LOCAL AND GLOBAL STABILITY LOCAL AND GLOBAL STABILITY OF TRUSS BRIDGES TAKING INTO OF TRUSS BRIDGES TAKING INTO

ACCOUNT CONSTRUCTION ACCOUNT CONSTRUCTION IMPERFECTIONSIMPERFECTIONS

Dr. Dr. Ionut RacanelIonut Racanel

Technical University of Civil Engineering, BucharestTechnical University of Civil Engineering, Bucharest

88The geometrical nonlinear analysis of semiThe geometrical nonlinear analysis of semi--trough truss bridges trough truss bridges -- Presentation of analyzed structuresPresentation of analyzed structures

Bridge over Bridge over JiuJiu CanalCanal

L=42.00 m

H=4

.60

mXY

Z

Typified deck from ISPCFTypified deck from ISPCF

L=55.00 m

H=8

.47

m

XY

Z

Bridge over Bridge over OltOlt riverriver

L=48.00 m

H=7

.20

mXY

Z

L=32.05 m

X

Z

Y

Steel deck on railway Steel deck on railway PodulPodul IloaieiIloaiei--HârlåuHârlåu

TT8.5 8.5 railway convoyrailway convoy

--Finite elements used in numerical analyses performed using Finite elements used in numerical analyses performed using LUSASLUSAS

finite element softwarefinite element software

The finite element BM3The finite element BM3

The finite element BS4The finite element BS4

The finite element BAR2The finite element BAR2

88Necessity of a nonlinear geometrical analysisNecessity of a nonlinear geometrical analysis Considered point

Considered element

-- Linear static analysis Linear static analysis →→ ideal structureideal structure

-- Geometrical nonlinear analysis Geometrical nonlinear analysis →→ structure with imperfectionsstructure with imperfections500/0 Le =

P- ∆ Curves

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Displacement [m]

λ

Ideal st ruct ure

St ruct . wih imperf .

P - ∆ C urves

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04 0.05

D isplacement [m]

λ

Ideal st ruct ure

St ruct . wit h imperf .

z

zi

y

yii

i WM

WM

AN

++=σ-- stresses:stresses:

Stress variation, σ

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0 5000 10000 15000 20000 25000

Stress, σ [daN/cm2 ]

λ

Ideal st ruct ure

St ruct . wit h imperf .

Stress variat io n, σ

0.00

0.50

1.00

1.50

2.00

2.50

0 500 1000 1500 2000 2500 3000

Stress, σ [daN / cm 2 ]

λ

Ideal st ruct ure

St ruct . wit h imperf .

77.6%77.6%

23.0%23.0%

-- displacementsdisplacements

88The buckling critical load of the compressed chordThe buckling critical load of the compressed chord

-- the first buckling the first buckling eigenmodeeigenmode

Bridge over Bridge over JiuJiu CanalCanal Typified deck from ISPCFTypified deck from ISPCF

Bridge over Bridge over OltOlt riverriver Steel deck on railway Steel deck on railway PodulPodul IloaieiIloaiei--HârlåuHârlåu

-- geometrical nonlinear analysisgeometrical nonlinear analysis

88The influence of shape of the construction imperfection on the sThe influence of shape of the construction imperfection on the stability of truss tability of truss

bridges upper chordbridges upper chord

ii

B

e

L

e

B

L

1 half1 half--wavewave

1S_SO1S_SO

1S_AS1S_ASB

e

L

ie

B

L

i

2 half2 half--waveswaves

2S_SO2S_SO

2S_AS2S_AS

ii

B

e

L

B

e

L

3 half3 half--waveswaves

3S_SO3S_SO

3S_AS3S_AS

P-∆ Curves

0

2

4

6

8

10

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

3D modelBifurcation pointσσcc

2.9965.0722Steel deck on railway Podul Iloaiei-Hîrlău

3.9998.2798Deck over Olt river

3.39896.8993Typified deck from ISPCF

2.9998.4136Bridge over Jiu Canal

λ, at reaching σcλStructure

P-∆ Curves

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1Displacement [m]

Tota

l loa

d fa

ctor

( λ)

1S_AS2S_AS3S_AS

The most disadvantageous case:The most disadvantageous case:

three halfthree half--waves in same direction 3S_ASwaves in same direction 3S_AS

Lxnee i

iπsin0=Hypotheses:Hypotheses:

)500/2000/(0 LLe ÷=

88The influence of the construction imperfection size on the stabiThe influence of the construction imperfection size on the stability of truss lity of truss

bridges upper chordbridges upper chordP-∆ Curves

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

e0=0e0=L/2000e0=L/1500e0=L/1000e0=L/750e0=L/500

88The influence of the main truss girder depth on the stability ofThe influence of the main truss girder depth on the stability of the upper chordthe upper chord

P-∆ Curves

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

H=4.20 mH=4.60 mH=5.70 mH=6.60 m

αα=(45=(45oo--6060oo))

88The influence of the reinforcement of the transverse UThe influence of the reinforcement of the transverse U--frames on the stability of frames on the stability of

upper chordupper chord

P-∆ Curves

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2

Displacement [m]

Tota

l loa

d fa

ctor

(λ)

Model without reinf.Model with reinf.

P-∆ Curves

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04 0.05

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Model without reinf.Model with reinf.

Hypotheses:Hypotheses:

500/0 Le =

HHminmin=4.20m 9.6%=4.20m 9.6%

HHmaxmax=6.60m 6.6%=6.60m 6.6%

88The influence of some types of wind bracing systems on the stabiThe influence of some types of wind bracing systems on the stability of the upper chordlity of the upper chord

P-∆ Curves

0

2

4

6

810

12

14

16

18

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Model withoutWBSModel with transv.beams

P-∆ Curves

05

101520253035404550

0 0.02 0.04 0.06 0.08

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

X form of WBS

P-∆ Curves

05

101520253035404550

0 0.1 0.2 0.3 0.4 0.5

Displacement [m]To

tal l

oad

fact

or ( λ

)

K form of WBS

-- size of initial deformationsize of initial deformation

-- deformed shapedeformed shape-- 3S_AS3S_AS

Hypotheses:Hypotheses:

500/0 Le =

SIMPLIFIED MODEL CONCERNING THE SIMPLIFIED MODEL CONCERNING THE STABILITY OF THE TRUSS BRIDGES UPPER STABILITY OF THE TRUSS BRIDGES UPPER

CHORDCHORD

88The lateral rigidity of the compressed chordThe lateral rigidity of the compressed chord

88Simplified models in the Simplified models in the literartureliterarture

StaticalStaticalschemesschemes

88Proposed simplified modelsProposed simplified models

X

Y

Upper chord

Springs Deformed shape

X

Y

Upper chord

Springs

Deformed shape

-- Evaluation of axial and rotational rigidity of springsEvaluation of axial and rotational rigidity of springs

1

1

1

1

1

1

1

1

Vertical

Cross beamBottom chord

Diagon

al

Cross beam

Vertical

1

1

Vertical

Cross beam

Bottom chord

1

1

X

Z

Y

X

Z

Y

X

Z

Y

Considered point1

1

1

1

1

1

1

1

1

1

Vertical

Cross beamBottom chord

Diagon

al

Cross beam

Vertical

1

1

1

1

1

1

1

1

Vertical

Cross beamBottom chord

Diagon

al

Cross beam

Vertical

1

1

Vertical

Cross beam

Bottom chord

1

1

X

Z

Y

X

Z

Y

X

Z

Y

Considered point

-- ComparaisonComparaison between the proposed model and models in literaturebetween the proposed model and models in literature

P-∆ Curves

0

1

2

3

4

5

6

7

8

9

10

-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Engesser (b)Proposed modelSpatial (a)Romanian norm

88Comparative study between simplified and spatial modelsComparative study between simplified and spatial models

P-∆ Curves

0

2

4

6

8

10

12

14

16

18

0 0.5 1 1.5 2

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Simplified model3D Model

-- ComparaisonComparaison of load factorsof load factors Hypotheses:Hypotheses: 500/0 Le =-- Size of initial deformationSize of initial deformation-- Shape of initial deformation 3S_ASShape of initial deformation 3S_AS

-- ComparaisonComparaison of the global rigidity (Current stiffness parameter of the global rigidity (Current stiffness parameter CSPCSP))

Stiffness variation

0

2

4

6

8

10

12

14

16

18

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

Current stiffness parameter, Cs

Tota

l loa

d fa

ctor

( λ)

Simplified model3D model

-- Influence of the springs length on the form of Influence of the springs length on the form of PP--∆ ∆ curvecurve

P-∆ Curves

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Simplified model

3D ModelSimplified model (3m)Simplified model (10m)

88The nonlinear The nonlinear behaviourbehaviour of the material in the study on simplified modelof the material in the study on simplified model0)(),( =−= peF κσκσ

( )21

23 J=σ

-- Von Von MisesMises yield criterionyield criterion

Curve for isotropic hardening

0

5000

10000

15000

20000

25000

30000

35000

40000

0 0.005 0.01 0.015 0.02 0.025 0.03

Strain ε (%)

Stre

ss σ

[tf/m

2 ]

σyielding

P-∆ Curve

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Displacement [m]

Tota

l loa

d fa

ctor

( λ)

Linear elastic material

Mat_neliniar

500/0 Le =Hypotheses:Hypotheses:σc=2400 daN/cm2

0evV +=

The finite element BSX4The finite element BSX4

88AlignementAlignement charts calculation 3D model charts calculation 3D model –– simplified modelsimplified model

α H

L

-- Equivalent relationshipsEquivalent relationships

stressesstresses

displacementsdisplacements

sps

cσσ

σ

≤

sps

c∆≤

∆

∆

Hypotheses:Hypotheses: 500/0 Le =

Shape of initial deformation 3S_ASShape of initial deformation 3S_AS

Equivalence coefficients of stresses, σ

y = -0.0063x2 + 0.0677x + 0.697R 2 = 0.9776

0.5

1

4.1 4.6 5.1 5.6 6.1 6.6 7.1

D eck depth, H [m]

σ

Equivalence coeffcients for displacements, ∆

y = -0.0082x2 + 0.0632x + 0.4734R2 = 0.9987

0.5

1

4.1 4.6 5.1 5.6 6.1 6.6 7.1

Deck depth, H [m]

∆

88CONCLUSIONSCONCLUSIONS

3D structures 3D structures without construction imperfectionswithout construction imperfections

-- lost of stability through bifurcation of equilibriumlost of stability through bifurcation of equilibrium

-- the deformed shape tend to a form having three halfthe deformed shape tend to a form having three half--waveswaves

3D structures 3D structures having construction imperfectionshaving construction imperfections

-- The shapeThe shape and and sizesize of the imperfection have a important influence on the stabilityof the imperfection have a important influence on the stability of the upper compressed of the upper compressed

chord of the truss steel bridgeschord of the truss steel bridges

-- The increasing of The increasing of main girders depthmain girders depth has an disadvantageous effect on the stability of the upper comhas an disadvantageous effect on the stability of the upper compressed pressed

chordchord

-- The presence of the The presence of the UU--frames reinforcement frames reinforcement has a has a favourablefavourable, but small influence on the stability of the upper , but small influence on the stability of the upper

chordchord

-- The The wind bracing systems wind bracing systems lead to a significant increasing of total load factor lead to a significant increasing of total load factor λλ which produce the instability which produce the instability

phenomena and lead also to a significant decreasing of the laphenomena and lead also to a significant decreasing of the lateral displacements of the chord teral displacements of the chord

(displacements are 100 times smaller in the case of a wind bra(displacements are 100 times smaller in the case of a wind bracing system having X form)cing system having X form)

ProposedProposed simplified modelssimplified models

-- The proposed simplified models offer better results concerning tThe proposed simplified models offer better results concerning the analysis of the stability of the top he analysis of the stability of the top

compressed chord of truss bridges than other models (ENGESSERcompressed chord of truss bridges than other models (ENGESSER, Romanian Norm 1911), in the same , Romanian Norm 1911), in the same

time time asumingasuming the effect of construction imperfectionsthe effect of construction imperfections

-- The lost of stability is produced by limitation of equilibrium bThe lost of stability is produced by limitation of equilibrium because of ecause of elastingelasting springssprings

-- The deformed shape has three halfThe deformed shape has three half--waves like in the case of 3D structurewaves like in the case of 3D structure

-- The springs length influences the stability of the upper compresThe springs length influences the stability of the upper compressed chord sed chord

-- Taking into account in the same time a initial deformation of thTaking into account in the same time a initial deformation of the chord and a nonlinear e chord and a nonlinear behaviourbehaviour of the of the

chord’s material lead to a severe reduction of the total load chord’s material lead to a severe reduction of the total load factorfactor

-- The simplified model can be used for design and checks of the elThe simplified model can be used for design and checks of the elements of truss steel bridges considering ements of truss steel bridges considering

the equivalence coefficients of stresses and displacements estthe equivalence coefficients of stresses and displacements established using the presented ablished using the presented alignementalignement chartscharts