dr hab. inż. Aleksander Urbański, prof. PKdr inż. Karol Ryż

Michał WszołekPrzemysław Milczarek

POLITECHNIKA KRAKOWSKA im. T. KościuszkiWydział Inżynierii LądowejInstytut Materiałów i Konstrukcji Budowlanych Katedra Budowy Mostów i Tuneli

Analysis of folded steel shell - earth bridge

Diploma work of: Under supervision of:

General informationAnalytical design methodsZSoil modeling of folded shellConceptionComputer modeling of a bridge and results

ConstructionExploitation

Conclusions

Contents

Examples

Examples, cont.

24m

11 000 kN

Is=0,95-1,00

φ=36o-45o

typical shapes soil parameters

Main components of the structure

Folded steel shell Soil

geometry (almost) arbitrary, min depth of backfill 0.5m:plane strain conditionformulas for: N,M –forces and bending moments for dead weihgt and moving load

Analytical method of design:Sundquist-Pettersson („swedish method”)- 2010

stresses in fill according to Boussinesq theoryq and P

moving loadSwedish Method, cont.

Swedish Method, cont. dead load, 2 phazes of construction:

1 phaze 2 phaze

Identification of folded shell model in ZSoil in Robot Milenium: how to deal with it in ZSoil ?

K[kN/m]

B[kNm]

H[kN/m]

9 module

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Γ

⋅

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

yxxyyyxxxy

EyyExx

hyyhxx

bxyxybyyyysbxxyybxxxx

kxyxykyyyyskxxyykxxxx

QyQxMxyMyyMxxNxyNyyNxx

ββκκκ

OO

OO

OO

=

plane stress orthotropic model in core layer

10 parameters:

2hXζ

2hXζ

2hYζ

2hYζ

h

kx

kxky

aaFEky =

κζζ kykxvEEGh YX ,,,1,2,1,0,

symetric reiforcements, different in x,y

Identification of folded shell model in ZSoil, cont.

εDσ ⋅=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⋅

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅−−−

⋅−⋅

⋅−

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

XY

YY

XX

XY

YY

XX

Gvv

Esvv

Evvv

E

γεε

σσσ

000

0211

2

0211

12211

11212

EEvv ⋅=

Robot Milenium => ZSoil, with help of MathCadGiven

G0 h⋅ kxyxy

G012

h3⋅ bxyxy

κhxx

G0 h⋅

1.

2. 1. E212

h3

1 v12 E2E1⋅−

⋅ 2 ky⋅y4⋅+ byyyy

2. E112

h3

1 v12 E2E1⋅−

⋅ 2 kx⋅x4⋅+ bxxxx

3. E1h

1 v12 E2E1⋅−

⋅ 2 kx⋅+ kxxxx

4. E2h

1 v12 E2E1⋅−

⋅ 2 ky⋅+ kyyyy

5. v1E2E1⋅ E1

h

1 v12 E2E1⋅−

⋅ 2 kx⋅+⎛⎜⎜⎝

⎞⎟⎟⎠

⋅ kxxyy

E1,E2, such as kx>0, ky>0:

G0 1630894847.794=

h 0.227 m=

κ 1.944= [-]

[m]

[kPa]

ζxx

[kPa]

[kPa]

[kNm]

[kNm]

[-]

[m2]

[m2]

h0.378=:=

ζxy

h0.378=:=

E1 2100=

E2 10=

x 0.007=

y 0.007=

kx 1233343.1=

ky 1172.1=

v1 0.0599=

Identification of folded shell model in ZSoil, cont.

Design conception of a bridge

ProfileSUPERCOR

Railway load(kl. 2)

R.C. foundation

10.013.0H=5.89

1.9

Backfill

5.0

381 140[mm]

7

Computer modeling in ZSoilGeneral assumptions

3D model, with materialsMohr-Coulomb (soil), elastic (shell, concrete)

small displacementuse of:

construction stages,contact (sheel-backfill),infinite elements,kinematic constraints

Nodes: 48880Nele : B8 - 37712

SQ4, CQ4 - 2304

embankment-old

rail-road bed

Model components

Continuum

embankment-backfill

oMPaE 36,175 == φ

oMPaE 41,250 == φ

concrete foundation

stone linning

Continuum

Model components, cont.

Infinite el.

Beam & truss

Model components, cont.

kinematicconstraint

Shellelastic, anizotropic

Contact

Model components, cont.

Existence functions

Results - final construction stage displacement, bending moments and forces in shell

Mxx

Nxx

Mxx

|u|

Ux Uy Uz IUImax [cm] 6,477 7,611 0,801

min [cm] -6,477 -4,596 -0,799

8,096

Results - construction stage,

uplift Uy=6,00cmmaximal displacement

Simulation of moving railway load

loads in ZSoil set on the top independently from FE discretizationset of 129 LTF to simulate moving load

Results – exploitation phazeDeformation under moving load

Ux Uy Uz IUI

max [cm] 0,778 0,635 0,708

min [cm] -0,286 -1,816 -0,7071,845

Results – exploitation phazeshell displ. at the top

left right

load placement load placement

Results – exploitation phazebending moment Mxx in shell

left right

load placement load placement

Persistent forcesNxx –after one run of load

Stress level in soil

Model Max. shell stress [MPa]Model 1 (basic) 246,36Model 2 (expanded) 168,63Model 3 (one shell) 129,81Swedish method 157,16

Model 1 Model 2

Model 3

ConclusionsResult comparison

Thank you for your attention

Michał WszołekPrzemysław Milczarek

Aleksander Urbański ( aurbansk@pk.edu.pl ) Karol Ryż