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Theory

As mentioned in EN 1992-1-1 chapter 7.4.1(3) calculated deformations should not exceed those that can be accommodated by other connected elements such as partitions, glazing, cladding, services or finishes. In some cases limitation may be required to ensure the proper functioning of machinery or apparatus supported by the structure, or to avoid ponding on flat roofs.

Generally two main situation are required to be checked:

• Total deflection - The appearance and general utility of the structure could be impaired when the calculated sag of a beam, slab or cantilever subjected to quasi-permanent loads exceeds span/250. The sag is assessed relative to the supports. Pre-camber may be used to compensate for some or all of the deflection but any upward deflection incorporated in the form-work should not generally exceed span/250
• Additional deflection - Deflections that could damage adjacent parts of the structure should be limited. For the deflection after construction, span/500 is normally an appropriate limit for quasi-permanent loads. Other limits may be considered, depending on the sensitivity of adjacent parts.

The calculation procedure used in new Deflection check can be described in the following steps:

1) Calculation of short-term stiffness - short-term stiffness is calculated using 28days E modulus for acting load

2) Calculation of long-term stiffness - long-term stiffness is calculated using effective E modulus based on creep coefficient for acting load

Unfortunately, for time being there is not possibility to distinguish between short-term and long-term part of the load in combination. Therefore some precondition has been established for determination of long-term part of the load. Long-term part of the load (LongTermPercentage) is estimated based on the type of combination for check. There are three main SLS combinations:

• 1) SLS characteristics - LongTermPercentage = 70 %
• 2) SLS frequent - LongTermPercentage = 85 %
• 3) SLS quasi-permanent- LongTermPercentage = 100 %

SCIA Engineer automatically use characteristic or quasi-permanent combination together in one step. Regardless on what combination (Characteristics, Quasi-permanent of Frequent) is selected. Characteristic combination is used to evaluation if crack appear. It affect final stiffness.

Selected combination is used as place holder for Load Cases, where appropriate coefficients are used during check (Characteristics, Quasi-permanent of Frequent). Recommended is to use the Result class with all SLS combinations from the project, or combination with all Load Cases.

3) Calculation of ratios - stiffness ratios are used as simplified method for calculation of particular deflections (see below). These values are calculated for each state according to points 1 and 2 above. Generally , the values are ratios of linear stiffness of concrete component only divided by resultant stiffness taking cracks into account.

ratio = Stiffness_lin / Stiffness_res

For example:

ratio_uz = EI_z,lin / EI_z,res

4) Calculation of particular component - Several particular components are needed for calculation of total and additional deflection

As mentioned beforehand, the short- and long-term stiffnesses are calculated using a so-called creep factor. This creep-factor is dependant on the relative humidity, outline of the cross-section, reinforcement percentage, concrete class, etc.This factor is used to divide the short-term stiffness and obtain the long-term stiffness, Thus by taken the concrete stiffness for short- and long-term and the representative compression strength the program calculates the stress and strain diagram.

Generally , the components calculated below can be graphically presented on the following figure

Linear (elastic) deflection - is the sum of short-term and long-term elastic deflection

δ_lin = δ_lin,s + δ_lin,l

Immediate deflection - to calculate the immediate deformation, the deformation of the permanent load is calculated using the short-term stress and strain diagram. Additionally by subtracting the immediate deformation from the total deformation, the programs calculates the additional deformation.

δ_imm = δ_lin,l ∙ ratio _s

Short-term deflection - is the multiplication of short-term elastic deflection and short-term ratio

δ_s = δ_lin,s ∙ ratio_s

Long-term deflection + creep - is the multiplication of long-term elastic deflection and long-term ratio

δ_l,creep = δ_lin,l ∙ ratio_l

Creep deflection - is calculated based on short and long term ratios

δ_creep = δ_lin,l ∙ (ratio_l - ratio_s)

Long-term deflection - is the difference between deflection caused by long-term + creep and creep parts of deflection

δ_l = δ_l,creep - δ_creep

Additional deflection - is the difference between sum of shot-term and long-term with creep towards immediate deflection

δ_add = δ_s + δ_l,creep - δ_imm

Total deflection - is the sum of short-term and long-term +creep deflection

δ_tot = δ_s + δ_l,creep

Deflections are calculated for selected SLS combination.

5) Check of deflections - as it was reported at the beginning of this chapter two deflections are required to be checked. At first the limit values has to be calculated for particular direction of deflections. These values are:

a) limit for total deflection

δ_tot,lim = L / 250

b) limit for additional deflection

δ_add,lim = L / 500

In formulas above, there is mentioned L value. This value corresponds to buckling length multiplied by β factor of the member in particular direction.

Finally the unity check can be calculated as follows:

Unity check = max (δ_tot / δ_tot,lim; δ_add / δ_add,lim)

Output values

There are presented the following output values:

• Unity check - unity check of check deflection(see above)
• δ_tot,y - total deflection in (y) direction
• δ_tot,z - total deflection in (z) direction
• δ_tot,lim,y - limit total deflection in (y) direction
• δ_tot,lim,z - limit total deflection in (z) direction
• δ_add,y - additional deflection in (y) direction
• δ_add,z - additional deflection in (z) direction
• δ_add,lim,z - limit additional deflection in (z) direction
• δ_add,lim,y - limit additional deflection in (y) direction

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