Models for Time and Temperature Dependent Effects
9.7 Computing Mechanical Strain in Concrete
⎣
1
ℬ𝑇(𝑇𝑖−1)√︁ℬ𝑐𝑐(𝑡𝑇𝑖−1)
− 1
ℬ𝑇(𝑇𝑖)√︁ℬ𝑐𝑐(𝑡𝑇𝑖)
⎤
⎦𝜎¯𝑐𝑖 (9.94) Here (ℬ𝑇,ℬ𝑐𝑐) are given by Eqs.(9.27,9.9), respectively; while (𝑡𝑇, 𝑇) are the tem-perature adjusted age according to Eq.(9.10) and the temtem-perature, respectively, at times indicated by the subscripts (‘𝑛’,‘𝑖’,‘𝑖 −1’). Like creep, the aging strain has to be evaluated for each principal direction at every integration point. As already pointed out in Subsections 9.3.3 and 9.5.4, the sustained stress ¯𝜎𝑐𝑖 for the generic time step will be taken as the average of the stresses at the times 𝑡𝑖−1 and 𝑡𝑖. Since the latter stress is not known yet when the aging strain is updated, an approximate value, determined as outlined in the next section, will instead be employed.
9.7 Computing Mechanical Strain in Concrete
So far, the stress ¯𝜎𝑐that enters the creep and aging strain increments for the time step considered is said to represent an average of the stress at the previous equilibrium state and the current stress. Since the latter stress in return is a function of the current instantaneous or mechanical strain𝜖𝑐, it follows from the strain decomposition in Eq.(9.1) that some kind of iteration in general becomes necessary in order to determine 𝜖𝑐. However, iteration at the integration point level is not desirable from
a computational point of view, and thus an approximate procedure to solve for 𝜖𝑐
without iteration will be outlined in this section.
By inserting the concrete strains from Eqs.(9.3,9.4) into Eq.(9.1) and rearranging terms, the following expression may result
𝜖𝑐+ Δ𝜖𝑐𝑐+ Δ𝜖𝑐𝑎 = 𝜖−𝜖𝑐𝑇 −𝜖𝑐𝑠−𝜖(𝑝)𝑐𝑐 −𝜖(𝑝)𝑐𝑎 (9.95) Here 𝜖 is the total strain, as derived from the nodal displacements, and strains with subscripts (‘𝑐𝑇’,‘𝑐𝑠’,‘𝑐𝑐’,‘𝑐𝑎’) are the thermal, shrinkage, creep and aging strains, re-spectively. Furthermore, superscript ‘(𝑝)’ refers to the previous equilibrium state, and Δ denotes increment for the time step considered. All quantities on the right hand side are known. The strain increments on the left hand side may be expressed through
Δ𝜖𝑐𝑐 = Δ𝜑𝑐 𝜎¯𝑐
𝐸𝑐28 − 𝜖(𝑝)𝑐𝑐𝑑ℬ𝑑 (9.96) Δ𝜖𝑐𝑎 = Δ𝜑𝑎 𝜎¯𝑐
𝐸𝑐28 (9.97)
where Eq.(9.96) follows from Eqs.(9.83,9.85,9.86). Here Δ𝜑𝑐 is the incremental creep coefficient for the time step, as a sum of the delayed elastic, basic flow and drying flow contributions, while the term−𝜖(𝑝)𝑐𝑐𝑑ℬ𝑑 expresses the recovery with time of the de-layed elastic creep strain component at the previous equilibrium state. Furthermore, Eq.(9.97) follows from Eq.(9.94), and here Δ𝜑𝑎 is the incremental aging coefficient.
Finally, 𝐸𝑐28 is the initial modulus of elasticity at the age of 28 days. Now the expression for the corresponding stress ¯𝜎𝑐 may conveniently be introduced, i.e.
¯ 𝜎𝑐 = 1
2
(︁𝜎˘𝑐 + 𝜎𝑐(𝑝))︁ (9.98) where (˘𝜎𝑐, 𝜎𝑐(𝑝)) are the current stress and the stress at the previous equilibrium state, respectively. The former is now equipped with an accent ‘ ˘ ’ to signify that it is a preliminary estimate of the current stress 𝜎𝑐, used here in the creep and aging strain computations. Note that ¯𝜎𝑐 is also entering the nonlinear creep coefficients (𝜑𝑓 𝑏1, 𝜑𝑓 𝑑2) in Eqs.(9.74,9.80) through the relative stress𝑠𝑎from Eq.(9.76). However, in these expressions 𝜎(𝑝)𝑐 will now instead be employed, so that Δ𝜑𝑐 in Eq.(9.96) can be considered as a known quantity. To proceed further, a stress-strain relationship has to be introduced. The initial assumption made here is that the redistribution of stresses due to combined creep and aging impliesunloading in concrete (and loading in reinforcing steel). Thus, the bilinear stress-strain relationship valid for concrete in unloading/reloading may be adopted, i.e.
˘
𝜎𝑐 = 𝐸𝑢𝜖𝑐 ; ˜𝜖(𝑐)𝑐 ≤𝜖𝑐≤˜𝜖(𝑡)𝑐 (9.99) where
𝐸𝑢 =
⎧
⎪⎨
⎪⎩
𝐸𝑢(𝑐) ; ˜𝜖(𝑐)𝑐 ≤𝜖𝑐 <0 𝐸𝑢(𝑡) ; 0≤𝜖𝑐≤˜𝜖(𝑡)𝑐
(9.100)
Here (𝐸𝑢(𝑐), 𝐸𝑢(𝑡)) are respectively the unloading/reloading moduli in compression and tension, as given by Eqs.(8.70,8.74) when unloading took place, and then adjusted for aging and temperature at the current time according to Eqs.(9.45,9.46). The strain limits (˜𝜖(𝑐)𝑐 ,˜𝜖(𝑡)𝑐 ) are approximations for the unknown actual strains at which the stress reaches the envelope curves in compression and tension. The following expressions for these limits are employed
˜𝜖(𝑐)𝑐 = 𝜎^𝑐(𝑐) 𝐸𝑢(𝑐)
(9.101)
˜
𝜖(𝑡)𝑐 = 𝜎^𝑐(𝑡) 𝐸𝑢(𝑡)
(9.102) where (^𝜎𝑐(𝑐),𝜎^(𝑡)𝑐 ) are the envelope stresses that correspond to the previously experi-enced extreme strains (^𝜖(𝑐)𝑐 ,^𝜖(𝑡)𝑐 ) in compression and tension, respectively. Note that (˜𝜖(𝑐)𝑐 ,˜𝜖(𝑡)𝑐 ) become different from (^𝜖(𝑐)𝑐 ,^𝜖(𝑡)𝑐 ) since (𝐸𝑢(𝑐), 𝐸𝑢(𝑡)) are adjusted for aging and temperature as mentioned. Now, by combining Eqs.(9.95-9.99), the following expres-sion for the current mechanical strain results
𝜖𝑐 =
𝜖−𝜖𝑐𝑇−𝜖𝑐𝑠 −𝜖(𝑝)𝑐𝑐 −𝜖(𝑝)𝑐𝑎 −1
2(Δ𝜑𝑐+ Δ𝜑𝑎)𝜎𝑐(𝑝)
𝐸𝑐28 +𝜖(𝑝)𝑐𝑐𝑑ℬ𝑑 1 + 1
2(Δ𝜑𝑐+ Δ𝜑𝑎) 𝐸𝑢 𝐸𝑐28
; ˜𝜖(𝑐)𝑐 ≤𝜖𝑐 ≤˜𝜖(𝑡)𝑐 (9.103) Thus, this expression is valid as long as 𝜖𝑐 falls within the range ˜𝜖(𝑐)𝑐 ≤ 𝜖𝑐 ≤𝜖˜(𝑡)𝑐 . In (rare) cases that do not fulfill this condition, the ˘𝜎𝑐-value has to be selected. Then the expression for 𝜖𝑐 becomes, using Eqs.(9.95-9.98) only
𝜖𝑐 = 𝜖−𝜖𝑐𝑇 −𝜖𝑐𝑠−𝜖(𝑝)𝑐𝑐 −𝜖(𝑝)𝑐𝑎 − 1
2(Δ𝜑𝑐+ Δ𝜑𝑎)𝜎˘𝑐+𝜎𝑐(𝑝)
𝐸𝑐28 +𝜖(𝑝)𝑐𝑐𝑑ℬ𝑑 (9.104) Based on the preceding expressions, a search procedure for finding the mechanical strain in each principal direction at an integration point has been developed. The steps are outlined in the following:
∙ For each principal direction 𝑗 that pertain to the previous equilibrium state, the following quantities are stored
𝜖(𝑝)𝑐𝑐𝑓 𝑗 𝜖(𝑝)𝑐𝑐𝑑𝑗 𝜖(𝑝)𝑐𝑎𝑗 ℬ𝜎𝑗(𝑝) 𝜎𝑐𝑗(𝑝) 𝐸𝑢28𝑗(𝑐) 𝐸𝑢28𝑗(𝑡) 𝜎^𝑐𝑗(𝑐) 𝜎^𝑐𝑗(𝑡) ;𝑗 = (^1,^2) Thus, the creep strain components due to irrecoverable flow (basic plus dry-ing) and delayed elasticity are stored separately, while the unloading/reloading moduli are represented by their equivalent 28 day-values. Furthermore, ℬ𝜎 ac-counts for the detrimental effect on compressive strength due to high sustained loading. The remaining quantities are explained previously in this section. In addition, the angle 𝛽𝑐1(𝑝) to the principal ^1-direction is also stored.
∙ Prior to entering the algorithm, the following quantities have been computed 𝜖𝑖 ;𝑖= (1,2)
𝛽𝑐1 𝜖𝑐𝑇 𝜖𝑐𝑠 Δ𝜑𝑐𝑑 ℬ𝑑 Δ𝜑𝑎
Δ𝜑𝑐𝑓 𝑗 𝐸𝑢𝑗(𝑐) 𝐸𝑢𝑗(𝑡) ˜𝜖(𝑐)𝑐𝑗 ˜𝜖(𝑡)𝑐𝑗 ;𝑗 = (^1,^2)
Here𝑖refers to the current principal directions, and𝛽𝑐1 is the angle to the first of these. Similar to strains, the incremental creep coefficients are now charac-terized by irrecoverable flow and delayed elastic components. The remaining quantities are explained previously in this section.
∙ Then compare current and previous directions according to the procedure in Subsection 8.1.6, i.e.
if
𝛽𝑐1(𝑝)− 𝜋
4 ≤ 𝛽𝑐1 ≤ 𝛽𝑐1(𝑝)+𝜋 4
then stored values for the previous directions 𝑗 are valid for the current directions 𝑖
𝑖/𝑗 = (1/^1,2/^2) else
then the correspondence between directions becomes 𝑖/𝑗 = (1/^2,2/^1)
end if
∙ Now search for the current mechanical strain𝜖𝑐𝑖in the principal 𝑖-direction and the corresponding stress ˘𝜎𝑐𝑖 by following the algorithm:
At start, assume tension state by computing 𝜖𝑐𝑖 from Eq.(9.103), inserting 𝐸𝑢𝑗 = 𝐸𝑢𝑗(𝑡).
if
0 ≤ 𝜖𝑐𝑖 ≤ ˜𝜖(𝑡)𝑐𝑗 then
˘
𝜎𝑐𝑖 = 𝐸𝑢𝑗(𝑡)𝜖𝑐𝑖 else if
𝜖𝑐𝑖 > ˜𝜖(𝑡)𝑐𝑗
then select
˘
𝜎𝑐𝑖 = ^𝜎𝑐𝑗(𝑡) and recompute 𝜖𝑐𝑖, using Eq.(9.104)
else passed to compression state
then recompute 𝜖𝑐𝑖 from Eq.(9.103), now inserting 𝐸𝑢𝑗 =𝐸𝑢𝑗(𝑐) if
˜
𝜖(𝑐)𝑐𝑗 ≤ 𝜖𝑐𝑖 ≤ 0 then
˘
𝜎𝑐𝑖 = 𝐸𝑢𝑗(𝑐)𝜖𝑐𝑖 else if
𝜖𝑐𝑖 < ˜𝜖(𝑐)𝑐𝑗 then select
˘
𝜎𝑐𝑖 = ^𝜎𝑐𝑗(𝑐) and recompute 𝜖𝑐𝑖, using Eq.(9.104) else passed to neutral state
then select
˘
𝜎𝑐𝑖 = 0 and recompute 𝜖𝑐𝑖, using Eq.(9.104) end if
end if
∙ Having determined ˘𝜎𝑐𝑖, the time dependent quantities (𝜖𝑐𝑐𝑓 𝑗, 𝜖𝑐𝑐𝑑𝑗, 𝜖𝑐𝑎𝑗,ℬ𝜎𝑗) can now be updated for the time step considered.
∙ Based on the computed mechanical strain𝜖𝑐𝑖, the final current stress 𝜎𝑐𝑖is then found according to the search procedure outlined in Section 8.1.6. If this stress belongs to a global equilibrium state, the aforementioned updated quantities are saved. Otherwise the values pertaining to the previous equilibrium state are retained for the mechanical strain computation in the next iteration cycle.
A consequence of adopting the form in Eq.(9.98) for the sustained stress ¯𝜎𝑐, is that the values (𝜎𝑐(𝑝),𝜎˘𝑐) at the beginning and end of the time step should refer to the same external load level. To comply with this, as already mentioned in Section 5.4, the prescribed sequence of the ‘neutral time’-parameter𝜆 will here be chosen so that typical short-time and long-time phenomena are handled in separate solution steps.
To be more specific; take a structure subjected to a sustained load 𝐴, followed by an additional load 𝐵. In this case two separate solutions should be made regarding the introduction of load 𝐵; one immediate to application that yields the long-time effect
of load𝐴, and one shortly after that includes the short-time response due to load 𝐵.
Also note that the introduction of (^𝜎(𝑐)𝑐 ,𝜎^𝑐(𝑡)) as limit-values for ˘𝜎𝑐, implies that this procedure in possible loading phases reduces to a simple step forward procedure, as adopted by e.g. Kang [22]. There the stress at the previous equilibrium state is used for the succeeding time step (i.e. ¯𝜎𝑐 = 𝜎(𝑝)𝑐 ). In general, such a procedure requires
‘small’ time steps in order to achieve acceptable accuracy.