On beam model for buckling (and post-buckling) analysis of ...

On beam model for buckling (and post-buckling)analysis of multilayered rubber bearings

Antonio D. Lanzo

Dipartimento di Strutture, Geotecnica e Geologia ApplicataUniversità 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 aimsbeam 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 instabilityphenomena in axial compression condition

introduction

general use of equivalent homogeneous beam models

simple and synthetic representation;

the specific design of m.e. bearings

introduction

a confused outline of models

several beam models have been suggested in literature;

several of these models come from the classic linear beam model, byadding ad hoc nonlinear terms;

that caused confused discussions on higher or lower reliability of themodels;

example

the discussion about the buckling load evaluation using Haringx or Engessermodels

introduction

example

introduction

example

introduction

example

introduction

aim of the work

to give a right framework to the problem and to revise some of the beammodels with their evaluation formula for the buckling of m.e. bearings

introduction

outlines

.A nonlinear Cosserat beam model, geometrically exact and completeof axial shear and flexural deformation, is presented

..On the basis of this model, the buckling and post-buckling behavior ofm.e. bearings is evaluated

...by the light of these results, the (only) buckling evaluation of Haringxand Engesser beam models is discussed

....at last, some considerations and investigations on a class of beammodels obtained as one-dimensional reduction of 3D solid ofhyperelastic nonlinear neo-hookean material, are carried out

introduction

outlines

introduction

outlines

introduction

outlines

introduction

summary

1 Koiter’s strategythe equilibrium bifurcation problemcritical and post-critical analysis

2 Cosserat’ beam modelgeneral relationsbuckling o m.e. bearingspost-buckling of m.e. bearingssome numerical results

3 neo-hookean constrained beam modelconstrained solids modelsthe frameworkreduced 1-D relations

Koiter’s strategy Cosserat’ beam neo-hookean beam bifurcation critical and post-critical analysis

Outline

the Koiter’s perturbation strategyasymptotic 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 uf [λ]

known (or extrapolated)

the branching equilibrium path

asymptotically reconstructed . . .

λd [ξ] = λb + λbξ + 12 λbξ

ud [ξ] = uf [λd [ξ]] + ξvb + 12 ξ

the Koiter’s perturbation strategybifurcation of equilibrium paths

fundamental and branching equilibrium paths define a bifurcation problem

critical and post-critical analysis

critical analysis

λb, vb bifurcation load and primary buckling mode, solutions of the criticalproblem

Π′′b vbδu = 0 ∀δu , ‖vb‖ = 1

post-critical analysis

λb initial post-critical slope of the branching path, evaluated by the ration of scalarcoefficients

λb = − 12

Π′′′b v3

Π′′′b ub v2

(λb = 0 in symmetric cases).

vb secondary buckling mode, solution of the constrained linear problem

Π′′b vbδu + Π′′′

b v2b δu = 0 , ∀δu , vb ⊥ vb

λb initial post-critical curvature, evaluated by the scalar coefficient

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

critical analysis

Π′′b vbδu = 0 ∀δu , ‖vb‖ = 1

λb = − 12

Π′′′b v3

Π′′′b ub v2

b v2b δu = 0 , ∀δu , vb ⊥ vb

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

critical analysis

Π′′b vbδu = 0 ∀δu , ‖vb‖ = 1

λb = − 12

Π′′′b v3

Π′′′b ub v2

b v2b δu = 0 , ∀δu , vb ⊥ vb

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

critical analysis

Π′′b vbδu = 0 ∀δu , ‖vb‖ = 1

λb = − 12

Π′′′b v3

Π′′′b ub v2

b v2b δu = 0 , ∀δu , vb ⊥ vb

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

critical analysis

Π′′b vbδu = 0 ∀δu , ‖vb‖ = 1

λb = − 12

Π′′′b v3

Π′′′b ub v2

b v2b δu = 0 , ∀δu , vb ⊥ vb

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

Koiter’s strategy Cosserat’ beam neo-hookean beam general buckling post-buckling numerical results

Outline

Cosserat’ beam modelgeneral relations

the kinematic

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

tension parameters

t = N a + T b , m = M b× 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

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

tension parameters

t = N a + T b , m = M b× a

M,s −(1 + u,s ) (−N sin θ + T cos θ)

r,s = (1 + ε) a + γ b , χ = θ,s

1 + ε = (1 + u,s ) cos θ − w ,s sin θ

γ = (1 + u,s ) sin θ + w ,s cos θ

χ = θ,s

tension parameters

t = N a + T b , m = M b× a

M,s −(1 + u,s ) (−N sin θ + T cos θ)

r,s = (1 + ε) a + γ b , χ = θ,s

1 + ε = (1 + u,s ) cos θ − w ,s sin θ

γ = (1 + u,s ) sin θ + w ,s cos θ

χ = θ,s

tension parameters

t = N a + T b , m = M b× a

M,s −(1 + u,s ) (−N sin θ + T cos θ)

linear elastic constitutive relations and strain energy

N = EA ε , T = GA γ , M = EJ χ

Φ[u] = 12

{EA ε2 + GA γ2 + EJ χ2

with EA, GA ed EJ axial, shear and flexural stiffness

r,s = (1 + ε) a + γ b , χ = θ,s

1 + ε = (1 + u,s ) cos θ − w ,s sin θ

γ = (1 + u,s ) sin θ + w ,s cos θ

χ = θ,s

tension parameters

t = N a + T b , m = M b× a

M,s −(1 + u,s ) (−N sin θ + T cos θ)

linear elastic constitutive relations and strain energy

N = EA ε , T = GA γ , M = EJ χ

Φ[u] = 12

{EA ε2 + GA γ2 + EJ χ2

with EA, GA ed EJ axial, shear and flexural stiffness

critical analysis of m.e. bearingsthe critical buckling problem

the fundamental path (axial compression)

No = −λ εo = uf [s],s No = EA εo

the critical equilibrium problem Π′′b vbδu = 0 ∀δu:

u,s = 0 || w ,s = −(1 + No

EA −NoGA

)θ || EJ θ,ss −No

(1 + No

EA −NoGA

)θ = 0

u,s = 0 || w ,s = −(1 + No

EA −NoGA

(1 + No

EA −NoGA

)θ = 0

u,s = 0 || w ,s = −(1 + No

EA −NoGA

(1 + No

EA −NoGA

)θ = 0

critical analysis of m.e. bearingsbifurcation load and primary buckling mode

the bifurcation load:

λb =−1 +

√1 + 4π2 EJ

l2(− 1

EA + 1GA

)2(− 1

EA + 1GA

)is obtained in closed analytical form, as solution of the 2nd degree algebraicequation:

λ2b(− 1

EA + 1GA

)+ λb −

l2EJ = 0

the primary buckling mode

u[s] = 0

w [s] = 12 −

12 cos

)θ[s] = − π

1− λbEA +

) sin(πs

)normalized according to ‖vb‖ ≡ w [l] = 1

λb =−1 +

√1 + 4π2 EJ

l2(− 1

EA + 1GA

)2(− 1

EA + 1GA

λ2b(− 1

EA + 1GA

)+ λb −

l2EJ = 0

u[s] = 0

w [s] = 12 −

12 cos

)θ[s] = − π

1− λbEA +

) sin(πs

λb =−1 +

√1 + 4π2 EJ

l2(− 1

EA + 1GA

)2(− 1

EA + 1GA

λ2b(− 1

EA + 1GA

)+ λb −

l2EJ = 0

u[s] = 0

w [s] = 12 −

12 cos

)θ[s] = − π

1− λbEA +

) sin(πs

observations on the critical behaviorHaringx, 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 theCosserat’ beam model λb in the limit condition EA→∞;

for real design of bearings, can be proved the following relation

λH < λb < λE

where λE is the Euler crical load (elastica model)

λH < λb < λE

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 centroidaxis of the beam

for a same linear constitutive relation assumption, the model cannot be equivalentto the Harigx beam model because of the different set of parameters referred to.

the Engesser beam model

the particular design of m.e. bearings suggest the use of Haringx-Cosserat beammodel for the buckling load evaluation.

post-critical analysis of m.e. bearingspost-critical slope, post-critical curvature and secondary buckling mode

the initial post-critical slope:

λb = − 12

Π′′′b v3

Π′′′b ub v2

the secondary buckling mode:

u,s = −(1− 2 λb

EA + 2 λbGA

)θ2 , w [s] = θ[s] = 0

using the orthogonality condition

vb ⊥ vb ⇔ w [l] w [l] = w [l] = 0

the initial post-critical curvature:

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

b= λb

(1− 4 λb

EA + 4 λbGA

)(1− 2 λb

EA + 2 λbGA

) (π4l

)2(1− λb

EA +λbGA

λb = − 12

Π′′′b v3

Π′′′b ub v2

u,s = −(1− 2 λb

EA + 2 λbGA

)θ2 , w [s] = θ[s] = 0

vb ⊥ vb ⇔ w [l] w [l] = w [l] = 0

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

b= λb

(1− 4 λb

EA + 4 λbGA

)(1− 2 λb

EA + 2 λbGA

) (π4l

)2(1− λb

EA +λbGA

λb = − 12

Π′′′b v3

Π′′′b ub v2

u,s = −(1− 2 λb

EA + 2 λbGA

)θ2 , w [s] = θ[s] = 0

vb ⊥ vb ⇔ w [l] w [l] = w [l] = 0

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

b= λb

(1− 4 λb

EA + 4 λbGA

)(1− 2 λb

EA + 2 λbGA

) (π4l

)2(1− λb

EA +λbGA

λb = − 12

Π′′′b v3

Π′′′b ub v2

u,s = −(1− 2 λb

EA + 2 λbGA

)θ2 , w [s] = θ[s] = 0

vb ⊥ vb ⇔ w [l] w [l] = w [l] = 0

λb = −Π′′′′

b v4b − 3Π′′

3Π′′′b uv2

b= λb

(1− 4 λb

EA + 4 λbGA

)(1− 2 λb

EA + 2 λbGA

) (π4l

)2(1− λb

EA +λbGA

post-critical analysis of m.e. bearingsobservations on post-critical behavior

a measure of the influence of the post-critical behavior

λb= 1

2 ×λb

λb× l2

can be proved that, for case of real technical interest, thisinfluence is of the stabilizing kind, i.e. ∆λ

λb> 0.

also can be proved that, for case of real technical interest, thiseffect is of very limited extend, being practically ∆λ

λb≈ 0.

that proves the usual idea that the stability analysis can beresolved by only computing the critical load of the problem, usedas load design for bearings

λb= 1

2 ×λb

λb× l2

λb> 0.

λb≈ 0.

λb= 1

2 ×λb

λb× l2

λb> 0.

λb≈ 0.

λb= 1

2 ×λb

λb× l2

λb> 0.

λb≈ 0.

λb= 1

2 ×λb

λb× l2

λb> 0.

λb≈ 0.

some numerical resultscritical 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 ≡2ν

1− 2ν, G =

E2(1 + ν)

σ ≡2ν

1− 2ν, G =

E2(1 + ν)

σ ≡2ν

1− 2ν, G =

E2(1 + ν)

σ ≡2ν

1− 2ν, G =

E2(1 + ν)

σ l λb (λH) λb12 ×

λbλb× l2

31 239702.740 (196377.662) 3.731762393 0.00748056461 115047.662 (95276.151) 1.550904150 0.025080537

10 133 46086.895 (39134.634) 0.441072514 0.088464588181 31014.407 (26751.477) 0.237507994 0.125441693241 20941.582 (18392.180) 0.122536612 0.169926248301 15142.905 (13516.232) 0.068599887 0.20521883031 240716.458 (197903.284) 3.670534423 0.00732684361 115607.287 (96048.934) 1.528870980 0.024604543

20 133 46378.632 (39483.359) 0.437059881 0.083348213181 31238.838 (27003.416) 0.236095988 0.123800069241 21114.699 (18576.310) 0.122242150 0.168128049301 15281.813 (13658.881) 0.068643984 0.20348415831 241605.049 (199223.465) 3.619327473 0.00719805661 116096.044 (96717.684) 1.510390500 0.024204800

100 133 46632.142 (39785.203) 0.433668053 0.082251780181 31433.540 (27221.533) 0.234894803 0.122407286241 21264.732 (18735.776) 0.121987954 0.166594676301 15402.161 (13782.468) 0.068679234 0.20199787031 241839.063 (199568.634) 3.606200579 0.00716501061 116224.499 (96892.537) 1.505645111 0.024102084

∞ 133 46698.582 (39864.134) 0.432793201 0.081969072181 31484.520 (27278.576) 0.234583838 0.122047298241 21303.993 (18777.488) 0.121921586 0.166197189301 15433.648 (13814.803) 0.068688011 0.201611519

G = 0.4407MPa

w/l (l=241)

w/l (l=31)

the numerical results confirm theprevious qualitative observations. Inparticular:

λb > λH , with 10 ≈ 15% maxdifferences;

the post-critical effect isstabilizing but of a limited extend.

w/l (l=241)

w/l (l=31)

w/l (l=241)

w/l (l=31)

w/l (l=241)

w/l (l=31)

Koiter’s strategy Cosserat’ beam neo-hookean beam constrained solids models framework reduced relations

Outline

1-D reduction of hyperelastic solids

beam models as 1-D reduction of 3D solids models of hyperelastic nonlinearconstitutive 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 ofrubber material

the 1-D reduction is made by a rigid representation of the kinematic ofcross-section beam

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

the particular design of m.e. bearings (and the recent use of FBR) give to themodel the ability of accurate representation of the behavior of m.e. bearings

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

x ′ = s + u[s] + z sin θ[s]

y ′ = y

z′ = w [s] + z cos θ[s]

a neo-hookean constrained beam modelthe framework

the model is obtained for the same kinematic (and static) framework of theCosserat’ beam model

using a neo-hookean constitutive model for the representation of rubber behavior

ϕ = 12 µ J1 = 1

2 µ tr(Ft F)

ϕ is the density funciont of the strain energyΦ[u,w , θ] =

(∫A ϕ[u,w , θ]dA

J1 i the first invariant (the trace) of Cauchy-Green C = Ft F deformationtensorF is the deformation gradient;µ is a constitutive parameter of the rubber (corresponds to the shearelasticity modulus G of the linear elastic relation)

using a constrain of incompressibility for the rubber, expressed on average for thewhole volume of the body∫

(det(F)− 1)

ds = 0

ϕ = 12 µ J1 = 1

2 µ tr(Ft F)

(det(F)− 1)

ds = 0

ϕ = 12 µ J1 = 1

2 µ tr(Ft F)

(det(F)− 1)

ds = 0

ϕ = 12 µ J1 = 1

2 µ tr(Ft F)

(det(F)− 1)

ds = 0

a neo-hookean constrained beam modelreduced relations

for the beam, the incompressibility constraint turns into a constraint on axialextension expressed by

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

as consequence, the shear deformation parameter γ is redefined into thefollowing

γ = (1 + u,s ) sin θ + w ,s cos θ =1

cos θ(sin θ + w ,s )

the strain energy is reduced into

Φ[u,w , θ] = 12

{µA γ2 + µJ χ2} ds

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 thecited model in the limit condition EA→ 0.

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

ε = (1 + u,s ) cos θ − w ,s sin θ − 1 = 0

γ = (1 + u,s ) sin θ + w ,s cos θ =1

Φ[u,w , θ] = 12

observations

conclusioni

the instability problem of m.e. bearings has been framed into thegeneral perturbation Koiter’s theoryby using an geometrically exact Cosserat’ beam model, bucklingand post-buckling behavior parameters have been obtained inanalytical closed formon the basis of these results, the use (restrained to the onlycritical analysis) of Haringx (and Engesser) beam model hasbeen revised and discussedthe Cosserat’ beam model with axial inestensibility has beenproved equivalent to 1-D reduction of solids of hyperelasticneo-hookean incompressibility material

conclusioni

thanks for your attention

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