The 5th PSU-UNS International Conference on Engineering and

Technology (ICET-2011), Phuket, May 2-3, 2011 Prince of Songkla University, Faculty of Engineering

Hat Yai, Songkhla, Thailand 90112

Abstract: Special finite elements (FEs) are used for

modeling of various phenomena in structural behavior.

The term "special" denotes FEs with specific

performances (stiffness parameters, behavior under load,

etc.) and particular role in modeling, what is their main

difference from usual FEs. There are many types of

special FE, but the following are often used: so-called

"nonlinear spring" FE, "gap" FE and "link" FE. On the

various examples it will be shown many advances in

application of these FE, especially in modeling of

boundary and interface conditions.

Key Words: FEM Modeling, Special FE, Boundary

and Interface Conditions

1. INTRODUCTION

FEM (Finite Element Method) has become a

dominant method of numerical modeling of complex

structural problems and has status of particular

"technology" for the structural analysis. CASA

(Computer Aided Structural Analysis) software based on

FEM has sophisticated options i.e. possibilities of

realistic simulation of structural behavior and analysis of

structural response (displacements and stresses) for

different actions (influences and forces).

Choice of FE type considerably determines the

quality of FEM solution. Figure 1 shows one of the many

possible classifications of FEs - classification based on

FE type and its role in modeling.

Fig. 1. Classification regards to the FEs type

Special FEs group are in use for modeling of:

various types of indirect connections between

standard FEs or between standard FEs and outer

structural nodes,

various types of shear, sliding or frictional

conditions between standard FEs,

changing of nature of connection between

standard FEs because of change in external load

conditions, etc.

Some well-known advantages of 3D FEs aren't so

expressed in case of application of special FEs together

with 1D and 2D standard FE for modeling.

2. INDIRECT CONNECTIONS IN FEM MODELS

It is well known that FEs can be connected directly

(when have common nodes) and indirectly. Indirect

connections can be consequence of eccentricity or

distance between nodes by which FEs are joined, Fig 2.

Fig. 2. Examples of indirect connection in models

The most evident examples are: connections between

FEs (beam/beam, plate/plate...) of different thickness,

connections with eccentricity between structural

elements with various functions in structural system

(wall/plate, column/beam, beam/plate...).

PRACTICAL ASPECTS OF THE SPECIAL

FINITE ELEMENTS APPLICATION

D. Kovacevic*, A. Antic, I. Budak University of Novi Sad, Faculty of Technical Sciences, Novi Sad, Serbia

*Authors to correspondence should be addressed via email: dusan@uns.ac.rs

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Mathematically models of such type of connections

can be described as so-called "interface conditions".

When these models are related with structural supports

then they are so-called "boundary conditions". Figure 3

shows typical "beam-plate" connection with

corresponding eccentricities "ex" and "ey".

еx

еy

h

d

w

Plate

Beam

k

ji

Fig. 3. "Beam-plate" connection with eccentricities

In the modeling of real structural behavior particular

importance has a phase of modeling of real behavior of

boundary and interface conditions. Special FEs make

possible modeling of the most variable structural support

conditions (various types of restrains) and structural

connection conditions (various types of interface

compatibility status).

3. SHEAR, FRICTIONAL AND ROLL

CONNECTIONS IN FEM MODELS

Many connections in civil and mechanical

engineering are based on shear or frictional stiffness and

sliding or rolling capability between two or more parts

made by one or more materials.

Fig. 4 shows some of these cases. Typical examples

of these connections are coupling of structural parts

made by different materials (reinforced concrete, steel-

concrete composite) and rolling connection (axle/bearing

joint), etc.

Fig. 4. Some shear/friction/sliding connections in models

Typical use of special FE (link FE especially) is

characteristic in this area of modeling. It is possible, by

convenient choice of shear/friction stiffness parameters

of special FE, to model various types of connections,

what will be illustrated by a few numerical examples.

4. CHANGING OF NATURE OF CONNECTION

IN FEM MODELS

This group comprises connections with variable

behavior under external loading. The most often

situations for modeling of such type of special FE are

cases of breakup and merge of model connections in

some load circumstances. Figure 5 shows example of

model's nodes which will be merged when distance

between them almost completely disappears (e.g. wood

insert between steel tube and concrete beam). Contrary,

when force (or stress) between merged joints (e.g. steel

tube hanging on concrete beam) reach some limit state

they will be detached.

Fig. 5. Connections which change nature for some load

or displacement circumstances

Modeling of these phenomena needs some

considerations in nonlinear theory domain. For that

reason it is needed to explain some details about

nonlinear structural analysis of boundary/interface

conditions.

There are many various classifications of nonlinear

phenomenon/effects. According to influence of

boundary/interface conditions to nonlinear behavior there

are:

phenomena of so-called "smooth" (continual)

nonlinearity and

phenomena of so-called "rough" (non-continual)

nonlinearity.

In general, behavior of boundary/interface conditions

can be described by application of some "smooth"

nonlinear model if these conditions are constant or

continually variable. If a change of boundary/interface

conditions is non-continual, it is necessary to apply some

"rough" nonlinear model.

Formulation of adequate FEM models for

discontinued nonlinearity is not only formal

(mathematical) problem, but it requires understanding of

the causes of discontinuity of nonlinear phenomenon.

For non-continual nonlinear problems, there are no

common solutions, and it is necessary to formulate

special techniques for every specific case of such

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nonlinear behavior. It is only unquestionable that one

incremental-iterative solution technique should use for

practical solving of nonlinear boundary/interface

condition problems. Fig. 6. shows schematic diagram of

incremental-iterative nonlinear solution.

ΔFi

Δuј

ΔR

ј

Δui

incremental-iterative solution

exact

solution

Fig. 6. Incremental solution of nonlinear problem with a

iterative improvement

4. NUMERICAL EXAMPLES

Following numerical examples could be illustrative

for presentation of advantages of use of special FEs in

numerical modeling of structural behavior.

Fig. 7 shows stress distribution for cantilever girder

which is made from beam and plate structural elements.

Model includes beam FEs, shell FEs and link FEs in

between. AxisVM® CASA software dialog box for

definition of link FE parameters is given on the Fig 8.

Fig. 7. Structural and FEs for beam-plate girder

Fig. 8. Definition of link FE parameters in AxisVM

®

It is possible to define perfect shear stiffness of

surface connections by choice of large values of all six

stiffness parameters. Three parameters are for axial

stiffness of connection and three are for flexural stiffness

of connection. Interface location parameter represents

position of common point between FEs which are joined

by link FE.

Reinforced concrete plate structural element, Fig. 9,

can be modeled by use of shell FEs for plate, beam FEs

for reinforcement and link FEs for connection between

them.

Fig. 9. RC plate FE model

Behavior of so-called "sandwich-plate" can be

modeled in similar way. Fig. 8. shows composite plate

formed by a three layers: steel foils (top and bottom face)

and polystyrene foam core (between steel foils).

Fig. 10. "Sandwich" composite plate FE model

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Stiffness and bearing capacity of this plate is based

on shear connection (so-called "bond") between foils and

core. Bond deterioration generates stiffness and bearing

capacity loss. Model contains top and bottom steel foil,

polystyrene foam core and link FE between foils and

core. Modeling of bond is performed by increasing or

decreasing stiffness parameters of link FEs. Figure

shows the stress peaks in zones with bond deterioration.

This bond behavior is modeled by small value of link

FEs shear component stiffness parameter.

Next two examples are dedicated on the explanation

of the possibilities of hinged (rolling) connections

modeling.

In the following numerical example one "externally"

prestressed beam is presented, Fig. 11.

Fig. 11. Externally pre-stressed beam with rollers in

cable/column joint

Prestressing force, from "external" cables (located

out of beam cross-section), causes lift of beam through

columns. High friction forces appear in the cable/column

joint area, as unwanted consequence. In order to decrease

friction, rollers are usually installed in cable/column

joint. Connection between rollers and cables is modeled

by link FE and by standard FE and results of

computation are given in Fig. 12.

The model with link FE gives more realistic moment

distribution and especially axial forces distribution. It is

well known that cable has constant axial force along its

length. Only with link FE it is possible to model the

phenomenon of friction between roller and cable.

Fig. 12. Flexural moments and axial forces for models

with link FE (up) and standard FE (down)

Next example shows possibility of modeling of rolling

connection in mechanical engineering structures. The

connection jib-axle is actually a cylindrical hinge which

allows only the rotation around the own axis. In Fig. 13

there is a model of this hinge with the distribution of link

FEs.

thinner part

of axle

thicker part

of axle

hinge part

of jib

thinner part

of axle FE

thicker part

of axle FE

hinge part

of jib FEssingle

link FE

Fig. 13. Modeling the hinged support by link FE

As it can be observed, link FEs radially join a node of

the FE axle and adjoining nodes of the shell FE of the jib

hinge.

From the six stiffness parameters of the link FE, only

the one related to the rotation around the own axle axis

has the zero value, and thus the ideal rotation without

friction can be modelled. All other parameters have the

value corresponding to friction and flexural or axial

stiffness. In such a manner, a simple and satisfactory

accurate model of this connection is obtained. In every

other case of modelling parasite forces/moments would

"appear" which not the real presentation of the hinge

behaviour is.

One of the causes of nonlinear behaviour in structural

systems is so-called "contact nonlinearity". This

nonlinear phenomenon appears due to changes in

structural systems according to "activation" or

"deactivation" of boundary/interface conditions. Usual

appearance of contact problem is due to loss of support

because of lifting of structural joint upwards or due to

realisation of connection between previously non-

connected structural joints. The first reason for contact

problem is in the domain of boundary conditions change,

while the second one is in the interface conditions

domain. Gap FE is used only in nonlinear analysis for

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modelling of changes in compatibility conditions of

displacements between structural elements.

Two beams in space with the gap located in the half

of their span, Fig. 14, is a numerical example which

illustrates application of gap FE in modelling of the

connection activation.

Fig. 14. Beams in space with 50mm gap in between

Structural behaviour is modelled by the gap FE with

stiffness of sAC=107kNm and sBC=10

-1kNm after the

contact and before the contact realisation and with length

according to geometry, Fig. 15.

Fig. 15. Gap FE for modelling of connection establishing

For load of P=50kN the system has displacement of

δNL=83.6mm. Fig. 16. shows system's nonlinear response

curve with one break point (P=21.4kN, δ=50.0mm)

which represents contact realisation. The first segment of

curve has a smaller slope than the second one which

represents the phase after the contact realisation, when

the stiffness increases.

Fig. 16. Nonlinear response of model with gap FE

5. CONCLUSIONS

The goal of this paper is emphasizing the possibilities

of creative use of special FE: nonlinear spring, gap and

link FE. Sometimes the modeling process needs a

nonstandard approach. The usual approach is oriented to

application of very complex and ineffective models

based on 3D FE even if these are not necessary. With

engineer-like approach, the use of special FE enables

relatively simple models, which correspond to requests

of everyday engineering/design practice.

6. REFERENCES

[1] Cook, R.D. "Finite Element Modeling for Stress

Analysis", John Wiley & Sons, Inc., 1995.

[2] Kovačević, D. "Numerical analysis and computation

of the jib structure of waterway bucket dredger -

Technical report", Ship Registry of Republic of

Serbia, 2009.

[3] Kovačević, D, Folić, R. "Some Aspects of FEM

Structural Modelling by Link FE", Proceedings of the

11th International Conference on Civil, Structural

and Environmental Engineering Computing -

CC2007, Malta, 2007. pp. 211-226.

[4] "AxisVM® 10.0 User's manual", InterCAD, 2009,

Budapest.

[5] Kovačević, D. "FEM Modeling in Structural

Analysis" (in Serbian), Građevinska knjiga, 2006,

Belgrade.

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