calculation of additional dynamic stresses in rebars …

Scientific Herald of the Voronezh State University of Architecture and Civil Engineering. Construction and Architecture

6

BUILDING STRUCTURES, BUILDINGS AND CONSTRUCTIONS

UDC 69.07

V. I. Kolhunov1, N. B. Androsova

2

CALCULATION OF ADDITIONAL DYNAMIC STRESSES IN REBARS

OF FLEXURAL REINFORCED CONCRETE STRUCTURAL ELEMENTS CAUSED

BY FRAGILE FAILURE OF TENSED CONCRETE

Southwest State University

Russia, Kursk, tel.: +7 (4712) 50-48-20, e-mail: [email protected]

1D.Sc. in Engineering, Prof., Member of the Russian Academy of Architecture and Construction Science,

Head of the Dept. of Unique Buildings and Structures

Orel State University

Russia, Orel, tel.: +7 (4862) 751-318

2PhD in Engineering, Assoc. Prof. of the Dept. of Building Constructions and Materials

Statement of the problem. The additional dynamic stresses are the result of a fragile failure of tensioned concrete

and the crack formation. The determination of the additional dynamic stresses is performed without involving

dynamic methods on the basis of the energy method taking into account the total energy constancy in an element

under load. In this case the method takes into account the hypothesis that in the first half-wave of oscillations of

tensioned rebars the increment of the stresses reaches its maximum value. The analytical models of

V. M. Bondarenko and Vl. I. Kolchunov have been used for the calculation of a reinforced concrete element under a

load at the moment of crack formation. On this basis, the authors have obtained an analytical expression for the

determination of the increment of additional dynamic stresses in a prestressed reinforced concrete element which can

be used in the assessment of survivability of reinforced concrete structural systems under a beyond limit states.

Results. The paper presents the method of calculation of additional dynamic stresses in rebars of a reinforced

concrete element under bending at the moment of a fragile failure of tensioned concrete during the crack formation.

Conclusions. The paper concerns the calculation of additional dynamic stresses in rebars of reinforced concrete

structures under bending at the moment of crack formation.

Keywords: structural safety, criterion of survivability, durability, extremal condition, additional dynamic forces.

Introduction

According to the current regulations in Russia and some European countries, the accidental

limit state design considering the failure of one of the element in a structural system should be

performed for prestressed reinforced concrete structural elements of buildings and structures of

higher level of importance. The abrupt failure of an element in a structural system causes the

© Kolhunov V. I., Androsova N. B., 2016

tel:+7(4862)75-13-18

Issue № 4(32), 2016 ISSN 2075-0811

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additional dynamic stresses and, in case of reinforced concrete structures, cracks in the

remained structural system. Moreover cracks also lead to additional stresses in rebars.

Suchactions should be especially considered for prestressed reinforced concrete structures in

which rebars have a high level of stresses yet before additional dynamic internal loading.

For the design of prestressed reinforced concrete structures the Russian regulations [1] and

Eurocode [2] establish the values of prestressing σsp (σp,max) as fractions of characteristic

strength of rebars. To enhance the effect of the prestressing the value σsp is usually taken as

close to the upper limit of a characteristic strength value of reinforcing steel. In European

countries the maximum stress applied to the tendon is determined as:

,max 1 2 0,1min ;p pk p kk f k f , (1)

where fpk is a characteristic tensile strength of prestressing steel; fp0,1k is a characteristic

0,1% proof-stress of prestressing steel; k1 = 0,8 and k2 = 0,9. The only limitation is the

satisfaction of the conditions of crack resistance of concrete in tension. At the same time,

for the structures of two-component materials, such as reinforced concrete, there is another

important limitation which establishes the upper limit of the prestressing values. The

condition is associated with a dynamic effect in a structural system of two-component

materials and with the consequent redistribution of stresses at abrupt failure of a fragile

component such as concrete in tension [3].

Literature review. In the last two or three decades, a number of studies has been performed

concerning not only structural safety problems but also the problem of structural survivability.

One of the first studies in the field of structural survivability of buildings and structures in

Russia was the research performed in the 1990s by scientists from the Russian Academy of

Architecture and Construction Sciences such as G. A. Geniev, [3, 4], V. I. Travush [5],

V. M. Bondarenko, V. I. Kolchunov [6, 7], N. I. Karpenko [8], and others. The issues related

to the additional dynamic stresses in the reinforcement of concrete elements at the sudden

brittle failure of the concrete are still fragmented [9, 10].

The assessment of dynamic stresses increments in a prestressed reinforced concrete

element in tension. For a more visual representation of the physical model of the static-

dynamic deformation of a two-component material, such as reinforced concrete element let us

use a very simple model for calculating a the load Ncrc causing crack formation. The crack

formation in reinforced concrete (two-component) element in tension, providing a brittle

fracture of concrete, causes the additional dynamic internal forces in rebars (see Fig. 1). Before


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crack formation at load Ntot = λcrcN total static force Ntot (resultant of imposed load and bond

stresses) in reinforced concrete element in tension is taken by concrete Ncb and rebars N

cs [9]:

sbcrc NNNN . (2)

At load higher than load causing crack formation Ncrc = λcrcN concrete in tension fails imme-

diately and tensile stresses of concrete transfer to rebars abruptly causing additional dynamic

internal force increasing up to Nd

sp from Nсsp [3, 9]. At the moment before crack formation in

tension area there are tensile forces in concrete Nb = Rbt * Abt and in rebars NS. When

Nb > Rbt * Abt concrete in tension fails immediately and tensile stresses of concrete transfer to

rebars abruptly causing additional dynamic internal force in rebars and longitudinal oscilla-

tion of rebars (see Fig. 1, b, c). The value of the additional dynamic internal force can be de-

termined on the basis of the energy method using the diagram σS-εS [9].

Fig. 1. Diagram of the stress state of concrete and reinforcement of an element in tension (a), a diagram

of the longitudinal internal force in rebars (b) and the diagram σS-εS for determining increments of dynamic

stresses in rebars at the moment of crack formation (c)

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Since at the moment of the brittle fracture of concrete the tensile stresses of concrete transfer

to rebars abruptly, the rebars are under longitudinal oscillation and internal dynamic force Nd

s

can be calculated as follows:

bsds NNN 2 , (3)

Ns = Ns(P) + Nsp, Ns(P) is a tensile force in rebars caused by imposed load P, Nsp is a pre-

stressing force.

Calculation model of static-dynamic deformation of a reinforced concrete element in

bending at the moment of the fragile failure of concrete. After the analysis of the static-

dynamic deformation of axially tensioned two-component reinforced concrete element at frag-

ile failure of concrete we move on to the analysis of reinforced concrete element in bending

(see Fig. 2, a, b). The physical models of reinforced concrete proposed by V. I. Bondarenko and

Vl. I. Kolchunov [11, 12] are taken as basic, and additional dynamic forces in rebars are pro-

posed to be calculated on the basis of energy method. Using the constants of fracture mechanics

and structural mechanics, the model proposed by V. I. Bondarenko and Vl. I. Kolchunov de-

scribes the stress-stain state of concrete at crack area more efficiently [13, 14].

The point of the method is that the analytical model includes two-console elements [11] that

provide the connection between potential energy of deformed reinforced concrete element and

compliance coefficient ξ of crack borders during crack formation and connection between the

compliance coefficient and traditional parameters of concrete E, G, ε under deformation.

The two-console element helps take into account the connection between stress-strain state of

concrete at prefracture zone and some constant ζbu depending on concrete parameters. Here-

with, the compliance of crack borders, which may be used for the evaluation of ζbu value, can

be defined by means of general methods of structural mechanics. Therefore, the two-console

element is used as an connection link between relations of solid mechanics and fracture me-

chanics. The compliance of a two-component element used for crack modeling is connected

with movements of a whole reinforced concrete element. Therefore, there is a methodological

connection between calculations performed in ULS and SLS design (between first and second

groups of limit states as adopted in the Russian regulations).

In general form the function of the compliance can be obtained from the determination of the

potential energy release rate:

dA

dV

dA

dW=

δA

δVδW

Abu

0lim

, (4)


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δV –– reduction of the potential energy of the body at crack extension increment δA; δW ––

the additional work done on the body at crack extension increment δA; A –– area of the result-

ing surface of a crack.

On the basis of the considerations set out above, the two-console model can be presented as

shown in the scheme in Fig. 2 c, d. The parameter tb (which stands for dimension of com-

pressed concrete near a crack), according to Saint Venant principle and experimental study of

a zone near rebars involving semi-analytical and analytical methods [13, 15], in first approx-

imation can be taken as one and a half diameter of rebar. Further, the tb value is taken from

refined solution of the problem of concrete-reinforcement bond. Tensile stresses in the regard-

ing sections are assumed to be distributed according to a quadratic parabola from the neutral

axis to the point where the stresses change sign. Herewith, the maximum values of tensile

stresses is limited to Rbt value. Therefore, the sizeable area of cross-section in compression

has the stress distribution close to the rectangular shape despite the distribution law at elastic

stage. Tensile stresses in the cross-section near reinforcement have triangular distribution.

The paper [10, 16, 17] shows that generalized relation "load –– displacement" for the assumed

two-console element is non-linear and load –– displacement curve can have a falling branch.

The area of diagram used for potential energy determination differs from 0,5P0e0. P0 is gener-

alized load and e0 is a generalized displacement. Integrals describing areas of such diagram

give the values close to (2/3) P0e0, therefore the potential energy expression can be represent-

ed as follows:

003

2ePV (5)

The compliance C of the element in general can be determined as follows:

00 PCe . (6)

With regard to the assumed two-console element subjected to five load factors (ΔT, P1, P2, q,

Mcon) we obtain the following equation:

5

1

2

3

1

=i

iii

ibu

A

PPC

A

CP=ζ . (7)

The algebraic transformation leads to the following expression:

crc

2con

crc

q

bt2

crc

II22

crc

II21

crc

I2bu

h

CM+

h

CPb+

h

CP+

h

CP+

h

C(ΔΔ

b=ζ 02

3

1

)h

MMC

h

PPC

h

PPC

h

ΔTΔTC

crc

concon

crcIII

crcII

crcI

0

22

11 , (8)

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ΔT is shearing forces at concrete-reinforcement contact area, P1, P2 are the result of compres-

sive and tensile forces along the height of a two-console element respectively, q is uniform

load along the height of the two-console element at the moment of crack formation equal to

σbt * b, Mcon is a fixed end moment in a two-console element.

When hcrc = 0, the shear force ΔT = Gτεqelbt, where hcrc is a crack length, Gτ is a concrete-

reinforcement mutual displacement modulus, εqel is a relative mutual displacement of concrete

and reinforcement at length t.

Internal forces Nbt in the area adjacent to a crack in a reinforced concrete bar at the moment of

crack formation can be calculated on the basis of a two-console element [11, 18]:

mbRmthbRtbσ+ΔT=N btcrcbtbtbt 3

20,5 (9)

b is the width of a reinforced concrete element; σS·AS is the tensile force in rebars at load

P = λcrcq; σSP·ASP is a pre-tensioning force; t = 1,5·d is a parameter characterizing a dimension

of the compressed concrete near crack (d is the diameter of main reinforcement); hcrc = h0 –

– d/2 – xcrc is the crack length; xcrc = ξ·h0 is the height of the compressed area of concrete in rein-

forced concrete element in bending at the moment of crack formation; ∆Т are shear forces in con-

crete-reinforcement contact area; m is a pre-failure zone; kbr is a critical stress intensity factor.

The design stress σbt,c at concrete-reinforcement contact area and shear force increment ∆Т

(see Fig. 2, d) in the first step of iteration can be defined as follows:

bcbt R

b

rr

21

,2

, (10)

bb RtrrT 2125,0 . (11)

The parameter В4 can be calculated as follows [12, 17]:

111

3

,

3

,4

kkBEkBkB

r

ubt

br

cbt

. (12)

The possible area of В4 change can be determined within the following limits:

btBeB

40 . (13)

If В4 < 0, then lnB4 that corresponds to the condition before crack formation does not exist

(acrc = 0).

If B4 = eB·t

b, then the distance between crack that corresponds to the condition when the dis-

tance between cracks is so small that the bond between concrete and reinforcement is equal to

zero and is thus negligible.


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If the condition (13) is not satisfied on the right, then the level of stresses σbt,c and forced ∆Т

should be decreased.

The functional value of the distance between cracks lcrc can be defined as follows:

B

tBBl

*4ln2

. (14)

Shear and normal stresses (τb(z) and σp) in rebars shown in Fig. 2, are not included in equilib-

rium equation since their projection on x-axis is equal to zero and in moment equilibrium

equation their arms are negligible. The dowel action of rebars is not considered either.

Fig. 2. Analytical schemes of reinforced concrete element in bending for additional dynamic stress calculation:

cross-section without cracks (a), cross-section with cracks (b); cross-section at the moment of crack formation

(c); a model of two-console element (d)

Calculation of stresses in rebars for n-times in a statically indeterminate system (with-

out cracks). Stage Ia of stress-strain state. Concrete stresses σbt approach the limit tensile

stress Rbt. Non-elastic strain develops in the tensile 0=N+N+N cnS,btb area of concrete,

concrete stress diagram at a tension area becomes curved and deformations approach the limit

values. A compressed area of concrete is mainly under elastic strain. Concrete stress diagram

at a compression area is close to a triangular shape (see Fig. 2, a).

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The internal force in rebars for the considered stage can be deduced from the equation of

force equilibrium along x-axis (∑X = 0):

; (15)

0=Aσ+xhbR+N SSbtb ; (16)

S

btbS

cnS, A

xhbRN=σ=σ

, (17)

xh

xhNMN bt

b

3

1

2/

0

0

, (18)

Nb and NS –– are respectively longitudinal internal forces of concrete and rebars in tension in

the static state.

Nb can be deduced from the equation of an equilibrium moment of about 0 point (∑M = 0):

03

2

20

0 =xh+xNxh

N+M bbt

. (19)

Calculation of stresses in rebars for n-times in a statically indeterminate system (with

cracks). Stage II of stress-strain. In a tension area at cross-section with crack tension forces

are taken by rebars and concrete in tension above cracks. In the zones between cracks the

concrete-reinforcement bond mainly remains and concrete works under tension. Further on, as

load increases, the crack reaches the neutral line and opens up, therefore the whole tension is

taken on by rebars (see Fig. 2,b). The internal force in rebars for considered stage can be ex-

pressed from the equation of force equilibrium along x-axis (∑X = 0):

0=N+N+N btSb ; (20)

btbS NN=N , (21)

Nb can be expressed from the equation of moment equilibrium about 0 point (∑M = 0):

xh

M=Nb

2/10

. (22)

Taking into account the relation for concrete-reinforcement bond at the moment of crack for-

mation, the internal force Ns in rebars can be calculated as follows:

mbR+mthbR+tbσΔT+

xh

M=N btcrcbt

'btS

3

20,5

2

10

. (23)

Using the diagram presented in Fig. 1, c and calculated value of internal forces in rebars, it is

possible to determine additional dynamic stresses in rebars of an element in flexure at the

moment of crack formation.


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An example of the calculation of additional dynamic stresses in rebars of a reinforced

concrete element under bending. The proposed method for the calculation of additional dy-

namic stresses in rebars has been tested in the calculation of a prestressed reinforced concrete

beam with a rectangular cross section [19].

The test sample represented a single-span prestressed reinforced concrete beam of 1200 mm

length with cross-sectional dimensions 60×140 mm and with concrete grade of B30. The beam

was reinforced with prestressed rebars of A500 grade, diameter of 6 mm with prestressing force

P = 11,69 kN. Rebars in a compression area are of A400 grade with the diameter of 6 m; trans-

verse rebars are of B500 grade with the diameter of 6 mm placed 100 mm away from each oth-

er. All the material grade are taken as adopted in the Russian regulations.

The calculation is performed with the following inputs: the height of cross-section is

h = 140 mm; the width of cross-section is b = 60 mm; a concrete cover in a tension area

aS = 15 mm; the diameter of prestressed rebars is d = 6 mm; the area of rebars is

AS = 0,283 cm2; Rs,ser = 540 МPа, RS = 435 МPа, ES = 20 * 10

4 МPа; concrete grade В30,

Rb = 17 МPа, Rb,ser = 22 МPа, Rbt = 1,2 МPа, Rbt,ser = 1,8 МPа, Eb = 32,5 * 103

МPа.

The parameters needed for the calculation of a tension force in concrete adjacent to a crack

are the effective height of a cross section: h0 = 125 mm; parameters characterizing the size of

a compression area of concrete adjacent to a crack: tb = 12 mm; t*

= 9 mm; the relative height

of a compressed area of concrete with cracks ξ = 0,217; k = 0,952; parameter В = 135,7;

σS = 267,54 MPa; σbt,c =42,7 MPa; shear force ΔT = 15,4 kN; coefficient B4 = 0,9999.

Checking the condition (13): 0<0,999<5,1. The condition is satisfied.

The design bending moment corresponding to crack formation is Mcrc = 2138 kN·m. Pre-

stressing force P1 = 11,69 kN.

n-times statically indeterminate system.

A compression force under concrete and tensile stresses in rebars can be deduced from equa-

tions (17), (18): Nb = 21868 N, =485,2 MPa.

(n-1)-times statically indeterminate system.

Tensile force in rebars can be expressed from equation (23): =40659 N.

Tensile force in concrete: Nb = 19175N. = 1437 MPa.

The values of stresses can be determined on the basis of the energy method using the diagram

σS-εS (see Fig. 3).

From the similarity of triangles with equal areas we can obtain: = 2388,8 MPa.

Issue № 4(32), 2016 ISSN 2075-0811

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Fig. 3. Diagram σS-εS for determination of dynamic stresses in rebars of (n-1)-times statically indeterminate system

Conclusions

1. The developed analytical relations can be used for the assessment of additional dynamic

stresses in rebars of prestressed reinforced concrete flexural elements;

2. The proposed relations can be used for the assessment of structural survivability parameters

for structures with prestressed flexural elements in states beyond the design basis.

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