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(1)
Long term deflection calculation - background
The deflection of a concrete slab is influenced by a range of factors, which include:
Tensile strength of the concrete
Compressive strength of the concrete
Aggregate type
Elastic modulus
Creep
shrinkage
ambient temperature
relative humidity
load intensity
load duration
age of concrete when loaded
cracking of section and extent of cracking
restraint to the slab due to other structure or support conditions
Given the nature of the above design parameters and the difficulty in accurately assessing them, the results of a deflection calculation should always be treated as an estimate, regardless of the complication of the methodology employed to calculate the deflection.
For most standard design situations, span-to-depth ratio-based deflection checks are satisfactory. Clause 7.4.2 of BS EN 1992-1-1:2004 notes that:
“Generally, it is not necessary to calculate the deflections explicitly as simple rules, for example limits to span/depth ratio may be formulated, which will be adequate for avoiding deflection problems in normal circumstances.”
However, in some cases, a more direct calculation may be desirable or may be required in cases where deflection limits other than those encompassed by the code rules would be required. This could result in the ability to specify thinner structural members or slabs than would be deemed to satisfy the span-to-depth criteria. Also, the earlier the concrete is loaded the greater the impact on the long-term deflection, so early striking by the contractor can lead to a requirement to check the deflections beyond the scope of a span-to-depth ratio check.
The rigorous calculation methods given in both BS 8110-2 and BS EN 1992-1-1 both essentially calculate the additional curvature in the concrete element due to loading, modified to account for creep, cracking and the curvature due to shrinkage, then calculating the resulting deflections in the concrete structure.
It should also be noted that despite the complexity of direct calculation of the deflections of a slab, the sensitivity of the calculation to the time dependent nature of the loading, along with the difficulty in determining accurately many of the parameters required for the calculation, means that measured calculated deflections remain an estimate for any given structure.
Of the many parameters involved in the deflection of a concrete slab, the most critical are the concrete tensile strength, the extent of cracking and the elastic modulus. Of these, the concrete properties are directly time varying, while the extent of cracking is a function of both the material properties and loading which are themselves time dependent.
The design codes give guidance on limiting deflections for structural elements. The scope of the guidance is limited and so it is up to the design engineer to determine the appropriate limits. Client requirements may require stricter limits on the deflections depending on the end use of the structure.
The calculation of the deflection of a concrete beam or slab element involves the assessment of the time dependent properties of the concrete itself, to account for the variation in the material over time, along with an assessment of the variations in loading during the structure’s serviceable life, particularly during the construction phase when the concrete is still undergoing curing. Particularly important is the determination of the extent of cracking which occurs at each stage, since cracking will modify the cross section and hence the stiffness of the section where it is cracked, leading to a reduction in the stiffness of the cracked cross section and hence an increase in the deflection. Cracking will only occur along part of the length of the element, and parts of the cross section will be in a state somewhere between fully uncracked and fully cracked. Since deflection is a serviceability state, elastic theory applies to the cross section whether it is cracked or not.
The time dependant properties of the material, loading and cross-sectional properties means that, unless significant simplifications are employed, a rigorous method of assessing deflection is not feasible using hand calculations. However, even when computer based calculations are used, which can range from the use of spreadsheets to advanced finite element methods which use iterative methods to determine the extent of cracking, the Concrete Centre Technical Report TR58 notes that calculated and actual deflections are highly likely to differ due to the difficulty in accurately assessing all of the involved parameters. TR58 advises that when reporting deflections, a reported range of +15% to -30% on the calculated values would be advisable, to account for the inherent variability in the calculations.
The underlying principle in calculating the deflection of concrete beams and slabs is to determine by calculation the resulting curvature of the structural element due to long term and short terms loads. For the basis of hand calculations, the codes permit the long- and short-term loads to be combined in superposition. However, for computer analysis, the recommendation is that the resulting curvature should be calculated for each combination of long- and short-term loads. This will involve an iterative calculation process and the cracking from one load combination will be present in those subsequent combinations, requiring the modification of the section properties due to cracking as well as a modification of the time dependent parameters at each stage of the calculation process.
Finite Element Analysis can be used for the design of concrete elements. However, where the analysis is 1st order linear elastic and based on the gross cross section, no account is taken of reinforcement nor the degree of cracking. Either the section properties have to be calculated and modified manually, or approximate methods employed which account for the modification of the material properties and change of cross section. One common method is to use a reduced Young’s modulus to make an estimate of the effects of cracking, creep and shrinkage. This method can be applied within the Masterframe FE module, where the facility to specify a reduced Young’s Modulus is provided such that serviceability load cases use the reduced E-value. For a more reliable estimate of the deflections of concrete element, it is necessary to use an analysis which automatically calculates the cracked section properties and assesses the effects of cracking on the element stiffness at each point in the loading history, incorporating the extent of cracking from the previous loading events and also modifying the material properties to account for the age of the concrete at the time of loading. Such an analysis requires an iterative computation method to account for the effects of time on the loading and material variations, resulting in a non-linear analysis. However, while this represents a significantly more detailed and so more reliable calculation of the deflections, the uncertainty over the material properties and loading means even a more detailed analysis remains an estimate of the deflections and recommendation to report a range of +15% to -30% on the calculated values remains.
Creep is the tendency for a material to undergo continuous and permanent deformation under conditions of permanent stress, even in cases where the stress is significantly less than the yield stress of the material. Therefore, creep causes an increase in strain, but without a corresponding change in stress. The rate at which the change in strain occurs is related to the materials physical properties, the stress levels and the duration of the stress. The rate of strain in creep is such that deformation of the structural element does not occur suddenly, but accumulates over time. Creep strains can be significant and so the influence of creep on the long-term deformation of the structure can also be significant.
In concrete elements such as beams and slabs, where, under bending, the resulting stress condition is such that there are clearly defined compression and tension regions in the slab, creep leads to additional strain in the compression zone of the element, which is turn results in additional deflections of the structural element.
Since creep is associated with increased strain but without a change in the stress over time, then this results in a modification in the Elastic modulus of the material over time, since Young’s modulus (E) is given by:
(1)
Thus, in concrete, the elastic modulus is also a time dependent property of the material, with a decrease in the E-value over time associated with creep of the material. However, concrete also gains strength over time due to the curing process, which sees a corresponding increase in the elastic modulus over time. Both the British Standard and Eurocodes use an effective elastic modulus to account for the time dependent nature of the Young’s modulus, utilising a creep coefficient to modify the short-term Elastic modulus. The effective modulus is given by:
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where
Ecm secant modulus of elasticity
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creep coefficient for concrete loaded at age t0 and for load duration t
The main parameters which affect the creep of the concrete are:
the mean compressive strength of the concrete
the age at loading
the cement type
the relative humidity
the duration of loading
the cross-sectional area and exposed perimeter of the section
temperature
stress limitation
The majority of the parameters affecting the creep coefficient .png){kind=small}
are themselves time dependent.
The effects of concrete grade, cement type, time and duration of loading, temperature and section geometry are taken into account when calculating the creep coefficient. In the Eurocode, the procedure is detailed in Annex B Section B1.
The Eurocode contains a limitation on the stress in the concrete under quasi-permanent loads which affects the creep coefficient used, requiring the modification of the creep coefficient to account for non-linear behaviour. Where the stress in the concrete is greater than 0.45* fck (t0 ) at an age t0 then a non-linear notional creep coefficient φnl(∞, t0) should be assessed and used in the modification of the Young’s Modulus E. Such high concrete stresses can occur in prestressed concrete.
The strain due to creep may be significantly larger than the elastic deformation of the structure that occurs due to the applied loading. Where elastic deformation due to loading is recovered with the removal of the load, this is not the case for deformations due to creep.
Shrinkage of concrete occurs due to a change in the volume the which then results in additional strain. The change in volume results from the loss of moisture during the curing and drying process. The amount of drying shrinkage that occurs in concrete structures depends on the constituent materials that make up the concrete, the proportions of the constituent materials, how the concrete is cured, environmental factors such as humidity and temperature and the degree of restraint to the structure.
When calculating the shrinkage, the following factors are taken into consideration in the equations:
cement type
water/cement ratio
aggregate type
age at exposure
surface area exposed
member size and shape
reinforcement quantities and position
relative humidity
curing
The shrinkage of the concrete is made up of two components, drying shrinkage and autogenous shrinkage. The total shrinkage is then given by:
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Where:
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– the total shrinkage strain
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– the drying shrinkage strain
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– the autogenous strain
Drying shrinkage
Drying shrinkage occurs due to the loss of water by capillary action. This leads to a compressive stress on the concrete which results in a contraction or loss of volume over time, resulting in the contraction of the concrete and subsequent strain. Since the migration of water through the hardened concrete is slow, drying shrinkage occurs over an extended period of time and so is a gradual but long-term effect. Structural elements which have a large surface area relative to their volume will undergo more significant loss of water and so have higher subsequent drying shrinkage. Where the structural elements have a high degree of restraint, more cracking will develop as a result of the drying shrinkage.
Autogenous shrinkage
Autogenous shrinkage occurs in early age concrete and is the result of pore water being drawn into the hydration process. As water is drawn in, the demand for more water results in fine capillaries being created in the concrete matrix. The surface tension within the capillaries results in a compressive stress in the concrete and a resulting strain and the formation of cracking.
Autogenous shrinkage tends to be significant in elements which have small cross sections and/or low water/cement ratios. Since it is linked to the hardening of the concrete, the majority of the autogenous shrinkage tends to occur early in the life of the concrete.
Predicting shrinkage
BS 8110-2 provides a method for predicting the effects of drying shrinkage at 6 months using a chart as given in Figure 7.2, Section 7. The range of applicability of this method is limited since it requires a limited range of values based on an assumed water content and assumed aggregate types. Figure 7.2 does not lend itself to automated calculation within software. However, similar predictions of drying shrinkage can be achieved using BS 5400-4.
EC2 provides a tabular method for estimating drying shrinkage in Table 3.2 of Section 3.1.4. However, a method to calculate the drying shrinkage is given in Appendix B.2 which, when combined with Clause 3.1.4(6) provides a method to estimate the strains due to drying and autogenous shrinkage by calculation.
Appendix B2 requires the following parameters to be defined to calculate the drying shrinkage:
the mean compressive strength fcm
determination of the cement class (S, N or R)
the relative ambient humidity RH
As noted in the Concrete Centre Technical Publication TR-58, drying shrinkage is the dominant factor affecting the shrinkage of the concrete. While shrinkage is less important than creep in the long-term deflection of the concrete, shrinkage can still account for up to 30% of the long-term deflection of the structure.
Once the creep and shrinkage effects have been calculated, their effects on the deflections of the slab have to be included in the calculation of the deflections of the slab. Within the FE analysis, this requires the modification of the material factors to account for creep and a subsequent modification of the curvature of the section due to shrinkage, but other factors must also be considered as part of the analysis. These factors include the tensile strength and its variation with time, the loading at relevant time intervals, and the extent of cracking over time which will modify the cross-sectional properties and so affect the stiffness of the structure.
Loading Sequence
The deflection of the structure is very significantly influenced by the extent of the cracking which occurs, which is in turn significantly influence by the loading on the slab and the material properties at the time of loading. In particular, loads applied when the concrete is early in age and the material properties are still developing can lead to significant cracking which will influence later performance.
The construction sequence will have a significant influence on the loading sequence and age of the concrete at the time of loading. The loading sequence will begin with the slab being struck which subjects an early age loading on relatively immature concrete. Various loads are then exerted on the slab during the construction of subsequent floor above where loads are transferred to lower structural elements due to the arrangement of propping. Then, finally, a structural element will be subjected to final construction and finishes and variable imposed loads during its serviceable life.
The effect of early age loading is more pronounced in the Eurocode than was the case with the British Standard method and so the Eurocode may produce a more critical design than would have been achieved with the British Standard.
Tensile Strength and degree of restraint
Since concrete will crack once the tensile stress in the structure exceeds the tensile strength of the concrete, which then affects the cross section, the tensile strength of the concrete is a significant factor in the long-term performance of the structure.
While the British Standard used a fixed tensile strength based on the concrete grade, the Eurocode allows for some modification for time with an increase in the tensile strength as the concrete compressive strength increases with time.
The tensile stress in the concrete is influenced by the restraint to the structure, since the restraint of the structure against shrinkage will induce tensile stresses in the structure which will then have an influence on the formation of cracking.
The restraint to the slab will have an influence on the effective tensile strength of the concrete. Where slabs are highly restrained, the shrinkage of the slab will induce tensile stresses which effectively reduces the tensile strength of the concrete. For high restraint situations, the concrete tensile strength fctm should be used, while for no restraint the tensile strength fctm,fl can be used. The Concrete Centre publication “How to design concrete structures using Eurocode 2” recommends that for low restraint, the average value of fctm and fctm,fl should be used, to take account of any unintentional restraint.
Cracking
The deflection of the structure is very closely linked with the formation of cracking, which is itself closely linked to the tensile stress and tensile strength of the concrete. The tensile stress in the structure is linked to the loading while the tensile strength is dependent on the material properties of the concrete and so both are time dependent properties.
Once cracking has occurred, the cross section in the vicinity of the crack is permanently altered, resulting in a reduced effective cross section and hence a reduction of stiffness. The presence of cracking can have a significant effect on the long-term subsequent distribution of forces in the structure in later loading events and so influence the performance and deflections of the structure. It is, therefore, important to identify the stages at which cracking occurs. Since cracking is dependent on loading and material properties, both of which are time dependent, crack formation and its effects are then also time dependent.
Reinforcement
The amount of reinforcement in a cross section, along with the relative quantities of reinforcement in the compression and tension zones can induce additional curvatures in the concrete as an result of the shrinkage of the concrete. In zones where the reinforcement is symmetrical, the strain due to curvature is uniform and so no curvature will be induced, but once the reinforcement becomes non-symmetrical, then an imbalance in the strains occurs which leads to an imbalance in the forces in the cross section which in turn leads to an induced curvature, with the increased curvature occurring on the side with the largest area of reinforcement. As a consequence of this, the larger amount of reinforcement on the tension side of the section will induce a curvature which acts to increase the deflection of the slab or beam.
The shrinkage curvature is calculated from:
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Where
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the curvature due to shrinkage
the free shrinkage strain
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the effective modular ratio
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the first moment of area of the reinforcement about the centroid of the section
I the second moment of area of the section
The effective modular ratio is calculated from:
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Where is calculated from Eq (3)
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.
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%.