Long-term concrete deflections are difficult to predict due to the influence of several non-linear behaviors that complicate the calculations. Historically, design engineers have relied on approximate methods, like deflection multipliers, to simplify the calculations and estimate long-term deflection. Although these methods reduce the calculation effort, they crudely approximate or even exclude some of the important non-linear behaviors and may predict unconservative deflections as a result.

The load history deflection analysis in RAM Concept follows a more rigorous approach that is based on detailed, time-dependent curvature calculations on cross sections. These rigorous calculations accounts for each of the following behaviors that influence long-term concrete deflections:

- Material Nonlinearity
- Early Age Concrete Strength
- Cracking
- Tension Stiffening
- Creep
- Shrinkage
- Internal Restraint to Shrinkage (from reinforcement)
- External Restraint to Shrinkage (from stiff supports)
- Load History

This article briefly discusses how each of these behaviors are included in RAM Concept’s load history deflection implementation.

## Material Nonlinearity

The modulus of elasticity is an important concrete property for long-term deflection calculations because it is directly associated with element stiffness. Although the stress-strain curve of concrete is non-linear (see figure below), in a finite element or frame analysis, the modulus of elasticity is normally assumed to be a constant value as prescribed by the governing building code or standard

RAM Concept uses the PCA concrete stress-strain curve in its cross-section calculations and in doing so considers the material non-linearity of concrete in the load history calculations. At each cross section, the non-linear stress-strain curve is used to determine the concrete material stress from the calculated long-term cross section strains. These stresses are integrated over the cross section to calculate forces resultants, which are calculated to be in equilibrium with the external loads acting on the cross section.

## Early Age Concrete Strength

Concrete compressive strength is an important property for accurate deflection prediction because it is used to determine other material properties like the modulus of elasticity (associated with member stiffness) and the modulus of rupture (associated with cracking). Although concrete strength increases over time, it is common to analyze and design concrete structures using a 28-day concrete strength (f’c). Using f’c in deflection calculations, however, can lead to unconservative deflection predictions, especially if construction loads are significant and cause cracking at an earlier age when the concrete strength is lower.

RAM Concept automatically adjusts the modulus of rupture to the actual time of loading to account for reduced strength at early age loading steps and more accurately calculate long-term deflections. These adjustments are calculated using equations referenced in the design code associated with the selected creep/shrinkage model.

## Cracking

Cracks initiate when an applied load or shrinkage strain produces a tensile stress that exceeds the cracking stress. When a crack forms, stress is relieved at the crack location and a redistribution of stress occurs, which may cause cracks to form in other areas. At cracked cross sections, there is a decrease in stiffness and an increase in cross section curvature, which leads to increased deflections.

RAM Concept accounts for cracking of sections subject to axial and flexure by comparing the calculated tensile stress at the extreme top and bottom fibers with the calculated modulus of rupture. The tensile stress calculations are based on transformed cross sections, which can contain concrete, mild-reinforcement, post-tensioning, or any combination thereof and consider long-term effects such as restraint to shrinkage (both internal and external) and creep. When the calculated tensile stress is less than the modulus of rupture, the section is considered uncracked. In this case, RAM Concept uses a concrete stress-strain curve that accounts for the ability of concrete to resist both tension and compression. When the tensile stress exceeds the modulus of rupture, the section is considered cracked. In this case, RAM Concept calculates cracked curvatures based upon a concrete stress-strain curve that has no tension capacity.

## Tension Stiffening

Tension stiffening refers to the ability of concrete to resist tension between cracks and the stiffness that is provided by this tension. This effect is reduced as the loading and cracking is increased. Normally, the mean curvature used for deflection calculations is interpolated between the calculated uncracked transformed curvature and the cracked curvature using a tension stiffening model. Examples of tension stiffening models used in practice include the models developed by Branson (see ACI 381-14 24.2.3.5), Bischoff (see ACI 318-19 24.2.3.5), and the model outlined in Eurocode 2-2004 (Equation 7.19).

RAM Concept uses the Eurocode 2 tension stiffening model for load history calculations with all building codes and creep/shrinkage models, with the following modifications:

- Mcr (gross section cracking moment) is replaced with fcr (concrete modulus of rupture), and Ma (gross section resultant moment) is replaced with fa (gross section top or bottom tensile stress). This modification is required because the original formula does not consider axial forces in combination with bending, which may be present and are especially important when post-tensioning is present.
- Beta is assumed equal to 1. Eurocode 2 states that b should be taken as 1 for short-term loading and 0.5 for long-term loading (see Clause 7.4.3). Some experts have concluded that beta is equivalent to reducing the cracking moment by about 30 percent and is an approximate way to account for shrinkage induced cracking (i.e. shrinkage restraint)
^{1}. Since this behavior is already accounted for in RAM Concept’s load history calculations (see “Internal Restraint to Shrinkage” and “External Restraint to Shrinkage” below), the program uses beta = 1 to avoid double counting that effect.

RAM Concept accounts for redistribution of forces using element stiffness adjustments and an iterative analysis. For each cracked cross section, RAM Concept calculates an elements stiffness adjustment using the ratio of the linear elastic section curvature to long-term section curvature and applies it to all finite elements that are tributary to the section. After the element stiffness adjustment is applied, another analysis is completed, and the cross-section strain/curvature calculations are repeated. This iterative process continues until the calculated deflections converge for the loading stage.

__Reference__

^{1} Gilbert, R.I and Ranzi, G., “Time-Dependent Behavior of Concrete Structures”, CRC Press, 2019.

## Creep

Creep accounts for the increase in concrete strain with time due to sustained loads. Typical concrete creep strains range between 2-4 times the elastic strain and are important for accurate prediction of long-term deflections as a result. Actual creep behavior is affected by the rate of load application and the variation of concrete strength over time.

RAM Concept calculates a differential creep strain over the duration of each loading stage using one of the following creep models that are implemented in the program: ACI 209R-92 (ECR Values), ACI 209.2R-08/GL 2000, AS 3600-2018, and Eurocode 2-2004. When calculating the strains and combining the effects over the load history, RAM Concept assumes that the creep strains are a linear factor of the elastic strain for a particular load and that creep strains of like or opposing signs can be superimposed. These assumptions are reasonable for typical service loads.

RAM Concept uses the age adjusted effective modulus method, which adjusts creep strains using an ageing coefficient, to approximate the effects that age and rate of loading have on creep behavior. The resulting modified creep strains are then used in the cross-section strain compatibility calculations.

## Shrinkage

Shrinkage is the reduction in concrete volume over time due to hydration of cement, loss of moisture, and other factors. It occurs independent of loading and is normally specified as a strain. In general, restraint to shrinkage (see following sections) causes time-dependent cracking and increases long-term deflection.

RAM Concept calculates a shrinkage strain for each loading stage using one of the following shrinkage models that are implemented in the program: ACI 209R-92 (ECR Values), ACI 209.2R-08/GL 2000, AS 3600-2018, and Eurocode 2-2004. The resulting shrinkage strains are then used in the cross-section calculations. Due to strain compatibility, the imposed shrinkage strain results in internal stresses in the reinforcement (normally compression) and concrete (normally tension) caused by shrinkage strains.

## Internal Restraint to Shrinkage

Internal restraint to shrinkage is due primarily to bonded reinforcement. As concrete shrinks, the reinforcement is compressed, and an equal and opposite tension force is induced in the concrete. When the reinforcement in the section is asymmetric (more bottom reinforcement than top reinforcement, or vice versa), this tension force is eccentric to the centroid of the concrete section and a curvature is imposed on the section. This curvature, also known as shrinkage warping, may cause cracking to occur and increase deflection as a result. In extreme cases, cracking due to shrinkage warping may occur in the absence of superimposed loads.

RAM Concept rigorously accounts for the shrinkage warping effect in the strain/curvature calculations at each cross section, considering all reinforcement intersecting the section. The approach is similar to the one outlined in Eurocode 2 -2004 Clause 7.4.3(6) Equation 7.21.

## External Restraint to Shrinkage

External restraint to shrinkage from stiff supports or adjacent slabs can lead to a build-up of tensile stress and increased cracking in concrete floors. Failure to account for this effect in long-term deflection calculations may lead to unconservative results.

RAM Concept accounts for external shrinkage restraint using an input External Shrinkage Restraint setting, which translates to a shrinkage restraint percentage. This percentage is multiped by the shrinkage strain at any loading stage to estimate a “restrained” shrinkage strain. This fictitious strain is then superimposed with the strains caused by loads and other effects and applied to the tension stress calculation used in the tension stiffening model but not the cross-section strain/curvature calculations. As such, the fictitious external restraint strain affects cracking prediction only but not the calculated curvatures directly.

## Load History

Load history is an important consideration for accurate deflection prediction, especially for structures subject to significant construction loads that cause early age cracking. RAM Concept accounts for load history using a series of user defined loading steps, which are mapped with a load combination and a load duration. Each loading step considers the effect of cracking, creep, and shrinkage from all previous steps. Creep strains are all linearly superimposed, and an unloading event is considered the same as a subsequent loading event of opposite sign as a previously applied loading. Once a cross section is determined to be cracked during a particular loading stage, it is assumed to be cracked for all future iterations and load history steps.

## Related Publications Authored by the RAM Concept Team

- Hirsch, J., “Accurate Long-Term Deflection Prediction in Flat Slabs Using Linear Elastic Global Analysis”,
*24th Biennial Conference of the Concrete Institute of Australia*, Sydney, Australia, 2009. - Hirsch, J., Calabrese, F., Connolly, E., and Bommer, A. “Practical Deflection Prediction of Concrete Slabs”, ACI SP284-18, March 2012.
- RAM Concept Manual, Chapter 70 “Load History Deflections”