Limitation of Stresses in Concrete According to Eurocode 2

When stresses in concrete in reinforced concrete beams with the reinforcement ratio being high due to the ultimate limit state are calculated – according to the linear elasticity theory – from strains of a characteristic combination or even a quasi-permanent one, in some cases compressive stresses in extreme concrete fibres can be even higher than the characteristic compressive strength of concrete. In the following report, it is postulated to calculate these stresses with the help of the non-linear stress-strain relationship for concrete. It has also been demonstrated that the requirements specified in points 7.2(1)P and 7.2(2) of the standard PNEN 1992-1-1:2008/NA:2010 Eurocode 2:Design of concrete structures. Part 1-1: General rules and rules for buildings, are useless and do not suit the limit state analysis, on which Eurocodes are based.


Introduction
Eurocode 2 [1] comprises regulations -not completely unambiguous ones -regarding stresses in concrete.Point 7.2(1)P of the norm [1] states 'the compressive stress in the concrete shall be limited in order to avoid longitudinal cracks, micro-cracks or high levels of creep, where they could result in unacceptable effects on the function of the structure'.Point 7.2(2) further specifies that 'longitudinal cracks can appear if the stress level under a characteristic combination of loads exceeds a critical value'.Such cracks can deteriorate the durability of a structure.If no other measures are taken, e.g. an increase in the cover of the reinforcement in the compressive zone or confinement of the concrete by transverse reinforcement, then it may appropriate to limit compressive stress to a value k1fck in areas exposed to the influence of environments of exposure classes XD, XF and XS.The recommended value is k1 = 0.6, but the norm [1] allows the specification of another value of this factor in a National Annex.The Polish National Annex NA.5 assumes that k1 = 1.0.
As noted by M. Knauff in [3], the above regulation is made ambiguous by the phrase: 'it may be appropriate (does it mean mandatory?) to limit the compressive stresses to a value k 1 f ck '.It is also uncertain whether an increase in the thickness of the cover up to the one required because of more unfavourable environments, such as XD, XF and XS, can be equated with the increase which abolishes the obligation to monitor compressive stresses in concrete.
Before Eurocode 2 [1] was adopted in Poland, when designing reinforced concrete structures according to the Polish standard PN-B-03264:1984 [2], it was necessary to check the ultimate limit state (ULT) as well as bendings and widths of cracks caused by quasipermanent loadings.Meanwhile, compressive stresses in concrete caused by a characteristic combination of loads were not monitored.This approach did not lead to any damage to a designed structure.

Calculation of Compressive Stresses in Compressed Concrete
As in [4], stresses in concrete for non-linear structural analysis were calculated as recommended by Eurocode 2 [1].By employing the stress-strain relation in concrete (3a -3c), given in point 3.1.5 of Eurocode 2 and shown in fig. 1, as well as the values illustrated in fig.2, two equilibrium equations in a cross-section being in the serviceability state can be written, i.e. the sum of projection forces onto the axis of a beam (1) and the sum of bending moments of the cross-section relative to the resultant force of the concrete's compression zone (2).
where: In line with the Bernoulli's hypothesis regarding flat cross-sections, the following are calculated from the relationships between strains: And afterwards, stresses σ s1 in reinforcement A s1 are derived: By substituting expressions (3) ÷ ( 6) to ( 1) and ( 2), we obtain a set of two equations with two unknowns x and ε cc .The problem can be solved numerically (integration leads to the formulation of implicit equations).In a programme written in the Fortran language, for a given ε cc ϵ (0, ε c1 ) that changes by 0.001ε c1 at a step, the subsequent values x ϵ (0, 0.5d) changing by 0.5d/1000 at a step are assumed.Then, it is checked whether the left side of equation ( 1) is equal to the right side, and when this agreement is sufficient, the appropriateness of the achieved values x and ε cc is verified with equation (2).If the verification fails, the programme resumes the search (at the values of ε cc and x at which it stopped) for a new pair of x and ε cc , again solving equation (1), then verifying whether a new pair of values satisfies equation (2).Integers present in expression (4) are calculated by the programme with the Simpson method.Having derived the value of ε cc from equations ( 1) and (2), equations (3a -3c) are used to determine the stress in the extreme compressive fibre of the cross-section σ c (ε c = ε cc ).

Examples
The solid reinforced concrete flooring shown in fig.3, of the thickness 0.2 m and the span 5.1 m, rests on the main T cross-section beam, which is a two-span continuous beam of the span length equal 8.1 m.The cross-section of the main beam: height h=0.70 m, width of the web of the T beam b=0.35 m, distance from the reinforcement centre of gravity to the tensile edge a1=0.05m; material: concrete C16/20, the secant modulus of elasticity of concrete Ecm = 29 GPa, design yield strength of reinforcement fyd = 350 MPa, the modulus of elasticity of reinforcing steel Es = 200 GPa.The live action of the warehouse equals 4.0 kN/m2, the value of the quasi-permanent coefficient of variable action is ψ2 = 0.8.The following were determined in the main beam above the support: design value of the applied bending moment MEd = 591.6 kNm and the cross-sectional area of reinforcement As1 = 34.00cm2.The planned reinforcement consisted of 7ϕ25 with As1 = 34.96cm2, which yields the reinforcement ratio ρL = As1/(bd) = 0.0154.

Fig. 3. Cross-sections of the floor
The moment from quasi-permanent actions is M = 430.65 kNm.Having solved a set of equations ( 1) and ( 2), the following were determined: x = 0.2535 m and ε cc = 0,0007011, and when ε cc was substituted to equations (3a -3c) the value σ c = 15.70 MPa was obtained.
It is worth underlining that calculations of widths of cracks of strains in compressed concrete and tensile steel at A s1in the cracked cross-section (being in phase II) -are traditionally made having assumed the linear elasticity theory (e.g.[5]).However, in crosssections exposed to very heavy actions, stresses in concrete are impossible to be calculated based on the assumption of their linear distribution, as this would generate absurd results, i.e. not only higher than the ones derived from the set of equations ( 1) and ( 2), but also, and not infrequently, even exceeding the characteristic compressive strength of concrete, as it is illustrated underneathtreating actions as short-term ones (immediately after the application of the moment M = 430.65 kNm), for which α e = E s /E c,eff = 200/29 = 6.90.An example of the difference in the distribution of stresses in the compression zone of concrete σ c calculated from the set of equations ( 1) and ( 2) and in agreement with the linear theory is shown in figure 2.
In the analysed cross-section, the moment of the characteristic combination was M = 464.11kNm.The stress calculated in line with the procedure described in point 2 was σ c = 16.58MPa, which is higher than f ck = 16.0MPa! Should we calculate these stresses in the elasticity range, as soon as the actions are applied (α e = 6.90), an even higher excess of the characteristic compressive strength of concrete would be achieved, which is shown below (x II = 0.238 m, I II = 0.00567 m 4 ): For the moment from the characteristic combination M = 464.11kNm, using the procedure described in point 2, the calculated stress was σ c = 19,91 MPa, which in this case does not exceed f ck = 32.0MPa.If these stresses are calculated in the elasticity range, for α e = 6.25, x II = 0.2141 m, I II = 0.00464 m 4 , then, naturally, higher values will be obtained from equation (12), i.e. σ c = 21.42MPa.

Conclusions
The above examples suggest that sometimes in a cross-section that fulfils the ultimate limit state and has cracks not exceeding the limit value compressive stresses in concrete caused by a characteristic combination exceed the characteristic compressive strength of concrete f ck .The ambiguously worded requirement in point 7.2(2) [1] -'it may be appropriate to limit the compressive stresses to a value 1 ck k f ' -does not resolve the dilemma whether a given cross-section satisfies the requirements set in Eurocode 2 [1].The above statement seems to be confounding and it can be claimed to be useless.Limiting stresses down to a level below the limit stresses would be more suitable for the method of designing allowable stresses rather than for the ultimate limit analysis, on which Eurocodes are based.
The excess stresses demonstrated in the discussed example arise solely from certain imperfections of the calculation models, which we have at our disposal.Limiting these stresses to the value in agreement with the original version of Eurocode 2 would be even more absurd, because than the limit stress in the first example would equal merely:

Fig. 1 .
Fig. 1.Schematic representation of the stress-strain relation in concrete for structural analysis

Fig. 2 .
Fig. 2. Stress-strain diagram in a bending cross-section in the serviceability state.For comparison, the picture shows the location of the cross-section centre of gravity in phase II -x II and the diagram of strains in concrete determined according to the linear theory -σ c (x II ).
S-P Seminar 2017, Theoretical Foundation of Civil Engineering7 of cracksin line with Eurocode 2is calculated for stresses even equal to f ck (in our case f ck = 16 MPa), and these are stresses triggered by a combination from quasipermanent loads, which causes smaller generalised internal strengths and stresses in a cross-section than a characteristic combination.Also, in the second caseif the original version of Eurocode 2 was binding in Polandstresses in concrete σ c = 19.91MPa would be above the recommended value: