STRUCTURAL CONCRETE SOFTWARE ADAPT-ABI. Addendum to Construction Phase Modeling and Analysis

Transcription

1 Add_v STRUCTURAL CONCRETE SOFTWARE ADAPT-ABI Addendum to Construction Phase Modeling and Analysis This supplemental reference manual is made available to users of ADAPT-ABI 2012 to help them understand the underlying modeling and analysis capabilities of the software. It references the previous, text-based INP file format used to define models. The current version of ABI uses a similar INP file format to send model information to the analysis engine. Copyright support@adaptsoft.com ADAPT Corporation, Redwood City, California, USA, Tel: +1 (650) Fax: +1 (650) ADAPT International Pvt. Ltd, Kolkata, India Tel: Fax:

2 TABLE OF CONTENTS 1. CHAPTER 3 ADDENDUM 1.1 ACI-209 Concrete Model (1992) AASHTO Concrete Model (1994) Eurocode 2 Concrete Model (2004) British Concrete Model Hong Kong Concrete Model Definition of Generic Material Properties Cable Stay Modeling APPENDIX A ADDENDUM Comments on the Tabular Output APPENDIX B ADDENDUM 3.1 ACI Committee 209 Recommendation (1992) Strength and stiffness Creep strain Shrinkage strain AASHTO Recommendations (1994) Strength and stiffness Creep strain Shrinkage strain Eurocode 2 Recommendation (2004) Strength and stiffness Creep strain Shrinkage strain British Code Recommendations Creep strain Shrinkage strain Hong Kong Code Recommendations Creep strain Shrinkage strain APPENDIX D ADDENDUM Comments on Camber Computations APPENDIX E 4.1 Cable Stay Modeling of Composite Constructions.53 1

3 1. CHAPTER 3 ADDENDUM 1.1 ACI-209 Concrete model (1992) Syntax: CONCRETE PARAMETERS N=? n M=ACI92 To be followed by several lines of concrete parameter specifications. N n = Total number of concrete parameter types; = Concrete model number. Syntax described refers to standard conditions for which creep correction factors are equal to 1. To describe conditions other than standard, the user should specify additional parameters such as: A=? B=? C=? D=? E=? F=? HM=? TH=? VS=? S=? P=? AC=? W=? CURE=? T0=? CC=? A = Strength factor [4.0]; AC = Air content (percent) [6]; B = Strength factor [0.85]; C = Creep factor [0.6]; CC = Cement content [0.015 lb/in 3, 4.15x10-7 kg/mm 3, 4.15x10-4 kg/cm 3 ]; CURE = Curing method of concrete (1 for moist, 2 for steam curing) [1]; D = Creep factor [10.0]; E = Shrinkage factor [1.0]; F = Shrinkage factor (35 for moist, 55 for steam cured concrete) [35]; HM = Ambient relative humidity (percent) [40]; P = Fine aggregate content (percent) [50]; S = Slump [2.65 in, 67 mm, 6.7 cm]; TH = Average member thickness [6 in, 152 mm, 15.2 cm]; T0 = Duration of curing (days) [7]; VS = Volume to surface ratio [1.5 in, 38mm, 3.8cm]; and, W = Density of concrete [8.6806x10-2 lb/in 3, x10-6 kg/mm 3, x10-3 kg/cm 3 ]. 2

4 The values in the brackets are default, standard condition values coded in the program. The detailed description of creep and shrinkage strain calculation, and application of the listed parameters are given in the Appendix B.2.3. Example: CONCRETE PARAMETERS N=1 1 M=ACI92 HM=60 First material type selected uses ACI-209 (1992) concrete model. The input value of humidity is 60% while all other concrete parameters that are not specified have default values. 1.2 AASHTO Concrete Model (1994) Syntax: CONCRETE PARAMETERS N=? n M=AASHTO To be followed by several lines of concrete parameter specifications. N n = Total number of concrete parameter types; = Concrete model number. This instruction means that the n material type selected uses AASHTO concrete model. Other concrete parameters are optional and user may specify as many of these as necessary. They are: A=? B=? C=? HM=? VS=? W=? CURE=? T0=? FPC=? A = Strength factor [4.0]; B = Strength factor [0.85]; C = Creep correction factor [1.56]; HM = Ambient relative humidity, (percent) [70]; VS = Volume-to-surface ratio [38 mm,3.8 cm, 1.5 in]; W = Density of concrete [2.4019x10-6 kg/mm 3, x10-3 kg/cm x10-2 psi]; CURE = Curing method (1 for moist, 2 for steam cured concrete) [1]; T0 = Duration of curing, (days) [7]; and FPC = The 28-day compressive strength of concrete [35 MPa, 350 kg/cm psi]. The values in the brackets are default, standard condition values coded in the program. 3

5 The detailed description of creep and shrinkage strain calculation, and application of the listed parameters are given in the Appendix B Eurocode 2 Concrete Model (2004) Syntax: CONCRETE PARAMETERS N =? n M=EC2 To be followed by several lines of concrete parameter specifications. N n = Total number of concrete parameter types; = Concrete model number; Concrete parameter specifications are optional. If user doesn t specify any concrete parameter, program will consider default, standard condition values. To describe nonstandard condition user can specify the following: HM=? AC= U=? CTYPE=? T0=? FPC=? Where, MPa]. HM = Ambient relative humidity, (percent) [70]; AC = Area of cross section [7.75x10-2 in 2, 50 mm 2, 0.5 cm 2 ]; U = Perimeter of section exposed to drying [3.937x10-2 in, 1 mm, 0.1cm]; CTYPE = Cement type (1 for Class S, 2 for Class N, 3 for Class R of cement) [2]; T0 = Duration of curing, (days) [7]; and, FPC = The 28-day compressive strength of concrete [5000 psi, 350 kg/cm 2, 35 The values in the brackets are default, standard condition values coded in the program. The detailed description of creep and shrinkage strain calculation, and application of the listed parameters are given in the Appendix B.2.5. NOTE: The ultimate creep and shrinkage coefficients are calculated internally by the program and therefore, in CONCRETE PROPERTIES data block, the user should use 1.0 for creep and shrinkage multipliers (Cr and Sr respectively) unless the user wants to have a different ultimate creep and shrinkage strains. In this case the user should input the ratio between the desired coefficient and the one calculated by the program. 4

6 1.4 British Concrete Model The time dependent material specifications can now be defined in accordance with the British code requirements under the CONCRETE PARAMETER command block. The PRESTRESSING STEEL as well as the CONCRETE PROPERTIES command block have been expanded for additional features. The syntax of these two command blocks are partially listed for a description of the new features. Syntax: CONCRETE PARAMETERS N=? n M=BS N n = Total number of concrete parameter types; = Concrete model number. For the British code requirements, the age at loadings and observation times are only internally generated by the program. Consequently, the G=? identifier described in the Concrete Material Generic Input line is not active. Following the Material Generic Input line, an additional line of input data in the following format is needed for the British model. Other parameters needed are defined under the SET command line (such as temperature) in which case the first occurrence of the SET command within the input file is used or under the MESH INPUT block (such as concrete 28 days strength, ultimate creep or creep scale factor, ultimate shrinkage or shrinkage scale factor, mild steel elastic modulus and percentage of mild steel reinforcement). Humidity=?, Thickness=?, Cement=?, WCratio=?, CemType=? Humidity = relative humidity in percent [70 %] Thickness Cement = effective thickness = sectional area divided by half perimeter = in., mm, cm [200 mm] = cement content = lb/yd3,kg/m3,kg/m3, [300 kg/m3] wcratio = water-cement ratio [0.5] CemType = RAPID for rapid hardening cement = PORTLAND for ordinary Portland cement [PORTLAND] Syntax: PRESTRESSING STEEL n Ep=?... R=? 5

7 or StressRatio=? Hours=? LossRatio=? Either R = Relaxation coefficient; or StressRatio = Initial stress to ultimate stress (Fpu) in the test used to determine the relaxation characteristics of the strand [0.80]; Hours = Number of hours of duration of test [1000]; LossRatio = Observed stress loss ratio in the test [0.045] The PRESTRESSING STEEL command has been expanded to allow the specification of the relaxation coefficient (R or C) directly through the R=? identifier or indirectly through the use of the StressLoss=?, Hours=? and LossRatio=? identifiers in which case the relaxation coefficient is internally computed using these three parameters. The command syntax as described in the manual remains unchanged with the above format needed when the indirect method is adopted. Syntax: CONCRETE PROPERTIES N=? n Fpc=? M =? Ac=? \ W=? CrScale=? ShScale=? or or or Weight=? (Cr=? or CrUltimate=?) (Sh=? or ShUltimate=?) Either or CrUltimate or Cr= Ultimate Creep Coefficient [0] CrScale= Scaling coefficient for ultimate creep computed from user specified concrete model [1] else Either or Ultimate creep coefficient assumed zero ShUltimate or Sr= Ultimate Shrinkage Coefficient [0] ShScale= scaling coefficient for ultimate shrinkage computed from user 6

8 else specified concrete model [1] Ultimate shrinkage coefficient assumed zero The CONCRETE PROPERTIES command has been modified and expanded to allow scaling and resetting of the creep and shrinkage ultimate coefficients. Either the W=? or Weight=? identifier can be used to define the unit weight of concrete. For the British creep model, the ultimate creep coefficient is reset by normalizing the 28 days internally generated value to the specified value given under the Cr=? and CrUltimate=? identifiers. The ultimate creep coefficient at other loading ages are scaled with the same factor. The ultimate creep coefficient can be scaled up or down by using the CrScale=? identifier instead. The same concept is applicable to the shrinkage time effects where the ultimate shrinkage coefficient is reset with the Sr=? or SrUltimate=? identifier and is scaled up or down with the SrScale=? identifier. For the British model only, the concrete ultimate strength at 28 days, the ultimate creep and shrinkage coefficients are used in conjunction with the data defined under the CONCRETE PARAMETER block to set up the material time dependent representation. 1.5 Hong Kong Concrete Model The syntax is similar to British concrete model. Syntax: CONCRETE PARAMETERS N=? n M=HK N n = Total number of concrete parameter types; = Concrete model number. Additional parameters that has to be specified are: Humidity=?, Thickness=?, Cement=?, WCratio=?, CemType=? Humidity = relative humidity in percent [70 %] Thickness = effective thickness = sectional area divided by half perimeter = in., mm, cm [200 mm] Cement = cement content = lb/yd3,kg/m3,kg/m3, [300 kg/m3] wcratio = water-cement ratio [0.5] CemType = RAPID for rapid hardening cement 7

9 = PORTLAND for ordinary Portland cement [PORTLAND] Syntax: PRESTRESSING STEEL n Ep=?... R=? or StressRatio=? Hours=? LossRatio=? Either R = Relaxation coefficient; or StressRatio = Initial stress to ultimate stress (Fpu) in the test used to determine the relaxation characteristics of the strand [0.80]; Hours = Number of hours of duration of test [1000]; LossRatio = Observed stress loss ratio in the test [0.045] Syntax: CONCRETE PROPERTIES N=? n Fpc=? M =? Ac=? \ W=? CrScale=? ShScale=? or or or Weight=? (Cr=? or CrUltimate=?) (Sh=? or ShUltimate=?) Either or CrUltimate or Cr= Ultimate Creep Coefficient [0] CrScale= Scaling coefficient for ultimate creep computed from user specified concrete model [1] else Either or Else Ultimate creep coefficient assumed zero ShUltimate or Sr= Ultimate Shrinkage Coefficient [0] ShScale= scaling coefficient for ultimate shrinkage computed from user specified concrete model [1] Ultimate shrinkage coefficient assumed zero For more detailed explanation refer to write-up for British concrete model 8

10 1.6 Definition of Generic Material Properties Besides concrete, the user can define other materials. These are defined in a separate data block called GENERIC PROPERTIES. The generic materials do no have time-dependent properties. Syntax: GENERIC PROPERTIES N=? n ES=Eg NAME=? Where, N = Total number of generic material types n = Generic material type number ES = Modulus of elasticity of generic material (Eg) NAME = Name of generic material Note: When defining generic material property for each frame element in the ELEMENTS data block use S =? instead of C =?. Example: The following is the example of input file for generic material: MESH INPUT NODES N=5 1 X=0.0 Y=0.0! 5 X=0 Y=200 G=1,5 ; Fixed base column CONCRETE PROPERTIES N=1 1 FPC=35.00 CR=0.0 SH=1.0 W=0 M=1 ; Laboratory model GENERIC PROPERTIES N=1 1 ES=35.00 NAME=STEEL SECTION PROPERTIES N=1 1 Area=7854 I=4.909E6 C=50, 50 ; Circular section ELEMENTS N=4 FRAME N=4 1,1,2 C=1 X=1 ST=1 Day=0 ; Laboratory model 2,2,3 C=1 X=1 ST=1 Day=0 3,3,4 C=1 X=1 ST=1 Day=0 4,4,5 S=1 X=1 ST=1 Day=0 ; Generic material property MESH COMPLETE 9

11 1.7 Cable Stay Modeling The following is an addendum to the User s Manual of ADAPT-ABI. It describes the commands specific to cable stay modeling Command Summary and Sequence The natural sequence of cable stay related commands within the body of the entire problem is given in the following. START TITLE UNITS CONCRETE PARAMETERS MESH INPUT NODES...blank line... SEQUENCE...blank line... CONCRETE PROPERTIES MILD STEEL PROPERTIES STAY MATERIAL PROPERTIES STAY ANALYSIS SECTION PROPERTIES OFFSET DATA ELEMENTS FRAME SPRINGS STAY ELEMENT...blank line... PRESTRESSING STEEL TENDON GEOMETRY...blank line... TRAVELERS MESH COMPLETE SET CHANGE STRUCTURE BUILD (Frame Element) RESTRAINTS...blank line... REMOVE ; Elements STRESS ; Tendons DE-STRESS ; Tendons STAY STRESS ; Stays MOVE ; Travelers CHANGE COMPLETE LOADING STAY=? ; Stay temperature 10

12 ...blank line... SOLVE F=? OUTPUT CAMBER STOP List of Commands The commands are organized in alphabetical order. LOADING Syntax : LOADING Stay= n1,n2,inc Temp=? n1 = first stay element number in series [ 0] n2 = last stay element in series [ 0] inc = stay element increment [ 0] Temp = uniform stay element temperature [20C, or 70F] Explanation : A detailed discussion of the LOADING command and the different load categories handled is provided in the user s manual. The following category is added to process temperature effects in cable stay elements. Uniform dead loading over the length of stay is specified under the STAY MATERIAL PROPERTIES command. Temperature effects can be individually specified for each stay element. The reference temperature for all stay elements is taken as the ambient temperature provided within the SET command. The stay element temperature is assumed uniform over the full cable length. SOLVE Syntax : SOLVE... F=? F= displacement increment scale factor [0.9] Explanation : The SOLVE command is expanded to control the solution convergence strategy. The features and options of the SOLVE command remain unchanged and are described in detail in the ABI manual. The cable stay element formulation allows geometrical as well as sag nonlinearities to be included or excluded independently in the solution. The displacement increment scale 11

13 factor is used to scale the full displacement vector by the value specified with the F=? identifier. A smaller F=? value will improve convergence. A larger F=? value will speed up the execution. A compromising F=? value should be used. The 0.9 default value have been found adequate for common problems. STAY ANALYSIS Syntax : STAY ANALYSIS Geometry={include][exclude}\ Sag={[include][exclude]} Geometry = geometrical nonlinearity option [exclude] Sag = sag inclusion option [exclude] Explanation : The STAY ANALYSIS command is used to control the type of analysis to be performed. The cable stay element formulation allows geometrical as well as sag nonlinearities to be included independently in the solution. Small displacement effects also known as geometrical nonlinearities considers the cable stay elements geometry and equilibrium in the deformed configuration. The Geometry= include identifier must be specified to activate this option. Similarly, the Sag= include identifier must be specified to invoke the sag nonlinearity. STAY ELEMENT Syntax : STAY ELEMENT N=? n,ni,nj Material=? N = number of stay elements [1] n = stay element number =< N [0] ni = stay element node I [0] nj = stay element node J [0] Material = material type number [0] Explanation : The STAY ELEMENT command is used to define all the cable stay elements used in modeling the structure. The stay element number n must be less than or equal to the total number of stay elements set on the STAY ELEMENTS command line. The stay element descriptions may be supplied in any order; however each stay element description must be specified or generated once. 12

14 STAY MATERIAL PROPERTIES Syntax : STAY MATERIAL PROPERTIES N=? n Ecable=? Area=? Weight=? Fpu=? Astay=? [[R=?]or [StressRatio=?][LossRatio=?][Hours=?]] N = number of cable materials [ 1] n = stay material number = < N [ 0] Ecable = cable elastic modulus [0.0] Area = cable area [0.0] Weight = stay unit weight per length [0.0] Fpu = cable ultimate stress [0.0] R = cable relaxation coefficient [0.0] Acable = cable thermal expansion coefficient [0.0] Hours = number of hours of duration of test [0.0] LossRatio = observed stress loss ratio in the test [ 0] StressRatio = initial stress to ultimate stress (Fpu) [0.8] in the test to determine the relaxation characteristics of the strand Explanation : The STAY MATERIAL PROPERTIES command is used to specify the material properties of the different stay elements defined in the structure. The stay material type number n must be less or equal to the total number of stay material types set on the STAY MATERIAL PROPERTIES command line. The cable modulus of elasticity Ecable is the stretched elastic modulus value independent of the cable sag. The unit length Weight is used to compute the equivalent modulus of elasticity and the equivalent dead load forces. The Ecable, Area, Fpu, R and Acable parameters relate to the structural resisting cable material of the stay element while the Weight parameter can implicitly account for any additional dead load per unit length over the stay. The stay material types may be supplied in any order; however each cable material type must be specified once. STAY STRESS Syntax : STAY STRESS N=n1,n2,inc / {[Ratio=?][StressTo=?][Force=?]} n1 = first stay element in series [0] n2 = last stay element in series [0] 13

15 inc = element increment [1] Ratio = stress ratio to ultimate stress [0] StressTo= actual stress [0] Force = actual stressing force [0] Explanation : The STAY STRESS command is used to install, stress, restress and remove cable stay elements. The stay elements geometry and material properties must have been input under the STAY MATERIAL PROPERTIES and STAY ELEMENT subcommands of the MESH INPUT command. A stay element is initially stressed by specifying its stressing force under this command. The stressing force can be set with either the Ratio=?, StressTo=? or Force=? identifier. The cable force is initialized to zero and subsequently overridden, when applicable, with the force derived from the stay element ultimate stress specification and its cross sectional area listed under the STAY MATERIAL PROPERTIES subcommand in conjunction with the Ratio=? value specified above. The latter cable force is superseded with the StressTo=? value, when specified, in conjunction with the cable cross sectional area. When applicable, the force defined under the Force=? identifier will take precedence over all previously derived cable forces. Subsequently, a single selection from the { [Ratio=?] [StressTo=?] [Force=?] } list will be sufficient to initialize the stay element stressing force. A stay element can be restressed by specifying a new stressing force under a subsequent application of this command. A cable stay can be removed entirely by the use of this command with a zero stressing force. Stay elements are internally assumed connected to active nodal points. Each end node of the stay element must be connected to at least a single frame element. The formulation of the cable stay element allows small displacements geometrical nonlinear analysis to be performed in which case this option must be invoked in the input data through the STAY ANALYSIS subcommand. Consequently, the stay element installation and subsequent equilibrium solutions will be both performed in the deformed configuration. Otherwise, the undeformed configuration will be considered. STAY ANALYSIS (UNDOCUMENTED) Syntax : STAY ANALYSIS Geometry { [include] [exclude] [incremental] } \ Sag= { [include] [exclude] [incremental] } Geometry = geometrical nonlinearity flag [exclude] 14

16 [exclude] Sag = sag nonlinearity flag Explanation : The STAY ANALYSIS command is used to control the type of analysis to be performed. The cable stay element formulation allows geometrical as well as sag nonlinearities to be included independently in the solution. Small displacement effects also known as geometrical nonlinearities considers the cable stay elements geometry and equilibrium in the deformed configuration. The Geometry= include identifier must be specified to activate this option. Similarly, the Sag= include identifier must be specified to activate the sag nonlinearity. The sag nonlinearities are implicitly accounted for with the use of an equivalent cable stay modulus of elasticity. The equivalent modulus of elasticity accounts for the change in sag configuration in addition to the cable stay extension. The use of the exclude and include options for both the GEOMETRY=? and the SAG=? identifiers is provided to perform a unique type of analysis for the whole input file. An incremental option is introduced to provide incremental control over the cable stay nonlinearities. When the incremental option is used, either the geometrical or the sag nonlinearity which are first initialized to true can be independently reset at every solution phase of the input file. A detailed explanation and syntax of the SOLVE command to account for these features is given next. SOLVE (UNDOCUMENTED) Syntax : SOLVE... Geometry= { [include] [exclude] [incremental] } \ Sag= { [include] [exclude] [incremental] } F=? [0.9] [previous] Geometry = geometrical nonlinearity flag [previous] Sag = sag nonlinearity flag F = displacement increment scale factor Explanation : The SOLVE command is expanded to control the type of analysis to be performed. The features and options of the SOLVE command remain unchanged and are described in detail in the ABI manual. The cable stay element formulation allows geometrical as well as sag nonlinearities to be included or excluded independently in the solution. When the incremental analysis is selected within the STAY ANALYSIS subcommand, the geometrical and sag nonlinearities may be then switched on and off 15

17 independently within the SOLVE command using the GEOMETRY=? and the SAG=? identifiers respectively. The displacement increment scale factor is used to scale the full displacement vector by the same value specified with the F=? identifier. A smaller F=? value will improve on convergence. A larger F=? value will speed up the execution. A compromising F=? value should be used. The 0.9 default value have been found adequate. 16

18 APPENDIX A ADDENDUM - Comments on the Tabular Output The following are comments on several of the data blocks in the tabular output of ADAPT- ABI PRIMARY ELEMENTS For each element, this data block lists the moments at the ends I and J, together with the shear and axial loading in the element. The actions listed are the integral of those acting on the concrete section only. Since the shear and axial loading within an element are assumed constant (actions are concentrated at the nodes when modeling), only one value for shear and axial loading is given. The important item to note for this data block is that in ADAPT-ABI output the actions listed are for the concrete section. This is explained in more detail next. Refer to Fig.1. Part (a) which shows an element with a prestressing tendon. The actions at node J of the element are shown as M, V and N (Actions are shown in the positive direction. Unlike the direction shown for P, the member is generally in compression). The actions on the face J of the member are broken to those due to the prestressing (p) and those due to concrete (c). These are shown in part (b) of the figure. Data block in the pre-capture mode (ADAPT-ABI) lists the actions on the concrete section. These are: Mc,Vc and Nc (refer to Fig. 1-b). Data block in the post-capture mode (ADAPT-Gen) lists M V and N (refer to Fig. 1- a) COMBINED ACTION AND FORCE OF ALL TENDONS This data block lists the actions at the centroid of the concrete portion of each face of an element due to the entire prestressing tendons contained within that element. An element with three tendons is shown in Fig. 2-a. Let the forces in the tendons be P1, P2 and P3. The tendon forces will have a resultant, which is shown as force P in Fig. 2-(b). The resultant force is reacted by an equal and opposite force on the concrete section. The components of the reactive force on the concrete section are shown in Fig. 2-(c) at node J. Obviously, the tendon force and the force on the concrete section will act in opposite directions, but for consistency with the rest of the formulations, actions are shown as vectors in the positive direction. The sign in front of the numerical value given in data blocks determines the direction. 17

19 110 TOTAL STATIC RESULTS FOR ELEMENTS This data block lists the resultant actions (moments, shear and axial loading) at the ends of an element (nodes I and J) due to the combined forces of the concrete section and prestressing. It simulates the free body diagram of an element from which the prestressing tendons are not removed. Consider Fig. 3-(a) which illustrates a beam resting freely on a frictionless bed. The beam is prestressed with a centroidal tendon. Fig. 3(b) shows the free body diagram of a section of this beam. The forces acting on the face of the element are: Tension P on the prestressing tendon Compression P on the concrete section The sum of total forces on the section is (P-P) = 0. Data block 110 will list zero for this condition. 18

20 3. APPENDIX B ADDENDUM 3.1 ACI Committee 209 Recommendation (1992) Strength and stiffness The variation of compressive strength of concrete with time is obtained from the following equation: f ' c t t a bt f ' bg= 28 c + b g (B.2.12) a and b are constants, f c (28) is the 28-day strength and t in days is the age of concrete. The initial modulus of elasticity is defined as: ' E t = x10 w f t (MPa) (B.2.13) c b g c b g w is the density of concrete in kilogram per cubic millimeter and f c(t) is the strength of the concrete in Newton per millimeter squared Creep strain Creep strain of concrete and its variation with time is defined as: v t = t C C t + D v u (B.2.14) for which: vu = Cr γ cr (B.2.15) γ = γ γ λ γ γ γ γ ψ γ α (B.2.16) cr la h vs s C C r = creep factor; = creep coefficient; 19

21 and, D t v u γ cr γ h γ la γ s γ vs γ α γ λ γ ψ = creep factor; = time in days after loading; = ultimate shrinkage strain; = creep correction factor; = creep correction factor for the effect of member size; = creep correction factor for the effect of loading age; = creep correction factor for the effect of slump of concrete; = creep correction factor for the effect of volume to surface ratio; = creep correction factor for the effect of air content; = creep correction factor for the effect of ambient relative humidity; = creep correction factor for the effect of fine aggregate content. The following describes each of the creep correction factors: Creep correction factor for the effect of member size γ h γ h = = R S T R S T 130. TH 2 in 1.17 TH = 3 in 1.11 TH = 4 in 1.04 TH = 5 in TH TH 6 in 130. TH 2 in 1.17 TH = 3 in 1.11 TH = 4 in 1.04 TH = 5 in TH TH 6 in, during the firs year after loading (B.2.17), ultimate values (B.2.18) TH is the average member thickness in inches. Creep correction factor for the effect of loading age γ la = R S T b lag bg la 125. t for moist cured concrete 113. t for steam cured concrete t la is loading age in days. (B.2.19) 20

22 Creep correction factor for the effect of slump of concrete γ s = S (B.2.20) S is the observed slump in inches. Creep correction factor for the effect of volume to surface ratio γ vs = VS. e (B.2.21) VS is the volume-surface ratio of the member in inches. Creep correction factor for the effect of air content γ α = AC 1.0 (B.2.22) AC is the air content in percent. Creep correction factor for the effect of ambient relative humidity γ λ = HM, for HM > 40 (B.2.23) HM is the relative humidity in percent. Creep correction factor for the effect of fine aggregate content γ ψ = P (B.2.24) 21

23 P in percentage is the ratio of the fine aggregate to total aggregate by weight Shrinkage strain Shrinkage of concrete is obtained from the following equation: bε sh gt = b g E t ε t + F E sh u (B.2.25) for which: bε sh g = Sr γ u sr (B.2.26) γ = γ λ γ γ γ γ ψ γ γ α (B.2.27) sr h vs s c and, ratio; E F Sr t γ sr γ c γ h γ s γ vs γ α γ λ humidity γ ψ = shrinkage factor; = shrinkage factor for the effect of curing type; = shrinkage coefficient; = time after the curing; = shrinkage correction factor; = shrinkage correction factor for the effect of cement type; = shrinkage correction factor for the effect of member size; = shrinkage correction factor for the effect of slump of concrete; = shrinkage correction factor for the effect of volume to surface = shrinkage correction factor for the effect of air content; = shrinkage correction factor for the effect of ambient relative = shrinkage correction factor for the effect of fine aggregate content; bε sh g = ultimate shrinkage strain. u The following describes each of the shrinkage correction factors: 22

24 Shrinkage correction factor for the effect of ambient relative humidity γ λ = R S T HM for 40 HM HM for 80 < HM 100 (B.2.28) HM in percentage is the ambient relative humidity. Shrinkage correction factor for the effect of member size γ h γ h = = R S T R S T 135. TH 2 in 1.25 TH = 3 in 1.17 TH = 4 in 1.08 TH = 5 in TH TH 6 in 135. TH 2 in 1.25 TH = 3 in 1.17 TH = 4 in 1.08 TH = 5 in TH TH 6 in TH is the average member thickness in inches., during the firs year after loading (B.2.29), ultimate values (B.2.30) Shrinkage correction factor for the effect of volume to surface ratio γ vs -0.12VS = 12e. (B.2.31) VS is the volume-surface ratio of the member in inches. Shrinkage correction factor for the effect of slump of concrete γ s = S (B.2.32) 23

25 S is the observed slump in inches. Shrinkage correction factor for the effect of fine aggregate content γ ψ = R S T P for P P for P > 50 (B.2.33) P in percentage is the ratio of the fine aggregate to total aggregate by weight. Shrinkage correction factor for the effect of cement type γ c = CC (B.2.34) CC is the cement content in pounds per cubic inch. Shrinkage correction factor for the effect of air content γ α = AC (B.2.35) AC is the air content in percent. 3.2 AASHTO Recommendations (1994) Strength and stiffness The variation of compressive strength of concrete with time is obtained from the following equation: f ' c t t a bt f ' bg= 28 c + b g (B.2.36) a and b are constants, f c (28) is the 28-day strength and t in days is the age of concrete. 24

26 The initial elastic modulus of elasticity is defined as: 1.5 ' E t = 0.043w f t (MPa) (B.2.37) c b g c b g w is the density of concrete in kilogram per cubic meter and f c(t) is the strength of the concrete in Newton per millimeter squared Creep strain The creep coefficient is defined as: b g F I b g HG K J b H Ct,t ckk t t t i = c f i i 10 + t t 0.6 i g C u 0.6 cr (B.2.38) for which: k k f c 62 = 42 + f ' c 45 + t e = V/S 26e b g + t b V/S g (B.2.39) (B.2.40) u C cr = ultimate creep coefficient c = creep correction factor H = ambient relative humidity in percent kc = factor for the effect of the volume-to-surface ratio kf = factor for the effect of concrete strength t = maturity of concrete in days u t i = age of concrete when the load is initially applied, ε cr is the ultimate creep strain V/S = volume-to-surface Shrinkage strain Shrinkage of concrete is obtained from the following equation: t d u ε sh = kk s h sh td + F ε (B.2.41) 25

27 F k h = Curing correction factor, = 35 for moist cure concrete = 55 for steam cured concrete = Humidity correction factor = R S T 143. HM = HM = HM = HM = HM = HM = 90 0 HM = 100 (B.2.42) k s = correction factor for the effect volume to surface ratio and drying time of concrete k s = 26e 45+ t d V/S b g + t d V / S 923 b g (B.2.43) t d u ε sh = drying time of concrete (days) = ultimate shrinkage strain 3.3 Eurocode 2 Recommendations (2004) Strength and stiffness Compressive strength of concrete at an age t can be calculated using the following equation: ' f t = e f c b g L F I HG K J NM s1-28 t 12 O QP ' c b g 28 (MPa) (B.2.44) t in days is the age of the concrete and s is a coefficient that is a function of cement type: s = 0.2 for Class R cement = 0.25 for Class N cement = 0.38 for Class S cement. Modulus of elasticity is defined as: 26

28 E Creep strain c ' 03 fc t = F H G. b g I KJ (MPa) (B.2.45) Creep coefficient and its variation with time is defined as: b g φ t,t = φ 0 β t,t 0 g (B.2.46) 0 c b t t 0 φ 0 = maturity of concrete = age of concrete when the load is initially applied. = notional creep coefficient b g b g = φ β f β t RH cm 0 and where where RH = for f 35 MPa (B.2.47) φ RH h0 F HG I RH φrh = α1 α h0 KJ for f cm > 35 MPa (B.2.48) βbfcmg= f (B.2.49) bg= 0 β t β cbt,t 0g = cm t L NM t-t0 β + t-t H 0 O QP β H.. 0 β = RH h f 35 cm (B.2.50) (B.2.51) b g cm (B.2.52) 18 b RHg h α3 α3 fcm (B.2.12) = H 0 α 1 L 35 = N M O Q P f cm 07. α 2 L 35 = N M O Q P f cm 02. α 3 L 35 = N M O Q P f cm 05. (B.2.53) 27

29 3.3.3 Shrinkage strain Shrinkage of concrete is obtained from the following equation: ε cs = ε cd + ε ca (B.2.54) ε cd is the drying shrinkage strain andε ca is the autogenous shrinkage strain Drying shrinkage strain is defined as: b g b g ε t = β t,t k ε (B.2.55) cd ds s h cd,0 k and, b g h = b R S T 100. h = h = h = h = 500 t t s β ds t,t s = 3 t t s h0 g (B.2.56) (B.2.57) h 0 is defined as: 2AC h0 = (B.2.58) U AC is the concrete cross section area and U is the perimeter exposed to drying. ε cd,0 α f = + α b ge 10 β (B.2.59) ds1 ds2 c ' RH where β RH is defined as: β RH L NM O QP F RH = H G 3 I 155. M1 K J P (B.2.60) 100 α ds1 and α ds2 are functions of cement type and RH is the relative humidity of the environment. 28

30 Cement class S: α ds1 = 3 α ds2 = 0.13 s = 0.38 Cement class N: α ds1 = 4 α ds2 = 0.12 s = 0.25 Cement class R: α ds1 = 6 α ds2 = 0.11 s = 0.20 The autogenous shrinkage strain is as follows: b g b g b g ε t = β t ε (B.2.61) ca as ca ε ca bg b f ck g = (B.2.62) b g 0.2t β as t = 1 e e 0.5 j (B.2.63) 3.4 British Code Recommendations Creep strain Creep coefficient and its variation with time is defined as: Ct b g = CKK r L mkkkk c e j s (B.2.64) C r = Ultimate creep coefficient; = depends on the relative humidity (Figure2.1); K L K m = depends on the hardening of the concrete at the age of loading (Figure 2.2); K c = depends on the composition of concrete (Figure 2.3); K e = depends on the effective thickness of the member (Figure 2.4); K j = defines the development of the time-dependent deformation with time (Figure 2.5);and K s = depends on the percent of reinforcement. 29

31 R T 1 1 = S E 1+ ρ E where ρ = steel ratio = A s /A c s c for plain concrete for reinforced concrete (B.2.65) The values of the coefficients are taken from the following diagrams: Figure 2.1. Coefficient K L (environmental thickness) Figure 2.2. Coefficient K m (hardening at the age of loading) 30

32 Figure 2.3. Coefficient K c (composition of the concrete) Figure 2.4. Coefficient K e (effective thickness) Figure 2.5. Coefficient K j (variation as a function of time) 31

33 3.4.2 Shrinkage strain Shrinkage of concrete can be obtained using the following equation: εbtg = SrKLKCKeKjKS (B.2.66) Sr = shrinkage coefficient; K L = depends on the relative humidity (Figure 2.6); K c = depends on the composition of concrete (Figure 2.3); K e = depends on the effective thickness of the member (Figure 2.7); K j = defines the development of the time-dependent deformation with time (Figure 2.5); and K s = depends on the percent of reinforcement (Equation B.2.65). Figure 2.6. Coefficient K L for shrinkage (environment) 32

34 Figure 2.7. Coefficient K e for shrinkage (effective thickness) 3.5 Hong Kong Code Recommendations Creep strain Creep coefficient and its variation with time is defined as: Ct b g = CKK r L mkkkk c e j s (B.2.67) C r = Ultimate creep coefficient; K L = depends on the relative humidity (Figure 2.1); K m = depends on the hardening of the concrete at the age of loading (Figure 2.2); K c = depends on the composition of concrete (Figure 2.3); K e = depends on the effective thickness of the member (Figure 2.4); K j = defines the development of the time-dependent deformation with time (Figure 2.5);and K s = depends on the percent of reinforcement (Equation B.2.65) Shrinkage strain Shrinkage of concrete can be obtained using the following equation: 33

35 εbtg = CS s rklkckekjks (B.2.68) C s = shrinkage coefficient for Hong Kong ; = 4.0; Sr = shrinkage coefficient; K L = depends on the relative humidity (Figure 2.6); K c = depends on the composition of concrete (Figure 2.3); K e = depends on the effective thickness of the member (Figure 2.7); K j = defines the development of the time-dependent deformation with time (Figure 2.5); and K s = depends on the percent of reinforcement (Equation B.2.65). 34

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