1 A Creep and shrinkage

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1 1 A Creep and shrinkage

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3 A1 Determination of creep and shrinkage values 1. Purpose of example It is necessary to determine the values of creep and shrinkage of concrete in a composite beam with cross-section shown in Figure A1.1 as follows: The values of the creep coefficient at t =, the final creep coefficient (, t ), and at t = 9 days which is denoted with (9, t ), The values of the total shrinkage strain at t = which is denoted with cs () (the final value) and at t = 9 days which is denoted with cs (9). 2. Cross-section b = Plate 2x16 Plate 4x12 Plate 2x16 Figure A1.1 Cross-section Input data Concrete strength class: C 2/25 f ck = 2, N/mm 2 fck 2, f cd = = = 1, N/mm 2 c 1,5 E = N/mm 2 Type of cement: N, strength class according to EN 197-1, 2,5 R = ds1 = 4 ds2 =,12 Relative humidity: inside conditions RH 5% First loading t = 28 days Beginning of drying t s = days Composite Structures according to Eurocode 4. Worked Examples.

4 4 A Creep and shrinkage 4. Creep coefficients 4.1 Determination of final creep coefficient For the calculation of the final creep coefficient (, t ) the following is valid: - the perimeter of that part which is exposed to drying, u u= 2 b u= 2 25 = 5 mm - the notional size of the cross-section, h 2 A h= c = = 16 u 5 mm = 16 - t = 28 days, - inside conditions, the ambient relative humidity RH 5 %, - the concrete strength class C 2/25, - the type of cement cement class N, strength class 2,5 R. The final value of creep coefficient (, t ) is determined using the nomogram shown in Figure.1, EN The process of determining the final value of the creep coefficient, taking into account these assumptions, is given in Figure A1.2: t S N 1 st step R 5 th step Figure A1.2 Method for determining the creep coefficient The value of the final creep coefficient found from Figure A1.2 is: t = (, t ) =, 4 th step 2 nd step rd step 1 7, 6, 5, 4,, 2, 1, (, t ) h [mm] t =, h = 16mm C2/25 C25/ C/7 C5/45 C4/5 C45/55 C5/6 C55/67 C6/75 C8/95 C7/85 C9/15

5 Example A Determination of creep coefficient at time t = 9 days The value of creep coefficient (t, t ) for some arbitrary time t can be calculated from: (, tt)= (, tt ) c is the notional creep coefficient, c (t, t ) is a coefficient to describe the development of creep with time after loading (at t =, c (t, t ) =, and at t =, c (t, t ) = 1), The value of is obtained as: = ( f ) ( t ) RH RH is a factor to allow for the effect of relative humidity on the notional creep coefficient and is calculated as follows: 1 RH / 1 RH = 1+ for f 5 N/mm 2,1 h 1 RH /1 RH = [1 + ],1 h 1 2 for f >5 N/mm 2 RH is the relative humidity of the ambient environment (in %), h is the notional size of the cross-section of the member (h in mm), h =2 / Ac u A c is the cross-sectional area of concrete (mm 2 ), u is the perimeter of the member in contact with the atmosphere (mm), (f ) is a factor to allow for the effect of concrete strength on the notional creep coefficient and is determined as follows: 16,8 ( f)= f f is the mean compressive cylinder strength of concrete at the age of 28 days (N/mm 2, f = f ck + 8 N/mm 2 ), (t ) is a factor to allow for the effect of concrete age at loading on the notional creep coefficient and is determined as follows:

6 6 A Creep and shrinkage ( )= 1 t (,1+ ),2 t The effect of the type of cement on the creep coefficient of concrete can be taken into account by modifying the age of loading t according to the following expression: 9 t = t [ + 1], T 1,2 2+t, T,5 days t,t is the factor that takes into account the development of concrete strength as a function of type of cement, is the temperature-adjusted age of concrete at loading in days. The effect of elevated or reduced temperatures within the range 8 C on the maturity of concrete can be taken into account by adjusting the concrete age according to the following expression: n (4/[27+ T( ti ) 1,65]) t T = e t i i=1 t T is the temperature-adjusted concrete age which replaces t in the corresponding expressions, T(t i ) is the temperature in C during the time period t i, t i is the number of days where a temperature T prevails. tt ( t t ), c (, )=[ ] H +( t t ) t is the age of concrete in days at the time considered (in days), t is the age of concrete at first loading (in days), t t is the non-adjusted duration of loading in days, H is the coefficient depending on the relative humidity (RH in %) and the notional member size (h in mm), and is estimated according to expressions: 18 H =1,5 [1+(,12 RH) ] h for f 5 N/mm 2 18 = 1,5 [1 + (, 12 RH) ] h for f > 5 N/mm 2 H

7 Example A1 7 i are correction factors which take into account the influence of the concrete strength according to the following expressions: =[5/ ] 1 f,7 =[5/ ] 2 f,2 =[5/ ] f,5 Thus, the mean compressive cylinder strength of concrete from Table.1, EN is: f = f + 8 N/mm 2 =2 + 8=28 N/mm 2 ck = (type of cement N) Correction factors which taken into account the influence of the concrete strength are: f = =,7,7 1 = [5 / ] [5 / 28] 1,17 f =,2,2 2 =[5/ ] =[5/28] 1,5 f,5,5 =[5/ ] =[5/28] =1,12 The factor to allow for the effect of relative humidity on the notional creep coefficient for f 5 N/mm 2 is: 1 RH / / 1 = 1+ = 1+ = 1,92 RH,1 h,1 16 The factor to allow for the effect of concrete strength on the notional creep coefficient is: 16,8 16,8 ( f)= = =,18 f 28 The effect of the type of cement on the creep coefficient of concrete can be taken into account by modifying the age of loading t according to the following expression, where t,t = t = 28 days:

8 8 A Creep and shrinkage 9 9 t = t [ + 1] = 28 [ + 1], T 1,2 1,2 2+t, T t =28 days,5 days The factor to allow for the effect of concrete age at loading on the notional creep coefficient is: 1 1 ( t )= = =,49 (,1+ ) (,1+ 28 ),2,2 t The coefficient depending on the relative humidity (RH in %) and the notional member size h for f 5 N/mm 2 is: RH h 18 H = 1,5 [1 + (,12 ) ] + 25 = 18 = 1,5 [1 + (,12 5) ] = The coefficient to describe the development of creep with time after loading is: tt ( t t ) (9 28),, c(, )=[ ] =[ ] =,52 H +( t t ) 49+(9 28) The notional creep coefficient is: = ( f ) ( t ) = 1, 92,18, 49 = 2, 99, RH The value of notional creep coefficient represents the value of the final creep coefficient (, t ) found from Figure A1.2. Thus, this result confirms the accuracy of the results obtained from the Figure A1.2 see Section 4.1. At t = 9 days the creep coefficient (t, t ) is: ( tt, )= ( tt, )=2,99,52=1,55 c 5. Shrinkage strains 5.1 Determination of final value of shrinkage strain The total shrinkage strain of concrete, cs, is composed of two components: cs () = cd () + ca ()

9 Example A1 9 cd () is the drying shrinkage strain, ca is the autogenous shrinkage strain (this develops during hardening of the concrete). The final value of the drying shrinkage strain cd () is: cd () = k h cd, k h is a coefficient depending on the notional size of the member h, Table A1.1, cd, is the nominal unrestrained drying shrinkage value, which can be taken from Table A1.2 or can be calculated by means of the following expression: f 1 6 cd, =, 85 [( ds1) exp( ds2 )] 1 RH RH RH 1 RH ( )=1,55 [1 ( ) ] f is the mean compressive cylinder strength of concrete at the age of 28 days (N/mm 2, f = f ck + 8 N/mm 2 ), dsi are factors which depend on the type of cement, RH is the ambient relative humidity (%). Table A1.1 Values for factor k h for calculation of final value of drying shrinkage strain h [mm] 1 1, 2,85,75 5,7 (1) h notional size of member (mm) (2) h = 2 x (cross-sectional area of concrete A c )/(perimeter of member in contact with atmosphere) k h

10 1 A Creep and shrinkage Table A1.2 Nominal unrestrained drying shrinkage values of cd, (in ) for concrete with cement class N Relative Humidity f ck,cy /f ck,cube (N/mm 2 ) Inside conditions, 5% Outside conditions, 8% 2/25,54, 4/5,42,24 6/75,,19 8/95,26,15 9/15,2,1 The final value of the drying shrinkage strain cd () are determined using Tables A1.1 and A1.2. The nominal unrestrained drying shrinkage value cd, according to Table A1.2 for concrete strength class C 2/25 and RH 5% is,54. The factor k h depending on the notional size of the member h according to Table A1.1 is: For h = 1 mm k h = 1, For h = 2 mm k h =,85 Linear interpolation: For h = 16 mm k h =, k h =,85+ (1,,85) 2 1 The final value of the drying shrinkage strain is: cd () = k h =, 91 5, 4 1 = 4, 91 1 cd, 4 4 The final value of the autogenous shrinkage strain is: ca () = 2,5 ( f - 1) 1 = 2,5 (25-1) 1 =,75 1 ck The total shrinkage strain cs () is: cs () = cd () + ca () cs () =, = 4, ,75 1 = 5,29 1

11 Example A Determination of shrinkage strain at time t = 9 days The total shrinkage strain at time t is calculated as: cs (t) = cd (t) + ca (t) The value of the drying shrinkage strain cd at time t is: cd (t) = ds( tt, s ) k h cd, ds ( t ts ) (, tts)= ( t t )+,4 h s t is the age of the concrete at the time considered, in days, t s is the age of the concrete in days at the beginning of drying shrinkage; normally this is at the end of the curing of the concrete. The value of the autogenous shrinkage strain ca at the age of concrete t, is given with the following expression: ca (t) = as( t) ca () as()=1 exp(,2 t t ), t in days ca () = 2,5 ( f 1) 1 ck 6 The drying shrinkage strain is: cd (t) = ds( tt, s ) k h cd, ds ( t ts ) (9 ) (, tts)= = =,52 ( t t ) +,4 h (9 ) +,4 16 s From Section 5.1 k h =,91. =,12 2 ds1 =4 ds

12 12 A Creep and shrinkage 6 cd, =, 85[( ds1) exp( ds2 f / 1)] 1 RH =,85[( ) exp(,12 28 / 1) ] 1 1,6 = 5, ( )=1,55 RH RH [1 ( RH /1) ] =1,55 [1 (5/1) ] =1,6 cd (t) = ( tt, ) k =,52,91 5,45 1 = 2,58 1 ds s h cd, 4 4 The autogenous shrinkage strain is: ca (t) = as( t) ca () ca () = 2,5 ( f 1) 1 = 2,5 (25-1) 1 =,75 1 ck as( t)=1 exp(,2 t )=1 exp(,2 9)=,85 ca (t) = as( t) ca () The total shrinkage strain is: cs (t) = cd (t) + ca (t) 5 5 =,85,75 1 =, cs(9) = 2,58 1 +,19 1 = 2, 89 1 =, Commentary The effects of time-dependent strains of concrete (creep and shrinkage) in the analysis of structural elements of composite structures are different according to whether they are observed in the level of cross-section or static system. The effects of creep and shrinkage of concrete produce internal forces and moments in cross-sections, and curvatures and longitudinal strains in members. The effects that occur in statically determinate systems are classified as primary effects. In statically indeterminate system, the primary effects of creep and shrinkage are associated with additional action effects, such that the total effects are compatible. These are classified as secondary effects and are considered as indirect actions which are sets of imposed deformations. Computational methods, principles and basic equations for estimating the timedependent strains of concrete are given in EN The determination of the final value of the creep coefficient (, t ) for concrete under normal environmental conditions is possible using nomograms. However, for the

13 Example A1 1 determination of the values of shrinkage strains we need to use the extensive numerical procedure.

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