revista-constructii.univ-ovidius.ro...Ovidius University Annals Series: Civil Engineering Volume 1,...

“OVIDIUS” UNIVERSITY OF CONSTANTZA UNIVERSITATEA „OVIDIUS” CONSTANŢA

“OVIDIUS” UNIVERSITY ANNALS - CONSTANTZA

Year IX (2007)

Series: CIVIL ENGINEERING

Volume 1

ANALELE

UNIVERSITĂŢII „OVIDIUS”CONSTANŢA ANUL IX

(2007)

Seria: CONSTRUCŢII Volumul 1

Ovidius University Press 2007

Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, 2007

ISSN-1584 - 5990 © 2000 Ovidius University Press

SECTION I

Structural Analysis and Reliability

The influence of the water level variation in reservoirs upon the earth dams strain state Virgil BREABAN Sunai GELMAMBET

7-14

The generalized eigenvalues method in the nonlinear dynamics analysis Gheorghe PICOL Mircea IEREMIA

15-22

Rotation Capacity of Reinforced Concrete Elements Bogdan HEGHEŞ Cornelia MĂGUREANU

23-28

Assessment of the Potential for Progressive Collapse in RC Frames Adrian IOANI Liviu CUCU Călin MIRCEA

29-36

Cracking of Reinforced Concrete Elements Laura- Catinca LEŢIA

37-44

A Short Introduction to Load Carrying Capacity for High Strength Concrete Cornelia MĂGUREANU Dumitru MOLDOVAN

45-52

Studies on the Modalities of Use of Sludge Resulting From the Lime Milk Neutralization ofAcid Waters Derived From the Pickling of Wire Obtained At S.C. Mechel Câmpia Turzii Daniela MANEA Claudiu ACIU Ofelia CORBU

53-58

Energy Conservation, an Essential Factor in Sustainable Construction Daniela MANEA Claudiu ACIU

59-64

Sustaining Systems for Underground Parking in Cluj - Napoca Augustin POPA Nicoleta Maria ILIEŞ

65-70

The Rayleigh Quotient, the Vector Iteration With Shift and the Rayleigh Product Daniela PREDA Florin MACAVEI

71-76

Calculation of deformation estimated value for protection harbor construction to seismicapplication shaped through stationary random process Isabella STAN Dragos VINTILA

77-80

Table of Contents / Ovidius University Annals Series: Civil Engineering 9, 179 - 181 (2007) 180

SECTION II Fluid Mechanics and Hydraulic Structures

Reactive centrifugal rotor – the analytical study of two applications Victor BENCHE Radu ŢÂRULECU Stelian ŢÂRULECU

83-86

Analogical electro hydrodynamic research on installations for launch subsonic constantdensity jets Victor BENCHE Virgil-Barbu UNGUREANU

87-92

The Safety of Concrete Structures from the Water Supply System, Undermined by the Errors and Careless in Design and Execution Olimpia BLAGOI Bogdan PATRAS Maricel GEORGESCU Marinela BARBUTA

93-98

Modeling, Simulation and Regulation of an Industrial Installation Intended for FieldIrrigation Using Attenuant Wastewater Adrian BOLMA Marian DORDESCU

99-106

Phased Execution of the Coastal Protection Works in the Southern Area of the Romanianseashore Romeo CIORTAN

107-112

A possible recovery system of the potential energy for the rain water in the case of high buildings Ovidiu Mihai CRĂCIUN Radu ŢÂRULECU

113-118

The Analysis of the Impact of Storage Lake on Environment Using the Chemical Characterization of the Water Resources. Case Study Bahlui Basin River Ion GIURMA Ioan CRĂCIUN Catrinel-Raluca GIURMA

119-124

The Multicriterial Decisional Management Within Irrigation Arrangements Gheorghe IORDACHE Marian DORDESCU

125-130

Protection Measures on the Algerian Coastline of the Mediterranean Sea Khoudir MEZOUAR

131-136

Shoreline Variation and Protection Measures on the Romanian Coast Line of the Black Sea –A Case Study for Mamaia Beach Khoudir MEZOUAR Romeo CIORTAN

137-144

Table of Contents / Ovidius University Annals Series: Civil Engineering 9, 179 - 181 (2007)

181

Explanatory Aspects of the Research Concerning the National Land Reclamation DigitalData Fund (FNDDIF) Irina STATE Tudor Viorel BLIDARU

145-150

Hydraulic Checking of a Sewerage Collector Gabriel TATU

151-154

The Increase of Strong Rainfall Concentrated on Small Areas as an Effect of ClimaticChanges Marius TELIŞCĂ Catrinel-Raluca GIURMA-HANDLEY Petru CERCEL

155-160

Energetic improvement of joinery embrasures Virgil-Barbu UNGUREANU

161-168

Analysis of heat exchangers obtained by division or multiplying of units Virgil-Barbu UNGUREANU Neculae ŞERBĂNOIU Maria MUREŞAN

169-176

Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, May 2007

ISSN 1584 - 5990 © 2000 Ovidius University Press

The influence of the water level variation in reservoirs upon the earth dams strain state

Virgil BREABAN a Sunai GELMAMBET a

a ”Ovidius” University of Constantza, Constantza, 8700, România

__________________________________________________________________________________________ Rezumat: În această lucrare este studiată şi prezentată influenţa variaţiei nivelului apei în lacul de acumulare asupra deformaţiilor barajelor de pământ. Pentru aceasta au fost realizate o serie de simulări numerice privind efectele unor variaţii bruşte sau lente ale nivelurilor în lacul de acumulare asupra stării de deformaţii şi eforturi în corpul barajelor de pământ. Analizele numerice s-au efectuat în ipoteza comportării neliniare a materialelor din corpul barajului, cu ajutorul programului cu elemente finite Cosmos 2.6. Compararea rezultatelor numerice cu măsurătorile din amplasamente validează modelele de calcul folosite şi permit evaluarea efectelor fenomenului studiat asupra siguranţei barajelor de pământ. În final, în urma analizei rezultatelor obţinute sunt prezentate o serie de concluzii cu privire la influenţa variaţiei nivelului apei în lac asupra deformaţiilor barajelor de pământ. Abstract: In this paper is presented and studied the influence of the water level variation in the reservoirs upon the earth dams strain state. For that there have been realized a series of numeric simulations about the effect of sudden or slow variations of the levels in the reservoir over the state of strains and stress in the body of the earth dams. The numerical analysis has been done in the hypothesis of nonlinear behaviour of materials in the dam body, using the finite element program Cosmos 2.6. Comparison between the numerical results and the local measurements made on dam validates the computational models used and allows the effects estimation upon the earth dams’ safety. Finally, are presented a series of conclusions about the influence of the variations of the water level in the reservoir over the earth dams strain state. Keywords: the variations of the water level in the reservoir, strains, earth dam. __________________________________________________________________________________________ 1. Introduction

Earth dams represent the most common and

the oldest category of all the dams. Almost 70% of the 46000 of great dams that are in the ICOLD system are embankment dams [3].

With all the spread and the age of earth dams, with all the remarkable scientific and technologic progresses realised in this domain, especially in the last five decades, the knowledge of the behaviour of the earth dams at sudden variations of the water level in the reservoir is not totally understood.

Because of these reasons, in this work is represented and studied with the help of numeric methods based on MEF [2], [6], the nonlinear behaviour of earth dams and the sudden variations of levels in the reservoir, adding to a better knowledge of the studied phenomena, the rise in performance and safety in the use of earth dams [1].

For the analysis of the state of stress and strains is important to take in consideration the hypothesis of sudden variation of water level in the reservoir. This sudden variation can appear in the situation in which is necessary of a rapid empting or the case of a flood wave [3]. The rapid empting of the dam may appear necessary for reasons of safety of the dam, urgent needs of the use of the water in the reservoir or other special situations. 2. Numerical Simulation

The numerical simulations for the case of sudden rise of the water level in the reservoir were done for Dopca dam., a lest affluent of the river Olt, at a distance of 1,5 km upstream of the town Dopca, a village in the town of Hoghiz, in the county of Brasov.

The Dopca dam is made of fillings, made of embankment from the materials extracted from the

The influence of the water … / Ovidius University Annals Series: Civil Engineering 9, 7-14 (2007)

8

lake ditch, with a reinforced concrete face, with a surface of 7800 m2, made on the upstream face with

the maximum height of 18,0 m and the length at the top of 175,0 m

Fig.1 The section of Dopca dam

The numerical simulations over the dam have

been done for two cases so that later, by comparing the results we can see the effect of the sudden variations of water in the lake. In both cases, the

simulations have been done with the help of the program of finite elements COSMOS 2.6 [4], [5]. The analyses done in the two cases have been nonlinear analysed and was used the Drucker-Prager model [8].

Fig.2 The section of Dopca dam, with the difference of the water level

In the case of the first simulation there has been

considered a rise in water level in the lake of 1m/day and in the second case a rise of 3m/day. The difference of level considered in the case of the sudden variation was of 10m and is presented in the fig. 2.

For the first case when the rise of level of water level was considered of 1m/day the filling had taken place in 10 days meaning 864000 seconds,

and in the second case the filling had taken place in 3,33 days meaning 288000 seconds.

To be sure that the results obtained represents a behavior close to the behavior of the real dam, the results have been compared with the real behavior of the dam in time.

The results that were obtained from the simulations in the two cases, in order to be seen and compared more easily were presented in the following drawings.

V. Breaban , S. Gelmambet / Ovidius University Annals Series: Civil Engineering 9, 7-14 (2007) 9

Fig.3. The numbering of nodes

Fig.4 The variation of the displacement in the y direction in node 1 in the case of sudden variation(cm)


Fig.5 The variation of the displacement in the y direction in node 1 in the case of no sudden variation (cm)

Fig.7 The variation of the displacement in the y direction in node 9 in the case of no sudden variation(cm)


10


Fig.10 The variation of the stress σx in node 26 in the case of sudden variation (Pa)

Fig.12 The variation of the stress τxy in node 26 in the case of sudden variation (Pa)

Fig.9 The variation of the displacement in the y direction in node 18 in the case of no sudden variation(cm)

Fig.11 The variation of the stress σx in node 26 in the case of no sudden variation (Pa)

Fig.13 The variation of the stress τxy in node 26 in the case of no sudden variation (Pa)

V. Breaban , S. Gelmambet / Ovidius University Annals Series: Civil Engineering 9, 7-14 (2007) 11

Fig.14 The diagram of stress τxy at step 100 in the case of sudden variation (Pa)

Fig.15 The diagram of stress τxy at step 100 in the case of no sudden variation (Pa)

Fig.16 The diagram of εx strains at step 100 in the case of sudden variation


12

Fig.17 The diagram of εx strains at step 100 in the case of no sudden variation

Fig.18 The situation schematics of the dam with disposal landmarks for the following in time

In the following figures are presented the data

obtained from the study in time of the dam are presented.

In fig. 18 is presented the plan of dispersion of the landmarks for the following in time of the Dopca dam is presented.

The landmark 41 coincides with node 1 of the finite elements mesh. The landmark 39 coincides with node 9 of the finite elements mesh. The landmark 46 coincides with node 18 of the finite elements mesh.

V. Breaban , S. Gelmambet / Ovidius University Annals Series: Civil Engineering 9, 7-14 (2007) 13 The marked values are form 1 Oct 2001 when the flow of water was 1m/day. The settlement for node 1 from the calculations shown in fig 5 is -5,75 mm.

The settlement for node 9 from the calculations shown in fig 7 is -3,27 mm. The settlement for node 18 from the calculations shown in fig 9 is -6,16 mm.

Fig.19 The values of settlement necessary for the comparison

In fig. 20 are presented the differences between the settlement obtained by the calculation and the settlement obtained by the measurement.

Fig.20 The comparison graphics between the calculation values and the measured values


14

3. Conclusions

We can see that in the case of sudden variations, the movements and strains on the horizontal of the dam modify significantly and are not to be neglected. For dams with reinforced concrete face, a problem is the deformation of the reinforced concrete face under the action of sudden variations of the level of water in the reservoir. Because of these movements (strains) modification appears to the profile of transversal sections and even longitudinal fissures witch can influence the resistance of the dam (infiltrations).

In the case of dams with reinforced concrete face, the effect of the sudden variation of water consists in the fact that the plastic strains are extremely big, especially in the center zone of the dam. The reinforced concrete face follows the strains of the filling of the dam produced by the pressure at the filling of the lake and it’s pulled to the center of the dam. Because of this the vertical ends of the central area of the reinforced concrete face have a tendency to close and the ends of perimeter areas open very much. In the case of the perimeter area there are three distinct components of the displacements: settlement on the normal face of the reinforced concrete face, openings on the normal direction of the end and displacements that form a tangent parallel.

Following the comparison of the results in the two cases, we can se that the influence of sudden variation of the water level over the results is very important. We can se that de difference of the displacement on the y (vertical) direction are very small, but the differences on the x (horizontal) direction are important and cant be ignored.

In the case of the stress we can see an increase of the values in the case of sudden variation in especially in the case of the stress σx and τxy. By comparing the results of the strains that are specific we can se that in the case of the strains the εy differences are very small like in the case of the displacements. The more important differences

appear in the case of the strains εx and in the special cases of strains γxy.

We also see that the horizontal strains can be registered at the ½ the height of the dam on the upstream prism. Therefore the biggest strains will be produced in the central area of the upstream prism.

The complex nature of the phenomena of behavior of embankment dams at the first filling of the reservoir imposes a careful study of them on the full period of the filling as well as in the first years of using. The comparison of the data obtained from the measures about the calculation from the design, the making of post analysis tests make the most direct methods for the understanding of the phenomena and the preventing of incidents or accidents. 4. References [1] Dibaj,M., Penzien, J., Nonlinear seismic response of earth structures, Report No. EERC 69-2 , Univ. of California, Berkeley, 1974. [2] Popovici, A., Dynamic analisys by numerical me thods, 1978, I.C.Bucuresti.. [3] Popovici, A., Dams for water accumulation, Vol.II, 2002, Editura Tehnică Bucuresti. [4] Gelmambet, S., Dam-foundation seismic interaction analysis, Simpozionul Concepţii Moderne în ingineria Amenajărilor Hidrotehnice, 13 mai 2005, Timişoara, Buletinul Ştiinţific al Universităţii „POLITEHNICA” din Timişoara, Seria Hidrotehnică, Tomul 49 (63), Fascicola 1, pag.46-53, Editura Politehnica, România 2005; [5] Gelmambet, S., Dam-reservoir seismic interaction analysis, The XXXth National Conference of Solid Mechanics Mecsol 2006 , 15-16 septembrie 2006, Constanta, Vol.9 pag.251-258. [6] Zienkiewicz, O.C. The finite element method in engineering science, McGraw-Hill, London, 1971. [7] Zienkiewicz,O.C., Bettess,P. Fluid-structure Dynamic interaction and wave forces; an introduction to numerical treatement, Int.J.Num.Meth.in Engng., Vol.13, 1978. [8] *** Cosmos/M Manual Teoretic, Structural Reasearch Corporation, Santa Monica USA, 1996.



The generalized eigenvalues method in the nonlinear dynamics analysis

Gheorghe PICOL a Mircea IEREMIA a a Technical University of Civil Engineering Bucharest, Bucharest,020396, România

__________________________________________________________________________________________ Rezumat: În analiza dinamică liniară, în condiţiile în care modelul de calcul are o lege constitutivă liniar-elastică, se poate folosi proprietatea de ortogonalitate a vectorilor proprii pentru rezolvarea ecuaţiilor de echilibru seismic. În acest caz, vectorii proprii sunt ortogonali în raport cu matricea maselor şi matricea rigidităţilor. Dacă se consideră amortizarea de tip Rayleigh, relaţia de ortogonalitate este valabilă şi pentru matricea de amortizare. Din punct de vedere matematic, sistemul de ecuaţii care caracterizează fenomenul se transformă în “n” ecuaţii decuplate , câte una pentru fiecare mod propriu de vibraţie.

În efectuarea unei analize dinamice incrementale neliniare, matricile maselor, de rigiditate şi de amortizare nu mai sunt neapărat matrici simetrice.În consecinţă relaţia de ortogonalitate a vectorilor proprii în raport cu matricile de rigiditate şi de amortizare nu mai este verificată şi ca urmare formele proprii de vibraţie nu se mai pot decupla.Se recurge la metoda numerică de rezolvare a valorilor şi vectorilor proprii generalizaţi. Abstract: In the linear dynamics analysis when the computation model has a linear-elastic constitutive law, it can be used the orthogonality property of the eigenvectors to solve the seismic equilibrium equations. In this case, the eigenvectors are orthogonal with respect to the mass matrix and stiffness matrix. If one considers the Rayleigh type damping, the orthogonality relationship holds for the damping matrix too. From the mathematical point of view the system of equations characterizing the phenomena becomes a set of “n” uncoupled equations, one equation for each mode. The achievement of a nonlinear dynamics analysis assumes the mass, rigidity and damping matrices are not necessary symmetric. Consequently, the orthogonality relationships of eigenvectors with respect to the rigidity and damping matrices are no longer true and the inner forms of vibrations cannot be decoupled. In this case one uses the numerical method of generalized eigenvalues and generalized eigenvectors.

Keywords: Dynamic,damping, eigenvalues, generalized eigenvectors. __________________________________________________________________________________________ 1. The problem-Generalitys

In the linear dynamics analysis when the

computation model has a linear-elastic constitutive law, the orthogonality property of the eigenvectors can be used to solve the seismic equilibrium equations. In this case, the eigenvectors are orthogonal with respect to the mass matrix and stiffness matrix. One neglects usually the damping matrix; if the damping is taken into account then the Rayleigh model is used where the damping matrix is a linear combination between the mass matrix and the stiffness matrix. Within these conditions the orthogonality relation is also true for the damping matrix. From the mathematical point

of view the system of equations of free or forced vibrations becomes a set of “n” independent equations, one equation for each mode.

The characteristics of an incremental nonlinear dynamics analysis are the following: -the history of the used excitation should be considered; - the numerical models must consider the rheological properties of real materials: elasticity, plasticity, viscosity ; - for the linearization, “stress- strain” curve is replaced by a polygonal line ; - for a material with different loading and unloading rigidity, the variation of rigidity is given by the variation of the modulus of elasticity/plasticity with

The generalized eigen. meth…/ Ovidius University Annals Series: Civil Engineering 9, 15 -22 (2007)

16

respect to strain value(elastic-plastic) at the corresponding moment of time ; - the damping effect which modifies the eigenfrequency of the damaged structure needs to be considered; - the stiffness matrix and the damping matrix are not constant anymore, they are functions of the total specific deformation; they are continuously degrading when loading ; - the mass, stiffness and damping matrices are non necessary symmetric ; - the orthogonality relatiohships of the eigenvectors with respect to the stiffness and damping matrices are not necessary true and as consequence the inner forms of vibrations cannot be detached; one uses the numerical method of generalized eigenvalues and generalized eigenvectors with solutions arising in the real field from the complex field.

In the sequel M, K are the mass matrix and the stiffness matrix of a model respectively , with N dynamical degree of freedom (DOF) and C the damping matrix; P(t) is the vector of loads. The

unknowns are u, •

u and ••

u - the vector of displacements, velocities and accelerations. Then the dynamic response of the body to an external excitation is modeled by a system of differential eqs. (1.1) .

( )tPuKuCuM =⋅+⋅+⋅•••

(1.1) One considers both the homogenous viscous

damped associated model and the homogeenous nondamped associated model, defined by eq.(1.2) and (1.3) respectively:

0uKuCuM =⋅+⋅+⋅•••

(1.2)

0uKuM =⋅+⋅••

(1.3)

The initial conditions of the dynamic response are given by the eqs.(1.4):

( ) ( ) 00 v0u;u0u ==•

, (1.4) where u0 şi v0 are the vectors of the initial dis-placements and the initial velocities.

To solve the equations (1.1)÷(1.4), there are several methods, each of them proceeding in several steps and some steps are common for two or more methods; each method has its own hypotheses, advantages and drawbacks. 2. Generalized eigenvalues method 2.1. The response in free vibrations with initial data

The system of homogenous differential eq.(1.2) by the transformations (2.1):

( ){ } ( )( )⎪⎭⎪⎬⎫

⎪⎩

⎪⎨⎧

=•

tututX , [ ] [ ] [ ][ ] [ ]⎥⎦

⎤⎢⎣

⎡−

=K0

0MB , [ ] [ ] [ ][ ] [ ] ⎥⎦

⎤⎢⎣

⎡−

−−=

0KKC

A , (2.1)

becomes equivalent with the system of differential equations of the first order

[ ] ( ) [ ] ( ){ } { }0tXAtXB =⋅−⎭⎬⎫

⎩⎨⎧⋅

• (2.2)

Looking for particular exponential solutions

of (2.2) one gets the relationships (2.3) and (2.4). det(β[A]-α[B])=0 (2.3) (β[A]-α[B]){V}={0} (2.4)

The eq. (2.3) is called the characteristic equation, the pairs ( ) ( )( ) { }0Cj,j 2 −∈βα and the vectors { } { }0CV nj −∈ , which satisfy (2.3) and (2.4) are called generalized eigenvalues and generalized eigenvector. With no loss of generality one assumes

.0,R ≥β∈β Usual eigenvalues is λ given by: βα=λ / (2.5)

The part α(j) of the generalized eigenvalue of

(2.4) has the form (2.6).

P. Gheorghe and M.Ieremia / Ovidius University Annals Series: Civil Engineering 9, 15 -22 (2007) 17 α(j) = αR(j)+i αI( j), (2.6) with αR(j) , ( ) .RjI ∈α As the roots of the characteristic equation are real or not we denote:

( ){ }( ){ }0j,nj1jI

0j,nj1jI

I2

I1

>α≤≤=

=α≤≤= (2.7)

If 1Ij∈ , then the generalized eigenvector is

a column of [ ] ( )RMVR n∈ .

If 2Ij∈ , then the real part and the imaginary part

of the generalized eigenvector are two columns of [VR].

Using the eqs. (2.1), and the initial data (1.4) one gets the coeficients μj from (2.8).

[ ] { } { }T00j u,vVR =μ⋅ (2.8)

The solution of the problem (1.2)÷(1.4) is given by (2.9), for any s, 1 ≤ s ≤ p:

( ) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( ) ( ) ,ej,spVR)tjsin1j,spVRj,spVR

tjcos1j,spVRj,spVR(etu

1

j

2

R

Ij

tjIj1j

IjI1jj

tjs

∑∑

∈

⋅λ+

∈+

⋅λ

⋅+⋅μ+⋅λ⋅++⋅μ−+⋅μ

+⋅λ⋅++⋅μ++⋅μ⋅= (2.9)

Here the generalized eigenvalues are λ(j):

( ) ( ) ( ) ( ) ( ) Rj,Rj,ijjj IRIR ∈λ∈λ⋅λ+λ=λ (2.10)

The functions tje ⋅λ , ( )( ) ( ) tjI Retjcos ⋅λ⋅⋅λ and ( )( ) ( ) tjI Retjsin ⋅λ⋅⋅λ from (2.9) are called

eigenfunctions. 2.2. The response in forced vibrations with piecewise linear exciting force

One assumes that{u(t)} is a solution of (1.1) and that the sequence of timestamps ti are ordered ascendingly, ti


18

3. The structural response in displacements, velocities and frequencies

Various significant cases from practice emmphisized the influence of structural damping, of the particularities of the chosen model and of the number of dynamical degrees of freedom.

In practical engineering applications we used the procedures DGGEVX and DGESV from LAPACK, DVOUT and DMOUT from ARPACK.

3.1. The undamped structural response

The frame with one opening of a bulding with

two floors is considered . The mass of the frame is supposed to be concentrated at the floor level and the floors move rigidly. The degree of freedom was indicated on figure. The response of the structure to the problem is required (1.1)÷(1.4) where the initial

conditions are null and the matrices [M],[C],[K] and the vector of forces {P} are given by:

[M] [C] [K] {P} 2.D0 0.D0 0.D0 0.D0 3.D0 -1.D0 0.0D0 0.D0 1.D0 0.D0 0.D0 -1.D0 1.D0 2.0D0

The exact displacements are given by (3.1).

( ){ } ( ) ( )( ) ( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛

⋅−−⋅−+=

3/2/tcos83/t2cos33/2/tcos43/t2cos1tu (3.1)

Table 1.Comp. btw. the analytic and DGGEVX freq.

Analytic Fortran90(DGGEVX) 2 1.4142135624...= .141421356D+01

1/ 2 0.7071067812= .707106781D+00

Table 2. Comparison between the analytic coefficients of the eigenfunctions and those obtained with DGGEVX

GLD1, DOF1 GLD2,DOF2 Eigenfunctions Analytic Fortran90(DGGEVX) Analytic Fotran90(DGGEVX)

1 3/1 3.3333333333333D-01 3/1− -3.3333333333333D-01 2 0 0.0000000000000D+00 0 0.0000000000000D+00 3 3/4− -1.3333333333333D+00 3/8− -2.6666666666667D+00 4 0 0.0000000000000D+00 0 0.0000000000000D+00

One can see from these tables that the approximation is as good as possible.

3.2. The effect of damping in modal analysis

One considers the system from fig. 1 in the

case of free and forced vibrations with initial data.

Fig.1. Free vibrating system with initial data

For undamping vibrations one considers the

matrices [M] , [K] and the vectors of the initial data {v0} and {u0}, given by (3.2) ,(3.2’).

[ ]

[ ]⎪⎪⎪⎪

⎭

⎪⎪⎪⎪

⎬

⎫

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

0D0.20D0.10D0.00D0.10D0.40D0.10D0.00D0.10D0.2

K

,0D5.00D0.00D0.00D0.00D0.10D0.00D0.00D0.00D5.0

M

(3.2)

{ } { } { }⎭⎬⎫

⎩⎨⎧

−−−==6

7,

3

1,

6

10u;00v (3.2’)

In the damping case one considers the damping

matrice (3.3) also.

P. Gheorghe and M.Ieremia / Ovidius University Annals Series: Civil Engineering 9, 15 -22 (2007) 19

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

5.000000001.0

C (3.3)

The vectors of the initial data are the same in both cases.

The analytic displacements are given by (3.4)

( ){ }( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⋅⋅−⋅⋅−⋅⋅−⋅⋅−⋅⋅

⋅⋅+⋅⋅−⋅⋅−=

t2cos21t2cos21t6cos61t2cos21t6cos61

t2cos21t2cos21t6cos61tu (3.4)

Table 3. Comparison between the analytic angular frequency and that obtained with DGGEVX

Analytic DGGEVX(undamped) DGGEVX(damped) ...4494897428.26 = .2449490D+01 .238949058D+01

...4142135624.12 = .1414214D+01 .145441134D+01 2 .2000000D+01 .195127286D+01

Table 4. Comparison between the undamped and damped eigenfunctions ( obtained with the proc. DGGEVX)

Undamped Damped kcos(.2449490D+01*t)*exp(.1857943D-14*t) cos(.2389491D+01*t)*exp(-.1093876D+00*t) sin(.2449490D+01*t)*exp(.1857943D-14*t) sin(.2389491D+01*t)*exp(-.1093876D+00*t) cos(.1414214D+01*t)*exp(.4906539D-15*t) cos(.1454411D+01*t)*exp(-.1449880D+00*t) sin(.1414214D+01*t)*exp(.4906539D-15*t) sin(.1454411D+01*t)*exp(-.1449880D+00*t) cos(.2000000D+01*t)*exp(.0000000D+00*t) cos(.1951273D+01*t)*exp(-.3456244D+00*t) sin(.2000000D+01*t)*exp(.0000000D+00*t) sin(.1951273D+01*t)*exp(-.3456244D+00*t)

Table 5. Comparison between the coefficients of the undamped and damped eigenfunctions

Eigen. Func. DOF Analytic Undamped Damped 1 1 6− -1.6666666666667D-01 -1.5324453745705D-01 2 1 6 1.6666666666667D-01 1.2964861943519D-01

1

3 1 6− -1.6666666666667D-01 -2.2799097435107D-02 1 0 6.9854441481544D-17 9.4031423163675D-02 2 0 -2.0789052147676D-17 -8.4284668222341D-02

2

3 0 8.9480597215092D-17 1.1683391713828D-01 1 1 2− -5.0000000000000D-01 -5.2811581389712D-01 2 1 2− -5.0000000000000D-01 -4.8113345857307D-01

3

3 1 2− -5.0000000000000D-01 -2.8695203010525D-01 1 0 3.7946038233531D-16 -2.2049260920957D-01 2 0 6.0827000183804D-17 -2.4145570810675D-01

4

3 0 2.9464341431205D-16 -4.4256711285042D-01 1 1 2 5.0000000000000D-01 5.1469368468751D-01 2 0 2.0004455055537D-16 1.8151505804552D-02

5

3 1 2− -5.0000000000000D-01 -8.5691553912631D-01 1 0 6.5959487396052D-18 9.2532881133750D-02 2 0 2.8818226857972D-30 2.5791898163980D-01

6

3 0 -6.5959487396111D-18 1.2418385824861D-02


20

3.3. The dynamic structural response to time variable loads

The dynamic response is considered of the frame from fig. 2, discretized with 4 beam elements.

The linear variation of the applied force is shown in figure 3. We have 9 DOF. In Table 6 a comparison between the first two DOF is shown.

Fig.2. Frame modelled with 4 beam elements Fig.3. The force function at node 2

Table 6. Comparison between displacements on the first two DOF Displ.DOF1(cm) Displ. DOF2(cm) Time,(s)

Paz,Leigh DGGEVX Paz,Leigh DGGEVX 0.00 0 0 0 0 0.03 0.12346 0.11855 -0.02569 -0.03626 0.04 0.16516 0.18429 -0.05606 -0.05369 0.05 0.20431 0.22177 -0.09073 -0.09270 --- --- --- --- ---

3.4. Extreme values of the structural response for forced vibrations. The damping and undamping case

Within the framework of the dynamics

analysis we propose to compute the frequencies, the structural response and the extreme values of the response for the sheared building with two floors from fig. 4. The structure is subjected to a force of

Fig.4. Frame with 2 levels subject to lateral forces

10kN applied suddenly to the second floor. We assume the elastic behaviour of the structure and we consider two kinds of damping:

a)no damping; b) 10% of the critical damping for each mode

Table 7.Comparison between the angular frequencies Paz,Leigh Fortran90

11.827 .1179679D+02 32.901 .3296284D+02

P. Gheorghe and M.Ieremia / Ovidius University Annals Series: Civil Engineering 9, 15 -22 (2007) 21Table 8. Comparison between the first two DOF

Displ.DOF1(cm) Time(s) Paz,Leigh DGGEVX

0.00 0 -.6938894D-17 0.01 2.05E-05 .2057270D-04 0.02 3.24E-04 .3251518D-03 0.03 0.00161 .1612764D-02 0.04 0.00493 .4953161D-02 0.05 0.0116 .1165484D-01 --- --- ---

0.39 0.40966 .4159244D+00

Displ.DOF2(cm) Time(s)

Paz,Leigh DGGEVX 0.00 0 .1387779D-16 0.01 0.00753 .7533334D-02 0.02 0.02963 .2963100D-01 0.03 0.06485 .6483580D-01 0.04 0.11093 .1108797D+00 0.05 0.16501 .1649071D+00 --- --- ---

0.39 0.52246 .5298873D+00

Table 9. Comparison between the eigenfunctions

Undamped Damped 10% cos(.1179679D+02*t)*exp(.0000000D+00*t) cos(.1173733D+02*t)*exp(-.1182954D+01*t) sin(.1179679D+02*t)*exp(.0000000D+00*t) sin(.1173733D+02*t)*exp(-.1182954D+01*t) cos(.3296284D+02*t)*exp(.0000000D+00*t) cos(.3279819D+02*t)*exp(-.3290506D+01*t) sin(.3296284D+02*t)*exp(.0000000D+00*t) sin(.3279819D+02*t)*exp(-.3290506D+01*t)

Next the maximum and minimum of the displacements and velocities are presented as well as the moments

when they are reached depending on DOF. a) Undamped

Dir, DOF umin at t umax at t vmin at t vmax at t 1 -.69D-17 .00D+00 .68D+00 .24D+00 -.49D+01 .39D+00 .60D+01 .14D+00 2 .00D+00 .00D+00 .11D+01 .28D+00 -.69D+01 .35D+00 .60D+01 .70D-01

b) Damped

Dir, DOF umin at t umax at t vmin at t vmax at t 1 .00D+00 .00D+00 .59D+00 .25D+00 -.29D+01 .39D+00 .48D+01 .14D+00 2 .00D+00 .00D+00 .92D+00 .28D+00 -.38D+01 .36D+00 .54D+01 .60D-01 Conclusions: 1. It is noted that the extreme values of the displacements and , respectively , the velocities, are reached at close moments of time in the case of the

undamping vibrations , on the direction of the same degree of dynamic freedom. 2. It is noted that the spectral values of the displacements decrease by aprox. 13% and those of the velocities by aprox.30% in the case of the damping vibrations compared to the undamping vibrations.


22

4. References [1] ANDERSON.E., Z.BAI, C.BISCHOF, S.BLACKFORD, J.DEMMEL, J. DONGARRA, J.DU CROZ, A.GREENBAUM, S.HAMMARLING, A. MCKENNEY and D.SORRENSEN, Lapack User’s Guide, Third Edition, SIAM, Philadelphia, 1999. [2] ATANASIU, G.M., Structural dynamics and stability , Iaşi, 1995 [3] BATHE, K. J. Finite Element Procedures, Prentice Hall., Engl.Chiffs, New Jersey, 1996. [4] GHEORGHE, P Tehnica alegerii pivotului în eliminarea gaussiană, Lucrările sesiunii ştiinţifice a catedrei de matematică, Universitatea Tehnică de Construcţii Bucureşti, 26 mai 2001 [5] GHEORGHE P., IEREMIA M., Numerical computation of the response of a structure in free vibrations with inital data , Constanta Maritime University , MECSOL ,9,2006

[6] GHEORGHE , P. Contribuţii la determinarea numerică a modurilor proprii de vibraţie ale unor structuri inginereşti de mari dimensiuni în analiza dinamică liniară şi neliniară , Teză de doctorat, UTCB, 13.12.2006 [7] LEHOUCH, R., SORRENSEN, D.C., VU, P.A. ARPACK: Fortran subroutines for solving large scale eigenvalue problems, Release 2.1. [8] MOLER, C.B. & STEWART, G.W. An Algorithm for Generalized Matrix Eigenvalue Problems, SIAM J. Numer. Anal. 10, pp.241-256 1973 [9] PAZ, M, , LEIGH,W. Structural Dynamics, Theory and Computation, Kluwer Academic Publishers, Boston-Dordrecht-London, 2004

Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, May. 2007

ISSN 1584 – 5990 © 2000 Ovidius University Press

Rotation Capacity of Reinforced Concrete Elements

Bogdan HEGHEŞ a Cornelia MĂGUREANU a a Technical University Cluj Napoca, Cluj Napoca, 400027, Romania

__________________________________________________________________________________________ Rezumat: Ductilitatea este o proprietate importantă pentru redistribuţia eforturilor şi prevenirea colapsului. Lucrarea prezintă o comparaţie între valorile de calcul ale ductilităţii şi cele obţinute utilizând valorile experimentale ale unor caracteristici de deformare. Elementele experimentale în număr de nouă, sunt grinzi încovoiate simplu armate realizate din betoane de înaltă rezistenţă de clasă C80/90. Ductilitatea a fost exprimată prin rotirea plastică a unui element în momentul formării articulaţiei plastice. Abstract: Ductility is an important property for redistribution of forces and prevention of progressive collapse. The ductility of structural members can be improved by confinement. For high strength concrete this is especially important due increased brittleness. This paper summarizes results from nine reinforced beams of high strength concrete. Ductility was explained by plastic rotation of the element, when in critical sections of the beams plastic hinges appear. Keywords: rotation capacity, high strength concrete, ductility. __________________________________________________________________________________________ 1. Introduction

The paper presents a comparison between the calculus values obtained through several standards codes and experimental values of the authors. 2. Experimental program

The experimental program contained a number of nine simple reinforced concrete beams,

tested at bending. The beams were realized with concrete class of C80/90, with constant length of L=3200mm and the section of 125×250mm. The longitudinal percentage of reinforcement was between 2.033-3.933%, and the transversal reinforcement was the same for all the beams, with stirrups Ø6/300mm.

All the beams were tested with an hydraulic press and loaded with two concentrated loads. (see Fig. 1).

Fig. 1. Schema de încărcare a grinzilor Both ends of the beams were free to rotate

under loading. At each increment of the forces, the strain on multiple heights of the section and the flexure of the beam were recorded. In Table 1 the compressive

Rotation capacity … / Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, 23-28 (2007)

24

strength of the concrete is presented at the date of the testing. Table 1. Compressive strength of the concrete

Beams Compressive strength fc,cube (MPa) FT5.1-1 78 FT5.2-1 91 I1-1, I1-2 92.4 I2-1, I2-2 85.1 I3-1, I3-2 84.9 I4-1 89.9

The longitudinal reinforcement percentage is

presented in Table 2. The longitudinal reinforcement was with steel type PC52 and the transversal reinforcement (stirrups) with steel type OB37. Table 2. Longitudinal reinforcement percentage

Beams p (%) FT5.1-1, FT5.2-1 2.033 I1-1 2.621 I1-2 2.654 I2-1 3.072 I2-2 2.990 I3-1, I3-2 3.357 I4-1 3.933

The differences between the beams of the

same series (i.e. I1-.., I2-.. ) came through the transversal dimensions deviations. Plastic rotation capability

The plastic rotation capability θpl, is defined by the difference between the total rotation θtot and elastic rotation θel: θpl =θtot – θel. (See Fig. 2)

Fig. 2. Total rotation

The following definitions are adopted, which apply universally to reinforced and prestressed concrete members:

• The total rotation θtot is taken as the "sum of angles made by the difference in tensile steel elongation and shortening of outermost compressive concrete fiber, where a section reaches nominal strength".

• The elastic rotation θel is taken as the "sum of angles made by the difference in tensile steel elongation and the shortening of the outermost compressive fibre for which neither the reinforcement nor prestress has reached its elastic limit."

• The plastic rotation θpl is taken as the "sum of additional deformations along the beam after yielding of either the ordinary or prestressed reinforcement and until a section reaches nominal strength" or, as previously shown, as the difference of the total rotation and the elastic rotation.

The plastic theory uses the reserves of plastic hinges of static undetermined structures which are capable of forming plastic hinges in the most stressed areas, and to redistribute the efforts at less stressed areas. This hypothesis presumes that the elements have sufficient plastic deformation capabilities.

To check the deformation capacity the required rotation Θreq has to be compared with the plastic rotation Θpl as follows:

plnec Θ≤Θ (1)

CEB-FIB Model Code 1990 The plastic rotation according to MC90:

( )∫ −⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

−=Θ pl

l

sysyk

srpl dafxd0 2

11 εεσδ (2)

where:

lpl – length of plastic hinge δ – the coefficient which taking into account the form of the stress-strain curve of the reinforcement in the inelastic range (δ ≈ 0,8) x – the depth of the compression zone d – the efficient height of the cross-section σsr1 – the steel stress in the crack the steel stress in the crack when the first crack forms as the characteristic concrete tensile strength is reached

B. Hegheş and C. Măgureanu / Ovidius University Annals Series: Civil Engineering 9, 23-28 (2007) 25 fyk – the characteristic steel yield stress εs2 – the steel strain of the cracked section εsy – the steel yield strain a – the abscissa

In order to facilitate practical applications, the abscissas Θpl and x/d represent the design values of the normalized neutral axis depth (Fig. 3)

Fig. 3. Plastic rotation according CEB-fib MC90 Eurocode 2 Dependent on the ductility class of steel,

normal (N) or high (H), the plastic rotation can be taken from Fig. 4. It can be seen that the plastic rotation Θpl for x/d ≤ 0,16 is limited for H-steel to Θpl = 20 mrad and for N-steel to Θpl = 10 mrad.

20=Θ pl mrad for 16.0/


26

The plastic rotation capacity can be obtained form Fig. 5.

Fig. 5. Plastic rotation according DIN 1045-1

The simplified expression reads:

3/1

, λεε

ββd

xsysu

sncappl−

−=Θ

∗

(6)

where: βn = 22.5 βs = 0.074 λ – shear slenderness; the distance between

M=0 şi Mmax after redistribution ε*su – steel strain at ultimate:

- steel failure:

( ) cukdx βε /13.04.0 +⋅

- concrete failure:

( ) ( )( )ε1//1/8.1 7.0 −dxdx

εsy – characteristic steel yield strain (=0.0025) εuk – characteristic steel strain at ultimate load

(= 0.05 – for high ductility steel) εcu – characteristic concrete strain at ultimate

load (=0.035 – for

B. Hegheş and C. Măgureanu / Ovidius University Annals Series: Civil Engineering 9, 23-28 (2007) 27

Fig. 6. Model for calculating the plastic rotation due to bending and model for a plastic hinge

3. Results and interpretation The experimental program consisted in testing

at bending of nine reinforced concrete beams.

In Table 3 the experimental data on the beams is

shown.

Table 3. Experimental results

p% kcr ky ku x/d Θpl_nec (mrad)

Θpl EC2 (mrad)

Θpl DIN (mrad)

Θpl MC90

(mrad)

FT 5.1-1 2.033 0.00394 0.00394 0.03314 0.1355 2.5167 20.0000 15.7800 15.8178 FT 5.2-1 2.033 0.00120 0.00438 0.02867 0.1162 3.3649 20.0000 15.7800 14.5266 I 1-1 2.621 0.00124 0.00528 0.03137 0.1475 2.1653 20.0000 15.7800 16.4929 I 1-2 2.654 0.00046 0.00455 0.03036 0.1493 1.9335 20.0000 15.7800 16.5665 I 2-1 3.072 0.00078 0.00590 0.04177 0.1877 1.8745 18.2633 16.2220 16.6596 I 2-2 2.990 0.00073 0.00555 0.03854 0.1827 1.2982 18.5805 16.5040 16.4065 I 3-1 3.357 0.00088 0.00496 0.04379 0.2057 2.2194 17.1449 15.2290 15.2868 I 3-2 3.357 0.00088 0.00496 0.03970 0.2057 2.2198 17.1449 15.2290 15.2868 I 4-1 3.933 0.00076 0.00577 0.04767 0.2274 3.2794 15.8541 14.0825 14.4795


28

4. Conclusions The comparison between the experimental

values and theoretical values is shown in Table 3. The data obtained from this experiment and the results of from the other authors leads us to the conclusion that actual standard codes are much to permissive regarding the plastic rotation.

The lack of an integrated and consistent concept for the development of non-linear calculation prevents a simplified calculation model for all kinds of concrete.

The number of experimental results is rather insufficient to compare the described models with the real structure behavior.

The future studies in our Reinforced and Prestressed Concrete Departement will be axed on a comparison of the same beams realized with steel type S500, others reinforcement percentage and beams with multiple openings.

5. Bibliography

[1] Magureanu Cornelia, Hegheş B, Experimental Study on Ductility Reinforced Concrete Beams Using High Strength Concrete, fib Congress, 2006, Napoli [2] Mark Rebentrost, Deformation Capacity and Moment Redistribution of Partially Prestressed Concrete Beams, PH.D. Thesys, 2003 [3] Carsten Ahner, Jochen Kliver, Development of a New Concept for the Rotation Capacity in DIN 1045, Part 1, Lacer 1998, pp 213-236 [4] DIN 1045-1, Tragwerke aus Beton, Stahlbeton und Spannbeton, Teil 1 [5] Eurocode 2, Design of reinforced and prestressed concrete structures.



Assessment of the Potential for Progressive Collapse in RC Frames

Adrian IOANI a Liviu CUCU a Călin MIRCEA a a Technical University Cluj Napoca, Cluj Napoca, 400020, Romania

__________________________________________________________________________________________ Rezumat: În lucrare sunt discutate preocupările actuale ale inginerilor structurişti pentru evitarea şi, în special, pentru reducerea riscului de cedare progresivă a structurilor supuse la sarcini catastrofice (anormale sau neobişnuite). Ca şi în proiectarea antiseismică, se urmăreşte ca structurile de beton armat să aibă un nivel adecvat de continuitate structurală, redundanţă, robusteţe şi ductilitate, astfel încât în condiţiile „pierderii” (cedării) unui element structural, să existe alte căi de transfer ale solicitării. Este prezentată metodologia de evaluare a riscului de cedare progresivă a unei structuri de beton armat dezvoltată de U.S. GSA (2003) şi rezultate care confirmă capacitatea intrinsecă a unei structuri proiectate antiseismic de a rezista la fenomenul de cedare progresivă. Abstract: In the paper, the concerns of structural engineers to avoid, and especially to mitigate the potential for progressive collapse of structures subjected to abnormal loads is discussed. As in the seismic design, reinforced concrete structures should be provided with an adequate level of structural continuity, redundancy, robustness and ductility, so that alternative load transfer paths can develop, following the loss of an individual member. The methodology developed by U.S. GSA (2003) for assessing the vulnerability of existing RC framed structures, as well as results that confirm the inherent capacity of such structures, seismically designed, to resist progressive collapse are presented. Keywords: RC frames, abnormal loads, progressive collapse, seismic design, assesememt of vulnerability. __________________________________________________________________________________________ 1. Introduction

Many structural collapses of important buildings concived in various structural solutions, tionalities and height regimes were registered in the last fifty years. Some of them had a local character, while other spread progressive to the scale of the full structure or large parts of it.

The main causes leading to a structural progressive collapse of buildings, seen as a chain reaction of failures that propagates throughout a portion of structure, disproportionate to the original local failure[1], are: fire, wind gusts, floods and human errors, impact by vehicles, but especially major earthquakes and blasts[2].

The concerns of the structural engineers to prevent or to mitigate the potential for progressive collapse have to be seen in correlation with the structural effects of the abnormal loads [1].

Considering the definition given in Section 2 of the GSA Guidelines [3], abnormal loads are: “other than conventional design loads (dead, live, wind,

seismic) for structure, such air blast pressures generated by an explosion or an impact by vehicles, etc.” The design philosophy of structures subjected abnormal loads is to prevent or mitigate damage, not necessarily to prevent the collapse initiation from a specific cause. This approach is similar to the concept adopted in any modern earthquake-resistant design codes.

If the progressive collapse prevention is associated to certain structural characteristics as an adequate level of continuity, redundancy and ductility so that alternative load transfer paths can develop following the loss of an individual member or a local failure, then it is obvious that these requirements are found in seismic design as well.

Structural response to blast, for instance, is related to the large variety of possible scenarios regarding the location of the detonation point, charge and design details. Due to the high intensity of the air pressure exerted in fractions of second, the localized failure of the direct exposed members (i.e., columns, walls, girders, floor systems) is most like to occur.

Assessment of … / Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, 29-36 (2007)

30

For this reason, in the assessment methodology for the potential progressive collapse, engineers should consider the loss of portions of the structure using different “missing column” or “missing beam” scenarios. Such checks are required, though the cause is not always specified (natural hazard or man-made hazard), in the currently used design codes for the reinforced concrete structures.

Thus, in the beam design, Section 15.4.2.1.5 of EC2 [4] requires that “reinforcement used should be continuous and able to resist accidental positive moments (settlement of the support, explosion etc.)”. The most recent Romanian Seismic Design Code P 100-1/2004 - Art. 4.1.1.2 [5], explicitly demands that “seismic design should provide the building structure with an adequate redundancy. In this manner, it is ensured that the failure of one single element or the failure of a structural link does not expose the structure to the loss of stability”.

Consequently, it seems natural at present for the engineers to use their creativity to find cost-effective solutions that make structures more resilient to both natural hazard (e.g. earthquakes) and man-made hazards (e.g. bomb blast, impact by vehicles) and, in consequence, the designed structural system will satisfy, at the same time, the requirements of lateral-load resistance and those of the prevention of the progressive collapse.

The study presents the methodology developed by U.S. GSA [3] for assessing the vulnerability of existing RC framed structures, as well as, results that confirm the inherent capacity of such structures seismically designed, to resist progressive collapse.

2. Progressive collapse of RC frames

It is known that the structures provided with

interior core structural walls for lateral–load resistance and ordinary moment frames (frames designed for gravity-loads) or flat-plate structures with interior core walls and, in general, ordinary moment frame structures have a limited capacity to redistribute loads and prevent progressive collapse.

Such a situation represents a consequence of the fact that gravity-load designed systems are not adequately reinforced and detailed to develop alternative load paths when a vertical support is removed due to a blast or an impact [1].

Fig. 1. Possible blast behavior of frame structures: a) earthquake resistant design b) gravity-load design.

As presented in Fig.1b, after the removal of an exterior column (“missing column” scenario) by the blast effect, the lack of bottom continuous reinforcement generates the flexural failure of the newly resulting two-bay frame beam. The potential of brittle failure by shear could also be induced by the lack of closely spaced stirrups at the ends of the frame beam (Fig. 1b).

This was the case of the Murrah Building, a nine story building with Ordinary Moment Frames designed for gravity loads [6]. When the blast effect abruptly removed column G20 by brisance, the transfer girder, which lost its support was unable to support the structure above the third floor. The type of damage that occurred and the resulting collapse of nearly half of the building indicate that progressive collapse extended the damage beyond that caused directly by the blast effect (Fig. 2) [6].

Fig. 2. Failure boundaries of floor slabs in Murrah Buildings [6].

A. Ioani et. al. / Ovidius University Annals Series: Civil Engineering 9, 29-36 (2007) 31

The report [6] underlined that the use of Special Moment Frames (SMF) rather than Ordinary Moment Frames would not completely eliminate the loss of some portions of the building, but the losses would be greatly reduced.

For this reason, an Executive Order of GSA (EO 12699/1990) has required that “new Federal buildings be designed to meet seismic requirements” and consequently, “these new buildings in areas of high seismicity may already provide suitable ductility for blast resistance at no additional cost by satisfying seismic design requirements” [6]. 3. Assesement of the potential for progressive collapse 3.1 Progressive collapse analysis

The U.S. General Services Administration (GSA) has developed the “Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects” – June 2003 [3] to ensure that the potential for progressive collapse is addressed in the design and construction of new buildings and major renovation projects. These guidelines provide a detailed methodology for minimizing the potential for progressive collapse in the design of new buildings and for assessing the vulnerability of existing buildings to progressive collapse.

Using a flow-chart methodology the Guidelines determine whether the building under consideration might be exempt from detailed analysis for progressive collapse. For example, a structure which:

- does not contain single point failure mechanism,

- does not possess atypical structural conditions, - is not over ten stories, - has public areas or parking areas controlled with

proper security systems, is designed consistently with at least Seismic Zone 3 or Seismic Design Category D or E requirements (see UBC – 1997 and IBC- 2000), is a candidate for automatic exemption from the consideration of progressive collapse.

If the existing construction is determined not to be exempt from further consideration for progressive collapse, the methodology presented in Section 4.2 or 5.2 [3] is applicable and shall be

executed, and the potential for progressive collapse determined in this process – whether low or high – must be quantified. 3.2 Missing column scenarios

The typical RC structural configuration, framed structures or flat plate structures, shall be considered individually in the following analysis scenarios:

1. The instantaneous loss of column for one floor above grade (1 story) located at or near the middle of the short side of the building - case C1, at or near the middle of the long side - case C2, and located at the corner of the building - case C3 (Fig. 3).

2. For facilities that have underground parking and/or uncontrolled public ground floor areas, the instantaneous loss of an interior column would also have to be considered.

Fig. 3. Missing column scenarios for exterior columns. 3.3 Loading assumptions

In the static analysis of each case, the vertical load applied downward to the structure is:

)25.0(2 LDLoad += (1) where D is the dead load and L the live load.

In the GSA criteria, live load is reduced to 25 % of the full design live load, admitting that the entire L value is less probable. At the same time, by multiplying the load combination by a factor of two, the Guidelines take into account – in a simplified approach – the dynamic amplification effect that occurs when a vertical support is instantaneously removed from the structure, and demands (QUD) in structural components are determine in terms of moments, axial forces, shear forces, etc.

The magnification effect of a static force when dynamically applied is termed impact factor by Popov


32

(1976) [7] or dynamic coefficient by authors and it is given by the expression:

Ψ⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ++= st

ststdyn P

h211PP (2)

If a load is applied to an elastic system

suddenly (instantaneously) and h = 0, Ψ is equivalent to twice the same load statically (gradually) applied, as in Eq. (2).

At the same time, strength increase factors are applied to the material properties in order to determine the ultimate strength capacity (QCE) of structural members (beams, columns etc.) under dynamic loads. The concrete compressive strength and the tensile or yield strength of the reinforcing steel are increased by a factor of 1.25, Table 4.2 of the GSA Guidelines [3], to account for strain rate effects and material over-strength.

3.4 Acceptance criteria

The GSA Guidelines – Section 4.1.2.4 – consider that local damage may occur and this is acceptable with the instantaneous removal of an exterior column, but the resulting structural collapse shall be limited to a reasonably sized area. In other words, the maximum allowable extents of the collapse shall be confined to whichever is smaller: the structural bays directly associated with the instantaneously removed vertical element or 1800 ft2 (167 m2) at the floor level directly above the removed vertical column.

Working with the results given by the linear elastic analysis (moment, shear, axial force), engineers shall identify the magnitude and distribution of potential areas of inelastic demands and thus, they will quantify the potential collapse areas.

The magnitude and distribution of these demands will be given by the concept of Demand – Capacity – Ratios defined as [3]:

CEUD QQ=DCR (3) where QUD - acting force(demand) in the component or connection (moment, axial force, shear and possible

combined forces) and QCE - expected ultimate un-factored capacity of the component or connection (Φ=1.0).

According to the GSA Guidelines, acceptance criteria, the allowable DCR values for structural elements are: DCR ≤ 2.0 for typical structural configurations and DCR ≤1.5 for atypical structural configurations. Using the DCR concept of linear elastic approach, structural elements that have DCR values exceeding the allowable magnitudes are considered to be severely damaged or collapsed.

It is underlined that if the DCR for any member is exceeded, based upon shear force, the member is to be regarded as a failed member. In addition, if the flexural DCR values for both ends of a member as well as the span itself are exceeded (creating the classical three hinged failure mechanism), the member is also to be seen as a failed member.For continuous elements, the flexural DCR value at an element section may exceed 1.0 because in this case flexural demand can be redistributed along the length of the element to sections that have reserve flexural capacity [1]. 4. Results and commentary 4.1 FEMA 277/1996 Report

In 1995, the Federal Emergency Management Agency (FEMA) deployed a Building Performance Assessment Team (BPAT) to investigate damage caused the terrorist attack against the Alfred P. Murrah Federal Building in Oklahoma City. From visual inspection and analysis of the damage that occurred in the Murrah Building as a result of a blast caused by a truck bomb, it is concluded that progressive collapse extended the damage beyond that caused directly by the blast. The main findings and conclusions are [6]:

- the loss of three columns and portions of some floors by direct effect of the blast accounted for only a small portion of the damage;

- most of the damage was caused by progressive collapse following loss of the columns;

- the nine-story frame type of the building was an Ordinary Moment Frame (OMF), i.e., a frame designed for gravity loads;

- if additional amounts and locations of reinforcing steel as for Special Moment Resisting Frame (SMRF) in seismic areas had been used, the Murrah Building would have had enough strength and ductility that about half of the damage would have been prevented;


- investigations to determine the cost of using SMRF rather than OMF were conducted and suggest that the average increase in cost is in the range of 1 to 2 percent of the total construction cost of the building;

- using reinforcement, connection and other details required by the design of frame structures or dual systems in areas of high seismic activity, will provide similar toughness and ductility in face of the blast;

- the most important aspect of using SMRF or Dual Systems is the ductility detailing (e.g., closed-hoop reinforcement to confine columns, continuous bars for continuity, beam-to-column connections to transfer forces through the joints, etc.);

- in areas of low seismic risk, incorporating the seismic details required for regions of high seismic risk can significantly improve the blast protection of the buildings.

4.2 Other studies

In a study upon redundancy and robustness of RC structures subjected to blast and earthquakes Mircea (2006)[2] makes a critical review of common structural types and shows that spatial frame structures have more redundancy potential than plane frame structures because more possibilities for load transfer are provided.

In general, frame structures and flat slab structures need supplementary lateral stiffness, usually provided by shear walls or vertical bracings. Even if walls are rigid and possess large masses, Crawford at all., cited in [2], reported significantly more column damage in blast tests on structure with columns and infill walls, in comparison with tests on structures without walls.

Baldrige and Humay (2003) conducted a progressive collapse analysis on a 12-story RC frame structure having five longitudinal bays of 7.3 m and three transversal bays of 7.3 m [1]. The model was designed to the older requirements of the Uniform Building Code (UBC-1991 edition). The required strength U to resist to a combination of dead load (D=2.0 kPa), live load (L= 2.4 kPa) and earthquake effect (E), considers [8]- for UBC Seismic Zone 2B (a moderate seismic zone)- a total equivalent seismic force having the magnitude of

GGFF tt 0732.00523.04.14.1* =⋅== .

The computer program ETABS was used to generate a 3-D model; case 1 investigated the structural effect of removal of an exterior column along the long side of the building, and case 2 examined the removal of a corner column, also at the ground floor. The removal of the column at the middle of the long side doubles the beam span at the first floor and the vertical forces of the magnitude 2(D+0.25L) generate a maximum positive moment in beam, over the removed column.

Following the GSA Guidelines, demands in structural components are assessed in beams at mid-span section and at column faces and the afferent DCR values are computed. All of the DCR values are below 1.0, except at the mid-span of the beam over the removed column (case 1) where a value of 1.02 for flexure was reported; the maximum DCR value in beams, for shear, was only 0.69.

Practically, the damaged structure remains in the elastic stage, no other structural member is expected to fail in shear or flexure and consequently, progressive collapse is not expected to occur [1].

Obviously, the American model [1], is seismically designed under similar or comparable gravity and seismic forces as a typical RC frame structure from Bucharest (Romania), where currently the total equivalent seismic force for such structure is F≈ 0.08G. The study [1] shows that RC frames designed for a moderate or high seismic intensity zone do not experience progressive collapse when are subjected to the removal of an external column. 4.3 Authors’ studies

In order to determine the inherent reserve capacity to progressive collapse of a RC structure erected in a high seismic zone of Romania, an investigation was conducted on a 13-story RC frame building designed according to the older requirements of the Romanian Seismic Design Code P100–92 [9]. One expects that for new buildings designed according to the provisions of the new P 100-1/2004, this analysis will be conservative.

The structure consists of five 6.0 m bays in the longitudinal direction and two 6.0 m bays in the transversal direction and has a story height of 2.75 m, except for the first two floors that are 3.60 m high [10]. In the design at the Ultimate Limit State, the Special Combination of loads according to the Romanian Standard STAS 10107/0A-77 (1977) is:


34

D+0.4L+E (4) meaning, for the design of the analyzed building, a combination of dead load D=2 kPa, live load L=2.4 kPa with a load factor of 0.4, and the earthquake effect (E).

The seismic analysis is performed for Bucharest, a seismic area of degree VIII on the MSK Intensity Scale (zone C on the Romanian zonation map with ks=PGA/g=0.2). For Romania, the seismic coefficient ks varies from 0.08 to a maximum value of 0.32. The magnitude of total equivalent seismic force S that enters the load combination given by Eq.(4) is calculated as follows [9]:

( ) GGTkS s 0945.0=⋅⋅⋅⋅⋅= εψβα (5)

Fig. 4. ROBOT Millennium model of a 13-story RC building: missing column scenarios.

The structural response of the model under the Special Combination of loads, and the behavior of the damaged structure (case C1, C2, C3 of the “missing column” scenarios) is determined with the 3-D linear elastic model, created and analyzed in the FEA program ROBOT Millennium, Version 19.0 (2006). The model generated by this computer program is shown in Fig. 4.

The material properties are given in Table 1. In the seismic design of the model, design values for strengths have been used.

In the progressive collapse analysis according to the GSA Guidelines provisions, the expected ultimate, un-factored, capacity of the structural elements was determined with the help of the characteristic (un-factored) values for the strengths, multiplied by the strength increase factor of 1.25 (Table 1).

Table 1. Strengths of materials for the model (MPa).

Seismic design

Progressive collapse analysis

Material Design values *

Characteristic un-factored

values

With 1.25

factor 5.12=cR 6.16)( ' =cck fR 20.75 Concrete

Bc20 95.0=tR 43.1=tkR 1.78 PC52 300=aR 345)( =yak fR 431 SteelOB37 210=aR 255)( =yak fR 318

• Rc (Rt) = design value of the compressive (tensile) strength of concrete;

• Ra = design value of the yield strength of reinforcement.

The removal of the column (Fig. 4) at the middle

of the short side – case C1 – doubles the beam span at the first floor and the vertical forces of the magnitude 2(D+ 0.25L)- see Eq.(1)- generate a maximum positive moment over the removed column, of 537.1 kNm (Fig. 5). If the bottom reinforcement in the beam is not continuous through the column joint as in the gravity-load designed frames, the positive moment capacity is limited to the cracking strength of the section and the failure in this case will be abrupt, leading to a brittle collapse (Fig. 1b).

In contrast, seismically designed frames used in the analyzed model having a large amount of bottom longitudinal reinforcement (As=9.64 cm2) in the beam, that means a reinforcement ratio ρ=0.0084, provides a positive flexural capacity over the “missing column”, and consequently, the beam has enough ductility to develop alternate load paths (Fig 1a).

The new bending moment and shear force diagrams generated in the damaged structure by the removal of the column (case C1) are shown, for the exterior transversal frame, in Fig. 5 and Fig. 6, respectively.

Following the GSA Guidelines (2003), demands in structural components (QUD) –see Eq. 3, assessed in


beams at mid-span and at column faces (Fig. 5), are compared to the expected ultimate beam capacities (QCE) from Eq. 3.

Following the procedure presented above, DCR values for significant beam sections are represented in Fig. 5 and Fig. 6, in brackets.

All of the DCR values for flexure are below 1.0, except at the mid-span of the beam over the removed column (Fig.5).

Fig. 5. Damaged structure (case C1): bending moments and DCR values ( ) in beams.

Because this value is only 1.015 and satisfies

the GSA Guidelines criteria (DCR≤2), the beam will have adequate reserve ductility for an efficient redistribution of loads and consequently, the flexural demand that exceeds 1.0 is redistributed to sections that have reserve flexural capacity .

Fig. 6. Damaged structure (case C1): shear forces and DCR values ( ) in beams.

The DCR values based upon shear (Fig. 6) are

well below 1.0, the maximum value being 0.67. The author’s results are similar with the results of

Baldrige & Humay (2003) [1] who reported a maximum DCR value for flexure of 1.02 and for shear of 0.69.

Even the differences between the computed deflections are small, being of only 26% [10], if one takes into consideration the differences between the models regarding the span length (6.0 m vs. 7.30 m) and beam dimensions (35×70 cm vs. 55.6×45.7 cm for the American model [1] ).

5. Conclusions

This study is in line with the trends of the specialized reference literature that aims at assessing the vulnerability of the existing structures subjected to abnormal or catastrophic loads produced by natural hazard (e.g. earthquakes) or by man-made hazards (terrorist attacks, impact by vehicles, bomb blast, etc).

The following conclusions can be reached based on this study:

1. Practically, due to economic constraints, it is

impossible to design an overall structure and each structural member individually so as to resist to abnormal loads or to prevent collapse initiation from a specific cause. More important is to stop or to limit the progression of the collapse and to reduce the extent of the damage and this should be the design philosophy assumed by engineers.

2. Many design codes (ACI 318, EC-2, P100-92, P 100-1/2004) require an adequate level of continuity, redundancy and ductility for the selected structural system. Interagency Security Committee (ISC) Security Criteria clearly requires all new constructed facilities to be designed with the intent of reducing the potential for progressive collapse, and the existing facilities to be evaluated to determine the potential for progressive collapse [3].

3. The GSA Guidelines [3] offer a realistic approach and performance criteria for these determinations.

4. The concept of DCR offers to engineers a valuable tool to identify the magnitude and distribution of potential areas of inelastic demands and thus, the extension of potential collapse zone can be evaluated and compared to the maximum allowable collapse area


36

resulting from the instantaneous removal of an exterior or interior column.

5. A typical medium-rise building (13 stories) having RC frames, seismically designed for the Bucharest – a zone of high seismic risk- does not experience progressive collapse [10] when subjected to different “missing column” scenarios, according to GSA Guidelines (2003). Similar results have been found by Baldrige & Humay (2003) [1] for a 12-story RC framed structure seismically designed for a moderate (Zone 2B) or a high seismic risk zone (Zone 4), according to the requirements of a Uniform Building Code (UBC-1991 edition).

6. For the Romanian zones of high seismic risks as the zone C (ks=0.20), zone B (ks=0.25) and zone A (ks=0.32) [9], further analyses will be developed by authors in order to determine the vulnerability to progressive collapse of other types of structural systems, including existing reinforced concrete frame structures of 7 to 9 story high, designed to the older requirements of Romanian Design Code P100-92 [9].

6. Acknowledgements

Part of this work is based on different

technical materials (books, reports, design codes) provided with generosity by the U.S. Federal Emergency Management Agency. The support of this organization is gratefully acknowledged.

7. References [1] Baldrige, S. M., and Humay, F. K., Preventing Progressive Collapse in Concrete Buildings, Concrete International, vol. 25, No. 11, Nov. 2005, pp. 73-79.

[2] Mircea, C., Risk factors in the redundancy and robustness of RC structures subjected to blast and earthquakes, ”Concrete Solutions”- Proceedings of The Second International Conference on Concrete Repair, St. Malo, June 2006, pp.782-792. [3]. U.S. General Services Administration (GSA), Progressive Collapse Analysis and Design Guidelines for New Federal Office Buildings and Major Modernization Projects, June 2003, 119 pp. [4]. European Committee of Standardization, EUROCODE 2: Design of Concrete Structures, Brussels, 1997, 160 pp.

1. [5]. Ministry of Public Works, P 100-1/2004, Seismic Design Code for Buildings (in Romanian), Bucharest, 2005, 410 pp. [6]. FEMA-277, The Oklahoma City Bombing: Improving Building Performance Through Multi-Hazard Mitigation, Federal Emergency Management Agency, Aug. 1996, 98 pp. [7]. Popov, E. E., Mechanics of Materials - second edition; Prentice/Hall International, Inc., London, 1976, 590 pp. [8]. ACI Committee 318, Building Code Requirements for Structural Concrete (ACI 318M-99) and Commentary (ACI 318RM-99), American Concrete Institute, Farmington Hills, Mich., 1999, 319 pp.

2. [9]. Ministry of Public Works, P100-92, Seismic Design Code for Buildings (in Romanian), Bucharest, 1992, 152 pp.

3. [10]. Ioani, A., Cucu, L.,and Mircea, C., Seismic design vs. progressive collapse :a reinforced concrete framed structure case study, Proceedings of the International Conference ISEC-4, Melbourne, Sept. 26-28, 2007 (in press).

Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, May. 2007


Cracking of Reinforced Concrete Elements

Laura- Catinca LEŢIA a a Technical University Cluj Napoca, Cluj Napoca, 400020, Romania

__________________________________________________________________________________________ Rezumat: Sunt prezentate relaţiile de calcul prevăzute în diferite norme, privind calculul deschiderii şi distanţei între fisuri, cu referire la betonul armat, realizat cu beton de înaltă rezistenţă. Se fac comparaţii cu valorile experimentale obţinute pe elemente de beton armat având ca variabilă procentul de armare. Abstract: There are presented the evaluation formulae from different norms, regarding the estimation of crack opening and spacing, which is referred to the reinforced concrete with high strength concrete. It is compared with the experimental results obtained on reinforced elements having as variable the reinforcement amount. Keywords: crack opening, crack spacing, high strength concrete. __________________________________________________________________________________________ 1. Foreword

Cracking of reinforced concrete elements is a

complex phenomenon; the causes of crack formation are different.

Assuming that under the loads action (such as tension, compression, torsion, bending, and share force) the crack formation in practically inevitable, the present norms are seeking mostly to confine this phenomenon to some values that are not affecting the behavior of the element or of the structure in service in a significant manner. The crack opening limitation has to take in account the cost of it, related to the concrete strength and the reinforcement yield point.

The studies and researches on high strength concrete (HSC) elements made until nowadays are showing that the north American and European norms, that allows us to estimate the crack opening and distance, are consistent for regular concrete, but they can not offer a correct image on the crack behavior of HSC elements. The cracking behavior for HSC is mostly influenced by the tension and compression strength, contraction and bond, etc.

2. The experimental program

The experimental program is based on

bending tests of eleven beams (three beams FT5 and two for each I beam). All beams had a width of

125 mm, height of 250 mm and span length L0=3000 mm.

The concrete strength at the day of testing is about 90 N/mm2 (C80/90).

The physical and mechanical characteristics of reinforcement are listed in Table 1.

Table 1. Mechanical characteristics of reinforcement

Reinforcement type

Longitudinal reinforcement

Transversal reinforcement

Nominal diameter (mm)

12, 14, 16 6

Yield point, fym (MPa)

320 210

The longitudinal reinforcement is made from

PC52 steel PC52. The transversal reinforcement (the stirrups) is made from OB37.

The longitudinal and transversal reinforcement coefficients are listed in Table 2. Table 2 The reinforcement coefficients

FT I The element 5 1 2 3 4

ρl=Asl/bw·d (%)

2.06 2.59 3.03 3.40 3.83

ρw=Asw/bw·s (%)

0.152

Cracking of Reinforced / Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, 37-44 (2007)

38

The loaing scheme is presented in Figure 1.

Fig. 1. The loading scheme All the beams were tested using hydraulic

testing system and loaded with the two equal concentrated loads, F. The distance between the two concentrated loads was kept equal to L1=1000 mm.

Both ends of the beam were free to rotate and translate under load. At each load increment, the mid-span deflection and all strain reading were recorded and the developing crack patterns marked an the beam surface. The concrete maximum compressive strain is recorded at the mid-span by strain gauges glued are placed on one of the beam side in the tensile zone. Beams are submitted to a growing monotonic loading until failure. The monotonic loading is applied 1/10 of the failure calculated force.

3. Formulas to estimate the crack opening and spacing

Sarkar, Adwan, Munday[2] researches show that at the reinforcement level the crack opening and spacing depend on the concrete strength and on reinforcement coefficient.

Generally, the first crack appear immediately after overtaking the I stage, when the cracks became visible, and their opening depend on the loads nature and intensity[9]. The estimation is compute in the IInd stage of service, because under this (service) loads action the elements are working in the cracked II stage [10].

The design of reinforced concrete elements is assuming that the concrete between two cracks is capable to undertake strain and it is uncracked [10].

The European norms (EC2, CEB- FIP) and the north American norms (ACI 318), but the Australian (AS 3600-Part2) and the Romanian norms (STAS10107/0-90) too recommend different relations to evaluate the crack behavior. The Romanian (STAS 100107/0-90) and rhe European (EC2) norms establish constructive conditions for crack control, such as the limitation of reinforcement bars spacing and/or diameter.

Fig. 2 Crack formation under load

L. C. Leţia / Ovidius University Annals Series: Civil Engineering 9, 37- 44 (2007) 39 EUROCODE 2 [3]

The crack control is analyzed in the 7.3 section of EC2, considering the Serviceability Limit States (SLS).

Taking in account the Exposure Class and type of element (reinforced concrete and prestressed elements with bonded or unbounded tendons) are established the maximum crack opening. The crack formation of reinforced elements subject to bending or shear is considered as being normal, but it cannot affect the normal behavior of the element or structure, or to create any discomfort.

The European norms establish a minimum reinforcement areas for crack control, denoted with As and witch depends on the mean value of the tension strength fct,eff= fctm [or it may be less, meaning fctm(t), if t where fcm - the mean concrete compressive strength at 28 days. fck - the characteristic compressive cylinder strength of concrete at 28 days.

For the crack control without special calculation,

EC2 establishes the maximum diameters and spacing of reinforcement.

The crack spacing is calculated using the equation:

( )cmsmmax,rk Sw ε−ε= ; (3)

where: cmsm ε−ε the difference between the mean reinforcement strain under the relevant combination of loads and the mean concrete strain between cracks, it may be computed using the formulas:

( )ss

s

eff,peeff,p

eff,ctts

cmsm E6,0

E

1f

kσ

≥

ρ⋅α+ρ

−σ

=ε−ε (4)

where: sσ - the stress in the tension reinforcement assuming a cracked section.

eff,c

p21s

eff,p A

'AA ξ+=ρ (5)

where: 1ξ - for the pretension elements, it represents the ratio the bond strength taking in account the different diameters of prestressed and reinforcing steel. Ap’ – aria of pre or post- tensioned As - the reinforcing area within the tension zone Ac,eff - the effective area of concrete in tension surrounding the reinforcement kt - factor depending on the duration of the load, taking the value 0,6 for short term loading, and 0,4 for long term loading.

eα - the ratio Es/Ecm (equivalence coefficient) Schematic the concrete area that is surrounding

the reinforcement area within the tensile zone may be represented as it follows:

Cracking of Reinforced / Ovidius University Annals Series: Civil Engineering Volume 1, Number 9, 37-44 (2007)

40

x

hct,eff

B εt

εt=0

A

dh

Fig. 3. Effective tension area

To determine the maximum spacing between cracks, Sr,max , if the bar spacing is less or equal to

)2/c(5 φ+ , it may be used the following expression:

eff,p4213max,r kkkckS ρ

φ+= (6)

where: φ - the bars diameter, that may be

considered an equivalent diameter eqφ , when the bars have a different diameters.

c – the cover to the longitudinal reinforcement k1 – the coefficient that takes account of the

bond properties of the bonded reinforcement =0,8 high bond bars =1,6 for bars with an effectively plain surface k2 – the coefficient that takes

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