prEN19921-1:2003(E (5) The values given in Figure 3. 1 are valid for ambient temperatures between-40oC and +40c and a mean relative humidity between rh= 40% and rH=100%. The following symbols are used p(oo, to) is the final creep coefficient is the age of the concrete at time of loading in days thsNR is the notional size 2Ac/u, where Ac is the concrete cross-sectional area and u is the perimeter of that part which is exposed to drying s is Class s, according to 3.1.2(6) N is Class N, according to 3.1.2(6) R is Class R, according to 3.1.2 (6) (6)The total shrinkage strain is composed of two components, the drying shrinkage strain and the autogenous shrinkage strain. The drying shrinkage strain develops slowly, since it is a function of the migration of the water through the hardened concrete. The autogenous shrinkage strain develops during hardening of the concrete: the major part therefore develops in the early days after casting. Autogenous shrinkage is a linear function of the concrete strength. It should be considered specifically when new concrete is cast against hardened concrete. Hence the values of the total shrinkage strain ecs follow from Ecs Ecd t Eca (3.8) where Ecs is the total shrinkage strain is the drying shrinkage strai eca is the autogenous shrinkage strain The final value of the drying shrinkage strain, Edd, oo is equal to k Ed, 0. Edd, o. may be taken from Table 3.2(expected mean values, with a coefficient of variation of about 30%) Note: The formula for Ecd. o is given in Annex B Table 3.2 Nominal unrestrained drying shrinkage values Ecd, o(in loo)for concrete Relative Humidity(in lo) (MPa) 80 90 100 0.64 0.60 0.50 0.31 0.17 40/50 0.51 0.48 0.40 0.14 60/75 0.41 0.38 0.32 0.20 0.11 8095 0.33 0.31 0.26 0.16 0.09 00000 90/105 0.30 0.28 0.23 0.15 0.05 The development of the drying shrinkage strain in time follows from ed(t)=Bs(t,ts)·kh·lodo (3.9) Note: Eedo is defined in Annex B there kn is a coefficient depending on the notional size ho according to Table 3.3 Table 3.3 Values for kn in Expression(3. 9) 100 1.0
prEN 1992-1-1:2003 (E) 31 (5) The values given in Figure 3.1 are valid for ambient temperatures between -40°C and +40°C and a mean relative humidity between RH = 40% and RH = 100%. The following symbols are used: ϕ (∞, t0) is the final creep coefficient t0 is the age of the concrete at time of loading in days h0 is the notional size = 2Ac /u, where Ac is the concrete cross-sectional area and u is the perimeter of that part which is exposed to drying S is Class S, according to 3.1.2 (6) N is Class N, according to 3.1.2 (6) R is Class R, according to 3.1.2 (6) (6) The total shrinkage strain is composed of two components, the drying shrinkage strain and the autogenous shrinkage strain. The drying shrinkage strain develops slowly, since it is a function of the migration of the water through the hardened concrete. The autogenous shrinkage strain develops during hardening of the concrete: the major part therefore develops in the early days after casting. Autogenous shrinkage is a linear function of the concrete strength. It should be considered specifically when new concrete is cast against hardened concrete. Hence the values of the total shrinkage strain εcs follow from εcs = εcd + εca (3.8) where: εcs is the total shrinkage strain εcd is the drying shrinkage strain εca is the autogenous shrinkage strain The final value of the drying shrinkage strain, εcd,∞ is equal to kh⋅εcd,0. εcd,0. may be taken from Table 3.2 (expected mean values, with a coefficient of variation of about 30%). Note: The formula for εcd,0 is given in Annex B. Table 3.2 Nominal unrestrained drying shrinkage values εcd,0 (in 0 /00) for concrete Relative Humidity (in 0 /0) fck/fck,cube (MPa) 20 40 60 80 90 100 20/25 0.64 0.60 0.50 0.31 0.17 0 40/50 0.51 0.48 0.40 0.25 0.14 0 60/75 0.41 0.38 0.32 0.20 0.11 0 80/95 0.33 0.31 0.26 0.16 0.09 0 90/105 0.30 0.28 0.23 0.15 0.05 0 The development of the drying shrinkage strain in time follows from: εcd(t) = βds(t, ts) ⋅ kh ⋅ εcd,0 (3.9) Note: εcd,0 is defined in Annex B. where kh is a coefficient depending on the notional size h0 according to Table 3.3 Table 3.3 Values for kh in Expression (3.9) h0 kh 100 1.0
prEN19921-1:2003 200 0.85 300 0.75 ≥500 Ba(t, t= (-t)+004h Where t is the age of the concrete at the moment considered, in days ts is the age of the concrete (days)at the beginning of drying shrinkage (or swelling ). Normally this is at the end of curing lo is the notional size(mm)of the cross-section 2Adu where Ac is the concrete cross- sectional area is the perimeter of that part of the cross section which is exposed to drying The autogenous shrinkage strain follows from Eca(t)=Bas(t) Eca(oo) (3.11) where ea(∞)=25(Gk-10)106 (3.12) and s()=1-exp(-0,2t05) (3.13 where t is given in days 3.1.5 Stress-strain relation for non-linear structural analysis (1) The relation between oc and ec shown in Figure 3.2(compressive stress and shortening strain shown as absolute values) for short term uniaxial loading is described by the Expression 1+(k-2 (3.14) where 7=E Ec1 is the strain at peak stress according to Table 3.1 k =1, 05 Ecm x/fcm (fom according to Table 3.1) Expression( 3. 14)is valid for o Ecl Cull where Ecu1 is the nominal ultimate strain (2)Other idealised stress-strain relations may be applied, if they adequately represent the behaviour of the concrete considered
prEN 1992-1-1:2003 (E) 32 200 300 ≥ 500 0.85 0.75 0.70 ( ) ( ) 3 s 0 s ds s 0,04 ( , ) t t h t t t t − + − β = (3.10) where: t is the age of the concrete at the moment considered, in days ts is the age of the concrete (days) at the beginning of drying shrinkage (or swelling). Normally this is at the end of curing. h0 is the notional size (mm) of the cross-section = 2Ac/u where: Ac is the concrete cross-sectional area u is the perimeter of that part of the cross section which is exposed to drying The autogenous shrinkage strain follows from: εca (t) = βas(t) εca(∞) (3.11) where: εca(∞) = 2,5 (fck ñ 10) 10-6 (3.12) and βas(t) =1 ñ exp (ñ 0,2t 0,5) (3.13) where t is given in days. 3.1.5 Stress-strain relation for non-linear structural analysis (1) The relation between σc and εc shown in Figure 3.2 (compressive stress and shortening strain shown as absolute values) for short term uniaxial loading is described by the Expression (3.14): ( ) k η kη η f σ 1 2 2 cm c + − − = (3.14) where: η = εc/εc1 εc1 is the strain at peak stress according to Table 3.1 k = 1,05 Ecm × |εc1| /fcm (fcm according to Table 3.1) Expression (3.14) is valid for 0 < |εc| < |εcu1| where εcu1 is the nominal ultimate strain. (2) Other idealised stress-strain relations may be applied, if they adequately represent the behaviour of the concrete considered
prEN19921-1:2003(E 0.4fm n Figure 3.2: Schematic representation of the stress-strain relation for structural analysis. 3.1.6 Design compressive and tensile strengths (1)P The value of the design compressive strength is defined as fed=ac ck/% (3.15) where r is the partial safety factor for concrete, see 2.4.2.4, and acc is the coefficient taking account of long term effects on the compressive strength and of unfavourable effects resulting from the way the load is applied Note: The value of ac for use in a Country should lie between 0, 8 and 1, 0 and may be found in its National Annex. The recommended value is 1 (2)P The value of the design tensile strength, fctd, is defined as fetd=act,0.05/%t where r is the partial safety factor for concrete, see 2.4.2.4, and xct is a coefficient taking account of long term effects on the tensile strength and of unfavourable effects, resulting from the way the load is applied Note: The value of aet for use in a Country may be found in its National Annex. The recommended value is 1,0 33
prEN 1992-1-1:2003 (E) 33 Figure 3.2: Schematic representation of the stress-strain relation for structural analysis. 3.1.6 Design compressive and tensile strengths (1)P The value of the design compressive strength is defined as fcd = αcc fck / γC (3.15) where: γC is the partial safety factor for concrete, see 2.4.2.4, and αcc is the coefficient taking account of long term effects on the compressive strength and of unfavourable effects resulting from the way the load is applied. Note: The value of αcc for use in a Country should lie between 0,8 and 1,0 and may be found in its National Annex. The recommended value is 1. (2)P The value of the design tensile strength, fctd, is defined as fctd = αct fctk,0,05 / γC (3.16) where: γC is the partial safety factor for concrete, see 2.4.2.4, and αct is a coefficient taking account of long term effects on the tensile strength and of unfavourable effects, resulting from the way the load is applied. Note: The value of αct for use in a Country may be found in its National Annex. The recommended value is 1,0. fcm 0,4 fcm ε c1 σc ε cu1 ε c tan α = Ecm α
prEN19921-1:2003(E) 3.1.7 Stress-strain relations for the design of cross-sections (1)For the design of cross-sections, the following stress-strain relationship may be used, see Figure 3. 3(compressive strain shown positive) for0≤ε。≤E (3.17) 。= ed for Ecr≤ Ec secur Where n is the exponent according to table 3.1 Ec2 is the strain at reaching the maximum strength according to table 3 Ecu2 is the ultimate strain according to Table 3.1 Figure 3.3: Parabola- rectangle diagram for concrete under compression. (2) Other simplified stress-strain relationships may be used if equivalent to or more conservative than the one defined in(1), for instance bi-linear according to Figure 3.4 (compressive stress and shortening strain shown as absolute values)with values of ec3 and Ecu3 according to table 3.1 34
prEN 1992-1-1:2003 (E) 34 3.1.7 Stress-strain relations for the design of cross-sections (1) For the design of cross-sections, the following stress-strain relationship may be used, see Figure 3.3 (compressive strain shown positive): c c2 n c2 c c cd 1 1 for 0 ε ε ε ε ≤ ≤ σ = f − − (3.17) c cd c2 c cu2 σ = f for ε ≤ ε ≤ ε (3.18) where: n is the exponent according to Table 3.1 εc2 is the strain at reaching the maximum strength according to Table 3.1 εcu2 is the ultimate strain according to Table 3.1 Figure 3.3: Parabola-rectangle diagram for concrete under compression. (2) Other simplified stress-strain relationships may be used if equivalent to or more conservative than the one defined in (1), for instance bi-linear according to Figure 3.4 (compressive stress and shortening strain shown as absolute values) with values of εc3 and εcu3 according to Table 3.1. fcd ε c2 σc ε cu2 0 ε c fck c3 ε 0 cu3 fcd ε σc ε c fck
prEN19921-1:2003(E Figure 3.4: Bi-linear stress-strain relation (3)A rectangular stress distribution(as given in Figure 3.5)may be assumed. The factor A, defining the effective height of the compression zone and the factor n, defining the effective strength, follow from 1=08 for fck≤50MPa (3.19 λ=0,8-(6k-50)/400for50<fk≤90MPa 7=10 for fck≤50MPa (3.21) 7=10-(f6k-50)/200for50<fak≤90MPa Note: If the width of the compression zone decreases in the direction of the extreme compression fibre, the value n fed should be reduced by 10% 巨F As F Figure 3.5: Rectangular stress distribution 3.1.8 Flexural tensile strength (1)The mean flexural tensile strength of reinforced concrete members depends on the mean axial tensile strength and the depth of the cross-section. The following relationship may be used fctm f= max ((1, 6-h/1000)fctm; fctm y Where h is the total member depth in mm fctm is the mean axial tensile strength following from Table 3.1 The relation given in Expression (3.23) also applies for the characteristic tensile strength values 3.1.9 Confined concrete (1) Confinement of concrete results in a modification of the effective stress-strain relationshi higher strength and higher critical strains are achieved. The other basic material characteristics may be considered as unaffected for design (2)In the absence of more precise data, the stress-strain relation shown in Figure 3.6 (compressive strain shown positive) may be used, with increased characteristic strength and strains according to fck.c= fck(1,000+ 5,0 o2/fck) for o2 <0,05fck (3.24) 35
prEN 1992-1-1:2003 (E) 35 Figure 3.4: Bi-linear stress-strain relation. (3) A rectangular stress distribution (as given in Figure 3.5) may be assumed. The factor λ, defining the effective height of the compression zone and the factor η, defining the effective strength, follow from: λ = 0,8 for fck ≤ 50 MPa (3.19) λ = 0,8 - (fck -50)/400 for 50 < fck ≤ 90 MPa (3.20) and η = 1,0 for fck ≤ 50 MPa (3.21) η = 1,0 - (fck -50)/200 for 50 < fck ≤ 90 MPa (3.22) Note: If the width of the compression zone decreases in the direction of the extreme compression fibre, the value η fcd should be reduced by 10%. Figure 3.5: Rectangular stress distribution 3.1.8 Flexural tensile strength (1) The mean flexural tensile strength of reinforced concrete members depends on the mean axial tensile strength and the depth of the cross-section. The following relationship may be used: fctm,fl = max {(1,6 - h/1000)fctm; fctm } (3.23) where: h is the total member depth in mm fctm is the mean axial tensile strength following from Table 3.1. The relation given in Expression (3.23) also applies for the characteristic tensile strength values. 3.1.9 Confined concrete (1) Confinement of concrete results in a modification of the effective stress-strain relationship: higher strength and higher critical strains are achieved. The other basic material characteristics may be considered as unaffected for design. (2) In the absence of more precise data, the stress-strain relation shown in Figure 3.6 (compressive strain shown positive) may be used, with increased characteristic strength and strains according to: fck,c = fck (1,000 + 5,0 σ2/fck) for σ2 ≤ 0,05fck (3.24) As d η fcd Fs λx εs x εcu3 Fc Ac