detailed calculations of main girder by means of grillage ... · influence line for lateral load...
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Detailed calculations of main girder by means
of grillage FEM model Bridges CE – educational materials for design exercise
Dr Mieszko KUŻAWA Wrocław, May 4th, 2016
Faculty of Civil Engineering
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Content of project report
1. Design assumptions
a) Objective of the project
b) Basic assumption concerning the design structure
• Theoretical length of span
• Type of construction of superstructure,
• Width of the roadway,
• Width of the sidewalks,
• Material of structure,
c) Scope of the project
• Range of conceptual design,
• Range of detailed design
d) Codes, regulations and literature
2. Technical description of entire structure
3. Initial calculations of main girder
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4. Detailed calculations of main girder of superstructure
• Cross section
• Lateral view / Longitudinal view and static system
A B C D
Basic dimensions of analysed bridge superstructure
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General geomerty modeling assumptions
• Model class (e1,s3) is applied – grillage system conssited of 1-dimensional bar elements in 3-dimensional space,
• Longitudinal elements are intended to represent T-shape main girder (stiffness of girder web as well as concrete deck is included),
• Transversal elements are supposed to model cross beams and concrete deck in transversal direction to longitudinal span axis.
• 2-dimensional shell elements are used to projecting the loads from deck surface to the bars.
• Geometrical characteristics of particular elements of FEM model are calculated according to centroid of its the cross-section and than offset vertical offset is applied for selected bar elements to properly represent stiffness of span.
• Geometrical characteristics of particular elements do not include the impact of reinforcing steel.
4.1. FEM model of bridge superstructure
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Discretization of multi-girder monolithic RC
superstructure for grillage FEM model
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• The definition of cross-sections
Main menu bar -> Geometry -> Properties - > Sections->
New section definition
Torsional stiffness of elements of superstructure
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Graphics of geometry model
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Properties of elements cross sections
Choose element-> clik left mouse button -> Object properties
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Limitations of applied modeling and analysis approach
• FEM model is evaluated using Linear Elastic type of Analysis.
• Cracking effects of concrete on bending stiffness changes of subsequent sections of main girder as well as on redistribution of internal forces is neglected.
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• Effective width effect aiming to represent flow of longitudinal axial forces in concrete deck is not considered.
Flow of compressive forces stream
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4.2. Lateral load distribution
KA,A
KA,B
KA,C KA,D
P = 1
max min
Influence Line for Lateral Load Distribution ILLLD
In initial calcuations characteristic values of ILLLD „i” ordinates were calculated using Courbon formula:
Ay
yy
nK
i
ji
ji
2,
1
where:
• yi – denotes location of girder,
• yj – denotes location of P force, • A=0 – parameter relataed to torsional stiffness
ILLLD „i” [-] – function specifying action of unit
force, located in subsequent points of cross
section of span, on investigated girder „i”.
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Cross-sectional deformation after loading
Symmetrical part of the cross-sectional deformation after loading
Asymmetric part of the cross-sectional deformation after loading
Źródło [5]
• Principle of the Courbon method
a) The cross section of the span has a vertical axis of symmetry.
b) Beam bending stiffness and their spacing are equal.
c) Problem is static, linear-elastic, the principle of rigid cross-section is valid.
d) In the analyzed cross-section of the span infinitely rigid cross member is located.
e) Mechanical model allowing to analyze the behavior of the cross-section of the span subjected to P force is assumed in the form of an infinitely stiff beam with elastic Winkler-type supports.
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• Symmetrical part of the cross-sectional deformation
The cross-section has moved evenly (translation) as a rigid body with a vector 𝑢(𝑠) which caused equal reactions in all elastic supports.
Conditions of equilibrium:
Symmetrical part of the cross-sectional deformation after loading
0
0
0
0
)(
M
H
n
PV s Symmetrical part of the cross-sectional deformation after
loading in real multi-girder bridge superstructures
ILLLD „A” [-]
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• Asymmetric part of the cross-sectional deformation
As a result of infinitely rigid cross beam the cross-section of the span rotated as a rigid body by an angle φ.
The rotation center is located on the vertical axis of symmetry of the system.
• Conditions of equilibrium:
Assuming hinged connection of girders with the cross beam!
• Deformation compatibility condition:
Asymmetric part of the cross-sectional deformation after loading
xPbbM
H
V
aa
1
)(
12
)(
20 20
0
0
1
)(
1
2
)(
2
1
)(
1
2
)(
2
bbb
u
b
u
tgaa
aa
Finally, the formula for reaction in edge springs if as follows:
2
2
2
1
2)(
22 bb
bxPa
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• In presented example rigid connection of girders with the cross beam can be assumed as well.
• Rotation of the cross beam will cause, in addition to the deflection, also the torsion of all main girders of φ angle.
• Conditions of equilibrium:
SSi
s
i MMyRM
H
V
60
0
0
1
0
Asymmetric part of the cross-sectional deformation after loading
y
z
u
• Uniform (pure) Torsion – ends are free to wrap
• Non-Uniform Torsion – warping deformation is constrained
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ω11
ω21
ω31 ω41
1 2 3 4
1 2
3 4
Analysed girder
ωij – denotes deflection of node „i” caused by load located at node „j”
Main menu bar -> Loads Types -> Add Load Case
Main menu bar -> Loads -> Loads Definition - > Node Tab
Cross beam
• Transversal deformation of analysed cross section
• Deformation of multi-girder bridge superstructure
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1 2 3 4
1 2 3 4
ω11
ω21
ω31 ω41
ω12 ω22
ω32 ω42
P1 • ω12 = P2 • ω21
P1 = 500 kN
P2 = 500 kN
On the basis of Maxwell-Betii reciprocal work theorem:
ω12 = ω21
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1 2 3 4
ω11/ω0 = kAA
ω21/ω0 = kAB
ω31/ω0 = kAC
ω41/ω0 = kAD
P1 = 1
A B C D
kij – denotes ordinate of ILLLD [-] function for girder „i” when load is located above girder „j”
• Obtaining of Influence Line of Lateral Load Distribution (ILLLD) function for girder A
ω0 = ω11 + ω21 + ω31 + ω41
ILLLD „A” [-]
max min
A B C D
• Applied load pattern it across the width of the span
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Exemplary combination of live loads in cross section
max
min
35,1F
35,1F
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Analysis of influence lines
of bending moments
4.3. Influence lines of global internal forces along
investigated girder
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• Load case for maximum bending moment in section x/Lt = 0.9 of girder A
Exemplary load pattern for extreme values of bending moments in selected section of girder A
γf max
γf max
γf max
γf min γf min
γf min
ILL
LD
„A
” [
-]
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• Load case for minimum bending moment in section x/Lt = 0.9 of girder A
Extreme values of bending moments in selected sections of girder A
γf min
γf min
γf min
γf max γf max
γf max
ILL
LD
„A
” [
-]
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Analysis of
influence lines
of shear forces
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Girder A
ILL
LD
„A
” [
-]
UDL loads patterns for unfavorable
possible load combinations acting on
girder A
IL M [m] / IL V [-]
MIN width
MAX width
A1 A2 A3 A4 A5 A7
A6 A8
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Selected simple load cases of UDL dead loads applied in MAX width
Selected combinations of simple load cases of UDL dead loads applied in
MAX width
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4.4. Loads
a) Dead loads
• Uniformly Distributed Loads (UDL), [kN/m2]
• Path Loads [kN/m] – cross beams:
characteristic value: Gk
design values: Gmax, Gmin
Bridge deck equipment
Left sidewalk
• characteristic value
gk
• design value
gmax, gmin
Bridge superstructure web of main girders + deck slab
Roadway
• characteristic value
gk
• design value
gmax, gmin
Right sidewalk
• characteristic value
gk
• design value
gmax, gmin
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Ca
rria
ge
wa
y w
idth
– w
Notional
Lane
Nr.
Notional
Lane
Nr.
Notional
Lane
Nr.
Remaining
area
Remaining
area
Load Model LM1:
• set of concentrated loads [kN] TS
• UDL [kN/m2] q
which cover most of the effects of the traffic of lorries and cars. This model should be used for general and local verifications.
ikiq Qikiq Q
ikqi q
kq q11
kq q22
rkqr q
b) Live loads
• Crowd of pedestrians – UDL [kN/m2] p
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• Application of moving Tandem System (TS) in FEM model
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Exemplary distribution of bending moments [kNm] in main girders corresponding to selected location of TS
4.4. Internal forces
a) Selected diagrams of M and V for selected load combinations for
different actions (dead loads, UDL live loads, TS moving load)
Show exemplary distributions of analysed internal forces (M & V) for 3x3 selected
combinations of investigated loads - 3 for dead loads + 3 for UDL live loads + 3 for TS
moving load.
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b) Partial envelopes of M and V corresponding to combinations for:
• Dead loads g,
• UDL live loads q & p
• TS live loads.
Show 3 partial envelopes of analysed internal forces (M & V) - for dead loads +
for UDL live loads + for TS moving load.
Exemplary envelope of bending moments [kNm] in main girders corresponding to prescribed route of TS
Route of TS
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b) Full envelopes of M and V
Envelope of bending moments M [kNm] in analysed main girder
Envelope of shear forces V [kNm] in analysed main girder
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Thank you for your
attention!