critical analysis
λb , v̇b bifurcation load and primary buckling mode, solutions of the critical
problem
Π00
b v̇b δu = 0 ∀δu , kv̇b k = 1
post-critical analysis
λ̇b initial post-critical slope of the branching path, evaluated by the ration of scalar
coefficients
Π000 v̇ 3
λ̇b = − 12 000b b 2 = 0
Πb ûb v̇b
(λ̇b = 0 in symmetric cases).
v̈b secondary buckling mode, solution of the constrained linear problem
000 2
Π00
b v̈b δu + Πb v̇b δu = 0 , ∀δu , v̇b ⊥ v̈b
λ̈b initial post-critical curvature, evaluated by the scalar coefficient
λ̈b = −
A. D. Lanzo (Università della Basilicata)
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
critical and post-critical analysis
critical analysis
λb , v̇b bifurcation load and primary buckling mode, solutions of the critical
problem
Π00
b v̇b δu = 0 ∀δu , kv̇b k = 1
post-critical analysis
λ̇b initial post-critical slope of the branching path, evaluated by the ration of scalar
coefficients
Π000 v̇ 3
λ̇b = − 12 000b b 2 = 0
Πb ûb v̇b
(λ̇b = 0 in symmetric cases).
v̈b secondary buckling mode, solution of the constrained linear problem
000 2
Π00
b v̈b δu + Πb v̇b δu = 0 , ∀δu , v̇b ⊥ v̈b
λ̈b initial post-critical curvature, evaluated by the scalar coefficient
λ̈b = −
A. D. Lanzo (Università della Basilicata)
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
critical and post-critical analysis
critical analysis
λb , v̇b bifurcation load and primary buckling mode, solutions of the critical
problem
Π00
b v̇b δu = 0 ∀δu , kv̇b k = 1
post-critical analysis
λ̇b initial post-critical slope of the branching path, evaluated by the ration of scalar
coefficients
Π000 v̇ 3
λ̇b = − 12 000b b 2 = 0
Πb ûb v̇b
(λ̇b = 0 in symmetric cases).
v̈b secondary buckling mode, solution of the constrained linear problem
000 2
Π00
b v̈b δu + Πb v̇b δu = 0 , ∀δu , v̇b ⊥ v̈b
λ̈b initial post-critical curvature, evaluated by the scalar coefficient
λ̈b = −
A. D. Lanzo (Università della Basilicata)
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
bifurcation critical and post-critical analysis
critical and post-critical analysis
critical analysis
λb , v̇b bifurcation load and primary buckling mode, solutions of the critical
problem
Π00
b v̇b δu = 0 ∀δu , kv̇b k = 1
post-critical analysis
λ̇b initial post-critical slope of the branching path, evaluated by the ration of scalar
coefficients
Π000 v̇ 3
λ̇b = − 12 000b b 2 = 0
Πb ûb v̇b
(λ̇b = 0 in symmetric cases).
v̈b secondary buckling mode, solution of the constrained linear problem
000 2
Π00
b v̈b δu + Πb v̇b δu = 0 , ∀δu , v̇b ⊥ v̈b
λ̈b initial post-critical curvature, evaluated by the scalar coefficient
λ̈b = −
A. D. Lanzo (Università della Basilicata)
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Outline
1
Koiter’s strategy
the equilibrium bifurcation problem
critical and post-critical analysis
2
Cosserat’ beam model
general relations
buckling o m.e. bearings
post-buckling of m.e. bearings
some numerical results
3
neo-hookean constrained beam model
constrained solids models
the framework
reduced 1-D relations
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Cosserat’ beam model
general relations
tension parameters
the kinematic
x0
=
s + u[s] + z sin θ[s]
0
=
y
z0
=
w[s] + z cos θ[s]
y
t=Na+T b ,
m=Mb×a
relations of static equilibrium
(+N cos θ + T sin θ) ,s = 0
(−N sin θ + T cos θ) ,s = 0
M,s −(1 + u,s ) (−N sin θ + T cos θ)
+w,s (N cos θ + T sin θ) = 0
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Cosserat’ beam model
general relations
tension parameters
deformation parameters
r,s = (1 + ε) a + γ b ,
χ = θ,s
relations of kinematical compatibility
1+ε
=
(1 + u,s ) cos θ − w,s sin θ
γ
=
(1 + u,s ) sin θ + w,s cos θ
χ
=
θ,s
t=Na+T b ,
m=Mb×a
relations of static equilibrium
(+N cos θ + T sin θ) ,s = 0
(−N sin θ + T cos θ) ,s = 0
M,s −(1 + u,s ) (−N sin θ + T cos θ)
+w,s (N cos θ + T sin θ) = 0
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Cosserat’ beam model
general relations
tension parameters
deformation parameters
r,s = (1 + ε) a + γ b ,
χ = θ,s
relations of kinematical compatibility
1+ε
=
(1 + u,s ) cos θ − w,s sin θ
γ
=
(1 + u,s ) sin θ + w,s cos θ
χ
=
θ,s
t=Na+T b ,
m=Mb×a
relations of static equilibrium
(+N cos θ + T sin θ) ,s = 0
(−N sin θ + T cos θ) ,s = 0
M,s −(1 + u,s ) (−N sin θ + T cos θ)
+w,s (N cos θ + T sin θ) = 0
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Cosserat’ beam model
general relations
tension parameters
deformation parameters
r,s = (1 + ε) a + γ b ,
χ = θ,s
relations of kinematical compatibility
1+ε
=
(1 + u,s ) cos θ − w,s sin θ
γ
=
(1 + u,s ) sin θ + w,s cos θ
χ
=
θ,s
t=Na+T b ,
m=Mb×a
relations of static equilibrium
(+N cos θ + T sin θ) ,s = 0
(−N sin θ + T cos θ) ,s = 0
M,s −(1 + u,s ) (−N sin θ + T cos θ)
+w,s (N cos θ + T sin θ) = 0
linear elastic constitutive relations and strain energy
N = EA ε , T = GA γ , M = EJ χ
Z n
o
Φ[u] = 21
EA ε2 + GA γ 2 + EJ χ2 ds
l
with EA, GA ed EJ axial, shear and flexural stiffness
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
Cosserat’ beam model
general relations
tension parameters
deformation parameters
r,s = (1 + ε) a + γ b ,
χ = θ,s
relations of kinematical compatibility
1+ε
=
(1 + u,s ) cos θ − w,s sin θ
γ
=
(1 + u,s ) sin θ + w,s cos θ
χ
=
θ,s
t=Na+T b ,
m=Mb×a
relations of static equilibrium
(+N cos θ + T sin θ) ,s = 0
(−N sin θ + T cos θ) ,s = 0
M,s −(1 + u,s ) (−N sin θ + T cos θ)
+w,s (N cos θ + T sin θ) = 0
linear elastic constitutive relations and strain energy
N = EA ε , T = GA γ , M = EJ χ
Z n
o
Φ[u] = 21
EA ε2 + GA γ 2 + EJ χ2 ds
l
with EA, GA ed EJ axial, shear and flexural stiffness
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
the critical buckling problem
the fundamental path (axial compression)
No = −λ
εo = u f [s],s
No = EA εo
the critical equilibrium problem Π00
b v̇b δu = 0 ∀δu:
No
No u̇,s = 0 || ẇ,s = − 1 + EA
− GA
θ̇ || EJ θ̇,ss −No 1 +
A. D. Lanzo (Università della Basilicata)
No
EA
−
No θ̇
GA
On beam model for buckling (and post-buckling) etc.
=0
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
the critical buckling problem
the fundamental path (axial compression)
No = −λ
εo = u f [s],s
No = EA εo
the critical equilibrium problem Π00
b v̇b δu = 0 ∀δu:
No
No u̇,s = 0 || ẇ,s = − 1 + EA
− GA
θ̇ || EJ θ̇,ss −No 1 +
A. D. Lanzo (Università della Basilicata)
No
EA
−
No θ̇
GA
On beam model for buckling (and post-buckling) etc.
=0
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
the critical buckling problem
the fundamental path (axial compression)
No = −λ
εo = u f [s],s
No = EA εo
the critical equilibrium problem Π00
b v̇b δu = 0 ∀δu:
No
No u̇,s = 0 || ẇ,s = − 1 + EA
− GA
θ̇ || EJ θ̇,ss −No 1 +
A. D. Lanzo (Università della Basilicata)
No
EA
−
No θ̇
GA
On beam model for buckling (and post-buckling) etc.
=0
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
bifurcation load and primary buckling mode
the bifurcation load:
−1 +
q
1 + 4π 2
λb =
EJ
l2
1
2 − EA
+
1
− EA
+
1
1
GA
GA
is obtained in closed analytical form, as solution of the 2nd degree algebraic
equation:
π2
1
1
λ2b − EA
+ GA
+ λb − 2 EJ = 0
l
the primary buckling mode
u̇[s] = 0
ẇ[s] =
1
2
−
π
θ̇[s] = − 2l
1
2
cos
1−
πs l
1
λb
EA
+
λb GA
sin
πs l
normalized according to kv̇b k ≡ ẇ[l] = 1
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
bifurcation load and primary buckling mode
the bifurcation load:
−1 +
q
1 + 4π 2
λb =
EJ
l2
1
2 − EA
+
1
− EA
+
1
1
GA
GA
is obtained in closed analytical form, as solution of the 2nd degree algebraic
equation:
π2
1
1
λ2b − EA
+ GA
+ λb − 2 EJ = 0
l
the primary buckling mode
u̇[s] = 0
ẇ[s] =
1
2
−
π
θ̇[s] = − 2l
1
2
cos
1−
πs l
1
λb
EA
+
λb GA
sin
πs l
normalized according to kv̇b k ≡ ẇ[l] = 1
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
critical analysis of m.e. bearings
bifurcation load and primary buckling mode
the bifurcation load:
−1 +
q
1 + 4π 2
λb =
EJ
l2
1
2 − EA
+
1
− EA
+
1
1
GA
GA
is obtained in closed analytical form, as solution of the 2nd degree algebraic
equation:
π2
1
1
λ2b − EA
+ GA
+ λb − 2 EJ = 0
l
the primary buckling mode
u̇[s] = 0
ẇ[s] =
1
2
−
π
θ̇[s] = − 2l
1
2
cos
1−
πs l
1
λb
EA
+
λb GA
sin
πs l
normalized according to kv̇b k ≡ ẇ[l] = 1
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx(1948) beam model
limited to the only critical analysis (correct up to the second asymptotic order)
and cannot be used in complete post-critical analysis;
coherent with Cosserat’ beam model in the limit condition of axial inextensibility;
the critical values obtained λH represent the limit values of that obtained from the
Cosserat’ beam model λb in the limit condition EA → ∞;
for real design of bearings, can be proved the following relation
λH < λ b < λ E
where λE is the Euler crical load (elastica model)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx(1948) beam model
limited to the only critical analysis (correct up to the second asymptotic order)
and cannot be used in complete post-critical analysis;
coherent with Cosserat’ beam model in the limit condition of axial inextensibility;
the critical values obtained λH represent the limit values of that obtained from the
Cosserat’ beam model λb in the limit condition EA → ∞;
for real design of bearings, can be proved the following relation
λH < λ b < λ E
where λE is the Euler crical load (elastica model)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx(1948) beam model
limited to the only critical analysis (correct up to the second asymptotic order)
and cannot be used in complete post-critical analysis;
coherent with Cosserat’ beam model in the limit condition of axial inextensibility;
the critical values obtained λH represent the limit values of that obtained from the
Cosserat’ beam model λb in the limit condition EA → ∞;
for real design of bearings, can be proved the following relation
λH < λ b < λ E
where λE is the Euler crical load (elastica model)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx(1948) beam model
limited to the only critical analysis (correct up to the second asymptotic order)
and cannot be used in complete post-critical analysis;
coherent with Cosserat’ beam model in the limit condition of axial inextensibility;
the critical values obtained λH represent the limit values of that obtained from the
Cosserat’ beam model λb in the limit condition EA → ∞;
for real design of bearings, can be proved the following relation
λH < λ b < λ E
where λE is the Euler crical load (elastica model)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx(1948) beam model
limited to the only critical analysis (correct up to the second asymptotic order)
and cannot be used in complete post-critical analysis;
coherent with Cosserat’ beam model in the limit condition of axial inextensibility;
the critical values obtained λH represent the limit values of that obtained from the
Cosserat’ beam model λb in the limit condition EA → ∞;
for real design of bearings, can be proved the following relation
λH < λ b < λ E
where λE is the Euler crical load (elastica model)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx beam model
the Engesser(1891) beam model
as Haringx model, limited to the only critical analysis
uses a different set of internal tension parameters, with T normal to the centroid
axis of the beam
for a same linear constitutive relation assumption, the model cannot be equivalent
to the Harigx beam model because of the different set of parameters referred to.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx beam model
the Engesser(1891) beam model
as Haringx model, limited to the only critical analysis
uses a different set of internal tension parameters, with T normal to the centroid
axis of the beam
for a same linear constitutive relation assumption, the model cannot be equivalent
to the Harigx beam model because of the different set of parameters referred to.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx beam model
the Engesser(1891) beam model
as Haringx model, limited to the only critical analysis
uses a different set of internal tension parameters, with T normal to the centroid
axis of the beam
for a same linear constitutive relation assumption, the model cannot be equivalent
to the Harigx beam model because of the different set of parameters referred to.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx beam model
the Engesser(1891) beam model
as Haringx model, limited to the only critical analysis
uses a different set of internal tension parameters, with T normal to the centroid
axis of the beam
for a same linear constitutive relation assumption, the model cannot be equivalent
to the Harigx beam model because of the different set of parameters referred to.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
observations on the critical behavior
Haringx, Engesser and Elastica beam models
the Haringx beam model
the Engesser beam model
the particular design of m.e. bearings suggest the use of Haringx-Cosserat beam
model for the buckling load evaluation.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
post-critical slope, post-critical curvature and secondary buckling mode
the initial post-critical slope:
λ̇b = − 12
3
Π000
b v̇b
2
Π000
b ûb v̇b
=0
the secondary buckling mode:
λ λ
ü,s = − 1 − 2 EAb + 2 GAb θ̇2 , ẅ[s] = θ̈[s] = 0
using the orthogonality condition
v̇b ⊥ v̈b ⇔ ẇ[l] ẅ[l] = ẅ[l] = 0
the initial post-critical curvature:
λ̈b = −
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
=
A. D. Lanzo (Università della Basilicata)
λb
λ
λ
λ
λb 1 − 4 EAb + 4 GAb
1 − 2 EAb + 2 GA
π 2
4l
1−
λb
EA
+
λb 2
GA
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
post-critical slope, post-critical curvature and secondary buckling mode
the initial post-critical slope:
λ̇b = − 12
3
Π000
b v̇b
2
Π000
b ûb v̇b
=0
the secondary buckling mode:
λ λ
ü,s = − 1 − 2 EAb + 2 GAb θ̇2 , ẅ[s] = θ̈[s] = 0
using the orthogonality condition
v̇b ⊥ v̈b ⇔ ẇ[l] ẅ[l] = ẅ[l] = 0
the initial post-critical curvature:
λ̈b = −
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
=
A. D. Lanzo (Università della Basilicata)
λb
λ
λ
λ
λb 1 − 4 EAb + 4 GAb
1 − 2 EAb + 2 GA
π 2
4l
1−
λb
EA
+
λb 2
GA
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
post-critical slope, post-critical curvature and secondary buckling mode
the initial post-critical slope:
λ̇b = − 12
3
Π000
b v̇b
2
Π000
b ûb v̇b
=0
the secondary buckling mode:
λ λ
ü,s = − 1 − 2 EAb + 2 GAb θ̇2 , ẅ[s] = θ̈[s] = 0
using the orthogonality condition
v̇b ⊥ v̈b ⇔ ẇ[l] ẅ[l] = ẅ[l] = 0
the initial post-critical curvature:
λ̈b = −
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
=
A. D. Lanzo (Università della Basilicata)
λb
λ
λ
λ
λb 1 − 4 EAb + 4 GAb
1 − 2 EAb + 2 GA
π 2
4l
1−
λb
EA
+
λb 2
GA
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
post-critical slope, post-critical curvature and secondary buckling mode
the initial post-critical slope:
λ̇b = − 12
3
Π000
b v̇b
2
Π000
b ûb v̇b
=0
the secondary buckling mode:
λ λ
ü,s = − 1 − 2 EAb + 2 GAb θ̇2 , ẅ[s] = θ̈[s] = 0
using the orthogonality condition
v̇b ⊥ v̈b ⇔ ẇ[l] ẅ[l] = ẅ[l] = 0
the initial post-critical curvature:
λ̈b = −
4
00 2
Π0000
b v̇b − 3Πb v̈b
2
3Π000
b û v̇b
=
A. D. Lanzo (Università della Basilicata)
λb
λ
λ
λ
λb 1 − 4 EAb + 4 GAb
1 − 2 EAb + 2 GA
π 2
4l
1−
λb
EA
+
λb 2
GA
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
observations on post-critical behavior
a measure of the influence of the post-critical behavior
∆λ
=
λb
1
2
×
λ̈b
× l2
λb
can be proved that, for case of real technical interest, this
influence is of the stabilizing kind, i.e. ∆λ
λb > 0.
also can be proved that, for case of real technical interest, this
effect is of very limited extend, being practically ∆λ
λb ≈ 0.
that proves the usual idea that the stability analysis can be
resolved by only computing the critical load of the problem, used
as load design for bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
observations on post-critical behavior
a measure of the influence of the post-critical behavior
∆λ
=
λb
1
2
×
λ̈b
× l2
λb
can be proved that, for case of real technical interest, this
influence is of the stabilizing kind, i.e. ∆λ
λb > 0.
also can be proved that, for case of real technical interest, this
effect is of very limited extend, being practically ∆λ
λb ≈ 0.
that proves the usual idea that the stability analysis can be
resolved by only computing the critical load of the problem, used
as load design for bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
observations on post-critical behavior
a measure of the influence of the post-critical behavior
∆λ
=
λb
1
2
×
λ̈b
× l2
λb
can be proved that, for case of real technical interest, this
influence is of the stabilizing kind, i.e. ∆λ
λb > 0.
also can be proved that, for case of real technical interest, this
effect is of very limited extend, being practically ∆λ
λb ≈ 0.
that proves the usual idea that the stability analysis can be
resolved by only computing the critical load of the problem, used
as load design for bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
observations on post-critical behavior
a measure of the influence of the post-critical behavior
∆λ
=
λb
1
2
×
λ̈b
× l2
λb
can be proved that, for case of real technical interest, this
influence is of the stabilizing kind, i.e. ∆λ
λb > 0.
also can be proved that, for case of real technical interest, this
effect is of very limited extend, being practically ∆λ
λb ≈ 0.
that proves the usual idea that the stability analysis can be
resolved by only computing the critical load of the problem, used
as load design for bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
post-critical analysis of m.e. bearings
observations on post-critical behavior
a measure of the influence of the post-critical behavior
∆λ
=
λb
1
2
×
λ̈b
× l2
λb
can be proved that, for case of real technical interest, this
influence is of the stabilizing kind, i.e. ∆λ
λb > 0.
also can be proved that, for case of real technical interest, this
effect is of very limited extend, being practically ∆λ
λb ≈ 0.
that proves the usual idea that the stability analysis can be
resolved by only computing the critical load of the problem, used
as load design for bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
of given geometry of the cross-section beam;
for different values of height (and then slenderness)
for different values of some constitutive parameters
σ≡
E
2ν
, G=
1 − 2ν
2(1 + ν)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
of given geometry of the cross-section beam;
for different values of height (and then slenderness)
for different values of some constitutive parameters
σ≡
E
2ν
, G=
1 − 2ν
2(1 + ν)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
of given geometry of the cross-section beam;
for different values of height (and then slenderness)
for different values of some constitutive parameters
σ≡
E
2ν
, G=
1 − 2ν
2(1 + ν)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
of given geometry of the cross-section beam;
for different values of height (and then slenderness)
for different values of some constitutive parameters
σ≡
E
2ν
, G=
1 − 2ν
2(1 + ν)
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
σ
10
20
100
∞
l
λb
(λH )
λ̈b
31
61
133
181
241
301
31
61
133
181
241
301
31
61
133
181
241
301
31
61
133
181
241
301
239702.740
115047.662
46086.895
31014.407
20941.582
15142.905
240716.458
115607.287
46378.632
31238.838
21114.699
15281.813
241605.049
116096.044
46632.142
31433.540
21264.732
15402.161
241839.063
116224.499
46698.582
31484.520
21303.993
15433.648
(196377.662)
(95276.151)
(39134.634)
(26751.477)
(18392.180)
(13516.232)
(197903.284)
(96048.934)
(39483.359)
(27003.416)
(18576.310)
(13658.881)
(199223.465)
(96717.684)
(39785.203)
(27221.533)
(18735.776)
(13782.468)
(199568.634)
(96892.537)
(39864.134)
(27278.576)
(18777.488)
(13814.803)
3.731762393
1.550904150
0.441072514
0.237507994
0.122536612
0.068599887
3.670534423
1.528870980
0.437059881
0.236095988
0.122242150
0.068643984
3.619327473
1.510390500
0.433668053
0.234894803
0.121987954
0.068679234
3.606200579
1.505645111
0.432793201
0.234583838
0.121921586
0.068688011
1 × λ̈b × l 2
λb
2
0.007480564
0.025080537
0.088464588
0.125441693
0.169926248
0.205218830
0.007326843
0.024604543
0.083348213
0.123800069
0.168128049
0.203484158
0.007198056
0.024204800
0.082251780
0.122407286
0.166594676
0.201997870
0.007165010
0.024102084
0.081969072
0.122047298
0.166197189
0.201611519
G = 0.4407MPa
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
b
w/l
(l=31)
the numerical results confirm the
previous qualitative observations. In
particular:
b
λb > λH , with 10 ≈ 15% max
differences;
w/l
(l=241)
A. D. Lanzo (Università della Basilicata)
the post-critical effect is
stabilizing but of a limited extend.
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
b
w/l
(l=31)
the numerical results confirm the
previous qualitative observations. In
particular:
b
λb > λH , with 10 ≈ 15% max
differences;
w/l
(l=241)
A. D. Lanzo (Università della Basilicata)
the post-critical effect is
stabilizing but of a limited extend.
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
b
w/l
(l=31)
the numerical results confirm the
previous qualitative observations. In
particular:
b
λb > λH , with 10 ≈ 15% max
differences;
w/l
(l=241)
A. D. Lanzo (Università della Basilicata)
the post-critical effect is
stabilizing but of a limited extend.
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
general buckling post-buckling numerical results
some numerical results
critical and post-critical behavior of homogenized bearings
b
w/l
(l=31)
the numerical results confirm the
previous qualitative observations. In
particular:
b
λb > λH , with 10 ≈ 15% max
differences;
w/l
(l=241)
A. D. Lanzo (Università della Basilicata)
the post-critical effect is
stabilizing but of a limited extend.
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
Outline
1
Koiter’s strategy
the equilibrium bifurcation problem
critical and post-critical analysis
2
Cosserat’ beam model
general relations
buckling o m.e. bearings
post-buckling of m.e. bearings
some numerical results
3
neo-hookean constrained beam model
constrained solids models
the framework
reduced 1-D relations
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
1-D reduction of hyperelastic solids
beam models as 1-D reduction of 3D solids models of hyperelastic nonlinear
constitutive relations have been suggested in literature
neo-hookean material (Ling 1995, Lanzo 2007)
Blatz and Ko material (D’Ambrosio 1995, Lanzo 2004)
take into account a more accurate representation of the nonlinear behavior of
rubber material
the 1-D reduction is made by a rigid representation of the kinematic of
cross-section beam
x0
=
s + u[s] + z sin θ[s]
y0
=
y
z0
=
w[s] + z cos θ[s]
the particular design of m.e. bearings (and the recent use of FBR) give to the
model the ability of accurate representation of the behavior of m.e. bearings
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
the framework
the model is obtained for the same kinematic (and static) framework of the
Cosserat’ beam model
using a neo-hookean constitutive model for the representation of rubber behavior
ϕ = 12 µ J1 = 21 µ tr Ft F
ϕ is the density
of the strain
energy
R funciont
R
Φ[u, w, θ] = l A ϕ[u, w, θ]dA ds
J1 i the first invariant (the trace) of Cauchy-Green C = Ft F deformation
tensor
F is the deformation gradient;
µ is a constitutive parameter of the rubber (corresponds to the shear
elasticity modulus G of the linear elastic relation)
using a constrain of incompressibility for the rubber, expressed on average for the
whole volume of the body
Z Z det F − 1 dA ds = 0
l
A
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
the framework
the model is obtained for the same kinematic (and static) framework of the
Cosserat’ beam model
using a neo-hookean constitutive model for the representation of rubber behavior
ϕ = 12 µ J1 = 21 µ tr Ft F
ϕ is the density
of the strain
energy
R funciont
R
Φ[u, w, θ] = l A ϕ[u, w, θ]dA ds
J1 i the first invariant (the trace) of Cauchy-Green C = Ft F deformation
tensor
F is the deformation gradient;
µ is a constitutive parameter of the rubber (corresponds to the shear
elasticity modulus G of the linear elastic relation)
using a constrain of incompressibility for the rubber, expressed on average for the
whole volume of the body
Z Z det F − 1 dA ds = 0
l
A
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
the framework
the model is obtained for the same kinematic (and static) framework of the
Cosserat’ beam model
using a neo-hookean constitutive model for the representation of rubber behavior
ϕ = 12 µ J1 = 21 µ tr Ft F
ϕ is the density
of the strain
energy
R funciont
R
Φ[u, w, θ] = l A ϕ[u, w, θ]dA ds
J1 i the first invariant (the trace) of Cauchy-Green C = Ft F deformation
tensor
F is the deformation gradient;
µ is a constitutive parameter of the rubber (corresponds to the shear
elasticity modulus G of the linear elastic relation)
using a constrain of incompressibility for the rubber, expressed on average for the
whole volume of the body
Z Z det F − 1 dA ds = 0
l
A
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
the framework
the model is obtained for the same kinematic (and static) framework of the
Cosserat’ beam model
using a neo-hookean constitutive model for the representation of rubber behavior
ϕ = 12 µ J1 = 21 µ tr Ft F
ϕ is the density
of the strain
energy
R funciont
R
Φ[u, w, θ] = l A ϕ[u, w, θ]dA ds
J1 i the first invariant (the trace) of Cauchy-Green C = Ft F deformation
tensor
F is the deformation gradient;
µ is a constitutive parameter of the rubber (corresponds to the shear
elasticity modulus G of the linear elastic relation)
using a constrain of incompressibility for the rubber, expressed on average for the
whole volume of the body
Z Z det F − 1 dA ds = 0
l
A
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
Koiter’s strategy Cosserat’ beam neo-hookean beam
constrained solids models framework reduced relations
a neo-hookean constrained beam model
reduced relations
for the beam, the incompressibility constraint turns into a constraint on axial
extension expressed by
ε = (1 + u,s ) cos θ − w,s sin θ − 1 = 0
as consequence, the shear deformation parameter γ is redefined into the
following
1
(sin θ + w,s )
γ = (1 + u,s ) sin θ + w,s cos θ =
cos θ
the strain energy is reduced into
Z
Φ[u, w, θ] = 12
µA γ 2 + µJ χ2 ds
l
observations
identifying shear and flexural stiffness parameters (GA, EJ) with (µA, µJ)
respectively, the obtained beam model corresponds exactly to the Cosserat’
beam model in the limit extensional constraint ε = 0
therefore, its results in buckling and post-buckling terms can deduced from the
cited model in the limit condition EA → 0.
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
conclusioni
the instability problem of m.e. bearings has been framed into the
general perturbation Koiter’s theory
by using an geometrically exact Cosserat’ beam model, buckling
and post-buckling behavior parameters have been obtained in
analytical closed form
on the basis of these results, the use (restrained to the only
critical analysis) of Haringx (and Engesser) beam model has
been revised and discussed
the Cosserat’ beam model with axial inestensibility has been
proved equivalent to 1-D reduction of solids of hyperelastic
neo-hookean incompressibility material
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
conclusioni
the instability problem of m.e. bearings has been framed into the
general perturbation Koiter’s theory
by using an geometrically exact Cosserat’ beam model, buckling
and post-buckling behavior parameters have been obtained in
analytical closed form
on the basis of these results, the use (restrained to the only
critical analysis) of Haringx (and Engesser) beam model has
been revised and discussed
the Cosserat’ beam model with axial inestensibility has been
proved equivalent to 1-D reduction of solids of hyperelastic
neo-hookean incompressibility material
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
conclusioni
the instability problem of m.e. bearings has been framed into the
general perturbation Koiter’s theory
by using an geometrically exact Cosserat’ beam model, buckling
and post-buckling behavior parameters have been obtained in
analytical closed form
on the basis of these results, the use (restrained to the only
critical analysis) of Haringx (and Engesser) beam model has
been revised and discussed
the Cosserat’ beam model with axial inestensibility has been
proved equivalent to 1-D reduction of solids of hyperelastic
neo-hookean incompressibility material
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
conclusioni
the instability problem of m.e. bearings has been framed into the
general perturbation Koiter’s theory
by using an geometrically exact Cosserat’ beam model, buckling
and post-buckling behavior parameters have been obtained in
analytical closed form
on the basis of these results, the use (restrained to the only
critical analysis) of Haringx (and Engesser) beam model has
been revised and discussed
the Cosserat’ beam model with axial inestensibility has been
proved equivalent to 1-D reduction of solids of hyperelastic
neo-hookean incompressibility material
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
conclusioni
the instability problem of m.e. bearings has been framed into the
general perturbation Koiter’s theory
by using an geometrically exact Cosserat’ beam model, buckling
and post-buckling behavior parameters have been obtained in
analytical closed form
on the basis of these results, the use (restrained to the only
critical analysis) of Haringx (and Engesser) beam model has
been revised and discussed
the Cosserat’ beam model with axial inestensibility has been
proved equivalent to 1-D reduction of solids of hyperelastic
neo-hookean incompressibility material
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.
thanks for your attention
A. D. Lanzo (Università della Basilicata)
On beam model for buckling (and post-buckling) etc.