The elastoplastic concrete strain influence on the cracking moment and deformation of rectangular reinforced concrete elements/Tampriai plastiniu deformaciju itaka gelzbetoniniu staciakampio skerspjuvio elementu pleisejimo momentui ir deformacijoms.

Augonis, M. ; Zadlauskas, S.

1. Introduction

When analysing the states of stresses of flexural reinforced concrete elements in the stages of failure, the operation of tensile concrete usually is not estimated [1-2] but its operation has the effect on the cracking moment and stiffness before the crack opening. The expressions of stress-strain relationship of compression concrete are defined by the rules of EC2 (Eurocode 2) as well as in literal sources [3-7]; however there are few sources about the stress-strain relationship of tensile concrete, especially in flexural elements.

The description and evaluation of compression concrete stress-strain relationship of flexural reinforced concrete elements when calculating their stiffness in different states is a complex problem which is usually solved using simplified methods. As the research shows [8], when calculations are made using the method of ultimate limit states, the coefficient values of concrete stress sheet have a significant effect on the change of the results according to different methods, and it is even more difficult to define the concrete stress-strain relationship before cracking.

One more important value subject to the concrete stress sheet is a limit relative height of compressive zone. In EC2 standards, this value depends only on reinforcement grade for lower class concrete in flexural reinforced concrete elements, i.e. concrete limit strain does not depend on its strength class in this case. However, when concrete class varies from C8/10 to C50/60 (more than 6 times), the concrete modulus of elasticity increases more than 1.5 times. If concrete strain characteristics change and reinforcement characteristics remain the same (in the same class), a limit relative height of compressive zone also changes. It can be concluded that a concrete stress-strain curve varies but, according to EC2, concrete stress diagram coefficients [lambda] and [eta] [8] are constant for the concrete of examined classes. In this case the construction technical regulations (STR) [2] evaluate concrete strength when calculating a limit relative height of compressive zone. On the ground of the research presented in literal sources [9-10], it can be stated that concrete limit strains also depend on concrete strength. Although this article does not discuss the ultimate limit state and concrete stress diagram in the failure stage; however the principle of description of concrete stresses is important.

For uncracked section, it is more important to know the tensile stress-strain relationship because the compressed concrete operates elastically in most cases. But the investigation of relationship of concrete tensile stress-strain is more complicated than the investigation of compression stress. EC2 does not describe that relationship and STR specifies the rectangular tensile stress diagram for the calculation of cracking moment. It is difficult to describe the relationship of concrete tensile stress and strain. Thus, the limit strain values of tensile concrete are used in some methods [11, 12].

One more method of problem solving (having evaluated the modulus of elasticity and the strain of concrete) is the description of stress-strain relationship of compression and tensile concrete on the ground of the relation of elastic and elastoplastic concrete strain [[lambda].sub.c] and [[lambda].sub.ct]. With regard to the experimental results [13], coefficient [[lambda].sub.c] varies from 1 to 0.15 for compression concrete and coefficient [[lambda].sub.ct] is equal to 0.5 for tensile concrete, approaching a short-term strength. Since coefficient [[lambda].sub.c] depends on loading time and extent (when time and loading extent increase, [[lambda].sub.c] decreases), it is taken that these coefficients are equal for both tensile and compression concrete [[lambda].sub.ct] = [[lambda].sub.c] = 0.5. In this case it is accepted that with the increase of loading, tensile and compression concrete change from the elastic to the plastic stage gradually, i.e. the coefficients vary gradually from [[lambda].sub.ct] = [[lambda].sub.c] = 1.0 to 0.5. Having accepted these initial conditions, it is possible to express the stress-strain relationship of concrete and to describe the equations of equilibrium of flexural reinforced concrete elements. The direct solution of the obtained equation system is quite complex; however, the solution can be found by the method of approximation. It is also possible to calculate the height of compressive zone and concrete strains using iterative procedure and dividing the element section into layers. Moreover, the solution can be found using the finite element method (FEM) by means of changing layers into finite bar elements. With the sufficient number of layers, the error is minor.

2. Concrete stress-strain relationship

Having accepted that the elastic part of both tensile and compression concrete ranges up to 0.4f.t and 0.4f the limit strains of elastic zone for tensile concrete [[epsilon].sub.ct,el] = 0.4 [f.sub.ct]/[E.sub.c] and for compression concrete [[epsilon].sub.c,el] = 0.4 [f.sub.c]/[E.sub.c]. According to the classic model, the coefficient of elastoplastic concrete strains which evaluates the relation of elastic and total strains is equal to [[lambda].sub.ct,lim] = [[epsilon].sub.ct,el]/([[epsilon].sub.ct,el] + [[epsilon].sub.ct,pl]) = 0.5 at the time of crack opening. The same coefficient of elastoplastic concrete strains can also be taken for compressive concrete zone, i.e. [[lambda].sub.c,lim] = 0.5. Thus, in the elastic zone the coefficient [[lambda].sub.ct] = [[lambda].sub.c] = 1 and when stresses (or strains) exceed the elasticity limit, the coefficient decreases until it becomes equal to 0.5 (when limit stresses are reached). On the assumption that the coefficient [[lambda].sub.ct] changes gradually from the elasticity limit to the concrete tensile strength limit, it can be expressed as follows:

[[lambda].sub.ct] = 1 - 0.5[[[epsilon].sub.ct-i] - [[epsilon].sub.ct,el]/[[epsilon].sub.ct,lim] - [[epsilon].sub.ct,el], (1)

where [[epsilon].sub.ct,lim] is limit strain of tensile concrete that conforms to tensile strength, i.e.:

[[epsilon].sub.ct,lim] = [[f.sub.ct]/[[lambda].sub.ct,lim][E.sub.c]] = [2[f.sub.ct]/[E.sub.c]]. (2)

Having the accurate data of stress-strain relationship and the strain modulus of tensile and compression concrete, it is possible to describe the variation of the modulus of strain more accurately when elastoplastic strains are applied in concrete. Having such relationship between stresses and strains of compression concrete, it can be compared with relationship according to EC2, and can be illustrated in the following way (Fig. 1).

[FIGURE 1 OMITTED]

The variation of coefficient [[lambda].sub.c] according to EC2 and the proposed model is presented in Fig. 2.

[FIGURE 2 OMITTED]

3. The state of stresses when concrete operates elastically in the compressive zone

In most cases, the strains in the compressive zone are elastic before the crack opening. When loading approaches the limit of element cracking, Hooke's law also holds in the tensile zone from the neutral axis to elastic strain limit [[epsilon].sub.ct,el] and the stress-strain relationship becomes nonlinear from [[epsilon].sub.ct,el] to [[epsilon].sub.ct] (Fig. 3).

[FIGURE 3 OMITTED]

Equality [[sigma].sub.ct] = [[lambda].sub.ct][E.sub.c][[epsilon].sub.ct] holds in this zone. Having stress-strain relationships in these three zones, we can write the equations of equilibrium of force projections and bending moments; therefore, we can obtain the compressive zone height of flexural element and the strains and stresses of edge layers. Thus, the resultant of rectangular section element in the compressive concrete zone following the assumption of flat sections is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [[epsilon].sub.ci] = [[epsilon].sub.cy]/x.

The resultant in the elastic part of tensile zone is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [x.sub.el] = [[epsilon].sub.ct,el]x/[[epsilon].sub.c] and [[epsilon].sub.ci] = [[epsilon].sub.ct,el]y/[x.sub.el].

The resultant in the elastoplastic part of tensile zone is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Having evaluated that [E.sup.*.sub.c] = [E.sub.c][[lambda].sub.ct] = [E.sub.c] x x (1.05[[[epsilon].sub.ci] - [[epsilon].sub.ct,el]/[[epsilon].sub.ct,lim] - [[epsilon].sub.ct,el]) and having indicated 1 + [[[epsilon].sub.ct,el]/[[epsilon].sub.c] = k (then [x.sub.pl] = h - x - [x.sub.ct,el] = h - kx), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Strains in elastoplastic zone can be expressed as follows:

[[epsilon].sub.ci] = [[epsilon].sub.ct,el] + ([[epsilon].sub.ct] - [[epsilon].sub.ct,el])[y/h - kx] (7)

Then the resultant is calculated:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Having evaluated that [[epsilon].sub.ct] = [[epsilon].sub.c](h - x)/x and having solved an integral expression, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Finally, the resultant in tensile reinforcement:

[F.sub.s] = [[epsilon].sub.s][E.sub.s][A.sub.s] = [[[epsilon].sub.c](d-x)/x][E.sub.s][A.sub.s] (10)

Bending moments around neutral element axis are calculated in an analogical manner. Bending moment from the resultant of compressive concrete zone:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Bending moment from the resultant of elastic part of tensile concrete zone:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Bending moment from the resultant of elastoplastic part of tensile concrete zone:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Bending moment from the resultant of tensile reinforcement:

[M.sub.s] = [[epsilon].sub.s][E.sub.s][A.sub.s](d - x) = [[[epsilon].sub.c][(d - x).sup.2]/x][E.sub.s][A.sub.s]. (14)

Having the expressions of resultants and bending moments produced by them, we find unknown values x and [[epsilon].sub.c] from the equation system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Basically, the calculation of strains, are made analogically when compressive concrete zone operates elastically and plastically, but equation members [F.sub.c] and [M.sub.c] become more complex functions in this case.

4. The cracking moment when concrete operates elastically in the compressive zone

The calculation of cracking moment is simpler because of known maximum tensile strain value [[epsilon].sub.ct] = [[epsilon].sub.ct,lim] before a crack opens. In this case, the value [lambda] and the elastoplastic strain could be expressed in another form:

[lambda] = 1 - 0.5[y/h - kx]; (16)

[[epsilon].sub.cl] = [[epsilon].sub.ct,el] + ([[epsilon].sub.ct,lim] - [[epsilon].sub.ct,el])[y/h - kx] (17)

So, the resultant values of the compression zone and elastic tensile zone are the same and the resultant of elastoplastic tensile zone after some changes according to Eqs. (16) and (17) will be:

[F.sub.ct,pl] = b[E.sub.c](h - kx)[5[[epsilon].sub.ct,el]/12] + [[epsilon].sub.ct,lim]/3]. (18)

The resultant of tensile reinforcement is:

[F.sub.s] = [[epsilon].sub.ct,lim](d - x)/h - x][E.sub.s][A.sub.s]. (19)

The compression zone height can be expressed from the forces equilibrium equation:

[Ax.sup.2] + Bx + C = 0, (20)

where A = [e.sub.1] - 0.5[[epsilon].sub.ct,lim] + [[[epsilon].sub.ct,el]/[[epsilon].sub.ct,lim]](0.5[[epsilon].sub.ct,el] - [e.sub.1]);

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When the compression zone height is known, the cracking moment is calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Section dimensions (b, h, d), the amount of reinforcement [A.sub.s], the properties of concrete ([[epsilon].sub.ct,el], [[epsilon].sub.ct,lim], [E.sub.c]) and reinforcement ([E.sub.s]) have the main influence on the cracking moment.

5. Layer calculation method of a flexural element with rectangular section

The solution of equations system (15) is complex because the equations in both first and second system expressions are not linear. Therefore, it is more convenient to find unknown values x and [[epsilon].sub.c] using iterative procedure and dividing a member into separate equal layers (Fig. 4).

[FIGURE 4 OMITTED]

In this case, when making equations of equilibrium it is more convenient to express the strains of layers according to the scheme of Fig. 5 because when the location of neutral axis is unknown it is difficult to find how many layers are included into compress and tensile zones.

[FIGURE 5 OMITTED]

According to the mentioned scheme, the strain of i layer can be found from the following equation:

[[epsilon].sub.c] + [[epsilon].sub.ct]/h] = [[epsilon].sub.ci] + [[epsilon].sub.ct]/[h.sub.i](i -1)) [right arrow] [[epsilon].sub.ci] = ([[epsilon].sub.c] + [[epsilon].sub.ct])([h.sub.i](i - 1))/[h.sup.*]] - [[epsilon].sub.ct], (22)

where [h.sub.i] is the height of i layer, i layer number.

Having expressed the strain of outer layer of tensile zone [[epsilon].sub.ct] = [[[epsilon].sub.c](h - x)/x], we obtain:

[[epsilon].sub.ci] = [[epsilon].sub.c][[h.sub.i](i - 1)]/[h.sup.*]] - [[epsilon].sub.c] [[h.sup.*] - x]/x(1 - [h.sub.i](1 - 1)[h.sup.*]) (23)

Then, the height of compressive zone can be expressed from the equation of equilibrium of force projections:

[n.summation over (i=1)][E.sub.ci][A.sub.ci][[epsilon].sub.ci] + [E.sub.s][A.sub.s][[epsilon].sub.s] = 0, (24)

where [E.sub.ci] = [[lambda].sub.ci][E.sub.c], [A.sub.ci] = [bh.sub.i].

Having evaluated expression (23) and the fact that [[epsilon].sub.s] = [[epsilon].sub.c][[a.sub.s]/[h.sup.*]] - [[epsilon].sub.c][[[h.sup.*] - x]/x] (1 - [a.sub.s]/[h.sup.*]), we can get the proportion H = [[h.sup.*] - x]/x from expression (24):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

where [a.sub.s] is distance from the edge of tensile concrete to the weight centre of tensile reinforcement.

Having obtained value H from the equation of equilibrium of bending moments, we can calculate the strain of compressive zone edge in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

In this case, it is not necessary to solve equation system but the solution is repeated in order to make layer Eci more precise.

This iterative solution can also be written in a matrix form:

F = E[epsilon]. (28)

If we divide element into 6 equal layers and mark the bottom tensile layer with number "1", and the reinforcement weight centre is between the axes of layers "1"and "2", then matrix E is equal to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

Displacement vector

[epsilon] = [{[[epsilon].sub.c1] [[epsilon].sub.c2] [[epsilon].sub.c3] [[epsilon].sub.c4] [[epsilon].sub.c5] [[epsilon].sub.c6] [[epsilon].sub.s]}.sup.T] and force vector is calculated - F = [{0 0 0 0 0 M 0}.sup.T].

When element is divided into quite small parts, the weight centre of tensile reinforcement will probably coincide with the axis of one of the layers. Let us suppose that it will coincide with the axis of layer "2". Then, member E(5, 2) of matrix E will be equal to ([E.sub.c2][A.sub.c2] + [E.sub.s][A.sub.s]) and member E(6, 2) - to ([E.sub.c2][A.sub.c2] + [E.sub.s][A.sub.s])hi. In this case, members of the final line and final column as well as vector members [epsilon](7) and F(7) will be equal to 0.

It is convenient to solve the discussed element divided into layers using the (FEM). In this instance, each layer will correspond to a bar element and stiffness of that layer (Fig. 6). In order to obtain a flat section strain, the external bending moment is added to the element with high stiffness which has the following characteristics [E.sub.st][I.sub.st]. Also the calculation can be made with reinforcement in the compression zone. In this case, the solution is the same as in the case of tensile reinforcement.

[FIGURE 6 OMITTED]

Having calculated a particular element of rectangular section (Eq. 15), the results of expression members are obtained similar according to expressions (3-4), (9-10) and (11-14), the layer method and FEM. The element calculation results (b = 0.2 m, h = 0.21 m, d = 0.185 m, [f.sub.ck] = 30 MPa, [f.sub.ct] = 2 MPa, As = 5 [cm.sup.2], [E.sub.c] = 30 GPa, [E.sub.s] = 200 GPa, [[epsilon].sub.ct,el] = 2.667e-5, [[epsilon].sub.ct,lim] = 1.333e-4), when 3 kNm bending moment is applied, are presented in Table.

The stress and strain relationship of the examined element when bending moment is equal to 3 kNm is presented in Fig. 7.

[FIGURE 7 OMITTED]

6. The comparison of the results of different methods

Comparing the proposed model with other methods laid down in STR, EC2, ACI [14] and the method presented in [12], different height of an element with the constant width 0.2 m was chosen. The differences of results could be seen in Fig 8. The comparison of the proposed method with EC2 and ACI codes is complicated because the influence of reinforcement is ignored. The cracking moment depends only on section dimensions in these codes.

[FIGURE 8 OMITTED]

The relationship between the cracking moment and reinforcement amount was calculated according to STR and the proposed model (Fig. 9).

To find out the influence of such parameters as the element dimensions and the reinforcement amount on the cracking moment, the rectangular section with different dimensions and reinforcement amount was calculated. The width varies from 0.1 m to 0.3 m, height - 0.2 m - 0.45 m and the cross-section of reinforcement - 2.5 [cm.sup.2] - 20 [cm.sup.2]. In order to eliminate the influence of width, values [M.sub.crc]/b and [I.sub.eff]/b were compared. The comparison of the average values of cracking moment calculated by expression (21) and STR is shown in Fig. 10, where differences vary from 5 to 15%. It can be seen from this figure that the influence of a section dimensions on the character of cracking moment variation is quite similar. The area of reinforcement to the inertia moment is not estimated in ACI and EC2 codes. Thus, the comparison with STR and the proposed method is not quite precise.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

For model testing, the cracking moment of element (b = 0.2 m, h = 0.5 m, d = 0.46 m, [A.sub.s] = 14.7 [cm.sup.2], C25/30, S400) was calculated and it was equal to 44.195 kNm. The cracking moment calculated according to STR is equal to 54.4 kNm, EC2 - 21.4 kNm; and ACI - 28.3 kNm. This quite significant difference of proposed method result compared with EC2 and ACI occurs because of the amount of tensile reinforcement.

7. Conclusions

It is convenient to calculate the cracking of flexural elements by applying the coefficients of elastoplastic strain of tensile and compression concrete. Moreover, using the proposed calculation model, it is not difficult to evaluate the coefficients of elastoplastic strain that are different for tensile and compression concrete, i. e. when [[lambda].sub.ct] [not equal to] [[lambda].sub.c]. In this case, concrete strengths that have essential effect on the limit strains when describing the cracking moments are also evaluated.

Since the stress sheet of tensile zone is not rectangular, the obtained values of cracking moments are slightly lower than those of classic calculation methods. The solutions of chosen iterative method of layers are quite precise and simple, therefore it is possible to avoid solving of more difficult integral equations. The FEM can also be used for the solution.

Received December 13, 2011

Accepted January 16, 2013

References

[1.] EN 1992-1-1:2004: E. Eurocode 2: Design of Concrete Structures--Part 1 - 1: General Rules and Rules for Buildings.--Brussels: European Committee for Standardization, 253p.

[2.] STR 2.05.05:2005 Design of Concrete and Reinforced Concrete Structures (in Lithuania).

[3.] Hamoush, S.; Abu-Lebdeh, T.; Cummins, T. 2010 Deflection behavior of concrete beams reinforced with PVA micro-fibers, Journal of Construction and Building Materials 24: 2285-2293. http://dx.doi.org/10.1016/j.conbuildmat.2010.04.027.

[4.] Kaklauskas, G.; Zamblauskait?, R.; Ba?inskas, D. 2005. Deformational analysis of prestressed high strength concrete members using flexural constitutive model, Journal of Civil Engineering and Management 11(2): 145-151. http://dx.doi.org/10.1080/13923730.2005.9636344.

[5.] KYip, W. 1998. Generic form of stress-strain equations for concrete, Journal of Cement and Concrete Composites 28: 499-508. http://dx.doi.org/10.1016/S0008-8846(98)00039-8.

[6.] Binici, B. 2005. An analytical model for stress-strain behavior of confined concrete, Journal of Engineering Structures 27: 1040-1051. http://dx.doi.org/10.1016/j.engstruct.2005.03.002.

[7.] Jiang, T; Teng, J.G. 2007. Analysis-oriented stress-strain models for FRP-confined concrete, Journal of Engineering Structures 29: 2968-2986. http://dx.doi.org/10.1016/j.engstruct.2007.01.010.

[8.] Dulinskas, E. J; Zabulionis, D.; Balevicius, R. 2007. On the equivalence of compressive concrete diagrams in analysis of flexural reinforced concrete elements. The 9th international conference "Modern building materials, structures and techniques": selected papers, May 16-18, 2007 Vilnius, 2: 523-530.

[9.] Tasdemir, M.A.; Tasdemir, C.; Akyuz, S.; Jefferson, A.D.; Lydon, F.; Barr, B.I.G. 1998. Evaluation of strains at peak stresses in concrete: a three-phase composite model approach, Journal of Cement and Concrete Composites 20: 301-318. http://dx.doi.org/10.1016/S0958-9465(98)00012-2.

[10.] Belen, G.F.; Fernando, M.A.; Diego, C.A.; Sindy, S.P. 2011. Stress-strain relationship in axial compression for concrete using recycled saturated coarse aggregate, Journal of Construction and Building Materials 25: 2335-2342. http://dx.doi.org/10.1016/j.conbuildmat.2010.11.031.

[11.] Valivonis, J.; Skuturna, T. 2007. Cracking and strength of reinforced concrete structures in flexure strengthened with carbon fibre laminates, Journal of Civil Engineering and Management XIII(4): 317-323.

[12.] Soranakom, C.; Mobasher, B. 2008. Correlation of tensile and flexural responses of strain softening And strain hardening cement composites, Journal of Cement and Concrete Composites 30: 465-477. http://dx.doi.org/10.1016/j.cemconcomp.2008.01.007.

[13.] Baikov, V.; Sigalov, E. 1991. Reinforced Concrete Structures, Moscow: Stroiizdat, 767p.

[14.] ACI 318M-05. Building code requirements for Structural concrete and Commentary. 2004.

M. Augonis, Kaunas University of Technology, Student? str. 48, 51367 Kaunas, Lithuania, E-mail: mindaugas.augonis@ktu.lt

S. Zadlauskas, Kaunas University of Technology, Student? str. 48, 51367Kaunas, Lithuania, E-mail: saulius.zadlauskas@gmail.com

http://dx.doi.org/10.5755/j01.mech.19.L3619

Table
The comparison of calculation results of presented example
according to different methods

 [F.sub.c], [F.sub.ct,el], [F.sub.ct,pl],
 kN kN kN

By (3-4), 21.308 3.589 13.222
 (9-10) and
 (11-14)
 expressions
By "Layer 21.35 2.957 13.885
 method" and
 "FEM" (21
 layers)
By "Layer 21.309 3.633 13.176
 method" and
 "FEM" (189
 layers)

 [F.sub.s], [M.sub.c], [M.sub.ct,el],
 kN kNm kNm

By (3-4), 4.5 1.553 0.107
 (9-10) and
 (11-14)
 expressions
By "Layer 4.51 1.553 0.079
 method" and
 "FEM" (21
 layers)
By "Layer 4.5 1.553 0.109
 method" and
 "FEM" (189
 layers)

 [M.sub.ct,pl], [M.sub.s],
 kNm kNm

By (3-4), 0.999 0.341
 (9-10) and
 (11-14)
 expressions
By "Layer 1.027 0.341
 method" and
 "FEM" (21
 layers)
By "Layer 0.997 0.341
 method" and
 "FEM" (189
 layers)