Materials and structures

Lateral deflections for the series 1B specimens Comparing the graphs presented in Figs 4 and 5 we can notice that in-plane shear behaviour of the series 1B specimens was more plastic tha

Vilnius Gediminas Technical University Lithuanian Academy of Sciences

Journal of Civil Engineering and Management

2004, Vol X, Supplement 1

Vilnius „Technika“ 2004

ISSN 1392-3730

EDITORIAL BOARDEditor-in-ChiefProf Edmundas K ZAVADSKAS, Lithuanian Academy of Sciences,Vilnius Gediminas Technical University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Editors

Managing editorAssoc Prof Darius BẰINSKAS, Vilnius Gediminas Technical University,

Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Dr Rogerio BAIRRAO, Portuguese National Laboratory for

Civil Engineering, Av Brasil, 101, 1700-066 Lisboa, Portugal

Prof Gyưrgy L BALÁZS, Budapest University of Technology

and Economics, Mûegyetem rkp.3, H-1111 Budapest,

Hun-gary

Assoc Prof Erik BEJDER, Aalborg University, Fibigerstraede

16, 9220 Aalborg, Denmark

Prof Adam BORKOWSKI, Institute of Fundamental

Techno-logical Research, Swiỉtokrzyska 21, 00-049 Warsaw, Poland

Prof Michá BOLTRYK, Biáystok Technical University,

Wiejska 45A, 15-351 Biáystok, Poland

Prof Patrick J DOWLING, Felow Royal Society, University

of Surrey, Guildford GU25XH, UK

Prof Aleksandr A GUSAKOV, Moscow State University of

Civil Engineering, Dorogomilevskaja, 5/114, 121059 Moscow,

Russia

Prof Boris V GUSEV, International and Russian Engineering

Academies, Tverskaja 11, 103905 Moscow, Russia

Assoc Prof Edward J JASELSKIS, Iowa State University,

Ames, IA 50011, USA

Prof Oleg KAPLIĐSKI, Poznan University of Technology,

Piotrovo 5, 60-965 Poznan, Poland

Prof Herbert A MANG, Austrian Academy of Sciences,

Vienna University of Technology, Karlsplatz 13, A-1040

Vienna, Austria

Prof Antanas ALIKONIS, Vilnius Gediminas Technical

Uni-versity, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Prof Juozas ATKOÈIÛNAS, Vilnius Gediminas Technical

University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Prof Algirdas E ÈIÞAS, Vilnius Gediminas Technical

Uni-versity, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Assoc Prof Juozas DELTUVA, Kaunas University of

Tech-nology, Studentø g 48, LT-3028 Kaunas, Lithuania

Prof Romualdas GINEVIÈIUS, Vilnius Gediminas Technical

University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Prof Arvydas JUODIS, Kaunas University of Technology,

Studentø g 48, LT-3028 Kaunas, Lithuania

Prof Pranciðkus JUÐKEVIÈIUS, Vilnius Gediminas

Techni-cal University, Saulëtekio al 11, LT-10223 Vilnius-40,

Lithuania

Prof Rimantas KẰIANAUSKAS, Lithuanian Academy of

Sci-ences, Vilnius Gediminas Technical University, Saulëtekio al.

11, LT-10223 Vilnius-40, Lithuania

Prof Gintaris KAKLAUSKAS, Vilnius Gediminas Technical

University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

International Editorial Board

Prof Rene MAQUOI, University of Liege, Building B52/3, Chemin des Chevreuils 1, B 4000 Liege, Belgium

Prof Yoshihiko OHAMA, Nihon University, Koriyama, Fukushima-Ken, 963-8642, Japan

Prof Friedel PELDSCHUS, Leipzig University of Applied Science, 132 Karl Liebknecht St, 04227 Leipzig, Germany Prof Karlis ROCENS, Latvian Academy of Sciences, Riga Technical University, Âzenes str 16, Riga, LV-1048 Latvia Prof Les RUDDOCK, University of Salford, Salford, Greater Manchester M5 4WT, UK

Prof Miroslaw J SKIBNIEWSKI, Purdue University, West Lafayette, Indiana 47907-1294, USA

Prof Martin SKITMORE, Queensland University of

Techno-logy, Brisbane QLD 4001, Australia Prof Zenon WASZCZYSZYN, Cracow University of Techno- logy, Warszawska 24, 31-155 Krakow, Poland

Prof Frank WERNER, Bauhaus University, Marienstrasse 5,

99423, Weimar, Germany Prof Nils-Erik WIBERG, Chalmers University of Technology,

SE - 412 96 Gưteborg, Sweden Prof Jiøí WITZANY, Czech Technical University, Prague, Thákurova 7, CZ 166 29 Praha 6, Czech Republic

Local Editorial Board

Prof Stanislovas KALANTA, Vilnius Gediminas Technical University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania Prof Ipolitas Z KAMAITIS, Lithuanian Academy of Sciences, Vilnius Gediminas Technical University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Prof Romualdas MẰIULAITIS, Vilnius Gediminas cal University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Techni-Prof Gediminas J MARÈIUKAITIS, Vilnius Gediminas nical University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Tech-Prof Josifas PARASONIS, Vilnius Gediminas Technical versity, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania Prof Vytautas STANKEVIÈIUS, Lithuanian Academy of Sciences, Lithuanian Institute of Architecture and Building Construction, Tunelio g 60, LT-3035 Kaunas, Lithuania Prof Vytautas J STAUSKIS, Vilnius Gediminas Technical University, Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania

Uni-d a sl a ir e t a M

t

S StruIcntuf orar mlMaeitcnhTneiccshao ldgPihe syisc,s C nstrucMitanaTeecmhnotl gya df

o

r

, y ti s e v i n U l a c i n c e T s

, y ti s e v i n U l a c i n c e T s a i m i d G s u i n li V

, 1 l a o i k t ë l u S

a i n u ti L , 0 - s u i n li V 3 2 1 - T

f o r

P A r t û r a s K K L A U S A S ,

, y ti s e v i n U l a c i n c e T s a i m i d G s u i n li V

, 1 l a o i k t ë l u S

a i n u ti L , 0 - s u i n li V 3 2 1 - T

Piotr Aliawdin1, Valery Simbirkin2, Vassili Toropov3

1University of Zielona Góra, Poland E-mail: P.Aliawdin@ib.uz.zgora.pl

2Belarussian Research Institute for Construction (BelNIIS), Minsk, Belarus E-mail: simbirkin@hotmail.com

3Altair Engineering, Coventry, UK E-mail: toropov@altair.com

Received 30 Apr 2004; accepted 7 June 2004

Abstract The paper presents results of large-scale tests carried out on masonry wall panels made of perforated bricks The specimens were subjected to in-plane: lateral loading combined with different levels of axial compression; concen- trated compressive load applied to the wall top at different distances from the wall edge Relationships between shear strength and deformability of masonry and compressive stresses perpendicular to the shear plane have been found An evaluation of strength of masonry under local compression is given depending on the position of the concentrated load relative to the wall edge Analysis of test results and comparison of calculation techniques adopted in different design codes is performed Behaviour of the test specimens is modelled using the finite element method.

Keywords: masonry structures, full-scale tests, shear, compression, strength, deformations.

1 Introduction

By the present time, an extensive theoretical and

experimental research has been carried out on the

behaviour of masonry structures made of solid clay

bricks, for instance [1–5] However, there are a few test

results for masonry structures made of perforated bricks

that are widely used in practice and have a number of

advantages

This study presents an experimental and analytical

research into the behaviour of masonry wall panels made

of perforated clay bricks The test specimens were

sub-jected to in-plane 1) local compressive force, and 2)

rack-ing shear force combined with vertical compression

For each loading type, two test series have been

devised In the local compression tests, position of the

applied force was changed In the shear tests, lateral force

was combined with different levels of axial compression

In the first case, vertical kinematic restraints were

in-stalled on the wall top to prevent in-plane rotation of the

walls The vertical pressure arising in this case varied

during the loading process and had the minimum value

In the second case, the lateral load was combined with

the given vertical compression

The loading of the specimens was increased

mono-tonically up to the total failure of the specimens The

resistance of the masonry walls to the predominant

ac-tion was evaluated with reference to the strength and

deformability

2 Properties of masonry and masonry materials

The following materials were used for producing thetest specimens:

• Clay bricks (length 250 mm, width 120 mm, height

88 mm) with vertical holes Each brick had 21 holeswhose cross-sections were square-shaped, 20x20 cm(volume of holes is 28 % of the gross volume) Brickgrade M150

• Dry pre-packed mortar mix, grade M100: Portlandcement of grade 500ÄÎ – 180 kg/t, lime – 50 kg/t,sand – 770 kg/t, water-retaining agent Valotsel

45000 – 0,3 κg/t

The strength properties of the brick and mortar weredetermined experimentally Their mean values are pre-sented in Table 1

Table 1 Brick and mortar strengths

a P M , h t g e r t s k c ir B h si ti r B y b ( e v is s e r p m o C

,]

6 [ 1 9 S B d r a n t S

) D x i d e p

s k c ir b g it s e t y b ( e li s n T

) g i d e r o f 6

,

a P M , h t g e r t s r a t r o M g it s e t y b ( e v is s e r p m o C

) m m 7 , 0 e is f o s e u

c hf rag m(bnytteositt hgraembairsc kn)ry

7 , 2 … 9 ,

4 P Aliawdin, et al / JOURNAL OF CIVIL ENGINEERING AND MANAGEMENT – 2004, Vol X, Suppl 1, 3–9

Strength and deformative properties of the masonry

under short-term compression were determined by tests

of five prismatic specimens having dimensions

lxhxt = 380×490×250 mm On all four vertical sides of

each specimen, displacement transducers were installed

over a gauge length of 200 mm They measured

longitu-dinal (vertical) and lateral (horizontal) deformations of

the masonry The strains measured in this way were used

to calculate the deformation modulus and the Poisson’s

ratio of the masonry

While testing the specimens, the mortar

compres-sive strength was checked Its mean value was 9,9 MPa

The tests showed that the masonry compressive

strength ranged between 8,4 and 11,1 MPa, and its value

averaged over strengths obtained for five specimens was

equal to σult= 9,3 MPa

Averaged curves for strains, secant deformation

modulus, and Poisson’s ratio of the masonry are

Fig 1 Dependences of strains ∑ , secant deformation

modu-lus Esec, and Poisson’s ratio upon stress level for masonry

under axial short-term compression

The initial modulus of elasticity of the masonry iscomputed according to [7] using the following logarith-mic stress-strain relation proposed by L I Onistchik:

The test specimens were divided into two series(Fig 2) The specimens of the first series (series 1A) weretested for incremental lateral load P, applied to the top

of the panel in its plane, combined with minimal verticalpressure that was necessary to prevent in-plane rotation

of the wall The vertical pressure was produced by springkinematic restraints on the wall top and varied duringloading so that detachment of the wall bottom from thefloor was not greater than 5 cm

Displacement transducers (LVDTs) were installedalong the wall height to measure lateral deflections dur-ing loading (Fig 2) In addition, displacement transduc-ers were used to measure translation of the horizontalsupport and detachment caused by a compliantly re-strained rotation of the wall in its plane Their readingswere taken into account for calculation of the “clear”lateral deflections by correcting the values obtained byLVDTs Th1…Th5

Unlike the first type specimens, specimens of theseries 1B were loaded, in addition to the lateral load P,with a vertical uniformly distributed load q equal to0,2Fk= 225 kN/m, where Fk is the ultimate failure load

in the pure compression case This load did not varyduring the testing The load P was applied to four toprows of bricks, and displacements were measured only

at one level (at a height of 1450 mm from the wall tom)

bot-The test showed that specimens of the series 1Acollapsed immediately after a zigzag crack has appeared

a) Test series 1A (three specimens)

b) Test series 1B (three specimens)

Fig 2 Shear test setup

along the wall diagonal connecting the lateral loading

point and the horizontal support (Fig 3, a) The failure

lateral load was equal to: 120,0 kN for the first

speci-men, 113,8 kN for the second specispeci-men, and 80,0 kN

for the third one Therefore, the failure lateral load

aver-aged over three these values was Pult= 104,6 kN At the

ultimate stage, average total value of the compressive

load q was equal to 118 kN

Experimental graphs showing the deforming process

of the series 1A specimens are presented in Fig 4

The walls of the series 1B having been tested for

combined shear and compression failed also with an

in-clined crack connecting the lateral loading point and the

horizontal support However, in this case some vertical

– LVDTs

Fig 4 Lateral deflections for the series 1A specimens:

a) distribution of displacements along the wall height;

b) load–displacement relationships

The lateral load-displacement relationship averaged

over results of three tests of the series 1B is shown in

Fig 5 Lateral deflections for the series 1B specimens

Comparing the graphs presented in Figs 4 and 5 we

can notice that in-plane shear behaviour of the series 1B

specimens was more plastic than the behaviour of the

series 1A specimens which deformed almost elastically

up to the failure (excepting a displacement leap observed

at the second loading stage) and collapsed in a brittle

mode Indeed, in the series 1A specimens the cracks were

not observed up to the failure, but cracks in the series

1B specimens appeared under the lateral load equal to

0,3 to 0,4 of the ultimate load However, the specimens

of the series 1A had a much lower rigidity than those of

the other test series Their failure occurred at lateral

deflections that were an order of magnitude higher than

ultimate deflections of the series 1B specimens

over, the compressive action on the masonry walls sulted in 84 % increase of the load-carrying capacity ofthe walls under lateral loading

re-Therefore, the effect of vertical compression leads

to a higher resistance of the masonry walls to shear loads,making their rigidity and load-carrying capacity higher.Behaviour of the test specimens is modelled on thefinite element basis using Software Stark_Es of theMicroFE family The wall panels are modelled withhighly accurate hybrid plane stress elements (mesh 30x30)derived using a Reissner functional [8] Second ordergeometrical effects and unilateral elastic supports aretaken into account As an example, Fig 6 shows someanalysis results for the specimens of series 1A

The test results presented above enable to draw anexperimental relationship between the shear strength andcompressive stress rate in masonry This relationship ispresented in Fig 7

As we can see in Fig 6, the masonry shear strengthdepends almost linearly upon the compressive stress level

óz

0 -0,40 -0,80 -1,20 -1,60 -2,00 -2,40 -2,80 -3,20 -3,60 -4,00 0,00 0,15 0,30 0,45 0,60 0,75 0,90 1,05 1,20 1,35 1,50

Fig 7 Relationships between masonry shear strength and

compressive stress level

Hence we can propose the following empirical formula

for approximate evaluation of the shear strength of

ma-sonry in a plane stress state for different levels of the

compressive stresses:

z ult

τ = ,0+0,28 , (2)where:

ult

τ is the masonry shear strength;

z

σ is the mean compressive stress perpendicular to

the shear plane;

0

,

ult

τ is the initial masonry shear strength, under

zero compressive stress

In equation (2), all magnitudes are in MPa

Equation (2) is valid for only the cases where the

compressive stress ⌠ does not exceed 0,2 of the ultimate

compressive strength

A similar relationship is given in Eurocode 6 [9] to

compute the masonry shear strength depending on the

compressive stress value In our case, this strength should

be determined using equation 3.3a [9] but its value must

be not higher than a value computed by equation 3.3c

[9] A graphical representation of the values calculated

by these equations for our cases is given in Fig 7 As

can be seen, equation 3.3a overestimates the shear

strength of masonry, but equation 3.3c provides a rather

high safety margin for the masonry shear strength

4 Response to local compression

For local compression tests of masonry walls, six

specimens were produced and stored analogously as

de-scribed in the previous section

The test specimens were tested to collapse for

con-centrated vertical load P applied incrementally at a

dis-tance 650 mm (series 2A) and 350 mm (series 2B) from

the wall edge, as shown in Fig 8 The bearing area was

10×12 = 120 cm2

Along the loading line on both sides of the

speci-mens, displacement transducers (Tv, Fig 8) were installed

at the middle height over a gauge length of 800 mm to

measure mean vertical strains

The tests showed that the specimens of both series

had the same failure mode – the failure was practically

brittle with formation of a local failure zone under the

bearing and a vertical crack along the loading line (Fig 9)

a) Test series 2A (three specimens)

b) Test series 2B (three specimens)

Fig 8 Local compression test setup

Fig 9 Failure pattern

Until the load reached the value P = 150 kN, the

mean vertical strains increased with loading almost

iden-tically for specimens of both series and had a slightly

non-linear kind (Fig 10) However, further loading caused

a deviation of the “load-strain” curve for series 2B from

the direct line and from the curve shown by the series

2A specimens After that, under the load 188 to 200 kN

the failure of the series 2B specimens occurred The mean

value of the failure load for these specimens was

192,7 kN The series 2A specimens showed a higher

load-bearing capacity equal to 220 to 256 kN with the mean

Fig 10 Experimental “load-strain” curves

At the failure moment, the mean value of the mid

height vertical strain was 50⋅105 and 35⋅105 for

speci-mens of the series 2A and 2B respectively As can be

seen from Fig 1a, such strains correspond to

compres-sive stresses not exceeding a half of the ultimate strength

of masonry in pure axial compression Thus the failure

of the specimens was local below the loaded area

The results presented enable to evaluate the effect

of increase of the masonry resistance to concentrated

compressive loads as compared with overall axial

com-pression case Table 2 presents values of the

enhance-ment factor for concentrated loads obtained

experimen-tally and calculated according to different building codes

Table 2 Local compression effect

Enhancement factor for concentrated

compressive loads Test series

As we can see from Table 2, all design codes

pro-vide a rather high safety margin for the compressive

strength of masonry subjected to concentrated loads In

addition, Russian code [10] defines the same

enhance-ment factor for both the test series and, in contrast to

Eurocode 6 [9] and Polish code [11], does not take into

account changes of the masonry local compressive

strength depending on the wall height

The ultimate stage of the wall behaviour is

mod-elled on the basis of the finite element method using

Software Stark_Es Results of the analysis are given inFig 11

The analysis shows that for specimens of the ent series under the ultimate failure load the maximumcompressive stresses below the loaded area (óz) have thesame ratio as the loads applied However, calculated ten-sile stresses in the orthogonal direction (óx), which havecaused the vertical crack formation in the test specimens,

differ-in the series 2B specimens are 1,25 times greater than differ-inthe series 2A specimens even under a smaller load Thisindicates that in the series 2B specimens local compres-sion (casing-type) effect is not so significant than in theother series specimens This fact is affirmed by the kind

of deformation distribution in the vicinity of the loadedarea – in the series 2A specimens the effective area isgreater than in the other specimens From the deformedshape presented in Fig 11 we can assume that the effec-

Fig 11 Results of finite element analysis (displacement scale 200:1)

b) Series 2B a) Series 2A

9tive area includes wall parts of 250 mm length for the

series 2A specimens and 200 mm for the series 2B

speci-mens to both sides from the loaded area (but not 120

mm as adopted in code [10] for both our cases) In this

case, the enhancement factor calculated by Eq (19) given

in [10] would be equal to 1,82 and 1,71 for specimens

of the first and the second series respectively These

values are much closer to the experimental ones than

those calculated according to [10] Therefore, the

ma-sonry resistance to concentrated compressive loads can

be evaluated sufficiently accurate by the finite element

analysis

5 Conclusions

1 Large-scale tests carried out on masonry wall

panels subjected to in-plane lateral (shear) loading

com-bined with different levels of axial compression show

that:

• Behaviour of masonry wall panels subjected to pure

shear is almost perfectly elastic, the failure occurs

in a brittle mode Compressive load affects the shear

behaviour of the masonry making it plastic

• Shear capacity of masonry walls increases by about

80% due to the action of axial compressive load

equal to 20% of the ultimate compressive strength;

the lateral rigidity of such walls can be of an order

of magnitude higher as compared with the walls

un-der pure shear

2 Local compression tests of masonry walls show

that resistance of masonry to concentrated compressive

load depends significantly on the distance from the wall

edge to the load position even if this distance 2,5 times

greater than the wall thickness This fact is not taken

into account in SNiP II-22-81 [10] A finite element

analysis can be used for strength evaluation for masonry

subjected to concentrated loads

Acknowledgement The authors are pleased to edge the support of INTAS under international project00-0600

5 Kubica, J.; Drobiec, Ù.; Jasiñski, R Study of secant formation modulus of masonry In: Proceedings of XLV Scientific Conference KILiW PAN i KN PZITB Wrocùaw- Krynica, 1999, p 133–140 (in Polish).

de-6 BRITISH STANDARD BS 3921: Specifications for clay bricks London: British Standards Institution, 2001 22 p.

7 Sementsov, S A On the method of selection of mic stress-strain relation using test data In: Strength and stability of large-panel structures, Vol 15 Moscow: Gosstroyizdat, 1962, p 303–309 (in Russian).

logarith-8 Semenov, V A.; Semenov, P J Highly accurate finite ements and their use in software MicroFE Residential Construction, 1998, No 8, p 18–22 (in Russian).

el-9 prEN 1996-1-1: Redraft 9A Eurocode 6: Design of sonry structures – Part 1-1: Common rules for reinforced and unreinforced masonry structures – European Commit- tee for Standardization, 2001 123 p.

ma-10 SNiP II-22-81 Masonry and reinforced masonry structures Design Code (ÑÍèÏ II-22-81 Moscow: Gosstroi USSR,

FE SOFTWARE ATENA APPLICATIONS TO NON-LINEAR ANALYSIS OF RC

BEAMS SUBJECTED TO HIGH TEMPERATURES

Darius Bacinskas1, Gintaris Kaklauskas2, Edgaras Geda3Dept of Bridges and Special Structures, Vilnius Gediminas Technical University, Saulëtekio al 11, LT-10233Vilnius-40, Lithuania E-mail: 1Darius.Bacinskas@st.vtu.lt, 2Gintaris.Kaklauskas@st.vtu.lt, 3egeda@salmija.lt

Received 15 Apr 2004; accepted 23 Feb 2004

Abstract Reinforced concrete structures subjected to fire will generally experience complex behaviour This paper presents a strategy of numerical simulation of reinforced concrete members exposed to high temperatures and subjected

to external loading Finite element modelling of full load – deflection behaviour of experimental reinforced concrete beams reported in the literature has been carried out by the FE software ATENA A constitutive model based on Eurocode

2 specifications has been used in the analysis Comparison of numerical simulation and test results have shown able accuracy.

reason-Keywords: reinforced concrete fire design, non-linear finite element analysis, fire tests, fire resistance, constitutive models of concrete and steel.

1 Introduction

There are many buildings and civil engineering

struc-tures (tunnels, high-rise buildings, bridges and viaducts,

containment shells, offshore platforms, airport runways

etc.) under construction which are at risk of fire A few

dramatic accidents in recent years have prompted

inves-tigations in the field of safety of reinforced concrete

struc-tures subjected to fire Fires in railway Channel Tunnel

(autumn 1996), in the road tunnels of Mont Blanc

(France/Italy 1999), in the television tower of Ostankino

(Moscow, 2000), in the Twin Towers (New York, 2001)

should be mentioned [1] In all cases, the load-bearing

capacity of structure in the actual fire conditions is of

primary importance for evacuation of persons and things,

as well as for safety of rescue teams

The analysis of the behaviour of load-bearing

mem-bers under high temperature conditions is very

compli-cated [2, 3] Various factors influencing the behaviour

of members need to be taken into account, including:

variation of member temperature with time, variation of

temperature over the cross-section and along the

mem-ber, temperature effects on material properties

(expan-sion, creep, reduction in strength and stiffness, spalling,

etc), material non-linearity, external restrains, section

shape, etc A parametric study of the influence of

differ-ent factors on the behaviour of RC beams and frames is

presented in [4]

Because of the no-linear nature of the problem,

closed-form solutions usually cannot be found and an

iterative approach is required [5] The non-linear

behaviour of a member under elevated temperature ditions can be simulated using the finite element method[6, 7] Because of increasing interest in the field of struc-tural fire protection, the number of existing softwarecapable to analysing the thermal response of materialsunder transient heating conditions is quite large [8, 9].The majority of these programmes was developed inprofessional software houses, such as DIANA [10],ATENA [11], ABAQUS, MSC.MARC, etc Suchprogrammes have many advantages including documen-tation, sophisticated non-linear material models, pre/post-processing facilities, etc

con-This paper presents a strategy of numerical tion of reinforced concrete members exposed to hightemperatures and subjected to external loading Finiteelement modelling of full load – deflection behaviour ofexperimental reinforced concrete beams reported in [12]has been carried out by the FE software ATENA A con-stitutive model based on Eurocode 2 specifications forfire design [13] has been used in the analysis Compari-son of numerical simulation and test results has beencarried out

simula-2 Reported fire tests of RC beams employed in thenumerical analysis

The present analysis employs experimental data [12]

of reinforced concrete beams subjected to external ing and elevated temperatures A total of 13 specimenswere cast and tested Except for TSB2-1, the other speci-mens were heated on three surfaces (the bottom and two

lateral surfaces) according to the same heating curve

Specimens TSB1-(0-6) were tested in the FT

(force-tem-perature) path to obtain failure temperatures under

dif-ferent applied load levels These specimens were first

loaded to a predetermined value, and then heated until

the specimens failed Specimens TSB2-(1-6) were tested

in the TF (temperature-force) path to obtain ultimate

bending moment resistances These specimens were first

heated up to a predetermined temperature, and then

loaded at a quicker rate until the specimens failed As

the loading time was very short compared to its heating

time, the thermal duration effect during loading can be

neglected Thus, the duration of thermal exposure

be-tween the FT and TF paths can be considered to be the

same

The specimens were 1300 mm long, 100 mm wide,

and 180 mm deep, with a 10 mm concrete cover all round

the section

The specimens were cast in two batches of normal

Portland cement (Standard grade China cement), natural

river sand and crushed limestone with 15 mm maximum

size The mean compressive cube strength of TSB2

se-ries is 29,45 MPa

Low-carbon plain steel bars with diameter 10 mm

and yield stress 270 MPa at room temperature were used

as tensile and compressive reinforcement, while those

with diameter 3,5 mm and yield stress 289 MPa at room

temperature were used as stirrups The specimen tensile

steel ratio was 0,95 % and the stirrup spacing was 80

mm The specimen dimensions, detailing and loading

po-sitions are shown in Fig 1

The specimens were compacted using a vibrating

rod and cured in a moist environment at 20 °C and 100 %

relative humidity for a period of 7 days after casting,

and then placed in a natural environment To reduce the

difference of the water content between specimens

aris-ing from a long test period, all specimens were tested

600 °C, respectively, and then subjected to external ing The experimental temperature distribution through-out the section of the beams TSB2-4 and TSB2-6 isshown in Fig 2 The experimental load-deflection dia-grams are presented in Fig 3 with the failure load speci-fied in Table 1

load-0 30 60 90 120 150 180

Table 1 Failure loads of test beams

D10 1

3 A Constitutive model applied in the analysis

The reliability of a fire analysis results is strongly

affected by the choice of the constitutive laws of

materi-als and the values of theirs parameters In the present FE

model the material properties are considered to be

tem-perature-dependent This section describes constitutive

models for concrete and steel assumed in the numerical

analysis The constitutive relationships are based on

Eurocode 2 specifications [13, 14]

3.1 Concrete

The constitutive model (material model) describes

the behaviour of heated and loaded concrete in

math-ematical terms It is based on the stress-strain

relation-ships of heated concrete The strain components can be

modelled using the superposition theory whereby the

to-tal strain is considered to be the sum of various strain

ε the thermal strain, εcr the creep strain, εtr the

tran-sient strain, θ the temperature, t the time, σ a stress,

σ the stress history

The superposition theory has been particularly

use-ful in the analysis of the strain components at high

tem-perature and has been found to be applicable

experimen-tally [3] Each of the terms of Eq 1 is briefly described

below

The EC2 model implicitly takes account of the

ef-fect of high-temperature creep Both the physical loss of

moisture and shrinkage at high temperature cause a

de-crease in the coefficient of expansion, but these effects

have not been considered in the present model The model

also does not attempt to model spalling, the concrete

cross-section being assumed to remain intact

3.1.1 Stress-strain relationships in compression and

tension

The stress-strain relationships of compressed

con-crete for different temperature levels are shown in Fig 4

The theoretical model of these relationships is given in

Fig 5 On the compression side, the curve consists of a

parabolic branch followed by a descending curve until

crushing occurs On the tension side, the curve consists

of a bilinear diagram An initial stiffness of concrete in

tension is equal to that in compression At tensile strains

greater than this value of εcr the concrete is assumed to

follow the descending branch of the stress-strain curve

Once tensile strains exceed εcu, the concrete in tension

is ignored, although it is still assumed to be capable of

carrying compression Once the concrete has crushed, it

is assumed to have no residual strength in either

com-pression or tension

Stress-strain behaviour of compressive concrete der normal conditions (θ=20oC) in ATENA is mod-elled by the EC2 [14] relationship the ascending branch

un-of which has the form

( ) ( ) η( η)η

σ

21C20C

20

2

−+

,

c c

elastic modulus of concrete

It should be noted that the stress-strain relationshipfor compressive concrete presented in Eurocode 2 forfire design of concrete structures [13] is different fromformula 2 The former relationship is not available on

Fig 4 Stress-strain relationship of concrete at different temperatures

the ATENA 2D user interface However, the shape of

the stress-strain relationship of the compressive concrete

does not have significant influence on the results of the

analysis Therefore, Eq (2) has been modified in order

to model temperature effects Thus the parameters

E from formula (2) corresponding to normal

con-ditions (θ=20oC) were replaced by respective

param-eters σc( )θ , f c( )θ , εc( )θ , εc0( )θ and E c( )θ taken

for given temperature θ Further the relationships for

( )θ

c

f , εc0( )θ and E c( )θ are briefly discussed

The variation of the relative compressive strength

( ) ( )20oC

c

f θ of concrete with siliceous and

calcare-ous aggregates under increasing temperatures is shown

in Fig 6 Similar relationship for strain εc0( )θ is

Fig 6 Relative compressive strength of concrete with

sili-ceous and calcareous aggregates at elevated temperatures

mum stress f c( )θ under increasing temperature

A relationship for E c( )θ is absent in Eurocode 2,

therefore it was taken from [15]:

As mentioned above, the behaviour of tensile crete was modelled by a bilinear diagram The currentmodel of tensile concrete is characterised by two mainfactors: tensile strength and the ultimate cracking strain.The reduction of tensile strength of concrete at hightemperatures is accounted for by the coefficient k t(θ),taken as [13]:

θθ

θ

θθ

Cº6000

Cº600C

º100500

1001

Cº100C

º200

,1

for k

for k

for k

t t

t

(8)

To the authors' knowledge, investigations regardingthe limit strain εcu( )θ of tensile concrete are practicallyabsent In reference [16] it is taken as 15εcr( )θ , where

3.1.2 Thermal strainThermal strain of concrete during heating is a simplefunction of temperature and its theoretical curve is plot-ted in Fig 8 The theoretical curve also includes dryingshrinkage, but despite this, the curve is justified for rapidheating during fire

3.1.3 Creep strainThe creep strain depends on concrete, the load, thetemperature and the time The following expression

is used to describe the creep of ordinary concrete:

20 04 , 3 6

310

θσθ

σ

cu

where εcr(σ,θ,t) is the creep strain, σ( )θ a stress of

concrete, σcu( )θ the ultimate compressive stress of

con-crete (Fig 5), θ is the temperature of concrete, t∆ the

time interval

3.1.4 Transient strain

Transient stress is the hindered part of thermal

expansion for loaded concrete structures exposed to

heat-ing It is an irreversible process and occurs only duringthe first heating The transient stress is found to be pro-portional to the thermal expansion and to the ratiobetween the compressive stress and strength at 20°C:

c tr

C

θσ

2035,2

where εtr( )σ,θ is the transient strain, ( ) ( )20oC

c f

θσ

is the ratio between the compressive stress and sive strength of the concrete at 20°C, εth the thermalexpansion

compres-3.2 ReinforcementThe constitutive model describes the behaviour ofheated and loaded steel in mathematical terms Since tran-sient strain does not exist for steel, the model is simplerthan for concrete and is described as the sum of threeterms [13]:

( ) th( ) cr( t)

cr tot ε σ,θ ε θ ε σ,θ,

where εtot is total strain, εcr( )σ,θ the stress relatedstrain, εth( )θ the thermal strain, εtot the total strain.The strength and deformation properties of reinforc-ing steel at elevated temperatures shall be obtained fromthe stress-strain relationships [13] specified in Fig 9 andTable 2

Fig 8 Thermal strain of concrete

Table 2 Stress-strain relationships for steel under a high temperature

−θε

=θ

sp s

s sy s

a a

b E

−θε

θε

−θε

−θ

=θσ

st u

st s sy

( )θε

=

sp sy s

sp sy

sp sy

f f E

f f c

2

2

Fig 9 Stress-strain relationship of steel

For a given steel temperature, the stress-strain curves

in Fig 9 are defined by three parameters:

– the slope of the linear elastic range E s( )θ for

reinforcement,

– the proportional limit f sp(θ),

– the maximum stress level f sy( )θ

Values for each of the three parameters for hot rolled

and cold worked steel are given in Fig 10–12 [13]

Fig 10 Relative maximum stress of hot-rolled and

cold-worked steel at elevated temperatures

Fig 11 Relative proportional limit of hot-rolled and

cold-worked steel at elevated temperatures

4 Numerical modelling of experimental beams4.1 FE package ATENA

ATENA is a commercial finite element softwarepackage developed for non-linear simulation of concreteand reinforced concrete structures Based on advancedmaterial models it can be used for realistic modellingthe structural response and behaviour

ATENA programme consists of solution core and theuser interface The solution core has got capabilities forthe 2D and 3D analysis of continuum structures It haslibraries of finite elements, material models and solutionmethods ATENA User Graphic Interface for 2D analysis

is a programme, which enables access to the ATENAsolution core It is limited to 2D graphical modellingand covers the state of plane stress, plain strain and ra-tional symmetry

A smeared approach is used to model the materialproperties, such as cracks This means that material prop-erties defined for a material point are valid within a cer-tain material volume, which is in this case associatedwith the entire finite element The constitutive model isbased on the stiffness and is described by the equation

of equilibrium in a material point The concrete modelscan include the following effects of concrete behaviour:non-linear behaviour in compression including harden-ing and softening, fracture of concrete in tension based

on the non-linear fracture mechanics, biaxial strengthfailure criterion, reduction of compressive strength aftercracking, tension stiffening effect, reduction of the shearstiffness after cracking (variable shear retention), fixeddirection crack model The discrete reinforcement is inthe uniaxial stress state and its constitutive law is a bi-linear stress-strain diagram The material matrix is de-rived using the non-linear elastic approach In this ap-proach the elastic constants are derived from astress-strain function

ATENA enables loading of the structure with ous actions: body forces, nodal or linear forces, supports,prescribed deformations, temperature, shrinkage, pre-

Fig 12 Relative elastic modulus of hot-rolled and worked steel at elevated temperatures

Temperature, °C

hot rolled cold worked

17stressing These loading cases are combined into load

steps, which are solved utilising advanced solution

meth-ods: Newton–Raphson, modified Newton–Raphson or

arc-length Secant, tangential or elastic material stiffness can

be employed in particular models Line-search method

with optional parameters accelerates the convergence of

solution, which is controlled by residual-based and

en-ergy-based criteria This is only a concise survey of

ATENA features All the described features support the

user by engineering analysis of connections between steel

and concrete and computer simulation of its behaviour

4.2 FE model of experimental beams

Load-deflection behaviour of the experimental beams

described in Section 2 have been analysed by the finite

element package ATENA The present report includes

results of modelling the three beams of the TF series, ie

TSB2-1, TSB2-4 and TSB2-6, first exposed to

tempera-tures 20, 400 and 600 °C, respectively, and then

sub-jected to external loading till failure

SBETA material model with parameters given in

Section 3 was applied for simulating the concrete

behaviour Reinforcement is modelled by a single straight

line in a discrete way („bar reinforcement“) Material of

reinforcement is represented by the bilinear model

The experimental temperature distribution

through-out the section of the beams TSB2-4 and TSB2-6 is

shown in Fig 2 In order to assess degrading material

properties due to high temperature effects, the beams

within the depth were divided into six macroelements

These macroelements were discretised by CCIsoQuad

type quadraliteral elements with rigid connections

be-tween the macroelements The temperatures and

respec-tive material properties in different macroelements were

assessed according to the experimental temperature

dia-grams from Fig 2 Standard Newton-Raphson solution

method was applied for non-linear analysis of

experi-mental beams FE model of TSB2 series experiexperi-mental

beams is presented in Fig 13

Fig 13 FE model of TSB2 series experimental beams

4.3 Analysis results

In this section, comparison of numerical modelling

with test data has been carried out The modelled

load-deflection diagrams are presented in Fig 8 along with

the experimental curves The modelling has included all

the stages of temperature and loading First, the beams

TSB2-4 and TSB2-6 were subjected to temperature of

400 and 600 °C, respectively As the temperatures wereincreasing from the bottom to the top, the beams havedeflected downwards The calculated deflections due totemperature effects only (no loading) are in a good agree-ment with the tests for the beam TSB2-6, but some dis-crepancies can be noted for the beam TSB2-4 With in-creasing load the experimental load-deflection diagrams(Figs 2, 14) can be roughly approximated by a bilineardiagram consisting of two lines: the first one describingpre-yielding and the second post-yielding behaviour Itcan be seen from Fig 14 that the shape of experimentalload-deflection diagrams has been qualitatively captured

in the finite element analysis Pre-yielding deflectionswere accurately modelled for the beam TSB2-1(t = 20 ºC), but were underestimated for the beam TSB2-

4 and overestimated for the beam TSB2-6 Agreement

of the ultimate load is within reasonable limits tion fields and cracking pattern of TSB2-4 beam at load

Deflec-P = 16 kN are shown in Fig 15

0 0,005 0,01 0,015 0,02 0,025

20 C temperature 400 C temperature 600 C temperature

20 C Atena 400 C Atena 600 C Atena

Fig 14 Calculated and experimental load-deflection grams

dia-Fig 15 Deflection fields and cracking pattern of TSB2-4 beam at load P = 16 kN

5 Concluding remarksLoad-deflection behaviour of reinforced concretebeams subjected to high temperatures (up to 600 °C) hasbeen modelled by the finite element package ATENA

A constitutive model based on specifications of Eurocode

2 has been used in the analysis Comparison of the

ex-perimental and modelling results has shown that ATENA

has satisfactorily captured the load-deflection behaviour

of the beams

6 Acknowledgment

The financial support under Framework 5 project

“Cost-effective, sustainable and innovative upgrading

methods for fire safety in existing tunnels” (UPTUN,

project No GRD1-2001-40739/UPTUN) provided by the

European Community is gratefully acknowledged

References

1 Felicetti, R.; Gambarova, P G and Meda, A Expertise

and Assesment of Structures after Fire In: Report in the

Meeting of fib Task Group 4.3.2 Guidelines for the

Struc-tural Design of Concrete Buildings Exposed to Fire,

Brus-sels, Nov 2002 15 p.

2 Khoury, G A.; Anderberg, Y.; Both, K.; Felinger, J.;

Majorana, C E and Hoj, N P Fire Design of Concrete:

Materials, Structures and Modelling In: Proc of the 1st

fib Congress Concrete Structures in 21 st Century, Osaka,

2002, p 99–118.

3 Khoury G A., Majorana C E., Pesavento F and Schrefler

B A Modelling of Heated Concrete Magazine of

Con-crete Research, Vol 54, No 2, 2002, p 77–101.

4 Riva, P Parametric Study on the Behaviour of RC Beams

and Frames under Fire Conditions In: Report in the

Meet-ing of fib Task Group 4.3.2 Guidelines for the Structural

Design of Concrete Buildings Exposed to Fire, Brussels,

Nov 2002 61 p.

5 Bazant, Z P and Kaplan, M F Concrete at High

Tem-peratures: Material Properties and Mathematical Models.

Longman Group Lt., 1996 412 p.

6 Mutoh, A and Yamazaki, N Non-linear Analysis of

Rein-forced Concrete Members under High Temperature In:

Proc of Conf DIANA Computational Mechanics ’94.

Kluwer Academic Publishers, 1994, p 45–55.

7 Bratina, S.; Planinc, I.; Saje, M and Turk, G Non-Linear Fire-Resistance Analysis of Reinforced Concrete Beams Structural Engineering and Mechanics, Vol 16, No 6, 2003,

p 695–712.

8 Sullivan, P J E.; Terro, M J and Morris, W A Critical Review of Fire-Dedicated Thermal Structural Computer Programs In: Applied Fire Science in Transition Series, Vol III Computer Applications in Fire Protection Engineer- ing Paul R DeCicco ed Baywood Publishing Company, Inc., 2001 p 5–27.

9 Wang, Y C Steel and Composite Structures Behaviour and Design for Fire Safety EF & N Spon, 2002 264 p.

10 de Witte, F C and Wijtze, P K DIANA – Finite Element Analysis Users Manual Release 8.1 Analysis Procedures TNO Building and Construction Research, Delft, 2002 580p.

11 Cervenka, V and Cervenka, J ATENA Program tation Part 2 ATENA 2D User Manual Prague, 2002.

Documen-138 p.

12 Shi, X.; Tan T.-H.; Tan, K.-H and Guo, Z Effect of Force– Temperature Paths on Behaviour of Reinforced Concrete Flexural Members Journal of Structural Engineering, Vol 128, No 3, March 2002, p 365–373.

13 prEN 1992-1-2 Eurocode2: Design of Concrete Structures

- Part 1.2: General Rules – Structural Fire Design pean Committee for Standartisation, Brussels, July 2001.

Euro-102 p.

14 prEN 19921 Eurocode2: Design of Concrete Structures Part 1: General Rules and Rules for Buildings European Committee for Standartisation, Brussels, Oct 2001 230 p.

-15. Iljin, N A Outcomes of fire effect on reinforcedconcrete structures (ỳĩựẻăảựòâèỮ ĩãắăâĩãĩ âĩẫảăéựò-âèỮ ắà ữăẻăẫĩáăòĩắắũă êĩắựòđóêỏèè) Moscow:Stroizdat, 1979 128 p (in Russian)

16 Cai, J.; Burgess, I and Plank, R A Generalised inforced Concrete Beam-Column Element Model for Fire Conditions Engineering Structures, Vol 25, No 6, 2003,

Keywords: thermal expansion, thermal strain, coefficient of linear thermal expansion, structual steel.

1 Introduction

The impact of elevated temperatures on structural

materials (including structural steels) results in a change

of their elastic and plastic behaviour The intensity of

such phenomena as creep and relaxation also increases

with temperature As results of our previous studies, such

phenomena have a considerable impact on structural

strength at fire temperatures

Furthermore, not only an absolute value of

tempera-ture is essential but also temperatempera-ture distribution with

time and rate of temperature increase are of vital

impor-tance

Our previous studies [1] concerning the impact of

rapid-heating conditions, like fire, on the properties of

reinforcing steel, also including its thermal strain, have

shown that:

• Such properties and the type of rupture are

influ-enced by temperature distribution during the test,

and in particular, by temperature increase rate

dT/dτ, what was found while testing steels at both

relatively slight and significant temperature increase

rates;

• Different grades of steels (including structural steels)

show some kind of inertia, which consists in a

par-tial or full inhibition of some processes leading to

the material rupture due to heating at a significant

rate as compared to the same processes at constant

temperatures or at a slight rate of temperature

in-crease;

• Thermal fields characterised by higher temperature

increase rates undoubtedly produce more favourable

effects in terms of the material strength, eg result

in higher critical temperatures (causing rupture)

Structural strength under fire conditions and fireresistance are calculated on the basis of well establishedmechanical and strength characteristics of building ma-terials

The nature of structural steels strain, being a result

of simultaneous impact of stresses and time-dependentthermal field during a fire, is still under examination.According to a proposal made by RILEM-COMMITEE 44-PHT, an international committee, totalstrain at elevated temperatures can be described by thefollowing constitutive equation for the material (steel):

σσσ

σεε

)(

1[002,0)(

−

y p

e p

τ

ε is creep strain (dependent on time τ) as described byDorn’s theory and Harmothy’s studies; also being thesubject of our earlier studies conducted at the AppliedMechanics Department (MSFS) under Z Bednarek’sguidance

The total strain of steel at elevated temperatures can

be calculated by summing up the thermal strain, the straincalculated from the Ramberg-Osgood equation and thecreep strain

This paper presents the results of studies of the firstcomponent of the steel strain model based on equation (1),

ie the thermal strain caused by linear expansion of steel

2 Model of thermal expansion of solid bodies

According to the microscopic description, the

ther-mal expansion of solid bodies can account for an

increase of the crystal lattice parameter (interatomic

dis-tances in a crystal) Some of these phenomena can also

account for defects in the crystal lattice – mainly

vacan-cies (the lack of atom in the place, which is assigned to

such atom)

As temperature rises, the amplitude of atoms

oscil-lations from their average equilibrium positions increases

Fig 1 Relation between force, potential energy and

inter-atomic distance r: r0, r1, r2 – average interatomic

dis-tances at increasingly elevated temperatures

The interatomic distance at temperature 0 °K is

con-stant and equal to r0

As temperature rises up to T1, the energy of atoms

in the crystal lattice increases resulting in their

oscilla-tions from their average equilibrium position r1 [2]

It can be shown that the average displacement of

the equilibrium position can be expressed as

2

K

T k b

x>= ⋅ ⋅

where <x> – average distance from r0, eg <x> = r1 – r0;

b – anharmonicity coefficient (determines the

de-viation of atom oscillations from harmonicity);

K – coefficient of quasi-elastic force acting between

atoms in the crystal lattice

(Fx = –Kx + bx2);

T – temperature; k – Boltzman constant

Thus, as temperature rises, the average interatomic

distance increases and the solid body expands

There is the following relation between the linearexpansion coefficient α and the anharmonicity coefficient:

0 2 0

1

r K

k b T r

∆

= 1,2 · 10–5T + 0 ,4 · 10–8T2– 2,416 · 10–4

20°C < T < 750 °C, (7a)

l l

∆

= 1,1 · 10–2 750°C < T < 860°C, (7b)

l l

∆

= 2 ⋅ 10 –5T + 6,2⋅ 10 –3 860 °C < T < 1200° C (7c)The linear expansion coefficient can be preciselydefined as:

p dT

T p dl

),((1

0

=

where p – constant pressure

At constant pressure, coefficient α is a dependant function

temperature-For practical purposes of making structural sis, the average based on the reference value of1,2 · 10–5(1/deg) for low-carbon steels is frequently as-sumed instead of an actual value of linear expansioncoefficient α at a given temperature There is no avail-able precise data on the linear expansion coefficient forstructural steels for the needs of a more detailed steelstrain analysis at elevated temperatures, including fireconditions characterised by a rapid increase in tempera-ture When searching through the publications available

analy-to us we have only found the data on American steelASTM A36 [7], austenitic steels S350GD, S355 andS460 [8] and formulae describing the relation betweencoefficient α and temperature as follows:

α = (0 ,0 0 4T + 12) · 10–6 (1/K) [9, 6], (9)

α= (6,1 + 0,0019T) · 10–6 inch/inch per degree [10].(10)

For the needs of further studies on individual

com-ponents of formula (1), which describes the strain of

structural steels at fire temperatures, the behaviour of

linear expansion coefficient for the steel, class AIII, grade

34GS, was examined in a linearly variable temperature

field at different heating rates

The tests were conducted under anisothermic

con-ditions (T≠const) for 4 different temperature increase

rates Fig 2 shows temperature-time distributions Under

fire conditions, the rate of temperature increase is

5 °C/min for a steel element covered by a good quality

fire insulation For uncovered structures, the rate of

tem-perature increase can reach 50°C/min The results of tests

are shown on Figs 3 and 4, below

Fig 2 Relation between temperature and time for

speci-mens heated at various temperature increase rates

Fig 3 Relation between strain and temperature for

speci-mens heated at various temperature increase rates

Below, we present a comparison of the curve taken

from ENV 1992-1-2/1995/ (curve a) with our curves

(curves b, c, d, e in Fig 3) describing the relation

between strain and temperature that we obtained from

∆ = 1,27 · 10–5T + 0,322 · 10–8T2– 6,65 · 10–4,

(11b)

d –

l l

∆ = 1,28 · 10–5T + 0,298 · 10–8T2 – 7,79 · 10–4,

(11c)

e –

l l

∆

= 1,28 · 10–5T + 0,244 · 10–8T2– 7,85 · 10–4

(11d)The points marked in Fig 3 to determine curves "b,

c, d and e" are measuring points obtained by the authorsfrom their own tests, whereas points on curve "a" werecalculated according to the formula 7a taken from thereferences

Below, we present a comparison of the curve takenfrom the references (curve “a”) with our curves (curves

“b, c, d, e” in Fig 4) describing the relation between

0,0E+00 2,0E-06 4,0E-06 6,0E-06 8,0E-06 1,0E-05 1,2E-05 1,4E-05 1,6E-05

thermal expansion coefficient a and temperature that weobtained by experiments:

c, d and e” are measuring points obtained by the authors

in their own tests, whereas points on curve “a” werecalculated according to formula (9) taken from the refer-ences

4 Conclusions

The objective of investigations was to determine and

conduct a comparative analysis of thermal strain and

ther-mal expansion coefficient for structural steels at

differ-ent temperature increase rates As the results of the tests

conducted at different heating rates on specimens made

of structural steel, class AIII, grade 34GS show, the

ther-mal strain of specimens is affected by the temperature

increase rate The higher the temperature increase rate,

the lower the thermal strain of specimen The thermal

expansion coefficient also changes in a similar way The

reason for such a behaviour of steel is its material

iner-tia which consists in a pariner-tial or full inhibition of some

processes leading to the material rupture and taking place

in steel due to a significant heating rate, as we have also

shown in our papers [1] and [11]

Linear expansion coefficient α(T) rises with

tem-perature As the regression analysis of the results,

ob-tained by the tests on linear expansion coefficient α at a

given heating rate shows, the best correlation degree was

obtained when approximating experimental data with

quadratic polynomials This paper includes the functions

that describe the relation between coefficient a and

tem-perature at different heating rates (formulae 12a, b, c,

2 Staub, F Metal Science, WNT Katowice 1994.

3 Lewis, K R Fire design of steel members, fire ing research report 2000/07 ISSN 1173–5996.

engineer-4 Böðvar, T High performance concrete Design guide lines, Department of fire safety engineering, Report 5008, Lund, 1998.

5 Burgon, B Elevated temperature and high strain rate erties of offshore steels, Steel Construction Institute, Off- shore Technology Report 2001, 020, Norwich.

prop-6 Alfawakhiri, F.; Sultan, M A.; MacKinnon, D H Fire Resistance of Loadbearing Steel-Stud Walls Protected with Gypsum Board: A Review, Fire Technology, Vol 35, No 4, 1999.

7 Skowroñski, W Theory of fire safety of steel structures, PWN 2001.

8 Outinen, J.; Kaitila, O.; Mäkeläinen, P High-temperature testing of structural steel and modelling of structures at fire temperatures Research report TKK-TER-23 Helsinki University of Technology, 2001.

9 Guy C Gosselin Structural fire protection- predictive methods, Building science inside 1987, Institute for Re- search in Construction, National Research Council Canada.

10 R.H.R Tide: Integrity of structural steel after exposure to fire, Engineering Journal /First Quarter, 1998.

11 Bednarek, Z Effects of increase of temperature on tural steel strength parameters as applied to the estimation

struc-of fire safety struc-of concrete construction Doctor Habilitatis thesis Vilnius: Technika, 1996, p 1–208.

SLIP OF “BULLDOG”-TYPE TOOTHED-PLATE CONNECTORS IN STEEL-TIMBER

JOINTS OF OPEN-WEB GIRDERS

Rimantas ÈechavièiusDept of Metal and Timber Structures, Vilnius Gediminas Technical University,Saulëtekio al 11, LT-10223 Vilnius-40, Lithuania E-mail: mktc@takas.lt

Received 4 June 2003; accepted 3 May 2004

1 Introduction

Toothed “Bulldog”-type plate connectors (DS

“Bull-dog”) are means of mechanical ties used in timber

struc-tures The main purpose of them is to increase the

tim-ber bearing area in structural joints and to diminish the

slip of feathered joints They could also allow to increase

considerably the bearing capacity of such joints and to

tie light steel-timber open-web girders (trusses) and

frames This is characteristic of “OPEN-WEB” trusses

having been produced since 1960 by the joint-stock

com-pany ”MacMillan”; these trusses can be used for

span-ning both small opespan-nings (l ≈ 4,5–9 m) and large (12–

120 m) ones (Fig 1)

The main advantages of such trusses are their small

weight and rational joint work of timber chords and the

network of metal tubes The main research on the

bear-ing capacity of toothed “Bulldog”-type connectors was

performed at Stevin-Laboratorium (Delft University of

Techology, Netherlands), Dannish Construction Research

Institute, Otto-Graf Institute (Stuttgart University,

Ger-many) [1–5] During these investigations the strength of

joints was analysed by J.H Blass, etc [6–7] The model

of calculating such joints presented in his work is

rec-ommended by the project of new Eurocode standards [8]

The slip of “Bulldog”-type plate connectors was

investi-gated by Y Hirashima [9] The results are presented in

Fig 2, where slipping of different joining means is

Keywords: composite structure, steel-timber joint "Bulldog"-type connector, slip, resistance test.

The majority of these results is obtained by gating separate joints But there is a lack of data con-cerning the slip of such joints in real steel-timber struc-tures where the redestribution of stresses amongindividual truss elements becomes clear

investi-The article presents the results of research on fouropen-web trusses with “Bulldog”-type connectors [10–12] Not only the strength of such joints and their slipbut also the stress redestribution among elements of thetruss were determined

Fig 1 Composite steel-timber open-web truss of Truss Joist MacMillan

Fig 2 Experimental load-slip curves for joints in tension

parallel to the grain: a – glued joints (12, 5·10³ mm²), b –

split ring (100 mm), c – double-sided toothed plate (¸ 62

mm) [15], d – dowel (¸ 14 mm), e – bolt (¸ 14 mm), f –

punched plate (0,1E5 mm²), nail (¸ 4,4 mm)

Fig 4 Truss testing scheme: a – general view of truss testing (SN-1-3); b – truss SN-1-1 testing diagram: 1 – truss SN-1-1; 2 – traverse; 3 – hinge; 4 – stiff support;

5 – jack; 6 – dynamometer; 7 steel spreader; 8 – timber pad; X – traverse braces; T1-T16 – electric strain resis- tance gauges; II.1 – II.7 – 0,01 mm accuracy dial gauges (deflection indicators); In.1–In 6 – displacement of ends

of pipe indicators with precision of 0,01 mm

Fig 3 Structure of SN-1 trusses: a – diagram for analysis;

b – structure of M6 joint; c – structure of M1 joint

Table 1 Schedule of materials for a SN-1 truss

elements of metal tubes are connected at 60° angle with

the upper and lower chords The tubes at connecting

points are flattened and a hole of 16,2 mm was drilled

In joints with one network element (M6 and M11), an

insertion was put The structure of these trusses and the

testing scheme are shown in Figs 3, 4 and Table 1

The trusses were tested at the laboratory of

build-ing structures of the VGTU The source of loadbuild-ing was

a hydraulic jack based on a rigid metal frame The

scheme of truss testing is shown in Fig 4 Strain gauges

(20 mm on metal and 50 mm on wooden basis) were

used only when testing SN-1-1 truss The vertical strains

of truss supports and lower chords joints as well as slip

strains of joints M1, M5, M6, M11 were measured by

indicators of 0,01 mm precision

For stability of experimental equipment in the plane

of bending moment, hinge supported horizontal wooden

squared beam connections were provided It was observed

during testing that the horizontal ties are free and they

do not hinder transferring vertical forces

3 Test results

It has been determined by testing steel-timber

con-nections [14, 15] that the characteristic value Rck  of truss

chord timber compressive strength along fibres is equal

to 38,61 MPa and characteristic volume weight rk = 434kg/m² Testing trusses lasted for 2–3 h During this timespan the strains of on average 21 devices were deter-mined at every stage of 15 loadings Loading duration

in separate stages was in the interval of 10–20 min pending on the necessity to rearrange either the devices(when strains were larger than the size of limit strains)

de-or the equipment of hde-orizontal braces Testing trusses isshown in Fig 5

The unit deformations of the truss SN-1-1 are shown

in Fig 6 The average strains in compressive truss bars1–7 and 5–10 under the loading of 80 kN (σc = 86,64MPa) and in the members in tension 1–6 and 5–11 un-der the loading of 110 kN (σc = 121,46 MPa) were close

to those calculated theoretically according to the mentally defined pipe compressive (Et) and tensionedbars elasticity models: Ec = 2,10·105 MPa, and

experi-Et = 2,12·105 MPa But from F = 85–90 kN loading thegrowth of strains of compressed pipes and from F = 110

kN the strains of tensioned pipes decreased considerablyand later have stopped almost entirely Thus at the in-crease of loading the stresses in these bars have notchanged, ie the stresses were redestributed among thetruss elements This phenomenon can be explained bythe data of Table 2: exactly at this time M-11 ir M-6joints slip deformations were larger than the allowable 2

Fig 5 General view of testing the open-web truss: a – test of truss SN-1-2; b – arrangement of test

devices in the truss SN 1–4

Table 2 Characteristics for serviceability limit state of “Bulldog”-type connectors in steel-to-timber joints

Impact kN Slip modulus according to LST EN 26891 [19],

K t ser

K e

K t u

Fig 6 Kinetics of strain in steel web members of SN-1-1

(Figs 3, 4) Tension members: 1 – 6 (T-9, T-10) and 5–11

(T-15, T-16); compression members: 1 – 7 (T-11, T-12)

and 5–10 (T-13, T-14); 1, 2 – strain of compression and

tension members, respectively

Fig 7 End displacements of web members of SN-1-2 truss (Figs 3, 4): dial gauges In.1 and In.4 for tensile member

1 – 6; In.2 and In.5 – for tensile member 5 – 11; In.3 and In.6 – for compressive struts 1 – 7 and 5 – 10, respectively

Fig 8 Views of joints M6 (In.1) (a) and M1 (In.3 and In.4) (b) of SN-1-4 truss after failure

mm limit (the total loading F reached 86,6 kN and

103,3 kN, and F2 for one DS was equal to 24,5 kN and

29,3 kN, respectively

It is clearly shown in Fig 7: the joint M6 (In.1) slip

strains were very similar to those of the joint M1 (In 3

and In 4) In Fig 8, deformations after failure of joints

M6 (In 1) and M1 (In 3 and In 4) are seen Maximal

bearing deformations of steel bolts M16 (dv = 15,9 mm)

reached 0,3 – 0,4 mm, and their bend 8,5 mm (SN-1-4)

and 14,8 mm (SN-1-3) In this picture the character of

bolt hole deformations is seen too The determined after

the failure measurements of bolt holes in upper and

bot-tom chords are presented in Table 3 It shows that the

direction (a) of hole maximal dimensions correlates well

with the force direction: in the girder SN-1-4 the

maxi-mal dimension of 19,0 mm of joint M6 is of a = 60°

direction, and M-1 is a maximal dimension (19,35 mm)

of a = 0° direction

4 The characteristic of DS “Bulldog” serviceability

limit state

This characteristic is presented in Figs 9, 10 and

Table 2 Here also the results of tests B-1 and B-2 of

metal-wood joints with “Bulldog”-type connectors are

shown

In this Table the theoretical moduli of the slip of

such joints were calculated according to European

In these formulas, due to a shortage of tests

cerning the humidity of timber, the influence of the

con-nection elements thickness and the number of

connec-tors in a joint has not been evaluated, as well as the

influence of the angle between the force and wood

fi-bres It was noted by H J Blass [16], too

Our investigations have disclosed that the bearing

capacity of DS “Bulldog” at the states of security and

serviceability (failure loading Fmax; force F2, when the

strain of the slip connector equals 2,0 mm; magnitude

of slip modulus at reaching the serviceability limits state

Ke

ser, connection static slip ms) depends on the angle (a)

between the force and wood fibres In Fig 9 we can see

that the dependence of slip modulus size on the impact

angle (a) is valid for the whole time span of the

connec-tor strain: from the initial impetus up to failure

Table 2 includes the DS “Bulldog” static slip

aver-age characteristics determined according to DAN-ENV

1995-1-1 [14]; in many cases they are larger (a/v

µs = 7,35 – 9,42 depending on a) than in these norms:

3 < µs < 6

It has been determined that the slip modulus Ke

ser

is by 1,12–1,4 times larger than that defined by [14]

depending on the angle between the force and wood

fibres

Fig 10 Relationship between carrying capacity of dog”-type connector in steel-to-timber joints and angle α between force F and grain direction: 1 – K t

“Bull-ser  – cal value of slip modulus [14]; 2 – slip modulus at ser- viceability limit state (K e

theoreti-ser ); 3 – force (F2) when tor slip equals 2 mm; 4 – maximum force (Fmax); 5 – stati- cal slip in steel-to-timber joints µ e [14]

connec-Fig 9 Variation of slip modulus of “Bulldog”-type timber connectors with relative force (F/Fmax) and angle ( α ) between force and timber grain directions

steel-5 Conclusions

1 The bearing capacity of steel-timber connectionswith “Bulldog”-type connectors depends, according to thestate of serviceability limits, on the angle between theforce and wood fibres

2 Experimental slip modulus Ke

ser is by 1,12–1,4times larger than that theoretically determined by experi-mental European standards Its value depends on theangle between the force and wood fibres

3 The static slip value µs with “Bulldog”-type nectors in steel-timber connections is much larger than

*1 dimensions were taken from the inner side of joint

*2 clockwise in the front side

Table 3 Dimensions of holes in chords of open-web girders after testing

that given in experimental European standards (Eurocode

5) Its magnitude also depends on the angle between the

force and wood fibres

4 Redistribution of stresses between the girder

web-members starts when the slip strains in steel-timber

con-nections with “Bulldog”-type connectors are near the limit

value (2 mm)

References

1 Kuipers, J and Kurstjens, P B J.: Creep and damage

re-search on timber joints Part one Rapport

4-86-15-HD-23 Stevin-Laboratorium Delft University of Technology,

Netherlands, 1986.

2 Kurstjens, P B J Creep and damage research on timber

joints Part two Rapport 25.4-89-15 C HD-24,

Stevin-Laboratorium, Delft University of Technology, Netherlands,

1989.

3 Kurstjens, P B J Creep and damage research on timber

joints Part three Rapport 25.4-90-12 C HD-26,

Stevin-Laboratorium, Delft University of Technology, Netherlands,

1990.

4 Kurstjens, P B J and Stolle, P Creep and damage

re-search on timber joints Part four Rapport 25.4-91-06/ C

HD-28, Stevin-Laboratorium, Delft University of

Technol-ogy, Netherlands, 1991.

5 Frech, P and Kolb, H Test of Bulldog-type connectors.

Test results H 30471 (Prüfung von Bulldog-Holzverbindern

Prüfzeugnis H 30471) Otto–Graf Institute of Stuttgart

University, 1971 (in German).

6 Blass, J H.; Ehlbeck, J and Schlager, M Characteristic

strength of toothed-plate connector joints Holz als

Roh-und Werkstoff, 51, 1993, p 395–399.

7 Blass, H J.; Aune, P.; Choo, B S.; Görlacher, R.; Griffiths,

D R.; Hilson, B O.; Racher, P and Steck, G Timber

Engineering Netherlands: Centrum Hout, 1995.

8 Eurocode 5 Design of timber structures Part: General rules

and rules for buildings ENV 1995–1–1 Brussels: CEN,

1993 133 p.

9 Hirashima, Y (1990) Lateral resistance of timber tor joints parallel to grain direction In: Proceedings of the International Engineering Conference, Vol 1: 254–261, Tokyo.

connec-10 Èechavièius, R Investigation of ring-toothed connectors

in metal-timber girders Research report of Technical tre for Timber Structures (Mokslo tiriamojo darbo ataskaita Dantytøjø sprausteliø tyrimai) Vilnius, 1999 93 p (in Lithuanian).

Cen-11 Ðliþys, M Application of ring-toothed connectors in timber girders (Dantytøjø sprausteliø panaudojimas) Vilnius, 1999 81 p (in Lithuanian).

metal-12 Narmontas, D.; Èechavièius, R.; Kudzys, A Behaviour of composite open-web trusses with toothed-plate connectors In: Proceedings of the International PhD Symposium in Civil Engineering, Institute of Structural Engineering Uni- versity of Agricultural Sciences, Vienna, Oct 5–7, 2000,

p 431–434.

13 Standard of Germany DIN 1052, Part 2: Timber tures design and construction (Deutsche Norm Holzbau- werke-Berechnung und Ausführung) Beuth Berlin, 1988.

struc-27 p (in German).

14 Standard of Lithuania LST EN 28970 Timber structures Testing of joints made with mechanical fasteners (Medinës konstrukcijos Sujungimø mechaninëms tvirtinimo detalëms bandymas) Requirements for wood density, 2000 4 p (in Lithuanian).

15 Standard of Lithuania LST EN 26891 Timber structures Joints made with mechanical fasteners (Medinës konstruk- cijos Sujungimai mechaninëmis tvirtinimo detalëmis) General principles for the determination of strength and deformation characteristics, 2000 6 p (in Lithuanian).

16 Blass, J H Joints of toothed-plate connectors In: Timber structures in limit state Introduction of Eurocode 5 Build- ings materials and dimensioning basis (Assemblages par crampons À: Structures en bois aux états limites) STEP1 Introduction à l’Eurocode 5 Matériaux et bases de calcul, Sedibois, Paris, 1996 517 p.

Wei Lu1, Pentti Mäkeläinen2, Jyrki Kesti3, Jukka Lindborg 4

1Steel Structures, Helsinki University of Technology, FIN-02015, Espoo, Finland E-mail: luwei@cc.hut.fi

2Steel Structures, Helsinki University of Technology, FIN-02015, Espoo, Finland E-mail: Pentti.Makelainen@hut.fi

3Rautaruukki Oyj, Construction Solutions / R &D, Helsinki, Finland E-mail: Jyrki.Kesti@rautaruukki.com

4Rautaruukki Oyj, Construction Solutions / R &D, Helsinki, Finland E-mail: Jukka.Lindborg@rautaruukki.com

Received 1 March 2004; accepted 18 May 2004

Abstract Cold-formed steel profiled sheeting is widely used for roof, floor system and wall cladding Due to the variety

of profiles available on the market, finding the optimum shapes is necessary In this paper, genetic algorithms are applied to optimise dimensions of cold-formed steel profiled sheeting The objective of the optimization is to obtain the optimum dimensions of profiled sheeting that has the minimum weight subjected to the given constraints Sheathings are designed in accordance with Eurocode 3, Part 1.3 With this optimization process, a set of easily accessed optimum sections may be provided for structural steel designers and steel manufacturers.

Keywords: cold-formed steel, profiled sheeting, optimization, genetic algorithm.

1 Introduction

Because of the high strength to weight ratio and ease

of assembly, the profiled sheeting has been widely used

for roofing, cladding and extended to floor systems in

building constructions Due to the variety of profiles

available on the market, finding the optimum shapes is

necessary

Genetic Algorithm (GA) is a general-purpose,

de-rivative-free, stochastic search algorithm [3, 6, 10] and

starts by randomly choosing an initial population that

consists of candidate solutions to the problem at hand

Each individual in the population is characterised by a

fixed length binary bit string, which is called

chromo-some These chromosomes are evaluated by means of a

fitness function Combining the fittest individuals from

the previous population, a new generation of

chromo-somes is created Evolutionary operators such as

selec-tion, crossover, and mutation are used to create this new

population Besides, Elitism, which is a method that

cop-ies the best chromosome or a few better chromosomes

to the new population, might be incorporated into the

algorithm to avoid losing the best individual This

pro-cess continues until the specified level of fitness is

reached

Normally, the objective for optimization is to

achieve maximum use of material by using

appropriat-ing profiles, for instance, to maximize the resistance of

sheeting subjected to bending stress [7] or to minimise

the weight of sheathing [11, 12] In this paper, GA-basedoptimization method is used to obtain the optimum shapeand dimension of roof sheathing that minimise the weightunder the given constraints, such as the geometric, stressand fabrication constraints Sheathings are designed inaccordance with Eurocode 3, Part 1.3 [5] Because ofthe many types of sheeting available and the diverse func-tional requirements and loading conditions that apply,design is generally based on experimental investigation.The analytical method can be used mostly for trapezoi-dal sheeting The GA-based design procedure is demon-strated with four design examples With this optimiza-tion process, a set of easily accessed optimum sectionsmay be provided for structural steel designers and steelmanufacturers

2 Description of optimum design problemThe minimum weight design can be expressed as:

L b A W Minimise =ρ⋅( g/ d)⋅ , (1)where W is the sheeting weight; L is the span of thesheeting; and bd is the notation width of the pitch asshown in Fig 1 Fig 1 also shows the dimensions of thesheeting for one fold, in which, bu and bp are notationwidths of the plane elements; hw is the height of theweb; Sw is the slant height of the web; and θ is the incli-nation of the web Except for Sw and bd, all other di-mensions shown in the figure are design variables

The shapes of the stiffeners on the flanges are shown

in Fig 2 The number of the stiffeners on the flange can

be zero, one or two The stiffeners are assumed to be

symmetric on the top of the flange When two stiffeners

appear, the sizes of them are the same

Fig 2 Types of flange stiffener

The dimensions of the upper flanges are shown in

Fig 3 The design variables are width and depth of the

stiffeners, x2 and x3, the position of the stiffeners, x1;

the inclination of the stiffener, θsu, and the number of

the stiffeners

Fig 3 Dimensions of the upper flange

According to the number of stiffeners on the web,

three cases can be classified: case (a) without stiffener,

case (b) with one stiffener and case (c) with two

ers as shown in Fig 4 In case (c), the size of the

stiffen-ers is assumed to be the same The dimensions of the

stiffeners on the web are shown in Fig 5, in which the

design variables are height and width of stiffeners bsw

and ssw1; positions of stiffeners, sw1 and sw2, and the

number of the stiffeners

The numbers and the dimensions of stiffeners on

the bottom flange may be different from those on the

Fig 1 Dimension of the cross-section for one fold

top flange Similarly to the top flange, the dimensions ofthe bottom flange are shown in Fig 6 The design vari-ables are the width and the height f the stiffeners, x9 and

x10, the position of the stiffeners, x8; the inclination ofthe stiffener, θsp, and the number of the stiffeners

Fig 4 Type of web stiffeners

Fig 5 Dimension of web with two stiffeners

Fig 6 Dimensions of the bottom flange

The constraints can be classified into three ries: the geometrical constraints, the strength constraintsand the fabrication constraints The geometrical limitsthat should be satisfied are taken from Eurocode 3,

33Part 1.3 These limits are listed in Table 1 as G1 and

G2 When designing sheeting, the following checks

should be carried out: bending resistance, shear

resis-tance, concentrated load resistance (crippling resistance),

interaction of bending and shear and/or crippling, and

stiffness of the sheeting Thus, the strength constraints

are given in Table 2 as SM1, SM2, SF3, SF4, SF5, SV6

and SMV7

Table 1 Geometrical constraints

Table 2 Strength constraints

The fabrication constraint in this analysis is defined

as to manufacture the profiled sheeting with actual

pro-vided strip width, ie

strip

where Ls is the total length of sheeting calculated by

using the cross-section dimensioned with the current

com-bination of design variables; and Lstrip is the length of

the provided strip width For the purpose of the

practi-cal application, the overlap length has been taken into

account in the calculation of Ls (Fig 7)

Fig 7 Overlap of two sheathings

3 GA-based design

Since GA is suitable for an unconstrained

optimiza-tion problem, the constrained problem can be transformed

to an unconstrained problem through a penalty function

A suitable penalty function must incur a positive for

in-feasible points and no penalty for in-feasible points In this

analysis, the quadratic penalty function is used, and the

corresponding unconstrained problem becomes:

,

)),0(max(

2 2 2

2 1

1

β

αΦ

⋅

⋅+

⋅

⋅+

nn KK

nn KK W Minimise

i

i

(3)

fold is calculated as dividing the required width of thestrip, Lstrip, by length of sheeting of each fold calculatedfrom the current combination of design variables, thus,the value of |Ls / Lstrip| is less than one And the value of

 ισ ϖαριεδ µορε ρεγυλαρλψ ωηεν χοµπαρινγ το ϖαλυε 〈.Τηερεφορε, τηε πεναλτψ ισ διϖιδεδ ιντο τωο τερµσ, ιε 〈ανδ 

In the above formula, nn1 is the coefficient thatmakes the values of W and 

the same order so as to avoid one value dominating theother KKi ≥ 0 are coefficients and the solution of thepenalty problem can be made arbitrarily close to thesolution of the original problem by choosing KKi suffi-ciently large [2]

Since GA is suitable to find the maximum value of

an optimization problem, thus, the above-mentioned constrained minimisation problem should be transformedinto maximisation problem by using the following for-mula [1]:

un-,

0

,

max

max max

ΦΦ

ΦΦΦΦ

if F

(4)where Φmax is average fitness, ie Φmax = ave(Φ) so thatthe individuals with fitness greater than or equal to thisvalue are discarded and with no chance to enter themating pool In GA terminology, F is called fitness func-tion, which is used in the reproduction stage

Fig 8 shows how the sheeting design is integratedinto the GA optimization process GA-based design startsfrom randomly generating an initial population that iscomposed of candidate solutions to the current problem.Each individual in the population is a bit string of fixedlength After decoding, these individuals that representthe dimensions of the sheeting are sent to the sheet de-sign programme, by which the resistances of the sheet-ing are calculated After that, the constraints are checkedand if the constraints are violated, the penalty is appliedand the fitness function is calculated After the evalua-tion of the fitness for each individual, a new generation

is created using such operators as selection, crossoverand mutation In order to keep the best individuals ineach generation, the elitism may also be used This pro-cess is continued until the specified stopping criteria aresatisfied

Compared to other search and optimization rithms, GA has the following features: GAs search a set

algo-of points in parallel, not only at a single point; GAs donot require derivative information or other auxiliaryknowledge Only the objective function and correspond-ing fitness affect the search direction; GAs use prob-ability rules; and GAs provide a number of potential

solutions to a given problem The final solution is left to

user

4 Examples

Fig 9 shows two-span roof sheathing with applied

loading The loading includes the permanent load such

as the self-weight of sheeting and insulations, which are

represented as g, and variable loads, in this case, snow

load, which is represented as s The inclination of the

whole sheeting is assumed to be zero

The load combination for the ultimate state design

according to Eurcode 1[4] can be calculated as:

k

G

q=1,35×( + )+1,5× (5)

in which 1,35 and 1,5 are partial safety factors for dead

load and variable load, respectively, under unfavorable

effects; Gk and Qk are characteristic values of dead load

and variable load; and w is the self-weight of the

sheet-ing

The yield strength of the steel is 350 N/mm2, the

elastic modulus is 210 000 N/mm2 and the density is 7850

kg/m3 The characteristic value of permanent load is

as-sumed to be 0,5 kN/m2 and that of variable load is 1,8

kN/m2 The thickness of the profile is 0,6 mm The

sup-port length is assumed to be 100 mm The length of

span is 4 m In addition, the minimum distance of the

Fig 8 GA-based sheeting design

parameters Randomly generating the initial population

Decoding

Sheeting design:

Gross section properties Effective section properties Moment resistance

Shear resistance Buckling resistance

Fitness evaluation

Checking the constraints and

calculating the normalised

constraints

Applying the penalty for

the violatedconstraints

Check if the max generation

is reached

Output the results

and stop

Apply the GA operators:

selection, crossover and mutation

Yes

No

stiffener from the nearest corner is set to 10 mm and theminimum distance between stiffeners is set to 10 mm

Fig 9 Loads applied to sheathing

Four design examples are demonstrated in this tion according to the GA-based design procedure men-tioned above The first example is to find the optimumdimensions of the profiled sheeting without any stiffen-ers The other three examples are with stiffeners on theflanges, with stiffeners on the webs and with no limita-tions, ie the stiffeners can be either on the flanges or onthe webs, or both or no stiffeners at all

35The GA, which is based on bit representation, two-

point crossover, bit-flip mutation, and tournament

selec-tion with elitism, is used to perform the optimizaselec-tion

The population size is set to at least twice of the length

of individual string Such parameters as the crossover

rate and the mutation rate in genetic algorithms are set

to 0,8 and 0,001, respectively The selection of these

parameters is based on previous research [8]

4.1 Profiles without stiffeners

The dimensions of the profile are shown in Fig 10

The design variables are the width of the top flange bu,

which is varied from 20 mm to 200 mm; the width of

the bottom flange bp, which is varied from 20 mm to

200 mm; the height of the profile hw, which is varied

from 20 to 170 mm and the inclination of the web θ,

which is varied between 45° to 90°

Fig 10 Dimensions of the profile without stiffeners

Each individual in the initial population can be

formed as concatenating the design variables end by end

and presenting them as a single string For each design

variable, the binary encoding method is used The

gen-eral formula for decoding design variable is [1]:

)(

min X X X X

where X is the decoded value of design variable; Xmax

and Xmin are the maximum and minimum value for the

given design variables; Xd is the decimal integer value

of the binary string; L is the string length corresponding

to each design variable

In the process of calculating the fitness function,

the values of KK1 and KK2 are set in the following way:

perform the optimization with initial value of KK1 = 10

and KK2 = 10; check the violation constraints afterwards

If constraints for the profile with minimum weight are

violated, the values of KK1 and KK2 are increased, for

instance, KK1 to 100 and KK2 to 100, until there is no

constraint violation for the profile of minimum weight

In this analysis, the value of KK1 is found as 1000 and

that of KK2 is as 100

The role of nn1 and nn2 in equation (3) is to make

the weight at the same order as penalty Three formulas

are used to define value of nni, ie case 1: L f−L c

2: L f−L cave

10 , and case 3: L fave−L cave

10 , in which Lf isthe order of weight of each individual; Lc is the order of

weight of individuals in a population and Lcave is theorder of average value of ∑i 

i))2,0

Table 3 also shows the length of sheathing and thepercentage value of the dominant constraints, ie the com-bination of bending and local crippling In addition, theaverage values of weight in 20 runs are also provided inthe Table

Table 3 Comparison of case 2 and case 3

Case 2 (kk 1 = 1000, kk 2 = 100) (kk 1 = 1000, kkCase 3 2 = 100)

[mm] [kg/mW12 ] SMF5 [mm] Ls [kg/mW12 ] 98,64 1500,54 12,87 99,33 1500,44 14,10 99,12 1500,36 14,21 97,89 1500,44 13,37 99,40 1500,19 13,19 99,09 1499,70 12,97 99,44 1500,43 13,54 95,28 1500,32 15,11 99,99 1499,98 13,37 100,16 1500,58 13,97 100,18 1499,86 13,58 99,73 1500,01 13,85 90,03 1500,43 15,64 99,20 1499,61 13,75 99,07 1500,50 12,77 95,93 1500,17 14,47 99,30 1500,14 13,82 95,76 1499,68 13,47 95,49 1500,23 14,46 99,99 1500,11 13,55 91,74 1500,35 14,68 97,31 1500,20 14,32 98,43 1500,27 13,67 98,64 1500,27 13,51 99,95 1500,15 13,75 99,67 1499,78 13,39 98,77 1500,02 14,30 97,50 1500,20 14,08 99,48 1499,73 13,18 98,53 1500,14 13,82 99,36 1499,77 13,48 99,70 1500,13 13,73 97,80 1500,26 13,58 97,32 1500,04 13,58 98,82 1499,71 13,97 96,17 1499,61 13,55 98,61 1499,94 13,33 99,44 1499,88 14,19 99,31 1500,41 13,94 95,45 1499,68 14,61

By running the program based on case 1, we foundout that the profile of minimum weight with no viola-tions of the inequality constraints can be found via in-creasing the value of KK1 gradually However, we can-not find the profiles that have the acceptable values ofstrip length via varying the value of KK2 This is due tothe fact the formula of defining nni in case 1 does notinclude the effect of the order of each individual Onlythe integer part is taken into account According to thedefinition of penalty for inequality constraints, the fea-sible individuals are kept with α = 0 Therefore, as the

optimization is preceded, the optimization is concentrated

to find the minimum weight among the individuals with

no constraints violation even the effect of the order is

not considered However, the value of β always appeared

in the formula for calculating fitness function As the

optimization proceeds, those individuals with lower value

of integer part rather than those with small constraints

violations are kept

When comparing the optimization results based on

case 2 and case 3, it can be seen from the Table that

both case 2 and case 3 give reasonable results

How-ever, the case 2 provides the least weight comparing to

case 3 As far as case 3 is concerned, it is only

neces-sary to calculate the order of the average weight and the

order of the average constraint Thus, the calculation

speed is improved when more design variables are

in-volved The analysis in this paper is based on case 2

The final optimum dimensions for the profile

with-out any stiffeners in 20 runs are shown in Fig 11 In the

figure, “n4-W1337” represents that the number of the

fold is 4 and the minimum weight is 13,37 kg/m2 The

weight of the best profile is 12,77 kg/m2 and the number

of fold is 6 The constraints for the optimum profiles are

(mm) (mm)

Fig 11 Optimum dimensions of the profile without

stiff-eners

The results in 20 runs can be classified into several

groups according to the numbers of folds The profiles

illustrated in Fig 11 are selected as the one with least

weight in each group By doing so, it is possible to

pro-vide more options for the manufacturers or designers

when the manufacture facilities and practical application

is taken into account For instance, roof sheeting can be

classified as “cold” roof, which has outer waterproof skin

with internal insulation if required, and “warm” roof,

which includes insulation and waterproofing For “warm”

roof, the main requirement of preventing penetration by

rainwater leads to shallow profiles with a sequence of

wide and narrow corrugations For “warm” roof, it

nor-mally has the wider flanges on the top so as to provide

sufficient support for the insulation

4.2 Profiles with stiffenersThe calculation is classified into three cases: pro-files with flange stiffeners, profiles with web stiffenersand profiles without any limitation Besides the designvariables provided in the profiles without stiffeners inthe previous section, the range of the following variablesare given before running the programme: the heights ofthe flange stiffeners are varied from 0 mm to 15 mm;the widths of the flange stiffeners are varied from 5 mm

to 15 mm; the inclinations of the flange stiffeners arevaried from 45° to 90° and the length of the web stiffen-ers are varied from 0 mm to 30 mm

The optimum dimensions for the above-mentionedthree cases are shown in Figs 12, 13, 14, respectively.Similarly, these figures also show the other possible pro-files with different numbers of folds The correspondingconstraints for these three cases are shown in Table 4

020406080100120140160

n6-W1136

(mm) (mm)

Fig 12 Optimum dimensions of profiles with flange stiffeners

020406080100120140160

(mm) (mm)

Fig 13 Optimum dimensions of profiles with web stiffeners

When comparing the optimum profiles shown inFig 11 to those in Fig 12, it can be seen that the profilewith stiffeners both on the flanges and on the webs hasthe minimum weight However, the other cases providedhere can give the alternatives when the cost, techniques

of manufacturing, and the practical applications of theprofiles are taken into account

Fig 14 Optimum dimensions of the profiles with no

limitations

Table 4 Values of constraints as percentage of the limits for the optimum profile for various cases

4.2 Comparison

Fig 15 shows the comparison of the weight of

opti-mized profiles for the case without any limitation for

web and flange stiffeners to some commercial profiles

It also shows the ratio of calculated strip width using

current dimensions to the provided strip width (1500 mm

here) It can be seen that optimized profile using GA

shows the lighter weight and more efficient use of

mate-rials

Fig 15 Comparisons with commercial profiles

5 Summary and future perspectives

As demonstrated in this paper, the Genetic

Algo-rithm (GA) can be used as an optimization tool to

ob-tain the optimum dimensions of the profiled sheeting

References

1 Adeli, H and Cheng, N T Integrated genetic algorithm for optimization of space structures Journal of Aerospace Engineering, 1993, Vol 6, No 4, p 315–328.

2 Bazaraa, M S.; Sherali, H D and Shetty, C M ear programming: theory and algorithms John Wiley & Sons, Inc., 1993, p 360–372.

Nonlin-3 Cogan, B The evolution of genetic algorithms Scientific Computing World, 2001, May/June, p 28–31.

4 ENV 1991-1 Eurocode 1: Basis of design and actions on structures, Part 1: Basis of design, 1994, p 45–53.

5 ENV 1993 Eurocode 3: Design of steel structures, Part 1.3: General rules Supplementary rules for cold-formed thin gauge members and sheeting, 1996.

6 Koumousis, V K and Georgion, P G Genetic algorithms

in discrete optimization of steel truss roofs Journal of Computing in Civil Engineering, 1994, Vol 8, p 309–325.

7 Lee, C L; Mioduchowski, A and Faulkner, M G mization of corrugated claddings Journal of Structural En- gineering, 1995, Vol 121, No 8, p 1190–1196.

Opti-8 Lu, W Optimum design of cold-formed steel purlins ing genetic algorithms, Publications, TKK-TER-25, Labo- ratory of steel structures, Helsinki University of Technol- ogy, 2003, p 59–79.

us-9 Michalewicz, Z Genetic Algorithms + Data Structures = Evolution Programs, Third, revised and Extended Edition, Springer, 1999, p 57–93.

10 Mitchell, M An introduction to genetic algorithms bridge (MA) MIT Press, 1998, p 1–31.

Cam-11 Nagy, Z V Evolution of optimum trapezoidal sheeting profile based on Eurocode, using finite strip method and genetic algorithm Proceedings of the third international conference on coupled instabilities in metal structures, Lisbon, Portgual, 21–23 Sept, 2000, p 643–650.

12 Seaburg, P A and Salmon, C G Minimum weight design

of light gage steel members Journal of Structural sion, 1971, Vol 97, No ST1, p 203–222.

Comercial profiles Opt profile (NL)

DEKORATYVINIO TANKAUS SILIKATINIO BETONO MIĐINIO SANDử SAVYBIử

ÁTAKA DIRBINIử KOKYBEI

Algimantas NaujokaitisStatybiniụ medợiagụ katedra, Vilniaus Gedimino technikos universitetas, Saulẻtekio al 11,

LT-10223 Vilnius-40, Lietuva El pađtas: naujok@st.vtu.lt

Áteikta 2003 08 28; priimta 2004 04 21

Santrauka Iđnagrinẻta dekoratyvinio tankaus silikatinio betono miđinio savybiụ priklausomybẻ nuo miđinio sandụ Darbo tikslas buvo parodyti, kokios sandụ savybẻs turi átakos tiksliụ matmenụ silikatiniụ dekoratyviniụ betonụ savybẻms Nustatyta, jog miđinio sutankinimo vienodumui, suformuoto dirbinio matmenụ tikslumui didợiausios átakos turi miđinio granuliometrinẻ sudẻtis Darbas atliktas naudojant naujo preso kompiuteryje tikslingai sukauptus duomenimis Tyrimams gamybinẻmis sàlygomis buvo naudoti praktiđkai neuợterđti priemaiđomis, vidutinio smulkumo ir smulkieji Giraitẻs telkinio kvarciniai smẻliai Parengta nauja miđiniụ su daợomaisiais pigmentais sudẻèiụ parinkimo metodika, ávertinanti riđiklio su pigmentu savybes Tyrimo duomenys naudojami tiksliụ matmenụ dekoratyviniụ dirbiniụ gamyboje.

Raktaợodợiai: sandai, silikatinis betonas, betono sudẻtis, smẻlis, grũdinẻ sudẻtis, pigmentai, smẻlio smulkumas, tiksliụ matmenụ dirbiniai, sutankinimo koeficientas.

1 Ávadas

Gaminant dekoratyviná silikatiná betonà visi jo

san-dai dalyvauja cheminẻse reakcijose ir turi átakos visoms

produkto savybẻms Pasikeitus vienam iđ sandụ,

pasikei-èia ir pagamintos medợiagos mechaninẻs bei fizikinẻs

sa-vybẻs Tai privalu ávertinti, parenkant silikatinẻs masẻs

sandụ sudẻtá, ypaè daợomojo pigmento rũđá ir kieká Đie

klausimai buvo sprendợiami empiriđkai, analizuojant

at-skirus sandus dalimis, o vẻliau sujungiant juos á sistemà

Akivaizdu, kad vienodomis gamybos sàlygomis, kai

sandụ savybẻs yra panađios, silikatinio betono

kokybi-niai rodikliai pirmiausia priklauso nuo silikatinẻs

cemen-tuojanèios medợiagos sudẻties Autorius daro prielaidà,

kad dekoratyvinis silikatinis betonas bũna geriausios

ko-kybẻs, kai sunaudojamas minimalus kalcitiniụ kalkiụ

kie-kis, galintis, naudojant daợomuosius pigmentus, susijungti

su kvarciniu smẻliu Idealiu atveju susidariusios

cemen-tuojanèios medợiagos kiekis priklausys nuo trijụ

veiks-niụ: naujadarụ sluoksnio storio, kvarcinio smẻlio

lygina-mojo pavirđiaus ir pigmento dispersiđkumo Ávertinus tai

parenkami smẻlio, kalkiụ ir pigmento kiekiai Reikia

áver-tinti ir norimo suformuoti pusfabrikaèio stiprá, kuris

pri-klauso nuo lyginamojo slẻgio á formavimo masữ,

slẻgi-mo trukmẻs, riđiklio ir kvarcinio smẻlio granuliometrinẻs

sudẻties, koloidiniụ daleliụ kiekio, drẻgmẻs kiekio

ma-sẻje Apskaièiuojami miđinio sandụ kiekiai ir gaminamas

miđinys Smẻlio, kurio grũdeliai yra ađtriabriauniai, su

nelygiu pavirđiumi, frakcijụ sankiba yra didesnẻ, nei

ap-valios formos grũdeliụ Pusfabrikaèio stipris priklauso nuoslẻgio vandens mikrokapiliaruose, kuriuos sudaro disper-sinẻs dalelẻs, susikaupusios tarp ávairaus dydợio smẻliodaleliụ Stiprio didinimas galimas didinant mikrokapilia-

rụ kieká miđinio struktũroje Tai pasiekiama, parenkantsmẻlio grũdinữ sudẻtá, didinant dispersiniụ ir riđamosiosmedợiagos daleliụ kieká

Pusfabrikaèio stipris dar priklauso nuo liniụ traukos jẻgụ, atsirandanèiụ ávairaus dydợio daleliụsusilietimo vietose, kai atstumas tarp daleliụ maợesnis uợ

tarpmoleku-jụ skersmená [1] Labai keièiasi kalkiniụ daleliụ dydis irkiekis masẻje Be to, á spalvotus dirbinius pridedamasmulkiadispersinio pigmento, kuris chemiđkai veikia mi-điná Kaip teigiama [2], daleliụ lyginamasis pavirđius yra

18 900 – 34 600 cm2/g Kalkiụ daleliụ skersmuo:

d = 6 ở 103 / (ρ Sp), mkm, (1)

ρ – Ca(OH)2 tankis; Sp – lyginamasis pavirđius, cm2/g.Dalelẻs skersmuo gali bũti nuo 1,5 mkm iki

210 mkm Taigi gali susidaryti pakankamai daug

kontak-tụ [2, 3] Negalima pamirđti, kad dalelẻs linkusios guliuoti Gesintụjụ kalkiụ masẻje yra rezervụ riđamajaimedợiagai atsirasti [4]

koa-Smẻlio grũdeliai daợnai yra ađtriabriauniai, tokie yra

ir nagrinẻjamos technologijos atveju Ađtrũs kampai didina pusfabrikaèio stiprá, taèiau priklauso nuo disper-siđkumo ir elektrostatinẻs sankibos [4]

pa-Diskutuojama dẻl tankiụ plonụ vandens plẻveliụ, suojant suriđanèiụ dispersines daleles [5, 6] Taèiau tokios

plẻvelẻs daợniausiai yra tik intarpai tarp daleliụ Iđskirtinữ

vietà, kaip manoma, turi koloidinẻs medợiagos, kuriụ

dalelẻs gali sudaryti tiltelius, jungianèius stambesnes

daleles, esanèias didesniu atstumu nei molekuliniụ jẻgụ

veikimo laukas [7]

Sutankintas pusfabrikatis sudaro pakankamai akytà

medợiagà, kurioje yra daug mikro- ir makrokapiliarụ,

ne-visiđkai uợpildytụ vandeniu Susidarữ tarp daleliụ

van-dens meniskai, turintys pakankamai laisvosios energijos,

sukelia átempimus, taèiau kartu stiprina pusfabrikatá [7, 8]

Maợesnis pigmentụ priedas turi teigiamos átakos

kal-cio hidrosilikatụ susidarymui, pagerẻja gaminiụ

stipru-mas ir jụ eksploatacinẻs savybẻs [9] Nustatyta, kad

pig-mentụ daợomàjà gebà lemia jụ smulkumas ir juose

esanèios daợomosios medợiagos kiekis Esant didesnẻms

điụ rodikliụ reikđmẻms intensyvesnẻ ir pigmentụ daợomoji

geba [10] Paợymẻtina iđskirtinẻ suodợiụ átaka silikatinio

akmens savybẻms, ypaè vandens ágeriamumui Đie

pig-mentai yra hidrofobiđki, yra didelis lyginamasis

pavir-đius, taèiau vandens ágeriamumas taip pat didelis

Mano-ma, kad prie pigmento daleliụ susidaro mikroporos dẻl

didelio hidrofobiđko pavirđiaus blogo sàlyèio su

silikati-nio akmens hidrosilikatais [11]

Iđanalizavus minẻtas teorijas, reikia pabrẻợti, jog

spalvotas silikatinis miđinys, iđ kurio formuojami

gami-niai, yra sudarytas iđ gamtinio grũdinio smẻlio,

disper-siđkos riđamosios medợiagos, taip pat ir gesintụjụ kalkiụ

bei pigmentụ, susidedanèiụ iđ gausybẻs smulkiụ daleliụ,

o smẻlyje yra labai maợụ kvarco grũdeliụ bei molio

mi-neralụ Miđinyje yra ir vandens bei oro burbulẻliụ, kuriụ

nepakanka uợpildyti formavimo metu susidariusioms

tuđ-tumoms Sutankinant silikatiná miđiná veikia ávairios

jẻ-gos, didinanèios jo stiprá: tai mechaninis grũdeliụ

sulipi-mas, molekuliniai sukibimo ryđiai vandens plẻveliụ

kapiliaruose ir tarpkoloidiniụ daleliụ sàveika Ypaè

di-delữ reikđmữ turi vanduo, sujungdamas koloidines

maợà-sias daleles su stambesniais smẻlio grũdeliais Sukibimo

jẻgụ dydis priklauso nuo sandụ savybiụ: smẻlio liometrinẻs sudẻties, grũdeliụ formos ir dydợio, sumaltosmẻlio kiekio, kalkiụ dispersiđkumo ir hidratacijos laips-nio, priemaiđụ sudẻties ir kiekio, pigmentụ kiekio ir sa-vybiụ, vandens kiekio Technologiniai preso ypatumai irgisvarbũs geram pusgaminio sutankinimui, nes privalu kuogeriau uợpildyti laisvà tũrá tarp smẻlio grũdeliụ, kad juosvienas nuo kito skirtụ ploniausi riđamosios medợiagossluoksniai Toks sutankinimas leidợia gauti tankụ ir stip-

granu-rụ silikatiná betonà

Darbo tikslas – iđtirti atskirụ sandụ átakà tiksliụ menụ dekoratyviniụ silikatiniụ betonụ ir plytụ gamybai.Atsiradus đalyje naujai technologinei árangai, yra gali-mybẻ gaminti didesnio santykinio tankio tiksliụ matme-

mat-nụ ávairios formos ir dydợio gaminius Iki điol mais technologiniais árenginiais negalima buvo tiksliaureguliuoti dirbiniụ matmenụ Suformuoti pusfabrikaèiaideformuojasi dẻl ávairiụ veiksniụ, taèiau gaminant tiks-liụ matmenụ dirbinius bũtina pagaminti kiek ámanomastipresná pusgaminá, maợiausiai paợeidợiamà kitose tech-nologinẻse operacijose Naujai iki điol đalyje nenaudotaitechnologinei presavimo árangai, kai naudojami vietiniaisandai, technologiniụ tyrimụ nẻra atlikta Reikẻjo iđnag-rinẻti điuos technologinius parametrus: formavimo miđi-nio sudẻties átakà; dvipusá slẻgimà á pusgaminá; smulkio-sios sandụ dalies kieká formavimo masẻje, miđiniolyginamojo pavirđiaus átakà, vandens kieká Pagrindinistyrimo tikslas – parinkti miđiná, norint gauti kokybiđkusdirbinius

naudoja-2. Tyrimụ metodikaTyrimams buvo naudotas dvipusio slẻgio hidraulinisautomatiđkai valdomas KSP 402 presas, kurio valdymosistema leidợia fiksuoti atskirụ operacijụ atlikimà irtechnologinius parametrus, árađant juos á valdymosistemos atmintá

Naudotas kvarcinis smẻlis iđ Giraitẻs telkinio.Cheminẻ jo sudẻtis: SiO2 82,6–91,48 %, Al2O3 3,2–4,19 %, CaO 2,8–4,5 % Grũdinẻ sudẻtis pateikiama

1 pav Sijojimas atliekamas pagal standarto EN 1015-1reikalavimus Dalis smẻlio buvo ápilta malant kalkes, jávadinsime maltu smẻliu Smẻlio smulkumas buvo nustato-mas AT-5 prietaisu Kalcitinẻs negesintosios antros rũđieskalkẻs – „Naujojo kalcito“ gamybos, jụ aktyvumas 65–

85 %, MgO – 1,2–1,5 % Jụ savybẻs tirtos pagal GOST

9179 metodikà Spalvà suteikiantis pigmentas – Bayerfirmos – 920, tankis 4,1 g/cm3, Fe2O3 yra 85–87 %.Silikatinio betono miđiniai buvo ruođiami naudojantsausas medợiagas, dozuojami pagal masữ Bandiniaiformuoti natũralaus dydợio (25ừ12ừ8,8 cm) Miđiniosudẻties, slẻgio dydợiui presavimo formoje, dirbiniosutankinimui, granuliometrinẻs sudẻties ir drẻgnio átakainustatyti bandiniai nebuvo kietinami Tyrimai atliktisuformavus bandinius Dalis jụ buvo kietinami irnustatomas galutinis gniuợdomasis bei lenkiamasis jụstipris, tankis ir vandens ágẻris

1 pav Smẻlio grũdinẻ sudẻtis: A – Giraitẻs telkinio smẻlis;

B – sijotas, geros grũdinẻs sudẻties smẻlis

Fig 1 Sieve graphical analysis of sand: A – sand from

the Giraitẻs deposit; B – sand riddle, granular structure of

Sietụ akuèiụ dydis, mm

3. Tyrimụ rezultatai

Pagrindiniai dekoratyviniụ silikatiniụ dirbiniụ

tiks-lumo technologiniai parametrai yra presavimo bũdas,

pre-savimo slẻgio dydis ir formavimo miđinio sudẻtis

Presa-vimo slẻgio dydá tiriamoje technologijoje galima keisti

nepriklausomai nuo kitụ technologiniụ parametrụ:

forma-vimo miđinio suslegiamumo, demferuojanèio veiksnio,

mi-đinio sudẻties Mimi-đinio granuliometrinẻ sudẻtis buvo

pasirinkta gera ir natũrali karjerinẻ Ji yra svarbi

dirbi-nio suformavimui, deformavimuisi nuo fizikiniụ ir kitụ

veiksniụ, todẻl buvo sudaryta naujos sudẻties jo

parinki-mo principinẻ metodika Siũlomas miđinio sudẻties

pa-rinkimo metodas Riđiklio kiekis P3 apskaièiuojamas taip:

P – 1 m3 sutankinto sauso formavimo miđinio masẻ, kg;

P1  – smẻlio masẻ 1 m3 sutankintame sausame

S1  – malto smẻlio lyginamasis pavirđius, m2/kg;

S2  – nemalto smẻlio lyginamasis pavirđius, m2/kg;

S3  – kalkiụ lyginamasis pavirđius, m2/kg;

S4  – pigmentụ lyginamasis pavirđius, m2/kg;

A  – kalkiụ aktyvumas, vieneto dalimis;

K11  – koeficientas, ávertinantis nemalto smẻlio daleliụ

pavirđiụ;

K21  – koeficientas, ávertinantis malto smẻlio daleliụ

pavirđiụ;

K5  –  koeficientas, ávertinantis pigmentụ savybes;

q – reikđmẻs, nustatomos pagal 2 pav reikđmes

1, 2 pav ir lentelẻje pateikiami duomenys

silikati-nio betono sudẻèiai parinkti pagal kalkiụ aktyviosios

da-lies masữ ir smẻlinẻs dada-lies dispersiđkumà Kiti

duome-nys apie sandus imami pagal savybiụ tyrimo reikđmes

Pigmentụ savybiụ koeficientụ (K11, K21) reikđmẻs

áverti-namos pagal gamintojo deklaracijas

2 pav Minimalus aktyvaus CaO kiekis miđinyje mai nuo smẻlio smulkumo

priklauso-Fig 2 Minimal amount of CaO in the mix depending on fine grained sand

Smẻlio tuđtymẻtumas ir lyginamasis pavirđius Sand voids and specific surface

ụ il e ũ r G

, o m s e s m m

o il ẻ m S - u t ẻ m y t đ u t

% ,s a m

si n it u i V ụ il e ũ r g

, o m s e s m m

si a m a i g y L

,s u iđ ri v a

m 2 / g 0

, 1 0 ,

5 , 0 0 ,

5 , 0 5 , 0 5 1 , 0 5 ,

5 0 , 0 5 1 ,

8 0 , 0 5 0 ,

Sudarant silikatinữ masữ kalkẻs sveriamos ne pagalbendrà masữ, o pagal aktyviosios dalies masữ, kuridalyvaus cheminẻje reakcijoje Be to, ávertinamakvarcinio (malto ir nemalto) smẻlio ir pigmento savybẻs.Esant tam paèiam kalkiụ aktyvumui, pagal siũlomàsudẻties parinkimo metodikà faktinis kalkiụ kiekispriklauso nuo jụ kokybẻs Naudojant đvieợiai iđdegtasdidelio aktyvumo kalkes su minimaliu priemaiđụ kiekiu,

jụ masẻ sumaợẻja Jei kalkẻs turi daug neiđdegusiokalkakmenio ir priemaiđụ ir buvo ilgai laikytos ore, jụmasẻ padidẻja Pakeitus nenutrũkstamai veikianèiusdozatorius á periodinio-porcijinio svẻrimo dozatorius,buvo galima gerokai tiksliau pasverti kalkes ir silikatinữriđamàjà medợiagà Sumaợẻjo kalkiụ sànaudos 1000 vnt.spalvotụjụ plytụ reikiamai stiprumo markei gauti Realiaitai pasiekiama tik naudojant elektroniná svẻrimo valdiklá.Slegiant tik preso puasonu iđ vienos pusẻs, slẻgissilikatinẻs masẻs pripildytoje presformoje pasiskirstonetolygiai [12] Miđinys susitankina prie formos sieneliụ,

o vidinẻje dalyje ir prieđingoje puasono pusẻje masẻsusitankina maợiausiai

0 0,001 0,002 0,003 0,004 0,005 0,006 0,007 0,008

Slegiant ið abiejø dirbinio pusiø dviem slëgimo dydá

reguliuojanèiais puasonais, dirbinio tankis skerspjûvyje

suvienodëja (2 pav)

3 paveiksle pateiktas slëgio dydþio pasiskirstymas

sutankintame silikatiniame betone Hidraulinis presas

slegia pradþioje apatinæ masës dalá, o po 0,5 s ásijungia

ir virðutinis puasonas 3 pav a) pateikta geresnë smëlio

grûdinë sudëtis, todël gaunamas tankiausias daleliø

iðsidëstymas, o pusfabrikaèio stipris bûna vienodesnis

negu 3 pav b), kur smëlio grûdinë sudëtis artima

natûraliai Miðinio struktûrà sudaro visi sandai, ji

priklauso nuo ðiø sandø iðsidëstymo ir uþimamo tûrio

Svarbiausi elementø parametrai yra tûris ir stambiøjø

daleliø vidutinis skersmuo, turintis átakos riðamosios

medþiagos kiekiui Vienodø stambiøjø daleliø didesnis

kiekis didina sistemos tuðtymëtumà, o ðioms tuðtymëms

uþpildyti sunaudojama daugiau riðamosios medþiagos

3 pav Silikatinio (nesukietinto) betono stipris gniuþdant,

MPa (slëgis formoje 18,6 MPa): a) – geros grûdinës

sudëties smëlis; b) – smëlio grûdinë sudëtis nëra pakankamai

gera (Giraitës telkinio smëlis sijotas per 20 mm akutës sietà)

Fig 3 Silicate concrete compressive strength: a – good

granular structure sand; b – sand granular structure is not

good enough (Giraitës bed sand sifted through the 20 mm

stitch bolter)

Tuðtymëtumui sumaþinti reikia smulkesniø

disper-siniø daleliø Koloidinës dalelës, maþesnës kaip 0,1 mkm,

yra labai svarbios [12] Padidëja kontaktø tarp stambiø

daleliø kiekis Pigmentai dekoratyviniame silikatiniame

miðinyje atlieka klijuojanèios medþiagos vaidmená ir

padidina pusfabrikaèio stiprá Buvo naudotas ávairios

sudëties kalkiø ir smëlio miðinys Ruoðiant toká miðiná

imamas vienodas pigmento kiekis ir keièiamas tik kalkiø

kieká permalant miðiná Ruoðiamas miðinys, kurio

aktyvumas – nuo 5 % iki 18 % Dispersiðkumas apytikriai

vienodas Maiðyta permalimo ir trynimo bûdu, o

antrajame variante pasverti komponentai sumaiðyti

priverstiniame maiðytuve 4 pav matyti, jog sandø sudëtis

pusfabrikaèio stipriui nëra labai svarbu, bet sumaiðymo

bûdas yra reikðmingas Sveriant sandus automatiðkai

reguliuojamomis svarstyklëmis, gaunami pakankamai

tikslûs jø kiekiai, todël praktikoje pasirenkami priverstinio

tipo maiðytuvai, uþtikrinantys vienodà sandø

pasiskirs-tymà miðinyje Miðinio daliø permalimas gamybos

sàlygomis yra sudëtingas, tam reikia dideliø energijos

sànaudø

Miðinio aktyvumo didinimas ekonomiðkai yranenaudingas, nes sunaudojami dideli riðamosios medþia-gos kiekiai ir pablogëja galutinio produkto atsparumasatmosferiniams veiksniams Todël praktiðkai pakanka 5,3–6,2 % miðinio aktyvumo

0,1 0,150,20,250,30,35

Miðinio aktyvumas, %

4 pav Miðinio sudëties átaka pusfabrikaèio stipriui: 1 – apskaièiuotos pagal (3) ir (4) formules; 2 – apskaièiuota pagal nepakeistà silikatinës masës paruoðimo schemà Fig 4 Influence of mix composition on the strength of half-finished product: 1 – composition according to for- mulae 3 and 4; 2 – composition under the application of non-modified silica paste preparation scheme

Silikatinio dekoratyvinio miðinio sutankinimas giai priklauso nuo smëlio grûdinës sudëties (5 pav)

tiesio-1 2 3 1,3

1,5 1,7 1,9 2,1 2,3

Fig 5 Influence of grain composition of sand on the paction of silica paste with pigment Activity: 1 % –  7,40%; 2 % – 5,30%; 3 % – 2,50%

com-Kuo daugiau miðinyje yra ávairiø frakcijos daleliø,tuo lengviau jis sutankinamas, tuo didesnis gaunamaspusfabrikaèio stipris Kalkiø ir pigmento smulkiadisper-sës dalelës kartu su vandeniu uþpildo poras tarpstambesniø grûdeliø, padidëja kontaktø kiekis tarp miðiniodaleliø, susidaro mikrokapiliarai, iðnaudojamos vandensfizikinës savybës didesniam pusfabrikaèio gniuþdomajamstipriui gauti Silikatinës masës formavimo drëgnis turi

43bûti proporcingas ðio miðinio lyginamajam pavirðiui Jis

nustatomas ne pagal smëlio frakcijos kieká, o pagal

smul-kiøjø daleliø masæ ir jø bendràjá lyginamàjá pavirðiø

(5 pav)

Miðinio drëgná charakterizuoja maksimalus jo

drëgnio imlumas Matome, kad kreivës 1, 2 ir 3 yra

vienodo pobûdþio, didëjant miðinio drëgniui pusfabrikaèio

stipris taip pat didëja Reguliuojamas masës presavimo

bûdas leidþia pasiekti pakankamà pusfabrikaèio mechaniná

gniuþdomàjá stiprá, esant 4,5–5,3 % formavimo masës

drëgniui Visais atvejais pusgaminio stiprio pagrindas yra

dispersinës dalies kiekis, kurio suriðimo procese dalyvauja

vanduo, esantis mikrokapiliaruose Sutankintas miðinys

su daþomaisiais pigmentais yra stipresnis

4 Iðvados

1. Bûtina suderinti dekoratyvinio tankaus silikatinio

betono sudëtiniø daliø savybes, norint pagaminti geros

kokybës dirbinius

2. Parenkant miðinio sudëtá, riðamosios medþiagos

ir kalkiø kiekis apskaièiuojamas ne pagal bendrà kalkiø

masæ, o tik pagal aktyviosios dalies masæ, susiejant jà su

kitø miðinio sandø savybëmis

3. Optimalus pigmentø kiekis, su riðamàja medþiaga

maiðant mineraliná pigmentà, kuris suteikia pageidaujamo

intensyvumo spalvà, parenkamas pagal kalkiø ir kitø

dispersiniø daleliø kieká Tai sudaro galimybæ taupyti

pigmentus (jø sunaudojama perpus arba net kelis kartus

maþiau), pagerinti dirbiniø kokybæ. Pusfabrikaèio stiprio

vienodumas dirbinio tûryje gaunamas slegiant paruoðtà

silikatinæ masæ vienodu slëgiu pagrindinëms plokðtumoms

prieðingomis kryptimis

4. Silikatinio betono pusfabrikaèio matmenø

tikslu-mui ir stipriui didþiausios átakos turi du pagrindiniai

veiksniai: silikatinës masës suspaudimo bûdas ir

dispersið-kosios dalies kiekis

5. Gaminant didelio matmenø tikslumo spalvotus

silikatinius dirbinius rekomenduojama naudoti minimalaus

tuðtymëtumo aðtriabriaunius smëlius, jø kiekius

apskai-èiuojant pagal siûlomà metodikà, tankinant dvipusio

prop-6 Hiese, W.  Collection of works (Baustoffkentnis Düsseldorf),

9 Karsten, R Constructional chemistry 9 (Bauchemie 9) Aufl., Verlag C F Müller Karlsruhe, 1999 16 p (in German).

10 Rade, D. Research of inorganic pigments and their use for the production of coloured silica articles (Einige Untersuchungen über dieVerwendung von Anorganischen Baupigmenten zur Herschtellung von Farbkalksandstein).

II JSD KB, Hannower, 1975 62 p (in German).

11 Hanssen, V Inorganic pigments for the production of silica bricks, Areas of application (Anorganische Bayer-Pigmente zur Einfarbung on Kalksandsteinen) Sparte AC Anw- endungstechnik 10/96 Leverkusen P, Bayer AG, 1999 27 p (in German).

12 Larrend, F The Influence of aggregate on the compressive

strength of normal and high-strength concrete ACI

Materi-als Journal, Vol 94, No 5, 1997, p 417–426.