ADVANCEDANALYTICALTECHNIQUES

Dmitry Maslov & Roman GuzeevInstitute Giprostroymost Saint-Petersburg

Russian Federation

PART I

…finding a problem…

Issue:The program we use may seem incapable of solving a problem

Reasons:► Excessive demand► Lack of knowledge

Solutions:► Find a different program► Expand the limits of existing software

3

All the commercial FE software is featured with non-linear analysis

Non-linear analysis is an iterative process involving the modelmodification according to immediate results

“Expand the limits” means a way of making any possible change in the model after results review

4

Manual result handling requires “the art of mouse clicks” for large models

XXI century information technologies hint at some automation5

There is a variety of instruments for building an automation facility

► VBScript or JavaScript

► MatLab or MathCad

► General purpose programming languages

Result postprocessing and interaction with solver functionality are theonly two things to be developed

6

Scripts:Easy to master but produce very slow code

Native code:Runs fast but requires programming skills to be developed

Engineers are not programmers, programmers are not familiar withStructural Mechanics

A balanced solution has to be found7

ORCODEN is the new name for Expression Converter program

New features :► Unicode text editor with syntax highlighting,

multiple undo, find and replace features,advanced clipboard facilities, popup hints,and context-sensitive help

► Script debug tools, breakpoints, watches, step-by-step tracing

► Resource-intensive GT STRUDL interaction is developed as built-infunctions which users treat as integral part of script language

8

Composite cross-section bridges

One of the simplest concepts causesadditional problems in its analysis

9

Construction Stages:

► Assembling of steel beams at construction site► Incremental launching using temporary and rolling piers► Deck concreting at bridge spans► Dismantling temporary piers► Deck concreting above piers► Making road surface

Service Actions: ► Carrying temporary loads► Suffering creep and shrinkage effects► Possible seismic events

10

Analytical model for composite section

N M

Such modeling produces a slightly incorrect diagrams

11

Construction steps for the analytical model:

► All deck members deactivated, pier joints declared supports, the structure carries the self weight of metal beam and the concreteof the first two divisions

Sq1Cq 2Cq

12

Construction steps for the analytical model:

► Truss members of the first two concreting divisions activated,temporary support joints made free, the structure carries thethird concreting division weight

3Cq

13

Construction steps for the analytical model:

► All members activated, the structure carries the remaining loads

IIq

14

Are the final displacements correct?

15

NO!

At the second stage we had to apply inverse joint reaction in the jointsmade free, which would make the concrete carry the self weight

1Cq

1R 2R 3R 4R16

Correct sequence

The joint reaction compensating for the stress can only be found afterthe first stage completed

Thus, the analysis after the first stage has to be suspended, reactionslisted and put into the model for stages 2 and 3

ORCODEN script performs the entire routine automatically from creating the model up to displaying the results

17

Displacements doubled in the “correct” model

18

So did stresses…

19

A new LOADING command would be useful

L

L

i JOINTSCOMPENSATING (FOR STRESS) LOADING FACTOR v

'a ' MEMBERSlist...list

⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬

⎩ ⎭⎩ ⎭

The command is supposed to calculate compensating joint forces beforethe user specifies STATUS FREE or INACTIVE MEMBERS/ELEMENTSfor the staged analysis

20

Plate deactivation problem

Concrete structural elements do not work in tension areas due to cracks

Tension areas are dependent on loading

Each loading has to be applied twice: in the ‘whole’ model and afterdeck deactivation

ORCODEN script performs the entire routine automatically aftersome modification of the ‘correct’ model

21

Displacement diagram after the deck deactivation

22

Stress diagram after the deck deactivation

23

According to our experience, it’s quite enough to find tension areas under dead load, deactivate the deck, and proceed with live loadanalysis…

24

Incremental launching

Consequent analysis of many steps for a mechanical system withunilateral constraints with initial gaps

25

Methods for finding solution:

► Disregarding “unilateracy” and/or gaps

26

Methods for finding solution:

► Traditional Rabinovitch algorithm

At each iteration all supports with negative reaction are detached (declared free joints), and detached supports with negative displacementare attached (declared support joints)

This algorithm may enter an endless loop and collapse at an iteration,even though the solution definitely exists

27

Methods for finding solution:

► Modified Rabinovitch algorithm

At each iteration only the support joint with maximal negative reactionis detached and the joint with maximal negative displacement isattached

The chance of looping and collapsing for this approach is way less thanfor traditional method

28

Methods for finding solution:

► Optimization problem

Quadratic programming approach: in the case of incremental launching,the cost function definition involves positive semi-definite matrix, thus not every algorithm is suitable

General optimization approach turns to either Rabinovitch algorithms

29

Methods for finding solution:

► Compression-only non-linear springs

Exact values for their stiffness have to be found. These cannot be orderslarger than the beam stiffness or the results will be absolutelyincorrect

We choose the modified Rabinovitch algorithm

30

Straight and plain models can be drawn in AutoCAD using annotativeblocks, then processed in Orcoden, which runs GT STRUDL and results inDXF file with force, moment, reaction, and stress diagrams

31

Curved and sophisticated models with thousands of joints and finiteelements still need GT STRUDL meshing facilities to be created

Joints: 17000Elements: 18500Steps: 200

32

PART II

…offering some science…

Creep and shrinkage basic assumptions

Concrete creep:

► Creep deformation is proportional to elastic deformation (linear creep)

► The ratio between creep and elastic deformation is a creep coefficient function

0( , )cr t tε

00

( , ) ( , )( )

cr

el

t t t tt

ε= ϕ

ε

( )el tε► creep deformation

► elastic deformation

► creep coefficient function

► concrete age at loading application moment

0( , )t tϕ

0t

34

Creep and shrinkage basic assumptions

Concrete shrinkage:

► Shrinkage deformation is time dependent function

( ) ( ) ( )sh sht f tε = ε ∞

► shrinkage deformation

► final shrinkage deformation

► shrinkage function

( )sh tε

( )shε ∞( )f t

35

Creep coefficient and shrinkage functionsfor composite bridges

(1 )( ) tkt e −αϕ = ϕ

( )1( ) ( ) tsh sht e −βε = ε ∞

1.6 1.8kϕ ≈ −final creep coefficient at t →∞

4( ) 2 10sh−ε ∞ = ×

final shrinkage deformation at t →∞36

Elastic deformation obtaining at moment t0

,0 0 / ( )el b bF E Aε =

t=t1…tn

► Incremental initial deformation calculation

► Member distortion calculation

► Bridge analysis on action of external loading and member distortion obtained

► Obtaining elastic deformation for the next step

Post-processing

, , , 1 ,i cr i sh i i el i sh i−∆ε = ∆ε + ∆ ε = ∆ϕ ε + ∆ ε

1 0i i id d l−= + ∆ε

, / ( )el i i b bF E Aε =

Step-by-step algorithm for creep and shrinkage problem

37

Seismic isolation and energy dissipation

Hydraulic damper Friction pendulumbearing

Anti-seismic devicesfor seismic isolation and energy

dissipation

►Rubber bearings with lead core

► Friction pendulum bearings

► Hydraulic dampers

► Steel hysteretic dampers

Main objectives of anti-seismic devices

►Increasing effective natural period

► Energy dissipation

38

Seismic isolation analysis

Direct approach

► Artificial accelerogramgeneration

► Nonlinear time history analysis

Simplified approach

► Response spectrum and damping scale factors are specified by code

► Response spectrum analysis .

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

Natural Period T, sec

β

Ground type II

Ground type I

39

Bridge analytical model with anti-seismic devices

ξeff(d)Κeff(d)

ξeff(d)Κeff(d)

ξeff(d)Κeff(d)

d

► The problem is that neither secant stiffness nor damping ratio which are dependent on amplitude can be determined in advance. ► We need an iterative process for determination of required parameters.

2

12

Deff

eff

EK d⎡ ⎤

ξ = ⎢ ⎥π ⎢ ⎥⎣ ⎦

- energy dissipated during vibration cycle

DE

40

Friction pendulum bearing Hydraulic damper

4D d sd bdE N d= µ

1 deff sd

bd

K NR d

⎛ ⎞µ= +⎜ ⎟

⎝ ⎠

max bdF C α= v0.15α ≈

max4D bdE F d≈

effbd

FKd

≈ max

41

The analysis of seismically isolated structure comes down to finding the solutionof non-linear equation with unknown displacements

1. Setup initial value of horizontal displacement d.2. Initial secant stiffness and damping ratio computation

1. Eigenproblem analysis2. Response spectrum analysis

Obtaining displacement from DBX

Adjust stiffness and damping ratio

1 5%i i

i

d dd

−−>

Postprocessing 42

Example of seismic isolation analysis

Friction pendulum bearingEffective curvature radius R=3.2mDynamic friction coefficient µ=0.055

Hydraulic damperVelocity exponent 0.15Maximal damper force Fmax=50 mton

max1 1 abeff

V FKV R d d

µ⎡ ⎤= + +⎢ ⎥⎣ ⎦

[ ]max4D abE V F d= µ +

Abutment Middle pier1 1 m

effm

VKV R d

µ⎡ ⎤= +⎢ ⎥⎣ ⎦

4D mE V d= µ

2

12

Deff

eff

EK d⎡ ⎤

ξ = ⎢ ⎥π ⎢ ⎥⎣ ⎦

Vab, Vm – support reaction caused by dead load 43

Response spectrum

10%ξ =

2%ξ =5%ξ =

10% 0.555%

η = ≥+ ξ

- damping scale factor

Peak ground acceleration 0.6g

44

Results of seismic isolation analysis

Effective natural period T=1.8sDesign horizontal seismic displacement ±430mmAcceleration of the bridge girder 0.56g<PGA=0.6g

45

Conclusions:

► No need to give up using the software if it seems incapable of something

► There is a way of going beyond the limits

Advice:

► Don’t trust the digit until it has been checked seven times

Questions:► ???

46

THANK YOU FOR YOUR

ATTENTION