Material Time-Dependent Properties at Sap 2000 [PDF]

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©COMPUTERS AND STRUCTURES, INC., NOVEMBER 2017

TECHNICAL NOTE MATERIAL TIME-DEPENDENT PROPERTIES

General Time-dependent behavior of certain materials may be considered during staged-construction analysis. For concrete materials this behavior includes creep, shrinkage, and age-dependent stiffness. For tendon materials this behavior includes stress relaxation, which is similar to concrete creep. Time-dependent behavior is available for: • •

Concrete frame objects Concrete shell objects using homogenous (thin or thick, not layered) section properties Tendon objects

NNN •

This Technical Note describes the specific formulations implemented in ETABS, SAP2000, and CSiBridge. Note that time-dependent material behavior may only be available for certain license levels in each of the products.

Staged Construction Analysis

To consider the time-dependent behavior of materials, the following steps are required: •

Specify the time-dependent properties for each affected material. This is the primary subject of this Technical Note.

•

For concrete creep and shrinkage, specify the notional size of the affected members. The notional size is specified in the frame section properties and shell/slab/wall section properties.

•

Define one or more staged construction load cases to specify the time sequence for analysis. o

By default, a staged-construction load case does not consider time-dependent material behavior. Select the option “Time1

General

2

Dependent Material Properties” under the Nonlinear Parameters for the load case. o

o

For each stage, specify the duration (time in days): 

If the duration is zero for a given stage, no timedependent change in material behavior is considered for that stage.



If the duration is positive, all specified stage operations (such as adding, removing, loading, and modifying objects) will be performed first, and occur instantaneously, before the time-dependent behavior is applied.

Specify the “Age at Add” for added objects, which indicates how old the material is before it becomes active in the model. This only affects concrete materials, and would typically express the age since casting a precast member or the age since pouring a cast-in-place member and until removing the forms.

NNN •

Staged-construction load cases can be sequenced together, in which case the durations are cumulative. Nonlinear static and nonlinear direct-integration load cases that are included in such a sequence are treated as zero-duration stages for the purpose of time-dependent material behavior.

For more information on staged-construction analysis, see the CSI Analysis Reference Manual, chapter “Nonlinear Static Analysis”.

Time Dependent Properties for Concrete Materials For concrete materials you may specify one or more of the following types of time-dependent behavior: •

Compressive Strength and Stiffness: This determines how the elastic moduli change with age. This behavior scales the elastic moduli specified for the material, whether isotropic, orthotropic, or anisotropic. It has no effect on the stress-strain curve used for nonlinear analysis.

General

3

The relationship between the mean elastic modulus at time 𝑡𝑡, 𝐸𝐸𝑐𝑐𝑐𝑐 (𝑡𝑡), and the specified elastic modulus, E, is: 𝐸𝐸𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝛽𝛽𝐸𝐸 (𝑡𝑡) 𝐸𝐸

(1.1)

where the time dependent elastic modulus coefficient 𝛽𝛽𝐸𝐸 (𝑡𝑡) is defined by the Time Dependent Type, as discussed in the following chapters. The elastic modulus coefficient is applied to both the elastic modulus and the shear modulus as specified in the material property definition. The mean concrete compressive strength at time 𝑡𝑡 is denoted 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) while the mean concrete compressive strength at 28 days is denoted 𝑓𝑓𝑐𝑐𝑐𝑐28 and the specified concrete compressive strength is 𝑓𝑓𝑐𝑐′ . The variation of concrete compressive strength over time may be viewed in the Material Property Time Dependent Plot form but is not used during analysis. Loads applied to older concrete will produce smaller instantaneous elastic strains than for younger concrete. Loads applied near zero age will result in unrealistic strains and deflections.

NNN •

Creep: This determines the change in strain with age under the action of stress. The creep strain at time t for a load 𝜎𝜎𝑐𝑐 applied at time 𝑡𝑡0 is: 𝜎𝜎 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) � 𝑐𝑐�𝐸𝐸

�

𝑐𝑐𝑐𝑐 (𝑡𝑡0 )

(1.2)

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) is the creep coefficient as defined by the Time Dependent Type, as discussed in the following chapters. Note that the creep strain is dependent on the stiffness at the time the load is applied, 𝐸𝐸𝑐𝑐𝑐𝑐 (𝑡𝑡0 ), which includes stiffness modifiers applied on the element and section properties. Every strain increment, whether due to an applied load or due to load redistribution caused by creep itself in a statically indeterminate structure, will increase with age due to creep. Creep behavior is additive as each strain increment ever experienced by the material maintains its own creep history.

•

Shrinkage: This determines the change in direct strains with time due to drying shrinkage. At time t, the strain due to drying shrinkage is denoted 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡). A positive value of 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) indicates swelling while a negative value indicates shrinkage. Shrinkage is independent of loading.

General

4

These three behaviors are independent, except that the instantaneous elastic strain used to compute creep does depend on the current modulus of elasticity, and this is affected by stiffness-aging if considered. The effects of the following items on the three time-dependent behaviors are not considered: • • • • •

Sustained loads Curing temperature Effect of high stress Temperature effects Variation of concrete strength due to confinement

Temperature effects are not considered explicitly but may be simulated in some cases by adjusting the duration of stages. Temperatures above 20°C would use a longer duration to cause greater aging, and temperatures below 20°C would use shorter durations.

NNN

Age at Add

Time-dependent concrete behavior is strongly affected by the age at which it begins the time-dependent behavior, particularly for age dependent stiffness. When adding objects during a staged-construction analysis, you specify an “Age at Add”, denoted 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 , which should be a realistic value. For precast members, this would be the age since casting at which the member is added to the structure. For a cast-in-place member, this would be the age since casting at which forms and supports are removed and the member begins to carry load.

Specifying zero or a very small value for 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 will result in unrealistically large strains if any load is applied to the member, since concrete is liquid at an early age. If you do specify zero, the analysis will set this to 0.001 days (1.44 minutes). This is still too small, but does provide some limit. A more realistic value would typically be on the order of one to ten days, or no less than a few hours (0.1 day) in the most extreme case. Alternatively, you can specify a small value for 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 provided that the member is fully supported and experiences no load before it is strong enough to have its forms removed and support load on its own.

General

5

The value of 𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 may be different for every member. It does not affect the time scale of the analysis, but rather refers to the history of the member before it becomes part of the structure.

Notional Size Notional size, denoted h, is used for creep and shrinkage time-dependent analysis to determine the rate moisture can move through a section. The notional size can be program-determined (auto) or directly specified by the user. When the notional size for a frame or shell/slab/wall section is programdetermined, the notional size is computed as:

where:

𝐴𝐴 𝑃𝑃

ℎ = 𝑆𝑆𝑆𝑆 ∙ 2 � �

(1.3)

h is the notional size SF is the notional size factor, equal to 1 unless specified by the user A is the cross-sectional area of the section P is the cross-sectional perimeter of the section

NNN

For a thin-walled frame section, h is on the order of the section thickness. For a circular section, h is equal to the radius. For a shell section, h is usually the thickness of the section. The notional size is twice the volume-to-surface ratio used in some building codes.

Strain Compatibility with Embedded Reinforcing Steel Bars For creep and shrinkage of concrete members, the effect of strain compatibility with embedded reinforcing steel bars is accounted for using the steel reinforcement correction factor, 𝑅𝑅𝑐𝑐𝑐𝑐 . 𝑅𝑅𝑐𝑐𝑐𝑐 is used to scale the incremental creep and shrinkage strain at each step. For a time step going from time 𝑡𝑡𝑖𝑖−1 to time 𝑖𝑖𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖 , the incremental creep strain 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) and incremental shrinkage strain 𝑖𝑖𝑖𝑖𝑖𝑖 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ), is given as: 𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) = 𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) ∙ [𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖 ) − 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 )] 𝜀𝜀𝑐𝑐𝑐𝑐

where:

𝑖𝑖𝑖𝑖𝑖𝑖 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) = 𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) ∙ [𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖 ) − 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 )]

(1.4) (1.5)

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖 ) and 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 ) are the total creep strains at time 𝑡𝑡𝑖𝑖 and 𝑡𝑡𝑖𝑖−1, respectively.

General

6

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖 ) and 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 ) are the total shrinkage strains at time 𝑡𝑡𝑖𝑖 and 𝑡𝑡𝑖𝑖−1 , respectively.

𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) is the steel reinforcement correction factor for a time step going from time 𝑡𝑡𝑖𝑖−1 to time 𝑡𝑡𝑖𝑖 . For each degree of freedom in the concrete frame section, the steel reinforcement correction factor is calculated as: 𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) = �1 + 𝑛𝑛(𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) ∙

𝜌𝜌𝑔𝑔𝑔𝑔𝑔𝑔

−1

��1 − 𝜌𝜌 �� 𝑔𝑔𝑔𝑔𝑔𝑔

(1.6)

where 𝜌𝜌𝑔𝑔𝑔𝑔𝑔𝑔 is the generalized reinforcing steel ratio given as the following: Axial and Shear: cross-sectional area 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 /𝐴𝐴𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 Torsion: torsion constant 𝐽𝐽𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 /𝐽𝐽𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑠𝑠 Bending: second moment of area 𝐼𝐼𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 /𝐼𝐼𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔

𝑛𝑛(𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) is the ratio between stiffness of the reinforcing steel and the stiffness of concrete for the respective degree of freedom, given as:

NNN 𝑛𝑛(𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) =

𝐸𝐸𝑠𝑠 0.5[𝐸𝐸𝑐𝑐𝑐𝑐 (𝑡𝑡)+𝐸𝐸𝑐𝑐𝑐𝑐 (𝑡𝑡−1)] 𝐺𝐺 0.5[𝐺𝐺𝑐𝑐𝑐𝑐 (𝑡𝑡)+𝐺𝐺𝑐𝑐𝑐𝑐 (𝑡𝑡−1)]

𝑎𝑎𝑎𝑎𝑖𝑖𝑎𝑎𝑎𝑎, 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑠𝑠ℎ𝑒𝑒𝑒𝑒𝑒𝑒, 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

(1.7)

where 𝐸𝐸𝑐𝑐𝑐𝑐 and 𝐺𝐺𝑐𝑐𝑐𝑐 are the time-dependent concrete elastic and shear moduli, as discussed in the CEB-FIP 1990 Concrete Materials section of this manual. Note that stiffness modifiers assigned to the section and element will be applied to the concrete elastic and shear moduli but will not be applied to the reinforcing steel elastic and shear moduli, 𝐸𝐸𝑠𝑠 and 𝐺𝐺𝑠𝑠 . For a shell/area/wall section, the steel reinforcement is assumed to be placed in two orthogonal directions in the plane of the shell element. The two directions are referred to as the local reinforcement axes, 1 and 2. The local reinforcement directions do not have to coincide with the local 1- and 2directions of the shell element. For shell/slab/wall elements, the area of concrete is assumed to be equal to the gross area of the section.

The steel reinforcement correction factor may be coupled in the membrane and plate directions. 𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) is given as a tensor related to the constitutive relation for concrete, 𝑪𝑪𝑐𝑐 , and constitutive relation for reinforcing steel scaled with the generalized reinforcing steel ratio in each direction, 𝑪𝑪𝑠𝑠 . 𝑅𝑅𝑐𝑐𝑐𝑐 (𝑡𝑡𝑖𝑖−1 , 𝑡𝑡𝑖𝑖 ) = [𝑪𝑪𝑠𝑠 + 𝑪𝑪𝑐𝑐 (𝑡𝑡𝑖𝑖 , 𝑡𝑡𝑖𝑖−1 )]−1 : 𝑪𝑪𝑐𝑐 (𝑡𝑡𝑖𝑖 , 𝑡𝑡𝑖𝑖−1 )

(1.8)

General

𝑪𝑪𝑐𝑐 (𝑡𝑡𝑖𝑖 , 𝑡𝑡𝑖𝑖−1 ) = 0.5 [𝛽𝛽𝐸𝐸 (𝑡𝑡𝑖𝑖 ) + 𝛽𝛽𝐸𝐸 (𝑡𝑡𝑖𝑖−1 )] 𝑪𝑪𝑐𝑐

7

(1.9)

where 𝑪𝑪𝑐𝑐 is the constitutive relation based on the specified concrete material properties and 𝛽𝛽𝐸𝐸 (𝑡𝑡𝑖𝑖 ) is the time dependent elastic modulus coefficient discussed in the CEB-FIP 1990 Concrete Materials section of this manual. For the membrane degrees of freedom, 𝑪𝑪𝑠𝑠 has the following form in the local reinforcement axis: 𝜌𝜌11 𝐸𝐸𝑠𝑠 𝑪𝑪𝑠𝑠,𝑚𝑚 = � 0 0

0 𝜌𝜌22 𝐸𝐸𝑠𝑠 0

0 0 � 0.5𝐺𝐺𝑠𝑠 𝜌𝜌12

(1.10)

where 𝜌𝜌11 , 𝜌𝜌22 , and 𝜌𝜌12 are the reinforcing steel ratio associated with the membrane f11, f22, and f12 directions, respectively, in the reinforcing local axes. For the plate being degrees of freedom, 𝑪𝑪𝑠𝑠 has the following form in the local reinforcement axis:

NNN 𝑪𝑪𝑠𝑠,𝑝𝑝𝑝𝑝𝑝𝑝

𝜌𝜌𝑚𝑚11 𝐸𝐸𝑠𝑠 =� 0 0

0

𝜌𝜌𝑚𝑚22 𝐸𝐸𝑠𝑠 0

0 0 � 0.5𝐺𝐺𝑠𝑠 𝜌𝜌𝑚𝑚12

(1.11)

where 𝜌𝜌𝑚𝑚11, 𝜌𝜌𝑚𝑚22, and 𝜌𝜌𝑚𝑚12 are the generalized reinforcing steel ratio associated with the out-of-plane bending m11, m22, and m12 directions, respectively, in the reinforcing local axes. For the plate shear degrees of freedom, 𝑪𝑪𝑠𝑠 has the following form in the local reinforcement axis: 𝑪𝑪𝑠𝑠,𝑝𝑝𝑝𝑝𝑝𝑝 = �

𝜌𝜌13 𝐺𝐺𝑠𝑠 0

0 � 𝜌𝜌23 𝐺𝐺𝑠𝑠

(1.12)

where 𝜌𝜌13 and 𝜌𝜌23 are the reinforcing steel ratio associated with the out-ofplane shear v13 and v23 directions, respectively, in the reinforcing local axes. The steel reinforcement correction factor used in this section is based on the following paper: Baker, W.F., Korista D.S., Noval, L.C., Pawlikowski, J., and Young B. (2007). “Creep and Shrinkage and the Design of Supertall Buildings – A Case Study: The Burj Debai Tower”. ACI Special Publications, 246, 133-148.

General

8

Time Dependent Properties for Steel Tendon Materials Stress relaxation is the change in strain with age under the action of stress and is similar to creep for concrete materials. The stress relaxation at time t for a load 𝜎𝜎𝑠𝑠 applied at time 𝑡𝑡0 is: 𝜎𝜎 𝜀𝜀𝑠𝑠𝑠𝑠 (𝜎𝜎𝑠𝑠 , 𝑡𝑡, 𝑡𝑡0 ) = 𝜙𝜙𝑠𝑠𝑠𝑠 (𝜎𝜎𝑠𝑠 , 𝑡𝑡, 𝑡𝑡0 ) � 𝑠𝑠�𝐸𝐸 � 𝑠𝑠

(1.13)

𝜙𝜙𝑠𝑠𝑠𝑠 (𝑡𝑡, 𝑡𝑡0 ) is the stress relaxation coefficient as defined by the Time Dependent Material Type, as discussed in the following chapters. Currently, stress relaxation for Tendon materials is only available for Time Dependent Material Type CEB-FIP 1990. Note that stiffness 𝐸𝐸𝑠𝑠 used to calculate the stress relaxation includes stiffness modifiers applied on the element and section properties. Similar to creep in concrete materials, every stress increment, whether due to an applied load or due to load redistribution caused by stress relaxation itself in a statically indeterminate structure, will cause a corresponding stress relaxation effect. The stress relaxation behavior is additive as each stress increment ever experienced by the material maintains its own stress relaxation history.

NNN

CEB-FIP 1990

9

CEB-FIP 1990 All equations presented in this section are intended for use with the units of Megapascal (MPa) and millimeters (mm) unless specifically marked with units.

Concrete Materials In this section, the units of time used are days unless otherwise stated specifically.

Input Parameters In addition to the elastic moduli E and concrete strength 𝑓𝑓𝑐𝑐′ specified in the material property definition, the following parameters are available: Affects

NNN Parameter

Stiffness

Cement Type Coefficient, s

Creep

Shrinkage

Y

Y

Y

Relative Humidity, RH (%) Shrinkage Coefficient, 𝛽𝛽

𝑠𝑠𝑠𝑠

Y

Shrinkage Start Age, 𝑡𝑡 (days) 𝑠𝑠

Y

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days is: 𝑓𝑓𝑐𝑐𝑐𝑐28 = 𝑓𝑓𝑐𝑐′ + 8 𝑀𝑀𝑀𝑀𝑀𝑀

(2.1)

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) 𝑓𝑓𝑐𝑐𝑐𝑐28

(2.2)

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡), the mean compressive strength of concrete at age t (days) is given as:

where:

0.5

𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 �𝑠𝑠 �1 − �28�𝑡𝑡�

��

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 2.1 s is the user-defined cement type coefficient parameter

𝛽𝛽𝐸𝐸 (𝑡𝑡), the elastic modulus coefficient at age t (days) is given as:

(2.3)

CEB-FIP 1990 10

𝛽𝛽𝐸𝐸 (𝑡𝑡) = [𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡)]0.5

(2.4)

where 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) is as defined in Eq. 2.3.

Creep

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: where:

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝜑𝜑0 ∙ 𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 )

(2.5)

𝜑𝜑0 = 𝜑𝜑𝑅𝑅𝑅𝑅 𝛽𝛽(𝑓𝑓𝑐𝑐𝑐𝑐28 ) 𝛽𝛽(𝑡𝑡0 )

(2.6)

𝜑𝜑𝑅𝑅𝑅𝑅 = 1 + �

1 − 𝑅𝑅𝑅𝑅�100

1� � 0.46 �ℎ�100� 3

𝛽𝛽(𝑓𝑓𝑐𝑐𝑐𝑐28 ) = 5.3�(0.1

𝑓𝑓𝑐𝑐𝑐𝑐28 )0.5

NNN 𝛽𝛽(𝑡𝑡0 ) = 1� (0.1 + 𝑡𝑡00.2 )

𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 ) = �

and:

0.3 (𝑡𝑡−𝑡𝑡0 ) � βH +(𝑡𝑡−𝑡𝑡0 )

18 𝛽𝛽𝐻𝐻 = 1.5 ℎ �1 + �1.2 𝑅𝑅𝑅𝑅�100� � + 250 ≤ 1500

(2.7) (2.8) (2.9) (2.10) (2.11)

h is the notional size in mm RH is the relative humidity in percentage (%) 𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 2.1

Shrinkage

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as: where:

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = 𝜀𝜀𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝛽𝛽𝑠𝑠 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 )

(2.12)

𝜀𝜀𝑠𝑠𝑠𝑠𝑠𝑠 = 𝜀𝜀𝑠𝑠 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) 𝛽𝛽𝑅𝑅𝑅𝑅

(2.13)

𝜀𝜀𝑠𝑠 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) = [160 + 10 𝛽𝛽𝑠𝑠𝑠𝑠 (9 − 0.1 𝑓𝑓𝑐𝑐𝑐𝑐28 )] × 10−6

(2.14)

CEB-FIP 1990 11 3

𝑅𝑅𝑅𝑅 𝛽𝛽𝑅𝑅𝑅𝑅 = −1.55 �1 − � �100� � 40% ≤ 𝑅𝑅𝑅𝑅 ≤ 99% −0.25 𝑅𝑅𝑅𝑅 ≥ 99% and:

𝛽𝛽𝑠𝑠 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) = �

0.5 (𝑡𝑡−𝑡𝑡𝑠𝑠 ) � 0.035 ℎ2 + (𝑡𝑡−𝑡𝑡𝑠𝑠 )

(2.15) (2.16)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 2.1 𝛽𝛽𝑠𝑠𝑠𝑠 is the user-defined shrinkage coefficient RH is the relative humidity in percentage (%) h is the notional size in mm

Steel Tendon Materials In this section, the units of time used are hours unless otherwise stated specifically.

Input Parameters

NNN

In addition to the characteristic tendon strength 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝 specified in the material property definition, the CEB-FIP relaxation class must be specified as either Class 1 or Class 2. The relaxation class affects the calculation of the stress relaxation coefficient for tendons.

Stress Relaxation

𝜙𝜙𝑠𝑠𝑠𝑠 (𝜎𝜎, 𝑡𝑡), the stress relaxation coefficient at time t (hours) for an stress, 𝜎𝜎𝑠𝑠 , applied at time 𝑡𝑡0 (days) is given as: where:

(𝑡𝑡−𝑡𝑡0 ) 𝑘𝑘 � 1000

𝜙𝜙𝑠𝑠𝑠𝑠 (𝜎𝜎𝑠𝑠 , 𝑡𝑡, 𝑡𝑡0 ) = 𝜙𝜙1000 ∙ �

(2.17)

𝜙𝜙1000 is the relaxation coefficient after 1000 hours and is dependent on the applied stress, 𝜎𝜎, normalized by the characteristic strength of the tendon, 𝑓𝑓𝑝𝑝𝑝𝑝𝑝𝑝 . 𝜙𝜙1000 is linearly interpolated between the following values: Stress level �𝝈𝝈�𝒇𝒇 0.60 0.65

𝒑𝒑𝒑𝒑𝒑𝒑

�

Class 1

𝝓𝝓𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

Class 2

0.04

0.01

0.06

0.013

CEB-FIP 1990 12

0.70

0.08

0.02

0.75

0.10

0.032

0.80

0.12

0.05

Relaxation Class 1 2

𝒌𝒌

𝑘𝑘 is dependent on the user specified relaxation class, as: 0.12 0.19

References The following lists the equation numbers in CEB-FIP 1990 which correspond to the equations used in this chapter: Equation number 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

CEB-FIP 1990 Reference Eq. 2.1-1 Eq. 2.1-53 Eq. 2.1-54 Eq. 2.1-58 Eq. 2.1-64 Eq. 2.1-65 Eq. 2.1-66 Eq. 2.1-67 Eq. 2.1-68

Equation number 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17

CEB-FIP 1990 Reference Eq. 2.1-70 Eq. 2.1-71 Eq. 2.1-74 Eq. 2.1-75 Eq. 2.1-76 Eqs. 2.1-77, 2.1-78 Eq. 2.1-79 Sec. 2.3.4.5

NNN

CEB-FIP 2010 13

CEB-FIP 2010 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli E and concrete strength 𝑓𝑓𝑐𝑐′ specified in the material property definition, the following parameters are available: Affects Parameter Stiffness Relative Humidity, RH (%)

Creep

Shrinkage

Y

Y

NNN Shrinkage Start Age, 𝑡𝑡 (days) Cement Type

Y

Y

Y

Lightweight Concrete *

Y

Y

Y

𝑠𝑠

Y

Lightweight Oven-dry Density 𝜌𝜌 (kg/m3) *

Y

* Lightweight Concrete Only

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days is:

where

𝑓𝑓𝑐𝑐𝑐𝑐28 = 𝑓𝑓𝑐𝑐′ + 8 𝑀𝑀𝑀𝑀𝑀𝑀

(3.1)

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) 𝑓𝑓𝑐𝑐𝑐𝑐28

(3.2)

𝑓𝑓𝑐𝑐′

is the specified concrete compressive strength

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡), the mean compressive strength of concrete at age t (days) is given as:

where:

0.5

𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 �𝑠𝑠 �1 − �28�𝑡𝑡�

��

(3.3)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 3.1 s is a coefficient based on the strength class of the cement, as given in the table below.

CEB-FIP 2010 14

After CEBFIP 2010 Table 5.1-9 𝒇𝒇𝒄𝒄𝒄𝒄 [MPa]

Strength class of cement

s

32.5N

0.38

32.5R, 42.5N

0.25

42.5R, 52.5N, 52.5R

0.20

> 60, Normal

All Classes

0.20

≤ 60, Lightweight

All Classes

0.25

> 60, Lightweight

All Classes

0.05

≤ 60, Normal

𝛽𝛽𝐸𝐸 (𝑡𝑡), the elastic modulus coefficient at age t (days) is given as: 𝛽𝛽𝐸𝐸 (𝑡𝑡) = [𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡)]0.5

(3.4)

where 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) is as defined in Eq. 3.3.

NNN

Creep

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝜂𝜂𝐸𝐸 ∙ [𝜑𝜑𝑏𝑏𝑏𝑏 (𝑡𝑡 − 𝑡𝑡0 ) + 𝜑𝜑𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡0 )]

𝜑𝜑𝑏𝑏𝑏𝑏 (𝑡𝑡 − 𝑡𝑡0 ) is the basic creep coefficient, given as: 𝜑𝜑𝑏𝑏𝑏𝑏 (𝑡𝑡 − 𝑡𝑡0 ) = (𝑓𝑓

1.8

𝑐𝑐𝑐𝑐28 )

where:

𝛽𝛽𝑏𝑏𝑏𝑏 (𝑡𝑡, 𝑡𝑡0 ) = ln ��

30

0.7

𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎

∙ 𝛽𝛽𝑏𝑏𝑏𝑏 (𝑡𝑡, 𝑡𝑡0 ) 2

+ 0.035� (𝑡𝑡 − 𝑡𝑡0 ) + 1�

(3.5)

(3.6) (3.7)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 3.1 ln( ) is the natural logarithm 𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎 is the load time adjusted for cement type (days), given as: 9

𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑡𝑡0 � 2+(𝑡𝑡

0

)1.2

𝛼𝛼

� ≥ 0.5

(3.8)

𝛼𝛼 is a coefficient based on the strength class of the cement, as given in the table below.

CEB-FIP 2010 15

Strength class of cement

𝜶𝜶

32.5N

-1

32.5R, 42.5N

0

42.5R, 52.5N, 52.5R

1

𝜑𝜑𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡0 ) is the drying creep coefficient, given as:

𝜑𝜑𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡0 ) = 𝛽𝛽𝑑𝑑𝑑𝑑 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) ∙ 𝛽𝛽(𝑅𝑅𝑅𝑅) ∙ 𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡0 ) ∙ 𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡0 ) 412

𝛽𝛽𝑑𝑑𝑑𝑑 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) = (𝑓𝑓 𝛽𝛽(𝑅𝑅𝑅𝑅) =

𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡0 ) =

𝑐𝑐𝑐𝑐28 )

1−𝑅𝑅𝑅𝑅�100 3

1.4

√0.001 ℎ 1

0.2

(3.12) 𝛾𝛾(𝑡𝑡0 )

NNN 𝛾𝛾(𝑡𝑡0 ) =

(𝑡𝑡−𝑡𝑡0 )

𝛽𝛽ℎ +(𝑡𝑡−𝑡𝑡0 1

2.3+3.5� �𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎

� )

𝛽𝛽ℎ = 1.5 ℎ + 250 𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 ≤ 1500 𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓

where:

𝛼𝛼𝑓𝑓𝑓𝑓𝑓𝑓 = �

35

𝑓𝑓𝑐𝑐𝑐𝑐28

0.5

�

(3.10) (3.11)

0.1+�𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎 �

𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡0 ) = �

(3.9)

(3.13) (3.14) (3.15) (3.16)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 3.1 RH is the relative humidity in percentage (%) h is the notional size in mm 𝑡𝑡0,𝑎𝑎𝑎𝑎𝑎𝑎 is the load time adjusted for cement type (days) from Eq. 3.12

𝜂𝜂𝐸𝐸 is the lightweight oven-dry density ratio modification factor. If normalweight concrete is used, the factor 𝜂𝜂𝐸𝐸 = 1. If lightweight concrete is used, 𝜂𝜂𝐸𝐸 given as: 2 𝜌𝜌 𝜂𝜂𝐸𝐸 = � �2200�

(3.17)

where 𝜌𝜌 is the user-defined lightweight oven-dry density in units of 𝑘𝑘𝑘𝑘/𝑚𝑚3

CEB-FIP 2010 16

Shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = 𝜂𝜂𝑠𝑠 ∙ [𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) + 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 )]

(3.18)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐0 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) ∙ 𝛽𝛽𝑏𝑏𝑏𝑏 (𝑡𝑡)

(3.19)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) is the basic shrinkage strain, given as: 0.1 𝑓𝑓

2.5

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐0 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) = −𝛼𝛼𝑏𝑏𝑏𝑏 �6+0.1𝑐𝑐𝑐𝑐28 � 𝑓𝑓 where:

𝛽𝛽𝑏𝑏𝑏𝑏 (𝑡𝑡) = 1 − 𝑒𝑒𝑒𝑒𝑒𝑒�−0.2 √𝑡𝑡�

𝑐𝑐𝑐𝑐28

× 10−6

(3.20) (3.21)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq 3.1 𝛼𝛼𝑏𝑏𝑏𝑏 is a coefficient based on the strength class of the cement, as given in the table below.

NNN 𝜶𝜶𝒃𝒃𝒃𝒃

Strength class of cement 32.5N

800

32.5R, 42.5N

700

42.5R, 52.5N, 52.5R

600

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) is the drying shrinkage strain, given as:

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐0 (𝑓𝑓𝑐𝑐𝑐𝑐 ) ∙ 𝛽𝛽𝑅𝑅𝑅𝑅 (𝑅𝑅𝑅𝑅) ∙ 𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 )

(3.22)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐0 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) = [(220 + 110𝛼𝛼𝑑𝑑𝑑𝑑1 ) 𝑒𝑒𝑒𝑒𝑒𝑒(−𝛼𝛼𝑑𝑑𝑑𝑑2 𝑓𝑓𝑐𝑐𝑐𝑐28 )] × 10−6 (3.23) 𝛽𝛽𝑅𝑅𝑅𝑅 =

3 −1.55 �1 − �𝑅𝑅𝑅𝑅�100� � 40% ≤ 𝑅𝑅𝑅𝑅 < 99% 𝛽𝛽𝑠𝑠1

𝛽𝛽𝑠𝑠1 = �𝑓𝑓

35

𝑐𝑐𝑐𝑐28

and:

0.1

�

𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) = �

−0.25 ≤ 1.0

(𝑡𝑡−𝑡𝑡𝑠𝑠 )

0.035 ℎ2 +(𝑡𝑡−𝑡𝑡𝑠𝑠

𝑅𝑅𝑅𝑅 ≥ 99% 𝛽𝛽𝑠𝑠1

0.5

� )

h is the notional size in mm RH is the relative humidity in percentage (%)

(3.24)

(3.25) (3.26)

CEB-FIP 2010 17

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength, given in Eq. 3.1 𝛽𝛽𝑠𝑠𝑠𝑠 is the user-defined shrinkage coefficient 𝛼𝛼𝑏𝑏𝑏𝑏1 is a coefficient based on the strength class of the cement, as given in the table below.

32.5N

𝜶𝜶𝒃𝒃𝒃𝒃𝒃𝒃

32.5R, 42.5N

4

42.5R, 52.5N, 52.5R

6

Strength class of cement

3

𝛼𝛼𝑏𝑏𝑏𝑏2 is a coefficient based on the strength class of the cement, as given in the table below. 𝜶𝜶𝒃𝒃𝒃𝒃𝒃𝒃

Strength class of cement

NNN 32.5N

0.013

32.5R, 42.5N, 42.5R, 52.5N, 52.5R

0.012

Lightweight Concrete Grade

𝜼𝜼

𝒔𝒔

𝜂𝜂𝑠𝑠 is the lightweight oven-dry density ratio modification factor. If normalweight concrete is used, the factor 𝜂𝜂𝑠𝑠 = 1. If lightweight concrete is used, 𝜂𝜂𝑠𝑠 given by the following table. LC8 to LC16

1.5

LC20 to LC80

1.2

References The following lists the equation numbers in CEB-FIP 2010 which correspond to the equations used in this chapter: Equation number 3.1 3.2 3.3 3.4

CEB-FIP 2010 Reference Eq. 5.1-1 Eq. 5.1-50 Eq. 5.1-51 Eq. 5.1-57

Equation number 3.14 3.15 3.16 3.17

CEB-FIP 2010 Reference Eq. 5.1-71b Eq. 5.1-71c Eq. 5.1-71d Eq. 5.1-72

CEB-FIP 2010 18

3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13

Eqs. 5.1-60, 5.1-72 Eqs. 5.1-64, 5.1-65 Eq. 5.1-66 Eq. 5.1-73 Eq. 5.1-67 Eq. 5.1-68 Eq. 5.1-69 Eq. 5.1-70 Eq. 5.1-71a

3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26

Eqs. 5.1-75, 5.1-84 Eq. 5.1-76 Eq. 5.1-78 Eq. 5.1-79 Eq. 5.1-77 Eq. 5.1-80 Eq. 5.1-81 Eq. 5.1-83 Eq. 5.1-82

NNN

ACI 209R-92 19

ACI 209R-92 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness Relative Humidity, RH (%)

Creep

Shrinkage

Y

Y

NNN

Y

𝑠𝑠

Shrinkage Start Age, 𝑡𝑡 (days)

Compressive Strength Factor, 𝑎𝑎

Compressive Strength Factor, 𝛽𝛽

Y Y

Curing Type

Y

Y

Slump, s (mm)

Y

Y

Fine-Aggregate Percentage, 𝜓𝜓 (%)

Y

Y

Y

Y

Air Content, 𝛼𝛼 (%)

3

Cement Content, c (lb/yd )

Y

Moist Curing Duration, 𝑡𝑡𝑐𝑐 (days) *

Y

* Moist Curing Duration defaults to 7 days and is currently not user-specified

Compressive Strength and Stiffness The mean compressive strength of concrete at an age of 28 days, 𝑓𝑓𝑐𝑐𝑐𝑐28, is taken to be equal to the specified concrete compressive strength, 𝑓𝑓𝑐𝑐′ . 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡), the mean compressive strength of concrete at age t (days) is given as: 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) =

𝑡𝑡 𝑎𝑎+𝛽𝛽𝛽𝛽

𝑓𝑓𝑐𝑐𝑚𝑚28

where 𝑎𝑎 and 𝛽𝛽 are user-defined compressive strength factors.

(4.1)

ACI 209R-92 20

𝛽𝛽𝐸𝐸 (𝑡𝑡), the elastic modulus coefficient at age t (days) is given as: 𝑡𝑡

0.5

𝛽𝛽𝐸𝐸 (𝑡𝑡) = �𝑎𝑎+𝛽𝛽𝛽𝛽�

(4.2)

where 𝑎𝑎 and 𝛽𝛽 are user-defined compressive strength factors.

Creep

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: (𝑡𝑡−𝑡𝑡 )0.60 0.60 � 0)

0 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝛾𝛾𝑢𝑢,𝑐𝑐 ∙ �10+(𝑡𝑡−𝑡𝑡

(4.3)

where 𝛾𝛾𝑢𝑢.𝑐𝑐 is the creep coefficient, given as:

𝛾𝛾𝑢𝑢,𝑐𝑐 = 𝛾𝛾𝑙𝑙𝑙𝑙,𝑐𝑐 ∙ 𝛾𝛾𝑅𝑅𝑅𝑅,𝑐𝑐 ∙ 𝛾𝛾𝑛𝑛,𝑐𝑐 ∙ 𝛾𝛾𝑠𝑠,𝑐𝑐 ∙ 𝛾𝛾𝜓𝜓,𝑐𝑐 ∙ 𝛾𝛾𝑎𝑎,𝑐𝑐

(4.4)

𝛾𝛾𝑙𝑙𝑙𝑙,𝑐𝑐 is the creep correction factor for loading age, given as: 1.0 (𝑡𝑡 𝑎𝑎1 0 )−𝑎𝑎2

𝑡𝑡0 ≤ 𝑡𝑡𝑙𝑙𝑙𝑙 𝑡𝑡0 > 𝑡𝑡𝑙𝑙𝑙𝑙

NNN 𝛾𝛾𝑙𝑙𝑙𝑙,𝑐𝑐 =

(4.5)

where the coefficients 𝑎𝑎1 , 𝑎𝑎2 , and 𝑎𝑎3 are given in the following table: 1.25

𝑎𝑎2

-0.118

𝑡𝑡 (days)

Steam cured

1.13

-0.094

3.67

𝑙𝑙𝑙𝑙

Moist cured

𝑎𝑎1

Concrete curing type

7

𝛾𝛾𝑅𝑅𝑅𝑅,𝑐𝑐 is the creep correction factor for ambient relative humidity, given as: 𝛾𝛾𝑅𝑅𝑅𝑅,𝑐𝑐 =

1.0 1.27 − 0.0067 𝑅𝑅𝑅𝑅

𝑅𝑅𝑅𝑅 ≤ 40 𝑅𝑅𝑅𝑅 > 40

(4.6)

where RH is the relative humidity in percentage (%). 𝛾𝛾𝑛𝑛,𝑐𝑐 is the creep correction factor for notional size, given as: 2 3

𝛾𝛾𝑛𝑛,𝑐𝑐 = [1 + 1.13 exp(−0.1065 ℎ)]

(4.7)

where h is the notional size in mm.

𝛾𝛾𝑠𝑠,𝑐𝑐 is the creep correction factor for concrete slump, given as: 𝛾𝛾𝑠𝑠,𝑐𝑐 = 0.82 + 0.00264 𝑠𝑠

where s is the slump in mm.

(4.8)

ACI 209R-92 21

𝛾𝛾𝜓𝜓,𝑐𝑐 is the creep correction factor for fine aggregate percentage, given as: 𝛾𝛾𝜓𝜓,𝑐𝑐 = 0.88 + 0.0024 𝜓𝜓

(4.9)

where 𝜓𝜓 is the ratio of the fine aggregate to total aggregate by weight, as a percentage. 𝛾𝛾𝑎𝑎,𝑐𝑐 is the creep correction factor for air content, given as: 𝛾𝛾𝑎𝑎,𝑐𝑐 = 0.46 + 0.09 𝛼𝛼 ≥ 1.0

(4.10)

where 𝛼𝛼 is the air content in percentage.

Shrinkage

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as: 𝜀𝜀𝑐𝑐𝑠𝑠 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) =

(𝑡𝑡−𝑡𝑡 )

𝑠𝑠 �35+(𝑡𝑡−𝑡𝑡 � 𝜀𝜀 ) 𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠

𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

NNN (𝑡𝑡−𝑡𝑡 )

𝑠𝑠 �55+(𝑡𝑡−𝑡𝑡 � 𝜀𝜀 ) 𝑐𝑐𝑐𝑐𝑐𝑐 𝑠𝑠

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

where 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 is the shrinkage coefficient, given as:

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 = −𝛾𝛾𝑐𝑐𝑐𝑐 ∙ 𝛾𝛾𝑅𝑅𝑅𝑅,𝑠𝑠 ∙ 𝛾𝛾ℎ,𝑠𝑠 ∙ 𝛾𝛾𝑠𝑠,𝑠𝑠 ∙ 𝛾𝛾𝜓𝜓,𝑠𝑠 ∙ 𝛾𝛾𝑐𝑐,𝑠𝑠 ∙ 𝛾𝛾𝑎𝑎,𝑠𝑠

(4.11)

(4.12)

𝛾𝛾𝑐𝑐𝑐𝑐 is the shrinkage correction factor for moist curing and is only applicable when the concrete curing type is specified as moist curing. 𝛾𝛾𝑐𝑐𝑐𝑐 is linearly interpolated between the following values: After ACI 209-R92 Table 2.5.3 𝑡𝑡𝑐𝑐 (days)

𝛾𝛾𝑐𝑐𝑐𝑐

3

1.1

7

1

14

0.93

28

0.86

90

0.75

1

1.2

𝛾𝛾𝑅𝑅𝑅𝑅,𝑠𝑠 is the shrinkage correction factor for ambient relative humidity, given as:

ACI 209R-92 22

1.0 𝛾𝛾𝑅𝑅𝑅𝑅,𝑠𝑠 = 1.40 − 0.0102 𝑅𝑅𝑅𝑅 3.0 − 0.030 𝑅𝑅𝑅𝑅

𝑅𝑅𝑅𝑅 < 40 40 ≤ 𝑅𝑅𝑅𝑅 ≤ 80 80 < 𝑅𝑅𝑅𝑅 ≤ 100

(4.13)

where RH is the relative humidity in percentage (%).

𝛾𝛾ℎ,𝑠𝑠 is the shrinkage correction factor for notional size, given as: 𝛾𝛾ℎ,𝑠𝑠 = 1.2 exp(−0.00236 ℎ)

(4.14)

where h is the notional size in mm.

𝛾𝛾𝑠𝑠,𝑠𝑠 is the shrinkage correction factor for concrete slump, given as: 𝛾𝛾𝑠𝑠,𝑠𝑠 = 0.89 + 0.00161 𝑠𝑠

(4.15)

where s is the slump in mm.

𝛾𝛾𝜓𝜓,𝑠𝑠 is the shrinkage correction factor for fine aggregate percentage, given as: 𝛾𝛾𝜓𝜓,𝑠𝑠 =

0.30 + 0.014 𝜓𝜓 𝜓𝜓 ≤ 50 0.90 + 0.002 𝜓𝜓 𝜓𝜓 > 50

NNN

(4.16)

where 𝜓𝜓 is the ratio of the fine aggregate to total aggregate by weight, as a percentage. 𝛾𝛾𝑐𝑐,𝑠𝑠 is the creep correction factor for cement content, given as: 𝛾𝛾𝑐𝑐,𝑠𝑠 = 0.75 + 0.00036 𝑐𝑐

(4.17)

where 𝑐𝑐 is the cement content in units of lb/yd3.

𝛾𝛾𝑎𝑎,𝑠𝑠 is the creep correction factor for air content, given as: 𝛾𝛾𝑎𝑎,𝑠𝑠 = 0.95 + 0.008 𝛼𝛼

(4.18)

where 𝛼𝛼 is the air content in percentage.

References

The following lists the equation numbers in ACI 209R-92 which correspond to the equations used in this chapter: Equation number 4.1 4.2 4.3 4.4

ACI 209R-92 Reference Eq. 2-1 Eqs. 2-1, 2.5 Eq. 2-8 Sec. 2.4-2.6

Equation number 4.10 4.11 4.12 4.13

ACI 209R-92 Reference Eq. 2-29 Eqs. 2-9, 2-10 Sec. 2.4-2.6 Eqs. 2-15, 2-16

ACI 209R-92 23

4.5 4.6 4.7 4.8 4.9

Eqs. 2-11, 2-12 Eq. 2-14 Eq. 2-21a Eq. 2-23a Eq. 2-25

4.14 4.15 4.16 4.17 4.18

Eq. 2-22a Eq. 2-24a Eqs. 2-26, 2-27 Eq. 2-28 Eq. 2-30

NNN

JTG D62-2004 24

JTG D62-2004 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness Relative Humidity, RH (%)

Creep

Shrinkage

Y

Y

NNN Shrinkage Start Age, 𝑡𝑡 (days) 𝑠𝑠

Y

Cement Type Coefficient, s *

Y

* CEB-FIP 1990 parameter used for compressive strength and stiffness

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days is: 𝑓𝑓𝑐𝑐𝑐𝑐28 = 𝑓𝑓𝑐𝑐′ + 8 𝑀𝑀𝑀𝑀𝑀𝑀

(5.1)

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength

Compressive strength and stiffness are calculated using the equations from CEB-FIP 1990, described in the CB-FIP Compressive Strength and Stiffness section of this document.

Creep 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) =

𝜑𝜑0 ∙ 𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 )

𝜑𝜑0 ∙ 𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 ) ∙ �

32.4 𝑓𝑓𝑐𝑐′

𝑓𝑓𝑐𝑐′ ≤ 32.4 𝑀𝑀𝑀𝑀𝑀𝑀

𝑓𝑓𝑐𝑐′ > 32.4 𝑀𝑀𝑀𝑀𝑀𝑀

(5.2)

JTG D62-2004 25

where: 𝜑𝜑0 = 𝜑𝜑𝑅𝑅𝑅𝑅 𝛽𝛽(𝑓𝑓𝑐𝑐𝑚𝑚28 ) 𝛽𝛽(𝑡𝑡0 ) 𝜑𝜑𝑅𝑅𝑅𝑅 = 1 + �

(5.3)

1 − 𝑅𝑅𝑅𝑅�100

1� � 0.46 �ℎ�100� 3

𝛽𝛽(𝑓𝑓𝑐𝑐𝑐𝑐28 ) = 5.3�(0.1

(5.4) (5.5)

𝑓𝑓𝑐𝑐𝑐𝑐28 )0.5

𝛽𝛽(𝑡𝑡0 ) = 1� (0.1 + 𝑡𝑡00.2 )

𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 ) = � and:

(5.6)

0.3 (𝑡𝑡−𝑡𝑡0 ) � βH +(𝑡𝑡−𝑡𝑡0 )

(5.7)

18 𝛽𝛽𝐻𝐻 = 1.5 ℎ �1 + �1.2 𝑅𝑅𝑅𝑅�100� � + 250 ≤ 1500

(5.8)

𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) h is the notional size in mm RH is the relative humidity in percentage (%) 𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 5.1

NNN

Shrinkage

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as:

where:

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) =

𝜀𝜀𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝛽𝛽𝑠𝑠 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 )

𝜀𝜀𝑠𝑠𝑠𝑠𝑠𝑠 ∙ 𝛽𝛽𝑠𝑠 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) ∙ �

32.4 𝑓𝑓𝑐𝑐′

𝜀𝜀𝑠𝑠𝑠𝑠𝑠𝑠 = 𝜀𝜀𝑠𝑠 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) 𝛽𝛽𝑅𝑅𝑅𝑅

𝑓𝑓𝑐𝑐′ ≤ 32.4 𝑀𝑀𝑀𝑀𝑀𝑀

𝑓𝑓𝑐𝑐′ > 32.4 𝑀𝑀𝑀𝑀𝑀𝑀

𝜀𝜀𝑠𝑠 (𝑓𝑓𝑐𝑐𝑐𝑐28 ) = [160 + 10 𝛽𝛽𝑠𝑠𝑠𝑠 (9 − 0.1 𝑓𝑓𝑐𝑐𝑐𝑐28 )] × 10−6 3 𝛽𝛽𝑅𝑅𝑅𝑅 = −1.55 �1 − �𝑅𝑅𝑅𝑅�100� �

and:

𝛽𝛽𝑠𝑠 (𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) = �

0.5 (𝑡𝑡−𝑡𝑡𝑠𝑠 ) � 0.035 ℎ2 + (𝑡𝑡−𝑡𝑡𝑠𝑠 )

(5.9)

(5.10) (5.11) (2.12) (2.13)

𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) 𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 2.1

JTG D62-2004 26

𝛽𝛽𝑠𝑠𝑠𝑠 , the shrinkage coefficient, is equal to 5.0 RH is the relative humidity in percentage (%) h is the notional size in mm

References The following lists the equation numbers in JTG D62-2004 which correspond to the equations used in this chapter: Equation number 5.1 5.2 5.3 5.4 5.5 5.6 5.7

JTG D62-2004 Reference CEB-FIP 90 Eq. 2.1-1 Eq. F.2.1-1 Eq. F.2.1-2 Eq. F.2.1-3 Eq. F.2.1-4 Eq. F.2.1-5 Eq. F.2.1-6

Equation number 5.8 5.9 5.10 5.11 5.12 5.13

JTG D62-2004 Reference Eq. F.2.1-7 Eq. F.1.1-1 Eq. F.1.1-2 Eq. F.1.1-3 Eq. F.1.1-4 Eq. F.1.1-5

NNN

Eurocode 2-2004 27

Eurocode 2-2004 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness Relative Humidity, RH (%)

Creep

Shrinkage

Y

Y

NNN Shrinkage Start Age, 𝑡𝑡 (days) 𝑠𝑠

Y

Cement Type

Y

Y

Y

Lightweight Concrete Grade *

Y

Y

Lightweight Oven-dry Density 𝜌𝜌 (kg/m3) *

Y

* Lightweight Concrete Only

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days is:

where

𝑓𝑓𝑐𝑐𝑐𝑐28 = 𝑓𝑓𝑐𝑐′ + 8 𝑀𝑀𝑀𝑀𝑀𝑀

(6.1)

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) 𝑓𝑓𝑐𝑐𝑐𝑐28

(6.2)

𝑓𝑓𝑐𝑐′

is the specified concrete compressive strength

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡), the mean compressive strength of concrete at age t (days) is given as:

where:

0.5

𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 �𝑠𝑠 �1 − �28�𝑡𝑡�

��

(6.3)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 6.1 s is a coefficient based on the strength class of the cement, as given in the table below.

Eurocode 2-2004 28

Cement Type

s

Class R

0.20

Class N

0.25

Class S

0.38

𝛽𝛽𝐸𝐸 (𝑡𝑡), the elastic modulus coefficient at age t (days) is given as: 𝛽𝛽𝐸𝐸 (𝑡𝑡) = [𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡)]0.3

(6.4)

where 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) is as defined in Eq. 3.3.

Creep

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝜂𝜂𝐸𝐸 ∙ 𝜂𝜂2 ∙ 𝜙𝜙𝑐𝑐,0 ∙ 𝛽𝛽(𝑡𝑡, 𝑡𝑡0 )

(6.5)

𝜑𝜑0 = 𝜑𝜑𝑅𝑅𝑅𝑅 𝛽𝛽(𝑓𝑓𝑐𝑐𝑐𝑐28 ) 𝛽𝛽(𝑡𝑡0 )

(6.6)

NNN

where:

𝜑𝜑𝑅𝑅𝑅𝑅 =

1+

�1 +

1 − 𝑅𝑅𝑅𝑅�100

1 0.1 (ℎ) �3 1 − 𝑅𝑅𝑅𝑅�100 𝛼𝛼1 � 𝛼𝛼2 1 0.1 (ℎ) �3

𝛽𝛽(𝑓𝑓𝑐𝑐𝑐𝑐28 ) = 16.8�(𝑓𝑓

0.5 𝑐𝑐𝑐𝑐28 )

𝑓𝑓𝑐𝑐𝑐𝑐28 ≤ 35𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑐𝑐𝑐𝑐28 > 35𝑀𝑀𝑀𝑀𝑀𝑀

𝛽𝛽(𝑡𝑡0 ) = 1� (0.1 + (𝑡𝑡�0 )0.2 ) 𝛽𝛽𝑐𝑐 (𝑡𝑡 − 𝑡𝑡0 ) = �

𝛼𝛼

𝛽𝛽𝐻𝐻 = � and:

0.3 (𝑡𝑡−𝑡𝑡0 ) � βH +(𝑡𝑡−𝑡𝑡0 )

1.5 ℎ[1 + (0.012 𝑅𝑅𝑅𝑅)18 ] +250 ≤ 1500 1.5 ℎ[1 + (0.012 𝑅𝑅𝑅𝑅)18 ] +250 𝛼𝛼3 ≤ 1500 𝛼𝛼3

(6.8) (6.9)

9 𝑡𝑡�0 = 𝑡𝑡0 � + 1� ≥ 0.5 2+(𝑡𝑡 )1.2 0

(6.7)

(6.10) (6.11) 𝑓𝑓𝑐𝑐𝑐𝑐28 ≤ 35𝑀𝑀𝑀𝑀𝑀𝑀

(6.12)

𝑓𝑓𝑐𝑐𝑐𝑐28 > 35𝑀𝑀𝑀𝑀𝑀𝑀

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 6.1 RH is the relative humidity in percentage (%)

Eurocode 2-2004 29

h is the notional size in mm 𝛼𝛼1 , 𝛼𝛼2 , and 𝛼𝛼3 are coefficients given as: 𝛼𝛼1 = �

35

𝑓𝑓𝑐𝑐𝑐𝑐28

0.7

�

; 𝛼𝛼2 = �

35

𝑓𝑓𝑐𝑐𝑐𝑐28

�

0.2

; 𝛼𝛼3 = �

35

𝑓𝑓𝑐𝑐𝑐𝑐28

0.5

�

(6.13)

𝑡𝑡�0 is the age of loading modified for the effect of type of cement. 𝛼𝛼 is a power for Eq. 6.13 which depends on the type of cement as: Cement Type Class R

𝜶𝜶 1

Class N

0

Class S

-1

𝜂𝜂𝐸𝐸 is the lightweight oven-dry density ratio modification factor. If normalweight concrete is used, the factor 𝜂𝜂𝐸𝐸 = 1. If lightweight concrete is used, 𝜂𝜂𝐸𝐸 given as:

NNN 2 𝜌𝜌 𝜂𝜂𝐸𝐸 = � �2200�

(6.14)

where 𝜌𝜌 is the user-defined lightweight oven-dry density in units of 𝑘𝑘𝑘𝑘/𝑚𝑚3

Lightweight Concrete Grade

𝜼𝜼

𝟐𝟐

𝜂𝜂2 is the lightweight aggregate creep strain modification factor. If normalweight concrete is used, the factor 𝜂𝜂2 = 1. If lightweight concrete is used, 𝜂𝜂2 given as: LC16/18 and lower

1.3

LC20/22 and higher

1.0

Shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = −𝜂𝜂3 ∙ [𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) + 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 )]

(6.15)

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝜀𝜀𝑐𝑐𝑐𝑐 (∞) ∙ 𝛽𝛽𝑎𝑎𝑎𝑎 (𝑡𝑡)

(6.16)

𝛽𝛽𝑎𝑎𝑎𝑎 (𝑡𝑡) = 1 − exp(−0.2 𝑡𝑡 0.5 )

(6.18)

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) is the autogenous shrinkage strain, given as: 𝜀𝜀𝑐𝑐𝑐𝑐 (∞) = 2.5 (𝑓𝑓𝑐𝑐′ − 10) × 10−6

(6.17)

Eurocode 2-2004 30

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) is the drying shrinkage strain, given as: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝜀𝜀𝑐𝑐𝑐𝑐,0 ∙ 𝑘𝑘ℎ ∙ 𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) 𝜀𝜀𝑐𝑐𝑐𝑐,0 =

(6.19)

0.85 𝛽𝛽𝑅𝑅𝑅𝑅 [(220 + 110 𝛼𝛼𝑑𝑑𝑑𝑑1 ) ∙ exp(−0.1 𝛼𝛼𝑑𝑑𝑑𝑑2 𝑓𝑓𝑐𝑐𝑐𝑐28 )] × 10−6

3 𝛽𝛽𝑅𝑅𝑅𝑅 = 1.55 �1 − �𝑅𝑅𝑅𝑅�100� �

where:

𝛽𝛽𝑑𝑑𝑑𝑑 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = �

𝑡𝑡−𝑡𝑡𝑠𝑠

0.5

�

0.04 √ℎ3 +(𝑡𝑡−𝑡𝑡𝑠𝑠 )

(6.20) (6.21) (6.22)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 2.1 𝑘𝑘ℎ is a coefficient dependent on the notional size, h, computed as a linear interpolation of the values given as:

NNN 1.0

200

0.85

300

0.75

≥ 500

0.70

𝒌𝒌

Notional size, h (mm) 100

𝒉𝒉

RH is the relative humidity in percentage (%) h is the notional size in mm 𝛼𝛼𝑑𝑑𝑑𝑑1 and 𝛼𝛼𝑑𝑑𝑑𝑑2 are coefficients given as: Cement Type Class R Class N Class S

𝜶𝜶𝒅𝒅𝒅𝒅𝒅𝒅

𝜶𝜶𝒅𝒅𝒅𝒅𝒅𝒅

3

0.13

5 4

0.11 0.12

𝜂𝜂3 is the lightweight aggregate concrete shrinkage strain modification factor. If normal-weight concrete is used, the factor 𝜂𝜂3 = 1. If lightweight concrete is used, 𝜂𝜂3 given by the following table. Lightweight Concrete Grade

𝜼𝜼𝟑𝟑

LC16/18 and lower

1.5

LC20/22 and higher

1.2

Eurocode 2-2004 31

References The following lists the equation numbers in Eurocode 2-2004 which correspond to the equations used in this chapter: Equation number 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Eurocode 2-2004 Reference Table 3.1 Eq. 3.1 Eq. 3.2 Eq. 3.5 Eqs. B.1, 11.3.3(1) Eq. B.2 Eqs. B.3a, B.3b Eq. B.4 Eq. B.5 Eq. B.9 Eq. B.7

Equation number 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22

Eurocode 2-2004 Reference Eqs. B.8a, B.8b Eq. B.8c Eq. 11.2 Eq. 3.8 Eq. 3.11 Eq. 3.12 Eq. 3.13 Eq. 3.9 Eq. B.11 Eq. B.12 Eq. 3.10

NNN

AS 3600-2009 32

AS 3600-2009 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness Compressive Strength Factor, 𝑎𝑎 *

Creep

Shrinkage

Y

NNN Compressive Strength Factor, 𝛽𝛽 *

Y

𝑏𝑏

Basic Creep Coefficient, 𝜑𝜑𝑐𝑐𝑐𝑐,

Y

Basic Drying Shrinkage Strain,

∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐,

Y

𝑏𝑏

Environment

Y Y

* ACI 209R-92 parameters used for compressive strength and stiffness

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days, 𝑓𝑓𝑐𝑐𝑐𝑐28, is taken to be equal to the specified concrete compressive strength, 𝑓𝑓𝑐𝑐′ .

Compressive strength and stiffness are calculated using the equations from ACI 290R-92, described in the ACI 209R-92 Compressive Strength and Stiffness section of this document.

Creep 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: where:

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝑘𝑘2 (𝑡𝑡 − 𝑡𝑡0 ) ∙ 𝑘𝑘3 ∙ 𝑘𝑘4 ∙ 𝑘𝑘5 ∙ 𝜑𝜑𝑐𝑐𝑐𝑐,𝑏𝑏

(7.1)

AS 3600-2009 33 𝛼𝛼 (𝑡𝑡−𝑡𝑡0 )0.8 +0.15 ℎ 0

𝑘𝑘2 (𝑡𝑡, 𝑡𝑡0 ) = (𝑡𝑡−𝑡𝑡2 )0.8

(7.2)

𝛼𝛼2 = 1.0 + 1.12 exp(−0.008 ℎ) 2.7

𝑘𝑘3 = 1+log 𝑘𝑘5 = and:

𝛼𝛼3 =

10 (𝑡𝑡0 )

1.0 (2.0 − 𝛼𝛼3 ) − 0.02(1.0 − 𝛼𝛼3 )𝑓𝑓𝑐𝑐′ 0.7 𝑘𝑘4 𝛼𝛼2

(7.3) (7.4) 𝑓𝑓𝑐𝑐′ ≤ 50 𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑐𝑐′ > 50 𝑀𝑀𝑀𝑀𝑀𝑀

(7.5) (7.6)

𝜑𝜑𝑐𝑐𝑐𝑐,𝑏𝑏 is the user-defined basic creep coefficient h is the notional size in mm 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) 𝑘𝑘4 is a factor depending on the user-defined environment parameter as given in the table below. 𝒌𝒌

NNN Environment

𝟒𝟒

Arid

0.7

Interior

0.65

Temperate Inland

0.6

Tropical / Near-coastal

0.5

Shrinkage

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with shrinkage start age 𝑡𝑡𝑠𝑠 = 0: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) = −�𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) + 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡)�

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) is the autogenous shrinkage strain, given as: ∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 ∙ [1.0 − exp(−0.1 𝑡𝑡)] ∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 = 50 × 10−6 ∙ (0.06 𝑓𝑓𝑐𝑐′ − 1.0)

(7.7)

(7.8) (7.9)

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) is the drying shrinkage strain, given as: 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑘𝑘1 (𝑡𝑡) ∙ 𝑘𝑘4 ∙ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐.𝑏𝑏 𝑘𝑘1 =

𝛼𝛼1 𝑡𝑡 0.8 0.8 𝑡𝑡 +0.15 ℎ

(7.10) (7.11)

AS 3600-2009 34

𝛼𝛼1 = 0.8 + 1.2 exp(−0.005 ℎ) where:

(7.12)

∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐.𝑏𝑏 = (1.0 − 0.008 𝑓𝑓𝑐𝑐′ ) ∙ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐.𝑏𝑏

(7.13)

h is the notional size in mm 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) ∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐.𝑏𝑏 is the user-defined parameter for basic drying shrinkage strain 𝑘𝑘4 is a factor depending on the user-defined environment parameter, defined in Section 7.1.3, Creep.

References

The following lists the equation numbers in AS 3600-2009 which correspond to the equations used in this chapter: Equation number 7.1 7.2 7.3 7.4 7.5 7.6 7.7

AS 3600-2009 Reference Eq. 3.1.8.3 Fig. 3.1.8.3(a) Fig. 3.1.8.3(a) Eq. 3.1.8.3 (Amdt #2) Eq. 3.1.8.3 Eq. 3.1.8.3 Eq. 3.1.7.2(1)

Equation number 7.8 7.9 7.10 7.11 7.12 7.13

AS 3600-2009 Reference Eq. 3.1.7.2(2) Eq. 3.1.7.2(3) Eq. 3.1.7.2(4) Fig. 3.1.7.2 Fig. 3.1.7.2 Eq. 3.1.7.2(5)

NNN

NZS 3101-2006 35

NZS 3101-2006 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness Δ𝑓𝑓 Mean Strength (MPa) *

Creep

Shrinkage

Y

NNN Cement Type **

Y

Y

Basic Creep Coefficient, 𝜑𝜑𝑐𝑐𝑐𝑐,

Y

Basic Drying Shrinkage Strain, 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐,

Y

Y

𝑏𝑏

Aggregate Type Factor, 𝑘𝑘6

𝑏𝑏

Relative Humidity, RH (%)

Y

* Parameter used for mean compressive strength ** CEB-FIP 2010 parameter used for compressive strength and stiffness

Compressive Strength and Stiffness The mean compressive strength of concrete at an age of 28 days is: where:

𝑓𝑓𝑐𝑐𝑐𝑐28 = 𝑓𝑓𝑐𝑐′ + Δ𝑓𝑓

(8.1)

𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength

Δ𝑓𝑓 is the user-defined parameter for mean compressive strength

Compressive strength and stiffness are calculated using the equations from the CEB-FIP 2010 Compressive Strength and Stiffness section of this document.

NZS 3101-2006 36

Creep 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: where:

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝑘𝑘2 (𝑡𝑡 − 𝑡𝑡0 ) ∙ 𝑘𝑘3 ∙ 𝑘𝑘4 ∙ 𝑘𝑘5 ∙ 𝑘𝑘6 ∙ 𝜑𝜑𝑐𝑐𝑐𝑐,𝑏𝑏 𝛼𝛼 (𝑡𝑡−𝑡𝑡0 )0.8 +0.15 ℎ 0

𝑘𝑘2 (𝑡𝑡, 𝑡𝑡0 ) = (𝑡𝑡−𝑡𝑡2 )0.8

(8.3)

𝛼𝛼2 = 1.0 + 1.12 exp(−0.008 ℎ) 𝑘𝑘3 = 𝑘𝑘5 =

(8.4)

2.7 1+log10 (𝑡𝑡0 )

(8.5) 𝑓𝑓𝑐𝑐′ ≤ 50 𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑐𝑐′ > 50 𝑀𝑀𝑀𝑀𝑀𝑀

1.0 (2.0 − 𝛼𝛼3 ) − 0.02(1.0 − 𝛼𝛼3 )𝑓𝑓𝑐𝑐′ 0.7 4 𝛼𝛼2

𝛼𝛼3 = 𝑘𝑘

NNN

(8.6) (8.7)

𝜑𝜑𝑐𝑐𝑐𝑐,𝑏𝑏 is the user-defined basic creep coefficient h is the notional size in mm 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) 𝑘𝑘6 is the user-defined parameter for the aggregate type factor 𝑘𝑘4 is a factor depending on the specified relative humidity (RH) value, linearly interpolated between the following values: After NZS 3101-2006 Table E.1 RH (%)

𝒌𝒌

and:

(8.2)

𝟒𝟒

40

0.74

50

0.68

60

0.61

70

0.50

80

0.39

90

0.21

NZS 3101-2006 37

Shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with shrinkage start age 𝑡𝑡𝑠𝑠 = 0: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) = −�𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) + 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡)�

(8.8)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) is the autogenous shrinkage strain, given as: ∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 ∙ [1.0 − exp(−0.1 𝑡𝑡)]

(8.9)

∗ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 = 50 × 10−6 ∙ (0.06 𝑓𝑓𝑐𝑐′ − 1.0)

(8.10)

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa)

𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) is the drying shrinkage strain, given as: 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑘𝑘1 (𝑡𝑡) ∙ 𝑘𝑘4 ∙ 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐.𝑏𝑏

(8.11)

𝛼𝛼1 = 0.8 + 1.2 exp(−0.005 ℎ)

(8.13)

𝑘𝑘1 =

𝑡𝑡 0.8

𝛼𝛼1 𝑡𝑡 0.8 +0.15 ℎ

(8.12)

NNN

where:

h is the notional size in mm 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength (MPa) 𝜀𝜀𝑐𝑐𝑐𝑐𝑐𝑐,𝑏𝑏 is the user-defined parameter for basic drying shrinkage strain 𝑘𝑘4 is a factor depending on the specified relative humidity (RH) value, defined in the NZS 3101-2006 Creep section of this manual.

References

The following lists the equation numbers in NZS 3101-2006 Amdt 3 (Appendix E) which correspond to the equations used in this chapter: Equation number 8.1 8.2 8.3 8.4 8.5 8.6 8.7

NZS 3101-2006 Reference CEB-FIP 2010 Eq. 5.1-1 Eq. E-7 Fig. E.2 Fig. E.2 Eq. E-7 Eq. E-7 Eq. E-7

Equation number 8.8 8.9 8.10 8.11 8.12 8.13

NZS 3101-2006 Reference Eq. E-1 Eq. E-2 Eq. E-3 Eq. E-4 Fig. E.1 Fig. E.1

GL2000 38

GL2000 All equations presented in this section are intended for use with the units of Megapascal (MPa), millimeters (mm), and days unless specifically marked with units.

Time Dependent Properties for Concrete Materials Input Parameters In addition to the elastic moduli and concrete strength, the following parameters are available: Affects Parameter Stiffness

Creep

Shrinkage

Relative Humidity, RH (%)

Y

Y

Shrinkage Start Age, 𝑡𝑡 (days)

Y

Y

𝑠𝑠

NNN Cement Type

Y

Strength Development Parameter, 𝑠𝑠 *

Y

Shrinkage Correction Term, 𝑘𝑘 *

Y

Y

* User-specified only when Cement Type is “User Defined”

Compressive Strength and Stiffness

The mean compressive strength of concrete at an age of 28 days is: 𝑓𝑓𝑐𝑐𝑐𝑐28 = 1.1𝑓𝑓𝑐𝑐′ + 5 𝑀𝑀𝑀𝑀𝑀𝑀

(9.1)

where 𝑓𝑓𝑐𝑐′ is the specified concrete compressive strength

𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡), the mean compressive strength of concrete at age t (days) is given as: 2 (𝑡𝑡) 𝑓𝑓𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝛽𝛽𝑐𝑐𝑐𝑐 𝑓𝑓𝑐𝑐𝑐𝑐28

where:

0.5 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 �0.5 𝑠𝑠 �1 − �28�𝑡𝑡� ��

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 9.1

(9.2) (9.3)

GL2000 39

s is a coefficient based on the strength class of the cement. If the cement type is specified, the coefficient s is as given in the table below. If the cement type is user-defined, s is the user-defined strength development parameter. After ACI 209.2R-28 Table A.14 Cement Type s Type I

0.335

Type II

0.40

Type III

0.13

𝛽𝛽𝐸𝐸 (𝑡𝑡), the elastic modulus coefficient at age t (days) is given as: 𝛽𝛽𝐸𝐸 (𝑡𝑡) =

�3500+4300𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡)�𝑓𝑓𝑐𝑐𝑐𝑐28 � �3500+4300�𝑓𝑓𝑐𝑐𝑐𝑐28 �

(9.4)

where 𝛽𝛽𝑐𝑐𝑐𝑐 (𝑡𝑡) is as defined in Eq. 9.3.

NNN

Creep

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: (𝑡𝑡−𝑡𝑡0 )0.3 7 0.5 (𝑡𝑡−𝑡𝑡0 ) 0.5 + � � � � 0.3 (𝑡𝑡−𝑡𝑡0 )+7 𝑡𝑡0 0 ) +14 0.5 (𝑡𝑡−𝑡𝑡0 ) 1.086ℎ2 ) �(𝑡𝑡−𝑡𝑡 )+0.03ℎ � 2� 0

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = Φ(𝑡𝑡𝑠𝑠 ) �2 (𝑡𝑡−𝑡𝑡

+ 2.5(1 −

(9.5)

where Φ(𝑡𝑡𝑐𝑐 ) is the correction for effect of drying prior to load application, given as:

and:

Φ(𝑡𝑡𝑠𝑠 ) = �1 − �(𝑡𝑡

0.5 0.5 (𝑡𝑡0 −𝑡𝑡𝑠𝑠 ) � � 2 0 −𝑡𝑡𝑠𝑠 )+0.03 ℎ

(9.6)

𝑡𝑡𝑠𝑠 is user-defined shrinkage start age in days h is the notional size in mm

Shrinkage

𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ), the shrinkage strain at time t (days) with a shrinkage start age 𝑡𝑡𝑠𝑠 (days) is given as: 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡, 𝑡𝑡𝑠𝑠 ) = −𝜀𝜀𝑠𝑠ℎ𝑢𝑢 ∙ 𝛽𝛽(𝑅𝑅𝑅𝑅) ∙ 𝛽𝛽(𝑡𝑡 − 𝑡𝑡𝑠𝑠 )

(9.7)

GL2000 40

𝜀𝜀𝑠𝑠ℎ𝑢𝑢 is the ultimate shrinkage, given as: where:

𝜀𝜀𝑠𝑠ℎ𝑢𝑢 = 900 × 10−6 ∙ 𝑘𝑘 ∙ �

30 � 𝑓𝑓𝑐𝑐𝑐𝑐28

(9.8)

𝑓𝑓𝑐𝑐𝑐𝑐28 is the 28 day mean compressive strength (MPa) from Eq. 9.1

k is a coefficient based on the strength class of the cement. If the cement type is specified, the coefficient k is as given in the table below. If the cement type is user-defined, s is the user-defined shrinkage correction term. After ACI 209.2R-28 Table A.14 Cement Type k Type I

1.0

Type II

0.75

NNN Type III

1.15

𝛽𝛽(𝑅𝑅𝑅𝑅) is the correction term for relative humidty: 4 𝛽𝛽(𝑅𝑅𝑅𝑅) = 1 − 1.18 �𝑅𝑅𝑅𝑅�100�

(9.9)

where RH is the relative humidity in percentage (%) 𝛽𝛽(𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) is the correction term for time of drying: where:

0.5 (𝑡𝑡−𝑡𝑡𝑠𝑠 ) � 2 𝑠𝑠 )+0.03 ℎ

𝛽𝛽(𝑡𝑡 − 𝑡𝑡𝑠𝑠 ) = �(𝑡𝑡−𝑡𝑡

(9.10)

h is the notional size in mm 𝑡𝑡𝑠𝑠 is the user-defined shrinkage start time in days

References

The following lists the equation numbers in ACI 209.2R-28 “Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete”, a report by ACI Committee 209, describing the Gardner and Lockman (2001) and Gardner (2004) model for creep and shrinkage. The equation numbers which correspond to the equations used in this chapter:

GL2000 41

Equation number 9.1 9.2 9.3 9.4 9.5

ACI 209.2R-28 Reference Eq. A-94 Eq. A-96 Eq. A-97 Eqs. A-94, A-95 Eq. A-103

Equation number 9.6 9.7 9.8 9.9 9.10

ACI 209.2R-28 Reference Eq. A-105 Eq. A-98 Eq. A-99 Eq. A-100 Eq. A-101

NNN

User-Defined 42

User-Defined For user-defined time dependent properties, the stiffness multiplier, creep, and shrinkage behavior can be defined. For this type of time dependent property, the stiffness, creep, and shrinkage behavior are independent from each other.

Time Dependent Properties for Concrete Materials Compressive Strength and Stiffness The time dependent elastic modulus coefficient, 𝛽𝛽𝐸𝐸 (𝑡𝑡), can be defined as a multi-linear relation with time in days in the Time Dependent Properties for Concrete – User Stiffness Curve form. 𝛽𝛽𝐸𝐸 (𝑡𝑡) is specified as parts of Age t (days) and an associated stiffness coefficient 𝛽𝛽𝐸𝐸 (𝑡𝑡). The values of 𝛽𝛽𝐸𝐸 (𝑡𝑡) are linearly interpolated between the specified values.

Creep

NNN

𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ), the creep coefficient at time t (days) for a load applied at time 𝑡𝑡0 (days) is given as: 𝜑𝜑𝑐𝑐 (𝑡𝑡, 𝑡𝑡0 ) = 𝑆𝑆𝑆𝑆(ℎ) ∙ 𝜑𝜑𝑐𝑐∗ (𝑡𝑡, 𝑡𝑡0 )

(10.1)

𝜑𝜑𝑐𝑐∗ (𝑡𝑡, 𝑡𝑡0 ) is the multi-linear user-defined age dependent basic creep coefficient curve specified in the Time Dependent Properties for Concrete – User Shrinkage Strain form. For each age at loading 𝑡𝑡0 , one set of basic creep coefficient curve data is defined, specified as pairs of Age t (days) and an associated basic creep coefficient 𝜑𝜑𝑐𝑐∗ (𝑡𝑡, 𝑡𝑡0 ). The values of 𝜑𝜑𝑐𝑐∗ (𝑡𝑡, 𝑡𝑡0 ) are linearly interpolated between the specified values. 𝑆𝑆𝑆𝑆(ℎ) is a scale factor dependent on the notional size can be specified with the following form: where:

𝑆𝑆𝑆𝑆 = 𝑎𝑎 + 𝑏𝑏 exp(− ℎ�ℎ ) 0

a, b, and ℎ0 are user defined coefficients in consistent units h is the notional size in consistent units

(10.2)

The creep coefficient is not dependent on the specified time dependent elastic modulus coefficient.

User-Defined 43

Shrinkage 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡), the shrinkage strain at time t (days) is defined as: ∗ 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) = 𝑆𝑆𝑆𝑆(ℎ) ∙ 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡)

(10.3)

∗ 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) is the multi-linear user-defined age dependent basic shrinkage strain curve specified in the Time Dependent Properties for Concrete – User ∗ Shrinkage Strain form. 𝜀𝜀𝑐𝑐𝑐𝑐 (𝑡𝑡) is specified as pairs of Age t (days) and an ∗ (𝑡𝑡). ∗ (𝑡𝑡) associated basic shrinkage strain 𝜀𝜀𝑐𝑐𝑐𝑐 The values of 𝜀𝜀𝑐𝑐𝑐𝑐 are linearly interpolated between the specified values.

𝑆𝑆𝑆𝑆(ℎ) is a scale factor dependent on the notional size can be specified with the following form: where:

𝑆𝑆𝑆𝑆 = 𝑎𝑎 + 𝑏𝑏 exp(− ℎ�ℎ ) 0

a, b, and ℎ0 are user defined coefficients in consistent units h is the notional size in consistent units

NNN

(10.4)