How to include creep in a concrete slender column in Member

When designing slender reinforced concrete elements, the effects of imperfections, second order, and creep on transverse deformation have to be considered. 

For a better understanding of the example on which the problem will be explained go through the tutorial Concrete slender column (EN).

The development of the transverse deformation of the compressed element is shown schematically in the figure above. The total load is composed of a long-term load F_LT and a short-term load F_V (variable load). Before loading begins, only the geometric imperfection e₀ forms the transverse deflection of the element. Once the element is loaded with the F_LT force, the transverse deformation increases to w_LT(t₀). Due to the creep, the transverse deflection will increase to w_LT(t_∞) in the <t₀;t_∞> time interval. The total transverse deflection at the end of the life of the structure (time t_∞) after the application of the short-term load F_V is then w_LT+V(t_∞). The second-order effect caused by this deflection governs the design of a slender compression member.

The individual components of the lateral deflection are shown schematically in the following figure.

Where:

e₀                  initial geometric imperfection defined by the design standard

e_2,LT(t₀)        second order effect from a permanent F_LT load, in time t₀. This deflection also includes the effect
                    of transverse loads or end moments. The value is the result of a GMNIA calculation in the member
                    (U_x or U_y displacement), where the initial imperfection is set to e₀

e_2,LT^CR(t_∞)    the increment to e_2,LT(t) that is caused by concrete creep in the time interval <t₀;t_∞>.

e_2,LT+V           second order effect at time t_∞ from constant (LT) and variable (V) loads. This value is automatically
                     taken into account by the program using the GMNIA calculation, where the imperfection is given by
                     e₀ + e_2,LT^CR(t_∞).

For the design of the compressed element, a value of e_2,LT^CR(t_∞) is required. As deflection e_2,LT^CR(t_∞) increases over time, deflection e_2,LT(t) will increase simultaneously. To accurately calculate the final value of e_2,LT^CR(t_∞), it would be necessary to use a time-dependent analysis (TDA). In the current version, the program does not calculate this automatically and it must be determined manually by an iterative procedure, which is discussed below.

The calculation steps in the member program are as follows:

1. GMNIA calculation of the response of the member to long-term loads F_LT with specified initial imperfection e₀.
2. Determination of total imperfection e₀ + e_2,LT^CR(t_∞) 
3. GMNIA calculation of the response of the member to total load F_LT + F_{V ,} with total imperfection e₀ + e_2,LT^CR(t_∞) specified in the program

Determination of the deflection e_2,LT^CR(t_∞):

For the total deflection from permanent F_LT loads at the end of life at time t_∞:

w_LT(t_∞) = e₀ + e_2,LT^CR(t_∞) + e_2,LT(t_∞)

Conservatively:

e_2,LT^CR(t_∞) = φ(t₀,t_∞) * e_2,LT(t_∞)       where φ(t₀,t_∞) is a creep coefficient

The value of e_2,LT(t_∞) is determined by a GMNIA calculation with the total specified imperfection e₀ + e_2,LT^CR(t_∞) = e₀ + φ(t₀,t_∞) * e_2,LT(t_∞). Clearly, for this simplified and conservative approach, the value of e_2,LT(t_∞) "depends on itself" and has to be determined by iteration.

You can iterate sequentially as shown below. The four steps of the iteration are shown. The variable labels are slightly different to keep the picture simple.

φ(t₀,t_∞) = φ
e_2,LT(t_∞) = e_2,LT,i
w_LT(t_∞) = w_LT,i

The video tutorial of gradual iteration described above is shown below. The excel file used in this tutorial is also attached.

Note: The load case LE4 contains only long-term loads (quasi-permanent combination) and is applied as a ULS load type. This means that the ULS material model is used to calculate the initial imperfection.

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