SLS analysis

SLS and ULS loads analysis

When looking at serviceability it is useful to be able to differentiate between short and long term conditions. Long term analysis takes account of creep, while short term analysis assumes that no creep takes place.

Long, short and long, short term analysis

When looking at serviceability it is useful to be able to differentiate between short and long term conditions. Long term analysis takes account of creep, while short term analysis assumes that no creep takes place.

The creep is defined by a creep coefficient $\varphi$ for concrete. It is assumed that the other materials are unaffected by creep. This coefficient is used to modify the material stress-strain curves.

In a long term analysis, the total strain is assumed to include the strain due to load plus an additional strain due to creep. In the linear case this can be written as

$$\varepsilon=\varepsilon_{load}+\varepsilon_{creep}=\frac{\sigma}{E}+\varphi\frac{\sigma}{E}$$

Rearranging this gives

$$\sigma=\varepsilon\frac{E}{(1+\varphi)}$$

AdSec takes these creep effects into account by modifying the effective elastic modulus

$$E_{creep}=\frac{E}{(1+\varphi)}$$

Resulting in stress-strain curves stretched along the strain axis.

Long and short term analysis is an option in AdSec to give a more detailed understand the serviceability behaviour of sections. Note: this is not available for all design codes.

Loading is defined in two stages. Firstly, long term loading, combined with a creep factor and then an additional short term loading.

In some circumstances, the long term loading is a permanent or quasi permanent loading, and the short term loading is an extreme event that happens after an extended period of time. However, in many cases short term loading will occur intermittently throughout the life of the section. The long and short term analysis option in AdSec will model the second case.

• Firstly, the cracking moment is calculated assuming the total long and short term axial load and moment direction, and short-term material properties.

• Secondly, the strains and stresses are found for the long term loads, and long term material properties. These strains are used to calculate the creep effects of long term loads where creep strain is: $\varepsilon_{creep}=-\varepsilon_{long}\frac{\varphi}{(1+\varphi)}$. In this analysis the BS8110 Pt.2 tension curve will use 0.55N/mm² as the maximum stress, if the section was deemed cracked under the total load. This will model the conservative assumption that, if cracked, this happened at an early stage of the sections life.

• The cracking moment is then recalculated, for the total load including the creep strain in the concrete calculated above. This will have the effect of slightly reducing the cracking moment if a compressive force has been acting on the section for a long time. This is the case, because the stress in the concrete will have reduced as the concrete creeps and more stress is transferred to the reinforcement.

• Finally, a short term analysis is performed for the total loads, using short term material properties and the calculated creep strain to include for the long term effects.

Note that if the same process is followed manually using sequential AdSec analyses the initial cracking moment will be calculated from the long term load only. This will give different results than the automated AdSec long and short term analysis in a small number of cases. The cases affected are where the BS8110 Pt.2 tension curve is selected, and the section is cracked under total load, but uncracked under the long term load, and the stress under long term load is between 0.55 and 1.0 N/mm² at the centroid of tension steel.

Some codes allow an intermediate term analysis, depending on the ratio

$$\frac{M_q}{M_g}$$

In this case

$$E_{inter}=E_{short}\frac{(M_q/M_g)}{1+(M_q/M_g)}+E_{long}\frac{1}{1+(M_q/M_g)}$$

$$E_{short}=E$$

$$E_{long}=\frac{E}{(1+\varphi)}=\frac{E}{2}$$

Tension in Concrete

Concrete exhibits a 4 phase behaviour in response to tension stresses.

• Low tension stress -- concrete tension stiffness similar to compression

• Cracking starts -- stiffness drops off as cracks form

• Cracks formed, cracks open up -- stiffness drops off more rapidly as cracks open up

• Fully cracked -- no residual stiffness left

This behaviour is complex as it is controlled by the reinforcement. The simplified means prescribed to deal with these phenomena vary from code to code.

All codes state that ultimate analysis and design should ignore the tension stiffening from the concrete. All codes will accept fully cracked section properties as a lower bound on stiffness.

Serviceability analysis is usually performed for stiffness, stress/strain checks, or crack width checks. Some codes imply a different tension stiffening method for crack width as opposed to the other checks. This may lead to a disparity in AdSec results between the cracking moment and the moment at which the crack width becomes > zero.

The code rules are developed for a rectangular section with uniaxial bending and one row of tension steel. However, the rules are not extended to sections made up of various zones of concrete, some with locked in strain planes. Because the tension stiffening is a function of the amount of damage cracking in the section, adjoining tensile zones need to be considered in evaluating the tension strength of a zone, as these may contain steel which will control the cracking.

BS8110 Pt 2 presents a stress/strain envelope which provide means of calculating an effective tensile Young's modulus for a linear tension stress/strain curve.

ICE Technical note 372 presents a more sophisticated envelope approach than BS8110 and is offered as an option in ADSEC.

BS5400 presents the same approach as BS8110 in Appendix A for stiffness calcs. But this is rarely used. Instead the main body of the code gives a crack width formula based on strains from an analysis with no tension stiffening. The crack width formula itself includes some terms to add back in an estimate of the contribution from tension stiffening. Ref BS5400 5.8.8.2 equation 25.

EC2 proposes 2 analyses, one with full tension stiffness and one with none. The final results are an interpolation between these results.

Recent research about the cracked stiffness of concrete has shown that the tension stiffness measured in the laboratory can only be retained for a very short time. This means that both the tension stiffening given in BS8110 and TN 372 is un-conservative for most building and bridge loadings. AdSec includes these findings for BS8110 and will give a smaller tension stiffness than previous versions.

EC2 tension stiffening

EC2 tension stiffening is described in Eurocode 2 section 7.4.3 equation 7.18. EC2 does not have a specific tension stiffening relationship used in analysis. Instead, two analyses are carried out assuming cracked and uncracked stiffness values, and the actual curvature & stiffness is an interpolation between the 2 results based on the amount of cracking predicted.

The cracking moment, $M_{cr}$, is defined as the moment when the stress in the outermost tensile element of an uncracked concrete section has reached $f_{ctm}$.

The tension stiffening options offered for EC2 in AdSec are zero tension, linear tension, and interpolated. Zero-tension stiffness will give a conservative, fully cracked lower bound. The linear tension stiffening uses the Elastic modulus of the concrete to produce a linear stress-strain relationship. This is for checking of the other results only and it is not appropriate to use this beyond the cracking moment. Note that the values in EC2 for serviceability are based on mean concrete properties rather than the characteristic values used for ultimate analysis and design. The interpolation depends on the amount of damage sustained by the section. This is calculated by AdSec based on the proximity of the applied loading to the cracking moment. But for sections which have been cracked in a previous load event the minimum value of $\zeta$ for use in equation 7.18 can be input. The default value of $\zeta_{min}$ is 0. To take account of the fast drop in tension stiffening following cracking, the value of $\beta$ in equation 7.19 defaults to 0.5.

AdSec does not use equation 7.19 to calculate the damage parameter $\zeta$. Instead $\zeta$ is calculated from the cracking strain

$$\varepsilon_{cc}=\frac{f_{ctm}}{Ecm}$$

and the most tensile strain $\varepsilon_{uncr}$ in the section under an uncracked analysis under full applied load.

The $E$ used for to determine $\varepsilon_{uncr}$ is short term (not modified for creep). For composite sections $\zeta_i$ is calculated for each component $i$ using the component material properties for $\varepsilon_{cc}$, and the most tensile strain on the component for $\varepsilon_{uncr}$. The highest value of $\zeta_i$ will be used for $\zeta$ in stiffness and cracking calculations.

Note engineering judgement should be used to assess if this approach fits the particular situation.

$$\zeta = \begin {cases} 1 - \beta(\frac{\varepsilon_{cc}}{\varepsilon_{uncr}})^2 \space \space \varepsilon_{uncr}>\varepsilon_{cc}\\ \zeta_{min} \space \space \space\space \space \space\space \space \space\space \space \space \space \space \space \varepsilon_{uncr} \space \leq \space \varepsilon_{cc} \end{cases}$$

If the EC2 interpolation is selected for the tension stiffness at serviceability, the properties which depend on the average behaviour along the element (e.g. stiffness, curvature and crack widths) are based on the interpolated strain plane. However, for moments greater than $M_{cr}$, the stresses output by AdSec for the interpolated tension stiffness are from the fully cracked analysis, because these represent the maximum stresses which occur at crack positions.

Stiffness

AdSec operates on strain, using non-linear materials. AdSec will show how the stiffness of the section changes with load and the effect of non-linear material behaviour. There are a number of ways in which the stiffness of a reinforced concrete section can be approximated. These are show in the diagram below. This diagram plots AdSec results along with the approximate stiffness values for comparison

For a symmetric section, symmetrically loaded, stiffness can be expressed as

$$EI=\frac{M}{K}$$

If there is an axial force, locked in strain plane, or pre-stress, there will be a residual curvature at zero moment.

This curvature can be called $K_0$ so AdSec uses

$$EI=\frac{M}{(K-K_0)}$$

The curvature at zero moment may not be in the same direction as the applied moment angle. To allow for this, the formula is further modified to give

$$EI=\frac{M}{[K-K_0(a_{appl}-a_{NA})]}$$

Where $a_{appl}$ is the angle of applied moment and $a_{NA}$ is the neutral axis angle from the $K_0$ calculation.