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