Slab deflection calculations in depth
Interrogating slab deflection calculations
Tekla Structural Designer's Slab Deflection Analysis is not a "black box" - the calculated results are all exposed for interrogation as required.
Several different Results views are available: Deflection, Extent of Cracking, Relative Stiffness and Effective Reinforcement. There is also a Composite modulus report available for each slab.
To help pull all the results together, the items noted above are tied together in the following way:
- Every element is part of a slab item. Each slab item can have different effective concrete properties.
The best way to understand this is to look at the summary of information from the Composite Modulus report for each event and consider things like:
• Adjusted event times due to temperature and cement class
• Adjusted creep properties due to the number of exposed faces
• Incremental loading factors
• The Composite Modulus Calculation
- Every shell element can have different Effective Reinforcement:
• This is determined automatically
• The Effective Reinforcement results view allows you to confirm the values used.
- Cracking and Stiffness Calculations for each event:
• This is dependent upon the effective concrete properties, effective reinforcement, and the forces that develop
• So this calculation is unique for each direction of each shell for each event.
• The stiffness of cracked sections is dependent on the degree of cracking - so the procedure is iterative (force -> stiffness -> new force, etc)
• At the converged conclusion of this you can see (and check):
- the extent of cracking via the Extent of Cracking results view
- the stiffnesses determined via the Relative Stiffness results view
Design codes typically provide a way of calculating an effective creep modulus at time, t for a constant load/stress applied at time t0. Typically this is presented as ϕ(t, t0).
An effective Young's Modulus is then calculated as Ec, eff (t, t0) = Ec,28 / [ 1 + ϕ(t, t0) ]
However, codes typically do not give guidance on how to deal with a loading history where loads vary over the time period being considered.
Technical Report 58 introduces guidance on this topic and proposes a method by which the loading history can be taken into account. (Reference Section 8.4.1, equation 8.37 and also the example on page 36):
n = event under consideration
wi = incremental load in event i (= load in event i - load in event (i-1)) (note that this will be a negative value when load is removed)
Eeff,i = Ec, eff (tend,n, ti) (i.e. covers period from start of event i to end of event n)
The above is logical when you consider a single member subjected to a constant loading arrangement that is increased or decreased at each event. However, when you consider an entire slab with many panels receiving different loading increments in different events it does not seem reasonable to consider all the panels together. Two examples of this are:
- Why should the addition of cladding loads affect internal panels to the same extent as edge panels?
- In a transfer slab why should panels that don't support columns be affected to the same extent as those which do?
The aim of TR58 is clear, in that loading on a span/panel is taken as an indication of stress. What this fails to consider is that loading on another span/panel can also induce stress (although in most situations this will be a secondary effect). It is also clear that you do not actually need "loads", you just need "relative loads" or some other measure of the relative work done in each event. With this in mind a more general approach has been developed where the relative "work done" by each panel is determined by considering the strain energy in each event:
- Calculate total strain energy 'Q0' for each Slab Item for a unit loadcase
• 'Q0' = sum of 2D Element strain energies
- For each Load Event 'i'
• Calculate incremental strain energy 'Qi' 'Qi' = sum of incremental 2D Element strain energies
• Calculate equivalent incremental load factor 'λi
' λi = Qi / Q0
- Ecomp can be established from the equation below where "incremental work done" replaces "incremental load" in the TR58 equation.
There is an array of intermediate values which lie behind the calculation of the composite modulus, Ecomp for each slab item, for each event. The composite modulus calculation is provided as an excel spreadsheet report.
You can either generate a report for a chosen slab, selected slabs or all slabs, dependant upon your selection method.
- To obtain slab modulus reports for all slab items, right click anywhere in a scene view and choose Export Eff. Modulus report to Excel > For all slab items
- To obtain slab modulus reports for selected slab items, select the slabs in the structure view regime, right click anywhere in the scene view and choose Export Eff. Modulus report to Excel > For selected slab items
- To obtain slab modulus reports for a chosen slab items, right click a slab panel in the structure view regime and choose Export Eff. Modulus report to Excel > For current slab items
A typical composite modulus report for a slab panel is shown below.
It should be noted that Tekla Structural Designer takes into account the cement class when determining the temperature adjusted age of loading so minor variations will occur.
The effective modulus is used in determining the properties for each load event.
Extent of cracking
The effective modulus, and the effective reinforcement are used to determine the cracked or uncracked state of each shell for each event, for each direction.
Eurocode 2 provides an expression that predicts the behavior between the cracked and uncracked states. This expression uses a Distribution factor, ζ that apportions the behavior between a fully-cracked state (±1.0) and an uncracked state (0) for interpolating the stiffness when a state of partial cracking exists.
ζ = 1 - ß (Mcr / M)2
- ß is a user defined value specified in the Event Sequences and is either: 1.0 for single short-term loading 0.5 for sustained loads or many cycles of repeated loading
- Mcr is the hogging (positive) or sagging (negative) cracking moment.
- Ma(+|-) is the relevant Wood-Armer moment in the direction for which the display is shown (X or Y). This is calculated from Mx, My & Mxy in the usual way, when determining the extent of cracking for a shell element for each iteration for each Event.
If you view Extent of Cracking results for a chosen result direction and cycle through the events you will see each of the FE elements shaded to indicate the extent of cracking.
The tooltip shows the applied moment for the Event in question, rather than the worst moment of all Events up to and including the one being looked at - this allows engineers to deduce that a greater level of cracking was caused by an earlier Event.
The tooltip lists:
- Ec28 : Edb*1.05*stiffness adjustment factor. (Where Edb = the short term modulus from the concrete materials database). The stiffness adjustment factor used is determined from the Slab Deflection ribbon > Settings >, Modification Factors page. Provided for information - not directly used in any analysis.
- Ec28+As : the short term modulus including for reinforcement. Provided for information - not directly used in any analysis.
- EST(i) : the short term modulus used in the instantaneous analysis (i.e. includes area of reinforcement and cracking if cracking has occurred) for the selected event.
- ELT,C(i) : the modulus used in the final iteration of long term deflection estimation (i.e. includes area of reinforcement and cracking if cracking has occurred, and effective creep) for the selected event.
- ELT,C&S(i) : ELT,C(i) with further adjustment to allow for effect of shrinkage (= ELT,C / multiplier) for the selected event. The shrinkage multiplier to determine the shrinkage contribution is determined for the chosen event based upon the ratio of the maximum panel Z deflection (including shrinkage) / total Z deflection (excluding shrinkage). This provides an indication of the overall effective stiffness adjustment. Provided for information - not directly used in any analysis.
Based on the modulus, E defined above, a number of ratios are provided in the tooltip for the chosen result direction.
- Short Term Uncracked = Ec28+As / Ec28
- Short Term + Cracking = EST / Ec28
- Creep + Cracking = ELT,C / Ec28
- Creep + Cracking + Shrinkage = ELT,C&S / Ec28
Effective reinforcement for each shell element is also required for the determination of the shell's effective properties at the end of each load event.
- Axt : X Top Effective Reinforcement.
- Dxt : Distance from the section centroid to the center of X top reinforcement.
- Axb : X Bottom Effective Reinforcement.
- Dxb : Distance from the section centroid to the center of X bottom reinforcement.
- Ayt : Y Top Effective Reinforcement.
- Dyt : Distance from the section centroid to the center of Y top reinforcement.
- Ayb : Y Bottom Effective Reinforcement.
- Dyb : Distance from the section centroid to the center of Y bottom reinforcement.
Shrinkage is the strain in hardened concrete that can occur due to moisture loss.
- Eurocode 2 provides a method to estimate shrinkage strains and curvatures based on exposed surface area, member size, relative humidity and reinforcement quantity and position.
- Asymmetry of reinforcement leads to curvature which leads to deflection. It is estimated this effect can contribute up to 30% to the long-term deflection.
- Technical Report 58 provides a theoretical method of estimating the additional shrinkage deflection effect in the analysis
At this time the TR58 method has not been implemented within Tekla Structural Designer. Shrinkage is taken into consideration using a multiplier, by making an overall adjustment to the total deflection (excluding shrinkage) in line with simpler adjustment proposals of the ACI code. This approach is in line with many other software products.
|Time||Multiplier for long-term deflections|
|5 years or more||2.0|
Note that we said shrinkage effects and not creep and shrinkage. Creep is dealt with rigorously in Tekla Structural Designer so we need to ascertain the proportional effect of shrinkage only. ACI 435 provides some indication of the separate contribution of creep and shrinkage.
Table 4.1 - Multipliers recommended by different authors
|Source||Modulus of rupture, psi||Immediate||Creep λc||Shrinkage λsh||Total λt|
|Sbarounis (1984)||7.5 √fc'||1.0||2.8||1.2||5.0|
|Branson (1977)||7.5 √fc'||1.0||2.0||
|Graham and Scanlon (1986b)||
|ACI Code||7.5 √fc'||1.0||2.0||3.0|
Comparing the different sources the ratio of shrinkage is as follows:
- Sbarounis 1.2 / 5 = 24%
- Branson 1 / 4 = 25%
- Graham and Scanlon 1 / 3.5 = 28%
(ignore higher modulus of rupture because reduced values are considered automatically in the cracked section analysis).
The above provides a shrinkage ratio of between 24% and 28%. Hence we recommend a value of between 20%-30% is used. A 25% default is provided via the Slab deflection ribbon > Options dialog and the Creep and Shrinkage page.
The total deflection due to shrinkage effect is determined based on an identified “Total Shrinkage Event” towards the end of the event sequence. The event sequence with the latest load start time is used for calculating the shrinkage adjustment. If multiple events exist with the latest load start time then the first one is considered.
A special note about the Final load event - It is perfectly ok to have multiple events at the same final time, you should not separate these events by a small number of days.
Using this “total shrinkage effect” we can then assign a proportion of the total shrinkage to each event.
With reference to earlier versions of ACI 435 (1966) on which the values in the graph above are based, additional values for 1 month and 3 years can be obtained. This follows the case where As’ = 0 (because compression steel is allowed for differently in the code). By comparison with the graph above, we can see closely matched values.
|Duration of loading||Factor F|
|As' = 0||As' = 0.5As||As' = As|
The values we have adopted for considering shrinkage effects are as tabulated below. The final column provides the proportion of the total shrinkage at a given time.
|Time||Long Term Effects Factor||Proportion of Total Shrinkage|
|5 years and above||2||1.00|
From the above, for any event, the end of event time is used to calculate a “Proportion of total shrinkage” using linear interpolation between the values discussed in the table above.
Deflection calculations on the Z deformation are then adjusted to account for shrinkage effects.
As an example, let’s assume the following event sequence.
Event 1 = 7 days, Event 2 = 10 days, Event 3 = 17 days, Event 4 = 20 days, Event 5 = 27 days, Event 6 = 2 months, Event 7 = 6 months, Event 8 = 1 year and Event 9 = 70 years
Assuming a shrinkage factor of 25% (user defined input value), a basic multiplier can be determined = 1/(1-25%) = 1.333
Event 9 Final event at 70 years analysis deflection (excluding shrinkage) = 32.4mm
Therefore, the Total deflection (including shrinkage) = 32.4 x 1.333 = 43.2 mm
Total Deflection from shrinkage alone is 43.2 - 32.4 = 10.8 mm
We can now apportion this deflection due to shrinkage, to each event based upon the event time and a proportion value.
At 0 days proportion of total shrinkage is 0, At 1 month proportion is 0.3. Therefore using linear interpolation between these values;
- Event 1 (7 days) Shrinkage multiplier = 0.3 * 7/30 = 0.07
- Event 2 (10 days) Shrinkage multiplier = 0.3 * 10/30 = 0.1
- Event 3 (17 days) Shrinkage multiplier = 0.3 * 17/30 = 0.17
- Event 4 (20 days) Shrinkage multiplier = 0.3 * 20/30 = 0.2
- Event 5 (27 days) Shrinkage multiplier = 0.3 * 27/30 = 0.27
At 1 month proportion of total shrinkage is 0.3, At 3 month proportion is 0.5. Therefore using linear interpolation
- Event 6 (2 months) Shrinkage multiplier = 0.4
At 6 month proportion of total shrinkage is 0.6
- Event 7 (6 months) Shrinkage Multiplier = 0.6
At 1 year proportion of total shrinkage is 0.7
- Event 8 (1 year) Shrinkage Multiplier = 0.7
At 70 years proportion of total shrinkage is 1.0
- Event 9 (70 years) Shrinkage Multiplier = 1.0
The Shrinkage deflection that occurs at each event is then the total shrinkage 10.8 mm x the shrinkage multiplier calculated above.
- Event 1 (7 days) Shrinkage = 0.07 x 10.8 = 0.76 mm
- Event 2 (10 days) Shrinkage = 0.1 x 10.8 = 1.08 mm
- Event 3 (17 days) Shrinkage = 0.17 x 10.8 = 1.84 mm
- Event 4 (20 days) Shrinkage = 0.2 x 10.8 = 2.16 mm
- Event 5 (27 days) Shrinkage = 0.27 x 10.8 = 2.92 mm
- Event 6 (2 months) Shrinkage = 0.4 x 10.8 = 4.32 mm
- Event 7 (6 months) Shrinkage = 0.6 x 10.8 = 6.48 mm
- Event 8 (1 year) Shrinkage = 0.7 x 10.8 = 7.56 mm
- Event 9 (70 years) Shrinkage = 1.0 x 10.8 = 10.8 mm
For each event, the total deflection (including shrinkage) reported in the Slab deflection view regime and the tooltips is the event analysis deflection (excluding shrinkage) + the proportion calculated above using the shrinkage multiplier.