Detailed calculations of main girder by means

of grillage FEM model Bridges CE – educational materials for design exercise

Dr Mieszko KUŻAWA Wrocław, December 9, 2014

Institute of Civil Engineering

Content of project report

1. General information

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

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

Geometry modeling assumptions

• Model class (e1+e2,s3) is applied – grillage system consited of 1-dimensional bar elements and 2-dimensional shell elements in 3-dimensional space,

• Longitudinal bar elements are intended to represent T-shape main girders properties (girder web as well as concrete deck is included),

• Transversal bar elements are supposed to model cross beams and deck slab properties in transversal direction of span.

• 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 each individual bar cross section and 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

Discretization of multi-girder monolithic RC

superstructure for grillage FEM model

• The definition of cross-sections

Main menu bar -> Geometry -> Properties - > Sections->

New section definition

Torsional stiffness of elements of superstructure

Graphics of geometry model

Element properties

Choose element-> Click left mouse button -> Object properties

Limitations of applied modeling and analysis approach

• FEM model is evaluated using Linear Elastic Analysis.

• Linear strain-stress dependency is applied, yielding effect is not considered.

• Cracking effects of concrete on stiffness changes of subsequent sections of main girder as well as on redistribution of internal forces is neglected.

• Effective width effect aiming to represent flow of longitudinal axial forces in concrete deck is not considered.

Flow of compressive forces stream

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”.

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 spacings are equal.

c) Problem is static, linear-elastic, the principle of planar cross-section before and after loading 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.

• 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 „i” [-]

• 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 mechanical 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

• 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

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

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

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 across the width of the spans

Exemplary combination of live loads in cross section

MAX

MIN

5,1F

5,1F

Analysis of influence lines

of bending moments

4.3. Influence lines of global internal forces along

investigated girder

• Load case for maximum bending moment in section x/Lt = 0.9 of girder A

Exempalary 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

” [

-]

• Load case for minimum bending moment in section x/Lt = 0.9 of girder A

γf min

γf min

γf min

γf max γf max

γf max

ILL

LD

„A

” [

-]

Analysis of

influence lines

of shear forces

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

Simple load cases of UDL permanent loads in MAX width

Selected combinations of simple load cases of UDL permanent loads in

MAX width

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

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

• Application of moving Tandem System (TS) in FEM model

4.4. Internal forces

a) Selected diagrams of M and V for various load combinations of

different actions (g, q, p, TS) presented separately for MAX and MIN

width of span

Distribution of bending moments [kNm] in main girders corresponding to selected location of TS

b) Partial envelopes of M and V (separated for MAX and MIN widths)

corresponding to combinations of:

• Permament loads g,

• UDL live loads q & p

• TS live loads.

Envelope of bending moments [kNm] in main girders corresponding to prescribed route of TS

Route of TS

b) Full envelopes of M and V in analysed main girder

Envelope of bending moments M [kNm] in analysed main girder

Envelope of shear forces V [kN] in analysed main girder

Thank you for your

attention!