Grillage Modelling Example

Unterweger H. ; page - 1 -

MODELLING OF BRIDGE DECKS INCLUDING CROSS SECTION DISTORTION USING SIMPLIFIED SPATIAL BEAM METHODS

Harald Unterweger Institute of Steel Structures, Technical University University Graz, Austria Keywords: Steel Bridges, Composite Bridges, Global Analysis, Bridge Deck Modelling Abstract: Nowadays Nowadays exact modelling of a complete steel or composite bridge structure - using finite elements - is possible, because of the rapid development of computer power. Nevertheless this leads in general to nonacceptable costs in practical work, due to the fact of the enormous amount of traffic load cases in complex structures and the often changes of the geometric proportions during the design process. Moreover the design procedures for members are based on nominal stresses without secondary stresses. Therefore simple universal methods for modelling the whole bridge structure are necessary, which can be combined with local finite element models if necessary. The paper gives some examples for global modelling of bridge decks under traffic loads, including complex structures and behaviour. The stresses are compared with results of more refinded calculations, using the finite element method.

1 INTR INTRO ODUCT DUCTIO ION N The main problem in the structural analysis of steel and composite bridges is the modelling of the bridge deck, consisting of an immense number of plates, stiffeners, cross beams and diaphragms acting together. together. The current design procedure in practice, shown in figure 1, is therefore a combination of different models on two different levels. On the one hand a global model consisting of the whole structure including its bearings and foundations if relevant and on the other hand local models for individual parts of the structure. P P CG 2 3 4 1 CG CB

m

m

CB 5 1

6

model

A2

A3

teff

4

t*

beff

m

σ2

5

6 P

P

P

P

M T

σ3 σ2 τ

m

σ5

CG 2

3 A2

M1 - loc

A3

4

shear lag

M - glob

CG

m

results

CG

buckling

M2 - loc

σ5 ∆σ5

fatigue M3 - loc

Fig. 1 Typical Bridge Modelling; Modelling; combination of of global (M-glob) and local models (Mi-loc).


In the global model the individual components can be summed up by “global elements” provided that they have an equivalent stiffness (e.g. longitudinal stiffeners of the slab “smeared”; beam elements representing a main girder consisting of flanges and web). The results of the global analysis are longitudinal and shear stresses of the deck cross section, which also act on the local models. They are the basis of the design of the individual i ndividual members. In practice the knowledge of the stresses in the edges (1 to 6 in Fig. 1) is sufficient. If plastic member resistances are taken into account (e.g. for the webs of girders) the stresses are summed up by internal forces for each individual girder. Local models involve only parts of the bridge. Due to their limited size the individual components are modelled in detail and all the relevant relevant effects, which are not visible in the t he global model, can be treated. tr eated. The boundary conditions and actions must consider the global behaviour of the structure. Therefore the interaction of data between local and global models is necessary. In general local models in practice can be devided into three groups: – local models for calculation of additional forces and stresses of the components, e.g. local bending of the bridge slab due to traffic loads – local models to determine the stress distribution in a component in detail, e.g. introduction of concentrated forces in the plates and stiffeners, e.g. surrounding of bearings or cable anchorages of stayed bridges – local models for determining the reduced resistances of components due to stability effects, e.g. buckling of webs, lateral torsional buckling of girder flanges – local models for f or design check of complex behaviour, behaviour, including simple design formulas based on nominal stresses of the global model, e.g. buckling of webs, fatigue verification of details 2 POSSIB POSSIBLE LE CONC CONCEPT EPTS S OF THE THE FUTUR FUTURE E 2.1 Global detailed model Due to the stormy developments in computer technology a detailed model using finite elements of the whole structure is possible. Because of the following aspects this procedure is not necessarily more reliable than the t he simplified procedure with additional local models: – the load models used in bridge design are also more or less estimations and simplifications of the reality, e.g. exact predictions of traffic loads in 100 years (lifetime of the structure) are hardly not possible; simplified temperature fields – correct selection of imperfections (geometric imperfections, residual stresses) for plate elements, which are important for stability effects, because they have have a wide scatter in reality – for a realistic determination of the stress fields all the individual i ndividual components components (e.g. stiffeners) in the regions of introduction of single loads (e.g. cross bracings) must be modelled in detail – finding out the relevant load combination, especially the traffic load configuration configuration - for all the individual elements - is very complex – dealing with secondary (geometric) stresses, usual neglected due to ductile behaviour of the elements, is not familiar for the designer – non compatibility with current simplified procedures for plate buckling and fatigue, based on nominal stresses (elimination of secondary stresses necessary) – enormous amount of data and computing time – results hardly to verify and difficulties to find errors, due to the great complexity However for special studies a detailed global model - at least in a limited region (submodel technique) - seems useful, as for instance: – limited study of sensible parts of a bridge, especially for limited accidental load cases, where nonlinear behaviour is important – assessment of existing structures. For a limited l imited amount of components, esecially if compared


and calibrated with tests and measurements – studies with regard to the evaluation evaluation and reasons of damages – calibration of simplified models with selected load cases 2.2 Integration of local and global models A combination of one or more global simple models with local models seems the best strategy for economic and confident solutions in the design process. The main advantages are: – this concept corresponds with the engeneering concept of the design process; starting with simple global models with reduced elements, which include only the relevant loading pathes. Therefore changes of the geometry of the structure, e.g. stiffness and number of cross frames are easy to do. Afterwards more detailed local models are used, e.g. for the determination of quantity and situation of stiffeners – enormous saving of data, because of the regular pattern of constructional details over the length of the bridge. Therefore the local models of a part of a bridge are with lit tle changes relevant levant for the whole structure, e.g. modelling local bending of the slab due to traffic load The accuracy of the concept mainly depends of the interaction of the global and local models. That means: – correct modelling of the stiffnesses of the local components in the global model – correct boundary conditions and global stresses for the local models Especially the interaction of data between different models should be the main software developments developments of the future. Following this concept means that also in the future the engineer makes the decisions of the individual models and the degree of detailing; but the time consuming and often faulty manual process of data change between the individual models is done by the computer. 2.3 Suggested solution Based on the proceedings in practical bridge design a general concept for all types of bridge decks is actually developed by the author. The main features are: – application of conventional conventional software software for spatial beams beams including shear shear deformations with additional moduls for automatic preprocessing of traffic load configurations and automatic superposition of load cases (e.g. [1]) – linear elastic analysis due to the enormous enormous amount of traffic load load cases, which allow allow superposition of individual load cases. Second order effects (e.g. arches, truss chords) can be approximately taken into account if the minimum axial forces of a first calculation are taken into account in the stiffness of the elements CB CG CG

main girder 1 local bending 1

distortion of bridge deck (deformation cross bracing)

CB

∆

warping (open sections)

slab - global bending effects

Fig. 2 Global analysis of bridge decks decks - effects effects which must must be considered. considered.


The main demand for modelling the global behaviour of a bridge deck is that the following effects can be taken into account (figure 2): – local bending of directly loaded main main girders, because of the distances distances between cross cross bracings – distortion of the bridge deck due to deformation deformation of the cross bracings bracings – global slab bending effects – warping of the the bridge deck, deck, which leads leads to restraint forces in the bearings The basic idea in global modelling is, that the structural detailing also depends on the type of action (figure 3). It is easier to make a suitable model only for one type t ype of action - limited to the relevant relevant loading paths - than for all types of action. For secondary actions, like temperatur gradient or slab weight the whole bridge deck acts as one beam. CG

system CG

CB

m a

b

action 1 Gslab

Gslab

x

x

c

m

CB

Model - G1 Gslab action 2

V

beam elements (I b)

y

b

y

∆T My

1 beam (Ib)

Model Model -- G2 G2

cross bracing

action 3 traffic P

beam a,c (Ib) beam b (Ib, It)

As, Ib

Myb

Myc

MT cross bracing (Ib, As)

∆

α a

c

Νdia=V/ sin α V

a

c

cross girders Ib

m y

Mya

∆

a

P

b c

Fig. 3 Global Modelling of the bridge deck; different different degree of detailing for different actions. actions. For traffic loads every every girder is modelled as a single beam. For the determination of the bending stiffness of the girders (Ib) the effective with of the deck plate (bottom plate) must be estimated, because it also depends on load configuration. In practice a constant value for all types of traffic load configurations - including shear lag and plate buckling effects if relev r elevant ant is used. Neglecting shear lag effects the effective with on either side of the girder is in general equal to half the distance to the next girder. The minimum value for the effective with is one third of the distance to the next girder in case of pure torsion of a box girder (model with two single girders), following folded plate theory. The torsion stiffness of box girders (I t) can be either modelled by a torsional beam in the center of the box or by split up equally (0,5 I t) to the outer girders (in cases of two or three girders), as suggested in [2].


The cross bracings and diaphragms are modelled by using beams including shear deformation (Ib, As). A comparison study on different bridge decks with exact FE - calculations showed that they are suitable to consider their deformation behaviour. Based on these studies the shear stiffnesses of the global elements representing the diaphragms are determined as shown in example 1. Important for practical work is, that for one type of action - especially traffic loads only one global model is used, because automatic superposition of the individual load cases than can be used (more complicate if traffic loads are split in bending and torsional part with different models, as suggested in the literature). The level of accuracy and degree of detailing of a model for one type of action should be evaluated for every component in an engeneering view. That means that if the calculated stresses of that model are only a little part of the design stresses the model error can be higher (e.g. stresses due to temperature are 13 % of the design stresses of a component ; if the model error for temperature stresses is 50 % this means only an error of 7 % of the design stresses, which may be tolerable due to the uncertanties in the load model). 3 GLOBAL GLOBAL BRIDGE BRIDGE DECK MODELLING MODELLING – EXAMPL EXAMPLES ES The aim of the three selected examples is to show the great efficiency of simple global models based on spatial beams. The main results for traffic loads are presented and compared with more exact solutions (FE - models with software package ABAQUS ABAQUS [5] ). 3.1 Example 1: Railway box girder bridge with deformable cross bracings Example 1 (figure 4) is a single span railway box girder bridge under live loads on one track (simplified load configuration as investigated in [3] ). The load case L can be devided into a symmetrical and antimetrical part. The behaviour under the symmetrical part (L1) is very simple, leading to bendig of the whole bridge deck, and is not shown in the following. The antimetrical load part (L2) was used for a study, where the number of cross bracings (one or three) and their stiffnesses was varied. The shear stiffness of the bracings is expressed in form of an equivalent thickness of a fully plate. 132 kN/ m 2,0 2,0 132 [kN/m] a

track 2

b

f2

c

load case L 66 kN/ m

CB

3,4 m

beff =2,0 6,0 m 2

z

CB 40,0 m

44 kN/ m

1u

1

2

model

L1 “symmetry”

beam element Ib= ∞ , As b c

L2 “antimetry”

b

z

3,0

MT = 264 kNm/ m

1

central beam It main girder Ib Fig. 4 Example 1: Single span span railway box box girder bridge under torsional torsional live loads; loads; system, load cases and simple beam model.


The simple model for global analysis is a grillage. It consists of the two main girders with bending stiffness I b based on an effective effective with of deck and bottom plate of one third of the girder distance (following folded plate theory). A third beam between the girders has only a torsional stiffness It and represents the torsional t orsional stiffness of the whole box section (St. ( St. Venant). Venant). The cross bracings connecting the girders are modelled by using beam elements with shear deformation. The determination of the axial forces in the diagonals is shown in figure 5. Due to the fact that the shear force in the equivalent equivalent beam element is twice t wice as in reality, the equivalent equivalent shear area, as recommended in the literature for truss elements, must also be doubled. The FEcalculations confirmed this assumption.

∆Mt = ∆P · a ∆P

∆P

TMt TMt = ∆P · a / 2 · a · b

Q*m 3,4

d TMt

2 a Q= ∆P

2 z

Mtl

Nd = Q*/ sin α =( ∆P / 2 ) / sin

∆P

Q = ∆P

global model

Nd = Q*/ sin α = ( Q / 2 ) / sin

Mtr 1 As

α

1

Ad

∆P

b

α

Nd

Q* = ∆P - TMt · b = ∆P / 2

A s

=

“conventional” value

 2 ⋅ b ⋅  

α

⋅ a-- ⋅ b ⋅ A d  2 ------------------------------ 3  G ⋅ d 

E

Fig. 5 Cross bracing of a box girder bridge; Determination of axial axial forces in the diagonal members (only due to ∆Mt ) and equivalent beam element for global analysis.

a.) a.)

b.) 2

σ su

2

σ

[kN/cm ]

su

-2 -1.5 -1 -0.5

-0.4 0

0.8

0 0.5 1

1.2

1.5 2

0.4

a

b.)

b exact / t* = 2 cm exact / t* = 0,5 cm exact / t* = 0,3 cm

c

[kN/cm ]

a

b

c

model / t* = 2 cm model / t* = 0,5 cm model / t* = 0,3 cm

Fig. 6 Example Example 1 - Normal Normal stresses stresses σ at the bottom of girder 1 due to load case L2 for system with a.) three cross bracings; bracings; b.) b.) one cross bracing. bracing.


cross bracings (axis b, c) shear stif.1.)

calculation model

t* = 2 cm

t* = 0,5 cm

t* = 0,3 cm

t* = 0,1 cm

w 1c [mm] load case L2 / L

cross bracing axial forces [kN]

σ1u - load case V [kN / cm2]

2.)

axis b

axis c

axis b

axis f2

axis c

model 1

1,80 / 38,9

276

306

6,62

8,50

8,59

“exact”

1,95 / 39,0

286

318

6,44

8,24

8,31

∆ [%]

- 7,7 / - 0,3

- 3,5 %

- 3,8 %

2,8 %

3,2 %

3,4 %

model 1

2,10 / 39,2

267

312

6,72

8,58

8,65

“exact”

2,17 / 39,2

279

322

6,50

8,28

8,34

∆ [%]

- 3,2 / - 0,2

- 4,3 %

- 3,1 %

3,4 %

3,6 %

3,7 %

model 1

2,35 / 39,4

260

314

6,81

8,65

8,71

“exact”

2,31 / 39,4

275

325

6,57

8,32

8,38

∆ [%]

1,7 / 0,1

- 5,5 %

-3,4 %

3,7 %

4,0 %

3,9 %

model 1

3,54 / 40,6

240

310

7,15

9,01

9,08

“exact”

2,97 / 40,0

260

328

6,77

8,53

8,56

∆ [%]

19,2 / 1,5

- 7,7 %

- 5,5 %

5,6 %

5,6 %

6,1 %

1.) equivalent equivalent thickness of a diaphragm, t* = 2 cm in i n axis a. 2.) midspan vertical deflection of main beam for load case L2 and L respectively. respectively. Table 1 Example 1- cross bracings in axis b and and c with different shear stiffnesses; stiffnesses; comparison of the main results. cross bracings (axis c) shear stif.

calculation model

t * = 2 cm

t* = 0,5 cm

w 1b axis b [mm] load case L2 / L

cross bracing force axis c [kN]

model 1

3,30 / 30,0

“exact”

σ1u - load case V [kN / cm2]

axis b

axis f2

axis c

675

8,06

8,49

7,12

3,38 / 30,1

690

7,89

8,23

6,86

∆ [%]

- 2,4 / - 0,3

- 2,2 %

2,2 %

3,2 %

3,8 %

model 1

4,03 / 30,7

655

8,23

8,73

7,44

“exact”

3,65 / 30,4

680

7,95

8,32

6,99

∆ [%]

10,4 / 1,3

- 3,7 %

3,5 %

4,9 %

6,4 %

Table 2 Example 1- only cross bracing in axis c with different different shear stiffness; stiffness; comparison of the main results. Some of the results of the simple grillage model in comparison with the refined FEcalculations, using shell elements for the plates of the structure, are presented in the following. In figure 6 the longitudinal normal stresses at the bottom of the girder (in axis 1u) due to the antimetrical load case are shown (bending and warping effects). The differencies seem high, but in addition with the symmetrical load case - shown in table 1 and 2 - the resulting stresses


of the simple model only differ from the exact values by 3 - 5 %. A similar tendency can be seen for the maximum vertical deflections of the girder (in axis 1) at midspan (for 3 cross bracings) and at axis b (for 1 cross bracing). The differencies in an extent of 2 - 10 % for the t he antimetrical load part L2 disappear for load case L (< 1 %). The comparison of the axial forces in the diagonals of the bracings of the simple model are also in accordance with the refined model (differencies in general 3 - 5 %). These results show that the simple beam model gives sufficient accuracy, also for cross bracings with high shear deformation. 3.3 Example 2: Plate girder bridge; Restraint forces due to warping of the cross section Example 2 (figure 7) deals with a highway composite plate girder bridge under onesided traffic loads, investigated in [4]. Due to warping of the open section nonnegligible horicontal restraint forces on the t he fixed bearings are introduced. 8,0 m DIN traffic loads (for MG 1 in m)

0,2

45 m Ah 1

2,0 m Ah

6,0 m MG 1

2

2 FL

m

LL moveable bearings

system DIN traffic loads

A1

A2

Model 1

beff

A1 My V

Ah = 0

y 1 beam (Iy)

1

2 beams (I y) A2

A1 My

2 beams

y

x

2

Model 2

y

2 beams + 1 CG

1

My

V 1

beam IT = ∞ no warping Ah = MT / hM

V 2

1 beam (I z) 2 beams (Iy) A1

1

M

V Mz A2

IT = 0

pin I = ∞

2

M hM

MT Ah

Model 3 space frame 1

hM My

y V 1

My z

V 2

hM beam Iz

Ix = ∞

2

Model 3- t

space frame including torsional stiffness Fig. 7 Example 2: Highway Highway composite plate plate girder bridge; Different Different global models models for the effect of horicontal restraint forces at the bearings due to nonsymmetric traffic loads.


To take this effect into account a more detailed model of the bridge deck, is necessary. In figure 7 three t hree different models with increasing accuracy are shown – model 1, a plane grillage - as used for example 1 - is not suitable to calculate the restraint forces at the bearings. – model 2 is a plane grillage - as in model 1 - with an additional crossgirder at the fixed bearings, having an infinite torsional stiffness. This cross girder represents the fixed bearings, preventing warping deformation. Knowing the height of the shear centre, the restraint forces can be calculated using the torsional moments of the cross girder. – model 3 is a spatial grillage consisting of the two main girders with vertical bending stiffness (Iy) and a third beam in axis of the shear centre, representing the horicontal stiffness of the bridge deck Iz. At the bearings cross frames connect this three beams. With this model the essential deformation behaviour of the bridge deck due to the restraint forces can be represented. – model 3- t includes also the torsional stiffness of the bridge deck. Every main girder gets half of the torsional stiffness (0,5 · I t) and additional cross girders across the bridge length are necessary. In figure 8 and table t able 3 the main results for the investigated traffic load case for all four models are presented and compared; on the one hand the horicontal restraint f orces at the fixed bearings and on the other hand the bending moments of the main girder 1.

model

horicontal bearing force (in axis FL)

main girder 1 moments My

AH [kN]

[%]

Mm [kN / m]

[%]

MFL

[%]

mod. 1

0

0

10240

100

0

0

mod. 2

1173

121

8663

84,6

-3155

30,8

mod. 3

1070

111

8799

85,9

-2881

28,1

mod. 3 - t

966

100

8570

83,7

-2600

25,4

Table 3 Example 2 - Comparison Comparison of the main results for the individual models. models.

My [ kNm ] -4000 -2000 0 2000 4000 6000

Modell 1 , LF

M0

Modell 2 , LF Modell 3 , lF

8000 10000 12000 LL

Modell 3- T ,

m

FL

Fig. 8 Example 2 - bending moments moments of the main girder 1 for the individual individual models.


From the engineering point of view for the design of the girders also model 1 seems sufficient, but for the design of the bearings the very high horicontal forces of about 1000 kN cannot be neglected. Model 3, including the horicontal bending stiffness of the bridge deck, gives realistic results and also allows considering the finite stiffness of bearings and piers. The simple model 2 overestimates the restraint forces and is suitable for a first estimation of this effect, but cannot consider finite stiffness of the bearings. The very high horicontal restraint forces on the fixed bearings due to nonsymmetric vertical traffic loads diminish mostly in the case of box girders, due to the small warping deformations. 3.4 Example 3: Central arch highway bridge Example 3 shows a very complex structure (figure 9), firstly studied due to a fatigue failure of the horicontal bracing [6, 7] and afterwards verified for the new traffic loads of Eurocode. It is a central arch highway bridge in Salzburg / Austria with non parallel bearings on the two abutments. The bridge deck consists consists of two main girders (3,7 m high) forming a box section and two secondary outside girders with reduced heights (1,2 m), supported by deformable cross bracings. The bridge deck also acts as a tie rod for the arch. The high deformation of the cross bracings leads also to nonnegligible global bending effects of the cross girders.

cross bracings c a

z

b d

10,3

6

06

133,5 m z detail A

1,57 horicontal bracing

1,64 horicontal bracing

15,0 11,2 0,75 12,5

3,0

12,5 m

0,75

7,2

1,95

1,20 3,70 1,95

7,2 c

11,2 a

b

Fig. 9 Example 3: Central Central arch highway highway bridge in Salzburg Salzburg / Austria.

d


To limit the computational effort a detailed FE- model (figure 10) deals for calibration of a simple spatial beam model, for selected simplified load cases. The beam model, shown in figure 10 and 11, consists of the four main girders with vertical bending stiffness I bz and cross section area A and a central beam with torsional stiffness I t and reduced horicontal bending stiffness of the whole bridge deck I by. The different heights of the neutral axis between inner and outer girders are taken into account using rigid vertical excenters (figure 11). The cross girders, including the girders girders at the cross bracings, bracings, are modelled as beams, beams, supported by the main gir-ders with their horicontal stiffnesses (Ib), considering an effective with of the deck plate. The determination of the equivalent stiffness of the cross bracing elements, neglecting the bending stiffness of the cross girder, is shown in figure 11. The high deformation of the diagonals connecting the inner girders with the arch hangers - only under symmetric loads - are modelled seperatly using a spring element (or equivalent short strut element with reduced area). Using this refined beam model for the cross bracings including the cross beams with their bending stiffnesses leads to accurate predictions of the global bending effects of the cross girders compared with the detailed FE - model. 4

y

6

2

z

It, Iby, Ibz Ibz, Iby

x

dI b Ibz bz z It, Iby

Iby, Ibz, It

0

2 4 cross girder Ib cross bracing elements Ib, As & cross girder Ib

c Ibz

a Ibz

beam model

FE - model

Fig. 10 Example Example 3 - overvie overview w of simple simple beam model model and refined refined FE - model. model. In the following only some limited results of member stresses, for simple beam model and refined FE - model, are shown. To show the global load bearing behaviour the results in general are presented in form of influence lines for different different members. In figure 12 the axial force of a diagonal of the horicontal bracing is investigated. In the beam model the diagonal force corresponds with the torsional moment of the central beam (formula of Bredt). The results show a very good agreement between simple beam model and FE - model. Due to the


nonparallel bearing axes the alternative model of replacing the torsional stiffness of the central beam by adding half of this value to the inner i nner box girder beams leads to nonsufficient nonsufficient results.

a.) P = 1,0 v

Pv = 1,0

cross girder waa Aeff

wcs d

wca c a

Pv = 1,0

was

c

d

b

z

Pv = 1,0

Aeff, Ib = 0

a

z

b

cross bracing symmetric load

cross bracing antimetric load z wa

equivalent global element

wc Ib

As= 48 cm

2

Ib arch hanger

b.)

As= 97 cm

2

It

Pv = 1,0 Aeff, Ib = 0

Pv = 1,0

wz

wz

spring c z equivalent global element a

cross bracing hanger loading

b z

cross girder Ib

c.)

global model cross bracing & girder

arch hanger A, Ib spring cz

Ib

Ib rigid

c

Aeff

Ib

Aeff a

Ib

It z

d

b

Fig. 11 Example Example 3 - determinati determination on of equiv equivalent alent stiffn stiffnesse essess of the cross cross bracing; bracing; a.) global global elements with effective shear area for cross bracing; b.) equivalent spring due to deformation of hanger arch action; c.) review of all global elements. In figure 13 the influence lines of the t he axial force of the outer strut of the cross bracing, supporting the outer main girder, is shown. These results also indicate the bending behaviour of the outer main girder. For comparison also the results with nondeformable cross bracings are shown. In figure 14 the vertical deformations of the cross section near midspan for simplified traffic loads on main girder b and a & b respectively respectively are shown. The results of the simple model seem sufficient. Therefore also the global bending moments of the cross girders are predictable.


c a

Nd = ηi · P

Nd

z b

P d

ηi FE - calculation Fig. 12 Example Example 3; Influenc Influencee lines in main main girder girder axes axes of axial axial force in the the diagonal diagonal of the horicontal bracing in the the area between axis 2 and 3.

b 6 d

rigid cross bracing

FE - calculation

Fig. 13 Example Example 3; Influenc Influencee lines in axis axis d of the axial axial force force in the outer outer strut strut of the cross cross bracing in axis 6. Summing up, it may be said that the simple spatial beam model shows for nearly all members sufficient accuracy and permits an automatic calculation of the enormous amount of traffic load cases, non practicable with the realistic FE - model.


beam model 46 mm beam model 43

Fig. 14 Example Example 3; vertical vertical deforma deformations tions of of the bridge bridge deck near near midspan midspan (axis 6) 6) due to vertical single loads on girder b and a & b (in axes 1 - 12) respectively. 4. CONCLUSIONS The main effort of this t his paper was to show that also complex bridge deck behaviour behaviour can be modelled using simple spatial beams with equivalent equivalent stiffnesses. The degree of detailing of the global model also depends on the type of action. It seems useful to work with different global models for the different type of actions (e.g. traffic loads, temperature, wind), which show only the relevant loading pathes of the structure. For estimation of the accuracy of the individual global models their portion on the overall design stresses of the individual members should be taken into account. That means that for nonrelevant actions the model error can be higher. The deformation of cross bracings can be taken into account using beam elements with shear deformation. For box girders the conventional conventional shear area of bracings must be doubled to predict the real behaviour (example 1). The warping effect of bridge decks can also be taken into account. In this case the grillage of the bridge deck must be extended by an additional beam in axis of the shear center, representing the horicontal bending stiffness of the bridge deck. Acknowledgement The author would like to thank his colleague Dr. Ofner for the FE- analysis. References [1] Program “RM SPACEFR SPACEFRAME”, AME”, TDV TDV- GmbH., Graz/ Austria [2] Hambly E.C.: Bridge Deck Behaviour,2 Behaviour,2nd nd edition, 1991. [3] Resinger F.: F.: Der dünnwandige, dünnwandige, einzellige Kastenträger Kastenträger mit einfachsymmetrischem, einfachsymmetrischem, ververformbarem Rechteckquerschnitt, doctor thesis, 1956. [4] Resinger F.: F.: Längszwängungen - eine Ursache von Brückenlagerschäden, Brückenlager schäden, in “Der Bauingenieur”, 46. Jahrgang, Heft 9, 1971. [5] Program “ABA “ABAQUS”, QUS”, Hibbit, Karlsson & Sorensen, Sorensen, Version Version 5.5, 1996. [6] Greiner R., Ofner R., Unterweger H.: Betriebsbeanspruchung Betriebsbeanspruchung des Torsionsverbandes orsionsverbandes einer Straßenbrücke - Analyse eines aktuellen Anwendungsfalles, in “Neue Entwicklungen im konstruktiven konstruktiven Ingenierbau”, Universität Universität Karlsruhe, 1994. [7] Unterweger H.: H.: Fatigue failure on the bracing of a steel arch highway bridge bridge - failure description, theoretical investigations and repair, in Proceedings of Int. Conference on Advances Advances in Steel Structures, Hongkong 1996.

Grillage Modelling Example

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