📄 Calculating deflections using FEA
Sign in

Calculating deflections using FEA

The calculation of the deflections of the structure using the rigorous method needs to account for changing material properties, loading events and the formation of cracking which leads to a change in the cross section and stiffness, which as can be seen from the discussion above, are time dependent events. The analysis, therefore, needs to be done in time steps to match significant time dependent events, such as changes in the loading on the structure, for example when the slab formwork is struck or when structure above is cast and force is applied to the structure below through the propping, taking into account the time dependent variation in the material properties at each calculation interval. The use of an iterative numerical solution allows for the non-linear material properties to be taken into account during the analysis.

At each stage of the analysis, the stress state of the structure, taking into account the effects of creep and shrinkage as well as the time dependent material properties, will determine whether or not cracking will occur. Cracking at a cross section will reduce the stiffness of the structure local to the cracking and this will result in a redistribution of the forces in the structure, which can lead to further cracking. The change in the cross section results in geometric non-linearity in the analysis. Therefore, at each time stage of the analysis, an iterative solution is required to determine the effects of cracking spatially over the structure.

When using a finite element model to calculate the deflection of a structural element, the structure is subdivided into finite elements as part of the modelling process and the elements make a suitable subdivision of the structure for assessing the extent of cracking. The stresses in each finite element are checked at the Gaussian points in the elements and each of the nodes associated with the element under consideration and, if the stresses are such that the cracking limit is exceeded, the properties of the element are modified to model the cracked section. The next iteration of the analysis is then carried out using the modified element properties and further cracking is determined. The process is repeated until no further cracking is determined to occur and the current stage of the analysis has converged to a solution.

Once cracking occurs at a cross section, that cross section will remain cracked. In terms of an FE analysis, this means that once the stresses in the element exceed the limit for cracking, the section properties of the element are modified and remain modified for all further calculation iterations.

After the analysis is complete for all loading events, the final deflection of the slab is calculated based on the cracked stiffness of the slab accounting for time-dependent properties such as material strength, creep and shrinkage. This is done by determining the curvature of the structure from which the final deflection can be calculated.

One of the significant factors that influences the stress in the concrete and so influences the point when cracking occurs, is the reinforcement provided in the slab. Thus, the deflection of the slab is influenced by the reinforcement from the ULS design. Hence, the ULS design must be completed as a first stage in the calculation of the deflection of the structure. This also means that any modification of the ultimate limit state design can also have an impact on the deflection analysis of the structure.

The iterative nature of the deflection design means it is computationally expensive and so can take a significant amount of time to run an analysis and, as such, modifications to the ULS design will mean that it is necessary to also rerun the deflection analysis. Therefore, it is necessary to have the ULS design as complete as possible before carrying out a deflection analysis to avoid losing time unnecessarily.


Deflection calculation accuracy

The calculation of the deflection of a structure depends on a number of factors. These factors include:

  • Elastic modulus
  • Tensile strength of the concrete
  • Creep
  • Shrinkage
  • Amount of reinforcement
  • Restraint to the structure
  • Aggregate properties
  • Cement type
  • Relative humidity
  • Loading
  • Temperature
  • Construction sequence
  • Age of concrete when loaded 

All of these factors are subject to inherent variability and many are also subject to variance with time. The properties of raw materials such as aggregate and cement are based on average material properties and the material used may not be fully known at the design stage. The properties of the concrete, which are dependent on both the raw materials and the mix proportions, vary with time, but are also affected by environmental factors such as temperature and humidity which not only vary from day to day, but also throughout the day. Construction sequences may not be fully known at design stage, but even where they are clearly laid out, variation of the timing on site is likely.

Many of the above factors are also influenced by others and so the factors are not wholly independent. As a result of the variability of the various factors and the difficulty in determining the relevant parameters at the design stage, the calculation of the deflections remains an estimate, even where an advanced and rigorous analysis method is employed. The published guidance advises that actual deflection may vary from calculated deflections in a range of +15% to -30%