Research ArticleStructural Phenomenon of Cement-Based CompositeElements in Ultimate Limit State

I Iskhakov and Y Ribakov

Department of Civil Engineering Ariel University 40700 Ariel Israel

Correspondence should be addressed to Y Ribakov ribakovarielacil

Received 7 March 2016 Revised 27 July 2016 Accepted 28 July 2016

Academic Editor Ana S Guimaraes

Copyright copy 2016 I Iskhakov and Y Ribakov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Cement-based composite materials have minimum of two components one of which has higher strength compared to the otherSuch materials include concrete reinforced concrete (RC) and ferrocement applied in single- or two-layer RC elements Thispaper discusses experimental and theoretical results obtained by the authors in the recent three decades The authors have payedattention to a structural phenomenon that many design features (parameters properties etc) at ultimate limit state (ULS) of astructure are twice higher (or lower) than at initial loading state This phenomenon is evident at material properties structures(or their elements) and static andor dynamic structural response The phenomenon is based on two ideas that were developedby first author quasi-isotropic state of a structure at ULS and minimax principle This phenomenon is supported by experimentaland theoretical results obtained for various structures like beams frames spatial structures and structural joints under staticorand dynamic loadings This study provides valuable indicators for experimentsrsquo planning and estimation of structural state Thephenomenon provides additional equation(s) for calculating parameters that are usually obtained experimentally and can lead todeveloping design concepts and RC theory in which the number of empirical design coefficients will be minimal

1 Introduction

Behavior of structures in certain cases is similar to basicprinciples in human society One of the main principles inhuman society is that all people are equal to each other Inother words if for example there are two persons 1198731 and1198732 then each of them is equal to the other and therefore 1198731is not greater and not less than 1198732 or

1198731 = 1198732 = 119873or 1198731 + 1198732 = 2119873 (1)

Let us assume that two persons1198731 and1198732 should carry119882 =50 kg Then following (1) each of these persons takes 25 kgthat is

1198822 = 1198821 = 25 kg1198821 + 1198822 = 119882 = 50 kg (2)

In other words119882 = 21198821 = 21198822 (3)

It will be demonstrated in this paper that a similar princi-ple is an objective property of a structure made of elastic-plastic or ideally elastic-plastic material in ultimate limitstate (ULS) from the load bearing capacity viewpoint Thisproperty is further called a ldquostructural phenomenonrdquo Theessence of this phenomenon is that values of some parametersof the structure that aremeasured experimentally under staticorand dynamic loadings increase or decrease by about a fac-tor of twoTherefore an alternative nameof this phenomenonis doubling of structural design parameter at ULS

A scheme of the principle for a general case is presentedin Figure 1 For example the figure presents dependence ofdominant vibrationmode frequency of a 1 10 scaledmodel ofa 24 times 24m reinforced concrete (RC) shell versus load ratio119902119902ul Here 119902 is the live load (the value of 119902 is varied from0 to 119902ul) and 119902ul is its ultimate value As it follows from thefigure at the ULS (119902 = 119902ul) the above-mentioned parameterdecreases twice relative to its initial value

Another example that demonstrates increase in structuralparameter is shown in Figure 2 A problem of increasing

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 4710752 9 pageshttpdxdoiorg10115520164710752

2 Advances in Materials Science and Engineering

10

15

20

25

00 02 04 06 08 10

K1

24 m

24 m

qqul

fI (1sec)

Figure 1 Dominant technical frequency119891119868 versus load ratio 119902119902ul

22

44

1

2

3

35

25

20

30

40

2 4 6 8 1000

12

w (cm)

120585 ()

Figure 2 24 times 24m shell deflection versus damping ratio forvarious ground motions 1 high-frequency earthquake 2 medium-frequency earthquake and 3 low-frequency earthquake

residual deflections in a 24 times 24m RC shell under staticload and additional vertical seismic excitation is consideredThe following earthquakes were selected Gazly (Uzbekistan)1976 San Fernando (USA) 1971 and Bucharest (Romania)1977 These earthquake records represent high- medium-and low-frequency groundmotions respectivelyThe depen-dence of residual deflections on damping ratio was studied(Table 1) The damping ratio 120585 varied from 0 to 12 Thestatic displacement value (hidden line in Figure 2) was22 cm For a high-frequency Gazly earthquake (curve 1in Figure 2) the residual deflections are accumulated veryintensively and for 120585 = 0 reached 44 cm which is two timeshigher compared to the static value A similar result wasobtained for a medium frequency earthquake (see Table 1and the corresponding curve 2 in the Figure) for whichthe residual deflection equals 36 cm However for a low-frequency earthquake (curve 3 in the Figure) the increasein residual deflections is lower It is because the dominant

Table 1 Deflections in a fill scale RC shell versus damping ratios

Earthquake typeDamping ratio

0 2 4 6 8 10 12Deflection cm

Low frequency 250 225 221 215 209 205 200Medium frequency 363 238 223 217 210 206 200High frequency 440 332 281 263 241 230 219

n

0

05

10

10

10

20mk

mmax

nk

m

qqul

nmin

f(m n) = 0

Figure 3 Relation between structural phenomenon and minimaxprinciple 119898 generalized ldquoincreasingrdquo parameter 119899 generalizedldquodecreasingrdquo parameter 119891(119898 119899) function relating the parameters(conditionally shown by a straight line) and119898119896 119899119896 arbitrary valuesof the parameters

frequency of the shell is high and comparable with that of ahigh-frequency earthquake

Comparing Figures 1 and 2 leads to conclusion thatdecreasing twice (Figure 1) and doubling (Figure 2) instructural parameters correspond to the ULS of the structureThese limit values for the same structure can be interrelatedby the minimax principle [1] Graphical interpretation of thisrelation is shown in Figure 3 The curve in plane 119899-119900-119902119902ulcorresponds to the dependence fromFigure 1 and the curve inplane119898-119900-119902119902ul to Figure 2Theminimax principle combinesthose curves and presents their structural interrelation asshown in the 119899-119900-119898 plane (it is assumed that this relation islinear) Thus if according to experimental data some factoris doubled it is logical to expect that another structuralparameter correspondingly decreases twice

Figure 4 shows the structural phenomenon evidence fora fixed concrete beam In elastic stage

1003816100381610038161003816119872minusel1003816100381610038161003816 = 119902el119871212 = 2119872+el (4)

where 119902el is a uniformly distributed load in elastic stage (119902el le119902el ul) and 119902el ul is the ultimate value of this load 119872minusel and119872+el are themaximumvalues of negative and positive bendingmoments respectively in beam elastic stage

Advances in Materials Science and Engineering 3

L

0 le q le qul

(a)

qL280 le q le qel ulMminus

el

M+el = 05

1003816100381610038161003816Mminusel1003816100381610038161003816

(b)

qulL28

q = qulMminuspl

M+pl =

100381610038161003816100381610038161003816Mminus

pl100381610038161003816100381610038161003816

(c)

Figure 4 Structural phenomenon evidence for a fixed concrete beam (a) static scheme (b) bending moments diagram in elastic stage and(c) bending moments diagram in ultimate limit state

In the ULS after the momentsrsquo redistribution

10038161003816100381610038161003816119872minuspl10038161003816100381610038161003816 = 119902ul119871216 = 119872+pl (5)

where 119902ul is a uniformly distributed load in ULS and119872minuspl and119872+pl are themaximumvalues of negative and positive bendingmoments respectively

In both cases1003816100381610038161003816119872minus1003816100381610038161003816 + 119872+ = 1199021198712

8 (6)

Thus

2 ge 1003816100381610038161003816119872minus1003816100381610038161003816119872+ ge 1 (7)

The examples that were presented in this chapter are notthe only ones Other relevant cases will be discussed below Itshould bementioned that as the structural phenomenon wasnot investigated previously there is a very limited numberof publications in which relevant experimental data wereobtained but not analyzed For example experimental results[2] show that using steel fibers in concrete increases trans-verse deformations about two times (from (016 020) 120576119888to 040 120576119888 where 120576119888 is the longitudinal deformation of thestructure) but it is not emphasized by the researchers

Four-point static tests in ultimate stages show that steelfibers increase the deflections in the middle-span of a highstrength concrete bending element approximately twice [3]the deflections in beams made of concrete class C140 were44 and 84mm without and with steel fibers respectivelyFor beams made of concrete class C200 adding steel fibershas increased the deflections from 33 to 59mm As in theprevious example the fact of doubling the middle-spandeflections was not emphasized

2 Minimax Principle in Ultimate LimitState of Structures

21 Short Review of Ultimate Limit State and MinimaxPrinciple The ultimate equilibrium method proposed by

Gvozdev in 1938 [5] was a logical development of basicexperimental and theoretical investigations carried out byoutstanding researchers as Galilei Coulomb Bach Graf andothers [6] This method is still used for calculating structuresat ultimate limit state (load bearing capacity)

Followingmodern design codes methods based on plas-tic analysis are suitable for checking structures at ultimatelimit state [7]Themethod is constantly developed For a spe-cific structure there is an exact correspondence between theexternal loadsrsquo system and internal forcesrsquo distribution Just inthis case the structure may transform into a mechanism [8]

Later the ultimate equilibrium method was enlarged forapplication in structures that aremade of concrete typemate-rials with downloading branch in the stress-strain diagram[9]

Following Gvozdev [5] there are three theorems of theultimate equilibrium method

(i) static theorem estimating the lower bound of theultimate loading capacity of the structure

(ii) kinematic theorem estimating the upper bound ofthe ultimate loading capacity of the structure

(iii) unity theorem that could directly obtain the ultimateloading capacity but does not allow finding themaximum static or minimum kinematic loads

It should be mentioned that the required ductility of an RCstructure or its element should be sufficient for the suggestedfailure mechanism

Series of experiments were performed in the previouscentury in order to overcome the problem of the unitytheorem and to find a more accurate value of the structuralload bearing capacity [10] The experimental investigationswere focused on studying the structures made of physicallynonlinear materials (ferrocement and reinforced concrete)The obtained experimental data allows calculation of the loadbearing capacity by static or kinematic theorems and realizesthe unity theorem

Another possible way for exact estimation of RC struc-turesrsquo load bearing capacity is simultaneous application of

4 Advances in Materials Science and Engineering

the static and kinematic methods at the ULS However thisway is not practical because it is impossible to estimate theconvergence of results obtained by these methods [1] It issuitable just for rather simple structural schemes (even forcontinuous beams it is not always applicable) Additionallyif condition (1) is not satisfied the fields of internal plasticforces 119865 and possible displacements 119908 should be changedbut convergence of this process is still not proved

Development of this idea yielded to minimax principleformulation [1] The essence of this principle is realizationof genuine bearing capacity of the structure (avoiding thelower andor upper bounds estimation) With this aim it isproposed to use both extreme properties of the structuresfailure load (static and kinematic) simultaneously in thecalculation process In this case just one calculation (staticor kinematic) is carried out Thus the minimax principlebecame indeed an apparatus for realization of the unitytheorem This apparatus logically unifies the extreme (staticand kinematic) theorems of the ultimate limit equilibriummethod

The authors have shown that the minimax term meansminimumofmaxima or the lowest board of upper bounds fora two-variable function of the structural bearing capacity [1]It should bementioned that at the same time themaximumorthe upper bound are found according to one of the variables(eg kinematic parameter) for a fixed value of the secondvariable (eg static parameter) After that the value of thesecond parameter is changed and the above described processis repeated

22 Quasi-Isotropy as a Limit State of RC Element Let usconsider a rectangular RC element section that containsreinforcing 119860 119904 in tensile zone and 1198601015840119904 in the compressionone 119860 119904 1198601015840119904 and the concrete compressed zone height119909 are unknown Two static equilibrium equations are notenough to determine the above specified unknowns Hencean additional condition should be found

A quasi-isotropic state requires minimal total reinforcingof the section that is (119860 119904 +1198601015840119904) rarr min It can be shown thatin this case the relative section compression zone depth 120596reaches its extremal value 05 (1 + 1198891015840119904119889) where 1198891015840119904 and 119889 arethe protective concrete layer for 1198601015840119904 and the section effectivedepth respectively In a common case of a section with avertical symmetry axis 120596extr = 1 minus 1198780(1198891198600) + 1198891015840119904(2119889) where1198780 is the first moment of an effective section and 1198600 is theeffective sectionrsquos area

Let us calculate for example an RC element with rect-angular section of 119887 times ℎ = 250 times 500mm 119889119904 = 1198891015840119904 =40mm steel design strength 119891sd = 350MPa concrete designstrength 119891cd = 10MPa and the external forces moment119872119889 = 250 kNm For the reinforcement sections in the tensileand compressed zones and the total reinforcement section asa function of the compressed zonersquos relative depth 120596extr =05(1 + 1198891015840119904119889) = 05(1 + 40460) = 0543

Graphical representation of the three above functions isgiven in Figure 5 where 21198891015840119904119889 le 120596 le 120596extr Increasing120596 yields increase of 119860 119904 and decrease of 1198601015840119904 The total

0 0

01

02

03

04

05

06

400

800

1200

1600

2000

2400

2800

3200

3600

054

3

07

Extremum (minimum) point

As(120596)

120596extr120596

As A998400s As + A998400

s (mm2)

(As + A998400s)(120596)

A998400s(120596)

Figure 5Minimization of total RC section reinforcement (119860 119904+1198601015840119904)

reinforcement (119860 119904+1198601015840119904) decreases until120596 reaches120596extr valueand then it increases In the same time 120596 gt 120596extr shouldbe avoided otherwise according to the concrete theory thesectionrsquos collapse will be brittleWhen (119860 119904+1198601015840119904) is minimumand 120596 = 120596extr the section is quasi-isotropic In other words aquasi-isotropic section stage yields optimal (minimum) totalreinforcement section

3 Analysis of the Structural PhenomenonBased on Original Experimental andTheoretical Data

The present study is based on selected experimental andtheoretical data that were reported by many researchers inthe last three decades of the previous century and till todayAs known very many researchers have investigated behaviorof structures under loads that increase from elastic state upto its failure In this case if a structure is symmetric andthe load is also symmetric usually structural parameters inelastic state increase or decrease twice at failure Thereforewe have presented and analyzed indeed those data

It is logical to analyze the structural phenomenon fromthe following three different groups of experiments

(i) investigation of structural concrete at material level(ii) behavior of RC structures and elements under static

loads(iii) response of RC structures and elements to dynamic

loads

31 Structural Concrete as Material Using high strengthconcrete in construction became very popular in recentdecades At the same time the following fact is evident inspite of the fact that concrete compressive strength increases

Advances in Materials Science and Engineering 5

with concrete class the mean value of concrete tensiledeformations 120576ctm reaches its maximum for concrete classC 70 and remains constant for higher concrete classes [7]Moreover the maximum value of 120576ctm is about 12 sdot 10minus4 andfor the lowest structural concrete class it is about 06 sdot 10minus4 Itis one of the structural phenomenon evidences as

120576ctm max120576ctm min= 2 (8)

As an extension of this idea let us discuss the relationbetween the displacements and Poisson deformations underultimate load and after unloading of two-layer beams consist-ing of normal and fibered high strength concrete in tensileand compression zones respectively [11] Following exper-imental results for full-scale 3m beams tested using four-point loading the mid span displacement under the ultimateload was 35mm and the residual deflection was about 18mmThe corresponding Poisson deformations were about 065 sdot10minus3 and 032 sdot 10minus3 A similar behavior was observed forhorizontal shear deformations between the concrete layersThe maximum deformations under the ultimate load were27 sdot 10minus3 whereas the residual ones were about 125 sdot 10minus3

Following modern codes ([7 12] etc) the maximumelastic deformations of normal strength concrete 120576119888 el ul areabout 1 sdot 10minus3 which corresponds to the initial modulusof elasticity for short term loading (without consideringdurability aspects) 1198641198881 = tg 1205721 (slope of line 119900119886 in Figure 6)The ultimate plastic deformation 120576119888 pl ul is 2 sdot 10minus3 If concretewould behave elastically up to 120576119888 pl ul (line 119900119887 in the figure) acorresponding modulus of elasticity is 1198641198882 = tg 1205722 It is easyto show that

1198641198881 = 21198641198882 (9)

It demonstrates again the same phenomenon that whenthe concrete element reaches the ultimate deformation itsstiffness characteristic decreases twice

As it was shown experimentally increasing the loadacting on cylindrical concrete specimens with different steelfibers contents (from 0 to 60 kgm3) yields increase inPoisson coefficient ] It was found that the optimal fibercontent is 30 kgm3 Following the experimental data inspecimens without fibers ]min 0 = 011 and ] = 022 Inspecimens with optimal fibers content ]min 30 = 022 and]max 30 = 044 The ratios

]min 30]min 0

= ]max 30]max 0

= 2]max 30]min 30

= ]max 0]min 0

= 2(10)

demonstrate that like in case of longitudinal deformationsalso for transverse ones the parameters are doubled

Concrete creep increases with time and correspondinglyyields a decrease in the modulus of elasticity ([7 12 13] etc)For example for normal strength concrete the initialmodulusof elasticity decreases twice during a five-year period [7]

119864119888 (119905 = 0)119864119888 (119905 = 5 years) = 2 (11)

1 2 350

a

1205721

1205722

120576c 103

b

120590c

fck

Figure 6 Influence of concrete element deformations on its stiffnessparameter

Additionally concrete creep depends on the compres-sion stress in service limit state surrounding environmenthumidity composition and class of concrete and so forthConsidering the concrete class only the concrete creepdepends on concrete creep coefficient120573119888119896 that decreases as theconcrete class increases Let us compare the concrete creepcoefficients for the lowest and the highest structural concreteclasses C 20 and C 90 respectively according to the equationavailable in Eurocode [7]

120573cm = 168119891cm2 (12)

where 119891cm is a mean compressive strength of concrete In thiscase the ratio between those coefficients is

120573cmC20

120573119888119896C90 = 33601697 = 198 (13)

In other words the relation between the concrete creepdeformations for high strength concrete (HSC) and normalstrength concrete (NSC) is

120576crNSC asymp 2120576crHSC (14)

Thus using HSC allows achieving significant decrease inconcrete creep deformations (up to two times)

32 Behavior of RC Structures and Elements under StaticLoads In the previous section examples of structural phe-nomenon for cement-based composite nonlinear materialwas discussed (concrete was considered as a private case ofsuch material) The present section focused on behavior ofRC elements from the viewpoint of statics A rectangularRC bending element section with double reinforcement of119860 119904 and 1198601015840119904 in tensile and compression zones respectivelyis considered If the element is in ULS the reinforcementsections are

1198601015840119904 = [119872119889 minus 119891cd1198871198892120596 (1 minus 05120596)][119891sd (119889 minus 1198891015840119904)]

119860 119904 = 1198601015840119904 + 119891cd119887119889120596119891sd (15)

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

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Nano

materials

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Journal ofNanomaterials

2 Advances in Materials Science and Engineering

10

15

20

25

00 02 04 06 08 10

K1

24 m

24 m

qqul

fI (1sec)

Figure 1 Dominant technical frequency119891119868 versus load ratio 119902119902ul

22

44

1

2

3

35

25

20

30

40

2 4 6 8 1000

12

w (cm)

120585 ()

Figure 2 24 times 24m shell deflection versus damping ratio forvarious ground motions 1 high-frequency earthquake 2 medium-frequency earthquake and 3 low-frequency earthquake

residual deflections in a 24 times 24m RC shell under staticload and additional vertical seismic excitation is consideredThe following earthquakes were selected Gazly (Uzbekistan)1976 San Fernando (USA) 1971 and Bucharest (Romania)1977 These earthquake records represent high- medium-and low-frequency groundmotions respectivelyThe depen-dence of residual deflections on damping ratio was studied(Table 1) The damping ratio 120585 varied from 0 to 12 Thestatic displacement value (hidden line in Figure 2) was22 cm For a high-frequency Gazly earthquake (curve 1in Figure 2) the residual deflections are accumulated veryintensively and for 120585 = 0 reached 44 cm which is two timeshigher compared to the static value A similar result wasobtained for a medium frequency earthquake (see Table 1and the corresponding curve 2 in the Figure) for whichthe residual deflection equals 36 cm However for a low-frequency earthquake (curve 3 in the Figure) the increasein residual deflections is lower It is because the dominant

Table 1 Deflections in a fill scale RC shell versus damping ratios

Earthquake typeDamping ratio

0 2 4 6 8 10 12Deflection cm

Low frequency 250 225 221 215 209 205 200Medium frequency 363 238 223 217 210 206 200High frequency 440 332 281 263 241 230 219

n

0

05

10

10

10

20mk

mmax

nk

m

qqul

nmin

f(m n) = 0

Figure 3 Relation between structural phenomenon and minimaxprinciple 119898 generalized ldquoincreasingrdquo parameter 119899 generalizedldquodecreasingrdquo parameter 119891(119898 119899) function relating the parameters(conditionally shown by a straight line) and119898119896 119899119896 arbitrary valuesof the parameters

frequency of the shell is high and comparable with that of ahigh-frequency earthquake

Comparing Figures 1 and 2 leads to conclusion thatdecreasing twice (Figure 1) and doubling (Figure 2) instructural parameters correspond to the ULS of the structureThese limit values for the same structure can be interrelatedby the minimax principle [1] Graphical interpretation of thisrelation is shown in Figure 3 The curve in plane 119899-119900-119902119902ulcorresponds to the dependence fromFigure 1 and the curve inplane119898-119900-119902119902ul to Figure 2Theminimax principle combinesthose curves and presents their structural interrelation asshown in the 119899-119900-119898 plane (it is assumed that this relation islinear) Thus if according to experimental data some factoris doubled it is logical to expect that another structuralparameter correspondingly decreases twice

Figure 4 shows the structural phenomenon evidence fora fixed concrete beam In elastic stage

1003816100381610038161003816119872minusel1003816100381610038161003816 = 119902el119871212 = 2119872+el (4)

where 119902el is a uniformly distributed load in elastic stage (119902el le119902el ul) and 119902el ul is the ultimate value of this load 119872minusel and119872+el are themaximumvalues of negative and positive bendingmoments respectively in beam elastic stage

Advances in Materials Science and Engineering 3

L

0 le q le qul

(a)

qL280 le q le qel ulMminus

el

M+el = 05

1003816100381610038161003816Mminusel1003816100381610038161003816

(b)

qulL28

q = qulMminuspl

M+pl =

100381610038161003816100381610038161003816Mminus

pl100381610038161003816100381610038161003816

(c)

Figure 4 Structural phenomenon evidence for a fixed concrete beam (a) static scheme (b) bending moments diagram in elastic stage and(c) bending moments diagram in ultimate limit state

In the ULS after the momentsrsquo redistribution

10038161003816100381610038161003816119872minuspl10038161003816100381610038161003816 = 119902ul119871216 = 119872+pl (5)

where 119902ul is a uniformly distributed load in ULS and119872minuspl and119872+pl are themaximumvalues of negative and positive bendingmoments respectively

In both cases1003816100381610038161003816119872minus1003816100381610038161003816 + 119872+ = 1199021198712

8 (6)

Thus

2 ge 1003816100381610038161003816119872minus1003816100381610038161003816119872+ ge 1 (7)

The examples that were presented in this chapter are notthe only ones Other relevant cases will be discussed below Itshould bementioned that as the structural phenomenon wasnot investigated previously there is a very limited numberof publications in which relevant experimental data wereobtained but not analyzed For example experimental results[2] show that using steel fibers in concrete increases trans-verse deformations about two times (from (016 020) 120576119888to 040 120576119888 where 120576119888 is the longitudinal deformation of thestructure) but it is not emphasized by the researchers

Four-point static tests in ultimate stages show that steelfibers increase the deflections in the middle-span of a highstrength concrete bending element approximately twice [3]the deflections in beams made of concrete class C140 were44 and 84mm without and with steel fibers respectivelyFor beams made of concrete class C200 adding steel fibershas increased the deflections from 33 to 59mm As in theprevious example the fact of doubling the middle-spandeflections was not emphasized

2 Minimax Principle in Ultimate LimitState of Structures

21 Short Review of Ultimate Limit State and MinimaxPrinciple The ultimate equilibrium method proposed by

Gvozdev in 1938 [5] was a logical development of basicexperimental and theoretical investigations carried out byoutstanding researchers as Galilei Coulomb Bach Graf andothers [6] This method is still used for calculating structuresat ultimate limit state (load bearing capacity)

Followingmodern design codes methods based on plas-tic analysis are suitable for checking structures at ultimatelimit state [7]Themethod is constantly developed For a spe-cific structure there is an exact correspondence between theexternal loadsrsquo system and internal forcesrsquo distribution Just inthis case the structure may transform into a mechanism [8]

Later the ultimate equilibrium method was enlarged forapplication in structures that aremade of concrete typemate-rials with downloading branch in the stress-strain diagram[9]

Following Gvozdev [5] there are three theorems of theultimate equilibrium method

(i) static theorem estimating the lower bound of theultimate loading capacity of the structure

(ii) kinematic theorem estimating the upper bound ofthe ultimate loading capacity of the structure

(iii) unity theorem that could directly obtain the ultimateloading capacity but does not allow finding themaximum static or minimum kinematic loads

It should be mentioned that the required ductility of an RCstructure or its element should be sufficient for the suggestedfailure mechanism

Series of experiments were performed in the previouscentury in order to overcome the problem of the unitytheorem and to find a more accurate value of the structuralload bearing capacity [10] The experimental investigationswere focused on studying the structures made of physicallynonlinear materials (ferrocement and reinforced concrete)The obtained experimental data allows calculation of the loadbearing capacity by static or kinematic theorems and realizesthe unity theorem

Another possible way for exact estimation of RC struc-turesrsquo load bearing capacity is simultaneous application of

4 Advances in Materials Science and Engineering

the static and kinematic methods at the ULS However thisway is not practical because it is impossible to estimate theconvergence of results obtained by these methods [1] It issuitable just for rather simple structural schemes (even forcontinuous beams it is not always applicable) Additionallyif condition (1) is not satisfied the fields of internal plasticforces 119865 and possible displacements 119908 should be changedbut convergence of this process is still not proved

Development of this idea yielded to minimax principleformulation [1] The essence of this principle is realizationof genuine bearing capacity of the structure (avoiding thelower andor upper bounds estimation) With this aim it isproposed to use both extreme properties of the structuresfailure load (static and kinematic) simultaneously in thecalculation process In this case just one calculation (staticor kinematic) is carried out Thus the minimax principlebecame indeed an apparatus for realization of the unitytheorem This apparatus logically unifies the extreme (staticand kinematic) theorems of the ultimate limit equilibriummethod

The authors have shown that the minimax term meansminimumofmaxima or the lowest board of upper bounds fora two-variable function of the structural bearing capacity [1]It should bementioned that at the same time themaximumorthe upper bound are found according to one of the variables(eg kinematic parameter) for a fixed value of the secondvariable (eg static parameter) After that the value of thesecond parameter is changed and the above described processis repeated

22 Quasi-Isotropy as a Limit State of RC Element Let usconsider a rectangular RC element section that containsreinforcing 119860 119904 in tensile zone and 1198601015840119904 in the compressionone 119860 119904 1198601015840119904 and the concrete compressed zone height119909 are unknown Two static equilibrium equations are notenough to determine the above specified unknowns Hencean additional condition should be found

A quasi-isotropic state requires minimal total reinforcingof the section that is (119860 119904 +1198601015840119904) rarr min It can be shown thatin this case the relative section compression zone depth 120596reaches its extremal value 05 (1 + 1198891015840119904119889) where 1198891015840119904 and 119889 arethe protective concrete layer for 1198601015840119904 and the section effectivedepth respectively In a common case of a section with avertical symmetry axis 120596extr = 1 minus 1198780(1198891198600) + 1198891015840119904(2119889) where1198780 is the first moment of an effective section and 1198600 is theeffective sectionrsquos area

Let us calculate for example an RC element with rect-angular section of 119887 times ℎ = 250 times 500mm 119889119904 = 1198891015840119904 =40mm steel design strength 119891sd = 350MPa concrete designstrength 119891cd = 10MPa and the external forces moment119872119889 = 250 kNm For the reinforcement sections in the tensileand compressed zones and the total reinforcement section asa function of the compressed zonersquos relative depth 120596extr =05(1 + 1198891015840119904119889) = 05(1 + 40460) = 0543

Graphical representation of the three above functions isgiven in Figure 5 where 21198891015840119904119889 le 120596 le 120596extr Increasing120596 yields increase of 119860 119904 and decrease of 1198601015840119904 The total

0 0

01

02

03

04

05

06

400

800

1200

1600

2000

2400

2800

3200

3600

054

3

07

Extremum (minimum) point

As(120596)

120596extr120596

As A998400s As + A998400

s (mm2)

(As + A998400s)(120596)

A998400s(120596)

Figure 5Minimization of total RC section reinforcement (119860 119904+1198601015840119904)

reinforcement (119860 119904+1198601015840119904) decreases until120596 reaches120596extr valueand then it increases In the same time 120596 gt 120596extr shouldbe avoided otherwise according to the concrete theory thesectionrsquos collapse will be brittleWhen (119860 119904+1198601015840119904) is minimumand 120596 = 120596extr the section is quasi-isotropic In other words aquasi-isotropic section stage yields optimal (minimum) totalreinforcement section

3 Analysis of the Structural PhenomenonBased on Original Experimental andTheoretical Data

The present study is based on selected experimental andtheoretical data that were reported by many researchers inthe last three decades of the previous century and till todayAs known very many researchers have investigated behaviorof structures under loads that increase from elastic state upto its failure In this case if a structure is symmetric andthe load is also symmetric usually structural parameters inelastic state increase or decrease twice at failure Thereforewe have presented and analyzed indeed those data

It is logical to analyze the structural phenomenon fromthe following three different groups of experiments

(i) investigation of structural concrete at material level(ii) behavior of RC structures and elements under static

loads(iii) response of RC structures and elements to dynamic

loads

31 Structural Concrete as Material Using high strengthconcrete in construction became very popular in recentdecades At the same time the following fact is evident inspite of the fact that concrete compressive strength increases

Advances in Materials Science and Engineering 5

with concrete class the mean value of concrete tensiledeformations 120576ctm reaches its maximum for concrete classC 70 and remains constant for higher concrete classes [7]Moreover the maximum value of 120576ctm is about 12 sdot 10minus4 andfor the lowest structural concrete class it is about 06 sdot 10minus4 Itis one of the structural phenomenon evidences as

120576ctm max120576ctm min= 2 (8)

As an extension of this idea let us discuss the relationbetween the displacements and Poisson deformations underultimate load and after unloading of two-layer beams consist-ing of normal and fibered high strength concrete in tensileand compression zones respectively [11] Following exper-imental results for full-scale 3m beams tested using four-point loading the mid span displacement under the ultimateload was 35mm and the residual deflection was about 18mmThe corresponding Poisson deformations were about 065 sdot10minus3 and 032 sdot 10minus3 A similar behavior was observed forhorizontal shear deformations between the concrete layersThe maximum deformations under the ultimate load were27 sdot 10minus3 whereas the residual ones were about 125 sdot 10minus3

Following modern codes ([7 12] etc) the maximumelastic deformations of normal strength concrete 120576119888 el ul areabout 1 sdot 10minus3 which corresponds to the initial modulusof elasticity for short term loading (without consideringdurability aspects) 1198641198881 = tg 1205721 (slope of line 119900119886 in Figure 6)The ultimate plastic deformation 120576119888 pl ul is 2 sdot 10minus3 If concretewould behave elastically up to 120576119888 pl ul (line 119900119887 in the figure) acorresponding modulus of elasticity is 1198641198882 = tg 1205722 It is easyto show that

1198641198881 = 21198641198882 (9)

It demonstrates again the same phenomenon that whenthe concrete element reaches the ultimate deformation itsstiffness characteristic decreases twice

As it was shown experimentally increasing the loadacting on cylindrical concrete specimens with different steelfibers contents (from 0 to 60 kgm3) yields increase inPoisson coefficient ] It was found that the optimal fibercontent is 30 kgm3 Following the experimental data inspecimens without fibers ]min 0 = 011 and ] = 022 Inspecimens with optimal fibers content ]min 30 = 022 and]max 30 = 044 The ratios

]min 30]min 0

= ]max 30]max 0

= 2]max 30]min 30

= ]max 0]min 0

= 2(10)

demonstrate that like in case of longitudinal deformationsalso for transverse ones the parameters are doubled

Concrete creep increases with time and correspondinglyyields a decrease in the modulus of elasticity ([7 12 13] etc)For example for normal strength concrete the initialmodulusof elasticity decreases twice during a five-year period [7]

119864119888 (119905 = 0)119864119888 (119905 = 5 years) = 2 (11)

1 2 350

a

1205721

1205722

120576c 103

b

120590c

fck

Figure 6 Influence of concrete element deformations on its stiffnessparameter

Additionally concrete creep depends on the compres-sion stress in service limit state surrounding environmenthumidity composition and class of concrete and so forthConsidering the concrete class only the concrete creepdepends on concrete creep coefficient120573119888119896 that decreases as theconcrete class increases Let us compare the concrete creepcoefficients for the lowest and the highest structural concreteclasses C 20 and C 90 respectively according to the equationavailable in Eurocode [7]

120573cm = 168119891cm2 (12)

where 119891cm is a mean compressive strength of concrete In thiscase the ratio between those coefficients is

120573cmC20

120573119888119896C90 = 33601697 = 198 (13)

In other words the relation between the concrete creepdeformations for high strength concrete (HSC) and normalstrength concrete (NSC) is

120576crNSC asymp 2120576crHSC (14)

Thus using HSC allows achieving significant decrease inconcrete creep deformations (up to two times)

32 Behavior of RC Structures and Elements under StaticLoads In the previous section examples of structural phe-nomenon for cement-based composite nonlinear materialwas discussed (concrete was considered as a private case ofsuch material) The present section focused on behavior ofRC elements from the viewpoint of statics A rectangularRC bending element section with double reinforcement of119860 119904 and 1198601015840119904 in tensile and compression zones respectivelyis considered If the element is in ULS the reinforcementsections are

1198601015840119904 = [119872119889 minus 119891cd1198871198892120596 (1 minus 05120596)][119891sd (119889 minus 1198891015840119904)]

119860 119904 = 1198601015840119904 + 119891cd119887119889120596119891sd (15)

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

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Nano

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Journal ofNanomaterials

Advances in Materials Science and Engineering 3

L

0 le q le qul

(a)

qL280 le q le qel ulMminus

el

M+el = 05

1003816100381610038161003816Mminusel1003816100381610038161003816

(b)

qulL28

q = qulMminuspl

M+pl =

100381610038161003816100381610038161003816Mminus

pl100381610038161003816100381610038161003816

(c)

Figure 4 Structural phenomenon evidence for a fixed concrete beam (a) static scheme (b) bending moments diagram in elastic stage and(c) bending moments diagram in ultimate limit state

In the ULS after the momentsrsquo redistribution

10038161003816100381610038161003816119872minuspl10038161003816100381610038161003816 = 119902ul119871216 = 119872+pl (5)

where 119902ul is a uniformly distributed load in ULS and119872minuspl and119872+pl are themaximumvalues of negative and positive bendingmoments respectively

In both cases1003816100381610038161003816119872minus1003816100381610038161003816 + 119872+ = 1199021198712

8 (6)

Thus

2 ge 1003816100381610038161003816119872minus1003816100381610038161003816119872+ ge 1 (7)

The examples that were presented in this chapter are notthe only ones Other relevant cases will be discussed below Itshould bementioned that as the structural phenomenon wasnot investigated previously there is a very limited numberof publications in which relevant experimental data wereobtained but not analyzed For example experimental results[2] show that using steel fibers in concrete increases trans-verse deformations about two times (from (016 020) 120576119888to 040 120576119888 where 120576119888 is the longitudinal deformation of thestructure) but it is not emphasized by the researchers

Four-point static tests in ultimate stages show that steelfibers increase the deflections in the middle-span of a highstrength concrete bending element approximately twice [3]the deflections in beams made of concrete class C140 were44 and 84mm without and with steel fibers respectivelyFor beams made of concrete class C200 adding steel fibershas increased the deflections from 33 to 59mm As in theprevious example the fact of doubling the middle-spandeflections was not emphasized

2 Minimax Principle in Ultimate LimitState of Structures

21 Short Review of Ultimate Limit State and MinimaxPrinciple The ultimate equilibrium method proposed by

Gvozdev in 1938 [5] was a logical development of basicexperimental and theoretical investigations carried out byoutstanding researchers as Galilei Coulomb Bach Graf andothers [6] This method is still used for calculating structuresat ultimate limit state (load bearing capacity)

Followingmodern design codes methods based on plas-tic analysis are suitable for checking structures at ultimatelimit state [7]Themethod is constantly developed For a spe-cific structure there is an exact correspondence between theexternal loadsrsquo system and internal forcesrsquo distribution Just inthis case the structure may transform into a mechanism [8]

Later the ultimate equilibrium method was enlarged forapplication in structures that aremade of concrete typemate-rials with downloading branch in the stress-strain diagram[9]

Following Gvozdev [5] there are three theorems of theultimate equilibrium method

(i) static theorem estimating the lower bound of theultimate loading capacity of the structure

(ii) kinematic theorem estimating the upper bound ofthe ultimate loading capacity of the structure

(iii) unity theorem that could directly obtain the ultimateloading capacity but does not allow finding themaximum static or minimum kinematic loads

It should be mentioned that the required ductility of an RCstructure or its element should be sufficient for the suggestedfailure mechanism

Series of experiments were performed in the previouscentury in order to overcome the problem of the unitytheorem and to find a more accurate value of the structuralload bearing capacity [10] The experimental investigationswere focused on studying the structures made of physicallynonlinear materials (ferrocement and reinforced concrete)The obtained experimental data allows calculation of the loadbearing capacity by static or kinematic theorems and realizesthe unity theorem

Another possible way for exact estimation of RC struc-turesrsquo load bearing capacity is simultaneous application of

4 Advances in Materials Science and Engineering

the static and kinematic methods at the ULS However thisway is not practical because it is impossible to estimate theconvergence of results obtained by these methods [1] It issuitable just for rather simple structural schemes (even forcontinuous beams it is not always applicable) Additionallyif condition (1) is not satisfied the fields of internal plasticforces 119865 and possible displacements 119908 should be changedbut convergence of this process is still not proved

Development of this idea yielded to minimax principleformulation [1] The essence of this principle is realizationof genuine bearing capacity of the structure (avoiding thelower andor upper bounds estimation) With this aim it isproposed to use both extreme properties of the structuresfailure load (static and kinematic) simultaneously in thecalculation process In this case just one calculation (staticor kinematic) is carried out Thus the minimax principlebecame indeed an apparatus for realization of the unitytheorem This apparatus logically unifies the extreme (staticand kinematic) theorems of the ultimate limit equilibriummethod

The authors have shown that the minimax term meansminimumofmaxima or the lowest board of upper bounds fora two-variable function of the structural bearing capacity [1]It should bementioned that at the same time themaximumorthe upper bound are found according to one of the variables(eg kinematic parameter) for a fixed value of the secondvariable (eg static parameter) After that the value of thesecond parameter is changed and the above described processis repeated

22 Quasi-Isotropy as a Limit State of RC Element Let usconsider a rectangular RC element section that containsreinforcing 119860 119904 in tensile zone and 1198601015840119904 in the compressionone 119860 119904 1198601015840119904 and the concrete compressed zone height119909 are unknown Two static equilibrium equations are notenough to determine the above specified unknowns Hencean additional condition should be found

A quasi-isotropic state requires minimal total reinforcingof the section that is (119860 119904 +1198601015840119904) rarr min It can be shown thatin this case the relative section compression zone depth 120596reaches its extremal value 05 (1 + 1198891015840119904119889) where 1198891015840119904 and 119889 arethe protective concrete layer for 1198601015840119904 and the section effectivedepth respectively In a common case of a section with avertical symmetry axis 120596extr = 1 minus 1198780(1198891198600) + 1198891015840119904(2119889) where1198780 is the first moment of an effective section and 1198600 is theeffective sectionrsquos area

Let us calculate for example an RC element with rect-angular section of 119887 times ℎ = 250 times 500mm 119889119904 = 1198891015840119904 =40mm steel design strength 119891sd = 350MPa concrete designstrength 119891cd = 10MPa and the external forces moment119872119889 = 250 kNm For the reinforcement sections in the tensileand compressed zones and the total reinforcement section asa function of the compressed zonersquos relative depth 120596extr =05(1 + 1198891015840119904119889) = 05(1 + 40460) = 0543

Graphical representation of the three above functions isgiven in Figure 5 where 21198891015840119904119889 le 120596 le 120596extr Increasing120596 yields increase of 119860 119904 and decrease of 1198601015840119904 The total

0 0

01

02

03

04

05

06

400

800

1200

1600

2000

2400

2800

3200

3600

054

3

07

Extremum (minimum) point

As(120596)

120596extr120596

As A998400s As + A998400

s (mm2)

(As + A998400s)(120596)

A998400s(120596)

Figure 5Minimization of total RC section reinforcement (119860 119904+1198601015840119904)

reinforcement (119860 119904+1198601015840119904) decreases until120596 reaches120596extr valueand then it increases In the same time 120596 gt 120596extr shouldbe avoided otherwise according to the concrete theory thesectionrsquos collapse will be brittleWhen (119860 119904+1198601015840119904) is minimumand 120596 = 120596extr the section is quasi-isotropic In other words aquasi-isotropic section stage yields optimal (minimum) totalreinforcement section

3 Analysis of the Structural PhenomenonBased on Original Experimental andTheoretical Data

The present study is based on selected experimental andtheoretical data that were reported by many researchers inthe last three decades of the previous century and till todayAs known very many researchers have investigated behaviorof structures under loads that increase from elastic state upto its failure In this case if a structure is symmetric andthe load is also symmetric usually structural parameters inelastic state increase or decrease twice at failure Thereforewe have presented and analyzed indeed those data

It is logical to analyze the structural phenomenon fromthe following three different groups of experiments

(i) investigation of structural concrete at material level(ii) behavior of RC structures and elements under static

loads(iii) response of RC structures and elements to dynamic

loads

31 Structural Concrete as Material Using high strengthconcrete in construction became very popular in recentdecades At the same time the following fact is evident inspite of the fact that concrete compressive strength increases

Advances in Materials Science and Engineering 5

with concrete class the mean value of concrete tensiledeformations 120576ctm reaches its maximum for concrete classC 70 and remains constant for higher concrete classes [7]Moreover the maximum value of 120576ctm is about 12 sdot 10minus4 andfor the lowest structural concrete class it is about 06 sdot 10minus4 Itis one of the structural phenomenon evidences as

120576ctm max120576ctm min= 2 (8)

As an extension of this idea let us discuss the relationbetween the displacements and Poisson deformations underultimate load and after unloading of two-layer beams consist-ing of normal and fibered high strength concrete in tensileand compression zones respectively [11] Following exper-imental results for full-scale 3m beams tested using four-point loading the mid span displacement under the ultimateload was 35mm and the residual deflection was about 18mmThe corresponding Poisson deformations were about 065 sdot10minus3 and 032 sdot 10minus3 A similar behavior was observed forhorizontal shear deformations between the concrete layersThe maximum deformations under the ultimate load were27 sdot 10minus3 whereas the residual ones were about 125 sdot 10minus3

Following modern codes ([7 12] etc) the maximumelastic deformations of normal strength concrete 120576119888 el ul areabout 1 sdot 10minus3 which corresponds to the initial modulusof elasticity for short term loading (without consideringdurability aspects) 1198641198881 = tg 1205721 (slope of line 119900119886 in Figure 6)The ultimate plastic deformation 120576119888 pl ul is 2 sdot 10minus3 If concretewould behave elastically up to 120576119888 pl ul (line 119900119887 in the figure) acorresponding modulus of elasticity is 1198641198882 = tg 1205722 It is easyto show that

1198641198881 = 21198641198882 (9)

It demonstrates again the same phenomenon that whenthe concrete element reaches the ultimate deformation itsstiffness characteristic decreases twice

As it was shown experimentally increasing the loadacting on cylindrical concrete specimens with different steelfibers contents (from 0 to 60 kgm3) yields increase inPoisson coefficient ] It was found that the optimal fibercontent is 30 kgm3 Following the experimental data inspecimens without fibers ]min 0 = 011 and ] = 022 Inspecimens with optimal fibers content ]min 30 = 022 and]max 30 = 044 The ratios

]min 30]min 0

= ]max 30]max 0

= 2]max 30]min 30

= ]max 0]min 0

= 2(10)

demonstrate that like in case of longitudinal deformationsalso for transverse ones the parameters are doubled

Concrete creep increases with time and correspondinglyyields a decrease in the modulus of elasticity ([7 12 13] etc)For example for normal strength concrete the initialmodulusof elasticity decreases twice during a five-year period [7]

119864119888 (119905 = 0)119864119888 (119905 = 5 years) = 2 (11)

1 2 350

a

1205721

1205722

120576c 103

b

120590c

fck

Figure 6 Influence of concrete element deformations on its stiffnessparameter

Additionally concrete creep depends on the compres-sion stress in service limit state surrounding environmenthumidity composition and class of concrete and so forthConsidering the concrete class only the concrete creepdepends on concrete creep coefficient120573119888119896 that decreases as theconcrete class increases Let us compare the concrete creepcoefficients for the lowest and the highest structural concreteclasses C 20 and C 90 respectively according to the equationavailable in Eurocode [7]

120573cm = 168119891cm2 (12)

where 119891cm is a mean compressive strength of concrete In thiscase the ratio between those coefficients is

120573cmC20

120573119888119896C90 = 33601697 = 198 (13)

In other words the relation between the concrete creepdeformations for high strength concrete (HSC) and normalstrength concrete (NSC) is

120576crNSC asymp 2120576crHSC (14)

Thus using HSC allows achieving significant decrease inconcrete creep deformations (up to two times)

32 Behavior of RC Structures and Elements under StaticLoads In the previous section examples of structural phe-nomenon for cement-based composite nonlinear materialwas discussed (concrete was considered as a private case ofsuch material) The present section focused on behavior ofRC elements from the viewpoint of statics A rectangularRC bending element section with double reinforcement of119860 119904 and 1198601015840119904 in tensile and compression zones respectivelyis considered If the element is in ULS the reinforcementsections are

1198601015840119904 = [119872119889 minus 119891cd1198871198892120596 (1 minus 05120596)][119891sd (119889 minus 1198891015840119904)]

119860 119904 = 1198601015840119904 + 119891cd119887119889120596119891sd (15)

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Advances in Materials Science and Engineering

the static and kinematic methods at the ULS However thisway is not practical because it is impossible to estimate theconvergence of results obtained by these methods [1] It issuitable just for rather simple structural schemes (even forcontinuous beams it is not always applicable) Additionallyif condition (1) is not satisfied the fields of internal plasticforces 119865 and possible displacements 119908 should be changedbut convergence of this process is still not proved

Development of this idea yielded to minimax principleformulation [1] The essence of this principle is realizationof genuine bearing capacity of the structure (avoiding thelower andor upper bounds estimation) With this aim it isproposed to use both extreme properties of the structuresfailure load (static and kinematic) simultaneously in thecalculation process In this case just one calculation (staticor kinematic) is carried out Thus the minimax principlebecame indeed an apparatus for realization of the unitytheorem This apparatus logically unifies the extreme (staticand kinematic) theorems of the ultimate limit equilibriummethod

The authors have shown that the minimax term meansminimumofmaxima or the lowest board of upper bounds fora two-variable function of the structural bearing capacity [1]It should bementioned that at the same time themaximumorthe upper bound are found according to one of the variables(eg kinematic parameter) for a fixed value of the secondvariable (eg static parameter) After that the value of thesecond parameter is changed and the above described processis repeated

22 Quasi-Isotropy as a Limit State of RC Element Let usconsider a rectangular RC element section that containsreinforcing 119860 119904 in tensile zone and 1198601015840119904 in the compressionone 119860 119904 1198601015840119904 and the concrete compressed zone height119909 are unknown Two static equilibrium equations are notenough to determine the above specified unknowns Hencean additional condition should be found

A quasi-isotropic state requires minimal total reinforcingof the section that is (119860 119904 +1198601015840119904) rarr min It can be shown thatin this case the relative section compression zone depth 120596reaches its extremal value 05 (1 + 1198891015840119904119889) where 1198891015840119904 and 119889 arethe protective concrete layer for 1198601015840119904 and the section effectivedepth respectively In a common case of a section with avertical symmetry axis 120596extr = 1 minus 1198780(1198891198600) + 1198891015840119904(2119889) where1198780 is the first moment of an effective section and 1198600 is theeffective sectionrsquos area

Let us calculate for example an RC element with rect-angular section of 119887 times ℎ = 250 times 500mm 119889119904 = 1198891015840119904 =40mm steel design strength 119891sd = 350MPa concrete designstrength 119891cd = 10MPa and the external forces moment119872119889 = 250 kNm For the reinforcement sections in the tensileand compressed zones and the total reinforcement section asa function of the compressed zonersquos relative depth 120596extr =05(1 + 1198891015840119904119889) = 05(1 + 40460) = 0543

Graphical representation of the three above functions isgiven in Figure 5 where 21198891015840119904119889 le 120596 le 120596extr Increasing120596 yields increase of 119860 119904 and decrease of 1198601015840119904 The total

0 0

01

02

03

04

05

06

400

800

1200

1600

2000

2400

2800

3200

3600

054

3

07

Extremum (minimum) point

As(120596)

120596extr120596

As A998400s As + A998400

s (mm2)

(As + A998400s)(120596)

A998400s(120596)

Figure 5Minimization of total RC section reinforcement (119860 119904+1198601015840119904)

reinforcement (119860 119904+1198601015840119904) decreases until120596 reaches120596extr valueand then it increases In the same time 120596 gt 120596extr shouldbe avoided otherwise according to the concrete theory thesectionrsquos collapse will be brittleWhen (119860 119904+1198601015840119904) is minimumand 120596 = 120596extr the section is quasi-isotropic In other words aquasi-isotropic section stage yields optimal (minimum) totalreinforcement section

3 Analysis of the Structural PhenomenonBased on Original Experimental andTheoretical Data

The present study is based on selected experimental andtheoretical data that were reported by many researchers inthe last three decades of the previous century and till todayAs known very many researchers have investigated behaviorof structures under loads that increase from elastic state upto its failure In this case if a structure is symmetric andthe load is also symmetric usually structural parameters inelastic state increase or decrease twice at failure Thereforewe have presented and analyzed indeed those data

It is logical to analyze the structural phenomenon fromthe following three different groups of experiments

(i) investigation of structural concrete at material level(ii) behavior of RC structures and elements under static

loads(iii) response of RC structures and elements to dynamic

loads

31 Structural Concrete as Material Using high strengthconcrete in construction became very popular in recentdecades At the same time the following fact is evident inspite of the fact that concrete compressive strength increases

Advances in Materials Science and Engineering 5

with concrete class the mean value of concrete tensiledeformations 120576ctm reaches its maximum for concrete classC 70 and remains constant for higher concrete classes [7]Moreover the maximum value of 120576ctm is about 12 sdot 10minus4 andfor the lowest structural concrete class it is about 06 sdot 10minus4 Itis one of the structural phenomenon evidences as

120576ctm max120576ctm min= 2 (8)

As an extension of this idea let us discuss the relationbetween the displacements and Poisson deformations underultimate load and after unloading of two-layer beams consist-ing of normal and fibered high strength concrete in tensileand compression zones respectively [11] Following exper-imental results for full-scale 3m beams tested using four-point loading the mid span displacement under the ultimateload was 35mm and the residual deflection was about 18mmThe corresponding Poisson deformations were about 065 sdot10minus3 and 032 sdot 10minus3 A similar behavior was observed forhorizontal shear deformations between the concrete layersThe maximum deformations under the ultimate load were27 sdot 10minus3 whereas the residual ones were about 125 sdot 10minus3

Following modern codes ([7 12] etc) the maximumelastic deformations of normal strength concrete 120576119888 el ul areabout 1 sdot 10minus3 which corresponds to the initial modulusof elasticity for short term loading (without consideringdurability aspects) 1198641198881 = tg 1205721 (slope of line 119900119886 in Figure 6)The ultimate plastic deformation 120576119888 pl ul is 2 sdot 10minus3 If concretewould behave elastically up to 120576119888 pl ul (line 119900119887 in the figure) acorresponding modulus of elasticity is 1198641198882 = tg 1205722 It is easyto show that

1198641198881 = 21198641198882 (9)

It demonstrates again the same phenomenon that whenthe concrete element reaches the ultimate deformation itsstiffness characteristic decreases twice

As it was shown experimentally increasing the loadacting on cylindrical concrete specimens with different steelfibers contents (from 0 to 60 kgm3) yields increase inPoisson coefficient ] It was found that the optimal fibercontent is 30 kgm3 Following the experimental data inspecimens without fibers ]min 0 = 011 and ] = 022 Inspecimens with optimal fibers content ]min 30 = 022 and]max 30 = 044 The ratios

]min 30]min 0

= ]max 30]max 0

= 2]max 30]min 30

= ]max 0]min 0

= 2(10)

demonstrate that like in case of longitudinal deformationsalso for transverse ones the parameters are doubled

Concrete creep increases with time and correspondinglyyields a decrease in the modulus of elasticity ([7 12 13] etc)For example for normal strength concrete the initialmodulusof elasticity decreases twice during a five-year period [7]

119864119888 (119905 = 0)119864119888 (119905 = 5 years) = 2 (11)

1 2 350

a

1205721

1205722

120576c 103

b

120590c

fck

Figure 6 Influence of concrete element deformations on its stiffnessparameter

Additionally concrete creep depends on the compres-sion stress in service limit state surrounding environmenthumidity composition and class of concrete and so forthConsidering the concrete class only the concrete creepdepends on concrete creep coefficient120573119888119896 that decreases as theconcrete class increases Let us compare the concrete creepcoefficients for the lowest and the highest structural concreteclasses C 20 and C 90 respectively according to the equationavailable in Eurocode [7]

120573cm = 168119891cm2 (12)

where 119891cm is a mean compressive strength of concrete In thiscase the ratio between those coefficients is

120573cmC20

120573119888119896C90 = 33601697 = 198 (13)

In other words the relation between the concrete creepdeformations for high strength concrete (HSC) and normalstrength concrete (NSC) is

120576crNSC asymp 2120576crHSC (14)

Thus using HSC allows achieving significant decrease inconcrete creep deformations (up to two times)

32 Behavior of RC Structures and Elements under StaticLoads In the previous section examples of structural phe-nomenon for cement-based composite nonlinear materialwas discussed (concrete was considered as a private case ofsuch material) The present section focused on behavior ofRC elements from the viewpoint of statics A rectangularRC bending element section with double reinforcement of119860 119904 and 1198601015840119904 in tensile and compression zones respectivelyis considered If the element is in ULS the reinforcementsections are

1198601015840119904 = [119872119889 minus 119891cd1198871198892120596 (1 minus 05120596)][119891sd (119889 minus 1198891015840119904)]

119860 119904 = 1198601015840119904 + 119891cd119887119889120596119891sd (15)

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

with concrete class the mean value of concrete tensiledeformations 120576ctm reaches its maximum for concrete classC 70 and remains constant for higher concrete classes [7]Moreover the maximum value of 120576ctm is about 12 sdot 10minus4 andfor the lowest structural concrete class it is about 06 sdot 10minus4 Itis one of the structural phenomenon evidences as

120576ctm max120576ctm min= 2 (8)

As an extension of this idea let us discuss the relationbetween the displacements and Poisson deformations underultimate load and after unloading of two-layer beams consist-ing of normal and fibered high strength concrete in tensileand compression zones respectively [11] Following exper-imental results for full-scale 3m beams tested using four-point loading the mid span displacement under the ultimateload was 35mm and the residual deflection was about 18mmThe corresponding Poisson deformations were about 065 sdot10minus3 and 032 sdot 10minus3 A similar behavior was observed forhorizontal shear deformations between the concrete layersThe maximum deformations under the ultimate load were27 sdot 10minus3 whereas the residual ones were about 125 sdot 10minus3

Following modern codes ([7 12] etc) the maximumelastic deformations of normal strength concrete 120576119888 el ul areabout 1 sdot 10minus3 which corresponds to the initial modulusof elasticity for short term loading (without consideringdurability aspects) 1198641198881 = tg 1205721 (slope of line 119900119886 in Figure 6)The ultimate plastic deformation 120576119888 pl ul is 2 sdot 10minus3 If concretewould behave elastically up to 120576119888 pl ul (line 119900119887 in the figure) acorresponding modulus of elasticity is 1198641198882 = tg 1205722 It is easyto show that

1198641198881 = 21198641198882 (9)

It demonstrates again the same phenomenon that whenthe concrete element reaches the ultimate deformation itsstiffness characteristic decreases twice

As it was shown experimentally increasing the loadacting on cylindrical concrete specimens with different steelfibers contents (from 0 to 60 kgm3) yields increase inPoisson coefficient ] It was found that the optimal fibercontent is 30 kgm3 Following the experimental data inspecimens without fibers ]min 0 = 011 and ] = 022 Inspecimens with optimal fibers content ]min 30 = 022 and]max 30 = 044 The ratios

]min 30]min 0

= ]max 30]max 0

= 2]max 30]min 30

= ]max 0]min 0

= 2(10)

demonstrate that like in case of longitudinal deformationsalso for transverse ones the parameters are doubled

Concrete creep increases with time and correspondinglyyields a decrease in the modulus of elasticity ([7 12 13] etc)For example for normal strength concrete the initialmodulusof elasticity decreases twice during a five-year period [7]

119864119888 (119905 = 0)119864119888 (119905 = 5 years) = 2 (11)

1 2 350

a

1205721

1205722

120576c 103

b

120590c

fck

Figure 6 Influence of concrete element deformations on its stiffnessparameter

Additionally concrete creep depends on the compres-sion stress in service limit state surrounding environmenthumidity composition and class of concrete and so forthConsidering the concrete class only the concrete creepdepends on concrete creep coefficient120573119888119896 that decreases as theconcrete class increases Let us compare the concrete creepcoefficients for the lowest and the highest structural concreteclasses C 20 and C 90 respectively according to the equationavailable in Eurocode [7]

120573cm = 168119891cm2 (12)

where 119891cm is a mean compressive strength of concrete In thiscase the ratio between those coefficients is

120573cmC20

120573119888119896C90 = 33601697 = 198 (13)

In other words the relation between the concrete creepdeformations for high strength concrete (HSC) and normalstrength concrete (NSC) is

120576crNSC asymp 2120576crHSC (14)

Thus using HSC allows achieving significant decrease inconcrete creep deformations (up to two times)

32 Behavior of RC Structures and Elements under StaticLoads In the previous section examples of structural phe-nomenon for cement-based composite nonlinear materialwas discussed (concrete was considered as a private case ofsuch material) The present section focused on behavior ofRC elements from the viewpoint of statics A rectangularRC bending element section with double reinforcement of119860 119904 and 1198601015840119904 in tensile and compression zones respectivelyis considered If the element is in ULS the reinforcementsections are

1198601015840119904 = [119872119889 minus 119891cd1198871198892120596 (1 minus 05120596)][119891sd (119889 minus 1198891015840119904)]

119860 119904 = 1198601015840119904 + 119891cd119887119889120596119891sd (15)

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

where 119872119889 is the external forces moment 119891cd and 119891sdare strength of the reinforced and reinforcing materialsrespectively 119887 and ℎ are the section dimensions 119889119904 and 1198891015840119904are concrete covering in the tensile and compression zonesrespectively 120596 = 119909119889 is the relative compression zone depth119909 is the compression zone depth and 119889 = ℎminus119889119904 is the effectivesection height

The reinforcement sections depend on 120596 that is a thirdunknown in (15) To obtain 120596 the minimax principle wasproposed [1 14]

(119860 119904 + 1198601015840119904) 997888rarr min (16)

that corresponds to effective section reinforcement Conse-quently

119889 (119860 119904 + 1198601015840119904)119889120596 = 0 (17)

Applying condition (17) to (15) the following extremumvalueof 120596 was obtained

120596extr = 05 (1 + 1198891015840119904119889 ) (18)

Taking into account that the order of 1198891015840119904 is lower than that of119889 the second term in the brackets can be neglected and theextremum value of the relative compression zone depth is

120596extr = 05ℎ = 2120596extr119889 (19)

It means that for optimal reinforcement of the consideredsection the concrete compression zone depth should be equalto half of the total section height In this case an anisotropicRC section becomes a quasi-isotropic one [14] In otherwords the structural phenomenon corresponds to optimaldesign concept of cement-based composite bending ele-ments Similar results were obtained for eccentrically loadedelements in compression and tensionwith large eccentricities

A sum of the bending moment taken by the compressedreinforcement Δ119872 and the maximal moment taken by thetensile reinforcement119872119904 max is equal to the external bendingmoment 119872119889 In order to carry the maximum value of thismoment 119872119889 max the section of 1198601015840119904 can be obtained from thecondition that the corresponding moment Δ119872max will notexceed that taken by the compressed concrete zone 119872119888 maxIn other words the force in compressed reinforcement is

1198601015840119904 max119891sd le 119909max119887119891cd (20)

In the ULS these forces are equal and therefore

Δ119872max = 119872119888 maxΔ119872max + 119872119888 max = 2119872119888 max (21)

Thus the maximum value of the compressed reinforcementsection corresponds to the condition when

119872119889 max = 2119872119888 max = 2119872119904 max (22)

At the same time (19) is valid that is brittle sectionfailure is avoided It can be concluded that an RC sectionthat behaves as quasi-isotropic one can be strengthenedby compression reinforcement up to achieving a doublemaximum moment bearing capacity of a section with singlereinforcement

The phenomenon related to doubling of various struc-tural parameters is also evident in problems of section designto shear forces [15] It is known that ultimate shear resistanceis determined as

119881119877119889 max = 119881119877119889119888 + 119881119877119889119904 (23)

where 119881119877119889119888 is the concrete shear bearing capacity and 119881119877119889119904 isshear resistance provided by shear reinforcement

Following modern codes [7 16]

119881119877119889119904 le 119881119877119889119888max119881119877119889119904 = 119881119877119889119888 (24)

Therefore the maximum section shear bearing capacity is

119881119877119889 max = 2119881119877119889119888 (25)

The main parameters affecting the section shear bearingcapacity are angles 120572 and 120579 defining respectively the linksrsquoinclination and the main compression stresses in concrete [715]Moreover relation between these angles is also importantThe angle 120572 is arbitrary selected by designer accordingto constructive and technological requirements whereas 120579depends on 120572

A theoretical value of 120579 can be obtained using minimaxprinciple [1] based on simultaneousminimization of ultimateexternal loading and maximization of internal stresses Theultimate shear force is maximized in the following additionalequation

119889119881119877119889 max119889120579 = 0 (26)

Thus the extreme value of 120579 that maximizes the sectionshear force is obtained as follows [1]

120579extr = 05120572120572 = 2120579extr (27)

Consequently when the section reaches its ultimate shearbearing capacity the angle of links 120572 selected at the designstage is equal to the doubled main compression forces angle120579extr33 Response of RC Structures and Elements to DynamicLoads Structural phenomenon is also evident in dynamicbehavior of buildings and proved by experimental data Forexample based on experimental results obtained for a full-scale three-story beamless precast framed RC building part[17] an unloaded frame had a dominant natural vibrationperiod 1198791 min = 06 s The frame was subjected to impulseload steps of 30 70 and 110 kN At the last step the frame

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

behaved as a geometrically and physically nonlinear systemand the corresponding dominant vibration period 1198791 max =12 s It follows that

1198791 max = 21198791 min (28)

Thus doubling of the dominant vibration period indicatesthat the structure reaches significant nonlinear deformationswhich characterizes the ultimate dynamic state of the struc-ture

The influence of gravitation stresses on RC sectionenergy dissipation under cyclic forces was examined [18]As the structural response in this case is nonsymmetricthe hysteretic loop area reduces relative to the case withoutconsidering the gravitation loading It was assumed in [18]that the maximum gravitation stress 120590119888119892 = 05119891119888119896 that isthe concrete reached the maximum elastic stress It was alsoassumed that under cyclic loading the section reached theplastic state The dissipated energy in this case is significantlylower than that obtained without considering the gravitationloads

119880tot asymp 2119880119892 tot (29)

where 119880tot and 119880119892 tot are the total section plastic energydissipation without and with gravitation loads respectively

As the section plastic energy dissipation decreases pro-portionally to the gravitation stresses the ductility factorcorrespondingly decreases twice

A six floor flat slab RC building with braced frame wasanalyzed [4] It was shown that disengagement of concretebraces leads to a system with variable stiffness Reduction ofmodulus of elasticity leads to unilateral disengagement Addi-tionally static loading countereffect appears due to dynamicloading These factors substantially reduce the seismic forces(about twice) giving rise to a static scheme that represents thestructurersquos most suitable response to a given earthquake

Seven possible structural static schemes running fromfully braced to unbraced were analyzed (Figure 7) [4] Thefirst and the last schemes are completely symmetric but thefirst is fully braced and the last is fully unbraced The fourthscheme represents a fully antisymmetric scheme Followingthe obtained results the dominant natural vibration periodscorresponding to those schemes were 1198791 = 0248 s 1198794 =0433 s and 1198797 = 1037 s Thus

11987941198791 = 04330248 = 175

11987971198794 = 10370433 = 239

(30)

It is evident that in this case also the average difference in thenatural vibration periods is 207

Following the results obtained in the same research [4]the structural phenomenon is evident not just for dynamicparameters but also for peak base shear forces which wereminimal for scheme 4 in Figure 7 1198811198894 = 5578 kN whereas1198811198891 = 10020 kN The ratio between these shear forces is 18which is about 2

1 4 7

Figure 7 Changes in the basic frame scheme under growinghorizontal dynamic loading (following [4])

Additionally a self-variable stiffness RC frame adaptsits response to an earthquake by using the basic concreteproperties [4] The frame selects a limit state with maximumenergy dissipation and the seismic forces effects decreaseabout twice The system has several seismic self-controlmodes (in terms of material structure and loading) whichare applied for adapting the frame to the given earthquake

As known it is impossible to test real full-scale structuresin ULS In most cases structural response in limit elasticstate is tested At the same time it is very important toknow structural dynamic parameters close to the ULSFor this reason modern numerical techniques are appliedExperimental results for the limit elastic state are effectivefor verification of numerically obtained initial structuralparameters The authors have shown that experimentallyobtained dominant vibration period for a full-scale 9-floorRC building was 072 s and the corresponding base shearforce (BSF) was 3400 kN [19] The investigated structure wasdesigned for an earthquake with peak ground accelerationPGA = 015 g Numerical analysis of the building subjected tothe same load like in the experiments showed that the BSFwas 3320 kN which is very close to the value obtained basedon experimental data

For adapting the building to a region with PGA = 03 g abase isolation system (BIS) was used The BIS was designedso that the BSF in the isolated building subjected to anearthquake with PGA = 03 g would be close to those ina fixed-base structure under an earthquake with PGA =015 g [19] Numerical results show that for the selectedreal earthquakes the average BSF in the isolated buildingis 3543 kN The corresponding dominant natural vibrationperiod of the isolated structure was 14 s that is it is abouttwo times higher relative to the fixed-base building in alimit elastic state (072 s) It was concluded that increasingthe dominant natural vibration period of the building from072 to 14 s (about twice) allows its adaptation to a zone with

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

0

R

t

TM

TH

RMRL

RH

TL

Figure 8 Structural dynamic response 119877 versus dominant vibra-tion period 119879H 119879M and 119879L dominant vibration periods for highmedium and low structural ductility respectively

higher seismicity (PGA = 03 g instead of 015 g) and keeps itsresponse in the same limits [19]

1198791 03 g1198791 015 g = 14 s

072 s asymp 2 = 03 g015 g (31)

Analysis of structural ductility enables us to prove theabove-mentioned conclusion As the input seismic energythat affects the building is independent of structural dynamicparameters especially of ductility hence according to Fig-ure 8 the energy (the area of the graph 119877119894 versus 119879119894 where119894 = H M and L) should be constant

119864119894 = int051198791198940

119877 (119879119894) 119889119879119894 = const 119894 = HM L (32)

Therefore if 119879119894 increases 119877(119879119894) decreases that is the forcethat acts on the building during an earthquake becomeslower

4 Conclusions

The present study is focused on analysis and discussionof available experimental and theoretical data from theviewpoint of a structural phenomenon It was shown thatthe phenomenon is valid for various design parameters atultimate limit state (ULS) of a structure or its elements It wasdemonstrated that the phenomenon is evident for materialproperties and structure (or its elements) as well as forstructural static andor dynamic response

The phenomenon is based on quasi-isotropic state ofa structure at ULS and minimax principle It is supportedby many experimental and theoretical results obtained fordifferent structures (beams frames spatial structures andstructural joints) under static orand dynamic loadings

The structural phenomenon enables us

(i) to predict the ULS of the building or appropriatesafety factor to this state

(ii) to assess the limit changes of strength and deforma-tion parameters in buildings before beginning theirreal design

(iii) to solve strengthening problems of a building

(iv) to carry out certification of a building (durabilityproblem)

(v) to find the limit values of steel fibers confining effectcompressed reinforcement section in the elementand so forth

(vi) to find the seismic resistance of a structure that isthe level of structural load bearing capacity under astrong earthquake

(vii) to reveal the stage when the structural static schemeis changed

The structural phenomenon discussed and analyzed inthe frame of the present study can be also applied for otherimportant issues in structural design For example one oflogical suggestions for selecting an upper limit for a numberof passive damping units in a structure is that maximumreduction in dynamic response of a building with effectivesupplemental passive devices is two times compared to theoriginal one (without dampers)

Thus the results of this study provide valuable indi-cators for experiments planning estimation of structuralstate (elastic elastic-plastic plastic or failure) evaluatingpossibilities of retrofitting and so forth From the math-ematical viewpoint the phenomenon provides additionalequation(s) that enable us to calculate parameters usuallyobtained experimentally or using some empirical coefficientsTherefore using this phenomenon can lead to developingproper design concepts and new RC theory in which thenumber of empirical design coefficients will be minimal

Competing Interests

The authors declare that they have no competing interests

References

[1] I Iskhakov and Y Ribakov Ultimate Equilibrium of RC Struc-tures using Mini-Max Principle Nova Science New York NYUSA 2014

[2] D J Dowrick Earthquake Resistant Design For Engineers andArchitects John Wiley amp Sons New York NY USA 1995

[3] J Magnuson and M Hallgern ldquoReinforced high strengthconcrete beams subjected to air blast loadingrdquo in Structuresunder Shock and Impact N Jones and C A Brebbia Eds vol8 WIT Press Southampton UK 8th edition 2004

[4] I Iskhakov and Y Ribakov ldquoSeismic response of semi-active friction-damped reinforced concrete structures with self-variable stiffnessrdquo Structural Design of Tall and Special Buildingsvol 17 no 2 pp 351ndash365 2008

[5] A A Gvozdev ldquoObtaining the destructive loads value forstatically indetermined systems under plastic deformationsrdquo inProceedings of the Conference on Plastic Deformations RussianAcademy of Sciences Moscow Russia 1938 (Russian)

[6] C Bach and O Graf Tests with Simply Supported QuadraticReinforced Concrete Plates vol 30 Deutscher Ausshuss furEisenbeton Berlin Germany 1915 (German)

[7] Eurocode 2 Design of Concrete StructuresmdashPart 1-1 GeneralRules and Rules for Buildings 2004

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 9

[8] M R Horne ldquoFundamental propositions in the plastic theoryof structuresrdquo Journal of the Institution of Civil Engineers vol34 no 6 pp 174ndash177 1950

[9] A C Palmer G Maier and D C Drucker ldquoNormality relationsand convexity of yield surfaces for unstable materials or struc-tural elementsrdquo ASME Journal of Applied Mechanic vol 34 no2 pp 464ndash470 1967

[10] I Iskhakov The Limit Equilibrium of Square Shells ConsideringTheir Deformation Scheme vol 2 of Stroitelnaya Mechanica iRaschyot Sooruzeniy Stroyizdat Moscow Russia 1972 (Rus-sian)

[11] I Iskhakov Y Ribakov K Holschemacher and T MuellerldquoExperimental investigation of full-scale two-layer reinforcedconcrete beamsrdquo Mechanics of Advanced Materials and Struc-tures vol 21 no 4 pp 273ndash283 2014

[12] Building Code Requirements for Structural Concrete (ACICommittee 318-05) 2005

[13] I Iskhakov and Y Ribakov ldquoA new concept for design of fiberedhigh strength reinforced concrete elements using ultimate limitstate methodrdquoMaterials and Design vol 51 pp 612ndash619 2013

[14] I Iskhakov and Y Ribakov ldquoOptimization of steel ratio incement based composite bending elementsrdquo in Proceedingsof the 13th International Conference lsquoMechanics of CompositeMaterials MSM 2004rsquo Riga Latvia May 2004

[15] I Iskhakov and Y Ribakov ldquoExact solution of shear problemfor inclined cracked bending reinforced concrete elementsrdquoMaterials and Design vol 57 pp 472ndash478 2014

[16] The Standards Institution of Israel ldquoSI 466 part 1 concrete codegeneral principlesrdquo 2012

[17] I Iskhakov and Y Ribakov ldquoSelecting the properties of a baseisolation system based on impulse testing of a three-storystructural partrdquo European Earthquake Engineering vol 1 pp38ndash42 2005

[18] I Iskhakov and Y Ribakov ldquoInfluence of various reinforcingarrangements and of gravitation stresses on hysteretic behaviorof RC sectionsrdquo European Earthquake Engineering vol 3 pp13ndash24 2005

[19] I Iskhakov and Y Ribakov ldquoExperimental and numericalinvestigation of a full-scale multistorey RC building underdynamic loadingrdquo The Structural Design of Tall and SpecialBuildings vol 14 no 4 pp 299ndash313 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials